LMFDB - Newform orbit 9360.2.k.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(4031,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 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("9360.4031");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.k (of order $2$, degree $1$, 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:	$74.7399762919$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_1 q^{5} - \beta_{10} q^{7}+O(q^{10}) $ q + b1 * q^5 - b10 * q^7	$ q + \beta_1 q^{5} - \beta_{10} q^{7} - \beta_{3} q^{11} - q^{13} + ( - \beta_{14} - \beta_{5} - \beta_1) q^{17} + \beta_{6} q^{19} + (\beta_{15} - \beta_{12}) q^{23} - q^{25} + ( - \beta_{8} - 2 \beta_{5} - 2 \beta_1) q^{29} + 2 \beta_{2} q^{31} - \beta_{12} q^{35} + (2 \beta_{11} - 3 \beta_{9} - \beta_{7} - 2) q^{37} + (\beta_{8} + 3 \beta_{5} + \beta_1) q^{41} + \beta_{6} q^{43} + (\beta_{15} + \beta_{12} + \cdots + \beta_{3}) q^{47}+ \cdots + (2 \beta_{11} - \beta_{9} - \beta_{7} - 2) q^{97}+O(q^{100}) $ q + b1 * q^5 - b10 * q^7 - b3 * q^11 - q^13 + (-b14 - b5 - b1) * q^17 + b6 * q^19 + (b15 - b12) * q^23 - q^25 + (-b8 - 2b5 - 2b1) * q^29 + 2b2 q^31 - b12 * q^35 + (2b11 - 3b9 - b7 - 2) * q^37 + (b8 + 3b5 + b1) q^41 + b6 * q^43 + (b15 + b12 + b4 + b3) * q^47 + (-b11 + b9 - b7 - 1) * q^49 + (-b14 + b5 - 3b1) q^53 - b2 * q^55 + (-b15 - b12 - b4 - b3) * q^59 + (b11 + 3b9 + b7) q^61 - b1 * q^65 + (-b13 + 2b10 + b6) q^67 + (-2b12 + 3b3) * q^71 - 2 * q^73 + (b14 - b8 - b5 - b1) * q^77 + (b13 - b10) * q^79 + (b15 - b12 - b4 + b3) * q^83 + (b11 - b9) * q^85 + (-b8 - b5 - 3b1) q^89 + b10 * q^91 + b4 * q^95 + (2b11 - b9 - b7 - 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{13} - 16 q^{25} - 8 q^{37} - 24 q^{49} - 24 q^{61} - 32 q^{73} + 8 q^{85} - 24 q^{97}+O(q^{100}) $ 16 * q - 16 * q^13 - 16 * q^25 - 8 * q^37 - 24 * q^49 - 24 * q^61 - 32 * q^73 + 8 * q^85 - 24 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7*x^12 + 40*x^8 - 112*x^4 + 256 :

$\beta_{1}$	$=$	$ ( \nu^{10} - 3\nu^{6} + 12\nu^{2} ) / 16 $ (v^10 - 3v^6 + 12v^2) / 16
$\beta_{2}$	$=$	$ ( - \nu^{15} + 2 \nu^{12} - \nu^{11} - 16 \nu^{9} + 10 \nu^{8} - 16 \nu^{7} + 48 \nu^{5} - 56 \nu^{4} + \cdots + 256 ) / 128 $ (-v^15 + 2v^12 - v^11 - 16v^9 + 10v^8 - 16v^7 + 48v^5 - 56v^4 + 16v^3 - 320v + 256) / 128
$\beta_{3}$	$=$	$ ( \nu^{15} + \nu^{14} + \nu^{11} - 3 \nu^{10} - 16 \nu^{9} + 16 \nu^{7} - 4 \nu^{6} + 48 \nu^{5} + \cdots - 320 \nu ) / 128 $ (v^15 + v^14 + v^11 - 3v^10 - 16v^9 + 16v^7 - 4v^6 + 48v^5 - 16v^3 + 96v^2 - 320v) / 128
$\beta_{4}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - \nu^{11} - 6 \nu^{10} + 16 \nu^{9} - 16 \nu^{7} - 8 \nu^{6} + \cdots + 320 \nu ) / 128 $ (-v^15 + 2v^14 - v^11 - 6v^10 + 16v^9 - 16v^7 - 8v^6 - 48v^5 + 16v^3 + 192v^2 + 320*v) / 128
$\beta_{5}$	$=$	$ ( \nu^{15} - 4 \nu^{14} - 6 \nu^{13} - 3 \nu^{11} + 12 \nu^{10} + 34 \nu^{9} - 4 \nu^{7} + \cdots + 256 \nu ) / 256 $ (v^15 - 4v^14 - 6v^13 - 3v^11 + 12v^10 + 34v^9 - 4v^7 - 112v^6 - 152v^5 - 32v^3 + 256v) / 256
$\beta_{6}$	$=$	$ ( - \nu^{15} - 4 \nu^{12} - \nu^{11} - 16 \nu^{9} - 20 \nu^{8} - 16 \nu^{7} + 48 \nu^{5} + 112 \nu^{4} + \cdots - 512 ) / 128 $ (-v^15 - 4v^12 - v^11 - 16v^9 - 20v^8 - 16v^7 + 48v^5 + 112v^4 + 16v^3 - 320v - 512) / 128
$\beta_{7}$	$=$	$ ( -\nu^{15} + 6\nu^{13} + 27\nu^{11} - 66\nu^{9} - 132\nu^{7} + 248\nu^{5} + 384\nu^{3} - 384\nu ) / 256 $ (-v^15 + 6v^13 + 27v^11 - 66v^9 - 132v^7 + 248v^5 + 384v^3 - 384*v) / 256
$\beta_{8}$	$=$	$ ( \nu^{15} + 6\nu^{13} - 27\nu^{11} - 66\nu^{9} + 132\nu^{7} + 248\nu^{5} - 384\nu^{3} - 384\nu ) / 256 $ (v^15 + 6v^13 - 27v^11 - 66v^9 + 132v^7 + 248v^5 - 384v^3 - 384*v) / 256
$\beta_{9}$	$=$	$ ( - \nu^{15} - 6 \nu^{13} - 16 \nu^{12} + 3 \nu^{11} + 34 \nu^{9} + 112 \nu^{8} + 4 \nu^{7} + \cdots + 768 ) / 256 $ (-v^15 - 6v^13 - 16v^12 + 3v^11 + 34v^9 + 112v^8 + 4v^7 - 152v^5 - 384v^4 + 32v^3 + 256v + 768) / 256
$\beta_{10}$	$=$	$ ( - \nu^{15} + \nu^{13} - 2 \nu^{12} + 5 \nu^{11} - 3 \nu^{9} + 6 \nu^{8} - 18 \nu^{7} - 4 \nu^{5} + \cdots + 64 ) / 64 $ (-v^15 + v^13 - 2v^12 + 5v^11 - 3v^9 + 6v^8 - 18v^7 - 4v^5 - 56v^4 + 8v^3 + 32*v + 64) / 64
$\beta_{11}$	$=$	$ ( -\nu^{15} - 3\nu^{13} + 9\nu^{11} + 9\nu^{9} - 30\nu^{7} - 52\nu^{5} + 120\nu^{3} + 96\nu ) / 64 $ (-v^15 - 3v^13 + 9v^11 + 9v^9 - 30v^7 - 52v^5 + 120v^3 + 96*v) / 64
$\beta_{12}$	$=$	$ ( - \nu^{15} - 2 \nu^{14} - \nu^{13} + 5 \nu^{11} + 10 \nu^{10} + 3 \nu^{9} - 18 \nu^{7} - 36 \nu^{6} + \cdots - 32 \nu ) / 64 $ (-v^15 - 2v^14 - v^13 + 5v^11 + 10v^10 + 3v^9 - 18v^7 - 36v^6 + 4v^5 + 8v^3 + 80v^2 - 32v) / 64
$\beta_{13}$	$=$	$ ( - \nu^{15} + \nu^{13} + 4 \nu^{12} + 5 \nu^{11} - 3 \nu^{9} - 12 \nu^{8} - 18 \nu^{7} - 4 \nu^{5} + \cdots - 128 ) / 64 $ (-v^15 + v^13 + 4v^12 + 5v^11 - 3v^9 - 12v^8 - 18v^7 - 4v^5 + 112v^4 + 8v^3 + 32*v - 128) / 64
$\beta_{14}$	$=$	$ ( -\nu^{15} + 3\nu^{13} + 9\nu^{11} - 9\nu^{9} - 30\nu^{7} + 52\nu^{5} + 120\nu^{3} - 96\nu ) / 64 $ (-v^15 + 3v^13 + 9v^11 - 9v^9 - 30v^7 + 52v^5 + 120v^3 - 96*v) / 64
$\beta_{15}$	$=$	$ ( \nu^{15} - 4 \nu^{14} + \nu^{13} - 5 \nu^{11} + 20 \nu^{10} - 3 \nu^{9} + 18 \nu^{7} - 72 \nu^{6} + \cdots + 32 \nu ) / 64 $ (v^15 - 4v^14 + v^13 - 5v^11 + 20v^10 - 3v^9 + 18v^7 - 72v^6 - 4v^5 - 8v^3 + 160v^2 + 32v) / 64

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

$\nu$	$=$	$ ( -\beta_{14} + \beta_{11} + 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{4} - 2\beta_{3} - 2\beta_{2} ) / 12 $ (-b14 + b11 + 2b8 + 2b7 - b6 + b4 - 2b3 - 2b2) / 12
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} + \beta_{12} - \beta_{8} - 3\beta_{5} + 2\beta_{4} + 2\beta_{3} - 3\beta_1 ) / 6 $ (b15 - b14 + b12 - b8 - 3b5 + 2b4 + 2b3 - 3b1) / 6
$\nu^{3}$	$=$	$ ( 2 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 5 \beta_{11} - 4 \beta_{10} + \cdots + 2 \beta_{2} ) / 12 $ (2b15 + 5b14 - 2b13 - 4b12 + 5b11 - 4b10 + 2b8 - 2b7 + b6 + b4 - 2b3 + 2b2) / 12
$\nu^{4}$	$=$	$ ( 3\beta_{13} - \beta_{11} - 3\beta_{10} + 3\beta_{9} + \beta_{7} + 2\beta_{6} - 2\beta_{2} + 12 ) / 6 $ (3b13 - b11 - 3b10 + 3b9 + b7 + 2b6 - 2*b2 + 12) / 6
$\nu^{5}$	$=$	$ ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - \beta_{11} - 4 \beta_{10} + \cdots - 2 \beta_{2} ) / 4 $ (-2b15 + b14 - 2b13 + 4b12 - b11 - 4b10 + 2b8 + 2b7 - b6 + b4 - 2b3 - 2b2) / 4
$\nu^{6}$	$=$	$ ( 3\beta_{15} - 7\beta_{14} + 3\beta_{12} - 7\beta_{8} - 21\beta_{5} - 2\beta_{4} - 2\beta_{3} - 9\beta_1 ) / 6 $ (3b15 - 7b14 + 3b12 - 7b8 - 21b5 - 2b4 - 2b3 - 9b1) / 6
$\nu^{7}$	$=$	$ ( 2 \beta_{15} + 19 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 19 \beta_{11} - 4 \beta_{10} + \cdots - 10 \beta_{2} ) / 12 $ (2b15 + 19b14 - 2b13 - 4b12 + 19b11 - 4b10 + 22b8 - 22b7 - 5b6 - 5b4 + 10b3 - 10b2) / 12
$\nu^{8}$	$=$	$ ( 13\beta_{13} - 7\beta_{11} - 13\beta_{10} + 21\beta_{9} + 7\beta_{7} - 2\beta_{6} + 2\beta_{2} - 36 ) / 6 $ (13b13 - 7b11 - 13b10 + 21b9 + 7b7 - 2b6 + 2*b2 - 36) / 6
$\nu^{9}$	$=$	$ ( - 18 \beta_{15} + 29 \beta_{14} - 18 \beta_{13} + 36 \beta_{12} - 29 \beta_{11} - 36 \beta_{10} + \cdots - 10 \beta_{2} ) / 12 $ (-18b15 + 29b14 - 18b13 + 36b12 - 29b11 - 36b10 - 22b8 - 22b7 - 5b6 + 5b4 - 10b3 - 10b2) / 12
$\nu^{10}$	$=$	$ ( -\beta_{15} - 3\beta_{14} - \beta_{12} - 3\beta_{8} - 9\beta_{5} - 10\beta_{4} - 10\beta_{3} + 35\beta_1 ) / 2 $ (-b15 - 3b14 - b12 - 3b8 - 9b5 - 10b4 - 10b3 + 35b1) / 2
$\nu^{11}$	$=$	$ ( - 18 \beta_{15} + 13 \beta_{14} + 18 \beta_{13} + 36 \beta_{12} + 13 \beta_{11} + 36 \beta_{10} + \cdots - 86 \beta_{2} ) / 12 $ (-18b15 + 13b14 + 18b13 + 36b12 + 13b11 + 36b10 + 10b8 - 10b7 - 43b6 - 43b4 + 86b3 - 86b2) / 12
$\nu^{12}$	$=$	$ ( 19\beta_{13} + 7\beta_{11} - 19\beta_{10} - 21\beta_{9} - 7\beta_{7} - 62\beta_{6} + 62\beta_{2} - 252 ) / 6 $ (19b13 + 7b11 - 19b10 - 21b9 - 7b7 - 62b6 + 62*b2 - 252) / 6
$\nu^{13}$	$=$	$ ( 50 \beta_{15} + 131 \beta_{14} + 50 \beta_{13} - 100 \beta_{12} - 131 \beta_{11} + 100 \beta_{10} + \cdots + 10 \beta_{2} ) / 12 $ (50b15 + 131b14 + 50b13 - 100b12 - 131b11 + 100b10 - 106b8 - 106b7 + 5b6 - 5b4 + 10b3 + 10b2) / 12
$\nu^{14}$	$=$	$ ( -93\beta_{15} + 41\beta_{14} - 93\beta_{12} + 41\beta_{8} + 123\beta_{5} - 34\beta_{4} - 34\beta_{3} + 567\beta_1 ) / 6 $ (-93b15 + 41b14 - 93b12 + 41b8 + 123b5 - 34b4 - 34b3 + 567b1) / 6
$\nu^{15}$	$=$	$ ( 6 \beta_{15} - 79 \beta_{14} - 6 \beta_{13} - 12 \beta_{12} - 79 \beta_{11} - 12 \beta_{10} + \cdots - 78 \beta_{2} ) / 4 $ (6b15 - 79b14 - 6b13 - 12b12 - 79b11 - 12b10 - 110b8 + 110b7 - 39b6 - 39b4 + 78b3 - 78b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9360\mathbb{Z}\right)^\times$.

$n$	$2081$	$2341$	$5617$	$5761$	$8191$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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} $

4031.1

0.481610	+	1.32968i
1.38588	−	0.281691i
−1.32968	−	0.481610i
−1.38588	+	0.281691i
0.281691	−	1.38588i
−0.481610	−	1.32968i
−0.281691	+	1.38588i
1.32968	+	0.481610i
1.32968	−	0.481610i
−0.281691	−	1.38588i
−0.481610	+	1.32968i
0.281691	+	1.38588i
−1.38588	−	0.281691i
−1.32968	+	0.481610i
1.38588	+	0.281691i
0.481610	−	1.32968i

−

1.00000i

−

4.21980i

4031.2

−

1.00000i

−

3.29472i

4031.3

−

1.00000i

−

1.82110i

4031.4

−

1.00000i

−

1.42187i

4031.5

−

1.00000i

1.42187i

4031.6

−

1.00000i

1.82110i

4031.7

−

1.00000i

3.29472i

4031.8

−

1.00000i

4.21980i

4031.9

1.00000i

−

4.21980i

4031.10

1.00000i

−

3.29472i

4031.11

1.00000i

−

1.82110i

4031.12

1.00000i

−

1.42187i

4031.13

1.00000i

1.42187i

4031.14

1.00000i

1.82110i

4031.15

1.00000i

3.29472i

4031.16

1.00000i

4.21980i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9360.2.k.a	✓	16
3.b	odd	2	1	inner	9360.2.k.a	✓	16
4.b	odd	2	1	inner	9360.2.k.a	✓	16
12.b	even	2	1	inner	9360.2.k.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9360.2.k.a	✓	16	1.a	even	1	1	trivial
9360.2.k.a	✓	16	3.b	odd	2	1	inner
9360.2.k.a	✓	16	4.b	odd	2	1	inner
9360.2.k.a	✓	16	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} + 34T_{7}^{6} + 353T_{7}^{4} + 1224T_{7}^{2} + 1296 $ T7^8 + 34*T7^6 + 353*T7^4 + 1224*T7^2 + 1296 acting on $S_{2}^{\mathrm{new}}(9360, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ (T^{8} + 34 T^{6} + \cdots + 1296)^{2} $ (T^8 + 34T^6 + 353T^4 + 1224*T^2 + 1296)^2
$11$	$ (T^{8} - 38 T^{6} + \cdots + 324)^{2} $ (T^8 - 38T^6 + 461T^4 - 1764*T^2 + 324)^2
$13$	$ (T + 1)^{16} $ (T + 1)^16
$17$	$ (T^{8} + 54 T^{6} + \cdots + 324)^{2} $ (T^8 + 54T^6 + 829T^4 + 2700*T^2 + 324)^2
$19$	$ (T^{8} + 68 T^{6} + \cdots + 20736)^{2} $ (T^8 + 68T^6 + 1412T^4 + 9792*T^2 + 20736)^2
$23$	$ (T^{8} - 94 T^{6} + \cdots + 5184)^{2} $ (T^8 - 94T^6 + 2609T^4 - 22608*T^2 + 5184)^2
$29$	$ (T^{8} + 108 T^{6} + \cdots + 5184)^{2} $ (T^8 + 108T^6 + 3316T^4 + 21600*T^2 + 5184)^2
$31$	$ (T^{8} + 152 T^{6} + \cdots + 82944)^{2} $ (T^8 + 152T^6 + 7376T^4 + 112896*T^2 + 82944)^2
$37$	$ (T^{4} + 2 T^{3} + \cdots + 612)^{4} $ (T^4 + 2T^3 - 101T^2 + 204*T + 612)^4
$41$	$ (T^{8} + 206 T^{6} + \cdots + 374544)^{2} $ (T^8 + 206T^6 + 10609T^4 + 165240*T^2 + 374544)^2
$43$	$ (T^{8} + 68 T^{6} + \cdots + 20736)^{2} $ (T^8 + 68T^6 + 1412T^4 + 9792*T^2 + 20736)^2
$47$	$ (T^{4} - 90 T^{2} + 648)^{4} $ (T^4 - 90*T^2 + 648)^4
$53$	$ (T^{8} + 150 T^{6} + \cdots + 324)^{2} $ (T^8 + 150T^6 + 829T^4 + 972*T^2 + 324)^2
$59$	$ (T^{4} - 90 T^{2} + 648)^{4} $ (T^4 - 90*T^2 + 648)^4
$61$	$ (T^{4} + 6 T^{3} + \cdots - 4496)^{4} $ (T^4 + 6T^3 - 167T^2 - 1752*T - 4496)^4
$67$	$ (T^{8} + 272 T^{6} + \cdots + 20736)^{2} $ (T^8 + 272T^6 + 10592T^4 + 39168*T^2 + 20736)^2
$71$	$ (T^{8} - 454 T^{6} + \cdots + 104898564)^{2} $ (T^8 - 454T^6 + 70349T^4 - 4547268*T^2 + 104898564)^2
$73$	$ (T + 2)^{16} $ (T + 2)^16
$79$	$ (T^{4} + 63 T^{2} + 648)^{4} $ (T^4 + 63*T^2 + 648)^4
$83$	$ (T^{8} - 188 T^{6} + \cdots + 82944)^{2} $ (T^8 - 188T^6 + 10436T^4 - 180864*T^2 + 82944)^2
$89$	$ (T^{8} + 78 T^{6} + \cdots + 1296)^{2} $ (T^8 + 78T^6 + 769T^4 + 2088*T^2 + 1296)^2
$97$	$ (T^{4} + 6 T^{3} + \cdots + 188)^{4} $ (T^4 + 6T^3 - 61T^2 + 28*T + 188)^4
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{5} - \beta_{10} q^{7}+O(q^{10}) \) q + b1 * q^5 - b10 * q^7	\( q + \beta_1 q^{5} - \beta_{10} q^{7} - \beta_{3} q^{11} - q^{13} + ( - \beta_{14} - \beta_{5} - \beta_1) q^{17} + \beta_{6} q^{19} + (\beta_{15} - \beta_{12}) q^{23} - q^{25} + ( - \beta_{8} - 2 \beta_{5} - 2 \beta_1) q^{29} + 2 \beta_{2} q^{31} - \beta_{12} q^{35} + (2 \beta_{11} - 3 \beta_{9} - \beta_{7} - 2) q^{37} + (\beta_{8} + 3 \beta_{5} + \beta_1) q^{41} + \beta_{6} q^{43} + (\beta_{15} + \beta_{12} + \cdots + \beta_{3}) q^{47}+ \cdots + (2 \beta_{11} - \beta_{9} - \beta_{7} - 2) q^{97}+O(q^{100}) \) q + b1 * q^5 - b10 * q^7 - b3 * q^11 - q^13 + (-b14 - b5 - b1) * q^17 + b6 * q^19 + (b15 - b12) * q^23 - q^25 + (-b8 - 2b5 - 2b1) * q^29 + 2b2 q^31 - b12 * q^35 + (2b11 - 3b9 - b7 - 2) * q^37 + (b8 + 3b5 + b1) q^41 + b6 * q^43 + (b15 + b12 + b4 + b3) * q^47 + (-b11 + b9 - b7 - 1) * q^49 + (-b14 + b5 - 3b1) q^53 - b2 * q^55 + (-b15 - b12 - b4 - b3) * q^59 + (b11 + 3b9 + b7) q^61 - b1 * q^65 + (-b13 + 2b10 + b6) q^67 + (-2b12 + 3b3) * q^71 - 2 * q^73 + (b14 - b8 - b5 - b1) * q^77 + (b13 - b10) * q^79 + (b15 - b12 - b4 + b3) * q^83 + (b11 - b9) * q^85 + (-b8 - b5 - 3b1) q^89 + b10 * q^91 + b4 * q^95 + (2b11 - b9 - b7 - 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{13} - 16 q^{25} - 8 q^{37} - 24 q^{49} - 24 q^{61} - 32 q^{73} + 8 q^{85} - 24 q^{97}+O(q^{100}) \) 16 * q - 16 * q^13 - 16 * q^25 - 8 * q^37 - 24 * q^49 - 24 * q^61 - 32 * q^73 + 8 * q^85 - 24 * q^97

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	9360.2.k.a

Level	$9360$
Weight	$2$
Character orbit	9360.k
Analytic conductor	$74.740$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(74.7399762919\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{14}\cdot 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{10} - 3\nu^{6} + 12\nu^{2} ) / 16 \) (v^10 - 3v^6 + 12v^2) / 16
\(\beta_{2}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{12} - \nu^{11} - 16 \nu^{9} + 10 \nu^{8} - 16 \nu^{7} + 48 \nu^{5} - 56 \nu^{4} + \cdots + 256 ) / 128 \) (-v^15 + 2v^12 - v^11 - 16v^9 + 10v^8 - 16v^7 + 48v^5 - 56v^4 + 16v^3 - 320v + 256) / 128
\(\beta_{3}\)	\(=\)	\( ( \nu^{15} + \nu^{14} + \nu^{11} - 3 \nu^{10} - 16 \nu^{9} + 16 \nu^{7} - 4 \nu^{6} + 48 \nu^{5} + \cdots - 320 \nu ) / 128 \) (v^15 + v^14 + v^11 - 3v^10 - 16v^9 + 16v^7 - 4v^6 + 48v^5 - 16v^3 + 96v^2 - 320v) / 128
\(\beta_{4}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - \nu^{11} - 6 \nu^{10} + 16 \nu^{9} - 16 \nu^{7} - 8 \nu^{6} + \cdots + 320 \nu ) / 128 \) (-v^15 + 2v^14 - v^11 - 6v^10 + 16v^9 - 16v^7 - 8v^6 - 48v^5 + 16v^3 + 192v^2 + 320*v) / 128
\(\beta_{5}\)	\(=\)	\( ( \nu^{15} - 4 \nu^{14} - 6 \nu^{13} - 3 \nu^{11} + 12 \nu^{10} + 34 \nu^{9} - 4 \nu^{7} + \cdots + 256 \nu ) / 256 \) (v^15 - 4v^14 - 6v^13 - 3v^11 + 12v^10 + 34v^9 - 4v^7 - 112v^6 - 152v^5 - 32v^3 + 256v) / 256
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} - 4 \nu^{12} - \nu^{11} - 16 \nu^{9} - 20 \nu^{8} - 16 \nu^{7} + 48 \nu^{5} + 112 \nu^{4} + \cdots - 512 ) / 128 \) (-v^15 - 4v^12 - v^11 - 16v^9 - 20v^8 - 16v^7 + 48v^5 + 112v^4 + 16v^3 - 320v - 512) / 128
\(\beta_{7}\)	\(=\)	\( ( -\nu^{15} + 6\nu^{13} + 27\nu^{11} - 66\nu^{9} - 132\nu^{7} + 248\nu^{5} + 384\nu^{3} - 384\nu ) / 256 \) (-v^15 + 6v^13 + 27v^11 - 66v^9 - 132v^7 + 248v^5 + 384v^3 - 384*v) / 256
\(\beta_{8}\)	\(=\)	\( ( \nu^{15} + 6\nu^{13} - 27\nu^{11} - 66\nu^{9} + 132\nu^{7} + 248\nu^{5} - 384\nu^{3} - 384\nu ) / 256 \) (v^15 + 6v^13 - 27v^11 - 66v^9 + 132v^7 + 248v^5 - 384v^3 - 384*v) / 256
\(\beta_{9}\)	\(=\)	\( ( - \nu^{15} - 6 \nu^{13} - 16 \nu^{12} + 3 \nu^{11} + 34 \nu^{9} + 112 \nu^{8} + 4 \nu^{7} + \cdots + 768 ) / 256 \) (-v^15 - 6v^13 - 16v^12 + 3v^11 + 34v^9 + 112v^8 + 4v^7 - 152v^5 - 384v^4 + 32v^3 + 256v + 768) / 256
\(\beta_{10}\)	\(=\)	\( ( - \nu^{15} + \nu^{13} - 2 \nu^{12} + 5 \nu^{11} - 3 \nu^{9} + 6 \nu^{8} - 18 \nu^{7} - 4 \nu^{5} + \cdots + 64 ) / 64 \) (-v^15 + v^13 - 2v^12 + 5v^11 - 3v^9 + 6v^8 - 18v^7 - 4v^5 - 56v^4 + 8v^3 + 32*v + 64) / 64
\(\beta_{11}\)	\(=\)	\( ( -\nu^{15} - 3\nu^{13} + 9\nu^{11} + 9\nu^{9} - 30\nu^{7} - 52\nu^{5} + 120\nu^{3} + 96\nu ) / 64 \) (-v^15 - 3v^13 + 9v^11 + 9v^9 - 30v^7 - 52v^5 + 120v^3 + 96*v) / 64
\(\beta_{12}\)	\(=\)	\( ( - \nu^{15} - 2 \nu^{14} - \nu^{13} + 5 \nu^{11} + 10 \nu^{10} + 3 \nu^{9} - 18 \nu^{7} - 36 \nu^{6} + \cdots - 32 \nu ) / 64 \) (-v^15 - 2v^14 - v^13 + 5v^11 + 10v^10 + 3v^9 - 18v^7 - 36v^6 + 4v^5 + 8v^3 + 80v^2 - 32v) / 64
\(\beta_{13}\)	\(=\)	\( ( - \nu^{15} + \nu^{13} + 4 \nu^{12} + 5 \nu^{11} - 3 \nu^{9} - 12 \nu^{8} - 18 \nu^{7} - 4 \nu^{5} + \cdots - 128 ) / 64 \) (-v^15 + v^13 + 4v^12 + 5v^11 - 3v^9 - 12v^8 - 18v^7 - 4v^5 + 112v^4 + 8v^3 + 32*v - 128) / 64
\(\beta_{14}\)	\(=\)	\( ( -\nu^{15} + 3\nu^{13} + 9\nu^{11} - 9\nu^{9} - 30\nu^{7} + 52\nu^{5} + 120\nu^{3} - 96\nu ) / 64 \) (-v^15 + 3v^13 + 9v^11 - 9v^9 - 30v^7 + 52v^5 + 120v^3 - 96*v) / 64
\(\beta_{15}\)	\(=\)	\( ( \nu^{15} - 4 \nu^{14} + \nu^{13} - 5 \nu^{11} + 20 \nu^{10} - 3 \nu^{9} + 18 \nu^{7} - 72 \nu^{6} + \cdots + 32 \nu ) / 64 \) (v^15 - 4v^14 + v^13 - 5v^11 + 20v^10 - 3v^9 + 18v^7 - 72v^6 - 4v^5 - 8v^3 + 160v^2 + 32v) / 64

\(\nu\)	\(=\)	\( ( -\beta_{14} + \beta_{11} + 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{4} - 2\beta_{3} - 2\beta_{2} ) / 12 \) (-b14 + b11 + 2b8 + 2b7 - b6 + b4 - 2b3 - 2b2) / 12
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} + \beta_{12} - \beta_{8} - 3\beta_{5} + 2\beta_{4} + 2\beta_{3} - 3\beta_1 ) / 6 \) (b15 - b14 + b12 - b8 - 3b5 + 2b4 + 2b3 - 3b1) / 6
\(\nu^{3}\)	\(=\)	\( ( 2 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 5 \beta_{11} - 4 \beta_{10} + \cdots + 2 \beta_{2} ) / 12 \) (2b15 + 5b14 - 2b13 - 4b12 + 5b11 - 4b10 + 2b8 - 2b7 + b6 + b4 - 2b3 + 2b2) / 12
\(\nu^{4}\)	\(=\)	\( ( 3\beta_{13} - \beta_{11} - 3\beta_{10} + 3\beta_{9} + \beta_{7} + 2\beta_{6} - 2\beta_{2} + 12 ) / 6 \) (3b13 - b11 - 3b10 + 3b9 + b7 + 2b6 - 2*b2 + 12) / 6
\(\nu^{5}\)	\(=\)	\( ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - \beta_{11} - 4 \beta_{10} + \cdots - 2 \beta_{2} ) / 4 \) (-2b15 + b14 - 2b13 + 4b12 - b11 - 4b10 + 2b8 + 2b7 - b6 + b4 - 2b3 - 2b2) / 4
\(\nu^{6}\)	\(=\)	\( ( 3\beta_{15} - 7\beta_{14} + 3\beta_{12} - 7\beta_{8} - 21\beta_{5} - 2\beta_{4} - 2\beta_{3} - 9\beta_1 ) / 6 \) (3b15 - 7b14 + 3b12 - 7b8 - 21b5 - 2b4 - 2b3 - 9b1) / 6
\(\nu^{7}\)	\(=\)	\( ( 2 \beta_{15} + 19 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 19 \beta_{11} - 4 \beta_{10} + \cdots - 10 \beta_{2} ) / 12 \) (2b15 + 19b14 - 2b13 - 4b12 + 19b11 - 4b10 + 22b8 - 22b7 - 5b6 - 5b4 + 10b3 - 10b2) / 12
\(\nu^{8}\)	\(=\)	\( ( 13\beta_{13} - 7\beta_{11} - 13\beta_{10} + 21\beta_{9} + 7\beta_{7} - 2\beta_{6} + 2\beta_{2} - 36 ) / 6 \) (13b13 - 7b11 - 13b10 + 21b9 + 7b7 - 2b6 + 2*b2 - 36) / 6
\(\nu^{9}\)	\(=\)	\( ( - 18 \beta_{15} + 29 \beta_{14} - 18 \beta_{13} + 36 \beta_{12} - 29 \beta_{11} - 36 \beta_{10} + \cdots - 10 \beta_{2} ) / 12 \) (-18b15 + 29b14 - 18b13 + 36b12 - 29b11 - 36b10 - 22b8 - 22b7 - 5b6 + 5b4 - 10b3 - 10b2) / 12
\(\nu^{10}\)	\(=\)	\( ( -\beta_{15} - 3\beta_{14} - \beta_{12} - 3\beta_{8} - 9\beta_{5} - 10\beta_{4} - 10\beta_{3} + 35\beta_1 ) / 2 \) (-b15 - 3b14 - b12 - 3b8 - 9b5 - 10b4 - 10b3 + 35b1) / 2
\(\nu^{11}\)	\(=\)	\( ( - 18 \beta_{15} + 13 \beta_{14} + 18 \beta_{13} + 36 \beta_{12} + 13 \beta_{11} + 36 \beta_{10} + \cdots - 86 \beta_{2} ) / 12 \) (-18b15 + 13b14 + 18b13 + 36b12 + 13b11 + 36b10 + 10b8 - 10b7 - 43b6 - 43b4 + 86b3 - 86b2) / 12
\(\nu^{12}\)	\(=\)	\( ( 19\beta_{13} + 7\beta_{11} - 19\beta_{10} - 21\beta_{9} - 7\beta_{7} - 62\beta_{6} + 62\beta_{2} - 252 ) / 6 \) (19b13 + 7b11 - 19b10 - 21b9 - 7b7 - 62b6 + 62*b2 - 252) / 6
\(\nu^{13}\)	\(=\)	\( ( 50 \beta_{15} + 131 \beta_{14} + 50 \beta_{13} - 100 \beta_{12} - 131 \beta_{11} + 100 \beta_{10} + \cdots + 10 \beta_{2} ) / 12 \) (50b15 + 131b14 + 50b13 - 100b12 - 131b11 + 100b10 - 106b8 - 106b7 + 5b6 - 5b4 + 10b3 + 10b2) / 12
\(\nu^{14}\)	\(=\)	\( ( -93\beta_{15} + 41\beta_{14} - 93\beta_{12} + 41\beta_{8} + 123\beta_{5} - 34\beta_{4} - 34\beta_{3} + 567\beta_1 ) / 6 \) (-93b15 + 41b14 - 93b12 + 41b8 + 123b5 - 34b4 - 34b3 + 567b1) / 6
\(\nu^{15}\)	\(=\)	\( ( 6 \beta_{15} - 79 \beta_{14} - 6 \beta_{13} - 12 \beta_{12} - 79 \beta_{11} - 12 \beta_{10} + \cdots - 78 \beta_{2} ) / 4 \) (6b15 - 79b14 - 6b13 - 12b12 - 79b11 - 12b10 - 110b8 + 110b7 - 39b6 - 39b4 + 78b3 - 78b2) / 4

\(n\)	\(2081\)	\(2341\)	\(5617\)	\(5761\)	\(8191\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 4031.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( (T^{8} + 34 T^{6} + \cdots + 1296)^{2} \) (T^8 + 34T^6 + 353T^4 + 1224*T^2 + 1296)^2
$11$	\( (T^{8} - 38 T^{6} + \cdots + 324)^{2} \) (T^8 - 38T^6 + 461T^4 - 1764*T^2 + 324)^2
$13$	\( (T + 1)^{16} \) (T + 1)^16
$17$	\( (T^{8} + 54 T^{6} + \cdots + 324)^{2} \) (T^8 + 54T^6 + 829T^4 + 2700*T^2 + 324)^2
$19$	\( (T^{8} + 68 T^{6} + \cdots + 20736)^{2} \) (T^8 + 68T^6 + 1412T^4 + 9792*T^2 + 20736)^2
$23$	\( (T^{8} - 94 T^{6} + \cdots + 5184)^{2} \) (T^8 - 94T^6 + 2609T^4 - 22608*T^2 + 5184)^2
$29$	\( (T^{8} + 108 T^{6} + \cdots + 5184)^{2} \) (T^8 + 108T^6 + 3316T^4 + 21600*T^2 + 5184)^2
$31$	\( (T^{8} + 152 T^{6} + \cdots + 82944)^{2} \) (T^8 + 152T^6 + 7376T^4 + 112896*T^2 + 82944)^2
$37$	\( (T^{4} + 2 T^{3} + \cdots + 612)^{4} \) (T^4 + 2T^3 - 101T^2 + 204*T + 612)^4
$41$	\( (T^{8} + 206 T^{6} + \cdots + 374544)^{2} \) (T^8 + 206T^6 + 10609T^4 + 165240*T^2 + 374544)^2
$43$	\( (T^{8} + 68 T^{6} + \cdots + 20736)^{2} \) (T^8 + 68T^6 + 1412T^4 + 9792*T^2 + 20736)^2
$47$	\( (T^{4} - 90 T^{2} + 648)^{4} \) (T^4 - 90*T^2 + 648)^4
$53$	\( (T^{8} + 150 T^{6} + \cdots + 324)^{2} \) (T^8 + 150T^6 + 829T^4 + 972*T^2 + 324)^2
$59$	\( (T^{4} - 90 T^{2} + 648)^{4} \) (T^4 - 90*T^2 + 648)^4
$61$	\( (T^{4} + 6 T^{3} + \cdots - 4496)^{4} \) (T^4 + 6T^3 - 167T^2 - 1752*T - 4496)^4
$67$	\( (T^{8} + 272 T^{6} + \cdots + 20736)^{2} \) (T^8 + 272T^6 + 10592T^4 + 39168*T^2 + 20736)^2
$71$	\( (T^{8} - 454 T^{6} + \cdots + 104898564)^{2} \) (T^8 - 454T^6 + 70349T^4 - 4547268*T^2 + 104898564)^2
$73$	\( (T + 2)^{16} \) (T + 2)^16
$79$	\( (T^{4} + 63 T^{2} + 648)^{4} \) (T^4 + 63*T^2 + 648)^4
$83$	\( (T^{8} - 188 T^{6} + \cdots + 82944)^{2} \) (T^8 - 188T^6 + 10436T^4 - 180864*T^2 + 82944)^2
$89$	\( (T^{8} + 78 T^{6} + \cdots + 1296)^{2} \) (T^8 + 78T^6 + 769T^4 + 2088*T^2 + 1296)^2
$97$	\( (T^{4} + 6 T^{3} + \cdots + 188)^{4} \) (T^4 + 6T^3 - 61T^2 + 28*T + 188)^4
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