Embedded newform 310.3.i.a.99.23

• Label: 310.3.i.a.99.23
• 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			99.23
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-0.246583 - 0.427094i) q^{3} -2.00000 q^{4} +(-0.926630 + 4.91339i) q^{5} +(0.604003 - 0.348721i) q^{6} +(4.59664 - 2.65387i) q^{7} -2.82843i q^{8} +(4.37839 - 7.58360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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
							\(3\)	−0.246583	−	0.427094i	−0.0821944	−	0.142365i	0.821998	−	0.569491i	\(-0.192859\pi\)
−0.904192	+	0.427126i	\(0.859526\pi\)
\(5\)	−0.926630	+	4.91339i	−0.185326	+	0.982677i
							\(7\)	4.59664	−	2.65387i	0.656663	−	0.379125i	−0.134341	−	0.990935i	\(-0.542892\pi\)
0.791004	+	0.611810i	\(0.209558\pi\)
\(11\)	11.2350	+	6.48654i	1.02137	+	0.589686i	0.914499	−	0.404588i	\(-0.132585\pi\)
							0.106866	+	0.994273i	\(0.465918\pi\)
\(13\)	−12.9840	+	22.4889i	−0.998766	+	1.72991i	−0.456375	+	0.889787i	\(0.650853\pi\)
							−0.542391	+	0.840126i	\(0.682481\pi\)
\(17\)	14.5007	+	25.1159i	0.852982	+	1.47741i	0.878505	+	0.477734i	\(0.158542\pi\)
							−0.0255228	+	0.999674i	\(0.508125\pi\)
\(19\)	−12.0811	−	20.9251i	−0.635849	−	1.10132i	−0.986335	−	0.164755i	\(-0.947317\pi\)
							0.350486	−	0.936568i	\(-0.386017\pi\)
\(23\)	−25.1660			−1.09417			−0.547087	−	0.837076i	\(-0.684263\pi\)
							−0.547087	+	0.837076i	\(0.684263\pi\)
\(29\)			39.5924i			1.36525i	0.730767	+	0.682627i	\(0.239162\pi\)
							−0.730767	+	0.682627i	\(0.760838\pi\)
\(31\)	14.7654	+	27.2577i	0.476305	+	0.879280i
							\(37\)	8.00412	+	13.8635i	0.216327	+	0.374690i	0.953682	−	0.300815i	\(-0.0972588\pi\)
−0.737355	+	0.675506i	\(0.763925\pi\)
\(41\)	26.8232	−	46.4591i	0.654224	−	1.13315i	−0.327864	−	0.944725i	\(-0.606329\pi\)
							0.982088	−	0.188424i	\(-0.0603380\pi\)
\(43\)	−0.675020	−	1.16917i	−0.0156981	−	0.0271900i	0.858070	−	0.513533i	\(-0.171664\pi\)
							−0.873768	+	0.486343i	\(0.838330\pi\)
\(47\)			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.99.23	yes	64
5.4	even	2	inner	310.3.i.a.99.10	✓	64
31.26	odd	6	inner	310.3.i.a.119.26	yes	64
155.119	odd	6	inner	310.3.i.a.119.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.10	✓	64	5.4	even	2	inner
310.3.i.a.99.23	yes	64	1.1	even	1	trivial
310.3.i.a.119.7	yes	64	155.119	odd	6	inner
310.3.i.a.119.26	yes	64	31.26	odd	6	inner