Embedded newform 100.10.e.e.7.22

• Label: 100.10.e.e.7.22
• Level: $100$
• Weight: $10$
• Character: 100.7
• Analytic conductor: $51.504$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	100.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(51.5035836164\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.22
Character	\(\chi\)	\(=\)	100.7
Dual form			100.10.e.e.43.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.5582 + 6.87337i) q^{2} +(-110.359 - 110.359i) q^{3} +(417.514 + 296.355i) q^{4} +(-1620.61 - 3137.69i) q^{6} +(-1215.70 + 1215.70i) q^{7} +(6963.90 + 9258.61i) q^{8} +4675.33i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 30 q^{2} + 6152 q^{6} + 17100 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.5582	+	6.87337i	0.952748	+	0.303763i
							\(3\)	−110.359	−	110.359i	−0.786617	−	0.786617i	0.194321	−	0.980938i	\(-0.437750\pi\)
−0.980938	+	0.194321i	\(0.937750\pi\)
\(5\)		0			0
							\(7\)	−1215.70	+	1215.70i	−0.191375	+	0.191375i	−0.796290	−	0.604915i	\(-0.793207\pi\)
0.604915	+	0.796290i	\(0.293207\pi\)
\(11\)			39038.7i			0.803948i	0.915651	+	0.401974i	\(0.131676\pi\)
							−0.915651	+	0.401974i	\(0.868324\pi\)
\(13\)	53500.0	−	53500.0i	0.519528	−	0.519528i	−0.397901	−	0.917428i	\(-0.630261\pi\)
							0.917428	+	0.397901i	\(0.130261\pi\)
\(17\)	−381766.	−	381766.i	−1.10861	−	1.10861i	−0.993334	−	0.115272i	\(-0.963226\pi\)
							−0.115272	−	0.993334i	\(-0.536774\pi\)
\(19\)	−575036.			−1.01229			−0.506144	−	0.862449i	\(-0.668929\pi\)
							−0.506144	+	0.862449i	\(0.668929\pi\)
\(23\)	1.06766e6	+	1.06766e6i	0.795533	+	0.795533i	0.982388	−	0.186855i	\(-0.0598294\pi\)
							−0.186855	+	0.982388i	\(0.559829\pi\)
\(29\)			3.51224e6i			0.922132i	0.887366	+	0.461066i	\(0.152533\pi\)
							−0.887366	+	0.461066i	\(0.847467\pi\)
\(31\)			4.45221e6i			0.865860i	0.901427	+	0.432930i	\(0.142520\pi\)
							−0.901427	+	0.432930i	\(0.857480\pi\)
\(37\)	−5.44413e6	−	5.44413e6i	−0.477552	−	0.477552i	0.426796	−	0.904348i	\(-0.359642\pi\)
							−0.904348	+	0.426796i	\(0.859642\pi\)
\(41\)	2.53455e6			0.140079			0.0700396	−	0.997544i	\(-0.477687\pi\)
							0.0700396	+	0.997544i	\(0.477687\pi\)
\(43\)	2.19281e7	+	2.19281e7i	0.978121	+	0.978121i	0.999766	−	0.0216445i	\(-0.00689021\pi\)
							−0.0216445	+	0.999766i	\(0.506890\pi\)
\(47\)	−1.86541e7	+	1.86541e7i	−0.557615	+	0.557615i	−0.928628	−	0.371013i	\(-0.879011\pi\)
							0.371013	+	0.928628i	\(0.379011\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.10.e.e.7.22		48
4.3	odd	2	inner	100.10.e.e.7.15		48
5.2	odd	4		20.10.e.b.3.10	yes	48
5.3	odd	4	inner	100.10.e.e.43.15		48
5.4	even	2		20.10.e.b.7.3	yes	48
20.3	even	4	inner	100.10.e.e.43.22		48
20.7	even	4		20.10.e.b.3.3	✓	48
20.19	odd	2		20.10.e.b.7.10	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.3	✓	48	20.7	even	4
20.10.e.b.3.10	yes	48	5.2	odd	4
20.10.e.b.7.3	yes	48	5.4	even	2
20.10.e.b.7.10	yes	48	20.19	odd	2
100.10.e.e.7.15		48	4.3	odd	2	inner
100.10.e.e.7.22		48	1.1	even	1	trivial
100.10.e.e.43.15		48	5.3	odd	4	inner
100.10.e.e.43.22		48	20.3	even	4	inner