Embedded newform 315.2.bq.a.164.11

• Label: 315.2.bq.a.164.11
• 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			164.11
Character	\(\chi\)	\(=\)	315.164
Dual form			315.2.bq.a.194.11

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.50968 q^{2} +(1.73201 - 0.0112588i) q^{3} +0.279131 q^{4} +(-1.98951 + 1.02071i) q^{5} +(-2.61479 + 0.0169972i) q^{6} +(-2.58044 - 0.584218i) q^{7} +2.59796 q^{8} +(2.99975 - 0.0390010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50968			−1.06750			−0.533752	−	0.845641i	\(-0.679218\pi\)
							−0.533752	+	0.845641i	\(0.679218\pi\)
\(3\)	1.73201	−	0.0112588i	0.999979	−	0.00650030i
							\(5\)	−1.98951	+	1.02071i	−0.889736	+	0.456476i
\(7\)	−2.58044	−	0.584218i	−0.975316	−	0.220814i
							\(11\)	−4.14823	+	2.39498i	−1.25074	+	0.722114i	−0.971256	−	0.238037i	\(-0.923496\pi\)
−0.279482	+	0.960151i	\(0.590163\pi\)
\(13\)	−3.45305	−	5.98085i	−0.957703	−	1.65879i	−0.728059	−	0.685515i	\(-0.759577\pi\)
							−0.229644	−	0.973275i	\(-0.573756\pi\)
\(17\)	−2.38509	−	1.37703i	−0.578469	−	0.333979i	0.182056	−	0.983288i	\(-0.441725\pi\)
							−0.760525	+	0.649309i	\(0.775058\pi\)
\(19\)	0.697344	−	0.402612i	0.159982	−	0.0923654i	−0.417872	−	0.908506i	\(-0.637224\pi\)
							0.577853	+	0.816141i	\(0.303891\pi\)
\(23\)	−3.25856	+	5.64398i	−0.679456	+	1.17685i	0.295689	+	0.955284i	\(0.404451\pi\)
							−0.975145	+	0.221568i	\(0.928883\pi\)
\(29\)	−3.73151	−	2.15439i	−0.692924	−	0.400060i	0.111782	−	0.993733i	\(-0.464344\pi\)
							−0.804707	+	0.593673i	\(0.797677\pi\)
\(31\)		−	0.0343870i		−	0.00617608i	−0.999995	−	0.00308804i	\(-0.999017\pi\)
							0.999995	−	0.00308804i	\(-0.000982955\pi\)
\(37\)	−7.77376	+	4.48818i	−1.27800	+	0.737853i	−0.976480	−	0.215607i	\(-0.930827\pi\)
							−0.301519	+	0.953460i	\(0.597494\pi\)
\(41\)	−1.61801	−	2.80248i	−0.252691	−	0.437673i	0.711575	−	0.702610i	\(-0.247982\pi\)
							−0.964266	+	0.264937i	\(0.914649\pi\)
\(43\)	−0.949361	−	0.548114i	−0.144776	−	0.0835866i	0.425862	−	0.904788i	\(-0.359971\pi\)
							−0.570638	+	0.821201i	\(0.693304\pi\)
\(47\)			2.42093i			0.353129i	0.984289	+	0.176565i	\(0.0564984\pi\)
							−0.984289	+	0.176565i	\(0.943502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.164.11	yes	88
3.2	odd	2		945.2.bq.a.899.34		88
5.4	even	2	inner	315.2.bq.a.164.34	yes	88
7.5	odd	6		315.2.u.a.299.34	yes	88
9.4	even	3		945.2.u.a.584.34		88
9.5	odd	6		315.2.u.a.59.11	✓	88
15.14	odd	2		945.2.bq.a.899.11		88
21.5	even	6		945.2.u.a.89.11		88
35.19	odd	6		315.2.u.a.299.11	yes	88
45.4	even	6		945.2.u.a.584.11		88
45.14	odd	6		315.2.u.a.59.34	yes	88
63.5	even	6	inner	315.2.bq.a.194.34	yes	88
63.40	odd	6		945.2.bq.a.719.11		88
105.89	even	6		945.2.u.a.89.34		88
315.194	even	6	inner	315.2.bq.a.194.11	yes	88
315.229	odd	6		945.2.bq.a.719.34		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.11	✓	88	9.5	odd	6
315.2.u.a.59.34	yes	88	45.14	odd	6
315.2.u.a.299.11	yes	88	35.19	odd	6
315.2.u.a.299.34	yes	88	7.5	odd	6
315.2.bq.a.164.11	yes	88	1.1	even	1	trivial
315.2.bq.a.164.34	yes	88	5.4	even	2	inner
315.2.bq.a.194.11	yes	88	315.194	even	6	inner
315.2.bq.a.194.34	yes	88	63.5	even	6	inner
945.2.u.a.89.11		88	21.5	even	6
945.2.u.a.89.34		88	105.89	even	6
945.2.u.a.584.11		88	45.4	even	6
945.2.u.a.584.34		88	9.4	even	3
945.2.bq.a.719.11		88	63.40	odd	6
945.2.bq.a.719.34		88	315.229	odd	6
945.2.bq.a.899.11		88	15.14	odd	2
945.2.bq.a.899.34		88	3.2	odd	2