Embedded newform 130.3.o.a.11.3

• Label: 130.3.o.a.11.3
• Level: $130$
• Weight: $3$
• Character: 130.11
• Analytic conductor: $3.542$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54224343668\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 3632*x^12 - 14876*x^11 + 62910*x^10 - 190580*x^9 + 559649*x^8 - 1237588*x^7 + 2523282*x^6 - 3897292*x^5 + 5230208*x^4 - 5098228*x^3 + 3909898*x^2 - 1850548*x + 404521
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			11.3
Root			\(0.500000 + 2.95698i\) of defining polynomial
Character	\(\chi\)	\(=\)	130.11
Dual form			130.3.o.a.71.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 - 1.36603i) q^{2} +(0.248375 + 0.430197i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(1.58114 + 1.58114i) q^{5} +(0.678572 - 0.181823i) q^{6} +(-3.32551 - 12.4110i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(4.37662 - 7.58053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{2} - 12 q^{6} + 16 q^{7} - 32 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	−	1.36603i	0.183013	−	0.683013i
							\(3\)	0.248375	+	0.430197i	0.0827915	+	0.143399i	0.904448	−	0.426584i	\(-0.140283\pi\)
−0.821656	+	0.569983i	\(0.806950\pi\)
\(5\)	1.58114	+	1.58114i	0.316228	+	0.316228i
							\(7\)	−3.32551	−	12.4110i	−0.475073	−	1.77300i	−0.621116	−	0.783719i	\(-0.713320\pi\)
0.146043	−	0.989278i	\(-0.453346\pi\)
\(11\)	6.17422	+	1.65438i	0.561292	+	0.150398i	0.528300	−	0.849058i	\(-0.322830\pi\)
							0.0329924	+	0.999456i	\(0.489496\pi\)
\(13\)	11.8845	−	5.26866i	0.914192	−	0.405282i
							\(17\)	−3.56159	−	2.05628i	−0.209505	−	0.120958i	0.391576	−	0.920146i	\(-0.371930\pi\)
−0.601081	+	0.799188i	\(0.705263\pi\)
\(19\)	−9.72165	+	2.60491i	−0.511666	+	0.137100i	−0.505409	−	0.862880i	\(-0.668658\pi\)
							−0.00625682	+	0.999980i	\(0.501992\pi\)
\(23\)	−25.1615	+	14.5270i	−1.09398	+	0.631609i	−0.934633	−	0.355614i	\(-0.884272\pi\)
							−0.159346	+	0.987223i	\(0.550938\pi\)
\(29\)	20.5917	+	35.6659i	0.710059	+	1.22986i	0.964835	+	0.262858i	\(0.0846649\pi\)
							−0.254776	+	0.967000i	\(0.582002\pi\)
\(31\)	12.4428	+	12.4428i	0.401379	+	0.401379i	0.878719	−	0.477340i	\(-0.158399\pi\)
							−0.477340	+	0.878719i	\(0.658399\pi\)
\(37\)	18.6637	+	5.00092i	0.504424	+	0.135160i	0.502053	−	0.864837i	\(-0.332578\pi\)
							0.00237179	+	0.999997i	\(0.499245\pi\)
\(41\)	−0.696348	+	2.59881i	−0.0169841	+	0.0633856i	−0.973898	−	0.226987i	\(-0.927113\pi\)
							0.956914	+	0.290372i	\(0.0937792\pi\)
\(43\)	−9.14607	−	5.28049i	−0.212699	−	0.122802i	0.389866	−	0.920872i	\(-0.372521\pi\)
							−0.602565	+	0.798070i	\(0.705855\pi\)
\(47\)	32.3417	−	32.3417i	0.688121	−	0.688121i	−0.273695	−	0.961816i	\(-0.588246\pi\)
							0.961816	+	0.273695i	\(0.0882459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.o.a.11.3	✓	16
13.6	odd	12	inner	130.3.o.a.71.3	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.o.a.11.3	✓	16	1.1	even	1	trivial
130.3.o.a.71.3	yes	16	13.6	odd	12	inner