Embedded newform 115.3.i.a.14.3

• Label: 115.3.i.a.14.3
• Level: $115$
• Weight: $3$
• Character: 115.14
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.3
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.909272 - 3.09670i) q^{2} +(2.61799 - 2.26850i) q^{3} +(-5.39774 + 3.46891i) q^{4} +(0.569021 - 4.96752i) q^{5} +(-9.40531 - 6.04443i) q^{6} +(-1.76617 + 3.86736i) q^{7} +(5.89367 + 5.10690i) q^{8} +(0.426936 - 2.96941i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.909272	−	3.09670i	−0.454636	−	1.54835i	−0.794135	−	0.607742i	\(-0.792076\pi\)
							0.339499	−	0.940606i	\(-0.389743\pi\)
\(3\)	2.61799	−	2.26850i	0.872662	−	0.756166i	−0.0983610	−	0.995151i	\(-0.531360\pi\)
							0.971023	+	0.238985i	\(0.0768145\pi\)
\(5\)	0.569021	−	4.96752i	0.113804	−	0.993503i
							\(7\)	−1.76617	+	3.86736i	−0.252309	+	0.552480i	−0.992827	−	0.119556i	\(-0.961853\pi\)
0.740518	+	0.672036i	\(0.234580\pi\)
\(11\)	3.33618	−	11.3620i	0.303289	−	1.03291i	−0.656998	−	0.753892i	\(-0.728174\pi\)
							0.960287	−	0.279015i	\(-0.0900079\pi\)
\(13\)	2.27436	−	1.03867i	0.174951	−	0.0798973i	−0.326014	−	0.945365i	\(-0.605706\pi\)
							0.500965	+	0.865467i	\(0.332979\pi\)
\(17\)	7.45436	+	4.79062i	0.438492	+	0.281801i	0.741203	−	0.671281i	\(-0.234256\pi\)
							−0.302712	+	0.953082i	\(0.597892\pi\)
\(19\)	13.9040	+	21.6351i	0.731790	+	1.13869i	0.985213	+	0.171332i	\(0.0548071\pi\)
							−0.253423	+	0.967355i	\(0.581557\pi\)
\(23\)	−10.5800	−	20.4221i	−0.460001	−	0.887919i
							\(29\)	9.51579	+	6.11543i	0.328131	+	0.210877i	0.694325	−	0.719662i	\(-0.255703\pi\)
−0.366194	+	0.930538i	\(0.619339\pi\)
\(31\)	19.8080	−	22.8597i	0.638968	−	0.737408i	−0.340224	−	0.940344i	\(-0.610503\pi\)
							0.979192	+	0.202936i	\(0.0650484\pi\)
\(37\)	8.48524	−	59.0162i	0.229331	−	1.59503i	−0.471607	−	0.881809i	\(-0.656326\pi\)
							0.700938	−	0.713223i	\(-0.252765\pi\)
\(41\)	−6.43831	−	44.7795i	−0.157032	−	1.09218i	−0.904065	−	0.427395i	\(-0.859431\pi\)
							0.747033	−	0.664787i	\(-0.231478\pi\)
\(43\)	−11.9620	−	13.8049i	−0.278186	−	0.321044i	0.599412	−	0.800440i	\(-0.295401\pi\)
							−0.877598	+	0.479397i	\(0.840856\pi\)
\(47\)			20.7777i			0.442079i	0.975265	+	0.221040i	\(0.0709450\pi\)
							−0.975265	+	0.221040i	\(0.929055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.i.a.14.3	✓	220
5.4	even	2	inner	115.3.i.a.14.20	yes	220
23.5	odd	22	inner	115.3.i.a.74.20	yes	220
115.74	odd	22	inner	115.3.i.a.74.3	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.i.a.14.3	✓	220	1.1	even	1	trivial
115.3.i.a.14.20	yes	220	5.4	even	2	inner
115.3.i.a.74.3	yes	220	115.74	odd	22	inner
115.3.i.a.74.20	yes	220	23.5	odd	22	inner