LMFDB - Newform orbit 1332.2.bt.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1332,2,Mod(145,1332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1332, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1332.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1332 = 2^{2} \cdot 3^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1332.bt (of order $9$ , 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:	$10.6360735492$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{9})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 9x^{10} - 14x^{9} + 69x^{8} - 72x^{7} + 151x^{6} - 78x^{5} + 180x^{4} - 66x^{3} + 117x^{2} + 27x + 9$ x^12 + 9x^10 - 14x^9 + 69x^8 - 72x^7 + 151x^6 - 78x^5 + 180x^4 - 66x^3 + 117x^2 + 27x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 444)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$ -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.

$f(q)$	$=$	$q + ( - \beta_{11} + \beta_{9} - 1) q^{5} + (\beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 2 \beta_1) q^{11} + (\beta_{11} + 2 \beta_{10} + \beta_{9} + \cdots - 1) q^{13}+ \cdots + ( - 3 \beta_{11} - 4 \beta_{10} + \cdots - 6) q^{97}+O(q^{100})$ q + (-b11 + b9 - 1) * q^5 + (b11 - b10 + b9 - b7 + b6 + 2b3 - b2) q^7 + (-b11 - b10 - b8 - b7 - 2b6 - b4 - 2b1) * q^11 + (b11 + 2b10 + b9 + b8 - b7 + b6 + b4 + b2 + b1 - 1) q^13 + (-b11 + b9 - b6 + b5 - b4 + 2b3 - b2 - b1 - 2) q^17 + (2b10 + b8 - 2b5 + b4 - b3 + 2b2 + 2b1 + 1) * q^19 + (b11 + 2b10 + 2b8 - 2b7 - b5 + b4 - b3 - b2) q^23 + (b11 + 2b10 - b9 + b8 + 2b6 - 2b4 + b3 - b2 - b1 - 1) q^25 + (-b10 + b8 - 3b6 - b5 - b4 - 2b3) * q^29 + (2b11 - b10 - 2b7 - b6 - b4 + 3b3 - b2 - b1 + 1) q^31 + (-b9 + b8 + 2b7 - b6 + 2b5 - 5b4 + b3 - b1 - 4) q^35 + (-b11 + 4b10 + 3b9 + b8 - b7 + b5 - b3 + 3b2 + 2b1) * q^37 + (2b11 - 2b9 + 2b8 - b7 - 3b5 + 2b4 - 2b2 + b1) * q^41 + (-4b10 - 2b8 - 2b7 - b6 - 2b5 - 2b4 + b3 - 3b2 - 3b1 + 1) q^43 + (b11 + 4b10 + 2b9 + 3b8 - 3b7 + b6 - b5 - b3 + b2 - 2) * q^47 + (2b11 - 3b9 + 2b8 + b7 + b5 - 4b4 + 3b3 - 2b2 - 2b1 - 1) q^49 + (b11 - b10 - 3b9 - b8 + b7 - 4b6 - 5b4 + 2b3 - b2 - 2b1 + 5) q^53 + (-b5 + 4b4 - 4b3 + b2 + b1) * q^55 + (2b11 - 2b10 + b9 - 2b8 - 6b7 + b6 - 3b4 + 4b3 - 2b2 - 4b1 + 3) * q^59 + (-2b11 - 2b10 - 2b8 + b7 - 3b6 + 2b5 - 6b4 + b2 - b1 - 1) * q^61 + (b11 + b10 + b8 + b6 + b5 + 3b3 + b1 - 3) q^65 + (-2b11 - b10 - 2b7 - 3b6 - 2b5 + 3b3 - 2b2 - 2b1 + 1) q^67 + (b10 + 2b9 - 6b7 - 2b6 - 3b5 + b4 - b3 + 3b2 - 3b1 + 1) * q^71 + (3b11 + b10 + b8 - 2b7 - b6 + b5 + 4b3 - 3b2 - 3b1 - 3) q^73 + (5b10 - b9 + b6 - b5 - 2b4 + 2b3 + b2 + b1 + 5) q^77 + (-5b11 + 3b10 - 3b9 + 4b7 - 5b6 + b5 - 8b3 + 4b2 + b1) q^79 + (2b11 + 4b10 + 5b9 + b8 + 2b6 + 3b5 - 5b4 + 4b3 + b2 + 2b1 - 4) * q^83 + (2b11 - 2b10 - 6b9 + b8 + 2b7 - 4b6 + b5 - 3b4 - 3b3) q^85 + (-b11 + 2b10 + 3b9 + 2b8 - 2b7 + 3b3 + 2b2 + 3b1) q^89 + (-2b11 - 5b10 + 3b9 - b8 - 2b6 + b5 - 3b4 + b3 - b2 - 2b1 - 1) * q^91 + (-b11 + 7b9 - b7 + 3b6 + 3b4 + 4b3 + b1 - 3) * q^95 + (-3b11 - 4b10 + 6b9 - 2b8 + 2b7 - 2b6 + 3b5 - b4 + 3b3 - 3b2 - 6) q^97
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 3 q^{5} - 6 q^{11} - 6 q^{13} - 12 q^{17} + 12 q^{19} - 15 q^{25} + 3 q^{29} - 36 q^{35} + 27 q^{37} - 18 q^{41} - 12 q^{43} - 9 q^{47} - 18 q^{49} + 36 q^{53} - 3 q^{55} + 6 q^{59} - 9 q^{61} - 30 q^{65}+ \cdots - 24 q^{97}+O(q^{100})$ 12 * q - 3 * q^5 - 6 * q^11 - 6 * q^13 - 12 * q^17 + 12 * q^19 - 15 * q^25 + 3 * q^29 - 36 * q^35 + 27 * q^37 - 18 * q^41 - 12 * q^43 - 9 * q^47 - 18 * q^49 + 36 * q^53 - 3 * q^55 + 6 * q^59 - 9 * q^61 - 30 * q^65 + 6 * q^67 - 3 * q^71 - 42 * q^73 + 51 * q^77 + 12 * q^79 - 9 * q^83 - 27 * q^85 + 27 * q^89 + 9 * q^91 + 6 * q^95 - 24 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} + 9x^{10} - 14x^{9} + 69x^{8} - 72x^{7} + 151x^{6} - 78x^{5} + 180x^{4} - 66x^{3} + 117x^{2} + 27x + 9$ x^12 + 9*x^10 - 14*x^9 + 69*x^8 - 72*x^7 + 151*x^6 - 78*x^5 + 180*x^4 - 66*x^3 + 117*x^2 + 27*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 22300 \nu^{11} - 39972 \nu^{10} - 207135 \nu^{9} - 16552 \nu^{8} - 1068399 \nu^{7} + \cdots - 1430838 ) / 5269059$ (-22300v^11 - 39972v^10 - 207135v^9 - 16552v^8 - 1068399v^7 - 831177v^6 - 1576951v^5 - 1652415v^4 - 5671629v^3 - 2275761v^2 - 5840550*v - 1430838) / 5269059
$\beta_{3}$	$=$	$( - 54202 \nu^{11} - 56620 \nu^{10} - 362478 \nu^{9} + 660840 \nu^{8} - 1457487 \nu^{7} + \cdots + 6547776 ) / 5269059$ (-54202v^11 - 56620v^10 - 362478v^9 + 660840v^8 - 1457487v^7 + 1717033v^6 + 987268v^5 + 6454315v^4 + 2293745v^3 + 8753631v^2 + 10148844*v + 6547776) / 5269059
$\beta_{4}$	$=$	$( 71196 \nu^{11} - 245497 \nu^{10} + 494337 \nu^{9} - 3002842 \nu^{8} + 7566829 \nu^{7} + \cdots + 2219844 ) / 5269059$ (71196v^11 - 245497v^10 + 494337v^9 - 3002842v^8 + 7566829v^7 - 17626894v^6 + 20172466v^5 - 22092027v^4 + 20840902v^3 - 19831914v^2 + 18268758*v + 2219844) / 5269059
$\beta_{5}$	$=$	$( 74027 \nu^{11} + 333687 \nu^{10} + 850507 \nu^{9} + 1595916 \nu^{8} + 1206871 \nu^{7} + \cdots + 966006 ) / 5269059$ (74027v^11 + 333687v^10 + 850507v^9 + 1595916v^8 + 1206871v^7 + 11479793v^6 - 1337096v^5 + 18140675v^4 - 816147v^3 + 22022940v^2 - 573510*v + 966006) / 5269059
$\beta_{6}$	$=$	$( - 107334 \nu^{11} + 74027 \nu^{10} - 632319 \nu^{9} + 2353183 \nu^{8} - 5810130 \nu^{7} + \cdots - 3471528 ) / 5269059$ (-107334v^11 + 74027v^10 - 632319v^9 + 2353183v^8 - 5810130v^7 + 8934919v^6 - 4727641v^5 + 7034956v^4 - 1179445v^3 + 6267897v^2 + 9464862*v - 3471528) / 5269059
$\beta_{7}$	$=$	$( 125253 \nu^{11} - 151332 \nu^{10} + 682852 \nu^{9} - 3373952 \nu^{8} + 7397665 \nu^{7} + \cdots - 2772270 ) / 5269059$ (125253v^11 - 151332v^10 + 682852v^9 - 3373952v^8 + 7397665v^7 - 14297782v^6 + 8748272v^5 - 18580135v^4 + 10210395v^3 - 23278908v^2 - 3625770*v - 2772270) / 5269059
$\beta_{8}$	$=$	$( 144076 \nu^{11} + 324620 \nu^{10} + 1890631 \nu^{9} + 1006066 \nu^{8} + 9884477 \nu^{7} + \cdots + 1707330 ) / 5269059$ (144076v^11 + 324620v^10 + 1890631v^9 + 1006066v^8 + 9884477v^7 + 3326194v^6 + 29713666v^5 + 3806993v^4 + 40862119v^3 + 12638790v^2 + 30236694*v + 1707330) / 5269059
$\beta_{9}$	$=$	$( - 158982 \nu^{11} + 22300 \nu^{10} - 1390866 \nu^{9} + 2432883 \nu^{8} - 10953206 \nu^{7} + \cdots + 1548036 ) / 5269059$ (-158982v^11 + 22300v^10 - 1390866v^9 + 2432883v^8 - 10953206v^7 + 12515103v^6 - 23175105v^5 + 13977547v^4 - 26964345v^3 + 16164441v^2 - 16325133*v + 1548036) / 5269059
$\beta_{10}$	$=$	$( - 200696 \nu^{11} - 199280 \nu^{10} - 1988619 \nu^{9} + 1276385 \nu^{8} - 12069988 \nu^{7} + \cdots - 1219512 ) / 5269059$ (-200696v^11 - 199280v^10 - 1988619v^9 + 1276385v^8 - 12069988v^7 + 5845576v^6 - 27487107v^5 + 8243112v^4 - 35685820v^3 + 3851688v^2 - 22225464*v - 1219512) / 5269059
$\beta_{11}$	$=$	$( 352831 \nu^{11} + 72400 \nu^{10} + 2638417 \nu^{9} - 5007488 \nu^{8} + 18310912 \nu^{7} + \cdots + 4112292 ) / 5269059$ (352831v^11 + 72400v^10 + 2638417v^9 - 5007488v^8 + 18310912v^7 - 18356789v^6 + 21266380v^5 - 15060578v^4 + 16312423v^3 - 16206723v^2 - 9762414*v + 4112292) / 5269059

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} - 2$ b11 - b10 + 2*b9 + b6 - b5 + b3 - 2
$\nu^{3}$	$=$	$\beta_{11} + 5 \beta_{10} + 2 \beta_{8} + \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \cdots + 2$ b11 + 5b10 + 2b8 + b7 + 3b6 + 2b5 + 3b4 - 2b3 - 4b2 - 4b1 + 2
$\nu^{4}$	$=$	$- 9 \beta_{11} - 10 \beta_{10} - 11 \beta_{9} - 7 \beta_{8} - 9 \beta_{7} - 19 \beta_{6} + \cdots + 5 \beta_1$ -9b11 - 10b10 - 11b9 - 7b8 - 9b7 - 19b6 - 2b5 - 7b4 - 3b3 + 5b1
$\nu^{5}$	$=$	$14 \beta_{11} - 31 \beta_{10} + 26 \beta_{9} - 10 \beta_{8} + 10 \beta_{7} + 25 \beta_{6} - 14 \beta_{5} + \cdots - 26$ 14b11 - 31b10 + 26b9 - 10b8 + 10b7 + 25b6 - 14b5 - 11b4 + 32b3 + 27b2 - 26
$\nu^{6}$	$=$	$25 \beta_{11} + 182 \beta_{10} + 83 \beta_{8} + 58 \beta_{7} + 90 \beta_{6} + 83 \beta_{5} + 99 \beta_{4} + \cdots + 92$ 25b11 + 182b10 + 83b8 + 58b7 + 90b6 + 83b5 + 99b4 - 65b3 - 64b2 - 64b1 + 92
$\nu^{7}$	$=$	$- 246 \beta_{11} - 262 \beta_{10} - 269 \beta_{9} - 154 \beta_{8} - 246 \beta_{7} - 559 \beta_{6} + \cdots + 233 \beta_1$ -246b11 - 262b10 - 269b9 - 154b8 - 246b7 - 559b6 - 92b5 - 205b4 - 108b3 + 233b1
$\nu^{8}$	$=$	$530 \beta_{11} - 895 \beta_{10} + 866 \beta_{9} - 262 \beta_{8} + 262 \beta_{7} + 868 \beta_{6} + \cdots - 866$ 530b11 - 895b10 + 866b9 - 262b8 + 262b7 + 868b6 - 530b5 - 338b4 + 971b3 + 669b2 - 866
$\nu^{9}$	$=$	$868 \beta_{11} + 5477 \beta_{10} + 2432 \beta_{8} + 1564 \beta_{7} + 2859 \beta_{6} + 2432 \beta_{5} + \cdots + 2660$ 868b11 + 5477b10 + 2432b8 + 1564b7 + 2859b6 + 2432b5 + 3045b4 - 1991b3 - 2188b2 - 2188b1 + 2660
$\nu^{10}$	$=$	$- 7665 \beta_{11} - 8347 \beta_{10} - 8372 \beta_{9} - 5047 \beta_{8} - 7665 \beta_{7} + \cdots + 6656 \beta_1$ -7665b11 - 8347b10 - 8372b9 - 5047b8 - 7665b7 - 17149b6 - 2618b5 - 6184b4 - 3300b3 + 6656b1
$\nu^{11}$	$=$	$15458 \beta_{11} - 27715 \beta_{10} + 26024 \beta_{9} - 8347 \beta_{8} + 8347 \beta_{7} + 25741 \beta_{6} + \cdots - 26024$ 15458b11 - 27715b10 + 26024b9 - 8347b8 + 8347b7 + 25741b6 - 15458b5 - 10283b4 + 29651b3 + 21084b2 - 26024

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times$ .

$n$	$667$	$1037$	$1297$
$\chi(n)$	$1$	$1$	$\beta_{4}$

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}

145.1

−0.130479	+	0.225997i
0.896524	−	1.55282i
0.735449	−	1.27384i
−0.561801	+	0.973068i
0.621301	+	1.07612i
−1.56099	−	2.70372i
−0.130479	−	0.225997i
0.896524	+	1.55282i
0.735449	+	1.27384i
−0.561801	−	0.973068i
0.621301	−	1.07612i
−1.56099	+	2.70372i

−1.68491

−

0.613258i

2.82666

1.02882i

145.2

0.245221

0.0892531i

−2.06061

−

0.750002i

181.1

−0.860729

0.722237i

1.42570

−

1.19631i

181.2

1.12677

−

0.945475i

−1.25205

1.05060i

793.1

−0.542127

3.07456i

−0.803091

4.55455i

793.2

0.215775

−

1.22372i

−0.136602

0.774709i

937.1

−1.68491

0.613258i

2.82666

−

1.02882i

937.2

0.245221

−

0.0892531i

−2.06061

0.750002i

1045.1

−0.860729

−

0.722237i

1.42570

1.19631i

1045.2

1.12677

0.945475i

−1.25205

−

1.05060i

1117.1

−0.542127

−

3.07456i

−0.803091

−

4.55455i

1117.2

0.215775

1.22372i

−0.136602

−

0.774709i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.f	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1332.2.bt.b		12
3.b	odd	2	1		444.2.u.a	✓	12
37.f	even	9	1	inner	1332.2.bt.b		12
111.p	odd	18	1		444.2.u.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
444.2.u.a	✓	12	3.b	odd	2	1
444.2.u.a	✓	12	111.p	odd	18	1
1332.2.bt.b		12	1.a	even	1	1	trivial
1332.2.bt.b		12	37.f	even	9	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{12} + 3 T_{5}^{11} + 12 T_{5}^{10} + 20 T_{5}^{9} + 9 T_{5}^{8} + 21 T_{5}^{7} + 70 T_{5}^{6} + \cdots + 9$ T5^12 + 3*T5^11 + 12*T5^10 + 20*T5^9 + 9*T5^8 + 21*T5^7 + 70*T5^6 + 51*T5^5 + 90*T5^4 + 102*T5^3 + 63*T5^2 - 54*T5 + 9 acting on $S_{2}^{\mathrm{new}}(1332, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12}$ T^12
$5$	$T^{12} + 3 T^{11} + \cdots + 9$ T^12 + 3T^11 + 12T^10 + 20T^9 + 9T^8 + 21T^7 + 70T^6 + 51T^5 + 90T^4 + 102T^3 + 63T^2 - 54*T + 9
$7$	$T^{12} + 9 T^{10} + \cdots + 5329$ T^12 + 9T^10 - 38T^9 - 135T^8 + 324T^7 + 1047T^6 - 594T^5 - 1620T^4 + 2410T^3 + 8397T^2 + 4599T + 5329
$11$	$T^{12} + 6 T^{11} + \cdots + 110889$ T^12 + 6T^11 + 45T^10 + 124T^9 + 651T^8 + 1548T^7 + 6373T^6 + 10872T^5 + 32328T^4 + 47934T^3 + 111213T^2 + 104895*T + 110889
$13$	$T^{12} + 6 T^{11} + \cdots + 81$ T^12 + 6T^11 - 6T^10 - 46T^9 + 186T^8 - 438T^7 + 1099T^6 - 1602T^5 + 846T^4 - 90T^3 + 648T^2 + 324*T + 81
$17$	$T^{12} + 12 T^{11} + \cdots + 59049$ T^12 + 12T^11 + 54T^10 + 24T^9 + 198T^8 + 4860T^7 + 21393T^6 + 47466T^5 + 93312T^4 + 137052T^3 + 144342T^2 + 118098*T + 59049
$19$	$T^{12} - 12 T^{11} + \cdots + 494209$ T^12 - 12T^11 + 15T^10 + 531T^9 - 1731T^8 - 9903T^7 + 45370T^6 - 47031T^5 + 651621T^4 + 161019T^3 + 1653267T^2 + 1324452*T + 494209
$23$	$T^{12} + 63 T^{10} + \cdots + 239121$ T^12 + 63T^10 + 220T^9 + 3534T^8 + 7083T^7 + 38527T^6 - 28572T^5 + 175248T^4 - 41025T^3 + 236124T^2 - 74817T + 239121
$29$	$T^{12} - 3 T^{11} + \cdots + 106929$ T^12 - 3T^11 + 54T^10 - 109T^9 + 1953T^8 - 3969T^7 + 31927T^6 - 47175T^5 + 341613T^4 - 564654T^3 + 1082223T^2 - 361989*T + 106929
$31$	$(T^{6} - 42 T^{4} + \cdots + 109)^{2}$ (T^6 - 42T^4 - 115T^3 + 210*T + 109)^2
$37$	$T^{12} + \cdots + 2565726409$ T^12 - 27T^11 + 372T^10 - 3645T^9 + 29445T^8 - 206334T^7 + 1305727T^6 - 7634358T^5 + 40310205T^4 - 184630185T^3 + 697187892T^2 - 1872286839*T + 2565726409
$41$	$T^{12} + 18 T^{11} + \cdots + 2152089$ T^12 + 18T^11 + 252T^10 + 2538T^9 + 19611T^8 + 123741T^7 + 620145T^6 + 2305179T^5 + 6006150T^4 + 10435635T^3 + 11462958T^2 + 7288056*T + 2152089
$43$	$(T^{6} + 6 T^{5} + \cdots + 3743)^{2}$ (T^6 + 6T^5 - 69T^4 - 378T^3 + 567T^2 + 3957*T + 3743)^2
$47$	$T^{12} + \cdots + 231983361$ T^12 + 9T^11 + 150T^10 + 667T^9 + 9903T^8 + 43203T^7 + 395473T^6 + 1170285T^5 + 8265771T^4 + 26510034T^3 + 101109447T^2 + 160519509*T + 231983361
$53$	$T^{12} - 36 T^{11} + \cdots + 6765201$ T^12 - 36T^11 + 693T^10 - 10468T^9 + 127674T^8 - 1112292T^7 + 6418162T^6 - 23209065T^5 + 49234752T^4 - 59326821T^3 + 58499091T^2 - 28254663*T + 6765201
$59$	$T^{12} + \cdots + 184932801$ T^12 - 6T^11 - 243T^10 + 2328T^9 + 21357T^8 - 387558T^7 + 2303100T^6 - 7074540T^5 + 37162800T^4 + 4603446T^3 + 238328973T^2 + 306956628*T + 184932801
$61$	$T^{12} + \cdots + 417425761$ T^12 + 9T^11 + 63T^10 - 135T^9 - 1422T^8 - 19755T^7 + 79912T^6 + 221310T^5 + 1806408T^4 - 16022880T^3 + 77966613T^2 - 273244194*T + 417425761
$67$	$T^{12} + \cdots + 143736121$ T^12 - 6T^11 + 42T^10 - 452T^9 + 4536T^8 - 46800T^7 + 406344T^6 - 2643732T^5 + 14037066T^4 - 56890640T^3 + 157277883T^2 - 234936444*T + 143736121
$71$	$T^{12} + \cdots + 231331178961$ T^12 + 3T^11 + 234T^10 - 102T^9 - 243T^8 + 4401T^7 - 907821T^6 - 4942863T^5 + 132942951T^4 + 479737458T^3 - 4795722936T^2 - 11051224713*T + 231331178961
$73$	$(T^{6} + 21 T^{5} + \cdots + 258371)^{2}$ (T^6 + 21T^5 + 12T^4 - 2061T^3 - 8505T^2 + 49098*T + 258371)^2
$79$	$T^{12} + \cdots + 336020786929$ T^12 - 12T^11 + 111T^10 + 592T^9 - 30636T^8 + 180036T^7 + 2195310T^6 - 36768177T^5 + 422343558T^4 - 1204878611T^3 - 1642831365T^2 + 28236451503*T + 336020786929
$83$	$T^{12} + \cdots + 1659992049$ T^12 + 9T^11 + 126T^10 + 1782T^9 + 6237T^8 - 19197T^7 + 148068T^6 - 4505949T^5 - 20095614T^4 + 154918332T^3 + 2407709853T^2 - 3088971288*T + 1659992049
$89$	$T^{12} + \cdots + 8131891329$ T^12 - 27T^11 + 474T^10 - 5996T^9 + 52143T^8 - 342033T^7 + 2072167T^6 - 9547179T^5 + 35255295T^4 - 209269326T^3 + 1820306934T^2 - 6526019313*T + 8131891329
$97$	$T^{12} + \cdots + 159693769$ T^12 + 24T^11 + 555T^10 + 3708T^9 + 39132T^8 - 5385T^7 + 2148001T^6 - 610908T^5 + 26906412T^4 - 44521047T^3 + 272256858T^2 - 209685741*T + 159693769
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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 145.2
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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