# Properties

 Label 99.8.d.a Level $99$ Weight $8$ Character orbit 99.d Analytic conductor $30.926$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,8,Mod(98,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("99.98");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.d (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: $$30.9261175229$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 1792 q^{4}+O(q^{10})$$ 28 * q + 1792 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 1792 q^{4} + 136936 q^{16} + 69696 q^{22} - 877308 q^{25} - 21616 q^{31} + 161304 q^{34} + 408464 q^{37} - 5023892 q^{49} + 439472 q^{55} + 2851344 q^{58} - 15376208 q^{64} - 17729872 q^{67} + 3438408 q^{70} + 54779784 q^{82} + 26540976 q^{88} - 19094256 q^{91} + 10561400 q^{97}+O(q^{100})$$ 28 * q + 1792 * q^4 + 136936 * q^16 + 69696 * q^22 - 877308 * q^25 - 21616 * q^31 + 161304 * q^34 + 408464 * q^37 - 5023892 * q^49 + 439472 * q^55 + 2851344 * q^58 - 15376208 * q^64 - 17729872 * q^67 + 3438408 * q^70 + 54779784 * q^82 + 26540976 * q^88 - 19094256 * q^91 + 10561400 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
98.1 −20.6546 0 298.612 288.556i 0 754.006i −3523.91 0 5960.01i
98.2 −20.6546 0 298.612 288.556i 0 754.006i −3523.91 0 5960.01i
98.3 −19.4265 0 249.387 529.951i 0 235.028i −2358.12 0 10295.1i
98.4 −19.4265 0 249.387 529.951i 0 235.028i −2358.12 0 10295.1i
98.5 −16.3395 0 138.981 19.0280i 0 1638.46i −179.422 0 310.908i
98.6 −16.3395 0 138.981 19.0280i 0 1638.46i −179.422 0 310.908i
98.7 −13.8089 0 62.6859 205.633i 0 449.641i 901.916 0 2839.57i
98.8 −13.8089 0 62.6859 205.633i 0 449.641i 901.916 0 2839.57i
98.9 −7.31902 0 −74.4319 81.1133i 0 179.404i 1481.60 0 593.670i
98.10 −7.31902 0 −74.4319 81.1133i 0 179.404i 1481.60 0 593.670i
98.11 −5.16121 0 −101.362 347.376i 0 1160.84i 1183.79 0 1792.88i
98.12 −5.16121 0 −101.362 347.376i 0 1160.84i 1183.79 0 1792.88i
98.13 −1.45890 0 −125.872 481.864i 0 1459.64i 370.374 0 702.992i
98.14 −1.45890 0 −125.872 481.864i 0 1459.64i 370.374 0 702.992i
98.15 1.45890 0 −125.872 481.864i 0 1459.64i −370.374 0 702.992i
98.16 1.45890 0 −125.872 481.864i 0 1459.64i −370.374 0 702.992i
98.17 5.16121 0 −101.362 347.376i 0 1160.84i −1183.79 0 1792.88i
98.18 5.16121 0 −101.362 347.376i 0 1160.84i −1183.79 0 1792.88i
98.19 7.31902 0 −74.4319 81.1133i 0 179.404i −1481.60 0 593.670i
98.20 7.31902 0 −74.4319 81.1133i 0 179.404i −1481.60 0 593.670i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 98.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.d.a 28
3.b odd 2 1 inner 99.8.d.a 28
11.b odd 2 1 inner 99.8.d.a 28
33.d even 2 1 inner 99.8.d.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.d.a 28 1.a even 1 1 trivial
99.8.d.a 28 3.b odd 2 1 inner
99.8.d.a 28 11.b odd 2 1 inner
99.8.d.a 28 33.d even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(99, [\chi])$$.