Embedded newform 315.2.u.a.59.9

• Label: 315.2.u.a.59.9
• Level: $315$
• Weight: $2$
• Character: 315.59
• 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.u (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			59.9
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00105 + 1.73387i) q^{2} +(0.113913 + 1.72830i) q^{3} +(-1.00421 - 1.73934i) q^{4} +(0.439789 + 2.19239i) q^{5} +(-3.11068 - 1.53261i) q^{6} +(2.40459 - 1.10361i) q^{7} +0.0168431 q^{8} +(-2.97405 + 0.393751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00105	+	1.73387i	−0.707850	+	1.22603i	0.257803	+	0.966197i	\(0.417001\pi\)
							−0.965653	+	0.259835i	\(0.916332\pi\)
\(3\)	0.113913	+	1.72830i	0.0657676	+	0.997835i
							\(5\)	0.439789	+	2.19239i	0.196680	+	0.980468i
\(7\)	2.40459	−	1.10361i	0.908850	−	0.417124i
							\(11\)			3.57984i			1.07936i	0.841870	+	0.539681i	\(0.181455\pi\)
−0.841870	+	0.539681i	\(0.818545\pi\)
\(13\)	−1.79302	+	3.10560i	−0.497293	+	0.861337i	−0.999995	−	0.00312250i	\(-0.999006\pi\)
							0.502702	+	0.864460i	\(0.332339\pi\)
\(17\)	−4.38458	−	2.53144i	−1.06342	−	0.613964i	−0.137042	−	0.990565i	\(-0.543759\pi\)
							−0.926376	+	0.376601i	\(0.877093\pi\)
\(19\)	4.31887	−	2.49350i	0.990817	−	0.572048i	0.0852986	−	0.996355i	\(-0.472816\pi\)
							0.905518	+	0.424307i	\(0.139482\pi\)
\(23\)	5.14597			1.07301			0.536505	−	0.843897i	\(-0.319744\pi\)
							0.536505	+	0.843897i	\(0.319744\pi\)
\(29\)	5.40087	−	3.11820i	1.00292	−	0.579034i	0.0938069	−	0.995590i	\(-0.470096\pi\)
							0.909110	+	0.416556i	\(0.136763\pi\)
\(31\)	−3.45097	+	1.99242i	−0.619813	+	0.357849i	−0.776796	−	0.629752i	\(-0.783156\pi\)
							0.156983	+	0.987601i	\(0.449823\pi\)
\(37\)	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	\(-0.908444\pi\)
							0.725136	+	0.688606i	\(0.241777\pi\)
\(43\)	−6.71768	+	3.87846i	−1.02444	+	0.591459i	−0.915386	−	0.402577i	\(-0.868114\pi\)
							−0.109051	+	0.994036i	\(0.534781\pi\)
\(47\)	−1.06848	−	0.616887i	−0.155854	−	0.0899822i	0.420045	−	0.907503i	\(-0.362014\pi\)
							−0.575899	+	0.817521i	\(0.695348\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.9	✓	88
3.2	odd	2		945.2.u.a.584.36		88
5.4	even	2	inner	315.2.u.a.59.36	yes	88
7.5	odd	6		315.2.bq.a.194.36	yes	88
9.2	odd	6		315.2.bq.a.164.9	yes	88
9.7	even	3		945.2.bq.a.899.36		88
15.14	odd	2		945.2.u.a.584.9		88
21.5	even	6		945.2.bq.a.719.9		88
35.19	odd	6		315.2.bq.a.194.9	yes	88
45.29	odd	6		315.2.bq.a.164.36	yes	88
45.34	even	6		945.2.bq.a.899.9		88
63.47	even	6	inner	315.2.u.a.299.36	yes	88
63.61	odd	6		945.2.u.a.89.9		88
105.89	even	6		945.2.bq.a.719.36		88
315.124	odd	6		945.2.u.a.89.36		88
315.299	even	6	inner	315.2.u.a.299.9	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.9	✓	88	1.1	even	1	trivial
315.2.u.a.59.36	yes	88	5.4	even	2	inner
315.2.u.a.299.9	yes	88	315.299	even	6	inner
315.2.u.a.299.36	yes	88	63.47	even	6	inner
315.2.bq.a.164.9	yes	88	9.2	odd	6
315.2.bq.a.164.36	yes	88	45.29	odd	6
315.2.bq.a.194.9	yes	88	35.19	odd	6
315.2.bq.a.194.36	yes	88	7.5	odd	6
945.2.u.a.89.9		88	63.61	odd	6
945.2.u.a.89.36		88	315.124	odd	6
945.2.u.a.584.9		88	15.14	odd	2
945.2.u.a.584.36		88	3.2	odd	2
945.2.bq.a.719.9		88	21.5	even	6
945.2.bq.a.719.36		88	105.89	even	6
945.2.bq.a.899.9		88	45.34	even	6
945.2.bq.a.899.36		88	9.7	even	3