Newform orbit 1184.2.g.b

• Label: 1184.2.g.b
• Level: $1184$
• Weight: $2$
• Character orbit: 1184.g
• Analytic conductor: $9.454$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1184 = 2^{5} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1184.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.45428759932\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 + 2*b * q^5 - 3 * q^9 - 2*b * q^13 + 4*b * q^17 - 11 * q^25 + 2*b * q^29 + (-3*b + 1) * q^37 - 10 * q^41 - 6*b * q^45 - 7 * q^49 - 14 * q^53 - 6*b * q^61 + 16 * q^65 + 6 * q^73 + 9 * q^81 - 32 * q^85 + 8*b * q^89 + 4*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} - 22 q^{25} + 2 q^{37} - 20 q^{41} - 14 q^{49} - 28 q^{53} + 32 q^{65} + 12 q^{73} + 18 q^{81} - 64 q^{85}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1184\mathbb{Z}\right)^\times\).

\(n\)	\(223\)	\(705\)	\(741\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

961.1

	−	1.00000i
		1.00000i

0

0

0

−

4.00000i

0

0

0

−3.00000

0

961.2

0

0

0

4.00000i

0

0

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
37.b	even	2	1	inner
148.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1184.2.g.b	✓	2
4.b	odd	2	1	CM	1184.2.g.b	✓	2
8.b	even	2	1		2368.2.g.d		2
8.d	odd	2	1		2368.2.g.d		2
37.b	even	2	1	inner	1184.2.g.b	✓	2
148.b	odd	2	1	inner	1184.2.g.b	✓	2
296.e	even	2	1		2368.2.g.d		2
296.h	odd	2	1		2368.2.g.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1184.2.g.b	✓	2	1.a	even	1	1	trivial
1184.2.g.b	✓	2	4.b	odd	2	1	CM
1184.2.g.b	✓	2	37.b	even	2	1	inner
1184.2.g.b	✓	2	148.b	odd	2	1	inner
2368.2.g.d		2	8.b	even	2	1
2368.2.g.d		2	8.d	odd	2	1
2368.2.g.d		2	296.e	even	2	1
2368.2.g.d		2	296.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1184, [\chi])\):

\( T_{3} \)

\( T_{5}^{2} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 16 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)