Embedded newform 315.2.bz.d.73.2

• Label: 315.2.bz.d.73.2
• Level: $315$
• Weight: $2$
• Character: 315.73
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.2
Character	\(\chi\)	\(=\)	315.73
Dual form			315.2.bz.d.82.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72112 - 0.461174i) q^{2} +(1.01754 + 0.587476i) q^{4} +(1.95031 - 1.09375i) q^{5} +(1.68076 - 2.04329i) q^{7} +(1.03952 + 1.03952i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 12 q^{5} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72112	−	0.461174i	−1.21702	−	0.326099i	−0.407507	−	0.913202i	\(-0.633602\pi\)
							−0.809512	+	0.587103i	\(0.800268\pi\)
\(3\)		0			0
							\(5\)	1.95031	−	1.09375i	0.872206	−	0.489138i
\(7\)	1.68076	−	2.04329i	0.635267	−	0.772293i
							\(11\)	−1.46711	+	2.54110i	−0.442349	+	0.766171i	−0.997863	−	0.0653360i	\(-0.979188\pi\)
0.555514	+	0.831507i	\(0.312521\pi\)
\(13\)	−0.187911	+	0.187911i	−0.0521172	+	0.0521172i	−0.732685	−	0.680568i	\(-0.761733\pi\)
							0.680568	+	0.732685i	\(0.261733\pi\)
\(17\)	3.24271	−	0.868882i	0.786473	−	0.210735i	0.156837	−	0.987625i	\(-0.449870\pi\)
							0.629637	+	0.776890i	\(0.283204\pi\)
\(19\)	1.81824	+	3.14928i	0.417132	+	0.722494i	0.995650	−	0.0931757i	\(-0.0297018\pi\)
							−0.578517	+	0.815670i	\(0.696369\pi\)
\(23\)	2.43175	−	9.07540i	0.507054	−	1.89235i	0.0591961	−	0.998246i	\(-0.481146\pi\)
							0.447858	−	0.894105i	\(-0.352187\pi\)
\(29\)		−	0.815162i		−	0.151372i	−0.997132	−	0.0756859i	\(-0.975885\pi\)
							0.997132	−	0.0756859i	\(-0.0241146\pi\)
\(31\)	−3.76330	−	2.17274i	−0.675908	−	0.390236i	0.122403	−	0.992480i	\(-0.460940\pi\)
							−0.798312	+	0.602245i	\(0.794273\pi\)
\(37\)	6.31853	+	1.69304i	1.03876	+	0.278335i	0.737599	−	0.675239i	\(-0.235959\pi\)
							0.301160	+	0.953574i	\(0.402626\pi\)
\(41\)			2.45734i			0.383772i	0.981417	+	0.191886i	\(0.0614603\pi\)
							−0.981417	+	0.191886i	\(0.938540\pi\)
\(43\)	−3.59998	−	3.59998i	−0.548992	−	0.548992i	0.377157	−	0.926149i	\(-0.376902\pi\)
							−0.926149	+	0.377157i	\(0.876902\pi\)
\(47\)	2.47795	−	9.24785i	0.361447	−	1.34894i	−0.510728	−	0.859743i	\(-0.670624\pi\)
							0.872174	−	0.489195i	\(-0.162709\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bz.d.73.2		32
3.2	odd	2		105.2.u.a.73.7	yes	32
5.2	odd	4	inner	315.2.bz.d.262.7		32
7.5	odd	6	inner	315.2.bz.d.208.7		32
15.2	even	4		105.2.u.a.52.2	✓	32
15.8	even	4		525.2.bc.e.157.7		32
15.14	odd	2		525.2.bc.e.493.2		32
21.2	odd	6		735.2.v.b.313.2		32
21.5	even	6		105.2.u.a.103.2	yes	32
21.11	odd	6		735.2.m.c.538.4		32
21.17	even	6		735.2.m.c.538.3		32
21.20	even	2		735.2.v.b.178.7		32
35.12	even	12	inner	315.2.bz.d.82.2		32
105.2	even	12		735.2.v.b.607.7		32
105.17	odd	12		735.2.m.c.97.4		32
105.32	even	12		735.2.m.c.97.3		32
105.47	odd	12		105.2.u.a.82.7	yes	32
105.62	odd	4		735.2.v.b.472.2		32
105.68	odd	12		525.2.bc.e.82.2		32
105.89	even	6		525.2.bc.e.418.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.2	✓	32	15.2	even	4
105.2.u.a.73.7	yes	32	3.2	odd	2
105.2.u.a.82.7	yes	32	105.47	odd	12
105.2.u.a.103.2	yes	32	21.5	even	6
315.2.bz.d.73.2		32	1.1	even	1	trivial
315.2.bz.d.82.2		32	35.12	even	12	inner
315.2.bz.d.208.7		32	7.5	odd	6	inner
315.2.bz.d.262.7		32	5.2	odd	4	inner
525.2.bc.e.82.2		32	105.68	odd	12
525.2.bc.e.157.7		32	15.8	even	4
525.2.bc.e.418.7		32	105.89	even	6
525.2.bc.e.493.2		32	15.14	odd	2
735.2.m.c.97.3		32	105.32	even	12
735.2.m.c.97.4		32	105.17	odd	12
735.2.m.c.538.3		32	21.17	even	6
735.2.m.c.538.4		32	21.11	odd	6
735.2.v.b.178.7		32	21.20	even	2
735.2.v.b.313.2		32	21.2	odd	6
735.2.v.b.472.2		32	105.62	odd	4
735.2.v.b.607.7		32	105.2	even	12