Embedded newform 414.3.h.a.229.13

• Label: 414.3.h.a.229.13
• 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			229.13
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(0.170504 - 2.99515i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-8.47335 - 4.89209i) q^{5} +(-3.78886 + 1.90907i) q^{6} +(8.68607 - 5.01491i) q^{7} +2.82843 q^{8} +(-8.94186 - 1.02137i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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
							\(3\)	0.170504	−	2.99515i	0.0568348	−	0.998384i
\(5\)	−8.47335	−	4.89209i	−1.69467	−	0.978419i								−0.950652	−	0.310260i	\(-0.899584\pi\)
							−0.744019	−	0.668158i	\(-0.767083\pi\)
\(7\)	8.68607	−	5.01491i	1.24087	−	0.716415i	0.271596	−	0.962411i	\(-0.412448\pi\)
							0.969271	+	0.245996i	\(0.0791151\pi\)
\(11\)	−12.3302	+	7.11882i	−1.12092	+	0.647165i	−0.941636	−	0.336632i	\(-0.890712\pi\)
							−0.179287	+	0.983797i	\(0.557379\pi\)
\(13\)	−6.00628	+	10.4032i	−0.462022	+	0.800245i	−0.999062	−	0.0433116i	\(-0.986209\pi\)
							0.537040	+	0.843557i	\(0.319543\pi\)
\(17\)		−	12.2722i		−	0.721895i	−0.932586	−	0.360948i	\(-0.882453\pi\)
							0.932586	−	0.360948i	\(-0.117547\pi\)
\(19\)		−	19.6611i		−	1.03480i	−0.855745	−	0.517398i	\(-0.826901\pi\)
							0.855745	−	0.517398i	\(-0.173099\pi\)
\(23\)	−2.70966	−	22.8398i	−0.117811	−	0.993036i
							\(29\)	18.9452	+	32.8140i	0.653281	+	1.13152i	0.982322	+	0.187200i	\(0.0599414\pi\)
−0.329041	+	0.944316i	\(0.606725\pi\)
\(31\)	17.8506	−	30.9182i	0.575827	−	0.997362i	−0.420124	−	0.907467i	\(-0.638013\pi\)
							0.995951	−	0.0898953i	\(-0.0286533\pi\)
\(37\)			18.1836i			0.491449i	0.969340	+	0.245724i	\(0.0790258\pi\)
							−0.969340	+	0.245724i	\(0.920974\pi\)
\(41\)	−12.3057	+	21.3141i	−0.300139	+	0.519855i	−0.976167	−	0.217020i	\(-0.930366\pi\)
							0.676029	+	0.736875i	\(0.263700\pi\)
\(43\)	6.01946	−	3.47534i	0.139987	−	0.0808218i	−0.428371	−	0.903603i	\(-0.640912\pi\)
							0.568358	+	0.822781i	\(0.307579\pi\)
\(47\)	14.0212	+	24.2854i	0.298323	+	0.516710i	0.975752	−	0.218877i	\(-0.0702394\pi\)
							−0.677430	+	0.735588i	\(0.736906\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.13	✓	96
3.2	odd	2		1242.3.h.a.91.26		96
9.2	odd	6		1242.3.h.a.505.25		96
9.7	even	3	inner	414.3.h.a.367.14	yes	96
23.22	odd	2	inner	414.3.h.a.229.14	yes	96
69.68	even	2		1242.3.h.a.91.25		96
207.137	even	6		1242.3.h.a.505.26		96
207.160	odd	6	inner	414.3.h.a.367.13	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.13	✓	96	1.1	even	1	trivial
414.3.h.a.229.14	yes	96	23.22	odd	2	inner
414.3.h.a.367.13	yes	96	207.160	odd	6	inner
414.3.h.a.367.14	yes	96	9.7	even	3	inner
1242.3.h.a.91.25		96	69.68	even	2
1242.3.h.a.91.26		96	3.2	odd	2
1242.3.h.a.505.25		96	9.2	odd	6
1242.3.h.a.505.26		96	207.137	even	6