LMFDB - Newform orbit 7500.2.a.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.a (trivial)

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:	yes
Analytic conductor:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$12$
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} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

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

$f(q)$	$=$	$ q + q^{3} + ( - \beta_{7} + 1) q^{7} + q^{9}+O(q^{10}) $ q + q^3 + (-b7 + 1) * q^7 + q^9	$ q + q^{3} + ( - \beta_{7} + 1) q^{7} + q^{9} + \beta_{8} q^{11} + \beta_{10} q^{13} + (\beta_{11} + \beta_{2} + 1) q^{17} + (\beta_{10} - \beta_{8} - \beta_{5} + \cdots + 1) q^{19}+ \cdots + \beta_{8} q^{99}+O(q^{100}) $ q + q^3 + (-b7 + 1) * q^7 + q^9 + b8 * q^11 + b10 * q^13 + (b11 + b2 + 1) * q^17 + (b10 - b8 - b5 + b2 + 1) * q^19 + (-b7 + 1) * q^21 + (b7 + b6 - b5 + 1) * q^23 + q^27 + (b3 - b2 + 1) * q^29 + (b11 + b10 - b9 - b8 - b7 + b6 - b3 + b2 - b1) * q^31 + b8 * q^33 + (-2b11 + b10 - b9 - b5 - b4 - b1 - 1) q^37 + b10 * q^39 + (b11 + b9 + b8 + b6 + b5 + b1 + 1) * q^41 + (-b11 - b10 + b9 + b8 - 2b6 + b5 + b4 + b3 - 2b2 + 3) * q^43 + (-b10 + b5 - b4 - b3 + 1) * q^47 + (-b11 - b10 + b9 - b8 - 2b7 - b6 + 3) q^49 + (b11 + b2 + 1) * q^51 + (b11 - b10 - b9 + b7 + 3b4 + b2 + b1 + 3) q^53 + (b10 - b8 - b5 + b2 + 1) * q^57 + (-b10 + 2b8 - b6 + 2b5 - 2b4 - b2 + 2b1 - 1) * q^59 + (-b10 - b7 - b6 - b3 - b2 + 3) * q^61 + (-b7 + 1) * q^63 + (b11 - b10 + b9 + b8 + b5 - b4 + b3 + 2b1 + 3) q^67 + (b7 + b6 - b5 + 1) * q^69 + (-2b11 - b10 - b8 - 2b7 - 2b6 - b5 + 3b4 + b3 - b2 - 2b1 + 2) q^71 + (-b10 - b7 + b4 - 2b3 - b1 + 2) q^73 + (-2b10 + 2b9 + 2b8 + b7 - b6 + b5 - 4b4 - 2b2 - 1) q^77 + (b11 - b9 + b6 + b5 + b2 + 2b1 + 1) q^79 + q^81 + (-b11 + b9 + b8 + 2b7 + b4 + b3 - b2 + 2) q^83 + (b3 - b2 + 1) * q^87 + (-2b9 - b8 - 2b7 + b6 - b5 - b4 - 2b3 + b2 - 2b1) * q^89 + (-2b11 + b10 + 2b9 - b8 + b7 - 2b6 + b5 + 2b4 + 3b3 - 2b2 - b1 + 3) * q^91 + (b11 + b10 - b9 - b8 - b7 + b6 - b3 + b2 - b1) * q^93 + (b11 + b10 + b8 + 2b7 + 2b6 + b5 - 3b4 - b3 + 2b1 - 1) * q^97 + b8 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6*x^11 - 11*x^10 + 94*x^9 + 27*x^8 - 460*x^7 + 55*x^6 + 812*x^5 - 127*x^4 - 512*x^3 + 40*x^2 + 72*x - 4 :

$\beta_{1}$	$=$	$ ( 1902023 \nu^{11} - 15346545 \nu^{10} + 3038089 \nu^{9} + 220227853 \nu^{8} - 329926105 \nu^{7} + \cdots - 427658178 ) / 168665204 $ (1902023v^11 - 15346545v^10 + 3038089v^9 + 220227853v^8 - 329926105v^7 - 909974979v^6 + 1944407257v^5 + 799087297v^4 - 3160553209v^3 + 477033175v^2 + 1283973348*v - 427658178) / 168665204
$\beta_{2}$	$=$	$ ( 573781 \nu^{11} + 249144 \nu^{10} - 30517201 \nu^{9} + 28884678 \nu^{8} + 364863104 \nu^{7} + \cdots + 193537713 ) / 42166301 $ (573781v^11 + 249144v^10 - 30517201v^9 + 28884678v^8 + 364863104v^7 - 383566399v^6 - 1429176716v^5 + 1551855632v^4 + 1644633434v^3 - 1558387780v^2 - 533575158*v + 193537713) / 42166301
$\beta_{3}$	$=$	$ ( 4446263 \nu^{11} - 31885257 \nu^{10} - 15534203 \nu^{9} + 460483725 \nu^{8} - 384922893 \nu^{7} + \cdots - 471915530 ) / 168665204 $ (4446263v^11 - 31885257v^10 - 15534203v^9 + 460483725v^8 - 384922893v^7 - 1955450995v^6 + 2622496817v^5 + 2115679805v^4 - 4596778413v^3 + 543616735v^2 + 2251682668*v - 471915530) / 168665204
$\beta_{4}$	$=$	$ ( - 391865 \nu^{11} + 2401909 \nu^{10} + 4018773 \nu^{9} - 37312825 \nu^{8} - 6589229 \nu^{7} + \cdots - 14339954 ) / 8877116 $ (-391865v^11 + 2401909v^10 + 4018773v^9 - 37312825v^8 - 6589229v^7 + 179797043v^6 - 35115263v^5 - 298851685v^4 + 57237739v^3 + 154217921v^2 - 9043168*v - 14339954) / 8877116
$\beta_{5}$	$=$	$ ( - 8420182 \nu^{11} + 49252135 \nu^{10} + 92384342 \nu^{9} - 744704695 \nu^{8} + \cdots - 334004182 ) / 168665204 $ (-8420182v^11 + 49252135v^10 + 92384342v^9 - 744704695v^8 - 195693826v^7 + 3339948103v^6 - 991891852v^5 - 4996541349v^4 + 2864832352v^3 + 2549576465v^2 - 2134099514*v - 334004182) / 168665204
$\beta_{6}$	$=$	$ ( - 8914978 \nu^{11} + 50577467 \nu^{10} + 111939570 \nu^{9} - 797245563 \nu^{8} + \cdots + 220528906 ) / 168665204 $ (-8914978v^11 + 50577467v^10 + 111939570v^9 - 797245563v^8 - 415423858v^7 + 3889973671v^6 - 89286696v^5 - 6961200305v^4 + 1445986192v^3 + 3891072669v^2 - 847833314*v + 220528906) / 168665204
$\beta_{7}$	$=$	$ ( - 2517509 \nu^{11} + 15676974 \nu^{10} + 25167924 \nu^{9} - 246751910 \nu^{8} - 30334377 \nu^{7} + \cdots - 138029335 ) / 42166301 $ (-2517509v^11 + 15676974v^10 + 25167924v^9 - 246751910v^8 - 30334377v^7 + 1223880325v^6 - 277813878v^5 - 2168660915v^4 + 455108743v^3 + 1335522859v^2 - 183251934*v - 138029335) / 42166301
$\beta_{8}$	$=$	$ ( - 5284275 \nu^{11} + 34049245 \nu^{10} + 40352913 \nu^{9} - 495464015 \nu^{8} + 90173559 \nu^{7} + \cdots + 357264474 ) / 84332602 $ (-5284275v^11 + 34049245v^10 + 40352913v^9 - 495464015v^8 + 90173559v^7 + 2100765259v^6 - 1072808513v^5 - 2447662093v^4 + 903666945v^3 + 283268561v^2 + 311995052*v + 357264474) / 84332602
$\beta_{9}$	$=$	$ ( 11430516 \nu^{11} - 55973135 \nu^{10} - 195330276 \nu^{9} + 909529415 \nu^{8} + 1380545088 \nu^{7} + \cdots + 843366226 ) / 168665204 $ (11430516v^11 - 55973135v^10 - 195330276v^9 + 909529415v^8 + 1380545088v^7 - 4533486639v^6 - 4245380734v^5 + 8480947257v^4 + 5630984674v^3 - 5809628005v^2 - 1986390078*v + 843366226) / 168665204
$\beta_{10}$	$=$	$ ( - 13569902 \nu^{11} + 91413839 \nu^{10} + 93467242 \nu^{9} - 1398305819 \nu^{8} + \cdots - 898531966 ) / 168665204 $ (-13569902v^11 + 91413839v^10 + 93467242v^9 - 1398305819v^8 + 463244558v^7 + 6739023411v^6 - 4315055896v^5 - 11511925689v^4 + 6443638716v^3 + 7278045201v^2 - 1706190394*v - 898531966) / 168665204
$\beta_{11}$	$=$	$ ( 26459466 \nu^{11} - 158726547 \nu^{10} - 270630514 \nu^{9} + 2376011335 \nu^{8} + \cdots + 260029058 ) / 168665204 $ (26459466v^11 - 158726547v^10 - 270630514v^9 + 2376011335v^8 + 452955598v^7 - 10563029603v^6 + 2595314028v^5 + 15054853845v^4 - 4362974516v^3 - 6271292845v^2 + 1093845846*v + 260029058) / 168665204

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

$\nu$	$=$	$ ( -\beta_{11} + 2\beta_{9} + \beta_{8} - 2\beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 + 2 ) / 5 $ (-b11 + 2b9 + b8 - 2b6 + b5 - b4 - 2*b2 + b1 + 2) / 5
$\nu^{2}$	$=$	$ ( - 4 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 2 \beta_{8} + \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + \cdots + 28 ) / 5 $ (-4b11 - 2b10 + 6b9 - 2b8 + b7 - 6b6 + 2b5 + 6b4 + b3 - 3b2 + 5*b1 + 28) / 5
$\nu^{3}$	$=$	$ ( - 14 \beta_{11} - 8 \beta_{10} + 32 \beta_{9} + 10 \beta_{8} + 14 \beta_{7} - 25 \beta_{6} + 17 \beta_{5} + \cdots + 48 ) / 5 $ (-14b11 - 8b10 + 32b9 + 10b8 + 14b7 - 25b6 + 17b5 - 9b4 - b3 - 23b2 + 28b1 + 48) / 5
$\nu^{4}$	$=$	$ ( - 74 \beta_{11} - 43 \beta_{10} + 126 \beta_{9} - 5 \beta_{8} + 54 \beta_{7} - 100 \beta_{6} + \cdots + 323 ) / 5 $ (-74b11 - 43b10 + 126b9 - 5b8 + 54b7 - 100b6 + 39b5 + 26b4 + 4b3 - 73b2 + 108*b1 + 323) / 5
$\nu^{5}$	$=$	$ ( - 312 \beta_{11} - 178 \beta_{10} + 574 \beta_{9} + 88 \beta_{8} + 324 \beta_{7} - 431 \beta_{6} + \cdots + 949 ) / 5 $ (-312b11 - 178b10 + 574b9 + 88b8 + 324b7 - 431b6 + 203b5 - 182b4 - 16b3 - 379b2 + 486*b1 + 949) / 5
$\nu^{6}$	$=$	$ ( - 1499 \beta_{11} - 814 \beta_{10} + 2379 \beta_{9} + 17 \beta_{8} + 1412 \beta_{7} - 1814 \beta_{6} + \cdots + 4733 ) / 5 $ (-1499b11 - 814b10 + 2379b9 + 17b8 + 1412b7 - 1814b6 + 540b5 - 559b4 - 68b3 - 1478b2 + 1961*b1 + 4733) / 5
$\nu^{7}$	$=$	$ ( - 6711 \beta_{11} - 3528 \beta_{10} + 10503 \beta_{9} + 627 \beta_{8} + 7104 \beta_{7} - 7924 \beta_{6} + \cdots + 17267 ) / 5 $ (-6711b11 - 3528b10 + 10503b9 + 627b8 + 7104b7 - 7924b6 + 2210b5 - 5136b4 - 601b3 - 6877b2 + 8405*b1 + 17267) / 5
$\nu^{8}$	$=$	$ ( - 31128 \beta_{11} - 15717 \beta_{10} + 44816 \beta_{9} - 298 \beta_{8} + 32406 \beta_{7} + \cdots + 77161 ) / 5 $ (-31128b11 - 15717b10 + 44816b9 - 298b8 + 32406b7 - 34139b6 + 5782b5 - 24183b4 - 3274b3 - 28976b2 + 35019*b1 + 77161) / 5
$\nu^{9}$	$=$	$ ( - 141469 \beta_{11} - 69421 \beta_{10} + 196849 \beta_{9} - 194 \beta_{8} + 154368 \beta_{7} + \cdots + 307868 ) / 5 $ (-141469b11 - 69421b10 + 196849b9 - 194b8 + 154368b7 - 149967b6 + 18264b5 - 136579b4 - 18267b3 - 130423b2 + 149322*b1 + 307868) / 5
$\nu^{10}$	$=$	$ ( - 648941 \beta_{11} - 309058 \beta_{10} + 855413 \beta_{9} - 30978 \beta_{8} + 711629 \beta_{7} + \cdots + 1329277 ) / 5 $ (-648941b11 - 309058b10 + 855413b9 - 30978b8 + 711629b7 - 655659b6 + 29115b5 - 665216b4 - 94011b3 - 567892b2 + 634165*b1 + 1329277) / 5
$\nu^{11}$	$=$	$ ( - 2958644 \beta_{11} - 1377117 \beta_{10} + 3769058 \beta_{9} - 169657 \beta_{8} + 3319976 \beta_{7} + \cdots + 5525588 ) / 5 $ (-2958644b11 - 1377117b10 + 3769058b9 - 169657b8 + 3319976b7 - 2896281b6 - 18872b5 - 3348094b4 - 474289b3 - 2536718b2 + 2722525*b1 + 5525588) / 5

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

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} $

1.1

4.54905
−2.34221
−1.20648
−0.446694
−0.879783
1.83182
0.434428
1.09160
3.89742
1.62661
0.0550688
−2.61083

1.00000

−4.13266

1.00000

1.2

1.00000

−3.54704

1.00000

1.3

1.00000

−1.57893

1.00000

1.4

1.00000

−1.31873

1.00000

1.5

1.00000

−0.957526

1.00000

1.6

1.00000

−0.595901

1.00000

1.7

1.00000

1.04684

1.00000

1.8

1.00000

2.44380

1.00000

1.9

1.00000

3.78808

1.00000

1.10

1.00000

3.80992

1.00000

1.11

1.00000

4.41540

1.00000

1.12

1.00000

4.62675

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$5$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7500.2.a.n		12
5.b	even	2	1		7500.2.a.m		12
5.c	odd	4	2		7500.2.d.g		24
25.d	even	5	2		1500.2.m.c		24
25.e	even	10	2		1500.2.m.d		24
25.f	odd	20	2		300.2.o.a	✓	24
25.f	odd	20	2		1500.2.o.c		24
75.l	even	20	2		900.2.w.c		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.2.o.a	✓	24	25.f	odd	20	2
900.2.w.c		24	75.l	even	20	2
1500.2.m.c		24	25.d	even	5	2
1500.2.m.d		24	25.e	even	10	2
1500.2.o.c		24	25.f	odd	20	2
7500.2.a.m		12	5.b	even	2	1
7500.2.a.n		12	1.a	even	1	1	trivial
7500.2.d.g		24	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} - 8 T_{7}^{11} - 24 T_{7}^{10} + 300 T_{7}^{9} + 10 T_{7}^{8} - 3768 T_{7}^{7} + 2289 T_{7}^{6} + \cdots + 13136 $ T7^12 - 8*T7^11 - 24*T7^10 + 300*T7^9 + 10*T7^8 - 3768*T7^7 + 2289*T7^6 + 19212*T7^5 - 7175*T7^4 - 44640*T7^3 - 6744*T7^2 + 30272*T7 + 13136 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7500))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T - 1)^{12} $ (T - 1)^12
$5$	$ T^{12} $ T^12
$7$	$ T^{12} - 8 T^{11} + \cdots + 13136 $ T^12 - 8T^11 - 24T^10 + 300T^9 + 10T^8 - 3768T^7 + 2289T^6 + 19212T^5 - 7175T^4 - 44640T^3 - 6744T^2 + 30272*T + 13136
$11$	$ T^{12} - 2 T^{11} + \cdots + 11216 $ T^12 - 2T^11 - 79T^10 + 130T^9 + 2180T^8 - 2922T^7 - 25266T^6 + 24138T^5 + 114645T^4 - 42780T^3 - 142804T^2 - 30832*T + 11216
$13$	$ T^{12} - 100 T^{10} + \cdots + 45125 $ T^12 - 100T^10 + 40T^9 + 3310T^8 - 2000T^7 - 42025T^6 + 21800T^5 + 187175T^4 - 41000T^3 - 260125T^2 - 45000T + 45125
$17$	$ T^{12} - 8 T^{11} + \cdots - 1075799 $ T^12 - 8T^11 - 99T^10 + 690T^9 + 4295T^8 - 20328T^7 - 101251T^6 + 194902T^5 + 1131890T^4 + 423720T^3 - 2618254T^2 - 3273818*T - 1075799
$19$	$ T^{12} - 10 T^{11} + \cdots + 50000 $ T^12 - 10T^11 - 85T^10 + 900T^9 + 2945T^8 - 28800T^7 - 63225T^6 + 383650T^5 + 897025T^4 - 1605500T^3 - 5362500T^2 - 3250000*T + 50000
$23$	$ T^{12} - 18 T^{11} + \cdots + 15235856 $ T^12 - 18T^11 + 11T^10 + 1530T^9 - 7740T^8 - 30698T^7 + 310054T^6 - 296908T^5 - 3222195T^4 + 9593360T^3 - 2800424T^2 - 17299328*T + 15235856
$29$	$ T^{12} - 8 T^{11} + \cdots - 8040059 $ T^12 - 8T^11 - 139T^10 + 950T^9 + 7155T^8 - 39458T^7 - 164201T^6 + 697152T^5 + 1526100T^4 - 5373620T^3 - 3651184T^2 + 15235032*T - 8040059
$31$	$ T^{12} + 2 T^{11} + \cdots + 5989136 $ T^12 + 2T^11 - 199T^10 - 40T^9 + 13690T^8 - 19738T^7 - 372156T^6 + 929412T^5 + 3661485T^4 - 12022460T^3 - 6230964T^2 + 29646192*T + 5989136
$37$	$ T^{12} - 4 T^{11} + \cdots + 69436261 $ T^12 - 4T^11 - 321T^10 + 1280T^9 + 36735T^8 - 151344T^7 - 1806129T^6 + 7893624T^5 + 33050690T^4 - 158406220T^3 - 36360416T^2 + 318432924*T + 69436261
$41$	$ T^{12} - 10 T^{11} + \cdots + 59575625 $ T^12 - 10T^11 - 205T^10 + 1950T^9 + 14595T^8 - 129200T^7 - 442450T^6 + 3374300T^5 + 5689275T^4 - 28208750T^3 - 35081625T^2 + 49848750*T + 59575625
$43$	$ T^{12} - 28 T^{11} + \cdots - 31628144 $ T^12 - 28T^11 + 86T^10 + 4120T^9 - 38345T^8 - 78528T^7 + 1986774T^6 - 3669708T^5 - 24491855T^4 + 56283880T^3 + 91425996T^2 - 50062848*T - 31628144
$47$	$ T^{12} - 22 T^{11} + \cdots + 834896 $ T^12 - 22T^11 + 26T^10 + 2580T^9 - 17420T^8 - 43222T^7 + 670449T^6 - 849772T^5 - 6770555T^4 + 17246320T^3 - 222004T^2 - 4201152*T + 834896
$53$	$ T^{12} + \cdots + 300763681 $ T^12 - 16T^11 - 271T^10 + 5310T^9 + 12895T^8 - 500156T^7 + 489901T^6 + 16126786T^5 - 18363930T^4 - 223936660T^3 + 70959334T^2 + 997363266*T + 300763681
$59$	$ T^{12} + \cdots - 866673664 $ T^12 + 2T^11 - 409T^10 - 330T^9 + 60555T^8 - 32078T^7 - 3856241T^6 + 7718152T^5 + 92907945T^4 - 323991720T^3 - 189582304T^2 + 1444159232*T - 866673664
$61$	$ T^{12} + \cdots + 4582416901 $ T^12 - 34T^11 + 199T^10 + 5030T^9 - 67205T^8 - 23104T^7 + 4016886T^6 - 14093136T^5 - 66403385T^4 + 418684210T^3 - 44054021T^2 - 3075579686*T + 4582416901
$67$	$ T^{12} + \cdots + 4750542736 $ T^12 - 32T^11 + 126T^10 + 5300T^9 - 49565T^8 - 237532T^7 + 3912794T^6 - 1759412T^5 - 111128475T^4 + 300987640T^3 + 841227356T^2 - 4379425312*T + 4750542736
$71$	$ T^{12} + \cdots + 5618722000 $ T^12 - 510T^10 - 290T^9 + 95440T^8 + 96250T^7 - 8046125T^6 - 10648150T^5 + 306792325T^4 + 511772500T^3 - 4440858500T^2 - 8944126000T + 5618722000
$73$	$ T^{12} + \cdots - 540439819 $ T^12 - 24T^11 - 191T^10 + 7760T^9 - 11870T^8 - 752584T^7 + 3307261T^6 + 23505904T^5 - 136209670T^4 - 189858360T^3 + 1586430889T^2 - 867360576*T - 540439819
$79$	$ T^{12} + \cdots + 2241715456 $ T^12 - 6T^11 - 411T^10 + 980T^9 + 66450T^8 + 87294T^7 - 4670604T^6 - 22694744T^5 + 80347385T^4 + 960597600T^3 + 3113091984T^2 + 4381631616*T + 2241715456
$83$	$ T^{12} + \cdots - 354151984 $ T^12 - 28T^11 + 56T^10 + 4120T^9 - 23930T^8 - 196508T^7 + 1385544T^6 + 4032672T^5 - 29541855T^4 - 32986080T^3 + 232385016T^2 + 47196672*T - 354151984
$89$	$ T^{12} + \cdots - 6367139875 $ T^12 - 10T^11 - 480T^10 + 4110T^9 + 79100T^8 - 436200T^7 - 6514225T^6 + 11314600T^5 + 249288875T^4 + 345801500T^3 - 2432776875T^2 - 7826798500*T - 6367139875
$97$	$ T^{12} + \cdots - 4393476419 $ T^12 - 16T^11 - 521T^10 + 9280T^9 + 69815T^8 - 1584216T^7 - 1772814T^6 + 98833736T^5 - 61934765T^4 - 2548037720T^3 + 2072387039T^2 + 23755582136*T - 4393476419
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\( q + q^{3} + ( - \beta_{7} + 1) q^{7} + q^{9}+O(q^{10}) \) q + q^3 + (-b7 + 1) * q^7 + q^9	\( q + q^{3} + ( - \beta_{7} + 1) q^{7} + q^{9} + \beta_{8} q^{11} + \beta_{10} q^{13} + (\beta_{11} + \beta_{2} + 1) q^{17} + (\beta_{10} - \beta_{8} - \beta_{5} + \cdots + 1) q^{19}+ \cdots + \beta_{8} q^{99}+O(q^{100}) \) q + q^3 + (-b7 + 1) * q^7 + q^9 + b8 * q^11 + b10 * q^13 + (b11 + b2 + 1) * q^17 + (b10 - b8 - b5 + b2 + 1) * q^19 + (-b7 + 1) * q^21 + (b7 + b6 - b5 + 1) * q^23 + q^27 + (b3 - b2 + 1) * q^29 + (b11 + b10 - b9 - b8 - b7 + b6 - b3 + b2 - b1) * q^31 + b8 * q^33 + (-2b11 + b10 - b9 - b5 - b4 - b1 - 1) q^37 + b10 * q^39 + (b11 + b9 + b8 + b6 + b5 + b1 + 1) * q^41 + (-b11 - b10 + b9 + b8 - 2b6 + b5 + b4 + b3 - 2b2 + 3) * q^43 + (-b10 + b5 - b4 - b3 + 1) * q^47 + (-b11 - b10 + b9 - b8 - 2b7 - b6 + 3) q^49 + (b11 + b2 + 1) * q^51 + (b11 - b10 - b9 + b7 + 3b4 + b2 + b1 + 3) q^53 + (b10 - b8 - b5 + b2 + 1) * q^57 + (-b10 + 2b8 - b6 + 2b5 - 2b4 - b2 + 2b1 - 1) * q^59 + (-b10 - b7 - b6 - b3 - b2 + 3) * q^61 + (-b7 + 1) * q^63 + (b11 - b10 + b9 + b8 + b5 - b4 + b3 + 2b1 + 3) q^67 + (b7 + b6 - b5 + 1) * q^69 + (-2b11 - b10 - b8 - 2b7 - 2b6 - b5 + 3b4 + b3 - b2 - 2b1 + 2) q^71 + (-b10 - b7 + b4 - 2b3 - b1 + 2) q^73 + (-2b10 + 2b9 + 2b8 + b7 - b6 + b5 - 4b4 - 2b2 - 1) q^77 + (b11 - b9 + b6 + b5 + b2 + 2b1 + 1) q^79 + q^81 + (-b11 + b9 + b8 + 2b7 + b4 + b3 - b2 + 2) q^83 + (b3 - b2 + 1) * q^87 + (-2b9 - b8 - 2b7 + b6 - b5 - b4 - 2b3 + b2 - 2b1) * q^89 + (-2b11 + b10 + 2b9 - b8 + b7 - 2b6 + b5 + 2b4 + 3b3 - 2b2 - b1 + 3) * q^91 + (b11 + b10 - b9 - b8 - b7 + b6 - b3 + b2 - b1) * q^93 + (b11 + b10 + b8 + 2b7 + 2b6 + b5 - 3b4 - b3 + 2b1 - 1) * q^97 + b8 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7500.2.a.n

Level	$7500$
Weight	$2$
Character orbit	7500.a
Self dual	yes
Analytic conductor	$59.888$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(59.8878015160\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 \) x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 5^{3} \)
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( ( 1902023 \nu^{11} - 15346545 \nu^{10} + 3038089 \nu^{9} + 220227853 \nu^{8} - 329926105 \nu^{7} + \cdots - 427658178 ) / 168665204 \) (1902023v^11 - 15346545v^10 + 3038089v^9 + 220227853v^8 - 329926105v^7 - 909974979v^6 + 1944407257v^5 + 799087297v^4 - 3160553209v^3 + 477033175v^2 + 1283973348*v - 427658178) / 168665204
\(\beta_{2}\)	\(=\)	\( ( 573781 \nu^{11} + 249144 \nu^{10} - 30517201 \nu^{9} + 28884678 \nu^{8} + 364863104 \nu^{7} + \cdots + 193537713 ) / 42166301 \) (573781v^11 + 249144v^10 - 30517201v^9 + 28884678v^8 + 364863104v^7 - 383566399v^6 - 1429176716v^5 + 1551855632v^4 + 1644633434v^3 - 1558387780v^2 - 533575158*v + 193537713) / 42166301
\(\beta_{3}\)	\(=\)	\( ( 4446263 \nu^{11} - 31885257 \nu^{10} - 15534203 \nu^{9} + 460483725 \nu^{8} - 384922893 \nu^{7} + \cdots - 471915530 ) / 168665204 \) (4446263v^11 - 31885257v^10 - 15534203v^9 + 460483725v^8 - 384922893v^7 - 1955450995v^6 + 2622496817v^5 + 2115679805v^4 - 4596778413v^3 + 543616735v^2 + 2251682668*v - 471915530) / 168665204
\(\beta_{4}\)	\(=\)	\( ( - 391865 \nu^{11} + 2401909 \nu^{10} + 4018773 \nu^{9} - 37312825 \nu^{8} - 6589229 \nu^{7} + \cdots - 14339954 ) / 8877116 \) (-391865v^11 + 2401909v^10 + 4018773v^9 - 37312825v^8 - 6589229v^7 + 179797043v^6 - 35115263v^5 - 298851685v^4 + 57237739v^3 + 154217921v^2 - 9043168*v - 14339954) / 8877116
\(\beta_{5}\)	\(=\)	\( ( - 8420182 \nu^{11} + 49252135 \nu^{10} + 92384342 \nu^{9} - 744704695 \nu^{8} + \cdots - 334004182 ) / 168665204 \) (-8420182v^11 + 49252135v^10 + 92384342v^9 - 744704695v^8 - 195693826v^7 + 3339948103v^6 - 991891852v^5 - 4996541349v^4 + 2864832352v^3 + 2549576465v^2 - 2134099514*v - 334004182) / 168665204
\(\beta_{6}\)	\(=\)	\( ( - 8914978 \nu^{11} + 50577467 \nu^{10} + 111939570 \nu^{9} - 797245563 \nu^{8} + \cdots + 220528906 ) / 168665204 \) (-8914978v^11 + 50577467v^10 + 111939570v^9 - 797245563v^8 - 415423858v^7 + 3889973671v^6 - 89286696v^5 - 6961200305v^4 + 1445986192v^3 + 3891072669v^2 - 847833314*v + 220528906) / 168665204
\(\beta_{7}\)	\(=\)	\( ( - 2517509 \nu^{11} + 15676974 \nu^{10} + 25167924 \nu^{9} - 246751910 \nu^{8} - 30334377 \nu^{7} + \cdots - 138029335 ) / 42166301 \) (-2517509v^11 + 15676974v^10 + 25167924v^9 - 246751910v^8 - 30334377v^7 + 1223880325v^6 - 277813878v^5 - 2168660915v^4 + 455108743v^3 + 1335522859v^2 - 183251934*v - 138029335) / 42166301
\(\beta_{8}\)	\(=\)	\( ( - 5284275 \nu^{11} + 34049245 \nu^{10} + 40352913 \nu^{9} - 495464015 \nu^{8} + 90173559 \nu^{7} + \cdots + 357264474 ) / 84332602 \) (-5284275v^11 + 34049245v^10 + 40352913v^9 - 495464015v^8 + 90173559v^7 + 2100765259v^6 - 1072808513v^5 - 2447662093v^4 + 903666945v^3 + 283268561v^2 + 311995052*v + 357264474) / 84332602
\(\beta_{9}\)	\(=\)	\( ( 11430516 \nu^{11} - 55973135 \nu^{10} - 195330276 \nu^{9} + 909529415 \nu^{8} + 1380545088 \nu^{7} + \cdots + 843366226 ) / 168665204 \) (11430516v^11 - 55973135v^10 - 195330276v^9 + 909529415v^8 + 1380545088v^7 - 4533486639v^6 - 4245380734v^5 + 8480947257v^4 + 5630984674v^3 - 5809628005v^2 - 1986390078*v + 843366226) / 168665204
\(\beta_{10}\)	\(=\)	\( ( - 13569902 \nu^{11} + 91413839 \nu^{10} + 93467242 \nu^{9} - 1398305819 \nu^{8} + \cdots - 898531966 ) / 168665204 \) (-13569902v^11 + 91413839v^10 + 93467242v^9 - 1398305819v^8 + 463244558v^7 + 6739023411v^6 - 4315055896v^5 - 11511925689v^4 + 6443638716v^3 + 7278045201v^2 - 1706190394*v - 898531966) / 168665204
\(\beta_{11}\)	\(=\)	\( ( 26459466 \nu^{11} - 158726547 \nu^{10} - 270630514 \nu^{9} + 2376011335 \nu^{8} + \cdots + 260029058 ) / 168665204 \) (26459466v^11 - 158726547v^10 - 270630514v^9 + 2376011335v^8 + 452955598v^7 - 10563029603v^6 + 2595314028v^5 + 15054853845v^4 - 4362974516v^3 - 6271292845v^2 + 1093845846*v + 260029058) / 168665204

\(\nu\)	\(=\)	\( ( -\beta_{11} + 2\beta_{9} + \beta_{8} - 2\beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 + 2 ) / 5 \) (-b11 + 2b9 + b8 - 2b6 + b5 - b4 - 2*b2 + b1 + 2) / 5
\(\nu^{2}\)	\(=\)	\( ( - 4 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 2 \beta_{8} + \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + \cdots + 28 ) / 5 \) (-4b11 - 2b10 + 6b9 - 2b8 + b7 - 6b6 + 2b5 + 6b4 + b3 - 3b2 + 5*b1 + 28) / 5
\(\nu^{3}\)	\(=\)	\( ( - 14 \beta_{11} - 8 \beta_{10} + 32 \beta_{9} + 10 \beta_{8} + 14 \beta_{7} - 25 \beta_{6} + 17 \beta_{5} + \cdots + 48 ) / 5 \) (-14b11 - 8b10 + 32b9 + 10b8 + 14b7 - 25b6 + 17b5 - 9b4 - b3 - 23b2 + 28b1 + 48) / 5
\(\nu^{4}\)	\(=\)	\( ( - 74 \beta_{11} - 43 \beta_{10} + 126 \beta_{9} - 5 \beta_{8} + 54 \beta_{7} - 100 \beta_{6} + \cdots + 323 ) / 5 \) (-74b11 - 43b10 + 126b9 - 5b8 + 54b7 - 100b6 + 39b5 + 26b4 + 4b3 - 73b2 + 108*b1 + 323) / 5
\(\nu^{5}\)	\(=\)	\( ( - 312 \beta_{11} - 178 \beta_{10} + 574 \beta_{9} + 88 \beta_{8} + 324 \beta_{7} - 431 \beta_{6} + \cdots + 949 ) / 5 \) (-312b11 - 178b10 + 574b9 + 88b8 + 324b7 - 431b6 + 203b5 - 182b4 - 16b3 - 379b2 + 486*b1 + 949) / 5
\(\nu^{6}\)	\(=\)	\( ( - 1499 \beta_{11} - 814 \beta_{10} + 2379 \beta_{9} + 17 \beta_{8} + 1412 \beta_{7} - 1814 \beta_{6} + \cdots + 4733 ) / 5 \) (-1499b11 - 814b10 + 2379b9 + 17b8 + 1412b7 - 1814b6 + 540b5 - 559b4 - 68b3 - 1478b2 + 1961*b1 + 4733) / 5
\(\nu^{7}\)	\(=\)	\( ( - 6711 \beta_{11} - 3528 \beta_{10} + 10503 \beta_{9} + 627 \beta_{8} + 7104 \beta_{7} - 7924 \beta_{6} + \cdots + 17267 ) / 5 \) (-6711b11 - 3528b10 + 10503b9 + 627b8 + 7104b7 - 7924b6 + 2210b5 - 5136b4 - 601b3 - 6877b2 + 8405*b1 + 17267) / 5
\(\nu^{8}\)	\(=\)	\( ( - 31128 \beta_{11} - 15717 \beta_{10} + 44816 \beta_{9} - 298 \beta_{8} + 32406 \beta_{7} + \cdots + 77161 ) / 5 \) (-31128b11 - 15717b10 + 44816b9 - 298b8 + 32406b7 - 34139b6 + 5782b5 - 24183b4 - 3274b3 - 28976b2 + 35019*b1 + 77161) / 5
\(\nu^{9}\)	\(=\)	\( ( - 141469 \beta_{11} - 69421 \beta_{10} + 196849 \beta_{9} - 194 \beta_{8} + 154368 \beta_{7} + \cdots + 307868 ) / 5 \) (-141469b11 - 69421b10 + 196849b9 - 194b8 + 154368b7 - 149967b6 + 18264b5 - 136579b4 - 18267b3 - 130423b2 + 149322*b1 + 307868) / 5
\(\nu^{10}\)	\(=\)	\( ( - 648941 \beta_{11} - 309058 \beta_{10} + 855413 \beta_{9} - 30978 \beta_{8} + 711629 \beta_{7} + \cdots + 1329277 ) / 5 \) (-648941b11 - 309058b10 + 855413b9 - 30978b8 + 711629b7 - 655659b6 + 29115b5 - 665216b4 - 94011b3 - 567892b2 + 634165*b1 + 1329277) / 5
\(\nu^{11}\)	\(=\)	\( ( - 2958644 \beta_{11} - 1377117 \beta_{10} + 3769058 \beta_{9} - 169657 \beta_{8} + 3319976 \beta_{7} + \cdots + 5525588 ) / 5 \) (-2958644b11 - 1377117b10 + 3769058b9 - 169657b8 + 3319976b7 - 2896281b6 - 18872b5 - 3348094b4 - 474289b3 - 2536718b2 + 2722525*b1 + 5525588) / 5

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T - 1)^{12} \) (T - 1)^12
$5$	\( T^{12} \) T^12
$7$	\( T^{12} - 8 T^{11} + \cdots + 13136 \) T^12 - 8T^11 - 24T^10 + 300T^9 + 10T^8 - 3768T^7 + 2289T^6 + 19212T^5 - 7175T^4 - 44640T^3 - 6744T^2 + 30272*T + 13136
$11$	\( T^{12} - 2 T^{11} + \cdots + 11216 \) T^12 - 2T^11 - 79T^10 + 130T^9 + 2180T^8 - 2922T^7 - 25266T^6 + 24138T^5 + 114645T^4 - 42780T^3 - 142804T^2 - 30832*T + 11216
$13$	\( T^{12} - 100 T^{10} + \cdots + 45125 \) T^12 - 100T^10 + 40T^9 + 3310T^8 - 2000T^7 - 42025T^6 + 21800T^5 + 187175T^4 - 41000T^3 - 260125T^2 - 45000T + 45125
$17$	\( T^{12} - 8 T^{11} + \cdots - 1075799 \) T^12 - 8T^11 - 99T^10 + 690T^9 + 4295T^8 - 20328T^7 - 101251T^6 + 194902T^5 + 1131890T^4 + 423720T^3 - 2618254T^2 - 3273818*T - 1075799
$19$	\( T^{12} - 10 T^{11} + \cdots + 50000 \) T^12 - 10T^11 - 85T^10 + 900T^9 + 2945T^8 - 28800T^7 - 63225T^6 + 383650T^5 + 897025T^4 - 1605500T^3 - 5362500T^2 - 3250000*T + 50000
$23$	\( T^{12} - 18 T^{11} + \cdots + 15235856 \) T^12 - 18T^11 + 11T^10 + 1530T^9 - 7740T^8 - 30698T^7 + 310054T^6 - 296908T^5 - 3222195T^4 + 9593360T^3 - 2800424T^2 - 17299328*T + 15235856
$29$	\( T^{12} - 8 T^{11} + \cdots - 8040059 \) T^12 - 8T^11 - 139T^10 + 950T^9 + 7155T^8 - 39458T^7 - 164201T^6 + 697152T^5 + 1526100T^4 - 5373620T^3 - 3651184T^2 + 15235032*T - 8040059
$31$	\( T^{12} + 2 T^{11} + \cdots + 5989136 \) T^12 + 2T^11 - 199T^10 - 40T^9 + 13690T^8 - 19738T^7 - 372156T^6 + 929412T^5 + 3661485T^4 - 12022460T^3 - 6230964T^2 + 29646192*T + 5989136
$37$	\( T^{12} - 4 T^{11} + \cdots + 69436261 \) T^12 - 4T^11 - 321T^10 + 1280T^9 + 36735T^8 - 151344T^7 - 1806129T^6 + 7893624T^5 + 33050690T^4 - 158406220T^3 - 36360416T^2 + 318432924*T + 69436261
$41$	\( T^{12} - 10 T^{11} + \cdots + 59575625 \) T^12 - 10T^11 - 205T^10 + 1950T^9 + 14595T^8 - 129200T^7 - 442450T^6 + 3374300T^5 + 5689275T^4 - 28208750T^3 - 35081625T^2 + 49848750*T + 59575625
$43$	\( T^{12} - 28 T^{11} + \cdots - 31628144 \) T^12 - 28T^11 + 86T^10 + 4120T^9 - 38345T^8 - 78528T^7 + 1986774T^6 - 3669708T^5 - 24491855T^4 + 56283880T^3 + 91425996T^2 - 50062848*T - 31628144
$47$	\( T^{12} - 22 T^{11} + \cdots + 834896 \) T^12 - 22T^11 + 26T^10 + 2580T^9 - 17420T^8 - 43222T^7 + 670449T^6 - 849772T^5 - 6770555T^4 + 17246320T^3 - 222004T^2 - 4201152*T + 834896
$53$	\( T^{12} + \cdots + 300763681 \) T^12 - 16T^11 - 271T^10 + 5310T^9 + 12895T^8 - 500156T^7 + 489901T^6 + 16126786T^5 - 18363930T^4 - 223936660T^3 + 70959334T^2 + 997363266*T + 300763681
$59$	\( T^{12} + \cdots - 866673664 \) T^12 + 2T^11 - 409T^10 - 330T^9 + 60555T^8 - 32078T^7 - 3856241T^6 + 7718152T^5 + 92907945T^4 - 323991720T^3 - 189582304T^2 + 1444159232*T - 866673664
$61$	\( T^{12} + \cdots + 4582416901 \) T^12 - 34T^11 + 199T^10 + 5030T^9 - 67205T^8 - 23104T^7 + 4016886T^6 - 14093136T^5 - 66403385T^4 + 418684210T^3 - 44054021T^2 - 3075579686*T + 4582416901
$67$	\( T^{12} + \cdots + 4750542736 \) T^12 - 32T^11 + 126T^10 + 5300T^9 - 49565T^8 - 237532T^7 + 3912794T^6 - 1759412T^5 - 111128475T^4 + 300987640T^3 + 841227356T^2 - 4379425312*T + 4750542736
$71$	\( T^{12} + \cdots + 5618722000 \) T^12 - 510T^10 - 290T^9 + 95440T^8 + 96250T^7 - 8046125T^6 - 10648150T^5 + 306792325T^4 + 511772500T^3 - 4440858500T^2 - 8944126000T + 5618722000
$73$	\( T^{12} + \cdots - 540439819 \) T^12 - 24T^11 - 191T^10 + 7760T^9 - 11870T^8 - 752584T^7 + 3307261T^6 + 23505904T^5 - 136209670T^4 - 189858360T^3 + 1586430889T^2 - 867360576*T - 540439819
$79$	\( T^{12} + \cdots + 2241715456 \) T^12 - 6T^11 - 411T^10 + 980T^9 + 66450T^8 + 87294T^7 - 4670604T^6 - 22694744T^5 + 80347385T^4 + 960597600T^3 + 3113091984T^2 + 4381631616*T + 2241715456
$83$	\( T^{12} + \cdots - 354151984 \) T^12 - 28T^11 + 56T^10 + 4120T^9 - 23930T^8 - 196508T^7 + 1385544T^6 + 4032672T^5 - 29541855T^4 - 32986080T^3 + 232385016T^2 + 47196672*T - 354151984
$89$	\( T^{12} + \cdots - 6367139875 \) T^12 - 10T^11 - 480T^10 + 4110T^9 + 79100T^8 - 436200T^7 - 6514225T^6 + 11314600T^5 + 249288875T^4 + 345801500T^3 - 2432776875T^2 - 7826798500*T - 6367139875
$97$	\( T^{12} + \cdots - 4393476419 \) T^12 - 16T^11 - 521T^10 + 9280T^9 + 69815T^8 - 1584216T^7 - 1772814T^6 + 98833736T^5 - 61934765T^4 - 2548037720T^3 + 2072387039T^2 + 23755582136*T - 4393476419
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