Embedded newform 315.2.cg.e.187.21

• Label: 315.2.cg.e.187.21
• Level: $315$
• Weight: $2$
• Character: 315.187
• 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.cg (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			187.21
Character	\(\chi\)	\(=\)	315.187
Dual form			315.2.cg.e.283.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0302234 + 0.112795i) q^{2} +(-1.53453 + 0.803249i) q^{3} +(1.72024 + 0.993182i) q^{4} +(-0.634953 + 2.14402i) q^{5} +(-0.0442239 - 0.197365i) q^{6} +(2.46509 - 0.960898i) q^{7} +(-0.329161 + 0.329161i) q^{8} +(1.70958 - 2.46523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{2} - 12 q^{3} - 24 q^{6} + 6 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0302234	+	0.112795i	−0.0213712	+	0.0797582i	−0.975788	−	0.218719i	\(-0.929812\pi\)
							0.954417	+	0.298477i	\(0.0964788\pi\)
\(3\)	−1.53453	+	0.803249i	−0.885963	+	0.463756i
							\(5\)	−0.634953	+	2.14402i	−0.283960	+	0.958836i
\(7\)	2.46509	−	0.960898i	0.931717	−	0.363185i
							\(11\)	−0.911859			−0.274936			−0.137468	−	0.990506i	\(-0.543896\pi\)
−0.137468	+	0.990506i	\(0.543896\pi\)
\(13\)	−1.53935	+	5.74495i	−0.426940	+	1.59336i	0.332709	+	0.943030i	\(0.392037\pi\)
							−0.759649	+	0.650333i	\(0.774629\pi\)
\(17\)	−3.29933	−	0.884052i	−0.800204	−	0.214414i	−0.164530	−	0.986372i	\(-0.552611\pi\)
							−0.635674	+	0.771958i	\(0.719278\pi\)
\(19\)	0.815184	−	1.41194i	0.187016	−	0.323921i	−0.757238	−	0.653139i	\(-0.773452\pi\)
							0.944254	+	0.329218i	\(0.106785\pi\)
\(23\)	−2.30853	+	2.30853i	−0.481363	+	0.481363i	−0.905567	−	0.424204i	\(-0.860554\pi\)
							0.424204	+	0.905567i	\(0.360554\pi\)
\(29\)	5.74244	+	3.31540i	1.06635	+	0.615655i	0.927181	−	0.374615i	\(-0.122225\pi\)
							0.139164	+	0.990269i	\(0.455558\pi\)
\(31\)	−1.00753	−	0.581697i	−0.180957	−	0.104476i	0.406785	−	0.913524i	\(-0.366650\pi\)
							−0.587742	+	0.809048i	\(0.699983\pi\)
\(37\)	3.49032	−	0.935230i	0.573806	−	0.153751i	0.0397644	−	0.999209i	\(-0.487339\pi\)
							0.534042	+	0.845458i	\(0.320673\pi\)
\(41\)	5.95218	−	3.43649i	0.929575	−	0.536690i	0.0428978	−	0.999079i	\(-0.486341\pi\)
							0.886677	+	0.462389i	\(0.153008\pi\)
\(43\)	−1.11001	−	4.14262i	−0.169275	−	0.631744i	−0.997456	−	0.0712831i	\(-0.977291\pi\)
							0.828181	−	0.560461i	\(-0.189376\pi\)
\(47\)	5.64317	+	1.51208i	0.823141	+	0.220560i	0.645719	−	0.763575i	\(-0.276558\pi\)
							0.177422	+	0.984135i	\(0.443224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.cg.e.187.21	yes	160
3.2	odd	2		945.2.cj.e.397.20		160
5.3	odd	4	inner	315.2.cg.e.313.20	yes	160
7.3	odd	6		315.2.bs.e.52.21	✓	160
9.4	even	3		315.2.bs.e.292.21	yes	160
9.5	odd	6		945.2.bv.e.712.20		160
15.8	even	4		945.2.cj.e.208.21		160
21.17	even	6		945.2.bv.e.262.20		160
35.3	even	12		315.2.bs.e.178.21	yes	160
45.13	odd	12		315.2.bs.e.103.21	yes	160
45.23	even	12		945.2.bv.e.523.20		160
63.31	odd	6	inner	315.2.cg.e.157.20	yes	160
63.59	even	6		945.2.cj.e.577.21		160
105.38	odd	12		945.2.bv.e.73.20		160
315.248	odd	12		945.2.cj.e.388.20		160
315.283	even	12	inner	315.2.cg.e.283.21	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bs.e.52.21	✓	160	7.3	odd	6
315.2.bs.e.103.21	yes	160	45.13	odd	12
315.2.bs.e.178.21	yes	160	35.3	even	12
315.2.bs.e.292.21	yes	160	9.4	even	3
315.2.cg.e.157.20	yes	160	63.31	odd	6	inner
315.2.cg.e.187.21	yes	160	1.1	even	1	trivial
315.2.cg.e.283.21	yes	160	315.283	even	12	inner
315.2.cg.e.313.20	yes	160	5.3	odd	4	inner
945.2.bv.e.73.20		160	105.38	odd	12
945.2.bv.e.262.20		160	21.17	even	6
945.2.bv.e.523.20		160	45.23	even	12
945.2.bv.e.712.20		160	9.5	odd	6
945.2.cj.e.208.21		160	15.8	even	4
945.2.cj.e.388.20		160	315.248	odd	12
945.2.cj.e.397.20		160	3.2	odd	2
945.2.cj.e.577.21		160	63.59	even	6