LMFDB - Newform orbit 76.2.i.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,2,Mod(5,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 16]))

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

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

chi := DirichletCharacter("76.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$76 = 2^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	76.i (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:	$0.606863055362$
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:	$x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161$ x^12 - 6x^11 - 3x^10 + 70x^9 - 15x^8 - 426x^7 + 64x^6 + 1659x^5 + 267x^4 - 3969x^3 - 2088x^2 + 4446*x + 4161
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + (\beta_{11} + \beta_{9} + \beta_{7} + \cdots - 1) q^{3} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{5} + (\beta_{9} + \beta_{6} + \cdots + \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \cdots + 1) q^{9}+ \cdots + (\beta_{11} + \beta_{10} + 5 \beta_{9} + \cdots - 1) q^{99}+O(q^{100})$ q + (b11 + b9 + b7 - b5 - 1) * q^3 + (-b9 - b7 - b6 + b4 - b3 + b2 + b1) * q^5 + (b9 + b6 - b5 + b3) * q^7 + (-b11 + b10 - b8 - b6 + 3b5 - b2 - b1 + 1) q^9 + (-b10 - b9 + b8 - b7) * q^11 + (-b10 + b8 + 2b7 + b6 - 2b5 - b4 - b1 - 1) * q^13 + (-b11 + b10 - 3b9 + b7 + 3b6 - 4b4 + b3 - b2 + 1) q^15 + (-b10 + 3b9 - b8 - 2b7 - b6 + 3b5 + b4 - b3 - b2 + b1 - 3) q^17 + (-b11 + b10 - b9 - b8 - 3b7 - b6 + 2b5 - b1 + 1) * q^19 + (-b9 + b8 + 2b7 + b6 - b5 + 2b4 + b3 + b2 - b1) * q^21 + (-b11 + 2b9 - b7 - b6 + 3b4 + b2 + b1 - 2) * q^23 + (b10 - 3b9 + 2b7 - 3b6 - 2b5 + b1) * q^25 + (b11 - b10 + b9 + b8 - 3b7 - 2b6 - b5 + b2 - 2) * q^27 + (b10 - b9 + 2b6 + 4b5 - 2b4 - b3 + 3) q^29 + (2b11 - b10 + 3b9 - 2b8 + b7 + 2b6 - 2b5 - b3 - b2 - 2) q^31 + (b11 - b10 + 3b9 - b8 + 4b7 + 2b6 - 5b5 - b4 + b3 - b2 + 2) * q^33 + (b11 - b9 + b8 + 2b5 + b4 - b3 + b1 + 2) q^35 + (b11 + b10 - b9 + 2b8 - b7 + 4b6 - b5 - 2b4 + b2 + b1 - 1) q^37 + (2b7 - 2b6 - b5 - b4 - 2b1 + 6) q^39 + (-b11 - 6b6 - b4 + b3 + b2 + 1) q^41 + (-b11 + b10 - b9 + 4b7 - 3b6 - b5 + 2b4 + b3 - b2 - b1 + 4) q^43 + (b11 - 2b10 + 4b9 - b8 - 8b7 + 7b4 - b3 + b2 - 1) * q^45 + (2b11 - b10 - 3b9 + b8 + b6 + 3b5 + 4b4 - b3 + b2 + 2b1 - 2) q^47 + (-b11 + 2b10 - 2b8 + b7 - b6 + b5 - b4 + b3 - b2 - b1 + 3) * q^49 + (-b11 + 3b10 + 5b9 - b8 + 4b7 + 4b6 - 4b5 - 5b4 - b3 + 3b1 - 5) q^51 + (b11 - b10 - 3b9 - 4b7 + 3b6 - 2b4 - b3 + b2 - 1) * q^53 + (3b9 + b8 + b7 + b6 + 3b5 - 4b4 + b3 + b2 - b1 - 3) q^55 + (-b10 - 4b9 + 2b8 - 3b7 - 2b6 - 3b5 + b3 + b2 + 2b1 - 3) * q^57 + (-b10 - 3b9 - 2b8 + 3b7 - 2b6 - b5 + 5b4 - 2b3 - 3) * q^59 + (2b11 - b10 + 4b9 + 4b4 - b3 - 3b2 - b1 - 4) * q^61 + (-b11 - b10 - 6b9 + b8 - 5b6 + 2b4 - b3 - b1 + 2) q^63 + (-2b11 + b10 + 7b9 - b8 - 2b7 + 5b5 - 5b4 + 2b3 - 2b2 - 2b1 - 4) * q^65 + (-b11 - 2b10 - b9 + b8 + 4b6 + 2b5 - 3b4 + 3b3 + b2 - b1 + 3) q^67 + (-3b11 + 3b10 - 8b9 + 3b8 + 4b7 + 4b6 - 4b5 - 4b4 + b3 + 3) * q^69 + (-3b11 + 3b10 - b9 + b8 - 3b6 + b5 - b1 + 3) q^71 + (b9 - b8 + 4b6 + 3b5 - b4 + b3 - b1 + 3) * q^73 + (-b11 - 3b10 + 3b9 - 4b8 - 7b7 + 2b6 + 9b5 + 3b4 - 3b2 - 2b1 + 2) q^75 + (-b10 + b9 - b8 - 4b7 + 3b6 - 2b5 - 3b4 - b2 + 2b1) q^77 + (-3b11 - b8 - b7 - 4b6 + 3b5 + b4 - b3 - 2b2 - b1 + 1) * q^79 + (b11 - b10 - 4b9 + 4b7 + 2b5 + b4 + b3 - b2 - b1 + 2) q^81 + (-b11 + 3b10 + b8 + 4b7 + b6 - b5 - 2b4 - b3 - 2b2 + 1) * q^83 + (-2b10 - 2b9 + 4b8 - b6 - 5b5 + b4 + 2b3 + 4b2 - 3) * q^85 + (2b11 + b9 - 8b7 - 5b6 + b5 + b4 - 4b3 + 2b2 + 4b1 - 5) * q^87 + (2b11 - b10 + b9 + 9b7 + b6 - 9b5 - 2b4 + 2b3 - b1 - 2) q^89 + (-b10 - 7b9 - 4b7 + 4b6 - 5b4 - b3 + 3b2 + b1 + 1) q^91 + (3b10 + 3b9 + b8 + 6b7 + b6 + 4b5 - 4b4 + b3 + 2b2 - 2b1 - 5) q^93 + (4b11 - b10 + 5b9 + b8 - b7 - 6b5 + 7b4 - b3 - b2 - 4) * q^95 + (b10 + 3b9 - 2b8 - b7 - 2b6 + b5 + 7b4 - 2b3 - 4b2 + 4b1 - 3) q^97 + (b11 + b10 + 5b9 + 6b7 - b6 + 2b4 + b3 + 2b2 - 2b1 - 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 3 q^{3} + 3 q^{7} - 3 q^{9} + 3 q^{11} - 9 q^{13} - 15 q^{15} - 3 q^{17} - 12 q^{19} - 15 q^{21} - 12 q^{23} - 18 q^{25} - 9 q^{27} + 27 q^{29} + 6 q^{31} + 48 q^{33} + 33 q^{35} - 12 q^{37} + 60 q^{39}+ \cdots - 6 q^{99}+O(q^{100})$ 12 * q - 3 * q^3 + 3 * q^7 - 3 * q^9 + 3 * q^11 - 9 * q^13 - 15 * q^15 - 3 * q^17 - 12 * q^19 - 15 * q^21 - 12 * q^23 - 18 * q^25 - 9 * q^27 + 27 * q^29 + 6 * q^31 + 48 * q^33 + 33 * q^35 - 12 * q^37 + 60 * q^39 + 3 * q^41 + 27 * q^43 + 24 * q^45 - 15 * q^47 + 9 * q^49 - 33 * q^51 - 21 * q^53 - 27 * q^55 - 42 * q^57 - 48 * q^59 - 6 * q^61 - 9 * q^63 - 33 * q^65 + 24 * q^67 - 33 * q^69 + 30 * q^73 + 42 * q^75 + 24 * q^77 + 3 * q^79 + 3 * q^81 + 3 * q^83 - 42 * q^85 - 18 * q^87 - 18 * q^89 - 24 * q^91 - 78 * q^93 + 9 * q^95 + 12 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161$ x^12 - 6*x^11 - 3*x^10 + 70*x^9 - 15*x^8 - 426*x^7 + 64*x^6 + 1659*x^5 + 267*x^4 - 3969*x^3 - 2088*x^2 + 4446*x + 4161 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 1861 \nu^{11} + 59764 \nu^{10} - 261706 \nu^{9} - 459237 \nu^{8} + 3512934 \nu^{7} + \cdots - 104999089 ) / 8740601$ (-1861v^11 + 59764v^10 - 261706v^9 - 459237v^8 + 3512934v^7 + 1223776v^6 - 19762014v^5 - 6600456v^4 + 62490886v^3 + 42461956v^2 - 98039407*v - 104999089) / 8740601
$\beta_{3}$	$=$	$( - 3014 \nu^{11} - 25876 \nu^{10} + 250822 \nu^{9} + 84243 \nu^{8} - 2839567 \nu^{7} + \cdots + 91337490 ) / 8740601$ (-3014v^11 - 25876v^10 + 250822v^9 + 84243v^8 - 2839567v^7 - 652331v^6 + 18054126v^5 + 5399484v^4 - 59109957v^3 - 38613723v^2 + 94201422*v + 91337490) / 8740601
$\beta_{4}$	$=$	$( - 7410 \nu^{11} + 92642 \nu^{10} - 242771 \nu^{9} - 590557 \nu^{8} + 2422014 \nu^{7} + \cdots - 39192874 ) / 8740601$ (-7410v^11 + 92642v^10 - 242771v^9 - 590557v^8 + 2422014v^7 + 2513886v^6 - 10863570v^5 - 10833438v^4 + 27062496v^3 + 32215080v^2 - 29279114*v - 39192874) / 8740601
$\beta_{5}$	$=$	$( 7410 \nu^{11} + 11132 \nu^{10} - 276099 \nu^{9} + 170744 \nu^{8} + 2370458 \nu^{7} + \cdots - 26703616 ) / 8740601$ (7410v^11 + 11132v^10 - 276099v^9 + 170744v^8 + 2370458v^7 - 1428976v^6 - 11344066v^5 + 2408876v^4 + 33428312v^3 + 6822812v^2 - 44659861*v - 26703616) / 8740601
$\beta_{6}$	$=$	$( 8405 \nu^{11} + 8018 \nu^{10} - 346679 \nu^{9} + 455105 \nu^{8} + 2627431 \nu^{7} + \cdots - 11328251 ) / 8740601$ (8405v^11 + 8018v^10 - 346679v^9 + 455105v^8 + 2627431v^7 - 3363517v^6 - 11636805v^5 + 9113704v^4 + 31536935v^3 - 4569699v^2 - 40761460*v - 11328251) / 8740601
$\beta_{7}$	$=$	$( 15815 \nu^{11} - 89341 \nu^{10} - 80323 \nu^{9} + 1088115 \nu^{8} - 105905 \nu^{7} + \cdots + 1553197 ) / 8740601$ (15815v^11 - 89341v^10 - 80323v^9 + 1088115v^8 - 105905v^7 - 5646270v^6 - 277950v^5 + 17338641v^4 + 8455587v^3 - 27341345v^2 - 22774844*v + 1553197) / 8740601
$\beta_{8}$	$=$	$( 16036 \nu^{11} - 88198 \nu^{10} + 124775 \nu^{9} + 357074 \nu^{8} - 2014841 \nu^{7} + \cdots + 33629265 ) / 8740601$ (16036v^11 - 88198v^10 + 124775v^9 + 357074v^8 - 2014841v^7 + 1083449v^6 + 10730801v^5 - 7009805v^4 - 37334292v^3 + 5511793v^2 + 68308502*v + 33629265) / 8740601
$\beta_{9}$	$=$	$( - 1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + \cdots - 745423 ) / 514153$ (-1506v^11 + 8283v^10 + 6301v^9 - 90477v^8 - 1422v^7 + 485184v^6 + 4698v^5 - 1477233v^4 - 388357v^3 + 2375103v^2 + 1084425*v - 745423) / 514153
$\beta_{10}$	$=$	$( 32367 \nu^{11} - 175660 \nu^{10} + 8466 \nu^{9} + 1222749 \nu^{8} - 536869 \nu^{7} + \cdots + 20539212 ) / 8740601$ (32367v^11 - 175660v^10 + 8466v^9 + 1222749v^8 - 536869v^7 - 4743060v^6 + 1737702v^5 + 11702325v^4 - 5650797v^3 - 17409849v^2 + 16526830*v + 20539212) / 8740601
$\beta_{11}$	$=$	$( 382 \nu^{11} - 2101 \nu^{10} + 162 \nu^{9} + 12670 \nu^{8} + 605 \nu^{7} - 42146 \nu^{6} + \cdots - 476163 ) / 80189$ (382v^11 - 2101v^10 + 162v^9 + 12670v^8 + 605v^7 - 42146v^6 - 39078v^5 + 48159v^4 + 203904v^3 + 142404v^2 - 273582*v - 476163) / 80189

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 3$ -b10 - b9 + b8 - b7 + b6 - b5 - b4 + b2 + b1 + 3
$\nu^{3}$	$=$	$\beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{4} + 3\beta _1 + 2$ b11 - 3b10 - b9 + b8 - 2b7 + b6 + 2b5 - 3b4 + 3*b1 + 2
$\nu^{4}$	$=$	$\beta_{11} - 9 \beta_{10} - 7 \beta_{9} + 6 \beta_{8} - 12 \beta_{7} + 8 \beta_{6} - 13 \beta_{4} + \cdots + 5$ b11 - 9b10 - 7b9 + 6b8 - 12b7 + 8b6 - 13b4 - 2b3 + 3b2 + 6*b1 + 5
$\nu^{5}$	$=$	$8 \beta_{11} - 28 \beta_{10} - 22 \beta_{9} + 10 \beta_{8} - 34 \beta_{7} + 11 \beta_{6} + 12 \beta_{5} + \cdots + 4$ 8b11 - 28b10 - 22b9 + 10b8 - 34b7 + 11b6 + 12b5 - 25b4 - 5b3 - 3b2 + 9*b1 + 4
$\nu^{6}$	$=$	$11 \beta_{11} - 62 \beta_{10} - 71 \beta_{9} + 26 \beta_{8} - 108 \beta_{7} + 26 \beta_{6} + 25 \beta_{5} + \cdots - 6$ 11b11 - 62b10 - 71b9 + 26b8 - 108b7 + 26b6 + 25b5 - 66b4 - 33b3 - 16b2 + 24*b1 - 6
$\nu^{7}$	$=$	$26 \beta_{11} - 152 \beta_{10} - 261 \beta_{9} + 42 \beta_{8} - 327 \beta_{7} - 12 \beta_{6} + 87 \beta_{5} + \cdots - 5$ 26b11 - 152b10 - 261b9 + 42b8 - 327b7 - 12b6 + 87b5 - 91b4 - 98b3 - 75b2 + 45*b1 - 5
$\nu^{8}$	$=$	$- 12 \beta_{11} - 288 \beta_{10} - 804 \beta_{9} + 76 \beta_{8} - 866 \beta_{7} - 236 \beta_{6} + \cdots + 18$ -12b11 - 288b10 - 804b9 + 76b8 - 866b7 - 236b6 + 233b5 - 51b4 - 362b3 - 270b2 + 141*b1 + 18
$\nu^{9}$	$=$	$- 236 \beta_{11} - 480 \beta_{10} - 2610 \beta_{9} + 88 \beta_{8} - 2462 \beta_{7} - 1334 \beta_{6} + \cdots + 384$ -236b11 - 480b10 - 2610b9 + 88b8 - 2462b7 - 1334b6 + 761b5 + 402b4 - 1062b3 - 762b2 + 417*b1 + 384
$\nu^{10}$	$=$	$- 1334 \beta_{11} - 495 \beta_{10} - 7824 \beta_{9} + 204 \beta_{8} - 6291 \beta_{7} - 5298 \beta_{6} + \cdots + 1889$ -1334b11 - 495b10 - 7824b9 + 204b8 - 6291b7 - 5298b6 + 2331b5 + 2500b4 - 3136b3 - 2046b2 + 1327*b1 + 1889
$\nu^{11}$	$=$	$- 5298 \beta_{11} + 678 \beta_{10} - 23140 \beta_{9} + 726 \beta_{8} - 16756 \beta_{7} - 18648 \beta_{6} + \cdots + 7815$ -5298b11 + 678b10 - 23140b9 + 726b8 - 16756b7 - 18648b6 + 7923b5 + 9189b4 - 8536b3 - 4679b2 + 3928*b1 + 7815

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$ .

$n$	$21$	$39$
$\chi(n)$	$-\beta_{7}$	$1$

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}

5.1

2.75227	−	0.342020i
−1.75227	−	0.342020i
2.26253	−	0.984808i
−1.26253	−	0.984808i
2.26253	+	0.984808i
−1.26253	+	0.984808i
−1.25236	+	0.642788i
2.25236	+	0.642788i
2.75227	+	0.342020i
−1.75227	+	0.342020i
−1.25236	−	0.642788i
2.25236	−	0.642788i

−0.467454

2.65106i

−0.982226

−

0.824185i

1.67233

2.89656i

−3.99055

−

1.45244i

5.2

0.314751

−

1.78504i

0.216181

0.181398i

−0.579936

−

1.00448i

−0.268219

−

0.0976237i

9.1

−0.834130

0.699919i

3.00735

1.09458i

0.278396

−

0.482195i

−0.315057

1.78678i

9.2

1.86622

−

1.56594i

−2.06765

−

0.752564i

−1.48413

2.57059i

0.509650

−

2.89037i

17.1

−0.834130

−

0.699919i

3.00735

−

1.09458i

0.278396

0.482195i

−0.315057

−

1.78678i

17.2

1.86622

1.56594i

−2.06765

0.752564i

−1.48413

−

2.57059i

0.509650

2.89037i

25.1

−2.83638

−

1.03236i

−0.658711

−

3.73574i

−0.0695116

0.120398i

4.68114

3.92794i

25.2

0.456991

0.166331i

0.485063

2.75093i

1.68285

−

2.91479i

−2.11696

−

1.77634i

61.1

−0.467454

−

2.65106i

−0.982226

0.824185i

1.67233

−

2.89656i

−3.99055

1.45244i

61.2

0.314751

1.78504i

0.216181

−

0.181398i

−0.579936

1.00448i

−0.268219

0.0976237i

73.1

−2.83638

1.03236i

−0.658711

3.73574i

−0.0695116

−

0.120398i

4.68114

−

3.92794i

73.2

0.456991

−

0.166331i

0.485063

−

2.75093i

1.68285

2.91479i

−2.11696

1.77634i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	76.2.i.a	✓	12
3.b	odd	2	1		684.2.bo.c		12
4.b	odd	2	1		304.2.u.e		12
19.e	even	9	1	inner	76.2.i.a	✓	12
19.e	even	9	1		1444.2.a.h		6
19.e	even	9	2		1444.2.e.g		12
19.f	odd	18	1		1444.2.a.g		6
19.f	odd	18	2		1444.2.e.h		12
57.l	odd	18	1		684.2.bo.c		12
76.k	even	18	1		5776.2.a.by		6
76.l	odd	18	1		304.2.u.e		12
76.l	odd	18	1		5776.2.a.bw		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.2.i.a	✓	12	1.a	even	1	1	trivial
76.2.i.a	✓	12	19.e	even	9	1	inner
304.2.u.e		12	4.b	odd	2	1
304.2.u.e		12	76.l	odd	18	1
684.2.bo.c		12	3.b	odd	2	1
684.2.bo.c		12	57.l	odd	18	1
1444.2.a.g		6	19.f	odd	18	1
1444.2.a.h		6	19.e	even	9	1
1444.2.e.g		12	19.e	even	9	2
1444.2.e.h		12	19.f	odd	18	2
5776.2.a.bw		6	76.l	odd	18	1
5776.2.a.by		6	76.k	even	18	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(76, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12} + 3 T^{11} + \cdots + 361$ T^12 + 3T^11 + 6T^10 + 18T^9 + 24T^8 + 75T^7 + 433T^6 + 195T^5 + 1137T^4 + 729T^3 + 39T^2 - 912*T + 361
$5$	$T^{12} + 9 T^{10} + \cdots + 729$ T^12 + 9T^10 - 19T^9 - 72T^8 + 36T^7 + 64T^6 + 1251T^5 + 7398T^4 + 9747T^3 + 4131T^2 - 2916T + 729
$7$	$T^{12} - 3 T^{11} + \cdots + 9$ T^12 - 3T^11 + 21T^10 - 22T^9 + 201T^8 - 186T^7 + 1141T^6 + 447T^5 + 1386T^4 - 366T^3 + 414T^2 + 54*T + 9
$11$	$T^{12} - 3 T^{11} + \cdots + 729$ T^12 - 3T^11 + 33T^10 + 2T^9 + 591T^8 - 300T^7 + 3331T^6 + 3231T^5 + 7452T^4 + 1890T^3 + 2430T^2 + 729
$13$	$T^{12} + 9 T^{11} + \cdots + 130321$ T^12 + 9T^11 + 45T^10 + 268T^9 + 1377T^8 + 4077T^7 + 20109T^6 + 152361T^5 + 639711T^4 + 1178665T^3 + 948708T^2 + 308655*T + 130321
$17$	$T^{12} + 3 T^{11} + \cdots + 8982009$ T^12 + 3T^11 + 9T^10 - 30T^9 - 1179T^8 + 999T^7 + 39861T^6 - 136809T^5 + 653913T^4 - 2532789T^3 + 5143824T^2 + 10924065*T + 8982009
$19$	$T^{12} + 12 T^{11} + \cdots + 47045881$ T^12 + 12T^11 + 33T^10 - 201T^9 - 1365T^8 + 627T^7 + 24358T^6 + 11913T^5 - 492765T^4 - 1378659T^3 + 4300593T^2 + 29713188*T + 47045881
$23$	$T^{12} + \cdots + 151585344$ T^12 + 12T^11 + 69T^10 + 251T^9 + 714T^8 - 3729T^7 - 40985T^6 - 36522T^5 + 856980T^4 + 4777056T^3 + 22161600T^2 + 75792672*T + 151585344
$29$	$T^{12} - 27 T^{11} + \cdots + 263169$ T^12 - 27T^11 + 369T^10 - 3023T^9 + 15840T^8 - 41454T^7 - 54107T^6 + 832302T^5 - 1363392T^4 - 6261165T^3 + 19768293T^2 - 872613*T + 263169
$31$	$T^{12} - 6 T^{11} + \cdots + 130321$ T^12 - 6T^11 + 126T^10 - 90T^9 + 8508T^8 - 14955T^7 + 205549T^6 - 321024T^5 + 3546369T^4 - 6277068T^3 + 19798323T^2 + 1584429*T + 130321
$37$	$(T^{6} + 6 T^{5} + \cdots + 11096)^{2}$ (T^6 + 6T^5 - 81T^4 - 639T^3 + 78T^2 + 7836*T + 11096)^2
$41$	$T^{12} + \cdots + 360278361$ T^12 - 3T^11 - 21T^10 + 910T^9 + 6492T^8 + 44274T^7 + 492832T^6 + 1806273T^5 + 3831138T^4 + 21232152T^3 + 23782896T^2 - 150671178*T + 360278361
$43$	$T^{12} - 27 T^{11} + \cdots + 1203409$ T^12 - 27T^11 + 309T^10 - 2142T^9 + 11874T^8 - 50418T^7 + 137242T^6 - 370035T^5 + 771306T^4 + 1881450T^3 + 18911724T^2 - 7325766*T + 1203409
$47$	$T^{12} + 15 T^{11} + \cdots + 729$ T^12 + 15T^11 - 51T^10 - 460T^9 + 32919T^8 + 417405T^7 + 1894231T^6 + 3504195T^5 + 3161673T^4 + 812565T^3 + 1256796T^2 + 10935*T + 729
$53$	$T^{12} + 21 T^{11} + \cdots + 95004009$ T^12 + 21T^11 + 177T^10 + 890T^9 + 3522T^8 + 4470T^7 + 42652T^6 + 666045T^5 + 1156302T^4 - 1832436T^3 + 34913754T^2 - 80003376*T + 95004009
$59$	$T^{12} + \cdots + 12621848409$ T^12 + 48T^11 + 1107T^10 + 16278T^9 + 183636T^8 + 1690956T^7 + 11066715T^6 + 40087386T^5 + 70948224T^4 + 282191040T^3 + 812641086T^2 - 3266938413*T + 12621848409
$61$	$T^{12} + \cdots + 1418049649$ T^12 + 6T^11 - 141T^10 - 560T^9 + 8838T^8 - 18414T^7 + 449985T^6 + 1422378T^5 + 9943830T^4 - 39761552T^3 + 276693828T^2 - 727194327*T + 1418049649
$67$	$T^{12} + \cdots + 6080256576$ T^12 - 24T^11 + 483T^10 - 6139T^9 + 59610T^8 - 406245T^7 + 2284291T^6 - 4719942T^5 - 32415444T^4 + 205544736T^3 + 19649952T^2 - 2720114784*T + 6080256576
$71$	$T^{12} + 45 T^{10} + \cdots + 16842816$ T^12 + 45T^10 - 487T^9 + 8442T^8 - 34011T^7 + 1016551T^6 - 5907222T^5 + 13286916T^4 - 17404416T^3 + 19478880T^2 + 43879968T + 16842816
$73$	$T^{12} - 30 T^{11} + \cdots + 36864$ T^12 - 30T^11 + 375T^10 - 2947T^9 + 21024T^8 - 109335T^7 + 259069T^6 - 66444T^5 + 211536T^4 + 89088T^3 + 78336T^2 - 82944*T + 36864
$79$	$T^{12} + \cdots + 1548343801$ T^12 - 3T^11 + 39T^10 + 259T^9 - 7938T^8 + 10548T^7 + 330321T^6 - 5080752T^5 + 80469864T^4 - 476164415T^3 + 1473881775T^2 - 2218221177*T + 1548343801
$83$	$T^{12} + \cdots + 11939714361$ T^12 - 3T^11 + 189T^10 - 114T^9 + 24957T^8 - 14472T^7 + 1386243T^6 + 866295T^5 + 53308368T^4 + 19603782T^3 + 958378392T^2 + 672659964*T + 11939714361
$89$	$T^{12} + \cdots + 7695324729$ T^12 + 18T^11 + 117T^10 + 2205T^9 + 26244T^8 - 24948T^7 + 1486998T^6 + 37993293T^5 + 125240742T^4 - 321329349T^3 + 1314529155T^2 - 2430102546*T + 7695324729
$97$	$T^{12} + \cdots + 16983563041$ T^12 - 12T^11 - 39T^10 + 2124T^9 - 19416T^8 - 14412T^7 + 4516399T^6 - 66305706T^5 + 595015248T^4 - 3870585684T^3 + 24477528468T^2 + 31607403735*T + 16983563041
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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 5.2
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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