# Properties

 Label 99.8.a.c Level $99$ Weight $8$ Character orbit 99.a Self dual yes Analytic conductor $30.926$ Analytic rank $1$ 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] = [99,8,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, 8, names="a")

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$30.9261175229$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{15})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 15$$ x^2 - 15 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{2} + (8 \beta - 52) q^{4} + (20 \beta + 235) q^{5} + ( - 82 \beta - 614) q^{7} + ( - 148 \beta - 240) q^{8}+O(q^{10})$$ q + (b + 4) * q^2 + (8*b - 52) * q^4 + (20*b + 235) * q^5 + (-82*b - 614) * q^7 + (-148*b - 240) * q^8 $$q + (\beta + 4) q^{2} + (8 \beta - 52) q^{4} + (20 \beta + 235) q^{5} + ( - 82 \beta - 614) q^{7} + ( - 148 \beta - 240) q^{8} + (315 \beta + 2140) q^{10} - 1331 q^{11} + ( - 518 \beta + 172) q^{13} + ( - 942 \beta - 7376) q^{14} + ( - 1856 \beta - 3184) q^{16} + ( - 3666 \beta + 4234) q^{17} + (2982 \beta - 17640) q^{19} + (840 \beta - 2620) q^{20} + ( - 1331 \beta - 5324) q^{22} + (4290 \beta + 30743) q^{23} + (9400 \beta + 1100) q^{25} + ( - 1900 \beta - 30392) q^{26} + ( - 648 \beta - 7432) q^{28} + ( - 11468 \beta - 89520) q^{29} + ( - 20210 \beta - 28583) q^{31} + (8336 \beta - 93376) q^{32} + ( - 10430 \beta - 203024) q^{34} + ( - 31550 \beta - 242690) q^{35} + (3748 \beta - 438849) q^{37} + ( - 5712 \beta + 108360) q^{38} + ( - 39580 \beta - 234000) q^{40} + (68870 \beta + 141808) q^{41} + ( - 12760 \beta + 137742) q^{43} + ( - 10648 \beta + 69212) q^{44} + (47903 \beta + 380372) q^{46} + ( - 15252 \beta - 831256) q^{47} + (100696 \beta - 43107) q^{49} + (38700 \beta + 568400) q^{50} + (28312 \beta - 257584) q^{52} + (66388 \beta - 808242) q^{53} + ( - 26620 \beta - 312785) q^{55} + (110552 \beta + 875520) q^{56} + ( - 135392 \beta - 1046160) q^{58} + ( - 147078 \beta + 1227065) q^{59} + (28900 \beta - 3009588) q^{61} + ( - 109423 \beta - 1326932) q^{62} + (177536 \beta + 534208) q^{64} + ( - 118290 \beta - 581180) q^{65} + (392590 \beta - 87349) q^{67} + (224504 \beta - 1979848) q^{68} + ( - 368890 \beta - 2863760) q^{70} + (452890 \beta + 575733) q^{71} + (195234 \beta + 442972) q^{73} + ( - 423857 \beta - 1530516) q^{74} + ( - 296184 \beta + 2348640) q^{76} + (109142 \beta + 817234) q^{77} + ( - 323896 \beta + 1900730) q^{79} + ( - 499840 \beta - 2975440) q^{80} + (417288 \beta + 4699432) q^{82} + ( - 175068 \beta + 1141458) q^{83} + ( - 776830 \beta - 3404210) q^{85} + (86702 \beta - 214632) q^{86} + (196988 \beta + 319440) q^{88} + (201740 \beta + 6740985) q^{89} + (303948 \beta + 2442952) q^{91} + (22864 \beta + 460564) q^{92} + ( - 892264 \beta - 4240144) q^{94} + (347970 \beta - 567000) q^{95} + ( - 174936 \beta - 34039) q^{97} + (359677 \beta + 5869332) q^{98}+O(q^{100})$$ q + (b + 4) * q^2 + (8*b - 52) * q^4 + (20*b + 235) * q^5 + (-82*b - 614) * q^7 + (-148*b - 240) * q^8 + (315*b + 2140) * q^10 - 1331 * q^11 + (-518*b + 172) * q^13 + (-942*b - 7376) * q^14 + (-1856*b - 3184) * q^16 + (-3666*b + 4234) * q^17 + (2982*b - 17640) * q^19 + (840*b - 2620) * q^20 + (-1331*b - 5324) * q^22 + (4290*b + 30743) * q^23 + (9400*b + 1100) * q^25 + (-1900*b - 30392) * q^26 + (-648*b - 7432) * q^28 + (-11468*b - 89520) * q^29 + (-20210*b - 28583) * q^31 + (8336*b - 93376) * q^32 + (-10430*b - 203024) * q^34 + (-31550*b - 242690) * q^35 + (3748*b - 438849) * q^37 + (-5712*b + 108360) * q^38 + (-39580*b - 234000) * q^40 + (68870*b + 141808) * q^41 + (-12760*b + 137742) * q^43 + (-10648*b + 69212) * q^44 + (47903*b + 380372) * q^46 + (-15252*b - 831256) * q^47 + (100696*b - 43107) * q^49 + (38700*b + 568400) * q^50 + (28312*b - 257584) * q^52 + (66388*b - 808242) * q^53 + (-26620*b - 312785) * q^55 + (110552*b + 875520) * q^56 + (-135392*b - 1046160) * q^58 + (-147078*b + 1227065) * q^59 + (28900*b - 3009588) * q^61 + (-109423*b - 1326932) * q^62 + (177536*b + 534208) * q^64 + (-118290*b - 581180) * q^65 + (392590*b - 87349) * q^67 + (224504*b - 1979848) * q^68 + (-368890*b - 2863760) * q^70 + (452890*b + 575733) * q^71 + (195234*b + 442972) * q^73 + (-423857*b - 1530516) * q^74 + (-296184*b + 2348640) * q^76 + (109142*b + 817234) * q^77 + (-323896*b + 1900730) * q^79 + (-499840*b - 2975440) * q^80 + (417288*b + 4699432) * q^82 + (-175068*b + 1141458) * q^83 + (-776830*b - 3404210) * q^85 + (86702*b - 214632) * q^86 + (196988*b + 319440) * q^88 + (201740*b + 6740985) * q^89 + (303948*b + 2442952) * q^91 + (22864*b + 460564) * q^92 + (-892264*b - 4240144) * q^94 + (347970*b - 567000) * q^95 + (-174936*b - 34039) * q^97 + (359677*b + 5869332) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} - 104 q^{4} + 470 q^{5} - 1228 q^{7} - 480 q^{8}+O(q^{10})$$ 2 * q + 8 * q^2 - 104 * q^4 + 470 * q^5 - 1228 * q^7 - 480 * q^8 $$2 q + 8 q^{2} - 104 q^{4} + 470 q^{5} - 1228 q^{7} - 480 q^{8} + 4280 q^{10} - 2662 q^{11} + 344 q^{13} - 14752 q^{14} - 6368 q^{16} + 8468 q^{17} - 35280 q^{19} - 5240 q^{20} - 10648 q^{22} + 61486 q^{23} + 2200 q^{25} - 60784 q^{26} - 14864 q^{28} - 179040 q^{29} - 57166 q^{31} - 186752 q^{32} - 406048 q^{34} - 485380 q^{35} - 877698 q^{37} + 216720 q^{38} - 468000 q^{40} + 283616 q^{41} + 275484 q^{43} + 138424 q^{44} + 760744 q^{46} - 1662512 q^{47} - 86214 q^{49} + 1136800 q^{50} - 515168 q^{52} - 1616484 q^{53} - 625570 q^{55} + 1751040 q^{56} - 2092320 q^{58} + 2454130 q^{59} - 6019176 q^{61} - 2653864 q^{62} + 1068416 q^{64} - 1162360 q^{65} - 174698 q^{67} - 3959696 q^{68} - 5727520 q^{70} + 1151466 q^{71} + 885944 q^{73} - 3061032 q^{74} + 4697280 q^{76} + 1634468 q^{77} + 3801460 q^{79} - 5950880 q^{80} + 9398864 q^{82} + 2282916 q^{83} - 6808420 q^{85} - 429264 q^{86} + 638880 q^{88} + 13481970 q^{89} + 4885904 q^{91} + 921128 q^{92} - 8480288 q^{94} - 1134000 q^{95} - 68078 q^{97} + 11738664 q^{98}+O(q^{100})$$ 2 * q + 8 * q^2 - 104 * q^4 + 470 * q^5 - 1228 * q^7 - 480 * q^8 + 4280 * q^10 - 2662 * q^11 + 344 * q^13 - 14752 * q^14 - 6368 * q^16 + 8468 * q^17 - 35280 * q^19 - 5240 * q^20 - 10648 * q^22 + 61486 * q^23 + 2200 * q^25 - 60784 * q^26 - 14864 * q^28 - 179040 * q^29 - 57166 * q^31 - 186752 * q^32 - 406048 * q^34 - 485380 * q^35 - 877698 * q^37 + 216720 * q^38 - 468000 * q^40 + 283616 * q^41 + 275484 * q^43 + 138424 * q^44 + 760744 * q^46 - 1662512 * q^47 - 86214 * q^49 + 1136800 * q^50 - 515168 * q^52 - 1616484 * q^53 - 625570 * q^55 + 1751040 * q^56 - 2092320 * q^58 + 2454130 * q^59 - 6019176 * q^61 - 2653864 * q^62 + 1068416 * q^64 - 1162360 * q^65 - 174698 * q^67 - 3959696 * q^68 - 5727520 * q^70 + 1151466 * q^71 + 885944 * q^73 - 3061032 * q^74 + 4697280 * q^76 + 1634468 * q^77 + 3801460 * q^79 - 5950880 * q^80 + 9398864 * q^82 + 2282916 * q^83 - 6808420 * q^85 - 429264 * q^86 + 638880 * q^88 + 13481970 * q^89 + 4885904 * q^91 + 921128 * q^92 - 8480288 * q^94 - 1134000 * q^95 - 68078 * q^97 + 11738664 * 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
 −3.87298 3.87298
−3.74597 0 −113.968 80.0807 0 21.1693 906.403 0 −299.980
1.2 11.7460 0 9.96773 389.919 0 −1249.17 −1386.40 0 4579.98
 $$n$$: 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 99.8.a.c 2
3.b odd 2 1 11.8.a.a 2
12.b even 2 1 176.8.a.d 2
15.d odd 2 1 275.8.a.a 2
21.c even 2 1 539.8.a.a 2
33.d even 2 1 121.8.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.a.a 2 3.b odd 2 1
99.8.a.c 2 1.a even 1 1 trivial
121.8.a.b 2 33.d even 2 1
176.8.a.d 2 12.b even 2 1
275.8.a.a 2 15.d odd 2 1
539.8.a.a 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 8T_{2} - 44$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 8T - 44$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 470T + 31225$$
$7$ $$T^{2} + 1228T - 26444$$
$11$ $$(T + 1331)^{2}$$
$13$ $$T^{2} - 344 T - 16069856$$
$17$ $$T^{2} - 8468 T - 788446604$$
$19$ $$T^{2} + 35280 T - 222369840$$
$23$ $$T^{2} - 61486 T - 159113951$$
$29$ $$T^{2} + 179040 T + 122928960$$
$31$ $$T^{2} + 57166 T - 23689658111$$
$37$ $$T^{2} + 877698 T + 191745594561$$
$41$ $$T^{2} - 283616 T - 264475105136$$
$43$ $$T^{2} - 275484 T + 9203802564$$
$47$ $$T^{2} + 1662512 T + 677029127296$$
$53$ $$T^{2} + 1616484 T + 388813137924$$
$59$ $$T^{2} - 2454130 T + 207772229185$$
$61$ $$T^{2} + 6019176 T + 9007507329744$$
$67$ $$T^{2} + 174698 T - 9239984638199$$
$71$ $$T^{2} - 1151466 T - 11975092638711$$
$73$ $$T^{2} - 885944 T - 2090754692576$$
$79$ $$T^{2} - 3801460 T - 2681742596060$$
$83$ $$T^{2} - 2282916 T - 536001911676$$
$89$ $$T^{2} - 13481970 T + 42998937114225$$
$97$ $$T^{2} + 68078 T - 1834997592239$$