Newform orbit 1805.1.h.b

• Label: 1805.1.h.b
• Level: $1805$
• Weight: $1$
• Character orbit: 1805.h
• Analytic conductor: $0.901$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{4}$
• CM discriminant: -95
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1805.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.900812347803\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 95)
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.475.1
Artin image:	$C_3\times D_8$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b1 * q^2 + b1 * q^3 + b2 * q^4 + (b2 + 1) * q^5 - 2*b2 * q^6 + b2 * q^9 + (-b3 - b1) * q^10 + b3 * q^12 + (b3 + b1) * q^13 + (b3 + b1) * q^15 + (b2 + 1) * q^16 - b3 * q^18 - q^20 + b2 * q^25 + 2 * q^26 + 2 * q^30 + (-b3 - b1) * q^32 + (-b2 - 1) * q^36 + b3 * q^37 - 2 * q^39 - q^45 + (b3 + b1) * q^48 + q^49 - b3 * q^50 - b1 * q^52 + (-b3 - b1) * q^53 - b1 * q^60 - q^64 + b3 * q^65 + (b3 + b1) * q^67 + (2*b2 + 2) * q^74 + b3 * q^75 + 2*b1 * q^78 + b2 * q^80 + (b2 + 1) * q^81 + b1 * q^90 + 2 * q^96 + b1 * q^97 - b1 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 2 q^{16} - 4 q^{20} - 2 q^{25} + 8 q^{26} + 8 q^{30} - 2 q^{36} - 8 q^{39} - 4 q^{45} + 4 q^{49} - 4 q^{64} + 4 q^{74} - 2 q^{80} + 2 q^{81} + 8 q^{96}+O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\)	\(362\)	\(1446\)
\(\chi(n)\)	\(-1\)	\(-\beta_{2}\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

69.1

0.707107	+	1.22474i
−0.707107	−	1.22474i
0.707107	−	1.22474i
−0.707107	+	1.22474i

−0.707107

−

1.22474i

0.707107

+

1.22474i

−0.500000

+

0.866025i

0.500000

+

0.866025i

1.00000

−

1.73205i

0

0

−0.500000

+

0.866025i

0.707107

−

1.22474i

69.2

0.707107

+

1.22474i

−0.707107

−

1.22474i

−0.500000

+

0.866025i

0.500000

+

0.866025i

1.00000

−

1.73205i

0

0

−0.500000

+

0.866025i

−0.707107

+

1.22474i

654.1

−0.707107

+

1.22474i

0.707107

−

1.22474i

−0.500000

−

0.866025i

0.500000

−

0.866025i

1.00000

+

1.73205i

0

0

−0.500000

−

0.866025i

0.707107

+

1.22474i

654.2

0.707107

−

1.22474i

−0.707107

+

1.22474i

−0.500000

−

0.866025i

0.500000

−

0.866025i

1.00000

+

1.73205i

0

0

−0.500000

−

0.866025i

−0.707107

−

1.22474i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
95.d	odd	2	1	CM by \(\Q(\sqrt{-95}) \)
5.b	even	2	1	inner
19.b	odd	2	1	inner
19.c	even	3	1	inner
19.d	odd	6	1	inner
95.h	odd	6	1	inner
95.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1805.1.h.b		4
5.b	even	2	1	inner	1805.1.h.b		4
19.b	odd	2	1	inner	1805.1.h.b		4
19.c	even	3	1		95.1.d.b	✓	2
19.c	even	3	1	inner	1805.1.h.b		4
19.d	odd	6	1		95.1.d.b	✓	2
19.d	odd	6	1	inner	1805.1.h.b		4
19.e	even	9	6		1805.1.o.b		12
19.f	odd	18	6		1805.1.o.b		12
57.f	even	6	1		855.1.g.c		2
57.h	odd	6	1		855.1.g.c		2
76.f	even	6	1		1520.1.m.b		2
76.g	odd	6	1		1520.1.m.b		2
95.d	odd	2	1	CM	1805.1.h.b		4
95.h	odd	6	1		95.1.d.b	✓	2
95.h	odd	6	1	inner	1805.1.h.b		4
95.i	even	6	1		95.1.d.b	✓	2
95.i	even	6	1	inner	1805.1.h.b		4
95.l	even	12	2		475.1.c.b		2
95.m	odd	12	2		475.1.c.b		2
95.o	odd	18	6		1805.1.o.b		12
95.p	even	18	6		1805.1.o.b		12
285.n	odd	6	1		855.1.g.c		2
285.q	even	6	1		855.1.g.c		2
380.p	odd	6	1		1520.1.m.b		2
380.s	even	6	1		1520.1.m.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.1.d.b	✓	2	19.c	even	3	1
95.1.d.b	✓	2	19.d	odd	6	1
95.1.d.b	✓	2	95.h	odd	6	1
95.1.d.b	✓	2	95.i	even	6	1
475.1.c.b		2	95.l	even	12	2
475.1.c.b		2	95.m	odd	12	2
855.1.g.c		2	57.f	even	6	1
855.1.g.c		2	57.h	odd	6	1
855.1.g.c		2	285.n	odd	6	1
855.1.g.c		2	285.q	even	6	1
1520.1.m.b		2	76.f	even	6	1
1520.1.m.b		2	76.g	odd	6	1
1520.1.m.b		2	380.p	odd	6	1
1520.1.m.b		2	380.s	even	6	1
1805.1.h.b		4	1.a	even	1	1	trivial
1805.1.h.b		4	5.b	even	2	1	inner
1805.1.h.b		4	19.b	odd	2	1	inner
1805.1.h.b		4	19.c	even	3	1	inner
1805.1.h.b		4	19.d	odd	6	1	inner
1805.1.h.b		4	95.d	odd	2	1	CM
1805.1.h.b		4	95.h	odd	6	1	inner
1805.1.h.b		4	95.i	even	6	1	inner
1805.1.o.b		12	19.e	even	9	6
1805.1.o.b		12	19.f	odd	18	6
1805.1.o.b		12	95.o	odd	18	6
1805.1.o.b		12	95.p	even	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 2T^{2} + 4 \)
$3$	\( T^{4} + 2T^{2} + 4 \)
$5$	\( (T^{2} - T + 1)^{2} \)
$7$	\( T^{4} \)
$11$	\( T^{4} \)