Embedded newform 57.3.k.b.52.2

• Label: 57.3.k.b.52.2
• Level: $57$
• Weight: $3$
• Character: 57.52
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	57.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.55313750685\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			52.2
Character	\(\chi\)	\(=\)	57.52
Dual form			57.3.k.b.34.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.691603 - 1.90016i) q^{2} +(1.70574 + 0.300767i) q^{3} +(-0.0681276 + 0.0571659i) q^{4} +(6.34346 + 5.32279i) q^{5} +(-0.608185 - 3.44919i) q^{6} +(-5.23790 - 9.07231i) q^{7} +(-6.84906 - 3.95431i) q^{8} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.691603	−	1.90016i	−0.345801	−	0.950082i	−0.983677	−	0.179942i	\(-0.942409\pi\)
							0.637876	−	0.770139i	\(-0.279813\pi\)
\(3\)	1.70574	+	0.300767i	0.568579	+	0.100256i
							\(5\)	6.34346	+	5.32279i	1.26869	+	1.06456i	0.994699	+	0.102827i	\(0.0327888\pi\)
0.273992	+	0.961732i	\(0.411656\pi\)
\(7\)	−5.23790	−	9.07231i	−0.748271	−	1.29604i	−0.948651	−	0.316326i	\(-0.897551\pi\)
							0.200379	−	0.979718i	\(-0.435783\pi\)
\(11\)	−6.84713	+	11.8596i	−0.622466	+	1.07814i	0.366559	+	0.930395i	\(0.380536\pi\)
							−0.989025	+	0.147749i	\(0.952797\pi\)
\(13\)	4.21615	−	0.743420i	0.324319	−	0.0571862i	−0.00911841	−	0.999958i	\(-0.502903\pi\)
							0.333437	+	0.942772i	\(0.391791\pi\)
\(17\)	16.6864	−	6.07334i	0.981551	−	0.357255i	0.199108	−	0.979978i	\(-0.436196\pi\)
							0.782443	+	0.622722i	\(0.213973\pi\)
\(19\)	−17.6770	+	6.96594i	−0.930368	+	0.366628i
							\(23\)	−25.9434	+	21.7691i	−1.12797	+	0.946482i	−0.998980	−	0.0451629i	\(-0.985619\pi\)
−0.128994	+	0.991645i	\(0.541175\pi\)
\(29\)	0.352595	−	0.968747i	0.0121584	−	0.0334051i	−0.933465	−	0.358669i	\(-0.883231\pi\)
							0.945623	+	0.325264i	\(0.105453\pi\)
\(31\)	13.2369	−	7.64232i	0.426996	−	0.246526i	−0.271070	−	0.962560i	\(-0.587377\pi\)
							0.698066	+	0.716033i	\(0.254044\pi\)
\(37\)		−	37.7418i		−	1.02005i	−0.860160	−	0.510024i	\(-0.829636\pi\)
							0.860160	−	0.510024i	\(-0.170364\pi\)
\(41\)	−24.9786	−	4.40440i	−0.609234	−	0.107424i	−0.139485	−	0.990224i	\(-0.544545\pi\)
							−0.469749	+	0.882800i	\(0.655656\pi\)
\(43\)	18.9491	+	15.9002i	0.440676	+	0.369771i	0.835962	−	0.548787i	\(-0.184910\pi\)
							−0.395286	+	0.918558i	\(0.629355\pi\)
\(47\)	30.1846	+	10.9863i	0.642226	+	0.233751i	0.642544	−	0.766249i	\(-0.277879\pi\)
							−0.000318180		1.00000i	\(0.500101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.52.2	yes	24
3.2	odd	2		171.3.ba.d.109.3		24
19.15	odd	18	inner	57.3.k.b.34.2	✓	24
57.53	even	18		171.3.ba.d.91.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.34.2	✓	24	19.15	odd	18	inner
57.3.k.b.52.2	yes	24	1.1	even	1	trivial
171.3.ba.d.91.3		24	57.53	even	18
171.3.ba.d.109.3		24	3.2	odd	2