Embedded newform 315.2.bq.a.164.15

• Label: 315.2.bq.a.164.15
• 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			164.15
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.20290 q^{2} +(1.27624 - 1.17099i) q^{3} -0.553033 q^{4} +(2.10508 - 0.754091i) q^{5} +(-1.53519 + 1.40858i) q^{6} +(2.51709 - 0.815010i) q^{7} +3.07104 q^{8} +(0.257567 - 2.98892i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20290			−0.850578			−0.425289	−	0.905058i	\(-0.639828\pi\)
							−0.425289	+	0.905058i	\(0.639828\pi\)
\(3\)	1.27624	−	1.17099i	0.736836	−	0.676071i
							\(5\)	2.10508	−	0.754091i	0.941419	−	0.337240i
\(7\)	2.51709	−	0.815010i	0.951372	−	0.308045i
							\(11\)	−4.19982	+	2.42477i	−1.26629	+	0.731094i	−0.974284	−	0.225322i	\(-0.927657\pi\)
−0.292008	+	0.956416i	\(0.594323\pi\)
\(13\)	0.669156	+	1.15901i	0.185590	+	0.321452i	0.943775	−	0.330588i	\(-0.107247\pi\)
							−0.758185	+	0.652040i	\(0.773914\pi\)
\(17\)	−0.153866	−	0.0888348i	−0.0373181	−	0.0215456i	0.481225	−	0.876597i	\(-0.340192\pi\)
							−0.518543	+	0.855052i	\(0.673525\pi\)
\(19\)	4.78531	−	2.76280i	1.09783	−	0.633830i	0.162177	−	0.986762i	\(-0.448148\pi\)
							0.935649	+	0.352931i	\(0.114815\pi\)
\(23\)	0.413408	−	0.716044i	0.0862016	−	0.149305i	−0.819701	−	0.572792i	\(-0.805860\pi\)
							0.905903	+	0.423486i	\(0.139194\pi\)
\(29\)	−6.88138	−	3.97297i	−1.27784	−	0.737762i	−0.301390	−	0.953501i	\(-0.597451\pi\)
							−0.976451	+	0.215739i	\(0.930784\pi\)
\(31\)			4.30832i			0.773797i	0.922122	+	0.386899i	\(0.126454\pi\)
							−0.922122	+	0.386899i	\(0.873546\pi\)
\(37\)	7.14549	−	4.12545i	1.17471	−	0.678220i	0.219926	−	0.975517i	\(-0.429418\pi\)
							0.954785	+	0.297297i	\(0.0960851\pi\)
\(41\)	−0.278013	−	0.481532i	−0.0434183	−	0.0752027i	0.843500	−	0.537130i	\(-0.180491\pi\)
							−0.886918	+	0.461927i	\(0.847158\pi\)
\(43\)	−4.31274	−	2.48996i	−0.657686	−	0.379715i	0.133709	−	0.991021i	\(-0.457311\pi\)
							−0.791395	+	0.611305i	\(0.790645\pi\)
\(47\)			7.92204i			1.15555i	0.816197	+	0.577774i	\(0.196078\pi\)
							−0.816197	+	0.577774i	\(0.803922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.15	yes	88
3.2	odd	2		945.2.bq.a.899.30		88
5.4	even	2	inner	315.2.bq.a.164.30	yes	88
7.5	odd	6		315.2.u.a.299.30	yes	88
9.4	even	3		945.2.u.a.584.30		88
9.5	odd	6		315.2.u.a.59.15	✓	88
15.14	odd	2		945.2.bq.a.899.15		88
21.5	even	6		945.2.u.a.89.15		88
35.19	odd	6		315.2.u.a.299.15	yes	88
45.4	even	6		945.2.u.a.584.15		88
45.14	odd	6		315.2.u.a.59.30	yes	88
63.5	even	6	inner	315.2.bq.a.194.30	yes	88
63.40	odd	6		945.2.bq.a.719.15		88
105.89	even	6		945.2.u.a.89.30		88
315.194	even	6	inner	315.2.bq.a.194.15	yes	88
315.229	odd	6		945.2.bq.a.719.30		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.15	✓	88	9.5	odd	6
315.2.u.a.59.30	yes	88	45.14	odd	6
315.2.u.a.299.15	yes	88	35.19	odd	6
315.2.u.a.299.30	yes	88	7.5	odd	6
315.2.bq.a.164.15	yes	88	1.1	even	1	trivial
315.2.bq.a.164.30	yes	88	5.4	even	2	inner
315.2.bq.a.194.15	yes	88	315.194	even	6	inner
315.2.bq.a.194.30	yes	88	63.5	even	6	inner
945.2.u.a.89.15		88	21.5	even	6
945.2.u.a.89.30		88	105.89	even	6
945.2.u.a.584.15		88	45.4	even	6
945.2.u.a.584.30		88	9.4	even	3
945.2.bq.a.719.15		88	63.40	odd	6
945.2.bq.a.719.30		88	315.229	odd	6
945.2.bq.a.899.15		88	15.14	odd	2
945.2.bq.a.899.30		88	3.2	odd	2