Embedded newform 100.4.l.b.3.5

• Label: 100.4.l.b.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.74601 - 0.677799i) q^{2} +(-7.00032 + 1.10874i) q^{3} +(7.08118 + 3.72249i) q^{4} +(4.93198 - 10.0337i) q^{5} +(19.9745 + 1.70019i) q^{6} +(-12.8348 - 12.8348i) q^{7} +(-16.9219 - 15.0216i) q^{8} +(22.0967 - 7.17965i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.74601	−	0.677799i	−0.970862	−	0.239638i
							\(3\)	−7.00032	+	1.10874i	−1.34721	+	0.213378i	−0.788015	−	0.615656i	\(-0.788891\pi\)
−0.559198	+	0.829034i	\(0.688891\pi\)
\(5\)	4.93198	−	10.0337i	0.441130	−	0.897443i
							\(7\)	−12.8348	−	12.8348i	−0.693016	−	0.693016i	0.269878	−	0.962894i	\(-0.413017\pi\)
−0.962894	+	0.269878i	\(0.913017\pi\)
\(11\)	18.4532	+	5.99580i	0.505804	+	0.164346i	0.550793	−	0.834642i	\(-0.314325\pi\)
							−0.0449892	+	0.998987i	\(0.514325\pi\)
\(13\)	−4.13507	+	8.11554i	−0.0882202	+	0.173142i	−0.930901	−	0.365272i	\(-0.880976\pi\)
							0.842681	+	0.538414i	\(0.180976\pi\)
\(17\)	−15.5131	+	97.9456i	−0.221322	+	1.39737i	0.587456	+	0.809256i	\(0.300129\pi\)
							−0.808778	+	0.588114i	\(0.799871\pi\)
\(19\)	−84.0005	+	61.0299i	−1.01427	+	0.736907i	−0.965099	−	0.261884i	\(-0.915656\pi\)
							−0.0491657	+	0.998791i	\(0.515656\pi\)
\(23\)	42.4318	+	83.2772i	0.384680	+	0.754978i	0.999431	−	0.0337441i	\(-0.0107431\pi\)
							−0.614750	+	0.788722i	\(0.710743\pi\)
\(29\)	−64.1481	+	88.2923i	−0.410759	+	0.565361i	−0.963403	−	0.268056i	\(-0.913619\pi\)
							0.552645	+	0.833417i	\(0.313619\pi\)
\(31\)	−30.1302	−	41.4706i	−0.174566	−	0.240269i	0.712765	−	0.701403i	\(-0.247443\pi\)
							−0.887330	+	0.461134i	\(0.847443\pi\)
\(37\)	129.559	+	66.0137i	0.575660	+	0.293313i	0.717480	−	0.696579i	\(-0.245295\pi\)
							−0.141820	+	0.989892i	\(0.545295\pi\)
\(41\)	106.834	+	328.801i	0.406943	+	1.25244i	0.919262	+	0.393646i	\(0.128786\pi\)
							−0.512320	+	0.858795i	\(0.671214\pi\)
\(43\)	−251.656	+	251.656i	−0.892494	+	0.892494i	−0.994757	−	0.102264i	\(-0.967391\pi\)
							0.102264	+	0.994757i	\(0.467391\pi\)
\(47\)	−64.7280	−	408.676i	−0.200884	−	1.26833i	−0.857649	−	0.514236i	\(-0.828076\pi\)
							0.656765	−	0.754095i	\(-0.271924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.5	✓	336
4.3	odd	2	inner	100.4.l.b.3.26	yes	336
25.17	odd	20	inner	100.4.l.b.67.26	yes	336
100.67	even	20	inner	100.4.l.b.67.5	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.5	✓	336	1.1	even	1	trivial
100.4.l.b.3.26	yes	336	4.3	odd	2	inner
100.4.l.b.67.5	yes	336	100.67	even	20	inner
100.4.l.b.67.26	yes	336	25.17	odd	20	inner