# Properties

 Label 99.4.a.a Level $99$ Weight $4$ Character orbit 99.a Self dual yes Analytic conductor $5.841$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,4,Mod(1,99)]

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([0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 99.a (trivial)

## 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: $$5.84118909057$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{4} + 4 q^{5} - 26 q^{7} - 15 q^{8}+O(q^{10})$$ q + q^2 - 7 * q^4 + 4 * q^5 - 26 * q^7 - 15 * q^8 $$q + q^{2} - 7 q^{4} + 4 q^{5} - 26 q^{7} - 15 q^{8} + 4 q^{10} - 11 q^{11} - 32 q^{13} - 26 q^{14} + 41 q^{16} - 74 q^{17} - 60 q^{19} - 28 q^{20} - 11 q^{22} + 182 q^{23} - 109 q^{25} - 32 q^{26} + 182 q^{28} + 90 q^{29} - 8 q^{31} + 161 q^{32} - 74 q^{34} - 104 q^{35} - 66 q^{37} - 60 q^{38} - 60 q^{40} - 422 q^{41} + 408 q^{43} + 77 q^{44} + 182 q^{46} + 506 q^{47} + 333 q^{49} - 109 q^{50} + 224 q^{52} - 348 q^{53} - 44 q^{55} + 390 q^{56} + 90 q^{58} + 200 q^{59} + 132 q^{61} - 8 q^{62} - 167 q^{64} - 128 q^{65} - 1036 q^{67} + 518 q^{68} - 104 q^{70} - 762 q^{71} - 542 q^{73} - 66 q^{74} + 420 q^{76} + 286 q^{77} - 550 q^{79} + 164 q^{80} - 422 q^{82} + 132 q^{83} - 296 q^{85} + 408 q^{86} + 165 q^{88} - 570 q^{89} + 832 q^{91} - 1274 q^{92} + 506 q^{94} - 240 q^{95} + 14 q^{97} + 333 q^{98}+O(q^{100})$$ q + q^2 - 7 * q^4 + 4 * q^5 - 26 * q^7 - 15 * q^8 + 4 * q^10 - 11 * q^11 - 32 * q^13 - 26 * q^14 + 41 * q^16 - 74 * q^17 - 60 * q^19 - 28 * q^20 - 11 * q^22 + 182 * q^23 - 109 * q^25 - 32 * q^26 + 182 * q^28 + 90 * q^29 - 8 * q^31 + 161 * q^32 - 74 * q^34 - 104 * q^35 - 66 * q^37 - 60 * q^38 - 60 * q^40 - 422 * q^41 + 408 * q^43 + 77 * q^44 + 182 * q^46 + 506 * q^47 + 333 * q^49 - 109 * q^50 + 224 * q^52 - 348 * q^53 - 44 * q^55 + 390 * q^56 + 90 * q^58 + 200 * q^59 + 132 * q^61 - 8 * q^62 - 167 * q^64 - 128 * q^65 - 1036 * q^67 + 518 * q^68 - 104 * q^70 - 762 * q^71 - 542 * q^73 - 66 * q^74 + 420 * q^76 + 286 * q^77 - 550 * q^79 + 164 * q^80 - 422 * q^82 + 132 * q^83 - 296 * q^85 + 408 * q^86 + 165 * q^88 - 570 * q^89 + 832 * q^91 - 1274 * q^92 + 506 * q^94 - 240 * q^95 + 14 * q^97 + 333 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −7.00000 4.00000 0 −26.0000 −15.0000 0 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.a.a 1
3.b odd 2 1 33.4.a.b 1
4.b odd 2 1 1584.4.a.l 1
5.b even 2 1 2475.4.a.e 1
11.b odd 2 1 1089.4.a.e 1
12.b even 2 1 528.4.a.h 1
15.d odd 2 1 825.4.a.f 1
15.e even 4 2 825.4.c.f 2
21.c even 2 1 1617.4.a.d 1
24.f even 2 1 2112.4.a.h 1
24.h odd 2 1 2112.4.a.u 1
33.d even 2 1 363.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 3.b odd 2 1
99.4.a.a 1 1.a even 1 1 trivial
363.4.a.d 1 33.d even 2 1
528.4.a.h 1 12.b even 2 1
825.4.a.f 1 15.d odd 2 1
825.4.c.f 2 15.e even 4 2
1089.4.a.e 1 11.b odd 2 1
1584.4.a.l 1 4.b odd 2 1
1617.4.a.d 1 21.c even 2 1
2112.4.a.h 1 24.f even 2 1
2112.4.a.u 1 24.h odd 2 1
2475.4.a.e 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 4$$
$7$ $$T + 26$$
$11$ $$T + 11$$
$13$ $$T + 32$$
$17$ $$T + 74$$
$19$ $$T + 60$$
$23$ $$T - 182$$
$29$ $$T - 90$$
$31$ $$T + 8$$
$37$ $$T + 66$$
$41$ $$T + 422$$
$43$ $$T - 408$$
$47$ $$T - 506$$
$53$ $$T + 348$$
$59$ $$T - 200$$
$61$ $$T - 132$$
$67$ $$T + 1036$$
$71$ $$T + 762$$
$73$ $$T + 542$$
$79$ $$T + 550$$
$83$ $$T - 132$$
$89$ $$T + 570$$
$97$ $$T - 14$$