Embedded newform 129.2.m.b.100.1

• Label: 129.2.m.b.100.1
• Level: $129$
• Weight: $2$
• Character: 129.100
• 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			100.1
Character	\(\chi\)	\(=\)	129.100
Dual form			129.2.m.b.40.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.26929 + 1.09283i) q^{2} +(0.0747301 + 0.997204i) q^{3} +(2.70841 - 3.39624i) q^{4} +(-1.86451 + 0.575125i) q^{5} +(-1.25936 - 2.18128i) q^{6} +(-1.60016 + 2.77155i) q^{7} +(-1.31371 + 5.75575i) q^{8} +(-0.988831 + 0.149042i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} + 4 q^{3} - 2 q^{4} + q^{5} - q^{6} - 16 q^{7} - 14 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.26929	+	1.09283i	−1.60463	+	0.772749i	−0.999723	−	0.0235473i	\(-0.992504\pi\)
							−0.604907	+	0.796296i	\(0.706790\pi\)
\(3\)	0.0747301	+	0.997204i	0.0431454	+	0.575736i
							\(5\)	−1.86451	+	0.575125i	−0.833834	+	0.257204i	−0.682138	−	0.731224i	\(-0.738950\pi\)
−0.151696	+	0.988427i	\(0.548473\pi\)
\(7\)	−1.60016	+	2.77155i	−0.604802	+	1.04755i	0.387281	+	0.921962i	\(0.373414\pi\)
							−0.992083	+	0.125586i	\(0.959919\pi\)
\(11\)	−2.48708	−	3.11871i	−0.749884	−	0.940325i	0.249724	−	0.968317i	\(-0.419660\pi\)
							−0.999608	+	0.0279921i	\(0.991089\pi\)
\(13\)	−2.72351	−	2.52705i	−0.755366	−	0.700878i	0.205264	−	0.978707i	\(-0.434195\pi\)
							−0.960631	+	0.277829i	\(0.910385\pi\)
\(17\)	4.46660	+	1.37776i	1.08331	+	0.334157i	0.784483	−	0.620151i	\(-0.212929\pi\)
							0.298827	+	0.954307i	\(0.403405\pi\)
\(19\)	−7.13439	−	1.07534i	−1.63674	−	0.246699i	−0.734714	−	0.678377i	\(-0.762684\pi\)
							−0.902028	+	0.431678i	\(0.857922\pi\)
\(23\)	−0.951682	+	2.42485i	−0.198439	+	0.505615i	−0.995107	−	0.0988022i	\(-0.968499\pi\)
							0.796668	+	0.604418i	\(0.206594\pi\)
\(29\)	−0.319731	+	4.26651i	−0.0593726	+	0.792272i	0.885459	+	0.464717i	\(0.153844\pi\)
							−0.944832	+	0.327555i	\(0.893775\pi\)
\(31\)	7.19970	+	4.90867i	1.29310	+	0.881624i	0.997286	−	0.0736204i	\(-0.0234553\pi\)
							0.295818	+	0.955244i	\(0.404408\pi\)
\(37\)	2.24583	+	3.88990i	0.369213	+	0.639495i	0.989443	−	0.144925i	\(-0.0462940\pi\)
							−0.620230	+	0.784420i	\(0.712961\pi\)
\(41\)	−1.16952	+	0.563212i	−0.182649	+	0.0879589i	−0.522973	−	0.852349i	\(-0.675177\pi\)
							0.340325	+	0.940308i	\(0.389463\pi\)
\(43\)	−5.29434	+	3.86911i	−0.807379	+	0.590034i
							\(47\)	−6.13115	+	7.68822i	−0.894320	+	1.12144i	0.0976814	+	0.995218i	\(0.468857\pi\)
−0.992002	+	0.126224i	\(0.959714\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.m.b.100.1	yes	48
3.2	odd	2		387.2.y.d.100.4		48
43.13	even	21		5547.2.a.bb.1.23		24
43.30	odd	42		5547.2.a.ba.1.2		24
43.40	even	21	inner	129.2.m.b.40.1	✓	48
129.83	odd	42		387.2.y.d.298.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.m.b.40.1	✓	48	43.40	even	21	inner
129.2.m.b.100.1	yes	48	1.1	even	1	trivial
387.2.y.d.100.4		48	3.2	odd	2
387.2.y.d.298.4		48	129.83	odd	42
5547.2.a.ba.1.2		24	43.30	odd	42
5547.2.a.bb.1.23		24	43.13	even	21