Embedded newform 130.3.o.a.41.1

• Label: 130.3.o.a.41.1
• Level: $130$
• Weight: $3$
• Character: 130.41
• 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			41.1
Root			\(0.500000 - 4.40016i\) of defining polynomial
Character	\(\chi\)	\(=\)	130.41
Dual form			130.3.o.a.111.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(-1.18833 - 2.05824i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-1.58114 + 1.58114i) q^{5} +(0.869917 + 3.24657i) q^{6} +(-6.47546 + 1.73509i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(1.67575 - 2.90249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{2} - 12 q^{6} + 16 q^{7} - 32 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−1.18833	−	2.05824i	−0.396109	−	0.686082i	0.597133	−	0.802142i	\(-0.296307\pi\)
−0.993242	+	0.116061i	\(0.962973\pi\)
\(5\)	−1.58114	+	1.58114i	−0.316228	+	0.316228i
							\(7\)	−6.47546	+	1.73509i	−0.925065	+	0.247871i	−0.689749	−	0.724048i	\(-0.742279\pi\)
−0.235316	+	0.971919i	\(0.575613\pi\)
\(11\)	−3.49343	+	13.0377i	−0.317585	+	1.18524i	0.603974	+	0.797004i	\(0.293583\pi\)
							−0.921559	+	0.388238i	\(0.873084\pi\)
\(13\)	−2.97144	+	12.6559i	−0.228572	+	0.973527i
							\(17\)	−4.37387	−	2.52526i	−0.257287	−	0.148545i	0.365809	−	0.930690i	\(-0.380792\pi\)
−0.623096	+	0.782145i	\(0.714126\pi\)
\(19\)	8.52657	+	31.8216i	0.448767	+	1.67482i	0.705794	+	0.708417i	\(0.250590\pi\)
							−0.257027	+	0.966404i	\(0.582743\pi\)
\(23\)	−20.9728	+	12.1086i	−0.911860	+	0.526463i	−0.881029	−	0.473062i	\(-0.843149\pi\)
							−0.0308309	+	0.999525i	\(0.509815\pi\)
\(29\)	−13.3320	−	23.0917i	−0.459725	−	0.796267i	0.539221	−	0.842164i	\(-0.318719\pi\)
							−0.998946	+	0.0458971i	\(0.985385\pi\)
\(31\)	1.02550	−	1.02550i	0.0330806	−	0.0330806i	−0.690373	−	0.723454i	\(-0.742554\pi\)
							0.723454	+	0.690373i	\(0.242554\pi\)
\(37\)	10.0968	−	37.6816i	0.272885	−	1.01842i	−0.684360	−	0.729144i	\(-0.739918\pi\)
							0.957245	−	0.289277i	\(-0.0934149\pi\)
\(41\)	−32.5360	−	8.71799i	−0.793560	−	0.212634i	−0.160806	−	0.986986i	\(-0.551409\pi\)
							−0.632755	+	0.774352i	\(0.718076\pi\)
\(43\)	58.5994	+	33.8324i	1.36278	+	0.786800i	0.989993	−	0.141118i	\(-0.0450697\pi\)
							0.372785	+	0.927918i	\(0.378403\pi\)
\(47\)	−54.2590	−	54.2590i	−1.15445	−	1.15445i	−0.985651	−	0.168797i	\(-0.946012\pi\)
							−0.168797	−	0.985651i	\(-0.553988\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	130.3.o.a.41.1	✓	16
13.7	odd	12	inner	130.3.o.a.111.1	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.3.o.a.41.1	✓	16	1.1	even	1	trivial
130.3.o.a.111.1	yes	16	13.7	odd	12	inner