Embedded newform 310.3.i.a.119.12

• Label: 310.3.i.a.119.12
• Level: $310$
• Weight: $3$
• Character: 310.119
• 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			119.12
Character	\(\chi\)	\(=\)	310.119
Dual form			310.3.i.a.99.28

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(1.14812 - 1.98861i) q^{3} -2.00000 q^{4} +(0.856935 + 4.92602i) q^{5} +(-2.81232 - 1.62369i) q^{6} +(6.29732 + 3.63576i) q^{7} +2.82843i q^{8} +(1.86363 + 3.22790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	1.14812	−	1.98861i	0.382708	−	0.662869i	−0.608741	−	0.793369i	\(-0.708325\pi\)
0.991448	+	0.130500i	\(0.0416583\pi\)
\(5\)	0.856935	+	4.92602i	0.171387	+	0.985204i
							\(7\)	6.29732	+	3.63576i	0.899617	+	0.519394i	0.877076	−	0.480352i	\(-0.159491\pi\)
0.0225410	+	0.999746i	\(0.492824\pi\)
\(11\)	−13.8507	+	7.99668i	−1.25915	+	0.726971i	−0.972909	−	0.231187i	\(-0.925739\pi\)
							−0.286241	+	0.958158i	\(0.592406\pi\)
\(13\)	5.06003	+	8.76422i	0.389233	+	0.674171i	0.992347	−	0.123484i	\(-0.0394068\pi\)
							−0.603114	+	0.797655i	\(0.706073\pi\)
\(17\)	−5.57904	+	9.66317i	−0.328179	+	0.568422i	−0.982150	−	0.188097i	\(-0.939768\pi\)
							0.653972	+	0.756519i	\(0.273101\pi\)
\(19\)	−11.0071	+	19.0648i	−0.579320	+	1.00341i	0.416237	+	0.909256i	\(0.363349\pi\)
							−0.995557	+	0.0941561i	\(0.969985\pi\)
\(23\)	18.3336			0.797112			0.398556	−	0.917144i	\(-0.369511\pi\)
							0.398556	+	0.917144i	\(0.369511\pi\)
\(29\)		−	45.4148i		−	1.56603i	−0.622004	−	0.783014i	\(-0.713681\pi\)
							0.622004	−	0.783014i	\(-0.286319\pi\)
\(31\)	29.8008	−	8.53879i	0.961317	−	0.275445i
							\(37\)	13.9504	−	24.1627i	0.377037	−	0.653047i	−0.613593	−	0.789623i	\(-0.710276\pi\)
0.990630	+	0.136576i	\(0.0436097\pi\)
\(41\)	−29.4427	−	50.9963i	−0.718115	−	1.24381i	−0.961746	−	0.273943i	\(-0.911672\pi\)
							0.243631	−	0.969868i	\(-0.421661\pi\)
\(43\)	−18.8087	+	32.5777i	−0.437412	+	0.757620i	−0.997489	−	0.0708206i	\(-0.977438\pi\)
							0.560077	+	0.828441i	\(0.310772\pi\)
\(47\)		−	89.9653i		−	1.91416i	−0.289831	−	0.957078i	\(-0.593599\pi\)
							0.289831	−	0.957078i	\(-0.406401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.119.12	yes	64
5.4	even	2	inner	310.3.i.a.119.21	yes	64
31.6	odd	6	inner	310.3.i.a.99.5	✓	64
155.99	odd	6	inner	310.3.i.a.99.28	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.5	✓	64	31.6	odd	6	inner
310.3.i.a.99.28	yes	64	155.99	odd	6	inner
310.3.i.a.119.12	yes	64	1.1	even	1	trivial
310.3.i.a.119.21	yes	64	5.4	even	2	inner