Embedded newform 414.3.h.a.229.11

• Label: 414.3.h.a.229.11
• Level: $414$
• Weight: $3$
• Character: 414.229
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			229.11
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-0.276911 + 2.98719i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-3.00967 - 1.73763i) q^{5} +(3.85435 - 1.77312i) q^{6} +(8.08407 - 4.66734i) q^{7} +2.82843 q^{8} +(-8.84664 - 1.65437i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707107	−	1.22474i	−0.353553	−	0.612372i
							\(3\)	−0.276911	+	2.98719i	−0.0923036	+	0.995731i
\(5\)	−3.00967	−	1.73763i	−0.601933	−	0.347526i								0.167869	−	0.985809i	\(-0.446312\pi\)
							−0.769802	+	0.638283i	\(0.779645\pi\)
\(7\)	8.08407	−	4.66734i	1.15487	−	0.666763i	0.204798	−	0.978804i	\(-0.434346\pi\)
							0.950068	+	0.312042i	\(0.101013\pi\)
\(11\)	−7.73685	+	4.46687i	−0.703350	+	0.406079i	−0.808594	−	0.588367i	\(-0.799771\pi\)
							0.105244	+	0.994446i	\(0.466438\pi\)
\(13\)	−3.57075	+	6.18473i	−0.274673	+	0.475748i	−0.970053	−	0.242895i	\(-0.921903\pi\)
							0.695379	+	0.718643i	\(0.255236\pi\)
\(17\)		−	24.2978i		−	1.42928i	−0.699492	−	0.714640i	\(-0.746591\pi\)
							0.699492	−	0.714640i	\(-0.253409\pi\)
\(19\)			10.3053i			0.542385i	0.962525	+	0.271192i	\(0.0874179\pi\)
							−0.962525	+	0.271192i	\(0.912582\pi\)
\(23\)	1.94523	+	22.9176i	0.0845752	+	0.996417i
							\(29\)	−27.0794	−	46.9028i	−0.933771	−	1.61734i	−0.776811	−	0.629733i	\(-0.783164\pi\)
−0.156959	−	0.987605i	\(-0.550169\pi\)
\(31\)	15.3602	−	26.6047i	0.495491	−	0.858216i	−0.504496	−	0.863414i	\(-0.668322\pi\)
							0.999986	+	0.00519879i	\(0.00165483\pi\)
\(37\)		−	38.8828i		−	1.05089i	−0.850829	−	0.525443i	\(-0.823899\pi\)
							0.850829	−	0.525443i	\(-0.176101\pi\)
\(41\)	0.766559	−	1.32772i	0.0186966	−	0.0323834i	−0.856526	−	0.516104i	\(-0.827382\pi\)
							0.875222	+	0.483721i	\(0.160715\pi\)
\(43\)	59.4115	−	34.3013i	1.38166	−	0.797704i	0.389307	−	0.921108i	\(-0.372715\pi\)
							0.992356	+	0.123405i	\(0.0393813\pi\)
\(47\)	−36.7333	−	63.6239i	−0.781559	−	1.35370i	−0.931033	−	0.364934i	\(-0.881091\pi\)
							0.149474	−	0.988766i	\(-0.452242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.11	✓	96
3.2	odd	2		1242.3.h.a.91.40		96
9.2	odd	6		1242.3.h.a.505.39		96
9.7	even	3	inner	414.3.h.a.367.12	yes	96
23.22	odd	2	inner	414.3.h.a.229.12	yes	96
69.68	even	2		1242.3.h.a.91.39		96
207.137	even	6		1242.3.h.a.505.40		96
207.160	odd	6	inner	414.3.h.a.367.11	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.11	✓	96	1.1	even	1	trivial
414.3.h.a.229.12	yes	96	23.22	odd	2	inner
414.3.h.a.367.11	yes	96	207.160	odd	6	inner
414.3.h.a.367.12	yes	96	9.7	even	3	inner
1242.3.h.a.91.39		96	69.68	even	2
1242.3.h.a.91.40		96	3.2	odd	2
1242.3.h.a.505.39		96	9.2	odd	6
1242.3.h.a.505.40		96	207.137	even	6