Embedded newform 310.3.i.a.99.9

• Label: 310.3.i.a.99.9
• 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			99.9
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.25

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-0.136432 - 0.236307i) q^{3} -2.00000 q^{4} +(-4.56478 - 2.04028i) q^{5} +(-0.334188 + 0.192944i) q^{6} +(-6.27915 + 3.62527i) q^{7} +2.82843i q^{8} +(4.46277 - 7.72975i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.136432	−	0.236307i	−0.0454773	−	0.0787689i	0.842391	−	0.538867i	\(-0.181148\pi\)
−0.887868	+	0.460098i	\(0.847814\pi\)
\(5\)	−4.56478	−	2.04028i	−0.912957	−	0.408057i
							\(7\)	−6.27915	+	3.62527i	−0.897022	+	0.517896i	−0.876233	−	0.481888i	\(-0.839951\pi\)
−0.0207892	+	0.999784i	\(0.506618\pi\)
\(11\)	3.86233	+	2.22992i	0.351121	+	0.202720i	0.665179	−	0.746684i	\(-0.268355\pi\)
							−0.314058	+	0.949404i	\(0.601689\pi\)
\(13\)	−4.00711	+	6.94051i	−0.308239	+	0.533886i	−0.977977	−	0.208712i	\(-0.933073\pi\)
							0.669738	+	0.742597i	\(0.266406\pi\)
\(17\)	12.4828	+	21.6208i	0.734280	+	1.27181i	0.955039	+	0.296481i	\(0.0958133\pi\)
							−0.220759	+	0.975328i	\(0.570853\pi\)
\(19\)	9.63037	+	16.6803i	0.506862	+	0.877910i	0.999968	+	0.00794154i	\(0.00252790\pi\)
							−0.493107	+	0.869969i	\(0.664139\pi\)
\(23\)	−18.9920			−0.825741			−0.412871	−	0.910790i	\(-0.635474\pi\)
							−0.412871	+	0.910790i	\(0.635474\pi\)
\(29\)			48.7604i			1.68139i	0.541507	+	0.840696i	\(0.317854\pi\)
							−0.541507	+	0.840696i	\(0.682146\pi\)
\(31\)	−28.7077	−	11.6992i	−0.926054	−	0.377392i
							\(37\)	−7.98508	−	13.8306i	−0.215813	−	0.373799i	0.737711	−	0.675117i	\(-0.235907\pi\)
−0.953524	+	0.301318i	\(0.902573\pi\)
\(41\)	4.42420	−	7.66294i	0.107907	−	0.186901i	−0.807015	−	0.590531i	\(-0.798918\pi\)
							0.914922	+	0.403630i	\(0.132252\pi\)
\(43\)	−26.1415	−	45.2784i	−0.607942	−	1.05299i	−0.991579	−	0.129503i	\(-0.958662\pi\)
							0.383637	−	0.923484i	\(-0.374671\pi\)
\(47\)			86.7712i			1.84620i	0.384565	+	0.923098i	\(0.374351\pi\)
							−0.384565	+	0.923098i	\(0.625649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.9	✓	64
5.4	even	2	inner	310.3.i.a.99.24	yes	64
31.26	odd	6	inner	310.3.i.a.119.8	yes	64
155.119	odd	6	inner	310.3.i.a.119.25	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.9	✓	64	1.1	even	1	trivial
310.3.i.a.99.24	yes	64	5.4	even	2	inner
310.3.i.a.119.8	yes	64	31.26	odd	6	inner
310.3.i.a.119.25	yes	64	155.119	odd	6	inner