Embedded newform 100.4.l.b.3.13

• Label: 100.4.l.b.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47086 + 2.41590i) q^{2} +(-3.25037 + 0.514808i) q^{3} +(-3.67314 - 7.10690i) q^{4} +(11.1335 - 1.02216i) q^{5} +(3.53711 - 8.60978i) q^{6} +(12.3581 + 12.3581i) q^{7} +(22.5722 + 1.57932i) q^{8} +(-15.3787 + 4.99683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47086	+	2.41590i	−0.520028	+	0.854149i
							\(3\)	−3.25037	+	0.514808i	−0.625534	+	0.0990748i	−0.461151	−	0.887322i	\(-0.652563\pi\)
−0.164383	+	0.986397i	\(0.552563\pi\)
\(5\)	11.1335	−	1.02216i	0.995812	−	0.0914251i
							\(7\)	12.3581	+	12.3581i	0.667276	+	0.667276i	0.957085	−	0.289809i	\(-0.0935917\pi\)
−0.289809	+	0.957085i	\(0.593592\pi\)
\(11\)	12.7856	+	4.15430i	0.350455	+	0.113870i	0.478954	−	0.877840i	\(-0.341016\pi\)
							−0.128499	+	0.991710i	\(0.541016\pi\)
\(13\)	−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	−0.855743	−	0.517400i	\(-0.826900\pi\)
							0.227638	+	0.973746i	\(0.426900\pi\)
\(23\)	26.8770	+	52.7492i	0.243663	+	0.478216i	0.980156	−	0.198230i	\(-0.0635193\pi\)
							−0.736492	+	0.676446i	\(0.763519\pi\)
\(29\)	−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	\(0.0264121\pi\)
							−0.229130	+	0.973396i	\(0.573588\pi\)
\(37\)	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	−0.378343	−	0.925666i	\(-0.623506\pi\)
							0.925666	+	0.378343i	\(0.123506\pi\)
\(47\)	31.3417	+	197.884i	0.0972693	+	0.614134i	0.987377	+	0.158385i	\(0.0506286\pi\)
							−0.890108	+	0.455749i	\(0.849371\pi\)

(See \(a_n\) instead)

Twists

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

    

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