# Properties

 Label 99.8.f.b Level $99$ Weight $8$ Character orbit 99.f Analytic conductor $30.926$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,8,Mod(37,99)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("99.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.f (of order $$5$$, degree $$4$$, minimal)

## 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: no Analytic conductor: $$30.9261175229$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 6 q^{2} + 160 q^{4} - 773 q^{5} + 1289 q^{7} - 2956 q^{8}+O(q^{10})$$ 28 * q + 6 * q^2 + 160 * q^4 - 773 * q^5 + 1289 * q^7 - 2956 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 6 q^{2} + 160 q^{4} - 773 q^{5} + 1289 q^{7} - 2956 q^{8} - 10640 q^{10} - 13209 q^{11} + 13499 q^{13} + 3318 q^{14} - 113196 q^{16} - 30296 q^{17} - 6858 q^{19} - 76725 q^{20} - 48859 q^{22} + 166984 q^{23} + 99832 q^{25} - 340424 q^{26} + 657387 q^{28} - 192014 q^{29} + 396068 q^{31} + 303498 q^{32} - 1229124 q^{34} + 725859 q^{35} + 811053 q^{37} - 1548798 q^{38} - 1427263 q^{40} - 1041497 q^{41} - 5265288 q^{43} + 5341085 q^{44} + 1198651 q^{46} - 1667505 q^{47} + 2968140 q^{49} - 11247432 q^{50} + 2325763 q^{52} + 1625736 q^{53} - 8338690 q^{55} + 18201906 q^{56} + 14965553 q^{58} + 2587454 q^{59} + 5801619 q^{61} - 15631121 q^{62} - 3732846 q^{64} + 15368174 q^{65} - 7141262 q^{67} - 17394545 q^{68} - 4329200 q^{70} - 3569199 q^{71} + 125008 q^{73} + 13691530 q^{74} - 19690428 q^{76} + 6038739 q^{77} + 17075485 q^{79} - 26606436 q^{80} + 19886007 q^{82} + 4645575 q^{83} + 12002715 q^{85} + 12296287 q^{86} - 50905990 q^{88} - 9601664 q^{89} - 14129477 q^{91} + 90776021 q^{92} + 1047679 q^{94} - 58935661 q^{95} - 22170261 q^{97} - 140532178 q^{98}+O(q^{100})$$ 28 * q + 6 * q^2 + 160 * q^4 - 773 * q^5 + 1289 * q^7 - 2956 * q^8 - 10640 * q^10 - 13209 * q^11 + 13499 * q^13 + 3318 * q^14 - 113196 * q^16 - 30296 * q^17 - 6858 * q^19 - 76725 * q^20 - 48859 * q^22 + 166984 * q^23 + 99832 * q^25 - 340424 * q^26 + 657387 * q^28 - 192014 * q^29 + 396068 * q^31 + 303498 * q^32 - 1229124 * q^34 + 725859 * q^35 + 811053 * q^37 - 1548798 * q^38 - 1427263 * q^40 - 1041497 * q^41 - 5265288 * q^43 + 5341085 * q^44 + 1198651 * q^46 - 1667505 * q^47 + 2968140 * q^49 - 11247432 * q^50 + 2325763 * q^52 + 1625736 * q^53 - 8338690 * q^55 + 18201906 * q^56 + 14965553 * q^58 + 2587454 * q^59 + 5801619 * q^61 - 15631121 * q^62 - 3732846 * q^64 + 15368174 * q^65 - 7141262 * q^67 - 17394545 * q^68 - 4329200 * q^70 - 3569199 * q^71 + 125008 * q^73 + 13691530 * q^74 - 19690428 * q^76 + 6038739 * q^77 + 17075485 * q^79 - 26606436 * q^80 + 19886007 * q^82 + 4645575 * q^83 + 12002715 * q^85 + 12296287 * q^86 - 50905990 * q^88 - 9601664 * q^89 - 14129477 * q^91 + 90776021 * q^92 + 1047679 * q^94 - 58935661 * q^95 - 22170261 * q^97 - 140532178 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −16.9629 12.3242i 0 96.2976 + 296.374i 380.493 276.444i 0 39.7826 + 122.438i 1189.75 3661.68i 0 −9861.22
37.2 −14.6841 10.6686i 0 62.2484 + 191.581i −425.412 + 309.080i 0 −184.854 568.921i 411.912 1267.74i 0 9544.22
37.3 −7.82727 5.68684i 0 −10.6283 32.7104i 100.934 73.3327i 0 −177.127 545.140i −485.517 + 1494.27i 0 −1207.07
37.4 3.24396 + 2.35687i 0 −34.5858 106.444i −354.050 + 257.233i 0 116.729 + 359.256i 297.283 914.942i 0 −1754.79
37.5 3.93160 + 2.85647i 0 −32.2561 99.2742i 221.480 160.915i 0 −146.721 451.561i 348.978 1074.04i 0 1330.42
37.6 12.1293 + 8.81245i 0 29.9064 + 92.0423i 3.10808 2.25815i 0 300.631 + 925.246i 144.646 445.175i 0 57.5987
37.7 17.1972 + 12.4945i 0 100.077 + 308.005i −190.798 + 138.623i 0 −503.289 1548.96i −1286.53 + 3959.52i 0 −5013.22
64.1 −4.95371 + 15.2460i 0 −104.346 75.8118i −99.6953 306.831i 0 803.319 + 583.645i 12.6937 9.22251i 0 5171.79
64.2 −2.75376 + 8.47521i 0 39.3082 + 28.5591i 42.2440 + 130.014i 0 29.1631 + 21.1883i −1273.10 + 924.960i 0 −1218.22
64.3 −0.813458 + 2.50357i 0 97.9481 + 71.1634i −78.7070 242.235i 0 −1135.08 824.681i −530.435 + 385.384i 0 670.476
64.4 0.970201 2.98597i 0 95.5794 + 69.4425i 100.399 + 308.998i 0 1340.18 + 973.701i 625.207 454.240i 0 1020.07
64.5 2.99223 9.20914i 0 27.6994 + 20.1248i 60.2993 + 185.582i 0 −765.654 556.280i 1270.94 923.389i 0 1889.48
64.6 4.06039 12.4966i 0 −36.1242 26.2458i −158.380 487.442i 0 846.952 + 615.347i 886.010 643.724i 0 −6734.46
64.7 6.47025 19.9134i 0 −251.124 182.452i 11.5844 + 35.6531i 0 80.4609 + 58.4583i −3089.84 + 2244.90i 0 784.928
82.1 −4.95371 15.2460i 0 −104.346 + 75.8118i −99.6953 + 306.831i 0 803.319 583.645i 12.6937 + 9.22251i 0 5171.79
82.2 −2.75376 8.47521i 0 39.3082 28.5591i 42.2440 130.014i 0 29.1631 21.1883i −1273.10 924.960i 0 −1218.22
82.3 −0.813458 2.50357i 0 97.9481 71.1634i −78.7070 + 242.235i 0 −1135.08 + 824.681i −530.435 385.384i 0 670.476
82.4 0.970201 + 2.98597i 0 95.5794 69.4425i 100.399 308.998i 0 1340.18 973.701i 625.207 + 454.240i 0 1020.07
82.5 2.99223 + 9.20914i 0 27.6994 20.1248i 60.2993 185.582i 0 −765.654 + 556.280i 1270.94 + 923.389i 0 1889.48
82.6 4.06039 + 12.4966i 0 −36.1242 + 26.2458i −158.380 + 487.442i 0 846.952 615.347i 886.010 + 643.724i 0 −6734.46
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.f.b 28
3.b odd 2 1 33.8.e.b 28
11.c even 5 1 inner 99.8.f.b 28
33.f even 10 1 363.8.a.r 14
33.h odd 10 1 33.8.e.b 28
33.h odd 10 1 363.8.a.o 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.e.b 28 3.b odd 2 1
33.8.e.b 28 33.h odd 10 1
99.8.f.b 28 1.a even 1 1 trivial
99.8.f.b 28 11.c even 5 1 inner
363.8.a.o 14 33.h odd 10 1
363.8.a.r 14 33.f even 10 1