Embedded newform 100.4.l.b.3.2

• Label: 100.4.l.b.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.76384 + 0.600991i) q^{2} +(3.69203 - 0.584760i) q^{3} +(7.27762 - 3.32209i) q^{4} +(-8.66395 + 7.06654i) q^{5} +(-9.85274 + 3.83506i) q^{6} +(6.66746 + 6.66746i) q^{7} +(-18.1176 + 13.5555i) q^{8} +(-12.3894 + 4.02556i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.76384	+	0.600991i	−0.977165	+	0.212483i
							\(3\)	3.69203	−	0.584760i	0.710531	−	0.112537i	0.209301	−	0.977851i	\(-0.432881\pi\)
0.501230	+	0.865314i	\(0.332881\pi\)
\(5\)	−8.66395	+	7.06654i	−0.774927	+	0.632051i
							\(7\)	6.66746	+	6.66746i	0.360009	+	0.360009i	0.863816	−	0.503807i	\(-0.168068\pi\)
−0.503807	+	0.863816i	\(0.668068\pi\)
\(11\)	34.5393	+	11.2225i	0.946726	+	0.307610i	0.741384	−	0.671081i	\(-0.234170\pi\)
							0.205341	+	0.978690i	\(0.434170\pi\)
\(13\)	2.33854	−	4.58964i	0.0498918	−	0.0979182i	−0.864714	−	0.502264i	\(-0.832501\pi\)
							0.914606	+	0.404345i	\(0.132501\pi\)
\(17\)	−18.6457	+	117.724i	−0.266015	+	1.67955i	0.386903	+	0.922121i	\(0.373545\pi\)
							−0.652917	+	0.757429i	\(0.726455\pi\)
\(19\)	−74.6143	+	54.2104i	−0.900931	+	0.654565i	−0.938705	−	0.344722i	\(-0.887973\pi\)
							0.0377739	+	0.999286i	\(0.487973\pi\)
\(23\)	76.7037	+	150.539i	0.695384	+	1.36477i	0.920619	+	0.390463i	\(0.127685\pi\)
							−0.225235	+	0.974304i	\(0.572315\pi\)
\(29\)	−1.43499	+	1.97510i	−0.00918869	+	0.0126471i	−0.813587	−	0.581444i	\(-0.802488\pi\)
							0.804398	+	0.594091i	\(0.202488\pi\)
\(31\)	30.0446	+	41.3528i	0.174070	+	0.239587i	0.887134	−	0.461512i	\(-0.152693\pi\)
							−0.713064	+	0.701099i	\(0.752693\pi\)
\(37\)	158.432	+	80.7253i	0.703949	+	0.358680i	0.769041	−	0.639199i	\(-0.220734\pi\)
							−0.0650920	+	0.997879i	\(0.520734\pi\)
\(41\)	−150.436	−	462.994i	−0.573028	−	1.76360i	−0.642797	−	0.766036i	\(-0.722226\pi\)
							0.0697693	−	0.997563i	\(-0.477774\pi\)
\(43\)	282.262	−	282.262i	1.00103	−	1.00103i	0.00103537	−	0.999999i	\(-0.499670\pi\)
							0.999999	−	0.00103537i	\(-0.000329570\pi\)
\(47\)	0.851165	+	5.37404i	0.00264160	+	0.0166784i	0.988974	−	0.148090i	\(-0.0473124\pi\)
							−0.986332	+	0.164768i	\(0.947312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.2	✓	336
4.3	odd	2	inner	100.4.l.b.3.33	yes	336
25.17	odd	20	inner	100.4.l.b.67.33	yes	336
100.67	even	20	inner	100.4.l.b.67.2	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.2	✓	336	1.1	even	1	trivial
100.4.l.b.3.33	yes	336	4.3	odd	2	inner
100.4.l.b.67.2	yes	336	100.67	even	20	inner
100.4.l.b.67.33	yes	336	25.17	odd	20	inner