Embedded newform 414.3.h.a.229.7

• Label: 414.3.h.a.229.7
• 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			229.7
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.45535 + 2.62335i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-6.63175 - 3.82884i) q^{5} +(4.24202 - 0.0725559i) q^{6} +(-9.09135 + 5.24889i) q^{7} +2.82843 q^{8} +(-4.76392 - 7.63578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.45535	+	2.62335i	−0.485116	+	0.874450i
\(5\)	−6.63175	−	3.82884i	−1.32635	−	0.765769i								−0.341617	−	0.939839i	\(-0.610975\pi\)
							−0.984733	+	0.174070i	\(0.944308\pi\)
\(7\)	−9.09135	+	5.24889i	−1.29876	+	0.749842i	−0.980191	−	0.198057i	\(-0.936537\pi\)
							−0.318573	+	0.947898i	\(0.603204\pi\)
\(11\)	−6.63227	+	3.82914i	−0.602934	+	0.348104i	−0.770195	−	0.637809i	\(-0.779841\pi\)
							0.167261	+	0.985913i	\(0.446508\pi\)
\(13\)	3.25235	−	5.63323i	0.250181	−	0.433326i	−0.713395	−	0.700762i	\(-0.752843\pi\)
							0.963575	+	0.267437i	\(0.0861766\pi\)
\(17\)		−	3.48310i		−	0.204888i	−0.994739	−	0.102444i	\(-0.967334\pi\)
							0.994739	−	0.102444i	\(-0.0326663\pi\)
\(19\)			17.2685i			0.908869i	0.890780	+	0.454434i	\(0.150159\pi\)
							−0.890780	+	0.454434i	\(0.849841\pi\)
\(23\)	14.6900	−	17.6975i	0.638697	−	0.769458i
							\(29\)	−2.15172	−	3.72690i	−0.0741974	−	0.128514i	0.826540	−	0.562879i	\(-0.190306\pi\)
−0.900737	+	0.434365i	\(0.856973\pi\)
\(31\)	−15.8006	+	27.3675i	−0.509697	+	0.882821i	0.490240	+	0.871588i	\(0.336909\pi\)
							−0.999937	+	0.0112335i	\(0.996424\pi\)
\(37\)			12.3638i			0.334157i	0.985944	+	0.167078i	\(0.0534333\pi\)
							−0.985944	+	0.167078i	\(0.946567\pi\)
\(41\)	31.3416	−	54.2853i	0.764430	−	1.32403i	−0.176118	−	0.984369i	\(-0.556354\pi\)
							0.940547	−	0.339662i	\(-0.110313\pi\)
\(43\)	11.5760	−	6.68344i	0.269210	−	0.155429i	−0.359318	−	0.933215i	\(-0.616991\pi\)
							0.628529	+	0.777786i	\(0.283657\pi\)
\(47\)	38.7644	+	67.1419i	0.824775	+	1.42855i	0.902091	+	0.431545i	\(0.142031\pi\)
							−0.0773169	+	0.997007i	\(0.524635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.7	✓	96
3.2	odd	2		1242.3.h.a.91.28		96
9.2	odd	6		1242.3.h.a.505.27		96
9.7	even	3	inner	414.3.h.a.367.8	yes	96
23.22	odd	2	inner	414.3.h.a.229.8	yes	96
69.68	even	2		1242.3.h.a.91.27		96
207.137	even	6		1242.3.h.a.505.28		96
207.160	odd	6	inner	414.3.h.a.367.7	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.7	✓	96	1.1	even	1	trivial
414.3.h.a.229.8	yes	96	23.22	odd	2	inner
414.3.h.a.367.7	yes	96	207.160	odd	6	inner
414.3.h.a.367.8	yes	96	9.7	even	3	inner
1242.3.h.a.91.27		96	69.68	even	2
1242.3.h.a.91.28		96	3.2	odd	2
1242.3.h.a.505.27		96	9.2	odd	6
1242.3.h.a.505.28		96	207.137	even	6