Embedded newform 525.2.bc.e.157.7

• Label: 525.2.bc.e.157.7
• Level: $525$
• Weight: $2$
• Character: 525.157
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
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			157.7
Character	\(\chi\)	\(=\)	525.157
Dual form			525.2.bc.e.418.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.461174 - 1.72112i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(-1.01754 - 0.587476i) q^{4} +1.78184i q^{6} +(-2.04329 - 1.68076i) q^{7} +(1.03952 - 1.03952i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.461174	−	1.72112i	0.326099	−	1.21702i	−0.587103	−	0.809512i	\(-0.699732\pi\)
							0.913202	−	0.407507i	\(-0.133602\pi\)
\(3\)	−0.965926	+	0.258819i	−0.557678	+	0.149429i
							\(5\)		0			0
\(7\)	−2.04329	−	1.68076i	−0.772293	−	0.635267i
							\(11\)	1.46711	−	2.54110i	0.442349	−	0.766171i	−0.555514	−	0.831507i	\(-0.687479\pi\)
0.997863	+	0.0653360i	\(0.0208119\pi\)
\(13\)	−0.187911	−	0.187911i	−0.0521172	−	0.0521172i	0.680568	−	0.732685i	\(-0.261733\pi\)
							−0.732685	+	0.680568i	\(0.761733\pi\)
\(17\)	0.868882	+	3.24271i	0.210735	+	0.786473i	0.987625	+	0.156837i	\(0.0501296\pi\)
							−0.776890	+	0.629637i	\(0.783204\pi\)
\(19\)	−1.81824	−	3.14928i	−0.417132	−	0.722494i	0.578517	−	0.815670i	\(-0.303631\pi\)
							−0.995650	+	0.0931757i	\(0.970298\pi\)
\(23\)	−9.07540	−	2.43175i	−1.89235	−	0.507054i	−0.998246	−	0.0591961i	\(-0.981146\pi\)
							−0.894105	−	0.447858i	\(-0.852187\pi\)
\(29\)		−	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	\(0.794273\pi\)
\(37\)	1.69304	−	6.31853i	0.278335	−	1.03876i	−0.675239	−	0.737599i	\(-0.735959\pi\)
							0.953574	−	0.301160i	\(-0.0973738\pi\)
\(41\)		−	2.45734i		−	0.383772i	−0.981417	−	0.191886i	\(-0.938540\pi\)
							0.981417	−	0.191886i	\(-0.0614603\pi\)
\(43\)	3.59998	−	3.59998i	0.548992	−	0.548992i	−0.377157	−	0.926149i	\(-0.623098\pi\)
							0.926149	+	0.377157i	\(0.123098\pi\)
\(47\)	9.24785	+	2.47795i	1.34894	+	0.361447i	0.859743	−	0.510728i	\(-0.170624\pi\)
							0.489195	+	0.872174i	\(0.337291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bc.e.157.7		32
5.2	odd	4		105.2.u.a.73.7	yes	32
5.3	odd	4	inner	525.2.bc.e.493.2		32
5.4	even	2		105.2.u.a.52.2	✓	32
7.5	odd	6	inner	525.2.bc.e.82.2		32
15.2	even	4		315.2.bz.d.73.2		32
15.14	odd	2		315.2.bz.d.262.7		32
35.2	odd	12		735.2.v.b.313.2		32
35.4	even	6		735.2.m.c.97.3		32
35.9	even	6		735.2.v.b.607.7		32
35.12	even	12		105.2.u.a.103.2	yes	32
35.17	even	12		735.2.m.c.538.3		32
35.19	odd	6		105.2.u.a.82.7	yes	32
35.24	odd	6		735.2.m.c.97.4		32
35.27	even	4		735.2.v.b.178.7		32
35.32	odd	12		735.2.m.c.538.4		32
35.33	even	12	inner	525.2.bc.e.418.7		32
35.34	odd	2		735.2.v.b.472.2		32
105.47	odd	12		315.2.bz.d.208.7		32
105.89	even	6		315.2.bz.d.82.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.2	✓	32	5.4	even	2
105.2.u.a.73.7	yes	32	5.2	odd	4
105.2.u.a.82.7	yes	32	35.19	odd	6
105.2.u.a.103.2	yes	32	35.12	even	12
315.2.bz.d.73.2		32	15.2	even	4
315.2.bz.d.82.2		32	105.89	even	6
315.2.bz.d.208.7		32	105.47	odd	12
315.2.bz.d.262.7		32	15.14	odd	2
525.2.bc.e.82.2		32	7.5	odd	6	inner
525.2.bc.e.157.7		32	1.1	even	1	trivial
525.2.bc.e.418.7		32	35.33	even	12	inner
525.2.bc.e.493.2		32	5.3	odd	4	inner
735.2.m.c.97.3		32	35.4	even	6
735.2.m.c.97.4		32	35.24	odd	6
735.2.m.c.538.3		32	35.17	even	12
735.2.m.c.538.4		32	35.32	odd	12
735.2.v.b.178.7		32	35.27	even	4
735.2.v.b.313.2		32	35.2	odd	12
735.2.v.b.472.2		32	35.34	odd	2
735.2.v.b.607.7		32	35.9	even	6