Embedded newform 414.3.h.a.229.2

• Label: 414.3.h.a.229.2
• 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			229.2
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-2.89884 - 0.772474i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(5.56174 + 3.21107i) q^{5} +(1.10371 + 4.09656i) q^{6} +(-7.31386 + 4.22266i) q^{7} +2.82843 q^{8} +(7.80657 + 4.47856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.89884	−	0.772474i	−0.966281	−	0.257491i
\(5\)	5.56174	+	3.21107i	1.11235	+	0.642214i								0.939436	−	0.342723i	\(-0.111349\pi\)
							0.172911	+	0.984937i	\(0.444683\pi\)
\(7\)	−7.31386	+	4.22266i	−1.04484	+	0.603237i	−0.921200	−	0.389090i	\(-0.872789\pi\)
							−0.123638	+	0.992327i	\(0.539456\pi\)
\(11\)	−14.0703	+	8.12347i	−1.27911	+	0.738497i	−0.976686	−	0.214675i	\(-0.931131\pi\)
							−0.302429	+	0.953172i	\(0.597798\pi\)
\(13\)	5.63383	−	9.75809i	0.433372	−	0.750622i	−0.563789	−	0.825919i	\(-0.690657\pi\)
							0.997161	+	0.0752966i	\(0.0239904\pi\)
\(17\)			0.718752i			0.0422796i	0.999777	+	0.0211398i	\(0.00672950\pi\)
							−0.999777	+	0.0211398i	\(0.993270\pi\)
\(19\)		−	17.8198i		−	0.937882i	−0.883229	−	0.468941i	\(-0.844636\pi\)
							0.883229	−	0.468941i	\(-0.155364\pi\)
\(23\)	11.6069	−	19.8565i	0.504647	−	0.863326i
							\(29\)	−25.3067	−	43.8324i	−0.872644	−	1.51146i	−0.859252	−	0.511553i	\(-0.829070\pi\)
−0.0133921	−	0.999910i	\(-0.504263\pi\)
\(31\)	−14.4165	+	24.9702i	−0.465049	+	0.805489i	−0.999204	−	0.0398977i	\(-0.987297\pi\)
							0.534154	+	0.845387i	\(0.320630\pi\)
\(37\)		−	11.2592i		−	0.304303i	−0.988357	−	0.152151i	\(-0.951380\pi\)
							0.988357	−	0.152151i	\(-0.0486202\pi\)
\(41\)	−10.1373	+	17.5583i	−0.247251	+	0.428250i	−0.962762	−	0.270351i	\(-0.912860\pi\)
							0.715511	+	0.698601i	\(0.246194\pi\)
\(43\)	−32.6026	+	18.8231i	−0.758199	+	0.437746i	−0.828649	−	0.559769i	\(-0.810890\pi\)
							0.0704498	+	0.997515i	\(0.477557\pi\)
\(47\)	−18.5261	−	32.0882i	−0.394172	−	0.682727i	0.598823	−	0.800882i	\(-0.295635\pi\)
							−0.992995	+	0.118155i	\(0.962302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.2	yes	96
3.2	odd	2		1242.3.h.a.91.31		96
9.2	odd	6		1242.3.h.a.505.32		96
9.7	even	3	inner	414.3.h.a.367.1	yes	96
23.22	odd	2	inner	414.3.h.a.229.1	✓	96
69.68	even	2		1242.3.h.a.91.32		96
207.137	even	6		1242.3.h.a.505.31		96
207.160	odd	6	inner	414.3.h.a.367.2	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.1	✓	96	23.22	odd	2	inner
414.3.h.a.229.2	yes	96	1.1	even	1	trivial
414.3.h.a.367.1	yes	96	9.7	even	3	inner
414.3.h.a.367.2	yes	96	207.160	odd	6	inner
1242.3.h.a.91.31		96	3.2	odd	2
1242.3.h.a.91.32		96	69.68	even	2
1242.3.h.a.505.31		96	207.137	even	6
1242.3.h.a.505.32		96	9.2	odd	6