Embedded newform 315.2.bq.a.164.16

• Label: 315.2.bq.a.164.16
• 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			164.16
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.16

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.932214 q^{2} +(-0.704225 + 1.58242i) q^{3} -1.13098 q^{4} +(1.20128 + 1.88598i) q^{5} +(0.656489 - 1.47516i) q^{6} +(0.405582 + 2.61448i) q^{7} +2.91874 q^{8} +(-2.00813 - 2.22877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.932214			−0.659175			−0.329587	−	0.944125i	\(-0.606910\pi\)
							−0.329587	+	0.944125i	\(0.606910\pi\)
\(3\)	−0.704225	+	1.58242i	−0.406585	+	0.913613i
							\(5\)	1.20128	+	1.88598i	0.537230	+	0.843436i
\(7\)	0.405582	+	2.61448i	0.153296	+	0.988180i
							\(11\)	−1.05849	+	0.611118i	−0.319146	+	0.184259i	−0.651012	−	0.759068i	\(-0.725655\pi\)
0.331866	+	0.943327i	\(0.392322\pi\)
\(13\)	1.31924	+	2.28500i	0.365892	+	0.633744i	0.988919	−	0.148456i	\(-0.0474305\pi\)
							−0.623026	+	0.782201i	\(0.714097\pi\)
\(17\)	−6.76575	−	3.90621i	−1.64093	−	0.947394i	−0.980502	−	0.196509i	\(-0.937040\pi\)
							−0.660433	−	0.750885i	\(-0.729627\pi\)
\(19\)	2.02157	−	1.16715i	0.463779	−	0.267763i	−0.249853	−	0.968284i	\(-0.580382\pi\)
							0.713632	+	0.700521i	\(0.247049\pi\)
\(23\)	−3.49143	+	6.04733i	−0.728013	+	1.26095i	0.229709	+	0.973259i	\(0.426222\pi\)
							−0.957722	+	0.287696i	\(0.907111\pi\)
\(29\)	−5.69293	−	3.28681i	−1.05715	−	0.610346i	−0.132508	−	0.991182i	\(-0.542303\pi\)
							−0.924643	+	0.380836i	\(0.875636\pi\)
\(31\)		−	4.13435i		−	0.742552i	−0.928523	−	0.371276i	\(-0.878920\pi\)
							0.928523	−	0.371276i	\(-0.121080\pi\)
\(37\)	0.395403	−	0.228286i	0.0650038	−	0.0375300i	−0.467146	−	0.884180i	\(-0.654718\pi\)
							0.532150	+	0.846650i	\(0.321384\pi\)
\(41\)	2.67616	+	4.63525i	0.417947	+	0.723905i	0.995733	−	0.0922834i	\(-0.0294166\pi\)
							−0.577786	+	0.816188i	\(0.696083\pi\)
\(43\)	6.97446	+	4.02671i	1.06360	+	0.614067i	0.926425	−	0.376480i	\(-0.122866\pi\)
							0.137171	+	0.990547i	\(0.456199\pi\)
\(47\)		−	4.57350i		−	0.667113i	−0.942730	−	0.333557i	\(-0.891751\pi\)
							0.942730	−	0.333557i	\(-0.108249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.16	yes	88
3.2	odd	2		945.2.bq.a.899.29		88
5.4	even	2	inner	315.2.bq.a.164.29	yes	88
7.5	odd	6		315.2.u.a.299.29	yes	88
9.4	even	3		945.2.u.a.584.29		88
9.5	odd	6		315.2.u.a.59.16	✓	88
15.14	odd	2		945.2.bq.a.899.16		88
21.5	even	6		945.2.u.a.89.16		88
35.19	odd	6		315.2.u.a.299.16	yes	88
45.4	even	6		945.2.u.a.584.16		88
45.14	odd	6		315.2.u.a.59.29	yes	88
63.5	even	6	inner	315.2.bq.a.194.29	yes	88
63.40	odd	6		945.2.bq.a.719.16		88
105.89	even	6		945.2.u.a.89.29		88
315.194	even	6	inner	315.2.bq.a.194.16	yes	88
315.229	odd	6		945.2.bq.a.719.29		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.16	✓	88	9.5	odd	6
315.2.u.a.59.29	yes	88	45.14	odd	6
315.2.u.a.299.16	yes	88	35.19	odd	6
315.2.u.a.299.29	yes	88	7.5	odd	6
315.2.bq.a.164.16	yes	88	1.1	even	1	trivial
315.2.bq.a.164.29	yes	88	5.4	even	2	inner
315.2.bq.a.194.16	yes	88	315.194	even	6	inner
315.2.bq.a.194.29	yes	88	63.5	even	6	inner
945.2.u.a.89.16		88	21.5	even	6
945.2.u.a.89.29		88	105.89	even	6
945.2.u.a.584.16		88	45.4	even	6
945.2.u.a.584.29		88	9.4	even	3
945.2.bq.a.719.16		88	63.40	odd	6
945.2.bq.a.719.29		88	315.229	odd	6
945.2.bq.a.899.16		88	15.14	odd	2
945.2.bq.a.899.29		88	3.2	odd	2