Embedded newform 315.2.bq.a.164.9

• Label: 315.2.bq.a.164.9
• Level: $315$
• Weight: $2$
• Character: 315.164
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bq (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			164.9
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.00210 q^{2} +(-1.55371 + 0.765499i) q^{3} +2.00841 q^{4} +(2.11856 + 0.715328i) q^{5} +(3.11068 - 1.53261i) q^{6} +(-2.15805 - 1.53063i) q^{7} -0.0168431 q^{8} +(1.82802 - 2.37873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00210			−1.41570			−0.707850	−	0.706363i	\(-0.750335\pi\)
							−0.707850	+	0.706363i	\(0.750335\pi\)
\(3\)	−1.55371	+	0.765499i	−0.897034	+	0.441961i
							\(5\)	2.11856	+	0.715328i	0.947450	+	0.319904i
\(7\)	−2.15805	−	1.53063i	−0.815665	−	0.578525i
							\(11\)	−3.10023	+	1.78992i	−0.934755	+	0.539681i	−0.888312	−	0.459240i	\(-0.848122\pi\)
−0.0464428	+	0.998921i	\(0.514789\pi\)
\(13\)	−1.79302	−	3.10560i	−0.497293	−	0.861337i	0.502702	−	0.864460i	\(-0.332339\pi\)
							−0.999995	+	0.00312250i	\(0.999006\pi\)
\(17\)	4.38458	+	2.53144i	1.06342	+	0.613964i	0.926376	−	0.376601i	\(-0.122907\pi\)
							0.137042	+	0.990565i	\(0.456241\pi\)
\(19\)	4.31887	−	2.49350i	0.990817	−	0.572048i	0.0852986	−	0.996355i	\(-0.472816\pi\)
							0.905518	+	0.424307i	\(0.139482\pi\)
\(23\)	2.57299	−	4.45654i	0.536505	−	0.929253i	−0.462584	−	0.886575i	\(-0.653078\pi\)
							0.999089	−	0.0426780i	\(-0.0135890\pi\)
\(29\)	5.40087	+	3.11820i	1.00292	+	0.579034i	0.909110	−	0.416556i	\(-0.136763\pi\)
							0.0938069	+	0.995590i	\(0.470096\pi\)
\(31\)		−	3.98484i		−	0.715698i	−0.933779	−	0.357849i	\(-0.883510\pi\)
							0.933779	−	0.357849i	\(-0.116490\pi\)
\(37\)	5.53411	−	3.19512i	0.909803	−	0.525275i	0.0294350	−	0.999567i	\(-0.490629\pi\)
							0.880368	+	0.474292i	\(0.157296\pi\)
\(41\)	1.49694	+	2.59277i	0.233782	+	0.404922i	0.958918	−	0.283683i	\(-0.0915565\pi\)
							−0.725136	+	0.688606i	\(0.758223\pi\)
\(43\)	6.71768	+	3.87846i	1.02444	+	0.591459i	0.915386	−	0.402577i	\(-0.131886\pi\)
							0.109051	+	0.994036i	\(0.465219\pi\)
\(47\)		−	1.23377i		−	0.179964i	−0.995943	−	0.0899822i	\(-0.971319\pi\)
							0.995943	−	0.0899822i	\(-0.0286810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.9	yes	88
3.2	odd	2		945.2.bq.a.899.36		88
5.4	even	2	inner	315.2.bq.a.164.36	yes	88
7.5	odd	6		315.2.u.a.299.36	yes	88
9.4	even	3		945.2.u.a.584.36		88
9.5	odd	6		315.2.u.a.59.9	✓	88
15.14	odd	2		945.2.bq.a.899.9		88
21.5	even	6		945.2.u.a.89.9		88
35.19	odd	6		315.2.u.a.299.9	yes	88
45.4	even	6		945.2.u.a.584.9		88
45.14	odd	6		315.2.u.a.59.36	yes	88
63.5	even	6	inner	315.2.bq.a.194.36	yes	88
63.40	odd	6		945.2.bq.a.719.9		88
105.89	even	6		945.2.u.a.89.36		88
315.194	even	6	inner	315.2.bq.a.194.9	yes	88
315.229	odd	6		945.2.bq.a.719.36		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.9	✓	88	9.5	odd	6
315.2.u.a.59.36	yes	88	45.14	odd	6
315.2.u.a.299.9	yes	88	35.19	odd	6
315.2.u.a.299.36	yes	88	7.5	odd	6
315.2.bq.a.164.9	yes	88	1.1	even	1	trivial
315.2.bq.a.164.36	yes	88	5.4	even	2	inner
315.2.bq.a.194.9	yes	88	315.194	even	6	inner
315.2.bq.a.194.36	yes	88	63.5	even	6	inner
945.2.u.a.89.9		88	21.5	even	6
945.2.u.a.89.36		88	105.89	even	6
945.2.u.a.584.9		88	45.4	even	6
945.2.u.a.584.36		88	9.4	even	3
945.2.bq.a.719.9		88	63.40	odd	6
945.2.bq.a.719.36		88	315.229	odd	6
945.2.bq.a.899.9		88	15.14	odd	2
945.2.bq.a.899.36		88	3.2	odd	2