Embedded newform 315.2.cb.a.13.11

• Label: 315.2.cb.a.13.11
• 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			13.11
Character	\(\chi\)	\(=\)	315.13
Dual form			315.2.cb.a.97.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66459 - 0.446025i) q^{2} +(-1.52423 + 0.822634i) q^{3} +(0.839860 + 0.484894i) q^{4} +(1.24718 - 1.85595i) q^{5} +(2.90413 - 0.689501i) q^{6} +(-2.31427 - 1.28225i) q^{7} +(1.25538 + 1.25538i) q^{8} +(1.64655 - 2.50776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 4 q^{2} - 2 q^{7} - 32 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66459	−	0.446025i	−1.17704	−	0.315387i	−0.383289	−	0.923629i	\(-0.625209\pi\)
							−0.793752	+	0.608242i	\(0.791875\pi\)
\(3\)	−1.52423	+	0.822634i	−0.880014	+	0.474948i
							\(5\)	1.24718	−	1.85595i	0.557755	−	0.830005i
\(7\)	−2.31427	−	1.28225i	−0.874710	−	0.484646i
							\(11\)	2.72143	+	4.71365i	0.820541	+	1.42122i	0.905280	+	0.424816i	\(0.139661\pi\)
−0.0847388	+	0.996403i	\(0.527006\pi\)
\(13\)	−0.478337	−	1.78518i	−0.132667	−	0.495119i	0.867330	−	0.497734i	\(-0.165834\pi\)
							−0.999997	+	0.00261470i	\(0.999168\pi\)
\(17\)	−3.23307	−	3.23307i	−0.784134	−	0.784134i	0.196391	−	0.980526i	\(-0.437078\pi\)
							−0.980526	+	0.196391i	\(0.937078\pi\)
\(19\)	−3.58101			−0.821539			−0.410770	−	0.911739i	\(-0.634740\pi\)
							−0.410770	+	0.911739i	\(0.634740\pi\)
\(23\)	−3.89718	+	1.04425i	−0.812618	+	0.217740i	−0.641116	−	0.767444i	\(-0.721528\pi\)
							−0.171501	+	0.985184i	\(0.554862\pi\)
\(29\)	−0.536529	+	0.309765i	−0.0996310	+	0.0575220i	−0.548988	−	0.835830i	\(-0.684987\pi\)
							0.449357	+	0.893352i	\(0.351653\pi\)
\(31\)	−8.04527	−	4.64494i	−1.44497	−	0.834256i	−0.446798	−	0.894635i	\(-0.647436\pi\)
							−0.998175	+	0.0603794i	\(0.980769\pi\)
\(37\)	−5.18951	+	5.18951i	−0.853151	+	0.853151i	−0.990520	−	0.137369i	\(-0.956135\pi\)
							0.137369	+	0.990520i	\(0.456135\pi\)
\(41\)	−3.79338	−	2.19011i	−0.592426	−	0.342038i	0.173630	−	0.984811i	\(-0.444450\pi\)
							−0.766056	+	0.642773i	\(0.777784\pi\)
\(43\)	−0.137334	+	0.512537i	−0.0209432	+	0.0781612i	−0.975607	−	0.219527i	\(-0.929549\pi\)
							0.954663	+	0.297688i	\(0.0962154\pi\)
\(47\)	−0.761523	+	2.84204i	−0.111079	+	0.414554i	−0.998964	−	0.0455126i	\(-0.985508\pi\)
							0.887884	+	0.460067i	\(0.152175\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.cb.a.13.11	✓	176
3.2	odd	2		945.2.ce.a.118.33		176
5.2	odd	4	inner	315.2.cb.a.202.34	yes	176
7.6	odd	2	inner	315.2.cb.a.13.12	yes	176
9.2	odd	6		945.2.ce.a.748.11		176
9.7	even	3	inner	315.2.cb.a.223.33	yes	176
15.2	even	4		945.2.ce.a.307.12		176
21.20	even	2		945.2.ce.a.118.34		176
35.27	even	4	inner	315.2.cb.a.202.33	yes	176
45.2	even	12		945.2.ce.a.937.34		176
45.7	odd	12	inner	315.2.cb.a.97.12	yes	176
63.20	even	6		945.2.ce.a.748.12		176
63.34	odd	6	inner	315.2.cb.a.223.34	yes	176
105.62	odd	4		945.2.ce.a.307.11		176
315.97	even	12	inner	315.2.cb.a.97.11	yes	176
315.272	odd	12		945.2.ce.a.937.33		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.cb.a.13.11	✓	176	1.1	even	1	trivial
315.2.cb.a.13.12	yes	176	7.6	odd	2	inner
315.2.cb.a.97.11	yes	176	315.97	even	12	inner
315.2.cb.a.97.12	yes	176	45.7	odd	12	inner
315.2.cb.a.202.33	yes	176	35.27	even	4	inner
315.2.cb.a.202.34	yes	176	5.2	odd	4	inner
315.2.cb.a.223.33	yes	176	9.7	even	3	inner
315.2.cb.a.223.34	yes	176	63.34	odd	6	inner
945.2.ce.a.118.33		176	3.2	odd	2
945.2.ce.a.118.34		176	21.20	even	2
945.2.ce.a.307.11		176	105.62	odd	4
945.2.ce.a.307.12		176	15.2	even	4
945.2.ce.a.748.11		176	9.2	odd	6
945.2.ce.a.748.12		176	63.20	even	6
945.2.ce.a.937.33		176	315.272	odd	12
945.2.ce.a.937.34		176	45.2	even	12