Embedded newform 130.3.t.b.59.6

• Label: 130.3.t.b.59.6
• 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			59.6
Character	\(\chi\)	\(=\)	130.59
Dual form			130.3.t.b.119.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(3.56714 + 2.05949i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(4.98861 + 0.337300i) q^{5} +(1.50765 - 5.62664i) q^{6} +(-2.94974 + 11.0086i) q^{7} +(2.00000 + 2.00000i) q^{8} +(3.98301 + 6.89878i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	3.56714	+	2.05949i	1.18905	+	0.686497i	0.958090	−	0.286466i	\(-0.0924807\pi\)
0.230958	+	0.972964i	\(0.425814\pi\)
\(5\)	4.98861	+	0.337300i	0.997722	+	0.0674600i
							\(7\)	−2.94974	+	11.0086i	−0.421391	+	1.57265i	0.350290	+	0.936641i	\(0.386083\pi\)
−0.771681	+	0.636010i	\(0.780584\pi\)
\(11\)	−3.67058	−	13.6988i	−0.333689	−	1.24534i	−0.905284	−	0.424808i	\(-0.860342\pi\)
							0.571595	−	0.820536i	\(-0.306325\pi\)
\(13\)	−0.399942	+	12.9938i	−0.0307648	+	0.999527i
							\(17\)	−1.99828	−	3.46112i	−0.117546	−	0.203595i	0.801249	−	0.598331i	\(-0.204169\pi\)
−0.918795	+	0.394736i	\(0.870836\pi\)
\(19\)	5.33451	−	19.9086i	0.280763	−	1.04782i	−0.671116	−	0.741352i	\(-0.734185\pi\)
							0.951880	−	0.306472i	\(-0.0991484\pi\)
\(23\)	17.1660	−	29.7323i	0.746347	−	1.29271i	−0.203216	−	0.979134i	\(-0.565139\pi\)
							0.949563	−	0.313577i	\(-0.101527\pi\)
\(29\)	8.69370	−	15.0579i	0.299783	−	0.519239i	−0.676303	−	0.736623i	\(-0.736419\pi\)
							0.976086	+	0.217384i	\(0.0697525\pi\)
\(31\)	−7.34836	−	7.34836i	−0.237044	−	0.237044i	0.578581	−	0.815625i	\(-0.303607\pi\)
							−0.815625	+	0.578581i	\(0.803607\pi\)
\(37\)	−47.2344	+	12.6564i	−1.27661	+	0.342065i	−0.832558	−	0.553938i	\(-0.813125\pi\)
							−0.444048	+	0.896003i	\(0.646458\pi\)
\(41\)	−54.2508	+	14.5365i	−1.32319	+	0.354548i	−0.850172	−	0.526504i	\(-0.823502\pi\)
							−0.473019	+	0.881052i	\(0.656836\pi\)
\(43\)	19.2854	+	33.4033i	0.448497	+	0.776820i	0.998288	−	0.0584820i	\(-0.0186260\pi\)
							−0.549791	+	0.835302i	\(0.685293\pi\)
\(47\)	−5.74456	−	5.74456i	−0.122225	−	0.122225i	0.643349	−	0.765573i	\(-0.277545\pi\)
							−0.765573	+	0.643349i	\(0.777545\pi\)

(See \(a_n\) instead)

Twists

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

    

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