Embedded newform 310.3.x.a.7.6

• Label: 310.3.x.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	310.7
Dual form			310.3.x.a.133.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642040 + 1.26007i) q^{2} +(-1.15774 + 1.78277i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(-2.91327 + 4.06360i) q^{5} +(-1.50310 - 2.60345i) q^{6} +(3.82473 + 9.96377i) q^{7} +(2.79360 - 0.442463i) q^{8} +(1.82273 + 4.09393i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 64 q^{2} - 4 q^{3} - 8 q^{5} - 8 q^{6} + 24 q^{7} + 128 q^{8}+O(q^{10}) \)

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
							\(3\)	−1.15774	+	1.78277i	−0.385915	+	0.594256i	−0.977230	−	0.212184i	\(-0.931942\pi\)
0.591315	+	0.806441i	\(0.298609\pi\)
\(5\)	−2.91327	+	4.06360i	−0.582655	+	0.812720i
							\(7\)	3.82473	+	9.96377i	0.546390	+	1.42340i	0.877396	+	0.479767i	\(0.159279\pi\)
−0.331006	+	0.943629i	\(0.607388\pi\)
\(11\)	0.371924	+	3.53862i	0.0338113	+	0.321693i	0.998334	+	0.0576941i	\(0.0183748\pi\)
							−0.964523	+	0.263999i	\(0.914959\pi\)
\(13\)	16.2017	+	0.849094i	1.24628	+	0.0653149i	0.664038	−	0.747699i	\(-0.268841\pi\)
							0.582245	+	0.813014i	\(0.302175\pi\)
\(17\)	−9.63012	+	7.79832i	−0.566478	+	0.458725i	−0.869274	−	0.494331i	\(-0.835413\pi\)
							0.302796	+	0.953055i	\(0.402080\pi\)
\(19\)	−19.6928	−	17.7315i	−1.03646	−	0.933237i	−0.0386457	−	0.999253i	\(-0.512304\pi\)
							−0.997819	+	0.0660164i	\(0.978971\pi\)
\(23\)	6.09865	−	0.965931i	0.265159	−	0.0419970i	−0.0224394	−	0.999748i	\(-0.507143\pi\)
							0.287598	+	0.957751i	\(0.407143\pi\)
\(29\)	−12.4976	+	4.06071i	−0.430951	+	0.140025i	−0.516457	−	0.856313i	\(-0.672749\pi\)
							0.0855057	+	0.996338i	\(0.472749\pi\)
\(31\)	16.8018	+	26.0519i	0.541992	+	0.840384i
							\(37\)	−0.971492	+	3.62566i	−0.0262565	+	0.0979907i	−0.977811	−	0.209490i	\(-0.932819\pi\)
0.951554	+	0.307481i	\(0.0994861\pi\)
\(41\)	−14.7617	−	3.13769i	−0.360040	−	0.0765290i	0.0243400	−	0.999704i	\(-0.492252\pi\)
							−0.384380	+	0.923175i	\(0.625585\pi\)
\(43\)	−2.31445	−	44.1624i	−0.0538245	−	1.02703i	−0.883394	−	0.468630i	\(-0.844748\pi\)
							0.829570	−	0.558403i	\(-0.188586\pi\)
\(47\)	77.7333	−	39.6071i	1.65390	−	0.842704i	0.657924	−	0.753084i	\(-0.271435\pi\)
							0.995976	−	0.0896202i	\(-0.0285653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.x.a.7.6	✓	256
5.3	odd	4	inner	310.3.x.a.193.6	yes	256
31.9	even	15	inner	310.3.x.a.257.6	yes	256
155.133	odd	60	inner	310.3.x.a.133.6	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.x.a.7.6	✓	256	1.1	even	1	trivial
310.3.x.a.133.6	yes	256	155.133	odd	60	inner
310.3.x.a.193.6	yes	256	5.3	odd	4	inner
310.3.x.a.257.6	yes	256	31.9	even	15	inner