Embedded newform 80.3.i.a.37.19

• Label: 80.3.i.a.37.19
• Level: $80$
• Weight: $3$
• Character: 80.37
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $44$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	80.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17984211488\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.19
Character	\(\chi\)	\(=\)	80.37
Dual form			80.3.i.a.13.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.83264 - 0.800895i) q^{2} -3.83124i q^{3} +(2.71713 - 2.93550i) q^{4} +(0.146974 + 4.99784i) q^{5} +(-3.06842 - 7.02129i) q^{6} +(-1.69668 + 1.69668i) q^{7} +(2.62850 - 7.55586i) q^{8} -5.67842 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 8 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.83264	−	0.800895i	0.916320	−	0.400448i
							\(3\)		−	3.83124i		−	1.27708i	−0.769588	−	0.638540i	\(-0.779539\pi\)
0.769588	−	0.638540i	\(-0.220461\pi\)
\(5\)	0.146974	+	4.99784i	0.0293949	+	0.999568i
							\(7\)	−1.69668	+	1.69668i	−0.242382	+	0.242382i	−0.817835	−	0.575453i	\(-0.804826\pi\)
0.575453	+	0.817835i	\(0.304826\pi\)
\(11\)	−6.09580	−	6.09580i	−0.554164	−	0.554164i	0.373476	−	0.927640i	\(-0.378166\pi\)
							−0.927640	+	0.373476i	\(0.878166\pi\)
\(13\)			14.9024i			1.14634i	0.819437	+	0.573169i	\(0.194286\pi\)
							−0.819437	+	0.573169i	\(0.805714\pi\)
\(17\)	10.0145	+	10.0145i	0.589087	+	0.589087i	0.937384	−	0.348297i	\(-0.113240\pi\)
							−0.348297	+	0.937384i	\(0.613240\pi\)
\(19\)	4.15403	+	4.15403i	0.218633	+	0.218633i	0.807922	−	0.589289i	\(-0.200592\pi\)
							−0.589289	+	0.807922i	\(0.700592\pi\)
\(23\)	31.1044	+	31.1044i	1.35237	+	1.35237i	0.883009	+	0.469356i	\(0.155514\pi\)
							0.469356	+	0.883009i	\(0.344486\pi\)
\(29\)	−38.9751	−	38.9751i	−1.34397	−	1.34397i	−0.892068	−	0.451901i	\(-0.850746\pi\)
							−0.451901	−	0.892068i	\(-0.649254\pi\)
\(31\)	−15.2616			−0.492309			−0.246154	−	0.969231i	\(-0.579167\pi\)
							−0.246154	+	0.969231i	\(0.579167\pi\)
\(37\)			10.0034i			0.270363i	0.990821	+	0.135181i	\(0.0431617\pi\)
							−0.990821	+	0.135181i	\(0.956838\pi\)
\(41\)		−	17.1555i		−	0.418427i	−0.977870	−	0.209213i	\(-0.932910\pi\)
							0.977870	−	0.209213i	\(-0.0670903\pi\)
\(43\)	−41.2482			−0.959261			−0.479630	−	0.877471i	\(-0.659229\pi\)
							−0.479630	+	0.877471i	\(0.659229\pi\)
\(47\)	−35.1314	−	35.1314i	−0.747476	−	0.747476i	0.226529	−	0.974004i	\(-0.427262\pi\)
							−0.974004	+	0.226529i	\(0.927262\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.3.i.a.37.19	yes	44
4.3	odd	2		320.3.i.a.177.19		44
5.2	odd	4		400.3.t.b.293.14		44
5.3	odd	4		80.3.t.a.53.9	yes	44
5.4	even	2		400.3.i.b.357.4		44
8.3	odd	2		640.3.i.a.97.4		44
8.5	even	2		640.3.i.b.97.19		44
16.3	odd	4		320.3.t.a.17.4		44
16.5	even	4		640.3.t.b.417.4		44
16.11	odd	4		640.3.t.a.417.19		44
16.13	even	4		80.3.t.a.77.9	yes	44
20.3	even	4		320.3.t.a.113.4		44
40.3	even	4		640.3.t.a.353.19		44
40.13	odd	4		640.3.t.b.353.4		44
80.3	even	4		320.3.i.a.273.4		44
80.13	odd	4	inner	80.3.i.a.13.19	✓	44
80.29	even	4		400.3.t.b.157.14		44
80.43	even	4		640.3.i.a.33.19		44
80.53	odd	4		640.3.i.b.33.4		44
80.77	odd	4		400.3.i.b.93.4		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.i.a.13.19	✓	44	80.13	odd	4	inner
80.3.i.a.37.19	yes	44	1.1	even	1	trivial
80.3.t.a.53.9	yes	44	5.3	odd	4
80.3.t.a.77.9	yes	44	16.13	even	4
320.3.i.a.177.19		44	4.3	odd	2
320.3.i.a.273.4		44	80.3	even	4
320.3.t.a.17.4		44	16.3	odd	4
320.3.t.a.113.4		44	20.3	even	4
400.3.i.b.93.4		44	80.77	odd	4
400.3.i.b.357.4		44	5.4	even	2
400.3.t.b.157.14		44	80.29	even	4
400.3.t.b.293.14		44	5.2	odd	4
640.3.i.a.33.19		44	80.43	even	4
640.3.i.a.97.4		44	8.3	odd	2
640.3.i.b.33.4		44	80.53	odd	4
640.3.i.b.97.19		44	8.5	even	2
640.3.t.a.353.19		44	40.3	even	4
640.3.t.a.417.19		44	16.11	odd	4
640.3.t.b.353.4		44	40.13	odd	4
640.3.t.b.417.4		44	16.5	even	4