Embedded newform 310.3.i.a.99.12

• Label: 310.3.i.a.99.12
• Level: $310$
• Weight: $3$
• Character: 310.99
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			99.12
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.28

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(1.19046 + 2.06194i) q^{3} -2.00000 q^{4} +(-4.71572 + 1.66192i) q^{5} +(2.91602 - 1.68357i) q^{6} +(9.16680 - 5.29245i) q^{7} +2.82843i q^{8} +(1.66560 - 2.88491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.41421i		−	0.707107i
							\(3\)	1.19046	+	2.06194i	0.396820	+	0.687313i	0.993332	−	0.115292i	\(-0.0367802\pi\)
−0.596511	+	0.802605i	\(0.703447\pi\)
\(5\)	−4.71572	+	1.66192i	−0.943144	+	0.332384i
							\(7\)	9.16680	−	5.29245i	1.30954	−	0.756065i	0.327523	−	0.944843i	\(-0.393786\pi\)
0.982020	+	0.188778i	\(0.0604528\pi\)
\(11\)	3.81391	+	2.20196i	0.346719	+	0.200178i	0.663239	−	0.748407i	\(-0.269181\pi\)
							−0.316520	+	0.948586i	\(0.602515\pi\)
\(13\)	−5.26107	+	9.11245i	−0.404698	+	0.700957i	−0.994286	−	0.106746i	\(-0.965957\pi\)
							0.589588	+	0.807704i	\(0.299290\pi\)
\(17\)	11.7042	+	20.2724i	0.688485	+	1.19249i	0.972328	+	0.233620i	\(0.0750573\pi\)
							−0.283843	+	0.958871i	\(0.591609\pi\)
\(19\)	−2.82401	−	4.89132i	−0.148632	−	0.257438i	0.782090	−	0.623165i	\(-0.214154\pi\)
							−0.930722	+	0.365727i	\(0.880820\pi\)
\(23\)	44.7869			1.94726			0.973629	−	0.228138i	\(-0.0732638\pi\)
							0.973629	+	0.228138i	\(0.0732638\pi\)
\(29\)		−	24.1112i		−	0.831420i	−0.909497	−	0.415710i	\(-0.863533\pi\)
							0.909497	−	0.415710i	\(-0.136467\pi\)
\(31\)	22.4072	−	21.4224i	0.722813	−	0.691044i
							\(37\)	30.2264	+	52.3537i	0.816931	+	1.41497i	0.907933	+	0.419115i	\(0.137660\pi\)
−0.0910020	+	0.995851i	\(0.529007\pi\)
\(41\)	−30.5166	+	52.8562i	−0.744307	+	1.28918i	0.206212	+	0.978507i	\(0.433886\pi\)
							−0.950518	+	0.310669i	\(0.899447\pi\)
\(43\)	−23.6323	−	40.9323i	−0.549588	−	0.951915i	−0.998303	−	0.0582398i	\(-0.981451\pi\)
							0.448714	−	0.893675i	\(-0.351882\pi\)
\(47\)		−	26.9063i		−	0.572474i	−0.958159	−	0.286237i	\(-0.907595\pi\)
							0.958159	−	0.286237i	\(-0.0924045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.12	✓	64
5.4	even	2	inner	310.3.i.a.99.21	yes	64
31.26	odd	6	inner	310.3.i.a.119.5	yes	64
155.119	odd	6	inner	310.3.i.a.119.28	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.12	✓	64	1.1	even	1	trivial
310.3.i.a.99.21	yes	64	5.4	even	2	inner
310.3.i.a.119.5	yes	64	31.26	odd	6	inner
310.3.i.a.119.28	yes	64	155.119	odd	6	inner