Embedded newform 315.2.bq.a.164.14

• Label: 315.2.bq.a.164.14
• 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			164.14
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.14

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.20693 q^{2} +(-0.344453 + 1.69745i) q^{3} -0.543308 q^{4} +(-2.01662 - 0.966042i) q^{5} +(0.415733 - 2.04872i) q^{6} +(-2.56388 + 0.653073i) q^{7} +3.06961 q^{8} +(-2.76270 - 1.16939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20693			−0.853432			−0.426716	−	0.904386i	\(-0.640330\pi\)
							−0.426716	+	0.904386i	\(0.640330\pi\)
\(3\)	−0.344453	+	1.69745i	−0.198870	+	0.980026i
							\(5\)	−2.01662	−	0.966042i	−0.901861	−	0.432027i
\(7\)	−2.56388	+	0.653073i	−0.969057	+	0.246838i
							\(11\)	4.47697	−	2.58478i	1.34986	−	0.779340i	0.361628	−	0.932323i	\(-0.382221\pi\)
0.988229	+	0.152983i	\(0.0488878\pi\)
\(13\)	−0.450567	−	0.780405i	−0.124965	−	0.216445i	0.796754	−	0.604303i	\(-0.206548\pi\)
							−0.921719	+	0.387858i	\(0.873215\pi\)
\(17\)	3.35120	+	1.93481i	0.812785	+	0.469261i	0.847922	−	0.530121i	\(-0.177854\pi\)
							−0.0351373	+	0.999382i	\(0.511187\pi\)
\(19\)	−0.999018	+	0.576783i	−0.229190	+	0.132323i	−0.610199	−	0.792249i	\(-0.708910\pi\)
							0.381008	+	0.924572i	\(0.375577\pi\)
\(23\)	2.81178	−	4.87014i	0.586296	−	1.01549i	−0.408417	−	0.912796i	\(-0.633919\pi\)
							0.994713	−	0.102699i	\(-0.0327477\pi\)
\(29\)	−0.212781	−	0.122849i	−0.0395125	−	0.0228126i	0.480114	−	0.877206i	\(-0.340595\pi\)
							−0.519626	+	0.854394i	\(0.673929\pi\)
\(31\)		−	3.32870i		−	0.597853i	−0.954276	−	0.298926i	\(-0.903371\pi\)
							0.954276	−	0.298926i	\(-0.0966285\pi\)
\(37\)	5.55371	−	3.20644i	0.913025	−	0.527135i	0.0316215	−	0.999500i	\(-0.489933\pi\)
							0.881403	+	0.472365i	\(0.156600\pi\)
\(41\)	−0.354801	−	0.614534i	−0.0554106	−	0.0959740i	0.836990	−	0.547219i	\(-0.184313\pi\)
							−0.892400	+	0.451245i	\(0.850980\pi\)
\(43\)	−8.57053	−	4.94820i	−1.30699	−	0.754593i	−0.325401	−	0.945576i	\(-0.605499\pi\)
							−0.981593	+	0.190983i	\(0.938832\pi\)
\(47\)		−	8.54044i		−	1.24575i	−0.782321	−	0.622876i	\(-0.785964\pi\)
							0.782321	−	0.622876i	\(-0.214036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.14	yes	88
3.2	odd	2		945.2.bq.a.899.31		88
5.4	even	2	inner	315.2.bq.a.164.31	yes	88
7.5	odd	6		315.2.u.a.299.31	yes	88
9.4	even	3		945.2.u.a.584.31		88
9.5	odd	6		315.2.u.a.59.14	✓	88
15.14	odd	2		945.2.bq.a.899.14		88
21.5	even	6		945.2.u.a.89.14		88
35.19	odd	6		315.2.u.a.299.14	yes	88
45.4	even	6		945.2.u.a.584.14		88
45.14	odd	6		315.2.u.a.59.31	yes	88
63.5	even	6	inner	315.2.bq.a.194.31	yes	88
63.40	odd	6		945.2.bq.a.719.14		88
105.89	even	6		945.2.u.a.89.31		88
315.194	even	6	inner	315.2.bq.a.194.14	yes	88
315.229	odd	6		945.2.bq.a.719.31		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.14	✓	88	9.5	odd	6
315.2.u.a.59.31	yes	88	45.14	odd	6
315.2.u.a.299.14	yes	88	35.19	odd	6
315.2.u.a.299.31	yes	88	7.5	odd	6
315.2.bq.a.164.14	yes	88	1.1	even	1	trivial
315.2.bq.a.164.31	yes	88	5.4	even	2	inner
315.2.bq.a.194.14	yes	88	315.194	even	6	inner
315.2.bq.a.194.31	yes	88	63.5	even	6	inner
945.2.u.a.89.14		88	21.5	even	6
945.2.u.a.89.31		88	105.89	even	6
945.2.u.a.584.14		88	45.4	even	6
945.2.u.a.584.31		88	9.4	even	3
945.2.bq.a.719.14		88	63.40	odd	6
945.2.bq.a.719.31		88	315.229	odd	6
945.2.bq.a.899.14		88	15.14	odd	2
945.2.bq.a.899.31		88	3.2	odd	2