Embedded newform 310.3.i.a.99.5

• Label: 310.3.i.a.99.5
• Level: $310$
• Weight: $3$
• Character: 310.99
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• 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			99.5
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.21

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-1.14812 - 1.98861i) q^{3} -2.00000 q^{4} +(3.83759 - 3.20514i) q^{5} +(-2.81232 + 1.62369i) q^{6} +(-6.29732 + 3.63576i) q^{7} +2.82843i q^{8} +(1.86363 - 3.22790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−1.14812	−	1.98861i	−0.382708	−	0.662869i	0.608741	−	0.793369i	\(-0.291675\pi\)
−0.991448	+	0.130500i	\(0.958342\pi\)
\(5\)	3.83759	−	3.20514i	0.767518	−	0.641027i
							\(7\)	−6.29732	+	3.63576i	−0.899617	+	0.519394i	−0.877076	−	0.480352i	\(-0.840509\pi\)
−0.0225410	+	0.999746i	\(0.507176\pi\)
\(11\)	−13.8507	−	7.99668i	−1.25915	−	0.726971i	−0.286241	−	0.958158i	\(-0.592406\pi\)
							−0.972909	+	0.231187i	\(0.925739\pi\)
\(13\)	−5.06003	+	8.76422i	−0.389233	+	0.674171i	−0.992347	−	0.123484i	\(-0.960593\pi\)
							0.603114	+	0.797655i	\(0.293927\pi\)
\(17\)	5.57904	+	9.66317i	0.328179	+	0.568422i	0.982150	−	0.188097i	\(-0.0602319\pi\)
							−0.653972	+	0.756519i	\(0.726899\pi\)
\(19\)	−11.0071	−	19.0648i	−0.579320	−	1.00341i	−0.995557	−	0.0941561i	\(-0.969985\pi\)
							0.416237	−	0.909256i	\(-0.363349\pi\)
\(23\)	−18.3336			−0.797112			−0.398556	−	0.917144i	\(-0.630489\pi\)
							−0.398556	+	0.917144i	\(0.630489\pi\)
\(29\)			45.4148i			1.56603i	0.622004	+	0.783014i	\(0.286319\pi\)
							−0.622004	+	0.783014i	\(0.713681\pi\)
\(31\)	29.8008	+	8.53879i	0.961317	+	0.275445i
							\(37\)	−13.9504	−	24.1627i	−0.377037	−	0.653047i	0.613593	−	0.789623i	\(-0.289724\pi\)
−0.990630	+	0.136576i	\(0.956390\pi\)
\(41\)	−29.4427	+	50.9963i	−0.718115	+	1.24381i	0.243631	+	0.969868i	\(0.421661\pi\)
							−0.961746	+	0.273943i	\(0.911672\pi\)
\(43\)	18.8087	+	32.5777i	0.437412	+	0.757620i	0.997489	−	0.0708206i	\(-0.0225618\pi\)
							−0.560077	+	0.828441i	\(0.689228\pi\)
\(47\)		−	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.99.5	✓	64
5.4	even	2	inner	310.3.i.a.99.28	yes	64
31.26	odd	6	inner	310.3.i.a.119.12	yes	64
155.119	odd	6	inner	310.3.i.a.119.21	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.5	✓	64	1.1	even	1	trivial
310.3.i.a.99.28	yes	64	5.4	even	2	inner
310.3.i.a.119.12	yes	64	31.26	odd	6	inner
310.3.i.a.119.21	yes	64	155.119	odd	6	inner