# Properties

 Label 99.8.p.a Level $99$ Weight $8$ Character orbit 99.p Analytic conductor $30.926$ Analytic rank $0$ Dimension $656$ 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(2,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(30))

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("99.2");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.p (of order $$30$$, degree $$8$$, 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: $$656$$ Relative dimension: $$82$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$656 q - 15 q^{2} + 33 q^{3} + 5117 q^{4} - 222 q^{5} + 1710 q^{6} - 5 q^{7} - 5725 q^{9}+O(q^{10})$$ 656 * q - 15 * q^2 + 33 * q^3 + 5117 * q^4 - 222 * q^5 + 1710 * q^6 - 5 * q^7 - 5725 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$656 q - 15 q^{2} + 33 q^{3} + 5117 q^{4} - 222 q^{5} + 1710 q^{6} - 5 q^{7} - 5725 q^{9} + 9393 q^{11} - 27942 q^{12} - 5 q^{13} - 9 q^{14} + 22800 q^{15} + 311549 q^{16} + 120870 q^{18} - 10490 q^{19} + 55671 q^{20} - 35516 q^{22} - 140658 q^{23} - 330890 q^{24} - 1192250 q^{25} - 23433 q^{27} - 20 q^{28} + 450390 q^{29} - 210220 q^{30} - 99330 q^{31} - 343484 q^{33} + 177704 q^{34} + 177022 q^{36} + 151770 q^{37} + 4281453 q^{38} + 3829465 q^{39} + 81915 q^{40} - 15 q^{41} - 591509 q^{42} + 4078212 q^{45} - 1300 q^{46} + 986667 q^{47} + 6404620 q^{48} - 8235433 q^{49} + 1171860 q^{50} - 53985 q^{51} - 5 q^{52} + 1333566 q^{55} + 2569734 q^{56} + 12786480 q^{57} - 1038899 q^{58} + 9215937 q^{59} + 14658901 q^{60} - 5 q^{61} - 459885 q^{63} - 35716364 q^{64} - 6836247 q^{66} - 8049662 q^{67} - 17851680 q^{68} + 23301904 q^{69} - 1622346 q^{70} + 28129145 q^{72} - 20 q^{73} - 15 q^{74} + 37489194 q^{75} - 46153767 q^{77} - 37220872 q^{78} - 5 q^{79} - 31488949 q^{81} - 5685788 q^{82} + 40067400 q^{83} + 36520905 q^{84} - 5 q^{85} + 21934239 q^{86} + 25796812 q^{88} + 6237200 q^{90} + 28718692 q^{91} - 50648541 q^{92} + 32879354 q^{93} - 5 q^{94} + 48932505 q^{95} + 157922135 q^{96} + 20930877 q^{97} + 165749363 q^{99}+O(q^{100})$$ 656 * q - 15 * q^2 + 33 * q^3 + 5117 * q^4 - 222 * q^5 + 1710 * q^6 - 5 * q^7 - 5725 * q^9 + 9393 * q^11 - 27942 * q^12 - 5 * q^13 - 9 * q^14 + 22800 * q^15 + 311549 * q^16 + 120870 * q^18 - 10490 * q^19 + 55671 * q^20 - 35516 * q^22 - 140658 * q^23 - 330890 * q^24 - 1192250 * q^25 - 23433 * q^27 - 20 * q^28 + 450390 * q^29 - 210220 * q^30 - 99330 * q^31 - 343484 * q^33 + 177704 * q^34 + 177022 * q^36 + 151770 * q^37 + 4281453 * q^38 + 3829465 * q^39 + 81915 * q^40 - 15 * q^41 - 591509 * q^42 + 4078212 * q^45 - 1300 * q^46 + 986667 * q^47 + 6404620 * q^48 - 8235433 * q^49 + 1171860 * q^50 - 53985 * q^51 - 5 * q^52 + 1333566 * q^55 + 2569734 * q^56 + 12786480 * q^57 - 1038899 * q^58 + 9215937 * q^59 + 14658901 * q^60 - 5 * q^61 - 459885 * q^63 - 35716364 * q^64 - 6836247 * q^66 - 8049662 * q^67 - 17851680 * q^68 + 23301904 * q^69 - 1622346 * q^70 + 28129145 * q^72 - 20 * q^73 - 15 * q^74 + 37489194 * q^75 - 46153767 * q^77 - 37220872 * q^78 - 5 * q^79 - 31488949 * q^81 - 5685788 * q^82 + 40067400 * q^83 + 36520905 * q^84 - 5 * q^85 + 21934239 * q^86 + 25796812 * q^88 + 6237200 * q^90 + 28718692 * q^91 - 50648541 * q^92 + 32879354 * q^93 - 5 * q^94 + 48932505 * q^95 + 157922135 * q^96 + 20930877 * q^97 + 165749363 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −14.9125 16.5620i 23.9237 + 40.1828i −38.5380 + 366.665i 93.4482 + 84.1412i 308.747 995.452i −449.229 1008.99i 4339.57 3152.88i −1042.32 + 1922.64i 2802.45i
2.2 −14.6325 16.2510i −20.0612 42.2439i −36.6065 + 348.288i 320.415 + 288.503i −392.962 + 944.149i −139.258 312.779i 3931.17 2856.16i −1382.10 + 1694.93i 9428.61i
2.3 −13.9932 15.5410i 23.0768 40.6751i −32.3340 + 307.638i −175.681 158.184i −955.050 + 210.537i 401.387 + 901.531i 3067.88 2228.95i −1121.92 1877.30i 4943.76i
2.4 −13.8477 15.3794i −6.18102 + 46.3551i −31.3882 + 298.638i −385.191 346.827i 798.506 546.849i 426.349 + 957.595i 2884.48 2095.69i −2110.59 573.044i 10726.7i
2.5 −13.7014 15.2170i −46.7619 0.572460i −30.4476 + 289.689i −28.5472 25.7040i 631.993 + 719.417i 317.879 + 713.967i 2704.95 1965.26i 2186.34 + 53.5386i 786.584i
2.6 −13.6703 15.1824i −36.3179 29.4620i −30.2485 + 287.795i −380.261 342.388i 49.1736 + 954.145i −597.873 1342.85i 2667.32 1937.92i 450.986 + 2140.00i 10453.8i
2.7 −13.3036 14.7751i −37.8231 + 27.5029i −27.9393 + 265.825i 161.103 + 145.057i 909.541 + 192.953i 66.4787 + 149.314i 2240.43 1627.77i 674.176 2080.49i 4310.09i
2.8 −13.1393 14.5927i 46.7654 0.0245390i −26.9252 + 256.176i −101.502 91.3931i −614.822 682.109i −218.528 490.822i 2058.64 1495.69i 2187.00 2.29515i 2682.03i
2.9 −12.9692 14.4038i 45.4800 10.8888i −25.8885 + 246.312i 373.691 + 336.473i −746.680 513.865i 518.645 + 1164.89i 1876.47 1363.34i 1949.87 990.443i 9746.34i
2.10 −12.3381 13.7029i 18.8615 42.7930i −22.1599 + 210.838i −25.2090 22.6983i −819.104 + 269.529i −262.264 589.054i 1253.06 910.399i −1475.49 1614.28i 625.491i
2.11 −11.8251 13.1331i 31.5876 + 34.4851i −19.2659 + 183.303i 79.1333 + 71.2519i 79.3692 822.635i 494.216 + 1110.03i 805.112 584.948i −191.441 + 2178.60i 1881.83i
2.12 −11.4504 12.7170i −23.9440 + 40.1707i −17.2298 + 163.931i −130.548 117.546i 785.019 155.476i −343.377 771.237i 509.933 370.488i −1040.37 1923.70i 3006.12i
2.13 −10.8812 12.0848i −44.3762 14.7565i −14.2624 + 135.697i 18.1792 + 16.3686i 304.538 + 696.848i −185.376 416.362i 111.101 80.7198i 1751.49 + 1309.67i 397.803i
2.14 −10.6294 11.8052i −29.0989 36.6095i −12.9977 + 123.665i −69.1594 62.2714i −122.877 + 732.655i 711.216 + 1597.42i −46.9542 + 34.1142i −493.513 + 2130.59i 1478.35i
2.15 −10.4489 11.6047i −11.7020 + 45.2776i −12.1094 + 115.213i 393.586 + 354.386i 647.705 337.303i −160.865 361.308i −153.527 + 111.544i −1913.13 1059.68i 8270.37i
2.16 −10.3935 11.5431i 42.3680 19.7979i −11.8396 + 112.646i 213.061 + 191.841i −668.878 283.289i −676.120 1518.59i −185.140 + 134.512i 1403.09 1677.59i 4453.28i
2.17 −10.1848 11.3113i 44.4772 + 14.4493i −10.8370 + 103.107i −336.520 303.004i −289.549 650.258i 41.7294 + 93.7257i −299.535 + 217.625i 1769.44 + 1285.33i 6892.50i
2.18 −10.1002 11.2174i 18.1091 + 43.1168i −10.4365 + 99.2966i 107.294 + 96.6082i 300.753 638.625i 45.1970 + 101.514i −343.838 + 249.813i −1531.12 + 1561.62i 2179.32i
2.19 −9.82354 10.9101i −13.2016 44.8633i −9.14969 + 87.0535i 189.746 + 170.848i −359.778 + 584.748i −35.8335 80.4833i −480.635 + 349.202i −1838.43 + 1184.54i 3748.49i
2.20 −8.13027 9.02958i −6.81500 46.2661i −2.05239 + 19.5272i −280.524 252.585i −362.356 + 437.693i −107.454 241.345i −1065.23 + 773.932i −2094.11 + 630.607i 4586.61i
See next 80 embeddings (of 656 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.82 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.d odd 10 1 inner
99.p even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.p.a 656
9.d odd 6 1 inner 99.8.p.a 656
11.d odd 10 1 inner 99.8.p.a 656
99.p even 30 1 inner 99.8.p.a 656

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.p.a 656 1.a even 1 1 trivial
99.8.p.a 656 9.d odd 6 1 inner
99.8.p.a 656 11.d odd 10 1 inner
99.8.p.a 656 99.p even 30 1 inner

## Hecke kernels

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