# Properties

 Label 100.9.b.a Level $100$ Weight $9$ Character orbit 100.b Self dual yes Analytic conductor $40.738$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,9,Mod(51,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("100.51");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.b (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$40.7378610061$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 16 q^{2} + 256 q^{4} - 4096 q^{8} + 6561 q^{9}+O(q^{10})$$ q - 16 * q^2 + 256 * q^4 - 4096 * q^8 + 6561 * q^9 $$q - 16 q^{2} + 256 q^{4} - 4096 q^{8} + 6561 q^{9} + 478 q^{13} + 65536 q^{16} + 63358 q^{17} - 104976 q^{18} - 7648 q^{26} - 1407838 q^{29} - 1048576 q^{32} - 1013728 q^{34} + 1679616 q^{36} - 925922 q^{37} + 3577922 q^{41} + 5764801 q^{49} + 122368 q^{52} + 9620638 q^{53} + 22525408 q^{58} + 20722082 q^{61} + 16777216 q^{64} + 16219648 q^{68} - 26873856 q^{72} + 54717118 q^{73} + 14814752 q^{74} + 43046721 q^{81} - 57246752 q^{82} - 30265918 q^{89} + 173379838 q^{97} - 92236816 q^{98}+O(q^{100})$$ q - 16 * q^2 + 256 * q^4 - 4096 * q^8 + 6561 * q^9 + 478 * q^13 + 65536 * q^16 + 63358 * q^17 - 104976 * q^18 - 7648 * q^26 - 1407838 * q^29 - 1048576 * q^32 - 1013728 * q^34 + 1679616 * q^36 - 925922 * q^37 + 3577922 * q^41 + 5764801 * q^49 + 122368 * q^52 + 9620638 * q^53 + 22525408 * q^58 + 20722082 * q^61 + 16777216 * q^64 + 16219648 * q^68 - 26873856 * q^72 + 54717118 * q^73 + 14814752 * q^74 + 43046721 * q^81 - 57246752 * q^82 - 30265918 * q^89 + 173379838 * q^97 - 92236816 * q^98

## Character values

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0
−16.0000 0 256.000 0 0 0 −4096.00 6561.00 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 $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.b.a 1
4.b odd 2 1 CM 100.9.b.a 1
5.b even 2 1 4.9.b.a 1
5.c odd 4 2 100.9.d.a 2
15.d odd 2 1 36.9.d.a 1
20.d odd 2 1 4.9.b.a 1
20.e even 4 2 100.9.d.a 2
40.e odd 2 1 64.9.c.a 1
40.f even 2 1 64.9.c.a 1
60.h even 2 1 36.9.d.a 1
80.k odd 4 2 256.9.d.a 2
80.q even 4 2 256.9.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.a 1 5.b even 2 1
4.9.b.a 1 20.d odd 2 1
36.9.d.a 1 15.d odd 2 1
36.9.d.a 1 60.h even 2 1
64.9.c.a 1 40.e odd 2 1
64.9.c.a 1 40.f even 2 1
100.9.b.a 1 1.a even 1 1 trivial
100.9.b.a 1 4.b odd 2 1 CM
100.9.d.a 2 5.c odd 4 2
100.9.d.a 2 20.e even 4 2
256.9.d.a 2 80.k odd 4 2
256.9.d.a 2 80.q even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$:

 $$T_{3}$$ T3 $$T_{13} - 478$$ T13 - 478

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 16$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 478$$
$17$ $$T - 63358$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 1407838$$
$31$ $$T$$
$37$ $$T + 925922$$
$41$ $$T - 3577922$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 9620638$$
$59$ $$T$$
$61$ $$T - 20722082$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 54717118$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 30265918$$
$97$ $$T - 173379838$$