# Properties

 Label 99.8.a.i Level $99$ Weight $8$ Character orbit 99.a Self dual yes Analytic conductor $30.926$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

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: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 382x^{3} + 558x^{2} + 23640x + 53488$$ x^5 - x^4 - 382*x^3 + 558*x^2 + 23640*x + 53488 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ Twist minimal: yes 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 2) q^{2} + ( - \beta_{3} + 4 \beta_1 + 35) q^{4} + (\beta_{4} - \beta_{3} + 13 \beta_1 + 105) q^{5} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 25 \beta_1 + 98) q^{7} + ( - \beta_{4} - 10 \beta_{3} - 5 \beta_{2} + 37 \beta_1 + 567) q^{8}+O(q^{10})$$ q + (b1 + 2) * q^2 + (-b3 + 4*b1 + 35) * q^4 + (b4 - b3 + 13*b1 + 105) * q^5 + (2*b4 - b3 - 3*b2 + 25*b1 + 98) * q^7 + (-b4 - 10*b3 - 5*b2 + 37*b1 + 567) * q^8 $$q + (\beta_1 + 2) q^{2} + ( - \beta_{3} + 4 \beta_1 + 35) q^{4} + (\beta_{4} - \beta_{3} + 13 \beta_1 + 105) q^{5} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 25 \beta_1 + 98) q^{7} + ( - \beta_{4} - 10 \beta_{3} - 5 \beta_{2} + 37 \beta_1 + 567) q^{8} + (3 \beta_{4} - 19 \beta_{3} + \beta_{2} + 245 \beta_1 + 2404) q^{10} - 1331 q^{11} + (29 \beta_{4} + 41 \beta_{3} + 28 \beta_{2} - 129 \beta_1 - 3399) q^{13} + ( - 5 \beta_{4} - 61 \beta_{3} + 25 \beta_{2} + 149 \beta_1 + 4182) q^{14} + ( - 34 \beta_{4} - 19 \beta_{3} - 26 \beta_{2} + 1182 \beta_1 + 3487) q^{16} + (29 \beta_{4} + 4 \beta_{3} + 37 \beta_{2} - 752 \beta_1 - 687) q^{17} + ( - 47 \beta_{4} + 44 \beta_{3} - 23 \beta_{2} + 1712 \beta_1 - 1399) q^{19} + ( - 131 \beta_{4} - 221 \beta_{3} - 83 \beta_{2} + 3559 \beta_1 + 32618) q^{20} + ( - 1331 \beta_1 - 2662) q^{22} + (85 \beta_{4} + 333 \beta_{3} - 96 \beta_{2} - 2229 \beta_1 - 8105) q^{23} + ( - 122 \beta_{4} - 112 \beta_{3} - 26 \beta_{2} + 6768 \beta_1 + 23621) q^{25} + (269 \beta_{4} + 655 \beta_{3} + 211 \beta_{2} - 7911 \beta_1 - 30640) q^{26} + ( - 237 \beta_{4} - 137 \beta_{3} - 101 \beta_{2} + 9637 \beta_1 + 27648) q^{28} + ( - 75 \beta_{4} + 438 \beta_{3} - 385 \beta_{2} + 2878 \beta_1 + 52829) q^{29} + (552 \beta_{4} + 354 \beta_{3} + 338 \beta_{2} + 4006 \beta_1 - 7608) q^{31} + ( - 131 \beta_{4} - 276 \beta_{3} + 497 \beta_{2} + 2795 \beta_1 + 123127) q^{32} + (268 \beta_{4} + 1146 \beta_{3} - 28 \beta_{2} - 1616 \beta_1 - 119566) q^{34} + ( - 448 \beta_{4} - 322 \beta_{3} - 350 \beta_{2} + 7266 \beta_1 + 195840) q^{35} + (1126 \beta_{4} - 486 \beta_{3} + 852 \beta_{2} + 2998 \beta_1 + 79812) q^{37} + ( - 236 \beta_{4} - 1678 \beta_{3} + 76 \beta_{2} - 3772 \beta_1 + 262826) q^{38} + ( - 1461 \beta_{4} - 3283 \beta_{3} - 1521 \beta_{2} + \cdots + 344466) q^{40}+ \cdots + ( - 12108 \beta_{4} + 2296 \beta_{3} + 2668 \beta_{2} + \cdots - 5272762) q^{98}+O(q^{100})$$ q + (b1 + 2) * q^2 + (-b3 + 4*b1 + 35) * q^4 + (b4 - b3 + 13*b1 + 105) * q^5 + (2*b4 - b3 - 3*b2 + 25*b1 + 98) * q^7 + (-b4 - 10*b3 - 5*b2 + 37*b1 + 567) * q^8 + (3*b4 - 19*b3 + b2 + 245*b1 + 2404) * q^10 - 1331 * q^11 + (29*b4 + 41*b3 + 28*b2 - 129*b1 - 3399) * q^13 + (-5*b4 - 61*b3 + 25*b2 + 149*b1 + 4182) * q^14 + (-34*b4 - 19*b3 - 26*b2 + 1182*b1 + 3487) * q^16 + (29*b4 + 4*b3 + 37*b2 - 752*b1 - 687) * q^17 + (-47*b4 + 44*b3 - 23*b2 + 1712*b1 - 1399) * q^19 + (-131*b4 - 221*b3 - 83*b2 + 3559*b1 + 32618) * q^20 + (-1331*b1 - 2662) * q^22 + (85*b4 + 333*b3 - 96*b2 - 2229*b1 - 8105) * q^23 + (-122*b4 - 112*b3 - 26*b2 + 6768*b1 + 23621) * q^25 + (269*b4 + 655*b3 + 211*b2 - 7911*b1 - 30640) * q^26 + (-237*b4 - 137*b3 - 101*b2 + 9637*b1 + 27648) * q^28 + (-75*b4 + 438*b3 - 385*b2 + 2878*b1 + 52829) * q^29 + (552*b4 + 354*b3 + 338*b2 + 4006*b1 - 7608) * q^31 + (-131*b4 - 276*b3 + 497*b2 + 2795*b1 + 123127) * q^32 + (268*b4 + 1146*b3 - 28*b2 - 1616*b1 - 119566) * q^34 + (-448*b4 - 322*b3 - 350*b2 + 7266*b1 + 195840) * q^35 + (1126*b4 - 486*b3 + 852*b2 + 2998*b1 + 79812) * q^37 + (-236*b4 - 1678*b3 + 76*b2 - 3772*b1 + 262826) * q^38 + (-1461*b4 - 3283*b3 - 1521*b2 + 33481*b1 + 344466) * q^40 + (627*b4 - 242*b3 + 1253*b2 - 3690*b1 + 309387) * q^41 + (-171*b4 - 1414*b3 - 1157*b2 - 8366*b1 + 49397) * q^43 + (1331*b3 - 5324*b1 - 46585) * q^44 + (289*b4 + 3267*b3 + 2751*b2 - 57229*b1 - 412764) * q^46 + (919*b4 + 71*b3 - 1552*b2 - 2719*b1 + 494601) * q^47 + (-366*b4 - 1708*b3 - 2234*b2 - 34884*b1 + 85723) * q^49 + (-704*b4 - 7700*b3 - 1136*b2 + 50887*b1 + 1134146) * q^50 + (-1137*b4 + 8703*b3 + 39*b2 - 104627*b1 - 949140) * q^52 + (1023*b4 + 3047*b3 + 926*b2 - 36099*b1 + 449895) * q^53 + (-1331*b4 + 1331*b3 - 17303*b1 - 139755) * q^55 + (-849*b4 - 3661*b3 - 4701*b2 + 42925*b1 + 1061700) * q^56 + (-1402*b4 - 4100*b3 + 4050*b2 - 7726*b1 + 495094) * q^58 + (-3638*b4 + 4870*b3 + 3068*b2 + 38362*b1 + 939130) * q^59 + (-593*b4 - 4025*b3 + 2688*b2 - 87823*b1 + 72323) * q^61 + (3914*b4 + 1498*b3 + 3054*b2 - 35370*b1 + 600036) * q^62 + (5540*b4 + 2951*b3 - 1820*b2 + 29536*b1 + 296179) * q^64 + (3848*b4 + 23278*b3 + 8414*b2 - 132182*b1 - 494884) * q^65 + (4950*b4 + 2028*b3 - 6886*b2 - 142796*b1 - 983414) * q^67 + (-1606*b4 + 7700*b3 + 2770*b2 - 169478*b1 - 540718) * q^68 + (-3514*b4 - 12698*b3 - 2198*b2 + 240990*b1 + 1565468) * q^70 + (-13407*b4 - 9231*b3 - 14448*b2 + 2407*b1 + 1122055) * q^71 + (8572*b4 - 332*b3 + 3664*b2 - 43988*b1 + 175786) * q^73 + (7426*b4 + 2606*b3 - 786*b2 + 165912*b1 + 740212) * q^74 + (3698*b4 - 11168*b3 - 7318*b2 + 245410*b1 + 302134) * q^76 + (-2662*b4 + 1331*b3 + 3993*b2 - 33275*b1 - 130438) * q^77 + (-21068*b4 - 13231*b3 + 7633*b2 + 77791*b1 + 604668) * q^79 + (1557*b4 - 40101*b3 - 5431*b2 + 314855*b1 + 2142926) * q^80 + (7278*b4 + 14768*b3 - 4966*b2 + 370370*b1 + 119274) * q^82 + (16732*b4 - 1134*b3 - 9542*b2 - 38562*b1 - 514064) * q^83 + (430*b4 + 27826*b3 + 5724*b2 - 143826*b1 - 910214) * q^85 + (-6726*b4 - 11688*b3 - 1154*b2 + 166046*b1 - 1116266) * q^86 + (1331*b4 + 13310*b3 + 6655*b2 - 49247*b1 - 754677) * q^88 + (3988*b4 - 46184*b3 + 12852*b2 + 99864*b1 + 855252) * q^89 + (-6780*b4 + 27856*b3 + 20180*b2 - 230024*b1 - 2595920) * q^91 + (4547*b4 + 61717*b3 + 13851*b2 - 546511*b1 - 9151306) * q^92 + (-2461*b4 - 12375*b3 + 15181*b2 + 418829*b1 + 492580) * q^94 + (9082*b4 - 40762*b3 + 340*b2 + 225146*b1 + 949310) * q^95 + (-24914*b4 + 36684*b3 - 41158*b2 + 132900*b1 + 596) * q^97 + (-12108*b4 + 2296*b3 + 2668*b2 + 149069*b1 - 5272762) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 8 q^{2} + 166 q^{4} + 500 q^{5} + 446 q^{7} + 2754 q^{8}+O(q^{10})$$ 5 * q + 8 * q^2 + 166 * q^4 + 500 * q^5 + 446 * q^7 + 2754 * q^8 $$5 q + 8 q^{2} + 166 q^{4} + 500 q^{5} + 446 q^{7} + 2754 q^{8} + 11516 q^{10} - 6655 q^{11} - 16666 q^{13} + 20516 q^{14} + 15010 q^{16} - 1906 q^{17} - 10446 q^{19} + 155572 q^{20} - 10648 q^{22} - 35468 q^{23} + 104239 q^{25} - 136396 q^{26} + 118456 q^{28} + 259062 q^{29} - 44932 q^{31} + 609010 q^{32} - 592888 q^{34} + 963800 q^{35} + 393978 q^{37} + 1319448 q^{38} + 1650684 q^{40} + 1554074 q^{41} + 263118 q^{43} - 220946 q^{44} - 1948268 q^{46} + 2481904 q^{47} + 498177 q^{49} + 5560984 q^{50} - 4530056 q^{52} + 2325840 q^{53} - 665500 q^{55} + 5221992 q^{56} + 2479968 q^{58} + 4613452 q^{59} + 529362 q^{61} + 3077192 q^{62} + 1437674 q^{64} - 2187496 q^{65} - 4612664 q^{67} - 2362916 q^{68} + 7327832 q^{70} + 5583864 q^{71} + 980054 q^{73} + 3387480 q^{74} + 1023396 q^{76} - 593626 q^{77} + 2804758 q^{79} + 10053364 q^{80} - 110080 q^{82} - 2451324 q^{83} - 4240456 q^{85} - 5937408 q^{86} - 3665574 q^{88} + 4025472 q^{89} - 12525436 q^{91} - 44606548 q^{92} + 1592764 q^{94} + 4273320 q^{95} - 234806 q^{97} - 26686536 q^{98}+O(q^{100})$$ 5 * q + 8 * q^2 + 166 * q^4 + 500 * q^5 + 446 * q^7 + 2754 * q^8 + 11516 * q^10 - 6655 * q^11 - 16666 * q^13 + 20516 * q^14 + 15010 * q^16 - 1906 * q^17 - 10446 * q^19 + 155572 * q^20 - 10648 * q^22 - 35468 * q^23 + 104239 * q^25 - 136396 * q^26 + 118456 * q^28 + 259062 * q^29 - 44932 * q^31 + 609010 * q^32 - 592888 * q^34 + 963800 * q^35 + 393978 * q^37 + 1319448 * q^38 + 1650684 * q^40 + 1554074 * q^41 + 263118 * q^43 - 220946 * q^44 - 1948268 * q^46 + 2481904 * q^47 + 498177 * q^49 + 5560984 * q^50 - 4530056 * q^52 + 2325840 * q^53 - 665500 * q^55 + 5221992 * q^56 + 2479968 * q^58 + 4613452 * q^59 + 529362 * q^61 + 3077192 * q^62 + 1437674 * q^64 - 2187496 * q^65 - 4612664 * q^67 - 2362916 * q^68 + 7327832 * q^70 + 5583864 * q^71 + 980054 * q^73 + 3387480 * q^74 + 1023396 * q^76 - 593626 * q^77 + 2804758 * q^79 + 10053364 * q^80 - 110080 * q^82 - 2451324 * q^83 - 4240456 * q^85 - 5937408 * q^86 - 3665574 * q^88 + 4025472 * q^89 - 12525436 * q^91 - 44606548 * q^92 + 1592764 * q^94 + 4273320 * q^95 - 234806 * q^97 - 26686536 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 382x^{3} + 558x^{2} + 23640x + 53488$$ :

 $$\beta_{1}$$ $$=$$ $$( 46\nu^{4} - 59\nu^{3} - 14435\nu^{2} + 54086\nu + 375282 ) / 19138$$ (46*v^4 - 59*v^3 - 14435*v^2 + 54086*v + 375282) / 19138 $$\beta_{2}$$ $$=$$ $$( 75\nu^{4} + 1776\nu^{3} - 20415\nu^{2} - 502182\nu + 460433 ) / 9569$$ (75*v^4 + 1776*v^3 - 20415*v^2 - 502182*v + 460433) / 9569 $$\beta_{3}$$ $$=$$ $$( 335\nu^{4} - 3550\nu^{3} - 129463\nu^{2} + 1014208\nu + 6063460 ) / 19138$$ (335*v^4 - 3550*v^3 - 129463*v^2 + 1014208*v + 6063460) / 19138 $$\beta_{4}$$ $$=$$ $$( 180\nu^{4} - 1479\nu^{3} - 77703\nu^{2} + 586080\nu + 4542224 ) / 9569$$ (180*v^4 - 1479*v^3 - 77703*v^2 + 586080*v + 4542224) / 9569
 $$\nu$$ $$=$$ $$( \beta_{4} - 2\beta_{3} - \beta_{2} + 10\beta _1 + 11 ) / 36$$ (b4 - 2*b3 - b2 + 10*b1 + 11) / 36 $$\nu^{2}$$ $$=$$ $$( -19\beta_{4} + 2\beta_{3} - 11\beta_{2} + 170\beta _1 + 5581 ) / 36$$ (-19*b4 + 2*b3 - 11*b2 + 170*b1 + 5581) / 36 $$\nu^{3}$$ $$=$$ $$( 347\beta_{4} - 634\beta_{3} - 113\beta_{2} + 2270\beta _1 - 2921 ) / 36$$ (347*b4 - 634*b3 - 113*b2 + 2270*b1 - 2921) / 36 $$\nu^{4}$$ $$=$$ $$( -2231\beta_{4} + 722\beta_{3} - 807\beta_{2} + 19826\beta _1 + 480321 ) / 12$$ (-2231*b4 + 722*b3 - 807*b2 + 19826*b1 + 480321) / 12

## 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
 −5.66065 11.5540 −18.2221 −2.78521 16.1140
−15.5299 0 113.178 −59.4610 0 −1115.32 230.190 0 923.423
1.2 −8.34840 0 −58.3042 113.711 0 1428.97 1555.34 0 −949.305
1.3 3.32273 0 −116.959 −363.595 0 −794.809 −813.934 0 −1208.13
1.4 8.09815 0 −62.4199 308.036 0 97.1968 −1542.05 0 2494.52
1.5 20.4574 0 290.506 501.309 0 829.954 3324.45 0 10255.5
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.i yes 5
3.b odd 2 1 99.8.a.h 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.a.h 5 3.b odd 2 1
99.8.a.i yes 5 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 8T_{2}^{4} - 371T_{2}^{3} + 1538T_{2}^{2} + 20636T_{2} - 71368$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 8 T^{4} - 371 T^{3} + \cdots - 71368$$
$3$ $$T^{5}$$
$5$ $$T^{5} - 500 T^{4} + \cdots - 379629184000$$
$7$ $$T^{5} + \cdots - 102186002826976$$
$11$ $$(T + 1331)^{5}$$
$13$ $$T^{5} + 16666 T^{4} + \cdots - 17\!\cdots\!76$$
$17$ $$T^{5} + 1906 T^{4} + \cdots + 66\!\cdots\!44$$
$19$ $$T^{5} + 10446 T^{4} + \cdots + 40\!\cdots\!44$$
$23$ $$T^{5} + 35468 T^{4} + \cdots - 10\!\cdots\!72$$
$29$ $$T^{5} - 259062 T^{4} + \cdots + 70\!\cdots\!64$$
$31$ $$T^{5} + 44932 T^{4} + \cdots - 47\!\cdots\!64$$
$37$ $$T^{5} - 393978 T^{4} + \cdots - 58\!\cdots\!28$$
$41$ $$T^{5} - 1554074 T^{4} + \cdots + 62\!\cdots\!80$$
$43$ $$T^{5} - 263118 T^{4} + \cdots - 99\!\cdots\!08$$
$47$ $$T^{5} - 2481904 T^{4} + \cdots - 26\!\cdots\!96$$
$53$ $$T^{5} - 2325840 T^{4} + \cdots + 30\!\cdots\!40$$
$59$ $$T^{5} - 4613452 T^{4} + \cdots + 22\!\cdots\!56$$
$61$ $$T^{5} - 529362 T^{4} + \cdots - 30\!\cdots\!44$$
$67$ $$T^{5} + 4612664 T^{4} + \cdots - 68\!\cdots\!84$$
$71$ $$T^{5} - 5583864 T^{4} + \cdots - 13\!\cdots\!60$$
$73$ $$T^{5} - 980054 T^{4} + \cdots - 59\!\cdots\!04$$
$79$ $$T^{5} - 2804758 T^{4} + \cdots - 47\!\cdots\!20$$
$83$ $$T^{5} + 2451324 T^{4} + \cdots + 25\!\cdots\!00$$
$89$ $$T^{5} - 4025472 T^{4} + \cdots + 44\!\cdots\!80$$
$97$ $$T^{5} + 234806 T^{4} + \cdots + 34\!\cdots\!84$$
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