Embedded newform 129.2.n.b.104.4

• Label: 129.2.n.b.104.4
• Level: $129$
• Weight: $2$
• Character: 129.104
• 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			104.4
Character	\(\chi\)	\(=\)	129.104
Dual form			129.2.n.b.98.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42761 - 0.687502i) q^{2} +(-0.989858 - 1.42133i) q^{3} +(0.318440 + 0.399311i) q^{4} +(1.37642 - 1.27713i) q^{5} +(0.435967 + 2.70964i) q^{6} +(-2.57128 - 1.48453i) q^{7} +(0.525101 + 2.30062i) q^{8} +(-1.04036 + 2.81383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42761	−	0.687502i	−1.00947	−	0.486137i	−0.145331	−	0.989383i	\(-0.546425\pi\)
							−0.864143	+	0.503246i	\(0.832139\pi\)
\(3\)	−0.989858	−	1.42133i	−0.571495	−	0.820606i
							\(5\)	1.37642	−	1.27713i	0.615553	−	0.571149i	−0.309542	−	0.950886i	\(-0.600176\pi\)
0.925095	+	0.379736i	\(0.123985\pi\)
\(7\)	−2.57128	−	1.48453i	−0.971854	−	0.561100i	−0.0720533	−	0.997401i	\(-0.522955\pi\)
							−0.899801	+	0.436300i	\(0.856288\pi\)
\(11\)	−0.999332	−	0.796941i	−0.301310	−	0.240287i	0.461150	−	0.887322i	\(-0.347437\pi\)
							−0.762460	+	0.647036i	\(0.776009\pi\)
\(13\)	−4.01235	−	1.23764i	−1.11282	−	0.343261i	−0.316780	−	0.948499i	\(-0.602602\pi\)
							−0.796044	+	0.605238i	\(0.793078\pi\)
\(17\)	−2.30773	+	2.48714i	−0.559707	+	0.603220i	−0.947982	−	0.318323i	\(-0.896880\pi\)
							0.388276	+	0.921543i	\(0.373071\pi\)
\(19\)	3.18927	+	1.25170i	0.731669	+	0.287159i	0.701779	−	0.712394i	\(-0.252389\pi\)
							0.0298893	+	0.999553i	\(0.490485\pi\)
\(23\)	1.42778	−	9.47271i	0.297713	−	1.97520i	0.0994979	−	0.995038i	\(-0.468276\pi\)
							0.198215	−	0.980159i	\(-0.436486\pi\)
\(29\)	2.61886	−	1.78551i	0.486310	−	0.331561i	−0.295219	−	0.955430i	\(-0.595393\pi\)
							0.781529	+	0.623869i	\(0.214440\pi\)
\(31\)	−0.593631	−	7.92146i	−0.106619	−	1.42274i	−0.752145	−	0.658997i	\(-0.770981\pi\)
							0.645526	−	0.763738i	\(-0.276638\pi\)
\(37\)	2.50219	−	1.44464i	0.411357	−	0.237497i	−0.280016	−	0.959995i	\(-0.590340\pi\)
							0.691373	+	0.722498i	\(0.257006\pi\)
\(41\)	−0.526991	+	1.09431i	−0.0823021	+	0.170902i	−0.938059	−	0.346477i	\(-0.887378\pi\)
							0.855756	+	0.517379i	\(0.173092\pi\)
\(43\)	−5.81331	−	3.03404i	−0.886522	−	0.462686i
							\(47\)	0.984132	−	0.784819i	0.143550	−	0.114478i	−0.549082	−	0.835769i	\(-0.685022\pi\)
0.692632	+	0.721291i	\(0.256451\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.n.b.104.4	yes	144
3.2	odd	2	inner	129.2.n.b.104.9	yes	144
43.12	odd	42	inner	129.2.n.b.98.9	yes	144
129.98	even	42	inner	129.2.n.b.98.4	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.n.b.98.4	✓	144	129.98	even	42	inner
129.2.n.b.98.9	yes	144	43.12	odd	42	inner
129.2.n.b.104.4	yes	144	1.1	even	1	trivial
129.2.n.b.104.9	yes	144	3.2	odd	2	inner