Embedded newform 756.2.cf.a.155.7

• Label: 756.2.cf.a.155.7
• 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			155.7
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37029 + 0.349726i) q^{2} +(-0.0159569 + 1.73198i) q^{3} +(1.75538 - 0.958451i) q^{4} +(1.66542 - 0.293658i) q^{5} +(-0.583852 - 2.37889i) q^{6} +(-0.642788 + 0.766044i) q^{7} +(-2.07019 + 1.92726i) q^{8} +(-2.99949 - 0.0552741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37029	+	0.349726i	−0.968941	+	0.247293i
							\(3\)	−0.0159569	+	1.73198i	−0.00921274	+	0.999958i
\(5\)	1.66542	−	0.293658i	0.744797	−	0.131328i								0.211644	−	0.977347i	\(-0.432118\pi\)
							0.533153	+	0.846019i	\(0.321007\pi\)
\(7\)	−0.642788	+	0.766044i	−0.242951	+	0.289538i
							\(11\)	0.286652	−	1.62568i	0.0864288	−	0.490162i	−0.910610	−	0.413266i	\(-0.864388\pi\)
0.997039	−	0.0768959i	\(-0.0245009\pi\)
\(13\)	−5.17731	−	1.88439i	−1.43593	−	0.522634i	−0.497302	−	0.867577i	\(-0.665676\pi\)
							−0.938624	+	0.344943i	\(0.887898\pi\)
\(17\)	−0.360829	+	0.208325i	−0.0875138	+	0.0505261i	−0.543118	−	0.839656i	\(-0.682756\pi\)
							0.455604	+	0.890182i	\(0.349423\pi\)
\(19\)	−5.70078	−	3.29135i	−1.30785	−	0.755087i	−0.326112	−	0.945331i	\(-0.605739\pi\)
							−0.981737	+	0.190244i	\(0.939072\pi\)
\(23\)	−6.97969	+	5.85666i	−1.45537	+	1.22120i	−0.526822	+	0.849976i	\(0.676617\pi\)
							−0.928544	+	0.371222i	\(0.878939\pi\)
\(29\)	1.76092	+	4.83807i	0.326994	+	0.898408i	0.988868	+	0.148793i	\(0.0475389\pi\)
							−0.661875	+	0.749615i	\(0.730239\pi\)
\(31\)	−0.668577	−	0.796779i	−0.120080	−	0.143106i	0.702655	−	0.711530i	\(-0.251998\pi\)
							−0.822735	+	0.568425i	\(0.807553\pi\)
\(37\)	1.71430	+	2.96926i	0.281829	+	0.488143i	0.971835	−	0.235661i	\(-0.0757254\pi\)
							−0.690006	+	0.723804i	\(0.742392\pi\)
\(41\)	−2.82971	+	7.77457i	−0.441927	+	1.21418i	0.496296	+	0.868153i	\(0.334693\pi\)
							−0.938223	+	0.346031i	\(0.887529\pi\)
\(43\)	−7.68893	−	1.35577i	−1.17255	−	0.206752i	−0.446751	−	0.894658i	\(-0.647419\pi\)
							−0.725800	+	0.687906i	\(0.758530\pi\)
\(47\)	9.84496	+	8.26090i	1.43603	+	1.20498i	0.942035	+	0.335514i	\(0.108910\pi\)
							0.493999	+	0.869462i	\(0.335535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.7	✓	648
4.3	odd	2	inner	756.2.cf.a.155.93	yes	648
27.23	odd	18	inner	756.2.cf.a.239.93	yes	648
108.23	even	18	inner	756.2.cf.a.239.7	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.7	✓	648	1.1	even	1	trivial
756.2.cf.a.155.93	yes	648	4.3	odd	2	inner
756.2.cf.a.239.7	yes	648	108.23	even	18	inner
756.2.cf.a.239.93	yes	648	27.23	odd	18	inner