Embedded newform 315.2.bq.a.164.10

• Label: 315.2.bq.a.164.10
• 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			164.10
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.10

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.70249 q^{2} +(-0.695317 - 1.58636i) q^{3} +0.898480 q^{4} +(1.25685 - 1.84941i) q^{5} +(1.18377 + 2.70076i) q^{6} +(-0.756494 - 2.53529i) q^{7} +1.87533 q^{8} +(-2.03307 + 2.20604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70249			−1.20384			−0.601922	−	0.798555i	\(-0.705598\pi\)
							−0.601922	+	0.798555i	\(0.705598\pi\)
\(3\)	−0.695317	−	1.58636i	−0.401441	−	0.915885i
							\(5\)	1.25685	−	1.84941i	0.562079	−	0.827084i
\(7\)	−0.756494	−	2.53529i	−0.285928	−	0.958251i
							\(11\)	4.83572	−	2.79191i	1.45803	−	0.841792i	0.459112	−	0.888379i	\(-0.348168\pi\)
0.998914	+	0.0465869i	\(0.0148344\pi\)
\(13\)	−1.03818	−	1.79817i	−0.287938	−	0.498724i	0.685379	−	0.728186i	\(-0.259637\pi\)
							−0.973318	+	0.229462i	\(0.926303\pi\)
\(17\)	2.46562	+	1.42353i	0.598001	+	0.345256i	0.768255	−	0.640144i	\(-0.221125\pi\)
							−0.170254	+	0.985400i	\(0.554459\pi\)
\(19\)	−2.21883	+	1.28104i	−0.509034	+	0.293891i	−0.732437	−	0.680835i	\(-0.761617\pi\)
							0.223402	+	0.974726i	\(0.428284\pi\)
\(23\)	−1.13577	+	1.96721i	−0.236824	+	0.410192i	−0.959801	−	0.280680i	\(-0.909440\pi\)
							0.722977	+	0.690872i	\(0.242773\pi\)
\(29\)	−8.84546	−	5.10693i	−1.64256	−	0.948333i	−0.979919	−	0.199397i	\(-0.936102\pi\)
							−0.662643	−	0.748936i	\(-0.730565\pi\)
\(31\)		−	2.49563i		−	0.448229i	−0.974563	−	0.224114i	\(-0.928051\pi\)
							0.974563	−	0.224114i	\(-0.0719489\pi\)
\(37\)	−5.94937	+	3.43487i	−0.978071	+	0.564689i	−0.901687	−	0.432389i	\(-0.857671\pi\)
							−0.0763837	+	0.997078i	\(0.524337\pi\)
\(41\)	2.24887	+	3.89516i	0.351215	+	0.608322i	0.986463	−	0.163986i	\(-0.0524353\pi\)
							−0.635248	+	0.772308i	\(0.719102\pi\)
\(43\)	3.55092	+	2.05013i	0.541511	+	0.312641i	0.745691	−	0.666292i	\(-0.232120\pi\)
							−0.204180	+	0.978933i	\(0.565453\pi\)
\(47\)		−	7.30173i		−	1.06507i	−0.846409	−	0.532533i	\(-0.821240\pi\)
							0.846409	−	0.532533i	\(-0.178760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.10	yes	88
3.2	odd	2		945.2.bq.a.899.35		88
5.4	even	2	inner	315.2.bq.a.164.35	yes	88
7.5	odd	6		315.2.u.a.299.35	yes	88
9.4	even	3		945.2.u.a.584.35		88
9.5	odd	6		315.2.u.a.59.10	✓	88
15.14	odd	2		945.2.bq.a.899.10		88
21.5	even	6		945.2.u.a.89.10		88
35.19	odd	6		315.2.u.a.299.10	yes	88
45.4	even	6		945.2.u.a.584.10		88
45.14	odd	6		315.2.u.a.59.35	yes	88
63.5	even	6	inner	315.2.bq.a.194.35	yes	88
63.40	odd	6		945.2.bq.a.719.10		88
105.89	even	6		945.2.u.a.89.35		88
315.194	even	6	inner	315.2.bq.a.194.10	yes	88
315.229	odd	6		945.2.bq.a.719.35		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.10	✓	88	9.5	odd	6
315.2.u.a.59.35	yes	88	45.14	odd	6
315.2.u.a.299.10	yes	88	35.19	odd	6
315.2.u.a.299.35	yes	88	7.5	odd	6
315.2.bq.a.164.10	yes	88	1.1	even	1	trivial
315.2.bq.a.164.35	yes	88	5.4	even	2	inner
315.2.bq.a.194.10	yes	88	315.194	even	6	inner
315.2.bq.a.194.35	yes	88	63.5	even	6	inner
945.2.u.a.89.10		88	21.5	even	6
945.2.u.a.89.35		88	105.89	even	6
945.2.u.a.584.10		88	45.4	even	6
945.2.u.a.584.35		88	9.4	even	3
945.2.bq.a.719.10		88	63.40	odd	6
945.2.bq.a.719.35		88	315.229	odd	6
945.2.bq.a.899.10		88	15.14	odd	2
945.2.bq.a.899.35		88	3.2	odd	2