Embedded newform 310.3.i.a.119.26

• Label: 310.3.i.a.119.26
• 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			119.26
Character	\(\chi\)	\(=\)	310.119
Dual form			310.3.i.a.99.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(0.246583 - 0.427094i) q^{3} -2.00000 q^{4} +(-3.79180 - 3.25918i) q^{5} +(0.604003 + 0.348721i) q^{6} +(-4.59664 - 2.65387i) q^{7} -2.82843i q^{8} +(4.37839 + 7.58360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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
							\(3\)	0.246583	−	0.427094i	0.0821944	−	0.142365i	−0.821998	−	0.569491i	\(-0.807141\pi\)
0.904192	+	0.427126i	\(0.140474\pi\)
\(5\)	−3.79180	−	3.25918i	−0.758360	−	0.651836i
							\(7\)	−4.59664	−	2.65387i	−0.656663	−	0.379125i	0.134341	−	0.990935i	\(-0.457108\pi\)
−0.791004	+	0.611810i	\(0.790442\pi\)
\(11\)	11.2350	−	6.48654i	1.02137	−	0.589686i	0.106866	−	0.994273i	\(-0.465918\pi\)
							0.914499	+	0.404588i	\(0.132585\pi\)
\(13\)	12.9840	+	22.4889i	0.998766	+	1.72991i	0.542391	+	0.840126i	\(0.317519\pi\)
							0.456375	+	0.889787i	\(0.349147\pi\)
\(17\)	−14.5007	+	25.1159i	−0.852982	+	1.47741i	0.0255228	+	0.999674i	\(0.491875\pi\)
							−0.878505	+	0.477734i	\(0.841458\pi\)
\(19\)	−12.0811	+	20.9251i	−0.635849	+	1.10132i	0.350486	+	0.936568i	\(0.386017\pi\)
							−0.986335	+	0.164755i	\(0.947317\pi\)
\(23\)	25.1660			1.09417			0.547087	−	0.837076i	\(-0.315737\pi\)
							0.547087	+	0.837076i	\(0.315737\pi\)
\(29\)		−	39.5924i		−	1.36525i	−0.730767	−	0.682627i	\(-0.760838\pi\)
							0.730767	−	0.682627i	\(-0.239162\pi\)
\(31\)	14.7654	−	27.2577i	0.476305	−	0.879280i
							\(37\)	−8.00412	+	13.8635i	−0.216327	+	0.374690i	−0.953682	−	0.300815i	\(-0.902741\pi\)
0.737355	+	0.675506i	\(0.236075\pi\)
\(41\)	26.8232	+	46.4591i	0.654224	+	1.13315i	0.982088	+	0.188424i	\(0.0603380\pi\)
							−0.327864	+	0.944725i	\(0.606329\pi\)
\(43\)	0.675020	−	1.16917i	0.0156981	−	0.0271900i	−0.858070	−	0.513533i	\(-0.828336\pi\)
							0.873768	+	0.486343i	\(0.161670\pi\)
\(47\)			40.6456i			0.864800i	0.901682	+	0.432400i	\(0.142333\pi\)
							−0.901682	+	0.432400i	\(0.857667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.119.26	yes	64
5.4	even	2	inner	310.3.i.a.119.7	yes	64
31.6	odd	6	inner	310.3.i.a.99.23	yes	64
155.99	odd	6	inner	310.3.i.a.99.10	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.10	✓	64	155.99	odd	6	inner
310.3.i.a.99.23	yes	64	31.6	odd	6	inner
310.3.i.a.119.7	yes	64	5.4	even	2	inner
310.3.i.a.119.26	yes	64	1.1	even	1	trivial