Embedded newform 315.2.bz.d.73.7

• Label: 315.2.bz.d.73.7
• 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			73.7
Character	\(\chi\)	\(=\)	315.73
Dual form			315.2.bz.d.82.7

$q$-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 \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.7	✓	32	15.2	even	4
105.2.u.a.73.2	yes	32	3.2	odd	2
105.2.u.a.82.2	yes	32	105.47	odd	12
105.2.u.a.103.7	yes	32	21.5	even	6
315.2.bz.d.73.7		32	1.1	even	1	trivial
315.2.bz.d.82.7		32	35.12	even	12	inner
315.2.bz.d.208.2		32	7.5	odd	6	inner
315.2.bz.d.262.2		32	5.2	odd	4	inner
525.2.bc.e.82.7		32	105.68	odd	12
525.2.bc.e.157.2		32	15.8	even	4
525.2.bc.e.418.2		32	105.89	even	6
525.2.bc.e.493.7		32	15.14	odd	2
735.2.m.c.97.13		32	105.32	even	12
735.2.m.c.97.14		32	105.17	odd	12
735.2.m.c.538.13		32	21.17	even	6
735.2.m.c.538.14		32	21.11	odd	6
735.2.v.b.178.2		32	21.20	even	2
735.2.v.b.313.7		32	21.2	odd	6
735.2.v.b.472.7		32	105.62	odd	4
735.2.v.b.607.2		32	105.2	even	12