Embedded newform 945.2.cz.a.41.7

• Label: 945.2.cz.a.41.7
• Level: $945$
• Weight: $2$
• Character: 945.41
• Analytic conductor: $7.546$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.cz (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.54586299101\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(48\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			41.7
Character	\(\chi\)	\(=\)	945.41
Dual form			945.2.cz.a.461.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.06315 + 0.363789i) q^{2} +(-0.333455 - 1.69965i) q^{3} +(2.24487 - 0.817065i) q^{4} +(0.766044 + 0.642788i) q^{5} +(1.30628 + 3.38533i) q^{6} +(1.06350 + 2.42260i) q^{7} +(-0.705652 + 0.407408i) q^{8} +(-2.77762 + 1.13351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 3 q^{3} - 6 q^{6} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(596\)	\(757\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{18}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06315	+	0.363789i	−1.45887	+	0.257238i	−0.846100	−	0.533024i	\(-0.821055\pi\)
							−0.612769	+	0.790262i	\(0.709944\pi\)
\(3\)	−0.333455	−	1.69965i	−0.192520	−	0.981293i
							\(5\)	0.766044	+	0.642788i	0.342585	+	0.287463i
\(7\)	1.06350	+	2.42260i	0.401965	+	0.915655i
							\(11\)	−0.974919	−	1.16186i	−0.293949	−	0.350315i	0.598776	−	0.800917i	\(-0.295654\pi\)
−0.892725	+	0.450602i	\(0.851210\pi\)
\(13\)	−2.33352	−	0.411462i	−0.647201	−	0.114119i	−0.159595	−	0.987183i	\(-0.551019\pi\)
							−0.487607	+	0.873063i	\(0.662130\pi\)
\(17\)	1.55506	−	2.69344i	0.377157	−	0.653256i	−0.613490	−	0.789702i	\(-0.710235\pi\)
							0.990647	+	0.136447i	\(0.0435683\pi\)
\(19\)	−2.22180	+	1.28276i	−0.509716	+	0.294285i	−0.732717	−	0.680533i	\(-0.761748\pi\)
							0.223001	+	0.974818i	\(0.428415\pi\)
\(23\)	−2.81577	−	7.73627i	−0.587129	−	1.61312i	−0.775727	−	0.631069i	\(-0.782617\pi\)
							0.188598	−	0.982054i	\(-0.439606\pi\)
\(29\)	5.12861	−	0.904313i	0.952359	−	0.167927i	0.324180	−	0.945995i	\(-0.394912\pi\)
							0.628179	+	0.778069i	\(0.283800\pi\)
\(31\)	1.18588	+	3.25818i	0.212990	+	0.585186i	0.999474	−	0.0324249i	\(-0.0103230\pi\)
							−0.786484	+	0.617611i	\(0.788101\pi\)
\(37\)	1.68743	−	2.92271i	0.277411	−	0.480491i	−0.693329	−	0.720621i	\(-0.743857\pi\)
							0.970741	+	0.240130i	\(0.0771902\pi\)
\(41\)	−0.127757	+	0.724549i	−0.0199524	+	0.113155i	−0.993157	−	0.116784i	\(-0.962742\pi\)
							0.973205	+	0.229939i	\(0.0738527\pi\)
\(43\)	7.27429	−	6.10385i	1.10932	−	0.930828i	0.111302	−	0.993787i	\(-0.464498\pi\)
							0.998016	+	0.0629582i	\(0.0200535\pi\)
\(47\)	5.43731	+	1.97902i	0.793114	+	0.288670i	0.706630	−	0.707583i	\(-0.250215\pi\)
							0.0864840	+	0.996253i	\(0.472437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.cz.a.41.7	✓	288
7.6	odd	2		945.2.cz.b.41.7	yes	288
27.2	odd	18		945.2.cz.b.461.7	yes	288
189.83	even	18	inner	945.2.cz.a.461.7	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.cz.a.41.7	✓	288	1.1	even	1	trivial
945.2.cz.a.461.7	yes	288	189.83	even	18	inner
945.2.cz.b.41.7	yes	288	7.6	odd	2
945.2.cz.b.461.7	yes	288	27.2	odd	18