Embedded newform 115.4.g.a.6.1

• Label: 115.4.g.a.6.1
• Level: $115$
• Weight: $4$
• Character: 115.6
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $110$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(110\)
Relative dimension:	\(11\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			6.1
Character	\(\chi\)	\(=\)	115.6
Dual form			115.4.g.a.96.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.30963 + 5.05739i) q^{2} +(-3.54080 - 1.03967i) q^{3} +(-15.0039 - 17.3154i) q^{4} +(4.20627 - 2.70320i) q^{5} +(13.4360 - 15.5060i) q^{6} +(-0.413013 + 2.87257i) q^{7} +(79.5474 - 23.3572i) q^{8} +(-11.2575 - 7.23474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.30963	+	5.05739i	−0.816578	+	1.78806i	−0.240529	+	0.970642i	\(0.577321\pi\)
							−0.576049	+	0.817415i	\(0.695406\pi\)
\(3\)	−3.54080	−	1.03967i	−0.681428	−	0.200085i	−0.0773421	−	0.997005i	\(-0.524643\pi\)
							−0.604086	+	0.796919i	\(0.706462\pi\)
\(5\)	4.20627	−	2.70320i	0.376220	−	0.241782i
							\(7\)	−0.413013	+	2.87257i	−0.0223006	+	0.155104i	−0.997929	−	0.0643254i	\(-0.979510\pi\)
0.975628	+	0.219429i	\(0.0704195\pi\)
\(11\)	12.3183	+	26.9733i	0.337647	+	0.739343i	0.999951	−	0.00989199i	\(-0.00314877\pi\)
							−0.662305	+	0.749235i	\(0.730421\pi\)
\(13\)	2.25124	+	15.6577i	0.0480294	+	0.334052i	0.999642	+	0.0267520i	\(0.00851645\pi\)
							−0.951613	+	0.307300i	\(0.900574\pi\)
\(17\)	70.6504	−	81.5349i	1.00796	−	1.16324i	0.0214078	−	0.999771i	\(-0.493185\pi\)
							0.986548	−	0.163472i	\(-0.0522694\pi\)
\(19\)	−18.4918	−	21.3407i	−0.223279	−	0.257678i	0.633047	−	0.774114i	\(-0.281804\pi\)
							−0.856326	+	0.516435i	\(0.827259\pi\)
\(23\)	99.7449	+	47.0953i	0.904271	+	0.426958i
							\(29\)	−132.925	+	153.404i	−0.851158	+	0.982288i	−0.999978	−	0.00657590i	\(-0.997907\pi\)
0.148821	+	0.988864i	\(0.452452\pi\)
\(31\)	173.575	−	50.9661i	1.00564	−	0.295283i	0.262873	−	0.964830i	\(-0.415330\pi\)
							0.742769	+	0.669547i	\(0.233512\pi\)
\(37\)	169.563	+	108.972i	0.753406	+	0.484185i	0.860112	−	0.510106i	\(-0.170394\pi\)
							−0.106705	+	0.994291i	\(0.534030\pi\)
\(41\)	212.862	−	136.798i	0.810814	−	0.521079i	−0.0683140	−	0.997664i	\(-0.521762\pi\)
							0.879128	+	0.476585i	\(0.158126\pi\)
\(43\)	395.764	+	116.207i	1.40357	+	0.412125i	0.893908	−	0.448250i	\(-0.147953\pi\)
							0.509661	+	0.860375i	\(0.329771\pi\)
\(47\)	−29.5595			−0.0917381			−0.0458690	−	0.998947i	\(-0.514606\pi\)
							−0.0458690	+	0.998947i	\(0.514606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.6.1	✓	110
23.4	even	11	inner	115.4.g.a.96.1	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.6.1	✓	110	1.1	even	1	trivial
115.4.g.a.96.1	yes	110	23.4	even	11	inner