Embedded newform 100.4.l.b.3.4

• Label: 100.4.l.b.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.74803 - 0.669566i) q^{2} +(8.37146 - 1.32591i) q^{3} +(7.10336 + 3.67998i) q^{4} +(-8.14258 - 7.66149i) q^{5} +(-23.8928 - 1.96161i) q^{6} +(-22.9461 - 22.9461i) q^{7} +(-17.0563 - 14.8689i) q^{8} +(42.6448 - 13.8562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.74803	−	0.669566i	−0.971576	−	0.236727i
							\(3\)	8.37146	−	1.32591i	1.61109	−	0.255171i	0.715026	−	0.699098i	\(-0.246415\pi\)
0.896063	+	0.443927i	\(0.146415\pi\)
\(5\)	−8.14258	−	7.66149i	−0.728294	−	0.685264i
							\(7\)	−22.9461	−	22.9461i	−1.23897	−	1.23897i	−0.960421	−	0.278551i	\(-0.910146\pi\)
−0.278551	−	0.960421i	\(-0.589854\pi\)
\(11\)	−12.7513	−	4.14314i	−0.349514	−	0.113564i	0.128998	−	0.991645i	\(-0.458824\pi\)
							−0.478512	+	0.878081i	\(0.658824\pi\)
\(13\)	−4.12738	+	8.10043i	−0.0880560	+	0.172820i	−0.930835	−	0.365441i	\(-0.880918\pi\)
							0.842779	+	0.538260i	\(0.180918\pi\)
\(17\)	−1.11392	+	7.03301i	−0.0158921	+	0.100339i	−0.994359	−	0.106063i	\(-0.966175\pi\)
							0.978467	+	0.206402i	\(0.0661754\pi\)
\(19\)	114.589	−	83.2537i	1.38361	−	1.00525i	0.387072	−	0.922049i	\(-0.373486\pi\)
							0.996533	−	0.0831986i	\(-0.0265136\pi\)
\(23\)	−51.2197	−	100.524i	−0.464350	−	0.911338i	−0.997850	−	0.0655381i	\(-0.979124\pi\)
							0.533500	−	0.845800i	\(-0.320876\pi\)
\(29\)	−10.2872	+	14.1591i	−0.0658717	+	0.0906646i	−0.840682	−	0.541529i	\(-0.817845\pi\)
							0.774810	+	0.632194i	\(0.217845\pi\)
\(31\)	102.553	+	141.151i	0.594160	+	0.817792i	0.995158	−	0.0982874i	\(-0.0313364\pi\)
							−0.400998	+	0.916079i	\(0.631336\pi\)
\(37\)	210.347	+	107.177i	0.934617	+	0.476211i	0.853848	−	0.520522i	\(-0.174263\pi\)
							0.0807685	+	0.996733i	\(0.474263\pi\)
\(41\)	−62.1042	−	191.137i	−0.236562	−	0.728063i	−0.996910	−	0.0785475i	\(-0.974972\pi\)
							0.760348	−	0.649515i	\(-0.225028\pi\)
\(43\)	367.024	−	367.024i	1.30164	−	1.30164i	0.374357	−	0.927284i	\(-0.377863\pi\)
							0.927284	−	0.374357i	\(-0.122137\pi\)
\(47\)	−46.7888	−	295.413i	−0.145210	−	0.916817i	−0.947470	−	0.319843i	\(-0.896370\pi\)
							0.802261	−	0.596973i	\(-0.203630\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.4	✓	336
4.3	odd	2	inner	100.4.l.b.3.27	yes	336
25.17	odd	20	inner	100.4.l.b.67.27	yes	336
100.67	even	20	inner	100.4.l.b.67.4	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.4	✓	336	1.1	even	1	trivial
100.4.l.b.3.27	yes	336	4.3	odd	2	inner
100.4.l.b.67.4	yes	336	100.67	even	20	inner
100.4.l.b.67.27	yes	336	25.17	odd	20	inner