Embedded newform 130.3.t.b.59.2

• Label: 130.3.t.b.59.2
• Level: $130$
• Weight: $3$
• Character: 130.59
• Analytic conductor: $3.542$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 130 = 2 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	130.t (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54224343668\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			59.2
Character	\(\chi\)	\(=\)	130.59
Dual form			130.3.t.b.119.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-3.60508 - 2.08139i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.72235 + 4.69399i) q^{5} +(-1.52369 + 5.68647i) q^{6} +(-1.33853 + 4.99548i) q^{7} +(2.00000 + 2.00000i) q^{8} +(4.16440 + 7.21295i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.366025	−	1.36603i	−0.183013	−	0.683013i
							\(3\)	−3.60508	−	2.08139i	−1.20169	−	0.693798i	−0.240762	−	0.970584i	\(-0.577397\pi\)
−0.960932	+	0.276786i	\(0.910731\pi\)
\(5\)	1.72235	+	4.69399i	0.344469	+	0.938798i
							\(7\)	−1.33853	+	4.99548i	−0.191219	+	0.713640i	0.801994	+	0.597332i	\(0.203773\pi\)
−0.993213	+	0.116308i	\(0.962894\pi\)
\(11\)	−0.867168	−	3.23632i	−0.0788335	−	0.294211i	0.915242	−	0.402905i	\(-0.132000\pi\)
							−0.994075	+	0.108695i	\(0.965333\pi\)
\(13\)	12.9265	+	1.38040i	0.994346	+	0.106185i
							\(17\)	11.3437	+	19.6479i	0.667276	+	1.15576i	0.978663	+	0.205473i	\(0.0658732\pi\)
−0.311387	+	0.950283i	\(0.600793\pi\)
\(19\)	−5.16642	+	19.2813i	−0.271917	+	1.01481i	0.685963	+	0.727636i	\(0.259381\pi\)
							−0.957880	+	0.287170i	\(0.907285\pi\)
\(23\)	−0.420908	+	0.729034i	−0.0183003	+	0.0316971i	−0.875031	−	0.484068i	\(-0.839159\pi\)
							0.856730	+	0.515765i	\(0.172492\pi\)
\(29\)	−17.0748	+	29.5744i	−0.588785	+	1.01981i	0.405606	+	0.914048i	\(0.367060\pi\)
							−0.994392	+	0.105758i	\(0.966273\pi\)
\(31\)	−12.9743	−	12.9743i	−0.418526	−	0.418526i	0.466169	−	0.884696i	\(-0.345634\pi\)
							−0.884696	+	0.466169i	\(0.845634\pi\)
\(37\)	−27.9928	+	7.50065i	−0.756562	+	0.202720i	−0.616427	−	0.787412i	\(-0.711420\pi\)
							−0.140135	+	0.990132i	\(0.544754\pi\)
\(41\)	59.0441	−	15.8208i	1.44010	−	0.385873i	0.547531	−	0.836785i	\(-0.315568\pi\)
							0.892568	+	0.450912i	\(0.148901\pi\)
\(43\)	2.39577	+	4.14960i	0.0557156	+	0.0965022i	0.892538	−	0.450972i	\(-0.148923\pi\)
							−0.836822	+	0.547474i	\(0.815589\pi\)
\(47\)	9.17171	+	9.17171i	0.195143	+	0.195143i	0.797914	−	0.602771i	\(-0.205937\pi\)
							−0.602771	+	0.797914i	\(0.705937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.59.2	yes	28
5.4	even	2		130.3.t.a.59.6	✓	28
13.2	odd	12		130.3.t.a.119.6	yes	28
65.54	odd	12	inner	130.3.t.b.119.2	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.59.6	✓	28	5.4	even	2
130.3.t.a.119.6	yes	28	13.2	odd	12
130.3.t.b.59.2	yes	28	1.1	even	1	trivial
130.3.t.b.119.2	yes	28	65.54	odd	12	inner