Embedded newform 115.5.i.a.14.1

• Label: 115.5.i.a.14.1
• Level: $115$
• Weight: $5$
• Character: 115.14
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $460$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(460\)
Relative dimension:	\(46\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.1
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.18300 - 7.43463i) q^{2} +(-2.64484 + 2.29177i) q^{3} +(-37.0481 + 23.8094i) q^{4} +(-13.3410 - 21.1428i) q^{5} +(22.8121 + 14.6605i) q^{6} +(-5.36934 + 11.7572i) q^{7} +(164.196 + 142.276i) q^{8} +(-9.78452 + 68.0528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.18300	−	7.43463i	−0.545751	−	1.85866i	−0.511570	−	0.859242i	\(-0.670936\pi\)
							−0.0341811	−	0.999416i	\(-0.510882\pi\)
\(3\)	−2.64484	+	2.29177i	−0.293871	+	0.254641i	−0.789300	−	0.614008i	\(-0.789556\pi\)
							0.495429	+	0.868648i	\(0.335011\pi\)
\(5\)	−13.3410	−	21.1428i	−0.533639	−	0.845712i
							\(7\)	−5.36934	+	11.7572i	−0.109578	+	0.239943i	−0.956475	−	0.291814i	\(-0.905741\pi\)
0.846897	+	0.531758i	\(0.178468\pi\)
\(11\)	12.5556	−	42.7604i	0.103765	−	0.353392i	−0.891200	−	0.453610i	\(-0.850136\pi\)
							0.994965	+	0.100218i	\(0.0319541\pi\)
\(13\)	−37.9152	+	17.3153i	−0.224350	+	0.102457i	−0.524419	−	0.851460i	\(-0.675718\pi\)
							0.300069	+	0.953918i	\(0.402990\pi\)
\(17\)	23.0850	+	14.8358i	0.0798789	+	0.0513350i	0.579971	−	0.814637i	\(-0.303064\pi\)
							−0.500092	+	0.865972i	\(0.666700\pi\)
\(19\)	−73.0536	−	113.674i	−0.202364	−	0.314885i	0.725207	−	0.688531i	\(-0.241744\pi\)
							−0.927572	+	0.373646i	\(0.878108\pi\)
\(23\)	480.488	−	221.296i	0.908296	−	0.418329i
							\(29\)	488.877	+	314.182i	0.581305	+	0.373582i	0.798000	−	0.602657i	\(-0.205891\pi\)
−0.216695	+	0.976239i	\(0.569528\pi\)
\(31\)	−183.573	+	211.854i	−0.191023	+	0.220452i	−0.843179	−	0.537632i	\(-0.819319\pi\)
							0.652157	+	0.758084i	\(0.273864\pi\)
\(37\)	313.759	−	2182.24i	0.229189	−	1.59404i	−0.472350	−	0.881411i	\(-0.656594\pi\)
							0.701539	−	0.712631i	\(-0.252497\pi\)
\(41\)	453.743	+	3155.85i	0.269924	+	1.87736i	0.449004	+	0.893530i	\(0.351779\pi\)
							−0.179080	+	0.983835i	\(0.557312\pi\)
\(43\)	1503.79	+	1735.46i	0.813297	+	0.938594i	0.999031	−	0.0440056i	\(-0.0140119\pi\)
							−0.185735	+	0.982600i	\(0.559466\pi\)
\(47\)			2530.33i			1.14546i	0.819743	+	0.572731i	\(0.194116\pi\)
							−0.819743	+	0.572731i	\(0.805884\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.1	✓	460
5.4	even	2	inner	115.5.i.a.14.46	yes	460
23.5	odd	22	inner	115.5.i.a.74.46	yes	460
115.74	odd	22	inner	115.5.i.a.74.1	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.1	✓	460	1.1	even	1	trivial
115.5.i.a.14.46	yes	460	5.4	even	2	inner
115.5.i.a.74.1	yes	460	115.74	odd	22	inner
115.5.i.a.74.46	yes	460	23.5	odd	22	inner