Embedded newform 129.2.n.b.29.7

• Label: 129.2.n.b.29.7
• Level: $129$
• Weight: $2$
• Character: 129.29
• Analytic conductor: $1.030$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.n (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03007018607\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			29.7
Character	\(\chi\)	\(=\)	129.29
Dual form			129.2.n.b.89.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.637600 - 0.307052i) q^{2} +(-1.26134 + 1.18702i) q^{3} +(-0.934726 + 1.17211i) q^{4} +(-1.99249 + 0.614601i) q^{5} +(-0.439758 + 1.14414i) q^{6} +(2.89916 + 1.67383i) q^{7} +(-0.551032 + 2.41423i) q^{8} +(0.181980 - 2.99448i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.637600	−	0.307052i	0.450852	−	0.217119i	−0.194658	−	0.980871i	\(-0.562360\pi\)
							0.645509	+	0.763753i	\(0.276645\pi\)
\(3\)	−1.26134	+	1.18702i	−0.728238	+	0.685325i
							\(5\)	−1.99249	+	0.614601i	−0.891068	+	0.274858i	−0.706290	−	0.707922i	\(-0.749633\pi\)
−0.184778	+	0.982780i	\(0.559157\pi\)
\(7\)	2.89916	+	1.67383i	1.09578	+	0.632648i	0.935109	−	0.354360i	\(-0.115301\pi\)
							0.160669	+	0.987008i	\(0.448635\pi\)
\(11\)	−2.61274	+	2.08359i	−0.787770	+	0.628225i	−0.932469	−	0.361250i	\(-0.882350\pi\)
							0.144699	+	0.989476i	\(0.453779\pi\)
\(13\)	2.41390	+	2.23977i	0.669495	+	0.621201i	0.939861	−	0.341557i	\(-0.110954\pi\)
							−0.270366	+	0.962758i	\(0.587145\pi\)
\(17\)	1.50773	−	4.88795i	0.365679	−	1.18550i	−0.566671	−	0.823944i	\(-0.691769\pi\)
							0.932350	−	0.361558i	\(-0.117755\pi\)
\(19\)	0.553315	−	3.67100i	0.126939	−	0.842186i	−0.831219	−	0.555946i	\(-0.812356\pi\)
							0.958158	−	0.286240i	\(-0.0924056\pi\)
\(23\)	5.58571	+	2.19223i	1.16470	+	0.457111i	0.867370	−	0.497663i	\(-0.165808\pi\)
							0.297331	+	0.954775i	\(0.403904\pi\)
\(29\)	−0.164314	+	2.19261i	−0.0305123	+	0.407158i	0.960946	+	0.276735i	\(0.0892525\pi\)
							−0.991458	+	0.130423i	\(0.958367\pi\)
\(31\)	6.48749	+	4.42310i	1.16519	+	0.794412i	0.982040	−	0.188674i	\(-0.0604189\pi\)
							0.183148	+	0.983085i	\(0.441371\pi\)
\(37\)	−1.81573	+	1.04831i	−0.298504	+	0.172341i	−0.641771	−	0.766897i	\(-0.721800\pi\)
							0.343267	+	0.939238i	\(0.388466\pi\)
\(41\)	2.79453	+	5.80290i	0.436432	+	0.906260i	0.996944	+	0.0781142i	\(0.0248899\pi\)
							−0.560513	+	0.828146i	\(0.689396\pi\)
\(43\)	−4.62574	−	4.64785i	−0.705419	−	0.708790i
							\(47\)	5.46298	+	4.35658i	0.796857	+	0.635472i	0.934883	−	0.354956i	\(-0.115504\pi\)
−0.138026	+	0.990429i	\(0.544076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.n.b.29.7	yes	144
3.2	odd	2	inner	129.2.n.b.29.6	✓	144
43.3	odd	42	inner	129.2.n.b.89.6	yes	144
129.89	even	42	inner	129.2.n.b.89.7	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.n.b.29.6	✓	144	3.2	odd	2	inner
129.2.n.b.29.7	yes	144	1.1	even	1	trivial
129.2.n.b.89.6	yes	144	43.3	odd	42	inner
129.2.n.b.89.7	yes	144	129.89	even	42	inner