# Properties

 Label 99.8.a.f Level $99$ Weight $8$ Character orbit 99.a Self dual yes Analytic conductor $30.926$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 510x^{2} - 1544x + 28880$$ x^4 - x^3 - 510*x^2 - 1544*x + 28880 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 4) q^{2} + (\beta_{2} - 2 \beta_1 + 142) q^{4} + ( - \beta_{3} - 10 \beta_1 - 74) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} - 32 \beta_1 + 230) q^{7} + (6 \beta_{3} - \beta_{2} + 126 \beta_1 - 646) q^{8}+O(q^{10})$$ q + (b1 - 4) * q^2 + (b2 - 2*b1 + 142) * q^4 + (-b3 - 10*b1 - 74) * q^5 + (-2*b3 - 2*b2 - 32*b1 + 230) * q^7 + (6*b3 - b2 + 126*b1 - 646) * q^8 $$q + (\beta_1 - 4) q^{2} + (\beta_{2} - 2 \beta_1 + 142) q^{4} + ( - \beta_{3} - 10 \beta_1 - 74) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} - 32 \beta_1 + 230) q^{7} + (6 \beta_{3} - \beta_{2} + 126 \beta_1 - 646) q^{8} + (14 \beta_{3} - 20 \beta_{2} - 70 \beta_1 - 2224) q^{10} - 1331 q^{11} + (15 \beta_{3} - 28 \beta_{2} + 206 \beta_1 - 514) q^{13} + (16 \beta_{3} - 54 \beta_{2} - 18 \beta_1 - 8844) q^{14} + ( - 90 \beta_{3} + 57 \beta_{2} - 398 \beta_1 + 16374) q^{16} + (21 \beta_{3} + 14 \beta_{2} - 842 \beta_1 - 8000) q^{17} + (43 \beta_{3} - 126 \beta_{2} + 810 \beta_1 - 3430) q^{19} + ( - 188 \beta_{3} + 50 \beta_{2} - 3740 \beta_1 + 1948) q^{20} + ( - 1331 \beta_1 + 5324) q^{22} + (149 \beta_{3} - 12 \beta_{2} - 3414 \beta_1 - 27864) q^{23} + (106 \beta_{3} + 280 \beta_{2} - 1260 \beta_1 + 18599) q^{25} + ( - 378 \beta_{3} + 328 \beta_{2} - 3710 \beta_1 + 56376) q^{26} + ( - 292 \beta_{3} + 344 \beta_{2} - 11432 \beta_1 + 5472) q^{28} + (349 \beta_{3} + 110 \beta_{2} + 3254 \beta_1 + 25452) q^{29} + (532 \beta_{3} + 744 \beta_{2} + 3832 \beta_1 - 7016) q^{31} + (834 \beta_{3} - 1113 \beta_{2} + 8222 \beta_1 - 86774) q^{32} + ( - 210 \beta_{3} - 618 \beta_{2} - 8564 \beta_1 - 183436) q^{34} + ( - 234 \beta_{3} + 780 \beta_{2} + 100 \beta_1 + 180884) q^{35} + ( - 610 \beta_{3} - 364 \beta_{2} + 8708 \beta_1 - 126982) q^{37} + ( - 1358 \beta_{3} + 1114 \beta_{2} - 17458 \beta_1 + 228932) q^{38} + (1140 \beta_{3} - 3010 \beta_{2} + 13740 \beta_1 - 673420) q^{40} + (1477 \beta_{3} - 310 \beta_{2} - 13602 \beta_1 - 180316) q^{41} + (119 \beta_{3} + 2814 \beta_{2} + 25034 \beta_1 - 56034) q^{43} + ( - 1331 \beta_{2} + 2662 \beta_1 - 189002) q^{44} + ( - 2158 \beta_{3} - 1936 \beta_{2} - 39660 \beta_1 - 757696) q^{46} + ( - 1677 \beta_{3} - 3428 \beta_{2} - 11674 \beta_1 - 496912) q^{47} + ( - 1336 \beta_{3} + 540 \beta_{2} + 4216 \beta_1 - 193675) q^{49} + (196 \beta_{3} + 80 \beta_{2} + 46015 \beta_1 - 419516) q^{50} + (5340 \beta_{3} - 3578 \beta_{2} + 69708 \beta_1 - 1121388) q^{52} + ( - 1883 \beta_{3} + 3908 \beta_{2} + 74378 \beta_1 - 250110) q^{53} + (1331 \beta_{3} + 13310 \beta_1 + 98494) q^{55} + (4104 \beta_{3} - 7096 \beta_{2} + 31824 \beta_1 - 1815952) q^{56} + ( - 4226 \beta_{3} + 6854 \beta_{2} + 36344 \beta_1 + 708708) q^{58} + ( - 582 \beta_{3} + 11944 \beta_{2} + 54068 \beta_1 - 348820) q^{59} + ( - 1057 \beta_{3} + 56 \beta_{2} - 66010 \beta_1 + 1016210) q^{61} + ( - 2984 \beta_{3} + 9896 \beta_{2} + 74184 \beta_1 + 929744) q^{62} + ( - 6834 \beta_{3} + 8153 \beta_{2} - 168510 \beta_1 + 414198) q^{64} + (1322 \beta_{3} - 5600 \beta_{2} + 100100 \beta_1 - 1679452) q^{65} + (3760 \beta_{3} - 1048 \beta_{2} - 36688 \beta_1 + 439412) q^{67} + ( - 3456 \beta_{3} - 13074 \beta_{2} - 159436 \beta_1 - 362636) q^{68} + (7956 \beta_{3} - 1460 \beta_{2} + 277180 \beta_1 - 757416) q^{70} + (7581 \beta_{3} - 4004 \beta_{2} - 62262 \beta_1 - 1378400) q^{71} + ( - 6122 \beta_{3} + 8684 \beta_{2} - 94460 \beta_1 + 1425918) q^{73} + (6356 \beta_{3} + 2244 \beta_{2} - 137150 \beta_1 + 2761808) q^{74} + (20192 \beta_{3} - 13796 \beta_{2} + 252152 \beta_1 - 4975208) q^{76} + (2662 \beta_{3} + 2662 \beta_{2} + 42592 \beta_1 - 306130) q^{77} + (16 \beta_{3} - 10722 \beta_{2} - 18508 \beta_1 - 1925610) q^{79} + ( - 9956 \beta_{3} + 15730 \beta_{2} - 543740 \beta_1 + 6158316) q^{80} + ( - 22538 \beta_{3} + 858 \beta_{2} - 278928 \beta_1 - 2737764) q^{82} + ( - 16670 \beta_{3} - 19220 \beta_{2} + 186716 \beta_1 - 909448) q^{83} + ( - 3556 \beta_{3} + 15300 \beta_{2} + 202160 \beta_1 + 1430656) q^{85} + (15218 \beta_{3} + 29038 \beta_{2} + 317602 \beta_1 + 6349644) q^{86} + ( - 7986 \beta_{3} + 1331 \beta_{2} - 167706 \beta_1 + 859826) q^{88} + ( - 6580 \beta_{3} + 18280 \beta_{2} - 107288 \beta_1 - 1121490) q^{89} + (6982 \beta_{3} - 26036 \beta_{2} + 243332 \beta_1 - 2325516) q^{91} + ( - 476 \beta_{3} - 61640 \beta_{2} - 572808 \beta_1 - 3274352) q^{92} + (2910 \beta_{3} - 31872 \beta_{2} - 877660 \beta_1 - 662912) q^{94} + (9130 \beta_{3} - 20900 \beta_{2} + 398540 \beta_1 - 5561860) q^{95} + (27384 \beta_{3} - 13116 \beta_{2} - 269112 \beta_1 + 414422) q^{97} + (21944 \beta_{3} - 8604 \beta_{2} - 90539 \beta_1 + 1828004) q^{98}+O(q^{100})$$ q + (b1 - 4) * q^2 + (b2 - 2*b1 + 142) * q^4 + (-b3 - 10*b1 - 74) * q^5 + (-2*b3 - 2*b2 - 32*b1 + 230) * q^7 + (6*b3 - b2 + 126*b1 - 646) * q^8 + (14*b3 - 20*b2 - 70*b1 - 2224) * q^10 - 1331 * q^11 + (15*b3 - 28*b2 + 206*b1 - 514) * q^13 + (16*b3 - 54*b2 - 18*b1 - 8844) * q^14 + (-90*b3 + 57*b2 - 398*b1 + 16374) * q^16 + (21*b3 + 14*b2 - 842*b1 - 8000) * q^17 + (43*b3 - 126*b2 + 810*b1 - 3430) * q^19 + (-188*b3 + 50*b2 - 3740*b1 + 1948) * q^20 + (-1331*b1 + 5324) * q^22 + (149*b3 - 12*b2 - 3414*b1 - 27864) * q^23 + (106*b3 + 280*b2 - 1260*b1 + 18599) * q^25 + (-378*b3 + 328*b2 - 3710*b1 + 56376) * q^26 + (-292*b3 + 344*b2 - 11432*b1 + 5472) * q^28 + (349*b3 + 110*b2 + 3254*b1 + 25452) * q^29 + (532*b3 + 744*b2 + 3832*b1 - 7016) * q^31 + (834*b3 - 1113*b2 + 8222*b1 - 86774) * q^32 + (-210*b3 - 618*b2 - 8564*b1 - 183436) * q^34 + (-234*b3 + 780*b2 + 100*b1 + 180884) * q^35 + (-610*b3 - 364*b2 + 8708*b1 - 126982) * q^37 + (-1358*b3 + 1114*b2 - 17458*b1 + 228932) * q^38 + (1140*b3 - 3010*b2 + 13740*b1 - 673420) * q^40 + (1477*b3 - 310*b2 - 13602*b1 - 180316) * q^41 + (119*b3 + 2814*b2 + 25034*b1 - 56034) * q^43 + (-1331*b2 + 2662*b1 - 189002) * q^44 + (-2158*b3 - 1936*b2 - 39660*b1 - 757696) * q^46 + (-1677*b3 - 3428*b2 - 11674*b1 - 496912) * q^47 + (-1336*b3 + 540*b2 + 4216*b1 - 193675) * q^49 + (196*b3 + 80*b2 + 46015*b1 - 419516) * q^50 + (5340*b3 - 3578*b2 + 69708*b1 - 1121388) * q^52 + (-1883*b3 + 3908*b2 + 74378*b1 - 250110) * q^53 + (1331*b3 + 13310*b1 + 98494) * q^55 + (4104*b3 - 7096*b2 + 31824*b1 - 1815952) * q^56 + (-4226*b3 + 6854*b2 + 36344*b1 + 708708) * q^58 + (-582*b3 + 11944*b2 + 54068*b1 - 348820) * q^59 + (-1057*b3 + 56*b2 - 66010*b1 + 1016210) * q^61 + (-2984*b3 + 9896*b2 + 74184*b1 + 929744) * q^62 + (-6834*b3 + 8153*b2 - 168510*b1 + 414198) * q^64 + (1322*b3 - 5600*b2 + 100100*b1 - 1679452) * q^65 + (3760*b3 - 1048*b2 - 36688*b1 + 439412) * q^67 + (-3456*b3 - 13074*b2 - 159436*b1 - 362636) * q^68 + (7956*b3 - 1460*b2 + 277180*b1 - 757416) * q^70 + (7581*b3 - 4004*b2 - 62262*b1 - 1378400) * q^71 + (-6122*b3 + 8684*b2 - 94460*b1 + 1425918) * q^73 + (6356*b3 + 2244*b2 - 137150*b1 + 2761808) * q^74 + (20192*b3 - 13796*b2 + 252152*b1 - 4975208) * q^76 + (2662*b3 + 2662*b2 + 42592*b1 - 306130) * q^77 + (16*b3 - 10722*b2 - 18508*b1 - 1925610) * q^79 + (-9956*b3 + 15730*b2 - 543740*b1 + 6158316) * q^80 + (-22538*b3 + 858*b2 - 278928*b1 - 2737764) * q^82 + (-16670*b3 - 19220*b2 + 186716*b1 - 909448) * q^83 + (-3556*b3 + 15300*b2 + 202160*b1 + 1430656) * q^85 + (15218*b3 + 29038*b2 + 317602*b1 + 6349644) * q^86 + (-7986*b3 + 1331*b2 - 167706*b1 + 859826) * q^88 + (-6580*b3 + 18280*b2 - 107288*b1 - 1121490) * q^89 + (6982*b3 - 26036*b2 + 243332*b1 - 2325516) * q^91 + (-476*b3 - 61640*b2 - 572808*b1 - 3274352) * q^92 + (2910*b3 - 31872*b2 - 877660*b1 - 662912) * q^94 + (9130*b3 - 20900*b2 + 398540*b1 - 5561860) * q^95 + (27384*b3 - 13116*b2 - 269112*b1 + 414422) * q^97 + (21944*b3 - 8604*b2 - 90539*b1 + 1828004) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 15 q^{2} + 565 q^{4} - 306 q^{5} + 890 q^{7} - 2457 q^{8}+O(q^{10})$$ 4 * q - 15 * q^2 + 565 * q^4 - 306 * q^5 + 890 * q^7 - 2457 * q^8 $$4 q - 15 q^{2} + 565 q^{4} - 306 q^{5} + 890 q^{7} - 2457 q^{8} - 8946 q^{10} - 5324 q^{11} - 1822 q^{13} - 35340 q^{14} + 65041 q^{16} - 32856 q^{17} - 12784 q^{19} + 4002 q^{20} + 19965 q^{22} - 114858 q^{23} + 72856 q^{25} + 221466 q^{26} + 10112 q^{28} + 104952 q^{29} - 24976 q^{31} - 337761 q^{32} - 741690 q^{34} + 722856 q^{35} - 498856 q^{37} + 897156 q^{38} - 2676930 q^{40} - 734556 q^{41} - 201916 q^{43} - 752015 q^{44} - 3068508 q^{46} - 1995894 q^{47} - 771024 q^{49} - 1632129 q^{50} - 4412266 q^{52} - 929970 q^{53} + 407286 q^{55} - 7224888 q^{56} + 2864322 q^{58} - 1353156 q^{59} + 3998774 q^{61} + 3783264 q^{62} + 1480129 q^{64} - 6612108 q^{65} + 1722008 q^{67} - 1596906 q^{68} - 2751024 q^{70} - 5571858 q^{71} + 5600528 q^{73} + 10907838 q^{74} - 19634884 q^{76} - 1184590 q^{77} - 7710226 q^{79} + 24073794 q^{80} - 11230842 q^{82} - 3431856 q^{83} + 5909484 q^{85} + 25687140 q^{86} + 3270267 q^{88} - 4611528 q^{89} - 9032696 q^{91} - 13608576 q^{92} - 3497436 q^{94} - 21828000 q^{95} + 1401692 q^{97} + 7230081 q^{98}+O(q^{100})$$ 4 * q - 15 * q^2 + 565 * q^4 - 306 * q^5 + 890 * q^7 - 2457 * q^8 - 8946 * q^10 - 5324 * q^11 - 1822 * q^13 - 35340 * q^14 + 65041 * q^16 - 32856 * q^17 - 12784 * q^19 + 4002 * q^20 + 19965 * q^22 - 114858 * q^23 + 72856 * q^25 + 221466 * q^26 + 10112 * q^28 + 104952 * q^29 - 24976 * q^31 - 337761 * q^32 - 741690 * q^34 + 722856 * q^35 - 498856 * q^37 + 897156 * q^38 - 2676930 * q^40 - 734556 * q^41 - 201916 * q^43 - 752015 * q^44 - 3068508 * q^46 - 1995894 * q^47 - 771024 * q^49 - 1632129 * q^50 - 4412266 * q^52 - 929970 * q^53 + 407286 * q^55 - 7224888 * q^56 + 2864322 * q^58 - 1353156 * q^59 + 3998774 * q^61 + 3783264 * q^62 + 1480129 * q^64 - 6612108 * q^65 + 1722008 * q^67 - 1596906 * q^68 - 2751024 * q^70 - 5571858 * q^71 + 5600528 * q^73 + 10907838 * q^74 - 19634884 * q^76 - 1184590 * q^77 - 7710226 * q^79 + 24073794 * q^80 - 11230842 * q^82 - 3431856 * q^83 + 5909484 * q^85 + 25687140 * q^86 + 3270267 * q^88 - 4611528 * q^89 - 9032696 * q^91 - 13608576 * q^92 - 3497436 * q^94 - 21828000 * q^95 + 1401692 * q^97 + 7230081 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 510x^{2} - 1544x + 28880$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6\nu - 254$$ v^2 - 6*v - 254 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 11\nu^{2} - 340\nu + 1352 ) / 6$$ (v^3 - 11*v^2 - 340*v + 1352) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6\beta _1 + 254$$ b2 + 6*b1 + 254 $$\nu^{3}$$ $$=$$ $$6\beta_{3} + 11\beta_{2} + 406\beta _1 + 1442$$ 6*b3 + 11*b2 + 406*b1 + 1442

## 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
 −17.6807 −11.0316 6.33327 23.3791
−21.6807 0 342.055 369.874 0 1000.53 −4640.87 0 −8019.14
1.2 −15.0316 0 97.9505 −367.278 0 −91.9512 451.694 0 5520.80
1.3 2.33327 0 −122.556 27.4165 0 860.612 −584.614 0 63.9699
1.4 19.3791 0 247.551 −336.012 0 −879.192 2316.79 0 −6511.62
 $$n$$: 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.f 4
3.b odd 2 1 33.8.a.e 4
12.b even 2 1 528.8.a.r 4
33.d even 2 1 363.8.a.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.e 4 3.b odd 2 1
99.8.a.f 4 1.a even 1 1 trivial
363.8.a.f 4 33.d even 2 1
528.8.a.r 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 15T_{2}^{3} - 426T_{2}^{2} - 5416T_{2} + 14736$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 15 T^{3} - 426 T^{2} + \cdots + 14736$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 306 T^{3} + \cdots + 1251456000$$
$7$ $$T^{4} - 890 T^{3} + \cdots + 69611139776$$
$11$ $$(T + 1331)^{4}$$
$13$ $$T^{4} + \cdots - 161846909982208$$
$17$ $$T^{4} + 32856 T^{3} + \cdots + 16\!\cdots\!76$$
$19$ $$T^{4} + 12784 T^{3} + \cdots + 43\!\cdots\!00$$
$23$ $$T^{4} + 114858 T^{3} + \cdots - 49\!\cdots\!52$$
$29$ $$T^{4} - 104952 T^{3} + \cdots + 66\!\cdots\!00$$
$31$ $$T^{4} + 24976 T^{3} + \cdots + 59\!\cdots\!52$$
$37$ $$T^{4} + 498856 T^{3} + \cdots + 37\!\cdots\!72$$
$41$ $$T^{4} + 734556 T^{3} + \cdots - 49\!\cdots\!64$$
$43$ $$T^{4} + 201916 T^{3} + \cdots - 18\!\cdots\!00$$
$47$ $$T^{4} + 1995894 T^{3} + \cdots - 25\!\cdots\!76$$
$53$ $$T^{4} + 929970 T^{3} + \cdots - 80\!\cdots\!56$$
$59$ $$T^{4} + 1353156 T^{3} + \cdots + 12\!\cdots\!40$$
$61$ $$T^{4} - 3998774 T^{3} + \cdots - 13\!\cdots\!48$$
$67$ $$T^{4} - 1722008 T^{3} + \cdots - 15\!\cdots\!96$$
$71$ $$T^{4} + 5571858 T^{3} + \cdots - 50\!\cdots\!80$$
$73$ $$T^{4} - 5600528 T^{3} + \cdots + 45\!\cdots\!12$$
$79$ $$T^{4} + 7710226 T^{3} + \cdots - 88\!\cdots\!40$$
$83$ $$T^{4} + 3431856 T^{3} + \cdots + 11\!\cdots\!72$$
$89$ $$T^{4} + 4611528 T^{3} + \cdots - 11\!\cdots\!60$$
$97$ $$T^{4} - 1401692 T^{3} + \cdots - 11\!\cdots\!36$$