Embedded newform 756.2.cf.a.155.6

• Label: 756.2.cf.a.155.6
• 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			155.6
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37092 + 0.347239i) q^{2} +(-1.36710 + 1.06350i) q^{3} +(1.75885 - 0.952074i) q^{4} +(-0.934734 + 0.164819i) q^{5} +(1.50490 - 1.93269i) q^{6} +(0.642788 - 0.766044i) q^{7} +(-2.08065 + 1.91596i) q^{8} +(0.737920 - 2.90783i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37092	+	0.347239i	−0.969388	+	0.245535i
							\(3\)	−1.36710	+	1.06350i	−0.789295	+	0.614014i
\(5\)	−0.934734	+	0.164819i	−0.418026	+	0.0737093i								−0.378705	−	0.925517i	\(-0.623631\pi\)
							−0.0393208	+	0.999227i	\(0.512519\pi\)
\(7\)	0.642788	−	0.766044i	0.242951	−	0.289538i
							\(11\)	−0.423225	+	2.40023i	−0.127607	+	0.723696i	0.852118	+	0.523350i	\(0.175318\pi\)
−0.979725	+	0.200346i	\(0.935793\pi\)
\(13\)	1.04505	+	0.380365i	0.289843	+	0.105494i	0.482850	−	0.875703i	\(-0.339602\pi\)
							−0.193007	+	0.981197i	\(0.561824\pi\)
\(17\)	−1.02602	+	0.592371i	−0.248845	+	0.143671i	−0.619235	−	0.785205i	\(-0.712557\pi\)
							0.370390	+	0.928876i	\(0.379224\pi\)
\(19\)	−1.39889	−	0.807649i	−0.320927	−	0.185287i	0.330879	−	0.943673i	\(-0.392655\pi\)
							−0.651806	+	0.758386i	\(0.725988\pi\)
\(23\)	−2.43230	+	2.04095i	−0.507171	+	0.425567i	−0.860132	−	0.510071i	\(-0.829619\pi\)
							0.352962	+	0.935638i	\(0.385175\pi\)
\(29\)	0.130718	+	0.359144i	0.0242737	+	0.0666913i	0.951238	−	0.308459i	\(-0.0998133\pi\)
							−0.926964	+	0.375150i	\(0.877591\pi\)
\(31\)	−4.46949	−	5.32654i	−0.802745	−	0.956674i	0.196973	−	0.980409i	\(-0.436889\pi\)
							−0.999718	+	0.0237348i	\(0.992444\pi\)
\(37\)	2.01728	+	3.49403i	0.331638	+	0.574414i	0.982833	−	0.184496i	\(-0.0590653\pi\)
							−0.651195	+	0.758910i	\(0.725732\pi\)
\(41\)	0.493777	−	1.35664i	0.0771150	−	0.211872i	−0.895145	−	0.445776i	\(-0.852928\pi\)
							0.972260	+	0.233904i	\(0.0751501\pi\)
\(43\)	2.68910	+	0.474160i	0.410083	+	0.0723088i	0.374884	−	0.927072i	\(-0.377682\pi\)
							0.0351990	+	0.999380i	\(0.488793\pi\)
\(47\)	−10.1964	−	8.55577i	−1.48729	−	1.24799i	−0.897946	−	0.440107i	\(-0.854941\pi\)
							−0.589347	−	0.807880i	\(-0.700615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.6	✓	648
4.3	odd	2	inner	756.2.cf.a.155.92	yes	648
27.23	odd	18	inner	756.2.cf.a.239.92	yes	648
108.23	even	18	inner	756.2.cf.a.239.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.6	✓	648	1.1	even	1	trivial
756.2.cf.a.155.92	yes	648	4.3	odd	2	inner
756.2.cf.a.239.6	yes	648	108.23	even	18	inner
756.2.cf.a.239.92	yes	648	27.23	odd	18	inner