LMFDB - Newform orbit 672.3.f.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,3,Mod(97,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.97"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	672.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3106737650$
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^{14} - 8 x^{13} - 57 x^{12} + 32 x^{11} + 466 x^{10} + 304 x^{9} + 3000 x^{8} + \cdots + 790321 $ x^16 + 2x^14 - 8x^13 - 57x^12 + 32x^11 + 466x^10 + 304x^9 + 3000x^8 + 1328x^7 - 11560x^6 + 22316x^5 - 23333x^4 - 87520x^3 - 135796x^2 - 88900x + 790321
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
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 + \beta_{6} q^{3} + \beta_{8} q^{5} + ( - \beta_{9} + \beta_{5} + \beta_1) q^{7} - 3 q^{9} + (\beta_{7} + \beta_{5}) q^{11} + ( - \beta_{11} + \beta_{8} + \beta_{3}) q^{13} + ( - \beta_{7} - \beta_1) q^{15}+ \cdots + ( - 3 \beta_{7} - 3 \beta_{5}) q^{99}+O(q^{100}) $ q + b6 * q^3 + b8 * q^5 + (-b9 + b5 + b1) * q^7 - 3 * q^9 + (b7 + b5) * q^11 + (-b11 + b8 + b3) * q^13 + (-b7 - b1) * q^15 + (b13 - b11 - b3) * q^17 + (-b10 + b9 + 2b6 - b5 - b1) q^19 + (b12 + b3 + 1) * q^21 + (b7 - 2b5 + b2 - b1) q^23 + (-b15 - b12 + b4 - 3) * q^25 - 3b6 q^27 + (-b15 - b13 + b12 + b3 - 8) * q^29 + (-b14 - b10 - 3b6) q^31 + (b13 + b8 - b3) * q^33 + (2b9 - 2b5 + 5b1) q^35 + (-b15 - 2b13 + 3b12 + 2b4 + 2b3 + 2) * q^37 + (b5 - b2 - b1) * q^39 + (3b13 - b11 - 4b8 + 3b3) q^41 + (-3b7 + b5 + b2 - 5b1) * q^43 - 3b8 q^45 + (b14 + 2b10 - 3b9 - 3b6 + 2b5 + 2b1) q^47 + (-b15 + b13 - b12 - 2b11 - 4b8 - b4 + 2b3 - 15) q^49 + (-2b7 - 2b5 - b2 + 3b1) q^51 + (-b15 - 3b13 + 5b12 + b4 + 3b3 - 12) q^53 + (b14 - b10 - 4b9 + 15b6 + b5 + b1) * q^55 + (-b15 - b12 - 8) * q^57 + (-b14 + 2b10 + 3b9 - b6) * q^59 + (2b13 - 3b11 - 3b8 - 5b3) * q^61 + (3b9 - 3b5 - 3b1) q^63 + (-2b15 - 2b12 + 5b4 - 16) q^65 + (5b7 + 9b5 - b2 - 7b1) q^67 + (-b13 - 3b11 + 4b8) * q^69 + (7b7 - 2b5 - 3b2 + b1) q^71 + (6b13 + 10b8 + 4b3) q^73 + (b14 + 3b10 - 4b9 + 3b5 + 3b1) * q^75 + (-b15 + b13 + b12 - 2b11 + 3b8 - b4 - 3b3 + 8) q^77 + (-2b7 - b5 + 2b2 + 13b1) q^79 + 9 * q^81 + (3b14 + 2b10 + 3b9 - 13b6 - 2b5 - 2b1) * q^83 + (-b15 - 4b13 + 7b12 - 2b4 + 4b3 - 4) * q^85 + (3b10 + 3b9 - 8b6) q^87 + (3b13 - 5b11 + 10b8 + 11b3) * q^89 + (-2b14 - b10 - 3b9 + 3b7 - 4b5 + 3b2 + 4b1) * q^91 + (-b15 - b13 + b12 + 3b4 + b3 + 6) q^93 + (-10b7 + 2b5 - 2b2 - 16b1) * q^95 + (4b13 - 2b11 + 8b8 - 2b3) * q^97 + (-3b7 - 3b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} + 24 q^{21} - 64 q^{25} - 128 q^{29} + 48 q^{37} - 256 q^{49} - 160 q^{53} - 144 q^{57} - 288 q^{65} + 128 q^{77} + 144 q^{81} - 16 q^{85} + 96 q^{93}+O(q^{100}) $ 16 * q - 48 * q^9 + 24 * q^21 - 64 * q^25 - 128 * q^29 + 48 * q^37 - 256 * q^49 - 160 * q^53 - 144 * q^57 - 288 * q^65 + 128 * q^77 + 144 * q^81 - 16 * q^85 + 96 * q^93

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 2 x^{14} - 8 x^{13} - 57 x^{12} + 32 x^{11} + 466 x^{10} + 304 x^{9} + 3000 x^{8} + \cdots + 790321 $ x^16 + 2*x^14 - 8*x^13 - 57*x^12 + 32*x^11 + 466*x^10 + 304*x^9 + 3000*x^8 + 1328*x^7 - 11560*x^6 + 22316*x^5 - 23333*x^4 - 87520*x^3 - 135796*x^2 - 88900*x + 790321 :

$\beta_{1}$	$=$	$ ( 12827995238497 \nu^{15} - 581190265302438 \nu^{14} + \cdots - 18\!\cdots\!83 ) / 66\!\cdots\!31 $ (12827995238497v^15 - 581190265302438v^14 + 1737559774441217v^13 - 3180484385351856v^12 + 2633733112382891v^11 + 29039772268083545v^10 - 85444707116929170v^9 - 67188609781839606v^8 + 683204853880188609v^7 - 2973986605133057123v^6 + 1735322209013429241v^5 + 12463695242476347423v^4 - 39601502364319381933v^3 + 68875998111592376644v^2 - 26048159145128777216*v - 180098559199799117983) / 66151241998754743831
$\beta_{2}$	$=$	$ ( 225970139316022 \nu^{15} + 821037119788618 \nu^{14} + \cdots - 56\!\cdots\!10 ) / 66\!\cdots\!31 $ (225970139316022v^15 + 821037119788618v^14 + 6209172670277582v^13 + 2754264413358832v^12 - 35059887452882164v^11 - 102707189280827664v^10 - 214594294540622338v^9 + 597091762388680376v^8 + 4876204147397252552v^7 + 5674267290438286048v^6 + 7546974672845795828v^5 - 9347929816377740782v^4 - 123055313254493157144v^3 - 53223350516417174796v^2 - 147957314137517144022*v - 563621686214008492210) / 66151241998754743831
$\beta_{3}$	$=$	$ ( 9811311006132 \nu^{15} - 118323800991498 \nu^{14} + 132224363315824 \nu^{13} + \cdots + 25\!\cdots\!96 ) / 16\!\cdots\!91 $ (9811311006132v^15 - 118323800991498v^14 + 132224363315824v^13 - 832477334805138v^12 + 1245535925530516v^11 + 2942556942038656v^10 + 2914938275423834v^9 - 34697616776804452v^8 + 29895971514625008v^7 - 497532611737162712v^6 + 74166678170662288v^5 - 474188824253538586v^4 - 2696550396151499562v^3 + 3924566477518416388v^2 - 5243154762311106844*v + 25982343432048887296) / 1613444926798896191
$\beta_{4}$	$=$	$ ( 44676200 \nu^{15} - 21022868 \nu^{14} + 318002448 \nu^{13} - 553890524 \nu^{12} + \cdots - 23181159999060 ) / 4949080935799 $ (44676200v^15 - 21022868v^14 + 318002448v^13 - 553890524v^12 - 1773143520v^11 - 258391024v^10 + 1729528776v^9 + 16311147808v^8 + 232204374608v^7 + 98722580104v^6 + 114369405344v^5 + 792517231512v^4 - 7522962436816v^3 + 1504441894212v^2 - 11914886408432*v - 23181159999060) / 4949080935799
$\beta_{5}$	$=$	$ ( 604553143311409 \nu^{15} + 718443434274215 \nu^{14} - 125615462650595 \nu^{13} + \cdots - 17\!\cdots\!36 ) / 66\!\cdots\!31 $ (604553143311409v^15 + 718443434274215v^14 - 125615462650595v^13 - 2650882015998645v^12 - 40591907623771769v^11 - 13364671429892703v^10 + 343095326420181150v^9 + 371789954966875042v^8 + 1467084543525808141v^7 + 2835869862346853779v^6 - 7288020297307872743v^5 + 2068147388382402223v^4 + 11531981831117812891v^3 - 107797860840394955403v^2 - 215691414335132209320*v - 174887696282180218736) / 66151241998754743831
$\beta_{6}$	$=$	$ ( 24794175248 \nu^{15} + 3163594257 \nu^{14} + 128697406654 \nu^{13} - 149540944872 \nu^{12} + \cdots - 41\!\cdots\!61 ) / 26\!\cdots\!47 $ (24794175248v^15 + 3163594257v^14 + 128697406654v^13 - 149540944872v^12 - 988956637100v^11 - 201559692944v^10 + 8514391322312v^9 + 6015278410378v^8 + 105394405423952v^7 + 90481462443496v^6 + 65108822623968v^5 + 783605359210704v^4 - 159640813387956v^3 - 862596014461513v^2 - 4324684791160230*v - 4198068038295061) / 2648910503293747
$\beta_{7}$	$=$	$ ( - 696083283744488 \nu^{15} + 278090520238218 \nu^{14} + \cdots - 23\!\cdots\!16 ) / 66\!\cdots\!31 $ (-696083283744488v^15 + 278090520238218v^14 - 1433945924728672v^13 + 6573541012695668v^12 + 33216208843230978v^11 - 31382142946947022v^10 - 307204687037369654v^9 - 147587680397808916v^8 - 1906648407153981030v^7 - 994458128504174262v^6 + 6720302171244843142v^5 - 12546371166039277016v^4 + 28866583455757807198v^3 + 33597872467953958944v^2 + 220473605947025149306*v - 235109792679524454916) / 66151241998754743831
$\beta_{8}$	$=$	$ ( - 17573995501919 \nu^{15} + 48573480362878 \nu^{14} - 79297748171381 \nu^{13} + \cdots - 78\!\cdots\!05 ) / 16\!\cdots\!91 $ (-17573995501919v^15 + 48573480362878v^14 - 79297748171381v^13 + 365445169906212v^12 + 292764681864037v^11 - 2314472696892089v^10 - 6822674206278134v^9 + 16616710625455228v^8 - 52464136563127843v^7 + 151443152111716901v^6 + 90494721187361487v^5 - 609753803613151745v^4 + 1461977600331898319v^3 + 961006216444950924v^2 - 56696290273675626*v - 7811189354901667805) / 1613444926798896191
$\beta_{9}$	$=$	$ ( - 119982199828042 \nu^{15} - 16910266735533 \nu^{14} - 935287779711267 \nu^{13} + \cdots - 26\!\cdots\!49 ) / 94\!\cdots\!33 $ (-119982199828042v^15 - 16910266735533v^14 - 935287779711267v^13 + 1928827412688198v^12 + 2615501486940016v^11 + 6975682549033102v^10 - 42919236057605063v^9 - 73130328305225903v^8 - 579275666220402230v^7 - 99528514650961070v^6 - 1467193008032885484v^5 - 669439572437359896v^4 + 35340472101006859v^3 - 10640573466085212728v^2 + 32246549117803088243*v - 26235310126494006349) / 9450177428393534833
$\beta_{10}$	$=$	$ ( 14\!\cdots\!36 \nu^{15} + \cdots - 22\!\cdots\!64 ) / 66\!\cdots\!31 $ (1494815560209336v^15 + 11597442963882262v^14 + 8404990584934241v^13 + 43305425455217717v^12 - 145541239837297770v^11 - 460285288072732988v^10 + 508325259774541627v^9 + 4631819093202100193v^8 + 9127983222731668888v^7 + 49645652222405128446v^6 + 34978454155633278002v^5 + 58297876441917900056v^4 + 249007256741457331407v^3 - 111589076625887458487v^2 - 618040404618309705599*v - 2231425907567711090264) / 66151241998754743831
$\beta_{11}$	$=$	$ ( 39630693572965 \nu^{15} - 1229380371689 \nu^{14} + 159532410095435 \nu^{13} + \cdots - 62\!\cdots\!72 ) / 16\!\cdots\!91 $ (39630693572965v^15 - 1229380371689v^14 + 159532410095435v^13 - 449855191219555v^12 - 1761139420526079v^11 - 326378139351765v^10 + 17398662922413126v^9 + 20627518214674920v^8 + 150367339748566889v^7 + 6135689372353043v^6 - 221253140981176281v^5 + 270784322949418951v^4 - 663721969656765865v^3 + 864930838681009221v^2 - 3808690954292896742*v - 625977732079095172) / 1613444926798896191
$\beta_{12}$	$=$	$ ( - 59705755476645 \nu^{15} + 34669631358846 \nu^{14} - 391910400736742 \nu^{13} + \cdots + 56\!\cdots\!02 ) / 16\!\cdots\!91 $ (-59705755476645v^15 + 34669631358846v^14 - 391910400736742v^13 + 641409991058804v^12 + 1624585751353186v^11 - 1201902525022636v^10 - 16647142947717587v^9 - 5380580158645443v^8 - 255826139025516196v^7 - 7126395295303178v^6 - 391418155467393308v^5 - 1902863981510212918v^4 + 1529051288018273746v^3 - 4188328461435773013v^2 + 13916874265025338698*v + 5631168445750302602) / 1613444926798896191
$\beta_{13}$	$=$	$ ( - 109029470992678 \nu^{15} - 128461943289595 \nu^{14} - 599997526477754 \nu^{13} + \cdots + 37\!\cdots\!33 ) / 16\!\cdots\!91 $ (-109029470992678v^15 - 128461943289595v^14 - 599997526477754v^13 + 243366587064953v^12 + 5283448917415514v^11 + 5013343171232330v^10 - 35837603161483274v^9 - 77626012663260624v^8 - 495045837992509346v^7 - 782779337258121412v^6 - 636096474705269258v^5 - 3239347226925132800v^4 - 1809059562465715288v^3 + 3512105987402207011v^2 + 25132585203527064228*v + 37453427320765722133) / 1613444926798896191
$\beta_{14}$	$=$	$ ( - 45\!\cdots\!50 \nu^{15} + \cdots + 28\!\cdots\!70 ) / 66\!\cdots\!31 $ (-4574875226080150v^15 - 11717160502604948v^14 - 19311302283106035v^13 - 27796870305108322v^12 + 323116846245661516v^11 + 402077884026471986v^10 - 1771010253284838689v^9 - 5419578159990802475v^8 - 21924345073099450874v^7 - 58615414491227403858v^6 + 772966219714635420v^5 - 148710496144128065792v^4 - 216575166205476192647v^3 + 296254993988734894937v^2 + 516421735573590864731*v + 2834262886133656204570) / 66151241998754743831
$\beta_{15}$	$=$	$ ( 17074128036347 \nu^{15} + 5538404533786 \nu^{14} + 69973947646466 \nu^{13} + \cdots - 96\!\cdots\!44 ) / 23\!\cdots\!13 $ (17074128036347v^15 + 5538404533786v^14 + 69973947646466v^13 - 150724439245314v^12 - 819387599821070v^11 - 286258993195464v^10 + 7253150200710045v^9 + 8403757202803023v^8 + 59200889494020668v^7 + 45855360276642118v^6 - 64801453181399748v^5 + 316038838289786252v^4 + 18349627440769330v^3 - 1196619597239145493v^2 - 5544136560961090918*v - 965935727024259544) / 230492132399842313

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

$\nu$	$=$	$ ( \beta_{14} + 4 \beta_{11} - \beta_{9} - 4 \beta_{8} + 4 \beta_{7} + \beta_{6} - 4 \beta_{5} + \cdots - 4 \beta_1 ) / 32 $ (b14 + 4b11 - b9 - 4b8 + 4b7 + b6 - 4b5 + b4 - 4b3 - 4b1) / 32
$\nu^{2}$	$=$	$ ( - \beta_{14} + 7 \beta_{13} - 8 \beta_{12} - \beta_{11} - 7 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots - 2 \beta_1 ) / 16 $ (-b14 + 7b13 - 8b12 - b11 - 7b9 + 3b8 - b7 + 7b6 + 4b5 - b4 - 5b3 + b2 - 2b1) / 16
$\nu^{3}$	$=$	$ ( - 3 \beta_{14} - 16 \beta_{13} - 12 \beta_{11} - 24 \beta_{10} - 21 \beta_{9} + 44 \beta_{8} + \cdots + 48 ) / 32 $ (-3b14 - 16b13 - 12b11 - 24b10 - 21b9 + 44b8 + 20b7 - 3b6 + 12b5 - b4 - 12b3 - 8b2 - 4b1 + 48) / 32
$\nu^{4}$	$=$	$ ( 2 \beta_{15} + 3 \beta_{14} + 11 \beta_{13} + 10 \beta_{12} - \beta_{11} + 2 \beta_{10} - 13 \beta_{9} + \cdots + 112 ) / 8 $ (2b15 + 3b14 + 11b13 + 10b12 - b11 + 2b10 - 13b9 + 11b8 + 7b7 + 127b6 + 2b5 + 5b4 - 9b3 - 7b2 + 36b1 + 112) / 8
$\nu^{5}$	$=$	$ ( - 10 \beta_{15} + 73 \beta_{14} + 11 \beta_{13} - 30 \beta_{12} + 67 \beta_{11} + 10 \beta_{10} + \cdots - 220 ) / 16 $ (-10b15 + 73b14 + 11b13 - 30b12 + 67b11 + 10b10 - 103b9 - 9b8 - 27b7 + 333b6 + 30b5 - 78b4 - 159b3 + 41b2 + 4*b1 - 220) / 16
$\nu^{6}$	$=$	$ ( - 38 \beta_{14} + 51 \beta_{13} - 193 \beta_{11} - 104 \beta_{10} - 466 \beta_{9} + 91 \beta_{8} + \cdots - 3304 ) / 16 $ (-38b14 + 51b13 - 193b11 - 104b10 - 466b9 + 91b8 - 455b7 + 162b6 - 240b5 + 4b4 - 193b3 + 15b2 - 210*b1 - 3304) / 16
$\nu^{7}$	$=$	$ ( - 784 \beta_{15} - 559 \beta_{14} - 910 \beta_{13} + 1680 \beta_{12} - 1074 \beta_{11} - 784 \beta_{10} + \cdots + 3024 ) / 32 $ (-784b15 - 559b14 - 910b13 + 1680b12 - 1074b11 - 784b10 - 1121b9 + 1886b8 - 1214b7 + 3361b6 + 2012b5 + 57b4 + 1382b3 - 462b2 + 2768*b1 + 3024) / 32
$\nu^{8}$	$=$	$ ( - 35 \beta_{15} + 264 \beta_{14} - 567 \beta_{13} + 789 \beta_{12} + 565 \beta_{11} + 35 \beta_{10} + \cdots - 1568 ) / 4 $ (-35b15 + 264b14 - 567b13 + 789b12 + 565b11 + 35b10 + 525b9 + 377b8 - 569b7 + 1078b6 - 909b5 - 58b4 + 165b3 - 159b2 + 661*b1 - 1568) / 4
$\nu^{9}$	$=$	$ ( 459 \beta_{14} + 7082 \beta_{13} + 2606 \beta_{11} + 8040 \beta_{10} - 1155 \beta_{9} - 16770 \beta_{8} + \cdots - 146112 ) / 32 $ (459b14 + 7082b13 + 2606b11 + 8040b10 - 1155b9 - 16770b8 - 20714b7 + 4827b6 - 10236b5 - 15443b4 + 2606b3 + 6110b2 + 1984*b1 - 146112) / 32
$\nu^{10}$	$=$	$ ( - 5316 \beta_{15} - 7547 \beta_{14} - 20810 \beta_{13} + 3828 \beta_{12} - 12042 \beta_{11} + \cdots - 146160 ) / 16 $ (-5316b15 - 7547b14 - 20810b13 + 3828b12 - 12042b11 - 5316b10 + 3719b9 - 18514b8 - 28532b7 - 155195b6 - 3020b5 - 5261b4 + 16110b3 - 252b2 - 4436*b1 - 146160) / 16
$\nu^{11}$	$=$	$ ( - 15158 \beta_{15} - 22159 \beta_{14} - 47799 \beta_{13} + 75878 \beta_{12} - 10699 \beta_{11} + \cdots + 10252 ) / 16 $ (-15158b15 - 22159b14 - 47799b13 + 75878b12 - 10699b11 + 15158b10 + 98037b9 - 7135b8 - 25535b7 - 93131b6 - 25506b5 + 44918b4 + 133247b3 - 38515b2 + 38252*b1 + 10252) / 16
$\nu^{12}$	$=$	$ ( 9675 \beta_{14} - 12547 \beta_{13} + 44757 \beta_{11} + 27988 \beta_{10} + 117137 \beta_{9} + \cdots + 122192 ) / 4 $ (9675b14 - 12547b13 + 44757b11 + 27988b10 + 117137b9 - 19663b8 + 1358b7 - 39737b6 - 37692b5 - 23386b4 + 44757b3 + 10362b2 - 16968*b1 + 122192) / 4
$\nu^{13}$	$=$	$ ( 374920 \beta_{15} - 212239 \beta_{14} + 122612 \beta_{13} - 782600 \beta_{12} - 16264 \beta_{11} + \cdots - 6119568 ) / 32 $ (374920b15 - 212239b14 + 122612b13 - 782600b12 - 16264b11 + 374920b10 + 994839b9 - 1778144b8 - 2264b7 - 6739487b6 - 1525836b5 - 501619b4 + 377800b3 + 470148b2 - 1584020*b1 - 6119568) / 32
$\nu^{14}$	$=$	$ ( - 162568 \beta_{15} - 974947 \beta_{14} - 20713 \beta_{13} - 620880 \beta_{12} - 1328377 \beta_{11} + \cdots + 9626736 ) / 16 $ (-162568b15 - 974947b14 - 20713b13 - 620880b12 - 1328377b11 + 162568b10 + 354067b9 - 2062533b8 + 1310119b7 - 9818235b6 + 2204388b5 + 860369b4 + 1635475b3 - 268127b2 - 150530*b1 + 9626736) / 16
$\nu^{15}$	$=$	$ ( 1233271 \beta_{14} - 4287824 \beta_{13} + 5841516 \beta_{11} + 1288536 \beta_{10} + 17210529 \beta_{9} + \cdots + 90847728 ) / 32 $ (1233271b14 - 4287824b13 + 5841516b11 + 1288536b10 + 17210529b9 + 2734132b8 + 13954716b7 - 7344361b6 + 1180996b5 + 9773721b4 + 5841516b3 - 4196088b2 - 7211180*b1 + 90847728) / 32

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} $

97.1

0.0550989	+	2.76316i
2.79294	−	1.11885i
−0.427517	−	2.97818i
1.71341	−	0.109119i
−2.42052	+	1.33386i
1.13462	+	1.75343i
−2.08583	−	0.105897i
−0.762206	−	1.53842i
−0.762206	+	1.53842i
−2.08583	+	0.105897i
1.13462	−	1.75343i
−2.42052	−	1.33386i
1.71341	+	0.109119i
−0.427517	+	2.97818i
2.79294	+	1.11885i
0.0550989	−	2.76316i

−

1.73205i

−

8.94195i

1.52790

6.83122i

−3.00000

97.2

−

1.73205i

−

5.15119i

−2.53356

−

6.52542i

−3.00000

97.3

−

1.73205i

−

2.97285i

3.90735

5.80798i

−3.00000

97.4

−

1.73205i

−

0.817910i

−6.47914

−

2.64968i

−3.00000

97.5

−

1.73205i

0.817910i

6.47914

−

2.64968i

−3.00000

97.6

−

1.73205i

2.97285i

−3.90735

5.80798i

−3.00000

97.7

−

1.73205i

5.15119i

2.53356

−

6.52542i

−3.00000

97.8

−

1.73205i

8.94195i

−1.52790

6.83122i

−3.00000

97.9

1.73205i

−

8.94195i

−1.52790

−

6.83122i

−3.00000

97.10

1.73205i

−

5.15119i

2.53356

6.52542i

−3.00000

97.11

1.73205i

−

2.97285i

−3.90735

−

5.80798i

−3.00000

97.12

1.73205i

−

0.817910i

6.47914

2.64968i

−3.00000

97.13

1.73205i

0.817910i

−6.47914

2.64968i

−3.00000

97.14

1.73205i

2.97285i

3.90735

−

5.80798i

−3.00000

97.15

1.73205i

5.15119i

−2.53356

6.52542i

−3.00000

97.16

1.73205i

8.94195i

1.52790

−

6.83122i

−3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.3.f.b	✓	16
3.b	odd	2	1		2016.3.f.g		16
4.b	odd	2	1	inner	672.3.f.b	✓	16
7.b	odd	2	1	inner	672.3.f.b	✓	16
8.b	even	2	1		1344.3.f.j		16
8.d	odd	2	1		1344.3.f.j		16
12.b	even	2	1		2016.3.f.g		16
21.c	even	2	1		2016.3.f.g		16
28.d	even	2	1	inner	672.3.f.b	✓	16
56.e	even	2	1		1344.3.f.j		16
56.h	odd	2	1		1344.3.f.j		16
84.h	odd	2	1		2016.3.f.g		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.3.f.b	✓	16	1.a	even	1	1	trivial
672.3.f.b	✓	16	4.b	odd	2	1	inner
672.3.f.b	✓	16	7.b	odd	2	1	inner
672.3.f.b	✓	16	28.d	even	2	1	inner
1344.3.f.j		16	8.b	even	2	1
1344.3.f.j		16	8.d	odd	2	1
1344.3.f.j		16	56.e	even	2	1
1344.3.f.j		16	56.h	odd	2	1
2016.3.f.g		16	3.b	odd	2	1
2016.3.f.g		16	12.b	even	2	1
2016.3.f.g		16	21.c	even	2	1
2016.3.f.g		16	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} + 116T_{5}^{6} + 3140T_{5}^{4} + 20800T_{5}^{2} + 12544 $ T5^8 + 116*T5^6 + 3140*T5^4 + 20800*T5^2 + 12544 acting on $S_{3}^{\mathrm{new}}(672, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{8} + 116 T^{6} + \cdots + 12544)^{2} $ (T^8 + 116T^6 + 3140T^4 + 20800*T^2 + 12544)^2
$7$	$ T^{16} + \cdots + 33232930569601 $ T^16 + 128T^14 + 8124T^12 + 294784T^10 + 10924550T^8 + 707776384T^6 + 46833243324T^4 + 1771684761728*T^2 + 33232930569601
$11$	$ (T^{8} - 308 T^{6} + \cdots + 18496)^{2} $ (T^8 - 308T^6 + 20820T^4 - 390368*T^2 + 18496)^2
$13$	$ (T^{8} + 704 T^{6} + \cdots + 58003456)^{2} $ (T^8 + 704T^6 + 139136T^4 + 5632000*T^2 + 58003456)^2
$17$	$ (T^{8} + 1300 T^{6} + \cdots + 2755830016)^{2} $ (T^8 + 1300T^6 + 527492T^4 + 76895552*T^2 + 2755830016)^2
$19$	$ (T^{8} + 1208 T^{6} + \cdots + 200704)^{2} $ (T^8 + 1208T^6 + 363920T^4 + 26222080*T^2 + 200704)^2
$23$	$ (T^{8} - 2788 T^{6} + \cdots + 25515589696)^{2} $ (T^8 - 2788T^6 + 2018772T^4 - 474407200*T^2 + 25515589696)^2
$29$	$ (T^{4} + 32 T^{3} + \cdots + 400624)^{4} $ (T^4 + 32T^3 - 1104T^2 - 18592*T + 400624)^4
$31$	$ (T^{8} + 2192 T^{6} + \cdots + 802816)^{2} $ (T^8 + 2192T^6 + 1084736T^4 + 10749952*T^2 + 802816)^2
$37$	$ (T^{4} - 12 T^{3} + \cdots - 184256)^{4} $ (T^4 - 12T^3 - 3052T^2 - 55392*T - 184256)^4
$41$	$ (T^{8} + 7348 T^{6} + \cdots + 39980802304)^{2} $ (T^8 + 7348T^6 + 11549636T^4 + 3262014272*T^2 + 39980802304)^2
$43$	$ (T^{8} + \cdots + 1854215996416)^{2} $ (T^8 - 6856T^6 + 14793552T^4 - 10449309952*T^2 + 1854215996416)^2
$47$	$ (T^{8} + 6368 T^{6} + \cdots + 554696704)^{2} $ (T^8 + 6368T^6 + 3696896T^4 + 543588352*T^2 + 554696704)^2
$53$	$ (T^{4} + 40 T^{3} + \cdots + 97648)^{4} $ (T^4 + 40T^3 - 4816T^2 - 89536*T + 97648)^4
$59$	$ (T^{8} + 6272 T^{6} + \cdots + 5398134784)^{2} $ (T^8 + 6272T^6 + 9981440T^4 + 1055555584*T^2 + 5398134784)^2
$61$	$ (T^{8} + \cdots + 81649874305024)^{2} $ (T^8 + 18032T^6 + 107183168T^4 + 222338117632*T^2 + 81649874305024)^2
$67$	$ (T^{8} + \cdots + 815571244171264)^{2} $ (T^8 - 28760T^6 + 262550160T^4 - 823007568896*T^2 + 815571244171264)^2
$71$	$ (T^{8} + \cdots + 492277878776896)^{2} $ (T^8 - 30628T^6 + 317278932T^4 - 1159201290784*T^2 + 492277878776896)^2
$73$	$ (T^{8} + \cdots + 13\!\cdots\!04)^{2} $ (T^8 + 37008T^6 + 463708224T^4 + 2079765061632*T^2 + 1386612638613504)^2
$79$	$ (T^{8} + \cdots + 511254788571136)^{2} $ (T^8 - 35160T^6 + 352588688T^4 - 1022359855104*T^2 + 511254788571136)^2
$83$	$ (T^{8} + \cdots + 242039913447424)^{2} $ (T^8 + 19168T^6 + 126970112T^4 + 327236354048*T^2 + 242039913447424)^2
$89$	$ (T^{8} + \cdots + 11\!\cdots\!24)^{2} $ (T^8 + 44308T^6 + 516994436T^4 + 1740995845952*T^2 + 1150107304685824)^2
$97$	$ (T^{8} + \cdots + 7895290740736)^{2} $ (T^8 + 17680T^6 + 83765312T^4 + 121315131392*T^2 + 7895290740736)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + \beta_{8} q^{5} + ( - \beta_{9} + \beta_{5} + \beta_1) q^{7} - 3 q^{9} + (\beta_{7} + \beta_{5}) q^{11} + ( - \beta_{11} + \beta_{8} + \beta_{3}) q^{13} + ( - \beta_{7} - \beta_1) q^{15}+ \cdots + ( - 3 \beta_{7} - 3 \beta_{5}) q^{99}+O(q^{100}) \) q + b6 * q^3 + b8 * q^5 + (-b9 + b5 + b1) * q^7 - 3 * q^9 + (b7 + b5) * q^11 + (-b11 + b8 + b3) * q^13 + (-b7 - b1) * q^15 + (b13 - b11 - b3) * q^17 + (-b10 + b9 + 2b6 - b5 - b1) q^19 + (b12 + b3 + 1) * q^21 + (b7 - 2b5 + b2 - b1) q^23 + (-b15 - b12 + b4 - 3) * q^25 - 3b6 q^27 + (-b15 - b13 + b12 + b3 - 8) * q^29 + (-b14 - b10 - 3b6) q^31 + (b13 + b8 - b3) * q^33 + (2b9 - 2b5 + 5b1) q^35 + (-b15 - 2b13 + 3b12 + 2b4 + 2b3 + 2) * q^37 + (b5 - b2 - b1) * q^39 + (3b13 - b11 - 4b8 + 3b3) q^41 + (-3b7 + b5 + b2 - 5b1) * q^43 - 3b8 q^45 + (b14 + 2b10 - 3b9 - 3b6 + 2b5 + 2b1) q^47 + (-b15 + b13 - b12 - 2b11 - 4b8 - b4 + 2b3 - 15) q^49 + (-2b7 - 2b5 - b2 + 3b1) q^51 + (-b15 - 3b13 + 5b12 + b4 + 3b3 - 12) q^53 + (b14 - b10 - 4b9 + 15b6 + b5 + b1) * q^55 + (-b15 - b12 - 8) * q^57 + (-b14 + 2b10 + 3b9 - b6) * q^59 + (2b13 - 3b11 - 3b8 - 5b3) * q^61 + (3b9 - 3b5 - 3b1) q^63 + (-2b15 - 2b12 + 5b4 - 16) q^65 + (5b7 + 9b5 - b2 - 7b1) q^67 + (-b13 - 3b11 + 4b8) * q^69 + (7b7 - 2b5 - 3b2 + b1) q^71 + (6b13 + 10b8 + 4b3) q^73 + (b14 + 3b10 - 4b9 + 3b5 + 3b1) * q^75 + (-b15 + b13 + b12 - 2b11 + 3b8 - b4 - 3b3 + 8) q^77 + (-2b7 - b5 + 2b2 + 13b1) q^79 + 9 * q^81 + (3b14 + 2b10 + 3b9 - 13b6 - 2b5 - 2b1) * q^83 + (-b15 - 4b13 + 7b12 - 2b4 + 4b3 - 4) * q^85 + (3b10 + 3b9 - 8b6) q^87 + (3b13 - 5b11 + 10b8 + 11b3) * q^89 + (-2b14 - b10 - 3b9 + 3b7 - 4b5 + 3b2 + 4b1) * q^91 + (-b15 - b13 + b12 + 3b4 + b3 + 6) q^93 + (-10b7 + 2b5 - 2b2 - 16b1) * q^95 + (4b13 - 2b11 + 8b8 - 2b3) * q^97 + (-3b7 - 3b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 24 q^{21} - 64 q^{25} - 128 q^{29} + 48 q^{37} - 256 q^{49} - 160 q^{53} - 144 q^{57} - 288 q^{65} + 128 q^{77} + 144 q^{81} - 16 q^{85} + 96 q^{93}+O(q^{100}) \) 16 * q - 48 * q^9 + 24 * q^21 - 64 * q^25 - 128 * q^29 + 48 * q^37 - 256 * q^49 - 160 * q^53 - 144 * q^57 - 288 * q^65 + 128 * q^77 + 144 * q^81 - 16 * q^85 + 96 * q^93

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	672.3.f.b

Level	$672$
Weight	$3$
Character orbit	672.f
Analytic conductor	$18.311$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(18.3106737650\)
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:	\( x^{16} + 2 x^{14} - 8 x^{13} - 57 x^{12} + 32 x^{11} + 466 x^{10} + 304 x^{9} + 3000 x^{8} + \cdots + 790321 \) x^16 + 2x^14 - 8x^13 - 57x^12 + 32x^11 + 466x^10 + 304x^9 + 3000x^8 + 1328x^7 - 11560x^6 + 22316x^5 - 23333x^4 - 87520x^3 - 135796x^2 - 88900x + 790321
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{30} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 12827995238497 \nu^{15} - 581190265302438 \nu^{14} + \cdots - 18\!\cdots\!83 ) / 66\!\cdots\!31 \) (12827995238497v^15 - 581190265302438v^14 + 1737559774441217v^13 - 3180484385351856v^12 + 2633733112382891v^11 + 29039772268083545v^10 - 85444707116929170v^9 - 67188609781839606v^8 + 683204853880188609v^7 - 2973986605133057123v^6 + 1735322209013429241v^5 + 12463695242476347423v^4 - 39601502364319381933v^3 + 68875998111592376644v^2 - 26048159145128777216*v - 180098559199799117983) / 66151241998754743831
\(\beta_{2}\)	\(=\)	\( ( 225970139316022 \nu^{15} + 821037119788618 \nu^{14} + \cdots - 56\!\cdots\!10 ) / 66\!\cdots\!31 \) (225970139316022v^15 + 821037119788618v^14 + 6209172670277582v^13 + 2754264413358832v^12 - 35059887452882164v^11 - 102707189280827664v^10 - 214594294540622338v^9 + 597091762388680376v^8 + 4876204147397252552v^7 + 5674267290438286048v^6 + 7546974672845795828v^5 - 9347929816377740782v^4 - 123055313254493157144v^3 - 53223350516417174796v^2 - 147957314137517144022*v - 563621686214008492210) / 66151241998754743831
\(\beta_{3}\)	\(=\)	\( ( 9811311006132 \nu^{15} - 118323800991498 \nu^{14} + 132224363315824 \nu^{13} + \cdots + 25\!\cdots\!96 ) / 16\!\cdots\!91 \) (9811311006132v^15 - 118323800991498v^14 + 132224363315824v^13 - 832477334805138v^12 + 1245535925530516v^11 + 2942556942038656v^10 + 2914938275423834v^9 - 34697616776804452v^8 + 29895971514625008v^7 - 497532611737162712v^6 + 74166678170662288v^5 - 474188824253538586v^4 - 2696550396151499562v^3 + 3924566477518416388v^2 - 5243154762311106844*v + 25982343432048887296) / 1613444926798896191
\(\beta_{4}\)	\(=\)	\( ( 44676200 \nu^{15} - 21022868 \nu^{14} + 318002448 \nu^{13} - 553890524 \nu^{12} + \cdots - 23181159999060 ) / 4949080935799 \) (44676200v^15 - 21022868v^14 + 318002448v^13 - 553890524v^12 - 1773143520v^11 - 258391024v^10 + 1729528776v^9 + 16311147808v^8 + 232204374608v^7 + 98722580104v^6 + 114369405344v^5 + 792517231512v^4 - 7522962436816v^3 + 1504441894212v^2 - 11914886408432*v - 23181159999060) / 4949080935799
\(\beta_{5}\)	\(=\)	\( ( 604553143311409 \nu^{15} + 718443434274215 \nu^{14} - 125615462650595 \nu^{13} + \cdots - 17\!\cdots\!36 ) / 66\!\cdots\!31 \) (604553143311409v^15 + 718443434274215v^14 - 125615462650595v^13 - 2650882015998645v^12 - 40591907623771769v^11 - 13364671429892703v^10 + 343095326420181150v^9 + 371789954966875042v^8 + 1467084543525808141v^7 + 2835869862346853779v^6 - 7288020297307872743v^5 + 2068147388382402223v^4 + 11531981831117812891v^3 - 107797860840394955403v^2 - 215691414335132209320*v - 174887696282180218736) / 66151241998754743831
\(\beta_{6}\)	\(=\)	\( ( 24794175248 \nu^{15} + 3163594257 \nu^{14} + 128697406654 \nu^{13} - 149540944872 \nu^{12} + \cdots - 41\!\cdots\!61 ) / 26\!\cdots\!47 \) (24794175248v^15 + 3163594257v^14 + 128697406654v^13 - 149540944872v^12 - 988956637100v^11 - 201559692944v^10 + 8514391322312v^9 + 6015278410378v^8 + 105394405423952v^7 + 90481462443496v^6 + 65108822623968v^5 + 783605359210704v^4 - 159640813387956v^3 - 862596014461513v^2 - 4324684791160230*v - 4198068038295061) / 2648910503293747
\(\beta_{7}\)	\(=\)	\( ( - 696083283744488 \nu^{15} + 278090520238218 \nu^{14} + \cdots - 23\!\cdots\!16 ) / 66\!\cdots\!31 \) (-696083283744488v^15 + 278090520238218v^14 - 1433945924728672v^13 + 6573541012695668v^12 + 33216208843230978v^11 - 31382142946947022v^10 - 307204687037369654v^9 - 147587680397808916v^8 - 1906648407153981030v^7 - 994458128504174262v^6 + 6720302171244843142v^5 - 12546371166039277016v^4 + 28866583455757807198v^3 + 33597872467953958944v^2 + 220473605947025149306*v - 235109792679524454916) / 66151241998754743831
\(\beta_{8}\)	\(=\)	\( ( - 17573995501919 \nu^{15} + 48573480362878 \nu^{14} - 79297748171381 \nu^{13} + \cdots - 78\!\cdots\!05 ) / 16\!\cdots\!91 \) (-17573995501919v^15 + 48573480362878v^14 - 79297748171381v^13 + 365445169906212v^12 + 292764681864037v^11 - 2314472696892089v^10 - 6822674206278134v^9 + 16616710625455228v^8 - 52464136563127843v^7 + 151443152111716901v^6 + 90494721187361487v^5 - 609753803613151745v^4 + 1461977600331898319v^3 + 961006216444950924v^2 - 56696290273675626*v - 7811189354901667805) / 1613444926798896191
\(\beta_{9}\)	\(=\)	\( ( - 119982199828042 \nu^{15} - 16910266735533 \nu^{14} - 935287779711267 \nu^{13} + \cdots - 26\!\cdots\!49 ) / 94\!\cdots\!33 \) (-119982199828042v^15 - 16910266735533v^14 - 935287779711267v^13 + 1928827412688198v^12 + 2615501486940016v^11 + 6975682549033102v^10 - 42919236057605063v^9 - 73130328305225903v^8 - 579275666220402230v^7 - 99528514650961070v^6 - 1467193008032885484v^5 - 669439572437359896v^4 + 35340472101006859v^3 - 10640573466085212728v^2 + 32246549117803088243*v - 26235310126494006349) / 9450177428393534833
\(\beta_{10}\)	\(=\)	\( ( 14\!\cdots\!36 \nu^{15} + \cdots - 22\!\cdots\!64 ) / 66\!\cdots\!31 \) (1494815560209336v^15 + 11597442963882262v^14 + 8404990584934241v^13 + 43305425455217717v^12 - 145541239837297770v^11 - 460285288072732988v^10 + 508325259774541627v^9 + 4631819093202100193v^8 + 9127983222731668888v^7 + 49645652222405128446v^6 + 34978454155633278002v^5 + 58297876441917900056v^4 + 249007256741457331407v^3 - 111589076625887458487v^2 - 618040404618309705599*v - 2231425907567711090264) / 66151241998754743831
\(\beta_{11}\)	\(=\)	\( ( 39630693572965 \nu^{15} - 1229380371689 \nu^{14} + 159532410095435 \nu^{13} + \cdots - 62\!\cdots\!72 ) / 16\!\cdots\!91 \) (39630693572965v^15 - 1229380371689v^14 + 159532410095435v^13 - 449855191219555v^12 - 1761139420526079v^11 - 326378139351765v^10 + 17398662922413126v^9 + 20627518214674920v^8 + 150367339748566889v^7 + 6135689372353043v^6 - 221253140981176281v^5 + 270784322949418951v^4 - 663721969656765865v^3 + 864930838681009221v^2 - 3808690954292896742*v - 625977732079095172) / 1613444926798896191
\(\beta_{12}\)	\(=\)	\( ( - 59705755476645 \nu^{15} + 34669631358846 \nu^{14} - 391910400736742 \nu^{13} + \cdots + 56\!\cdots\!02 ) / 16\!\cdots\!91 \) (-59705755476645v^15 + 34669631358846v^14 - 391910400736742v^13 + 641409991058804v^12 + 1624585751353186v^11 - 1201902525022636v^10 - 16647142947717587v^9 - 5380580158645443v^8 - 255826139025516196v^7 - 7126395295303178v^6 - 391418155467393308v^5 - 1902863981510212918v^4 + 1529051288018273746v^3 - 4188328461435773013v^2 + 13916874265025338698*v + 5631168445750302602) / 1613444926798896191
\(\beta_{13}\)	\(=\)	\( ( - 109029470992678 \nu^{15} - 128461943289595 \nu^{14} - 599997526477754 \nu^{13} + \cdots + 37\!\cdots\!33 ) / 16\!\cdots\!91 \) (-109029470992678v^15 - 128461943289595v^14 - 599997526477754v^13 + 243366587064953v^12 + 5283448917415514v^11 + 5013343171232330v^10 - 35837603161483274v^9 - 77626012663260624v^8 - 495045837992509346v^7 - 782779337258121412v^6 - 636096474705269258v^5 - 3239347226925132800v^4 - 1809059562465715288v^3 + 3512105987402207011v^2 + 25132585203527064228*v + 37453427320765722133) / 1613444926798896191
\(\beta_{14}\)	\(=\)	\( ( - 45\!\cdots\!50 \nu^{15} + \cdots + 28\!\cdots\!70 ) / 66\!\cdots\!31 \) (-4574875226080150v^15 - 11717160502604948v^14 - 19311302283106035v^13 - 27796870305108322v^12 + 323116846245661516v^11 + 402077884026471986v^10 - 1771010253284838689v^9 - 5419578159990802475v^8 - 21924345073099450874v^7 - 58615414491227403858v^6 + 772966219714635420v^5 - 148710496144128065792v^4 - 216575166205476192647v^3 + 296254993988734894937v^2 + 516421735573590864731*v + 2834262886133656204570) / 66151241998754743831
\(\beta_{15}\)	\(=\)	\( ( 17074128036347 \nu^{15} + 5538404533786 \nu^{14} + 69973947646466 \nu^{13} + \cdots - 96\!\cdots\!44 ) / 23\!\cdots\!13 \) (17074128036347v^15 + 5538404533786v^14 + 69973947646466v^13 - 150724439245314v^12 - 819387599821070v^11 - 286258993195464v^10 + 7253150200710045v^9 + 8403757202803023v^8 + 59200889494020668v^7 + 45855360276642118v^6 - 64801453181399748v^5 + 316038838289786252v^4 + 18349627440769330v^3 - 1196619597239145493v^2 - 5544136560961090918*v - 965935727024259544) / 230492132399842313

\(\nu\)	\(=\)	\( ( \beta_{14} + 4 \beta_{11} - \beta_{9} - 4 \beta_{8} + 4 \beta_{7} + \beta_{6} - 4 \beta_{5} + \cdots - 4 \beta_1 ) / 32 \) (b14 + 4b11 - b9 - 4b8 + 4b7 + b6 - 4b5 + b4 - 4b3 - 4b1) / 32
\(\nu^{2}\)	\(=\)	\( ( - \beta_{14} + 7 \beta_{13} - 8 \beta_{12} - \beta_{11} - 7 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots - 2 \beta_1 ) / 16 \) (-b14 + 7b13 - 8b12 - b11 - 7b9 + 3b8 - b7 + 7b6 + 4b5 - b4 - 5b3 + b2 - 2b1) / 16
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{14} - 16 \beta_{13} - 12 \beta_{11} - 24 \beta_{10} - 21 \beta_{9} + 44 \beta_{8} + \cdots + 48 ) / 32 \) (-3b14 - 16b13 - 12b11 - 24b10 - 21b9 + 44b8 + 20b7 - 3b6 + 12b5 - b4 - 12b3 - 8b2 - 4b1 + 48) / 32
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} + 3 \beta_{14} + 11 \beta_{13} + 10 \beta_{12} - \beta_{11} + 2 \beta_{10} - 13 \beta_{9} + \cdots + 112 ) / 8 \) (2b15 + 3b14 + 11b13 + 10b12 - b11 + 2b10 - 13b9 + 11b8 + 7b7 + 127b6 + 2b5 + 5b4 - 9b3 - 7b2 + 36b1 + 112) / 8
\(\nu^{5}\)	\(=\)	\( ( - 10 \beta_{15} + 73 \beta_{14} + 11 \beta_{13} - 30 \beta_{12} + 67 \beta_{11} + 10 \beta_{10} + \cdots - 220 ) / 16 \) (-10b15 + 73b14 + 11b13 - 30b12 + 67b11 + 10b10 - 103b9 - 9b8 - 27b7 + 333b6 + 30b5 - 78b4 - 159b3 + 41b2 + 4*b1 - 220) / 16
\(\nu^{6}\)	\(=\)	\( ( - 38 \beta_{14} + 51 \beta_{13} - 193 \beta_{11} - 104 \beta_{10} - 466 \beta_{9} + 91 \beta_{8} + \cdots - 3304 ) / 16 \) (-38b14 + 51b13 - 193b11 - 104b10 - 466b9 + 91b8 - 455b7 + 162b6 - 240b5 + 4b4 - 193b3 + 15b2 - 210*b1 - 3304) / 16
\(\nu^{7}\)	\(=\)	\( ( - 784 \beta_{15} - 559 \beta_{14} - 910 \beta_{13} + 1680 \beta_{12} - 1074 \beta_{11} - 784 \beta_{10} + \cdots + 3024 ) / 32 \) (-784b15 - 559b14 - 910b13 + 1680b12 - 1074b11 - 784b10 - 1121b9 + 1886b8 - 1214b7 + 3361b6 + 2012b5 + 57b4 + 1382b3 - 462b2 + 2768*b1 + 3024) / 32
\(\nu^{8}\)	\(=\)	\( ( - 35 \beta_{15} + 264 \beta_{14} - 567 \beta_{13} + 789 \beta_{12} + 565 \beta_{11} + 35 \beta_{10} + \cdots - 1568 ) / 4 \) (-35b15 + 264b14 - 567b13 + 789b12 + 565b11 + 35b10 + 525b9 + 377b8 - 569b7 + 1078b6 - 909b5 - 58b4 + 165b3 - 159b2 + 661*b1 - 1568) / 4
\(\nu^{9}\)	\(=\)	\( ( 459 \beta_{14} + 7082 \beta_{13} + 2606 \beta_{11} + 8040 \beta_{10} - 1155 \beta_{9} - 16770 \beta_{8} + \cdots - 146112 ) / 32 \) (459b14 + 7082b13 + 2606b11 + 8040b10 - 1155b9 - 16770b8 - 20714b7 + 4827b6 - 10236b5 - 15443b4 + 2606b3 + 6110b2 + 1984*b1 - 146112) / 32
\(\nu^{10}\)	\(=\)	\( ( - 5316 \beta_{15} - 7547 \beta_{14} - 20810 \beta_{13} + 3828 \beta_{12} - 12042 \beta_{11} + \cdots - 146160 ) / 16 \) (-5316b15 - 7547b14 - 20810b13 + 3828b12 - 12042b11 - 5316b10 + 3719b9 - 18514b8 - 28532b7 - 155195b6 - 3020b5 - 5261b4 + 16110b3 - 252b2 - 4436*b1 - 146160) / 16
\(\nu^{11}\)	\(=\)	\( ( - 15158 \beta_{15} - 22159 \beta_{14} - 47799 \beta_{13} + 75878 \beta_{12} - 10699 \beta_{11} + \cdots + 10252 ) / 16 \) (-15158b15 - 22159b14 - 47799b13 + 75878b12 - 10699b11 + 15158b10 + 98037b9 - 7135b8 - 25535b7 - 93131b6 - 25506b5 + 44918b4 + 133247b3 - 38515b2 + 38252*b1 + 10252) / 16
\(\nu^{12}\)	\(=\)	\( ( 9675 \beta_{14} - 12547 \beta_{13} + 44757 \beta_{11} + 27988 \beta_{10} + 117137 \beta_{9} + \cdots + 122192 ) / 4 \) (9675b14 - 12547b13 + 44757b11 + 27988b10 + 117137b9 - 19663b8 + 1358b7 - 39737b6 - 37692b5 - 23386b4 + 44757b3 + 10362b2 - 16968*b1 + 122192) / 4
\(\nu^{13}\)	\(=\)	\( ( 374920 \beta_{15} - 212239 \beta_{14} + 122612 \beta_{13} - 782600 \beta_{12} - 16264 \beta_{11} + \cdots - 6119568 ) / 32 \) (374920b15 - 212239b14 + 122612b13 - 782600b12 - 16264b11 + 374920b10 + 994839b9 - 1778144b8 - 2264b7 - 6739487b6 - 1525836b5 - 501619b4 + 377800b3 + 470148b2 - 1584020*b1 - 6119568) / 32
\(\nu^{14}\)	\(=\)	\( ( - 162568 \beta_{15} - 974947 \beta_{14} - 20713 \beta_{13} - 620880 \beta_{12} - 1328377 \beta_{11} + \cdots + 9626736 ) / 16 \) (-162568b15 - 974947b14 - 20713b13 - 620880b12 - 1328377b11 + 162568b10 + 354067b9 - 2062533b8 + 1310119b7 - 9818235b6 + 2204388b5 + 860369b4 + 1635475b3 - 268127b2 - 150530*b1 + 9626736) / 16
\(\nu^{15}\)	\(=\)	\( ( 1233271 \beta_{14} - 4287824 \beta_{13} + 5841516 \beta_{11} + 1288536 \beta_{10} + 17210529 \beta_{9} + \cdots + 90847728 ) / 32 \) (1233271b14 - 4287824b13 + 5841516b11 + 1288536b10 + 17210529b9 + 2734132b8 + 13954716b7 - 7344361b6 + 1180996b5 + 9773721b4 + 5841516b3 - 4196088b2 - 7211180*b1 + 90847728) / 32

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{8} + 116 T^{6} + \cdots + 12544)^{2} \) (T^8 + 116T^6 + 3140T^4 + 20800*T^2 + 12544)^2
$7$	\( T^{16} + \cdots + 33232930569601 \) T^16 + 128T^14 + 8124T^12 + 294784T^10 + 10924550T^8 + 707776384T^6 + 46833243324T^4 + 1771684761728*T^2 + 33232930569601
$11$	\( (T^{8} - 308 T^{6} + \cdots + 18496)^{2} \) (T^8 - 308T^6 + 20820T^4 - 390368*T^2 + 18496)^2
$13$	\( (T^{8} + 704 T^{6} + \cdots + 58003456)^{2} \) (T^8 + 704T^6 + 139136T^4 + 5632000*T^2 + 58003456)^2
$17$	\( (T^{8} + 1300 T^{6} + \cdots + 2755830016)^{2} \) (T^8 + 1300T^6 + 527492T^4 + 76895552*T^2 + 2755830016)^2
$19$	\( (T^{8} + 1208 T^{6} + \cdots + 200704)^{2} \) (T^8 + 1208T^6 + 363920T^4 + 26222080*T^2 + 200704)^2
$23$	\( (T^{8} - 2788 T^{6} + \cdots + 25515589696)^{2} \) (T^8 - 2788T^6 + 2018772T^4 - 474407200*T^2 + 25515589696)^2
$29$	\( (T^{4} + 32 T^{3} + \cdots + 400624)^{4} \) (T^4 + 32T^3 - 1104T^2 - 18592*T + 400624)^4
$31$	\( (T^{8} + 2192 T^{6} + \cdots + 802816)^{2} \) (T^8 + 2192T^6 + 1084736T^4 + 10749952*T^2 + 802816)^2
$37$	\( (T^{4} - 12 T^{3} + \cdots - 184256)^{4} \) (T^4 - 12T^3 - 3052T^2 - 55392*T - 184256)^4
$41$	\( (T^{8} + 7348 T^{6} + \cdots + 39980802304)^{2} \) (T^8 + 7348T^6 + 11549636T^4 + 3262014272*T^2 + 39980802304)^2
$43$	\( (T^{8} + \cdots + 1854215996416)^{2} \) (T^8 - 6856T^6 + 14793552T^4 - 10449309952*T^2 + 1854215996416)^2
$47$	\( (T^{8} + 6368 T^{6} + \cdots + 554696704)^{2} \) (T^8 + 6368T^6 + 3696896T^4 + 543588352*T^2 + 554696704)^2
$53$	\( (T^{4} + 40 T^{3} + \cdots + 97648)^{4} \) (T^4 + 40T^3 - 4816T^2 - 89536*T + 97648)^4
$59$	\( (T^{8} + 6272 T^{6} + \cdots + 5398134784)^{2} \) (T^8 + 6272T^6 + 9981440T^4 + 1055555584*T^2 + 5398134784)^2
$61$	\( (T^{8} + \cdots + 81649874305024)^{2} \) (T^8 + 18032T^6 + 107183168T^4 + 222338117632*T^2 + 81649874305024)^2
$67$	\( (T^{8} + \cdots + 815571244171264)^{2} \) (T^8 - 28760T^6 + 262550160T^4 - 823007568896*T^2 + 815571244171264)^2
$71$	\( (T^{8} + \cdots + 492277878776896)^{2} \) (T^8 - 30628T^6 + 317278932T^4 - 1159201290784*T^2 + 492277878776896)^2
$73$	\( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) (T^8 + 37008T^6 + 463708224T^4 + 2079765061632*T^2 + 1386612638613504)^2
$79$	\( (T^{8} + \cdots + 511254788571136)^{2} \) (T^8 - 35160T^6 + 352588688T^4 - 1022359855104*T^2 + 511254788571136)^2
$83$	\( (T^{8} + \cdots + 242039913447424)^{2} \) (T^8 + 19168T^6 + 126970112T^4 + 327236354048*T^2 + 242039913447424)^2
$89$	\( (T^{8} + \cdots + 11\!\cdots\!24)^{2} \) (T^8 + 44308T^6 + 516994436T^4 + 1740995845952*T^2 + 1150107304685824)^2
$97$	\( (T^{8} + \cdots + 7895290740736)^{2} \) (T^8 + 17680T^6 + 83765312T^4 + 121315131392*T^2 + 7895290740736)^2
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