# Properties

 Label 99.8.f.c 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 22 q^{2} - 692 q^{4} + 777 q^{5} - 83 q^{7} - 3568 q^{8}+O(q^{10})$$ 28 * q + 22 * q^2 - 692 * q^4 + 777 * q^5 - 83 * q^7 - 3568 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 22 q^{2} - 692 q^{4} + 777 q^{5} - 83 q^{7} - 3568 q^{8} - 1352 q^{10} + 18851 q^{11} + 14635 q^{13} - 62912 q^{14} - 36148 q^{16} + 10338 q^{17} + 90342 q^{19} + 289719 q^{20} - 202339 q^{22} - 138508 q^{23} - 174068 q^{25} - 466130 q^{26} - 640233 q^{28} + 487290 q^{29} - 388352 q^{31} - 2726322 q^{32} - 1543708 q^{34} - 742659 q^{35} + 670209 q^{37} - 2282072 q^{38} + 4641741 q^{40} + 2721357 q^{41} - 199784 q^{43} + 5887437 q^{44} + 3389447 q^{46} - 2112053 q^{47} + 3927748 q^{49} + 3663090 q^{50} - 4343053 q^{52} + 1743854 q^{53} - 8387710 q^{55} + 6307818 q^{56} + 546605 q^{58} + 218820 q^{59} - 743157 q^{61} - 3996183 q^{62} - 21725666 q^{64} - 4895794 q^{65} - 3235686 q^{67} - 961671 q^{68} + 31155916 q^{70} - 9848757 q^{71} - 6667768 q^{73} + 31115192 q^{74} - 6535748 q^{76} - 8320585 q^{77} - 10121851 q^{79} - 44592526 q^{80} + 24927115 q^{82} - 13288897 q^{83} - 28050005 q^{85} - 20937351 q^{86} + 39687014 q^{88} + 40241460 q^{89} + 8733043 q^{91} - 11607651 q^{92} - 7877385 q^{94} - 14371097 q^{95} + 30518151 q^{97} + 16954218 q^{98}+O(q^{100})$$ 28 * q + 22 * q^2 - 692 * q^4 + 777 * q^5 - 83 * q^7 - 3568 * q^8 - 1352 * q^10 + 18851 * q^11 + 14635 * q^13 - 62912 * q^14 - 36148 * q^16 + 10338 * q^17 + 90342 * q^19 + 289719 * q^20 - 202339 * q^22 - 138508 * q^23 - 174068 * q^25 - 466130 * q^26 - 640233 * q^28 + 487290 * q^29 - 388352 * q^31 - 2726322 * q^32 - 1543708 * q^34 - 742659 * q^35 + 670209 * q^37 - 2282072 * q^38 + 4641741 * q^40 + 2721357 * q^41 - 199784 * q^43 + 5887437 * q^44 + 3389447 * q^46 - 2112053 * q^47 + 3927748 * q^49 + 3663090 * q^50 - 4343053 * q^52 + 1743854 * q^53 - 8387710 * q^55 + 6307818 * q^56 + 546605 * q^58 + 218820 * q^59 - 743157 * q^61 - 3996183 * q^62 - 21725666 * q^64 - 4895794 * q^65 - 3235686 * q^67 - 961671 * q^68 + 31155916 * q^70 - 9848757 * q^71 - 6667768 * q^73 + 31115192 * q^74 - 6535748 * q^76 - 8320585 * q^77 - 10121851 * q^79 - 44592526 * q^80 + 24927115 * q^82 - 13288897 * q^83 - 28050005 * q^85 - 20937351 * q^86 + 39687014 * q^88 + 40241460 * q^89 + 8733043 * q^91 - 11607651 * q^92 - 7877385 * q^94 - 14371097 * q^95 + 30518151 * q^97 + 16954218 * 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 −15.4308 11.2111i 0 72.8662 + 224.259i −0.205524 + 0.149322i 0 242.486 + 746.295i 635.379 1955.49i 0 4.84548
37.2 −8.90518 6.46999i 0 −2.11276 6.50241i 328.988 239.024i 0 −385.790 1187.34i −458.645 + 1411.56i 0 −4476.18
37.3 −6.67280 4.84807i 0 −18.5317 57.0347i −269.190 + 195.578i 0 −36.4929 112.314i −479.095 + 1474.50i 0 2744.43
37.4 2.14212 + 1.55634i 0 −37.3877 115.067i −121.253 + 88.0954i 0 231.374 + 712.095i 203.727 627.009i 0 −396.845
37.5 8.43584 + 6.12900i 0 −5.95538 18.3288i 186.461 135.472i 0 −43.6776 134.426i 474.540 1460.49i 0 2403.26
37.6 12.1414 + 8.82122i 0 30.0447 + 92.4681i −158.638 + 115.257i 0 −372.740 1147.18i 142.714 439.227i 0 −2942.80
37.7 18.2616 + 13.2678i 0 117.897 + 362.849i 299.083 217.296i 0 233.965 + 720.070i −1768.39 + 5442.55i 0 8344.78
64.1 −6.71359 + 20.6623i 0 −278.304 202.200i 94.4035 + 290.544i 0 945.004 + 686.586i 3796.54 2758.35i 0 −6637.09
64.2 −3.92793 + 12.0889i 0 −27.1593 19.7324i 17.6507 + 54.3232i 0 −962.124 699.024i −971.059 + 705.516i 0 −726.039
64.3 −2.98016 + 9.17200i 0 28.3100 + 20.5684i −33.3401 102.610i 0 1247.17 + 906.120i −1271.70 + 923.944i 0 1040.50
64.4 0.907839 2.79404i 0 96.5717 + 70.1634i −61.3322 188.761i 0 −198.219 144.015i 587.936 427.160i 0 −583.086
64.5 1.78222 5.48510i 0 76.6441 + 55.6852i 142.765 + 439.384i 0 −506.310 367.856i 1039.27 755.075i 0 2664.50
64.6 5.89947 18.1567i 0 −191.308 138.993i −133.778 411.726i 0 −1338.24 972.291i −1675.32 + 1217.19i 0 −8264.81
64.7 6.06001 18.6508i 0 −207.575 150.812i 96.8862 + 298.185i 0 902.102 + 655.415i −2039.90 + 1482.08i 0 6148.53
82.1 −6.71359 20.6623i 0 −278.304 + 202.200i 94.4035 290.544i 0 945.004 686.586i 3796.54 + 2758.35i 0 −6637.09
82.2 −3.92793 12.0889i 0 −27.1593 + 19.7324i 17.6507 54.3232i 0 −962.124 + 699.024i −971.059 705.516i 0 −726.039
82.3 −2.98016 9.17200i 0 28.3100 20.5684i −33.3401 + 102.610i 0 1247.17 906.120i −1271.70 923.944i 0 1040.50
82.4 0.907839 + 2.79404i 0 96.5717 70.1634i −61.3322 + 188.761i 0 −198.219 + 144.015i 587.936 + 427.160i 0 −583.086
82.5 1.78222 + 5.48510i 0 76.6441 55.6852i 142.765 439.384i 0 −506.310 + 367.856i 1039.27 + 755.075i 0 2664.50
82.6 5.89947 + 18.1567i 0 −191.308 + 138.993i −133.778 + 411.726i 0 −1338.24 + 972.291i −1675.32 1217.19i 0 −8264.81
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.c 28
3.b odd 2 1 33.8.e.a 28
11.c even 5 1 inner 99.8.f.c 28
33.f even 10 1 363.8.a.p 14
33.h odd 10 1 33.8.e.a 28
33.h odd 10 1 363.8.a.q 14

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