LMFDB - Newform orbit 116.2.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,2,Mod(25,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 8]))

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

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

chi := DirichletCharacter("116.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$116 = 2^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	116.g (of order $7$ , degree $6$ , 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:	$0.926264663447$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{7})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 5 x^{11} + 12 x^{10} - 16 x^{9} + 22 x^{8} + 28 x^{7} + 71 x^{6} + 154 x^{5} + 442 x^{4} + \cdots + 841$ x^12 - 5x^11 + 12x^10 - 16x^9 + 22x^8 + 28x^7 + 71x^6 + 154x^5 + 442x^4 + 555x^3 + 1573x^2 + 1363*x + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \cdots - \beta_{2}) q^{5} + (\beta_{8} - \beta_1) q^{7} + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_{5}) q^{9} + (\beta_{11} - \beta_{9} + 2 \beta_{8} + \cdots + 2) q^{11}+ \cdots + (\beta_{10} + \beta_{9} + 6 \beta_{8} + \cdots - 2) q^{99}+O(q^{100})$ q + (-b8 - b1) * q^3 + (b6 + b5 + b4 + b3 - b2) * q^5 + (b8 - b1) * q^7 + (-b11 + b10 - b8 - b6 - 2b5) q^9 + (b11 - b9 + 2b8 + 3b7 + 2b6 - b5 - 3b4 - b3 - b2 + 2) * q^11 + (b11 - b8 - b7 - 3b6 + 2b5 + b4 - b2 - 3) * q^13 + (-2b11 + 2b9 + b8 - 2b7 + 2b4 + 3b2 + b1 + 1) q^15 + (-b10 - b9 + b8 + b7 + 2b6 - 2b4 - b3 + b2) * q^17 + (2b11 - 2b8 - 3b7 + 2b5 + 2b4 + b3 + 2b1) * q^19 + (-b11 - b10 - b8 - b6 - 3b5) q^21 + (-b11 - b10 + b9 - 2b8 - b7 - b4 + b3 + b2 - 2) q^23 + (3b11 - b9 + b8 + 3b7 + b6 + 2b5 - b4 - b2 + 3b1) * q^25 + (b9 - b7 - 3b6 + b5 + b4 - b3 + 2b2 + b1 - 1) * q^27 + (-2b11 - b8 - 2b7 - 3b6 + b5 + 3b4 - b3 - 3) * q^29 + (b11 + b10 - b9 - 3b7 + 2b5 + b4 - b3 - 3) * q^31 + (-b11 - b9 + 5b8 + 8b7 + 4b6 - 5b5 - 5b4 - 2b3 - b2 - b1) * q^33 + (-2b11 + 2b9 + 3b8 - 2b5 - 2b4 + b2 - b1 + 3) q^35 + (b11 + 2b10 + 2b8 + 2b6 - 4b5 - 2b4 + 2) q^37 + (b11 - 3b9 + 5b8 + 2b6 - b5 - 5b4 + 2b3 - 3b2 + b1) * q^39 + (2b10 + 2b9 - b8 - b7 - 4b6 + 4b4 + 2b3 + b2 - 1) q^41 + (b11 + b10 - b9 + 3b7 - 2b5 + b4 - b3 - b2) * q^43 + (b11 - b10 + 2b9 - 8b8 - 4b7 - 2b6 + 9b5 + 4b4 + 2b3 - b2 - b1 - 2) q^45 + (-b11 - 2b10 - b9 - 6b8 - 5b7 - 2b6 + 5b5 + 5b4 - b3 + b2 - 2b1 - 2) q^47 + (-b11 - 3b10 - b8 - b6 + 2b5) * q^49 + (-2b11 - 2b10 + 6b7 + 6b6 - 4b5 - 2b3 - 2b1 + 2) q^51 + (3b7 + 4b6 + 3) * q^53 + (2b11 + 3b10 - 3b8 - 2b7 - 2b6 - 3b5 + 2b3 + b2 + b1 + 5) q^55 + (-b8 - b7 + 4b6 - 4b4 + 3b3 - 3b1 + 6) * q^57 + (2b3 - 4b2 - 2b1 + 4) q^59 + (b11 - b10 + 3b8 - 2b7 - 2b6 + 4b5 + b3 - 2b2 - b1 - 3) q^61 + (b11 + b10 - b9 - b7 - 4b6 + 4b5 + 3b4 - b3 - 1) q^63 + (-3b11 + 2b10 + 2b8 - 2b7 - 2b6 - 6b5 - 3b3 + 5b2 + 3) * q^65 + (b11 + b10 + 4b9 + b8 + b6 - 3b5) * q^67 + (b10 - 2b9 - b8 - 4b7 + 2b6 + b5 + 4b4 - 2b3 + b1 + 2) q^69 + (-b11 + b9 - 2b8 + 5b7 - 2b6 + b5 - 5b4 + b3 + b2 - 2) * q^71 + (2b11 - 2b9 + 5b8 + 4b7 - 2b5 + 2b4 - 2b2 + 5) q^73 + (-2b10 - 2b9 + 7b8 + 7b7 + 10b6 - 10b4 + b3 - 5b2 - 3b1 + 3) * q^75 + (-b11 + b9 + 3b8 + 2b7 + 2b6 - 3b5 - 3b4 + b2 - b1) q^77 + (-3b11 + b10 - 9b8 - 9b6 - b5 + 6b4 - 6) * q^79 + (-7b8 - 6b7 + 6b5 + 5b4 - b2 - b1 - 7) * q^81 + (-2b9 + 3b7 + 2b6 - 2b5 - 3b3 - 2b2) * q^83 + (-b9 - 3b7 - 4b6 - 3b5 - 3b4 - 2b3 + b2 - b1 - 3) q^85 + (2b11 + 2b9 + b8 - 4b7 - 4b6 + 10b5 + 6b4 + 2b3 + 2b2 + 3b1 - 6) q^87 + (2b11 + 2b10 + 2b9 + 3b6 - b5 - 3b4 + 3b3 - b2 + 4b1) q^89 + (b11 - b9 + b8 + 2b7 + 2b6 - b5 - b4 + 2b3 - b2 + b1) q^91 + (-2b11 - 2b10 + 2b9 + 3b8 + 4b7 - 6b5 - 4b4 + 2b3 + 3b2 + b1 + 3) q^93 + (-4b10 - 3b9 + 3b8 + 3b6 - b5 + 9b4 - 9) q^95 + (4b9 - b8 - 13b7 - b6 + b5 + b4 + 4b2) q^97 + (b10 + b9 + 6b8 + 6b7 - b6 + b4 - 2b3 - b2 + 3b1 - 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 3 q^{3} - q^{5} - 7 q^{7} + q^{9} + 3 q^{11} - 21 q^{13} + 29 q^{15} - 10 q^{17} + 17 q^{19} + 3 q^{21} - 19 q^{23} - 3 q^{25} + 9 q^{27} - 5 q^{29} - 27 q^{31} - 47 q^{33} + 27 q^{35} - 3 q^{37}+ \cdots - 16 q^{99}+O(q^{100})$ 12 * q - 3 * q^3 - q^5 - 7 * q^7 + q^9 + 3 * q^11 - 21 * q^13 + 29 * q^15 - 10 * q^17 + 17 * q^19 + 3 * q^21 - 19 * q^23 - 3 * q^25 + 9 * q^27 - 5 * q^29 - 27 * q^31 - 47 * q^33 + 27 * q^35 - 3 * q^37 - 37 * q^39 - 2 * q^41 - 9 * q^43 + 22 * q^45 + 19 * q^47 + 17 * q^49 + 2 * q^51 + 22 * q^53 + 49 * q^55 + 30 * q^57 + 36 * q^59 - 33 * q^61 + 9 * q^63 + 38 * q^65 - 3 * q^67 + 47 * q^69 - 35 * q^71 + 34 * q^73 - 44 * q^75 - 27 * q^77 - 19 * q^79 - 39 * q^81 - q^83 - 34 * q^85 - 25 * q^87 - 11 * q^89 - 23 * q^91 + 7 * q^93 - 105 * q^95 + 38 * q^97 - 16 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 5 x^{11} + 12 x^{10} - 16 x^{9} + 22 x^{8} + 28 x^{7} + 71 x^{6} + 154 x^{5} + 442 x^{4} + \cdots + 841$ x^12 - 5*x^11 + 12*x^10 - 16*x^9 + 22*x^8 + 28*x^7 + 71*x^6 + 154*x^5 + 442*x^4 + 555*x^3 + 1573*x^2 + 1363*x + 841 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 2666113593 \nu^{11} - 103378288493 \nu^{10} + 730364594638 \nu^{9} + \cdots - 87085829035390 ) / 87597662955238$ (-2666113593v^11 - 103378288493v^10 + 730364594638v^9 - 2617095642752v^8 + 5599337776959v^7 - 7237868754197v^6 - 1125438433581v^5 - 5468629429730v^4 - 16215452878554v^3 - 40201374965761v^2 - 32642709144147*v - 87085829035390) / 87597662955238
$\beta_{3}$	$=$	$( 76569689033 \nu^{11} - 329670261352 \nu^{10} + 1009877030172 \nu^{9} + \cdots + 235137749968678 ) / 175195325910476$ (76569689033v^11 - 329670261352v^10 + 1009877030172v^9 - 2534683875493v^8 + 5937771552943v^7 - 4222797428935v^6 + 17848379235141v^5 + 21800134881322v^4 + 60004518163979v^3 + 116923700403773v^2 + 278755901368884*v + 235137749968678) / 175195325910476
$\beta_{4}$	$=$	$( - 80877989929 \nu^{11} + 560470291448 \nu^{10} - 1931233965956 \nu^{9} + \cdots + 92982388988572 ) / 175195325910476$ (-80877989929v^11 + 560470291448v^10 - 1931233965956v^9 + 4061042157113v^8 - 6120903487387v^7 + 2137065928721v^6 + 2550547281141v^5 - 27706532831152v^4 - 1323469693285v^3 + 5743160943041v^2 - 97558850127494*v + 92982388988572) / 175195325910476
$\beta_{5}$	$=$	$( 154305296456 \nu^{11} - 821175970637 \nu^{10} + 2060526694663 \nu^{9} + \cdots + 132826473739629 ) / 175195325910476$ (154305296456v^11 - 821175970637v^10 + 2060526694663v^9 - 2816519381170v^8 + 3263504728095v^7 + 5395187031600v^6 + 6126189805294v^5 + 21010022307646v^4 + 66720959492669v^3 + 2460296246703v^2 + 229622143247462*v + 132826473739629) / 175195325910476
$\beta_{6}$	$=$	$( - 182774232211 \nu^{11} + 793118206506 \nu^{10} - 1236048179004 \nu^{9} + \cdots - 277769000014142 ) / 175195325910476$ (-182774232211v^11 + 793118206506v^10 - 1236048179004v^9 - 60033760267v^8 + 761390672941v^7 - 10993309460293v^6 - 7557189998617v^5 - 36266040787968v^4 - 50805505575209v^3 - 128354244159169v^2 - 218025552980484*v - 277769000014142) / 175195325910476
$\beta_{7}$	$=$	$( 103550331790 \nu^{11} - 520417772543 \nu^{10} + 1139225692987 \nu^{9} + \cdots + 108496393085623 ) / 87597662955238$ (103550331790v^11 - 520417772543v^10 + 1139225692987v^9 - 926440714002v^8 - 338988343372v^7 + 8498747067079v^6 + 114204802893v^5 + 14821312662079v^4 + 40300617221450v^3 + 41254981264896v^2 + 122683296939909*v + 108496393085623) / 87597662955238
$\beta_{8}$	$=$	$( - 230492168800 \nu^{11} + 1461546568214 \nu^{10} - 4597897267111 \nu^{9} + \cdots + 9060333114751 ) / 175195325910476$ (-230492168800v^11 + 1461546568214v^10 - 4597897267111v^9 + 8640803697714v^8 - 11626543588058v^7 - 362765391425v^6 - 8068386070687v^5 - 25282044594087v^4 - 66178229616421v^3 - 41121882413590v^2 - 218153905521527*v + 9060333114751) / 175195325910476
$\beta_{9}$	$=$	$( - 232650030836 \nu^{11} + 1290368348160 \nu^{10} - 3776871348421 \nu^{9} + \cdots - 303156139498967 ) / 175195325910476$ (-232650030836v^11 + 1290368348160v^10 - 3776871348421v^9 + 6876271584442v^8 - 10339421199676v^7 - 4070087137165v^6 - 2597056853055v^5 - 56224736736655v^4 - 110634963517615v^3 - 146585928434596v^2 - 481974990630683*v - 303156139498967) / 175195325910476
$\beta_{10}$	$=$	$( 309085724214 \nu^{11} - 1831991241511 \nu^{10} + 4952928996914 \nu^{9} + \cdots + 193843913960800 ) / 175195325910476$ (309085724214v^11 - 1831991241511v^10 + 4952928996914v^9 - 6555715874458v^8 + 6091015334975v^7 + 8296557914113v^6 + 10213749401113v^5 + 35699308993179v^4 + 86801271270410v^3 + 144410276000873v^2 + 323221159189151*v + 193843913960800) / 175195325910476
$\beta_{11}$	$=$	$( - 513040509027 \nu^{11} + 2862030349339 \nu^{10} - 7361090329451 \nu^{9} + \cdots - 456441142019925 ) / 175195325910476$ (-513040509027v^11 + 2862030349339v^10 - 7361090329451v^9 + 9241023461691v^8 - 8279881332238v^7 - 18521231188795v^6 - 19092932552985v^5 - 58191312841708v^4 - 236701374049456v^3 - 159970660325402v^2 - 805530273676988*v - 456441142019925) / 175195325910476

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} - 4\beta_{5}$ -b11 - b10 - b8 - b6 - 4*b5
$\nu^{3}$	$=$	$- 6 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} - 10 \beta_{5} - 4 \beta_{4} + \cdots + 1$ -6b11 - 6b10 + 2b9 + b7 - b6 - 10b5 - 4b4 + b3 + b2 - 4b1 + 1
$\nu^{4}$	$=$	$- 13 \beta_{11} - 11 \beta_{10} + 13 \beta_{9} + 8 \beta_{8} + 13 \beta_{7} - 26 \beta_{5} - 26 \beta_{4} + \cdots + 8$ -13b11 - 11b10 + 13b9 + 8b8 + 13b7 - 26b5 - 26b4 + 11b3 + 5b2 - 8b1 + 8
$\nu^{5}$	$=$	$- 29 \beta_{11} - 16 \beta_{10} + 50 \beta_{9} + 34 \beta_{8} + 63 \beta_{7} + 14 \beta_{6} - 63 \beta_{5} + \cdots + 14$ -29b11 - 16b10 + 50b9 + 34b8 + 63b7 + 14b6 - 63b5 - 63b4 + 50b3 + 29b2 - 16*b1 + 14
$\nu^{6}$	$=$	$- 49 \beta_{11} + 115 \beta_{9} + 153 \beta_{8} + 229 \beta_{7} + 76 \beta_{6} - 125 \beta_{5} + \cdots - 49 \beta_1$ -49b11 + 115b9 + 153b8 + 229b7 + 76b6 - 125b5 - 153b4 + 142b3 + 115b2 - 49b1
$\nu^{7}$	$=$	$192 \beta_{10} + 192 \beta_{9} + 536 \beta_{8} + 536 \beta_{7} + 262 \beta_{6} - 262 \beta_{4} + \cdots - 147$ 192b10 + 192b9 + 536b8 + 536b7 + 262b6 - 262b4 + 344b3 + 334b2 - 152*b1 - 147
$\nu^{8}$	$=$	$606 \beta_{11} + 1214 \beta_{10} + 1498 \beta_{8} + 920 \beta_{7} + 920 \beta_{6} + 1396 \beta_{5} + \cdots - 790$ 606b11 + 1214b10 + 1498b8 + 920b7 + 920b6 + 1396b5 + 606b3 + 608b2 - 299*b1 - 790
$\nu^{9}$	$=$	$3829 \beta_{11} + 5015 \beta_{10} - 2134 \beta_{9} + 2729 \beta_{8} + 2729 \beta_{6} + 8366 \beta_{5} + \cdots - 3032$ 3829b11 + 5015b10 - 2134b9 + 2729b8 + 2729b6 + 8366b5 + 3032*b4 - 3032
$\nu^{10}$	$=$	$16110 \beta_{11} + 16110 \beta_{10} - 12910 \beta_{9} - 10231 \beta_{7} + 5253 \beta_{6} + 34746 \beta_{5} + \cdots - 10231$ 16110b11 + 16110b10 - 12910b9 - 10231b7 + 5253b6 + 34746b5 + 18636b4 - 6861b3 - 6049b2 + 3200b1 - 10231
$\nu^{11}$	$=$	$52909 \beta_{11} + 41607 \beta_{10} - 52909 \beta_{9} - 28208 \beta_{8} - 61701 \beta_{7} + 114610 \beta_{5} + \cdots - 28208$ 52909b11 + 41607b10 - 52909b9 - 28208b8 - 61701b7 + 114610b5 + 77350b4 - 41607b3 - 34443b2 + 18466b1 - 28208

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$ .

$n$	$59$	$89$
$\chi(n)$	$1$	$\beta_{7}$

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}

25.1

2.94643	−	1.41893i
−1.32294	+	0.637094i
1.30120	−	1.63166i
−0.523722	+	0.656727i
1.30120	+	1.63166i
−0.523722	−	0.656727i
0.465490	+	2.03945i
−0.366459	−	1.60556i
2.94643	+	1.41893i
−1.32294	−	0.637094i
0.465490	−	2.03945i
−0.366459	+	1.60556i

−2.04546

0.985042i

−0.299629

1.31276i

−3.84740

1.85281i

1.34313

−

1.68423i

25.2

2.22391

−

1.07098i

−0.768902

3.36878i

0.421971

−

0.203210i

1.92831

−

2.41802i

45.1

−1.92469

2.41349i

−3.16565

−

1.52449i

−0.677712

0.849824i

−1.45292

−

6.36565i

45.2

−0.0997674

0.125104i

1.58623

0.763888i

1.14721

−

1.43856i

0.661865

2.89982i

49.1

−1.92469

−

2.41349i

−3.16565

1.52449i

−0.677712

−

0.849824i

−1.45292

6.36565i

49.2

−0.0997674

−

0.125104i

1.58623

−

0.763888i

1.14721

1.43856i

0.661865

−

2.89982i

53.1

−0.242969

−

1.06452i

2.52737

3.16923i

−0.688011

−

3.01437i

1.62874

−

0.784360i

53.2

0.588980

2.58049i

−0.379425

−

0.475783i

0.143938

0.630635i

−3.60913

1.73806i

65.1

−2.04546

−

0.985042i

−0.299629

−

1.31276i

−3.84740

−

1.85281i

1.34313

1.68423i

65.2

2.22391

1.07098i

−0.768902

−

3.36878i

0.421971

0.203210i

1.92831

2.41802i

81.1

−0.242969

1.06452i

2.52737

−

3.16923i

−0.688011

3.01437i

1.62874

0.784360i

81.2

0.588980

−

2.58049i

−0.379425

0.475783i

0.143938

−

0.630635i

−3.60913

−

1.73806i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	116.2.g.b	✓	12
3.b	odd	2	1		1044.2.u.c		12
4.b	odd	2	1		464.2.u.g		12
29.d	even	7	1	inner	116.2.g.b	✓	12
29.d	even	7	1		3364.2.a.m		6
29.e	even	14	1		3364.2.a.p		6
29.f	odd	28	2		3364.2.c.j		12
87.j	odd	14	1		1044.2.u.c		12
116.j	odd	14	1		464.2.u.g		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
116.2.g.b	✓	12	1.a	even	1	1	trivial
116.2.g.b	✓	12	29.d	even	7	1	inner
464.2.u.g		12	4.b	odd	2	1
464.2.u.g		12	116.j	odd	14	1
1044.2.u.c		12	3.b	odd	2	1
1044.2.u.c		12	87.j	odd	14	1
3364.2.a.m		6	29.d	even	7	1
3364.2.a.p		6	29.e	even	14	1
3364.2.c.j		12	29.f	odd	28	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{12} + 3 T_{3}^{11} + 7 T_{3}^{10} + T_{3}^{9} + 17 T_{3}^{8} - 21 T_{3}^{7} - 55 T_{3}^{6} + \cdots + 64$ T3^12 + 3*T3^11 + 7*T3^10 + T3^9 + 17*T3^8 - 21*T3^7 - 55*T3^6 + 560*T3^5 + 2448*T3^4 + 2248*T3^3 + 2912*T3^2 + 544*T3 + 64 acting on $S_{2}^{\mathrm{new}}(116, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12} + 3 T^{11} + \cdots + 64$ T^12 + 3T^11 + 7T^10 + T^9 + 17T^8 - 21T^7 - 55T^6 + 560T^5 + 2448T^4 + 2248T^3 + 2912T^2 + 544T + 64
$5$	$T^{12} + T^{11} + \cdots + 5041$ T^12 + T^11 + 7T^10 + 34T^9 + 146T^8 + 455T^7 + 1198T^6 - 3591T^5 + 2421T^4 - 2395T^3 + 11914T^2 + 8520T + 5041
$7$	$T^{12} + 7 T^{11} + \cdots + 64$ T^12 + 7T^11 + 23T^10 + 49T^9 + 95T^8 - 35T^7 + 603T^6 + 70T^5 + 380T^4 - 392T^3 + 368T^2 - 224*T + 64
$11$	$T^{12} - 3 T^{11} + \cdots + 46656$ T^12 - 3T^11 + 13T^10 - 191T^9 + 1353T^8 - 5523T^7 + 16465T^6 - 37800T^5 + 70704T^4 - 104760T^3 + 121824T^2 - 101088*T + 46656
$13$	$T^{12} + 21 T^{11} + \cdots + 1$ T^12 + 21T^11 + 235T^10 + 1750T^9 + 8906T^8 + 29351T^7 + 58850T^6 + 61985T^5 + 27437T^4 + 39725T^3 + 142074T^2 + 168*T + 1
$17$	$(T^{6} + 5 T^{5} + \cdots + 344)^{2}$ (T^6 + 5T^5 - 30T^4 - 245T^3 - 480T^2 - 92*T + 344)^2
$19$	$T^{12} + \cdots + 285745216$ T^12 - 17T^11 + 195T^10 - 1899T^9 + 14779T^8 - 88263T^7 + 435639T^6 - 1844934T^5 + 7329032T^4 - 28802520T^3 + 98653600T^2 - 224214656*T + 285745216
$23$	$T^{12} + 19 T^{11} + \cdots + 1032256$ T^12 + 19T^11 + 217T^10 + 1667T^9 + 8307T^8 + 26481T^7 + 66991T^6 + 163450T^5 + 292596T^4 + 263656T^3 + 641200T^2 + 849376*T + 1032256
$29$	$T^{12} + \cdots + 594823321$ T^12 + 5T^11 + 38T^10 + 122T^9 - 807T^8 - 7727T^7 - 65408T^6 - 224083T^5 - 678687T^4 + 2975458T^3 + 26876678T^2 + 102555745*T + 594823321
$31$	$T^{12} + 27 T^{11} + \cdots + 53824$ T^12 + 27T^11 + 451T^10 + 5077T^9 + 39435T^8 + 213577T^7 + 806751T^6 + 2086434T^5 + 3563072T^4 + 3701864T^3 + 1858240T^2 - 44544*T + 53824
$37$	$T^{12} + \cdots + 161620369$ T^12 + 3T^11 + 91T^10 + 260T^9 + 5232T^8 + 25417T^7 + 209798T^6 + 603435T^5 + 1542693T^4 - 3868521T^3 - 17771096T^2 + 20620486*T + 161620369
$41$	$(T^{6} + T^{5} - 142 T^{4} + \cdots - 1624)^{2}$ (T^6 + T^5 - 142T^4 + 167T^3 + 5320T^2 - 16436T - 1624)^2
$43$	$T^{12} + 9 T^{11} + \cdots + 11235904$ T^12 + 9T^11 + 105T^10 + 307T^9 + 2693T^8 - 3353T^7 + 36401T^6 - 191744T^5 + 542384T^4 - 2636152T^3 + 12599776T^2 - 19910880*T + 11235904
$47$	$T^{12} + \cdots + 16533845056$ T^12 - 19T^11 + 201T^10 - 891T^9 + 3151T^8 - 10941T^7 + 101571T^6 - 402150T^5 + 15410984T^4 - 104579928T^3 + 762910048T^2 - 1600613632*T + 16533845056
$53$	$(T^{6} - 11 T^{5} + \cdots + 841)^{2}$ (T^6 - 11T^5 + 79T^4 - 155T^3 + 529T^2 - 1073*T + 841)^2
$59$	$(T^{6} - 18 T^{5} + \cdots + 19264)^{2}$ (T^6 - 18T^5 - 4T^4 + 1576T^3 - 8176T^2 + 6048*T + 19264)^2
$61$	$T^{12} + \cdots + 2812499089$ T^12 + 33T^11 + 635T^10 + 8290T^9 + 79934T^8 + 580223T^7 + 3210222T^6 + 13633865T^5 + 46227581T^4 + 141101161T^3 + 496466906T^2 + 1620900612*T + 2812499089
$67$	$T^{12} + \cdots + 5027377216$ T^12 + 3T^11 + 147T^10 + 673T^9 + 59T^8 - 38843T^7 + 511127T^6 - 21508214T^5 + 334224476T^4 - 1548239944T^3 + 4274825968T^2 - 6193322592*T + 5027377216
$71$	$T^{12} + 35 T^{11} + \cdots + 96983104$ T^12 + 35T^11 + 869T^10 + 12215T^9 + 143641T^8 + 984739T^7 + 5772705T^6 + 9529912T^5 + 11387440T^4 - 380535400T^3 + 920337888T^2 - 510953632*T + 96983104
$73$	$T^{12} - 34 T^{11} + \cdots + 37442161$ T^12 - 34T^11 + 455T^10 - 3074T^9 + 22399T^8 - 197778T^7 + 1329959T^6 - 7140098T^5 + 36432969T^4 - 56393128T^3 + 384073389T^2 - 149866548*T + 37442161
$79$	$T^{12} + \cdots + 397313387584$ T^12 + 19T^11 + 287T^10 + 6903T^9 + 92335T^8 + 269647T^7 + 10561335T^6 + 103761574T^5 + 509910040T^4 + 2121072696T^3 + 11993177952T^2 + 11880422144*T + 397313387584
$83$	$T^{12} + T^{11} + \cdots + 13601344$ T^12 + T^11 - 81T^10 - 597T^9 + 4953T^8 + 141225T^7 + 1592809T^6 + 6207544T^5 + 16109676T^4 + 9475328T^3 + 21933344T^2 - 15991168T + 13601344
$89$	$T^{12} + \cdots + 876692881$ T^12 + 11T^11 + 117T^10 + 2916T^9 + 61918T^8 + 719491T^7 + 5245212T^6 + 28546453T^5 + 138995989T^4 + 470879049T^3 + 754887090T^2 + 153966800*T + 876692881
$97$	$T^{12} + \cdots + 1819798302001$ T^12 - 38T^11 + 923T^10 - 18748T^9 + 363897T^8 - 6668578T^7 + 119676411T^6 - 1522723258T^5 + 15122854093T^4 - 104602882088T^3 + 518394725523T^2 - 1466375402990*T + 1819798302001
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 25.2
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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