# Properties

 Label 100.2.c.a Level 100 Weight 2 Character orbit 100.c Analytic conductor 0.799 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 100.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.798504020213$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + 2 i q^{7} - q^{9} -2 i q^{13} -6 i q^{17} + 4 q^{19} -4 q^{21} -6 i q^{23} + 4 i q^{27} -6 q^{29} -4 q^{31} + 2 i q^{37} + 4 q^{39} + 6 q^{41} + 10 i q^{43} -6 i q^{47} + 3 q^{49} + 12 q^{51} + 6 i q^{53} + 8 i q^{57} -12 q^{59} + 2 q^{61} -2 i q^{63} + 2 i q^{67} + 12 q^{69} -12 q^{71} -2 i q^{73} -8 q^{79} -11 q^{81} -6 i q^{83} -12 i q^{87} + 6 q^{89} + 4 q^{91} -8 i q^{93} + 2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 8q^{19} - 8q^{21} - 12q^{29} - 8q^{31} + 8q^{39} + 12q^{41} + 6q^{49} + 24q^{51} - 24q^{59} + 4q^{61} + 24q^{69} - 24q^{71} - 16q^{79} - 22q^{81} + 12q^{89} + 8q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 2.00000i 0 −1.00000 0
49.2 0 2.00000i 0 0 0 2.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.2.c.a 2
3.b odd 2 1 900.2.d.c 2
4.b odd 2 1 400.2.c.b 2
5.b even 2 1 inner 100.2.c.a 2
5.c odd 4 1 20.2.a.a 1
5.c odd 4 1 100.2.a.a 1
7.b odd 2 1 4900.2.e.f 2
8.b even 2 1 1600.2.c.d 2
8.d odd 2 1 1600.2.c.e 2
12.b even 2 1 3600.2.f.j 2
15.d odd 2 1 900.2.d.c 2
15.e even 4 1 180.2.a.a 1
15.e even 4 1 900.2.a.b 1
20.d odd 2 1 400.2.c.b 2
20.e even 4 1 80.2.a.b 1
20.e even 4 1 400.2.a.c 1
35.c odd 2 1 4900.2.e.f 2
35.f even 4 1 980.2.a.h 1
35.f even 4 1 4900.2.a.e 1
35.k even 12 2 980.2.i.c 2
35.l odd 12 2 980.2.i.i 2
40.e odd 2 1 1600.2.c.e 2
40.f even 2 1 1600.2.c.d 2
40.i odd 4 1 320.2.a.f 1
40.i odd 4 1 1600.2.a.c 1
40.k even 4 1 320.2.a.a 1
40.k even 4 1 1600.2.a.w 1
45.k odd 12 2 1620.2.i.h 2
45.l even 12 2 1620.2.i.b 2
55.e even 4 1 2420.2.a.a 1
60.h even 2 1 3600.2.f.j 2
60.l odd 4 1 720.2.a.h 1
60.l odd 4 1 3600.2.a.be 1
65.f even 4 1 3380.2.f.b 2
65.h odd 4 1 3380.2.a.c 1
65.k even 4 1 3380.2.f.b 2
80.i odd 4 1 1280.2.d.c 2
80.j even 4 1 1280.2.d.g 2
80.s even 4 1 1280.2.d.g 2
80.t odd 4 1 1280.2.d.c 2
85.f odd 4 1 5780.2.c.a 2
85.g odd 4 1 5780.2.a.f 1
85.i odd 4 1 5780.2.c.a 2
95.g even 4 1 7220.2.a.f 1
105.k odd 4 1 8820.2.a.g 1
120.q odd 4 1 2880.2.a.f 1
120.w even 4 1 2880.2.a.m 1
140.j odd 4 1 3920.2.a.h 1
220.i odd 4 1 9680.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 5.c odd 4 1
80.2.a.b 1 20.e even 4 1
100.2.a.a 1 5.c odd 4 1
100.2.c.a 2 1.a even 1 1 trivial
100.2.c.a 2 5.b even 2 1 inner
180.2.a.a 1 15.e even 4 1
320.2.a.a 1 40.k even 4 1
320.2.a.f 1 40.i odd 4 1
400.2.a.c 1 20.e even 4 1
400.2.c.b 2 4.b odd 2 1
400.2.c.b 2 20.d odd 2 1
720.2.a.h 1 60.l odd 4 1
900.2.a.b 1 15.e even 4 1
900.2.d.c 2 3.b odd 2 1
900.2.d.c 2 15.d odd 2 1
980.2.a.h 1 35.f even 4 1
980.2.i.c 2 35.k even 12 2
980.2.i.i 2 35.l odd 12 2
1280.2.d.c 2 80.i odd 4 1
1280.2.d.c 2 80.t odd 4 1
1280.2.d.g 2 80.j even 4 1
1280.2.d.g 2 80.s even 4 1
1600.2.a.c 1 40.i odd 4 1
1600.2.a.w 1 40.k even 4 1
1600.2.c.d 2 8.b even 2 1
1600.2.c.d 2 40.f even 2 1
1600.2.c.e 2 8.d odd 2 1
1600.2.c.e 2 40.e odd 2 1
1620.2.i.b 2 45.l even 12 2
1620.2.i.h 2 45.k odd 12 2
2420.2.a.a 1 55.e even 4 1
2880.2.a.f 1 120.q odd 4 1
2880.2.a.m 1 120.w even 4 1
3380.2.a.c 1 65.h odd 4 1
3380.2.f.b 2 65.f even 4 1
3380.2.f.b 2 65.k even 4 1
3600.2.a.be 1 60.l odd 4 1
3600.2.f.j 2 12.b even 2 1
3600.2.f.j 2 60.h even 2 1
3920.2.a.h 1 140.j odd 4 1
4900.2.a.e 1 35.f even 4 1
4900.2.e.f 2 7.b odd 2 1
4900.2.e.f 2 35.c odd 2 1
5780.2.a.f 1 85.g odd 4 1
5780.2.c.a 2 85.f odd 4 1
5780.2.c.a 2 85.i odd 4 1
7220.2.a.f 1 95.g even 4 1
8820.2.a.g 1 105.k odd 4 1
9680.2.a.ba 1 220.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 10 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$1 + 14 T^{2} + 1849 T^{4}$$
$47$ $$1 - 58 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 130 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$
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