Embedded newform 83.7.b.a.82.1

• Label: 83.7.b.a.82.1
• Level: $83$
• Weight: $7$
• Character: 83.82
• Self dual: yes
• Analytic conductor: $19.094$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -83
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	83.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(19.0944889404\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			82.1
Character	\(\chi\)	\(=\)	83.82

$q$-expansion

\(f(q)\)

\(=\)

\(q-29.0000 q^{3} +64.0000 q^{4} -61.0000 q^{7} +112.000 q^{9} +587.000 q^{11} -1856.00 q^{12} +4096.00 q^{16} -4201.00 q^{17} +1769.00 q^{21} +8066.00 q^{23} +15625.0 q^{25} +17893.0 q^{27} -3904.00 q^{28} +46703.0 q^{29} +40907.0 q^{31} -17023.0 q^{33} +7168.00 q^{36} +10919.0 q^{37} +5042.00 q^{41} +37568.0 q^{44} -118784. q^{48} -113928. q^{49} +121829. q^{51} -188917. q^{59} +435287. q^{61} -6832.00 q^{63} +262144. q^{64} -268864. q^{68} -233914. q^{69} -453125. q^{75} -35807.0 q^{77} -600545. q^{81} -571787. q^{83} +113216. q^{84} -1.35439e6 q^{87} +516224. q^{92} -1.18630e6 q^{93} +65744.0 q^{99} +O(q^{100})\)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(3\)	−29.0000	−1.07407	−0.537037	−	0.843559i	\(-0.680456\pi\)
			−0.537037	+	0.843559i	\(0.680456\pi\)
\(5\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	−61.0000	−0.177843	−0.0889213	−	0.996039i	\(-0.528342\pi\)
			−0.0889213	+	0.996039i	\(0.528342\pi\)
\(11\)	587.000	0.441022	0.220511	−	0.975385i	\(-0.429228\pi\)
			0.220511	+	0.975385i	\(0.429228\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	−4201.00	−0.855078	−0.427539	−	0.903997i	\(-0.640619\pi\)
			−0.427539	+	0.903997i	\(0.640619\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	8066.00	0.662941	0.331470	−	0.943466i	\(-0.392455\pi\)
			0.331470	+	0.943466i	\(0.392455\pi\)
\(29\)	46703.0	1.91492	0.957460	−	0.288565i	\(-0.0931781\pi\)
			0.957460	+	0.288565i	\(0.0931781\pi\)
\(31\)	40907.0	1.37313	0.686566	−	0.727067i	\(-0.259117\pi\)
			0.686566	+	0.727067i	\(0.259117\pi\)
\(37\)	10919.0	0.215565	0.107782	−	0.994175i	\(-0.465625\pi\)
			0.107782	+	0.994175i	\(0.465625\pi\)
\(41\)	5042.00	0.0731562	0.0365781	−	0.999331i	\(-0.488354\pi\)
			0.0365781	+	0.999331i	\(0.488354\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	−188917.	−0.919846	−0.459923	−	0.887959i	\(-0.652123\pi\)
			−0.459923	+	0.887959i	\(0.652123\pi\)
\(61\)	435287.	1.91772	0.958862	−	0.283872i	\(-0.0916191\pi\)
			0.958862	+	0.283872i	\(0.0916191\pi\)
\(67\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(79\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(83\)	−571787.	−1.00000
			\(89\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(97\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.7.b.a.82.1	✓	1
83.82	odd	2	CM	83.7.b.a.82.1	✓	1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.7.b.a.82.1	✓	1	1.1	even	1	trivial
83.7.b.a.82.1	✓	1	83.82	odd	2	CM