LMFDB - Newform orbit 672.4.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,4,Mod(337,672)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("672.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 5 x^{18} - 26 x^{17} + 122 x^{16} + 124 x^{15} - 276 x^{14} - 1376 x^{13} + \cdots + 1073741824 $ x^20 - 2x^19 - 5x^18 - 26x^17 + 122x^16 + 124x^15 - 276x^14 - 1376x^13 - 3344x^12 + 15424x^11 - 1920x^10 + 123392x^9 - 214016x^8 - 704512x^7 - 1130496x^6 + 4063232x^5 + 31981568x^4 - 54525952x^3 - 83886080x^2 - 268435456*x + 1073741824
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{60}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} - \beta_1 q^{5} + 7 q^{7} - 9 q^{9}+O(q^{10}) $ q + b2 * q^3 - b1 * q^5 + 7 * q^7 - 9 * q^9	$ q + \beta_{2} q^{3} - \beta_1 q^{5} + 7 q^{7} - 9 q^{9} + (\beta_{6} - 2 \beta_{2}) q^{11} + ( - \beta_{9} - 2 \beta_{2}) q^{13} + (\beta_{16} - 3) q^{15} + (\beta_{5} - 3) q^{17} + (\beta_{12} + 5 \beta_{2}) q^{19} + 7 \beta_{2} q^{21} + ( - \beta_{19} + 12) q^{23} + ( - \beta_{17} + \beta_{16} + \cdots - 42) q^{25}+ \cdots + ( - 9 \beta_{6} + 18 \beta_{2}) q^{99}+O(q^{100}) $ q + b2 * q^3 - b1 * q^5 + 7 * q^7 - 9 * q^9 + (b6 - 2b2) q^11 + (-b9 - 2b2) q^13 + (b16 - 3) * q^15 + (b5 - 3) * q^17 + (b12 + 5b2) q^19 + 7b2 q^21 + (-b19 + 12) * q^23 + (-b17 + b16 + b15 - 42) * q^25 - 9b2 q^27 + (-b12 + b11 + b2 + 3b1) q^29 + (-b17 + b16 - b13 - b5 - 13) * q^31 + (b19 + b18 + 20) * q^33 - 7b1 q^35 + (b10 - b9 + b7 + 4b6 + 5b2 - 4b1) q^37 + (-b19 + b15 + b14 + b13 + 16) * q^39 + (-b19 - 3b18 + b17 - b16 - b15 - b14 - b13 + 2b5 + b3 - 13) * q^41 + (-b12 + b11 - b10 + 2b9 - 2b6 - b4 - 12b2 - 2b1) * q^43 + 9b1 q^45 + (b19 + b17 - b16 - 3b15 - b14 + b13 + b5 + 2b3 - 1) * q^47 + 49 * q^49 + (b12 - b9 + b8 - b7 - 3b2 - b1) q^51 + (-b12 - 2b9 - b8 - 4b6 - b4 + 11b2 - 7b1) * q^53 + (b19 + b18 - 3b16 - b15 + 2b14 + 4b13 + 3b5 + 3b3 - 4) q^55 + (-b18 - b17 + b15 - b14 - b5 + b3 - 45) * q^57 + (2b12 + b11 + 2b10 + 2b9 - b8 - 2b6 + b4 - 30b2 + 4b1) * q^59 + (-b12 - b11 + 2b10 + 3b9 + 2b7 - 4b6 - b4 - 50b2 + 6b1) * q^61 - 63 * q^63 + (4b19 + b18 + b16 + b14 - 4b13 + b5 + b3 + 88) * q^65 + (2b11 - b10 + 2b8 - 2b7 + 4b6 + 34b2 + 18b1) * q^67 + (2b12 - b11 + b10 - b8 + 5b6 - b4 + 12b2 - 4b1) * q^69 + (-3b19 - 2b18 - b17 - 6b16 + b15 - b13 - 3b5 + 2b3 + 33) q^71 + (-3b19 - b18 + 2b16 + b15 - b13 + 2b5 + b3 - 140) q^73 + (-3b12 + b11 + 2b9 + b8 - b7 - 2b6 - b4 - 39b2 + 8b1) q^75 + (7b6 - 14b2) * q^77 + (-2b19 - b17 + 10b16 - b15 - 2b14 + 3b13 + 7b5 + 4b3 - 147) * q^79 + 81 * q^81 + (-2b12 + 5b11 + 2b10 + 2b9 - b8 - 4b7 - 2b6 - b4 - 4b2 - 4b1) * q^83 + (-4b12 + 5b11 + 2b10 + 6b9 - 6b6 - b4 + 7b2 + 18b1) q^85 + (-2b19 + b18 + 3b17 - b16 + 3b5 - 1) q^87 + (-b19 - 3b18 + 3b17 - 6b16 - 3b15 + 4b14 - b13 - b5 + 3b3 + 12) * q^89 + (-7b9 - 14b2) * q^91 + (2b11 - 2b9 - b8 - 2b7 + 2b6 - 2b4 - 9b2 + 10b1) q^93 + (-3b19 - 3b18 + 3b17 + 19b16 + b15 - 3b14 - b13 + 5b5 + b3 + 60) * q^95 + (-b19 + b18 + 3b17 + 10b16 + b15 - 2b14 - 3b13 + 3b3 + 221) q^97 + (-9b6 + 18b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 140 q^{7} - 180 q^{9}+O(q^{10}) $ 20 * q + 140 * q^7 - 180 * q^9	$ 20 q + 140 q^{7} - 180 q^{9} - 60 q^{15} - 52 q^{17} + 244 q^{23} - 844 q^{25} - 264 q^{31} + 396 q^{33} + 312 q^{39} - 236 q^{41} + 980 q^{49} - 72 q^{55} - 912 q^{57} - 1260 q^{63} + 1744 q^{65} + 636 q^{71} - 2784 q^{73} - 2872 q^{79} + 1620 q^{81} + 220 q^{89} + 1240 q^{95} + 4400 q^{97}+O(q^{100}) $ 20 * q + 140 * q^7 - 180 * q^9 - 60 * q^15 - 52 * q^17 + 244 * q^23 - 844 * q^25 - 264 * q^31 + 396 * q^33 + 312 * q^39 - 236 * q^41 + 980 * q^49 - 72 * q^55 - 912 * q^57 - 1260 * q^63 + 1744 * q^65 + 636 * q^71 - 2784 * q^73 - 2872 * q^79 + 1620 * q^81 + 220 * q^89 + 1240 * q^95 + 4400 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2 x^{19} - 5 x^{18} - 26 x^{17} + 122 x^{16} + 124 x^{15} - 276 x^{14} - 1376 x^{13} + \cdots + 1073741824 $ x^20 - 2*x^19 - 5*x^18 - 26*x^17 + 122*x^16 + 124*x^15 - 276*x^14 - 1376*x^13 - 3344*x^12 + 15424*x^11 - 1920*x^10 + 123392*x^9 - 214016*x^8 - 704512*x^7 - 1130496*x^6 + 4063232*x^5 + 31981568*x^4 - 54525952*x^3 - 83886080*x^2 - 268435456*x + 1073741824 :

$\beta_{1}$	$=$	$ ( - 249239 \nu^{19} - 9170499 \nu^{18} - 76180607 \nu^{17} + 361395123 \nu^{16} + \cdots + 64\!\cdots\!60 ) / 154267331592192 $ (-249239v^19 - 9170499v^18 - 76180607v^17 + 361395123v^16 + 415516840v^15 + 102571018v^14 - 8076761720v^13 + 3626671812v^12 + 39649206112v^11 - 12520659504v^10 + 240130318208v^9 - 1235771452288v^8 + 204671621120v^7 - 9834948232192v^6 + 12641407373312v^5 + 82539077779456v^4 - 84223732154368v^3 - 326477670252544v^2 - 1337098988683264*v + 6492443440578560) / 154267331592192
$\beta_{2}$	$=$	$ ( 10567955 \nu^{19} - 35602606 \nu^{18} - 39792079 \nu^{17} - 378375750 \nu^{16} + \cdots - 39\!\cdots\!56 ) / 822759101825024 $ (10567955v^19 - 35602606v^18 - 39792079v^17 - 378375750v^16 + 857936990v^15 + 931239268v^14 - 1563305308v^13 - 885932416v^12 - 81947838256v^11 + 167021695296v^10 - 125038508160v^9 + 1453449232896v^8 - 1231343410176v^7 - 7215125635072v^6 - 3311153430528v^5 - 20249624969216v^4 + 305852764913664v^3 - 490834835275776v^2 - 192571150172160*v - 3988038464045056) / 822759101825024
$\beta_{3}$	$=$	$ ( - 202665 \nu^{19} - 14089318 \nu^{18} + 11395197 \nu^{17} + 96736322 \nu^{16} + \cdots + 11\!\cdots\!60 ) / 5687274897408 $ (-202665v^19 - 14089318v^18 + 11395197v^17 + 96736322v^16 + 211117158v^15 - 763902732v^14 - 2248790892v^13 + 3015992256v^12 + 8193803664v^11 + 51532465472v^10 - 43421509248v^9 - 176669105664v^8 - 1159072951296v^7 - 1031382294528v^6 + 12185911050240v^5 + 18258007162880v^4 - 33671172587520v^3 - 301881213583360v^2 + 3684628955136*v + 1181244519874560) / 5687274897408
$\beta_{4}$	$=$	$ ( 135994909 \nu^{19} - 787430514 \nu^{18} + 12407585951 \nu^{17} + 2076769958 \nu^{16} + \cdots + 33\!\cdots\!24 ) / 24\!\cdots\!72 $ (135994909v^19 - 787430514v^18 + 12407585951v^17 + 2076769958v^16 + 47364462018v^15 - 75137431076v^14 - 202000027364v^13 + 1316116003200v^12 + 1040346202928v^11 + 14252539436736v^10 - 51486551937920v^9 - 40027596042752v^8 - 69830251496448v^7 + 163209461473280v^6 + 3598048635305984v^5 - 2375881360932864v^4 - 8310679485808640v^3 - 28619453096787968v^2 + 213594179295510528*v + 335017194368794624) / 2468277305475072
$\beta_{5}$	$=$	$ ( - 68772 \nu^{19} + 1552331 \nu^{18} - 1949478 \nu^{17} - 8969215 \nu^{16} + \cdots - 117029713477632 ) / 710909362176 $ (-68772v^19 + 1552331v^18 - 1949478v^17 - 8969215v^16 - 16014546v^15 + 93281862v^14 + 212366652v^13 - 590095596v^12 - 344336208v^11 - 5125373872v^10 + 9488773632v^9 + 11329546368v^8 + 92392854528v^7 + 100286696448v^6 - 1449889640448v^5 - 713698754560v^4 + 3092927545344v^3 + 30227736559616v^2 - 43217873534976*v - 117029713477632) / 710909362176
$\beta_{6}$	$=$	$ ( 10885131 \nu^{19} - 27521502 \nu^{18} + 232133689 \nu^{17} - 901697014 \nu^{16} + \cdots - 12\!\cdots\!12 ) / 94933742518272 $ (10885131v^19 - 27521502v^18 + 232133689v^17 - 901697014v^16 + 322202414v^15 - 1292562108v^14 + 13639990788v^13 + 14264327424v^12 - 139114104368v^11 + 175510406976v^10 - 956129378944v^9 + 3763837471232v^8 - 634707037184v^7 + 7169160486912v^6 - 5842551521280v^5 - 174958784020480v^4 + 452310723461120v^3 - 35604460994560v^2 + 3665175789961216*v - 12263289794854912) / 94933742518272
$\beta_{7}$	$=$	$ ( 448208679 \nu^{19} + 1034629674 \nu^{18} - 19502268083 \nu^{17} - 36011697742 \nu^{16} + \cdots - 44\!\cdots\!96 ) / 24\!\cdots\!72 $ (448208679v^19 + 1034629674v^18 - 19502268083v^17 - 36011697742v^16 + 17724887654v^15 + 381332818452v^14 + 159182150676v^13 - 3081301669632v^12 - 1763041071728v^11 - 10566824204736v^10 + 53773600717184v^9 + 74380770724352v^8 + 186199320564736v^7 - 1001814637682688v^6 - 4529584351395840v^5 + 4689692138405888v^4 + 24344746669899776v^3 + 39684888231673856v^2 - 284463129927090176*v - 447922539537104896) / 2468277305475072
$\beta_{8}$	$=$	$ ( - 77198039 \nu^{19} + 643792422 \nu^{18} - 1075345597 \nu^{17} + 6355490174 \nu^{16} + \cdots + 10\!\cdots\!00 ) / 352611043639296 $ (-77198039v^19 + 643792422v^18 - 1075345597v^17 + 6355490174v^16 - 12599192742v^15 + 12466684684v^14 + 29350267372v^13 + 118543826496v^12 + 1256115282800v^11 - 4182073950528v^10 + 3025233390208v^9 - 29027817419264v^8 + 44682625649664v^7 + 93783782481920v^6 - 65538017542144v^5 + 760389568561152v^4 - 4541254138855424v^3 + 6874067869106176v^2 - 7923456213516288*v + 107994731354521600) / 352611043639296
$\beta_{9}$	$=$	$ ( - 104055827 \nu^{19} - 80412826 \nu^{18} + 2982879391 \nu^{17} - 888846802 \nu^{16} + \cdots - 45\!\cdots\!04 ) / 411379550912512 $ (-104055827v^19 - 80412826v^18 + 2982879391v^17 - 888846802v^16 - 6293067598v^15 - 41886116404v^14 + 122096534268v^13 + 222138996256v^12 - 154542481104v^11 + 590491677504v^10 - 8664621149568v^9 + 4807168043520v^8 - 34269223130112v^7 + 204792003936256v^6 + 235310339637248v^5 - 1498432539852800v^4 - 1596266681729024v^3 + 1582487826333696v^2 + 47273598754226176*v - 45552267851464704) / 411379550912512
$\beta_{10}$	$=$	$ ( - 44941207 \nu^{19} + 319710534 \nu^{18} - 317005309 \nu^{17} + 2683807326 \nu^{16} + \cdots + 39\!\cdots\!36 ) / 176305521819648 $ (-44941207v^19 + 319710534v^18 - 317005309v^17 + 2683807326v^16 - 6956340646v^15 + 12825179084v^14 + 1543821548v^13 + 56860920000v^12 + 378625683056v^11 - 1333050694464v^10 + 1262767844992v^9 - 11295935853056v^8 + 19956892429312v^7 + 22204836044800v^6 - 19894624206848v^5 + 214916617207808v^4 - 1682841687031808v^3 + 5817610453123072v^2 - 6213611143823360*v + 39627131491188736) / 176305521819648
$\beta_{11}$	$=$	$ ( - 26115319 \nu^{19} - 17003346 \nu^{18} + 226602675 \nu^{17} - 66446698 \nu^{16} + \cdots - 14\!\cdots\!64 ) / 94933742518272 $ (-26115319v^19 - 17003346v^18 + 226602675v^17 - 66446698v^16 - 1599657494v^15 - 4701785572v^14 + 9982337804v^13 + 7099780896v^12 + 50045726448v^11 - 81775917504v^10 - 434894395776v^9 - 2310117176832v^8 - 103974335488v^7 + 16305342447616v^6 + 52635255783424v^5 - 188669288382464v^4 - 723220596523008v^3 + 476570217086976v^2 + 668790312402944*v - 1477826171633664) / 94933742518272
$\beta_{12}$	$=$	$ ( 238410449 \nu^{19} + 718435510 \nu^{18} - 8207601221 \nu^{17} + 9821924366 \nu^{16} + \cdots + 23\!\cdots\!44 ) / 822759101825024 $ (238410449v^19 + 718435510v^18 - 8207601221v^17 + 9821924366v^16 + 11290429482v^15 + 118379531308v^14 - 280593528244v^13 - 527365171264v^12 + 1441833089776v^11 - 4494021887296v^10 + 29857710984832v^9 - 50583697592832v^8 + 112048889719808v^7 - 477627790270464v^6 - 491407645949952v^5 + 4463492083482624v^4 + 1169148622667776v^3 + 8209608216674304v^2 - 127149972649934848*v + 230594015546834944) / 822759101825024
$\beta_{13}$	$=$	$ ( 2372275 \nu^{19} - 10488814 \nu^{18} + 9104593 \nu^{17} - 25546758 \nu^{16} + \cdots - 209083705589760 ) / 5687274897408 $ (2372275v^19 - 10488814v^18 + 9104593v^17 - 25546758v^16 + 154532062v^15 - 339732124v^14 - 227818076v^13 + 3405169536v^12 - 11059478576v^11 + 35177847104v^10 - 69328385152v^9 + 310367212032v^8 - 500690025472v^7 - 1387557658624v^6 + 4145202151424v^5 - 6301718151168v^4 + 16098725986304v^3 - 164895706316800v^2 + 435397024808960*v - 209083705589760) / 5687274897408
$\beta_{14}$	$=$	$ ( - 2797375 \nu^{19} - 7895890 \nu^{18} + 14918939 \nu^{17} + 103151350 \nu^{16} + \cdots + 860759563173888 ) / 5687274897408 $ (-2797375v^19 - 7895890v^18 + 14918939v^17 + 103151350v^16 + 10838234v^15 - 624954404v^14 - 907460692v^13 + 2435107296v^12 + 10284383216v^11 + 35578589504v^10 - 25407754112v^9 - 285391517184v^8 - 738784756736v^7 + 22325690368v^6 + 9749757804544v^5 + 1320198209536v^4 - 49469222027264v^3 - 220051680526336v^2 + 148867744006144*v + 860759563173888) / 5687274897408
$\beta_{15}$	$=$	$ ( 3017837 \nu^{19} - 27312378 \nu^{18} - 11097025 \nu^{17} + 155360078 \nu^{16} + \cdots + 25\!\cdots\!52 ) / 5687274897408 $ (3017837v^19 - 27312378v^18 - 11097025v^17 + 155360078v^16 + 415564946v^15 - 425733044v^14 - 5561053636v^13 + 6166924320v^12 + 5189231024v^11 + 72848155968v^10 + 33597166720v^9 - 274212711936v^8 - 904260051968v^7 - 5498544521216v^6 + 17496898355200v^5 + 33960772370432v^4 - 29150950719488v^3 - 468283639726080v^2 - 193542584532992*v + 2580248109514752) / 5687274897408
$\beta_{16}$	$=$	$ ( - 1024917 \nu^{19} + 751938 \nu^{18} + 1597593 \nu^{17} + 23152362 \nu^{16} + \cdots + 215348787806208 ) / 1895758299136 $ (-1024917v^19 + 751938v^18 + 1597593v^17 + 23152362v^16 - 35219922v^15 - 123191676v^14 - 12650748v^13 + 275677056v^12 + 5151510480v^11 - 3870979776v^10 + 3698697600v^9 - 97117644288v^8 - 64318245888v^7 + 495805341696v^6 + 822640558080v^5 + 297144680448v^4 - 21405552869376v^3 + 6044415688704v^2 + 6132458323968*v + 215348787806208) / 1895758299136
$\beta_{17}$	$=$	$ ( 101797 \nu^{19} + 27810 \nu^{18} - 759857 \nu^{17} - 1519646 \nu^{16} + 2251594 \nu^{15} + \cdots + 1907538001920 ) / 177727340544 $ (101797v^19 + 27810v^18 - 759857v^17 - 1519646v^16 + 2251594v^15 + 21185444v^14 - 33194708v^13 - 15629520v^12 - 254539088v^11 - 213221760v^10 + 2602940288v^9 + 3768883200v^8 + 11609142272v^7 - 90829336576v^6 - 64797196288v^5 + 291173761024v^4 + 1619550273536v^3 + 305871716352v^2 - 12138198335488*v + 1907538001920) / 177727340544
$\beta_{18}$	$=$	$ ( - 5692025 \nu^{19} + 21035450 \nu^{18} + 402637 \nu^{17} + 28868706 \nu^{16} + \cdots - 34404298653696 ) / 5687274897408 $ (-5692025v^19 + 21035450v^18 + 402637v^17 + 28868706v^16 - 452436986v^15 + 147301556v^14 + 2409039700v^13 - 4133408064v^12 + 23356219792v^11 - 73008488128v^10 + 91646262656v^9 - 511008861696v^8 + 673817670656v^7 + 4524788129792v^6 - 9456951672832v^5 + 890601209856v^4 - 88552783740928v^3 + 371191508369408v^2 - 324748366053376*v - 34404298653696) / 5687274897408
$\beta_{19}$	$=$	$ ( - 847982 \nu^{19} - 1675015 \nu^{18} + 2115040 \nu^{17} + 36344691 \nu^{16} + \cdots + 379966093000704 ) / 710909362176 $ (-847982v^19 - 1675015v^18 + 2115040v^17 + 36344691v^16 - 769826v^15 - 180476158v^14 - 376473620v^13 + 783815340v^12 + 4684136944v^11 + 2328552752v^10 + 4477780736v^9 - 103967011968v^8 - 181116402688v^7 + 78578889728v^6 + 2593983967232v^5 + 2751109447680v^4 - 24595025428480v^3 - 36662407069696v^2 + 27822865252352*v + 379966093000704) / 710909362176

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{19} + \beta_{18} - 2 \beta_{16} - 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \cdots + 20 ) / 192 $ (b19 + b18 - 2b16 - 2b11 - b10 + b9 + b7 - 2b6 - 3b5 + b4 - 3b3 + 12b2 - b1 + 20) / 192
$\nu^{2}$	$=$	$ ( - \beta_{17} + 2 \beta_{16} - 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 45 ) / 64 $ (-b17 + 2b16 - 2b14 + 2b12 + 2b10 + 2b9 - 2b8 + b7 + 2b6 - 2b5 + 5*b2 + 45) / 64
$\nu^{3}$	$=$	$ ( - 5 \beta_{19} - 5 \beta_{18} + 6 \beta_{17} + 20 \beta_{16} + 6 \beta_{14} - 24 \beta_{13} + \cdots + 1124 ) / 192 $ (-5b19 - 5b18 + 6b17 + 20b16 + 6b14 - 24b13 + 10b12 - 14b11 - b10 + 21b9 - 2b8 + 9b7 + 28b6 - 27b5 - 5b4 - 9b3 - 18b2 + 19*b1 + 1124) / 192
$\nu^{4}$	$=$	$ ( 14 \beta_{19} + 22 \beta_{18} - 3 \beta_{17} - 46 \beta_{16} - 4 \beta_{15} - 6 \beta_{14} - 2 \beta_{12} + \cdots - 421 ) / 64 $ (14b19 + 22b18 - 3b17 - 46b16 - 4b15 - 6b14 - 2b12 - 12b11 + 2b10 - 14b9 + 2b8 + 5b7 + 38b6 - 20b5 + 4b4 + 2b3 + 165b2 + 28b1 - 421) / 64
$\nu^{5}$	$=$	$ ( 117 \beta_{19} - 75 \beta_{18} + 54 \beta_{17} - 108 \beta_{16} - 120 \beta_{15} - 30 \beta_{14} + \cdots - 1752 ) / 192 $ (117b19 - 75b18 + 54b17 - 108b16 - 120b15 - 30b14 - 120b13 + 118b12 + 62b11 + 61b10 - 65b9 - 14b8 + 43b7 + 284b6 - 165b5 + 41b4 + 33b3 - 462b2 - 975*b1 - 1752) / 192
$\nu^{6}$	$=$	$ ( 34 \beta_{19} - 22 \beta_{18} - 49 \beta_{17} + 38 \beta_{16} + 44 \beta_{15} - 66 \beta_{14} + \cdots + 5345 ) / 64 $ (34b19 - 22b18 - 49b17 + 38b16 + 44b15 - 66b14 - 144b13 + 42b12 - 52b11 - 38b10 - 46b9 + 70b8 - 65b7 + 58b6 + 4b5 - 8b4 + 6b3 + 3331b2 - 1448*b1 + 5345) / 64
$\nu^{7}$	$=$	$ ( 359 \beta_{19} + 455 \beta_{18} - 1242 \beta_{17} - 1028 \beta_{16} + 312 \beta_{15} + 150 \beta_{14} + \cdots - 65156 ) / 192 $ (359b19 + 455b18 - 1242b17 - 1028b16 + 312b15 + 150b14 - 264b13 + 514b12 + 202b11 - 289b10 - 2283b9 + 214b8 - 507b7 + 4228b6 - 567b5 + 211b4 - 981b3 - 10686b2 - 1853*b1 - 65156) / 192
$\nu^{8}$	$=$	$ ( 434 \beta_{19} - 758 \beta_{18} - 155 \beta_{17} + 642 \beta_{16} - 188 \beta_{15} - 366 \beta_{14} + \cdots + 80435 ) / 64 $ (434b19 - 758b18 - 155b17 + 642b16 - 188b15 - 366b14 + 144b13 - 218b12 - 276b11 + 666b10 - 1734b9 - 278b8 + 17b7 + 2270b6 + 704b5 - 124b4 + 558b3 - 36447b2 - 7260*b1 + 80435) / 64
$\nu^{9}$	$=$	$ ( 4637 \beta_{19} + 4061 \beta_{18} + 6054 \beta_{17} - 628 \beta_{16} + 2328 \beta_{15} + 2682 \beta_{14} + \cdots + 179728 ) / 192 $ (4637b19 + 4061b18 + 6054b17 - 628b16 + 2328b15 + 2682b14 + 1704b13 + 9582b12 - 4274b11 - 6907b10 + 3463b9 - 678b8 - 5165b7 + 13876b6 + 4299b5 - 1127b4 - 4551b3 - 18726b2 - 49063*b1 + 179728) / 192
$\nu^{10}$	$=$	$ ( - 4182 \beta_{19} + 7282 \beta_{18} - 5177 \beta_{17} + 7158 \beta_{16} + 1804 \beta_{15} + \cdots - 113927 ) / 64 $ (-4182b19 + 7282b18 - 5177b17 + 7158b16 + 1804b15 + 4958b14 + 11088b13 + 7402b12 - 2756b11 + 5138b10 - 6438b9 - 5786b8 - 2497b7 + 8074b6 - 1364b5 + 2624b4 + 1006b3 + 56507b2 - 39552*b1 - 113927) / 64
$\nu^{11}$	$=$	$ ( - 70049 \beta_{19} - 28097 \beta_{18} + 26046 \beta_{17} + 269404 \beta_{16} + 45048 \beta_{15} + \cdots - 1130332 ) / 192 $ (-70049b19 - 28097b18 + 26046b17 + 269404b16 + 45048b15 + 20550b14 + 15672b13 + 23634b12 - 35174b11 + 1367b10 + 29629b9 - 3258b8 + 60709b7 + 102628b6 - 1839b5 + 20395b4 + 15171b3 - 2705526b2 - 325909*b1 - 1130332) / 192
$\nu^{12}$	$=$	$ ( - 4182 \beta_{19} + 34306 \beta_{18} + 43725 \beta_{17} + 114706 \beta_{16} + 17092 \beta_{15} + \cdots - 5933157 ) / 64 $ (-4182b19 + 34306b18 + 43725b17 + 114706b16 + 17092b15 + 1170b14 + 32976b13 + 13350b12 - 36804b11 + 12002b10 + 7666b9 + 18218b8 - 29375b7 + 175790b6 + 28376b5 - 10564b4 + 4102b3 + 2186361b2 + 229116*b1 - 5933157) / 64
$\nu^{13}$	$=$	$ ( - 8507 \beta_{19} + 229957 \beta_{18} - 216642 \beta_{17} - 441412 \beta_{16} - 185544 \beta_{15} + \cdots - 6708472 ) / 192 $ (-8507b19 + 229957b18 - 216642b17 - 441412b16 - 185544b15 + 413754b14 + 649704b13 + 686446b12 + 153022b11 + 185549b10 + 288255b9 + 117658b8 + 163299b7 + 1297924b6 + 273459b5 + 22801b4 + 462369b3 - 833646b2 + 329617*b1 - 6708472) / 192
$\nu^{14}$	$=$	$ ( 68994 \beta_{19} - 191606 \beta_{18} + 160223 \beta_{17} - 141274 \beta_{16} + 293676 \beta_{15} + \cdots + 100625 ) / 64 $ (68994b19 - 191606b18 + 160223b17 - 141274b16 + 293676b15 + 236990b14 - 237936b13 - 312438b12 - 245172b11 + 205386b10 + 255522b9 + 216326b8 + 291711b7 - 394982b6 + 704804b5 + 258920b4 + 44486b3 + 1002355b2 - 2937304*b1 + 100625) / 64
$\nu^{15}$	$=$	$ ( 2049879 \beta_{19} - 2335113 \beta_{18} - 2365338 \beta_{17} + 1589292 \beta_{16} - 729576 \beta_{15} + \cdots - 474540180 ) / 192 $ (2049879b19 - 2335113b18 - 2365338b17 + 1589292b16 - 729576b15 - 3089466b14 - 165384b13 + 3325906b12 + 495722b11 - 2427761b10 + 1621957b9 + 1164166b8 - 2450123b7 + 5539604b6 + 8174601b5 + 1664435b4 + 1998651b3 + 84826098b2 + 19798707*b1 - 474540180) / 192
$\nu^{16}$	$=$	$ ( - 929742 \beta_{19} - 3694134 \beta_{18} - 4003595 \beta_{17} - 1513822 \beta_{16} - 27676 \beta_{15} + \cdots - 304116733 ) / 64 $ (-929742b19 - 3694134b18 - 4003595b17 - 1513822b16 - 27676b15 + 1293618b14 + 169488b13 + 1580166b12 - 259988b11 + 3034986b10 + 1641578b9 - 1245430b8 - 858335b7 - 3176674b6 + 3885184b5 - 1968300b4 + 1653454b3 - 102035679b2 - 5859948*b1 - 304116733) / 64
$\nu^{17}$	$=$	$ ( - 2456563 \beta_{19} + 4158797 \beta_{18} + 14825622 \beta_{17} - 40809044 \beta_{16} + \cdots + 534670336 ) / 192 $ (-2456563b19 + 4158797b18 + 14825622b17 - 40809044b16 + 21910296b15 + 10960506b14 - 10451928b13 - 18566738b12 + 1502830b11 - 16849387b10 + 18200823b9 - 1277798b8 - 10585197b7 + 10613524b6 + 60551163b5 - 1724087b4 + 13122921b3 - 1406413782b2 + 114689353*b1 + 534670336) / 192
$\nu^{18}$	$=$	$ ( 5084218 \beta_{19} - 3499262 \beta_{18} + 24572471 \beta_{17} - 12173930 \beta_{16} - 37050356 \beta_{15} + \cdots - 3142526039 ) / 64 $ (5084218b19 - 3499262b18 + 24572471b17 - 12173930b16 - 37050356b15 + 8089758b14 + 23869008b13 - 6969430b12 + 58780b11 - 7618686b10 - 34541846b9 - 2837082b8 - 34656801b7 - 44798806b6 + 23852348b5 + 9195408b4 + 21476414b3 + 39991115b2 + 119559248*b1 - 3142526039) / 64
$\nu^{19}$	$=$	$ ( - 322266865 \beta_{19} - 62122897 \beta_{18} + 27380814 \beta_{17} + 233109308 \beta_{16} + \cdots - 13877927372 ) / 192 $ (-322266865b19 - 62122897b18 + 27380814b17 + 233109308b16 + 98293560b15 + 4747398b14 + 288665016b13 + 143150226b12 + 173715898b11 - 22214713b10 + 260578381b9 + 13411206b8 - 75641147b7 - 871824380b6 + 118362945b5 - 49296389b4 + 250978803b3 - 8653045734b2 - 1844956933*b1 - 13877927372) / 192

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

337.1

0.404977	−	2.79928i
2.75494	−	0.640567i
−0.878338	+	2.68859i
1.92113	+	2.07587i
−2.80433	+	0.368457i
−1.78337	−	2.19536i
2.78865	+	0.472666i
2.32756	−	1.60701i
−2.07092	+	1.92647i
−1.66031	−	2.28984i
−1.66031	+	2.28984i
−2.07092	−	1.92647i
2.32756	+	1.60701i
2.78865	−	0.472666i
−1.78337	+	2.19536i
−2.80433	−	0.368457i
1.92113	−	2.07587i
−0.878338	−	2.68859i
2.75494	+	0.640567i
0.404977	+	2.79928i

−

3.00000i

−

19.4431i

7.00000

−9.00000

337.2

−

3.00000i

−

19.2168i

7.00000

−9.00000

337.3

−

3.00000i

−

8.95699i

7.00000

−9.00000

337.4

−

3.00000i

−

6.44878i

7.00000

−9.00000

337.5

−

3.00000i

−

5.44298i

7.00000

−9.00000

337.6

−

3.00000i

−

4.34928i

7.00000

−9.00000

337.7

−

3.00000i

8.98099i

7.00000

−9.00000

337.8

−

3.00000i

14.0008i

7.00000

−9.00000

337.9

−

3.00000i

14.7392i

7.00000

−9.00000

337.10

−

3.00000i

16.1369i

7.00000

−9.00000

337.11

3.00000i

−

16.1369i

7.00000

−9.00000

337.12

3.00000i

−

14.7392i

7.00000

−9.00000

337.13

3.00000i

−

14.0008i

7.00000

−9.00000

337.14

3.00000i

−

8.98099i

7.00000

−9.00000

337.15

3.00000i

4.34928i

7.00000

−9.00000

337.16

3.00000i

5.44298i

7.00000

−9.00000

337.17

3.00000i

6.44878i

7.00000

−9.00000

337.18

3.00000i

8.95699i

7.00000

−9.00000

337.19

3.00000i

19.2168i

7.00000

−9.00000

337.20

3.00000i

19.4431i

7.00000

−9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.4.c.b		20
3.b	odd	2	1		2016.4.c.f		20
4.b	odd	2	1		168.4.c.b	✓	20
8.b	even	2	1	inner	672.4.c.b		20
8.d	odd	2	1		168.4.c.b	✓	20
12.b	even	2	1		504.4.c.e		20
24.f	even	2	1		504.4.c.e		20
24.h	odd	2	1		2016.4.c.f		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.c.b	✓	20	4.b	odd	2	1
168.4.c.b	✓	20	8.d	odd	2	1
504.4.c.e		20	12.b	even	2	1
504.4.c.e		20	24.f	even	2	1
672.4.c.b		20	1.a	even	1	1	trivial
672.4.c.b		20	8.b	even	2	1	inner
2016.4.c.f		20	3.b	odd	2	1
2016.4.c.f		20	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{20} + 1672 T_{5}^{18} + 1173484 T_{5}^{16} + 450983936 T_{5}^{14} + 103974448816 T_{5}^{12} + \cdots + 23\!\cdots\!84 $ T5^20 + 1672*T5^18 + 1173484*T5^16 + 450983936*T5^14 + 103974448816*T5^12 + 14850485033600*T5^10 + 1316041065605696*T5^8 + 70903953752226816*T5^6 + 2212040377493473280*T5^4 + 36053012128106496000*T5^2 + 233465691124629520384 acting on $S_{4}^{\mathrm{new}}(672, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ (T^{2} + 9)^{10} $ (T^2 + 9)^10
$5$	$ T^{20} + \cdots + 23\!\cdots\!84 $ T^20 + 1672T^18 + 1173484T^16 + 450983936T^14 + 103974448816T^12 + 14850485033600T^10 + 1316041065605696T^8 + 70903953752226816T^6 + 2212040377493473280T^4 + 36053012128106496000*T^2 + 233465691124629520384
$7$	$ (T - 7)^{20} $ (T - 7)^20
$11$	$ T^{20} + \cdots + 75\!\cdots\!00 $ T^20 + 15704T^18 + 102365708T^16 + 360879005312T^14 + 751563079243696T^12 + 946283918948315008T^10 + 703836288585119300672T^8 + 285648037673111796358144T^6 + 53220218147801329796122624T^4 + 3586257170236870779544289280*T^2 + 75730207643638981857168998400
$13$	$ T^{20} + \cdots + 52\!\cdots\!96 $ T^20 + 30352T^18 + 378661408T^16 + 2522123049728T^14 + 9742795424557312T^12 + 22147646137863249920T^10 + 28688501139509429583872T^8 + 19667401526399964284977152T^6 + 6605281544145016059925102592T^4 + 1005735925459832055914123231232*T^2 + 52926420921772555195431723728896
$17$	$ (T^{10} + \cdots + 21\!\cdots\!84)^{2} $ (T^10 + 26T^9 - 27018T^8 - 1042544T^7 + 216516980T^6 + 11020076040T^5 - 483856853416T^4 - 31730435368448T^3 + 33437020420992T^2 + 19627436397294464*T + 214673556973158784)^2
$19$	$ T^{20} + \cdots + 13\!\cdots\!44 $ T^20 + 85032T^18 + 2913629936T^16 + 51555146674944T^14 + 502593396637372160T^12 + 2681355887327122171904T^10 + 7431098407644204581392384T^8 + 9590307493610794391611375616T^6 + 5096445183238242813793433288704T^4 + 849872064139339837902314524377088*T^2 + 13603926435376030683628870297452544
$23$	$ (T^{10} + \cdots + 21\!\cdots\!44)^{2} $ (T^10 - 122T^9 - 66962T^8 + 5378832T^7 + 1438397972T^6 - 56184979624T^5 - 9069661448488T^4 + 271994671101312T^3 + 10871709816030208T^2 - 353209000085663744*T + 2151576659642834944)^2
$29$	$ T^{20} + \cdots + 33\!\cdots\!56 $ T^20 + 240904T^18 + 23101918832T^16 + 1133131490677504T^14 + 30818881976695512832T^12 + 482725360040045108332544T^10 + 4411408252611545248406933504T^8 + 23036133944651755275203872292864T^6 + 64121058620616299474950358906699776T^4 + 81829886633073010331346805446326353920*T^2 + 33813408983542721808644005531958384787456
$31$	$ (T^{10} + \cdots - 37\!\cdots\!76)^{2} $ (T^10 + 132T^9 - 160996T^8 - 15830304T^7 + 7577264624T^6 + 557219392576T^5 - 86429722728896T^4 - 8861733857202176T^3 - 240236250109337600T^2 - 2063454147402268672*T - 3794846278609534976)^2
$37$	$ T^{20} + \cdots + 49\!\cdots\!36 $ T^20 + 603000T^18 + 150788047472T^16 + 20413778962129152T^14 + 1639355826401973485312T^12 + 80523922770033086806292480T^10 + 2412287126608547822183861948416T^8 + 42965548476823184111454104380768256T^6 + 436731799036549540312372334876975693824T^4 + 2318127051915186126693583859240108805849088*T^2 + 4953187904643516770204555381134047858273550336
$41$	$ (T^{10} + \cdots - 59\!\cdots\!64)^{2} $ (T^10 + 118T^9 - 494618T^8 - 3763152T^7 + 86885022068T^6 - 6866917858120T^5 - 5681166207892456T^4 + 721734865492681728T^3 + 98760119364419019136T^2 - 11208973857905539557248*T - 591413176650464925467264)^2
$43$	$ T^{20} + \cdots + 10\!\cdots\!24 $ T^20 + 960088T^18 + 385099349968T^16 + 84337937601503744T^14 + 11062576589358244008448T^12 + 895560438978848949342113792T^10 + 44207510122426547534721028136960T^8 + 1251493829237416703730393547924439040T^6 + 17431105756444474830695431695017370779648T^4 + 79078391008983206403654404117288743337459712*T^2 + 105480414758180014565006528186555007676919578624
$47$	$ (T^{10} + \cdots - 23\!\cdots\!64)^{2} $ (T^10 - 594720T^8 - 11066112T^7 + 106955230464T^6 + 3778246176768T^5 - 5689913169641472T^4 - 220978359456104448T^3 + 80168961288050049024T^2 + 2801469846826248044544T - 239869143704974041022464)^2
$53$	$ T^{20} + \cdots + 48\!\cdots\!64 $ T^20 + 1326152T^18 + 682357605488T^16 + 178101820224362240T^14 + 25803820796538167017216T^12 + 2109084466299814330872678400T^10 + 93055355507477760034515565776896T^8 + 1936096636138377418084363291067416576T^6 + 13387030805141108062561738745187452059648T^4 + 16620312583551693978676301041719376204529664*T^2 + 4856515072038801159178648885069123510824075264
$59$	$ T^{20} + \cdots + 11\!\cdots\!56 $ T^20 + 2345568T^18 + 2279077190912T^16 + 1197213431704092672T^14 + 370350801501044591427584T^12 + 68569689407045661131056087040T^10 + 7336419344508466743877009980522496T^8 + 408914412289684249210521659858279727104T^6 + 9299570809383861386488231594817012119896064T^4 + 59248263644510407993267700627837841199285141504*T^2 + 111551138823487271013112373598136122647278105133056
$61$	$ T^{20} + \cdots + 19\!\cdots\!00 $ T^20 + 2784272T^18 + 3245293256288T^16 + 2061011555618727680T^14 + 775227470344628033949952T^12 + 174972316792752861708535816192T^10 + 22694457943272255586812485394563072T^8 + 1487880183559674995282599228417257766912T^6 + 35456578525817810277679285544348235285397504T^4 + 279573733792701452794199767472289390277825658880*T^2 + 192781819955599575310378457368547799133910885990400
$67$	$ T^{20} + \cdots + 25\!\cdots\!36 $ T^20 + 2330504T^18 + 2112676052432T^16 + 996473453892591104T^14 + 270311946917113731842560T^12 + 43497624298198022657184182272T^10 + 4110025122397273892163692106358784T^8 + 217795193506061979135905564089714081792T^6 + 5905779196134163274315463227149479168507904T^4 + 70639865494680874410412878550598634786607595520*T^2 + 254495572374770040856726080017109136400758405595136
$71$	$ (T^{10} + \cdots - 71\!\cdots\!36)^{2} $ (T^10 - 318T^9 - 1729162T^8 + 425926608T^7 + 1042245030804T^6 - 208094892640632T^5 - 257405923022296072T^4 + 41711694901881813120T^3 + 23501452134348460137472T^2 - 1993523475029812389040128*T - 715398606192172063919579136)^2
$73$	$ (T^{10} + \cdots + 70\!\cdots\!96)^{2} $ (T^10 + 1392T^9 - 3708T^8 - 600849792T^7 - 93669708960T^6 + 88925205680640T^5 + 12730232714760576T^4 - 5507382524080601088T^3 - 473290732495963051776T^2 + 101570488497077602037760*T + 7018866357169674911671296)^2
$79$	$ (T^{10} + \cdots + 18\!\cdots\!80)^{2} $ (T^10 + 1436T^9 - 1685576T^8 - 3270761088T^7 - 84838875840T^6 + 1564835933359872T^5 + 301091963185313280T^4 - 257645441457425055744T^3 - 49680863460570938277888T^2 + 10074422440376508450078720*T + 1881646443399124516004167680)^2
$83$	$ T^{20} + \cdots + 11\!\cdots\!56 $ T^20 + 6876256T^18 + 19014911371456T^16 + 27158487478869733376T^14 + 21318376366888501142302720T^12 + 8916826738298971924021297676288T^10 + 1705147218301039157585174635944869888T^8 + 88097506088419718523977179026163828260864T^6 + 1444720746285012514925959202751590743263936512T^4 + 7464817808057589641026155868964864984193567817728*T^2 + 11864347855688613013276594326011058763552272947347456
$89$	$ (T^{10} + \cdots + 11\!\cdots\!20)^{2} $ (T^10 - 110T^9 - 4501082T^8 - 418854768T^7 + 6886897227572T^6 + 1718073871978472T^5 - 3877003615721339752T^4 - 1253784372025831113984T^3 + 601850237658369533283712T^2 + 185810317386725208004336000*T + 11469692042233047574326670720)^2
$97$	$ (T^{10} + \cdots + 11\!\cdots\!72)^{2} $ (T^10 - 2200T^9 - 2125676T^8 + 6497925376T^7 - 1527723150752T^6 - 1847766853389056T^5 + 86402872867081088T^4 + 138696912126784905216T^3 + 19081871524261469600000T^2 + 862685905062247060211712*T + 11881671701938118407631872)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - \beta_1 q^{5} + 7 q^{7} - 9 q^{9}+O(q^{10}) \) q + b2 * q^3 - b1 * q^5 + 7 * q^7 - 9 * q^9	\( q + \beta_{2} q^{3} - \beta_1 q^{5} + 7 q^{7} - 9 q^{9} + (\beta_{6} - 2 \beta_{2}) q^{11} + ( - \beta_{9} - 2 \beta_{2}) q^{13} + (\beta_{16} - 3) q^{15} + (\beta_{5} - 3) q^{17} + (\beta_{12} + 5 \beta_{2}) q^{19} + 7 \beta_{2} q^{21} + ( - \beta_{19} + 12) q^{23} + ( - \beta_{17} + \beta_{16} + \cdots - 42) q^{25}+ \cdots + ( - 9 \beta_{6} + 18 \beta_{2}) q^{99}+O(q^{100}) \) q + b2 * q^3 - b1 * q^5 + 7 * q^7 - 9 * q^9 + (b6 - 2b2) q^11 + (-b9 - 2b2) q^13 + (b16 - 3) * q^15 + (b5 - 3) * q^17 + (b12 + 5b2) q^19 + 7b2 q^21 + (-b19 + 12) * q^23 + (-b17 + b16 + b15 - 42) * q^25 - 9b2 q^27 + (-b12 + b11 + b2 + 3b1) q^29 + (-b17 + b16 - b13 - b5 - 13) * q^31 + (b19 + b18 + 20) * q^33 - 7b1 q^35 + (b10 - b9 + b7 + 4b6 + 5b2 - 4b1) q^37 + (-b19 + b15 + b14 + b13 + 16) * q^39 + (-b19 - 3b18 + b17 - b16 - b15 - b14 - b13 + 2b5 + b3 - 13) * q^41 + (-b12 + b11 - b10 + 2b9 - 2b6 - b4 - 12b2 - 2b1) * q^43 + 9b1 q^45 + (b19 + b17 - b16 - 3b15 - b14 + b13 + b5 + 2b3 - 1) * q^47 + 49 * q^49 + (b12 - b9 + b8 - b7 - 3b2 - b1) q^51 + (-b12 - 2b9 - b8 - 4b6 - b4 + 11b2 - 7b1) * q^53 + (b19 + b18 - 3b16 - b15 + 2b14 + 4b13 + 3b5 + 3b3 - 4) q^55 + (-b18 - b17 + b15 - b14 - b5 + b3 - 45) * q^57 + (2b12 + b11 + 2b10 + 2b9 - b8 - 2b6 + b4 - 30b2 + 4b1) * q^59 + (-b12 - b11 + 2b10 + 3b9 + 2b7 - 4b6 - b4 - 50b2 + 6b1) * q^61 - 63 * q^63 + (4b19 + b18 + b16 + b14 - 4b13 + b5 + b3 + 88) * q^65 + (2b11 - b10 + 2b8 - 2b7 + 4b6 + 34b2 + 18b1) * q^67 + (2b12 - b11 + b10 - b8 + 5b6 - b4 + 12b2 - 4b1) * q^69 + (-3b19 - 2b18 - b17 - 6b16 + b15 - b13 - 3b5 + 2b3 + 33) q^71 + (-3b19 - b18 + 2b16 + b15 - b13 + 2b5 + b3 - 140) q^73 + (-3b12 + b11 + 2b9 + b8 - b7 - 2b6 - b4 - 39b2 + 8b1) q^75 + (7b6 - 14b2) * q^77 + (-2b19 - b17 + 10b16 - b15 - 2b14 + 3b13 + 7b5 + 4b3 - 147) * q^79 + 81 * q^81 + (-2b12 + 5b11 + 2b10 + 2b9 - b8 - 4b7 - 2b6 - b4 - 4b2 - 4b1) * q^83 + (-4b12 + 5b11 + 2b10 + 6b9 - 6b6 - b4 + 7b2 + 18b1) q^85 + (-2b19 + b18 + 3b17 - b16 + 3b5 - 1) q^87 + (-b19 - 3b18 + 3b17 - 6b16 - 3b15 + 4b14 - b13 - b5 + 3b3 + 12) * q^89 + (-7b9 - 14b2) * q^91 + (2b11 - 2b9 - b8 - 2b7 + 2b6 - 2b4 - 9b2 + 10b1) q^93 + (-3b19 - 3b18 + 3b17 + 19b16 + b15 - 3b14 - b13 + 5b5 + b3 + 60) * q^95 + (-b19 + b18 + 3b17 + 10b16 + b15 - 2b14 - 3b13 + 3b3 + 221) q^97 + (-9b6 + 18b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 140 q^{7} - 180 q^{9}+O(q^{10}) \) 20 * q + 140 * q^7 - 180 * q^9	\( 20 q + 140 q^{7} - 180 q^{9} - 60 q^{15} - 52 q^{17} + 244 q^{23} - 844 q^{25} - 264 q^{31} + 396 q^{33} + 312 q^{39} - 236 q^{41} + 980 q^{49} - 72 q^{55} - 912 q^{57} - 1260 q^{63} + 1744 q^{65} + 636 q^{71} - 2784 q^{73} - 2872 q^{79} + 1620 q^{81} + 220 q^{89} + 1240 q^{95} + 4400 q^{97}+O(q^{100}) \) 20 * q + 140 * q^7 - 180 * q^9 - 60 * q^15 - 52 * q^17 + 244 * q^23 - 844 * q^25 - 264 * q^31 + 396 * q^33 + 312 * q^39 - 236 * q^41 + 980 * q^49 - 72 * q^55 - 912 * q^57 - 1260 * q^63 + 1744 * q^65 + 636 * q^71 - 2784 * q^73 - 2872 * q^79 + 1620 * q^81 + 220 * q^89 + 1240 * q^95 + 4400 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	672.4.c.b

Level	$672$
Weight	$4$
Character orbit	672.c
Analytic conductor	$39.649$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(39.6492835239\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 2 x^{19} - 5 x^{18} - 26 x^{17} + 122 x^{16} + 124 x^{15} - 276 x^{14} - 1376 x^{13} + \cdots + 1073741824 \) x^20 - 2x^19 - 5x^18 - 26x^17 + 122x^16 + 124x^15 - 276x^14 - 1376x^13 - 3344x^12 + 15424x^11 - 1920x^10 + 123392x^9 - 214016x^8 - 704512x^7 - 1130496x^6 + 4063232x^5 + 31981568x^4 - 54525952x^3 - 83886080x^2 - 268435456*x + 1073741824
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{60}\cdot 3^{10} \)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 249239 \nu^{19} - 9170499 \nu^{18} - 76180607 \nu^{17} + 361395123 \nu^{16} + \cdots + 64\!\cdots\!60 ) / 154267331592192 \) (-249239v^19 - 9170499v^18 - 76180607v^17 + 361395123v^16 + 415516840v^15 + 102571018v^14 - 8076761720v^13 + 3626671812v^12 + 39649206112v^11 - 12520659504v^10 + 240130318208v^9 - 1235771452288v^8 + 204671621120v^7 - 9834948232192v^6 + 12641407373312v^5 + 82539077779456v^4 - 84223732154368v^3 - 326477670252544v^2 - 1337098988683264*v + 6492443440578560) / 154267331592192
\(\beta_{2}\)	\(=\)	\( ( 10567955 \nu^{19} - 35602606 \nu^{18} - 39792079 \nu^{17} - 378375750 \nu^{16} + \cdots - 39\!\cdots\!56 ) / 822759101825024 \) (10567955v^19 - 35602606v^18 - 39792079v^17 - 378375750v^16 + 857936990v^15 + 931239268v^14 - 1563305308v^13 - 885932416v^12 - 81947838256v^11 + 167021695296v^10 - 125038508160v^9 + 1453449232896v^8 - 1231343410176v^7 - 7215125635072v^6 - 3311153430528v^5 - 20249624969216v^4 + 305852764913664v^3 - 490834835275776v^2 - 192571150172160*v - 3988038464045056) / 822759101825024
\(\beta_{3}\)	\(=\)	\( ( - 202665 \nu^{19} - 14089318 \nu^{18} + 11395197 \nu^{17} + 96736322 \nu^{16} + \cdots + 11\!\cdots\!60 ) / 5687274897408 \) (-202665v^19 - 14089318v^18 + 11395197v^17 + 96736322v^16 + 211117158v^15 - 763902732v^14 - 2248790892v^13 + 3015992256v^12 + 8193803664v^11 + 51532465472v^10 - 43421509248v^9 - 176669105664v^8 - 1159072951296v^7 - 1031382294528v^6 + 12185911050240v^5 + 18258007162880v^4 - 33671172587520v^3 - 301881213583360v^2 + 3684628955136*v + 1181244519874560) / 5687274897408
\(\beta_{4}\)	\(=\)	\( ( 135994909 \nu^{19} - 787430514 \nu^{18} + 12407585951 \nu^{17} + 2076769958 \nu^{16} + \cdots + 33\!\cdots\!24 ) / 24\!\cdots\!72 \) (135994909v^19 - 787430514v^18 + 12407585951v^17 + 2076769958v^16 + 47364462018v^15 - 75137431076v^14 - 202000027364v^13 + 1316116003200v^12 + 1040346202928v^11 + 14252539436736v^10 - 51486551937920v^9 - 40027596042752v^8 - 69830251496448v^7 + 163209461473280v^6 + 3598048635305984v^5 - 2375881360932864v^4 - 8310679485808640v^3 - 28619453096787968v^2 + 213594179295510528*v + 335017194368794624) / 2468277305475072
\(\beta_{5}\)	\(=\)	\( ( - 68772 \nu^{19} + 1552331 \nu^{18} - 1949478 \nu^{17} - 8969215 \nu^{16} + \cdots - 117029713477632 ) / 710909362176 \) (-68772v^19 + 1552331v^18 - 1949478v^17 - 8969215v^16 - 16014546v^15 + 93281862v^14 + 212366652v^13 - 590095596v^12 - 344336208v^11 - 5125373872v^10 + 9488773632v^9 + 11329546368v^8 + 92392854528v^7 + 100286696448v^6 - 1449889640448v^5 - 713698754560v^4 + 3092927545344v^3 + 30227736559616v^2 - 43217873534976*v - 117029713477632) / 710909362176
\(\beta_{6}\)	\(=\)	\( ( 10885131 \nu^{19} - 27521502 \nu^{18} + 232133689 \nu^{17} - 901697014 \nu^{16} + \cdots - 12\!\cdots\!12 ) / 94933742518272 \) (10885131v^19 - 27521502v^18 + 232133689v^17 - 901697014v^16 + 322202414v^15 - 1292562108v^14 + 13639990788v^13 + 14264327424v^12 - 139114104368v^11 + 175510406976v^10 - 956129378944v^9 + 3763837471232v^8 - 634707037184v^7 + 7169160486912v^6 - 5842551521280v^5 - 174958784020480v^4 + 452310723461120v^3 - 35604460994560v^2 + 3665175789961216*v - 12263289794854912) / 94933742518272
\(\beta_{7}\)	\(=\)	\( ( 448208679 \nu^{19} + 1034629674 \nu^{18} - 19502268083 \nu^{17} - 36011697742 \nu^{16} + \cdots - 44\!\cdots\!96 ) / 24\!\cdots\!72 \) (448208679v^19 + 1034629674v^18 - 19502268083v^17 - 36011697742v^16 + 17724887654v^15 + 381332818452v^14 + 159182150676v^13 - 3081301669632v^12 - 1763041071728v^11 - 10566824204736v^10 + 53773600717184v^9 + 74380770724352v^8 + 186199320564736v^7 - 1001814637682688v^6 - 4529584351395840v^5 + 4689692138405888v^4 + 24344746669899776v^3 + 39684888231673856v^2 - 284463129927090176*v - 447922539537104896) / 2468277305475072
\(\beta_{8}\)	\(=\)	\( ( - 77198039 \nu^{19} + 643792422 \nu^{18} - 1075345597 \nu^{17} + 6355490174 \nu^{16} + \cdots + 10\!\cdots\!00 ) / 352611043639296 \) (-77198039v^19 + 643792422v^18 - 1075345597v^17 + 6355490174v^16 - 12599192742v^15 + 12466684684v^14 + 29350267372v^13 + 118543826496v^12 + 1256115282800v^11 - 4182073950528v^10 + 3025233390208v^9 - 29027817419264v^8 + 44682625649664v^7 + 93783782481920v^6 - 65538017542144v^5 + 760389568561152v^4 - 4541254138855424v^3 + 6874067869106176v^2 - 7923456213516288*v + 107994731354521600) / 352611043639296
\(\beta_{9}\)	\(=\)	\( ( - 104055827 \nu^{19} - 80412826 \nu^{18} + 2982879391 \nu^{17} - 888846802 \nu^{16} + \cdots - 45\!\cdots\!04 ) / 411379550912512 \) (-104055827v^19 - 80412826v^18 + 2982879391v^17 - 888846802v^16 - 6293067598v^15 - 41886116404v^14 + 122096534268v^13 + 222138996256v^12 - 154542481104v^11 + 590491677504v^10 - 8664621149568v^9 + 4807168043520v^8 - 34269223130112v^7 + 204792003936256v^6 + 235310339637248v^5 - 1498432539852800v^4 - 1596266681729024v^3 + 1582487826333696v^2 + 47273598754226176*v - 45552267851464704) / 411379550912512
\(\beta_{10}\)	\(=\)	\( ( - 44941207 \nu^{19} + 319710534 \nu^{18} - 317005309 \nu^{17} + 2683807326 \nu^{16} + \cdots + 39\!\cdots\!36 ) / 176305521819648 \) (-44941207v^19 + 319710534v^18 - 317005309v^17 + 2683807326v^16 - 6956340646v^15 + 12825179084v^14 + 1543821548v^13 + 56860920000v^12 + 378625683056v^11 - 1333050694464v^10 + 1262767844992v^9 - 11295935853056v^8 + 19956892429312v^7 + 22204836044800v^6 - 19894624206848v^5 + 214916617207808v^4 - 1682841687031808v^3 + 5817610453123072v^2 - 6213611143823360*v + 39627131491188736) / 176305521819648
\(\beta_{11}\)	\(=\)	\( ( - 26115319 \nu^{19} - 17003346 \nu^{18} + 226602675 \nu^{17} - 66446698 \nu^{16} + \cdots - 14\!\cdots\!64 ) / 94933742518272 \) (-26115319v^19 - 17003346v^18 + 226602675v^17 - 66446698v^16 - 1599657494v^15 - 4701785572v^14 + 9982337804v^13 + 7099780896v^12 + 50045726448v^11 - 81775917504v^10 - 434894395776v^9 - 2310117176832v^8 - 103974335488v^7 + 16305342447616v^6 + 52635255783424v^5 - 188669288382464v^4 - 723220596523008v^3 + 476570217086976v^2 + 668790312402944*v - 1477826171633664) / 94933742518272
\(\beta_{12}\)	\(=\)	\( ( 238410449 \nu^{19} + 718435510 \nu^{18} - 8207601221 \nu^{17} + 9821924366 \nu^{16} + \cdots + 23\!\cdots\!44 ) / 822759101825024 \) (238410449v^19 + 718435510v^18 - 8207601221v^17 + 9821924366v^16 + 11290429482v^15 + 118379531308v^14 - 280593528244v^13 - 527365171264v^12 + 1441833089776v^11 - 4494021887296v^10 + 29857710984832v^9 - 50583697592832v^8 + 112048889719808v^7 - 477627790270464v^6 - 491407645949952v^5 + 4463492083482624v^4 + 1169148622667776v^3 + 8209608216674304v^2 - 127149972649934848*v + 230594015546834944) / 822759101825024
\(\beta_{13}\)	\(=\)	\( ( 2372275 \nu^{19} - 10488814 \nu^{18} + 9104593 \nu^{17} - 25546758 \nu^{16} + \cdots - 209083705589760 ) / 5687274897408 \) (2372275v^19 - 10488814v^18 + 9104593v^17 - 25546758v^16 + 154532062v^15 - 339732124v^14 - 227818076v^13 + 3405169536v^12 - 11059478576v^11 + 35177847104v^10 - 69328385152v^9 + 310367212032v^8 - 500690025472v^7 - 1387557658624v^6 + 4145202151424v^5 - 6301718151168v^4 + 16098725986304v^3 - 164895706316800v^2 + 435397024808960*v - 209083705589760) / 5687274897408
\(\beta_{14}\)	\(=\)	\( ( - 2797375 \nu^{19} - 7895890 \nu^{18} + 14918939 \nu^{17} + 103151350 \nu^{16} + \cdots + 860759563173888 ) / 5687274897408 \) (-2797375v^19 - 7895890v^18 + 14918939v^17 + 103151350v^16 + 10838234v^15 - 624954404v^14 - 907460692v^13 + 2435107296v^12 + 10284383216v^11 + 35578589504v^10 - 25407754112v^9 - 285391517184v^8 - 738784756736v^7 + 22325690368v^6 + 9749757804544v^5 + 1320198209536v^4 - 49469222027264v^3 - 220051680526336v^2 + 148867744006144*v + 860759563173888) / 5687274897408
\(\beta_{15}\)	\(=\)	\( ( 3017837 \nu^{19} - 27312378 \nu^{18} - 11097025 \nu^{17} + 155360078 \nu^{16} + \cdots + 25\!\cdots\!52 ) / 5687274897408 \) (3017837v^19 - 27312378v^18 - 11097025v^17 + 155360078v^16 + 415564946v^15 - 425733044v^14 - 5561053636v^13 + 6166924320v^12 + 5189231024v^11 + 72848155968v^10 + 33597166720v^9 - 274212711936v^8 - 904260051968v^7 - 5498544521216v^6 + 17496898355200v^5 + 33960772370432v^4 - 29150950719488v^3 - 468283639726080v^2 - 193542584532992*v + 2580248109514752) / 5687274897408
\(\beta_{16}\)	\(=\)	\( ( - 1024917 \nu^{19} + 751938 \nu^{18} + 1597593 \nu^{17} + 23152362 \nu^{16} + \cdots + 215348787806208 ) / 1895758299136 \) (-1024917v^19 + 751938v^18 + 1597593v^17 + 23152362v^16 - 35219922v^15 - 123191676v^14 - 12650748v^13 + 275677056v^12 + 5151510480v^11 - 3870979776v^10 + 3698697600v^9 - 97117644288v^8 - 64318245888v^7 + 495805341696v^6 + 822640558080v^5 + 297144680448v^4 - 21405552869376v^3 + 6044415688704v^2 + 6132458323968*v + 215348787806208) / 1895758299136
\(\beta_{17}\)	\(=\)	\( ( 101797 \nu^{19} + 27810 \nu^{18} - 759857 \nu^{17} - 1519646 \nu^{16} + 2251594 \nu^{15} + \cdots + 1907538001920 ) / 177727340544 \) (101797v^19 + 27810v^18 - 759857v^17 - 1519646v^16 + 2251594v^15 + 21185444v^14 - 33194708v^13 - 15629520v^12 - 254539088v^11 - 213221760v^10 + 2602940288v^9 + 3768883200v^8 + 11609142272v^7 - 90829336576v^6 - 64797196288v^5 + 291173761024v^4 + 1619550273536v^3 + 305871716352v^2 - 12138198335488*v + 1907538001920) / 177727340544
\(\beta_{18}\)	\(=\)	\( ( - 5692025 \nu^{19} + 21035450 \nu^{18} + 402637 \nu^{17} + 28868706 \nu^{16} + \cdots - 34404298653696 ) / 5687274897408 \) (-5692025v^19 + 21035450v^18 + 402637v^17 + 28868706v^16 - 452436986v^15 + 147301556v^14 + 2409039700v^13 - 4133408064v^12 + 23356219792v^11 - 73008488128v^10 + 91646262656v^9 - 511008861696v^8 + 673817670656v^7 + 4524788129792v^6 - 9456951672832v^5 + 890601209856v^4 - 88552783740928v^3 + 371191508369408v^2 - 324748366053376*v - 34404298653696) / 5687274897408
\(\beta_{19}\)	\(=\)	\( ( - 847982 \nu^{19} - 1675015 \nu^{18} + 2115040 \nu^{17} + 36344691 \nu^{16} + \cdots + 379966093000704 ) / 710909362176 \) (-847982v^19 - 1675015v^18 + 2115040v^17 + 36344691v^16 - 769826v^15 - 180476158v^14 - 376473620v^13 + 783815340v^12 + 4684136944v^11 + 2328552752v^10 + 4477780736v^9 - 103967011968v^8 - 181116402688v^7 + 78578889728v^6 + 2593983967232v^5 + 2751109447680v^4 - 24595025428480v^3 - 36662407069696v^2 + 27822865252352*v + 379966093000704) / 710909362176

\(\nu\)	\(=\)	\( ( \beta_{19} + \beta_{18} - 2 \beta_{16} - 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \cdots + 20 ) / 192 \) (b19 + b18 - 2b16 - 2b11 - b10 + b9 + b7 - 2b6 - 3b5 + b4 - 3b3 + 12b2 - b1 + 20) / 192
\(\nu^{2}\)	\(=\)	\( ( - \beta_{17} + 2 \beta_{16} - 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 45 ) / 64 \) (-b17 + 2b16 - 2b14 + 2b12 + 2b10 + 2b9 - 2b8 + b7 + 2b6 - 2b5 + 5*b2 + 45) / 64
\(\nu^{3}\)	\(=\)	\( ( - 5 \beta_{19} - 5 \beta_{18} + 6 \beta_{17} + 20 \beta_{16} + 6 \beta_{14} - 24 \beta_{13} + \cdots + 1124 ) / 192 \) (-5b19 - 5b18 + 6b17 + 20b16 + 6b14 - 24b13 + 10b12 - 14b11 - b10 + 21b9 - 2b8 + 9b7 + 28b6 - 27b5 - 5b4 - 9b3 - 18b2 + 19*b1 + 1124) / 192
\(\nu^{4}\)	\(=\)	\( ( 14 \beta_{19} + 22 \beta_{18} - 3 \beta_{17} - 46 \beta_{16} - 4 \beta_{15} - 6 \beta_{14} - 2 \beta_{12} + \cdots - 421 ) / 64 \) (14b19 + 22b18 - 3b17 - 46b16 - 4b15 - 6b14 - 2b12 - 12b11 + 2b10 - 14b9 + 2b8 + 5b7 + 38b6 - 20b5 + 4b4 + 2b3 + 165b2 + 28b1 - 421) / 64
\(\nu^{5}\)	\(=\)	\( ( 117 \beta_{19} - 75 \beta_{18} + 54 \beta_{17} - 108 \beta_{16} - 120 \beta_{15} - 30 \beta_{14} + \cdots - 1752 ) / 192 \) (117b19 - 75b18 + 54b17 - 108b16 - 120b15 - 30b14 - 120b13 + 118b12 + 62b11 + 61b10 - 65b9 - 14b8 + 43b7 + 284b6 - 165b5 + 41b4 + 33b3 - 462b2 - 975*b1 - 1752) / 192
\(\nu^{6}\)	\(=\)	\( ( 34 \beta_{19} - 22 \beta_{18} - 49 \beta_{17} + 38 \beta_{16} + 44 \beta_{15} - 66 \beta_{14} + \cdots + 5345 ) / 64 \) (34b19 - 22b18 - 49b17 + 38b16 + 44b15 - 66b14 - 144b13 + 42b12 - 52b11 - 38b10 - 46b9 + 70b8 - 65b7 + 58b6 + 4b5 - 8b4 + 6b3 + 3331b2 - 1448*b1 + 5345) / 64
\(\nu^{7}\)	\(=\)	\( ( 359 \beta_{19} + 455 \beta_{18} - 1242 \beta_{17} - 1028 \beta_{16} + 312 \beta_{15} + 150 \beta_{14} + \cdots - 65156 ) / 192 \) (359b19 + 455b18 - 1242b17 - 1028b16 + 312b15 + 150b14 - 264b13 + 514b12 + 202b11 - 289b10 - 2283b9 + 214b8 - 507b7 + 4228b6 - 567b5 + 211b4 - 981b3 - 10686b2 - 1853*b1 - 65156) / 192
\(\nu^{8}\)	\(=\)	\( ( 434 \beta_{19} - 758 \beta_{18} - 155 \beta_{17} + 642 \beta_{16} - 188 \beta_{15} - 366 \beta_{14} + \cdots + 80435 ) / 64 \) (434b19 - 758b18 - 155b17 + 642b16 - 188b15 - 366b14 + 144b13 - 218b12 - 276b11 + 666b10 - 1734b9 - 278b8 + 17b7 + 2270b6 + 704b5 - 124b4 + 558b3 - 36447b2 - 7260*b1 + 80435) / 64
\(\nu^{9}\)	\(=\)	\( ( 4637 \beta_{19} + 4061 \beta_{18} + 6054 \beta_{17} - 628 \beta_{16} + 2328 \beta_{15} + 2682 \beta_{14} + \cdots + 179728 ) / 192 \) (4637b19 + 4061b18 + 6054b17 - 628b16 + 2328b15 + 2682b14 + 1704b13 + 9582b12 - 4274b11 - 6907b10 + 3463b9 - 678b8 - 5165b7 + 13876b6 + 4299b5 - 1127b4 - 4551b3 - 18726b2 - 49063*b1 + 179728) / 192
\(\nu^{10}\)	\(=\)	\( ( - 4182 \beta_{19} + 7282 \beta_{18} - 5177 \beta_{17} + 7158 \beta_{16} + 1804 \beta_{15} + \cdots - 113927 ) / 64 \) (-4182b19 + 7282b18 - 5177b17 + 7158b16 + 1804b15 + 4958b14 + 11088b13 + 7402b12 - 2756b11 + 5138b10 - 6438b9 - 5786b8 - 2497b7 + 8074b6 - 1364b5 + 2624b4 + 1006b3 + 56507b2 - 39552*b1 - 113927) / 64
\(\nu^{11}\)	\(=\)	\( ( - 70049 \beta_{19} - 28097 \beta_{18} + 26046 \beta_{17} + 269404 \beta_{16} + 45048 \beta_{15} + \cdots - 1130332 ) / 192 \) (-70049b19 - 28097b18 + 26046b17 + 269404b16 + 45048b15 + 20550b14 + 15672b13 + 23634b12 - 35174b11 + 1367b10 + 29629b9 - 3258b8 + 60709b7 + 102628b6 - 1839b5 + 20395b4 + 15171b3 - 2705526b2 - 325909*b1 - 1130332) / 192
\(\nu^{12}\)	\(=\)	\( ( - 4182 \beta_{19} + 34306 \beta_{18} + 43725 \beta_{17} + 114706 \beta_{16} + 17092 \beta_{15} + \cdots - 5933157 ) / 64 \) (-4182b19 + 34306b18 + 43725b17 + 114706b16 + 17092b15 + 1170b14 + 32976b13 + 13350b12 - 36804b11 + 12002b10 + 7666b9 + 18218b8 - 29375b7 + 175790b6 + 28376b5 - 10564b4 + 4102b3 + 2186361b2 + 229116*b1 - 5933157) / 64
\(\nu^{13}\)	\(=\)	\( ( - 8507 \beta_{19} + 229957 \beta_{18} - 216642 \beta_{17} - 441412 \beta_{16} - 185544 \beta_{15} + \cdots - 6708472 ) / 192 \) (-8507b19 + 229957b18 - 216642b17 - 441412b16 - 185544b15 + 413754b14 + 649704b13 + 686446b12 + 153022b11 + 185549b10 + 288255b9 + 117658b8 + 163299b7 + 1297924b6 + 273459b5 + 22801b4 + 462369b3 - 833646b2 + 329617*b1 - 6708472) / 192
\(\nu^{14}\)	\(=\)	\( ( 68994 \beta_{19} - 191606 \beta_{18} + 160223 \beta_{17} - 141274 \beta_{16} + 293676 \beta_{15} + \cdots + 100625 ) / 64 \) (68994b19 - 191606b18 + 160223b17 - 141274b16 + 293676b15 + 236990b14 - 237936b13 - 312438b12 - 245172b11 + 205386b10 + 255522b9 + 216326b8 + 291711b7 - 394982b6 + 704804b5 + 258920b4 + 44486b3 + 1002355b2 - 2937304*b1 + 100625) / 64
\(\nu^{15}\)	\(=\)	\( ( 2049879 \beta_{19} - 2335113 \beta_{18} - 2365338 \beta_{17} + 1589292 \beta_{16} - 729576 \beta_{15} + \cdots - 474540180 ) / 192 \) (2049879b19 - 2335113b18 - 2365338b17 + 1589292b16 - 729576b15 - 3089466b14 - 165384b13 + 3325906b12 + 495722b11 - 2427761b10 + 1621957b9 + 1164166b8 - 2450123b7 + 5539604b6 + 8174601b5 + 1664435b4 + 1998651b3 + 84826098b2 + 19798707*b1 - 474540180) / 192
\(\nu^{16}\)	\(=\)	\( ( - 929742 \beta_{19} - 3694134 \beta_{18} - 4003595 \beta_{17} - 1513822 \beta_{16} - 27676 \beta_{15} + \cdots - 304116733 ) / 64 \) (-929742b19 - 3694134b18 - 4003595b17 - 1513822b16 - 27676b15 + 1293618b14 + 169488b13 + 1580166b12 - 259988b11 + 3034986b10 + 1641578b9 - 1245430b8 - 858335b7 - 3176674b6 + 3885184b5 - 1968300b4 + 1653454b3 - 102035679b2 - 5859948*b1 - 304116733) / 64
\(\nu^{17}\)	\(=\)	\( ( - 2456563 \beta_{19} + 4158797 \beta_{18} + 14825622 \beta_{17} - 40809044 \beta_{16} + \cdots + 534670336 ) / 192 \) (-2456563b19 + 4158797b18 + 14825622b17 - 40809044b16 + 21910296b15 + 10960506b14 - 10451928b13 - 18566738b12 + 1502830b11 - 16849387b10 + 18200823b9 - 1277798b8 - 10585197b7 + 10613524b6 + 60551163b5 - 1724087b4 + 13122921b3 - 1406413782b2 + 114689353*b1 + 534670336) / 192
\(\nu^{18}\)	\(=\)	\( ( 5084218 \beta_{19} - 3499262 \beta_{18} + 24572471 \beta_{17} - 12173930 \beta_{16} - 37050356 \beta_{15} + \cdots - 3142526039 ) / 64 \) (5084218b19 - 3499262b18 + 24572471b17 - 12173930b16 - 37050356b15 + 8089758b14 + 23869008b13 - 6969430b12 + 58780b11 - 7618686b10 - 34541846b9 - 2837082b8 - 34656801b7 - 44798806b6 + 23852348b5 + 9195408b4 + 21476414b3 + 39991115b2 + 119559248*b1 - 3142526039) / 64
\(\nu^{19}\)	\(=\)	\( ( - 322266865 \beta_{19} - 62122897 \beta_{18} + 27380814 \beta_{17} + 233109308 \beta_{16} + \cdots - 13877927372 ) / 192 \) (-322266865b19 - 62122897b18 + 27380814b17 + 233109308b16 + 98293560b15 + 4747398b14 + 288665016b13 + 143150226b12 + 173715898b11 - 22214713b10 + 260578381b9 + 13411206b8 - 75641147b7 - 871824380b6 + 118362945b5 - 49296389b4 + 250978803b3 - 8653045734b2 - 1844956933*b1 - 13877927372) / 192

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 337.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( (T^{2} + 9)^{10} \) (T^2 + 9)^10
$5$	\( T^{20} + \cdots + 23\!\cdots\!84 \) T^20 + 1672T^18 + 1173484T^16 + 450983936T^14 + 103974448816T^12 + 14850485033600T^10 + 1316041065605696T^8 + 70903953752226816T^6 + 2212040377493473280T^4 + 36053012128106496000*T^2 + 233465691124629520384
$7$	\( (T - 7)^{20} \) (T - 7)^20
$11$	\( T^{20} + \cdots + 75\!\cdots\!00 \) T^20 + 15704T^18 + 102365708T^16 + 360879005312T^14 + 751563079243696T^12 + 946283918948315008T^10 + 703836288585119300672T^8 + 285648037673111796358144T^6 + 53220218147801329796122624T^4 + 3586257170236870779544289280*T^2 + 75730207643638981857168998400
$13$	\( T^{20} + \cdots + 52\!\cdots\!96 \) T^20 + 30352T^18 + 378661408T^16 + 2522123049728T^14 + 9742795424557312T^12 + 22147646137863249920T^10 + 28688501139509429583872T^8 + 19667401526399964284977152T^6 + 6605281544145016059925102592T^4 + 1005735925459832055914123231232*T^2 + 52926420921772555195431723728896
$17$	\( (T^{10} + \cdots + 21\!\cdots\!84)^{2} \) (T^10 + 26T^9 - 27018T^8 - 1042544T^7 + 216516980T^6 + 11020076040T^5 - 483856853416T^4 - 31730435368448T^3 + 33437020420992T^2 + 19627436397294464*T + 214673556973158784)^2
$19$	\( T^{20} + \cdots + 13\!\cdots\!44 \) T^20 + 85032T^18 + 2913629936T^16 + 51555146674944T^14 + 502593396637372160T^12 + 2681355887327122171904T^10 + 7431098407644204581392384T^8 + 9590307493610794391611375616T^6 + 5096445183238242813793433288704T^4 + 849872064139339837902314524377088*T^2 + 13603926435376030683628870297452544
$23$	\( (T^{10} + \cdots + 21\!\cdots\!44)^{2} \) (T^10 - 122T^9 - 66962T^8 + 5378832T^7 + 1438397972T^6 - 56184979624T^5 - 9069661448488T^4 + 271994671101312T^3 + 10871709816030208T^2 - 353209000085663744*T + 2151576659642834944)^2
$29$	\( T^{20} + \cdots + 33\!\cdots\!56 \) T^20 + 240904T^18 + 23101918832T^16 + 1133131490677504T^14 + 30818881976695512832T^12 + 482725360040045108332544T^10 + 4411408252611545248406933504T^8 + 23036133944651755275203872292864T^6 + 64121058620616299474950358906699776T^4 + 81829886633073010331346805446326353920*T^2 + 33813408983542721808644005531958384787456
$31$	\( (T^{10} + \cdots - 37\!\cdots\!76)^{2} \) (T^10 + 132T^9 - 160996T^8 - 15830304T^7 + 7577264624T^6 + 557219392576T^5 - 86429722728896T^4 - 8861733857202176T^3 - 240236250109337600T^2 - 2063454147402268672*T - 3794846278609534976)^2
$37$	\( T^{20} + \cdots + 49\!\cdots\!36 \) T^20 + 603000T^18 + 150788047472T^16 + 20413778962129152T^14 + 1639355826401973485312T^12 + 80523922770033086806292480T^10 + 2412287126608547822183861948416T^8 + 42965548476823184111454104380768256T^6 + 436731799036549540312372334876975693824T^4 + 2318127051915186126693583859240108805849088*T^2 + 4953187904643516770204555381134047858273550336
$41$	\( (T^{10} + \cdots - 59\!\cdots\!64)^{2} \) (T^10 + 118T^9 - 494618T^8 - 3763152T^7 + 86885022068T^6 - 6866917858120T^5 - 5681166207892456T^4 + 721734865492681728T^3 + 98760119364419019136T^2 - 11208973857905539557248*T - 591413176650464925467264)^2
$43$	\( T^{20} + \cdots + 10\!\cdots\!24 \) T^20 + 960088T^18 + 385099349968T^16 + 84337937601503744T^14 + 11062576589358244008448T^12 + 895560438978848949342113792T^10 + 44207510122426547534721028136960T^8 + 1251493829237416703730393547924439040T^6 + 17431105756444474830695431695017370779648T^4 + 79078391008983206403654404117288743337459712*T^2 + 105480414758180014565006528186555007676919578624
$47$	\( (T^{10} + \cdots - 23\!\cdots\!64)^{2} \) (T^10 - 594720T^8 - 11066112T^7 + 106955230464T^6 + 3778246176768T^5 - 5689913169641472T^4 - 220978359456104448T^3 + 80168961288050049024T^2 + 2801469846826248044544T - 239869143704974041022464)^2
$53$	\( T^{20} + \cdots + 48\!\cdots\!64 \) T^20 + 1326152T^18 + 682357605488T^16 + 178101820224362240T^14 + 25803820796538167017216T^12 + 2109084466299814330872678400T^10 + 93055355507477760034515565776896T^8 + 1936096636138377418084363291067416576T^6 + 13387030805141108062561738745187452059648T^4 + 16620312583551693978676301041719376204529664*T^2 + 4856515072038801159178648885069123510824075264
$59$	\( T^{20} + \cdots + 11\!\cdots\!56 \) T^20 + 2345568T^18 + 2279077190912T^16 + 1197213431704092672T^14 + 370350801501044591427584T^12 + 68569689407045661131056087040T^10 + 7336419344508466743877009980522496T^8 + 408914412289684249210521659858279727104T^6 + 9299570809383861386488231594817012119896064T^4 + 59248263644510407993267700627837841199285141504*T^2 + 111551138823487271013112373598136122647278105133056
$61$	\( T^{20} + \cdots + 19\!\cdots\!00 \) T^20 + 2784272T^18 + 3245293256288T^16 + 2061011555618727680T^14 + 775227470344628033949952T^12 + 174972316792752861708535816192T^10 + 22694457943272255586812485394563072T^8 + 1487880183559674995282599228417257766912T^6 + 35456578525817810277679285544348235285397504T^4 + 279573733792701452794199767472289390277825658880*T^2 + 192781819955599575310378457368547799133910885990400
$67$	\( T^{20} + \cdots + 25\!\cdots\!36 \) T^20 + 2330504T^18 + 2112676052432T^16 + 996473453892591104T^14 + 270311946917113731842560T^12 + 43497624298198022657184182272T^10 + 4110025122397273892163692106358784T^8 + 217795193506061979135905564089714081792T^6 + 5905779196134163274315463227149479168507904T^4 + 70639865494680874410412878550598634786607595520*T^2 + 254495572374770040856726080017109136400758405595136
$71$	\( (T^{10} + \cdots - 71\!\cdots\!36)^{2} \) (T^10 - 318T^9 - 1729162T^8 + 425926608T^7 + 1042245030804T^6 - 208094892640632T^5 - 257405923022296072T^4 + 41711694901881813120T^3 + 23501452134348460137472T^2 - 1993523475029812389040128*T - 715398606192172063919579136)^2
$73$	\( (T^{10} + \cdots + 70\!\cdots\!96)^{2} \) (T^10 + 1392T^9 - 3708T^8 - 600849792T^7 - 93669708960T^6 + 88925205680640T^5 + 12730232714760576T^4 - 5507382524080601088T^3 - 473290732495963051776T^2 + 101570488497077602037760*T + 7018866357169674911671296)^2
$79$	\( (T^{10} + \cdots + 18\!\cdots\!80)^{2} \) (T^10 + 1436T^9 - 1685576T^8 - 3270761088T^7 - 84838875840T^6 + 1564835933359872T^5 + 301091963185313280T^4 - 257645441457425055744T^3 - 49680863460570938277888T^2 + 10074422440376508450078720*T + 1881646443399124516004167680)^2
$83$	\( T^{20} + \cdots + 11\!\cdots\!56 \) T^20 + 6876256T^18 + 19014911371456T^16 + 27158487478869733376T^14 + 21318376366888501142302720T^12 + 8916826738298971924021297676288T^10 + 1705147218301039157585174635944869888T^8 + 88097506088419718523977179026163828260864T^6 + 1444720746285012514925959202751590743263936512T^4 + 7464817808057589641026155868964864984193567817728*T^2 + 11864347855688613013276594326011058763552272947347456
$89$	\( (T^{10} + \cdots + 11\!\cdots\!20)^{2} \) (T^10 - 110T^9 - 4501082T^8 - 418854768T^7 + 6886897227572T^6 + 1718073871978472T^5 - 3877003615721339752T^4 - 1253784372025831113984T^3 + 601850237658369533283712T^2 + 185810317386725208004336000*T + 11469692042233047574326670720)^2
$97$	\( (T^{10} + \cdots + 11\!\cdots\!72)^{2} \) (T^10 - 2200T^9 - 2125676T^8 + 6497925376T^7 - 1527723150752T^6 - 1847766853389056T^5 + 86402872867081088T^4 + 138696912126784905216T^3 + 19081871524261469600000T^2 + 862685905062247060211712*T + 11881671701938118407631872)^2
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