Embedded newform 414.2.m.a.13.4

• Label: 414.2.m.a.13.4
• Level: $414$
• Weight: $2$
• Character: 414.13
• Analytic conductor: $3.306$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.30580664368\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	414.13
Dual form			414.2.m.a.223.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 - 0.945001i) q^{2} +(-1.08222 + 1.35233i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(0.204372 + 0.287001i) q^{5} +(0.923996 + 1.46500i) q^{6} +(-0.726527 - 2.99479i) q^{7} +(-0.841254 + 0.540641i) q^{8} +(-0.657606 - 2.92704i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 12 q^{2} + 12 q^{4} - 13 q^{7} + 24 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.327068	−	0.945001i	0.231272	−	0.668216i
							\(3\)	−1.08222	+	1.35233i	−0.624819	+	0.780769i
\(5\)	0.204372	+	0.287001i	0.0913980	+	0.128351i								0.857743	−	0.514079i	\(-0.171866\pi\)
							−0.766345	+	0.642429i	\(0.777927\pi\)
\(7\)	−0.726527	−	2.99479i	−0.274602	−	1.13192i	−0.927304	−	0.374309i	\(-0.877880\pi\)
							0.652703	−	0.757614i	\(-0.273635\pi\)
\(11\)	4.81224	+	0.927484i	1.45095	+	0.279647i	0.853009	−	0.521896i	\(-0.174775\pi\)
							0.597937	+	0.801543i	\(0.295987\pi\)
\(13\)	−0.235775	+	0.971876i	−0.0653921	+	0.269550i	−0.995117	−	0.0987058i	\(-0.968530\pi\)
							0.929725	+	0.368256i	\(0.120045\pi\)
\(17\)	−0.863401	−	6.00509i	−0.209406	−	1.45645i	−0.775103	−	0.631835i	\(-0.782302\pi\)
							0.565698	−	0.824613i	\(-0.308607\pi\)
\(19\)	0.209251	−	1.45537i	0.0480055	−	0.333885i	−0.951639	−	0.307219i	\(-0.900602\pi\)
							0.999644	−	0.0266666i	\(-0.00848924\pi\)
\(23\)	−1.29451	−	4.61782i	−0.269924	−	0.962882i
							\(29\)	7.50349	−	5.90081i	1.39336	−	1.09575i	0.410410	−	0.911901i	\(-0.365386\pi\)
0.982954	−	0.183853i	\(-0.0588569\pi\)
\(31\)	0.0965191	+	2.02619i	0.0173353	+	0.363914i	0.990822	+	0.135171i	\(0.0431584\pi\)
							−0.973487	+	0.228743i	\(0.926539\pi\)
\(37\)	3.58775	+	7.85607i	0.589822	+	1.29153i	0.935550	+	0.353195i	\(0.114905\pi\)
							−0.345727	+	0.938335i	\(0.612368\pi\)
\(41\)	−5.48637	−	7.70453i	−0.856827	−	1.20325i	−0.977580	−	0.210563i	\(-0.932470\pi\)
							0.120753	−	0.992683i	\(-0.461469\pi\)
\(43\)	−0.473921	+	9.94883i	−0.0722723	+	1.51718i	0.613691	+	0.789546i	\(0.289684\pi\)
							−0.685963	+	0.727636i	\(0.740619\pi\)
\(47\)	−0.280611	−	0.486032i	−0.0409313	−	0.0708951i	0.844834	−	0.535029i	\(-0.179699\pi\)
							−0.885765	+	0.464133i	\(0.846366\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.2.m.a.13.4	✓	240
9.7	even	3	inner	414.2.m.a.151.8	yes	240
23.16	even	11	inner	414.2.m.a.85.8	yes	240
207.16	even	33	inner	414.2.m.a.223.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.m.a.13.4	✓	240	1.1	even	1	trivial
414.2.m.a.85.8	yes	240	23.16	even	11	inner
414.2.m.a.151.8	yes	240	9.7	even	3	inner
414.2.m.a.223.4	yes	240	207.16	even	33	inner