LMFDB - Newform orbit 2112.2.h.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1759,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2112.1759");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.h (of order $2$, degree $1$, 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:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 5 x^{14} + 20 x^{13} - 49 x^{12} + 56 x^{11} + 174 x^{10} - 572 x^{9} + 120 x^{8} + \cdots + 433 $ x^16 - 2x^15 + 5x^14 + 20x^13 - 49x^12 + 56x^11 + 174x^10 - 572x^9 + 120x^8 + 1148x^7 - 2520x^6 - 296x^5 + 4721x^4 - 1622x^3 - 1429x^2 + 788*x + 433
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + q^{3} + \beta_{11} q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10}) $ q + q^3 + b11 * q^5 + b2 * q^7 + q^9	$ q + q^{3} + \beta_{11} q^{5} + \beta_{2} q^{7} + q^{9} + (\beta_{9} - 1) q^{11} + \beta_{8} q^{13} + \beta_{11} q^{15} + \beta_{6} q^{17} + (\beta_{13} + \beta_{6}) q^{19} + \beta_{2} q^{21} + (\beta_{7} + \beta_{5}) q^{23} + ( - \beta_{12} - \beta_{9} + \beta_{4} - 1) q^{25} + q^{27} - \beta_{10} q^{29} + (\beta_{14} - \beta_{11}) q^{31} + (\beta_{9} - 1) q^{33} + (\beta_{13} - \beta_{12} + \beta_{9} - 1) q^{35} + (\beta_{14} + \beta_{5}) q^{37} + \beta_{8} q^{39} - \beta_{15} q^{41} + ( - \beta_{13} + \beta_{6}) q^{43} + \beta_{11} q^{45} + (\beta_{14} + 2 \beta_{11}) q^{47} - \beta_{4} q^{49} + \beta_{6} q^{51} + (2 \beta_{14} + \beta_{11} - \beta_{7}) q^{53} + (\beta_{11} - \beta_{8} + \cdots - 2 \beta_{2}) q^{55}+ \cdots + (\beta_{9} - 1) q^{99}+O(q^{100}) $ q + q^3 + b11 * q^5 + b2 * q^7 + q^9 + (b9 - 1) * q^11 + b8 * q^13 + b11 * q^15 + b6 * q^17 + (b13 + b6) * q^19 + b2 * q^21 + (b7 + b5) * q^23 + (-b12 - b9 + b4 - 1) * q^25 + q^27 - b10 * q^29 + (b14 - b11) * q^31 + (b9 - 1) * q^33 + (b13 - b12 + b9 - 1) * q^35 + (b14 + b5) * q^37 + b8 * q^39 - b15 * q^41 + (-b13 + b6) * q^43 + b11 * q^45 + (b14 + 2b11) q^47 - b4 * q^49 + b6 * q^51 + (2b14 + b11 - b7) q^53 + (b11 - b8 - b7 - b5 - 2b2) q^55 + (b13 + b6) * q^57 + (-b12 - b9 + b4 + b3 - 5) * q^59 + (2b2 + b1) q^61 + b2 * q^63 + (b15 - 2b12 + 2b9 - b6 - 2) * q^65 + (b12 + b9 + b4 + 4) * q^67 + (b7 + b5) * q^69 + (b14 - b7) * q^71 + (b15 - b13 - b12 + b9 - b6 - 1) * q^73 + (-b12 - b9 + b4 - 1) * q^75 + (-b14 + 2b11 + b8 - b2) q^77 + (-b8 + b2 - b1) * q^79 + q^81 + (b15 - b12 + b9 + b6 - 1) * q^83 + (-b10 + b8 + b2) * q^85 - b10 * q^87 + (-b4 + b3 + 2) * q^89 + (3b12 + 3b9 - 2b4 + b3) q^91 + (b14 - b11) * q^93 + (-b10 - 3b2 - b1) q^95 + (-2b12 - 2b9 - b4 - 1) * q^97 + (b9 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 16 * q^9	$ 16 q + 16 q^{3} + 16 q^{9} - 8 q^{11} - 16 q^{25} + 16 q^{27} - 8 q^{33} - 80 q^{59} + 64 q^{67} - 16 q^{75} + 16 q^{81} + 32 q^{89} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^11 - 16 * q^25 + 16 * q^27 - 8 * q^33 - 80 * q^59 + 64 * q^67 - 16 * q^75 + 16 * q^81 + 32 * q^89 - 16 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2 x^{15} + 5 x^{14} + 20 x^{13} - 49 x^{12} + 56 x^{11} + 174 x^{10} - 572 x^{9} + 120 x^{8} + \cdots + 433 $ x^16 - 2*x^15 + 5*x^14 + 20*x^13 - 49*x^12 + 56*x^11 + 174*x^10 - 572*x^9 + 120*x^8 + 1148*x^7 - 2520*x^6 - 296*x^5 + 4721*x^4 - 1622*x^3 - 1429*x^2 + 788*x + 433 :

$\beta_{1}$	$=$	$ ( - 34\!\cdots\!17 \nu^{15} + \cdots + 32\!\cdots\!04 ) / 45\!\cdots\!16 $ (-3411519755021917v^15 + 25218117466846640v^14 - 166238613719899032v^13 + 104922309992602242v^12 + 106288529379921305v^11 - 3550592557690042514v^10 + 3032219299984278283v^9 + 1593030849689862526v^8 - 25579982020477094049v^7 + 42873099076883482646v^6 + 25814541166549802255v^5 - 89186693273137852522v^4 + 195080131081974397546v^3 - 9486742388439306218v^2 - 133996866940967143547*v + 323244951745472925104) / 45685202299231782716
$\beta_{2}$	$=$	$ ( 11\!\cdots\!83 \nu^{15} + \cdots + 14\!\cdots\!04 ) / 45\!\cdots\!16 $ (11574739710081783v^15 + 31542989549717200v^14 - 43197660227947108v^13 + 606443555736740118v^12 + 308487044449543533v^11 - 1158049318712634426v^10 + 6620875964566407315v^9 - 1959280175143383414v^8 - 20082368054369625293v^7 + 27562561529026760818v^6 - 19304513468337724437v^5 - 108238153128821151826v^4 + 84568257250429215590v^3 + 19290289365664567938v^2 - 52547799385417368107*v + 148055173394829966904) / 45685202299231782716
$\beta_{3}$	$=$	$ ( 11\!\cdots\!06 \nu^{15} + \cdots - 38\!\cdots\!79 ) / 17\!\cdots\!66 $ (1140125766349406v^15 - 4367617546824849v^14 - 197183259870740v^13 + 30022482828924986v^12 - 158173701797230294v^11 - 10776238979066195v^10 + 416055175740255026v^9 - 1674891184372108413v^8 + 190061002689498806v^7 + 4862953742390552803v^6 - 5814109328492353890v^5 + 340228636362324399v^4 + 22436736761513794752v^3 - 4987739011983929514v^2 - 14869703130547301562*v - 3823376651677252079) / 1757123165355068566
$\beta_{4}$	$=$	$ ( - 26\!\cdots\!10 \nu^{15} + \cdots - 88\!\cdots\!17 ) / 17\!\cdots\!66 $ (-2695955060801510v^15 + 3648835634283717v^14 - 11477433752833372v^13 - 62281015650186122v^12 + 90516883745466238v^11 - 97741927011845753v^10 - 571483007952477098v^9 + 1226555094546014657v^8 + 494032734220193906v^7 - 3259136877343240311v^6 + 5894147590599933834v^5 + 5095484001938383309v^4 - 13348509975648757728v^3 + 1213668426945726994v^2 + 8452099174337807954*v - 8855765297464677017) / 1757123165355068566
$\beta_{5}$	$=$	$ ( 63\!\cdots\!81 \nu^{15} + \cdots + 63\!\cdots\!44 ) / 35\!\cdots\!32 $ (6362588869820581v^15 - 19063311774243996v^14 + 29953228320444548v^13 + 107493144403898662v^12 - 523673628324225525v^11 + 340290449377091346v^10 + 873018366997406489v^9 - 5745272895431736830v^8 + 1755303578395086237v^7 + 9172394045768786898v^6 - 23802516418019644811v^5 + 8405121262820736378v^4 + 47841649303051007298v^3 - 20617508090195046562v^2 + 1963355353139054003*v + 6380017355763765544) / 3514246330710137132
$\beta_{6}$	$=$	$ ( 42\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!33 ) / 22\!\cdots\!58 $ (42062200207606612v^15 - 66493356316015995v^14 + 225778070242334298v^13 + 937368216274372802v^12 - 1570539176450854128v^11 + 3101315941930414481v^10 + 8284076220069362254v^9 - 20067430137555293875v^8 + 10937593427005479408v^7 + 42511763539658651789v^6 - 108859731116777625032v^5 + 9129109876533571899v^4 + 135605166802313500722v^3 - 165596671181047425326v^2 - 2188306075681915558*v + 31972362819830644433) / 22842601149615891358
$\beta_{7}$	$=$	$ ( - 250973189 \nu^{15} + 1154940404 \nu^{14} - 2423159572 \nu^{13} - 1370245474 \nu^{12} + \cdots - 594916780180 ) / 104154789211 $ (-250973189v^15 + 1154940404v^14 - 2423159572v^13 - 1370245474v^12 + 25113894805v^11 - 38837348066v^10 - 2760428749v^9 + 253467181310v^8 - 333579911501v^7 - 211051473346v^6 + 1225991204783v^5 - 1335092443430v^4 - 1487733042786v^3 + 2567868225822v^2 - 298723636215*v - 594916780180) / 104154789211
$\beta_{8}$	$=$	$ ( 14\!\cdots\!69 \nu^{15} + \cdots + 74\!\cdots\!28 ) / 45\!\cdots\!16 $ (142487390425027769v^15 - 329287467641625288v^14 + 736540770497388376v^13 + 2654878104268383710v^12 - 7956531511818141829v^11 + 7995173026699626378v^10 + 24073441202354112017v^9 - 88377832298453972054v^8 + 19044438541335806509v^7 + 190288222187924112786v^6 - 366382723933506713011v^5 - 50639231129949486830v^4 + 830066158310951450286v^3 - 167168894096700070566v^2 - 533624559431962819185*v + 74212439894842452328) / 45685202299231782716
$\beta_{9}$	$=$	$ ( - 67\!\cdots\!08 \nu^{15} + \cdots - 77\!\cdots\!37 ) / 20\!\cdots\!78 $ (-6701410765864608v^15 + 20397507351839063v^14 - 48275267124211374v^13 - 90056873346665050v^12 + 456837063109262652v^11 - 692973263371677665v^10 - 565387499172981542v^9 + 4873086269443131607v^8 - 4426665294965668284v^7 - 5053558582117961913v^6 + 22778652450722967180v^5 - 15908019658506505439v^4 - 28197494178908763438v^3 + 32502048532392926410v^2 - 9964556405655829162*v - 7759583114818798837) / 2076600104510535578
$\beta_{10}$	$=$	$ ( 63\!\cdots\!09 \nu^{15} + \cdots + 12\!\cdots\!88 ) / 17\!\cdots\!66 $ (6349114021265209v^15 - 12398321458211784v^14 + 29112569996758046v^13 + 127714532428272680v^12 - 306664443443721877v^11 + 288412211047750906v^10 + 1128500431801786735v^9 - 3480861813162239036v^8 + 250808920977944233v^7 + 7725304242280913068v^6 - 14143792471177397323v^5 - 4234814238148221182v^4 + 29814661243738397372v^3 - 5177523217354105724v^2 - 18941935548763488607*v + 1250918694264577088) / 1757123165355068566
$\beta_{11}$	$=$	$ ( - 16\!\cdots\!93 \nu^{15} + \cdots - 16\!\cdots\!28 ) / 35\!\cdots\!32 $ (-16609076345849793v^15 + 43777706567834420v^14 - 109614877257798068v^13 - 268451204999532446v^12 + 982344604604297537v^11 - 1530004100726979842v^10 - 2116215153570410069v^9 + 10775043896635596134v^8 - 8563271175014306681v^7 - 15450404472982565026v^6 + 51485835461026702735v^5 - 23489162306911701370v^4 - 69012678892821242922v^3 + 72033002771164191794v^2 - 3717987832020587847*v - 16195965593270251328) / 3514246330710137132
$\beta_{12}$	$=$	$ ( 11\!\cdots\!56 \nu^{15} + \cdots + 73\!\cdots\!61 ) / 22\!\cdots\!58 $ (111550879420235656v^15 - 292983570349548421v^14 + 654624629375306434v^13 + 1876539693245210668v^12 - 7022490804557082040v^11 + 8466219722131675361v^10 + 15218981538250111326v^9 - 76980975366990988559v^8 + 42911383223128781120v^7 + 122751408952353446869v^6 - 341674328313058912824v^5 + 114532532653511592607v^4 + 570969820816643665374v^3 - 393605588975953109468v^2 - 58381597757886918742*v + 73477060311685045661) / 22842601149615891358
$\beta_{13}$	$=$	$ ( - 16\!\cdots\!70 \nu^{15} + \cdots - 20\!\cdots\!51 ) / 22\!\cdots\!58 $ (-167697984468815770v^15 + 441386680282489493v^14 - 1077470600807150056v^13 - 2910360544454317218v^12 + 10445298055451247154v^11 - 16215955494739336141v^10 - 24031921424473814138v^9 + 115664814917559906273v^8 - 97921438492302334730v^7 - 177984861917444515559v^6 + 577870605226179815278v^5 - 286117525562097876335v^4 - 768335194753343861452v^3 + 952522406033919619010v^2 - 53310807513569673806*v - 208981489155345308951) / 22842601149615891358
$\beta_{14}$	$=$	$ ( 33\!\cdots\!93 \nu^{15} + \cdots + 14\!\cdots\!20 ) / 35\!\cdots\!32 $ (33980931764191393v^15 - 62240396148230256v^14 + 171993593601197780v^13 + 701814995202085890v^12 - 1523144458129482905v^11 + 2046308592924876482v^10 + 5875358106164794749v^9 - 18189859139642029302v^8 + 4830250631624342257v^7 + 34414605294202320866v^6 - 83280654881583067399v^5 - 4674140963452736550v^4 + 133608261297018280546v^3 - 60541906052245491782v^2 - 4887015750460537105*v + 14180984115307910620) / 3514246330710137132
$\beta_{15}$	$=$	$ ( - 18\!\cdots\!77 \nu^{15} + \cdots - 35\!\cdots\!78 ) / 11\!\cdots\!79 $ (-181882803471002077v^15 + 316671046966123234v^14 - 952207769537059555v^13 - 3640877615609526631v^12 + 7105855601592527295v^11 - 10313486800028212216v^10 - 30477259374679535090v^9 + 86723307005154113145v^8 - 14723960228640721403v^7 - 166033459453260524561v^6 + 389088981450540478065v^5 + 85377665264369441912v^4 - 615994711730995487999v^3 + 148007820979410893503v^2 + 44965632989574769596*v - 35724605451299759778) / 11421300574807945679

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

$\nu$	$=$	$ ( 2\beta_{14} - 2\beta_{11} - 2\beta_{10} + 4\beta_{9} - \beta_{8} - \beta_{7} - 4\beta_{6} - 2\beta_{2} + \beta_1 ) / 16 $ (2b14 - 2b11 - 2b10 + 4b9 - b8 - b7 - 4b6 - 2b2 + b1) / 16
$\nu^{2}$	$=$	$ ( 2 \beta_{14} - 16 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots - 12 ) / 16 $ (2b14 - 16b12 - 6b11 + 2b10 - 4b9 + 3b8 - b7 - 4b6 + 4b5 + 2b4 + 2b3 + 2*b2 + b1 - 12) / 16
$\nu^{3}$	$=$	$ ( - 2 \beta_{14} + \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - \beta_{10} - 12 \beta_{9} + 4 \beta_{8} + \cdots - 37 ) / 8 $ (-2b14 + b13 - 4b12 - 2b11 - b10 - 12b9 + 4b8 + 4b7 + 2b6 - 2b5 + 3b4 + 2b3 + 9b2 + 3b1 - 37) / 8
$\nu^{4}$	$=$	$ ( - 8 \beta_{15} - 34 \beta_{14} - 12 \beta_{13} + 24 \beta_{12} + 38 \beta_{11} - 6 \beta_{10} + \cdots + 56 ) / 16 $ (-8b15 - 34b14 - 12b13 + 24b12 + 38b11 - 6b10 - 44b9 + 17b8 + 29b7 + 44b6 + 4b5 + 10b4 - 2b3 + 2b2 - b1 + 56) / 16
$\nu^{5}$	$=$	$ ( - 12 \beta_{15} - 14 \beta_{14} - 22 \beta_{13} + 164 \beta_{12} + 122 \beta_{11} - 32 \beta_{10} + \cdots + 450 ) / 16 $ (-12b15 - 14b14 - 22b13 + 164b12 + 122b11 - 32b10 + 48b9 - 47b8 - 3b7 + 4b6 - 56b5 - 52b4 + 6b3 - 92b2 - 27*b1 + 450) / 16
$\nu^{6}$	$=$	$ ( 12 \beta_{15} + 82 \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + \cdots + 346 ) / 8 $ (12b15 + 82b14 + 12b13 + 12b12 - 8b11 - 10b10 + 184b9 - 71b8 - 91b7 - 56b6 + 10b5 - 99b4 + b3 - 186b2 - 21b1 + 346) / 8
$\nu^{7}$	$=$	$ ( 164 \beta_{15} + 510 \beta_{14} + 372 \beta_{13} - 820 \beta_{12} - 974 \beta_{11} + 158 \beta_{10} + \cdots - 1308 ) / 16 $ (164b15 + 510b14 + 372b13 - 820b12 - 974b11 + 158b10 + 400b9 - 111b8 - 399b7 - 248b6 + 160b5 - 4b4 + 92b3 + 94b2 + 75*b1 - 1308) / 16
$\nu^{8}$	$=$	$ ( 200 \beta_{15} + 10 \beta_{14} + 460 \beta_{13} - 2024 \beta_{12} - 1846 \beta_{11} + 310 \beta_{10} + \cdots - 7032 ) / 16 $ (200b15 + 10b14 + 460b13 - 2024b12 - 1846b11 + 310b10 - 1276b9 + 1195b8 + 415b7 + 380b6 + 580b5 + 1622b4 - 206b3 + 2270b2 + 573*b1 - 7032) / 16
$\nu^{9}$	$=$	$ ( - 474 \beta_{15} - 1736 \beta_{14} - 893 \beta_{13} + 626 \beta_{12} + 1714 \beta_{11} - 65 \beta_{10} + \cdots - 2705 ) / 8 $ (-474b15 - 1736b14 - 893b13 + 626b12 + 1714b11 - 65b10 - 2830b9 + 1381b8 + 1557b7 + 712b6 - 332b5 + 1920b4 - 245b3 + 2717b2 + 284*b1 - 2705) / 8
$\nu^{10}$	$=$	$ ( - 3296 \beta_{15} - 7374 \beta_{14} - 8152 \beta_{13} + 15720 \beta_{12} + 20966 \beta_{11} + \cdots + 32664 ) / 16 $ (-3296b15 - 7374b14 - 8152b13 + 15720b12 + 20966b11 - 2550b10 - 4020b9 - 1733b8 + 4703b7 - 636b6 - 5120b5 - 4168b4 - 184b3 - 6790b2 - 2047*b1 + 32664) / 16
$\nu^{11}$	$=$	$ ( - 1232 \beta_{15} + 5018 \beta_{14} - 3410 \beta_{13} + 24504 \beta_{12} + 16742 \beta_{11} + \cdots + 108042 ) / 16 $ (-1232b15 + 5018b14 - 3410b13 + 24504b12 + 16742b11 - 3636b10 + 28204b9 - 25903b8 - 9867b7 - 7264b6 - 5172b5 - 38170b4 + 5984b3 - 51088b2 - 9263*b1 + 108042) / 16
$\nu^{12}$	$=$	$ ( 2988 \beta_{15} + 8169 \beta_{14} + 6680 \beta_{13} - 6435 \beta_{12} - 14826 \beta_{11} + 706 \beta_{10} + \cdots + 2363 ) / 2 $ (2988b15 + 8169b14 + 6680b13 - 6435b12 - 14826b11 + 706b10 + 11385b9 - 4920b8 - 6137b7 - 628b6 + 3225b5 - 6708b4 + 1362b3 - 8606b2 - 770*b1 + 2363) / 2
$\nu^{13}$	$=$	$ ( 57452 \beta_{15} + 109778 \beta_{14} + 144572 \beta_{13} - 281068 \beta_{12} - 369122 \beta_{11} + \cdots - 689884 ) / 16 $ (57452b15 + 109778b14 + 144572b13 - 281068b12 - 369122b11 + 32210b10 + 18768b9 + 101107b8 - 55837b7 + 44560b6 + 92880b5 + 164972b4 - 20444b3 + 224650b2 + 48641*b1 - 689884) / 16
$\nu^{14}$	$=$	$ ( - 39032 \beta_{15} - 180622 \beta_{14} - 82928 \beta_{13} - 268976 \beta_{12} + 101698 \beta_{11} + \cdots - 1660468 ) / 16 $ (-39032b15 - 180622b14 - 82928b13 - 268976b12 + 101698b11 + 39146b10 - 602052b9 + 506935b8 + 191643b7 + 73268b6 - 10308b5 + 758674b4 - 128470b3 + 984714b2 + 156189*b1 - 1660468) / 16
$\nu^{15}$	$=$	$ ( - 259938 \beta_{15} - 608946 \beta_{14} - 617433 \beta_{13} + 672706 \beta_{12} + 1437426 \beta_{11} + \cdots + 482809 ) / 8 $ (-259938b15 - 608946b14 - 617433b13 + 672706b12 + 1437426b11 - 49647b10 - 753554b9 + 177792b8 + 405084b7 - 88284b6 - 338454b5 + 231165b4 - 57976b3 + 276855b2 + 13797*b1 + 482809) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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} $

1759.1

1.11858	+	2.43543i
−0.425330	+	0.151831i
0.947487	+	1.75599i
0.691227	−	2.19508i
−1.76521	+	0.136903i
0.626493	−	0.442020i
1.51042	+	0.0694892i
−1.70368	+	0.651747i
1.51042	−	0.0694892i
−1.70368	−	0.651747i
−1.76521	−	0.136903i
0.626493	+	0.442020i
0.947487	−	1.75599i
0.691227	+	2.19508i
1.11858	−	2.43543i
−0.425330	−	0.151831i

1.00000

−

3.78801i

−3.68457

1.00000

1759.2

1.00000

−

3.78801i

3.68457

1.00000

1759.3

1.00000

−

2.39924i

−1.09614

1.00000

1759.4

1.00000

−

2.39924i

1.09614

1.00000

1759.5

1.00000

−

1.66719i

−3.05522

1.00000

1759.6

1.00000

−

1.66719i

3.05522

1.00000

1759.7

1.00000

−

1.05596i

−1.97182

1.00000

1759.8

1.00000

−

1.05596i

1.97182

1.00000

1759.9

1.00000

1.05596i

−1.97182

1.00000

1759.10

1.00000

1.05596i

1.97182

1.00000

1759.11

1.00000

1.66719i

−3.05522

1.00000

1759.12

1.00000

1.66719i

3.05522

1.00000

1759.13

1.00000

2.39924i

−1.09614

1.00000

1759.14

1.00000

2.39924i

1.09614

1.00000

1759.15

1.00000

3.78801i

−3.68457

1.00000

1759.16

1.00000

3.78801i

3.68457

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
11.b	odd	2	1	inner
88.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2112.2.h.d	yes	16
4.b	odd	2	1		2112.2.h.c	✓	16
8.b	even	2	1		2112.2.h.c	✓	16
8.d	odd	2	1	inner	2112.2.h.d	yes	16
11.b	odd	2	1	inner	2112.2.h.d	yes	16
44.c	even	2	1		2112.2.h.c	✓	16
88.b	odd	2	1		2112.2.h.c	✓	16
88.g	even	2	1	inner	2112.2.h.d	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2112.2.h.c	✓	16	4.b	odd	2	1
2112.2.h.c	✓	16	8.b	even	2	1
2112.2.h.c	✓	16	44.c	even	2	1
2112.2.h.c	✓	16	88.b	odd	2	1
2112.2.h.d	yes	16	1.a	even	1	1	trivial
2112.2.h.d	yes	16	8.d	odd	2	1	inner
2112.2.h.d	yes	16	11.b	odd	2	1	inner
2112.2.h.d	yes	16	88.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2112, [\chi])$:

$ T_{5}^{8} + 24T_{5}^{6} + 164T_{5}^{4} + 384T_{5}^{2} + 256 $

$ T_{59}^{4} + 20T_{59}^{3} - 12T_{59}^{2} - 1984T_{59} - 8384 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T - 1)^{16} $ (T - 1)^16
$5$	$ (T^{8} + 24 T^{6} + \cdots + 256)^{2} $ (T^8 + 24T^6 + 164T^4 + 384*T^2 + 256)^2
$7$	$ (T^{8} - 28 T^{6} + \cdots + 592)^{2} $ (T^8 - 28T^6 + 248T^4 - 752*T^2 + 592)^2
$11$	$ (T^{8} + 4 T^{7} + \cdots + 14641)^{2} $ (T^8 + 4T^7 + 16T^6 + 68T^5 + 174T^4 + 748T^3 + 1936T^2 + 5324*T + 14641)^2
$13$	$ (T^{8} - 88 T^{6} + \cdots + 9472)^{2} $ (T^8 - 88T^6 + 2276T^4 - 17600*T^2 + 9472)^2
$17$	$ (T^{8} + 52 T^{6} + \cdots + 2368)^{2} $ (T^8 + 52T^6 + 740T^4 + 2528*T^2 + 2368)^2
$19$	$ (T^{8} + 116 T^{6} + \cdots + 2368)^{2} $ (T^8 + 116T^6 + 4100T^4 + 43552*T^2 + 2368)^2
$23$	$ (T^{8} + 72 T^{6} + \cdots + 16384)^{2} $ (T^8 + 72T^6 + 1220T^4 + 7680*T^2 + 16384)^2
$29$	$ (T^{8} - 164 T^{6} + \cdots + 1600768)^{2} $ (T^8 - 164T^6 + 9236T^4 - 210304*T^2 + 1600768)^2
$31$	$ (T^{8} + 104 T^{6} + \cdots + 1024)^{2} $ (T^8 + 104T^6 + 2640T^4 + 17408*T^2 + 1024)^2
$37$	$ (T^{8} + 120 T^{6} + \cdots + 173056)^{2} $ (T^8 + 120T^6 + 4304T^4 + 49152*T^2 + 173056)^2
$41$	$ (T^{8} + 268 T^{6} + \cdots + 9699328)^{2} $ (T^8 + 268T^6 + 23924T^4 + 837632*T^2 + 9699328)^2
$43$	$ (T^{8} + 212 T^{6} + \cdots + 3241792)^{2} $ (T^8 + 212T^6 + 15620T^4 + 444064*T^2 + 3241792)^2
$47$	$ (T^{8} + 128 T^{6} + \cdots + 87616)^{2} $ (T^8 + 128T^6 + 4692T^4 + 42464*T^2 + 87616)^2
$53$	$ (T^{8} + 264 T^{6} + \cdots + 3748096)^{2} $ (T^8 + 264T^6 + 22916T^4 + 696960*T^2 + 3748096)^2
$59$	$ (T^{4} + 20 T^{3} + \cdots - 8384)^{4} $ (T^4 + 20T^3 - 12T^2 - 1984*T - 8384)^4
$61$	$ (T^{8} - 536 T^{6} + \cdots + 190993408)^{2} $ (T^8 - 536T^6 + 98948T^4 - 7475200*T^2 + 190993408)^2
$67$	$ (T^{4} - 16 T^{3} + \cdots + 256)^{4} $ (T^4 - 16T^3 - 12T^2 + 320*T + 256)^4
$71$	$ (T^{8} + 112 T^{6} + \cdots + 64)^{2} $ (T^8 + 112T^6 + 2868T^4 + 2272*T^2 + 64)^2
$73$	$ (T^{8} + 352 T^{6} + \cdots + 2424832)^{2} $ (T^8 + 352T^6 + 22016T^4 + 450560*T^2 + 2424832)^2
$79$	$ (T^{8} - 428 T^{6} + \cdots + 52925392)^{2} $ (T^8 - 428T^6 + 61208T^4 - 3359152*T^2 + 52925392)^2
$83$	$ (T^{8} + 352 T^{6} + \cdots + 2424832)^{2} $ (T^8 + 352T^6 + 36416T^4 + 1126400*T^2 + 2424832)^2
$89$	$ (T^{2} - 4 T - 44)^{8} $ (T^2 - 4*T - 44)^8
$97$	$ (T^{4} + 4 T^{3} + \cdots + 256)^{4} $ (T^4 + 4T^3 - 232T^2 + 320*T + 256)^4
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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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + q^{3} + \beta_{11} q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10}) \) q + q^3 + b11 * q^5 + b2 * q^7 + q^9	\( q + q^{3} + \beta_{11} q^{5} + \beta_{2} q^{7} + q^{9} + (\beta_{9} - 1) q^{11} + \beta_{8} q^{13} + \beta_{11} q^{15} + \beta_{6} q^{17} + (\beta_{13} + \beta_{6}) q^{19} + \beta_{2} q^{21} + (\beta_{7} + \beta_{5}) q^{23} + ( - \beta_{12} - \beta_{9} + \beta_{4} - 1) q^{25} + q^{27} - \beta_{10} q^{29} + (\beta_{14} - \beta_{11}) q^{31} + (\beta_{9} - 1) q^{33} + (\beta_{13} - \beta_{12} + \beta_{9} - 1) q^{35} + (\beta_{14} + \beta_{5}) q^{37} + \beta_{8} q^{39} - \beta_{15} q^{41} + ( - \beta_{13} + \beta_{6}) q^{43} + \beta_{11} q^{45} + (\beta_{14} + 2 \beta_{11}) q^{47} - \beta_{4} q^{49} + \beta_{6} q^{51} + (2 \beta_{14} + \beta_{11} - \beta_{7}) q^{53} + (\beta_{11} - \beta_{8} + \cdots - 2 \beta_{2}) q^{55}+ \cdots + (\beta_{9} - 1) q^{99}+O(q^{100}) \) q + q^3 + b11 * q^5 + b2 * q^7 + q^9 + (b9 - 1) * q^11 + b8 * q^13 + b11 * q^15 + b6 * q^17 + (b13 + b6) * q^19 + b2 * q^21 + (b7 + b5) * q^23 + (-b12 - b9 + b4 - 1) * q^25 + q^27 - b10 * q^29 + (b14 - b11) * q^31 + (b9 - 1) * q^33 + (b13 - b12 + b9 - 1) * q^35 + (b14 + b5) * q^37 + b8 * q^39 - b15 * q^41 + (-b13 + b6) * q^43 + b11 * q^45 + (b14 + 2b11) q^47 - b4 * q^49 + b6 * q^51 + (2b14 + b11 - b7) q^53 + (b11 - b8 - b7 - b5 - 2b2) q^55 + (b13 + b6) * q^57 + (-b12 - b9 + b4 + b3 - 5) * q^59 + (2b2 + b1) q^61 + b2 * q^63 + (b15 - 2b12 + 2b9 - b6 - 2) * q^65 + (b12 + b9 + b4 + 4) * q^67 + (b7 + b5) * q^69 + (b14 - b7) * q^71 + (b15 - b13 - b12 + b9 - b6 - 1) * q^73 + (-b12 - b9 + b4 - 1) * q^75 + (-b14 + 2b11 + b8 - b2) q^77 + (-b8 + b2 - b1) * q^79 + q^81 + (b15 - b12 + b9 + b6 - 1) * q^83 + (-b10 + b8 + b2) * q^85 - b10 * q^87 + (-b4 + b3 + 2) * q^89 + (3b12 + 3b9 - 2b4 + b3) q^91 + (b14 - b11) * q^93 + (-b10 - 3b2 - b1) q^95 + (-2b12 - 2b9 - b4 - 1) * q^97 + (b9 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 16 * q^9	\( 16 q + 16 q^{3} + 16 q^{9} - 8 q^{11} - 16 q^{25} + 16 q^{27} - 8 q^{33} - 80 q^{59} + 64 q^{67} - 16 q^{75} + 16 q^{81} + 32 q^{89} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^11 - 16 * q^25 + 16 * q^27 - 8 * q^33 - 80 * q^59 + 64 * q^67 - 16 * q^75 + 16 * q^81 + 32 * q^89 - 16 * q^97 - 8 * q^99

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	2112.2.h.d

Level	$2112$
Weight	$2$
Character orbit	2112.h
Analytic conductor	$16.864$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(16.8644049069\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2 x^{15} + 5 x^{14} + 20 x^{13} - 49 x^{12} + 56 x^{11} + 174 x^{10} - 572 x^{9} + 120 x^{8} + \cdots + 433 \) x^16 - 2x^15 + 5x^14 + 20x^13 - 49x^12 + 56x^11 + 174x^10 - 572x^9 + 120x^8 + 1148x^7 - 2520x^6 - 296x^5 + 4721x^4 - 1622x^3 - 1429x^2 + 788*x + 433
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{27} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 34\!\cdots\!17 \nu^{15} + \cdots + 32\!\cdots\!04 ) / 45\!\cdots\!16 \) (-3411519755021917v^15 + 25218117466846640v^14 - 166238613719899032v^13 + 104922309992602242v^12 + 106288529379921305v^11 - 3550592557690042514v^10 + 3032219299984278283v^9 + 1593030849689862526v^8 - 25579982020477094049v^7 + 42873099076883482646v^6 + 25814541166549802255v^5 - 89186693273137852522v^4 + 195080131081974397546v^3 - 9486742388439306218v^2 - 133996866940967143547*v + 323244951745472925104) / 45685202299231782716
\(\beta_{2}\)	\(=\)	\( ( 11\!\cdots\!83 \nu^{15} + \cdots + 14\!\cdots\!04 ) / 45\!\cdots\!16 \) (11574739710081783v^15 + 31542989549717200v^14 - 43197660227947108v^13 + 606443555736740118v^12 + 308487044449543533v^11 - 1158049318712634426v^10 + 6620875964566407315v^9 - 1959280175143383414v^8 - 20082368054369625293v^7 + 27562561529026760818v^6 - 19304513468337724437v^5 - 108238153128821151826v^4 + 84568257250429215590v^3 + 19290289365664567938v^2 - 52547799385417368107*v + 148055173394829966904) / 45685202299231782716
\(\beta_{3}\)	\(=\)	\( ( 11\!\cdots\!06 \nu^{15} + \cdots - 38\!\cdots\!79 ) / 17\!\cdots\!66 \) (1140125766349406v^15 - 4367617546824849v^14 - 197183259870740v^13 + 30022482828924986v^12 - 158173701797230294v^11 - 10776238979066195v^10 + 416055175740255026v^9 - 1674891184372108413v^8 + 190061002689498806v^7 + 4862953742390552803v^6 - 5814109328492353890v^5 + 340228636362324399v^4 + 22436736761513794752v^3 - 4987739011983929514v^2 - 14869703130547301562*v - 3823376651677252079) / 1757123165355068566
\(\beta_{4}\)	\(=\)	\( ( - 26\!\cdots\!10 \nu^{15} + \cdots - 88\!\cdots\!17 ) / 17\!\cdots\!66 \) (-2695955060801510v^15 + 3648835634283717v^14 - 11477433752833372v^13 - 62281015650186122v^12 + 90516883745466238v^11 - 97741927011845753v^10 - 571483007952477098v^9 + 1226555094546014657v^8 + 494032734220193906v^7 - 3259136877343240311v^6 + 5894147590599933834v^5 + 5095484001938383309v^4 - 13348509975648757728v^3 + 1213668426945726994v^2 + 8452099174337807954*v - 8855765297464677017) / 1757123165355068566
\(\beta_{5}\)	\(=\)	\( ( 63\!\cdots\!81 \nu^{15} + \cdots + 63\!\cdots\!44 ) / 35\!\cdots\!32 \) (6362588869820581v^15 - 19063311774243996v^14 + 29953228320444548v^13 + 107493144403898662v^12 - 523673628324225525v^11 + 340290449377091346v^10 + 873018366997406489v^9 - 5745272895431736830v^8 + 1755303578395086237v^7 + 9172394045768786898v^6 - 23802516418019644811v^5 + 8405121262820736378v^4 + 47841649303051007298v^3 - 20617508090195046562v^2 + 1963355353139054003*v + 6380017355763765544) / 3514246330710137132
\(\beta_{6}\)	\(=\)	\( ( 42\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!33 ) / 22\!\cdots\!58 \) (42062200207606612v^15 - 66493356316015995v^14 + 225778070242334298v^13 + 937368216274372802v^12 - 1570539176450854128v^11 + 3101315941930414481v^10 + 8284076220069362254v^9 - 20067430137555293875v^8 + 10937593427005479408v^7 + 42511763539658651789v^6 - 108859731116777625032v^5 + 9129109876533571899v^4 + 135605166802313500722v^3 - 165596671181047425326v^2 - 2188306075681915558*v + 31972362819830644433) / 22842601149615891358
\(\beta_{7}\)	\(=\)	\( ( - 250973189 \nu^{15} + 1154940404 \nu^{14} - 2423159572 \nu^{13} - 1370245474 \nu^{12} + \cdots - 594916780180 ) / 104154789211 \) (-250973189v^15 + 1154940404v^14 - 2423159572v^13 - 1370245474v^12 + 25113894805v^11 - 38837348066v^10 - 2760428749v^9 + 253467181310v^8 - 333579911501v^7 - 211051473346v^6 + 1225991204783v^5 - 1335092443430v^4 - 1487733042786v^3 + 2567868225822v^2 - 298723636215*v - 594916780180) / 104154789211
\(\beta_{8}\)	\(=\)	\( ( 14\!\cdots\!69 \nu^{15} + \cdots + 74\!\cdots\!28 ) / 45\!\cdots\!16 \) (142487390425027769v^15 - 329287467641625288v^14 + 736540770497388376v^13 + 2654878104268383710v^12 - 7956531511818141829v^11 + 7995173026699626378v^10 + 24073441202354112017v^9 - 88377832298453972054v^8 + 19044438541335806509v^7 + 190288222187924112786v^6 - 366382723933506713011v^5 - 50639231129949486830v^4 + 830066158310951450286v^3 - 167168894096700070566v^2 - 533624559431962819185*v + 74212439894842452328) / 45685202299231782716
\(\beta_{9}\)	\(=\)	\( ( - 67\!\cdots\!08 \nu^{15} + \cdots - 77\!\cdots\!37 ) / 20\!\cdots\!78 \) (-6701410765864608v^15 + 20397507351839063v^14 - 48275267124211374v^13 - 90056873346665050v^12 + 456837063109262652v^11 - 692973263371677665v^10 - 565387499172981542v^9 + 4873086269443131607v^8 - 4426665294965668284v^7 - 5053558582117961913v^6 + 22778652450722967180v^5 - 15908019658506505439v^4 - 28197494178908763438v^3 + 32502048532392926410v^2 - 9964556405655829162*v - 7759583114818798837) / 2076600104510535578
\(\beta_{10}\)	\(=\)	\( ( 63\!\cdots\!09 \nu^{15} + \cdots + 12\!\cdots\!88 ) / 17\!\cdots\!66 \) (6349114021265209v^15 - 12398321458211784v^14 + 29112569996758046v^13 + 127714532428272680v^12 - 306664443443721877v^11 + 288412211047750906v^10 + 1128500431801786735v^9 - 3480861813162239036v^8 + 250808920977944233v^7 + 7725304242280913068v^6 - 14143792471177397323v^5 - 4234814238148221182v^4 + 29814661243738397372v^3 - 5177523217354105724v^2 - 18941935548763488607*v + 1250918694264577088) / 1757123165355068566
\(\beta_{11}\)	\(=\)	\( ( - 16\!\cdots\!93 \nu^{15} + \cdots - 16\!\cdots\!28 ) / 35\!\cdots\!32 \) (-16609076345849793v^15 + 43777706567834420v^14 - 109614877257798068v^13 - 268451204999532446v^12 + 982344604604297537v^11 - 1530004100726979842v^10 - 2116215153570410069v^9 + 10775043896635596134v^8 - 8563271175014306681v^7 - 15450404472982565026v^6 + 51485835461026702735v^5 - 23489162306911701370v^4 - 69012678892821242922v^3 + 72033002771164191794v^2 - 3717987832020587847*v - 16195965593270251328) / 3514246330710137132
\(\beta_{12}\)	\(=\)	\( ( 11\!\cdots\!56 \nu^{15} + \cdots + 73\!\cdots\!61 ) / 22\!\cdots\!58 \) (111550879420235656v^15 - 292983570349548421v^14 + 654624629375306434v^13 + 1876539693245210668v^12 - 7022490804557082040v^11 + 8466219722131675361v^10 + 15218981538250111326v^9 - 76980975366990988559v^8 + 42911383223128781120v^7 + 122751408952353446869v^6 - 341674328313058912824v^5 + 114532532653511592607v^4 + 570969820816643665374v^3 - 393605588975953109468v^2 - 58381597757886918742*v + 73477060311685045661) / 22842601149615891358
\(\beta_{13}\)	\(=\)	\( ( - 16\!\cdots\!70 \nu^{15} + \cdots - 20\!\cdots\!51 ) / 22\!\cdots\!58 \) (-167697984468815770v^15 + 441386680282489493v^14 - 1077470600807150056v^13 - 2910360544454317218v^12 + 10445298055451247154v^11 - 16215955494739336141v^10 - 24031921424473814138v^9 + 115664814917559906273v^8 - 97921438492302334730v^7 - 177984861917444515559v^6 + 577870605226179815278v^5 - 286117525562097876335v^4 - 768335194753343861452v^3 + 952522406033919619010v^2 - 53310807513569673806*v - 208981489155345308951) / 22842601149615891358
\(\beta_{14}\)	\(=\)	\( ( 33\!\cdots\!93 \nu^{15} + \cdots + 14\!\cdots\!20 ) / 35\!\cdots\!32 \) (33980931764191393v^15 - 62240396148230256v^14 + 171993593601197780v^13 + 701814995202085890v^12 - 1523144458129482905v^11 + 2046308592924876482v^10 + 5875358106164794749v^9 - 18189859139642029302v^8 + 4830250631624342257v^7 + 34414605294202320866v^6 - 83280654881583067399v^5 - 4674140963452736550v^4 + 133608261297018280546v^3 - 60541906052245491782v^2 - 4887015750460537105*v + 14180984115307910620) / 3514246330710137132
\(\beta_{15}\)	\(=\)	\( ( - 18\!\cdots\!77 \nu^{15} + \cdots - 35\!\cdots\!78 ) / 11\!\cdots\!79 \) (-181882803471002077v^15 + 316671046966123234v^14 - 952207769537059555v^13 - 3640877615609526631v^12 + 7105855601592527295v^11 - 10313486800028212216v^10 - 30477259374679535090v^9 + 86723307005154113145v^8 - 14723960228640721403v^7 - 166033459453260524561v^6 + 389088981450540478065v^5 + 85377665264369441912v^4 - 615994711730995487999v^3 + 148007820979410893503v^2 + 44965632989574769596*v - 35724605451299759778) / 11421300574807945679

\(\nu\)	\(=\)	\( ( 2\beta_{14} - 2\beta_{11} - 2\beta_{10} + 4\beta_{9} - \beta_{8} - \beta_{7} - 4\beta_{6} - 2\beta_{2} + \beta_1 ) / 16 \) (2b14 - 2b11 - 2b10 + 4b9 - b8 - b7 - 4b6 - 2b2 + b1) / 16
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{14} - 16 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots - 12 ) / 16 \) (2b14 - 16b12 - 6b11 + 2b10 - 4b9 + 3b8 - b7 - 4b6 + 4b5 + 2b4 + 2b3 + 2*b2 + b1 - 12) / 16
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{14} + \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - \beta_{10} - 12 \beta_{9} + 4 \beta_{8} + \cdots - 37 ) / 8 \) (-2b14 + b13 - 4b12 - 2b11 - b10 - 12b9 + 4b8 + 4b7 + 2b6 - 2b5 + 3b4 + 2b3 + 9b2 + 3b1 - 37) / 8
\(\nu^{4}\)	\(=\)	\( ( - 8 \beta_{15} - 34 \beta_{14} - 12 \beta_{13} + 24 \beta_{12} + 38 \beta_{11} - 6 \beta_{10} + \cdots + 56 ) / 16 \) (-8b15 - 34b14 - 12b13 + 24b12 + 38b11 - 6b10 - 44b9 + 17b8 + 29b7 + 44b6 + 4b5 + 10b4 - 2b3 + 2b2 - b1 + 56) / 16
\(\nu^{5}\)	\(=\)	\( ( - 12 \beta_{15} - 14 \beta_{14} - 22 \beta_{13} + 164 \beta_{12} + 122 \beta_{11} - 32 \beta_{10} + \cdots + 450 ) / 16 \) (-12b15 - 14b14 - 22b13 + 164b12 + 122b11 - 32b10 + 48b9 - 47b8 - 3b7 + 4b6 - 56b5 - 52b4 + 6b3 - 92b2 - 27*b1 + 450) / 16
\(\nu^{6}\)	\(=\)	\( ( 12 \beta_{15} + 82 \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + \cdots + 346 ) / 8 \) (12b15 + 82b14 + 12b13 + 12b12 - 8b11 - 10b10 + 184b9 - 71b8 - 91b7 - 56b6 + 10b5 - 99b4 + b3 - 186b2 - 21b1 + 346) / 8
\(\nu^{7}\)	\(=\)	\( ( 164 \beta_{15} + 510 \beta_{14} + 372 \beta_{13} - 820 \beta_{12} - 974 \beta_{11} + 158 \beta_{10} + \cdots - 1308 ) / 16 \) (164b15 + 510b14 + 372b13 - 820b12 - 974b11 + 158b10 + 400b9 - 111b8 - 399b7 - 248b6 + 160b5 - 4b4 + 92b3 + 94b2 + 75*b1 - 1308) / 16
\(\nu^{8}\)	\(=\)	\( ( 200 \beta_{15} + 10 \beta_{14} + 460 \beta_{13} - 2024 \beta_{12} - 1846 \beta_{11} + 310 \beta_{10} + \cdots - 7032 ) / 16 \) (200b15 + 10b14 + 460b13 - 2024b12 - 1846b11 + 310b10 - 1276b9 + 1195b8 + 415b7 + 380b6 + 580b5 + 1622b4 - 206b3 + 2270b2 + 573*b1 - 7032) / 16
\(\nu^{9}\)	\(=\)	\( ( - 474 \beta_{15} - 1736 \beta_{14} - 893 \beta_{13} + 626 \beta_{12} + 1714 \beta_{11} - 65 \beta_{10} + \cdots - 2705 ) / 8 \) (-474b15 - 1736b14 - 893b13 + 626b12 + 1714b11 - 65b10 - 2830b9 + 1381b8 + 1557b7 + 712b6 - 332b5 + 1920b4 - 245b3 + 2717b2 + 284*b1 - 2705) / 8
\(\nu^{10}\)	\(=\)	\( ( - 3296 \beta_{15} - 7374 \beta_{14} - 8152 \beta_{13} + 15720 \beta_{12} + 20966 \beta_{11} + \cdots + 32664 ) / 16 \) (-3296b15 - 7374b14 - 8152b13 + 15720b12 + 20966b11 - 2550b10 - 4020b9 - 1733b8 + 4703b7 - 636b6 - 5120b5 - 4168b4 - 184b3 - 6790b2 - 2047*b1 + 32664) / 16
\(\nu^{11}\)	\(=\)	\( ( - 1232 \beta_{15} + 5018 \beta_{14} - 3410 \beta_{13} + 24504 \beta_{12} + 16742 \beta_{11} + \cdots + 108042 ) / 16 \) (-1232b15 + 5018b14 - 3410b13 + 24504b12 + 16742b11 - 3636b10 + 28204b9 - 25903b8 - 9867b7 - 7264b6 - 5172b5 - 38170b4 + 5984b3 - 51088b2 - 9263*b1 + 108042) / 16
\(\nu^{12}\)	\(=\)	\( ( 2988 \beta_{15} + 8169 \beta_{14} + 6680 \beta_{13} - 6435 \beta_{12} - 14826 \beta_{11} + 706 \beta_{10} + \cdots + 2363 ) / 2 \) (2988b15 + 8169b14 + 6680b13 - 6435b12 - 14826b11 + 706b10 + 11385b9 - 4920b8 - 6137b7 - 628b6 + 3225b5 - 6708b4 + 1362b3 - 8606b2 - 770*b1 + 2363) / 2
\(\nu^{13}\)	\(=\)	\( ( 57452 \beta_{15} + 109778 \beta_{14} + 144572 \beta_{13} - 281068 \beta_{12} - 369122 \beta_{11} + \cdots - 689884 ) / 16 \) (57452b15 + 109778b14 + 144572b13 - 281068b12 - 369122b11 + 32210b10 + 18768b9 + 101107b8 - 55837b7 + 44560b6 + 92880b5 + 164972b4 - 20444b3 + 224650b2 + 48641*b1 - 689884) / 16
\(\nu^{14}\)	\(=\)	\( ( - 39032 \beta_{15} - 180622 \beta_{14} - 82928 \beta_{13} - 268976 \beta_{12} + 101698 \beta_{11} + \cdots - 1660468 ) / 16 \) (-39032b15 - 180622b14 - 82928b13 - 268976b12 + 101698b11 + 39146b10 - 602052b9 + 506935b8 + 191643b7 + 73268b6 - 10308b5 + 758674b4 - 128470b3 + 984714b2 + 156189*b1 - 1660468) / 16
\(\nu^{15}\)	\(=\)	\( ( - 259938 \beta_{15} - 608946 \beta_{14} - 617433 \beta_{13} + 672706 \beta_{12} + 1437426 \beta_{11} + \cdots + 482809 ) / 8 \) (-259938b15 - 608946b14 - 617433b13 + 672706b12 + 1437426b11 - 49647b10 - 753554b9 + 177792b8 + 405084b7 - 88284b6 - 338454b5 + 231165b4 - 57976b3 + 276855b2 + 13797*b1 + 482809) / 8

\(n\)	\(133\)	\(1409\)	\(1729\)	\(2047\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1759.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T - 1)^{16} \) (T - 1)^16
$5$	\( (T^{8} + 24 T^{6} + \cdots + 256)^{2} \) (T^8 + 24T^6 + 164T^4 + 384*T^2 + 256)^2
$7$	\( (T^{8} - 28 T^{6} + \cdots + 592)^{2} \) (T^8 - 28T^6 + 248T^4 - 752*T^2 + 592)^2
$11$	\( (T^{8} + 4 T^{7} + \cdots + 14641)^{2} \) (T^8 + 4T^7 + 16T^6 + 68T^5 + 174T^4 + 748T^3 + 1936T^2 + 5324*T + 14641)^2
$13$	\( (T^{8} - 88 T^{6} + \cdots + 9472)^{2} \) (T^8 - 88T^6 + 2276T^4 - 17600*T^2 + 9472)^2
$17$	\( (T^{8} + 52 T^{6} + \cdots + 2368)^{2} \) (T^8 + 52T^6 + 740T^4 + 2528*T^2 + 2368)^2
$19$	\( (T^{8} + 116 T^{6} + \cdots + 2368)^{2} \) (T^8 + 116T^6 + 4100T^4 + 43552*T^2 + 2368)^2
$23$	\( (T^{8} + 72 T^{6} + \cdots + 16384)^{2} \) (T^8 + 72T^6 + 1220T^4 + 7680*T^2 + 16384)^2
$29$	\( (T^{8} - 164 T^{6} + \cdots + 1600768)^{2} \) (T^8 - 164T^6 + 9236T^4 - 210304*T^2 + 1600768)^2
$31$	\( (T^{8} + 104 T^{6} + \cdots + 1024)^{2} \) (T^8 + 104T^6 + 2640T^4 + 17408*T^2 + 1024)^2
$37$	\( (T^{8} + 120 T^{6} + \cdots + 173056)^{2} \) (T^8 + 120T^6 + 4304T^4 + 49152*T^2 + 173056)^2
$41$	\( (T^{8} + 268 T^{6} + \cdots + 9699328)^{2} \) (T^8 + 268T^6 + 23924T^4 + 837632*T^2 + 9699328)^2
$43$	\( (T^{8} + 212 T^{6} + \cdots + 3241792)^{2} \) (T^8 + 212T^6 + 15620T^4 + 444064*T^2 + 3241792)^2
$47$	\( (T^{8} + 128 T^{6} + \cdots + 87616)^{2} \) (T^8 + 128T^6 + 4692T^4 + 42464*T^2 + 87616)^2
$53$	\( (T^{8} + 264 T^{6} + \cdots + 3748096)^{2} \) (T^8 + 264T^6 + 22916T^4 + 696960*T^2 + 3748096)^2
$59$	\( (T^{4} + 20 T^{3} + \cdots - 8384)^{4} \) (T^4 + 20T^3 - 12T^2 - 1984*T - 8384)^4
$61$	\( (T^{8} - 536 T^{6} + \cdots + 190993408)^{2} \) (T^8 - 536T^6 + 98948T^4 - 7475200*T^2 + 190993408)^2
$67$	\( (T^{4} - 16 T^{3} + \cdots + 256)^{4} \) (T^4 - 16T^3 - 12T^2 + 320*T + 256)^4
$71$	\( (T^{8} + 112 T^{6} + \cdots + 64)^{2} \) (T^8 + 112T^6 + 2868T^4 + 2272*T^2 + 64)^2
$73$	\( (T^{8} + 352 T^{6} + \cdots + 2424832)^{2} \) (T^8 + 352T^6 + 22016T^4 + 450560*T^2 + 2424832)^2
$79$	\( (T^{8} - 428 T^{6} + \cdots + 52925392)^{2} \) (T^8 - 428T^6 + 61208T^4 - 3359152*T^2 + 52925392)^2
$83$	\( (T^{8} + 352 T^{6} + \cdots + 2424832)^{2} \) (T^8 + 352T^6 + 36416T^4 + 1126400*T^2 + 2424832)^2
$89$	\( (T^{2} - 4 T - 44)^{8} \) (T^2 - 4*T - 44)^8
$97$	\( (T^{4} + 4 T^{3} + \cdots + 256)^{4} \) (T^4 + 4T^3 - 232T^2 + 320*T + 256)^4
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