Embedded newform 315.2.u.a.59.8

• Label: 315.2.u.a.59.8
• 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			59.8
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02260 + 1.77120i) q^{2} +(1.55827 - 0.756177i) q^{3} +(-1.09143 - 1.89041i) q^{4} +(-0.00270873 - 2.23607i) q^{5} +(-0.254147 + 3.53327i) q^{6} +(-1.69883 - 2.02829i) q^{7} +0.373988 q^{8} +(1.85639 - 2.35665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02260	+	1.77120i	−0.723089	+	1.25243i	0.236667	+	0.971591i	\(0.423945\pi\)
							−0.959756	+	0.280836i	\(0.909388\pi\)
\(3\)	1.55827	−	0.756177i	0.899666	−	0.436579i
							\(5\)	−0.00270873	−	2.23607i	−0.00121138	−	0.999999i
\(7\)	−1.69883	−	2.02829i	−0.642097	−	0.766623i
							\(11\)			2.19336i			0.661324i	0.943749	+	0.330662i	\(0.107272\pi\)
−0.943749	+	0.330662i	\(0.892728\pi\)
\(13\)	1.60580	−	2.78133i	0.445369	−	0.771402i	−0.552709	−	0.833375i	\(-0.686406\pi\)
							0.998078	+	0.0619725i	\(0.0197391\pi\)
\(17\)	−0.719457	−	0.415379i	−0.174494	−	0.100744i	0.410209	−	0.911991i	\(-0.365456\pi\)
							−0.584703	+	0.811247i	\(0.698789\pi\)
\(19\)	4.98256	−	2.87668i	1.14308	−	0.659956i	0.195886	−	0.980627i	\(-0.437242\pi\)
							0.947191	+	0.320671i	\(0.103908\pi\)
\(23\)	1.14504			0.238757			0.119378	−	0.992849i	\(-0.461910\pi\)
							0.119378	+	0.992849i	\(0.461910\pi\)
\(29\)	−0.421739	+	0.243491i	−0.0783150	+	0.0452152i	−0.538646	−	0.842532i	\(-0.681064\pi\)
							0.460331	+	0.887747i	\(0.347731\pi\)
\(31\)	3.48175	−	2.01019i	0.625341	−	0.361041i	−0.153605	−	0.988132i	\(-0.549088\pi\)
							0.778945	+	0.627092i	\(0.215755\pi\)
\(37\)	−1.45808	+	0.841821i	−0.239706	+	0.138395i	−0.615042	−	0.788494i	\(-0.710861\pi\)
							0.375335	+	0.926889i	\(0.377528\pi\)
\(41\)	−5.85132	+	10.1348i	−0.913823	+	1.58279i	−0.105207	+	0.994450i	\(0.533550\pi\)
							−0.808616	+	0.588337i	\(0.799783\pi\)
\(43\)	−4.22719	+	2.44057i	−0.644640	+	0.372183i	−0.786400	−	0.617718i	\(-0.788057\pi\)
							0.141760	+	0.989901i	\(0.454724\pi\)
\(47\)	9.50350	+	5.48685i	1.38623	+	0.800339i	0.992888	−	0.119054i	\(-0.0379862\pi\)
							0.393340	+	0.919393i	\(0.371320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.8	✓	88
3.2	odd	2		945.2.u.a.584.37		88
5.4	even	2	inner	315.2.u.a.59.37	yes	88
7.5	odd	6		315.2.bq.a.194.37	yes	88
9.2	odd	6		315.2.bq.a.164.8	yes	88
9.7	even	3		945.2.bq.a.899.37		88
15.14	odd	2		945.2.u.a.584.8		88
21.5	even	6		945.2.bq.a.719.8		88
35.19	odd	6		315.2.bq.a.194.8	yes	88
45.29	odd	6		315.2.bq.a.164.37	yes	88
45.34	even	6		945.2.bq.a.899.8		88
63.47	even	6	inner	315.2.u.a.299.37	yes	88
63.61	odd	6		945.2.u.a.89.8		88
105.89	even	6		945.2.bq.a.719.37		88
315.124	odd	6		945.2.u.a.89.37		88
315.299	even	6	inner	315.2.u.a.299.8	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.8	✓	88	1.1	even	1	trivial
315.2.u.a.59.37	yes	88	5.4	even	2	inner
315.2.u.a.299.8	yes	88	315.299	even	6	inner
315.2.u.a.299.37	yes	88	63.47	even	6	inner
315.2.bq.a.164.8	yes	88	9.2	odd	6
315.2.bq.a.164.37	yes	88	45.29	odd	6
315.2.bq.a.194.8	yes	88	35.19	odd	6
315.2.bq.a.194.37	yes	88	7.5	odd	6
945.2.u.a.89.8		88	63.61	odd	6
945.2.u.a.89.37		88	315.124	odd	6
945.2.u.a.584.8		88	15.14	odd	2
945.2.u.a.584.37		88	3.2	odd	2
945.2.bq.a.719.8		88	21.5	even	6
945.2.bq.a.719.37		88	105.89	even	6
945.2.bq.a.899.8		88	45.34	even	6
945.2.bq.a.899.37		88	9.7	even	3