# Properties

 Label 99.6.a.g Level $99$ Weight $6$ Character orbit 99.a Self dual yes Analytic conductor $15.878$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 11) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - 4 \beta_{2} - 2 \beta_1 + 28) q^{4} + (\beta_{2} - 3 \beta_1 - 8) q^{5} + ( - 10 \beta_{2} - 10 \beta_1 + 28) q^{7} + ( - 26 \beta_{2} + 188) q^{8}+O(q^{10})$$ q - b2 * q^2 + (-4*b2 - 2*b1 + 28) * q^4 + (b2 - 3*b1 - 8) * q^5 + (-10*b2 - 10*b1 + 28) * q^7 + (-26*b2 + 188) * q^8 $$q - \beta_{2} q^{2} + ( - 4 \beta_{2} - 2 \beta_1 + 28) q^{4} + (\beta_{2} - 3 \beta_1 - 8) q^{5} + ( - 10 \beta_{2} - 10 \beta_1 + 28) q^{7} + ( - 26 \beta_{2} + 188) q^{8} + ( - 9 \beta_{2} + 14 \beta_1 - 138) q^{10} - 121 q^{11} + (70 \beta_{2} + 6 \beta_1 + 162) q^{13} + ( - 138 \beta_{2} + 20 \beta_1 + 340) q^{14} + ( - 164 \beta_{2} + 12 \beta_1 + 664) q^{16} + ( - 132 \beta_{2} + 20 \beta_1 - 362) q^{17} + ( - 60 \beta_{2} - 20 \beta_1 + 460) q^{19} + (168 \beta_{2} + 22 \beta_1 + 1160) q^{20} + 121 \beta_{2} q^{22} + (21 \beta_{2} - 167 \beta_1 + 1022) q^{23} + (265 \beta_{2} + 85 \beta_1 - 19) q^{25} + (160 \beta_{2} + 116 \beta_1 - 4044) q^{26} + ( - 432 \beta_{2} - 36 \beta_1 + 7904) q^{28} + (182 \beta_{2} + 46 \beta_1 + 1142) q^{29} + ( - 101 \beta_{2} + 23 \beta_1 - 1366) q^{31} + ( - 404 \beta_{2} - 376 \beta_1 + 4136) q^{32} + ( - 26 \beta_{2} - 344 \beta_1 + 8440) q^{34} + (818 \beta_{2} + 306 \beta_1 + 8076) q^{35} + (937 \beta_{2} + 197 \beta_1 + 5908) q^{37} + ( - 840 \beta_{2} - 40 \beta_1 + 3080) q^{38} + ( - 46 \beta_{2} - 200 \beta_1 - 5092) q^{40} + (1378 \beta_{2} - 94 \beta_1 - 1998) q^{41} + (1190 \beta_{2} - 370 \beta_1 - 8736) q^{43} + (484 \beta_{2} + 242 \beta_1 - 3388) q^{44} + ( - 2107 \beta_{2} + 710 \beta_1 - 5602) q^{46} + (600 \beta_{2} + 424 \beta_1 + 5744) q^{47} + (340 \beta_{2} + 740 \beta_1 + 16177) q^{49} + (1674 \beta_{2} + 190 \beta_1 - 13690) q^{50} + (3256 \beta_{2} - 336 \beta_1 - 11768) q^{52} + (476 \beta_{2} + 540 \beta_1 - 16862) q^{53} + ( - 121 \beta_{2} + 363 \beta_1 + 968) q^{55} + ( - 5468 \beta_{2} - 1360 \beta_1 + 14104) q^{56} + ( - 92 \beta_{2} + 180 \beta_1 - 9724) q^{58} + ( - 3141 \beta_{2} - 249 \beta_1 + 1246) q^{59} + ( - 1466 \beta_{2} - 1346 \beta_1 + 6162) q^{61} + (1123 \beta_{2} - 294 \beta_1 + 6658) q^{62} + ( - 3136 \beta_{2} + 312 \beta_1 - 6784) q^{64} + (264 \beta_{2} - 1616 \beta_1 + 2556) q^{65} + ( - 6575 \beta_{2} - 907 \beta_1 - 15918) q^{67} + ( - 6728 \beta_{2} + 684 \beta_1 + 4200) q^{68} + ( - 2662 \beta_{2} + 412 \beta_1 - 41124) q^{70} + (3935 \beta_{2} + 891 \beta_1 - 13094) q^{71} + (5370 \beta_{2} - 1174 \beta_1 + 5142) q^{73} + ( - 781 \beta_{2} + 1086 \beta_1 - 51098) q^{74} + ( - 4800 \beta_{2} - 880 \beta_1 + 34640) q^{76} + (1210 \beta_{2} + 1210 \beta_1 - 3388) q^{77} + (3902 \beta_{2} + 454 \beta_1 + 41716) q^{79} + ( - 1868 \beta_{2} + 4 \beta_1 - 39560) q^{80} + (6852 \beta_{2} + 3132 \beta_1 - 85124) q^{82} + ( - 3150 \beta_{2} - 3750 \beta_1 + 47976) q^{83} + ( - 3310 \beta_{2} + 2434 \beta_1 - 34680) q^{85} + (10906 \beta_{2} + 3860 \beta_1 - 81020) q^{86} + (3146 \beta_{2} - 22748) q^{88} + (5167 \beta_{2} + 1523 \beta_1 + 35608) q^{89} + (6840 \beta_{2} - 3512 \beta_1 - 36544) q^{91} + (1472 \beta_{2} - 1710 \beta_1 + 112176) q^{92} + ( - 376 \beta_{2} - 496 \beta_1 - 24976) q^{94} + (1680 \beta_{2} - 40 \beta_1 + 7400) q^{95} + (123 \beta_{2} + 2143 \beta_1 + 3228) q^{97} + ( - 9637 \beta_{2} - 2280 \beta_1 - 1160) q^{98}+O(q^{100})$$ q - b2 * q^2 + (-4*b2 - 2*b1 + 28) * q^4 + (b2 - 3*b1 - 8) * q^5 + (-10*b2 - 10*b1 + 28) * q^7 + (-26*b2 + 188) * q^8 + (-9*b2 + 14*b1 - 138) * q^10 - 121 * q^11 + (70*b2 + 6*b1 + 162) * q^13 + (-138*b2 + 20*b1 + 340) * q^14 + (-164*b2 + 12*b1 + 664) * q^16 + (-132*b2 + 20*b1 - 362) * q^17 + (-60*b2 - 20*b1 + 460) * q^19 + (168*b2 + 22*b1 + 1160) * q^20 + 121*b2 * q^22 + (21*b2 - 167*b1 + 1022) * q^23 + (265*b2 + 85*b1 - 19) * q^25 + (160*b2 + 116*b1 - 4044) * q^26 + (-432*b2 - 36*b1 + 7904) * q^28 + (182*b2 + 46*b1 + 1142) * q^29 + (-101*b2 + 23*b1 - 1366) * q^31 + (-404*b2 - 376*b1 + 4136) * q^32 + (-26*b2 - 344*b1 + 8440) * q^34 + (818*b2 + 306*b1 + 8076) * q^35 + (937*b2 + 197*b1 + 5908) * q^37 + (-840*b2 - 40*b1 + 3080) * q^38 + (-46*b2 - 200*b1 - 5092) * q^40 + (1378*b2 - 94*b1 - 1998) * q^41 + (1190*b2 - 370*b1 - 8736) * q^43 + (484*b2 + 242*b1 - 3388) * q^44 + (-2107*b2 + 710*b1 - 5602) * q^46 + (600*b2 + 424*b1 + 5744) * q^47 + (340*b2 + 740*b1 + 16177) * q^49 + (1674*b2 + 190*b1 - 13690) * q^50 + (3256*b2 - 336*b1 - 11768) * q^52 + (476*b2 + 540*b1 - 16862) * q^53 + (-121*b2 + 363*b1 + 968) * q^55 + (-5468*b2 - 1360*b1 + 14104) * q^56 + (-92*b2 + 180*b1 - 9724) * q^58 + (-3141*b2 - 249*b1 + 1246) * q^59 + (-1466*b2 - 1346*b1 + 6162) * q^61 + (1123*b2 - 294*b1 + 6658) * q^62 + (-3136*b2 + 312*b1 - 6784) * q^64 + (264*b2 - 1616*b1 + 2556) * q^65 + (-6575*b2 - 907*b1 - 15918) * q^67 + (-6728*b2 + 684*b1 + 4200) * q^68 + (-2662*b2 + 412*b1 - 41124) * q^70 + (3935*b2 + 891*b1 - 13094) * q^71 + (5370*b2 - 1174*b1 + 5142) * q^73 + (-781*b2 + 1086*b1 - 51098) * q^74 + (-4800*b2 - 880*b1 + 34640) * q^76 + (1210*b2 + 1210*b1 - 3388) * q^77 + (3902*b2 + 454*b1 + 41716) * q^79 + (-1868*b2 + 4*b1 - 39560) * q^80 + (6852*b2 + 3132*b1 - 85124) * q^82 + (-3150*b2 - 3750*b1 + 47976) * q^83 + (-3310*b2 + 2434*b1 - 34680) * q^85 + (10906*b2 + 3860*b1 - 81020) * q^86 + (3146*b2 - 22748) * q^88 + (5167*b2 + 1523*b1 + 35608) * q^89 + (6840*b2 - 3512*b1 - 36544) * q^91 + (1472*b2 - 1710*b1 + 112176) * q^92 + (-376*b2 - 496*b1 - 24976) * q^94 + (1680*b2 - 40*b1 + 7400) * q^95 + (123*b2 + 2143*b1 + 3228) * q^97 + (-9637*b2 - 2280*b1 - 1160) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8}+O(q^{10})$$ 3 * q + 84 * q^4 - 24 * q^5 + 84 * q^7 + 564 * q^8 $$3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8} - 414 q^{10} - 363 q^{11} + 486 q^{13} + 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} + 23712 q^{28} + 3426 q^{29} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 17724 q^{37} + 9240 q^{38} - 15276 q^{40} - 5994 q^{41} - 26208 q^{43} - 10164 q^{44} - 16806 q^{46} + 17232 q^{47} + 48531 q^{49} - 41070 q^{50} - 35304 q^{52} - 50586 q^{53} + 2904 q^{55} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} + 18486 q^{61} + 19974 q^{62} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} - 123372 q^{70} - 39282 q^{71} + 15426 q^{73} - 153294 q^{74} + 103920 q^{76} - 10164 q^{77} + 125148 q^{79} - 118680 q^{80} - 255372 q^{82} + 143928 q^{83} - 104040 q^{85} - 243060 q^{86} - 68244 q^{88} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} - 74928 q^{94} + 22200 q^{95} + 9684 q^{97} - 3480 q^{98}+O(q^{100})$$ 3 * q + 84 * q^4 - 24 * q^5 + 84 * q^7 + 564 * q^8 - 414 * q^10 - 363 * q^11 + 486 * q^13 + 1020 * q^14 + 1992 * q^16 - 1086 * q^17 + 1380 * q^19 + 3480 * q^20 + 3066 * q^23 - 57 * q^25 - 12132 * q^26 + 23712 * q^28 + 3426 * q^29 - 4098 * q^31 + 12408 * q^32 + 25320 * q^34 + 24228 * q^35 + 17724 * q^37 + 9240 * q^38 - 15276 * q^40 - 5994 * q^41 - 26208 * q^43 - 10164 * q^44 - 16806 * q^46 + 17232 * q^47 + 48531 * q^49 - 41070 * q^50 - 35304 * q^52 - 50586 * q^53 + 2904 * q^55 + 42312 * q^56 - 29172 * q^58 + 3738 * q^59 + 18486 * q^61 + 19974 * q^62 - 20352 * q^64 + 7668 * q^65 - 47754 * q^67 + 12600 * q^68 - 123372 * q^70 - 39282 * q^71 + 15426 * q^73 - 153294 * q^74 + 103920 * q^76 - 10164 * q^77 + 125148 * q^79 - 118680 * q^80 - 255372 * q^82 + 143928 * q^83 - 104040 * q^85 - 243060 * q^86 - 68244 * q^88 + 106824 * q^89 - 109632 * q^91 + 336528 * q^92 - 74928 * q^94 + 22200 * q^95 + 9684 * q^97 - 3480 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x - 38$$ :

 $$\beta_{1}$$ $$=$$ $$3\nu - 1$$ 3*v - 1 $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 3\nu - 34 ) / 3$$ (v^2 - 3*v - 34) / 3
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 3$$ (b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + \beta _1 + 35$$ 3*b2 + b1 + 35

## 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
 −6.29828 8.04796 −0.749680
−8.18772 0 35.0388 59.8722 0 145.071 −24.8808 0 −490.217
1.2 −2.20859 0 −27.1221 −75.2230 0 −225.525 130.577 0 166.137
1.3 10.3963 0 76.0833 −8.64919 0 164.454 458.304 0 −89.9197
 $$n$$: 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.g 3
3.b odd 2 1 11.6.a.b 3
11.b odd 2 1 1089.6.a.r 3
12.b even 2 1 176.6.a.i 3
15.d odd 2 1 275.6.a.b 3
15.e even 4 2 275.6.b.b 6
21.c even 2 1 539.6.a.e 3
24.f even 2 1 704.6.a.t 3
24.h odd 2 1 704.6.a.q 3
33.d even 2 1 121.6.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 3.b odd 2 1
99.6.a.g 3 1.a even 1 1 trivial
121.6.a.d 3 33.d even 2 1
176.6.a.i 3 12.b even 2 1
275.6.a.b 3 15.d odd 2 1
275.6.b.b 6 15.e even 4 2
539.6.a.e 3 21.c even 2 1
704.6.a.q 3 24.h odd 2 1
704.6.a.t 3 24.f even 2 1
1089.6.a.r 3 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 90T_{2} - 188$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 90T - 188$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 24 T^{2} + \cdots - 38954$$
$7$ $$T^{3} - 84 T^{2} + \cdots + 5380448$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} - 486 T^{2} + \cdots + 164136608$$
$17$ $$T^{3} + 1086 T^{2} + \cdots - 331752056$$
$19$ $$T^{3} - 1380 T^{2} + \cdots + 57024000$$
$23$ $$T^{3} + \cdots + 17004325928$$
$29$ $$T^{3} + \cdots + 4029189120$$
$31$ $$T^{3} + \cdots + 1094344400$$
$37$ $$T^{3} + \cdots + 541788167034$$
$41$ $$T^{3} + \cdots - 201929821568$$
$43$ $$T^{3} + \cdots - 2443875098544$$
$47$ $$T^{3} + \cdots + 70174939136$$
$53$ $$T^{3} + \cdots + 1850911309656$$
$59$ $$T^{3} + \cdots - 7759637437060$$
$61$ $$T^{3} + \cdots + 15233874751008$$
$67$ $$T^{3} + \cdots - 147288561330212$$
$71$ $$T^{3} + \cdots + 1290398551704$$
$73$ $$T^{3} + \cdots - 34539701265952$$
$79$ $$T^{3} + \cdots + 1279883216320$$
$83$ $$T^{3} + \cdots + 411597824719824$$
$89$ $$T^{3} + \cdots + 90320980174650$$
$97$ $$T^{3} + \cdots - 10221902527106$$