Embedded newform 912.2.cn.a.451.22

• Label: 912.2.cn.a.451.22
• 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			451.22
Character	\(\chi\)	\(=\)	912.451
Dual form			912.2.cn.a.91.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.922087 + 1.07227i) q^{2} +(0.819152 - 0.573576i) q^{3} +(-0.299511 - 1.97745i) q^{4} +(-3.12494 + 0.273397i) q^{5} +(-0.140303 + 1.40724i) q^{6} +(-0.765704 - 1.32624i) q^{7} +(2.39652 + 1.50222i) q^{8} +(0.342020 - 0.939693i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.922087	+	1.07227i	−0.652014	+	0.758207i
							\(3\)	0.819152	−	0.573576i	0.472938	−	0.331155i
\(5\)	−3.12494	+	0.273397i	−1.39752	+	0.122267i								−0.760877	−	0.648896i	\(-0.775231\pi\)
							−0.636638	+	0.771163i	\(0.719676\pi\)
\(7\)	−0.765704	−	1.32624i	−0.289409	−	0.501271i	0.684260	−	0.729238i	\(-0.260125\pi\)
							−0.973669	+	0.227967i	\(0.926792\pi\)
\(11\)	4.47165	−	1.19817i	1.34825	−	0.361263i	0.488763	−	0.872417i	\(-0.337448\pi\)
							0.859490	+	0.511153i	\(0.170782\pi\)
\(13\)	2.92323	+	2.04687i	0.810757	+	0.567698i	0.903796	−	0.427964i	\(-0.140769\pi\)
							−0.0930385	+	0.995663i	\(0.529658\pi\)
\(17\)	−3.10768	+	1.13110i	−0.753722	+	0.274332i	−0.690171	−	0.723646i	\(-0.742465\pi\)
							−0.0635509	+	0.997979i	\(0.520243\pi\)
\(19\)	−2.98183	−	3.17941i	−0.684079	−	0.729408i
							\(23\)	−3.02258	+	2.53625i	−0.630252	+	0.528844i	−0.901007	−	0.433804i	\(-0.857171\pi\)
0.270755	+	0.962648i	\(0.412727\pi\)
\(29\)	−6.18095	+	2.88222i	−1.14777	+	0.535215i	−0.900983	−	0.433854i	\(-0.857153\pi\)
							−0.246790	+	0.969069i	\(0.579376\pi\)
\(31\)	−4.89081	−	8.47113i	−0.878415	−	1.52146i	−0.853079	−	0.521781i	\(-0.825268\pi\)
							−0.0253362	−	0.999679i	\(-0.508066\pi\)
\(37\)	−4.45970	−	4.45970i	−0.733170	−	0.733170i	0.238077	−	0.971246i	\(-0.423483\pi\)
							−0.971246	+	0.238077i	\(0.923483\pi\)
\(41\)	−0.872660	+	4.94910i	−0.136287	+	0.772920i	0.837668	+	0.546179i	\(0.183918\pi\)
							−0.973955	+	0.226741i	\(0.927193\pi\)
\(43\)	−0.665935	−	7.61167i	−0.101554	−	1.16077i	−0.860442	−	0.509548i	\(-0.829813\pi\)
							0.758888	−	0.651221i	\(-0.225743\pi\)
\(47\)	4.37554	−	12.0217i	0.638239	−	1.75355i	−0.0189503	−	0.999820i	\(-0.506032\pi\)
							0.657189	−	0.753726i	\(-0.271745\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cn.a.451.22	yes	960
16.11	odd	4	inner	912.2.cn.a.907.77	yes	960
19.15	odd	18	inner	912.2.cn.a.547.77	yes	960
304.91	even	36	inner	912.2.cn.a.91.22	✓	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.cn.a.91.22	✓	960	304.91	even	36	inner
912.2.cn.a.451.22	yes	960	1.1	even	1	trivial
912.2.cn.a.547.77	yes	960	19.15	odd	18	inner
912.2.cn.a.907.77	yes	960	16.11	odd	4	inner