Embedded newform 912.2.bq.a.733.9

• Label: 912.2.bq.a.733.9
• Level: $912$
• Weight: $2$
• Character: 912.733
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			733.9
Character	\(\chi\)	\(=\)	912.733
Dual form			912.2.bq.a.805.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34077 - 0.449829i) q^{2} +(0.965926 - 0.258819i) q^{3} +(1.59531 + 1.20623i) q^{4} +(-0.0729229 - 0.272152i) q^{5} +(-1.41150 - 0.0874854i) q^{6} -2.02738i q^{7} +(-1.59634 - 2.33489i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(799\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34077	−	0.449829i	−0.948065	−	0.318077i
							\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
\(5\)	−0.0729229	−	0.272152i	−0.0326121	−	0.121710i								0.947701	−	0.319160i	\(-0.103401\pi\)
							−0.980313	+	0.197450i	\(0.936734\pi\)
\(7\)		−	2.02738i		−	0.766279i	−0.923690	−	0.383140i	\(-0.874843\pi\)
							0.923690	−	0.383140i	\(-0.125157\pi\)
\(11\)	−2.20071	+	2.20071i	−0.663540	+	0.663540i	−0.956213	−	0.292672i	\(-0.905455\pi\)
							0.292672	+	0.956213i	\(0.405455\pi\)
\(13\)	−1.75811	+	6.56137i	−0.487613	+	1.81980i	0.0803819	+	0.996764i	\(0.474386\pi\)
							−0.567995	+	0.823032i	\(0.692281\pi\)
\(17\)	−3.67234	+	6.36067i	−0.890673	+	1.54269i	−0.0516018	+	0.998668i	\(0.516433\pi\)
							−0.839071	+	0.544022i	\(0.816901\pi\)
\(19\)	−0.537964	+	4.32557i	−0.123417	+	0.992355i
							\(23\)	2.12540	−	1.22710i	0.443177	−	0.255868i	−0.261767	−	0.965131i	\(-0.584305\pi\)
0.704944	+	0.709263i	\(0.250972\pi\)
\(29\)	−0.771663	+	2.87988i	−0.143294	+	0.534781i	0.856531	+	0.516095i	\(0.172615\pi\)
							−0.999825	+	0.0186859i	\(0.994052\pi\)
\(31\)	5.40546			0.970849			0.485425	−	0.874279i	\(-0.338665\pi\)
							0.485425	+	0.874279i	\(0.338665\pi\)
\(37\)	−1.00577	+	1.00577i	−0.165347	+	0.165347i	−0.784931	−	0.619584i	\(-0.787301\pi\)
							0.619584	+	0.784931i	\(0.287301\pi\)
\(41\)	−3.53774	−	2.04251i	−0.552502	−	0.318987i	0.197629	−	0.980277i	\(-0.436676\pi\)
							−0.750130	+	0.661290i	\(0.770009\pi\)
\(43\)	1.71362	+	6.39532i	0.261325	+	0.975277i	0.964462	+	0.264223i	\(0.0851154\pi\)
							−0.703137	+	0.711054i	\(0.748218\pi\)
\(47\)	−0.508934	−	0.881500i	−0.0742357	−	0.128580i	0.826518	−	0.562910i	\(-0.190318\pi\)
							−0.900754	+	0.434330i	\(0.856985\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.bq.a.733.9	yes	320
16.5	even	4	inner	912.2.bq.a.277.61	✓	320
19.7	even	3	inner	912.2.bq.a.349.61	yes	320
304.197	even	12	inner	912.2.bq.a.805.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bq.a.277.61	✓	320	16.5	even	4	inner
912.2.bq.a.349.61	yes	320	19.7	even	3	inner
912.2.bq.a.733.9	yes	320	1.1	even	1	trivial
912.2.bq.a.805.9	yes	320	304.197	even	12	inner