Embedded newform 100.4.l.b.3.20

• Label: 100.4.l.b.3.20
• 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			3.20
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.195449 + 2.82167i) q^{2} +(-6.08610 + 0.963944i) q^{3} +(-7.92360 - 1.10299i) q^{4} +(-4.53820 + 10.2179i) q^{5} +(-1.53040 - 17.3614i) q^{6} +(1.35326 + 1.35326i) q^{7} +(4.66092 - 22.1422i) q^{8} +(10.4330 - 3.38987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.195449	+	2.82167i	−0.0691017	+	0.997610i
							\(3\)	−6.08610	+	0.963944i	−1.17127	+	0.185511i	−0.711605	−	0.702580i	\(-0.752031\pi\)
−0.459667	+	0.888091i	\(0.652031\pi\)
\(5\)	−4.53820	+	10.2179i	−0.405909	+	0.913913i
							\(7\)	1.35326	+	1.35326i	0.0730690	+	0.0730690i	0.742697	−	0.669628i	\(-0.233546\pi\)
−0.669628	+	0.742697i	\(0.733546\pi\)
\(11\)	4.66383	+	1.51537i	0.127836	+	0.0415365i	0.372236	−	0.928138i	\(-0.378591\pi\)
							−0.244400	+	0.969674i	\(0.578591\pi\)
\(13\)	36.7096	−	72.0467i	0.783186	−	1.53709i	−0.0592193	−	0.998245i	\(-0.518861\pi\)
							0.842405	−	0.538844i	\(-0.181139\pi\)
\(17\)	5.61082	−	35.4253i	0.0800484	−	0.505406i	−0.914790	−	0.403931i	\(-0.867644\pi\)
							0.994838	−	0.101475i	\(-0.0323563\pi\)
\(19\)	−101.728	+	73.9099i	−1.22832	+	0.892426i	−0.996763	−	0.0803961i	\(-0.974381\pi\)
							−0.231555	+	0.972822i	\(0.574381\pi\)
\(23\)	5.95649	+	11.6903i	0.0540006	+	0.105982i	0.916430	−	0.400196i	\(-0.131058\pi\)
							−0.862429	+	0.506178i	\(0.831058\pi\)
\(29\)	−75.2231	+	103.536i	−0.481675	+	0.662969i	−0.978826	−	0.204696i	\(-0.934380\pi\)
							0.497151	+	0.867664i	\(0.334380\pi\)
\(31\)	−140.213	−	192.987i	−0.812356	−	1.11811i	−0.990956	−	0.134191i	\(-0.957157\pi\)
							0.178599	−	0.983922i	\(-0.442843\pi\)
\(37\)	−225.386	−	114.840i	−1.00144	−	0.510259i	−0.125199	−	0.992132i	\(-0.539957\pi\)
							−0.876241	+	0.481872i	\(0.839957\pi\)
\(41\)	−46.7170	−	143.780i	−0.177950	−	0.547675i	0.821806	−	0.569768i	\(-0.192967\pi\)
							−0.999756	+	0.0220931i	\(0.992967\pi\)
\(43\)	−46.6004	+	46.6004i	−0.165267	+	0.165267i	−0.784896	−	0.619628i	\(-0.787283\pi\)
							0.619628	+	0.784896i	\(0.287283\pi\)
\(47\)	−32.1916	−	203.250i	−0.0999069	−	0.630787i	−0.985932	−	0.167149i	\(-0.946544\pi\)
							0.886025	−	0.463638i	\(-0.153456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.20	✓	336
4.3	odd	2	inner	100.4.l.b.3.34	yes	336
25.17	odd	20	inner	100.4.l.b.67.34	yes	336
100.67	even	20	inner	100.4.l.b.67.20	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.20	✓	336	1.1	even	1	trivial
100.4.l.b.3.34	yes	336	4.3	odd	2	inner
100.4.l.b.67.20	yes	336	100.67	even	20	inner
100.4.l.b.67.34	yes	336	25.17	odd	20	inner