Embedded newform 756.2.cf.a.155.8

• Label: 756.2.cf.a.155.8
• Level: $756$
• Weight: $2$
• Character: 756.155
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• 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			155.8
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36715 + 0.361799i) q^{2} +(0.921295 - 1.46670i) q^{3} +(1.73820 - 0.989266i) q^{4} +(2.55429 - 0.450391i) q^{5} +(-0.728898 + 2.33853i) q^{6} +(-0.642788 + 0.766044i) q^{7} +(-2.01847 + 1.98136i) q^{8} +(-1.30243 - 2.70253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36715	+	0.361799i	−0.966722	+	0.255830i
							\(3\)	0.921295	−	1.46670i	0.531910	−	0.846801i
\(5\)	2.55429	−	0.450391i	1.14231	−	0.201421i								0.429697	−	0.902973i	\(-0.358621\pi\)
							0.712618	+	0.701553i	\(0.247509\pi\)
\(7\)	−0.642788	+	0.766044i	−0.242951	+	0.289538i
							\(11\)	−0.225653	+	1.27974i	−0.0680370	+	0.385857i	0.931707	+	0.363212i	\(0.118320\pi\)
−0.999744	+	0.0226450i	\(0.992791\pi\)
\(13\)	−1.34120	−	0.488155i	−0.371981	−	0.135390i	0.149264	−	0.988797i	\(-0.452310\pi\)
							−0.521244	+	0.853408i	\(0.674532\pi\)
\(17\)	5.84783	−	3.37624i	1.41831	−	0.818859i	0.422156	−	0.906523i	\(-0.361274\pi\)
							0.996150	+	0.0876640i	\(0.0279402\pi\)
\(19\)	1.71792	+	0.991842i	0.394118	+	0.227544i	0.683943	−	0.729536i	\(-0.260264\pi\)
							−0.289825	+	0.957080i	\(0.593597\pi\)
\(23\)	6.10253	−	5.12063i	1.27246	−	1.06772i	0.278229	−	0.960515i	\(-0.410253\pi\)
							0.994236	−	0.107210i	\(-0.0341917\pi\)
\(29\)	1.39733	+	3.83914i	0.259478	+	0.712910i	0.999200	+	0.0399971i	\(0.0127349\pi\)
							−0.739722	+	0.672913i	\(0.765043\pi\)
\(31\)	−5.10836	−	6.08791i	−0.917489	−	1.09342i	−0.995337	−	0.0964558i	\(-0.969249\pi\)
							0.0778481	−	0.996965i	\(-0.475195\pi\)
\(37\)	0.524608	+	0.908647i	0.0862450	+	0.149381i	0.905921	−	0.423447i	\(-0.139180\pi\)
							−0.819676	+	0.572827i	\(0.805847\pi\)
\(41\)	2.29970	−	6.31837i	0.359153	−	0.986763i	−0.620172	−	0.784466i	\(-0.712937\pi\)
							0.979324	−	0.202297i	\(-0.0648407\pi\)
\(43\)	−0.995918	−	0.175607i	−0.151876	−	0.0267799i	0.0971931	−	0.995266i	\(-0.469014\pi\)
							−0.249069	+	0.968486i	\(0.580125\pi\)
\(47\)	0.969357	+	0.813387i	0.141395	+	0.118645i	0.710742	−	0.703453i	\(-0.248359\pi\)
							−0.569347	+	0.822098i	\(0.692804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.8	✓	648
4.3	odd	2	inner	756.2.cf.a.155.95	yes	648
27.23	odd	18	inner	756.2.cf.a.239.95	yes	648
108.23	even	18	inner	756.2.cf.a.239.8	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.8	✓	648	1.1	even	1	trivial
756.2.cf.a.155.95	yes	648	4.3	odd	2	inner
756.2.cf.a.239.8	yes	648	108.23	even	18	inner
756.2.cf.a.239.95	yes	648	27.23	odd	18	inner