Embedded newform 57.3.k.b.10.3

• Label: 57.3.k.b.10.3
• 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			10.3
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.b.40.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.637756 + 0.112454i) q^{2} +(-1.11334 - 1.32683i) q^{3} +(-3.36468 + 1.22464i) q^{4} +(-6.57652 - 2.39366i) q^{5} +(0.859247 + 0.720994i) q^{6} +(-1.12773 + 1.95329i) q^{7} +(4.25147 - 2.45459i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.637756	+	0.112454i	−0.318878	+	0.0562268i	−0.330796	−	0.943702i	\(-0.607317\pi\)
							0.0119182	+	0.999929i	\(0.496206\pi\)
\(3\)	−1.11334	−	1.32683i	−0.371114	−	0.442276i
							\(5\)	−6.57652	−	2.39366i	−1.31530	−	0.478732i	−0.413354	−	0.910570i	\(-0.635643\pi\)
−0.901951	+	0.431839i	\(0.857865\pi\)
\(7\)	−1.12773	+	1.95329i	−0.161105	+	0.279042i	−0.935265	−	0.353948i	\(-0.884839\pi\)
							0.774160	+	0.632989i	\(0.218172\pi\)
\(11\)	−5.72877	−	9.92252i	−0.520797	−	0.902048i	−0.999708	−	0.0241837i	\(-0.992301\pi\)
							0.478910	−	0.877864i	\(-0.341032\pi\)
\(13\)	−4.27586	+	5.09578i	−0.328913	+	0.391983i	−0.905004	−	0.425402i	\(-0.860133\pi\)
							0.576092	+	0.817385i	\(0.304577\pi\)
\(17\)	−2.16104	−	12.2558i	−0.127120	−	0.720932i	−0.980026	−	0.198868i	\(-0.936273\pi\)
							0.852907	−	0.522064i	\(-0.174838\pi\)
\(19\)	12.4838	+	14.3232i	0.657044	+	0.753852i
							\(23\)	−20.3143	+	7.39381i	−0.883232	+	0.321470i	−0.743513	−	0.668721i	\(-0.766842\pi\)
−0.139719	+	0.990191i	\(0.544620\pi\)
\(29\)	−6.16622	−	1.08727i	−0.212628	−	0.0374921i	0.0663192	−	0.997798i	\(-0.478874\pi\)
							−0.278948	+	0.960306i	\(0.589986\pi\)
\(31\)	−44.9654	−	25.9608i	−1.45050	−	0.837444i	−0.451986	−	0.892025i	\(-0.649284\pi\)
							−0.998509	+	0.0545810i	\(0.982618\pi\)
\(37\)		−	47.8984i		−	1.29455i	−0.762256	−	0.647276i	\(-0.775908\pi\)
							0.762256	−	0.647276i	\(-0.224092\pi\)
\(41\)	4.93331	+	5.87929i	0.120325	+	0.143397i	0.822844	−	0.568267i	\(-0.192386\pi\)
							−0.702519	+	0.711665i	\(0.747942\pi\)
\(43\)	−49.4611	−	18.0024i	−1.15026	−	0.418659i	−0.304650	−	0.952464i	\(-0.598539\pi\)
							−0.845608	+	0.533805i	\(0.820762\pi\)
\(47\)	−12.5636	+	71.2519i	−0.267311	+	1.51600i	0.495061	+	0.868858i	\(0.335146\pi\)
							−0.762372	+	0.647139i	\(0.775965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.10.3	✓	24
3.2	odd	2		171.3.ba.d.10.2		24
19.2	odd	18	inner	57.3.k.b.40.3	yes	24
57.2	even	18		171.3.ba.d.154.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.10.3	✓	24	1.1	even	1	trivial
57.3.k.b.40.3	yes	24	19.2	odd	18	inner
171.3.ba.d.10.2		24	3.2	odd	2
171.3.ba.d.154.2		24	57.2	even	18