Embedded newform 130.3.t.b.89.2

• Label: 130.3.t.b.89.2
• Level: $130$
• Weight: $3$
• Character: 130.89
• 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			89.2
Character	\(\chi\)	\(=\)	130.89
Dual form			130.3.t.b.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(-2.11647 + 1.22195i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-3.39373 + 3.67186i) q^{5} +(-3.33842 + 0.894526i) q^{6} +(-2.98209 + 0.799048i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-1.51370 + 2.62180i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	+	0.366025i	0.683013	+	0.183013i
							\(3\)	−2.11647	+	1.22195i	−0.705491	+	0.407315i	−0.809389	−	0.587273i	\(-0.800202\pi\)
0.103899	+	0.994588i	\(0.466868\pi\)
\(5\)	−3.39373	+	3.67186i	−0.678746	+	0.734373i
							\(7\)	−2.98209	+	0.799048i	−0.426013	+	0.114150i	−0.465454	−	0.885072i	\(-0.654109\pi\)
0.0394412	+	0.999222i	\(0.487442\pi\)
\(11\)	0.0167932	+	0.00449973i	0.00152666	+	0.000409066i	0.259582	−	0.965721i	\(-0.416415\pi\)
							−0.258056	+	0.966130i	\(0.583082\pi\)
\(13\)	−5.70790	+	11.6799i	−0.439069	+	0.898453i
							\(17\)	2.27820	−	3.94596i	0.134012	−	0.232115i	−0.791208	−	0.611547i	\(-0.790547\pi\)
0.925219	+	0.379432i	\(0.123881\pi\)
\(19\)	15.2168	−	4.07734i	0.800886	−	0.214597i	0.164913	−	0.986308i	\(-0.447266\pi\)
							0.635973	+	0.771711i	\(0.280599\pi\)
\(23\)	16.6963	+	28.9188i	0.725924	+	1.25734i	0.958592	+	0.284781i	\(0.0919210\pi\)
							−0.232668	+	0.972556i	\(0.574746\pi\)
\(29\)	−5.13949	−	8.90186i	−0.177224	−	0.306961i	0.763705	−	0.645566i	\(-0.223378\pi\)
							−0.940929	+	0.338605i	\(0.890045\pi\)
\(31\)	30.4309	+	30.4309i	0.981643	+	0.981643i	0.999835	−	0.0181916i	\(-0.00579088\pi\)
							−0.0181916	+	0.999835i	\(0.505791\pi\)
\(37\)	14.7271	−	54.9624i	0.398030	−	1.48547i	−0.418526	−	0.908205i	\(-0.637453\pi\)
							0.816557	−	0.577265i	\(-0.195880\pi\)
\(41\)	−13.4117	+	50.0532i	−0.327115	+	1.22081i	0.585054	+	0.810995i	\(0.301073\pi\)
							−0.912169	+	0.409815i	\(0.865593\pi\)
\(43\)	22.3872	−	38.7757i	0.520632	−	0.901761i	−0.479080	−	0.877771i	\(-0.659030\pi\)
							0.999712	−	0.0239899i	\(-0.00763694\pi\)
\(47\)	26.3101	+	26.3101i	0.559790	+	0.559790i	0.929248	−	0.369457i	\(-0.120456\pi\)
							−0.369457	+	0.929248i	\(0.620456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.t.b.89.2	yes	28
5.4	even	2		130.3.t.a.89.6	yes	28
13.6	odd	12		130.3.t.a.19.6	✓	28
65.19	odd	12	inner	130.3.t.b.19.2	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.t.a.19.6	✓	28	13.6	odd	12
130.3.t.a.89.6	yes	28	5.4	even	2
130.3.t.b.19.2	yes	28	65.19	odd	12	inner
130.3.t.b.89.2	yes	28	1.1	even	1	trivial