Embedded newform 130.3.t.b.59.4

• Label: 130.3.t.b.59.4
• 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			59.4
Character	\(\chi\)	\(=\)	130.59
Dual form			130.3.t.b.119.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(0.787112 + 0.454439i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-4.77553 + 1.48134i) q^{5} +(0.332673 - 1.24155i) q^{6} +(-3.19988 + 11.9421i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-4.08697 - 7.07884i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	0.787112	+	0.454439i	0.262371	+	0.151480i	0.625415	−	0.780292i	\(-0.284930\pi\)
−0.363045	+	0.931772i	\(0.618263\pi\)
\(5\)	−4.77553	+	1.48134i	−0.955105	+	0.296268i
							\(7\)	−3.19988	+	11.9421i	−0.457125	+	1.70601i	0.224639	+	0.974442i	\(0.427880\pi\)
−0.681764	+	0.731572i	\(0.738787\pi\)
\(11\)	4.17241	+	15.5716i	0.379310	+	1.41560i	0.846944	+	0.531681i	\(0.178440\pi\)
							−0.467635	+	0.883922i	\(0.654894\pi\)
\(13\)	−11.0274	−	6.88456i	−0.848259	−	0.529582i
							\(17\)	5.88525	+	10.1936i	0.346191	+	0.599621i	0.985569	−	0.169271i	\(-0.0541415\pi\)
−0.639378	+	0.768892i	\(0.720808\pi\)
\(19\)	3.13113	−	11.6856i	0.164797	−	0.615029i	−0.833270	−	0.552867i	\(-0.813534\pi\)
							0.998066	−	0.0621621i	\(-0.0197996\pi\)
\(23\)	−11.0226	+	19.0917i	−0.479244	+	0.830075i	−0.999717	−	0.0238035i	\(-0.992422\pi\)
							0.520473	+	0.853878i	\(0.325756\pi\)
\(29\)	−14.2283	+	24.6442i	−0.490632	+	0.849799i	−0.999942	−	0.0107842i	\(-0.996567\pi\)
							0.509310	+	0.860583i	\(0.329901\pi\)
\(31\)	11.4919	+	11.4919i	0.370706	+	0.370706i	0.867734	−	0.497028i	\(-0.165576\pi\)
							−0.497028	+	0.867734i	\(0.665576\pi\)
\(37\)	11.4335	−	3.06360i	0.309014	−	0.0828001i	−0.100979	−	0.994889i	\(-0.532198\pi\)
							0.409994	+	0.912088i	\(0.365531\pi\)
\(41\)	−2.73714	+	0.733413i	−0.0667594	+	0.0178881i	−0.292044	−	0.956405i	\(-0.594335\pi\)
							0.225285	+	0.974293i	\(0.427669\pi\)
\(43\)	−21.2673	−	36.8360i	−0.494587	−	0.856650i	0.505393	−	0.862889i	\(-0.331347\pi\)
							−0.999981	+	0.00623874i	\(0.998014\pi\)
\(47\)	−31.6084	−	31.6084i	−0.672520	−	0.672520i	0.285777	−	0.958296i	\(-0.407748\pi\)
							−0.958296	+	0.285777i	\(0.907748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.59.4	yes	28
5.4	even	2		130.3.t.a.59.4	✓	28
13.2	odd	12		130.3.t.a.119.4	yes	28
65.54	odd	12	inner	130.3.t.b.119.4	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.59.4	✓	28	5.4	even	2
130.3.t.a.119.4	yes	28	13.2	odd	12
130.3.t.b.59.4	yes	28	1.1	even	1	trivial
130.3.t.b.119.4	yes	28	65.54	odd	12	inner