Embedded newform 57.3.k.a.10.2

• Label: 57.3.k.a.10.2
• Level: $57$
• Weight: $3$
• Character: 57.10
• 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			10.2
Root			\(-1.57028i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.a.40.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.54643 - 0.272677i) q^{2} +(1.11334 + 1.32683i) q^{3} +(-1.44168 + 0.524730i) q^{4} +(4.30798 + 1.56798i) q^{5} +(2.08350 + 1.74826i) q^{6} +(3.52548 - 6.10631i) q^{7} +(-7.52600 + 4.34514i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.54643	−	0.272677i	0.773214	−	0.136339i	0.226899	−	0.973918i	\(-0.427141\pi\)
							0.546315	+	0.837580i	\(0.316030\pi\)
\(3\)	1.11334	+	1.32683i	0.371114	+	0.442276i
							\(5\)	4.30798	+	1.56798i	0.861597	+	0.313596i	0.734759	−	0.678328i	\(-0.237295\pi\)
0.126837	+	0.991924i	\(0.459517\pi\)
\(7\)	3.52548	−	6.10631i	0.503640	−	0.872330i	−0.496351	−	0.868122i	\(-0.665327\pi\)
							0.999991	−	0.00420801i	\(-0.00133946\pi\)
\(11\)	−9.95549	−	17.2434i	−0.905045	−	1.56758i	−0.820857	−	0.571134i	\(-0.806504\pi\)
							−0.0841879	−	0.996450i	\(-0.526830\pi\)
\(13\)	−9.57673	+	11.4131i	−0.736671	+	0.877931i	−0.996136	−	0.0878217i	\(-0.972009\pi\)
							0.259465	+	0.965753i	\(0.416454\pi\)
\(17\)	−1.49716	−	8.49083i	−0.0880684	−	0.499461i	−0.996652	−	0.0817580i	\(-0.973947\pi\)
							0.908584	−	0.417703i	\(-0.137165\pi\)
\(19\)	12.7546	+	14.0826i	0.671297	+	0.741189i
							\(23\)	3.73381	−	1.35900i	0.162340	−	0.0590868i	−0.259572	−	0.965724i	\(-0.583581\pi\)
0.421912	+	0.906637i	\(0.361359\pi\)
\(29\)	−29.3393	−	5.17330i	−1.01170	−	0.178390i	−0.356859	−	0.934158i	\(-0.616152\pi\)
							−0.654839	+	0.755768i	\(0.727264\pi\)
\(31\)	48.6897	+	28.1110i	1.57063	+	0.906806i	0.996091	+	0.0883318i	\(0.0281536\pi\)
							0.574543	+	0.818474i	\(0.305180\pi\)
\(37\)		−	4.47920i		−	0.121060i	−0.998166	−	0.0605298i	\(-0.980721\pi\)
							0.998166	−	0.0605298i	\(-0.0192790\pi\)
\(41\)	17.9903	+	21.4400i	0.438788	+	0.522927i	0.939436	−	0.342724i	\(-0.111349\pi\)
							−0.500648	+	0.865651i	\(0.666905\pi\)
\(43\)	45.0597	+	16.4004i	1.04790	+	0.381405i	0.807870	−	0.589361i	\(-0.200620\pi\)
							0.240031	+	0.970765i	\(0.422843\pi\)
\(47\)	−5.21692	+	29.5866i	−0.110998	+	0.629502i	0.877656	+	0.479291i	\(0.159106\pi\)
							−0.988654	+	0.150211i	\(0.952005\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.10.2	✓	18
3.2	odd	2		171.3.ba.c.10.2		18
19.2	odd	18	inner	57.3.k.a.40.2	yes	18
57.2	even	18		171.3.ba.c.154.2		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.10.2	✓	18	1.1	even	1	trivial
57.3.k.a.40.2	yes	18	19.2	odd	18	inner
171.3.ba.c.10.2		18	3.2	odd	2
171.3.ba.c.154.2		18	57.2	even	18