Embedded newform 115.3.i.a.14.17

• Label: 115.3.i.a.14.17
• 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			14.17
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.636390 + 2.16735i) q^{2} +(2.13320 - 1.84843i) q^{3} +(-0.927382 + 0.595992i) q^{4} +(-1.62206 + 4.72958i) q^{5} +(5.36373 + 3.44706i) q^{6} +(1.73429 - 3.79756i) q^{7} +(4.94659 + 4.28624i) q^{8} +(-0.146981 + 1.02227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.636390	+	2.16735i	0.318195	+	1.08367i	0.950956	+	0.309326i	\(0.100103\pi\)
							−0.632761	+	0.774347i	\(0.718078\pi\)
\(3\)	2.13320	−	1.84843i	0.711066	−	0.616142i	−0.222338	−	0.974970i	\(-0.571369\pi\)
							0.933404	+	0.358827i	\(0.116823\pi\)
\(5\)	−1.62206	+	4.72958i	−0.324411	+	0.945916i
							\(7\)	1.73429	−	3.79756i	0.247756	−	0.542509i	−0.744368	−	0.667769i	\(-0.767249\pi\)
0.992124	+	0.125260i	\(0.0399766\pi\)
\(11\)	−1.82373	+	6.21105i	−0.165794	+	0.564641i	0.834121	+	0.551581i	\(0.185975\pi\)
							−0.999915	+	0.0130597i	\(0.995843\pi\)
\(13\)	21.2946	−	9.72490i	1.63804	−	0.748069i	0.638280	−	0.769804i	\(-0.279646\pi\)
							0.999764	+	0.0217350i	\(0.00691902\pi\)
\(17\)	−12.5837	−	8.08708i	−0.740221	−	0.475711i	0.115397	−	0.993319i	\(-0.463186\pi\)
							−0.855618	+	0.517609i	\(0.826822\pi\)
\(19\)	−18.1039	−	28.1703i	−0.952839	−	1.48265i	−0.874097	−	0.485751i	\(-0.838546\pi\)
							−0.0787411	−	0.996895i	\(-0.525090\pi\)
\(23\)	−21.4881	+	8.20120i	−0.934267	+	0.356574i
							\(29\)	−25.7683	−	16.5603i	−0.888562	−	0.571044i	0.0148159	−	0.999890i	\(-0.495284\pi\)
−0.903377	+	0.428846i	\(0.858920\pi\)
\(31\)	3.98541	−	4.59941i	0.128562	−	0.148368i	−0.687819	−	0.725882i	\(-0.741432\pi\)
							0.816381	+	0.577514i	\(0.195977\pi\)
\(37\)	5.03026	−	34.9862i	0.135953	−	0.945574i	−0.801632	−	0.597818i	\(-0.796035\pi\)
							0.937585	−	0.347756i	\(-0.113056\pi\)
\(41\)	−9.40205	−	65.3927i	−0.229318	−	1.59494i	−0.700992	−	0.713170i	\(-0.747259\pi\)
							0.471673	−	0.881773i	\(-0.343650\pi\)
\(43\)	29.2400	+	33.7448i	0.680000	+	0.784762i	0.985906	−	0.167300i	\(-0.0535048\pi\)
							−0.305906	+	0.952062i	\(0.598959\pi\)
\(47\)			46.6855i			0.993309i	0.867948	+	0.496655i	\(0.165438\pi\)
							−0.867948	+	0.496655i	\(0.834562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.i.a.14.17	yes	220
5.4	even	2	inner	115.3.i.a.14.6	✓	220
23.5	odd	22	inner	115.3.i.a.74.6	yes	220
115.74	odd	22	inner	115.3.i.a.74.17	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.i.a.14.6	✓	220	5.4	even	2	inner
115.3.i.a.14.17	yes	220	1.1	even	1	trivial
115.3.i.a.74.6	yes	220	23.5	odd	22	inner
115.3.i.a.74.17	yes	220	115.74	odd	22	inner