LMFDB - Newform orbit 6975.2.a.ck

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6975,2,Mod(1,6975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6975, 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("6975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.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:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 152x^{12} - 571x^{10} + 1130x^{8} - 1138x^{6} + 492x^{4} - 43x^{2} + 1 $ x^16 - 20x^14 + 152x^12 - 571x^10 + 1130x^8 - 1138x^6 + 492x^4 - 43*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 1395)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{2} + ( - \beta_{11} - \beta_{10} + \beta_1) q^{4} + ( - \beta_{12} - \beta_{9}) q^{7} + (\beta_{8} - \beta_{7} + \cdots - 2 \beta_{2}) q^{8}+O(q^{10}) $ q - b2 * q^2 + (-b11 - b10 + b1) * q^4 + (-b12 - b9) * q^7 + (b8 - b7 + 2b4 - 2b2) * q^8	$ q - \beta_{2} q^{2} + ( - \beta_{11} - \beta_{10} + \beta_1) q^{4} + ( - \beta_{12} - \beta_{9}) q^{7} + (\beta_{8} - \beta_{7} + \cdots - 2 \beta_{2}) q^{8}+ \cdots + ( - 3 \beta_{8} - 2 \beta_{7} + \cdots + 5 \beta_{2}) q^{98}+O(q^{100}) $ q - b2 * q^2 + (-b11 - b10 + b1) * q^4 + (-b12 - b9) * q^7 + (b8 - b7 + 2b4 - 2b2) * q^8 + (b15 + b14 - b13) * q^11 + (b6 + b3) * q^13 + (b14 + b13 + b5) * q^14 + (-2b11 - 4b10 + b1 + 1) * q^16 + (2b7 + b4 - b2) q^17 + (b11 - 2b10 - b1) q^19 + (-2b12 - b6 + 2b3) * q^22 + (2b8 + b4 - 3b2) * q^23 + (-2b15 - b13) q^26 + b12 * q^28 + (-b15 + b14 + 2b13 + 2b5) * q^29 - q^31 + (-b7 + 2b4 - 6b2) * q^32 + (-2b11 - b10 - b1 + 1) q^34 + (-2b12 + 2b9 - b3) * q^37 + (-b8 - 2b7 + b4 + b2) q^38 + (-2b14 + 2b13 + 5b5) q^41 + (-2b9 + b6) q^43 + (5b15 + 3b14 - 3b13 - 8b5) * q^44 + (-4b11 - 3b10 + 3b1 + 3) q^46 + (2b8 + 2b7 - b4 - b2) * q^47 + (3b11 + 3b10 + 2b1 + 2) q^49 + (2b12 - 2b9 - b3) * q^52 + (-2b8 - 2b7 - b4 + 3b2) q^53 + (-b15 - 3b14 - 2b13 - b5) * q^56 + (-4b9 + b6 - b3) q^58 + (2b15 - b13 + b5) q^59 + (4b10 + 4b1 + 4) * q^61 + b2 * q^62 + (-4b11 - 3b10 + 5b1 + 8) q^64 + (-4b12 - 2b9 + b6 + b3) * q^67 + (2b8 - 2b7 + b4 - 3b2) q^68 + (b15 + b14 - 4b13 - 3b5) * q^71 + (2b12 - 2b9 - b3) * q^73 + (3b15 + b14 - 4b5) * q^74 + (-2b11 + 3b1 - 2) * q^76 + (-4b4 - 2b2) * q^77 + (2b11 + b10 + b1 + 5) q^79 + (2b12 - 3b9 - 4b3) q^82 + (3b4 - b2) q^83 + (-5b15 - b14 + 3b13 + 6b5) q^86 + (-4b12 + 10b9 - 3b6 + 2b3) * q^88 + (-2b14 + 3b13) * q^89 + (-4b11 - b10 - 5b1 - 1) * q^91 + (-2b7 + 5b4 - 11b2) q^92 + (5b10 - b1 + 1) q^94 + (3b12 + 3b9 + b6 - b3) * q^97 + (-3b8 - 2b7 - 6b4 + 5b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4}+O(q^{10}) $ 16 * q + 12 * q^4	$ 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) $ 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 20x^{14} + 152x^{12} - 571x^{10} + 1130x^{8} - 1138x^{6} + 492x^{4} - 43x^{2} + 1 $ x^16 - 20*x^14 + 152*x^12 - 571*x^10 + 1130*x^8 - 1138*x^6 + 492*x^4 - 43*x^2 + 1 :

$\beta_{1}$	$=$	$ ( -\nu^{14} + 17\nu^{12} - 95\nu^{10} + 174\nu^{8} + 154\nu^{6} - 798\nu^{4} + 620\nu^{2} - 27 ) / 4 $ (-v^14 + 17v^12 - 95v^10 + 174v^8 + 154v^6 - 798v^4 + 620v^2 - 27) / 4
$\beta_{2}$	$=$	$ ( 6\nu^{14} - 109\nu^{12} + 708\nu^{10} - 2047\nu^{8} + 2449\nu^{6} - 431\nu^{4} - 751\nu^{2} + 41 ) / 8 $ (6v^14 - 109v^12 + 708v^10 - 2047v^8 + 2449v^6 - 431v^4 - 751*v^2 + 41) / 8
$\beta_{3}$	$=$	$ ( 10\nu^{15} - 175\nu^{13} + 1052\nu^{11} - 2509\nu^{9} + 1123\nu^{7} + 3827\nu^{5} - 3981\nu^{3} + 307\nu ) / 8 $ (10v^15 - 175v^13 + 1052v^11 - 2509v^9 + 1123v^7 + 3827v^5 - 3981v^3 + 307v) / 8
$\beta_{4}$	$=$	$ ( -4\nu^{14} + 81\nu^{12} - 626\nu^{10} + 2401\nu^{8} - 4871\nu^{6} + 5033\nu^{4} - 2163\nu^{2} + 95 ) / 8 $ (-4v^14 + 81v^12 - 626v^10 + 2401v^8 - 4871v^6 + 5033v^4 - 2163*v^2 + 95) / 8
$\beta_{5}$	$=$	$ ( -7\nu^{15} + 133\nu^{13} - 933\nu^{11} + 3100\nu^{9} - 5044\nu^{7} + 3624\nu^{5} - 790\nu^{3} - 27\nu ) / 8 $ (-7v^15 + 133v^13 - 933v^11 + 3100v^9 - 5044v^7 + 3624v^5 - 790v^3 - 27v) / 8
$\beta_{6}$	$=$	$ ( -11\nu^{15} + 201\nu^{13} - 1321\nu^{11} + 3924\nu^{9} - 5108\nu^{7} + 1896\nu^{5} + 538\nu^{3} + 65\nu ) / 8 $ (-11v^15 + 201v^13 - 1321v^11 + 3924v^9 - 5108v^7 + 1896v^5 + 538v^3 + 65v) / 8
$\beta_{7}$	$=$	$ ( -18\nu^{14} + 337\nu^{12} - 2304\nu^{10} + 7311\nu^{8} - 10865\nu^{6} + 6247\nu^{4} - 417\nu^{2} - 13 ) / 8 $ (-18v^14 + 337v^12 - 2304v^10 + 7311v^8 - 10865v^6 + 6247v^4 - 417*v^2 - 13) / 8
$\beta_{8}$	$=$	$ ( 9\nu^{14} - 168\nu^{12} + 1145\nu^{10} - 3629\nu^{8} + 5427\nu^{6} - 3245\nu^{4} + 361\nu^{2} - 14 ) / 4 $ (9v^14 - 168v^12 + 1145v^10 - 3629v^8 + 5427v^6 - 3245v^4 + 361*v^2 - 14) / 4
$\beta_{9}$	$=$	$ ( 5\nu^{15} - 115\nu^{13} + 1039\nu^{11} - 4744\nu^{9} + 11560\nu^{7} - 14268\nu^{5} + 7110\nu^{3} - 331\nu ) / 8 $ (5v^15 - 115v^13 + 1039v^11 - 4744v^9 + 11560v^7 - 14268v^5 + 7110v^3 - 331v) / 8
$\beta_{10}$	$=$	$ ( -19\nu^{14} + 359\nu^{12} - 2493\nu^{10} + 8130\nu^{8} - 12758\nu^{6} + 8470\nu^{4} - 1504\nu^{2} + 67 ) / 8 $ (-19v^14 + 359v^12 - 2493v^10 + 8130v^8 - 12758v^6 + 8470v^4 - 1504*v^2 + 67) / 8
$\beta_{11}$	$=$	$ 3\nu^{14} - 57\nu^{12} + 400\nu^{10} - 1331\nu^{8} + 2176\nu^{6} - 1590\nu^{4} + 383\nu^{2} - 16 $ 3v^14 - 57v^12 + 400v^10 - 1331v^8 + 2176v^6 - 1590v^4 + 383*v^2 - 16
$\beta_{12}$	$=$	$ ( 21\nu^{15} - 406\nu^{13} + 2929\nu^{11} - 10183\nu^{9} + 17937\nu^{7} - 15099\nu^{5} + 4899\nu^{3} - 192\nu ) / 8 $ (21v^15 - 406v^13 + 2929v^11 - 10183v^9 + 17937v^7 - 15099v^5 + 4899v^3 - 192v) / 8
$\beta_{13}$	$=$	$ ( -21\nu^{15} + 416\nu^{13} - 3117\nu^{11} + 11473\nu^{9} - 22031\nu^{7} + 21141\nu^{5} - 8305\nu^{3} + 414\nu ) / 8 $ (-21v^15 + 416v^13 - 3117v^11 + 11473v^9 - 22031v^7 + 21141v^5 - 8305v^3 + 414v) / 8
$\beta_{14}$	$=$	$ ( 17\nu^{15} - 326\nu^{13} + 2321\nu^{11} - 7899\nu^{9} + 13421\nu^{7} - 10599\nu^{5} + 3115\nu^{3} - 200\nu ) / 4 $ (17v^15 - 326v^13 + 2321v^11 - 7899v^9 + 13421v^7 - 10599v^5 + 3115v^3 - 200v) / 4
$\beta_{15}$	$=$	$ ( -40\nu^{15} + 775\nu^{13} - 5610\nu^{11} + 19603\nu^{9} - 34789\nu^{7} + 29611\nu^{5} - 9801\nu^{3} + 449\nu ) / 8 $ (-40v^15 + 775v^13 - 5610v^11 + 19603v^9 - 34789v^7 + 29611v^5 - 9801v^3 + 449v) / 8

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

$\nu$	$=$	$ ( \beta_{12} - 3\beta_{9} + \beta_{6} - 5\beta_{5} - 3\beta_{3} ) / 10 $ (b12 - 3b9 + b6 - 5b5 - 3*b3) / 10
$\nu^{2}$	$=$	$ ( -\beta_{11} - 2\beta_{10} + \beta_{7} + \beta_{2} + \beta _1 + 4 ) / 2 $ (-b11 - 2*b10 + b7 + b2 + b1 + 4) / 2
$\nu^{3}$	$=$	$ ( 10\beta_{15} + 5\beta_{14} - 10\beta_{13} + 8\beta_{12} - 19\beta_{9} + 8\beta_{6} - 25\beta_{5} - 14\beta_{3} ) / 10 $ (10b15 + 5b14 - 10b13 + 8b12 - 19b9 + 8b6 - 25b5 - 14b3) / 10
$\nu^{4}$	$=$	$ ( -9\beta_{11} - 14\beta_{10} - 2\beta_{8} + 5\beta_{7} - 4\beta_{4} + 13\beta_{2} + 9\beta _1 + 16 ) / 2 $ (-9b11 - 14b10 - 2b8 + 5b7 - 4b4 + 13b2 + 9*b1 + 16) / 2
$\nu^{5}$	$=$	$ ( 115 \beta_{15} + 55 \beta_{14} - 95 \beta_{13} + 74 \beta_{12} - 122 \beta_{9} + 59 \beta_{6} + \cdots - 82 \beta_{3} ) / 10 $ (115b15 + 55b14 - 95b13 + 74b12 - 122b9 + 59b6 - 180b5 - 82b3) / 10
$\nu^{6}$	$=$	$ ( -72\beta_{11} - 102\beta_{10} - 21\beta_{8} + 33\beta_{7} - 47\beta_{4} + 118\beta_{2} + 67\beta _1 + 88 ) / 2 $ (-72b11 - 102b10 - 21b8 + 33b7 - 47b4 + 118b2 + 67*b1 + 88) / 2
$\nu^{7}$	$=$	$ ( 1035 \beta_{15} + 500 \beta_{14} - 775 \beta_{13} + 617 \beta_{12} - 846 \beta_{9} + \cdots - 561 \beta_{3} ) / 10 $ (1035b15 + 500b14 - 775b13 + 617b12 - 846b9 + 442b6 - 1410b5 - 561b3) / 10
$\nu^{8}$	$=$	$ ( -563\beta_{11} - 770\beta_{10} - 184\beta_{8} + 245\beta_{7} - 426\beta_{4} + 983\beta_{2} + 499\beta _1 + 584 ) / 2 $ (-563b11 - 770b10 - 184b8 + 245b7 - 426b4 + 983b2 + 499*b1 + 584) / 2
$\nu^{9}$	$=$	$ ( 8605 \beta_{15} + 4200 \beta_{14} - 6155 \beta_{13} + 4931 \beta_{12} - 6218 \beta_{9} + \cdots - 4133 \beta_{3} ) / 10 $ (8605b15 + 4200b14 - 6155b13 + 4931b12 - 6218b9 + 3386b6 - 11180b5 - 4133b3) / 10
$\nu^{10}$	$=$	$ - 2196 \beta_{11} - 2964 \beta_{10} - 760 \beta_{8} + 946 \beta_{7} - 1775 \beta_{4} + 3965 \beta_{2} + \cdots + 2127 $ -2196b11 - 2964b10 - 760b8 + 946b7 - 1775b4 + 3965b2 + 1897*b1 + 2127
$\nu^{11}$	$=$	$ ( 69310 \beta_{15} + 34035 \beta_{14} - 48570 \beta_{13} + 38868 \beta_{12} - 47259 \beta_{9} + \cdots - 31514 \beta_{3} ) / 10 $ (69310b15 + 34035b14 - 48570b13 + 38868b12 - 47259b9 + 26268b6 - 88485b5 - 31514b3) / 10
$\nu^{12}$	$=$	$ ( - 34323 \beta_{11} - 46096 \beta_{10} - 12229 \beta_{8} + 14792 \beta_{7} - 28611 \beta_{4} + \cdots + 32322 ) / 2 $ (-34323b11 - 46096b10 - 12229b8 + 14792b7 - 28611b4 + 63073b2 + 29288*b1 + 32322) / 2
$\nu^{13}$	$=$	$ ( 550635 \beta_{15} + 271230 \beta_{14} - 382405 \beta_{13} + 305219 \beta_{12} - 365392 \beta_{9} + \cdots - 244137 \beta_{3} ) / 10 $ (550635b15 + 271230b14 - 382405b13 + 305219b12 - 365392b9 + 205054b6 - 698200b5 - 244137b3) / 10
$\nu^{14}$	$=$	$ ( - 268739 \beta_{11} - 360228 \beta_{10} - 97147 \beta_{8} + 116066 \beta_{7} - 227307 \beta_{4} + \cdots + 250170 ) / 2 $ (-268739b11 - 360228b10 - 97147b8 + 116066b7 - 227307b4 + 498351b2 + 228040*b1 + 250170) / 2
$\nu^{15}$	$=$	$ ( 4347435 \beta_{15} + 2144615 \beta_{14} - 3007405 \beta_{13} + 2395152 \beta_{12} + \cdots - 1905186 \beta_{3} ) / 10 $ (4347435b15 + 2144615b14 - 3007405b13 + 2395152b12 - 2848586b9 + 1605537b6 - 5497510b5 - 1905186b3) / 10

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

−2.80230
2.80230
−0.199216
0.199216
−1.82625
1.82625
1.13362
−1.13362
−1.54642
1.54642
0.257202
−0.257202
1.11940
−1.11940
−1.94332
1.94332

−2.72869

5.44577

−0.363635

−9.40246

1.2

−2.72869

5.44577

0.363635

−9.40246

1.3

−1.33304

−0.223003

−2.26227

2.96335

1.4

−1.33304

−0.223003

2.26227

2.96335

1.5

−1.23705

−0.469715

−4.86027

3.05515

1.6

−1.23705

−0.469715

4.86027

3.05515

1.7

−0.496936

−1.75305

−1.76854

1.86503

1.8

−0.496936

−1.75305

1.76854

1.86503

1.9

0.496936

−1.75305

−1.76854

−1.86503

1.10

0.496936

−1.75305

1.76854

−1.86503

1.11

1.23705

−0.469715

−4.86027

−3.05515

1.12

1.23705

−0.469715

4.86027

−3.05515

1.13

1.33304

−0.223003

−2.26227

−2.96335

1.14

1.33304

−0.223003

2.26227

−2.96335

1.15

2.72869

5.44577

−0.363635

9.40246

1.16

2.72869

5.44577

0.363635

9.40246

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$5$	$ -1 $
$31$	$ +1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6975.2.a.ck		16
3.b	odd	2	1	inner	6975.2.a.ck		16
5.b	even	2	1	inner	6975.2.a.ck		16
5.c	odd	4	2		1395.2.c.g	✓	16
15.d	odd	2	1	inner	6975.2.a.ck		16
15.e	even	4	2		1395.2.c.g	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1395.2.c.g	✓	16	5.c	odd	4	2
1395.2.c.g	✓	16	15.e	even	4	2
6975.2.a.ck		16	1.a	even	1	1	trivial
6975.2.a.ck		16	3.b	odd	2	1	inner
6975.2.a.ck		16	5.b	even	2	1	inner
6975.2.a.ck		16	15.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6975))$:

$ T_{2}^{8} - 11T_{2}^{6} + 30T_{2}^{4} - 27T_{2}^{2} + 5 $

$ T_{7}^{8} - 32T_{7}^{6} + 215T_{7}^{4} - 406T_{7}^{2} + 50 $

$ T_{11}^{8} - 80T_{11}^{6} + 2248T_{11}^{4} - 26208T_{11}^{2} + 108160 $

$ T_{13}^{8} - 90T_{13}^{6} + 2746T_{13}^{4} - 31656T_{13}^{2} + 96800 $

$ T_{17}^{8} - 92T_{17}^{6} + 2820T_{17}^{4} - 31488T_{17}^{2} + 81920 $

$ T_{29}^{8} - 118T_{29}^{6} + 4602T_{29}^{4} - 66152T_{29}^{2} + 295840 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 11 T^{6} + 30 T^{4} + \cdots + 5)^{2} $ (T^8 - 11T^6 + 30T^4 - 27*T^2 + 5)^2
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} - 32 T^{6} + \cdots + 50)^{2} $ (T^8 - 32T^6 + 215T^4 - 406*T^2 + 50)^2
$11$	$ (T^{8} - 80 T^{6} + \cdots + 108160)^{2} $ (T^8 - 80T^6 + 2248T^4 - 26208*T^2 + 108160)^2
$13$	$ (T^{8} - 90 T^{6} + \cdots + 96800)^{2} $ (T^8 - 90T^6 + 2746T^4 - 31656*T^2 + 96800)^2
$17$	$ (T^{8} - 92 T^{6} + \cdots + 81920)^{2} $ (T^8 - 92T^6 + 2820T^4 - 31488*T^2 + 81920)^2
$19$	$ (T^{4} - 45 T^{2} + 52 T - 4)^{4} $ (T^4 - 45T^2 + 52T - 4)^4
$23$	$ (T^{8} - 120 T^{6} + \cdots + 54080)^{2} $ (T^8 - 120T^6 + 4388T^4 - 48048*T^2 + 54080)^2
$29$	$ (T^{8} - 118 T^{6} + \cdots + 295840)^{2} $ (T^8 - 118T^6 + 4602T^4 - 66152*T^2 + 295840)^2
$31$	$ (T + 1)^{16} $ (T + 1)^16
$37$	$ (T^{8} - 174 T^{6} + \cdots + 231200)^{2} $ (T^8 - 174T^6 + 8730T^4 - 140264*T^2 + 231200)^2
$41$	$ (T^{8} - 342 T^{6} + \cdots + 48841000)^{2} $ (T^8 - 342T^6 + 43117T^4 - 2383108*T^2 + 48841000)^2
$43$	$ (T^{8} - 144 T^{6} + \cdots + 80000)^{2} $ (T^8 - 144T^6 + 4808T^4 - 40416*T^2 + 80000)^2
$47$	$ (T^{8} - 196 T^{6} + \cdots + 81920)^{2} $ (T^8 - 196T^6 + 10436T^4 - 132608*T^2 + 81920)^2
$53$	$ (T^{8} - 148 T^{6} + \cdots + 619520)^{2} $ (T^8 - 148T^6 + 6596T^4 - 112832*T^2 + 619520)^2
$59$	$ (T^{8} - 180 T^{6} + \cdots + 156250)^{2} $ (T^8 - 180T^6 + 9875T^4 - 168750*T^2 + 156250)^2
$61$	$ (T^{4} - 16 T^{3} + \cdots - 2048)^{4} $ (T^4 - 16T^3 - 96T^2 + 1792*T - 2048)^4
$67$	$ (T^{8} - 414 T^{6} + \cdots + 2420000)^{2} $ (T^8 - 414T^6 + 53666T^4 - 2178576*T^2 + 2420000)^2
$71$	$ (T^{8} - 296 T^{6} + \cdots + 5550250)^{2} $ (T^8 - 296T^6 + 28339T^4 - 939978*T^2 + 5550250)^2
$73$	$ (T^{8} - 206 T^{6} + \cdots + 96800)^{2} $ (T^8 - 206T^6 + 9818T^4 - 70024*T^2 + 96800)^2
$79$	$ (T^{4} - 18 T^{3} + 62 T^{2} + \cdots + 8)^{4} $ (T^4 - 18T^3 + 62T^2 - 48*T + 8)^4
$83$	$ (T^{8} - 212 T^{6} + \cdots + 5000000)^{2} $ (T^8 - 212T^6 + 15732T^4 - 476832*T^2 + 5000000)^2
$89$	$ (T^{8} - 374 T^{6} + \cdots + 160)^{2} $ (T^8 - 374T^6 + 18082T^4 - 232752*T^2 + 160)^2
$97$	$ (T^{8} - 386 T^{6} + \cdots + 2121800)^{2} $ (T^8 - 386T^6 + 37033T^4 - 1071004*T^2 + 2121800)^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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + ( - \beta_{11} - \beta_{10} + \beta_1) q^{4} + ( - \beta_{12} - \beta_{9}) q^{7} + (\beta_{8} - \beta_{7} + \cdots - 2 \beta_{2}) q^{8}+O(q^{10}) \) q - b2 * q^2 + (-b11 - b10 + b1) * q^4 + (-b12 - b9) * q^7 + (b8 - b7 + 2b4 - 2b2) * q^8	\( q - \beta_{2} q^{2} + ( - \beta_{11} - \beta_{10} + \beta_1) q^{4} + ( - \beta_{12} - \beta_{9}) q^{7} + (\beta_{8} - \beta_{7} + \cdots - 2 \beta_{2}) q^{8}+ \cdots + ( - 3 \beta_{8} - 2 \beta_{7} + \cdots + 5 \beta_{2}) q^{98}+O(q^{100}) \) q - b2 * q^2 + (-b11 - b10 + b1) * q^4 + (-b12 - b9) * q^7 + (b8 - b7 + 2b4 - 2b2) * q^8 + (b15 + b14 - b13) * q^11 + (b6 + b3) * q^13 + (b14 + b13 + b5) * q^14 + (-2b11 - 4b10 + b1 + 1) * q^16 + (2b7 + b4 - b2) q^17 + (b11 - 2b10 - b1) q^19 + (-2b12 - b6 + 2b3) * q^22 + (2b8 + b4 - 3b2) * q^23 + (-2b15 - b13) q^26 + b12 * q^28 + (-b15 + b14 + 2b13 + 2b5) * q^29 - q^31 + (-b7 + 2b4 - 6b2) * q^32 + (-2b11 - b10 - b1 + 1) q^34 + (-2b12 + 2b9 - b3) * q^37 + (-b8 - 2b7 + b4 + b2) q^38 + (-2b14 + 2b13 + 5b5) q^41 + (-2b9 + b6) q^43 + (5b15 + 3b14 - 3b13 - 8b5) * q^44 + (-4b11 - 3b10 + 3b1 + 3) q^46 + (2b8 + 2b7 - b4 - b2) * q^47 + (3b11 + 3b10 + 2b1 + 2) q^49 + (2b12 - 2b9 - b3) * q^52 + (-2b8 - 2b7 - b4 + 3b2) q^53 + (-b15 - 3b14 - 2b13 - b5) * q^56 + (-4b9 + b6 - b3) q^58 + (2b15 - b13 + b5) q^59 + (4b10 + 4b1 + 4) * q^61 + b2 * q^62 + (-4b11 - 3b10 + 5b1 + 8) q^64 + (-4b12 - 2b9 + b6 + b3) * q^67 + (2b8 - 2b7 + b4 - 3b2) q^68 + (b15 + b14 - 4b13 - 3b5) * q^71 + (2b12 - 2b9 - b3) * q^73 + (3b15 + b14 - 4b5) * q^74 + (-2b11 + 3b1 - 2) * q^76 + (-4b4 - 2b2) * q^77 + (2b11 + b10 + b1 + 5) q^79 + (2b12 - 3b9 - 4b3) q^82 + (3b4 - b2) q^83 + (-5b15 - b14 + 3b13 + 6b5) q^86 + (-4b12 + 10b9 - 3b6 + 2b3) * q^88 + (-2b14 + 3b13) * q^89 + (-4b11 - b10 - 5b1 - 1) * q^91 + (-2b7 + 5b4 - 11b2) q^92 + (5b10 - b1 + 1) q^94 + (3b12 + 3b9 + b6 - b3) * q^97 + (-3b8 - 2b7 - 6b4 + 5b2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4}+O(q^{10}) \) 16 * q + 12 * q^4	\( 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) \) 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

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	6975.2.a.ck

Level	$6975$
Weight	$2$
Character orbit	6975.a
Self dual	yes
Analytic conductor	$55.696$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	yes
Analytic conductor:	\(55.6956554098\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 20x^{14} + 152x^{12} - 571x^{10} + 1130x^{8} - 1138x^{6} + 492x^{4} - 43x^{2} + 1 \) x^16 - 20x^14 + 152x^12 - 571x^10 + 1130x^8 - 1138x^6 + 492x^4 - 43*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{6}\cdot 5 \)
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{14} + 17\nu^{12} - 95\nu^{10} + 174\nu^{8} + 154\nu^{6} - 798\nu^{4} + 620\nu^{2} - 27 ) / 4 \) (-v^14 + 17v^12 - 95v^10 + 174v^8 + 154v^6 - 798v^4 + 620v^2 - 27) / 4
\(\beta_{2}\)	\(=\)	\( ( 6\nu^{14} - 109\nu^{12} + 708\nu^{10} - 2047\nu^{8} + 2449\nu^{6} - 431\nu^{4} - 751\nu^{2} + 41 ) / 8 \) (6v^14 - 109v^12 + 708v^10 - 2047v^8 + 2449v^6 - 431v^4 - 751*v^2 + 41) / 8
\(\beta_{3}\)	\(=\)	\( ( 10\nu^{15} - 175\nu^{13} + 1052\nu^{11} - 2509\nu^{9} + 1123\nu^{7} + 3827\nu^{5} - 3981\nu^{3} + 307\nu ) / 8 \) (10v^15 - 175v^13 + 1052v^11 - 2509v^9 + 1123v^7 + 3827v^5 - 3981v^3 + 307v) / 8
\(\beta_{4}\)	\(=\)	\( ( -4\nu^{14} + 81\nu^{12} - 626\nu^{10} + 2401\nu^{8} - 4871\nu^{6} + 5033\nu^{4} - 2163\nu^{2} + 95 ) / 8 \) (-4v^14 + 81v^12 - 626v^10 + 2401v^8 - 4871v^6 + 5033v^4 - 2163*v^2 + 95) / 8
\(\beta_{5}\)	\(=\)	\( ( -7\nu^{15} + 133\nu^{13} - 933\nu^{11} + 3100\nu^{9} - 5044\nu^{7} + 3624\nu^{5} - 790\nu^{3} - 27\nu ) / 8 \) (-7v^15 + 133v^13 - 933v^11 + 3100v^9 - 5044v^7 + 3624v^5 - 790v^3 - 27v) / 8
\(\beta_{6}\)	\(=\)	\( ( -11\nu^{15} + 201\nu^{13} - 1321\nu^{11} + 3924\nu^{9} - 5108\nu^{7} + 1896\nu^{5} + 538\nu^{3} + 65\nu ) / 8 \) (-11v^15 + 201v^13 - 1321v^11 + 3924v^9 - 5108v^7 + 1896v^5 + 538v^3 + 65v) / 8
\(\beta_{7}\)	\(=\)	\( ( -18\nu^{14} + 337\nu^{12} - 2304\nu^{10} + 7311\nu^{8} - 10865\nu^{6} + 6247\nu^{4} - 417\nu^{2} - 13 ) / 8 \) (-18v^14 + 337v^12 - 2304v^10 + 7311v^8 - 10865v^6 + 6247v^4 - 417*v^2 - 13) / 8
\(\beta_{8}\)	\(=\)	\( ( 9\nu^{14} - 168\nu^{12} + 1145\nu^{10} - 3629\nu^{8} + 5427\nu^{6} - 3245\nu^{4} + 361\nu^{2} - 14 ) / 4 \) (9v^14 - 168v^12 + 1145v^10 - 3629v^8 + 5427v^6 - 3245v^4 + 361*v^2 - 14) / 4
\(\beta_{9}\)	\(=\)	\( ( 5\nu^{15} - 115\nu^{13} + 1039\nu^{11} - 4744\nu^{9} + 11560\nu^{7} - 14268\nu^{5} + 7110\nu^{3} - 331\nu ) / 8 \) (5v^15 - 115v^13 + 1039v^11 - 4744v^9 + 11560v^7 - 14268v^5 + 7110v^3 - 331v) / 8
\(\beta_{10}\)	\(=\)	\( ( -19\nu^{14} + 359\nu^{12} - 2493\nu^{10} + 8130\nu^{8} - 12758\nu^{6} + 8470\nu^{4} - 1504\nu^{2} + 67 ) / 8 \) (-19v^14 + 359v^12 - 2493v^10 + 8130v^8 - 12758v^6 + 8470v^4 - 1504*v^2 + 67) / 8
\(\beta_{11}\)	\(=\)	\( 3\nu^{14} - 57\nu^{12} + 400\nu^{10} - 1331\nu^{8} + 2176\nu^{6} - 1590\nu^{4} + 383\nu^{2} - 16 \) 3v^14 - 57v^12 + 400v^10 - 1331v^8 + 2176v^6 - 1590v^4 + 383*v^2 - 16
\(\beta_{12}\)	\(=\)	\( ( 21\nu^{15} - 406\nu^{13} + 2929\nu^{11} - 10183\nu^{9} + 17937\nu^{7} - 15099\nu^{5} + 4899\nu^{3} - 192\nu ) / 8 \) (21v^15 - 406v^13 + 2929v^11 - 10183v^9 + 17937v^7 - 15099v^5 + 4899v^3 - 192v) / 8
\(\beta_{13}\)	\(=\)	\( ( -21\nu^{15} + 416\nu^{13} - 3117\nu^{11} + 11473\nu^{9} - 22031\nu^{7} + 21141\nu^{5} - 8305\nu^{3} + 414\nu ) / 8 \) (-21v^15 + 416v^13 - 3117v^11 + 11473v^9 - 22031v^7 + 21141v^5 - 8305v^3 + 414v) / 8
\(\beta_{14}\)	\(=\)	\( ( 17\nu^{15} - 326\nu^{13} + 2321\nu^{11} - 7899\nu^{9} + 13421\nu^{7} - 10599\nu^{5} + 3115\nu^{3} - 200\nu ) / 4 \) (17v^15 - 326v^13 + 2321v^11 - 7899v^9 + 13421v^7 - 10599v^5 + 3115v^3 - 200v) / 4
\(\beta_{15}\)	\(=\)	\( ( -40\nu^{15} + 775\nu^{13} - 5610\nu^{11} + 19603\nu^{9} - 34789\nu^{7} + 29611\nu^{5} - 9801\nu^{3} + 449\nu ) / 8 \) (-40v^15 + 775v^13 - 5610v^11 + 19603v^9 - 34789v^7 + 29611v^5 - 9801v^3 + 449v) / 8

\(\nu\)	\(=\)	\( ( \beta_{12} - 3\beta_{9} + \beta_{6} - 5\beta_{5} - 3\beta_{3} ) / 10 \) (b12 - 3b9 + b6 - 5b5 - 3*b3) / 10
\(\nu^{2}\)	\(=\)	\( ( -\beta_{11} - 2\beta_{10} + \beta_{7} + \beta_{2} + \beta _1 + 4 ) / 2 \) (-b11 - 2*b10 + b7 + b2 + b1 + 4) / 2
\(\nu^{3}\)	\(=\)	\( ( 10\beta_{15} + 5\beta_{14} - 10\beta_{13} + 8\beta_{12} - 19\beta_{9} + 8\beta_{6} - 25\beta_{5} - 14\beta_{3} ) / 10 \) (10b15 + 5b14 - 10b13 + 8b12 - 19b9 + 8b6 - 25b5 - 14b3) / 10
\(\nu^{4}\)	\(=\)	\( ( -9\beta_{11} - 14\beta_{10} - 2\beta_{8} + 5\beta_{7} - 4\beta_{4} + 13\beta_{2} + 9\beta _1 + 16 ) / 2 \) (-9b11 - 14b10 - 2b8 + 5b7 - 4b4 + 13b2 + 9*b1 + 16) / 2
\(\nu^{5}\)	\(=\)	\( ( 115 \beta_{15} + 55 \beta_{14} - 95 \beta_{13} + 74 \beta_{12} - 122 \beta_{9} + 59 \beta_{6} + \cdots - 82 \beta_{3} ) / 10 \) (115b15 + 55b14 - 95b13 + 74b12 - 122b9 + 59b6 - 180b5 - 82b3) / 10
\(\nu^{6}\)	\(=\)	\( ( -72\beta_{11} - 102\beta_{10} - 21\beta_{8} + 33\beta_{7} - 47\beta_{4} + 118\beta_{2} + 67\beta _1 + 88 ) / 2 \) (-72b11 - 102b10 - 21b8 + 33b7 - 47b4 + 118b2 + 67*b1 + 88) / 2
\(\nu^{7}\)	\(=\)	\( ( 1035 \beta_{15} + 500 \beta_{14} - 775 \beta_{13} + 617 \beta_{12} - 846 \beta_{9} + \cdots - 561 \beta_{3} ) / 10 \) (1035b15 + 500b14 - 775b13 + 617b12 - 846b9 + 442b6 - 1410b5 - 561b3) / 10
\(\nu^{8}\)	\(=\)	\( ( -563\beta_{11} - 770\beta_{10} - 184\beta_{8} + 245\beta_{7} - 426\beta_{4} + 983\beta_{2} + 499\beta _1 + 584 ) / 2 \) (-563b11 - 770b10 - 184b8 + 245b7 - 426b4 + 983b2 + 499*b1 + 584) / 2
\(\nu^{9}\)	\(=\)	\( ( 8605 \beta_{15} + 4200 \beta_{14} - 6155 \beta_{13} + 4931 \beta_{12} - 6218 \beta_{9} + \cdots - 4133 \beta_{3} ) / 10 \) (8605b15 + 4200b14 - 6155b13 + 4931b12 - 6218b9 + 3386b6 - 11180b5 - 4133b3) / 10
\(\nu^{10}\)	\(=\)	\( - 2196 \beta_{11} - 2964 \beta_{10} - 760 \beta_{8} + 946 \beta_{7} - 1775 \beta_{4} + 3965 \beta_{2} + \cdots + 2127 \) -2196b11 - 2964b10 - 760b8 + 946b7 - 1775b4 + 3965b2 + 1897*b1 + 2127
\(\nu^{11}\)	\(=\)	\( ( 69310 \beta_{15} + 34035 \beta_{14} - 48570 \beta_{13} + 38868 \beta_{12} - 47259 \beta_{9} + \cdots - 31514 \beta_{3} ) / 10 \) (69310b15 + 34035b14 - 48570b13 + 38868b12 - 47259b9 + 26268b6 - 88485b5 - 31514b3) / 10
\(\nu^{12}\)	\(=\)	\( ( - 34323 \beta_{11} - 46096 \beta_{10} - 12229 \beta_{8} + 14792 \beta_{7} - 28611 \beta_{4} + \cdots + 32322 ) / 2 \) (-34323b11 - 46096b10 - 12229b8 + 14792b7 - 28611b4 + 63073b2 + 29288*b1 + 32322) / 2
\(\nu^{13}\)	\(=\)	\( ( 550635 \beta_{15} + 271230 \beta_{14} - 382405 \beta_{13} + 305219 \beta_{12} - 365392 \beta_{9} + \cdots - 244137 \beta_{3} ) / 10 \) (550635b15 + 271230b14 - 382405b13 + 305219b12 - 365392b9 + 205054b6 - 698200b5 - 244137b3) / 10
\(\nu^{14}\)	\(=\)	\( ( - 268739 \beta_{11} - 360228 \beta_{10} - 97147 \beta_{8} + 116066 \beta_{7} - 227307 \beta_{4} + \cdots + 250170 ) / 2 \) (-268739b11 - 360228b10 - 97147b8 + 116066b7 - 227307b4 + 498351b2 + 228040*b1 + 250170) / 2
\(\nu^{15}\)	\(=\)	\( ( 4347435 \beta_{15} + 2144615 \beta_{14} - 3007405 \beta_{13} + 2395152 \beta_{12} + \cdots - 1905186 \beta_{3} ) / 10 \) (4347435b15 + 2144615b14 - 3007405b13 + 2395152b12 - 2848586b9 + 1605537b6 - 5497510b5 - 1905186b3) / 10

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 11 T^{6} + 30 T^{4} + \cdots + 5)^{2} \) (T^8 - 11T^6 + 30T^4 - 27*T^2 + 5)^2
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} - 32 T^{6} + \cdots + 50)^{2} \) (T^8 - 32T^6 + 215T^4 - 406*T^2 + 50)^2
$11$	\( (T^{8} - 80 T^{6} + \cdots + 108160)^{2} \) (T^8 - 80T^6 + 2248T^4 - 26208*T^2 + 108160)^2
$13$	\( (T^{8} - 90 T^{6} + \cdots + 96800)^{2} \) (T^8 - 90T^6 + 2746T^4 - 31656*T^2 + 96800)^2
$17$	\( (T^{8} - 92 T^{6} + \cdots + 81920)^{2} \) (T^8 - 92T^6 + 2820T^4 - 31488*T^2 + 81920)^2
$19$	\( (T^{4} - 45 T^{2} + 52 T - 4)^{4} \) (T^4 - 45T^2 + 52T - 4)^4
$23$	\( (T^{8} - 120 T^{6} + \cdots + 54080)^{2} \) (T^8 - 120T^6 + 4388T^4 - 48048*T^2 + 54080)^2
$29$	\( (T^{8} - 118 T^{6} + \cdots + 295840)^{2} \) (T^8 - 118T^6 + 4602T^4 - 66152*T^2 + 295840)^2
$31$	\( (T + 1)^{16} \) (T + 1)^16
$37$	\( (T^{8} - 174 T^{6} + \cdots + 231200)^{2} \) (T^8 - 174T^6 + 8730T^4 - 140264*T^2 + 231200)^2
$41$	\( (T^{8} - 342 T^{6} + \cdots + 48841000)^{2} \) (T^8 - 342T^6 + 43117T^4 - 2383108*T^2 + 48841000)^2
$43$	\( (T^{8} - 144 T^{6} + \cdots + 80000)^{2} \) (T^8 - 144T^6 + 4808T^4 - 40416*T^2 + 80000)^2
$47$	\( (T^{8} - 196 T^{6} + \cdots + 81920)^{2} \) (T^8 - 196T^6 + 10436T^4 - 132608*T^2 + 81920)^2
$53$	\( (T^{8} - 148 T^{6} + \cdots + 619520)^{2} \) (T^8 - 148T^6 + 6596T^4 - 112832*T^2 + 619520)^2
$59$	\( (T^{8} - 180 T^{6} + \cdots + 156250)^{2} \) (T^8 - 180T^6 + 9875T^4 - 168750*T^2 + 156250)^2
$61$	\( (T^{4} - 16 T^{3} + \cdots - 2048)^{4} \) (T^4 - 16T^3 - 96T^2 + 1792*T - 2048)^4
$67$	\( (T^{8} - 414 T^{6} + \cdots + 2420000)^{2} \) (T^8 - 414T^6 + 53666T^4 - 2178576*T^2 + 2420000)^2
$71$	\( (T^{8} - 296 T^{6} + \cdots + 5550250)^{2} \) (T^8 - 296T^6 + 28339T^4 - 939978*T^2 + 5550250)^2
$73$	\( (T^{8} - 206 T^{6} + \cdots + 96800)^{2} \) (T^8 - 206T^6 + 9818T^4 - 70024*T^2 + 96800)^2
$79$	\( (T^{4} - 18 T^{3} + 62 T^{2} + \cdots + 8)^{4} \) (T^4 - 18T^3 + 62T^2 - 48*T + 8)^4
$83$	\( (T^{8} - 212 T^{6} + \cdots + 5000000)^{2} \) (T^8 - 212T^6 + 15732T^4 - 476832*T^2 + 5000000)^2
$89$	\( (T^{8} - 374 T^{6} + \cdots + 160)^{2} \) (T^8 - 374T^6 + 18082T^4 - 232752*T^2 + 160)^2
$97$	\( (T^{8} - 386 T^{6} + \cdots + 2121800)^{2} \) (T^8 - 386T^6 + 37033T^4 - 1071004*T^2 + 2121800)^2
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\( p \)	Sign
\(3\)	\( +1 \)
\(5\)	\( -1 \)
\(31\)	\( +1 \)