Embedded newform 129.4.i.a.4.7

• Label: 129.4.i.a.4.7
• Level: $129$
• Weight: $4$
• Character: 129.4
• Analytic conductor: $7.611$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	129.i (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.61124639074\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.7
Character	\(\chi\)	\(=\)	129.4
Dual form			129.4.i.a.97.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.934501 - 1.17183i) q^{2} +(1.87047 + 2.34549i) q^{3} +(1.28028 + 5.60928i) q^{4} +(-5.12960 - 2.47028i) q^{5} +4.49647 q^{6} +3.27932 q^{7} +(18.5727 + 8.94412i) q^{8} +(-2.00269 + 8.77435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 2 q^{2} - 33 q^{3} - 54 q^{4} + 12 q^{5} + 6 q^{6} - 118 q^{7} - 74 q^{8} - 99 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.934501	−	1.17183i	0.330396	−	0.414303i	−0.588691	−	0.808358i	\(-0.700357\pi\)
							0.919087	+	0.394055i	\(0.128928\pi\)
\(3\)	1.87047	+	2.34549i	0.359972	+	0.451391i
							\(5\)	−5.12960	−	2.47028i	−0.458805	−	0.220949i	0.190181	−	0.981749i	\(-0.439092\pi\)
−0.648986	+	0.760800i	\(0.724807\pi\)
\(7\)	3.27932			0.177067			0.0885333	−	0.996073i	\(-0.471782\pi\)
							0.0885333	+	0.996073i	\(0.471782\pi\)
\(11\)	−5.19167	+	22.7462i	−0.142304	+	0.623476i	0.852592	+	0.522577i	\(0.175029\pi\)
							−0.994897	+	0.100899i	\(0.967828\pi\)
\(13\)	67.4367	+	32.4758i	1.43874	+	0.692859i	0.980600	−	0.196022i	\(-0.0628023\pi\)
							0.458138	+	0.888881i	\(0.348517\pi\)
\(17\)	−81.1741	+	39.0914i	−1.15809	+	0.557709i	−0.911456	−	0.411399i	\(-0.865040\pi\)
							−0.246639	+	0.969107i	\(0.579326\pi\)
\(19\)	30.5411	+	133.809i	0.368769	+	1.61568i	0.730165	+	0.683271i	\(0.239443\pi\)
							−0.361396	+	0.932412i	\(0.617700\pi\)
\(23\)	23.7552	−	104.079i	0.215361	−	0.943560i	−0.745495	−	0.666512i	\(-0.767787\pi\)
							0.960856	−	0.277048i	\(-0.0893561\pi\)
\(29\)	72.0864	−	90.3935i	0.461590	−	0.578816i	−0.495499	−	0.868608i	\(-0.665015\pi\)
							0.957089	+	0.289793i	\(0.0935864\pi\)
\(31\)	151.623	−	190.129i	0.878461	−	1.10155i	−0.115661	−	0.993289i	\(-0.536899\pi\)
							0.994122	−	0.108266i	\(-0.0345298\pi\)
\(37\)	292.282			1.29867			0.649336	−	0.760502i	\(-0.275047\pi\)
							0.649336	+	0.760502i	\(0.275047\pi\)
\(41\)	131.742	−	165.199i	0.501821	−	0.629263i	−0.464818	−	0.885406i	\(-0.653880\pi\)
							0.966639	+	0.256143i	\(0.0824518\pi\)
\(43\)	−281.353	−	18.6393i	−0.997813	−	0.0661040i
							\(47\)	−112.022	−	490.800i	−0.347661	−	1.52320i	−0.782475	−	0.622682i	\(-0.786043\pi\)
0.434814	−	0.900520i	\(-0.356814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.4.i.a.4.7	✓	66
43.11	even	7	inner	129.4.i.a.97.7	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.4.i.a.4.7	✓	66	1.1	even	1	trivial
129.4.i.a.97.7	yes	66	43.11	even	7	inner