Newform orbit 400.2.u.d

• Label: 400.2.u.d
• Level: $400$
• Weight: $2$
• Character orbit: 400.u
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.u (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.58140625.2
Defining polynomial:	\( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 5 \)
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2}) q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{7} + 3 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} - 4 q^{7} - q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) :

\(\nu\)	\(=\)	\( ( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 5 \)
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} - 3\beta_{4} + 2\beta_{3} + 11\beta_{2} + 6\beta _1 - 4 ) / 5 \)
\(\nu^{3}\)	\(=\)	\( ( -\beta_{7} + 16\beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{3} + 17\beta_{2} - 3\beta _1 + 2 ) / 5 \)
\(\nu^{4}\)	\(=\)	\( ( 13\beta_{7} - 3\beta_{6} - 6\beta_{5} + 13\beta_{4} - 7\beta_{3} - 6\beta_{2} - 6\beta _1 + 14 ) / 5 \)
\(\nu^{5}\)	\(=\)	\( ( 16\beta_{7} - 6\beta_{6} - 22\beta_{5} + 6\beta_{4} + 51\beta_{3} + 43\beta_{2} + 8\beta _1 - 67 ) / 5 \)
\(\nu^{6}\)	\(=\)	\( ( -23\beta_{7} + 63\beta_{6} + 46\beta_{5} + 12\beta_{4} + 52\beta_{3} + 156\beta_{2} - 34\beta _1 - 144 ) / 5 \)
\(\nu^{7}\)	\(=\)	\( ( -31\beta_{7} - 9\beta_{6} + 137\beta_{5} + 84\beta_{4} - 31\beta_{3} - 33\beta_{2} - 168\beta _1 + 62 ) / 5 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(1\)	\(-\beta_{3}\)	\(1\)

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.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

81.1

−0.983224	−	0.644389i
1.17421	+	0.0566033i
1.66637	−	0.917186i
−0.357358	+	1.86824i
1.66637	+	0.917186i
−0.357358	−	1.86824i
−0.983224	+	0.644389i
1.17421	−	0.0566033i

0

−1.09089

+

0.792578i

0

−2.15743

−

0.587785i

0

0.833366

0

−0.365190

+

1.12394i

0

81.2

0

2.39991

−

1.74363i

0

2.15743

−

0.587785i

0

−1.83337

0

1.79224

−

5.51595i

0

161.1

0

−0.529876

+

1.63079i

0

2.02373

+

0.951057i

0

2.77447

0

0.0483405

+

0.0351215i

0

161.2

0

0.720859

−

2.21858i

0

−2.02373

+

0.951057i

0

−3.77447

0

−1.97539

−

1.43521i

0

241.1

0

−0.529876

−

1.63079i

0

2.02373

−

0.951057i

0

2.77447

0

0.0483405

−

0.0351215i

0

241.2

0

0.720859

+

2.21858i

0

−2.02373

−

0.951057i

0

−3.77447

0

−1.97539

+

1.43521i

0

321.1

0

−1.09089

−

0.792578i

0

−2.15743

+

0.587785i

0

0.833366

0

−0.365190

−

1.12394i

0

321.2

0

2.39991

+

1.74363i

0

2.15743

+

0.587785i

0

−1.83337

0

1.79224

+

5.51595i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.2.u.d		8
4.b	odd	2	1		50.2.d.b	✓	8
12.b	even	2	1		450.2.h.e		8
20.d	odd	2	1		250.2.d.d		8
20.e	even	4	2		250.2.e.c		16
25.d	even	5	1	inner	400.2.u.d		8
25.d	even	5	1		10000.2.a.t		4
25.e	even	10	1		10000.2.a.x		4
100.h	odd	10	1		250.2.d.d		8
100.h	odd	10	1		1250.2.a.f		4
100.j	odd	10	1		50.2.d.b	✓	8
100.j	odd	10	1		1250.2.a.l		4
100.l	even	20	2		250.2.e.c		16
100.l	even	20	2		1250.2.b.e		8
300.n	even	10	1		450.2.h.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
50.2.d.b	✓	8	4.b	odd	2	1
50.2.d.b	✓	8	100.j	odd	10	1
250.2.d.d		8	20.d	odd	2	1
250.2.d.d		8	100.h	odd	10	1
250.2.e.c		16	20.e	even	4	2
250.2.e.c		16	100.l	even	20	2
400.2.u.d		8	1.a	even	1	1	trivial
400.2.u.d		8	25.d	even	5	1	inner
450.2.h.e		8	12.b	even	2	1
450.2.h.e		8	300.n	even	10	1
1250.2.a.f		4	100.h	odd	10	1
1250.2.a.l		4	100.j	odd	10	1
1250.2.b.e		8	100.l	even	20	2
10000.2.a.t		4	25.d	even	5	1
10000.2.a.x		4	25.e	even	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 3T_{3}^{7} + 8T_{3}^{6} - 6T_{3}^{5} + 25T_{3}^{4} + 24T_{3}^{3} + 128T_{3}^{2} + 192T_{3} + 256 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	T^8 - 3*T^7 + 8*T^6 - 6*T^5 + 25*T^4 + 24*T^3 + 128*T^2 + 192*T + 256
$5$	T^8 - 15*T^6 + 105*T^4 - 375*T^2 + 625
$7$	\( (T^{4} + 2 T^{3} - 11 T^{2} - 12 T + 16)^{2} \)
$11$	T^8 + T^7 - 3*T^6 - 7*T^5 + 105*T^4 - 128*T^3 + 352*T^2 + 64*T + 256