Embedded newform 57.3.k.a.22.1

• Label: 57.3.k.a.22.1
• Level: $57$
• Weight: $3$
• Character: 57.22
• 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			22.1
Root			\(2.85524i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.22
Dual form			57.3.k.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83532 + 2.18724i) q^{2} +(0.592396 + 1.62760i) q^{3} +(-0.721060 - 4.08934i) q^{4} +(-0.943682 + 5.35188i) q^{5} +(-4.64718 - 1.69144i) q^{6} +(-1.78953 - 3.09955i) q^{7} +(0.376890 + 0.217598i) q^{8} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83532	+	2.18724i	−0.917658	+	1.09362i	0.0776614	+	0.996980i	\(0.475255\pi\)
							−0.995319	+	0.0966422i	\(0.969190\pi\)
\(3\)	0.592396	+	1.62760i	0.197465	+	0.542532i
							\(5\)	−0.943682	+	5.35188i	−0.188736	+	1.07038i	0.732324	+	0.680957i	\(0.238436\pi\)
−0.921060	+	0.389420i	\(0.872675\pi\)
\(7\)	−1.78953	−	3.09955i	−0.255647	−	0.442793i	0.709424	−	0.704782i	\(-0.248955\pi\)
							−0.965071	+	0.261989i	\(0.915622\pi\)
\(11\)	−0.576461	+	0.998460i	−0.0524055	+	0.0907691i	−0.891038	−	0.453928i	\(-0.850022\pi\)
							0.838633	+	0.544697i	\(0.183356\pi\)
\(13\)	−2.47810	+	6.80853i	−0.190623	+	0.523733i	−0.997779	−	0.0666047i	\(-0.978783\pi\)
							0.807156	+	0.590338i	\(0.201006\pi\)
\(17\)	19.7466	+	16.5694i	1.16156	+	0.974668i	0.999926	−	0.0121824i	\(-0.00387789\pi\)
							0.161638	+	0.986850i	\(0.448322\pi\)
\(19\)	13.6977	+	13.1672i	0.720930	+	0.693008i
							\(23\)	1.76845	+	10.0294i	0.0768889	+	0.436059i	0.998814	+	0.0486904i	\(0.0155048\pi\)
−0.921925	+	0.387368i	\(0.873384\pi\)
\(29\)	−35.1809	−	41.9270i	−1.21313	−	1.44576i	−0.860083	−	0.510153i	\(-0.829589\pi\)
							−0.353051	−	0.935604i	\(-0.614856\pi\)
\(31\)	29.5929	−	17.0855i	0.954610	−	0.551145i	0.0601003	−	0.998192i	\(-0.480858\pi\)
							0.894510	+	0.447048i	\(0.147525\pi\)
\(37\)			68.0288i			1.83862i	0.393537	+	0.919309i	\(0.371251\pi\)
							−0.393537	+	0.919309i	\(0.628749\pi\)
\(41\)	−8.55768	−	23.5120i	−0.208724	−	0.573464i	0.790516	−	0.612441i	\(-0.209812\pi\)
							−0.999240	+	0.0389769i	\(0.987590\pi\)
\(43\)	1.46750	−	8.32261i	0.0341279	−	0.193549i	−0.962977	−	0.269582i	\(-0.913114\pi\)
							0.997105	+	0.0760332i	\(0.0242255\pi\)
\(47\)	66.6334	−	55.9121i	1.41773	−	1.18962i	0.465190	−	0.885211i	\(-0.345986\pi\)
							0.952542	−	0.304408i	\(-0.0984586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.22.1	yes	18
3.2	odd	2		171.3.ba.c.136.3		18
19.13	odd	18	inner	57.3.k.a.13.1	✓	18
57.32	even	18		171.3.ba.c.127.3		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.13.1	✓	18	19.13	odd	18	inner
57.3.k.a.22.1	yes	18	1.1	even	1	trivial
171.3.ba.c.127.3		18	57.32	even	18
171.3.ba.c.136.3		18	3.2	odd	2