Embedded newform 129.2.j.a.113.5

• Label: 129.2.j.a.113.5
• Level: $129$
• Weight: $2$
• Character: 129.113
• Analytic conductor: $1.030$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03007018607\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			113.5
Character	\(\chi\)	\(=\)	129.113
Dual form			129.2.j.a.8.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.178779 - 0.783281i) q^{2} +(-1.72901 - 0.102631i) q^{3} +(1.22037 - 0.587699i) q^{4} +(2.53156 + 3.17448i) q^{5} +(0.228721 + 1.37265i) q^{6} -3.17594i q^{7} +(-1.68036 - 2.10711i) q^{8} +(2.97893 + 0.354901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.178779	−	0.783281i	−0.126416	−	0.553863i	−0.997977	−	0.0635767i	\(-0.979749\pi\)
							0.871561	−	0.490287i	\(-0.163108\pi\)
\(3\)	−1.72901	−	0.102631i	−0.998243	−	0.0592542i
							\(5\)	2.53156	+	3.17448i	1.13215	+	1.41967i	0.893785	+	0.448495i	\(0.148040\pi\)
0.238364	+	0.971176i	\(0.423389\pi\)
\(7\)		−	3.17594i		−	1.20039i	−0.799852	−	0.600197i	\(-0.795089\pi\)
							0.799852	−	0.600197i	\(-0.204911\pi\)
\(11\)	0.516214	−	1.07193i	0.155644	−	0.323199i	−0.808536	−	0.588447i	\(-0.799740\pi\)
							0.964180	+	0.265248i	\(0.0854539\pi\)
\(13\)	0.593805	+	0.744608i	0.164692	+	0.206517i	0.857329	−	0.514770i	\(-0.172123\pi\)
							−0.692637	+	0.721287i	\(0.743551\pi\)
\(17\)	−0.109436	−	0.0872725i	−0.0265422	−	0.0211667i	0.610129	−	0.792302i	\(-0.291118\pi\)
							−0.636671	+	0.771136i	\(0.719689\pi\)
\(19\)	−0.157259	−	0.326551i	−0.0360776	−	0.0749159i	0.882151	−	0.470967i	\(-0.156095\pi\)
							−0.918229	+	0.396051i	\(0.870380\pi\)
\(23\)	−3.80396	+	7.89901i	−0.793181	+	1.64706i	−0.0311525	+	0.999515i	\(0.509918\pi\)
							−0.762029	+	0.647543i	\(0.775797\pi\)
\(29\)	−1.25733	−	5.50874i	−0.233481	−	1.02295i	−0.946728	−	0.322034i	\(-0.895633\pi\)
							0.713247	−	0.700913i	\(-0.247224\pi\)
\(31\)	1.51188	+	6.62398i	0.271542	+	1.18970i	0.908194	+	0.418550i	\(0.137462\pi\)
							−0.636652	+	0.771151i	\(0.719681\pi\)
\(37\)		−	1.16757i		−	0.191947i	−0.995384	−	0.0959733i	\(-0.969404\pi\)
							0.995384	−	0.0959733i	\(-0.0305963\pi\)
\(41\)	−6.60740	+	1.50809i	−1.03190	+	0.235525i	−0.704757	−	0.709449i	\(-0.748944\pi\)
							−0.327145	+	0.944974i	\(0.606087\pi\)
\(43\)	−5.93308	−	2.79260i	−0.904786	−	0.425867i
							\(47\)	−0.952840	−	1.97859i	−0.138986	−	0.288607i	0.819845	−	0.572586i	\(-0.194060\pi\)
−0.958831	+	0.283979i	\(0.908345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.j.a.113.5	yes	72
3.2	odd	2	inner	129.2.j.a.113.8	yes	72
43.8	odd	14	inner	129.2.j.a.8.8	yes	72
129.8	even	14	inner	129.2.j.a.8.5	✓	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.j.a.8.5	✓	72	129.8	even	14	inner
129.2.j.a.8.8	yes	72	43.8	odd	14	inner
129.2.j.a.113.5	yes	72	1.1	even	1	trivial
129.2.j.a.113.8	yes	72	3.2	odd	2	inner