Embedded newform 115.5.i.a.14.6

• Label: 115.5.i.a.14.6
• 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			14.6
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77123 - 6.03227i) q^{2} +(11.0627 - 9.58586i) q^{3} +(-19.7910 + 12.7189i) q^{4} +(-22.7723 - 10.3161i) q^{5} +(-77.4191 - 49.7542i) q^{6} +(-22.6376 + 49.5694i) q^{7} +(35.7565 + 30.9832i) q^{8} +(18.9665 - 131.915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77123	−	6.03227i	−0.442809	−	1.50807i	−0.814751	−	0.579811i	\(-0.803126\pi\)
							0.371942	−	0.928256i	\(-0.378692\pi\)
\(3\)	11.0627	−	9.58586i	1.22919	−	1.06510i	0.233495	−	0.972358i	\(-0.424984\pi\)
							0.995691	−	0.0927377i	\(-0.0295618\pi\)
\(5\)	−22.7723	−	10.3161i	−0.910892	−	0.412644i
							\(7\)	−22.6376	+	49.5694i	−0.461991	+	1.01162i	0.525038	+	0.851079i	\(0.324051\pi\)
−0.987029	+	0.160541i	\(0.948676\pi\)
\(11\)	−49.4626	+	168.454i	−0.408782	+	1.39218i	0.455975	+	0.889993i	\(0.349291\pi\)
							−0.864757	+	0.502191i	\(0.832528\pi\)
\(13\)	131.873	−	60.2243i	0.780313	−	0.356357i	0.0148937	−	0.999889i	\(-0.495259\pi\)
							0.765419	+	0.643532i	\(0.222532\pi\)
\(17\)	−390.320	−	250.843i	−1.35059	−	0.867970i	−0.352882	−	0.935668i	\(-0.614798\pi\)
							−0.997706	+	0.0676978i	\(0.978435\pi\)
\(19\)	−194.981	−	303.397i	−0.540114	−	0.840434i	0.458726	−	0.888578i	\(-0.348306\pi\)
							−0.998840	+	0.0481436i	\(0.984669\pi\)
\(23\)	−322.255	−	419.515i	−0.609178	−	0.793033i
							\(29\)	−996.759	−	640.578i	−1.18521	−	0.761687i	−0.208871	−	0.977943i	\(-0.566979\pi\)
−0.976337	+	0.216257i	\(0.930615\pi\)
\(31\)	777.249	−	896.994i	0.808792	−	0.933396i	−0.190037	−	0.981777i	\(-0.560861\pi\)
							0.998829	+	0.0483810i	\(0.0154062\pi\)
\(37\)	−135.176	+	940.170i	−0.0987407	+	0.686757i	0.878982	+	0.476855i	\(0.158223\pi\)
							−0.977723	+	0.209901i	\(0.932686\pi\)
\(41\)	40.6406	+	282.662i	0.0241764	+	0.168151i	0.998333	−	0.0577251i	\(-0.0183847\pi\)
							−0.974156	+	0.225876i	\(0.927476\pi\)
\(43\)	−678.470	−	782.997i	−0.366939	−	0.423470i	0.542014	−	0.840370i	\(-0.317662\pi\)
							−0.908953	+	0.416900i	\(0.863117\pi\)
\(47\)		−	466.197i		−	0.211044i	−0.994417	−	0.105522i	\(-0.966349\pi\)
							0.994417	−	0.105522i	\(-0.0336514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.6	✓	460
5.4	even	2	inner	115.5.i.a.14.41	yes	460
23.5	odd	22	inner	115.5.i.a.74.41	yes	460
115.74	odd	22	inner	115.5.i.a.74.6	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.6	✓	460	1.1	even	1	trivial
115.5.i.a.14.41	yes	460	5.4	even	2	inner
115.5.i.a.74.6	yes	460	115.74	odd	22	inner
115.5.i.a.74.41	yes	460	23.5	odd	22	inner