Embedded newform 310.3.i.a.119.32

• Label: 310.3.i.a.119.32
• Level: $310$
• Weight: $3$
• Character: 310.119
• 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			119.32
Character	\(\chi\)	\(=\)	310.119
Dual form			310.3.i.a.99.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(2.70805 - 4.69048i) q^{3} -2.00000 q^{4} +(2.12665 + 4.52519i) q^{5} +(6.63334 + 3.82976i) q^{6} +(-7.85437 - 4.53472i) q^{7} -2.82843i q^{8} +(-10.1671 - 17.6099i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

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

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

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	0.0786852	−	0.996900i	\(-0.474928\pi\)
0.823998	−	0.566593i	\(-0.191739\pi\)
\(5\)	2.12665	+	4.52519i	0.425330	+	0.905038i
							\(7\)	−7.85437	−	4.53472i	−1.12205	−	0.647817i	−0.180129	−	0.983643i	\(-0.557651\pi\)
−0.941924	+	0.335826i	\(0.890985\pi\)
\(11\)	11.8842	−	6.86133i	1.08038	−	0.623758i	0.149381	−	0.988780i	\(-0.452272\pi\)
							0.930999	+	0.365022i	\(0.118939\pi\)
\(13\)	−10.6610	−	18.4653i	−0.820074	−	1.42041i	−0.905626	−	0.424077i	\(-0.860599\pi\)
							0.0855520	−	0.996334i	\(-0.472735\pi\)
\(17\)	2.30156	−	3.98641i	0.135386	−	0.234495i	−0.790359	−	0.612644i	\(-0.790106\pi\)
							0.925745	+	0.378149i	\(0.123439\pi\)
\(19\)	−7.69675	+	13.3312i	−0.405092	+	0.701640i	−0.994332	−	0.106318i	\(-0.966094\pi\)
							0.589240	+	0.807958i	\(0.299427\pi\)
\(23\)	34.1036			1.48277			0.741384	−	0.671082i	\(-0.234170\pi\)
							0.741384	+	0.671082i	\(0.234170\pi\)
\(29\)		−	19.5719i		−	0.674894i	−0.941345	−	0.337447i	\(-0.890437\pi\)
							0.941345	−	0.337447i	\(-0.109563\pi\)
\(31\)	1.23547	−	30.9754i	0.0398540	−	0.999206i
							\(37\)	24.4492	−	42.3472i	0.660789	−	1.14452i	−0.319620	−	0.947546i	\(-0.603555\pi\)
0.980409	−	0.196974i	\(-0.0631114\pi\)
\(41\)	18.9881	+	32.8883i	0.463123	+	0.802153i	0.999115	−	0.0420705i	\(-0.0133954\pi\)
							−0.535991	+	0.844223i	\(0.680062\pi\)
\(43\)	−40.0166	+	69.3107i	−0.930618	+	1.61188i	−0.148351	+	0.988935i	\(0.547396\pi\)
							−0.782267	+	0.622943i	\(0.785937\pi\)
\(47\)		−	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.119.32	yes	64
5.4	even	2	inner	310.3.i.a.119.1	yes	64
31.6	odd	6	inner	310.3.i.a.99.17	yes	64
155.99	odd	6	inner	310.3.i.a.99.16	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.16	✓	64	155.99	odd	6	inner
310.3.i.a.99.17	yes	64	31.6	odd	6	inner
310.3.i.a.119.1	yes	64	5.4	even	2	inner
310.3.i.a.119.32	yes	64	1.1	even	1	trivial