Embedded newform 310.3.i.a.99.8

• Label: 310.3.i.a.99.8
• 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			99.8
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.24

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-0.404044 - 0.699825i) q^{3} -2.00000 q^{4} +(0.104343 + 4.99891i) q^{5} +(-0.989702 + 0.571405i) q^{6} +(0.505559 - 0.291885i) q^{7} +2.82843i q^{8} +(4.17350 - 7.22871i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.404044	−	0.699825i	−0.134681	−	0.233275i	0.790794	−	0.612082i	\(-0.209668\pi\)
−0.925476	+	0.378807i	\(0.876334\pi\)
\(5\)	0.104343	+	4.99891i	0.0208685	+	0.999782i
							\(7\)	0.505559	−	0.291885i	0.0722228	−	0.0416978i	−0.463454	−	0.886121i	\(-0.653390\pi\)
0.535677	+	0.844423i	\(0.320057\pi\)
\(11\)	−17.8799	−	10.3230i	−1.62545	−	0.938452i	−0.985428	−	0.170093i	\(-0.945593\pi\)
							−0.640019	−	0.768359i	\(-0.721073\pi\)
\(13\)	−4.10600	+	7.11180i	−0.315846	+	0.547062i	−0.979617	−	0.200874i	\(-0.935622\pi\)
							0.663771	+	0.747936i	\(0.268955\pi\)
\(17\)	−9.34812	−	16.1914i	−0.549890	−	0.952437i	−0.998282	−	0.0585997i	\(-0.981336\pi\)
							0.448392	−	0.893837i	\(-0.351997\pi\)
\(19\)	−7.11815	−	12.3290i	−0.374640	−	0.648895i	0.615633	−	0.788033i	\(-0.288900\pi\)
							−0.990273	+	0.139138i	\(0.955567\pi\)
\(23\)	−5.80600			−0.252435			−0.126217	−	0.992003i	\(-0.540284\pi\)
							−0.126217	+	0.992003i	\(0.540284\pi\)
\(29\)		−	14.8232i		−	0.511144i	−0.966790	−	0.255572i	\(-0.917736\pi\)
							0.966790	−	0.255572i	\(-0.0822637\pi\)
\(31\)	−28.8449	−	11.3566i	−0.930480	−	0.366342i
							\(37\)	36.1292	+	62.5776i	0.976465	+	1.69129i	0.675013	+	0.737806i	\(0.264138\pi\)
0.301452	+	0.953481i	\(0.402529\pi\)
\(41\)	1.22669	−	2.12468i	0.0299192	−	0.0518216i	−0.850678	−	0.525687i	\(-0.823808\pi\)
							0.880597	+	0.473865i	\(0.157142\pi\)
\(43\)	−23.5875	−	40.8548i	−0.548547	−	0.950111i	−0.998374	−	0.0569955i	\(-0.981848\pi\)
							0.449828	−	0.893115i	\(-0.351485\pi\)
\(47\)			76.7458i			1.63289i	0.577424	+	0.816444i	\(0.304058\pi\)
							−0.577424	+	0.816444i	\(0.695942\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.8	✓	64
5.4	even	2	inner	310.3.i.a.99.25	yes	64
31.26	odd	6	inner	310.3.i.a.119.9	yes	64
155.119	odd	6	inner	310.3.i.a.119.24	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.8	✓	64	1.1	even	1	trivial
310.3.i.a.99.25	yes	64	5.4	even	2	inner
310.3.i.a.119.9	yes	64	31.26	odd	6	inner
310.3.i.a.119.24	yes	64	155.119	odd	6	inner