Newform orbit 3800.1.y.a

• Label: 3800.1.y.a
• Level: $3800$
• Weight: $1$
• Character orbit: 3800.y
• Analytic conductor: $1.896$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -95
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3800.y (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.89644704801\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.0.115520.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\)	\(401\)	\(951\)	\(1901\)	\(1977\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(i\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1443.1

	−	1.00000i
		1.00000i

−1.00000

−1.00000

+

1.00000i

1.00000

0

1.00000

−

1.00000i

0

−1.00000

−

1.00000i

0

2507.1

−1.00000

−1.00000

−

1.00000i

1.00000

0

1.00000

+

1.00000i

0

−1.00000

1.00000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
95.d	odd	2	1	CM by \(\Q(\sqrt{-95}) \)
40.k	even	4	1	inner
760.y	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3800.1.y.a	✓	2
5.b	even	2	1		3800.1.y.d	yes	2
5.c	odd	4	1		3800.1.y.b	yes	2
5.c	odd	4	1		3800.1.y.c	yes	2
8.d	odd	2	1		3800.1.y.b	yes	2
19.b	odd	2	1		3800.1.y.d	yes	2
40.e	odd	2	1		3800.1.y.c	yes	2
40.k	even	4	1	inner	3800.1.y.a	✓	2
40.k	even	4	1		3800.1.y.d	yes	2
95.d	odd	2	1	CM	3800.1.y.a	✓	2
95.g	even	4	1		3800.1.y.b	yes	2
95.g	even	4	1		3800.1.y.c	yes	2
152.b	even	2	1		3800.1.y.c	yes	2
760.p	even	2	1		3800.1.y.b	yes	2
760.y	odd	4	1	inner	3800.1.y.a	✓	2
760.y	odd	4	1		3800.1.y.d	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3800.1.y.a	✓	2	1.a	even	1	1	trivial
3800.1.y.a	✓	2	40.k	even	4	1	inner
3800.1.y.a	✓	2	95.d	odd	2	1	CM
3800.1.y.a	✓	2	760.y	odd	4	1	inner
3800.1.y.b	yes	2	5.c	odd	4	1
3800.1.y.b	yes	2	8.d	odd	2	1
3800.1.y.b	yes	2	95.g	even	4	1
3800.1.y.b	yes	2	760.p	even	2	1
3800.1.y.c	yes	2	5.c	odd	4	1
3800.1.y.c	yes	2	40.e	odd	2	1
3800.1.y.c	yes	2	95.g	even	4	1
3800.1.y.c	yes	2	152.b	even	2	1
3800.1.y.d	yes	2	5.b	even	2	1
3800.1.y.d	yes	2	19.b	odd	2	1
3800.1.y.d	yes	2	40.k	even	4	1
3800.1.y.d	yes	2	760.y	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \)

\( T_{13}^{2} - 2T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} + 2T + 2 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)