# Properties

 Label 100.2.e.b Level $100$ Weight $2$ Character orbit 100.e Analytic conductor $0.799$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,2,Mod(7,100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("100.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.798504020213$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + 2 i q^{4} + (2 i - 2) q^{8} - 3 i q^{9} +O(q^{10})$$ q + (i + 1) * q^2 + 2*i * q^4 + (2*i - 2) * q^8 - 3*i * q^9 $$q + (i + 1) q^{2} + 2 i q^{4} + (2 i - 2) q^{8} - 3 i q^{9} + ( - i + 1) q^{13} - 4 q^{16} + ( - 3 i - 3) q^{17} + ( - 3 i + 3) q^{18} + 2 q^{26} + 4 i q^{29} + ( - 4 i - 4) q^{32} - 6 i q^{34} + 6 q^{36} + (7 i + 7) q^{37} - 8 q^{41} + 7 i q^{49} + (2 i + 2) q^{52} + (9 i - 9) q^{53} + (4 i - 4) q^{58} + 12 q^{61} - 8 i q^{64} + ( - 6 i + 6) q^{68} + (6 i + 6) q^{72} + ( - 11 i + 11) q^{73} + 14 i q^{74} - 9 q^{81} + ( - 8 i - 8) q^{82} - 16 i q^{89} + ( - 13 i - 13) q^{97} + (7 i - 7) q^{98} +O(q^{100})$$ q + (i + 1) * q^2 + 2*i * q^4 + (2*i - 2) * q^8 - 3*i * q^9 + (-i + 1) * q^13 - 4 * q^16 + (-3*i - 3) * q^17 + (-3*i + 3) * q^18 + 2 * q^26 + 4*i * q^29 + (-4*i - 4) * q^32 - 6*i * q^34 + 6 * q^36 + (7*i + 7) * q^37 - 8 * q^41 + 7*i * q^49 + (2*i + 2) * q^52 + (9*i - 9) * q^53 + (4*i - 4) * q^58 + 12 * q^61 - 8*i * q^64 + (-6*i + 6) * q^68 + (6*i + 6) * q^72 + (-11*i + 11) * q^73 + 14*i * q^74 - 9 * q^81 + (-8*i - 8) * q^82 - 16*i * q^89 + (-13*i - 13) * q^97 + (7*i - 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^8 $$2 q + 2 q^{2} - 4 q^{8} + 2 q^{13} - 8 q^{16} - 6 q^{17} + 6 q^{18} + 4 q^{26} - 8 q^{32} + 12 q^{36} + 14 q^{37} - 16 q^{41} + 4 q^{52} - 18 q^{53} - 8 q^{58} + 24 q^{61} + 12 q^{68} + 12 q^{72} + 22 q^{73} - 18 q^{81} - 16 q^{82} - 26 q^{97} - 14 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^8 + 2 * q^13 - 8 * q^16 - 6 * q^17 + 6 * q^18 + 4 * q^26 - 8 * q^32 + 12 * q^36 + 14 * q^37 - 16 * q^41 + 4 * q^52 - 18 * q^53 - 8 * q^58 + 24 * q^61 + 12 * q^68 + 12 * q^72 + 22 * q^73 - 18 * q^81 - 16 * q^82 - 26 * q^97 - 14 * q^98

## Character values

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.00000i − 1.00000i
1.00000 + 1.00000i 0 2.00000i 0 0 0 −2.00000 + 2.00000i 3.00000i 0
43.1 1.00000 1.00000i 0 2.00000i 0 0 0 −2.00000 2.00000i 3.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.2.e.b 2
3.b odd 2 1 900.2.k.c 2
4.b odd 2 1 CM 100.2.e.b 2
5.b even 2 1 20.2.e.a 2
5.c odd 4 1 20.2.e.a 2
5.c odd 4 1 inner 100.2.e.b 2
8.b even 2 1 1600.2.n.h 2
8.d odd 2 1 1600.2.n.h 2
12.b even 2 1 900.2.k.c 2
15.d odd 2 1 180.2.k.c 2
15.e even 4 1 180.2.k.c 2
15.e even 4 1 900.2.k.c 2
20.d odd 2 1 20.2.e.a 2
20.e even 4 1 20.2.e.a 2
20.e even 4 1 inner 100.2.e.b 2
35.c odd 2 1 980.2.k.a 2
35.f even 4 1 980.2.k.a 2
35.i odd 6 2 980.2.x.c 4
35.j even 6 2 980.2.x.d 4
35.k even 12 2 980.2.x.c 4
35.l odd 12 2 980.2.x.d 4
40.e odd 2 1 320.2.n.e 2
40.f even 2 1 320.2.n.e 2
40.i odd 4 1 320.2.n.e 2
40.i odd 4 1 1600.2.n.h 2
40.k even 4 1 320.2.n.e 2
40.k even 4 1 1600.2.n.h 2
60.h even 2 1 180.2.k.c 2
60.l odd 4 1 180.2.k.c 2
60.l odd 4 1 900.2.k.c 2
80.i odd 4 1 1280.2.o.j 2
80.j even 4 1 1280.2.o.g 2
80.k odd 4 1 1280.2.o.g 2
80.k odd 4 1 1280.2.o.j 2
80.q even 4 1 1280.2.o.g 2
80.q even 4 1 1280.2.o.j 2
80.s even 4 1 1280.2.o.j 2
80.t odd 4 1 1280.2.o.g 2
140.c even 2 1 980.2.k.a 2
140.j odd 4 1 980.2.k.a 2
140.p odd 6 2 980.2.x.d 4
140.s even 6 2 980.2.x.c 4
140.w even 12 2 980.2.x.d 4
140.x odd 12 2 980.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 5.b even 2 1
20.2.e.a 2 5.c odd 4 1
20.2.e.a 2 20.d odd 2 1
20.2.e.a 2 20.e even 4 1
100.2.e.b 2 1.a even 1 1 trivial
100.2.e.b 2 4.b odd 2 1 CM
100.2.e.b 2 5.c odd 4 1 inner
100.2.e.b 2 20.e even 4 1 inner
180.2.k.c 2 15.d odd 2 1
180.2.k.c 2 15.e even 4 1
180.2.k.c 2 60.h even 2 1
180.2.k.c 2 60.l odd 4 1
320.2.n.e 2 40.e odd 2 1
320.2.n.e 2 40.f even 2 1
320.2.n.e 2 40.i odd 4 1
320.2.n.e 2 40.k even 4 1
900.2.k.c 2 3.b odd 2 1
900.2.k.c 2 12.b even 2 1
900.2.k.c 2 15.e even 4 1
900.2.k.c 2 60.l odd 4 1
980.2.k.a 2 35.c odd 2 1
980.2.k.a 2 35.f even 4 1
980.2.k.a 2 140.c even 2 1
980.2.k.a 2 140.j odd 4 1
980.2.x.c 4 35.i odd 6 2
980.2.x.c 4 35.k even 12 2
980.2.x.c 4 140.s even 6 2
980.2.x.c 4 140.x odd 12 2
980.2.x.d 4 35.j even 6 2
980.2.x.d 4 35.l odd 12 2
980.2.x.d 4 140.p odd 6 2
980.2.x.d 4 140.w even 12 2
1280.2.o.g 2 80.j even 4 1
1280.2.o.g 2 80.k odd 4 1
1280.2.o.g 2 80.q even 4 1
1280.2.o.g 2 80.t odd 4 1
1280.2.o.j 2 80.i odd 4 1
1280.2.o.j 2 80.k odd 4 1
1280.2.o.j 2 80.q even 4 1
1280.2.o.j 2 80.s even 4 1
1600.2.n.h 2 8.b even 2 1
1600.2.n.h 2 8.d odd 2 1
1600.2.n.h 2 40.i odd 4 1
1600.2.n.h 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 2T + 2$$
$17$ $$T^{2} + 6T + 18$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 14T + 98$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 18T + 162$$
$59$ $$T^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 22T + 242$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} + 26T + 338$$