Embedded newform 756.2.cf.a.659.57

• Label: 756.2.cf.a.659.57
• Level: $756$
• Weight: $2$
• Character: 756.659
• 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			659.57
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.152848 + 1.40593i) q^{2} +(0.0477819 + 1.73139i) q^{3} +(-1.95328 + 0.429787i) q^{4} +(-0.988005 + 2.71452i) q^{5} +(-2.42691 + 0.331818i) q^{6} +(0.984808 + 0.173648i) q^{7} +(-0.902804 - 2.68047i) q^{8} +(-2.99543 + 0.165458i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.152848	+	1.40593i	0.108080	+	0.994142i
							\(3\)	0.0477819	+	1.73139i	0.0275869	+	0.999619i
\(5\)	−0.988005	+	2.71452i	−0.441849	+	1.21397i								0.496425	+	0.868080i	\(0.334646\pi\)
							−0.938274	+	0.345892i	\(0.887576\pi\)
\(7\)	0.984808	+	0.173648i	0.372222	+	0.0656328i
							\(11\)	−3.71947	+	1.35378i	−1.12146	+	0.408179i	−0.835185	−	0.549969i	\(-0.814640\pi\)
−0.286276	+	0.958147i	\(0.592417\pi\)
\(13\)	0.633183	−	0.531304i	0.175613	−	0.147357i	−0.550744	−	0.834674i	\(-0.685656\pi\)
							0.726358	+	0.687317i	\(0.241212\pi\)
\(17\)	−0.212149	+	0.122484i	−0.0514536	+	0.0297067i	−0.525506	−	0.850790i	\(-0.676124\pi\)
							0.474053	+	0.880497i	\(0.342791\pi\)
\(19\)	6.07372	+	3.50666i	1.39341	+	0.804484i	0.993691	−	0.112156i	\(-0.0357755\pi\)
							0.399716	+	0.916639i	\(0.369109\pi\)
\(23\)	−1.46788	−	8.32475i	−0.306074	−	1.73583i	−0.618407	−	0.785858i	\(-0.712222\pi\)
							0.312334	−	0.949972i	\(-0.398889\pi\)
\(29\)	−6.46796	+	7.70821i	−1.20107	+	1.43138i	−0.327381	+	0.944892i	\(0.606166\pi\)
							−0.873688	+	0.486486i	\(0.838278\pi\)
\(31\)	−1.64886	+	0.290738i	−0.296143	+	0.0522181i	−0.319746	−	0.947503i	\(-0.603598\pi\)
							0.0236024	+	0.999721i	\(0.492486\pi\)
\(37\)	4.90834	+	8.50150i	0.806926	+	1.39764i	0.914983	+	0.403493i	\(0.132204\pi\)
							−0.108057	+	0.994145i	\(0.534463\pi\)
\(41\)	−0.575510	−	0.685866i	−0.0898796	−	0.107114i	0.719230	−	0.694772i	\(-0.244495\pi\)
							−0.809110	+	0.587658i	\(0.800050\pi\)
\(43\)	0.711282	+	1.95423i	0.108470	+	0.298018i	0.982038	−	0.188684i	\(-0.0604222\pi\)
							−0.873568	+	0.486702i	\(0.838200\pi\)
\(47\)	1.44871	−	8.21605i	0.211316	−	1.19843i	−0.675869	−	0.737022i	\(-0.736232\pi\)
							0.887186	−	0.461413i	\(-0.152657\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.57	yes	648
4.3	odd	2	inner	756.2.cf.a.659.5	yes	648
27.5	odd	18	inner	756.2.cf.a.491.5	✓	648
108.59	even	18	inner	756.2.cf.a.491.57	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.5	✓	648	27.5	odd	18	inner
756.2.cf.a.491.57	yes	648	108.59	even	18	inner
756.2.cf.a.659.5	yes	648	4.3	odd	2	inner
756.2.cf.a.659.57	yes	648	1.1	even	1	trivial