Embedded newform 315.2.bq.a.164.6

• Label: 315.2.bq.a.164.6
• 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			164.6
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.6

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.16166 q^{2} +(-1.44589 + 0.953629i) q^{3} +2.67276 q^{4} +(-2.00646 + 0.986977i) q^{5} +(3.12551 - 2.06142i) q^{6} +(2.53153 - 0.769008i) q^{7} -1.45428 q^{8} +(1.18118 - 2.75768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16166			−1.52852			−0.764261	−	0.644907i	\(-0.776896\pi\)
							−0.764261	+	0.644907i	\(0.776896\pi\)
\(3\)	−1.44589	+	0.953629i	−0.834784	+	0.550578i
							\(5\)	−2.00646	+	0.986977i	−0.897316	+	0.441390i
\(7\)	2.53153	−	0.769008i	0.956827	−	0.290658i
							\(11\)	0.971237	−	0.560744i	0.292839	−	0.169071i	−0.346383	−	0.938093i	\(-0.612590\pi\)
0.639221	+	0.769023i	\(0.279257\pi\)
\(13\)	0.883246	+	1.52983i	0.244968	+	0.424298i	0.962123	−	0.272617i	\(-0.0878891\pi\)
							−0.717154	+	0.696914i	\(0.754556\pi\)
\(17\)	1.91907	+	1.10798i	0.465443	+	0.268724i	0.714330	−	0.699809i	\(-0.246731\pi\)
							−0.248887	+	0.968533i	\(0.580065\pi\)
\(19\)	−0.946699	+	0.546577i	−0.217188	+	0.125393i	−0.604647	−	0.796493i	\(-0.706686\pi\)
							0.387460	+	0.921887i	\(0.373353\pi\)
\(23\)	−2.25898	+	3.91267i	−0.471030	+	0.815847i	−0.999451	−	0.0331351i	\(-0.989451\pi\)
							0.528421	+	0.848982i	\(0.322784\pi\)
\(29\)	−2.20570	−	1.27346i	−0.409588	−	0.236476i	0.281025	−	0.959701i	\(-0.409326\pi\)
							−0.690613	+	0.723225i	\(0.742659\pi\)
\(31\)			9.04083i			1.62378i	0.583810	+	0.811890i	\(0.301561\pi\)
							−0.583810	+	0.811890i	\(0.698439\pi\)
\(37\)	−9.01769	+	5.20637i	−1.48250	+	0.855921i	−0.999803	−	0.0198664i	\(-0.993676\pi\)
							−0.482696	+	0.875788i	\(0.660343\pi\)
\(41\)	4.76108	+	8.24644i	0.743557	+	1.28788i	0.950866	+	0.309602i	\(0.100196\pi\)
							−0.207310	+	0.978275i	\(0.566471\pi\)
\(43\)	−0.210104	−	0.121303i	−0.0320405	−	0.0184986i	0.483894	−	0.875127i	\(-0.339222\pi\)
							−0.515935	+	0.856628i	\(0.672555\pi\)
\(47\)		−	6.08788i		−	0.888008i	−0.896025	−	0.444004i	\(-0.853558\pi\)
							0.896025	−	0.444004i	\(-0.146442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.6	yes	88
3.2	odd	2		945.2.bq.a.899.39		88
5.4	even	2	inner	315.2.bq.a.164.39	yes	88
7.5	odd	6		315.2.u.a.299.39	yes	88
9.4	even	3		945.2.u.a.584.39		88
9.5	odd	6		315.2.u.a.59.6	✓	88
15.14	odd	2		945.2.bq.a.899.6		88
21.5	even	6		945.2.u.a.89.6		88
35.19	odd	6		315.2.u.a.299.6	yes	88
45.4	even	6		945.2.u.a.584.6		88
45.14	odd	6		315.2.u.a.59.39	yes	88
63.5	even	6	inner	315.2.bq.a.194.39	yes	88
63.40	odd	6		945.2.bq.a.719.6		88
105.89	even	6		945.2.u.a.89.39		88
315.194	even	6	inner	315.2.bq.a.194.6	yes	88
315.229	odd	6		945.2.bq.a.719.39		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.6	✓	88	9.5	odd	6
315.2.u.a.59.39	yes	88	45.14	odd	6
315.2.u.a.299.6	yes	88	35.19	odd	6
315.2.u.a.299.39	yes	88	7.5	odd	6
315.2.bq.a.164.6	yes	88	1.1	even	1	trivial
315.2.bq.a.164.39	yes	88	5.4	even	2	inner
315.2.bq.a.194.6	yes	88	315.194	even	6	inner
315.2.bq.a.194.39	yes	88	63.5	even	6	inner
945.2.u.a.89.6		88	21.5	even	6
945.2.u.a.89.39		88	105.89	even	6
945.2.u.a.584.6		88	45.4	even	6
945.2.u.a.584.39		88	9.4	even	3
945.2.bq.a.719.6		88	63.40	odd	6
945.2.bq.a.719.39		88	315.229	odd	6
945.2.bq.a.899.6		88	15.14	odd	2
945.2.bq.a.899.39		88	3.2	odd	2