Embedded newform 129.2.n.b.116.3

• Label: 129.2.n.b.116.3
• Level: $129$
• Weight: $2$
• Character: 129.116
• 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			116.3
Character	\(\chi\)	\(=\)	129.116
Dual form			129.2.n.b.119.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.427041 + 1.87099i) q^{2} +(-1.37920 - 1.04776i) q^{3} +(-1.51629 - 0.730208i) q^{4} +(-2.44115 - 0.367944i) q^{5} +(2.54932 - 2.13303i) q^{6} +(-3.43034 - 1.98051i) q^{7} +(-0.379355 + 0.475696i) q^{8} +(0.804395 + 2.89015i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.427041	+	1.87099i	−0.301963	+	1.32299i	0.565196	+	0.824957i	\(0.308801\pi\)
							−0.867159	+	0.498031i	\(0.834057\pi\)
\(3\)	−1.37920	−	1.04776i	−0.796283	−	0.604925i
							\(5\)	−2.44115	−	0.367944i	−1.09171	−	0.164549i	−0.421581	−	0.906791i	\(-0.638525\pi\)
−0.670133	+	0.742241i	\(0.733763\pi\)
\(7\)	−3.43034	−	1.98051i	−1.29655	−	0.748563i	−0.316742	−	0.948512i	\(-0.602589\pi\)
							−0.979806	+	0.199949i	\(0.935922\pi\)
\(11\)	−1.56164	−	3.24278i	−0.470853	−	0.977736i	−0.992231	−	0.124410i	\(-0.960296\pi\)
							0.521378	−	0.853326i	\(-0.325418\pi\)
\(13\)	1.33644	+	3.40518i	0.370661	+	0.944428i	0.987540	+	0.157370i	\(0.0503015\pi\)
							−0.616879	+	0.787058i	\(0.711603\pi\)
\(17\)	0.256603	+	1.70245i	0.0622355	+	0.412905i	0.998156	+	0.0606951i	\(0.0193317\pi\)
							−0.935921	+	0.352210i	\(0.885430\pi\)
\(19\)	2.42012	−	0.181363i	0.555213	−	0.0416074i	0.205830	−	0.978588i	\(-0.434010\pi\)
							0.349382	+	0.936980i	\(0.386391\pi\)
\(23\)	−3.99059	+	5.85311i	−0.832095	+	1.22046i	0.140990	+	0.990011i	\(0.454971\pi\)
							−0.973085	+	0.230448i	\(0.925981\pi\)
\(29\)	−6.03082	−	5.59578i	−1.11989	−	1.03911i	−0.998920	−	0.0464547i	\(-0.985208\pi\)
							−0.120974	−	0.992656i	\(-0.538602\pi\)
\(31\)	−1.86493	+	0.575256i	−0.334952	+	0.103319i	−0.457669	−	0.889123i	\(-0.651316\pi\)
							0.122717	+	0.992442i	\(0.460839\pi\)
\(37\)	−3.03244	+	1.75078i	−0.498529	+	0.287826i	−0.728106	−	0.685464i	\(-0.759599\pi\)
							0.229577	+	0.973291i	\(0.426266\pi\)
\(41\)	4.00465	+	0.914036i	0.625422	+	0.142748i	0.523475	−	0.852041i	\(-0.324636\pi\)
							0.101947	+	0.994790i	\(0.467493\pi\)
\(43\)	−2.98916	+	5.83652i	−0.455843	+	0.890060i
							\(47\)	3.99769	−	8.30128i	0.583123	−	1.21087i	−0.375669	−	0.926754i	\(-0.622587\pi\)
0.958791	−	0.284112i	\(-0.0916988\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.n.b.116.3	✓	144
3.2	odd	2	inner	129.2.n.b.116.10	yes	144
43.33	odd	42	inner	129.2.n.b.119.10	yes	144
129.119	even	42	inner	129.2.n.b.119.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.n.b.116.3	✓	144	1.1	even	1	trivial
129.2.n.b.116.10	yes	144	3.2	odd	2	inner
129.2.n.b.119.3	yes	144	129.119	even	42	inner
129.2.n.b.119.10	yes	144	43.33	odd	42	inner