Embedded newform 57.4.i.b.25.5

• Label: 57.4.i.b.25.5
• Level: $57$
• Weight: $4$
• Character: 57.25
• Analytic conductor: $3.363$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	57.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.36310887033\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			25.5
Character	\(\chi\)	\(=\)	57.25
Dual form			57.4.i.b.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57889 - 2.16395i) q^{2} +(2.81908 + 1.02606i) q^{3} +(0.578834 - 3.28273i) q^{4} +(-1.62337 - 9.20660i) q^{5} +(9.49045 - 3.45424i) q^{6} +(10.2044 - 17.6746i) q^{7} +(7.85512 + 13.6055i) q^{8} +(6.89440 + 5.78509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{2} + 9 q^{4} + 12 q^{5} - 9 q^{6} - 48 q^{7} - 57 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.57889	−	2.16395i	0.911777	−	0.765072i	−0.0606793	−	0.998157i	\(-0.519327\pi\)
							0.972456	+	0.233086i	\(0.0748823\pi\)
\(3\)	2.81908	+	1.02606i	0.542532	+	0.197465i
							\(5\)	−1.62337	−	9.20660i	−0.145199	−	0.823463i	−0.967207	−	0.253988i	\(-0.918258\pi\)
0.822009	−	0.569475i	\(-0.192853\pi\)
\(7\)	10.2044	−	17.6746i	0.550987	−	0.954337i	−0.447217	−	0.894426i	\(-0.647585\pi\)
							0.998204	−	0.0599116i	\(-0.0190819\pi\)
\(11\)	16.0000	+	27.7127i	0.438561	+	0.759610i	0.997579	−	0.0695460i	\(-0.0221551\pi\)
							−0.559018	+	0.829156i	\(0.688822\pi\)
\(13\)	−67.0703	+	24.4116i	−1.43092	+	0.520813i	−0.937195	−	0.348805i	\(-0.886587\pi\)
							−0.493725	+	0.869618i	\(0.664365\pi\)
\(17\)	−60.7411	+	50.9678i	−0.866581	+	0.727148i	−0.963375	−	0.268157i	\(-0.913585\pi\)
							0.0967945	+	0.995304i	\(0.469141\pi\)
\(19\)	−69.0817	−	45.6807i	−0.834128	−	0.551572i
							\(23\)	−15.7121	+	89.1075i	−0.142443	+	0.807835i	0.826942	+	0.562288i	\(0.190079\pi\)
−0.969385	+	0.245547i	\(0.921032\pi\)
\(29\)	16.0172	+	13.4401i	0.102563	+	0.0860606i	0.692627	−	0.721296i	\(-0.256453\pi\)
							−0.590064	+	0.807356i	\(0.700898\pi\)
\(31\)	67.4581	−	116.841i	0.390833	−	0.676943i	−0.601727	−	0.798702i	\(-0.705520\pi\)
							0.992560	+	0.121759i	\(0.0388536\pi\)
\(37\)	235.126			1.04472			0.522358	−	0.852726i	\(-0.325053\pi\)
							0.522358	+	0.852726i	\(0.325053\pi\)
\(41\)	−303.396	−	110.427i	−1.15567	−	0.420630i	−0.308121	−	0.951347i	\(-0.599700\pi\)
							−0.847549	+	0.530717i	\(0.821923\pi\)
\(43\)	−59.8743	−	339.564i	−0.212343	−	1.20426i	−0.885458	−	0.464719i	\(-0.846155\pi\)
							0.673115	−	0.739538i	\(-0.264956\pi\)
\(47\)	232.960	+	195.477i	0.722993	+	0.606663i	0.928212	−	0.372053i	\(-0.121346\pi\)
							−0.205218	+	0.978716i	\(0.565790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.4.i.b.25.5	yes	36
3.2	odd	2		171.4.u.c.82.2		36
19.4	even	9		1083.4.a.s.1.13		18
19.15	odd	18		1083.4.a.t.1.6		18
19.16	even	9	inner	57.4.i.b.16.5	✓	36
57.35	odd	18		171.4.u.c.73.2		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.4.i.b.16.5	✓	36	19.16	even	9	inner
57.4.i.b.25.5	yes	36	1.1	even	1	trivial
171.4.u.c.73.2		36	57.35	odd	18
171.4.u.c.82.2		36	3.2	odd	2
1083.4.a.s.1.13		18	19.4	even	9
1083.4.a.t.1.6		18	19.15	odd	18