# Properties

 Label 99.4.a.c Level $99$ Weight $4$ Character orbit 99.a Self dual yes Analytic conductor $5.841$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + (4 \beta + 10) q^{7} + ( - 10 \beta + 6) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b - 4) * q^4 + (8*b - 1) * q^5 + (4*b + 10) * q^7 + (-10*b + 6) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + (4 \beta + 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 9 \beta + 25) q^{10} + 11 q^{11} + (20 \beta + 40) q^{13} + (6 \beta + 2) q^{14} + (32 \beta - 4) q^{16} + (12 \beta + 62) q^{17} + ( - 60 \beta + 36) q^{19} + ( - 30 \beta - 44) q^{20} + (11 \beta - 11) q^{22} + ( - 36 \beta + 49) q^{23} + ( - 16 \beta + 68) q^{25} + (20 \beta + 20) q^{26} + ( - 36 \beta - 64) q^{28} + ( - 56 \beta - 72) q^{29} + ( - 28 \beta - 17) q^{31} + (44 \beta + 52) q^{32} + (50 \beta - 26) q^{34} + (76 \beta + 86) q^{35} + (8 \beta + 27) q^{37} + (96 \beta - 216) q^{38} + (58 \beta - 246) q^{40} + ( - 4 \beta - 268) q^{41} + (16 \beta - 30) q^{43} + ( - 22 \beta - 44) q^{44} + (85 \beta - 157) q^{46} + ( - 120 \beta + 136) q^{47} + (80 \beta - 195) q^{49} + (84 \beta - 116) q^{50} + ( - 160 \beta - 280) q^{52} + ( - 56 \beta + 246) q^{53} + (88 \beta - 11) q^{55} + ( - 76 \beta - 60) q^{56} + ( - 16 \beta - 96) q^{58} + ( - 132 \beta - 317) q^{59} + ( - 184 \beta + 420) q^{61} + (11 \beta - 67) q^{62} + ( - 248 \beta + 112) q^{64} + (300 \beta + 440) q^{65} + (20 \beta + 377) q^{67} + ( - 172 \beta - 320) q^{68} + (10 \beta + 142) q^{70} + (76 \beta + 339) q^{71} + (468 \beta - 200) q^{73} + (19 \beta - 3) q^{74} + (168 \beta + 216) q^{76} + (44 \beta + 110) q^{77} + ( - 656 \beta + 158) q^{79} + ( - 64 \beta + 772) q^{80} + ( - 264 \beta + 256) q^{82} + (120 \beta - 234) q^{83} + (484 \beta + 226) q^{85} + ( - 46 \beta + 78) q^{86} + ( - 110 \beta + 66) q^{88} + ( - 328 \beta + 921) q^{89} + (360 \beta + 640) q^{91} + (46 \beta + 20) q^{92} + (256 \beta - 496) q^{94} + (348 \beta - 1476) q^{95} + ( - 144 \beta + 1097) q^{97} + ( - 275 \beta + 435) q^{98}+O(q^{100})$$ q + (b - 1) * q^2 + (-2*b - 4) * q^4 + (8*b - 1) * q^5 + (4*b + 10) * q^7 + (-10*b + 6) * q^8 + (-9*b + 25) * q^10 + 11 * q^11 + (20*b + 40) * q^13 + (6*b + 2) * q^14 + (32*b - 4) * q^16 + (12*b + 62) * q^17 + (-60*b + 36) * q^19 + (-30*b - 44) * q^20 + (11*b - 11) * q^22 + (-36*b + 49) * q^23 + (-16*b + 68) * q^25 + (20*b + 20) * q^26 + (-36*b - 64) * q^28 + (-56*b - 72) * q^29 + (-28*b - 17) * q^31 + (44*b + 52) * q^32 + (50*b - 26) * q^34 + (76*b + 86) * q^35 + (8*b + 27) * q^37 + (96*b - 216) * q^38 + (58*b - 246) * q^40 + (-4*b - 268) * q^41 + (16*b - 30) * q^43 + (-22*b - 44) * q^44 + (85*b - 157) * q^46 + (-120*b + 136) * q^47 + (80*b - 195) * q^49 + (84*b - 116) * q^50 + (-160*b - 280) * q^52 + (-56*b + 246) * q^53 + (88*b - 11) * q^55 + (-76*b - 60) * q^56 + (-16*b - 96) * q^58 + (-132*b - 317) * q^59 + (-184*b + 420) * q^61 + (11*b - 67) * q^62 + (-248*b + 112) * q^64 + (300*b + 440) * q^65 + (20*b + 377) * q^67 + (-172*b - 320) * q^68 + (10*b + 142) * q^70 + (76*b + 339) * q^71 + (468*b - 200) * q^73 + (19*b - 3) * q^74 + (168*b + 216) * q^76 + (44*b + 110) * q^77 + (-656*b + 158) * q^79 + (-64*b + 772) * q^80 + (-264*b + 256) * q^82 + (120*b - 234) * q^83 + (484*b + 226) * q^85 + (-46*b + 78) * q^86 + (-110*b + 66) * q^88 + (-328*b + 921) * q^89 + (360*b + 640) * q^91 + (46*b + 20) * q^92 + (256*b - 496) * q^94 + (348*b - 1476) * q^95 + (-144*b + 1097) * q^97 + (-275*b + 435) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8 $$2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28} - 144 q^{29} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} + 54 q^{37} - 432 q^{38} - 492 q^{40} - 536 q^{41} - 60 q^{43} - 88 q^{44} - 314 q^{46} + 272 q^{47} - 390 q^{49} - 232 q^{50} - 560 q^{52} + 492 q^{53} - 22 q^{55} - 120 q^{56} - 192 q^{58} - 634 q^{59} + 840 q^{61} - 134 q^{62} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 284 q^{70} + 678 q^{71} - 400 q^{73} - 6 q^{74} + 432 q^{76} + 220 q^{77} + 316 q^{79} + 1544 q^{80} + 512 q^{82} - 468 q^{83} + 452 q^{85} + 156 q^{86} + 132 q^{88} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} - 992 q^{94} - 2952 q^{95} + 2194 q^{97} + 870 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8 + 50 * q^10 + 22 * q^11 + 80 * q^13 + 4 * q^14 - 8 * q^16 + 124 * q^17 + 72 * q^19 - 88 * q^20 - 22 * q^22 + 98 * q^23 + 136 * q^25 + 40 * q^26 - 128 * q^28 - 144 * q^29 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 + 54 * q^37 - 432 * q^38 - 492 * q^40 - 536 * q^41 - 60 * q^43 - 88 * q^44 - 314 * q^46 + 272 * q^47 - 390 * q^49 - 232 * q^50 - 560 * q^52 + 492 * q^53 - 22 * q^55 - 120 * q^56 - 192 * q^58 - 634 * q^59 + 840 * q^61 - 134 * q^62 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 284 * q^70 + 678 * q^71 - 400 * q^73 - 6 * q^74 + 432 * q^76 + 220 * q^77 + 316 * q^79 + 1544 * q^80 + 512 * q^82 - 468 * q^83 + 452 * q^85 + 156 * q^86 + 132 * q^88 + 1842 * q^89 + 1280 * q^91 + 40 * q^92 - 992 * q^94 - 2952 * q^95 + 2194 * q^97 + 870 * 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
 −1.73205 1.73205
−2.73205 0 −0.535898 −14.8564 0 3.07180 23.3205 0 40.5885
1.2 0.732051 0 −7.46410 12.8564 0 16.9282 −11.3205 0 9.41154
 $$n$$: 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.c 2
3.b odd 2 1 11.4.a.a 2
4.b odd 2 1 1584.4.a.bc 2
5.b even 2 1 2475.4.a.q 2
11.b odd 2 1 1089.4.a.v 2
12.b even 2 1 176.4.a.i 2
15.d odd 2 1 275.4.a.b 2
15.e even 4 2 275.4.b.c 4
21.c even 2 1 539.4.a.e 2
24.f even 2 1 704.4.a.n 2
24.h odd 2 1 704.4.a.p 2
33.d even 2 1 121.4.a.c 2
33.f even 10 4 121.4.c.f 8
33.h odd 10 4 121.4.c.c 8
39.d odd 2 1 1859.4.a.a 2
132.d odd 2 1 1936.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 3.b odd 2 1
99.4.a.c 2 1.a even 1 1 trivial
121.4.a.c 2 33.d even 2 1
121.4.c.c 8 33.h odd 10 4
121.4.c.f 8 33.f even 10 4
176.4.a.i 2 12.b even 2 1
275.4.a.b 2 15.d odd 2 1
275.4.b.c 4 15.e even 4 2
539.4.a.e 2 21.c even 2 1
704.4.a.n 2 24.f even 2 1
704.4.a.p 2 24.h odd 2 1
1089.4.a.v 2 11.b odd 2 1
1584.4.a.bc 2 4.b odd 2 1
1859.4.a.a 2 39.d odd 2 1
1936.4.a.w 2 132.d odd 2 1
2475.4.a.q 2 5.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T - 191$$
$7$ $$T^{2} - 20T + 52$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} - 80T + 400$$
$17$ $$T^{2} - 124T + 3412$$
$19$ $$T^{2} - 72T - 9504$$
$23$ $$T^{2} - 98T - 1487$$
$29$ $$T^{2} + 144T - 4224$$
$31$ $$T^{2} + 34T - 2063$$
$37$ $$T^{2} - 54T + 537$$
$41$ $$T^{2} + 536T + 71776$$
$43$ $$T^{2} + 60T + 132$$
$47$ $$T^{2} - 272T - 24704$$
$53$ $$T^{2} - 492T + 51108$$
$59$ $$T^{2} + 634T + 48217$$
$61$ $$T^{2} - 840T + 74832$$
$67$ $$T^{2} - 754T + 140929$$
$71$ $$T^{2} - 678T + 97593$$
$73$ $$T^{2} + 400T - 617072$$
$79$ $$T^{2} - 316 T - 1266044$$
$83$ $$T^{2} + 468T + 11556$$
$89$ $$T^{2} - 1842 T + 525489$$
$97$ $$T^{2} - 2194 T + 1141201$$