Embedded newform 310.3.x.a.7.13

• Label: 310.3.x.a.7.13
• Level: $310$
• Weight: $3$
• Character: 310.7
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			7.13
Character	\(\chi\)	\(=\)	310.7
Dual form			310.3.x.a.133.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642040 + 1.26007i) q^{2} +(1.39985 - 2.15559i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(0.828904 + 4.93081i) q^{5} +(1.81744 + 3.14789i) q^{6} +(-2.99737 - 7.80840i) q^{7} +(2.79360 - 0.442463i) q^{8} +(0.973668 + 2.18689i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 64 q^{2} - 4 q^{3} - 8 q^{5} - 8 q^{6} + 24 q^{7} + 128 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.642040	+	1.26007i	−0.321020	+	0.630037i
							\(3\)	1.39985	−	2.15559i	0.466618	−	0.718529i	−0.524515	−	0.851401i	\(-0.675753\pi\)
0.991133	+	0.132872i	\(0.0424200\pi\)
\(5\)	0.828904	+	4.93081i	0.165781	+	0.986163i
							\(7\)	−2.99737	−	7.80840i	−0.428195	−	1.11549i	−0.962760	−	0.270359i	\(-0.912858\pi\)
0.534565	−	0.845128i	\(-0.320476\pi\)
\(11\)	0.382381	+	3.63812i	0.0347620	+	0.330738i	0.998058	+	0.0622957i	\(0.0198422\pi\)
							−0.963296	+	0.268442i	\(0.913491\pi\)
\(13\)	20.6100	+	1.08012i	1.58538	+	0.0830865i	0.824523	−	0.565828i	\(-0.191444\pi\)
							0.760861	+	0.648915i	\(0.224777\pi\)
\(17\)	4.34140	−	3.51560i	0.255377	−	0.206800i	−0.493079	−	0.869984i	\(-0.664129\pi\)
							0.748456	+	0.663184i	\(0.230795\pi\)
\(19\)	−3.40748	−	3.06811i	−0.179341	−	0.161479i	0.574571	−	0.818455i	\(-0.305169\pi\)
							−0.753912	+	0.656975i	\(0.771836\pi\)
\(23\)	34.6866	−	5.49382i	1.50811	−	0.238862i	0.653023	−	0.757338i	\(-0.273501\pi\)
							0.855092	+	0.518476i	\(0.173501\pi\)
\(29\)	9.74857	−	3.16750i	0.336158	−	0.109224i	−0.136074	−	0.990699i	\(-0.543449\pi\)
							0.472232	+	0.881474i	\(0.343449\pi\)
\(31\)	10.3846	−	29.2089i	0.334987	−	0.942223i
							\(37\)	−6.06838	+	22.6475i	−0.164010	+	0.612094i	0.834154	+	0.551531i	\(0.185956\pi\)
−0.998164	+	0.0605630i	\(0.980710\pi\)
\(41\)	63.5134	+	13.5002i	1.54911	+	0.329273i	0.901526	−	0.432725i	\(-0.142448\pi\)
							0.647580	+	0.761997i	\(0.275781\pi\)
\(43\)	−0.845453	−	16.1322i	−0.0196617	−	0.375167i	−0.990684	−	0.136179i	\(-0.956518\pi\)
							0.971023	−	0.238988i	\(-0.0768157\pi\)
\(47\)	−16.0679	+	8.18698i	−0.341869	+	0.174191i	−0.616492	−	0.787361i	\(-0.711447\pi\)
							0.274623	+	0.961552i	\(0.411447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.x.a.7.13	✓	256
5.3	odd	4	inner	310.3.x.a.193.13	yes	256
31.9	even	15	inner	310.3.x.a.257.13	yes	256
155.133	odd	60	inner	310.3.x.a.133.13	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.x.a.7.13	✓	256	1.1	even	1	trivial
310.3.x.a.133.13	yes	256	155.133	odd	60	inner
310.3.x.a.193.13	yes	256	5.3	odd	4	inner
310.3.x.a.257.13	yes	256	31.9	even	15	inner