Embedded newform 315.2.bv.a.23.17

• Label: 315.2.bv.a.23.17
• 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			23.17
Character	\(\chi\)	\(=\)	315.23
Dual form			315.2.bv.a.137.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.477311 - 0.477311i) q^{2} +(-1.72825 + 0.114750i) q^{3} -1.54435i q^{4} +(0.814847 - 2.08231i) q^{5} +(0.879681 + 0.770138i) q^{6} +(-2.51050 - 0.835101i) q^{7} +(-1.69176 + 1.69176i) q^{8} +(2.97366 - 0.396634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.477311	−	0.477311i	−0.337510	−	0.337510i	0.517920	−	0.855429i	\(-0.326707\pi\)
							−0.855429	+	0.517920i	\(0.826707\pi\)
\(3\)	−1.72825	+	0.114750i	−0.997803	+	0.0662512i
							\(5\)	0.814847	−	2.08231i	0.364411	−	0.931238i
\(7\)	−2.51050	−	0.835101i	−0.948879	−	0.315639i
							\(11\)	0.137097	+	0.0791529i	0.0413362	+	0.0238655i	0.520526	−	0.853846i	\(-0.325736\pi\)
−0.479189	+	0.877711i	\(0.659069\pi\)
\(13\)	−0.549513	+	0.147242i	−0.152408	+	0.0408375i	−0.334216	−	0.942496i	\(-0.608472\pi\)
							0.181809	+	0.983334i	\(0.441805\pi\)
\(17\)	−1.11235	+	4.15136i	−0.269786	+	1.00685i	0.689470	+	0.724314i	\(0.257843\pi\)
							−0.959256	+	0.282539i	\(0.908823\pi\)
\(19\)	−3.32280	−	1.91842i	−0.762303	−	0.440116i	0.0678188	−	0.997698i	\(-0.478396\pi\)
							−0.830122	+	0.557582i	\(0.811729\pi\)
\(23\)	−1.02485	−	0.274607i	−0.213695	−	0.0572594i	0.150383	−	0.988628i	\(-0.451949\pi\)
							−0.364078	+	0.931368i	\(0.618616\pi\)
\(29\)	2.59211	+	4.48966i	0.481342	+	0.833709i	0.999771	−	0.0214119i	\(-0.00681614\pi\)
							−0.518429	+	0.855121i	\(0.673483\pi\)
\(31\)	−5.05866			−0.908562			−0.454281	−	0.890858i	\(-0.650104\pi\)
							−0.454281	+	0.890858i	\(0.650104\pi\)
\(37\)	−3.10392	−	11.5840i	−0.510282	−	1.90440i	−0.417404	−	0.908721i	\(-0.637060\pi\)
							−0.0928775	−	0.995678i	\(-0.529606\pi\)
\(41\)	−4.20028	−	2.42503i	−0.655974	−	0.378727i	0.134768	−	0.990877i	\(-0.456971\pi\)
							−0.790741	+	0.612151i	\(0.790305\pi\)
\(43\)	−0.495431	+	1.84897i	−0.0755525	+	0.281966i	−0.993358	−	0.115064i	\(-0.963293\pi\)
							0.917805	+	0.397030i	\(0.129959\pi\)
\(47\)	−2.76501	−	2.76501i	−0.403318	−	0.403318i	0.476083	−	0.879400i	\(-0.342056\pi\)
							−0.879400	+	0.476083i	\(0.842056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bv.a.23.17	✓	176
3.2	odd	2		945.2.by.a.233.28		176
5.2	odd	4	inner	315.2.bv.a.212.28	yes	176
7.4	even	3		315.2.bx.a.158.28	yes	176
9.2	odd	6		315.2.bx.a.128.28	yes	176
9.7	even	3		945.2.ca.a.548.17		176
15.2	even	4		945.2.by.a.422.17		176
21.11	odd	6		945.2.ca.a.368.17		176
35.32	odd	12		315.2.bx.a.32.28	yes	176
45.2	even	12		315.2.bx.a.2.28	yes	176
45.7	odd	12		945.2.ca.a.737.17		176
63.11	odd	6	inner	315.2.bv.a.263.28	yes	176
63.25	even	3		945.2.by.a.683.17		176
105.32	even	12		945.2.ca.a.557.17		176
315.137	even	12	inner	315.2.bv.a.137.17	yes	176
315.277	odd	12		945.2.by.a.872.28		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.17	✓	176	1.1	even	1	trivial
315.2.bv.a.137.17	yes	176	315.137	even	12	inner
315.2.bv.a.212.28	yes	176	5.2	odd	4	inner
315.2.bv.a.263.28	yes	176	63.11	odd	6	inner
315.2.bx.a.2.28	yes	176	45.2	even	12
315.2.bx.a.32.28	yes	176	35.32	odd	12
315.2.bx.a.128.28	yes	176	9.2	odd	6
315.2.bx.a.158.28	yes	176	7.4	even	3
945.2.by.a.233.28		176	3.2	odd	2
945.2.by.a.422.17		176	15.2	even	4
945.2.by.a.683.17		176	63.25	even	3
945.2.by.a.872.28		176	315.277	odd	12
945.2.ca.a.368.17		176	21.11	odd	6
945.2.ca.a.548.17		176	9.7	even	3
945.2.ca.a.557.17		176	105.32	even	12
945.2.ca.a.737.17		176	45.7	odd	12