Embedded newform 57.3.k.a.52.3

• Label: 57.3.k.a.52.3
• Level: $57$
• Weight: $3$
• Character: 57.52
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $18$
• CM: no
• 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:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial:	x^18 + 48*x^16 + 936*x^14 + 9539*x^12 + 54576*x^10 + 176517*x^8 + 313396*x^6 + 277917*x^4 + 96435*x^2 + 8427
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			52.3
Root			\(-3.54770i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.52
Dual form			57.3.k.a.34.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21338 + 3.33374i) q^{2} +(-1.70574 - 0.300767i) q^{3} +(-6.57738 + 5.51907i) q^{4} +(-0.783175 - 0.657162i) q^{5} +(-1.06703 - 6.05144i) q^{6} +(4.99091 + 8.64451i) q^{7} +(-14.0905 - 8.13515i) q^{8} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21338	+	3.33374i	0.606692	+	1.66687i	0.737404	+	0.675452i	\(0.236051\pi\)
							−0.130712	+	0.991420i	\(0.541726\pi\)
\(3\)	−1.70574	−	0.300767i	−0.568579	−	0.100256i
							\(5\)	−0.783175	−	0.657162i	−0.156635	−	0.131432i	0.561102	−	0.827747i	\(-0.310378\pi\)
−0.717737	+	0.696314i	\(0.754822\pi\)
\(7\)	4.99091	+	8.64451i	0.712987	+	1.23493i	0.963731	+	0.266877i	\(0.0859916\pi\)
							−0.250743	+	0.968054i	\(0.580675\pi\)
\(11\)	7.98270	−	13.8264i	0.725700	−	1.25695i	−0.232986	−	0.972480i	\(-0.574849\pi\)
							0.958685	−	0.284469i	\(-0.0918172\pi\)
\(13\)	16.0613	−	2.83203i	1.23548	−	0.217849i	0.482503	−	0.875894i	\(-0.339728\pi\)
							0.752978	+	0.658046i	\(0.228617\pi\)
\(17\)	−10.7243	+	3.90334i	−0.630843	+	0.229608i	−0.637598	−	0.770369i	\(-0.720072\pi\)
							0.00675521	+	0.999977i	\(0.497850\pi\)
\(19\)	−13.5828	+	13.2856i	−0.714884	+	0.699243i
							\(23\)	3.44165	−	2.88789i	0.149637	−	0.125560i	−0.564896	−	0.825162i	\(-0.691084\pi\)
0.714533	+	0.699602i	\(0.246639\pi\)
\(29\)	12.7897	−	35.1393i	0.441023	−	1.21170i	−0.497797	−	0.867293i	\(-0.665858\pi\)
							0.938820	−	0.344407i	\(-0.111920\pi\)
\(31\)	1.66652	−	0.962164i	0.0537586	−	0.0310376i	−0.472880	−	0.881127i	\(-0.656786\pi\)
							0.526638	+	0.850089i	\(0.323452\pi\)
\(37\)		−	23.7084i		−	0.640767i	−0.947288	−	0.320383i	\(-0.896188\pi\)
							0.947288	−	0.320383i	\(-0.103812\pi\)
\(41\)	−57.7632	−	10.1852i	−1.40886	−	0.248420i	−0.583080	−	0.812415i	\(-0.698153\pi\)
							−0.825778	+	0.563995i	\(0.809264\pi\)
\(43\)	61.4011	+	51.5217i	1.42793	+	1.19818i	0.946922	+	0.321463i	\(0.104175\pi\)
							0.481011	+	0.876715i	\(0.340270\pi\)
\(47\)	41.7263	+	15.1871i	0.887793	+	0.323130i	0.745350	−	0.666673i	\(-0.232282\pi\)
							0.142443	+	0.989803i	\(0.454504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.52.3	yes	18
3.2	odd	2		171.3.ba.c.109.1		18
19.15	odd	18	inner	57.3.k.a.34.3	✓	18
57.53	even	18		171.3.ba.c.91.1		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.34.3	✓	18	19.15	odd	18	inner
57.3.k.a.52.3	yes	18	1.1	even	1	trivial
171.3.ba.c.91.1		18	57.53	even	18
171.3.ba.c.109.1		18	3.2	odd	2