Embedded newform 310.2.s.a.77.3

• Label: 310.2.s.a.77.3
• Level: $310$
• Weight: $2$
• Character: 310.77
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.s (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47536246266\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			77.3
Character	\(\chi\)	\(=\)	310.77
Dual form			310.2.s.a.153.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987688 - 0.156434i) q^{2} +(-0.102776 - 0.648899i) q^{3} +(0.951057 + 0.309017i) q^{4} +(1.87383 - 1.22015i) q^{5} +0.656988i q^{6} +(-1.32682 + 0.676049i) q^{7} +(-0.891007 - 0.453990i) q^{8} +(2.44266 - 0.793669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.987688	−	0.156434i	−0.698401	−	0.110616i
							\(3\)	−0.102776	−	0.648899i	−0.0593375	−	0.374642i	−0.999431	−	0.0337258i	\(-0.989263\pi\)
0.940094	−	0.340916i	\(-0.110737\pi\)
\(5\)	1.87383	−	1.22015i	0.838003	−	0.545666i
							\(7\)	−1.32682	+	0.676049i	−0.501491	+	0.255522i	−0.686388	−	0.727236i	\(-0.740805\pi\)
0.184897	+	0.982758i	\(0.440805\pi\)
\(11\)	1.63188	+	0.530231i	0.492031	+	0.159871i	0.544515	−	0.838751i	\(-0.316714\pi\)
							−0.0524838	+	0.998622i	\(0.516714\pi\)
\(13\)	−0.252292	−	1.59291i	−0.0699732	−	0.441793i	−0.997656	−	0.0684247i	\(-0.978203\pi\)
							0.927683	−	0.373368i	\(-0.121797\pi\)
\(17\)	0.717641	+	0.365656i	0.174053	+	0.0886847i	0.538847	−	0.842404i	\(-0.318860\pi\)
							−0.364793	+	0.931089i	\(0.618860\pi\)
\(19\)	−1.81540	−	2.49869i	−0.416482	−	0.573238i	0.548302	−	0.836280i	\(-0.315274\pi\)
							−0.964784	+	0.263042i	\(0.915274\pi\)
\(23\)	−0.620993	+	1.21877i	−0.129486	+	0.254131i	−0.946643	−	0.322285i	\(-0.895549\pi\)
							0.817157	+	0.576416i	\(0.195549\pi\)
\(29\)	3.25244	−	2.36304i	0.603964	−	0.438805i	−0.243320	−	0.969946i	\(-0.578236\pi\)
							0.847284	+	0.531141i	\(0.178236\pi\)
\(31\)	4.85430	−	2.72687i	0.871857	−	0.489760i
							\(37\)	5.96824	−	5.96824i	0.981173	−	0.981173i	−0.0186527	−	0.999826i	\(-0.505938\pi\)
0.999826	+	0.0186527i	\(0.00593769\pi\)
\(41\)	−0.954273	+	0.693320i	−0.149032	+	0.108278i	−0.659803	−	0.751439i	\(-0.729360\pi\)
							0.510770	+	0.859717i	\(0.329360\pi\)
\(43\)	−5.53364	−	0.876443i	−0.843873	−	0.133656i	−0.280495	−	0.959856i	\(-0.590499\pi\)
							−0.563378	+	0.826199i	\(0.690499\pi\)
\(47\)	0.433984	+	2.74007i	0.0633031	+	0.399680i	0.998912	+	0.0466410i	\(0.0148517\pi\)
							−0.935609	+	0.353039i	\(0.885148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.2.s.a.77.3	✓	128
5.3	odd	4	inner	310.2.s.a.263.6	yes	128
31.29	odd	10	inner	310.2.s.a.277.6	yes	128
155.153	even	20	inner	310.2.s.a.153.3	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.s.a.77.3	✓	128	1.1	even	1	trivial
310.2.s.a.153.3	yes	128	155.153	even	20	inner
310.2.s.a.263.6	yes	128	5.3	odd	4	inner
310.2.s.a.277.6	yes	128	31.29	odd	10	inner