Embedded newform 100.4.l.b.3.14

• Label: 100.4.l.b.3.14
• 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			3.14
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41772 + 2.44746i) q^{2} +(3.68208 - 0.583183i) q^{3} +(-3.98013 - 6.93964i) q^{4} +(-7.22245 + 8.53442i) q^{5} +(-3.79284 + 9.83853i) q^{6} +(-12.1513 - 12.1513i) q^{7} +(22.6272 + 0.0972760i) q^{8} +(-12.4610 + 4.04881i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41772	+	2.44746i	−0.501241	+	0.865308i
							\(3\)	3.68208	−	0.583183i	0.708616	−	0.112234i	0.208285	−	0.978068i	\(-0.433212\pi\)
0.500331	+	0.865834i	\(0.333212\pi\)
\(5\)	−7.22245	+	8.53442i	−0.645995	+	0.763342i
							\(7\)	−12.1513	−	12.1513i	−0.656108	−	0.656108i	0.298349	−	0.954457i	\(-0.403564\pi\)
−0.954457	+	0.298349i	\(0.903564\pi\)
\(11\)	−31.9656	−	10.3862i	−0.876180	−	0.284688i	−0.163810	−	0.986492i	\(-0.552378\pi\)
							−0.712370	+	0.701804i	\(0.752378\pi\)
\(13\)	−36.7231	+	72.0731i	−0.783473	+	1.53765i	0.0585946	+	0.998282i	\(0.481338\pi\)
							−0.842068	+	0.539371i	\(0.818662\pi\)
\(17\)	15.9264	−	100.555i	0.227219	−	1.43461i	−0.565364	−	0.824841i	\(-0.691264\pi\)
							0.792583	−	0.609764i	\(-0.208736\pi\)
\(19\)	−19.8414	+	14.4156i	−0.239576	+	0.174062i	−0.701094	−	0.713069i	\(-0.747305\pi\)
							0.461519	+	0.887131i	\(0.347305\pi\)
\(23\)	−40.2074	−	78.9115i	−0.364514	−	0.715399i	0.633797	−	0.773500i	\(-0.281496\pi\)
							−0.998311	+	0.0581007i	\(0.981496\pi\)
\(29\)	29.2764	−	40.2954i	0.187465	−	0.258023i	−0.704932	−	0.709275i	\(-0.749022\pi\)
							0.892397	+	0.451252i	\(0.149022\pi\)
\(31\)	186.351	+	256.490i	1.07967	+	1.48603i	0.859888	+	0.510482i	\(0.170533\pi\)
							0.219777	+	0.975550i	\(0.429467\pi\)
\(37\)	13.8526	+	7.05827i	0.0615502	+	0.0313614i	0.484495	−	0.874794i	\(-0.339003\pi\)
							−0.422945	+	0.906156i	\(0.639003\pi\)
\(41\)	87.0628	+	267.952i	0.331632	+	1.02066i	0.968357	+	0.249568i	\(0.0802888\pi\)
							−0.636725	+	0.771091i	\(0.719711\pi\)
\(43\)	−309.787	+	309.787i	−1.09865	+	1.09865i	−0.104084	+	0.994568i	\(0.533191\pi\)
							−0.994568	+	0.104084i	\(0.966809\pi\)
\(47\)	−15.9103	−	100.454i	−0.0493777	−	0.311759i	−0.999999	−	0.00140532i	\(-0.999553\pi\)
							0.950621	−	0.310353i	\(-0.100447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.14	✓	336
4.3	odd	2	inner	100.4.l.b.3.41	yes	336
25.17	odd	20	inner	100.4.l.b.67.41	yes	336
100.67	even	20	inner	100.4.l.b.67.14	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.14	✓	336	1.1	even	1	trivial
100.4.l.b.3.41	yes	336	4.3	odd	2	inner
100.4.l.b.67.14	yes	336	100.67	even	20	inner
100.4.l.b.67.41	yes	336	25.17	odd	20	inner