Embedded newform 315.2.bs.e.292.7

• Label: 315.2.bs.e.292.7
• Level: $315$
• Weight: $2$
• Character: 315.292
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			292.7
Character	\(\chi\)	\(=\)	315.292
Dual form			315.2.bs.e.178.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48808 - 1.48808i) q^{2} +(-1.69041 - 0.377509i) q^{3} +2.42876i q^{4} +(0.500247 + 2.17939i) q^{5} +(1.95370 + 3.07723i) q^{6} +(-2.43052 - 1.04527i) q^{7} +(0.638021 - 0.638021i) q^{8} +(2.71497 + 1.27629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 18 q^{3} - 6 q^{5} + 24 q^{6} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48808	−	1.48808i	−1.05223	−	1.05223i	−0.998559	−	0.0536717i	\(-0.982908\pi\)
							−0.0536717	−	0.998559i	\(-0.517092\pi\)
\(3\)	−1.69041	−	0.377509i	−0.975959	−	0.217955i
							\(5\)	0.500247	+	2.17939i	0.223717	+	0.974654i
\(7\)	−2.43052	−	1.04527i	−0.918649	−	0.395074i
							\(11\)	0.0441776	−	0.0765178i	0.0133200	−	0.0230710i	−0.859289	−	0.511491i	\(-0.829093\pi\)
0.872609	+	0.488420i	\(0.162427\pi\)
\(13\)	5.14229	−	1.37787i	1.42622	−	0.382153i	0.538531	−	0.842606i	\(-0.318979\pi\)
							0.887684	+	0.460452i	\(0.152313\pi\)
\(17\)	−3.40854	−	0.913316i	−0.826693	−	0.221512i	−0.179422	−	0.983772i	\(-0.557423\pi\)
							−0.647270	+	0.762260i	\(0.724090\pi\)
\(19\)	2.57895	−	4.46688i	0.591652	−	1.02477i	−0.402358	−	0.915482i	\(-0.631809\pi\)
							0.994010	−	0.109289i	\(-0.0348574\pi\)
\(23\)	3.58010	+	0.959285i	0.746503	+	0.200025i	0.611966	−	0.790884i	\(-0.290379\pi\)
							0.134536	+	0.990909i	\(0.457045\pi\)
\(29\)	−0.609023	+	0.351619i	−0.113093	+	0.0652941i	−0.555479	−	0.831530i	\(-0.687465\pi\)
							0.442387	+	0.896824i	\(0.354132\pi\)
\(31\)		−	4.87601i		−	0.875758i	−0.899034	−	0.437879i	\(-0.855730\pi\)
							0.899034	−	0.437879i	\(-0.144270\pi\)
\(37\)	8.44603	−	2.26311i	1.38852	−	0.372053i	0.514311	−	0.857604i	\(-0.328048\pi\)
							0.874208	+	0.485551i	\(0.161381\pi\)
\(41\)	−2.57843	−	1.48865i	−0.402682	−	0.232489i	0.284958	−	0.958540i	\(-0.408020\pi\)
							−0.687641	+	0.726051i	\(0.741354\pi\)
\(43\)	2.22267	+	0.595563i	0.338954	+	0.0908225i	0.424281	−	0.905530i	\(-0.360527\pi\)
							−0.0853269	+	0.996353i	\(0.527193\pi\)
\(47\)	4.06198	−	4.06198i	0.592501	−	0.592501i	−0.345805	−	0.938306i	\(-0.612394\pi\)
							0.938306	+	0.345805i	\(0.112394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bs.e.292.7	yes	160
3.2	odd	2		945.2.bv.e.712.34		160
5.3	odd	4	inner	315.2.bs.e.103.7	yes	160
7.3	odd	6		315.2.cg.e.157.34	yes	160
9.2	odd	6		945.2.cj.e.397.34		160
9.7	even	3		315.2.cg.e.187.7	yes	160
15.8	even	4		945.2.bv.e.523.34		160
21.17	even	6		945.2.cj.e.577.7		160
35.3	even	12		315.2.cg.e.283.7	yes	160
45.38	even	12		945.2.cj.e.208.7		160
45.43	odd	12		315.2.cg.e.313.34	yes	160
63.38	even	6		945.2.bv.e.262.34		160
63.52	odd	6	inner	315.2.bs.e.52.7	✓	160
105.38	odd	12		945.2.cj.e.388.34		160
315.38	odd	12		945.2.bv.e.73.34		160
315.178	even	12	inner	315.2.bs.e.178.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bs.e.52.7	✓	160	63.52	odd	6	inner
315.2.bs.e.103.7	yes	160	5.3	odd	4	inner
315.2.bs.e.178.7	yes	160	315.178	even	12	inner
315.2.bs.e.292.7	yes	160	1.1	even	1	trivial
315.2.cg.e.157.34	yes	160	7.3	odd	6
315.2.cg.e.187.7	yes	160	9.7	even	3
315.2.cg.e.283.7	yes	160	35.3	even	12
315.2.cg.e.313.34	yes	160	45.43	odd	12
945.2.bv.e.73.34		160	315.38	odd	12
945.2.bv.e.262.34		160	63.38	even	6
945.2.bv.e.523.34		160	15.8	even	4
945.2.bv.e.712.34		160	3.2	odd	2
945.2.cj.e.208.7		160	45.38	even	12
945.2.cj.e.388.34		160	105.38	odd	12
945.2.cj.e.397.34		160	9.2	odd	6
945.2.cj.e.577.7		160	21.17	even	6