Embedded newform 315.2.u.a.59.4

• Label: 315.2.u.a.59.4
• 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			59.4
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23016 + 2.13070i) q^{2} +(0.771307 - 1.55083i) q^{3} +(-2.02658 - 3.51014i) q^{4} +(-2.13577 + 0.662193i) q^{5} +(2.35553 + 3.55119i) q^{6} +(2.39641 + 1.12126i) q^{7} +5.05144 q^{8} +(-1.81017 - 2.39234i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23016	+	2.13070i	−0.869854	+	1.50663i	−0.00770834	+	0.999970i	\(0.502454\pi\)
							−0.862146	+	0.506661i	\(0.830880\pi\)
\(3\)	0.771307	−	1.55083i	0.445314	−	0.895374i
							\(5\)	−2.13577	+	0.662193i	−0.955144	+	0.296142i
\(7\)	2.39641	+	1.12126i	0.905757	+	0.423798i
							\(11\)		−	2.66331i		−	0.803019i	−0.915855	−	0.401510i	\(-0.868486\pi\)
0.915855	−	0.401510i	\(-0.131514\pi\)
\(13\)	1.60728	−	2.78389i	0.445779	−	0.772113i	−0.552327	−	0.833628i	\(-0.686260\pi\)
							0.998106	+	0.0615151i	\(0.0195932\pi\)
\(17\)	2.53088	+	1.46120i	0.613828	+	0.354394i	0.774462	−	0.632620i	\(-0.218021\pi\)
							−0.160634	+	0.987014i	\(0.551354\pi\)
\(19\)	5.06365	−	2.92350i	1.16168	−	0.670697i	0.209975	−	0.977707i	\(-0.432662\pi\)
							0.951706	+	0.307009i	\(0.0993283\pi\)
\(23\)	2.55295			0.532327			0.266164	−	0.963928i	\(-0.414244\pi\)
							0.266164	+	0.963928i	\(0.414244\pi\)
\(29\)	4.19625	−	2.42271i	0.779224	−	0.449885i	−0.0569312	−	0.998378i	\(-0.518132\pi\)
							0.836155	+	0.548493i	\(0.184798\pi\)
\(31\)	−6.12985	+	3.53907i	−1.10095	+	0.635636i	−0.936471	−	0.350744i	\(-0.885929\pi\)
							−0.164482	+	0.986380i	\(0.552595\pi\)
\(37\)	1.59927	−	0.923338i	0.262918	−	0.151796i	−0.362747	−	0.931888i	\(-0.618161\pi\)
							0.625665	+	0.780092i	\(0.284828\pi\)
\(41\)	5.20058	−	9.00767i	0.812194	−	1.40676i	−0.0991306	−	0.995074i	\(-0.531606\pi\)
							0.911325	−	0.411688i	\(-0.135061\pi\)
\(43\)	−2.40794	+	1.39022i	−0.367207	+	0.212007i	−0.672237	−	0.740336i	\(-0.734667\pi\)
							0.305031	+	0.952343i	\(0.401333\pi\)
\(47\)	−5.23424	−	3.02199i	−0.763492	−	0.440803i	0.0670559	−	0.997749i	\(-0.478639\pi\)
							−0.830548	+	0.556947i	\(0.811973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.4	✓	88
3.2	odd	2		945.2.u.a.584.41		88
5.4	even	2	inner	315.2.u.a.59.41	yes	88
7.5	odd	6		315.2.bq.a.194.41	yes	88
9.2	odd	6		315.2.bq.a.164.4	yes	88
9.7	even	3		945.2.bq.a.899.41		88
15.14	odd	2		945.2.u.a.584.4		88
21.5	even	6		945.2.bq.a.719.4		88
35.19	odd	6		315.2.bq.a.194.4	yes	88
45.29	odd	6		315.2.bq.a.164.41	yes	88
45.34	even	6		945.2.bq.a.899.4		88
63.47	even	6	inner	315.2.u.a.299.41	yes	88
63.61	odd	6		945.2.u.a.89.4		88
105.89	even	6		945.2.bq.a.719.41		88
315.124	odd	6		945.2.u.a.89.41		88
315.299	even	6	inner	315.2.u.a.299.4	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.4	✓	88	1.1	even	1	trivial
315.2.u.a.59.41	yes	88	5.4	even	2	inner
315.2.u.a.299.4	yes	88	315.299	even	6	inner
315.2.u.a.299.41	yes	88	63.47	even	6	inner
315.2.bq.a.164.4	yes	88	9.2	odd	6
315.2.bq.a.164.41	yes	88	45.29	odd	6
315.2.bq.a.194.4	yes	88	35.19	odd	6
315.2.bq.a.194.41	yes	88	7.5	odd	6
945.2.u.a.89.4		88	63.61	odd	6
945.2.u.a.89.41		88	315.124	odd	6
945.2.u.a.584.4		88	15.14	odd	2
945.2.u.a.584.41		88	3.2	odd	2
945.2.bq.a.719.4		88	21.5	even	6
945.2.bq.a.719.41		88	105.89	even	6
945.2.bq.a.899.4		88	45.34	even	6
945.2.bq.a.899.41		88	9.7	even	3