# Properties

 Label 100.9.d.a Level $100$ Weight $9$ Character orbit 100.d Analytic conductor $40.738$ 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] = [100,9,Mod(99,100)]

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, 1]))

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

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

chi := DirichletCharacter("100.99");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.d (of order $$2$$, degree $$1$$, not 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: $$40.7378610061$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 \beta q^{2} - 256 q^{4} + 2048 \beta q^{8} - 6561 q^{9} +O(q^{10})$$ q - 8*b * q^2 - 256 * q^4 + 2048*b * q^8 - 6561 * q^9 $$q - 8 \beta q^{2} - 256 q^{4} + 2048 \beta q^{8} - 6561 q^{9} - 239 \beta q^{13} + 65536 q^{16} + 31679 \beta q^{17} + 52488 \beta q^{18} - 7648 q^{26} + 1407838 q^{29} - 524288 \beta q^{32} + 1013728 q^{34} + 1679616 q^{36} - 462961 \beta q^{37} + 3577922 q^{41} - 5764801 q^{49} + 61184 \beta q^{52} - 4810319 \beta q^{53} - 11262704 \beta q^{58} + 20722082 q^{61} - 16777216 q^{64} - 8109824 \beta q^{68} - 13436928 \beta q^{72} - 27358559 \beta q^{73} - 14814752 q^{74} + 43046721 q^{81} - 28623376 \beta q^{82} + 30265918 q^{89} + 86689919 \beta q^{97} + 46118408 \beta q^{98} +O(q^{100})$$ q - 8*b * q^2 - 256 * q^4 + 2048*b * q^8 - 6561 * q^9 - 239*b * q^13 + 65536 * q^16 + 31679*b * q^17 + 52488*b * q^18 - 7648 * q^26 + 1407838 * q^29 - 524288*b * q^32 + 1013728 * q^34 + 1679616 * q^36 - 462961*b * q^37 + 3577922 * q^41 - 5764801 * q^49 + 61184*b * q^52 - 4810319*b * q^53 - 11262704*b * q^58 + 20722082 * q^61 - 16777216 * q^64 - 8109824*b * q^68 - 13436928*b * q^72 - 27358559*b * q^73 - 14814752 * q^74 + 43046721 * q^81 - 28623376*b * q^82 + 30265918 * q^89 + 86689919*b * q^97 + 46118408*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 512 q^{4} - 13122 q^{9}+O(q^{10})$$ 2 * q - 512 * q^4 - 13122 * q^9 $$2 q - 512 q^{4} - 13122 q^{9} + 131072 q^{16} - 15296 q^{26} + 2815676 q^{29} + 2027456 q^{34} + 3359232 q^{36} + 7155844 q^{41} - 11529602 q^{49} + 41444164 q^{61} - 33554432 q^{64} - 29629504 q^{74} + 86093442 q^{81} + 60531836 q^{89}+O(q^{100})$$ 2 * q - 512 * q^4 - 13122 * q^9 + 131072 * q^16 - 15296 * q^26 + 2815676 * q^29 + 2027456 * q^34 + 3359232 * q^36 + 7155844 * q^41 - 11529602 * q^49 + 41444164 * q^61 - 33554432 * q^64 - 29629504 * q^74 + 86093442 * q^81 + 60531836 * q^89

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.00000i − 1.00000i
16.0000i 0 −256.000 0 0 0 4096.00i −6561.00 0
99.2 16.0000i 0 −256.000 0 0 0 4096.00i −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 $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 256$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 228484$$
$17$ $$T^{2} + 4014236164$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 1407838)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 857331550084$$
$41$ $$(T - 3577922)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 92556675527044$$
$59$ $$T^{2}$$
$61$ $$(T - 20722082)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 29\!\cdots\!24$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 30265918)^{2}$$
$97$ $$T^{2} + 30\!\cdots\!44$$