LMFDB - Newform orbit 1568.2.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(31,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("1568.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 224)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{9} q^{3} + \beta_{8} q^{5} + ( - \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) $ q + b9 * q^3 + b8 * q^5 + (-b4 - b2) * q^9	\( q + \beta_{9} q^{3} + \beta_{8} q^{5} + ( - \beta_{4} - \beta_{2}) q^{9} - \beta_1 q^{11} + (\beta_{14} + \beta_{13} + \beta_{7}) q^{13} + (\beta_{6} - \beta_{5}) q^{15} + ( - \beta_{14} + 2 \beta_{8} + 2 \beta_{7}) q^{17} + (\beta_{15} + \beta_{11}) q^{19} + ( - 2 \beta_{5} - \beta_{3} + \beta_1) q^{23} + ( - \beta_{10} + \beta_{4} + \beta_{2} - 1) q^{25} + (\beta_{15} + \beta_{12}) q^{27} + ( - 2 \beta_{10} - 2) q^{29} + ( - \beta_{12} + 2 \beta_{9}) q^{31} + \beta_{13} q^{33} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{37} + (\beta_{6} + 4 \beta_1) q^{39} + ( - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{7}) q^{41} + (2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{43} + (\beta_{14} + \beta_{8} + \beta_{7}) q^{45} + ( - \beta_{15} + 2 \beta_{11}) q^{47} + ( - 4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{51} + ( - 2 \beta_{2} + 2) q^{53} + ( - \beta_{15} - \beta_{12}) q^{55} + ( - 3 \beta_{10} - 4) q^{57} + ( - \beta_{12} - 3 \beta_{9}) q^{59} - \beta_{8} q^{61} + ( - 3 \beta_{4} + 4 \beta_{2}) q^{65} + (2 \beta_{6} - 5 \beta_1) q^{67} + (\beta_{14} + \beta_{13} - 4 \beta_{7}) q^{69} + (\beta_{6} - \beta_{5} - 3 \beta_{3}) q^{71} + ( - \beta_{14} - 4 \beta_{8} - 4 \beta_{7}) q^{73} + ( - \beta_{15} - 3 \beta_{11}) q^{75} + ( - \beta_{5} - 5 \beta_{3} + 5 \beta_1) q^{79} + (\beta_{10} - \beta_{4} - 3 \beta_{2} + 3) q^{81} + ( - 2 \beta_{15} - 2 \beta_{12} + \cdots - \beta_{9}) q^{83}+ \cdots + (2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) q + b9 * q^3 + b8 * q^5 + (-b4 - b2) * q^9 - b1 * q^11 + (b14 + b13 + b7) * q^13 + (b6 - b5) * q^15 + (-b14 + 2b8 + 2b7) * q^17 + (b15 + b11) * q^19 + (-2b5 - b3 + b1) q^23 + (-b10 + b4 + b2 - 1) * q^25 + (b15 + b12) * q^27 + (-2b10 - 2) q^29 + (-b12 + 2b9) q^31 + b13 * q^33 + (-2b4 + 2b2) * q^37 + (b6 + 4b1) q^39 + (-2b14 - 2b13 - 2b7) q^41 + (2b6 - 2b5 + b3) * q^43 + (b14 + b8 + b7) * q^45 + (-b15 + 2b11) q^47 + (-4b5 + 4b3 - 4b1) q^51 + (-2b2 + 2) q^53 + (-b15 - b12) * q^55 + (-3b10 - 4) q^57 + (-b12 - 3b9) q^59 - b8 * q^61 + (-3b4 + 4b2) * q^65 + (2b6 - 5b1) * q^67 + (b14 + b13 - 4b7) q^69 + (b6 - b5 - 3b3) q^71 + (-b14 - 4b8 - 4b7) * q^73 + (-b15 - 3b11) q^75 + (-b5 - 5b3 + 5b1) * q^79 + (b10 - b4 - 3b2 + 3) q^81 + (-2b15 - 2b12 + b11 - b9) * q^83 + 8 * q^85 + (2b12 - 6b9) * q^87 - b13 * q^89 + (-4b4 - 8b2) * q^93 + (-b6 + 4b1) q^95 + (-b14 - b13 + 6b7) q^97 + (2b6 - 2b5 + b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 8 q^{25} - 32 q^{29} + 16 q^{37} + 16 q^{53} - 64 q^{57} + 32 q^{65} + 24 q^{81} + 128 q^{85} - 64 q^{93}+O(q^{100}) $ 16 * q - 8 * q^9 - 8 * q^25 - 32 * q^29 + 16 * q^37 + 16 * q^53 - 64 * q^57 + 32 * q^65 + 24 * q^81 + 128 * q^85 - 64 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ 2\zeta_{48}^{4} $ 2*v^4
$\beta_{2}$	$=$	$ \zeta_{48}^{8} $ v^8
$\beta_{3}$	$=$	$ 2\zeta_{48}^{12} $ 2*v^12
$\beta_{4}$	$=$	$ 2\zeta_{48}^{14} + 2\zeta_{48}^{2} $ 2v^14 + 2v^2
$\beta_{5}$	$=$	$ -2\zeta_{48}^{14} + 2\zeta_{48}^{2} $ -2v^14 + 2v^2
$\beta_{6}$	$=$	$ -2\zeta_{48}^{14} + 2\zeta_{48}^{10} + 2\zeta_{48}^{6} $ -2v^14 + 2v^10 + 2*v^6
$\beta_{7}$	$=$	$ -\zeta_{48}^{15} + \zeta_{48}^{13} - \zeta_{48}^{9} - \zeta_{48}^{5} + \zeta_{48}^{3} $ -v^15 + v^13 - v^9 - v^5 + v^3
$\beta_{8}$	$=$	$ \zeta_{48}^{15} - \zeta_{48}^{11} + \zeta_{48}^{9} - \zeta_{48}^{7} + \zeta_{48}^{5} - \zeta_{48} $ v^15 - v^11 + v^9 - v^7 + v^5 - v
$\beta_{9}$	$=$	$ -\zeta_{48}^{13} - \zeta_{48}^{11} - \zeta_{48}^{7} + \zeta_{48}^{3} + \zeta_{48} $ -v^13 - v^11 - v^7 + v^3 + v
$\beta_{10}$	$=$	$ -2\zeta_{48}^{10} + 2\zeta_{48}^{6} + 2\zeta_{48}^{2} $ -2v^10 + 2v^6 + 2*v^2
$\beta_{11}$	$=$	$ \zeta_{48}^{15} - \zeta_{48}^{11} - \zeta_{48}^{9} - \zeta_{48}^{7} - \zeta_{48}^{5} + \zeta_{48} $ v^15 - v^11 - v^9 - v^7 - v^5 + v
$\beta_{12}$	$=$	$ 2\zeta_{48}^{13} + 2\zeta_{48}^{11} - 2\zeta_{48}^{7} - 2\zeta_{48}^{3} + 2\zeta_{48} $ 2v^13 + 2v^11 - 2v^7 - 2v^3 + 2*v
$\beta_{13}$	$=$	$ 2\zeta_{48}^{15} + 2\zeta_{48}^{11} + 2\zeta_{48}^{9} - 2\zeta_{48}^{7} - 2\zeta_{48}^{5} - 2\zeta_{48} $ 2v^15 + 2v^11 + 2v^9 - 2v^7 - 2v^5 - 2v
$\beta_{14}$	$=$	$ 2\zeta_{48}^{13} - 2\zeta_{48}^{11} + 2\zeta_{48}^{7} + 2\zeta_{48}^{3} + 2\zeta_{48} $ 2v^13 - 2v^11 + 2v^7 + 2v^3 + 2*v
$\beta_{15}$	$=$	$ -2\zeta_{48}^{15} - 2\zeta_{48}^{11} + 2\zeta_{48}^{9} + 2\zeta_{48}^{7} - 2\zeta_{48}^{5} - 2\zeta_{48} $ -2v^15 - 2v^11 + 2v^9 + 2v^7 - 2v^5 - 2v

Display $\zeta_{48}^j$ in terms of $\beta_i$

$\zeta_{48}$	$=$	$ ( \beta_{14} + \beta_{12} + 2\beta_{9} - 2\beta_{8} - 2\beta_{7} ) / 8 $ (b14 + b12 + 2b9 - 2b8 - 2*b7) / 8
$\zeta_{48}^{2}$	$=$	$ ( \beta_{5} + \beta_{4} ) / 4 $ (b5 + b4) / 4
$\zeta_{48}^{3}$	$=$	$ ( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - 2\beta_{11} + 2\beta_{9} + 2\beta_{7} ) / 8 $ (-b15 + b14 + b13 - b12 - 2b11 + 2b9 + 2*b7) / 8
$\zeta_{48}^{4}$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\zeta_{48}^{5}$	$=$	$ ( -\beta_{15} - \beta_{13} - 2\beta_{11} + 2\beta_{8} ) / 8 $ (-b15 - b13 - 2b11 + 2b8) / 8
$\zeta_{48}^{6}$	$=$	$ ( \beta_{10} + \beta_{6} - \beta_{5} ) / 4 $ (b10 + b6 - b5) / 4
$\zeta_{48}^{7}$	$=$	$ ( \beta_{14} - \beta_{12} - 2\beta_{9} - 2\beta_{8} - 2\beta_{7} ) / 8 $ (b14 - b12 - 2b9 - 2b8 - 2*b7) / 8
$\zeta_{48}^{8}$	$=$	$ \beta_{2} $ b2
$\zeta_{48}^{9}$	$=$	$ ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2\beta_{11} + 2\beta_{9} - 2\beta_{7} ) / 8 $ (b15 + b14 + b13 + b12 - 2b11 + 2b9 - 2*b7) / 8
$\zeta_{48}^{10}$	$=$	$ ( -\beta_{10} + \beta_{6} + \beta_{4} ) / 4 $ (-b10 + b6 + b4) / 4
$\zeta_{48}^{11}$	$=$	$ ( -\beta_{15} + \beta_{13} - 2\beta_{11} - 2\beta_{8} ) / 8 $ (-b15 + b13 - 2b11 - 2b8) / 8
$\zeta_{48}^{12}$	$=$	$ ( \beta_{3} ) / 2 $ (b3) / 2
$\zeta_{48}^{13}$	$=$	$ ( \beta_{14} + \beta_{12} - 2\beta_{9} + 2\beta_{8} + 2\beta_{7} ) / 8 $ (b14 + b12 - 2b9 + 2b8 + 2*b7) / 8
$\zeta_{48}^{14}$	$=$	$ ( -\beta_{5} + \beta_{4} ) / 4 $ (-b5 + b4) / 4
$\zeta_{48}^{15}$	$=$	$ ( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + 2\beta_{11} - 2\beta_{9} - 2\beta_{7} ) / 8 $ (-b15 + b14 + b13 - b12 + 2b11 - 2b9 - 2*b7) / 8

Display $\beta_i$ in terms of $\zeta_{48}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$\beta_{2}$

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} $

31.1

0.608761	+	0.793353i
−0.991445	−	0.130526i
−0.130526	+	0.991445i
−0.793353	+	0.608761i
0.793353	−	0.608761i
0.130526	−	0.991445i
0.991445	+	0.130526i
−0.608761	−	0.793353i
0.608761	−	0.793353i
−0.991445	+	0.130526i
−0.130526	−	0.991445i
−0.793353	−	0.608761i
0.793353	+	0.608761i
0.130526	+	0.991445i
0.991445	−	0.130526i
−0.608761	+	0.793353i

−1.30656

2.26303i

−0.937379

0.541196i

−1.91421

−

3.31552i

31.2

−1.30656

2.26303i

0.937379

−

0.541196i

−1.91421

−

3.31552i

31.3

−0.541196

0.937379i

−2.26303

1.30656i

0.914214

1.58346i

31.4

−0.541196

0.937379i

2.26303

−

1.30656i

0.914214

1.58346i

31.5

0.541196

−

0.937379i

−2.26303

1.30656i

0.914214

1.58346i

31.6

0.541196

−

0.937379i

2.26303

−

1.30656i

0.914214

1.58346i

31.7

1.30656

−

2.26303i

−0.937379

0.541196i

−1.91421

−

3.31552i

31.8

1.30656

−

2.26303i

0.937379

−

0.541196i

−1.91421

−

3.31552i

607.1

−1.30656

−

2.26303i

−0.937379

−

0.541196i

−1.91421

3.31552i

607.2

−1.30656

−

2.26303i

0.937379

0.541196i

−1.91421

3.31552i

607.3

−0.541196

−

0.937379i

−2.26303

−

1.30656i

0.914214

−

1.58346i

607.4

−0.541196

−

0.937379i

2.26303

1.30656i

0.914214

−

1.58346i

607.5

0.541196

0.937379i

−2.26303

−

1.30656i

0.914214

−

1.58346i

607.6

0.541196

0.937379i

2.26303

1.30656i

0.914214

−

1.58346i

607.7

1.30656

2.26303i

−0.937379

−

0.541196i

−1.91421

3.31552i

607.8

1.30656

2.26303i

0.937379

0.541196i

−1.91421

3.31552i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
28.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1568.2.p.a		16
4.b	odd	2	1	inner	1568.2.p.a		16
7.b	odd	2	1	inner	1568.2.p.a		16
7.c	even	3	1		224.2.f.a	✓	8
7.c	even	3	1	inner	1568.2.p.a		16
7.d	odd	6	1		224.2.f.a	✓	8
7.d	odd	6	1	inner	1568.2.p.a		16
21.g	even	6	1		2016.2.b.b		8
21.h	odd	6	1		2016.2.b.b		8
28.d	even	2	1	inner	1568.2.p.a		16
28.f	even	6	1		224.2.f.a	✓	8
28.f	even	6	1	inner	1568.2.p.a		16
28.g	odd	6	1		224.2.f.a	✓	8
28.g	odd	6	1	inner	1568.2.p.a		16
56.j	odd	6	1		448.2.f.d		8
56.k	odd	6	1		448.2.f.d		8
56.m	even	6	1		448.2.f.d		8
56.p	even	6	1		448.2.f.d		8
84.j	odd	6	1		2016.2.b.b		8
84.n	even	6	1		2016.2.b.b		8
112.u	odd	12	1		1792.2.e.f		8
112.u	odd	12	1		1792.2.e.g		8
112.v	even	12	1		1792.2.e.f		8
112.v	even	12	1		1792.2.e.g		8
112.w	even	12	1		1792.2.e.f		8
112.w	even	12	1		1792.2.e.g		8
112.x	odd	12	1		1792.2.e.f		8
112.x	odd	12	1		1792.2.e.g		8
168.s	odd	6	1		4032.2.b.p		8
168.v	even	6	1		4032.2.b.p		8
168.ba	even	6	1		4032.2.b.p		8
168.be	odd	6	1		4032.2.b.p		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.f.a	✓	8	7.c	even	3	1
224.2.f.a	✓	8	7.d	odd	6	1
224.2.f.a	✓	8	28.f	even	6	1
224.2.f.a	✓	8	28.g	odd	6	1
448.2.f.d		8	56.j	odd	6	1
448.2.f.d		8	56.k	odd	6	1
448.2.f.d		8	56.m	even	6	1
448.2.f.d		8	56.p	even	6	1
1568.2.p.a		16	1.a	even	1	1	trivial
1568.2.p.a		16	4.b	odd	2	1	inner
1568.2.p.a		16	7.b	odd	2	1	inner
1568.2.p.a		16	7.c	even	3	1	inner
1568.2.p.a		16	7.d	odd	6	1	inner
1568.2.p.a		16	28.d	even	2	1	inner
1568.2.p.a		16	28.f	even	6	1	inner
1568.2.p.a		16	28.g	odd	6	1	inner
1792.2.e.f		8	112.u	odd	12	1
1792.2.e.f		8	112.v	even	12	1
1792.2.e.f		8	112.w	even	12	1
1792.2.e.f		8	112.x	odd	12	1
1792.2.e.g		8	112.u	odd	12	1
1792.2.e.g		8	112.v	even	12	1
1792.2.e.g		8	112.w	even	12	1
1792.2.e.g		8	112.x	odd	12	1
2016.2.b.b		8	21.g	even	6	1
2016.2.b.b		8	21.h	odd	6	1
2016.2.b.b		8	84.j	odd	6	1
2016.2.b.b		8	84.n	even	6	1
4032.2.b.p		8	168.s	odd	6	1
4032.2.b.p		8	168.v	even	6	1
4032.2.b.p		8	168.ba	even	6	1
4032.2.b.p		8	168.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 8T_{3}^{6} + 56T_{3}^{4} + 64T_{3}^{2} + 64 $ T3^8 + 8*T3^6 + 56*T3^4 + 64*T3^2 + 64 acting on $S_{2}^{\mathrm{new}}(1568, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} $ (T^8 + 8T^6 + 56T^4 + 64*T^2 + 64)^2
$5$	$ (T^{8} - 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} $ (T^8 - 8T^6 + 56T^4 - 64*T^2 + 64)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{4} - 4 T^{2} + 16)^{4} $ (T^4 - 4*T^2 + 16)^4
$13$	$ (T^{4} + 40 T^{2} + 392)^{4} $ (T^4 + 40*T^2 + 392)^4
$17$	$ (T^{8} - 64 T^{6} + \cdots + 262144)^{2} $ (T^8 - 64T^6 + 3584T^4 - 32768*T^2 + 262144)^2
$19$	$ (T^{8} + 40 T^{6} + \cdots + 153664)^{2} $ (T^8 + 40T^6 + 1208T^4 + 15680*T^2 + 153664)^2
$23$	$ (T^{8} - 72 T^{6} + \cdots + 614656)^{2} $ (T^8 - 72T^6 + 4400T^4 - 56448*T^2 + 614656)^2
$29$	$ (T^{2} + 4 T - 28)^{8} $ (T^2 + 4*T - 28)^8
$31$	$ (T^{8} + 64 T^{6} + \cdots + 262144)^{2} $ (T^8 + 64T^6 + 3584T^4 + 32768*T^2 + 262144)^2
$37$	$ (T^{4} - 4 T^{3} + \cdots + 784)^{4} $ (T^4 - 4T^3 + 44T^2 + 112*T + 784)^4
$41$	$ (T^{4} + 160 T^{2} + 6272)^{4} $ (T^4 + 160*T^2 + 6272)^4
$43$	$ (T^{4} + 72 T^{2} + 784)^{4} $ (T^4 + 72*T^2 + 784)^4
$47$	$ (T^{8} + 64 T^{6} + \cdots + 262144)^{2} $ (T^8 + 64T^6 + 3584T^4 + 32768*T^2 + 262144)^2
$53$	$ (T^{2} - 2 T + 4)^{8} $ (T^2 - 2*T + 4)^8
$59$	$ (T^{8} + 104 T^{6} + \cdots + 5345344)^{2} $ (T^8 + 104T^6 + 8504T^4 + 240448*T^2 + 5345344)^2
$61$	$ (T^{8} - 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} $ (T^8 - 8T^6 + 56T^4 - 64*T^2 + 64)^2
$67$	$ (T^{8} - 264 T^{6} + \cdots + 21381376)^{2} $ (T^8 - 264T^6 + 65072T^4 - 1220736*T^2 + 21381376)^2
$71$	$ (T^{4} + 88 T^{2} + 784)^{4} $ (T^4 + 88*T^2 + 784)^4
$73$	$ (T^{8} - 160 T^{6} + \cdots + 16384)^{2} $ (T^8 - 160T^6 + 25472T^4 - 20480*T^2 + 16384)^2
$79$	$ (T^{8} - 216 T^{6} + \cdots + 71639296)^{2} $ (T^8 - 216T^6 + 38192T^4 - 1828224*T^2 + 71639296)^2
$83$	$ (T^{4} - 136 T^{2} + 392)^{4} $ (T^4 - 136*T^2 + 392)^4
$89$	$ (T^{8} - 32 T^{6} + \cdots + 16384)^{2} $ (T^8 - 32T^6 + 896T^4 - 4096*T^2 + 16384)^2
$97$	$ (T^{4} + 320 T^{2} + 25088)^{4} $ (T^4 + 320*T^2 + 25088)^4
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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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{3} + \beta_{8} q^{5} + ( - \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) q + b9 * q^3 + b8 * q^5 + (-b4 - b2) * q^9	\( q + \beta_{9} q^{3} + \beta_{8} q^{5} + ( - \beta_{4} - \beta_{2}) q^{9} - \beta_1 q^{11} + (\beta_{14} + \beta_{13} + \beta_{7}) q^{13} + (\beta_{6} - \beta_{5}) q^{15} + ( - \beta_{14} + 2 \beta_{8} + 2 \beta_{7}) q^{17} + (\beta_{15} + \beta_{11}) q^{19} + ( - 2 \beta_{5} - \beta_{3} + \beta_1) q^{23} + ( - \beta_{10} + \beta_{4} + \beta_{2} - 1) q^{25} + (\beta_{15} + \beta_{12}) q^{27} + ( - 2 \beta_{10} - 2) q^{29} + ( - \beta_{12} + 2 \beta_{9}) q^{31} + \beta_{13} q^{33} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{37} + (\beta_{6} + 4 \beta_1) q^{39} + ( - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{7}) q^{41} + (2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{43} + (\beta_{14} + \beta_{8} + \beta_{7}) q^{45} + ( - \beta_{15} + 2 \beta_{11}) q^{47} + ( - 4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{51} + ( - 2 \beta_{2} + 2) q^{53} + ( - \beta_{15} - \beta_{12}) q^{55} + ( - 3 \beta_{10} - 4) q^{57} + ( - \beta_{12} - 3 \beta_{9}) q^{59} - \beta_{8} q^{61} + ( - 3 \beta_{4} + 4 \beta_{2}) q^{65} + (2 \beta_{6} - 5 \beta_1) q^{67} + (\beta_{14} + \beta_{13} - 4 \beta_{7}) q^{69} + (\beta_{6} - \beta_{5} - 3 \beta_{3}) q^{71} + ( - \beta_{14} - 4 \beta_{8} - 4 \beta_{7}) q^{73} + ( - \beta_{15} - 3 \beta_{11}) q^{75} + ( - \beta_{5} - 5 \beta_{3} + 5 \beta_1) q^{79} + (\beta_{10} - \beta_{4} - 3 \beta_{2} + 3) q^{81} + ( - 2 \beta_{15} - 2 \beta_{12} + \cdots - \beta_{9}) q^{83}+ \cdots + (2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) q + b9 * q^3 + b8 * q^5 + (-b4 - b2) * q^9 - b1 * q^11 + (b14 + b13 + b7) * q^13 + (b6 - b5) * q^15 + (-b14 + 2b8 + 2b7) * q^17 + (b15 + b11) * q^19 + (-2b5 - b3 + b1) q^23 + (-b10 + b4 + b2 - 1) * q^25 + (b15 + b12) * q^27 + (-2b10 - 2) q^29 + (-b12 + 2b9) q^31 + b13 * q^33 + (-2b4 + 2b2) * q^37 + (b6 + 4b1) q^39 + (-2b14 - 2b13 - 2b7) q^41 + (2b6 - 2b5 + b3) * q^43 + (b14 + b8 + b7) * q^45 + (-b15 + 2b11) q^47 + (-4b5 + 4b3 - 4b1) q^51 + (-2b2 + 2) q^53 + (-b15 - b12) * q^55 + (-3b10 - 4) q^57 + (-b12 - 3b9) q^59 - b8 * q^61 + (-3b4 + 4b2) * q^65 + (2b6 - 5b1) * q^67 + (b14 + b13 - 4b7) q^69 + (b6 - b5 - 3b3) q^71 + (-b14 - 4b8 - 4b7) * q^73 + (-b15 - 3b11) q^75 + (-b5 - 5b3 + 5b1) * q^79 + (b10 - b4 - 3b2 + 3) q^81 + (-2b15 - 2b12 + b11 - b9) * q^83 + 8 * q^85 + (2b12 - 6b9) * q^87 - b13 * q^89 + (-4b4 - 8b2) * q^93 + (-b6 + 4b1) q^95 + (-b14 - b13 + 6b7) q^97 + (2b6 - 2b5 + b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 8 q^{25} - 32 q^{29} + 16 q^{37} + 16 q^{53} - 64 q^{57} + 32 q^{65} + 24 q^{81} + 128 q^{85} - 64 q^{93}+O(q^{100}) \) 16 * q - 8 * q^9 - 8 * q^25 - 32 * q^29 + 16 * q^37 + 16 * q^53 - 64 * q^57 + 32 * q^65 + 24 * q^81 + 128 * q^85 - 64 * q^93

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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Twists

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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	1568.2.p.a

Level	$1568$
Weight	$2$
Character orbit	1568.p
Analytic conductor	$12.521$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(12.5205430369\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{48})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{8} + 1 \) x^16 - x^8 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( 2\zeta_{48}^{4} \) 2*v^4
\(\beta_{2}\)	\(=\)	\( \zeta_{48}^{8} \) v^8
\(\beta_{3}\)	\(=\)	\( 2\zeta_{48}^{12} \) 2*v^12
\(\beta_{4}\)	\(=\)	\( 2\zeta_{48}^{14} + 2\zeta_{48}^{2} \) 2v^14 + 2v^2
\(\beta_{5}\)	\(=\)	\( -2\zeta_{48}^{14} + 2\zeta_{48}^{2} \) -2v^14 + 2v^2
\(\beta_{6}\)	\(=\)	\( -2\zeta_{48}^{14} + 2\zeta_{48}^{10} + 2\zeta_{48}^{6} \) -2v^14 + 2v^10 + 2*v^6
\(\beta_{7}\)	\(=\)	\( -\zeta_{48}^{15} + \zeta_{48}^{13} - \zeta_{48}^{9} - \zeta_{48}^{5} + \zeta_{48}^{3} \) -v^15 + v^13 - v^9 - v^5 + v^3
\(\beta_{8}\)	\(=\)	\( \zeta_{48}^{15} - \zeta_{48}^{11} + \zeta_{48}^{9} - \zeta_{48}^{7} + \zeta_{48}^{5} - \zeta_{48} \) v^15 - v^11 + v^9 - v^7 + v^5 - v
\(\beta_{9}\)	\(=\)	\( -\zeta_{48}^{13} - \zeta_{48}^{11} - \zeta_{48}^{7} + \zeta_{48}^{3} + \zeta_{48} \) -v^13 - v^11 - v^7 + v^3 + v
\(\beta_{10}\)	\(=\)	\( -2\zeta_{48}^{10} + 2\zeta_{48}^{6} + 2\zeta_{48}^{2} \) -2v^10 + 2v^6 + 2*v^2
\(\beta_{11}\)	\(=\)	\( \zeta_{48}^{15} - \zeta_{48}^{11} - \zeta_{48}^{9} - \zeta_{48}^{7} - \zeta_{48}^{5} + \zeta_{48} \) v^15 - v^11 - v^9 - v^7 - v^5 + v
\(\beta_{12}\)	\(=\)	\( 2\zeta_{48}^{13} + 2\zeta_{48}^{11} - 2\zeta_{48}^{7} - 2\zeta_{48}^{3} + 2\zeta_{48} \) 2v^13 + 2v^11 - 2v^7 - 2v^3 + 2*v
\(\beta_{13}\)	\(=\)	\( 2\zeta_{48}^{15} + 2\zeta_{48}^{11} + 2\zeta_{48}^{9} - 2\zeta_{48}^{7} - 2\zeta_{48}^{5} - 2\zeta_{48} \) 2v^15 + 2v^11 + 2v^9 - 2v^7 - 2v^5 - 2v
\(\beta_{14}\)	\(=\)	\( 2\zeta_{48}^{13} - 2\zeta_{48}^{11} + 2\zeta_{48}^{7} + 2\zeta_{48}^{3} + 2\zeta_{48} \) 2v^13 - 2v^11 + 2v^7 + 2v^3 + 2*v
\(\beta_{15}\)	\(=\)	\( -2\zeta_{48}^{15} - 2\zeta_{48}^{11} + 2\zeta_{48}^{9} + 2\zeta_{48}^{7} - 2\zeta_{48}^{5} - 2\zeta_{48} \) -2v^15 - 2v^11 + 2v^9 + 2v^7 - 2v^5 - 2v

\(\zeta_{48}\)	\(=\)	\( ( \beta_{14} + \beta_{12} + 2\beta_{9} - 2\beta_{8} - 2\beta_{7} ) / 8 \) (b14 + b12 + 2b9 - 2b8 - 2*b7) / 8
\(\zeta_{48}^{2}\)	\(=\)	\( ( \beta_{5} + \beta_{4} ) / 4 \) (b5 + b4) / 4
\(\zeta_{48}^{3}\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - 2\beta_{11} + 2\beta_{9} + 2\beta_{7} ) / 8 \) (-b15 + b14 + b13 - b12 - 2b11 + 2b9 + 2*b7) / 8
\(\zeta_{48}^{4}\)	\(=\)	\( ( \beta_1 ) / 2 \) (b1) / 2
\(\zeta_{48}^{5}\)	\(=\)	\( ( -\beta_{15} - \beta_{13} - 2\beta_{11} + 2\beta_{8} ) / 8 \) (-b15 - b13 - 2b11 + 2b8) / 8
\(\zeta_{48}^{6}\)	\(=\)	\( ( \beta_{10} + \beta_{6} - \beta_{5} ) / 4 \) (b10 + b6 - b5) / 4
\(\zeta_{48}^{7}\)	\(=\)	\( ( \beta_{14} - \beta_{12} - 2\beta_{9} - 2\beta_{8} - 2\beta_{7} ) / 8 \) (b14 - b12 - 2b9 - 2b8 - 2*b7) / 8
\(\zeta_{48}^{8}\)	\(=\)	\( \beta_{2} \) b2
\(\zeta_{48}^{9}\)	\(=\)	\( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2\beta_{11} + 2\beta_{9} - 2\beta_{7} ) / 8 \) (b15 + b14 + b13 + b12 - 2b11 + 2b9 - 2*b7) / 8
\(\zeta_{48}^{10}\)	\(=\)	\( ( -\beta_{10} + \beta_{6} + \beta_{4} ) / 4 \) (-b10 + b6 + b4) / 4
\(\zeta_{48}^{11}\)	\(=\)	\( ( -\beta_{15} + \beta_{13} - 2\beta_{11} - 2\beta_{8} ) / 8 \) (-b15 + b13 - 2b11 - 2b8) / 8
\(\zeta_{48}^{12}\)	\(=\)	\( ( \beta_{3} ) / 2 \) (b3) / 2
\(\zeta_{48}^{13}\)	\(=\)	\( ( \beta_{14} + \beta_{12} - 2\beta_{9} + 2\beta_{8} + 2\beta_{7} ) / 8 \) (b14 + b12 - 2b9 + 2b8 + 2*b7) / 8
\(\zeta_{48}^{14}\)	\(=\)	\( ( -\beta_{5} + \beta_{4} ) / 4 \) (-b5 + b4) / 4
\(\zeta_{48}^{15}\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + 2\beta_{11} - 2\beta_{9} - 2\beta_{7} ) / 8 \) (-b15 + b14 + b13 - b12 + 2b11 - 2b9 - 2*b7) / 8

\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\beta_{2}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} \) (T^8 + 8T^6 + 56T^4 + 64*T^2 + 64)^2
$5$	\( (T^{8} - 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} \) (T^8 - 8T^6 + 56T^4 - 64*T^2 + 64)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{4} - 4 T^{2} + 16)^{4} \) (T^4 - 4*T^2 + 16)^4
$13$	\( (T^{4} + 40 T^{2} + 392)^{4} \) (T^4 + 40*T^2 + 392)^4
$17$	\( (T^{8} - 64 T^{6} + \cdots + 262144)^{2} \) (T^8 - 64T^6 + 3584T^4 - 32768*T^2 + 262144)^2
$19$	\( (T^{8} + 40 T^{6} + \cdots + 153664)^{2} \) (T^8 + 40T^6 + 1208T^4 + 15680*T^2 + 153664)^2
$23$	\( (T^{8} - 72 T^{6} + \cdots + 614656)^{2} \) (T^8 - 72T^6 + 4400T^4 - 56448*T^2 + 614656)^2
$29$	\( (T^{2} + 4 T - 28)^{8} \) (T^2 + 4*T - 28)^8
$31$	\( (T^{8} + 64 T^{6} + \cdots + 262144)^{2} \) (T^8 + 64T^6 + 3584T^4 + 32768*T^2 + 262144)^2
$37$	\( (T^{4} - 4 T^{3} + \cdots + 784)^{4} \) (T^4 - 4T^3 + 44T^2 + 112*T + 784)^4
$41$	\( (T^{4} + 160 T^{2} + 6272)^{4} \) (T^4 + 160*T^2 + 6272)^4
$43$	\( (T^{4} + 72 T^{2} + 784)^{4} \) (T^4 + 72*T^2 + 784)^4
$47$	\( (T^{8} + 64 T^{6} + \cdots + 262144)^{2} \) (T^8 + 64T^6 + 3584T^4 + 32768*T^2 + 262144)^2
$53$	\( (T^{2} - 2 T + 4)^{8} \) (T^2 - 2*T + 4)^8
$59$	\( (T^{8} + 104 T^{6} + \cdots + 5345344)^{2} \) (T^8 + 104T^6 + 8504T^4 + 240448*T^2 + 5345344)^2
$61$	\( (T^{8} - 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} \) (T^8 - 8T^6 + 56T^4 - 64*T^2 + 64)^2
$67$	\( (T^{8} - 264 T^{6} + \cdots + 21381376)^{2} \) (T^8 - 264T^6 + 65072T^4 - 1220736*T^2 + 21381376)^2
$71$	\( (T^{4} + 88 T^{2} + 784)^{4} \) (T^4 + 88*T^2 + 784)^4
$73$	\( (T^{8} - 160 T^{6} + \cdots + 16384)^{2} \) (T^8 - 160T^6 + 25472T^4 - 20480*T^2 + 16384)^2
$79$	\( (T^{8} - 216 T^{6} + \cdots + 71639296)^{2} \) (T^8 - 216T^6 + 38192T^4 - 1828224*T^2 + 71639296)^2
$83$	\( (T^{4} - 136 T^{2} + 392)^{4} \) (T^4 - 136*T^2 + 392)^4
$89$	\( (T^{8} - 32 T^{6} + \cdots + 16384)^{2} \) (T^8 - 32T^6 + 896T^4 - 4096*T^2 + 16384)^2
$97$	\( (T^{4} + 320 T^{2} + 25088)^{4} \) (T^4 + 320*T^2 + 25088)^4
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