Embedded newform 83.7.d.a.2.10

• Label: 83.7.d.a.2.10
• Level: $83$
• Weight: $7$
• Character: 83.2
• Analytic conductor: $19.094$
• Analytic rank: $0$
• Dimension: $1640$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	83.d (of order \(82\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0944889404\)
Analytic rank:	\(0\)
Dimension:	\(1640\)
Relative dimension:	\(41\) over \(\Q(\zeta_{82})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{82}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	83.2
Dual form			83.7.d.a.42.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.29306 - 0.356211i) q^{2} +(-40.0220 + 16.1303i) q^{3} +(22.4218 + 1.72142i) q^{4} +(101.756 + 171.160i) q^{5} +(377.672 - 135.643i) q^{6} +(-487.620 - 154.317i) q^{7} +(383.512 + 44.2745i) q^{8} +(816.310 - 785.620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1640 q - 41 q^{2} - 59 q^{3} + 1109 q^{4} - 41 q^{5} - 41 q^{6} - 303 q^{7} - 41 q^{8} - 11296 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/83\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{82}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.29306	−	0.356211i	−1.16163	−	0.0445264i	−0.549648	−	0.835396i	\(-0.685238\pi\)
							−0.611984	+	0.790870i	\(0.709628\pi\)
\(3\)	−40.0220	+	16.1303i	−1.48229	+	0.597418i	−0.965896	−	0.258930i	\(-0.916630\pi\)
							−0.516398	+	0.856348i	\(0.672728\pi\)
\(5\)	101.756	+	171.160i	0.814045	+	1.36928i	0.926760	+	0.375654i	\(0.122582\pi\)
							−0.112715	+	0.993627i	\(0.535955\pi\)
\(7\)	−487.620	−	154.317i	−1.42163	−	0.449903i	−0.509738	−	0.860330i	\(-0.670258\pi\)
							−0.911893	+	0.410427i	\(0.865380\pi\)
\(11\)	−692.571	−	908.568i	−0.520339	−	0.682621i	0.458161	−	0.888869i	\(-0.348508\pi\)
							−0.978500	+	0.206249i	\(0.933874\pi\)
\(13\)	−2185.12	+	423.780i	−0.994595	+	0.192890i	−0.661453	−	0.749987i	\(-0.730060\pi\)
							−0.333142	+	0.942877i	\(0.608109\pi\)
\(17\)	3226.78	−	4982.72i	0.656783	−	1.01419i	−0.340479	−	0.940252i	\(-0.610589\pi\)
							0.997263	−	0.0739389i	\(-0.0235570\pi\)
\(19\)	757.525	−	3237.16i	0.110442	−	0.471958i	−0.889446	−	0.457040i	\(-0.848910\pi\)
							0.999889	−	0.0149183i	\(-0.00474883\pi\)
\(23\)	3764.00	−	4223.53i	0.309361	−	0.347130i	−0.571416	−	0.820660i	\(-0.693606\pi\)
							0.880778	+	0.473530i	\(0.157020\pi\)
\(29\)	23392.2	+	11582.2i	0.959128	+	0.474894i	0.854405	−	0.519607i	\(-0.173922\pi\)
							0.104722	+	0.994501i	\(0.466605\pi\)
\(31\)	1097.41	+	9505.94i	0.0368371	+	0.319088i	0.998925	+	0.0463491i	\(0.0147587\pi\)
							−0.962088	+	0.272739i	\(0.912071\pi\)
\(37\)	−67421.7	−	64886.9i	−1.33105	−	1.28101i	−0.931239	−	0.364410i	\(-0.881271\pi\)
							−0.399811	−	0.916597i	\(-0.630924\pi\)
\(41\)	4777.91	+	124649.i	0.0693244	+	1.80858i	0.459861	+	0.887991i	\(0.347899\pi\)
							−0.390536	+	0.920588i	\(0.627710\pi\)
\(43\)	−64560.9	+	28942.2i	−0.812015	+	0.364020i	−0.772354	−	0.635193i	\(-0.780921\pi\)
							−0.0396615	+	0.999213i	\(0.512628\pi\)
\(47\)	−36983.3	−	44844.9i	−0.356215	−	0.431936i	0.562536	−	0.826773i	\(-0.309826\pi\)
							−0.918751	+	0.394836i	\(0.870801\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.7.d.a.2.10	✓	1640
83.42	odd	82	inner	83.7.d.a.42.10	yes	1640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.7.d.a.2.10	✓	1640	1.1	even	1	trivial
83.7.d.a.42.10	yes	1640	83.42	odd	82	inner