Embedded newform 115.5.i.a.14.14

• Label: 115.5.i.a.14.14
• 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			14.14
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04916 - 3.57310i) q^{2} +(12.0617 - 10.4515i) q^{3} +(1.79375 - 1.15277i) q^{4} +(24.8811 - 2.43500i) q^{5} +(-49.9988 - 32.1323i) q^{6} +(-30.9341 + 67.7361i) q^{7} +(-51.0308 - 44.2185i) q^{8} +(24.7226 - 171.949i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04916	−	3.57310i	−0.262289	−	0.893275i	−0.980345	−	0.197290i	\(-0.936786\pi\)
							0.718056	−	0.695985i	\(-0.245032\pi\)
\(3\)	12.0617	−	10.4515i	1.34018	−	1.16128i	0.367270	−	0.930114i	\(-0.380293\pi\)
							0.972915	−	0.231162i	\(-0.0742529\pi\)
\(5\)	24.8811	−	2.43500i	0.995245	−	0.0974001i
							\(7\)	−30.9341	+	67.7361i	−0.631307	+	1.38237i	0.275695	+	0.961245i	\(0.411092\pi\)
−0.907003	+	0.421125i	\(0.861635\pi\)
\(11\)	40.9337	−	139.408i	0.338295	−	1.15213i	−0.598170	−	0.801369i	\(-0.704105\pi\)
							0.936466	−	0.350759i	\(-0.114076\pi\)
\(13\)	−163.566	+	74.6983i	−0.967849	+	0.442002i	−0.835675	−	0.549225i	\(-0.814923\pi\)
							−0.132174	+	0.991227i	\(0.542196\pi\)
\(17\)	264.369	+	169.899i	0.914771	+	0.587887i	0.911136	−	0.412106i	\(-0.135207\pi\)
							0.00363471	+	0.999993i	\(0.498843\pi\)
\(19\)	−156.032	−	242.790i	−0.432221	−	0.672549i	0.555010	−	0.831844i	\(-0.312714\pi\)
							−0.987231	+	0.159295i	\(0.949078\pi\)
\(23\)	308.170	−	429.968i	0.582552	−	0.812794i
							\(29\)	335.735	+	215.764i	0.399209	+	0.256556i	0.724807	−	0.688952i	\(-0.241929\pi\)
−0.325598	+	0.945508i	\(0.605565\pi\)
\(31\)	−90.7960	+	104.784i	−0.0944808	+	0.109037i	−0.801021	−	0.598636i	\(-0.795709\pi\)
							0.706540	+	0.707673i	\(0.250255\pi\)
\(37\)	−171.750	+	1194.55i	−0.125457	+	0.872571i	0.825754	+	0.564030i	\(0.190750\pi\)
							−0.951211	+	0.308541i	\(0.900159\pi\)
\(41\)	195.839	+	1362.09i	0.116501	+	0.810285i	0.961360	+	0.275295i	\(0.0887755\pi\)
							−0.844858	+	0.534990i	\(0.820315\pi\)
\(43\)	1835.59	+	2118.39i	0.992749	+	1.14569i	0.989329	+	0.145699i	\(0.0465432\pi\)
							0.00342021	+	0.999994i	\(0.498911\pi\)
\(47\)			969.135i			0.438721i	0.975644	+	0.219361i	\(0.0703971\pi\)
							−0.975644	+	0.219361i	\(0.929603\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.14	✓	460
5.4	even	2	inner	115.5.i.a.14.33	yes	460
23.5	odd	22	inner	115.5.i.a.74.33	yes	460
115.74	odd	22	inner	115.5.i.a.74.14	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.14	✓	460	1.1	even	1	trivial
115.5.i.a.14.33	yes	460	5.4	even	2	inner
115.5.i.a.74.14	yes	460	115.74	odd	22	inner
115.5.i.a.74.33	yes	460	23.5	odd	22	inner