Embedded newform 315.2.bv.a.23.6

• Label: 315.2.bv.a.23.6
• Level: $315$
• Weight: $2$
• Character: 315.23
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.6
Character	\(\chi\)	\(=\)	315.23
Dual form			315.2.bv.a.137.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57216 - 1.57216i) q^{2} +(0.440043 + 1.67522i) q^{3} +2.94339i q^{4} +(0.402568 + 2.19953i) q^{5} +(1.94190 - 3.32554i) q^{6} +(2.54364 - 0.727930i) q^{7} +(1.48316 - 1.48316i) q^{8} +(-2.61272 + 1.47434i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57216	−	1.57216i	−1.11169	−	1.11169i	−0.992922	−	0.118764i	\(-0.962107\pi\)
							−0.118764	−	0.992922i	\(-0.537893\pi\)
\(3\)	0.440043	+	1.67522i	0.254059	+	0.967189i
							\(5\)	0.402568	+	2.19953i	0.180034	+	0.983660i
\(7\)	2.54364	−	0.727930i	0.961407	−	0.275132i
							\(11\)	−1.58491	−	0.915048i	−0.477868	−	0.275897i	0.241659	−	0.970361i	\(-0.422308\pi\)
−0.719528	+	0.694464i	\(0.755642\pi\)
\(13\)	−4.38884	+	1.17599i	−1.21725	+	0.326160i	−0.809601	−	0.586980i	\(-0.800317\pi\)
							−0.407645	+	0.913140i	\(0.633650\pi\)
\(17\)	−1.34642	+	5.02493i	−0.326556	+	1.21872i	0.586182	+	0.810179i	\(0.300630\pi\)
							−0.912738	+	0.408545i	\(0.866036\pi\)
\(19\)	5.05610	+	2.91914i	1.15995	+	0.669697i	0.951292	−	0.308292i	\(-0.0997575\pi\)
							0.208657	+	0.977989i	\(0.433091\pi\)
\(23\)	−5.58069	−	1.49534i	−1.16365	−	0.311800i	−0.375230	−	0.926932i	\(-0.622436\pi\)
							−0.788424	+	0.615132i	\(0.789103\pi\)
\(29\)	0.693450	+	1.20109i	0.128770	+	0.223037i	0.923200	−	0.384319i	\(-0.125564\pi\)
							−0.794430	+	0.607356i	\(0.792230\pi\)
\(31\)	6.18812			1.11142			0.555710	−	0.831376i	\(-0.312446\pi\)
							0.555710	+	0.831376i	\(0.312446\pi\)
\(37\)	−0.258254	−	0.963817i	−0.0424567	−	0.158451i	0.941443	−	0.337172i	\(-0.109470\pi\)
							−0.983900	+	0.178722i	\(0.942804\pi\)
\(41\)	3.68494	+	2.12750i	0.575491	+	0.332260i	0.759340	−	0.650695i	\(-0.225522\pi\)
							−0.183848	+	0.982955i	\(0.558856\pi\)
\(43\)	−2.30682	+	8.60918i	−0.351787	+	1.31289i	0.532692	+	0.846309i	\(0.321180\pi\)
							−0.884480	+	0.466579i	\(0.845486\pi\)
\(47\)	2.56685	+	2.56685i	0.374414	+	0.374414i	0.869082	−	0.494668i	\(-0.164710\pi\)
							−0.494668	+	0.869082i	\(0.664710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bv.a.23.6	✓	176
3.2	odd	2		945.2.by.a.233.39		176
5.2	odd	4	inner	315.2.bv.a.212.39	yes	176
7.4	even	3		315.2.bx.a.158.39	yes	176
9.2	odd	6		315.2.bx.a.128.39	yes	176
9.7	even	3		945.2.ca.a.548.6		176
15.2	even	4		945.2.by.a.422.6		176
21.11	odd	6		945.2.ca.a.368.6		176
35.32	odd	12		315.2.bx.a.32.39	yes	176
45.2	even	12		315.2.bx.a.2.39	yes	176
45.7	odd	12		945.2.ca.a.737.6		176
63.11	odd	6	inner	315.2.bv.a.263.39	yes	176
63.25	even	3		945.2.by.a.683.6		176
105.32	even	12		945.2.ca.a.557.6		176
315.137	even	12	inner	315.2.bv.a.137.6	yes	176
315.277	odd	12		945.2.by.a.872.39		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.6	✓	176	1.1	even	1	trivial
315.2.bv.a.137.6	yes	176	315.137	even	12	inner
315.2.bv.a.212.39	yes	176	5.2	odd	4	inner
315.2.bv.a.263.39	yes	176	63.11	odd	6	inner
315.2.bx.a.2.39	yes	176	45.2	even	12
315.2.bx.a.32.39	yes	176	35.32	odd	12
315.2.bx.a.128.39	yes	176	9.2	odd	6
315.2.bx.a.158.39	yes	176	7.4	even	3
945.2.by.a.233.39		176	3.2	odd	2
945.2.by.a.422.6		176	15.2	even	4
945.2.by.a.683.6		176	63.25	even	3
945.2.by.a.872.39		176	315.277	odd	12
945.2.ca.a.368.6		176	21.11	odd	6
945.2.ca.a.548.6		176	9.7	even	3
945.2.ca.a.557.6		176	105.32	even	12
945.2.ca.a.737.6		176	45.7	odd	12