Embedded newform 315.2.u.a.59.7

• Label: 315.2.u.a.59.7
• 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			59.7
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03259 + 1.78850i) q^{2} +(-1.10896 - 1.33049i) q^{3} +(-1.13248 - 1.96151i) q^{4} +(1.44626 + 1.70539i) q^{5} +(3.52468 - 0.609527i) q^{6} +(-0.162008 - 2.64079i) q^{7} +0.547180 q^{8} +(-0.540404 + 2.95093i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03259	+	1.78850i	−0.730150	+	1.26466i	0.226668	+	0.973972i	\(0.427217\pi\)
							−0.956819	+	0.290685i	\(0.906117\pi\)
\(3\)	−1.10896	−	1.33049i	−0.640260	−	0.768158i
							\(5\)	1.44626	+	1.70539i	0.646786	+	0.762672i
\(7\)	−0.162008	−	2.64079i	−0.0612334	−	0.998123i
							\(11\)		−	3.41166i		−	1.02865i	−0.857595	−	0.514326i	\(-0.828042\pi\)
0.857595	−	0.514326i	\(-0.171958\pi\)
\(13\)	2.21501	−	3.83651i	0.614334	−	1.06406i	−0.376167	−	0.926552i	\(-0.622758\pi\)
							0.990501	−	0.137505i	\(-0.0439085\pi\)
\(17\)	4.95478	+	2.86064i	1.20171	+	0.693808i	0.960935	−	0.276773i	\(-0.0892651\pi\)
							0.240775	+	0.970581i	\(0.422598\pi\)
\(19\)	−1.64398	+	0.949154i	−0.377156	+	0.217751i	−0.676580	−	0.736369i	\(-0.736539\pi\)
							0.299424	+	0.954120i	\(0.403205\pi\)
\(23\)	3.00745			0.627096			0.313548	−	0.949572i	\(-0.398482\pi\)
							0.313548	+	0.949572i	\(0.398482\pi\)
\(29\)	4.89207	−	2.82444i	0.908434	−	0.524485i	0.0285071	−	0.999594i	\(-0.490925\pi\)
							0.879927	+	0.475109i	\(0.157591\pi\)
\(31\)	3.08100	−	1.77882i	0.553365	−	0.319485i	−0.197113	−	0.980381i	\(-0.563157\pi\)
							0.750478	+	0.660895i	\(0.229823\pi\)
\(37\)	2.19991	−	1.27012i	0.361663	−	0.208806i	−0.308147	−	0.951339i	\(-0.599709\pi\)
							0.669810	+	0.742533i	\(0.266376\pi\)
\(41\)	−2.85028	+	4.93682i	−0.445138	+	0.771002i	−0.998062	−	0.0622299i	\(-0.980179\pi\)
							0.552924	+	0.833232i	\(0.313512\pi\)
\(43\)	2.49748	−	1.44192i	0.380862	−	0.219891i	−0.297331	−	0.954774i	\(-0.596097\pi\)
							0.678193	+	0.734884i	\(0.262763\pi\)
\(47\)	−7.69425	−	4.44228i	−1.12232	−	0.647973i	−0.180329	−	0.983606i	\(-0.557716\pi\)
							−0.941993	+	0.335634i	\(0.891050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.7	✓	88
3.2	odd	2		945.2.u.a.584.38		88
5.4	even	2	inner	315.2.u.a.59.38	yes	88
7.5	odd	6		315.2.bq.a.194.38	yes	88
9.2	odd	6		315.2.bq.a.164.7	yes	88
9.7	even	3		945.2.bq.a.899.38		88
15.14	odd	2		945.2.u.a.584.7		88
21.5	even	6		945.2.bq.a.719.7		88
35.19	odd	6		315.2.bq.a.194.7	yes	88
45.29	odd	6		315.2.bq.a.164.38	yes	88
45.34	even	6		945.2.bq.a.899.7		88
63.47	even	6	inner	315.2.u.a.299.38	yes	88
63.61	odd	6		945.2.u.a.89.7		88
105.89	even	6		945.2.bq.a.719.38		88
315.124	odd	6		945.2.u.a.89.38		88
315.299	even	6	inner	315.2.u.a.299.7	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.7	✓	88	1.1	even	1	trivial
315.2.u.a.59.38	yes	88	5.4	even	2	inner
315.2.u.a.299.7	yes	88	315.299	even	6	inner
315.2.u.a.299.38	yes	88	63.47	even	6	inner
315.2.bq.a.164.7	yes	88	9.2	odd	6
315.2.bq.a.164.38	yes	88	45.29	odd	6
315.2.bq.a.194.7	yes	88	35.19	odd	6
315.2.bq.a.194.38	yes	88	7.5	odd	6
945.2.u.a.89.7		88	63.61	odd	6
945.2.u.a.89.38		88	315.124	odd	6
945.2.u.a.584.7		88	15.14	odd	2
945.2.u.a.584.38		88	3.2	odd	2
945.2.bq.a.719.7		88	21.5	even	6
945.2.bq.a.719.38		88	105.89	even	6
945.2.bq.a.899.7		88	45.34	even	6
945.2.bq.a.899.38		88	9.7	even	3