Newform orbit 100.2.e.a

• Label: 100.2.e.a
• Level: $100$
• Weight: $2$
• Character orbit: 100.e
• Analytic conductor: $0.799$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -20
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	100.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.798504020213\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

7.1

		1.00000i
	−	1.00000i

−1.00000

+

1.00000i

−1.00000

−

1.00000i

−

2.00000i

0

2.00000

3.00000

−

3.00000i

2.00000

+

2.00000i

−

1.00000i

0

43.1

−1.00000

−

1.00000i

−1.00000

+

1.00000i

2.00000i

0

2.00000

3.00000

+

3.00000i

2.00000

−

2.00000i

1.00000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)
5.c	odd	4	1	inner
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	100.2.e.a	✓	2
3.b	odd	2	1		900.2.k.e		2
4.b	odd	2	1		100.2.e.c	yes	2
5.b	even	2	1		100.2.e.c	yes	2
5.c	odd	4	1	inner	100.2.e.a	✓	2
5.c	odd	4	1		100.2.e.c	yes	2
8.b	even	2	1		1600.2.n.l		2
8.d	odd	2	1		1600.2.n.b		2
12.b	even	2	1		900.2.k.a		2
15.d	odd	2	1		900.2.k.a		2
15.e	even	4	1		900.2.k.a		2
15.e	even	4	1		900.2.k.e		2
20.d	odd	2	1	CM	100.2.e.a	✓	2
20.e	even	4	1	inner	100.2.e.a	✓	2
20.e	even	4	1		100.2.e.c	yes	2
40.e	odd	2	1		1600.2.n.l		2
40.f	even	2	1		1600.2.n.b		2
40.i	odd	4	1		1600.2.n.b		2
40.i	odd	4	1		1600.2.n.l		2
40.k	even	4	1		1600.2.n.b		2
40.k	even	4	1		1600.2.n.l		2
60.h	even	2	1		900.2.k.e		2
60.l	odd	4	1		900.2.k.a		2
60.l	odd	4	1		900.2.k.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
100.2.e.a	✓	2	1.a	even	1	1	trivial
100.2.e.a	✓	2	5.c	odd	4	1	inner
100.2.e.a	✓	2	20.d	odd	2	1	CM
100.2.e.a	✓	2	20.e	even	4	1	inner
100.2.e.c	yes	2	4.b	odd	2	1
100.2.e.c	yes	2	5.b	even	2	1
100.2.e.c	yes	2	5.c	odd	4	1
100.2.e.c	yes	2	20.e	even	4	1
900.2.k.a		2	12.b	even	2	1
900.2.k.a		2	15.d	odd	2	1
900.2.k.a		2	15.e	even	4	1
900.2.k.a		2	60.l	odd	4	1
900.2.k.e		2	3.b	odd	2	1
900.2.k.e		2	15.e	even	4	1
900.2.k.e		2	60.h	even	2	1
900.2.k.e		2	60.l	odd	4	1
1600.2.n.b		2	8.d	odd	2	1
1600.2.n.b		2	40.f	even	2	1
1600.2.n.b		2	40.i	odd	4	1
1600.2.n.b		2	40.k	even	4	1
1600.2.n.l		2	8.b	even	2	1
1600.2.n.l		2	40.e	odd	2	1
1600.2.n.l		2	40.i	odd	4	1
1600.2.n.l		2	40.k	even	4	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 2 \)
$3$	\( T^{2} + 2T + 2 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 6T + 18 \)
$11$	\( T^{2} \)