Embedded newform 310.3.i.a.99.20

• Label: 310.3.i.a.99.20
• 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			99.20
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-1.41600 - 2.45259i) q^{3} -2.00000 q^{4} +(-0.0417218 - 4.99983i) q^{5} +(3.46848 - 2.00253i) q^{6} +(0.711149 - 0.410582i) q^{7} -2.82843i q^{8} +(0.489879 - 0.848494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.41600	−	2.45259i	−0.472001	−	0.817529i	0.527486	−	0.849564i	\(-0.323135\pi\)
−0.999487	+	0.0320347i	\(0.989801\pi\)
\(5\)	−0.0417218	−	4.99983i	−0.00834435	−	0.999965i
							\(7\)	0.711149	−	0.410582i	0.101593	−	0.0586546i	−0.448343	−	0.893862i	\(-0.647986\pi\)
0.549936	+	0.835207i	\(0.314652\pi\)
\(11\)	−6.49906	−	3.75223i	−0.590824	−	0.341112i	0.174599	−	0.984640i	\(-0.444137\pi\)
							−0.765423	+	0.643527i	\(0.777470\pi\)
\(13\)	−12.5415	+	21.7226i	−0.964733	+	1.67097i	−0.254402	+	0.967099i	\(0.581879\pi\)
							−0.710331	+	0.703868i	\(0.751455\pi\)
\(17\)	−7.77837	−	13.4725i	−0.457551	−	0.792502i	0.541280	−	0.840842i	\(-0.317940\pi\)
							−0.998831	+	0.0483409i	\(0.984607\pi\)
\(19\)	11.9347	+	20.6716i	0.628144	+	1.08798i	0.987924	+	0.154940i	\(0.0495184\pi\)
							−0.359780	+	0.933037i	\(0.617148\pi\)
\(23\)	−12.4457			−0.541118			−0.270559	−	0.962703i	\(-0.587209\pi\)
							−0.270559	+	0.962703i	\(0.587209\pi\)
\(29\)		−	16.3903i		−	0.565183i	−0.959240	−	0.282592i	\(-0.908806\pi\)
							0.959240	−	0.282592i	\(-0.0911941\pi\)
\(31\)	−30.9975	−	0.395949i	−0.999918	−	0.0127725i
							\(37\)	−12.9346	−	22.4033i	−0.349583	−	0.605496i	0.636592	−	0.771201i	\(-0.280343\pi\)
−0.986175	+	0.165705i	\(0.947010\pi\)
\(41\)	8.35237	−	14.4667i	0.203716	−	0.352847i	−0.746007	−	0.665938i	\(-0.768031\pi\)
							0.949723	+	0.313092i	\(0.101365\pi\)
\(43\)	−0.465143	−	0.805650i	−0.0108173	−	0.0187361i	0.860566	−	0.509339i	\(-0.170110\pi\)
							−0.871383	+	0.490603i	\(0.836777\pi\)
\(47\)			9.31396i			0.198169i	0.995079	+	0.0990847i	\(0.0315915\pi\)
							−0.995079	+	0.0990847i	\(0.968409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.20	yes	64
5.4	even	2	inner	310.3.i.a.99.13	✓	64
31.26	odd	6	inner	310.3.i.a.119.29	yes	64
155.119	odd	6	inner	310.3.i.a.119.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.13	✓	64	5.4	even	2	inner
310.3.i.a.99.20	yes	64	1.1	even	1	trivial
310.3.i.a.119.4	yes	64	155.119	odd	6	inner
310.3.i.a.119.29	yes	64	31.26	odd	6	inner