Embedded newform 912.2.cn.a.451.73

• Label: 912.2.cn.a.451.73
• Level: $912$
• Weight: $2$
• Character: 912.451
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $960$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.cn (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(960\)
Relative dimension:	\(80\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			451.73
Character	\(\chi\)	\(=\)	912.451
Dual form			912.2.cn.a.91.73

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36897 - 0.354847i) q^{2} +(-0.819152 + 0.573576i) q^{3} +(1.74817 - 0.971551i) q^{4} +(3.14735 - 0.275358i) q^{5} +(-0.917864 + 1.07588i) q^{6} +(-0.694115 - 1.20224i) q^{7} +(2.04844 - 1.95036i) q^{8} +(0.342020 - 0.939693i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(229\)	\(305\)	\(799\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36897	−	0.354847i	0.968009	−	0.250915i
							\(3\)	−0.819152	+	0.573576i	−0.472938	+	0.331155i
\(5\)	3.14735	−	0.275358i	1.40754	−	0.123144i								0.642084	−	0.766634i	\(-0.278070\pi\)
							0.765454	+	0.643490i	\(0.222514\pi\)
\(7\)	−0.694115	−	1.20224i	−0.262351	−	0.454405i	0.704515	−	0.709689i	\(-0.251164\pi\)
							−0.966866	+	0.255284i	\(0.917831\pi\)
\(11\)	1.29491	−	0.346971i	0.390431	−	0.104616i	−0.0582626	−	0.998301i	\(-0.518556\pi\)
							0.448694	+	0.893686i	\(0.351889\pi\)
\(13\)	4.16655	+	2.91745i	1.15559	+	0.809155i	0.984376	−	0.176078i	\(-0.0563411\pi\)
							0.171217	+	0.985233i	\(0.445230\pi\)
\(17\)	−6.46157	+	2.35182i	−1.56716	+	0.570400i	−0.972363	−	0.233476i	\(-0.924990\pi\)
							−0.594798	+	0.803875i	\(0.702768\pi\)
\(19\)	−4.35861	−	0.0502942i	−0.999933	−	0.0115383i
							\(23\)	5.23295	−	4.39097i	1.09115	−	0.915581i	0.0943482	−	0.995539i	\(-0.469923\pi\)
0.996798	+	0.0799587i	\(0.0254789\pi\)
\(29\)	−1.24034	+	0.578380i	−0.230326	+	0.107403i	−0.534355	−	0.845260i	\(-0.679446\pi\)
							0.304030	+	0.952663i	\(0.401668\pi\)
\(31\)	−4.91591	−	8.51461i	−0.882924	−	1.52927i	−0.848075	−	0.529876i	\(-0.822239\pi\)
							−0.0348483	−	0.999393i	\(-0.511095\pi\)
\(37\)	3.20263	+	3.20263i	0.526510	+	0.526510i	0.919530	−	0.393020i	\(-0.128570\pi\)
							−0.393020	+	0.919530i	\(0.628570\pi\)
\(41\)	−1.95339	+	11.0782i	−0.305069	+	1.73013i	0.318113	+	0.948053i	\(0.396951\pi\)
							−0.623182	+	0.782077i	\(0.714160\pi\)
\(43\)	0.289997	+	3.31468i	0.0442241	+	0.505484i	0.985712	+	0.168438i	\(0.0538722\pi\)
							−0.941488	+	0.337046i	\(0.890572\pi\)
\(47\)	−1.99696	+	5.48660i	−0.291286	+	0.800302i	0.704593	+	0.709612i	\(0.251130\pi\)
							−0.995879	+	0.0906906i	\(0.971093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cn.a.451.73	yes	960
16.11	odd	4	inner	912.2.cn.a.907.11	yes	960
19.15	odd	18	inner	912.2.cn.a.547.11	yes	960
304.91	even	36	inner	912.2.cn.a.91.73	✓	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.cn.a.91.73	✓	960	304.91	even	36	inner
912.2.cn.a.451.73	yes	960	1.1	even	1	trivial
912.2.cn.a.547.11	yes	960	19.15	odd	18	inner
912.2.cn.a.907.11	yes	960	16.11	odd	4	inner