# Properties

 Label 100.9.b.b Level $100$ Weight $9$ Character orbit 100.b Analytic conductor $40.738$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $4$

# 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: 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) 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.

 $$f(q)$$ $$=$$ $$q + 8 \beta q^{2} + 79 \beta q^{3} - 256 q^{4} - 2528 q^{6} + 961 \beta q^{7} - 2048 \beta q^{8} - 18403 q^{9} +O(q^{10})$$ q + 8*b * q^2 + 79*b * q^3 - 256 * q^4 - 2528 * q^6 + 961*b * q^7 - 2048*b * q^8 - 18403 * q^9 $$q + 8 \beta q^{2} + 79 \beta q^{3} - 256 q^{4} - 2528 q^{6} + 961 \beta q^{7} - 2048 \beta q^{8} - 18403 q^{9} - 20224 \beta q^{12} - 30752 q^{14} + 65536 q^{16} - 147224 \beta q^{18} - 303676 q^{21} - 105601 \beta q^{23} + 647168 q^{24} - 935518 \beta q^{27} - 246016 \beta q^{28} - 20642 q^{29} + 524288 \beta q^{32} + 4711168 q^{36} - 5419198 q^{41} - 2429408 \beta q^{42} + 1259759 \beta q^{43} + 3379232 q^{46} + 4809121 \beta q^{47} + 5177344 \beta q^{48} + 2070717 q^{49} + 29936576 q^{54} + 7872512 q^{56} - 165136 \beta q^{58} - 11061598 q^{61} - 17685283 \beta q^{63} - 16777216 q^{64} - 10124879 \beta q^{67} + 33369916 q^{69} + 37689344 \beta q^{72} + 174881605 q^{81} - 43353584 \beta q^{82} + 15442319 \beta q^{83} + 77741056 q^{84} - 40312288 q^{86} - 1630718 \beta q^{87} + 106804798 q^{89} + 27033856 \beta q^{92} - 153891872 q^{94} - 165675008 q^{96} + 16565736 \beta q^{98} +O(q^{100})$$ q + 8*b * q^2 + 79*b * q^3 - 256 * q^4 - 2528 * q^6 + 961*b * q^7 - 2048*b * q^8 - 18403 * q^9 - 20224*b * q^12 - 30752 * q^14 + 65536 * q^16 - 147224*b * q^18 - 303676 * q^21 - 105601*b * q^23 + 647168 * q^24 - 935518*b * q^27 - 246016*b * q^28 - 20642 * q^29 + 524288*b * q^32 + 4711168 * q^36 - 5419198 * q^41 - 2429408*b * q^42 + 1259759*b * q^43 + 3379232 * q^46 + 4809121*b * q^47 + 5177344*b * q^48 + 2070717 * q^49 + 29936576 * q^54 + 7872512 * q^56 - 165136*b * q^58 - 11061598 * q^61 - 17685283*b * q^63 - 16777216 * q^64 - 10124879*b * q^67 + 33369916 * q^69 + 37689344*b * q^72 + 174881605 * q^81 - 43353584*b * q^82 + 15442319*b * q^83 + 77741056 * q^84 - 40312288 * q^86 - 1630718*b * q^87 + 106804798 * q^89 + 27033856*b * q^92 - 153891872 * q^94 - 165675008 * q^96 + 16565736*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 512 q^{4} - 5056 q^{6} - 36806 q^{9}+O(q^{10})$$ 2 * q - 512 * q^4 - 5056 * q^6 - 36806 * q^9 $$2 q - 512 q^{4} - 5056 q^{6} - 36806 q^{9} - 61504 q^{14} + 131072 q^{16} - 607352 q^{21} + 1294336 q^{24} - 41284 q^{29} + 9422336 q^{36} - 10838396 q^{41} + 6758464 q^{46} + 4141434 q^{49} + 59873152 q^{54} + 15745024 q^{56} - 22123196 q^{61} - 33554432 q^{64} + 66739832 q^{69} + 349763210 q^{81} + 155482112 q^{84} - 80624576 q^{86} + 213609596 q^{89} - 307783744 q^{94} - 331350016 q^{96}+O(q^{100})$$ 2 * q - 512 * q^4 - 5056 * q^6 - 36806 * q^9 - 61504 * q^14 + 131072 * q^16 - 607352 * q^21 + 1294336 * q^24 - 41284 * q^29 + 9422336 * q^36 - 10838396 * q^41 + 6758464 * q^46 + 4141434 * q^49 + 59873152 * q^54 + 15745024 * q^56 - 22123196 * q^61 - 33554432 * q^64 + 66739832 * q^69 + 349763210 * q^81 + 155482112 * q^84 - 80624576 * q^86 + 213609596 * q^89 - 307783744 * q^94 - 331350016 * q^96

## 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
 − 1.00000i 1.00000i
16.0000i 158.000i −256.000 0 −2528.00 1922.00i 4096.00i −18403.0 0
51.2 16.0000i 158.000i −256.000 0 −2528.00 1922.00i 4096.00i −18403.0 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
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 100.9.b.b 2
4.b odd 2 1 inner 100.9.b.b 2
5.b even 2 1 inner 100.9.b.b 2
5.c odd 4 1 20.9.d.a 1
5.c odd 4 1 20.9.d.b yes 1
20.d odd 2 1 CM 100.9.b.b 2
20.e even 4 1 20.9.d.a 1
20.e even 4 1 20.9.d.b yes 1
40.i odd 4 1 320.9.h.a 1
40.i odd 4 1 320.9.h.b 1
40.k even 4 1 320.9.h.a 1
40.k even 4 1 320.9.h.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.d.a 1 5.c odd 4 1
20.9.d.a 1 20.e even 4 1
20.9.d.b yes 1 5.c odd 4 1
20.9.d.b yes 1 20.e even 4 1
100.9.b.b 2 1.a even 1 1 trivial
100.9.b.b 2 4.b odd 2 1 inner
100.9.b.b 2 5.b even 2 1 inner
100.9.b.b 2 20.d odd 2 1 CM
320.9.h.a 1 40.i odd 4 1
320.9.h.a 1 40.k even 4 1
320.9.h.b 1 40.i odd 4 1
320.9.h.b 1 40.k even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 24964$$ T3^2 + 24964 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 256$$
$3$ $$T^{2} + 24964$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3694084$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 44606284804$$
$29$ $$(T + 20642)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 5419198)^{2}$$
$43$ $$T^{2} + 6347970952324$$
$47$ $$T^{2} + 92510579170564$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 11061598)^{2}$$
$67$ $$T^{2} + 410052699058564$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 953860864391044$$
$89$ $$(T - 106804798)^{2}$$
$97$ $$T^{2}$$