Embedded newform 315.2.bv.a.23.2

• Label: 315.2.bv.a.23.2
• Level: $315$
• Weight: $2$
• Character: 315.23
• 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.bv (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			23.2
Character	\(\chi\)	\(=\)	315.23
Dual form			315.2.bv.a.137.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77797 - 1.77797i) q^{2} +(0.265860 - 1.71153i) q^{3} +4.32238i q^{4} +(-0.447954 + 2.19074i) q^{5} +(-3.51574 + 2.57035i) q^{6} +(1.37680 + 2.25930i) q^{7} +(4.12912 - 4.12912i) q^{8} +(-2.85864 - 0.910053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77797	−	1.77797i	−1.25722	−	1.25722i	−0.952416	−	0.304801i	\(-0.901410\pi\)
							−0.304801	−	0.952416i	\(-0.598590\pi\)
\(3\)	0.265860	−	1.71153i	0.153494	−	0.988150i
							\(5\)	−0.447954	+	2.19074i	−0.200331	+	0.979728i
\(7\)	1.37680	+	2.25930i	0.520381	+	0.853934i
							\(11\)	−0.726422	−	0.419400i	−0.219025	−	0.126454i	0.386474	−	0.922300i	\(-0.373693\pi\)
−0.605499	+	0.795846i	\(0.707026\pi\)
\(13\)	5.08982	−	1.36381i	1.41166	−	0.378254i	0.529144	−	0.848532i	\(-0.322513\pi\)
							0.882518	+	0.470279i	\(0.155847\pi\)
\(17\)	0.961886	−	3.58981i	0.233292	−	0.870656i	−0.745620	−	0.666372i	\(-0.767846\pi\)
							0.978911	−	0.204285i	\(-0.0654869\pi\)
\(19\)	2.60684	+	1.50506i	0.598051	+	0.345285i	0.768275	−	0.640120i	\(-0.221116\pi\)
							−0.170223	+	0.985406i	\(0.554449\pi\)
\(23\)	7.56038	+	2.02580i	1.57645	+	0.422408i	0.937824	−	0.347112i	\(-0.112837\pi\)
							0.638624	+	0.769519i	\(0.279504\pi\)
\(29\)	0.209921	+	0.363594i	0.0389814	+	0.0675177i	0.884858	−	0.465861i	\(-0.154255\pi\)
							−0.845876	+	0.533379i	\(0.820922\pi\)
\(31\)	−1.19799			−0.215165			−0.107582	−	0.994196i	\(-0.534311\pi\)
							−0.107582	+	0.994196i	\(0.534311\pi\)
\(37\)	1.56418	+	5.83758i	0.257149	+	0.959693i	0.966882	+	0.255223i	\(0.0821487\pi\)
							−0.709733	+	0.704470i	\(0.751185\pi\)
\(41\)	9.09472	+	5.25084i	1.42036	+	0.820044i	0.996329	−	0.0856054i	\(-0.0272825\pi\)
							0.424028	+	0.905649i	\(0.360616\pi\)
\(43\)	1.86638	−	6.96543i	0.284621	−	1.06222i	−0.664495	−	0.747292i	\(-0.731353\pi\)
							0.949116	−	0.314926i	\(-0.101980\pi\)
\(47\)	−2.94938	−	2.94938i	−0.430212	−	0.430212i	0.458489	−	0.888700i	\(-0.348391\pi\)
							−0.888700	+	0.458489i	\(0.848391\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bv.a.23.2	✓	176
3.2	odd	2		945.2.by.a.233.43		176
5.2	odd	4	inner	315.2.bv.a.212.43	yes	176
7.4	even	3		315.2.bx.a.158.43	yes	176
9.2	odd	6		315.2.bx.a.128.43	yes	176
9.7	even	3		945.2.ca.a.548.2		176
15.2	even	4		945.2.by.a.422.2		176
21.11	odd	6		945.2.ca.a.368.2		176
35.32	odd	12		315.2.bx.a.32.43	yes	176
45.2	even	12		315.2.bx.a.2.43	yes	176
45.7	odd	12		945.2.ca.a.737.2		176
63.11	odd	6	inner	315.2.bv.a.263.43	yes	176
63.25	even	3		945.2.by.a.683.2		176
105.32	even	12		945.2.ca.a.557.2		176
315.137	even	12	inner	315.2.bv.a.137.2	yes	176
315.277	odd	12		945.2.by.a.872.43		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.2	✓	176	1.1	even	1	trivial
315.2.bv.a.137.2	yes	176	315.137	even	12	inner
315.2.bv.a.212.43	yes	176	5.2	odd	4	inner
315.2.bv.a.263.43	yes	176	63.11	odd	6	inner
315.2.bx.a.2.43	yes	176	45.2	even	12
315.2.bx.a.32.43	yes	176	35.32	odd	12
315.2.bx.a.128.43	yes	176	9.2	odd	6
315.2.bx.a.158.43	yes	176	7.4	even	3
945.2.by.a.233.43		176	3.2	odd	2
945.2.by.a.422.2		176	15.2	even	4
945.2.by.a.683.2		176	63.25	even	3
945.2.by.a.872.43		176	315.277	odd	12
945.2.ca.a.368.2		176	21.11	odd	6
945.2.ca.a.548.2		176	9.7	even	3
945.2.ca.a.557.2		176	105.32	even	12
945.2.ca.a.737.2		176	45.7	odd	12