Embedded newform 57.3.k.a.52.2

• Label: 57.3.k.a.52.2
• Level: $57$
• Weight: $3$
• Character: 57.52
• 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			52.2
Root			\(-0.707729i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.52
Dual form			57.3.k.a.34.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.242057 + 0.665047i) q^{2} +(-1.70574 - 0.300767i) q^{3} +(2.68048 - 2.24919i) q^{4} +(5.34756 + 4.48713i) q^{5} +(-0.212862 - 1.20720i) q^{6} +(-0.504300 - 0.873473i) q^{7} +(4.59629 + 2.65367i) q^{8} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.242057	+	0.665047i	0.121029	+	0.332524i	0.985382	−	0.170362i	\(-0.0544938\pi\)
							−0.864353	+	0.502886i	\(0.832272\pi\)
\(3\)	−1.70574	−	0.300767i	−0.568579	−	0.100256i
							\(5\)	5.34756	+	4.48713i	1.06951	+	0.897426i	0.995008	−	0.0997943i	\(-0.0318185\pi\)
0.0745031	+	0.997221i	\(0.476263\pi\)
\(7\)	−0.504300	−	0.873473i	−0.0720429	−	0.124782i	0.827754	−	0.561092i	\(-0.189619\pi\)
							−0.899797	+	0.436310i	\(0.856285\pi\)
\(11\)	3.17026	−	5.49105i	0.288205	−	0.499186i	−0.685176	−	0.728377i	\(-0.740275\pi\)
							0.973381	+	0.229191i	\(0.0736081\pi\)
\(13\)	−18.8398	+	3.32196i	−1.44921	+	0.255535i	−0.842203	−	0.539160i	\(-0.818742\pi\)
							−0.607009	+	0.794695i	\(0.707631\pi\)
\(17\)	−17.6556	+	6.42612i	−1.03857	+	0.378007i	−0.804336	−	0.594175i	\(-0.797479\pi\)
							−0.234229	+	0.972181i	\(0.575257\pi\)
\(19\)	−3.75932	−	18.6244i	−0.197859	−	0.980231i
							\(23\)	4.24742	−	3.56401i	0.184670	−	0.154957i	−0.545766	−	0.837938i	\(-0.683761\pi\)
0.730436	+	0.682981i	\(0.239317\pi\)
\(29\)	−15.9133	+	43.7213i	−0.548733	+	1.50763i	0.286690	+	0.958023i	\(0.407445\pi\)
							−0.835423	+	0.549608i	\(0.814777\pi\)
\(31\)	−11.5290	+	6.65625i	−0.371902	+	0.214718i	−0.674289	−	0.738468i	\(-0.735550\pi\)
							0.302387	+	0.953185i	\(0.402216\pi\)
\(37\)		−	33.7511i		−	0.912193i	−0.889930	−	0.456096i	\(-0.849247\pi\)
							0.889930	−	0.456096i	\(-0.150753\pi\)
\(41\)	14.6890	+	2.59006i	0.358268	+	0.0631723i	0.349885	−	0.936793i	\(-0.386221\pi\)
							0.00838270	+	0.999965i	\(0.497332\pi\)
\(43\)	54.4444	+	45.6843i	1.26615	+	1.06242i	0.994999	+	0.0998873i	\(0.0318482\pi\)
							0.271149	+	0.962537i	\(0.412596\pi\)
\(47\)	−61.8359	−	22.5064i	−1.31566	−	0.478860i	−0.413594	−	0.910462i	\(-0.635727\pi\)
							−0.902064	+	0.431601i	\(0.857949\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.52.2	yes	18
3.2	odd	2		171.3.ba.c.109.2		18
19.15	odd	18	inner	57.3.k.a.34.2	✓	18
57.53	even	18		171.3.ba.c.91.2		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.34.2	✓	18	19.15	odd	18	inner
57.3.k.a.52.2	yes	18	1.1	even	1	trivial
171.3.ba.c.91.2		18	57.53	even	18
171.3.ba.c.109.2		18	3.2	odd	2