Embedded newform 310.3.i.a.99.17

• Label: 310.3.i.a.99.17
• Level: $310$
• Weight: $3$
• Character: 310.99
• 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			99.17
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-2.70805 - 4.69048i) q^{3} -2.00000 q^{4} +(2.85561 - 4.10433i) q^{5} +(6.63334 - 3.82976i) q^{6} +(7.85437 - 4.53472i) q^{7} -2.82843i q^{8} +(-10.1671 + 17.6099i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.70805	−	4.69048i	−0.902683	−	1.56349i	−0.823998	−	0.566593i	\(-0.808261\pi\)
−0.0786852	−	0.996900i	\(-0.525072\pi\)
\(5\)	2.85561	−	4.10433i	0.571121	−	0.820866i
							\(7\)	7.85437	−	4.53472i	1.12205	−	0.647817i	0.180129	−	0.983643i	\(-0.442349\pi\)
0.941924	+	0.335826i	\(0.109015\pi\)
\(11\)	11.8842	+	6.86133i	1.08038	+	0.623758i	0.930999	−	0.365022i	\(-0.118939\pi\)
							0.149381	+	0.988780i	\(0.452272\pi\)
\(13\)	10.6610	−	18.4653i	0.820074	−	1.42041i	−0.0855520	−	0.996334i	\(-0.527265\pi\)
							0.905626	−	0.424077i	\(-0.139401\pi\)
\(17\)	−2.30156	−	3.98641i	−0.135386	−	0.234495i	0.790359	−	0.612644i	\(-0.209894\pi\)
							−0.925745	+	0.378149i	\(0.876561\pi\)
\(19\)	−7.69675	−	13.3312i	−0.405092	−	0.701640i	0.589240	−	0.807958i	\(-0.299427\pi\)
							−0.994332	+	0.106318i	\(0.966094\pi\)
\(23\)	−34.1036			−1.48277			−0.741384	−	0.671082i	\(-0.765830\pi\)
							−0.741384	+	0.671082i	\(0.765830\pi\)
\(29\)			19.5719i			0.674894i	0.941345	+	0.337447i	\(0.109563\pi\)
							−0.941345	+	0.337447i	\(0.890437\pi\)
\(31\)	1.23547	+	30.9754i	0.0398540	+	0.999206i
							\(37\)	−24.4492	−	42.3472i	−0.660789	−	1.14452i	−0.980409	−	0.196974i	\(-0.936889\pi\)
0.319620	−	0.947546i	\(-0.396445\pi\)
\(41\)	18.9881	−	32.8883i	0.463123	−	0.802153i	−0.535991	−	0.844223i	\(-0.680062\pi\)
							0.999115	+	0.0420705i	\(0.0133954\pi\)
\(43\)	40.0166	+	69.3107i	0.930618	+	1.61188i	0.782267	+	0.622943i	\(0.214063\pi\)
							0.148351	+	0.988935i	\(0.452604\pi\)
\(47\)		−	0.952699i		−	0.0202702i	−0.999949	−	0.0101351i	\(-0.996774\pi\)
							0.999949	−	0.0101351i	\(-0.00322616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.17	yes	64
5.4	even	2	inner	310.3.i.a.99.16	✓	64
31.26	odd	6	inner	310.3.i.a.119.32	yes	64
155.119	odd	6	inner	310.3.i.a.119.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.16	✓	64	5.4	even	2	inner
310.3.i.a.99.17	yes	64	1.1	even	1	trivial
310.3.i.a.119.1	yes	64	155.119	odd	6	inner
310.3.i.a.119.32	yes	64	31.26	odd	6	inner