Embedded newform 129.2.m.b.103.1

• Label: 129.2.m.b.103.1
• Level: $129$
• Weight: $2$
• Character: 129.103
• Analytic conductor: $1.030$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.m (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03007018607\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			103.1
Character	\(\chi\)	\(=\)	129.103
Dual form			129.2.m.b.124.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500155 - 2.19132i) q^{2} +(0.955573 - 0.294755i) q^{3} +(-2.74980 + 1.32423i) q^{4} +(0.873904 - 2.22667i) q^{5} +(-1.12384 - 1.94654i) q^{6} +(-0.751867 + 1.30227i) q^{7} +(1.47434 + 1.84876i) q^{8} +(0.826239 - 0.563320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} + 4 q^{3} - 2 q^{4} + q^{5} - q^{6} - 16 q^{7} - 14 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500155	−	2.19132i	−0.353663	−	1.54950i	−0.768649	−	0.639671i	\(-0.779071\pi\)
							0.414986	−	0.909828i	\(-0.363786\pi\)
\(3\)	0.955573	−	0.294755i	0.551700	−	0.170177i
							\(5\)	0.873904	−	2.22667i	0.390822	−	0.995797i	−0.591115	−	0.806587i	\(-0.701312\pi\)
0.981937	−	0.189210i	\(-0.0605927\pi\)
\(7\)	−0.751867	+	1.30227i	−0.284179	+	0.492212i	−0.972410	−	0.233279i	\(-0.925054\pi\)
							0.688231	+	0.725492i	\(0.258388\pi\)
\(11\)	−0.755238	−	0.363704i	−0.227713	−	0.109661i	0.316549	−	0.948576i	\(-0.397476\pi\)
							−0.544262	+	0.838915i	\(0.683190\pi\)
\(13\)	1.17429	−	0.176995i	0.325689	−	0.0490897i	0.0158372	−	0.999875i	\(-0.494959\pi\)
							0.309852	+	0.950785i	\(0.399721\pi\)
\(17\)	2.76679	+	7.04966i	0.671044	+	1.70979i	0.703867	+	0.710332i	\(0.251455\pi\)
							−0.0328220	+	0.999461i	\(0.510449\pi\)
\(19\)	−0.395164	−	0.269418i	−0.0906568	−	0.0618088i	0.517143	−	0.855899i	\(-0.326995\pi\)
							−0.607800	+	0.794090i	\(0.707948\pi\)
\(23\)	−0.516586	−	6.89336i	−0.107716	−	1.43736i	−0.745169	−	0.666875i	\(-0.767631\pi\)
							0.637454	−	0.770489i	\(-0.279988\pi\)
\(29\)	5.34249	+	1.64794i	0.992075	+	0.306015i	0.747969	−	0.663734i	\(-0.231029\pi\)
							0.244106	+	0.969749i	\(0.421505\pi\)
\(31\)	1.97992	−	1.83710i	0.355604	−	0.329952i	−0.482060	−	0.876138i	\(-0.660111\pi\)
							0.837664	+	0.546186i	\(0.183921\pi\)
\(37\)	1.64929	+	2.85666i	0.271142	+	0.469631i	0.969154	−	0.246454i	\(-0.0792655\pi\)
							−0.698013	+	0.716085i	\(0.745932\pi\)
\(41\)	−2.80172	−	12.2751i	−0.437555	−	1.91705i	−0.396854	−	0.917882i	\(-0.629898\pi\)
							−0.0407012	−	0.999171i	\(-0.512959\pi\)
\(43\)	1.04534	+	6.47358i	0.159413	+	0.987212i
							\(47\)	−10.0684	+	4.84869i	−1.46863	+	0.707255i	−0.985716	−	0.168414i	\(-0.946135\pi\)
−0.482913	+	0.875668i	\(0.660421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.m.b.103.1	✓	48
3.2	odd	2		387.2.y.d.361.4		48
43.9	even	21		5547.2.a.bb.1.20		24
43.34	odd	42		5547.2.a.ba.1.5		24
43.38	even	21	inner	129.2.m.b.124.1	yes	48
129.38	odd	42		387.2.y.d.253.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.m.b.103.1	✓	48	1.1	even	1	trivial
129.2.m.b.124.1	yes	48	43.38	even	21	inner
387.2.y.d.253.4		48	129.38	odd	42
387.2.y.d.361.4		48	3.2	odd	2
5547.2.a.ba.1.5		24	43.34	odd	42
5547.2.a.bb.1.20		24	43.9	even	21