LMFDB - Newform orbit 3900.2.cd.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3900,2,Mod(901,3900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("3900.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.cd (of order $6$, degree $2$, 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:	$31.1416567883$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
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} + 26x^{10} + 239x^{8} + 924x^{6} + 1407x^{4} + 538x^{2} + 49 $ x^12 + 26x^10 + 239x^8 + 924x^6 + 1407x^4 + 538*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 780)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{6} + 1) q^{3} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{7}+ \cdots - \beta_{6} q^{9}+O(q^{10}) $ q + (-b6 + 1) * q^3 + (-b11 - b10 + b9 - b6 + b5 - b2 + 1) * q^7 - b6 * q^9	$ q + ( - \beta_{6} + 1) q^{3} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) $ q + (-b6 + 1) * q^3 + (-b11 - b10 + b9 - b6 + b5 - b2 + 1) * q^7 - b6 * q^9 + (-b11 + b10 + b8) * q^11 + (-b10 + b3) * q^13 + (b10 + b9 - 2b8 + b7 + 2b6 - b5 + b4 - b3 - b2 + 2b1) q^17 + (b6 - b5 - 1) * q^19 + (-b11 - b10 + b8 - 2b6 + b5 - b4 - b1) q^21 + (-b9 - b8 + b7 - b5 + b4 - b2 + 1) * q^23 - q^27 + (b11 + b10 - b7 + b6 - b5 + 2b2 + b1) q^29 + (-b10 - b9 - 2b8 + 2b7 - b5 - b3 - b2 + b1) * q^31 + (b10 - b9 - b5 + b4 + 1) * q^33 + (-b11 + b10 + b9 + b7 - b5 + b4 - 2b3 - b2 + 2b1 - 1) * q^37 + (-b10 - b8 + b7 + b6 - b4 - b2 + b1 - 1) * q^39 + (-b11 + b10 - 3b8 + b7 + 3b6 - b5 - 2b3 - b2 + 2b1 + 2) * q^41 + (b11 - 2b8 + 4b6 - b5 + b4 - b2 + 2b1 + 1) q^43 + (-2b11 - b10 + b9 - 2b4 - b1) * q^47 + (-2b11 - 2b10 + b8 - 2b7 - 3b6 - 2b1 + 3) q^49 + (-b11 + b9 - b8 + b4 - b3 - b2 + b1 + 1) * q^51 + (2b8 + 2b3 + 2b2 - 2b1) * q^53 + (-b8 + 2b6 - b5) q^57 + (2b11 + 3b10 - 3b9 + b7 + 3b6 - 2b5 + b4 + b3 + 3b2 - 3) * q^59 + (b11 - 4b8 + 3b7 + 6b6 - 2b5 + b4 - 3b3 - 3b2 + 6b1 - 1) q^61 + (-b9 + b8 - b6 - b4 + b2 - b1 - 1) * q^63 + (-2b9 - b8 + b7 - b5 - 2b4 - 2b3 + 4b2 - 3b1 - 1) q^67 + (b11 - b10 - b9 - 2b8 + b7 - b5 - b3) q^69 + (-b7 - b6 - b3 + 4b2 + 1) q^71 + (-b11 + b9 + 3b8 - 2b7 + 2b6 + 2b5 - b4 + b3 + b2 - b1 - 2) * q^73 + (-b11 - b10 + 2b9 + b8 + b4 + b3 - b2 - 1) q^77 + (-2b11 + 2b9 - 3b8 + b5 + 2b4 - 2b3 - 2b2 + 2b1 + 1) q^79 + (b6 - 1) * q^81 + (-b11 + b10 + 2b9 + 2b7 + b5 - b4 - b3 - b2 + 3b1 - 2) q^83 + (b10 + b9 - b7 + b6 + b4 + b3 - b2 + 2b1 + 1) q^87 + (-2b9 + b8 + b7 - b6 - b5 - 2b4 - 2b3 + b1 - 2) q^89 + (b11 - 4b9 - 2b8 + b7 - 2b6 - 4b5 - b4 - b3 + 3b2 - 2b1 + 2) * q^91 + (b11 - b10 - b9 - 2b8 + b7 - b5 - b4 - 2b3 + b1 - 1) * q^93 + (b11 - 2b6 - b5 - b4 + 3b2 + 5) * q^97 + (b11 - b9 - b8 - b5 + b4 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9}+O(q^{10}) $ 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9	$ 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9} + 6 q^{11} - 8 q^{13} + 2 q^{17} - 12 q^{19} + 4 q^{23} - 12 q^{27} - 2 q^{29} + 6 q^{33} - 6 q^{37} - 4 q^{39} + 36 q^{41} + 16 q^{43} + 4 q^{49} + 4 q^{51} - 18 q^{59} + 22 q^{61} - 6 q^{63} + 6 q^{67} - 4 q^{69} + 6 q^{71} - 24 q^{77} - 4 q^{79} - 6 q^{81} + 2 q^{87} + 2 q^{91} - 6 q^{93} + 48 q^{97}+O(q^{100}) $ 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9 + 6 * q^11 - 8 * q^13 + 2 * q^17 - 12 * q^19 + 4 * q^23 - 12 * q^27 - 2 * q^29 + 6 * q^33 - 6 * q^37 - 4 * q^39 + 36 * q^41 + 16 * q^43 + 4 * q^49 + 4 * q^51 - 18 * q^59 + 22 * q^61 - 6 * q^63 + 6 * q^67 - 4 * q^69 + 6 * q^71 - 24 * q^77 - 4 * q^79 - 6 * q^81 + 2 * q^87 + 2 * q^91 - 6 * q^93 + 48 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 26x^{10} + 239x^{8} + 924x^{6} + 1407x^{4} + 538x^{2} + 49 $ x^12 + 26*x^10 + 239*x^8 + 924*x^6 + 1407*x^4 + 538*x^2 + 49 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{6} + 13\nu^{4} + 35\nu^{2} + 4\nu + 7 ) / 8 $ (v^6 + 13v^4 + 35v^2 + 4*v + 7) / 8
$\beta_{3}$	$=$	$ ( - \nu^{11} + \nu^{10} - 25 \nu^{9} + 21 \nu^{8} - 218 \nu^{7} + 134 \nu^{6} - 790 \nu^{5} + \cdots - 287 ) / 112 $ (-v^11 + v^10 - 25v^9 + 21v^8 - 218v^7 + 134v^6 - 790v^5 + 226v^4 - 1181v^3 - 143v^2 - 569*v - 287) / 112
$\beta_{4}$	$=$	$ ( - 3 \nu^{10} - 2 \nu^{9} - 77 \nu^{8} - 42 \nu^{7} - 682 \nu^{6} - 282 \nu^{5} - 2414 \nu^{4} + \cdots - 469 ) / 112 $ (-3v^10 - 2v^9 - 77v^8 - 42v^7 - 682v^6 - 282v^5 - 2414v^4 - 606v^3 - 2987v^2 - 92v - 469) / 112
$\beta_{5}$	$=$	$ ( \nu^{11} + 3 \nu^{10} + 25 \nu^{9} + 77 \nu^{8} + 218 \nu^{7} + 696 \nu^{6} + 790 \nu^{5} + 2624 \nu^{4} + \cdots + 875 ) / 112 $ (v^11 + 3v^10 + 25v^9 + 77v^8 + 218v^7 + 696v^6 + 790v^5 + 2624v^4 + 1181v^3 + 3757v^2 + 625v + 875) / 112
$\beta_{6}$	$=$	$ ( -\nu^{11} - 26\nu^{9} - 239\nu^{7} - 917\nu^{5} - 1316\nu^{3} - 293\nu + 28 ) / 56 $ (-v^11 - 26v^9 - 239v^7 - 917v^5 - 1316v^3 - 293*v + 28) / 56
$\beta_{7}$	$=$	$ ( - 2 \nu^{11} - \nu^{10} - 53 \nu^{9} - 28 \nu^{8} - 492 \nu^{7} - 274 \nu^{6} - 1870 \nu^{5} + \cdots - 420 ) / 112 $ (-2v^11 - v^10 - 53v^9 - 28v^8 - 492v^7 - 274v^6 - 1870v^5 - 1108v^4 - 2550v^3 - 1705v^2 - 457v - 420) / 112
$\beta_{8}$	$=$	$ ( \nu^{11} - 3 \nu^{10} + 25 \nu^{9} - 77 \nu^{8} + 218 \nu^{7} - 696 \nu^{6} + 790 \nu^{5} - 2624 \nu^{4} + \cdots - 763 ) / 112 $ (v^11 - 3v^10 + 25v^9 - 77v^8 + 218v^7 - 696v^6 + 790v^5 - 2624v^4 + 1181v^3 - 3757v^2 + 625v - 763) / 112
$\beta_{9}$	$=$	$ ( - 2 \nu^{11} - \nu^{10} - 49 \nu^{9} - 28 \nu^{8} - 408 \nu^{7} - 274 \nu^{6} - 1306 \nu^{5} + \cdots - 252 ) / 112 $ (-2v^11 - v^10 - 49v^9 - 28v^8 - 408v^7 - 274v^6 - 1306v^5 - 1108v^4 - 1282v^3 - 1649v^2 + 119v - 252) / 112
$\beta_{10}$	$=$	$ ( - 2 \nu^{11} + \nu^{10} - 49 \nu^{9} + 28 \nu^{8} - 408 \nu^{7} + 274 \nu^{6} - 1306 \nu^{5} + \cdots + 252 ) / 112 $ (-2v^11 + v^10 - 49v^9 + 28v^8 - 408v^7 + 274v^6 - 1306v^5 + 1108v^4 - 1282v^3 + 1649v^2 + 119v + 252) / 112
$\beta_{11}$	$=$	$ ( - 2 \nu^{11} + 2 \nu^{10} - 51 \nu^{9} + 49 \nu^{8} - 450 \nu^{7} + 408 \nu^{6} - 1588 \nu^{5} + \cdots + 217 ) / 112 $ (-2v^11 + 2v^10 - 51v^9 + 49v^8 - 450v^7 + 408v^6 - 1588v^5 + 1306v^4 - 1888v^3 + 1338v^2 + 27*v + 217) / 112

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} - \beta_{2} + \beta _1 - 4 $ -b10 + b9 - b8 - b3 - b2 + b1 - 4
$\nu^{3}$	$=$	$ 2\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + 2\beta_{4} + \beta_{3} + \beta_{2} - 8\beta _1 + 1 $ 2b11 + b10 - b9 + b8 - 2b7 + 2b4 + b3 + b2 - 8b1 + 1
$\nu^{4}$	$=$	$ - 2 \beta_{11} + 10 \beta_{10} - 8 \beta_{9} + 8 \beta_{8} + 2 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} + \cdots + 31 $ -2b11 + 10b10 - 8b9 + 8b8 + 2b5 + 2b4 + 10b3 + 6b2 - 8*b1 + 31
$\nu^{5}$	$=$	$ - 26 \beta_{11} - 12 \beta_{10} + 14 \beta_{9} - 8 \beta_{8} + 24 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + \cdots - 18 $ -26b11 - 12b10 + 14b9 - 8b8 + 24b7 + 4b6 + 4b5 - 26b4 - 12b3 - 12b2 + 69*b1 - 18
$\nu^{6}$	$=$	$ 26 \beta_{11} - 95 \beta_{10} + 69 \beta_{9} - 69 \beta_{8} - 26 \beta_{5} - 26 \beta_{4} - 95 \beta_{3} + \cdots - 270 $ 26b11 - 95b10 + 69b9 - 69b8 - 26b5 - 26b4 - 95b3 - 35b2 + 65*b1 - 270
$\nu^{7}$	$=$	$ 276 \beta_{11} + 125 \beta_{10} - 151 \beta_{9} + 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + \cdots + 215 $ 276b11 + 125b10 - 151b9 + 57b8 - 234b7 - 76b6 - 60b5 + 276b4 + 117b3 + 117b2 - 612*b1 + 215
$\nu^{8}$	$=$	$ - 268 \beta_{11} + 904 \beta_{10} - 636 \beta_{9} + 620 \beta_{8} + 260 \beta_{5} + 268 \beta_{4} + \cdots + 2449 $ -268b11 + 904b10 - 636b9 + 620b8 + 260b5 + 268b4 + 880b3 + 200b2 - 540*b1 + 2449
$\nu^{9}$	$=$	$ - 2764 \beta_{11} - 1236 \beta_{10} + 1528 \beta_{9} - 372 \beta_{8} + 2136 \beta_{7} + 1032 \beta_{6} + \cdots - 2280 $ -2764b11 - 1236b10 + 1528b9 - 372b8 + 2136b7 + 1032b6 + 696b5 - 2764b4 - 1068b3 - 1068b2 + 5501*b1 - 2280
$\nu^{10}$	$=$	$ 2596 \beta_{11} - 8657 \beta_{10} + 6061 \beta_{9} - 5669 \beta_{8} - 2372 \beta_{5} - 2596 \beta_{4} + \cdots - 22596 $ 2596b11 - 8657b10 + 6061b9 - 5669b8 - 2372b5 - 2596b4 - 8041b3 - 1009b2 + 4525*b1 - 22596
$\nu^{11}$	$=$	$ 27110 \beta_{11} + 11949 \beta_{10} - 15161 \beta_{9} + 2069 \beta_{8} - 18986 \beta_{7} - 12392 \beta_{6} + \cdots + 23113 $ 27110b11 + 11949b10 - 15161b9 + 2069b8 - 18986b7 - 12392b6 - 7424b5 + 27110b4 + 9493b3 + 9493b2 - 49796*b1 + 23113

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$\beta_{6}$	$1$	$1$	$1$

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} $

901.1

	−	2.03646i
		3.11341i
		1.66294i
	−	2.98167i
	−	0.366223i
		0.608003i
		2.03646i
	−	3.11341i
	−	1.66294i
		2.98167i
		0.366223i
	−	0.608003i

0.500000

−

0.866025i

−2.65726

1.53417i

−0.500000

−

0.866025i

901.2

0.500000

−

0.866025i

−1.43544

0.828751i

−0.500000

−

0.866025i

901.3

0.500000

−

0.866025i

−0.380104

0.219453i

−0.500000

−

0.866025i

901.4

0.500000

−

0.866025i

0.681305

−

0.393352i

−0.500000

−

0.866025i

901.5

0.500000

−

0.866025i

2.47173

−

1.42706i

−0.500000

−

0.866025i

901.6

0.500000

−

0.866025i

4.31977

−

2.49402i

−0.500000

−

0.866025i

2701.1

0.500000

0.866025i

−2.65726

−

1.53417i

−0.500000

0.866025i

2701.2

0.500000

0.866025i

−1.43544

−

0.828751i

−0.500000

0.866025i

2701.3

0.500000

0.866025i

−0.380104

−

0.219453i

−0.500000

0.866025i

2701.4

0.500000

0.866025i

0.681305

0.393352i

−0.500000

0.866025i

2701.5

0.500000

0.866025i

2.47173

1.42706i

−0.500000

0.866025i

2701.6

0.500000

0.866025i

4.31977

2.49402i

−0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3900.2.cd.o		12
5.b	even	2	1		3900.2.cd.n		12
5.c	odd	4	2		780.2.bv.a	✓	24
13.e	even	6	1	inner	3900.2.cd.o		12
15.e	even	4	2		2340.2.cr.b		24
65.l	even	6	1		3900.2.cd.n		12
65.r	odd	12	2		780.2.bv.a	✓	24
195.bf	even	12	2		2340.2.cr.b		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
780.2.bv.a	✓	24	5.c	odd	4	2
780.2.bv.a	✓	24	65.r	odd	12	2
2340.2.cr.b		24	15.e	even	4	2
2340.2.cr.b		24	195.bf	even	12	2
3900.2.cd.n		12	5.b	even	2	1
3900.2.cd.n		12	65.l	even	6	1
3900.2.cd.o		12	1.a	even	1	1	trivial
3900.2.cd.o		12	13.e	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3900, [\chi])$:

$ T_{7}^{12} - 6 T_{7}^{11} - 5 T_{7}^{10} + 102 T_{7}^{9} + 100 T_{7}^{8} - 960 T_{7}^{7} - 43 T_{7}^{6} + \cdots + 625 $

$ T_{11}^{12} - 6 T_{11}^{11} - 14 T_{11}^{10} + 156 T_{11}^{9} + 384 T_{11}^{8} - 1776 T_{11}^{7} + \cdots + 3136 $

$ T_{17}^{12} - 2 T_{17}^{11} + 48 T_{17}^{10} + 48 T_{17}^{9} + 1630 T_{17}^{8} + 906 T_{17}^{7} + \cdots + 11236 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$5$	$ T^{12} $ T^12
$7$	$ T^{12} - 6 T^{11} + \cdots + 625 $ T^12 - 6T^11 - 5T^10 + 102T^9 + 100T^8 - 960T^7 - 43T^6 + 3990T^5 + 3136T^4 - 4140T^3 - 525T^2 + 1500*T + 625
$11$	$ T^{12} - 6 T^{11} + \cdots + 3136 $ T^12 - 6T^11 - 14T^10 + 156T^9 + 384T^8 - 1776T^7 - 1800T^6 + 10752T^5 + 5408T^4 - 50592T^3 + 62432T^2 - 22848*T + 3136
$13$	$ T^{12} + 8 T^{11} + \cdots + 4826809 $ T^12 + 8T^11 + 31T^10 + 104T^9 + 560T^8 + 2198T^7 + 7601T^6 + 28574T^5 + 94640T^4 + 228488T^3 + 885391T^2 + 2970344*T + 4826809
$17$	$ T^{12} - 2 T^{11} + \cdots + 11236 $ T^12 - 2T^11 + 48T^10 + 48T^9 + 1630T^8 + 906T^7 + 16640T^6 + 42084T^5 + 116340T^4 + 134852T^3 + 124928T^2 + 42612*T + 11236
$19$	$ T^{12} + 12 T^{11} + \cdots + 784 $ T^12 + 12T^11 + 52T^10 + 48T^9 - 212T^8 - 348T^7 + 776T^6 + 1056T^5 - 1904T^4 - 1296T^3 + 4224T^2 - 3024*T + 784
$23$	$ T^{12} - 4 T^{11} + \cdots + 11669056 $ T^12 - 4T^11 + 74T^10 - 32T^9 + 3016T^8 - 1448T^7 + 58200T^6 + 36496T^5 + 674848T^4 + 201024T^3 + 3632416T^2 + 2732800*T + 11669056
$29$	$ T^{12} + 2 T^{11} + \cdots + 4 $ T^12 + 2T^11 + 88T^10 + 444T^9 + 7462T^8 + 24878T^7 + 110940T^6 - 63376T^5 + 41656T^4 - 1636T^3 + 416T^2 + 4*T + 4
$31$	$ T^{12} + 162 T^{10} + \cdots + 96138025 $ T^12 + 162T^10 + 10279T^8 + 324428T^6 + 5270191T^4 + 39924802*T^2 + 96138025
$37$	$ T^{12} + \cdots + 1033236736 $ T^12 + 6T^11 - 116T^10 - 768T^9 + 11904T^8 + 132048T^7 + 186240T^6 - 2639712T^5 - 6939904T^4 + 63889920T^3 + 442403072T^2 + 1087752960*T + 1033236736
$41$	$ T^{12} - 36 T^{11} + \cdots + 2044900 $ T^12 - 36T^11 + 488T^10 - 2016T^9 - 11866T^8 + 84858T^7 + 674424T^6 - 6712164T^5 + 20337240T^4 - 20493588T^3 + 1816312T^2 + 6941220*T + 2044900
$43$	$ T^{12} - 16 T^{11} + \cdots + 290521 $ T^12 - 16T^11 + 187T^10 - 1200T^9 + 6412T^8 - 23374T^7 + 87273T^6 - 250420T^5 + 741520T^4 - 1334566T^3 + 1988963T^2 - 846230*T + 290521
$47$	$ T^{12} + 224 T^{10} + \cdots + 22372900 $ T^12 + 224T^10 + 16496T^8 + 484728T^6 + 5215092T^4 + 19483588*T^2 + 22372900
$53$	$ (T^{6} - 104 T^{4} + \cdots + 640)^{2} $ (T^6 - 104T^4 + 32T^3 + 1312T^2 - 1856T + 640)^2
$59$	$ T^{12} + \cdots + 26814062500 $ T^12 + 18T^11 - 72T^10 - 3240T^9 + 8286T^8 + 348570T^7 - 287668T^6 - 21030120T^5 + 50950800T^4 + 540958500T^3 - 941745000T^2 - 8179312500*T + 26814062500
$61$	$ T^{12} + \cdots + 1838265625 $ T^12 - 22T^11 + 489T^10 - 6546T^9 + 101290T^8 - 1173354T^7 + 12553313T^6 - 91629402T^5 + 528805050T^4 - 1886794658T^3 + 4794662489T^2 - 3320325750*T + 1838265625
$67$	$ T^{12} + \cdots + 8640260209 $ T^12 - 6T^11 - 213T^10 + 1350T^9 + 40696T^8 - 41940T^7 - 1950367T^6 + 1384566T^5 + 69465736T^4 - 37703160T^3 - 879778313T^2 + 368093880*T + 8640260209
$71$	$ T^{12} + \cdots + 434889316 $ T^12 - 6T^11 - 202T^10 + 1284T^9 + 32894T^8 - 238014T^7 - 1528860T^6 + 14476812T^5 + 58280304T^4 - 750862668T^3 + 1681128712T^2 + 1569930828*T + 434889316
$73$	$ T^{12} + 266 T^{10} + \cdots + 24552025 $ T^12 + 266T^10 + 21299T^8 + 711036T^6 + 10319259T^4 + 55178782*T^2 + 24552025
$79$	$ (T^{6} + 2 T^{5} + \cdots + 18385)^{2} $ (T^6 + 2T^5 - 245T^4 - 212T^3 + 9043T^2 + 25638*T + 18385)^2
$83$	$ T^{12} + \cdots + 5489031744 $ T^12 + 360T^10 + 49500T^8 + 3239568T^6 + 102076848T^4 + 1379213568*T^2 + 5489031744
$89$	$ T^{12} + \cdots + 1758795844 $ T^12 - 162T^10 + 21130T^8 - 39378T^7 - 721360T^6 + 1350096T^5 + 19060720T^4 - 17336460T^3 - 210640232T^2 + 142169820*T + 1758795844
$97$	$ T^{12} - 48 T^{11} + \cdots + 305809 $ T^12 - 48T^11 + 967T^10 - 9552T^9 + 36000T^8 + 117750T^7 - 975783T^6 - 6448764T^5 + 80513960T^4 - 277735530T^3 + 360996947T^2 - 18079782*T + 305809
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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:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{6} + 1) q^{3} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{7}+ \cdots - \beta_{6} q^{9}+O(q^{10}) \) q + (-b6 + 1) * q^3 + (-b11 - b10 + b9 - b6 + b5 - b2 + 1) * q^7 - b6 * q^9	\( q + ( - \beta_{6} + 1) q^{3} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) q + (-b6 + 1) * q^3 + (-b11 - b10 + b9 - b6 + b5 - b2 + 1) * q^7 - b6 * q^9 + (-b11 + b10 + b8) * q^11 + (-b10 + b3) * q^13 + (b10 + b9 - 2b8 + b7 + 2b6 - b5 + b4 - b3 - b2 + 2b1) q^17 + (b6 - b5 - 1) * q^19 + (-b11 - b10 + b8 - 2b6 + b5 - b4 - b1) q^21 + (-b9 - b8 + b7 - b5 + b4 - b2 + 1) * q^23 - q^27 + (b11 + b10 - b7 + b6 - b5 + 2b2 + b1) q^29 + (-b10 - b9 - 2b8 + 2b7 - b5 - b3 - b2 + b1) * q^31 + (b10 - b9 - b5 + b4 + 1) * q^33 + (-b11 + b10 + b9 + b7 - b5 + b4 - 2b3 - b2 + 2b1 - 1) * q^37 + (-b10 - b8 + b7 + b6 - b4 - b2 + b1 - 1) * q^39 + (-b11 + b10 - 3b8 + b7 + 3b6 - b5 - 2b3 - b2 + 2b1 + 2) * q^41 + (b11 - 2b8 + 4b6 - b5 + b4 - b2 + 2b1 + 1) q^43 + (-2b11 - b10 + b9 - 2b4 - b1) * q^47 + (-2b11 - 2b10 + b8 - 2b7 - 3b6 - 2b1 + 3) q^49 + (-b11 + b9 - b8 + b4 - b3 - b2 + b1 + 1) * q^51 + (2b8 + 2b3 + 2b2 - 2b1) * q^53 + (-b8 + 2b6 - b5) q^57 + (2b11 + 3b10 - 3b9 + b7 + 3b6 - 2b5 + b4 + b3 + 3b2 - 3) * q^59 + (b11 - 4b8 + 3b7 + 6b6 - 2b5 + b4 - 3b3 - 3b2 + 6b1 - 1) q^61 + (-b9 + b8 - b6 - b4 + b2 - b1 - 1) * q^63 + (-2b9 - b8 + b7 - b5 - 2b4 - 2b3 + 4b2 - 3b1 - 1) q^67 + (b11 - b10 - b9 - 2b8 + b7 - b5 - b3) q^69 + (-b7 - b6 - b3 + 4b2 + 1) q^71 + (-b11 + b9 + 3b8 - 2b7 + 2b6 + 2b5 - b4 + b3 + b2 - b1 - 2) * q^73 + (-b11 - b10 + 2b9 + b8 + b4 + b3 - b2 - 1) q^77 + (-2b11 + 2b9 - 3b8 + b5 + 2b4 - 2b3 - 2b2 + 2b1 + 1) q^79 + (b6 - 1) * q^81 + (-b11 + b10 + 2b9 + 2b7 + b5 - b4 - b3 - b2 + 3b1 - 2) q^83 + (b10 + b9 - b7 + b6 + b4 + b3 - b2 + 2b1 + 1) q^87 + (-2b9 + b8 + b7 - b6 - b5 - 2b4 - 2b3 + b1 - 2) q^89 + (b11 - 4b9 - 2b8 + b7 - 2b6 - 4b5 - b4 - b3 + 3b2 - 2b1 + 2) * q^91 + (b11 - b10 - b9 - 2b8 + b7 - b5 - b4 - 2b3 + b1 - 1) * q^93 + (b11 - 2b6 - b5 - b4 + 3b2 + 5) * q^97 + (b11 - b9 - b8 - b5 + b4 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9	\( 12 q + 6 q^{3} + 6 q^{7} - 6 q^{9} + 6 q^{11} - 8 q^{13} + 2 q^{17} - 12 q^{19} + 4 q^{23} - 12 q^{27} - 2 q^{29} + 6 q^{33} - 6 q^{37} - 4 q^{39} + 36 q^{41} + 16 q^{43} + 4 q^{49} + 4 q^{51} - 18 q^{59} + 22 q^{61} - 6 q^{63} + 6 q^{67} - 4 q^{69} + 6 q^{71} - 24 q^{77} - 4 q^{79} - 6 q^{81} + 2 q^{87} + 2 q^{91} - 6 q^{93} + 48 q^{97}+O(q^{100}) \) 12 * q + 6 * q^3 + 6 * q^7 - 6 * q^9 + 6 * q^11 - 8 * q^13 + 2 * q^17 - 12 * q^19 + 4 * q^23 - 12 * q^27 - 2 * q^29 + 6 * q^33 - 6 * q^37 - 4 * q^39 + 36 * q^41 + 16 * q^43 + 4 * q^49 + 4 * q^51 - 18 * q^59 + 22 * q^61 - 6 * q^63 + 6 * q^67 - 4 * q^69 + 6 * q^71 - 24 * q^77 - 4 * q^79 - 6 * q^81 + 2 * q^87 + 2 * q^91 - 6 * q^93 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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	3900.2.cd.o

Level	$3900$
Weight	$2$
Character orbit	3900.cd
Analytic conductor	$31.142$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(31.1416567883\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 26x^{10} + 239x^{8} + 924x^{6} + 1407x^{4} + 538x^{2} + 49 \) x^12 + 26x^10 + 239x^8 + 924x^6 + 1407x^4 + 538*x^2 + 49
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 780)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + 13\nu^{4} + 35\nu^{2} + 4\nu + 7 ) / 8 \) (v^6 + 13v^4 + 35v^2 + 4*v + 7) / 8
\(\beta_{3}\)	\(=\)	\( ( - \nu^{11} + \nu^{10} - 25 \nu^{9} + 21 \nu^{8} - 218 \nu^{7} + 134 \nu^{6} - 790 \nu^{5} + \cdots - 287 ) / 112 \) (-v^11 + v^10 - 25v^9 + 21v^8 - 218v^7 + 134v^6 - 790v^5 + 226v^4 - 1181v^3 - 143v^2 - 569*v - 287) / 112
\(\beta_{4}\)	\(=\)	\( ( - 3 \nu^{10} - 2 \nu^{9} - 77 \nu^{8} - 42 \nu^{7} - 682 \nu^{6} - 282 \nu^{5} - 2414 \nu^{4} + \cdots - 469 ) / 112 \) (-3v^10 - 2v^9 - 77v^8 - 42v^7 - 682v^6 - 282v^5 - 2414v^4 - 606v^3 - 2987v^2 - 92v - 469) / 112
\(\beta_{5}\)	\(=\)	\( ( \nu^{11} + 3 \nu^{10} + 25 \nu^{9} + 77 \nu^{8} + 218 \nu^{7} + 696 \nu^{6} + 790 \nu^{5} + 2624 \nu^{4} + \cdots + 875 ) / 112 \) (v^11 + 3v^10 + 25v^9 + 77v^8 + 218v^7 + 696v^6 + 790v^5 + 2624v^4 + 1181v^3 + 3757v^2 + 625v + 875) / 112
\(\beta_{6}\)	\(=\)	\( ( -\nu^{11} - 26\nu^{9} - 239\nu^{7} - 917\nu^{5} - 1316\nu^{3} - 293\nu + 28 ) / 56 \) (-v^11 - 26v^9 - 239v^7 - 917v^5 - 1316v^3 - 293*v + 28) / 56
\(\beta_{7}\)	\(=\)	\( ( - 2 \nu^{11} - \nu^{10} - 53 \nu^{9} - 28 \nu^{8} - 492 \nu^{7} - 274 \nu^{6} - 1870 \nu^{5} + \cdots - 420 ) / 112 \) (-2v^11 - v^10 - 53v^9 - 28v^8 - 492v^7 - 274v^6 - 1870v^5 - 1108v^4 - 2550v^3 - 1705v^2 - 457v - 420) / 112
\(\beta_{8}\)	\(=\)	\( ( \nu^{11} - 3 \nu^{10} + 25 \nu^{9} - 77 \nu^{8} + 218 \nu^{7} - 696 \nu^{6} + 790 \nu^{5} - 2624 \nu^{4} + \cdots - 763 ) / 112 \) (v^11 - 3v^10 + 25v^9 - 77v^8 + 218v^7 - 696v^6 + 790v^5 - 2624v^4 + 1181v^3 - 3757v^2 + 625v - 763) / 112
\(\beta_{9}\)	\(=\)	\( ( - 2 \nu^{11} - \nu^{10} - 49 \nu^{9} - 28 \nu^{8} - 408 \nu^{7} - 274 \nu^{6} - 1306 \nu^{5} + \cdots - 252 ) / 112 \) (-2v^11 - v^10 - 49v^9 - 28v^8 - 408v^7 - 274v^6 - 1306v^5 - 1108v^4 - 1282v^3 - 1649v^2 + 119v - 252) / 112
\(\beta_{10}\)	\(=\)	\( ( - 2 \nu^{11} + \nu^{10} - 49 \nu^{9} + 28 \nu^{8} - 408 \nu^{7} + 274 \nu^{6} - 1306 \nu^{5} + \cdots + 252 ) / 112 \) (-2v^11 + v^10 - 49v^9 + 28v^8 - 408v^7 + 274v^6 - 1306v^5 + 1108v^4 - 1282v^3 + 1649v^2 + 119v + 252) / 112
\(\beta_{11}\)	\(=\)	\( ( - 2 \nu^{11} + 2 \nu^{10} - 51 \nu^{9} + 49 \nu^{8} - 450 \nu^{7} + 408 \nu^{6} - 1588 \nu^{5} + \cdots + 217 ) / 112 \) (-2v^11 + 2v^10 - 51v^9 + 49v^8 - 450v^7 + 408v^6 - 1588v^5 + 1306v^4 - 1888v^3 + 1338v^2 + 27*v + 217) / 112

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} - \beta_{2} + \beta _1 - 4 \) -b10 + b9 - b8 - b3 - b2 + b1 - 4
\(\nu^{3}\)	\(=\)	\( 2\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + 2\beta_{4} + \beta_{3} + \beta_{2} - 8\beta _1 + 1 \) 2b11 + b10 - b9 + b8 - 2b7 + 2b4 + b3 + b2 - 8b1 + 1
\(\nu^{4}\)	\(=\)	\( - 2 \beta_{11} + 10 \beta_{10} - 8 \beta_{9} + 8 \beta_{8} + 2 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} + \cdots + 31 \) -2b11 + 10b10 - 8b9 + 8b8 + 2b5 + 2b4 + 10b3 + 6b2 - 8*b1 + 31
\(\nu^{5}\)	\(=\)	\( - 26 \beta_{11} - 12 \beta_{10} + 14 \beta_{9} - 8 \beta_{8} + 24 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + \cdots - 18 \) -26b11 - 12b10 + 14b9 - 8b8 + 24b7 + 4b6 + 4b5 - 26b4 - 12b3 - 12b2 + 69*b1 - 18
\(\nu^{6}\)	\(=\)	\( 26 \beta_{11} - 95 \beta_{10} + 69 \beta_{9} - 69 \beta_{8} - 26 \beta_{5} - 26 \beta_{4} - 95 \beta_{3} + \cdots - 270 \) 26b11 - 95b10 + 69b9 - 69b8 - 26b5 - 26b4 - 95b3 - 35b2 + 65*b1 - 270
\(\nu^{7}\)	\(=\)	\( 276 \beta_{11} + 125 \beta_{10} - 151 \beta_{9} + 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + \cdots + 215 \) 276b11 + 125b10 - 151b9 + 57b8 - 234b7 - 76b6 - 60b5 + 276b4 + 117b3 + 117b2 - 612*b1 + 215
\(\nu^{8}\)	\(=\)	\( - 268 \beta_{11} + 904 \beta_{10} - 636 \beta_{9} + 620 \beta_{8} + 260 \beta_{5} + 268 \beta_{4} + \cdots + 2449 \) -268b11 + 904b10 - 636b9 + 620b8 + 260b5 + 268b4 + 880b3 + 200b2 - 540*b1 + 2449
\(\nu^{9}\)	\(=\)	\( - 2764 \beta_{11} - 1236 \beta_{10} + 1528 \beta_{9} - 372 \beta_{8} + 2136 \beta_{7} + 1032 \beta_{6} + \cdots - 2280 \) -2764b11 - 1236b10 + 1528b9 - 372b8 + 2136b7 + 1032b6 + 696b5 - 2764b4 - 1068b3 - 1068b2 + 5501*b1 - 2280
\(\nu^{10}\)	\(=\)	\( 2596 \beta_{11} - 8657 \beta_{10} + 6061 \beta_{9} - 5669 \beta_{8} - 2372 \beta_{5} - 2596 \beta_{4} + \cdots - 22596 \) 2596b11 - 8657b10 + 6061b9 - 5669b8 - 2372b5 - 2596b4 - 8041b3 - 1009b2 + 4525*b1 - 22596
\(\nu^{11}\)	\(=\)	\( 27110 \beta_{11} + 11949 \beta_{10} - 15161 \beta_{9} + 2069 \beta_{8} - 18986 \beta_{7} - 12392 \beta_{6} + \cdots + 23113 \) 27110b11 + 11949b10 - 15161b9 + 2069b8 - 18986b7 - 12392b6 - 7424b5 + 27110b4 + 9493b3 + 9493b2 - 49796*b1 + 23113

\(n\)	\(301\)	\(1301\)	\(1951\)	\(3277\)
\(\chi(n)\)	\(\beta_{6}\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 901.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$5$	\( T^{12} \) T^12
$7$	\( T^{12} - 6 T^{11} + \cdots + 625 \) T^12 - 6T^11 - 5T^10 + 102T^9 + 100T^8 - 960T^7 - 43T^6 + 3990T^5 + 3136T^4 - 4140T^3 - 525T^2 + 1500*T + 625
$11$	\( T^{12} - 6 T^{11} + \cdots + 3136 \) T^12 - 6T^11 - 14T^10 + 156T^9 + 384T^8 - 1776T^7 - 1800T^6 + 10752T^5 + 5408T^4 - 50592T^3 + 62432T^2 - 22848*T + 3136
$13$	\( T^{12} + 8 T^{11} + \cdots + 4826809 \) T^12 + 8T^11 + 31T^10 + 104T^9 + 560T^8 + 2198T^7 + 7601T^6 + 28574T^5 + 94640T^4 + 228488T^3 + 885391T^2 + 2970344*T + 4826809
$17$	\( T^{12} - 2 T^{11} + \cdots + 11236 \) T^12 - 2T^11 + 48T^10 + 48T^9 + 1630T^8 + 906T^7 + 16640T^6 + 42084T^5 + 116340T^4 + 134852T^3 + 124928T^2 + 42612*T + 11236
$19$	\( T^{12} + 12 T^{11} + \cdots + 784 \) T^12 + 12T^11 + 52T^10 + 48T^9 - 212T^8 - 348T^7 + 776T^6 + 1056T^5 - 1904T^4 - 1296T^3 + 4224T^2 - 3024*T + 784
$23$	\( T^{12} - 4 T^{11} + \cdots + 11669056 \) T^12 - 4T^11 + 74T^10 - 32T^9 + 3016T^8 - 1448T^7 + 58200T^6 + 36496T^5 + 674848T^4 + 201024T^3 + 3632416T^2 + 2732800*T + 11669056
$29$	\( T^{12} + 2 T^{11} + \cdots + 4 \) T^12 + 2T^11 + 88T^10 + 444T^9 + 7462T^8 + 24878T^7 + 110940T^6 - 63376T^5 + 41656T^4 - 1636T^3 + 416T^2 + 4*T + 4
$31$	\( T^{12} + 162 T^{10} + \cdots + 96138025 \) T^12 + 162T^10 + 10279T^8 + 324428T^6 + 5270191T^4 + 39924802*T^2 + 96138025
$37$	\( T^{12} + \cdots + 1033236736 \) T^12 + 6T^11 - 116T^10 - 768T^9 + 11904T^8 + 132048T^7 + 186240T^6 - 2639712T^5 - 6939904T^4 + 63889920T^3 + 442403072T^2 + 1087752960*T + 1033236736
$41$	\( T^{12} - 36 T^{11} + \cdots + 2044900 \) T^12 - 36T^11 + 488T^10 - 2016T^9 - 11866T^8 + 84858T^7 + 674424T^6 - 6712164T^5 + 20337240T^4 - 20493588T^3 + 1816312T^2 + 6941220*T + 2044900
$43$	\( T^{12} - 16 T^{11} + \cdots + 290521 \) T^12 - 16T^11 + 187T^10 - 1200T^9 + 6412T^8 - 23374T^7 + 87273T^6 - 250420T^5 + 741520T^4 - 1334566T^3 + 1988963T^2 - 846230*T + 290521
$47$	\( T^{12} + 224 T^{10} + \cdots + 22372900 \) T^12 + 224T^10 + 16496T^8 + 484728T^6 + 5215092T^4 + 19483588*T^2 + 22372900
$53$	\( (T^{6} - 104 T^{4} + \cdots + 640)^{2} \) (T^6 - 104T^4 + 32T^3 + 1312T^2 - 1856T + 640)^2
$59$	\( T^{12} + \cdots + 26814062500 \) T^12 + 18T^11 - 72T^10 - 3240T^9 + 8286T^8 + 348570T^7 - 287668T^6 - 21030120T^5 + 50950800T^4 + 540958500T^3 - 941745000T^2 - 8179312500*T + 26814062500
$61$	\( T^{12} + \cdots + 1838265625 \) T^12 - 22T^11 + 489T^10 - 6546T^9 + 101290T^8 - 1173354T^7 + 12553313T^6 - 91629402T^5 + 528805050T^4 - 1886794658T^3 + 4794662489T^2 - 3320325750*T + 1838265625
$67$	\( T^{12} + \cdots + 8640260209 \) T^12 - 6T^11 - 213T^10 + 1350T^9 + 40696T^8 - 41940T^7 - 1950367T^6 + 1384566T^5 + 69465736T^4 - 37703160T^3 - 879778313T^2 + 368093880*T + 8640260209
$71$	\( T^{12} + \cdots + 434889316 \) T^12 - 6T^11 - 202T^10 + 1284T^9 + 32894T^8 - 238014T^7 - 1528860T^6 + 14476812T^5 + 58280304T^4 - 750862668T^3 + 1681128712T^2 + 1569930828*T + 434889316
$73$	\( T^{12} + 266 T^{10} + \cdots + 24552025 \) T^12 + 266T^10 + 21299T^8 + 711036T^6 + 10319259T^4 + 55178782*T^2 + 24552025
$79$	\( (T^{6} + 2 T^{5} + \cdots + 18385)^{2} \) (T^6 + 2T^5 - 245T^4 - 212T^3 + 9043T^2 + 25638*T + 18385)^2
$83$	\( T^{12} + \cdots + 5489031744 \) T^12 + 360T^10 + 49500T^8 + 3239568T^6 + 102076848T^4 + 1379213568*T^2 + 5489031744
$89$	\( T^{12} + \cdots + 1758795844 \) T^12 - 162T^10 + 21130T^8 - 39378T^7 - 721360T^6 + 1350096T^5 + 19060720T^4 - 17336460T^3 - 210640232T^2 + 142169820*T + 1758795844
$97$	\( T^{12} - 48 T^{11} + \cdots + 305809 \) T^12 - 48T^11 + 967T^10 - 9552T^9 + 36000T^8 + 117750T^7 - 975783T^6 - 6448764T^5 + 80513960T^4 - 277735530T^3 + 360996947T^2 - 18079782*T + 305809
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