Embedded newform 315.2.bv.a.23.5

• Label: 315.2.bv.a.23.5
• 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			23.5
Character	\(\chi\)	\(=\)	315.23
Dual form			315.2.bv.a.137.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69531 - 1.69531i) q^{2} +(1.72524 - 0.153414i) q^{3} +3.74812i q^{4} +(1.48276 - 1.67374i) q^{5} +(-3.18490 - 2.66473i) q^{6} +(1.46770 + 2.20133i) q^{7} +(2.96360 - 2.96360i) q^{8} +(2.95293 - 0.529352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69531	−	1.69531i	−1.19876	−	1.19876i	−0.974538	−	0.224224i	\(-0.928015\pi\)
							−0.224224	−	0.974538i	\(-0.571985\pi\)
\(3\)	1.72524	−	0.153414i	0.996070	−	0.0885735i
							\(5\)	1.48276	−	1.67374i	0.663111	−	0.748521i
\(7\)	1.46770	+	2.20133i	0.554738	+	0.832025i
							\(11\)	3.23691	+	1.86883i	0.975964	+	0.563473i	0.901049	−	0.433717i	\(-0.142798\pi\)
0.0749150	+	0.997190i	\(0.476131\pi\)
\(13\)	−3.47861	+	0.932090i	−0.964792	+	0.258515i	−0.706628	−	0.707586i	\(-0.749784\pi\)
							−0.258165	+	0.966101i	\(0.583118\pi\)
\(17\)	0.159742	−	0.596165i	0.0387431	−	0.144591i	−0.943845	−	0.330389i	\(-0.892820\pi\)
							0.982588	+	0.185797i	\(0.0594868\pi\)
\(19\)	4.50566	+	2.60134i	1.03367	+	0.596789i	0.918033	−	0.396503i	\(-0.129776\pi\)
							0.115635	+	0.993292i	\(0.463110\pi\)
\(23\)	−2.45412	−	0.657579i	−0.511719	−	0.137115i	−0.00628535	−	0.999980i	\(-0.502001\pi\)
							−0.505433	+	0.862866i	\(0.668667\pi\)
\(29\)	−0.111818	−	0.193674i	−0.0207640	−	0.0359644i	0.855457	−	0.517874i	\(-0.173277\pi\)
							−0.876221	+	0.481910i	\(0.839943\pi\)
\(31\)	−8.97572			−1.61209			−0.806043	−	0.591857i	\(-0.798395\pi\)
							−0.806043	+	0.591857i	\(0.798395\pi\)
\(37\)	−1.36755	−	5.10378i	−0.224824	−	0.839056i	−0.982475	−	0.186395i	\(-0.940319\pi\)
							0.757650	−	0.652661i	\(-0.226347\pi\)
\(41\)	−9.82481	−	5.67235i	−1.53438	−	0.885873i	−0.999152	−	0.0411629i	\(-0.986894\pi\)
							−0.535224	−	0.844710i	\(-0.679773\pi\)
\(43\)	−0.153652	+	0.573436i	−0.0234316	+	0.0874481i	−0.976652	−	0.214830i	\(-0.931080\pi\)
							0.953220	+	0.302278i	\(0.0977470\pi\)
\(47\)	−3.98940	−	3.98940i	−0.581914	−	0.581914i	0.353515	−	0.935429i	\(-0.384986\pi\)
							−0.935429	+	0.353515i	\(0.884986\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bv.a.23.5	✓	176
3.2	odd	2		945.2.by.a.233.40		176
5.2	odd	4	inner	315.2.bv.a.212.40	yes	176
7.4	even	3		315.2.bx.a.158.40	yes	176
9.2	odd	6		315.2.bx.a.128.40	yes	176
9.7	even	3		945.2.ca.a.548.5		176
15.2	even	4		945.2.by.a.422.5		176
21.11	odd	6		945.2.ca.a.368.5		176
35.32	odd	12		315.2.bx.a.32.40	yes	176
45.2	even	12		315.2.bx.a.2.40	yes	176
45.7	odd	12		945.2.ca.a.737.5		176
63.11	odd	6	inner	315.2.bv.a.263.40	yes	176
63.25	even	3		945.2.by.a.683.5		176
105.32	even	12		945.2.ca.a.557.5		176
315.137	even	12	inner	315.2.bv.a.137.5	yes	176
315.277	odd	12		945.2.by.a.872.40		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bv.a.23.5	✓	176	1.1	even	1	trivial
315.2.bv.a.137.5	yes	176	315.137	even	12	inner
315.2.bv.a.212.40	yes	176	5.2	odd	4	inner
315.2.bv.a.263.40	yes	176	63.11	odd	6	inner
315.2.bx.a.2.40	yes	176	45.2	even	12
315.2.bx.a.32.40	yes	176	35.32	odd	12
315.2.bx.a.128.40	yes	176	9.2	odd	6
315.2.bx.a.158.40	yes	176	7.4	even	3
945.2.by.a.233.40		176	3.2	odd	2
945.2.by.a.422.5		176	15.2	even	4
945.2.by.a.683.5		176	63.25	even	3
945.2.by.a.872.40		176	315.277	odd	12
945.2.ca.a.368.5		176	21.11	odd	6
945.2.ca.a.548.5		176	9.7	even	3
945.2.ca.a.557.5		176	105.32	even	12
945.2.ca.a.737.5		176	45.7	odd	12