Newform orbit 100.10.c.a

• Label: 100.10.c.a
• Level: $100$
• Weight: $10$
• Character orbit: 100.c
• Analytic conductor: $51.504$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(51.5035836164\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 114 \beta q^{3} + 3164 \beta q^{7} - 32301 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64602 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(1\)	\(-1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

	−	1.00000i
		1.00000i

0

−

228.000i

0

0

0

−

6328.00i

0

−32301.0

0

49.2

0

228.000i

0

0

0

6328.00i

0

−32301.0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	100.10.c.a		2
4.b	odd	2	1		400.10.c.a		2
5.b	even	2	1	inner	100.10.c.a		2
5.c	odd	4	1		4.10.a.a	✓	1
5.c	odd	4	1		100.10.a.a		1
15.e	even	4	1		36.10.a.b		1
20.d	odd	2	1		400.10.c.a		2
20.e	even	4	1		16.10.a.a		1
20.e	even	4	1		400.10.a.k		1
35.f	even	4	1		196.10.a.a		1
35.k	even	12	2		196.10.e.b		2
35.l	odd	12	2		196.10.e.a		2
40.i	odd	4	1		64.10.a.a		1
40.k	even	4	1		64.10.a.i		1
45.k	odd	12	2		324.10.e.e		2
45.l	even	12	2		324.10.e.b		2
60.l	odd	4	1		144.10.a.j		1
80.i	odd	4	1		256.10.b.j		2
80.j	even	4	1		256.10.b.b		2
80.s	even	4	1		256.10.b.b		2
80.t	odd	4	1		256.10.b.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.10.a.a	✓	1	5.c	odd	4	1
16.10.a.a		1	20.e	even	4	1
36.10.a.b		1	15.e	even	4	1
64.10.a.a		1	40.i	odd	4	1
64.10.a.i		1	40.k	even	4	1
100.10.a.a		1	5.c	odd	4	1
100.10.c.a		2	1.a	even	1	1	trivial
100.10.c.a		2	5.b	even	2	1	inner
144.10.a.j		1	60.l	odd	4	1
196.10.a.a		1	35.f	even	4	1
196.10.e.a		2	35.l	odd	12	2
196.10.e.b		2	35.k	even	12	2
256.10.b.b		2	80.j	even	4	1
256.10.b.b		2	80.s	even	4	1
256.10.b.j		2	80.i	odd	4	1
256.10.b.j		2	80.t	odd	4	1
324.10.e.b		2	45.l	even	12	2
324.10.e.e		2	45.k	odd	12	2
400.10.a.k		1	20.e	even	4	1
400.10.c.a		2	4.b	odd	2	1
400.10.c.a		2	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 51984 \) acting on \(S_{10}^{\mathrm{new}}(100, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 51984 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 40043584 \)
$11$	\( (T + 30420)^{2} \)