Embedded newform 315.2.bv.a.23.10

• Label: 315.2.bv.a.23.10
• 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			23.10
Character	\(\chi\)	\(=\)	315.23
Dual form			315.2.bv.a.137.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14944 - 1.14944i) q^{2} +(0.768701 + 1.55213i) q^{3} +0.642437i q^{4} +(-0.945461 - 2.02635i) q^{5} +(0.900504 - 2.66766i) q^{6} +(-1.70140 + 2.02614i) q^{7} +(-1.56044 + 1.56044i) q^{8} +(-1.81820 + 2.38624i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14944	−	1.14944i	−0.812779	−	0.812779i	0.172271	−	0.985050i	\(-0.444890\pi\)
							−0.985050	+	0.172271i	\(0.944890\pi\)
\(3\)	0.768701	+	1.55213i	0.443810	+	0.896121i
							\(5\)	−0.945461	−	2.02635i	−0.422823	−	0.906212i
\(7\)	−1.70140	+	2.02614i	−0.643067	+	0.765810i
							\(11\)	−4.00472	−	2.31213i	−1.20747	−	0.697133i	−0.245264	−	0.969456i	\(-0.578875\pi\)
−0.962206	+	0.272323i	\(0.912208\pi\)
\(13\)	−1.75941	+	0.471432i	−0.487972	+	0.130752i	−0.494412	−	0.869228i	\(-0.664617\pi\)
							0.00644023	+	0.999979i	\(0.497950\pi\)
\(17\)	−0.0796730	+	0.297344i	−0.0193235	+	0.0721164i	−0.974914	−	0.222580i	\(-0.928552\pi\)
							0.955591	+	0.294697i	\(0.0952187\pi\)
\(19\)	−0.824691	−	0.476135i	−0.189197	−	0.109233i	0.402410	−	0.915460i	\(-0.368173\pi\)
							−0.591607	+	0.806227i	\(0.701506\pi\)
\(23\)	−0.827238	−	0.221658i	−0.172491	−	0.0462188i	0.171540	−	0.985177i	\(-0.445126\pi\)
							−0.344031	+	0.938958i	\(0.611792\pi\)
\(29\)	−0.974351	−	1.68763i	−0.180932	−	0.313384i	0.761266	−	0.648440i	\(-0.224578\pi\)
							−0.942198	+	0.335056i	\(0.891245\pi\)
\(31\)	−8.03229			−1.44264			−0.721321	−	0.692601i	\(-0.756465\pi\)
							−0.721321	+	0.692601i	\(0.756465\pi\)
\(37\)	2.98039	+	11.1230i	0.489973	+	1.82860i	0.556540	+	0.830821i	\(0.312129\pi\)
							−0.0665669	+	0.997782i	\(0.521205\pi\)
\(41\)	2.09426	+	1.20912i	0.327069	+	0.188833i	0.654539	−	0.756028i	\(-0.272863\pi\)
							−0.327470	+	0.944862i	\(0.606196\pi\)
\(43\)	1.63758	−	6.11153i	0.249729	−	0.932000i	−0.721219	−	0.692707i	\(-0.756418\pi\)
							0.970947	−	0.239293i	\(-0.0769156\pi\)
\(47\)	−1.60268	−	1.60268i	−0.233775	−	0.233775i	0.580491	−	0.814266i	\(-0.302861\pi\)
							−0.814266	+	0.580491i	\(0.802861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bv.a.23.10	✓	176
3.2	odd	2		945.2.by.a.233.35		176
5.2	odd	4	inner	315.2.bv.a.212.35	yes	176
7.4	even	3		315.2.bx.a.158.35	yes	176
9.2	odd	6		315.2.bx.a.128.35	yes	176
9.7	even	3		945.2.ca.a.548.10		176
15.2	even	4		945.2.by.a.422.10		176
21.11	odd	6		945.2.ca.a.368.10		176
35.32	odd	12		315.2.bx.a.32.35	yes	176
45.2	even	12		315.2.bx.a.2.35	yes	176
45.7	odd	12		945.2.ca.a.737.10		176
63.11	odd	6	inner	315.2.bv.a.263.35	yes	176
63.25	even	3		945.2.by.a.683.10		176
105.32	even	12		945.2.ca.a.557.10		176
315.137	even	12	inner	315.2.bv.a.137.10	yes	176
315.277	odd	12		945.2.by.a.872.35		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.10	✓	176	1.1	even	1	trivial
315.2.bv.a.137.10	yes	176	315.137	even	12	inner
315.2.bv.a.212.35	yes	176	5.2	odd	4	inner
315.2.bv.a.263.35	yes	176	63.11	odd	6	inner
315.2.bx.a.2.35	yes	176	45.2	even	12
315.2.bx.a.32.35	yes	176	35.32	odd	12
315.2.bx.a.128.35	yes	176	9.2	odd	6
315.2.bx.a.158.35	yes	176	7.4	even	3
945.2.by.a.233.35		176	3.2	odd	2
945.2.by.a.422.10		176	15.2	even	4
945.2.by.a.683.10		176	63.25	even	3
945.2.by.a.872.35		176	315.277	odd	12
945.2.ca.a.368.10		176	21.11	odd	6
945.2.ca.a.548.10		176	9.7	even	3
945.2.ca.a.557.10		176	105.32	even	12
945.2.ca.a.737.10		176	45.7	odd	12