Embedded newform 115.3.i.a.14.6

• Label: 115.3.i.a.14.6
• 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			14.6
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.636390 - 2.16735i) q^{2} +(-2.13320 + 1.84843i) q^{3} +(-0.927382 + 0.595992i) q^{4} +(-2.88883 - 4.08101i) q^{5} +(5.36373 + 3.44706i) q^{6} +(-1.73429 + 3.79756i) q^{7} +(-4.94659 - 4.28624i) q^{8} +(-0.146981 + 1.02227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

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	\(-0.899897\pi\)
							0.632761	−	0.774347i	\(-0.281922\pi\)
\(3\)	−2.13320	+	1.84843i	−0.711066	+	0.616142i	−0.933404	−	0.358827i	\(-0.883177\pi\)
							0.222338	+	0.974970i	\(0.428631\pi\)
\(5\)	−2.88883	−	4.08101i	−0.577766	−	0.816203i
							\(7\)	−1.73429	+	3.79756i	−0.247756	+	0.542509i	−0.992124	−	0.125260i	\(-0.960023\pi\)
0.744368	+	0.667769i	\(0.232751\pi\)
\(11\)	−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	−0.999764	−	0.0217350i	\(-0.993081\pi\)
							−0.638280	+	0.769804i	\(0.720354\pi\)
\(17\)	12.5837	+	8.08708i	0.740221	+	0.475711i	0.855618	−	0.517609i	\(-0.173178\pi\)
							−0.115397	+	0.993319i	\(0.536814\pi\)
\(19\)	−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	\(0.203965\pi\)
							−0.937585	+	0.347756i	\(0.886944\pi\)
\(41\)	−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	0.305906	−	0.952062i	\(-0.401041\pi\)
							−0.985906	+	0.167300i	\(0.946495\pi\)
\(47\)		−	46.6855i		−	0.993309i	−0.867948	−	0.496655i	\(-0.834562\pi\)
							0.867948	−	0.496655i	\(-0.165438\pi\)

(See \(a_n\) instead)

Twists

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

    

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