Properties

 Label 99.8.a.b 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: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ 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 $$\beta = \frac{1}{2}(1 + 3\sqrt{97})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 90) q^{4} + (14 \beta + 90) q^{5} + ( - 42 \beta - 188) q^{7} + (37 \beta - 218) q^{8} +O(q^{10})$$ q - b * q^2 + (b + 90) * q^4 + (14*b + 90) * q^5 + (-42*b - 188) * q^7 + (37*b - 218) * q^8 $$q - \beta q^{2} + (\beta + 90) q^{4} + (14 \beta + 90) q^{5} + ( - 42 \beta - 188) q^{7} + (37 \beta - 218) q^{8} + ( - 104 \beta - 3052) q^{10} - 1331 q^{11} + (38 \beta - 6642) q^{13} + (230 \beta + 9156) q^{14} + (53 \beta - 19586) q^{16} + ( - 180 \beta + 5218) q^{17} + (2372 \beta + 5912) q^{19} + (1364 \beta + 11152) q^{20} + 1331 \beta q^{22} + (2050 \beta + 5808) q^{23} + (2716 \beta - 27297) q^{25} + (6604 \beta - 8284) q^{26} + ( - 4010 \beta - 26076) q^{28} + (4564 \beta - 9938) q^{29} + (184 \beta - 24112) q^{31} + (14797 \beta + 16350) q^{32} + ( - 5038 \beta + 39240) q^{34} + ( - 7000 \beta - 145104) q^{35} + (13104 \beta + 130494) q^{37} + ( - 8284 \beta - 517096) q^{38} + (796 \beta + 93304) q^{40} + ( - 29712 \beta - 363062) q^{41} + ( - 7216 \beta - 848440) q^{43} + ( - 1331 \beta - 119790) q^{44} + ( - 7858 \beta - 446900) q^{46} + ( - 42642 \beta + 612368) q^{47} + (17556 \beta - 403647) q^{49} + (24581 \beta - 592088) q^{50} + ( - 3184 \beta - 589496) q^{52} + (26758 \beta + 1064818) q^{53} + ( - 18634 \beta - 119790) q^{55} + (646 \beta - 297788) q^{56} + (5374 \beta - 994952) q^{58} + ( - 76380 \beta - 425476) q^{59} + ( - 172662 \beta - 444666) q^{61} + (23928 \beta - 40112) q^{62} + ( - 37931 \beta - 718738) q^{64} + ( - 89036 \beta - 481804) q^{65} + (2392 \beta - 1631172) q^{67} + ( - 11162 \beta + 430380) q^{68} + (152104 \beta + 1526000) q^{70} + ( - 3574 \beta + 2749544) q^{71} + ( - 118912 \beta - 2665950) q^{73} + ( - 143598 \beta - 2856672) q^{74} + (221764 \beta + 1049176) q^{76} + (55902 \beta + 250228) q^{77} + ( - 43494 \beta - 746548) q^{79} + ( - 268692 \beta - 1600984) q^{80} + (392774 \beta + 6477216) q^{82} + (255344 \beta - 4553116) q^{83} + (54332 \beta - 79740) q^{85} + (855656 \beta + 1573088) q^{86} + ( - 49247 \beta + 290158) q^{88} + (72992 \beta - 3441562) q^{89} + (270224 \beta + 900768) q^{91} + (192358 \beta + 969620) q^{92} + ( - 569726 \beta + 9295956) q^{94} + (329456 \beta + 7771424) q^{95} + ( - 481620 \beta + 5189498) q^{97} + (386091 \beta - 3827208) q^{98} +O(q^{100})$$ q - b * q^2 + (b + 90) * q^4 + (14*b + 90) * q^5 + (-42*b - 188) * q^7 + (37*b - 218) * q^8 + (-104*b - 3052) * q^10 - 1331 * q^11 + (38*b - 6642) * q^13 + (230*b + 9156) * q^14 + (53*b - 19586) * q^16 + (-180*b + 5218) * q^17 + (2372*b + 5912) * q^19 + (1364*b + 11152) * q^20 + 1331*b * q^22 + (2050*b + 5808) * q^23 + (2716*b - 27297) * q^25 + (6604*b - 8284) * q^26 + (-4010*b - 26076) * q^28 + (4564*b - 9938) * q^29 + (184*b - 24112) * q^31 + (14797*b + 16350) * q^32 + (-5038*b + 39240) * q^34 + (-7000*b - 145104) * q^35 + (13104*b + 130494) * q^37 + (-8284*b - 517096) * q^38 + (796*b + 93304) * q^40 + (-29712*b - 363062) * q^41 + (-7216*b - 848440) * q^43 + (-1331*b - 119790) * q^44 + (-7858*b - 446900) * q^46 + (-42642*b + 612368) * q^47 + (17556*b - 403647) * q^49 + (24581*b - 592088) * q^50 + (-3184*b - 589496) * q^52 + (26758*b + 1064818) * q^53 + (-18634*b - 119790) * q^55 + (646*b - 297788) * q^56 + (5374*b - 994952) * q^58 + (-76380*b - 425476) * q^59 + (-172662*b - 444666) * q^61 + (23928*b - 40112) * q^62 + (-37931*b - 718738) * q^64 + (-89036*b - 481804) * q^65 + (2392*b - 1631172) * q^67 + (-11162*b + 430380) * q^68 + (152104*b + 1526000) * q^70 + (-3574*b + 2749544) * q^71 + (-118912*b - 2665950) * q^73 + (-143598*b - 2856672) * q^74 + (221764*b + 1049176) * q^76 + (55902*b + 250228) * q^77 + (-43494*b - 746548) * q^79 + (-268692*b - 1600984) * q^80 + (392774*b + 6477216) * q^82 + (255344*b - 4553116) * q^83 + (54332*b - 79740) * q^85 + (855656*b + 1573088) * q^86 + (-49247*b + 290158) * q^88 + (72992*b - 3441562) * q^89 + (270224*b + 900768) * q^91 + (192358*b + 969620) * q^92 + (-569726*b + 9295956) * q^94 + (329456*b + 7771424) * q^95 + (-481620*b + 5189498) * q^97 + (386091*b - 3827208) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 181 q^{4} + 194 q^{5} - 418 q^{7} - 399 q^{8}+O(q^{10})$$ 2 * q - q^2 + 181 * q^4 + 194 * q^5 - 418 * q^7 - 399 * q^8 $$2 q - q^{2} + 181 q^{4} + 194 q^{5} - 418 q^{7} - 399 q^{8} - 6208 q^{10} - 2662 q^{11} - 13246 q^{13} + 18542 q^{14} - 39119 q^{16} + 10256 q^{17} + 14196 q^{19} + 23668 q^{20} + 1331 q^{22} + 13666 q^{23} - 51878 q^{25} - 9964 q^{26} - 56162 q^{28} - 15312 q^{29} - 48040 q^{31} + 47497 q^{32} + 73442 q^{34} - 297208 q^{35} + 274092 q^{37} - 1042476 q^{38} + 187404 q^{40} - 755836 q^{41} - 1704096 q^{43} - 240911 q^{44} - 901658 q^{46} + 1182094 q^{47} - 789738 q^{49} - 1159595 q^{50} - 1182176 q^{52} + 2156394 q^{53} - 258214 q^{55} - 594930 q^{56} - 1984530 q^{58} - 927332 q^{59} - 1061994 q^{61} - 56296 q^{62} - 1475407 q^{64} - 1052644 q^{65} - 3259952 q^{67} + 849598 q^{68} + 3204104 q^{70} + 5495514 q^{71} - 5450812 q^{73} - 5856942 q^{74} + 2320116 q^{76} + 556358 q^{77} - 1536590 q^{79} - 3470660 q^{80} + 13347206 q^{82} - 8850888 q^{83} - 105148 q^{85} + 4001832 q^{86} + 531069 q^{88} - 6810132 q^{89} + 2071760 q^{91} + 2131598 q^{92} + 18022186 q^{94} + 15872304 q^{95} + 9897376 q^{97} - 7268325 q^{98}+O(q^{100})$$ 2 * q - q^2 + 181 * q^4 + 194 * q^5 - 418 * q^7 - 399 * q^8 - 6208 * q^10 - 2662 * q^11 - 13246 * q^13 + 18542 * q^14 - 39119 * q^16 + 10256 * q^17 + 14196 * q^19 + 23668 * q^20 + 1331 * q^22 + 13666 * q^23 - 51878 * q^25 - 9964 * q^26 - 56162 * q^28 - 15312 * q^29 - 48040 * q^31 + 47497 * q^32 + 73442 * q^34 - 297208 * q^35 + 274092 * q^37 - 1042476 * q^38 + 187404 * q^40 - 755836 * q^41 - 1704096 * q^43 - 240911 * q^44 - 901658 * q^46 + 1182094 * q^47 - 789738 * q^49 - 1159595 * q^50 - 1182176 * q^52 + 2156394 * q^53 - 258214 * q^55 - 594930 * q^56 - 1984530 * q^58 - 927332 * q^59 - 1061994 * q^61 - 56296 * q^62 - 1475407 * q^64 - 1052644 * q^65 - 3259952 * q^67 + 849598 * q^68 + 3204104 * q^70 + 5495514 * q^71 - 5450812 * q^73 - 5856942 * q^74 + 2320116 * q^76 + 556358 * q^77 - 1536590 * q^79 - 3470660 * q^80 + 13347206 * q^82 - 8850888 * q^83 - 105148 * q^85 + 4001832 * q^86 + 531069 * q^88 - 6810132 * q^89 + 2071760 * q^91 + 2131598 * q^92 + 18022186 * q^94 + 15872304 * q^95 + 9897376 * q^97 - 7268325 * 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
 5.42443 −4.42443
−15.2733 0 105.273 303.826 0 −829.478 347.112 0 −4640.42
1.2 14.2733 0 75.7267 −109.826 0 411.478 −746.112 0 −1567.58
 $$n$$: 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.b 2
3.b odd 2 1 33.8.a.c 2
12.b even 2 1 528.8.a.h 2
33.d even 2 1 363.8.a.c 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 218$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(99))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 218$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 194T - 33368$$
$7$ $$T^{2} + 418T - 341312$$
$11$ $$(T + 1331)^{2}$$
$13$ $$T^{2} + 13246 T + 43548976$$
$17$ $$T^{2} - 10256 T + 19225084$$
$19$ $$T^{2} - 14196 T - 1177576704$$
$23$ $$T^{2} - 13666 T - 870505736$$
$29$ $$T^{2} + 15312 T - 4487554116$$
$31$ $$T^{2} + 48040 T + 569571328$$
$37$ $$T^{2} - 274092 T - 18695152476$$
$41$ $$T^{2} + 755836 T - 49849727804$$
$43$ $$T^{2} + 1704096 T + 714621373632$$
$47$ $$T^{2} - 1182094 T - 47516184584$$
$53$ $$T^{2} - 2156394 T + 1006243830216$$
$59$ $$T^{2} + 927332 T - 1058263475744$$
$61$ $$T^{2} + 1061994 T - 6224547468744$$
$67$ $$T^{2} + 3259952 T + 2655573007408$$
$71$ $$T^{2} - 5495514 T + 7547380719912$$
$73$ $$T^{2} + 5450812 T + 4341768952708$$
$79$ $$T^{2} + 1536590 T + 177407563168$$
$83$ $$T^{2} + 8850888 T + 5354532740304$$
$89$ $$T^{2} + 6810132 T + 10431675116388$$
$97$ $$T^{2} - 9897376 T - 26135282253956$$