Embedded newform 315.2.bs.e.103.13

• Label: 315.2.bs.e.103.13
• Level: $315$
• Weight: $2$
• Character: 315.103
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• 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			103.13
Character	\(\chi\)	\(=\)	315.103
Dual form			315.2.bs.e.52.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.690838 + 0.690838i) q^{2} +(0.0911190 - 1.72965i) q^{3} +1.04549i q^{4} +(1.77477 - 1.36022i) q^{5} +(1.13196 + 1.25786i) q^{6} +(-0.704044 - 2.55036i) q^{7} +(-2.10394 - 2.10394i) q^{8} +(-2.98339 - 0.315208i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 18 q^{3} - 6 q^{5} + 24 q^{6} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.690838	+	0.690838i	−0.488496	+	0.488496i	−0.907832	−	0.419335i	\(-0.862263\pi\)
							0.419335	+	0.907832i	\(0.362263\pi\)
\(3\)	0.0911190	−	1.72965i	0.0526076	−	0.998615i
							\(5\)	1.77477	−	1.36022i	0.793700	−	0.608310i
\(7\)	−0.704044	−	2.55036i	−0.266104	−	0.963944i
							\(11\)	0.816710	−	1.41458i	0.246247	−	0.426513i	−0.716234	−	0.697860i	\(-0.754136\pi\)
0.962482	+	0.271347i	\(0.0874691\pi\)
\(13\)	−0.137573	−	0.513429i	−0.0381558	−	0.142400i	0.944220	−	0.329315i	\(-0.106818\pi\)
							−0.982376	+	0.186915i	\(0.940151\pi\)
\(17\)	0.265268	−	0.989992i	0.0643369	−	0.240108i	−0.926268	−	0.376866i	\(-0.877002\pi\)
							0.990604	+	0.136758i	\(0.0436683\pi\)
\(19\)	−2.99374	+	5.18531i	−0.686811	+	1.18959i	0.286053	+	0.958214i	\(0.407657\pi\)
							−0.972864	+	0.231377i	\(0.925677\pi\)
\(23\)	1.92122	−	7.17009i	0.400602	−	1.49507i	−0.411423	−	0.911444i	\(-0.634968\pi\)
							0.812025	−	0.583622i	\(-0.198365\pi\)
\(29\)	5.82340	−	3.36214i	1.08138	−	0.624334i	0.150110	−	0.988669i	\(-0.452037\pi\)
							0.931268	+	0.364335i	\(0.118704\pi\)
\(31\)		−	8.35659i		−	1.50089i	−0.660934	−	0.750444i	\(-0.729840\pi\)
							0.660934	−	0.750444i	\(-0.270160\pi\)
\(37\)	1.47305	+	5.49748i	0.242167	+	0.903780i	0.974786	+	0.223141i	\(0.0716311\pi\)
							−0.732619	+	0.680639i	\(0.761702\pi\)
\(41\)	6.13816	+	3.54387i	0.958619	+	0.553459i	0.895748	−	0.444563i	\(-0.146641\pi\)
							0.0628714	+	0.998022i	\(0.479974\pi\)
\(43\)	−1.98612	+	7.41230i	−0.302880	+	1.13036i	0.631874	+	0.775071i	\(0.282286\pi\)
							−0.934755	+	0.355294i	\(0.884381\pi\)
\(47\)	1.39656	+	1.39656i	0.203709	+	0.203709i	0.801587	−	0.597878i	\(-0.203989\pi\)
							−0.597878	+	0.801587i	\(0.703989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bs.e.103.13	yes	160
3.2	odd	2		945.2.bv.e.523.28		160
5.2	odd	4	inner	315.2.bs.e.292.13	yes	160
7.3	odd	6		315.2.cg.e.283.13	yes	160
9.2	odd	6		945.2.cj.e.208.13		160
9.7	even	3		315.2.cg.e.313.28	yes	160
15.2	even	4		945.2.bv.e.712.28		160
21.17	even	6		945.2.cj.e.388.28		160
35.17	even	12		315.2.cg.e.157.28	yes	160
45.2	even	12		945.2.cj.e.397.28		160
45.7	odd	12		315.2.cg.e.187.13	yes	160
63.38	even	6		945.2.bv.e.73.28		160
63.52	odd	6	inner	315.2.bs.e.178.13	yes	160
105.17	odd	12		945.2.cj.e.577.13		160
315.52	even	12	inner	315.2.bs.e.52.13	✓	160
315.227	odd	12		945.2.bv.e.262.28		160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bs.e.52.13	✓	160	315.52	even	12	inner
315.2.bs.e.103.13	yes	160	1.1	even	1	trivial
315.2.bs.e.178.13	yes	160	63.52	odd	6	inner
315.2.bs.e.292.13	yes	160	5.2	odd	4	inner
315.2.cg.e.157.28	yes	160	35.17	even	12
315.2.cg.e.187.13	yes	160	45.7	odd	12
315.2.cg.e.283.13	yes	160	7.3	odd	6
315.2.cg.e.313.28	yes	160	9.7	even	3
945.2.bv.e.73.28		160	63.38	even	6
945.2.bv.e.262.28		160	315.227	odd	12
945.2.bv.e.523.28		160	3.2	odd	2
945.2.bv.e.712.28		160	15.2	even	4
945.2.cj.e.208.13		160	9.2	odd	6
945.2.cj.e.388.28		160	21.17	even	6
945.2.cj.e.397.28		160	45.2	even	12
945.2.cj.e.577.13		160	105.17	odd	12