# Properties

 Label 99.8.f.d Level $99$ Weight $8$ Character orbit 99.f Analytic conductor $30.926$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# 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: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 1260 q^{4} - 3902 q^{7}+O(q^{10})$$ 56 * q - 1260 * q^4 - 3902 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 1260 q^{4} - 3902 q^{7} + 20008 q^{10} - 18158 q^{13} + 54064 q^{16} - 111812 q^{19} + 16106 q^{22} - 78708 q^{25} + 69378 q^{28} - 189868 q^{31} + 1561016 q^{34} - 795786 q^{37} - 1589918 q^{40} + 355744 q^{43} + 2273926 q^{46} - 4728752 q^{49} - 438038 q^{52} + 3414760 q^{55} + 4154034 q^{58} + 1604778 q^{61} + 9561960 q^{64} - 18424972 q^{67} - 5609964 q^{70} + 17372 q^{73} + 8053960 q^{76} - 15803082 q^{79} + 7756570 q^{82} - 21218378 q^{85} + 37593088 q^{88} + 18308922 q^{91} + 68235162 q^{94} + 34988922 q^{97}+O(q^{100})$$ 56 * q - 1260 * q^4 - 3902 * q^7 + 20008 * q^10 - 18158 * q^13 + 54064 * q^16 - 111812 * q^19 + 16106 * q^22 - 78708 * q^25 + 69378 * q^28 - 189868 * q^31 + 1561016 * q^34 - 795786 * q^37 - 1589918 * q^40 + 355744 * q^43 + 2273926 * q^46 - 4728752 * q^49 - 438038 * q^52 + 3414760 * q^55 + 4154034 * q^58 + 1604778 * q^61 + 9561960 * q^64 - 18424972 * q^67 - 5609964 * q^70 + 17372 * q^73 + 8053960 * q^76 - 15803082 * q^79 + 7756570 * q^82 - 21218378 * q^85 + 37593088 * q^88 + 18308922 * q^91 + 68235162 * q^94 + 34988922 * q^97

## 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 −16.2902 11.8355i 0 85.7365 + 263.870i 21.6619 15.7383i 0 176.688 + 543.790i 929.915 2861.99i 0 −539.148
37.2 −14.6343 10.6324i 0 61.5594 + 189.460i −142.204 + 103.317i 0 −426.560 1312.82i 398.052 1225.08i 0 3179.57
37.3 −10.8132 7.85623i 0 15.6502 + 48.1663i 386.114 280.528i 0 41.4630 + 127.610i −319.496 + 983.309i 0 −6379.01
37.4 −10.1078 7.34373i 0 8.68266 + 26.7225i −401.640 + 291.809i 0 534.412 + 1644.75i −385.706 + 1187.08i 0 6202.66
37.5 −7.75695 5.63576i 0 −11.1456 34.3026i 103.730 75.3643i 0 −38.0759 117.185i −486.116 + 1496.11i 0 −1229.36
37.6 −4.74849 3.44998i 0 −28.9084 88.9708i −314.728 + 228.663i 0 −446.571 1374.41i −401.838 + 1236.73i 0 2283.36
37.7 −1.39520 1.01367i 0 −38.6351 118.907i 60.6814 44.0876i 0 164.506 + 506.297i −134.842 + 415.001i 0 −129.353
37.8 1.39520 + 1.01367i 0 −38.6351 118.907i −60.6814 + 44.0876i 0 164.506 + 506.297i 134.842 415.001i 0 −129.353
37.9 4.74849 + 3.44998i 0 −28.9084 88.9708i 314.728 228.663i 0 −446.571 1374.41i 401.838 1236.73i 0 2283.36
37.10 7.75695 + 5.63576i 0 −11.1456 34.3026i −103.730 + 75.3643i 0 −38.0759 117.185i 486.116 1496.11i 0 −1229.36
37.11 10.1078 + 7.34373i 0 8.68266 + 26.7225i 401.640 291.809i 0 534.412 + 1644.75i 385.706 1187.08i 0 6202.66
37.12 10.8132 + 7.85623i 0 15.6502 + 48.1663i −386.114 + 280.528i 0 41.4630 + 127.610i 319.496 983.309i 0 −6379.01
37.13 14.6343 + 10.6324i 0 61.5594 + 189.460i 142.204 103.317i 0 −426.560 1312.82i −398.052 + 1225.08i 0 3179.57
37.14 16.2902 + 11.8355i 0 85.7365 + 263.870i −21.6619 + 15.7383i 0 176.688 + 543.790i −929.915 + 2861.99i 0 −539.148
64.1 −6.77090 + 20.8387i 0 −284.852 206.957i −58.5463 180.187i 0 −846.159 614.770i 3972.44 2886.14i 0 4151.28
64.2 −5.41946 + 16.6794i 0 −145.277 105.550i 15.0294 + 46.2557i 0 950.111 + 690.296i 731.727 531.631i 0 −852.968
64.3 −5.18142 + 15.9468i 0 −123.898 90.0173i 117.380 + 361.258i 0 150.428 + 109.292i 341.116 247.835i 0 −6369.09
64.4 −4.06180 + 12.5009i 0 −36.2213 26.3163i −52.8659 162.704i 0 −783.627 569.338i −885.042 + 643.021i 0 2248.69
64.5 −3.21612 + 9.89821i 0 15.9231 + 11.5688i −154.473 475.419i 0 455.594 + 331.009i −1243.47 + 903.434i 0 5202.60
64.6 −1.80708 + 5.56161i 0 75.8882 + 55.1360i 114.934 + 353.730i 0 −1344.69 976.976i −1049.35 + 762.395i 0 −2175.00
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.f.d 56
3.b odd 2 1 inner 99.8.f.d 56
11.c even 5 1 inner 99.8.f.d 56
33.h odd 10 1 inner 99.8.f.d 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.f.d 56 1.a even 1 1 trivial
99.8.f.d 56 3.b odd 2 1 inner
99.8.f.d 56 11.c even 5 1 inner
99.8.f.d 56 33.h odd 10 1 inner