Embedded newform 912.2.cn.a.451.78

• Label: 912.2.cn.a.451.78
• 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			451.78
Character	\(\chi\)	\(=\)	912.451
Dual form			912.2.cn.a.91.78

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40840 + 0.128131i) q^{2} +(0.819152 - 0.573576i) q^{3} +(1.96716 + 0.360920i) q^{4} +(1.82976 - 0.160083i) q^{5} +(1.22718 - 0.702864i) q^{6} +(1.50734 + 2.61079i) q^{7} +(2.72430 + 0.760374i) q^{8} +(0.342020 - 0.939693i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40840	+	0.128131i	0.995887	+	0.0906026i
							\(3\)	0.819152	−	0.573576i	0.472938	−	0.331155i
\(5\)	1.82976	−	0.160083i	0.818293	−	0.0715914i								0.329681	−	0.944092i	\(-0.393059\pi\)
							0.488612	+	0.872501i	\(0.337503\pi\)
\(7\)	1.50734	+	2.61079i	0.569721	+	0.986785i	0.996593	+	0.0824735i	\(0.0262820\pi\)
							−0.426872	+	0.904312i	\(0.640385\pi\)
\(11\)	−1.39981	+	0.375079i	−0.422059	+	0.113090i	−0.463596	−	0.886046i	\(-0.653441\pi\)
							0.0415369	+	0.999137i	\(0.486775\pi\)
\(13\)	−0.659953	−	0.462104i	−0.183038	−	0.128165i	0.478466	−	0.878106i	\(-0.341193\pi\)
							−0.661504	+	0.749941i	\(0.730082\pi\)
\(17\)	−4.63683	+	1.68767i	−1.12460	+	0.409320i	−0.836328	−	0.548230i	\(-0.815302\pi\)
							−0.288269	+	0.957550i	\(0.593080\pi\)
\(19\)	−1.84234	−	3.95041i	−0.422663	−	0.906287i
							\(23\)	0.682678	−	0.572835i	0.142348	−	0.119444i	−0.568833	−	0.822453i	\(-0.692605\pi\)
0.711181	+	0.703009i	\(0.248161\pi\)
\(29\)	−5.81280	+	2.71055i	−1.07941	+	0.503337i	−0.879218	−	0.476420i	\(-0.841934\pi\)
							−0.200192	+	0.979757i	\(0.564156\pi\)
\(31\)	1.18353	+	2.04993i	0.212568	+	0.368179i	0.952518	−	0.304484i	\(-0.0984839\pi\)
							−0.739949	+	0.672663i	\(0.765151\pi\)
\(37\)	−3.83497	−	3.83497i	−0.630466	−	0.630466i	0.317719	−	0.948185i	\(-0.397083\pi\)
							−0.948185	+	0.317719i	\(0.897083\pi\)
\(41\)	−0.0654086	+	0.370951i	−0.0102151	+	0.0579328i	−0.989489	−	0.144607i	\(-0.953808\pi\)
							0.979274	+	0.202539i	\(0.0649194\pi\)
\(43\)	−1.03818	−	11.8665i	−0.158321	−	1.80962i	−0.495959	−	0.868346i	\(-0.665183\pi\)
							0.337638	−	0.941276i	\(-0.390372\pi\)
\(47\)	2.17301	−	5.97031i	0.316967	−	0.870859i	−0.674237	−	0.738515i	\(-0.735527\pi\)
							0.991204	−	0.132344i	\(-0.0422504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cn.a.451.78	yes	960
16.11	odd	4	inner	912.2.cn.a.907.20	yes	960
19.15	odd	18	inner	912.2.cn.a.547.20	yes	960
304.91	even	36	inner	912.2.cn.a.91.78	✓	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.cn.a.91.78	✓	960	304.91	even	36	inner
912.2.cn.a.451.78	yes	960	1.1	even	1	trivial
912.2.cn.a.547.20	yes	960	19.15	odd	18	inner
912.2.cn.a.907.20	yes	960	16.11	odd	4	inner