Embedded newform 100.4.l.b.3.12

• Label: 100.4.l.b.3.12
• 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			3.12
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53298 - 2.37697i) q^{2} +(9.34224 - 1.47967i) q^{3} +(-3.29997 + 7.28767i) q^{4} +(-7.08642 + 8.64770i) q^{5} +(-17.8386 - 19.9379i) q^{6} +(22.1183 + 22.1183i) q^{7} +(22.3814 - 3.32790i) q^{8} +(59.4095 - 19.3033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53298	−	2.37697i	−0.541989	−	0.840386i
							\(3\)	9.34224	−	1.47967i	1.79792	−	0.284762i	0.834155	−	0.551531i	\(-0.185956\pi\)
0.963761	+	0.266769i	\(0.0859561\pi\)
\(5\)	−7.08642	+	8.64770i	−0.633828	+	0.773474i
							\(7\)	22.1183	+	22.1183i	1.19428	+	1.19428i	0.975854	+	0.218424i	\(0.0700917\pi\)
0.218424	+	0.975854i	\(0.429908\pi\)
\(11\)	−10.8595	−	3.52848i	−0.297661	−	0.0967160i	0.156379	−	0.987697i	\(-0.450018\pi\)
							−0.454040	+	0.890981i	\(0.650018\pi\)
\(13\)	−16.7509	+	32.8754i	−0.357373	+	0.701385i	−0.997776	−	0.0666544i	\(-0.978768\pi\)
							0.640403	+	0.768039i	\(0.278768\pi\)
\(17\)	8.15984	−	51.5192i	0.116415	−	0.735014i	−0.858562	−	0.512709i	\(-0.828642\pi\)
							0.974977	−	0.222305i	\(-0.0713581\pi\)
\(19\)	12.5903	−	9.14736i	0.152021	−	0.110450i	−0.509175	−	0.860663i	\(-0.670049\pi\)
							0.661196	+	0.750213i	\(0.270049\pi\)
\(23\)	−36.9231	−	72.4657i	−0.334739	−	0.656963i	0.660877	−	0.750494i	\(-0.270184\pi\)
							−0.995616	+	0.0935313i	\(0.970184\pi\)
\(29\)	−1.27848	+	1.75968i	−0.00818650	+	0.0112677i	−0.813091	−	0.582137i	\(-0.802217\pi\)
							0.804904	+	0.593405i	\(0.202217\pi\)
\(31\)	−121.693	−	167.496i	−0.705057	−	0.970427i	−0.999889	−	0.0148785i	\(-0.995264\pi\)
							0.294833	−	0.955549i	\(-0.404736\pi\)
\(37\)	310.380	+	158.147i	1.37909	+	0.702680i	0.977063	−	0.212949i	\(-0.0683068\pi\)
							0.402024	+	0.915629i	\(0.368307\pi\)
\(41\)	−68.4243	−	210.588i	−0.260636	−	0.802155i	−0.992667	−	0.120883i	\(-0.961427\pi\)
							0.732031	−	0.681272i	\(-0.238573\pi\)
\(43\)	−106.774	+	106.774i	−0.378673	+	0.378673i	−0.870623	−	0.491951i	\(-0.836284\pi\)
							0.491951	+	0.870623i	\(0.336284\pi\)
\(47\)	−4.64599	−	29.3336i	−0.0144189	−	0.0910373i	0.979429	−	0.201787i	\(-0.0646748\pi\)
							−0.993848	+	0.110750i	\(0.964675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.12	✓	336
4.3	odd	2	inner	100.4.l.b.3.18	yes	336
25.17	odd	20	inner	100.4.l.b.67.18	yes	336
100.67	even	20	inner	100.4.l.b.67.12	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.12	✓	336	1.1	even	1	trivial
100.4.l.b.3.18	yes	336	4.3	odd	2	inner
100.4.l.b.67.12	yes	336	100.67	even	20	inner
100.4.l.b.67.18	yes	336	25.17	odd	20	inner