Embedded newform 315.2.cb.a.13.9

• Label: 315.2.cb.a.13.9
• Level: $315$
• Weight: $2$
• Character: 315.13
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			13.9
Character	\(\chi\)	\(=\)	315.13
Dual form			315.2.cb.a.97.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73908 - 0.465984i) q^{2} +(-1.63791 - 0.563241i) q^{3} +(1.07519 + 0.620762i) q^{4} +(-2.23593 + 0.0247189i) q^{5} +(2.58599 + 1.74276i) q^{6} +(-1.24960 + 2.33206i) q^{7} +(0.965609 + 0.965609i) q^{8} +(2.36552 + 1.84508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 4 q^{2} - 2 q^{7} - 32 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73908	−	0.465984i	−1.22971	−	0.329500i	−0.415245	−	0.909710i	\(-0.636304\pi\)
							−0.814467	+	0.580209i	\(0.802971\pi\)
\(3\)	−1.63791	−	0.563241i	−0.945650	−	0.325187i
							\(5\)	−2.23593	+	0.0247189i	−0.999939	+	0.0110546i
\(7\)	−1.24960	+	2.33206i	−0.472304	+	0.881436i
							\(11\)	−0.941623	−	1.63094i	−0.283910	−	0.491747i	0.688434	−	0.725299i	\(-0.258298\pi\)
−0.972344	+	0.233552i	\(0.924965\pi\)
\(13\)	0.604322	+	2.25536i	0.167609	+	0.625524i	0.997693	+	0.0678865i	\(0.0216256\pi\)
							−0.830084	+	0.557638i	\(0.811708\pi\)
\(17\)	−4.80500	−	4.80500i	−1.16538	−	1.16538i	−0.983279	−	0.182105i	\(-0.941709\pi\)
							−0.182105	−	0.983279i	\(-0.558291\pi\)
\(19\)	4.78276			1.09724			0.548620	−	0.836072i	\(-0.315153\pi\)
							0.548620	+	0.836072i	\(0.315153\pi\)
\(23\)	−1.17709	+	0.315401i	−0.245441	+	0.0657656i	−0.379442	−	0.925216i	\(-0.623884\pi\)
							0.134001	+	0.990981i	\(0.457217\pi\)
\(29\)	3.04168	−	1.75611i	0.564825	−	0.326102i	−0.190255	−	0.981735i	\(-0.560931\pi\)
							0.755080	+	0.655633i	\(0.227598\pi\)
\(31\)	−2.07363	−	1.19721i	−0.372435	−	0.215026i	0.302087	−	0.953280i	\(-0.402317\pi\)
							−0.674522	+	0.738255i	\(0.735650\pi\)
\(37\)	6.27745	−	6.27745i	1.03201	−	1.03201i	0.0325351	−	0.999471i	\(-0.489642\pi\)
							0.999471	−	0.0325351i	\(-0.0103581\pi\)
\(41\)	8.25634	+	4.76680i	1.28942	+	0.744449i	0.978551	−	0.206003i	\(-0.0660456\pi\)
							0.310872	+	0.950452i	\(0.399379\pi\)
\(43\)	0.808038	−	3.01564i	0.123225	−	0.459880i	−0.876546	−	0.481319i	\(-0.840158\pi\)
							0.999770	+	0.0214384i	\(0.00682458\pi\)
\(47\)	0.988021	−	3.68735i	0.144118	−	0.537855i	−0.855675	−	0.517513i	\(-0.826858\pi\)
							0.999793	−	0.0203417i	\(-0.00647540\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.cb.a.13.9	✓	176
3.2	odd	2		945.2.ce.a.118.36		176
5.2	odd	4	inner	315.2.cb.a.202.35	yes	176
7.6	odd	2	inner	315.2.cb.a.13.10	yes	176
9.2	odd	6		945.2.ce.a.748.9		176
9.7	even	3	inner	315.2.cb.a.223.36	yes	176
15.2	even	4		945.2.ce.a.307.10		176
21.20	even	2		945.2.ce.a.118.35		176
35.27	even	4	inner	315.2.cb.a.202.36	yes	176
45.2	even	12		945.2.ce.a.937.35		176
45.7	odd	12	inner	315.2.cb.a.97.10	yes	176
63.20	even	6		945.2.ce.a.748.10		176
63.34	odd	6	inner	315.2.cb.a.223.35	yes	176
105.62	odd	4		945.2.ce.a.307.9		176
315.97	even	12	inner	315.2.cb.a.97.9	yes	176
315.272	odd	12		945.2.ce.a.937.36		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.cb.a.13.9	✓	176	1.1	even	1	trivial
315.2.cb.a.13.10	yes	176	7.6	odd	2	inner
315.2.cb.a.97.9	yes	176	315.97	even	12	inner
315.2.cb.a.97.10	yes	176	45.7	odd	12	inner
315.2.cb.a.202.35	yes	176	5.2	odd	4	inner
315.2.cb.a.202.36	yes	176	35.27	even	4	inner
315.2.cb.a.223.35	yes	176	63.34	odd	6	inner
315.2.cb.a.223.36	yes	176	9.7	even	3	inner
945.2.ce.a.118.35		176	21.20	even	2
945.2.ce.a.118.36		176	3.2	odd	2
945.2.ce.a.307.9		176	105.62	odd	4
945.2.ce.a.307.10		176	15.2	even	4
945.2.ce.a.748.9		176	9.2	odd	6
945.2.ce.a.748.10		176	63.20	even	6
945.2.ce.a.937.35		176	45.2	even	12
945.2.ce.a.937.36		176	315.272	odd	12