Embedded newform 100.4.l.b.3.7

• Label: 100.4.l.b.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60734 + 1.09625i) q^{2} +(-2.84160 + 0.450065i) q^{3} +(5.59648 - 5.71659i) q^{4} +(7.98147 + 7.82918i) q^{5} +(6.91564 - 4.28857i) q^{6} +(-22.0571 - 22.0571i) q^{7} +(-8.32514 + 21.0403i) q^{8} +(-17.8064 + 5.78565i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60734	+	1.09625i	−0.921835	+	0.387582i
							\(3\)	−2.84160	+	0.450065i	−0.546866	+	0.0866150i	−0.423753	−	0.905778i	\(-0.639288\pi\)
−0.123112	+	0.992393i	\(0.539288\pi\)
\(5\)	7.98147	+	7.82918i	0.713885	+	0.700263i
							\(7\)	−22.0571	−	22.0571i	−1.19097	−	1.19097i	−0.976795	−	0.214175i	\(-0.931294\pi\)
−0.214175	−	0.976795i	\(-0.568706\pi\)
\(11\)	41.7842	+	13.5765i	1.14531	+	0.372133i	0.819374	−	0.573259i	\(-0.194321\pi\)
							0.325934	+	0.945392i	\(0.394321\pi\)
\(13\)	20.5289	−	40.2902i	0.437976	−	0.859577i	−0.561508	−	0.827471i	\(-0.689779\pi\)
							0.999484	−	0.0321056i	\(-0.0102213\pi\)
\(17\)	16.0885	−	101.579i	0.229531	−	1.44920i	−0.556412	−	0.830906i	\(-0.687822\pi\)
							0.785944	−	0.618298i	\(-0.212178\pi\)
\(19\)	100.152	−	72.7650i	1.20929	−	0.878602i	0.214125	−	0.976806i	\(-0.431310\pi\)
							0.995166	+	0.0982045i	\(0.0313099\pi\)
\(23\)	−11.8576	−	23.2718i	−0.107499	−	0.210978i	0.830990	−	0.556287i	\(-0.187774\pi\)
							−0.938489	+	0.345308i	\(0.887774\pi\)
\(29\)	14.7433	−	20.2924i	0.0944057	−	0.129938i	−0.759201	−	0.650856i	\(-0.774410\pi\)
							0.853607	+	0.520918i	\(0.174410\pi\)
\(31\)	−34.6756	−	47.7269i	−0.200901	−	0.276516i	0.696665	−	0.717396i	\(-0.254666\pi\)
							−0.897566	+	0.440880i	\(0.854666\pi\)
\(37\)	−29.1700	−	14.8628i	−0.129609	−	0.0660389i	0.387984	−	0.921666i	\(-0.373172\pi\)
							−0.517592	+	0.855627i	\(0.673172\pi\)
\(41\)	−128.756	−	396.271i	−0.490448	−	1.50944i	−0.823932	−	0.566688i	\(-0.808224\pi\)
							0.333484	−	0.942756i	\(-0.391776\pi\)
\(43\)	−67.9934	+	67.9934i	−0.241137	+	0.241137i	−0.817320	−	0.576183i	\(-0.804541\pi\)
							0.576183	+	0.817320i	\(0.304541\pi\)
\(47\)	−58.3074	−	368.139i	−0.180958	−	1.14252i	−0.896202	−	0.443647i	\(-0.853684\pi\)
							0.715244	−	0.698875i	\(-0.246316\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.7	✓	336
4.3	odd	2	inner	100.4.l.b.3.36	yes	336
25.17	odd	20	inner	100.4.l.b.67.36	yes	336
100.67	even	20	inner	100.4.l.b.67.7	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.7	✓	336	1.1	even	1	trivial
100.4.l.b.3.36	yes	336	4.3	odd	2	inner
100.4.l.b.67.7	yes	336	100.67	even	20	inner
100.4.l.b.67.36	yes	336	25.17	odd	20	inner