Embedded newform 315.2.bq.a.164.18

• Label: 315.2.bq.a.164.18
• Level: $315$
• Weight: $2$
• Character: 315.164
• 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.bq (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			164.18
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.18

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.586009 q^{2} +(-1.67705 - 0.433007i) q^{3} -1.65659 q^{4} +(-2.03395 - 0.928995i) q^{5} +(0.982768 + 0.253746i) q^{6} +(-1.03863 - 2.43336i) q^{7} +2.14280 q^{8} +(2.62501 + 1.45235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.586009			−0.414371			−0.207186	−	0.978302i	\(-0.566430\pi\)
							−0.207186	+	0.978302i	\(0.566430\pi\)
\(3\)	−1.67705	−	0.433007i	−0.968247	−	0.249996i
							\(5\)	−2.03395	−	0.928995i	−0.909612	−	0.415459i
\(7\)	−1.03863	−	2.43336i	−0.392566	−	0.919724i
							\(11\)	−2.03579	+	1.17536i	−0.613814	+	0.354386i	−0.774457	−	0.632627i	\(-0.781977\pi\)
0.160643	+	0.987013i	\(0.448643\pi\)
\(13\)	2.24605	+	3.89027i	0.622942	+	1.07897i	0.988935	+	0.148349i	\(0.0473960\pi\)
							−0.365993	+	0.930618i	\(0.619271\pi\)
\(17\)	−0.0921548	−	0.0532056i	−0.0223508	−	0.0129043i	0.488783	−	0.872405i	\(-0.337441\pi\)
							−0.511134	+	0.859501i	\(0.670774\pi\)
\(19\)	1.23839	−	0.714984i	0.284106	−	0.164029i	−0.351175	−	0.936310i	\(-0.614218\pi\)
							0.635281	+	0.772281i	\(0.280884\pi\)
\(23\)	−3.91755	+	6.78539i	−0.816865	+	1.41485i	0.0911157	+	0.995840i	\(0.470957\pi\)
							−0.907981	+	0.419012i	\(0.862377\pi\)
\(29\)	3.63409	+	2.09814i	0.674834	+	0.389616i	0.797906	−	0.602782i	\(-0.205941\pi\)
							−0.123072	+	0.992398i	\(0.539275\pi\)
\(31\)		−	5.26645i		−	0.945882i	−0.881094	−	0.472941i	\(-0.843192\pi\)
							0.881094	−	0.472941i	\(-0.156808\pi\)
\(37\)	6.99610	−	4.03920i	1.15015	−	0.664040i	0.201227	−	0.979545i	\(-0.435507\pi\)
							0.948924	+	0.315504i	\(0.102174\pi\)
\(41\)	−3.30164	−	5.71861i	−0.515630	−	0.893097i	−0.999835	−	0.0181429i	\(-0.994225\pi\)
							0.484205	−	0.874954i	\(-0.339109\pi\)
\(43\)	9.00855	+	5.20109i	1.37379	+	0.793158i	0.991403	−	0.130844i	\(-0.0417686\pi\)
							0.382388	+	0.924002i	\(0.375102\pi\)
\(47\)			3.40836i			0.497161i	0.968611	+	0.248580i	\(0.0799640\pi\)
							−0.968611	+	0.248580i	\(0.920036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.18	yes	88
3.2	odd	2		945.2.bq.a.899.27		88
5.4	even	2	inner	315.2.bq.a.164.27	yes	88
7.5	odd	6		315.2.u.a.299.27	yes	88
9.4	even	3		945.2.u.a.584.27		88
9.5	odd	6		315.2.u.a.59.18	✓	88
15.14	odd	2		945.2.bq.a.899.18		88
21.5	even	6		945.2.u.a.89.18		88
35.19	odd	6		315.2.u.a.299.18	yes	88
45.4	even	6		945.2.u.a.584.18		88
45.14	odd	6		315.2.u.a.59.27	yes	88
63.5	even	6	inner	315.2.bq.a.194.27	yes	88
63.40	odd	6		945.2.bq.a.719.18		88
105.89	even	6		945.2.u.a.89.27		88
315.194	even	6	inner	315.2.bq.a.194.18	yes	88
315.229	odd	6		945.2.bq.a.719.27		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.18	✓	88	9.5	odd	6
315.2.u.a.59.27	yes	88	45.14	odd	6
315.2.u.a.299.18	yes	88	35.19	odd	6
315.2.u.a.299.27	yes	88	7.5	odd	6
315.2.bq.a.164.18	yes	88	1.1	even	1	trivial
315.2.bq.a.164.27	yes	88	5.4	even	2	inner
315.2.bq.a.194.18	yes	88	315.194	even	6	inner
315.2.bq.a.194.27	yes	88	63.5	even	6	inner
945.2.u.a.89.18		88	21.5	even	6
945.2.u.a.89.27		88	105.89	even	6
945.2.u.a.584.18		88	45.4	even	6
945.2.u.a.584.27		88	9.4	even	3
945.2.bq.a.719.18		88	63.40	odd	6
945.2.bq.a.719.27		88	315.229	odd	6
945.2.bq.a.899.18		88	15.14	odd	2
945.2.bq.a.899.27		88	3.2	odd	2