Embedded newform 115.5.i.a.14.7

• Label: 115.5.i.a.14.7
• 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			14.7
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75464 - 5.97575i) q^{2} +(-7.06070 + 6.11814i) q^{3} +(-19.1708 + 12.3203i) q^{4} +(24.2582 + 6.04485i) q^{5} +(48.9494 + 31.4579i) q^{6} +(-12.9566 + 28.3710i) q^{7} +(31.9516 + 27.6862i) q^{8} +(0.894466 - 6.22115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75464	−	5.97575i	−0.438660	−	1.49394i	−0.821546	−	0.570142i	\(-0.806888\pi\)
							0.382886	−	0.923795i	\(-0.374930\pi\)
\(3\)	−7.06070	+	6.11814i	−0.784523	+	0.679793i	−0.952005	−	0.306082i	\(-0.900982\pi\)
							0.167482	+	0.985875i	\(0.446436\pi\)
\(5\)	24.2582	+	6.04485i	0.970328	+	0.241794i
							\(7\)	−12.9566	+	28.3710i	−0.264420	+	0.579000i	−0.994544	−	0.104315i	\(-0.966735\pi\)
0.730124	+	0.683315i	\(0.239462\pi\)
\(11\)	66.4971	−	226.468i	0.549562	−	1.87164i	0.0629566	−	0.998016i	\(-0.479947\pi\)
							0.486606	−	0.873622i	\(-0.338235\pi\)
\(13\)	18.2210	−	8.32127i	0.107817	−	0.0492383i	−0.360776	−	0.932652i	\(-0.617488\pi\)
							0.468593	+	0.883414i	\(0.344761\pi\)
\(17\)	−357.085	−	229.485i	−1.23559	−	0.794065i	−0.250837	−	0.968029i	\(-0.580706\pi\)
							−0.984752	+	0.173964i	\(0.944342\pi\)
\(19\)	81.2312	+	126.398i	0.225017	+	0.350134i	0.935345	−	0.353738i	\(-0.115089\pi\)
							−0.710327	+	0.703871i	\(0.751453\pi\)
\(23\)	373.280	−	374.837i	0.705633	−	0.708577i
							\(29\)	−1179.76	−	758.184i	−1.40280	−	0.901527i	−0.402898	−	0.915245i	\(-0.631997\pi\)
−0.999906	+	0.0137176i	\(0.995633\pi\)
\(31\)	−637.576	+	735.802i	−0.663450	+	0.765662i	−0.983337	−	0.181793i	\(-0.941810\pi\)
							0.319886	+	0.947456i	\(0.396355\pi\)
\(37\)	78.6895	−	547.297i	0.0574795	−	0.399779i	−0.940688	−	0.339272i	\(-0.889819\pi\)
							0.998168	−	0.0605068i	\(-0.0192717\pi\)
\(41\)	−329.660	−	2292.83i	−0.196109	−	1.36397i	−0.815441	−	0.578841i	\(-0.803505\pi\)
							0.619331	−	0.785130i	\(-0.287404\pi\)
\(43\)	−822.855	−	949.626i	−0.445027	−	0.513589i	0.488271	−	0.872692i	\(-0.337628\pi\)
							−0.933298	+	0.359104i	\(0.883082\pi\)
\(47\)		−	3579.71i		−	1.62051i	−0.586076	−	0.810256i	\(-0.699328\pi\)
							0.586076	−	0.810256i	\(-0.300672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.7	✓	460
5.4	even	2	inner	115.5.i.a.14.40	yes	460
23.5	odd	22	inner	115.5.i.a.74.40	yes	460
115.74	odd	22	inner	115.5.i.a.74.7	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.7	✓	460	1.1	even	1	trivial
115.5.i.a.14.40	yes	460	5.4	even	2	inner
115.5.i.a.74.7	yes	460	115.74	odd	22	inner
115.5.i.a.74.40	yes	460	23.5	odd	22	inner