Embedded newform 100.4.l.b.3.42

• Label: 100.4.l.b.3.42
• Level: $100$
• Weight: $4$
• Character: 100.3
• Analytic conductor: $5.900$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.90019100057\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(42\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			3.42
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.81905 + 0.230079i) q^{2} +(3.25037 - 0.514808i) q^{3} +(7.89413 + 1.29721i) q^{4} +(11.1335 - 1.02216i) q^{5} +(9.28141 - 0.703429i) q^{6} +(-12.3581 - 12.3581i) q^{7} +(21.9555 + 5.47318i) q^{8} +(-15.3787 + 4.99683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.81905	+	0.230079i	0.996686	+	0.0813453i
							\(3\)	3.25037	−	0.514808i	0.625534	−	0.0990748i	0.164383	−	0.986397i	\(-0.447437\pi\)
0.461151	+	0.887322i	\(0.347437\pi\)
\(5\)	11.1335	−	1.02216i	0.995812	−	0.0914251i
							\(7\)	−12.3581	−	12.3581i	−0.667276	−	0.667276i	0.289809	−	0.957085i	\(-0.406408\pi\)
−0.957085	+	0.289809i	\(0.906408\pi\)
\(11\)	−12.7856	−	4.15430i	−0.350455	−	0.113870i	0.128499	−	0.991710i	\(-0.458984\pi\)
							−0.478954	+	0.877840i	\(0.658984\pi\)
\(13\)	−16.0576	+	31.5148i	−0.342583	+	0.672357i	−0.996444	−	0.0842578i	\(-0.973148\pi\)
							0.653861	+	0.756615i	\(0.273148\pi\)
\(17\)	−5.29959	+	33.4603i	−0.0756083	+	0.477372i	0.920610	+	0.390484i	\(0.127692\pi\)
							−0.996218	+	0.0868880i	\(0.972308\pi\)
\(19\)	52.0191	−	37.7941i	0.628106	−	0.456345i	−0.227638	−	0.973746i	\(-0.573100\pi\)
							0.855743	+	0.517400i	\(0.173100\pi\)
\(23\)	−26.8770	−	52.7492i	−0.243663	−	0.478216i	0.736492	−	0.676446i	\(-0.236481\pi\)
							−0.980156	+	0.198230i	\(0.936481\pi\)
\(29\)	−168.687	+	232.178i	−1.08015	+	1.48671i	−0.220804	+	0.975318i	\(0.570868\pi\)
							−0.859351	+	0.511387i	\(0.829132\pi\)
\(31\)	−132.459	−	182.314i	−0.767430	−	1.05628i	−0.996559	−	0.0828807i	\(-0.973588\pi\)
							0.229130	−	0.973396i	\(-0.426412\pi\)
\(37\)	157.002	+	79.9967i	0.697596	+	0.355443i	0.766557	−	0.642177i	\(-0.221968\pi\)
							−0.0689611	+	0.997619i	\(0.521968\pi\)
\(41\)	−35.2622	−	108.526i	−0.134318	−	0.413388i	0.861165	−	0.508325i	\(-0.169735\pi\)
							−0.995483	+	0.0949368i	\(0.969735\pi\)
\(43\)	−154.329	+	154.329i	−0.547323	+	0.547323i	−0.925666	−	0.378343i	\(-0.876494\pi\)
							0.378343	+	0.925666i	\(0.376494\pi\)
\(47\)	−31.3417	−	197.884i	−0.0972693	−	0.614134i	−0.987377	−	0.158385i	\(-0.949371\pi\)
							0.890108	−	0.455749i	\(-0.150629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.42	yes	336
4.3	odd	2	inner	100.4.l.b.3.13	✓	336
25.17	odd	20	inner	100.4.l.b.67.13	yes	336
100.67	even	20	inner	100.4.l.b.67.42	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.13	✓	336	4.3	odd	2	inner
100.4.l.b.3.42	yes	336	1.1	even	1	trivial
100.4.l.b.67.13	yes	336	25.17	odd	20	inner
100.4.l.b.67.42	yes	336	100.67	even	20	inner