Embedded newform 315.2.bs.e.52.9

• Label: 315.2.bs.e.52.9
• Level: $315$
• Weight: $2$
• Character: 315.52
• 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.bs (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			52.9
Character	\(\chi\)	\(=\)	315.52
Dual form			315.2.bs.e.103.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21497 - 1.21497i) q^{2} +(-1.20986 - 1.23945i) q^{3} +0.952318i q^{4} +(-0.123740 - 2.23264i) q^{5} +(-0.0359577 + 2.97585i) q^{6} +(0.124742 - 2.64281i) q^{7} +(-1.27291 + 1.27291i) q^{8} +(-0.0724885 + 2.99912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 18 q^{3} - 6 q^{5} + 24 q^{6} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21497	−	1.21497i	−0.859116	−	0.859116i	0.132118	−	0.991234i	\(-0.457822\pi\)
							−0.991234	+	0.132118i	\(0.957822\pi\)
\(3\)	−1.20986	−	1.23945i	−0.698512	−	0.715599i
							\(5\)	−0.123740	−	2.23264i	−0.0553384	−	0.998468i
\(7\)	0.124742	−	2.64281i	0.0471481	−	0.998888i
							\(11\)	−2.07383	−	3.59199i	−0.625284	−	1.08302i	−0.988486	−	0.151314i	\(-0.951650\pi\)
0.363201	−	0.931711i	\(-0.381684\pi\)
\(13\)	0.943605	−	3.52158i	0.261709	−	0.976711i	−0.702525	−	0.711659i	\(-0.747944\pi\)
							0.964234	−	0.265052i	\(-0.0853892\pi\)
\(17\)	−0.771188	−	2.87811i	−0.187041	−	0.698045i	−0.994184	−	0.107690i	\(-0.965655\pi\)
							0.807144	−	0.590355i	\(-0.201012\pi\)
\(19\)	3.95135	+	6.84394i	0.906502	+	1.57011i	0.818888	+	0.573953i	\(0.194591\pi\)
							0.0876137	+	0.996155i	\(0.472076\pi\)
\(23\)	1.52889	+	5.70591i	0.318797	+	1.18977i	0.920403	+	0.390972i	\(0.127861\pi\)
							−0.601606	+	0.798793i	\(0.705472\pi\)
\(29\)	−2.46538	−	1.42339i	−0.457810	−	0.264317i	0.253313	−	0.967384i	\(-0.418480\pi\)
							−0.711123	+	0.703067i	\(0.751813\pi\)
\(31\)			4.62735i			0.831097i	0.909571	+	0.415548i	\(0.136410\pi\)
							−0.909571	+	0.415548i	\(0.863590\pi\)
\(37\)	1.19282	−	4.45165i	0.196098	−	0.731847i	−0.795882	−	0.605452i	\(-0.792993\pi\)
							0.991980	−	0.126395i	\(-0.0403408\pi\)
\(41\)	3.84106	−	2.21763i	0.599872	−	0.346336i	−0.169119	−	0.985596i	\(-0.554092\pi\)
							0.768991	+	0.639259i	\(0.220759\pi\)
\(43\)	1.27772	+	4.76853i	0.194851	+	0.727194i	0.992305	+	0.123815i	\(0.0395129\pi\)
							−0.797454	+	0.603379i	\(0.793820\pi\)
\(47\)	−0.613183	+	0.613183i	−0.0894420	+	0.0894420i	−0.750412	−	0.660970i	\(-0.770145\pi\)
							0.660970	+	0.750412i	\(0.270145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bs.e.52.9	✓	160
3.2	odd	2		945.2.bv.e.262.32		160
5.3	odd	4	inner	315.2.bs.e.178.9	yes	160
7.5	odd	6		315.2.cg.e.187.9	yes	160
9.4	even	3		315.2.cg.e.157.32	yes	160
9.5	odd	6		945.2.cj.e.577.9		160
15.8	even	4		945.2.bv.e.73.32		160
21.5	even	6		945.2.cj.e.397.32		160
35.33	even	12		315.2.cg.e.313.32	yes	160
45.13	odd	12		315.2.cg.e.283.9	yes	160
45.23	even	12		945.2.cj.e.388.32		160
63.5	even	6		945.2.bv.e.712.32		160
63.40	odd	6	inner	315.2.bs.e.292.9	yes	160
105.68	odd	12		945.2.cj.e.208.9		160
315.68	odd	12		945.2.bv.e.523.32		160
315.103	even	12	inner	315.2.bs.e.103.9	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bs.e.52.9	✓	160	1.1	even	1	trivial
315.2.bs.e.103.9	yes	160	315.103	even	12	inner
315.2.bs.e.178.9	yes	160	5.3	odd	4	inner
315.2.bs.e.292.9	yes	160	63.40	odd	6	inner
315.2.cg.e.157.32	yes	160	9.4	even	3
315.2.cg.e.187.9	yes	160	7.5	odd	6
315.2.cg.e.283.9	yes	160	45.13	odd	12
315.2.cg.e.313.32	yes	160	35.33	even	12
945.2.bv.e.73.32		160	15.8	even	4
945.2.bv.e.262.32		160	3.2	odd	2
945.2.bv.e.523.32		160	315.68	odd	12
945.2.bv.e.712.32		160	63.5	even	6
945.2.cj.e.208.9		160	105.68	odd	12
945.2.cj.e.388.32		160	45.23	even	12
945.2.cj.e.397.32		160	21.5	even	6
945.2.cj.e.577.9		160	9.5	odd	6