Embedded newform 315.2.bz.d.82.7

• Label: 315.2.bz.d.82.7
• Level: $315$
• Weight: $2$
• Character: 315.82
• 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			82.7
Character	\(\chi\)	\(=\)	315.82
Dual form			315.2.bz.d.73.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.24814 - 0.602389i) q^{2} +(2.95923 - 1.70851i) q^{4} +(2.22726 + 0.198269i) q^{5} +(-2.59417 + 0.519864i) q^{7} +(2.33208 - 2.33208i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 12 q^{5} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.24814	−	0.602389i	1.58968	−	0.425953i	0.647774	−	0.761833i	\(-0.275700\pi\)
							0.941905	+	0.335880i	\(0.109034\pi\)
\(3\)		0			0
							\(5\)	2.22726	+	0.198269i	0.996061	+	0.0886685i
\(7\)	−2.59417	+	0.519864i	−0.980506	+	0.196490i
							\(11\)	1.76389	+	3.05515i	0.531834	+	0.921163i	0.999309	+	0.0371569i	\(0.0118301\pi\)
−0.467476	+	0.884006i	\(0.654837\pi\)
\(13\)	−4.49057	−	4.49057i	−1.24546	−	1.24546i	−0.957704	−	0.287756i	\(-0.907091\pi\)
							−0.287756	−	0.957704i	\(-0.592909\pi\)
\(17\)	−1.79795	−	0.481759i	−0.436066	−	0.116844i	0.0341060	−	0.999418i	\(-0.489142\pi\)
							−0.470172	+	0.882575i	\(0.655808\pi\)
\(19\)	0.0699116	−	0.121090i	0.0160388	−	0.0277800i	−0.857895	−	0.513826i	\(-0.828228\pi\)
							0.873933	+	0.486046i	\(0.161561\pi\)
\(23\)	−0.997072	−	3.72112i	−0.207904	−	0.775908i	−0.988545	−	0.150928i	\(-0.951774\pi\)
							0.780641	−	0.624980i	\(-0.214893\pi\)
\(29\)			2.01969i			0.375047i	0.982260	+	0.187524i	\(0.0600462\pi\)
							−0.982260	+	0.187524i	\(0.939954\pi\)
\(31\)	−4.56612	+	2.63625i	−0.820100	+	0.473485i	−0.850451	−	0.526054i	\(-0.823671\pi\)
							0.0303510	+	0.999539i	\(0.490337\pi\)
\(37\)	5.61323	−	1.50406i	0.922810	−	0.247266i	0.234024	−	0.972231i	\(-0.424811\pi\)
							0.688786	+	0.724965i	\(0.258144\pi\)
\(41\)			0.903323i			0.141075i	0.997509	+	0.0705377i	\(0.0224715\pi\)
							−0.997509	+	0.0705377i	\(0.977529\pi\)
\(43\)	2.38469	−	2.38469i	0.363663	−	0.363663i	−0.501497	−	0.865159i	\(-0.667217\pi\)
							0.865159	+	0.501497i	\(0.167217\pi\)
\(47\)	0.639474	+	2.38655i	0.0932768	+	0.348114i	0.996753	−	0.0805254i	\(-0.0256598\pi\)
							−0.903476	+	0.428639i	\(0.858993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bz.d.82.7		32
3.2	odd	2		105.2.u.a.82.2	yes	32
5.3	odd	4	inner	315.2.bz.d.208.2		32
7.3	odd	6	inner	315.2.bz.d.262.2		32
15.2	even	4		525.2.bc.e.418.2		32
15.8	even	4		105.2.u.a.103.7	yes	32
15.14	odd	2		525.2.bc.e.82.7		32
21.2	odd	6		735.2.m.c.97.14		32
21.5	even	6		735.2.m.c.97.13		32
21.11	odd	6		735.2.v.b.472.7		32
21.17	even	6		105.2.u.a.52.7	✓	32
21.20	even	2		735.2.v.b.607.2		32
35.3	even	12	inner	315.2.bz.d.73.7		32
105.17	odd	12		525.2.bc.e.493.7		32
105.23	even	12		735.2.m.c.538.13		32
105.38	odd	12		105.2.u.a.73.2	yes	32
105.53	even	12		735.2.v.b.178.2		32
105.59	even	6		525.2.bc.e.157.2		32
105.68	odd	12		735.2.m.c.538.14		32
105.83	odd	4		735.2.v.b.313.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.7	✓	32	21.17	even	6
105.2.u.a.73.2	yes	32	105.38	odd	12
105.2.u.a.82.2	yes	32	3.2	odd	2
105.2.u.a.103.7	yes	32	15.8	even	4
315.2.bz.d.73.7		32	35.3	even	12	inner
315.2.bz.d.82.7		32	1.1	even	1	trivial
315.2.bz.d.208.2		32	5.3	odd	4	inner
315.2.bz.d.262.2		32	7.3	odd	6	inner
525.2.bc.e.82.7		32	15.14	odd	2
525.2.bc.e.157.2		32	105.59	even	6
525.2.bc.e.418.2		32	15.2	even	4
525.2.bc.e.493.7		32	105.17	odd	12
735.2.m.c.97.13		32	21.5	even	6
735.2.m.c.97.14		32	21.2	odd	6
735.2.m.c.538.13		32	105.23	even	12
735.2.m.c.538.14		32	105.68	odd	12
735.2.v.b.178.2		32	105.53	even	12
735.2.v.b.313.7		32	105.83	odd	4
735.2.v.b.472.7		32	21.11	odd	6
735.2.v.b.607.2		32	21.20	even	2