Embedded newform 115.2.l.a.17.2

• Label: 115.2.l.a.17.2
• Level: $115$
• Weight: $2$
• Character: 115.17
• Analytic conductor: $0.918$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.l (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.918279623245\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(10\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			17.2
Character	\(\chi\)	\(=\)	115.17
Dual form			115.2.l.a.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75302 + 0.653844i) q^{2} +(0.975483 - 0.532654i) q^{3} +(1.13408 - 0.982690i) q^{4} +(0.872739 - 2.05872i) q^{5} +(-1.36177 + 1.57157i) q^{6} +(-1.17575 - 1.57062i) q^{7} +(0.447790 - 0.820066i) q^{8} +(-0.954075 + 1.48457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 18 q^{2} - 14 q^{3} - 22 q^{5} - 36 q^{6} - 22 q^{7} - 26 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75302	+	0.653844i	−1.23958	+	0.462338i	−0.881868	−	0.471496i	\(-0.843714\pi\)
							−0.357707	+	0.933834i	\(0.616441\pi\)
\(3\)	0.975483	−	0.532654i	0.563195	−	0.307528i	−0.172314	−	0.985042i	\(-0.555125\pi\)
							0.735510	+	0.677514i	\(0.236943\pi\)
\(5\)	0.872739	−	2.05872i	0.390301	−	0.920687i
							\(7\)	−1.17575	−	1.57062i	−0.444393	−	0.593639i	0.521285	−	0.853383i	\(-0.325453\pi\)
−0.965678	+	0.259743i	\(0.916362\pi\)
\(11\)	5.87966	−	2.68515i	1.77279	−	0.809604i	0.793092	−	0.609102i	\(-0.208470\pi\)
							0.979693	−	0.200502i	\(-0.0642572\pi\)
\(13\)	0.651910	+	0.488014i	0.180807	+	0.135351i	0.685848	−	0.727745i	\(-0.259432\pi\)
							−0.505041	+	0.863096i	\(0.668522\pi\)
\(17\)	−0.200268	−	2.80012i	−0.0485722	−	0.679128i	−0.962160	−	0.272486i	\(-0.912154\pi\)
							0.913588	−	0.406642i	\(-0.133300\pi\)
\(19\)	2.85710	+	3.29726i	0.655463	+	0.756444i	0.982029	−	0.188730i	\(-0.0604371\pi\)
							−0.326566	+	0.945174i	\(0.605892\pi\)
\(23\)	−4.75948	+	0.589343i	−0.992421	+	0.122887i
							\(29\)	0.106957	+	0.0926789i	0.0198614	+	0.0172100i	0.664735	−	0.747079i	\(-0.268544\pi\)
−0.644873	+	0.764289i	\(0.723090\pi\)
\(31\)	−7.54846	+	2.21643i	−1.35574	+	0.398082i	−0.877261	−	0.480014i	\(-0.840631\pi\)
							−0.478483	+	0.878097i	\(0.658813\pi\)
\(37\)	−0.927669	+	0.201802i	−0.152508	+	0.0331761i	−0.288171	−	0.957579i	\(-0.593047\pi\)
							0.135663	+	0.990755i	\(0.456684\pi\)
\(41\)	0.739301	−	0.475120i	0.115459	−	0.0742012i	−0.481637	−	0.876371i	\(-0.659957\pi\)
							0.597096	+	0.802170i	\(0.296321\pi\)
\(43\)	4.42359	+	8.10121i	0.674592	+	1.23542i	0.960797	+	0.277254i	\(0.0894243\pi\)
							−0.286205	+	0.958168i	\(0.592394\pi\)
\(47\)	6.29926	+	6.29926i	0.918842	+	0.918842i	0.996945	−	0.0781033i	\(-0.0248864\pi\)
							−0.0781033	+	0.996945i	\(0.524886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.l.a.17.2	✓	200
5.2	odd	4		575.2.r.b.293.2		200
5.3	odd	4	inner	115.2.l.a.63.9	yes	200
5.4	even	2		575.2.r.b.132.9		200
23.19	odd	22	inner	115.2.l.a.42.9	yes	200
115.19	odd	22		575.2.r.b.157.2		200
115.42	even	44		575.2.r.b.318.9		200
115.88	even	44	inner	115.2.l.a.88.2	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.l.a.17.2	✓	200	1.1	even	1	trivial
115.2.l.a.42.9	yes	200	23.19	odd	22	inner
115.2.l.a.63.9	yes	200	5.3	odd	4	inner
115.2.l.a.88.2	yes	200	115.88	even	44	inner
575.2.r.b.132.9		200	5.4	even	2
575.2.r.b.157.2		200	115.19	odd	22
575.2.r.b.293.2		200	5.2	odd	4
575.2.r.b.318.9		200	115.42	even	44