Embedded newform 57.3.k.b.10.1

• Label: 57.3.k.b.10.1
• Level: $57$
• Weight: $3$
• Character: 57.10
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $24$
• 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:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.b.40.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75842 + 0.662712i) q^{2} +(-1.11334 - 1.32683i) q^{3} +(9.92780 - 3.61342i) q^{4} +(-0.758325 - 0.276008i) q^{5} +(5.06371 + 4.24896i) q^{6} +(-5.10471 + 8.84162i) q^{7} +(-21.6978 + 12.5272i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.75842	+	0.662712i	−1.87921	+	0.331356i	−0.991611	−	0.129257i	\(-0.958741\pi\)
							−0.887601	+	0.460613i	\(0.847630\pi\)
\(3\)	−1.11334	−	1.32683i	−0.371114	−	0.442276i
							\(5\)	−0.758325	−	0.276008i	−0.151665	−	0.0552015i	0.265072	−	0.964229i	\(-0.414604\pi\)
−0.416737	+	0.909027i	\(0.636826\pi\)
\(7\)	−5.10471	+	8.84162i	−0.729245	+	1.26309i	0.227958	+	0.973671i	\(0.426795\pi\)
							−0.957203	+	0.289418i	\(0.906538\pi\)
\(11\)	7.18408	+	12.4432i	0.653098	+	1.13120i	0.982367	+	0.186963i	\(0.0598643\pi\)
							−0.329269	+	0.944236i	\(0.606802\pi\)
\(13\)	−2.56999	+	3.06280i	−0.197692	+	0.235600i	−0.855779	−	0.517342i	\(-0.826921\pi\)
							0.658087	+	0.752942i	\(0.271366\pi\)
\(17\)	2.00780	+	11.3868i	0.118106	+	0.669812i	0.985165	+	0.171607i	\(0.0548960\pi\)
							−0.867059	+	0.498205i	\(0.833993\pi\)
\(19\)	−13.0262	+	13.8318i	−0.685590	+	0.727988i
							\(23\)	15.8782	−	5.77919i	0.690357	−	0.251269i	0.0270687	−	0.999634i	\(-0.491383\pi\)
0.663288	+	0.748364i	\(0.269160\pi\)
\(29\)	−34.6065	−	6.10206i	−1.19333	−	0.210416i	−0.458515	−	0.888687i	\(-0.651619\pi\)
							−0.734812	+	0.678271i	\(0.762730\pi\)
\(31\)	32.6597	+	18.8561i	1.05354	+	0.608261i	0.923638	−	0.383266i	\(-0.125201\pi\)
							0.129901	+	0.991527i	\(0.458534\pi\)
\(37\)			11.2648i			0.304453i	0.988346	+	0.152227i	\(0.0486443\pi\)
							−0.988346	+	0.152227i	\(0.951356\pi\)
\(41\)	−24.1933	−	28.8324i	−0.590080	−	0.703230i	0.385541	−	0.922691i	\(-0.374015\pi\)
							−0.975621	+	0.219460i	\(0.929570\pi\)
\(43\)	−3.41774	−	1.24396i	−0.0794824	−	0.0289292i	0.301973	−	0.953316i	\(-0.402355\pi\)
							−0.381455	+	0.924387i	\(0.624577\pi\)
\(47\)	1.47054	−	8.33987i	0.0312882	−	0.177444i	−0.965159	−	0.261664i	\(-0.915729\pi\)
							0.996447	+	0.0842200i	\(0.0268399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.10.1	✓	24
3.2	odd	2		171.3.ba.d.10.4		24
19.2	odd	18	inner	57.3.k.b.40.1	yes	24
57.2	even	18		171.3.ba.d.154.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.10.1	✓	24	1.1	even	1	trivial
57.3.k.b.40.1	yes	24	19.2	odd	18	inner
171.3.ba.d.10.4		24	3.2	odd	2
171.3.ba.d.154.4		24	57.2	even	18