Embedded newform 315.2.bq.a.164.12

• Label: 315.2.bq.a.164.12
• 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			164.12
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.12

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.37324 q^{2} +(-1.73021 + 0.0799234i) q^{3} -0.114213 q^{4} +(0.214801 - 2.22573i) q^{5} +(2.37599 - 0.109754i) q^{6} +(1.26290 + 2.32489i) q^{7} +2.90332 q^{8} +(2.98722 - 0.276568i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37324			−0.971027			−0.485513	−	0.874229i	\(-0.661367\pi\)
							−0.485513	+	0.874229i	\(0.661367\pi\)
\(3\)	−1.73021	+	0.0799234i	−0.998935	+	0.0461438i
							\(5\)	0.214801	−	2.22573i	0.0960619	−	0.995375i
\(7\)	1.26290	+	2.32489i	0.477330	+	0.878724i
							\(11\)	−1.16486	+	0.672529i	−0.351217	+	0.202775i	−0.665221	−	0.746646i	\(-0.731663\pi\)
0.314004	+	0.949422i	\(0.398329\pi\)
\(13\)	−1.39214	−	2.41125i	−0.386109	−	0.668761i	0.605813	−	0.795607i	\(-0.292848\pi\)
							−0.991922	+	0.126846i	\(0.959515\pi\)
\(17\)	−4.41188	−	2.54720i	−1.07004	−	0.617787i	−0.141847	−	0.989889i	\(-0.545304\pi\)
							−0.928192	+	0.372101i	\(0.878637\pi\)
\(19\)	−5.17926	+	2.99025i	−1.18820	+	0.686010i	−0.957898	−	0.287109i	\(-0.907306\pi\)
							−0.230306	+	0.973118i	\(0.573973\pi\)
\(23\)	4.47192	−	7.74559i	0.932460	−	1.61507i	0.153358	−	0.988171i	\(-0.450991\pi\)
							0.779102	−	0.626897i	\(-0.215675\pi\)
\(29\)	−0.962825	−	0.555887i	−0.178792	−	0.103226i	0.407933	−	0.913012i	\(-0.366250\pi\)
							−0.586725	+	0.809786i	\(0.699583\pi\)
\(31\)			2.30606i			0.414180i	0.978322	+	0.207090i	\(0.0663994\pi\)
							−0.978322	+	0.207090i	\(0.933601\pi\)
\(37\)	−3.04415	+	1.75754i	−0.500455	+	0.288938i	−0.728902	−	0.684618i	\(-0.759969\pi\)
							0.228446	+	0.973557i	\(0.426635\pi\)
\(41\)	−5.49846	−	9.52360i	−0.858714	−	1.48734i	−0.873156	−	0.487441i	\(-0.837930\pi\)
							0.0144414	−	0.999896i	\(-0.495403\pi\)
\(43\)	−3.55310	−	2.05138i	−0.541843	−	0.312833i	0.203983	−	0.978974i	\(-0.434611\pi\)
							−0.745825	+	0.666141i	\(0.767945\pi\)
\(47\)			7.94322i			1.15864i	0.815101	+	0.579319i	\(0.196681\pi\)
							−0.815101	+	0.579319i	\(0.803319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.12	yes	88
3.2	odd	2		945.2.bq.a.899.33		88
5.4	even	2	inner	315.2.bq.a.164.33	yes	88
7.5	odd	6		315.2.u.a.299.33	yes	88
9.4	even	3		945.2.u.a.584.33		88
9.5	odd	6		315.2.u.a.59.12	✓	88
15.14	odd	2		945.2.bq.a.899.12		88
21.5	even	6		945.2.u.a.89.12		88
35.19	odd	6		315.2.u.a.299.12	yes	88
45.4	even	6		945.2.u.a.584.12		88
45.14	odd	6		315.2.u.a.59.33	yes	88
63.5	even	6	inner	315.2.bq.a.194.33	yes	88
63.40	odd	6		945.2.bq.a.719.12		88
105.89	even	6		945.2.u.a.89.33		88
315.194	even	6	inner	315.2.bq.a.194.12	yes	88
315.229	odd	6		945.2.bq.a.719.33		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.12	✓	88	9.5	odd	6
315.2.u.a.59.33	yes	88	45.14	odd	6
315.2.u.a.299.12	yes	88	35.19	odd	6
315.2.u.a.299.33	yes	88	7.5	odd	6
315.2.bq.a.164.12	yes	88	1.1	even	1	trivial
315.2.bq.a.164.33	yes	88	5.4	even	2	inner
315.2.bq.a.194.12	yes	88	315.194	even	6	inner
315.2.bq.a.194.33	yes	88	63.5	even	6	inner
945.2.u.a.89.12		88	21.5	even	6
945.2.u.a.89.33		88	105.89	even	6
945.2.u.a.584.12		88	45.4	even	6
945.2.u.a.584.33		88	9.4	even	3
945.2.bq.a.719.12		88	63.40	odd	6
945.2.bq.a.719.33		88	315.229	odd	6
945.2.bq.a.899.12		88	15.14	odd	2
945.2.bq.a.899.33		88	3.2	odd	2