Embedded newform 115.3.i.a.14.13

• Label: 115.3.i.a.14.13
• 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			14.13
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.245567 + 0.836325i) q^{2} +(-2.37235 + 2.05565i) q^{3} +(2.72588 - 1.75182i) q^{4} +(4.86304 + 1.16224i) q^{5} +(-2.30176 - 1.47925i) q^{6} +(2.30176 - 5.04015i) q^{7} +(4.76942 + 4.13272i) q^{8} +(0.121499 - 0.845043i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.245567	+	0.836325i	0.122784	+	0.418163i	0.997827	−	0.0658810i	\(-0.0209858\pi\)
							−0.875044	+	0.484044i	\(0.839168\pi\)
\(3\)	−2.37235	+	2.05565i	−0.790783	+	0.685217i	−0.953479	−	0.301459i	\(-0.902526\pi\)
							0.162697	+	0.986676i	\(0.447981\pi\)
\(5\)	4.86304	+	1.16224i	0.972609	+	0.232449i
							\(7\)	2.30176	−	5.04015i	0.328822	−	0.720021i	−0.670947	−	0.741506i	\(-0.734112\pi\)
0.999769	+	0.0214848i	\(0.00683935\pi\)
\(11\)	−4.65972	+	15.8695i	−0.423611	+	1.44269i	0.420880	+	0.907116i	\(0.361721\pi\)
							−0.844491	+	0.535570i	\(0.820097\pi\)
\(13\)	−0.727473	+	0.332226i	−0.0559595	+	0.0255558i	−0.443198	−	0.896424i	\(-0.646156\pi\)
							0.387239	+	0.921979i	\(0.373429\pi\)
\(17\)	13.8472	+	8.89908i	0.814543	+	0.523475i	0.880332	−	0.474358i	\(-0.157320\pi\)
							−0.0657885	+	0.997834i	\(0.520956\pi\)
\(19\)	−4.56516	−	7.10352i	−0.240271	−	0.373870i	0.700086	−	0.714059i	\(-0.253145\pi\)
							−0.940357	+	0.340189i	\(0.889509\pi\)
\(23\)	−4.85780	−	22.4811i	−0.211209	−	0.977441i
							\(29\)	−20.8719	−	13.4135i	−0.719719	−	0.462536i	0.128820	−	0.991668i	\(-0.458881\pi\)
−0.848539	+	0.529132i	\(0.822517\pi\)
\(31\)	1.31745	−	1.52042i	0.0424985	−	0.0490459i	−0.734102	−	0.679039i	\(-0.762397\pi\)
							0.776601	+	0.629993i	\(0.216942\pi\)
\(37\)	0.957842	−	6.66194i	0.0258876	−	0.180052i	−0.972775	−	0.231751i	\(-0.925554\pi\)
							0.998663	+	0.0516988i	\(0.0164636\pi\)
\(41\)	−1.79945	−	12.5154i	−0.0438890	−	0.305255i	−0.999928	−	0.0120146i	\(-0.996176\pi\)
							0.956039	−	0.293240i	\(-0.0947335\pi\)
\(43\)	−40.3379	−	46.5524i	−0.938090	−	1.08261i	−0.996439	−	0.0843177i	\(-0.973129\pi\)
							0.0583486	−	0.998296i	\(-0.481417\pi\)
\(47\)			58.7938i			1.25093i	0.780252	+	0.625466i	\(0.215091\pi\)
							−0.780252	+	0.625466i	\(0.784909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.i.a.14.13	yes	220
5.4	even	2	inner	115.3.i.a.14.10	✓	220
23.5	odd	22	inner	115.3.i.a.74.10	yes	220
115.74	odd	22	inner	115.3.i.a.74.13	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.i.a.14.10	✓	220	5.4	even	2	inner
115.3.i.a.14.13	yes	220	1.1	even	1	trivial
115.3.i.a.74.10	yes	220	23.5	odd	22	inner
115.3.i.a.74.13	yes	220	115.74	odd	22	inner