Embedded newform 310.3.i.a.99.10

• Label: 310.3.i.a.99.10
• Level: $310$
• Weight: $3$
• Character: 310.99
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• 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			99.10
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.26

$q$-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 \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

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

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	0.904192	−	0.427126i	\(-0.140474\pi\)
−0.821998	+	0.569491i	\(0.807141\pi\)
\(5\)	−3.79180	+	3.25918i	−0.758360	+	0.651836i
							\(7\)	−4.59664	+	2.65387i	−0.656663	+	0.379125i	−0.791004	−	0.611810i	\(-0.790442\pi\)
0.134341	+	0.990935i	\(0.457108\pi\)
\(11\)	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	\(-0.349147\pi\)
							0.542391	−	0.840126i	\(-0.317519\pi\)
\(17\)	−14.5007	−	25.1159i	−0.852982	−	1.47741i	−0.878505	−	0.477734i	\(-0.841458\pi\)
							0.0255228	−	0.999674i	\(-0.491875\pi\)
\(19\)	−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	\(-0.315737\pi\)
							0.547087	+	0.837076i	\(0.315737\pi\)
\(29\)			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	0.737355	−	0.675506i	\(-0.236075\pi\)
−0.953682	+	0.300815i	\(0.902741\pi\)
\(41\)	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	0.873768	−	0.486343i	\(-0.161670\pi\)
							−0.858070	+	0.513533i	\(0.828336\pi\)
\(47\)		−	40.6456i		−	0.864800i	−0.901682	−	0.432400i	\(-0.857667\pi\)
							0.901682	−	0.432400i	\(-0.142333\pi\)

(See \(a_n\) instead)

Twists

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

    

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