Embedded newform 315.2.u.a.59.13

• Label: 315.2.u.a.59.13
• 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			59.13
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.665500 + 1.15268i) q^{2} +(-1.68054 - 0.419267i) q^{3} +(0.114220 + 0.197834i) q^{4} +(-1.38997 + 1.75157i) q^{5} +(1.60168 - 1.65810i) q^{6} +(1.25852 + 2.32726i) q^{7} -2.96605 q^{8} +(2.64843 + 1.40919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.665500	+	1.15268i	−0.470580	+	0.815068i	−0.999434	−	0.0336450i	\(-0.989288\pi\)
							0.528854	+	0.848713i	\(0.322622\pi\)
\(3\)	−1.68054	−	0.419267i	−0.970260	−	0.242064i
							\(5\)	−1.38997	+	1.75157i	−0.621612	+	0.783325i
\(7\)	1.25852	+	2.32726i	0.475676	+	0.879620i
							\(11\)		−	1.52676i		−	0.460335i	−0.973151	−	0.230167i	\(-0.926073\pi\)
0.973151	−	0.230167i	\(-0.0739274\pi\)
\(13\)	0.272143	−	0.471366i	0.0754789	−	0.130733i	−0.825816	−	0.563940i	\(-0.809285\pi\)
							0.901294	+	0.433207i	\(0.142618\pi\)
\(17\)	−4.41967	−	2.55170i	−1.07193	−	0.618877i	−0.143219	−	0.989691i	\(-0.545745\pi\)
							−0.928707	+	0.370814i	\(0.879079\pi\)
\(19\)	−5.57286	+	3.21749i	−1.27850	+	0.738143i	−0.976573	−	0.215188i	\(-0.930964\pi\)
							−0.301928	+	0.953331i	\(0.597630\pi\)
\(23\)	−1.59765			−0.333133			−0.166567	−	0.986030i	\(-0.553268\pi\)
							−0.166567	+	0.986030i	\(0.553268\pi\)
\(29\)	0.841685	−	0.485947i	0.156297	−	0.0902381i	−0.419812	−	0.907611i	\(-0.637904\pi\)
							0.576109	+	0.817373i	\(0.304571\pi\)
\(31\)	−3.25769	+	1.88083i	−0.585098	+	0.337807i	−0.763157	−	0.646213i	\(-0.776352\pi\)
							0.178059	+	0.984020i	\(0.443018\pi\)
\(37\)	6.59583	−	3.80810i	1.08435	−	0.626048i	0.152282	−	0.988337i	\(-0.451338\pi\)
							0.932066	+	0.362289i	\(0.118005\pi\)
\(41\)	−1.77448	+	3.07349i	−0.277128	+	0.479999i	−0.970670	−	0.240417i	\(-0.922716\pi\)
							0.693542	+	0.720416i	\(0.256049\pi\)
\(43\)	1.37551	−	0.794149i	0.209763	−	0.121107i	−0.391438	−	0.920204i	\(-0.628022\pi\)
							0.601201	+	0.799098i	\(0.294689\pi\)
\(47\)	11.0266	+	6.36621i	1.60839	+	0.928607i	0.989730	+	0.142953i	\(0.0456597\pi\)
							0.618665	+	0.785655i	\(0.287674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.13	✓	88
3.2	odd	2		945.2.u.a.584.32		88
5.4	even	2	inner	315.2.u.a.59.32	yes	88
7.5	odd	6		315.2.bq.a.194.32	yes	88
9.2	odd	6		315.2.bq.a.164.13	yes	88
9.7	even	3		945.2.bq.a.899.32		88
15.14	odd	2		945.2.u.a.584.13		88
21.5	even	6		945.2.bq.a.719.13		88
35.19	odd	6		315.2.bq.a.194.13	yes	88
45.29	odd	6		315.2.bq.a.164.32	yes	88
45.34	even	6		945.2.bq.a.899.13		88
63.47	even	6	inner	315.2.u.a.299.32	yes	88
63.61	odd	6		945.2.u.a.89.13		88
105.89	even	6		945.2.bq.a.719.32		88
315.124	odd	6		945.2.u.a.89.32		88
315.299	even	6	inner	315.2.u.a.299.13	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.13	✓	88	1.1	even	1	trivial
315.2.u.a.59.32	yes	88	5.4	even	2	inner
315.2.u.a.299.13	yes	88	315.299	even	6	inner
315.2.u.a.299.32	yes	88	63.47	even	6	inner
315.2.bq.a.164.13	yes	88	9.2	odd	6
315.2.bq.a.164.32	yes	88	45.29	odd	6
315.2.bq.a.194.13	yes	88	35.19	odd	6
315.2.bq.a.194.32	yes	88	7.5	odd	6
945.2.u.a.89.13		88	63.61	odd	6
945.2.u.a.89.32		88	315.124	odd	6
945.2.u.a.584.13		88	15.14	odd	2
945.2.u.a.584.32		88	3.2	odd	2
945.2.bq.a.719.13		88	21.5	even	6
945.2.bq.a.719.32		88	105.89	even	6
945.2.bq.a.899.13		88	45.34	even	6
945.2.bq.a.899.32		88	9.7	even	3