# Properties

 Label 99.6.a.b Level $99$ Weight $6$ Character orbit 99.a Self dual yes Analytic conductor $15.878$ Analytic rank $0$ 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] = [99,6,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, 6, names="a")

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$15.8779981615$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 28 q^{4} - 46 q^{5} + 148 q^{7} - 120 q^{8}+O(q^{10})$$ q + 2 * q^2 - 28 * q^4 - 46 * q^5 + 148 * q^7 - 120 * q^8 $$q + 2 q^{2} - 28 q^{4} - 46 q^{5} + 148 q^{7} - 120 q^{8} - 92 q^{10} - 121 q^{11} + 574 q^{13} + 296 q^{14} + 656 q^{16} + 722 q^{17} + 2160 q^{19} + 1288 q^{20} - 242 q^{22} + 2536 q^{23} - 1009 q^{25} + 1148 q^{26} - 4144 q^{28} - 4650 q^{29} + 5032 q^{31} + 5152 q^{32} + 1444 q^{34} - 6808 q^{35} + 8118 q^{37} + 4320 q^{38} + 5520 q^{40} + 5138 q^{41} + 8304 q^{43} + 3388 q^{44} + 5072 q^{46} - 24728 q^{47} + 5097 q^{49} - 2018 q^{50} - 16072 q^{52} + 28746 q^{53} + 5566 q^{55} - 17760 q^{56} - 9300 q^{58} + 5860 q^{59} - 53658 q^{61} + 10064 q^{62} - 10688 q^{64} - 26404 q^{65} + 30908 q^{67} - 20216 q^{68} - 13616 q^{70} + 69648 q^{71} - 18446 q^{73} + 16236 q^{74} - 60480 q^{76} - 17908 q^{77} - 25300 q^{79} - 30176 q^{80} + 10276 q^{82} + 17556 q^{83} - 33212 q^{85} + 16608 q^{86} + 14520 q^{88} - 132570 q^{89} + 84952 q^{91} - 71008 q^{92} - 49456 q^{94} - 99360 q^{95} + 70658 q^{97} + 10194 q^{98}+O(q^{100})$$ q + 2 * q^2 - 28 * q^4 - 46 * q^5 + 148 * q^7 - 120 * q^8 - 92 * q^10 - 121 * q^11 + 574 * q^13 + 296 * q^14 + 656 * q^16 + 722 * q^17 + 2160 * q^19 + 1288 * q^20 - 242 * q^22 + 2536 * q^23 - 1009 * q^25 + 1148 * q^26 - 4144 * q^28 - 4650 * q^29 + 5032 * q^31 + 5152 * q^32 + 1444 * q^34 - 6808 * q^35 + 8118 * q^37 + 4320 * q^38 + 5520 * q^40 + 5138 * q^41 + 8304 * q^43 + 3388 * q^44 + 5072 * q^46 - 24728 * q^47 + 5097 * q^49 - 2018 * q^50 - 16072 * q^52 + 28746 * q^53 + 5566 * q^55 - 17760 * q^56 - 9300 * q^58 + 5860 * q^59 - 53658 * q^61 + 10064 * q^62 - 10688 * q^64 - 26404 * q^65 + 30908 * q^67 - 20216 * q^68 - 13616 * q^70 + 69648 * q^71 - 18446 * q^73 + 16236 * q^74 - 60480 * q^76 - 17908 * q^77 - 25300 * q^79 - 30176 * q^80 + 10276 * q^82 + 17556 * q^83 - 33212 * q^85 + 16608 * q^86 + 14520 * q^88 - 132570 * q^89 + 84952 * q^91 - 71008 * q^92 - 49456 * q^94 - 99360 * q^95 + 70658 * q^97 + 10194 * 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
2.00000 0 −28.0000 −46.0000 0 148.000 −120.000 0 −92.0000
 $$n$$: 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 99.6.a.b 1
3.b odd 2 1 33.6.a.a 1
11.b odd 2 1 1089.6.a.d 1
12.b even 2 1 528.6.a.i 1
15.d odd 2 1 825.6.a.b 1
33.d even 2 1 363.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 3.b odd 2 1
99.6.a.b 1 1.a even 1 1 trivial
363.6.a.c 1 33.d even 2 1
528.6.a.i 1 12.b even 2 1
825.6.a.b 1 15.d odd 2 1
1089.6.a.d 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 46$$
$7$ $$T - 148$$
$11$ $$T + 121$$
$13$ $$T - 574$$
$17$ $$T - 722$$
$19$ $$T - 2160$$
$23$ $$T - 2536$$
$29$ $$T + 4650$$
$31$ $$T - 5032$$
$37$ $$T - 8118$$
$41$ $$T - 5138$$
$43$ $$T - 8304$$
$47$ $$T + 24728$$
$53$ $$T - 28746$$
$59$ $$T - 5860$$
$61$ $$T + 53658$$
$67$ $$T - 30908$$
$71$ $$T - 69648$$
$73$ $$T + 18446$$
$79$ $$T + 25300$$
$83$ $$T - 17556$$
$89$ $$T + 132570$$
$97$ $$T - 70658$$