Embedded newform 57.3.k.a.13.2

• Label: 57.3.k.a.13.2
• Level: $57$
• Weight: $3$
• Character: 57.13
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $18$
• 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			13.2
Root			\(0.358663i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.13
Dual form			57.3.k.a.22.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.230544 + 0.274751i) q^{2} +(0.592396 - 1.62760i) q^{3} +(0.672255 - 3.81255i) q^{4} +(0.125095 + 0.709447i) q^{5} +(0.583757 - 0.212470i) q^{6} +(1.07822 - 1.86753i) q^{7} +(2.44493 - 1.41158i) q^{8} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.230544	+	0.274751i	0.115272	+	0.137376i	0.820595	−	0.571511i	\(-0.193642\pi\)
							−0.705323	+	0.708886i	\(0.749198\pi\)
\(3\)	0.592396	−	1.62760i	0.197465	−	0.542532i
							\(5\)	0.125095	+	0.709447i	0.0250189	+	0.141889i	0.994758	−	0.102252i	\(-0.0326049\pi\)
−0.969740	+	0.244142i	\(0.921494\pi\)
\(7\)	1.07822	−	1.86753i	0.154032	−	0.266791i	−0.778674	−	0.627428i	\(-0.784108\pi\)
							0.932706	+	0.360638i	\(0.117441\pi\)
\(11\)	8.08873	+	14.0101i	0.735339	+	1.27364i	0.954574	+	0.297973i	\(0.0963104\pi\)
							−0.219235	+	0.975672i	\(0.570356\pi\)
\(13\)	1.98473	+	5.45301i	0.152672	+	0.419462i	0.992324	−	0.123662i	\(-0.0394637\pi\)
							−0.839653	+	0.543124i	\(0.817242\pi\)
\(17\)	−8.34542	+	7.00264i	−0.490907	+	0.411920i	−0.854351	−	0.519696i	\(-0.826045\pi\)
							0.363444	+	0.931616i	\(0.381601\pi\)
\(19\)	−18.5899	−	3.92617i	−0.978417	−	0.206641i
							\(23\)	−4.80653	+	27.2592i	−0.208979	+	1.18518i	0.682074	+	0.731283i	\(0.261078\pi\)
−0.891054	+	0.453898i	\(0.850033\pi\)
\(29\)	4.49593	−	5.35805i	0.155032	−	0.184760i	−0.682938	−	0.730477i	\(-0.739298\pi\)
							0.837970	+	0.545716i	\(0.183742\pi\)
\(31\)	20.9272	+	12.0823i	0.675071	+	0.389752i	0.797995	−	0.602664i	\(-0.205894\pi\)
							−0.122924	+	0.992416i	\(0.539227\pi\)
\(37\)		−	25.4790i		−	0.688621i	−0.938856	−	0.344311i	\(-0.888113\pi\)
							0.938856	−	0.344311i	\(-0.111887\pi\)
\(41\)	10.3019	−	28.3043i	0.251266	−	0.690348i	−0.748367	−	0.663284i	\(-0.769162\pi\)
							0.999634	−	0.0270640i	\(-0.00861577\pi\)
\(43\)	0.808058	+	4.58272i	0.0187920	+	0.106575i	0.992761	−	0.120106i	\(-0.0383235\pi\)
							−0.973969	+	0.226681i	\(0.927212\pi\)
\(47\)	−41.4983	−	34.8212i	−0.882942	−	0.740876i	0.0838398	−	0.996479i	\(-0.473282\pi\)
							−0.966782	+	0.255603i	\(0.917726\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.13.2	✓	18
3.2	odd	2		171.3.ba.c.127.2		18
19.3	odd	18	inner	57.3.k.a.22.2	yes	18
57.41	even	18		171.3.ba.c.136.2		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.13.2	✓	18	1.1	even	1	trivial
57.3.k.a.22.2	yes	18	19.3	odd	18	inner
171.3.ba.c.127.2		18	3.2	odd	2
171.3.ba.c.136.2		18	57.41	even	18