Embedded newform 315.2.cb.a.13.14

• Label: 315.2.cb.a.13.14
• 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			13.14
Character	\(\chi\)	\(=\)	315.13
Dual form			315.2.cb.a.97.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44975 - 0.388459i) q^{2} +(0.181894 + 1.72247i) q^{3} +(0.218820 + 0.126336i) q^{4} +(-2.12297 - 0.702131i) q^{5} +(0.405409 - 2.56781i) q^{6} +(1.49535 - 2.18265i) q^{7} +(1.85442 + 1.85442i) q^{8} +(-2.93383 + 0.626616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 4 q^{2} - 2 q^{7} - 32 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44975	−	0.388459i	−1.02513	−	0.274682i	−0.293190	−	0.956054i	\(-0.594717\pi\)
							−0.731937	+	0.681372i	\(0.761384\pi\)
\(3\)	0.181894	+	1.72247i	0.105017	+	0.994470i
							\(5\)	−2.12297	−	0.702131i	−0.949422	−	0.314003i
\(7\)	1.49535	−	2.18265i	0.565188	−	0.824962i
							\(11\)	1.50520	+	2.60709i	0.453836	+	0.786066i	0.998620	−	0.0525096i	\(-0.0167220\pi\)
−0.544785	+	0.838576i	\(0.683389\pi\)
\(13\)	−1.45210	−	5.41932i	−0.402741	−	1.50305i	−0.808184	−	0.588930i	\(-0.799549\pi\)
							0.405443	−	0.914120i	\(-0.367117\pi\)
\(17\)	−1.04847	−	1.04847i	−0.254291	−	0.254291i	0.568437	−	0.822727i	\(-0.307548\pi\)
							−0.822727	+	0.568437i	\(0.807548\pi\)
\(19\)	7.21291			1.65475			0.827377	−	0.561646i	\(-0.189832\pi\)
							0.827377	+	0.561646i	\(0.189832\pi\)
\(23\)	5.45189	−	1.46083i	1.13680	−	0.304604i	0.359134	−	0.933286i	\(-0.383072\pi\)
							0.777663	+	0.628682i	\(0.216405\pi\)
\(29\)	4.88990	−	2.82318i	0.908031	−	0.524252i	0.0282341	−	0.999601i	\(-0.491012\pi\)
							0.879797	+	0.475349i	\(0.157678\pi\)
\(31\)	4.98189	+	2.87630i	0.894775	+	0.516598i	0.875501	−	0.483216i	\(-0.160531\pi\)
							0.0192734	+	0.999814i	\(0.493865\pi\)
\(37\)	2.33216	−	2.33216i	0.383404	−	0.383404i	−0.488923	−	0.872327i	\(-0.662610\pi\)
							0.872327	+	0.488923i	\(0.162610\pi\)
\(41\)	−3.05504	−	1.76383i	−0.477117	−	0.275464i	0.242097	−	0.970252i	\(-0.422165\pi\)
							−0.719214	+	0.694788i	\(0.755498\pi\)
\(43\)	−0.793762	+	2.96236i	−0.121048	+	0.451755i	−0.999668	−	0.0257652i	\(-0.991798\pi\)
							0.878621	+	0.477521i	\(0.158464\pi\)
\(47\)	1.57778	−	5.88834i	0.230142	−	0.858903i	−0.750137	−	0.661283i	\(-0.770012\pi\)
							0.980279	−	0.197620i	\(-0.0633211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.cb.a.13.14	yes	176
3.2	odd	2		945.2.ce.a.118.32		176
5.2	odd	4	inner	315.2.cb.a.202.32	yes	176
7.6	odd	2	inner	315.2.cb.a.13.13	✓	176
9.2	odd	6		945.2.ce.a.748.13		176
9.7	even	3	inner	315.2.cb.a.223.31	yes	176
15.2	even	4		945.2.ce.a.307.14		176
21.20	even	2		945.2.ce.a.118.31		176
35.27	even	4	inner	315.2.cb.a.202.31	yes	176
45.2	even	12		945.2.ce.a.937.31		176
45.7	odd	12	inner	315.2.cb.a.97.13	yes	176
63.20	even	6		945.2.ce.a.748.14		176
63.34	odd	6	inner	315.2.cb.a.223.32	yes	176
105.62	odd	4		945.2.ce.a.307.13		176
315.97	even	12	inner	315.2.cb.a.97.14	yes	176
315.272	odd	12		945.2.ce.a.937.32		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.cb.a.13.13	✓	176	7.6	odd	2	inner
315.2.cb.a.13.14	yes	176	1.1	even	1	trivial
315.2.cb.a.97.13	yes	176	45.7	odd	12	inner
315.2.cb.a.97.14	yes	176	315.97	even	12	inner
315.2.cb.a.202.31	yes	176	35.27	even	4	inner
315.2.cb.a.202.32	yes	176	5.2	odd	4	inner
315.2.cb.a.223.31	yes	176	9.7	even	3	inner
315.2.cb.a.223.32	yes	176	63.34	odd	6	inner
945.2.ce.a.118.31		176	21.20	even	2
945.2.ce.a.118.32		176	3.2	odd	2
945.2.ce.a.307.13		176	105.62	odd	4
945.2.ce.a.307.14		176	15.2	even	4
945.2.ce.a.748.13		176	9.2	odd	6
945.2.ce.a.748.14		176	63.20	even	6
945.2.ce.a.937.31		176	45.2	even	12
945.2.ce.a.937.32		176	315.272	odd	12