Embedded newform 756.2.cf.a.239.11

• Label: 756.2.cf.a.239.11
• Level: $756$
• Weight: $2$
• Character: 756.239
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $648$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.cf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(108\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			239.11
Character	\(\chi\)	\(=\)	756.239
Dual form			756.2.cf.a.155.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36197 + 0.380829i) q^{2} +(-0.829350 - 1.52058i) q^{3} +(1.70994 - 1.03736i) q^{4} +(1.49277 + 0.263215i) q^{5} +(1.70863 + 1.75515i) q^{6} +(-0.642788 - 0.766044i) q^{7} +(-1.93384 + 2.06404i) q^{8} +(-1.62436 + 2.52220i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36197	+	0.380829i	−0.963060	+	0.269287i
							\(3\)	−0.829350	−	1.52058i	−0.478826	−	0.877910i
\(5\)	1.49277	+	0.263215i	0.667586	+	0.117713i								0.497162	−	0.867658i	\(-0.334375\pi\)
							0.170423	+	0.985371i	\(0.445486\pi\)
\(7\)	−0.642788	−	0.766044i	−0.242951	−	0.289538i
							\(11\)	−0.184270	−	1.04505i	−0.0555596	−	0.315094i	0.944344	−	0.328959i	\(-0.106698\pi\)
−0.999904	+	0.0138646i	\(0.995587\pi\)
\(13\)	5.49027	−	1.99830i	1.52273	−	0.554228i	0.560900	−	0.827883i	\(-0.310455\pi\)
							0.961828	+	0.273655i	\(0.0882328\pi\)
\(17\)	1.26901	+	0.732663i	0.307780	+	0.177697i	0.645933	−	0.763394i	\(-0.276469\pi\)
							−0.338153	+	0.941091i	\(0.609802\pi\)
\(19\)	−0.226881	+	0.130990i	−0.0520501	+	0.0300511i	−0.525799	−	0.850609i	\(-0.676234\pi\)
							0.473749	+	0.880660i	\(0.342900\pi\)
\(23\)	3.99480	+	3.35204i	0.832974	+	0.698948i	0.955972	−	0.293459i	\(-0.0948063\pi\)
							−0.122998	+	0.992407i	\(0.539251\pi\)
\(29\)	0.556693	−	1.52950i	0.103375	−	0.284021i	−0.877212	−	0.480103i	\(-0.840599\pi\)
							0.980588	+	0.196081i	\(0.0628217\pi\)
\(31\)	4.42218	−	5.27015i	0.794247	−	0.946547i	−0.205236	−	0.978713i	\(-0.565796\pi\)
							0.999483	+	0.0321660i	\(0.0102405\pi\)
\(37\)	1.93917	−	3.35873i	0.318797	−	0.552172i	−0.661440	−	0.749998i	\(-0.730055\pi\)
							0.980237	+	0.197825i	\(0.0633879\pi\)
\(41\)	−0.957056	−	2.62949i	−0.149467	−	0.410657i	0.842252	−	0.539084i	\(-0.181229\pi\)
							−0.991719	+	0.128427i	\(0.959007\pi\)
\(43\)	−4.92105	+	0.867713i	−0.750453	+	0.132325i	−0.535776	−	0.844360i	\(-0.679981\pi\)
							−0.214677	+	0.976685i	\(0.568870\pi\)
\(47\)	−7.62824	+	6.40086i	−1.11269	+	0.933661i	−0.998212	−	0.0597647i	\(-0.980965\pi\)
							−0.114481	+	0.993425i	\(0.536521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.239.11	yes	648
4.3	odd	2	inner	756.2.cf.a.239.75	yes	648
27.20	odd	18	inner	756.2.cf.a.155.75	yes	648
108.47	even	18	inner	756.2.cf.a.155.11	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.11	✓	648	108.47	even	18	inner
756.2.cf.a.155.75	yes	648	27.20	odd	18	inner
756.2.cf.a.239.11	yes	648	1.1	even	1	trivial
756.2.cf.a.239.75	yes	648	4.3	odd	2	inner