# Properties

 Label 99.8.j.a Level $99$ Weight $8$ Character orbit 99.j Analytic conductor $30.926$ Analytic rank $0$ Dimension $112$ 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(8,99)]

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([5, 3]))

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

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

chi := DirichletCharacter("99.8");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.j (of order $$10$$, 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: $$112$$ Relative dimension: $$28$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 1792 q^{4}+O(q^{10})$$ 112 * q - 1792 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 1792 q^{4} - 134096 q^{16} + 401484 q^{22} - 68552 q^{25} + 1493020 q^{28} - 398144 q^{31} - 729944 q^{34} + 685476 q^{37} - 399360 q^{40} - 1410880 q^{46} + 2923872 q^{49} + 6472520 q^{52} + 1445488 q^{55} + 13215936 q^{58} - 7843440 q^{61} - 12806712 q^{64} + 1864032 q^{67} - 1233728 q^{70} + 53841940 q^{73} - 53845440 q^{79} - 36360204 q^{82} + 41703500 q^{85} + 21474024 q^{88} + 27611736 q^{91} - 94707560 q^{94} - 27695460 q^{97}+O(q^{100})$$ 112 * q - 1792 * q^4 - 134096 * q^16 + 401484 * q^22 - 68552 * q^25 + 1493020 * q^28 - 398144 * q^31 - 729944 * q^34 + 685476 * q^37 - 399360 * q^40 - 1410880 * q^46 + 2923872 * q^49 + 6472520 * q^52 + 1445488 * q^55 + 13215936 * q^58 - 7843440 * q^61 - 12806712 * q^64 + 1864032 * q^67 - 1233728 * q^70 + 53841940 * q^73 - 53845440 * q^79 - 36360204 * q^82 + 41703500 * q^85 + 21474024 * q^88 + 27611736 * q^91 - 94707560 * q^94 - 27695460 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −17.5311 + 12.7371i 0 105.552 324.855i −159.065 + 218.934i 0 −1136.60 369.305i 1430.14 + 4401.52i 0 5864.17i
8.2 −16.8891 + 12.2707i 0 95.1191 292.746i −112.238 + 154.483i 0 −161.473 52.4657i 1159.98 + 3570.05i 0 3986.32i
8.3 −16.1346 + 11.7225i 0 83.3550 256.540i 233.551 321.456i 0 1297.08 + 421.447i 873.541 + 2688.48i 0 7924.37i
8.4 −14.7879 + 10.7440i 0 63.6930 196.027i 194.917 268.280i 0 −72.5991 23.5889i 441.228 + 1357.96i 0 6061.48i
8.5 −12.2382 + 8.89156i 0 31.1592 95.8980i 48.6403 66.9477i 0 −28.8309 9.36773i −126.993 390.846i 0 1251.81i
8.6 −11.6523 + 8.46591i 0 24.5509 75.5599i −325.140 + 447.517i 0 1250.84 + 406.423i −216.093 665.067i 0 7967.22i
8.7 −11.3691 + 8.26015i 0 21.4727 66.0862i −96.0085 + 132.144i 0 553.787 + 179.936i −254.099 782.037i 0 2295.41i
8.8 −10.9960 + 7.98908i 0 17.5330 53.9610i 47.9158 65.9504i 0 −905.136 294.096i −299.308 921.176i 0 1108.00i
8.9 −7.62018 + 5.53639i 0 −12.1386 + 37.3586i −208.552 + 287.047i 0 −1358.87 441.525i −486.897 1498.52i 0 3341.97i
8.10 −6.50881 + 4.72893i 0 −19.5523 + 60.1758i 255.658 351.883i 0 −495.513 161.002i −475.531 1463.54i 0 3499.33i
8.11 −4.82032 + 3.50217i 0 −28.5839 + 87.9721i −17.1626 + 23.6223i 0 1682.17 + 546.570i −405.983 1249.49i 0 173.973i
8.12 −4.02244 + 2.92248i 0 −31.9150 + 98.2243i −62.7155 + 86.3205i 0 522.505 + 169.772i −355.346 1093.64i 0 530.504i
8.13 −3.14320 + 2.28367i 0 −34.8896 + 107.379i 310.082 426.791i 0 354.776 + 115.274i −289.230 890.158i 0 2049.61i
8.14 −1.51130 + 1.09802i 0 −38.4758 + 118.416i −76.1409 + 104.799i 0 −748.574 243.227i −145.766 448.620i 0 241.988i
8.15 1.51130 1.09802i 0 −38.4758 + 118.416i 76.1409 104.799i 0 −748.574 243.227i 145.766 + 448.620i 0 241.988i
8.16 3.14320 2.28367i 0 −34.8896 + 107.379i −310.082 + 426.791i 0 354.776 + 115.274i 289.230 + 890.158i 0 2049.61i
8.17 4.02244 2.92248i 0 −31.9150 + 98.2243i 62.7155 86.3205i 0 522.505 + 169.772i 355.346 + 1093.64i 0 530.504i
8.18 4.82032 3.50217i 0 −28.5839 + 87.9721i 17.1626 23.6223i 0 1682.17 + 546.570i 405.983 + 1249.49i 0 173.973i
8.19 6.50881 4.72893i 0 −19.5523 + 60.1758i −255.658 + 351.883i 0 −495.513 161.002i 475.531 + 1463.54i 0 3499.33i
8.20 7.62018 5.53639i 0 −12.1386 + 37.3586i 208.552 287.047i 0 −1358.87 441.525i 486.897 + 1498.52i 0 3341.97i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.28 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.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.j.a 112
3.b odd 2 1 inner 99.8.j.a 112
11.d odd 10 1 inner 99.8.j.a 112
33.f even 10 1 inner 99.8.j.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.j.a 112 1.a even 1 1 trivial
99.8.j.a 112 3.b odd 2 1 inner
99.8.j.a 112 11.d odd 10 1 inner
99.8.j.a 112 33.f even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(99, [\chi])$$.