Embedded newform 315.2.bx.a.2.8

• Label: 315.2.bx.a.2.8
• Level: $315$
• Weight: $2$
• Character: 315.2
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bx (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			2.8
Character	\(\chi\)	\(=\)	315.2
Dual form			315.2.bx.a.158.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95085 - 0.522728i) q^{2} +(-1.45630 - 0.937654i) q^{3} +(1.80051 + 1.03953i) q^{4} +(2.23387 + 0.0991923i) q^{5} +(2.35088 + 2.59047i) q^{6} +(-1.87306 - 1.86860i) q^{7} +(-0.112896 - 0.112896i) q^{8} +(1.24161 + 2.73101i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{2} - 2 q^{3} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95085	−	0.522728i	−1.37946	−	0.369625i	−0.508534	−	0.861042i	\(-0.669812\pi\)
							−0.870924	+	0.491418i	\(0.836479\pi\)
\(3\)	−1.45630	−	0.937654i	−0.840794	−	0.541355i
							\(5\)	2.23387	+	0.0991923i	0.999016	+	0.0443602i
\(7\)	−1.87306	−	1.86860i	−0.707950	−	0.706263i
							\(11\)			5.34196i			1.61066i	0.592826	+	0.805330i	\(0.298012\pi\)
−0.592826	+	0.805330i	\(0.701988\pi\)
\(13\)	1.95482	+	0.523792i	0.542169	+	0.145274i	0.519502	−	0.854469i	\(-0.326118\pi\)
							0.0226672	+	0.999743i	\(0.492784\pi\)
\(17\)	−0.564743	−	0.151323i	−0.136970	−	0.0367011i	0.189682	−	0.981845i	\(-0.439254\pi\)
							−0.326653	+	0.945144i	\(0.605921\pi\)
\(19\)	0.863778	+	0.498703i	0.198164	+	0.114410i	0.595799	−	0.803134i	\(-0.296836\pi\)
							−0.397635	+	0.917544i	\(0.630169\pi\)
\(23\)	4.67937	+	4.67937i	0.975716	+	0.975716i	0.999712	−	0.0239961i	\(-0.00763894\pi\)
							−0.0239961	+	0.999712i	\(0.507639\pi\)
\(29\)	2.71972	−	4.71069i	0.505039	−	0.874752i	−0.494945	−	0.868925i	\(-0.664812\pi\)
							0.999983	−	0.00582778i	\(-0.00185505\pi\)
\(31\)	2.53588	−	4.39228i	0.455458	−	0.788876i	−0.543256	−	0.839567i	\(-0.682809\pi\)
							0.998714	+	0.0506904i	\(0.0161422\pi\)
\(37\)	2.84074	−	0.761173i	0.467014	−	0.125136i	−0.0176355	−	0.999844i	\(-0.505614\pi\)
							0.484649	+	0.874708i	\(0.338947\pi\)
\(41\)	8.92697	−	5.15399i	1.39416	−	0.804917i	0.400385	−	0.916347i	\(-0.368876\pi\)
							0.993772	+	0.111429i	\(0.0355429\pi\)
\(43\)	−1.09907	−	4.10178i	−0.167606	−	0.625515i	−0.997693	−	0.0678816i	\(-0.978376\pi\)
							0.830087	−	0.557634i	\(-0.188291\pi\)
\(47\)	−4.23162	−	1.13386i	−0.617245	−	0.165390i	−0.0633700	−	0.997990i	\(-0.520185\pi\)
							−0.553875	+	0.832600i	\(0.686851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bx.a.2.8	yes	176
3.2	odd	2		945.2.ca.a.737.37		176
5.3	odd	4	inner	315.2.bx.a.128.8	yes	176
7.4	even	3		315.2.bv.a.137.37	yes	176
9.4	even	3		945.2.by.a.422.37		176
9.5	odd	6		315.2.bv.a.212.8	yes	176
15.8	even	4		945.2.ca.a.548.37		176
21.11	odd	6		945.2.by.a.872.8		176
35.18	odd	12		315.2.bv.a.263.8	yes	176
45.13	odd	12		945.2.by.a.233.8		176
45.23	even	12		315.2.bv.a.23.37	✓	176
63.4	even	3		945.2.ca.a.557.37		176
63.32	odd	6	inner	315.2.bx.a.32.8	yes	176
105.53	even	12		945.2.by.a.683.37		176
315.158	even	12	inner	315.2.bx.a.158.8	yes	176
315.193	odd	12		945.2.ca.a.368.37		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.37	✓	176	45.23	even	12
315.2.bv.a.137.37	yes	176	7.4	even	3
315.2.bv.a.212.8	yes	176	9.5	odd	6
315.2.bv.a.263.8	yes	176	35.18	odd	12
315.2.bx.a.2.8	yes	176	1.1	even	1	trivial
315.2.bx.a.32.8	yes	176	63.32	odd	6	inner
315.2.bx.a.128.8	yes	176	5.3	odd	4	inner
315.2.bx.a.158.8	yes	176	315.158	even	12	inner
945.2.by.a.233.8		176	45.13	odd	12
945.2.by.a.422.37		176	9.4	even	3
945.2.by.a.683.37		176	105.53	even	12
945.2.by.a.872.8		176	21.11	odd	6
945.2.ca.a.368.37		176	315.193	odd	12
945.2.ca.a.548.37		176	15.8	even	4
945.2.ca.a.557.37		176	63.4	even	3
945.2.ca.a.737.37		176	3.2	odd	2