Embedded newform 756.2.cf.a.155.19

• Label: 756.2.cf.a.155.19
• 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			155.19
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18703 + 0.768741i) q^{2} +(-0.163462 - 1.72432i) q^{3} +(0.818074 - 1.82504i) q^{4} +(-0.0623853 + 0.0110002i) q^{5} +(1.51959 + 1.92116i) q^{6} +(-0.642788 + 0.766044i) q^{7} +(0.431902 + 2.79526i) q^{8} +(-2.94656 + 0.563722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18703	+	0.768741i	−0.839356	+	0.543582i
							\(3\)	−0.163462	−	1.72432i	−0.0943749	−	0.995537i
\(5\)	−0.0623853	+	0.0110002i	−0.0278996	+	0.00491945i								−0.187581	−	0.982249i	\(-0.560065\pi\)
							0.159681	+	0.987169i	\(0.448953\pi\)
\(7\)	−0.642788	+	0.766044i	−0.242951	+	0.289538i
							\(11\)	−0.360487	+	2.04442i	−0.108691	+	0.616416i	0.880991	+	0.473133i	\(0.156877\pi\)
−0.989682	+	0.143283i	\(0.954234\pi\)
\(13\)	6.32079	+	2.30058i	1.75307	+	0.638066i	0.999807	−	0.0196513i	\(-0.00625560\pi\)
							0.753265	+	0.657717i	\(0.228478\pi\)
\(17\)	−4.89322	+	2.82510i	−1.18678	+	0.685188i	−0.957573	−	0.288189i	\(-0.906947\pi\)
							−0.229207	+	0.973378i	\(0.573613\pi\)
\(19\)	−5.29333	−	3.05610i	−1.21437	−	0.701118i	−0.250664	−	0.968074i	\(-0.580649\pi\)
							−0.963709	+	0.266956i	\(0.913982\pi\)
\(23\)	−1.36503	+	1.14540i	−0.284629	+	0.238832i	−0.773912	−	0.633293i	\(-0.781703\pi\)
							0.489283	+	0.872125i	\(0.337258\pi\)
\(29\)	0.325222	+	0.893540i	0.0603922	+	0.165926i	0.966219	−	0.257721i	\(-0.0829717\pi\)
							−0.905827	+	0.423648i	\(0.860749\pi\)
\(31\)	3.21586	+	3.83251i	0.577585	+	0.688339i	0.973169	−	0.230091i	\(-0.0739023\pi\)
							−0.395584	+	0.918430i	\(0.629458\pi\)
\(37\)	0.445984	+	0.772468i	0.0733194	+	0.126993i	0.900354	−	0.435158i	\(-0.143307\pi\)
							−0.827035	+	0.562151i	\(0.809974\pi\)
\(41\)	−2.23548	+	6.14193i	−0.349123	+	0.959208i	0.633524	+	0.773723i	\(0.281608\pi\)
							−0.982647	+	0.185485i	\(0.940614\pi\)
\(43\)	8.36979	+	1.47582i	1.27638	+	0.225060i	0.770443	−	0.637509i	\(-0.220035\pi\)
							0.505939	+	0.862570i	\(0.331146\pi\)
\(47\)	2.85818	+	2.39830i	0.416908	+	0.349827i	0.826985	−	0.562223i	\(-0.190054\pi\)
							−0.410077	+	0.912051i	\(0.634498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.19	✓	648
4.3	odd	2	inner	756.2.cf.a.155.104	yes	648
27.23	odd	18	inner	756.2.cf.a.239.104	yes	648
108.23	even	18	inner	756.2.cf.a.239.19	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.19	✓	648	1.1	even	1	trivial
756.2.cf.a.155.104	yes	648	4.3	odd	2	inner
756.2.cf.a.239.19	yes	648	108.23	even	18	inner
756.2.cf.a.239.104	yes	648	27.23	odd	18	inner