Embedded newform 100.2.l.b.23.9

• Label: 100.2.l.b.23.9
• Level: $100$
• Weight: $2$
• Character: 100.23
• Analytic conductor: $0.799$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.798504020213\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(12\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			23.9
Character	\(\chi\)	\(=\)	100.23
Dual form			100.2.l.b.87.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.868976 - 1.11574i) q^{2} +(-0.956383 - 1.87701i) q^{3} +(-0.489760 - 1.93911i) q^{4} +(0.727343 + 2.11447i) q^{5} +(-2.92533 - 0.563998i) q^{6} +(-0.0806175 - 0.0806175i) q^{7} +(-2.58913 - 1.13859i) q^{8} +(-0.845131 + 1.16322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 10 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.868976	−	1.11574i	0.614459	−	0.788949i
							\(3\)	−0.956383	−	1.87701i	−0.552168	−	1.08369i	−0.983400	−	0.181450i	\(-0.941921\pi\)
0.431232	−	0.902241i	\(-0.358079\pi\)
\(5\)	0.727343	+	2.11447i	0.325278	+	0.945619i
							\(7\)	−0.0806175	−	0.0806175i	−0.0304705	−	0.0304705i	0.691707	−	0.722178i	\(-0.256859\pi\)
−0.722178	+	0.691707i	\(0.756859\pi\)
\(11\)	1.81801	+	2.50228i	0.548151	+	0.754465i	0.989760	−	0.142743i	\(-0.0455921\pi\)
							−0.441609	+	0.897208i	\(0.645592\pi\)
\(13\)	−0.504198	−	3.18338i	−0.139839	−	0.882911i	−0.953461	−	0.301516i	\(-0.902507\pi\)
							0.813622	−	0.581395i	\(-0.197493\pi\)
\(17\)	4.92677	+	2.51032i	1.19492	+	0.608841i	0.934261	−	0.356590i	\(-0.116061\pi\)
							0.260657	+	0.965431i	\(0.416061\pi\)
\(19\)	2.00242	+	6.16282i	0.459387	+	1.41385i	0.865907	+	0.500206i	\(0.166742\pi\)
							−0.406520	+	0.913642i	\(0.633258\pi\)
\(23\)	0.711667	−	4.49329i	0.148393	−	0.936916i	−0.795330	−	0.606177i	\(-0.792702\pi\)
							0.943722	−	0.330739i	\(-0.107298\pi\)
\(29\)	−5.80662	−	1.88668i	−1.07826	−	0.350348i	−0.284562	−	0.958658i	\(-0.591848\pi\)
							−0.793700	+	0.608309i	\(0.791848\pi\)
\(31\)	−1.98751	+	0.645780i	−0.356967	+	0.115986i	−0.482010	−	0.876166i	\(-0.660093\pi\)
							0.125043	+	0.992151i	\(0.460093\pi\)
\(37\)	−4.46869	+	0.707771i	−0.734648	+	0.116357i	−0.512538	−	0.858665i	\(-0.671295\pi\)
							−0.222110	+	0.975022i	\(0.571295\pi\)
\(41\)	2.59234	+	1.88344i	0.404855	+	0.294145i	0.771516	−	0.636210i	\(-0.219499\pi\)
							−0.366660	+	0.930355i	\(0.619499\pi\)
\(43\)	−9.16551	+	9.16551i	−1.39773	+	1.39773i	−0.591209	+	0.806518i	\(0.701349\pi\)
							−0.806518	+	0.591209i	\(0.798651\pi\)
\(47\)	6.28004	−	3.19984i	0.916038	−	0.466744i	0.0686029	−	0.997644i	\(-0.478146\pi\)
							0.847435	+	0.530900i	\(0.178146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.2.l.b.23.9	yes	96
3.2	odd	2		900.2.bj.d.523.4		96
4.3	odd	2	inner	100.2.l.b.23.6	✓	96
5.2	odd	4		500.2.l.e.407.10		96
5.3	odd	4		500.2.l.d.407.3		96
5.4	even	2		500.2.l.f.343.4		96
12.11	even	2		900.2.bj.d.523.7		96
20.3	even	4		500.2.l.d.407.1		96
20.7	even	4		500.2.l.e.407.12		96
20.19	odd	2		500.2.l.f.343.7		96
25.9	even	10		500.2.l.d.43.1		96
25.12	odd	20	inner	100.2.l.b.87.6	yes	96
25.13	odd	20		500.2.l.f.207.7		96
25.16	even	5		500.2.l.e.43.12		96
75.62	even	20		900.2.bj.d.487.7		96
100.59	odd	10		500.2.l.d.43.3		96
100.63	even	20		500.2.l.f.207.4		96
100.87	even	20	inner	100.2.l.b.87.9	yes	96
100.91	odd	10		500.2.l.e.43.10		96
300.287	odd	20		900.2.bj.d.487.4		96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.l.b.23.6	✓	96	4.3	odd	2	inner
100.2.l.b.23.9	yes	96	1.1	even	1	trivial
100.2.l.b.87.6	yes	96	25.12	odd	20	inner
100.2.l.b.87.9	yes	96	100.87	even	20	inner
500.2.l.d.43.1		96	25.9	even	10
500.2.l.d.43.3		96	100.59	odd	10
500.2.l.d.407.1		96	20.3	even	4
500.2.l.d.407.3		96	5.3	odd	4
500.2.l.e.43.10		96	100.91	odd	10
500.2.l.e.43.12		96	25.16	even	5
500.2.l.e.407.10		96	5.2	odd	4
500.2.l.e.407.12		96	20.7	even	4
500.2.l.f.207.4		96	100.63	even	20
500.2.l.f.207.7		96	25.13	odd	20
500.2.l.f.343.4		96	5.4	even	2
500.2.l.f.343.7		96	20.19	odd	2
900.2.bj.d.487.4		96	300.287	odd	20
900.2.bj.d.487.7		96	75.62	even	20
900.2.bj.d.523.4		96	3.2	odd	2
900.2.bj.d.523.7		96	12.11	even	2