LMFDB - Newform orbit 64.19.c.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [64,19,Mod(63,64)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 19, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("64.63"); S:= CuspForms(chi, 19); N := Newforms(S);

Level:	$N$	$=$	$64 = 2^{6}$
Weight:	$k$	$=$	$19$
Character orbit:	$[\chi]$	$=$	64.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-860880] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$131.447128134$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 3 x^{7} - 10969 x^{6} - 835887 x^{5} + 20786973 x^{4} - 17082875145 x^{3} + \cdots + 71\!\cdots\!30$ x^8 - 3x^7 - 10969x^6 - 835887x^5 + 20786973x^4 - 17082875145x^3 - 2740871824195x^2 + 109263554839035*x + 71754930984829630
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{112}\cdot 3^{6}$
Twist minimal:	no (minimal twist has level 4)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + (\beta_{2} - 107610) q^{5} + (\beta_{3} + 202 \beta_1) q^{7} + (\beta_{4} - 15 \beta_{2} - 91192695) q^{9} + (\beta_{6} + 12 \beta_{3} - 5621 \beta_1) q^{11} + ( - \beta_{5} - 12 \beta_{4} + \cdots + 832347286) q^{13}+ \cdots + ( - 22498458 \beta_{7} + \cdots + 8858850972369 \beta_1) q^{99}+O(q^{100})$ q - b1 * q^3 + (b2 - 107610) * q^5 + (b3 + 202b1) q^7 + (b4 - 15b2 - 91192695) q^9 + (b6 + 12b3 - 5621b1) * q^11 + (-b5 - 12b4 - 163b2 + 832347286) * q^13 + (b7 + 11b6 + 288b3 - 15746b1) q^15 + (34b5 - 187b4 + 7293b2 + 26731772706) q^17 + (-14b7 - 7b6 + 4782b3 + 697137b1) * q^19 + (-285b5 - 1796b4 - 139188b2 + 96477792384) q^21 + (67b7 - 183b6 + 31374b3 + 7022902b1) * q^23 + (730b5 + 2490b4 - 734750b2 + 66115716555) q^25 + (-42b7 - 777b6 + 108090b3 + 5836416b1) * q^27 + (1680b5 + 26424b4 - 5073599b2 - 1404757071114) q^29 + (-749b7 + 1673b6 + 67847b3 - 174475408b1) * q^31 + (-12900b5 + 13375b4 - 22382433b2 - 2692742303040) q^33 + (2450b7 + 15120b6 - 857790b3 - 1144052200b1) * q^35 + (17131b5 - 115548b4 - 37619195b2 + 19860871102054) q^37 + (180b7 - 7596b6 - 2882997b3 - 4293300394b1) * q^39 + (46870b5 - 419590b4 - 34019214b2 + 56438748090642) q^41 + (-15108b7 - 129198b6 - 2767660b3 - 12717599419b1) * q^43 + (-182115b5 - 63820b4 - 147891b2 - 49285299775530) q^45 + (26425b7 - 29645b6 + 6213207b3 - 35016979852b1) * q^47 + (150800b5 + 3026676b4 + 393829268b2 - 281400141696335) q^49 + (11696b7 + 641665b6 + 31642236b3 - 87902616040b1) * q^51 + (399769b5 + 1535108b4 + 1093662725b2 - 152048095010874) q^53 + (-104244b7 + 660156b6 + 56823773b3 - 180379364326b1) * q^55 + (-1355940b5 - 9681075b4 + 1152647469b2 + 332695422273600) q^57 + (178486b7 - 1979138b6 - 16540530b3 - 311326185067b1) * q^59 + (1386935b5 + 2607660b4 + 823615829b2 - 23074114388042) q^61 + (-85281b7 - 4178811b6 - 278386920b3 - 505603992558b1) * q^63 + (1604210b5 + 11919430b4 - 1930456146b2 - 708786381448380) q^65 + (-472860b7 + 3406815b6 - 411526760b3 - 660868346531b1) * q^67 + (-6922335b5 - 56337596b4 - 8619440364b2 + 3354907573017984) q^69 + (1290581b7 + 12978799b6 + 285663774b3 - 340650033294b1) * q^71 + (9127856b5 - 14765883b4 - 14658485227b2 + 3936795810844754) q^73 + (-815850b7 - 8820b6 + 1153899990b3 + 683353038825b1) * q^75 + (-1064435b5 + 225744500b4 - 23923830140b2 - 22601398495486080) q^77 + (-1569464b7 - 12795232b6 - 87871238b3 + 2187232017892b1) * q^79 + (-23617980b5 + 184881531b4 - 62098533b2 - 32558341623482223) q^81 + (4049094b7 - 17162726b6 - 926493810b3 + 4988027145763b1) * q^83 + (48578265b5 - 385150980b4 + 70994533374b2 + 25582545236684300) q^85 + (-5818967b7 - 49894117b6 + 4007508768b3 + 9759336659518b1) * q^87 + (-26022200b5 - 716644339b4 + 47484172413b2 + 20250158128730994) q^89 + (2158786b7 + 43096676b6 + 3759170818b3 + 12261187651744b1) * q^91 + (-38375940b5 + 48531680b4 + 43064517504b2 - 83519859542545920) q^93 + (13491652b7 + 205094732b6 - 13814042169b3 + 9255577453758b1) * q^95 + (109516154b5 + 468975873b4 + 247041871673b2 - 151872094112970814) q^97 + (-22498458b7 - 36568074b6 - 19477626786b3 + 8858850972369b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 860880 q^{5} - 729541560 q^{9} + 6658778288 q^{13} + 213854181648 q^{17} + 771822339072 q^{21} + 528925732440 q^{25} - 11238056568912 q^{29} - 21541938424320 q^{33} + 158886968816432 q^{37} + 451509984725136 q^{41}+ \cdots - 12\!\cdots\!12 q^{97}+O(q^{100})$ 8 * q - 860880 * q^5 - 729541560 * q^9 + 6658778288 * q^13 + 213854181648 * q^17 + 771822339072 * q^21 + 528925732440 * q^25 - 11238056568912 * q^29 - 21541938424320 * q^33 + 158886968816432 * q^37 + 451509984725136 * q^41 - 394282398204240 * q^45 - 2251201133570680 * q^49 - 1216384760086992 * q^53 + 2661563378188800 * q^57 - 184592915104336 * q^61 - 5670291051587040 * q^65 + 26839260584143872 * q^69 + 31494366486758032 * q^73 - 180811187963888640 * q^77 - 260466732987857784 * q^81 + 204660361893474400 * q^85 + 162001265029847952 * q^89 - 668158876340367360 * q^93 - 1214976752903766512 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 3 x^{7} - 10969 x^{6} - 835887 x^{5} + 20786973 x^{4} - 17082875145 x^{3} + \cdots + 71\!\cdots\!30$ x^8 - 3*x^7 - 10969*x^6 - 835887*x^5 + 20786973*x^4 - 17082875145*x^3 - 2740871824195*x^2 + 109263554839035*x + 71754930984829630 :

$\beta_{1}$	$=$	$( 349 \nu^{7} + 53248 \nu^{6} + 525867 \nu^{5} + 212100462 \nu^{4} - 9178232109 \nu^{3} + \cdots - 18\!\cdots\!30 ) / 5772436045824$ (349v^7 + 53248v^6 + 525867v^5 + 212100462v^4 - 9178232109v^3 - 5640029463756v^2 - 1862342586261691*v - 183524160977116930) / 5772436045824
$\beta_{2}$	$=$	$( 203 \nu^{7} - 24576 \nu^{6} + 1893869 \nu^{5} + 118153602 \nu^{4} - 88763350907 \nu^{3} + \cdots + 32\!\cdots\!10 ) / 120259084288$ (203v^7 - 24576v^6 + 1893869v^5 + 118153602v^4 - 88763350907v^3 - 1984220300436v^2 - 142517595284445*v + 32331435548100210) / 120259084288
$\beta_{3}$	$=$	$( 164099 \nu^{7} + 21180416 \nu^{6} + 1883223669 \nu^{5} - 147902791470 \nu^{4} + \cdots - 10\!\cdots\!90 ) / 1443109011456$ (164099v^7 + 21180416v^6 + 1883223669v^5 - 147902791470v^4 + 5429670825165v^3 - 1690414361102772v^2 - 808518865001245157*v - 104224999886983859390) / 1443109011456
$\beta_{4}$	$=$	$( - 1281 \nu^{7} + 6627328 \nu^{6} + 155568089 \nu^{5} - 70792780870 \nu^{4} + \cdots - 98\!\cdots\!78 ) / 120259084288$ (-1281v^7 + 6627328v^6 + 155568089v^5 - 70792780870v^4 - 9322771666159v^3 - 1792216960398532v^2 - 106797677251726345*v - 9855166832108006678) / 120259084288
$\beta_{5}$	$=$	$( 103453 \nu^{7} - 1875968 \nu^{6} - 1446588053 \nu^{5} - 71155841554 \nu^{4} + \cdots + 13\!\cdots\!38 ) / 30064771072$ (103453v^7 - 1875968v^6 - 1446588053v^5 - 71155841554v^4 + 11939137481747v^3 - 940760129993164v^2 - 762870984582072315*v + 13471894074972282238) / 30064771072
$\beta_{6}$	$=$	$( 10150111 \nu^{7} - 135335936 \nu^{6} + 74463550713 \nu^{5} + 9702533103354 \nu^{4} + \cdots - 16\!\cdots\!10 ) / 2886218022912$ (10150111v^7 - 135335936v^6 + 74463550713v^5 + 9702533103354v^4 + 1463180979799857v^3 - 522433687782171972v^2 - 28173630976421209897*v - 1677439805022069271510) / 2886218022912
$\beta_{7}$	$=$	$( 81903265 \nu^{7} + 625807360 \nu^{6} - 985551982713 \nu^{5} - 91240663441914 \nu^{4} + \cdots + 67\!\cdots\!30 ) / 2886218022912$ (81903265v^7 + 625807360v^6 - 985551982713v^5 - 91240663441914v^4 + 326045729382735v^3 - 1012788725136342204v^2 + 150452357897355346601*v + 6774323677785624764630) / 2886218022912

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

$\nu$	$=$	$( \beta_{7} + \beta_{6} - 8\beta_{5} - 16\beta_{4} - 15\beta_{3} - 1872\beta_{2} + 5444\beta _1 + 100663296 ) / 268435456$ (b7 + b6 - 8b5 - 16b4 - 15b3 - 1872b2 + 5444*b1 + 100663296) / 268435456
$\nu^{2}$	$=$	$( - 7 \beta_{7} - 1031 \beta_{6} - 8 \beta_{5} - 8208 \beta_{4} + 22633 \beta_{3} + \cdots + 736419119104 ) / 268435456$ (-7b7 - 1031b6 - 8b5 - 8208b4 + 22633b3 + 235696b2 + 13116196*b1 + 736419119104) / 268435456
$\nu^{3}$	$=$	$( 1171 \beta_{7} + 78995 \beta_{6} + 4712 \beta_{5} - 678704 \beta_{4} + 2464611 \beta_{3} + \cdots + 87456573554688 ) / 268435456$ (1171b7 + 78995b6 + 4712b5 - 678704b4 + 2464611b3 - 213822704b2 - 4197800436*b1 + 87456573554688) / 268435456
$\nu^{4}$	$=$	$( - 938323 \beta_{7} + 6058669 \beta_{6} - 7867688 \beta_{5} - 98957904 \beta_{4} + \cdots + 56\!\cdots\!48 ) / 268435456$ (-938323b7 + 6058669b6 - 7867688b5 - 98957904b4 - 714478627b3 + 2686343536b2 + 1787126832884*b1 + 5634313890562048) / 268435456
$\nu^{5}$	$=$	$( - 161202677 \beta_{7} + 765179403 \beta_{6} - 1308895448 \beta_{5} - 4418474416 \beta_{4} + \cdots + 44\!\cdots\!56 ) / 268435456$ (-161202677b7 + 765179403b6 - 1308895448b5 - 4418474416b4 + 100098903387b3 + 1011477244560b2 - 111635355151892*b1 + 4455715642074464256) / 268435456
$\nu^{6}$	$=$	$( 10493167501 \beta_{7} - 104457993331 \beta_{6} - 179426196648 \beta_{5} + 439762740912 \beta_{4} + \cdots + 68\!\cdots\!04 ) / 268435456$ (10493167501b7 - 104457993331b6 - 179426196648b5 + 439762740912b4 - 498592258371b3 - 257559634006416b2 + 19569010040785524*b1 + 686495826536462024704) / 268435456
$\nu^{7}$	$=$	$( 4466073973275 \beta_{7} + 1854603989531 \beta_{6} - 8565850414808 \beta_{5} + \cdots + 41\!\cdots\!88 ) / 268435456$ (4466073973275b7 + 1854603989531b6 - 8565850414808b5 - 236172164611504b4 + 709994261598315b3 + 24336306125655696b2 + 666924688032548396*b1 + 41018050070719431180288) / 268435456

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$ .

$n$	$5$	$63$
$\chi(n)$	$1$	$-1$

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}

63.1

130.763	+	7.78570i
−116.282	−	46.4306i
−67.8767	+	106.586i
54.8957	−	117.008i
54.8957	+	117.008i
−67.8767	−	106.586i
−116.282	+	46.4306i
130.763	−	7.78570i

−

30905.2i

−991387.

6.49702e7i

−5.67711e8

63.2

−

27088.6i

2.11965e6

−

3.35455e7i

−3.46370e8

63.3

−

14439.2i

−2.88086e6

−

4.40005e7i

1.78931e8

63.4

−

4128.11i

1.32216e6

−

1.88890e7i

3.70379e8

63.5

4128.11i

1.32216e6

1.88890e7i

3.70379e8

63.6

14439.2i

−2.88086e6

4.40005e7i

1.78931e8

63.7

27088.6i

2.11965e6

3.35455e7i

−3.46370e8

63.8

30905.2i

−991387.

−

6.49702e7i

−5.67711e8

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	64.19.c.e		8
4.b	odd	2	1	inner	64.19.c.e		8
8.b	even	2	1		4.19.b.a	✓	8
8.d	odd	2	1		4.19.b.a	✓	8
24.f	even	2	1		36.19.d.c		8
24.h	odd	2	1		36.19.d.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.19.b.a	✓	8	8.b	even	2	1
4.19.b.a	✓	8	8.d	odd	2	1
36.19.d.c		8	24.f	even	2	1
36.19.d.c		8	24.h	odd	2	1
64.19.c.e		8	1.a	even	1	1	trivial
64.19.c.e		8	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{8} + 1914452736 T_{3}^{6} + \cdots + 24\!\cdots\!00$ T3^8 + 1914452736*T3^6 + 1085323270307143680*T3^4 + 164067554187992950568386560*T3^2 + 2490124129707690259281917824204800 acting on $S_{19}^{\mathrm{new}}(64, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} + \cdots + 24\!\cdots\!00$ T^8 + 1914452736T^6 + 1085323270307143680T^4 + 164067554187992950568386560*T^2 + 2490124129707690259281917824204800
$5$	$(T^{4} + \cdots + 80\!\cdots\!00)^{2}$ (T^4 + 430440T^3 - 7668986667560T^2 + 1022055002686308000*T + 8004119036805021618250000)^2
$7$	$T^{8} + \cdots + 32\!\cdots\!00$ T^8 + 7639254958427136T^6 + 17699248655874169060827709440000T^4 + 14584157526474928790758978634307230421221376000*T^2 + 3281171419443909620442303369504724656656975192634949632000000
$11$	$T^{8} + \cdots + 16\!\cdots\!00$ T^8 + 30912043823670516480T^6 + 313697860044877916110922557539674972160T^4 + 1234201284535410109607483175681019891942994348083956940800*T^2 + 1612956298413015554571871360922916166639722438348243230991153782084075520000
$13$	$(T^{4} + \cdots - 57\!\cdots\!60)^{2}$ (T^4 - 3329389144T^3 - 55029888719359807656T^2 - 118629856404136652211555090016*T - 57752794406539638395031653128268856560)^2
$17$	$(T^{4} + \cdots - 57\!\cdots\!80)^{2}$ (T^4 - 106927090824T^3 - 31272662623293716552936T^2 + 1761002373861825865109300392004064*T - 5709746358487200936988122581265066519837680)^2
$19$	$T^{8} + \cdots + 11\!\cdots\!00$ T^8 + 556092972313025917858560T^6 + 83597641727689254469190124388902102223478415360T^4 + 3539122147143049456856500894046328167910364384003291838219097813811200*T^2 + 116463862934450992562574941567889687308142540778505033198924784561510912406574617067520000
$23$	$T^{8} + \cdots + 21\!\cdots\!00$ T^8 + 17467835592077581362195456T^6 + 68005041928572066462038581595901925645975142400000T^4 + 60538874462554796380633228004830680582616135527308245026853005987843932160*T^2 + 2177134852282426898131908756321389709479557839845061312617947433522083466690939488047893525299200
$29$	$(T^{4} + \cdots - 48\!\cdots\!16)^{2}$ (T^4 + 5619028284456T^3 - 435747640491268824152974760T^2 - 4727952808174172379702462990836571537504*T - 4868765916345711931098397292176793621513941980133616)^2
$31$	$T^{8} + \cdots + 54\!\cdots\!00$ T^8 + 1292671105352276510761205760T^6 + 380337484866703786687988971491353123321063800370626560T^4 + 10123419620146855302491039129789176732543601131797566202744224739223508562739200*T^2 + 5480074540869583647278073316997418450033128221632890392936529823229434205840485275057568933551800320000
$37$	$(T^{4} + \cdots - 30\!\cdots\!40)^{2}$ (T^4 - 79443484408216T^3 - 18855148053268624346776411176T^2 + 1663951596935900837564601864415973353233056*T - 30379224491021136438292506770447133325001450382263129840)^2
$41$	$(T^{4} + \cdots - 21\!\cdots\!44)^{2}$ (T^4 - 225754992362568T^3 - 89817353107803354937246851176T^2 + 29323803715688450775847007031285845997221088*T - 2113959084412965748980372393070779051412862937507185640944)^2
$43$	$T^{8} + \cdots + 49\!\cdots\!00$ T^8 + 1276412676197337138455288384256T^6 + 474924328065440044143127618896202358179450588532700758630400T^4 + 42975207966673548714793730011836668927629328231653900373408058858885478987132313665536000*T^2 + 496575232523299525187582805961788298664498146203981479217364425110370347603869575177108443870715170967649517568000000
$47$	$T^{8} + \cdots + 70\!\cdots\!00$ T^8 + 4039877971354659745835104874496T^6 + 4974695717737988072969155159959258858199345416827786640752640T^4 + 1744548184658003774833231915539718034552097301392022219474264396232207203789239202307112960*T^2 + 70957216148369940614719537144750676462579195157952833146697375692181708535814587597653519976273940261656040594787532800
$53$	$(T^{4} + \cdots + 18\!\cdots\!20)^{2}$ (T^4 + 608192380043496T^3 - 12909811458216522174227162542376T^2 - 9730644339290656922299996041042428963027162976*T + 18167584879021086724651509610721541173052017571814797161704720)^2
$59$	$T^{8} + \cdots + 61\!\cdots\!00$ T^8 + 372355632468773948305158094713600T^6 + 39861762466942015402438365700441481452353474428562619540600872960T^4 + 987228670000868855526859434619410008770899387444794711294198506394193185914211917262126501068800*T^2 + 6164332723066951770883136229010932385390170242620933493646687062726423794082558390055023800125703118647166566451354397573120000
$61$	$(T^{4} + \cdots + 14\!\cdots\!56)^{2}$ (T^4 + 92296457552168T^3 - 45636386094166191299587520855976T^2 - 7937139706035533088950340694601724822884795488*T + 141049873100211471577118558471750726544788546132218541774442256)^2
$67$	$T^{8} + \cdots + 66\!\cdots\!00$ T^8 + 2918909096566538820322624008051456T^6 + 2195012810764111122320052603072811926294871324802972273181880442880T^4 + 233976899424666795365293236261952621741234344859255662815340224466986352873548582359234847267880960*T^2 + 6660940463389417223517297996145597762902008917789245544574077615411984280454915501049271942721777070247025758208759870201318604800
$71$	$T^{8} + \cdots + 84\!\cdots\!00$ T^8 + 8818792517474451897574225679354880T^6 + 19460746819372537200333066343131001335515598002185590820247562485760T^4 + 9649899355947669766267758003298413612659674927835050826682013396023071164829412595356136813835059200*T^2 + 847094661877057305505428738220288778530275398239163020525464786692207994465995094977065181183764828639158681067236821503699845120000
$73$	$(T^{4} + \cdots + 16\!\cdots\!60)^{2}$ (T^4 - 15747183243379016T^3 - 3363936369955401144064744242668136T^2 + 16027021937238546952204880659822403996614834215136*T + 160815952619860603019189081172826632554939079107101728803709510160)^2
$79$	$T^{8} + \cdots + 26\!\cdots\!00$ T^8 + 18546399127759610293109334492917760T^6 + 65102746186944961294580550709984566690898948322963192816243268648960T^4 + 77287065161050903229505550690337839206024789538973474764614719509983274436341588935580456129344307200*T^2 + 26921573334839213770771106204461704158141780183986853134086167325522463754756814229232205183763161895482497122869720064712842936320000
$83$	$T^{8} + \cdots + 18\!\cdots\!00$ T^8 + 96131084756775858305509515155755776T^6 + 3258366957830010603594323371560868774603805719020766516624178640936960T^4 + 44545501255514383700533685986181942678088848303574105534160878097072536363774844180031515878340621762560*T^2 + 188707892071879114297098717747991055975721288324732113112692959595883609467565062177000984446933747983201262917671218573373387404266700800
$89$	$(T^{4} + \cdots + 45\!\cdots\!04)^{2}$ (T^4 - 81000632514923976T^3 - 172145371010834081229525625312724840T^2 + 3564235037834761841982077743208567807287752808399584*T + 4578094686670045809402180022611257703693645093879840110008076310578704)^2
$97$	$(T^{4} + \cdots + 26\!\cdots\!80)^{2}$ (T^4 + 607488376451883256T^3 - 629660680894200615248618543864931816T^2 - 262037067245898769285321315892800940455930432565474336*T + 26424434551587191813205148003252429232789992285784653407407711987595280)^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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 63.8
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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