Embedded newform 414.2.p.a.11.14

• Label: 414.2.p.a.11.14
• Level: $414$
• Weight: $2$
• Character: 414.11
• Analytic conductor: $3.306$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.30580664368\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(24\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			11.14
Character	\(\chi\)	\(=\)	414.11
Dual form			414.2.p.a.113.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.998867 - 0.0475819i) q^{2} +(-1.50108 - 0.864152i) q^{3} +(0.995472 - 0.0950560i) q^{4} +(1.85977 + 1.77329i) q^{5} +(-1.54050 - 0.791749i) q^{6} +(-0.441159 - 2.28895i) q^{7} +(0.989821 - 0.142315i) q^{8} +(1.50648 + 2.59432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3} - 24 q^{4} + 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.998867	−	0.0475819i	0.706306	−	0.0336455i
							\(3\)	−1.50108	−	0.864152i	−0.866649	−	0.498918i
\(5\)	1.85977	+	1.77329i	0.831715	+	0.793039i								0.980642	−	0.195809i	\(-0.0627334\pi\)
							−0.148927	+	0.988848i	\(0.547582\pi\)
\(7\)	−0.441159	−	2.28895i	−0.166742	−	0.865142i	−0.964885	−	0.262673i	\(-0.915396\pi\)
							0.798142	−	0.602469i	\(-0.205816\pi\)
\(11\)	3.18441	−	1.64168i	0.960135	−	0.494984i	0.0944262	−	0.995532i	\(-0.469898\pi\)
							0.865709	+	0.500548i	\(0.166868\pi\)
\(13\)	−1.34397	−	0.259030i	−0.372751	−	0.0718419i	−0.000564707	−	1.00000i	\(-0.500180\pi\)
							−0.372187	+	0.928158i	\(0.621392\pi\)
\(17\)	0.749738	+	1.64170i	0.181838	+	0.398170i	0.978498	−	0.206259i	\(-0.0661288\pi\)
							−0.796659	+	0.604428i	\(0.793402\pi\)
\(19\)	2.19897	+	1.00423i	0.504477	+	0.230387i	0.651363	−	0.758766i	\(-0.274198\pi\)
							−0.146886	+	0.989153i	\(0.546925\pi\)
\(23\)	4.69983	−	0.954757i	0.979983	−	0.199080i
							\(29\)	0.682783	−	7.15043i	0.126790	−	1.32780i	−0.678360	−	0.734729i	\(-0.737309\pi\)
0.805150	−	0.593071i	\(-0.202085\pi\)
\(31\)	1.71935	−	1.35211i	0.308805	−	0.242847i	−0.451714	−	0.892163i	\(-0.649187\pi\)
							0.760519	+	0.649316i	\(0.224945\pi\)
\(37\)	−1.75011	+	5.96034i	−0.287717	+	0.979873i	0.681120	+	0.732172i	\(0.261493\pi\)
							−0.968836	+	0.247701i	\(0.920325\pi\)
\(41\)	−7.43305	+	7.79556i	−1.16085	+	1.21746i	−0.188969	+	0.981983i	\(0.560514\pi\)
							−0.971879	+	0.235479i	\(0.924334\pi\)
\(43\)	−3.73494	+	4.74936i	−0.569573	+	0.724270i	−0.982452	−	0.186515i	\(-0.940281\pi\)
							0.412880	+	0.910786i	\(0.364523\pi\)
\(47\)	−10.6535	−	6.15079i	−1.55397	−	0.897185i	−0.997813	−	0.0661071i	\(-0.978942\pi\)
							−0.556157	−	0.831077i	\(-0.687725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.2.p.a.11.14	✓	480
9.5	odd	6	inner	414.2.p.a.149.24	yes	480
23.21	odd	22	inner	414.2.p.a.389.24	yes	480
207.113	even	66	inner	414.2.p.a.113.14	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.p.a.11.14	✓	480	1.1	even	1	trivial
414.2.p.a.113.14	yes	480	207.113	even	66	inner
414.2.p.a.149.24	yes	480	9.5	odd	6	inner
414.2.p.a.389.24	yes	480	23.21	odd	22	inner