Embedded newform 756.2.cf.a.659.54

• Label: 756.2.cf.a.659.54
• 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			659.54
Character	\(\chi\)	\(=\)	756.659
Dual form			756.2.cf.a.491.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00333790 + 1.41421i) q^{2} +(0.524281 - 1.65080i) q^{3} +(-1.99998 - 0.00944097i) q^{4} +(0.407793 - 1.12040i) q^{5} +(2.33282 + 0.746953i) q^{6} +(-0.984808 - 0.173648i) q^{7} +(0.0200272 - 2.82836i) q^{8} +(-2.45026 - 1.73096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00333790	+	1.41421i	−0.00236025	+	0.999997i
							\(3\)	0.524281	−	1.65080i	0.302694	−	0.953088i
\(5\)	0.407793	−	1.12040i	0.182371	−	0.501059i								−0.814495	−	0.580170i	\(-0.802986\pi\)
							0.996866	+	0.0791113i	\(0.0252082\pi\)
\(7\)	−0.984808	−	0.173648i	−0.372222	−	0.0656328i
							\(11\)	−4.49837	+	1.63727i	−1.35631	+	0.493656i	−0.914911	−	0.403655i	\(-0.867740\pi\)
−0.441398	+	0.897311i	\(0.645517\pi\)
\(13\)	−2.88619	+	2.42180i	−0.800484	+	0.671686i	−0.948316	−	0.317326i	\(-0.897215\pi\)
							0.147832	+	0.989012i	\(0.452771\pi\)
\(17\)	−0.870855	+	0.502788i	−0.211213	+	0.121944i	−0.601875	−	0.798590i	\(-0.705580\pi\)
							0.390662	+	0.920534i	\(0.372246\pi\)
\(19\)	−2.65646	−	1.53371i	−0.609434	−	0.351857i	0.163310	−	0.986575i	\(-0.447783\pi\)
							−0.772744	+	0.634718i	\(0.781116\pi\)
\(23\)	1.53285	+	8.69325i	0.319622	+	1.81267i	0.545045	+	0.838407i	\(0.316513\pi\)
							−0.225423	+	0.974261i	\(0.572376\pi\)
\(29\)	2.25978	−	2.69310i	0.419630	−	0.500096i	−0.514271	−	0.857628i	\(-0.671937\pi\)
							0.933901	+	0.357532i	\(0.116382\pi\)
\(31\)	−9.67703	+	1.70632i	−1.73805	+	0.306464i	−0.950715	−	0.310066i	\(-0.899649\pi\)
							−0.787331	+	0.616530i	\(0.788538\pi\)
\(37\)	2.69649	+	4.67047i	0.443301	+	0.767820i	0.997932	−	0.0642765i	\(-0.0204740\pi\)
							−0.554631	+	0.832096i	\(0.687141\pi\)
\(41\)	−4.07778	−	4.85971i	−0.636843	−	0.758959i	0.347025	−	0.937856i	\(-0.387192\pi\)
							−0.983868	+	0.178896i	\(0.942747\pi\)
\(43\)	0.956453	+	2.62783i	0.145858	+	0.400741i	0.991011	−	0.133784i	\(-0.0427129\pi\)
							−0.845153	+	0.534525i	\(0.820491\pi\)
\(47\)	2.24195	−	12.7147i	0.327022	−	1.85463i	−0.168048	−	0.985779i	\(-0.553746\pi\)
							0.495069	−	0.868853i	\(-0.335143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.659.54	yes	648
4.3	odd	2	inner	756.2.cf.a.659.7	yes	648
27.5	odd	18	inner	756.2.cf.a.491.7	✓	648
108.59	even	18	inner	756.2.cf.a.491.54	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.491.7	✓	648	27.5	odd	18	inner
756.2.cf.a.491.54	yes	648	108.59	even	18	inner
756.2.cf.a.659.7	yes	648	4.3	odd	2	inner
756.2.cf.a.659.54	yes	648	1.1	even	1	trivial