LMFDB - Newform orbit 2548.2.j.r

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(1145,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2548.1145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.j (of order $3$ , degree $2$ , not 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:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 10 x^{10} - 8 x^{9} + 80 x^{8} - 56 x^{7} + 220 x^{6} - 240 x^{5} + 484 x^{4} - 336 x^{3} + \cdots + 4$ x^12 + 10x^10 - 8x^9 + 80x^8 - 56x^7 + 220x^6 - 240x^5 + 484x^4 - 336x^3 + 216x^2 - 32x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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.

$f(q)$	$=$	$q + (\beta_{7} + \beta_{2} + \beta_1) q^{3} + ( - \beta_{10} + 2 \beta_{6} - 2) q^{5} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \cdots - 1) q^{9} + ( - \beta_{10} + \beta_{6} + \cdots + 2 \beta_1) q^{11}+ \cdots + (\beta_{9} + 8 \beta_{5} - 4 \beta_{4} + \cdots - 8) q^{99}+O(q^{100})$ q + (b7 + b2 + b1) * q^3 + (-b10 + 2b6 - 2) q^5 + (-b10 + b8 + b6 + b1 - 1) * q^9 + (-b10 + b6 - b3 + 2b2 + 2b1) * q^11 - q^13 + (b9 + 3b5 - b4 - b3 - 2b2 + 1) * q^15 + (-b11 - b8 - 2b6 - b4 - b2 - b1) q^17 + (-b11 + 2b10 + b9 + b8 + 2b7 + b6 + 2b5 - b1 - 1) q^19 + (-2b11 + b10 + 2b9 - b1) * q^23 + (3b10 - 3b6 + 3b3 + b2 + b1) q^25 + (2b9 + 3b5 - 2b4 - 2) q^27 + (2b5 - 2b3 + 1) * q^29 + (b11 - b10 - b7 - b6 - b3) * q^31 + (-b11 - b10 + b9 + b8 + 2b7 + 5b6 + 2b5 + 3b1 - 5) * q^33 + (2b11 - b10 - 2b9 - 2b8 - 2b7 + 3b6 - 2b5 - 3) * q^37 + (-b7 - b2 - b1) * q^39 + (-b9 - b5 - 2b4 - 2b3 + 2b2 + 3) q^41 + (-2b9 + 2b4 + 1) * q^43 + (-b11 + 2b10 - 3b8 - 5b6 - 3b4 + 2b3 - 3b2 - 3b1) q^45 + (-b11 + b9 + b8 + 2b7 + 5b6 + 2b5 - b1 - 5) q^47 + (2b11 - b10 - 2b9 - b8 - 8b7 - 2b6 - 8b5 - 3b1 + 2) * q^51 + (-3b10 - 2b7 + 2b6 - 3b3 + 3b2 + 3b1) * q^53 + (2b5 - 2b4 - 3b2 - 2) q^55 + (-2b9 + b5 - b4 + 3b3 - 2b2 - 2) q^57 + (-b10 + 3b8 - b7 - 2b6 + 3b4 - b3 + b2 + b1) q^59 + (b11 + 2b10 - b9 - b8 - 2b7 + 2b6 - 2b5 - 2) * q^61 + (b10 - 2b6 + 2) q^65 + (-b11 + b8 - 3b7 + b4 + 2b2 + 2b1) q^67 + (-3b9 - 5b5 + b4 + 4b3 - b2) q^69 + (b9 - 2b5 + 2b4 + 3b3 - 2) q^71 + (3b11 - b8 + 2b7 - 3b6 - b4 + 3b2 + 3b1) q^73 + (3b11 - 4b10 - 3b9 - 3b8 - 6b7 + 6b6 - 6b5 - 2b1 - 6) * q^75 + (b8 - 4b7 + 2b6 - 4b5 - 2) q^79 + (b10 - 2b8 - 6b7 - 2b6 - 2b4 + b3 - 3b2 - 3b1) * q^81 + (-2b5 + 2b4 + 3b3 + 2b2 - 4) * q^83 + (2b9 + 4b5 + 4b4 - 3b3 + 4b2 + 3) q^85 + (-2b11 + 2b10 + 3b7 - 4b6 + 2b3 + b2 + b1) q^87 + (2b11 + 2b10 - 2b9 - b8 - 5b7 + 4b6 - 5b5 - 3b1 - 4) q^89 + (-2b11 + 3b10 + 2b9 + 2b7 - 3b6 + 2b5 - 2b1 + 3) q^93 + (2b11 - 6b7 + b6 - 2b2 - 2b1) * q^95 + (2b9 - b5 - 3b4 + b2 + 4) * q^97 + (b9 + 8b5 - 4b4 + b3 - 4b2 - 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 10 q^{5} - 6 q^{9} + 4 q^{11} - 12 q^{13} + 8 q^{15} - 16 q^{17} - 10 q^{19} + 2 q^{23} - 12 q^{25} - 24 q^{27} + 4 q^{29} - 6 q^{31} - 28 q^{33} - 16 q^{37} + 16 q^{41} + 12 q^{43} - 34 q^{45} - 30 q^{47}+ \cdots - 104 q^{99}+O(q^{100})$ 12 * q - 10 * q^5 - 6 * q^9 + 4 * q^11 - 12 * q^13 + 8 * q^15 - 16 * q^17 - 10 * q^19 + 2 * q^23 - 12 * q^25 - 24 * q^27 + 4 * q^29 - 6 * q^31 - 28 * q^33 - 16 * q^37 + 16 * q^41 + 12 * q^43 - 34 * q^45 - 30 * q^47 + 12 * q^51 + 6 * q^53 - 32 * q^55 - 24 * q^57 - 8 * q^59 - 16 * q^61 + 10 * q^65 + 8 * q^69 - 14 * q^73 - 28 * q^75 - 14 * q^79 - 14 * q^81 - 28 * q^83 + 48 * q^85 - 24 * q^87 - 30 * q^89 + 16 * q^93 + 10 * q^95 + 44 * q^97 - 104 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} + 10 x^{10} - 8 x^{9} + 80 x^{8} - 56 x^{7} + 220 x^{6} - 240 x^{5} + 484 x^{4} - 336 x^{3} + \cdots + 4$ x^12 + 10*x^10 - 8*x^9 + 80*x^8 - 56*x^7 + 220*x^6 - 240*x^5 + 484*x^4 - 336*x^3 + 216*x^2 - 32*x + 4 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 379000 \nu^{11} + 322300 \nu^{10} + 3778645 \nu^{9} + 152800 \nu^{8} + 28166440 \nu^{7} + \cdots - 12684696 ) / 86544668$ (379000v^11 + 322300v^10 + 3778645v^9 + 152800v^8 + 28166440v^7 + 4327072v^6 + 68974400v^5 - 25453960v^4 + 147974882v^3 + 10721360v^2 - 1728720*v - 12684696) / 86544668
$\beta_{3}$	$=$	$( - 3905120 \nu^{11} - 7830811 \nu^{10} - 41548809 \nu^{9} - 37653720 \nu^{8} - 264964348 \nu^{7} + \cdots + 1118396056 ) / 432723340$ (-3905120v^11 - 7830811v^10 - 41548809v^9 - 37653720v^8 - 264964348v^7 - 290522622v^6 - 602456824v^5 + 43991346v^4 - 335361882v^3 - 207884308v^2 + 32244016*v + 1118396056) / 432723340
$\beta_{4}$	$=$	$( 3705810 \nu^{11} + 4692768 \nu^{10} + 38183017 \nu^{9} + 13824960 \nu^{8} + 266775954 \nu^{7} + \cdots + 270746912 ) / 216361670$ (3705810v^11 + 4692768v^10 + 38183017v^9 + 13824960v^8 + 266775954v^7 + 120064991v^6 + 637429512v^5 - 174283368v^4 + 882781546v^3 + 138125424v^2 - 21835548*v + 270746912) / 216361670
$\beta_{5}$	$=$	$( 11205 \nu^{11} + 10949 \nu^{10} + 114461 \nu^{9} + 15880 \nu^{8} + 824777 \nu^{7} + 187358 \nu^{6} + \cdots + 510976 ) / 347290$ (11205v^11 + 10949v^10 + 114461v^9 + 15880v^8 + 824777v^7 + 187358v^6 + 2005116v^5 - 683794v^4 + 3045328v^3 + 347652v^2 - 55654*v + 510976) / 347290
$\beta_{6}$	$=$	$( - 3171174 \nu^{11} - 379000 \nu^{10} - 32034040 \nu^{9} + 21590747 \nu^{8} - 253846720 \nu^{7} + \cdots + 103206288 ) / 86544668$ (-3171174v^11 - 379000v^10 - 32034040v^9 + 21590747v^8 - 253846720v^7 + 149419304v^6 - 701985352v^5 + 692107360v^4 - 1509394256v^3 + 917539582v^2 - 695694944*v + 103206288) / 86544668
$\beta_{7}$	$=$	$( 27192621 \nu^{11} - 7411620 \nu^{10} + 262540674 \nu^{9} - 293907002 \nu^{8} + 2147759760 \nu^{7} + \cdots - 826492776 ) / 432723340$ (27192621v^11 - 7411620v^10 + 262540674v^9 - 293907002v^8 + 2147759760v^7 - 2056338684v^6 + 5742246638v^5 - 7801088064v^4 + 13509795300v^3 - 10902283748v^2 + 5597355288*v - 826492776) / 432723340
$\beta_{8}$	$=$	$( - 29498324 \nu^{11} - 4899106 \nu^{10} - 291648120 \nu^{9} + 197195993 \nu^{8} + \cdots + 139153820 ) / 432723340$ (-29498324v^11 - 4899106v^10 - 291648120v^9 + 197195993v^8 - 2284521748v^7 + 1320236584v^6 - 6008662636v^5 + 6423390232v^4 - 12671186152v^3 + 7672711674v^2 - 4561916576*v + 139153820) / 432723340
$\beta_{9}$	$=$	$( - 15940620 \nu^{11} - 13784186 \nu^{10} - 161115064 \nu^{9} - 8253520 \nu^{8} + \cdots - 608382084 ) / 216361670$ (-15940620v^11 - 13784186v^10 - 161115064v^9 - 8253520v^8 - 1183392788v^7 - 178289157v^6 - 2895561024v^5 + 1059533356v^4 - 4914958702v^3 - 455869368v^2 + 73440136*v - 608382084) / 216361670
$\beta_{10}$	$=$	$( - 35536656 \nu^{11} - 2624119 \nu^{10} - 364956680 \nu^{9} + 245386747 \nu^{8} + \cdots + 201658580 ) / 432723340$ (-35536656v^11 - 2624119v^10 - 364956680v^9 + 245386747v^8 - 2920168012v^7 + 1740177496v^6 - 8358574604v^5 + 7596062558v^4 - 18207025768v^3 + 11096043566v^2 - 8916475484*v + 201658580) / 432723340
$\beta_{11}$	$=$	$( 52697053 \nu^{11} - 16024810 \nu^{10} + 514656112 \nu^{9} - 582294921 \nu^{8} + \cdots - 1617413908 ) / 432723340$ (52697053v^11 - 16024810v^10 + 514656112v^9 - 582294921v^8 + 4219807680v^7 - 4149316202v^6 + 11491856774v^5 - 15595168232v^4 + 26645163240v^3 - 23001774114v^2 + 10957605904*v - 1617413908) / 432723340

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{10} + \beta_{8} - 4\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1$ b10 + b8 - 4*b6 + b4 + b3 - b2 - b1
$\nu^{3}$	$=$	$\beta_{9} + 2\beta_{5} - \beta_{4} + 6\beta_{2} + 2$ b9 + 2b5 - b4 + 6b2 + 2
$\nu^{4}$	$=$	$-6\beta_{10} - 9\beta_{8} - 2\beta_{7} + 25\beta_{6} - 2\beta_{5} + 8\beta _1 - 25$ -6b10 - 9b8 - 2b7 + 25b6 - 2b5 + 8b1 - 25
$\nu^{5}$	$=$	$- 9 \beta_{11} + 2 \beta_{10} + 13 \beta_{8} + 21 \beta_{7} - 22 \beta_{6} + 13 \beta_{4} + \cdots - 42 \beta_1$ -9b11 + 2b10 + 13b8 + 21b7 - 22b6 + 13b4 + 2b3 - 42b2 - 42*b1
$\nu^{6}$	$=$	$4\beta_{9} + 28\beta_{5} - 74\beta_{4} - 40\beta_{3} + 68\beta_{2} + 176$ 4b9 + 28b5 - 74b4 - 40b3 + 68*b2 + 176
$\nu^{7}$	$=$	$70 \beta_{11} - 28 \beta_{10} - 70 \beta_{9} - 130 \beta_{8} - 178 \beta_{7} + 216 \beta_{6} + \cdots - 216$ 70b11 - 28b10 - 70b9 - 130b8 - 178b7 + 216b6 - 178b5 + 314b1 - 216
$\nu^{8}$	$=$	$- 60 \beta_{11} + 286 \beta_{10} + 594 \beta_{8} + 292 \beta_{7} - 1308 \beta_{6} + 594 \beta_{4} + \cdots - 590 \beta_1$ -60b11 + 286b10 + 594b8 + 292b7 - 1308b6 + 594b4 + 286b3 - 590b2 - 590*b1
$\nu^{9}$	$=$	$534\beta_{9} + 1436\beta_{5} - 1190\beta_{4} - 304\beta_{3} + 2432\beta_{2} + 2020$ 534b9 + 1436b5 - 1190b4 - 304b3 + 2432*b2 + 2020
$\nu^{10}$	$=$	$656 \beta_{11} - 2128 \beta_{10} - 656 \beta_{9} - 4754 \beta_{8} - 2732 \beta_{7} + 10022 \beta_{6} + \cdots - 10022$ 656b11 - 2128b10 - 656b9 - 4754b8 - 2732b7 + 10022b6 - 2732b5 + 5108b1 - 10022
$\nu^{11}$	$=$	$- 4098 \beta_{11} + 2980 \beta_{10} + 10466 \beta_{8} + 11478 \beta_{7} - 18252 \beta_{6} + \cdots - 19228 \beta_1$ -4098b11 + 2980b10 + 10466b8 + 11478b7 - 18252b6 + 10466b4 + 2980b3 - 19228b2 - 19228*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$ .

$n$	$197$	$885$	$1275$
$\chi(n)$	$1$	$-1 + \beta_{6}$	$1$

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}

1145.1

0.652846	+	1.13076i
0.0779999	+	0.135100i
1.23667	+	2.14198i
0.379586	+	0.657462i
−1.43795	−	2.49061i
−0.909151	−	1.57470i
0.652846	−	1.13076i
0.0779999	−	0.135100i
1.23667	−	2.14198i
0.379586	−	0.657462i
−1.43795	+	2.49061i
−0.909151	+	1.57470i

−1.35995

2.35551i

−1.92853

−

3.34030i

−2.19894

−

3.80868i

1145.2

−0.785107

1.35984i

0.299524

0.518790i

0.267215

0.462830i

1145.3

−0.529566

0.917235i

−2.04647

−

3.54459i

0.939121

1.62660i

1145.4

0.327521

−

0.567283i

0.369059

0.639229i

1.28546

2.22648i

1145.5

0.730846

−

1.26586i

−0.163891

−

0.283868i

0.431728

0.747776i

1145.6

1.61626

−

2.79944i

−1.52970

−

2.64951i

−3.72458

−

6.45116i

1353.1

−1.35995

−

2.35551i

−1.92853

3.34030i

−2.19894

3.80868i

1353.2

−0.785107

−

1.35984i

0.299524

−

0.518790i

0.267215

−

0.462830i

1353.3

−0.529566

−

0.917235i

−2.04647

3.54459i

0.939121

−

1.62660i

1353.4

0.327521

0.567283i

0.369059

−

0.639229i

1.28546

−

2.22648i

1353.5

0.730846

1.26586i

−0.163891

0.283868i

0.431728

−

0.747776i

1353.6

1.61626

2.79944i

−1.52970

2.64951i

−3.72458

6.45116i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.2.j.r		12
7.b	odd	2	1		2548.2.j.s		12
7.c	even	3	1		2548.2.a.s	yes	6
7.c	even	3	1	inner	2548.2.j.r		12
7.d	odd	6	1		2548.2.a.r	✓	6
7.d	odd	6	1		2548.2.j.s		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2548.2.a.r	✓	6	7.d	odd	6	1
2548.2.a.s	yes	6	7.c	even	3	1
2548.2.j.r		12	1.a	even	1	1	trivial
2548.2.j.r		12	7.c	even	3	1	inner
2548.2.j.s		12	7.b	odd	2	1
2548.2.j.s		12	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2548, [\chi])$ :

T_{3}^{12} + 12 T_{3}^{10} + 8 T_{3}^{9} + 116 T_{3}^{8} + 56 T_{3}^{7} + 324 T_{3}^{6} + 80 T_{3}^{5} + \cdots + 196

T_{5}^{12} + 10 T_{5}^{11} + 71 T_{5}^{10} + 274 T_{5}^{9} + 808 T_{5}^{8} + 1178 T_{5}^{7} + 1381 T_{5}^{6} + \cdots + 49

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12} + 12 T^{10} + \cdots + 196$ T^12 + 12T^10 + 8T^9 + 116T^8 + 56T^7 + 324T^6 + 80T^5 + 648T^4 + 112T^3 + 456T^2 - 112T + 196
$5$	$T^{12} + 10 T^{11} + \cdots + 49$ T^12 + 10T^11 + 71T^10 + 274T^9 + 808T^8 + 1178T^7 + 1381T^6 - 654T^5 + 1958T^4 - 394T^3 + 365T^2 + 42*T + 49
$7$	$T^{12}$ T^12
$11$	$T^{12} - 4 T^{11} + \cdots + 6724$ T^12 - 4T^11 + 46T^10 + 16T^9 + 786T^8 + 1268T^7 + 13208T^6 + 32192T^5 + 88120T^4 + 89672T^3 + 89908T^2 - 20664*T + 6724
$13$	$(T + 1)^{12}$ (T + 1)^12
$17$	$T^{12} + 16 T^{11} + \cdots + 64516$ T^12 + 16T^11 + 208T^10 + 1432T^9 + 9384T^8 + 39352T^7 + 215188T^6 + 706560T^5 + 2710400T^4 + 2108528T^3 + 1209872T^2 + 327152*T + 64516
$19$	$T^{12} + 10 T^{11} + \cdots + 13300609$ T^12 + 10T^11 + 117T^10 + 534T^9 + 4158T^8 + 15482T^7 + 100085T^6 + 244930T^5 + 1070136T^4 + 1688706T^3 + 7613127T^2 + 9183146*T + 13300609
$23$	$T^{12} - 2 T^{11} + \cdots + 23804641$ T^12 - 2T^11 + 97T^10 - 118T^9 + 7034T^8 - 7394T^7 + 189945T^6 + 127722T^5 + 3370782T^4 - 309130T^3 + 10235157T^2 + 4556986*T + 23804641
$29$	$(T^{6} - 2 T^{5} - 57 T^{4} + \cdots + 25)^{2}$ (T^6 - 2T^5 - 57T^4 + 100T^3 + 127T^2 - 130*T + 25)^2
$31$	$T^{12} + 6 T^{11} + \cdots + 7921$ T^12 + 6T^11 + 55T^10 + 294T^9 + 2084T^8 + 9422T^7 + 34609T^6 + 84466T^5 + 157142T^4 + 184246T^3 + 150953T^2 + 39338*T + 7921
$37$	$T^{12} + \cdots + 2907150724$ T^12 + 16T^11 + 266T^10 + 2072T^9 + 20890T^8 + 121892T^7 + 1054448T^6 + 4473912T^5 + 27945440T^4 + 70924680T^3 + 441915748T^2 + 908194792*T + 2907150724
$41$	$(T^{6} - 8 T^{5} + \cdots - 76816)^{2}$ (T^6 - 8T^5 - 152T^4 + 712T^3 + 7800T^2 - 7200*T - 76816)^2
$43$	$(T^{6} - 6 T^{5} - 105 T^{4} + \cdots + 73)^{2}$ (T^6 - 6T^5 - 105T^4 + 460T^3 + 2047T^2 + 1114*T + 73)^2
$47$	$T^{12} + \cdots + 209699361$ T^12 + 30T^11 + 581T^10 + 6930T^9 + 61574T^8 + 377286T^7 + 1783405T^6 + 5810622T^5 + 16281688T^4 + 33196902T^3 + 82013823T^2 + 124160094*T + 209699361
$53$	$T^{12} + \cdots + 1070532961$ T^12 - 6T^11 + 177T^10 - 1090T^9 + 22938T^8 - 119382T^7 + 1164041T^6 - 1626210T^5 + 18351630T^4 + 19586578T^3 + 343010805T^2 + 520428414*T + 1070532961
$59$	$T^{12} + \cdots + 368793616$ T^12 + 8T^11 + 246T^10 + 1288T^9 + 35896T^8 + 175776T^7 + 2878416T^6 + 9460784T^5 + 142475480T^4 + 417688768T^3 + 3444966512T^2 - 1101080544*T + 368793616
$61$	$T^{12} + \cdots + 1193840704$ T^12 + 16T^11 + 278T^10 + 1984T^9 + 21160T^8 + 107648T^7 + 1096216T^6 + 3681600T^5 + 26128736T^4 + 44261504T^3 + 404898272T^2 + 633545472*T + 1193840704
$67$	$T^{12} + \cdots + 397444096$ T^12 + 176T^10 - 704T^9 + 23168T^8 - 92160T^7 + 1537984T^6 - 7884800T^5 + 75106816T^4 - 249899008T^3 + 756862976T^2 - 602226688T + 397444096
$71$	$(T^{6} - 212 T^{4} + \cdots - 15842)^{2}$ (T^6 - 212T^4 - 136T^3 + 6754T^2 - 7476T - 15842)^2
$73$	$T^{12} + 14 T^{11} + \cdots + 1896129$ T^12 + 14T^11 + 357T^10 + 1922T^9 + 54102T^8 + 312950T^7 + 4439261T^6 - 214146T^5 + 13849384T^4 + 776358T^3 + 40011327T^2 - 8551170*T + 1896129
$79$	$T^{12} + 14 T^{11} + \cdots + 703921$ T^12 + 14T^11 + 229T^10 + 1178T^9 + 12546T^8 + 36246T^7 + 533861T^6 + 618174T^5 + 8033442T^4 - 1149870T^3 + 96648197T^2 - 8247370*T + 703921
$83$	$(T^{6} + 14 T^{5} + \cdots - 38089)^{2}$ (T^6 + 14T^5 - 139T^4 - 984T^3 + 7305T^2 + 1666*T - 38089)^2
$89$	$T^{12} + \cdots + 73501174321$ T^12 + 30T^11 + 865T^10 + 13842T^9 + 255950T^8 + 3420566T^7 + 47438833T^6 + 418929130T^5 + 3132356636T^4 + 11403741442T^3 + 32983093871T^2 - 34234270414*T + 73501174321
$97$	$(T^{6} - 22 T^{5} + \cdots - 233)^{2}$ (T^6 - 22T^5 + 11T^4 + 1732T^3 - 2317T^2 - 33526*T - 233)^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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1145.6
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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