Embedded newform 315.2.bv.a.23.12

• Label: 315.2.bv.a.23.12
• 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			23.12
Character	\(\chi\)	\(=\)	315.23
Dual form			315.2.bv.a.137.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08313 - 1.08313i) q^{2} +(1.10330 - 1.33519i) q^{3} +0.346328i q^{4} +(2.23510 - 0.0656858i) q^{5} +(-2.64119 + 0.251168i) q^{6} +(-2.63546 - 0.233113i) q^{7} +(-1.79114 + 1.79114i) q^{8} +(-0.565465 - 2.94623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08313	−	1.08313i	−0.765886	−	0.765886i	0.211493	−	0.977379i	\(-0.432167\pi\)
							−0.977379	+	0.211493i	\(0.932167\pi\)
\(3\)	1.10330	−	1.33519i	0.636990	−	0.770872i
							\(5\)	2.23510	−	0.0656858i	0.999568	−	0.0293756i
\(7\)	−2.63546	−	0.233113i	−0.996111	−	0.0881084i
							\(11\)	−4.30782	−	2.48712i	−1.29886	−	0.749896i	−0.318651	−	0.947872i	\(-0.603230\pi\)
−0.980207	+	0.197976i	\(0.936563\pi\)
\(13\)	1.91315	−	0.512626i	0.530611	−	0.142177i	0.0164406	−	0.999865i	\(-0.494767\pi\)
							0.514170	+	0.857688i	\(0.328100\pi\)
\(17\)	1.22771	−	4.58187i	0.297763	−	1.11127i	−0.641235	−	0.767345i	\(-0.721578\pi\)
							0.938998	−	0.343922i	\(-0.111756\pi\)
\(19\)	4.32733	+	2.49839i	0.992758	+	0.573169i	0.906098	−	0.423069i	\(-0.139047\pi\)
							0.0866604	+	0.996238i	\(0.472381\pi\)
\(23\)	−7.14955	−	1.91572i	−1.49078	−	0.399454i	−0.580782	−	0.814059i	\(-0.697253\pi\)
							−0.910002	+	0.414605i	\(0.863920\pi\)
\(29\)	0.380013	+	0.658203i	0.0705667	+	0.122225i	0.899150	−	0.437641i	\(-0.144186\pi\)
							−0.828583	+	0.559866i	\(0.810853\pi\)
\(31\)	5.59577			1.00503			0.502515	−	0.864569i	\(-0.332408\pi\)
							0.502515	+	0.864569i	\(0.332408\pi\)
\(37\)	−1.59004	−	5.93412i	−0.261401	−	0.975563i	−0.964416	−	0.264388i	\(-0.914830\pi\)
							0.703015	−	0.711175i	\(-0.251837\pi\)
\(41\)	8.97390	+	5.18108i	1.40149	+	0.809149i	0.994545	−	0.104305i	\(-0.0332617\pi\)
							0.406942	+	0.913454i	\(0.366595\pi\)
\(43\)	−2.17648	+	8.12274i	−0.331910	+	1.23871i	0.575270	+	0.817964i	\(0.304897\pi\)
							−0.907180	+	0.420743i	\(0.861770\pi\)
\(47\)	1.38667	+	1.38667i	0.202267	+	0.202267i	0.800971	−	0.598704i	\(-0.204317\pi\)
							−0.598704	+	0.800971i	\(0.704317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bv.a.23.12	✓	176
3.2	odd	2		945.2.by.a.233.33		176
5.2	odd	4	inner	315.2.bv.a.212.33	yes	176
7.4	even	3		315.2.bx.a.158.33	yes	176
9.2	odd	6		315.2.bx.a.128.33	yes	176
9.7	even	3		945.2.ca.a.548.12		176
15.2	even	4		945.2.by.a.422.12		176
21.11	odd	6		945.2.ca.a.368.12		176
35.32	odd	12		315.2.bx.a.32.33	yes	176
45.2	even	12		315.2.bx.a.2.33	yes	176
45.7	odd	12		945.2.ca.a.737.12		176
63.11	odd	6	inner	315.2.bv.a.263.33	yes	176
63.25	even	3		945.2.by.a.683.12		176
105.32	even	12		945.2.ca.a.557.12		176
315.137	even	12	inner	315.2.bv.a.137.12	yes	176
315.277	odd	12		945.2.by.a.872.33		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.12	✓	176	1.1	even	1	trivial
315.2.bv.a.137.12	yes	176	315.137	even	12	inner
315.2.bv.a.212.33	yes	176	5.2	odd	4	inner
315.2.bv.a.263.33	yes	176	63.11	odd	6	inner
315.2.bx.a.2.33	yes	176	45.2	even	12
315.2.bx.a.32.33	yes	176	35.32	odd	12
315.2.bx.a.128.33	yes	176	9.2	odd	6
315.2.bx.a.158.33	yes	176	7.4	even	3
945.2.by.a.233.33		176	3.2	odd	2
945.2.by.a.422.12		176	15.2	even	4
945.2.by.a.683.12		176	63.25	even	3
945.2.by.a.872.33		176	315.277	odd	12
945.2.ca.a.368.12		176	21.11	odd	6
945.2.ca.a.548.12		176	9.7	even	3
945.2.ca.a.557.12		176	105.32	even	12
945.2.ca.a.737.12		176	45.7	odd	12