Embedded newform 57.3.k.a.10.1

• Label: 57.3.k.a.10.1
• 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			10.1
Root			\(1.45784i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.a.40.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.43569 + 0.253151i) q^{2} +(1.11334 + 1.32683i) q^{3} +(-1.76165 + 0.641190i) q^{4} +(3.63888 + 1.32445i) q^{5} +(-1.93430 - 1.62307i) q^{6} +(-6.51740 + 11.2885i) q^{7} +(7.41696 - 4.28219i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.43569	+	0.253151i	−0.717844	+	0.126575i	−0.520627	−	0.853784i	\(-0.674302\pi\)
							−0.197218	+	0.980360i	\(0.563191\pi\)
\(3\)	1.11334	+	1.32683i	0.371114	+	0.442276i
							\(5\)	3.63888	+	1.32445i	0.727777	+	0.264889i	0.679224	−	0.733931i	\(-0.262317\pi\)
0.0485534	+	0.998821i	\(0.484539\pi\)
\(7\)	−6.51740	+	11.2885i	−0.931057	+	1.61264i	−0.149538	+	0.988756i	\(0.547779\pi\)
							−0.781519	+	0.623882i	\(0.785555\pi\)
\(11\)	4.25948	+	7.37764i	0.387226	+	0.670695i	0.992075	−	0.125645i	\(-0.0401001\pi\)
							−0.604850	+	0.796340i	\(0.706767\pi\)
\(13\)	2.30709	−	2.74948i	0.177468	−	0.211499i	−0.669976	−	0.742383i	\(-0.733696\pi\)
							0.847444	+	0.530884i	\(0.178140\pi\)
\(17\)	−4.38825	−	24.8870i	−0.258132	−	1.46394i	−0.787903	−	0.615799i	\(-0.788833\pi\)
							0.529771	−	0.848141i	\(-0.322278\pi\)
\(19\)	16.8721	−	8.73687i	0.888004	−	0.459835i
							\(23\)	6.42443	−	2.33830i	0.279323	−	0.101665i	−0.198560	−	0.980089i	\(-0.563627\pi\)
0.477883	+	0.878423i	\(0.341404\pi\)
\(29\)	43.3089	+	7.63653i	1.49341	+	0.263329i	0.859923	−	0.510423i	\(-0.170511\pi\)
							0.633488	+	0.773752i	\(0.281622\pi\)
\(31\)	−0.623261	−	0.359840i	−0.0201052	−	0.0116077i	0.489914	−	0.871771i	\(-0.337028\pi\)
							−0.510019	+	0.860163i	\(0.670362\pi\)
\(37\)			61.8811i			1.67246i	0.548376	+	0.836232i	\(0.315246\pi\)
							−0.548376	+	0.836232i	\(0.684754\pi\)
\(41\)	3.33462	+	3.97404i	0.0813321	+	0.0969278i	0.805176	−	0.593036i	\(-0.202071\pi\)
							−0.723844	+	0.689964i	\(0.757626\pi\)
\(43\)	23.2868	+	8.47571i	0.541554	+	0.197109i	0.598290	−	0.801280i	\(-0.295847\pi\)
							−0.0567360	+	0.998389i	\(0.518069\pi\)
\(47\)	4.46348	−	25.3137i	0.0949677	−	0.538589i	−0.899790	−	0.436324i	\(-0.856280\pi\)
							0.994757	−	0.102264i	\(-0.0326088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.10.1	✓	18
3.2	odd	2		171.3.ba.c.10.3		18
19.2	odd	18	inner	57.3.k.a.40.1	yes	18
57.2	even	18		171.3.ba.c.154.3		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.10.1	✓	18	1.1	even	1	trivial
57.3.k.a.40.1	yes	18	19.2	odd	18	inner
171.3.ba.c.10.3		18	3.2	odd	2
171.3.ba.c.154.3		18	57.2	even	18