# Properties

 Label 99.8.f.a Level $99$ Weight $8$ Character orbit 99.f Analytic conductor $30.926$ Analytic rank $0$ Dimension $24$ 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: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{5})$$ Twist minimal: no (minimal twist has level 11) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 3 q^{2} - 505 q^{4} + 72 q^{5} + 68 q^{7} + 4545 q^{8}+O(q^{10})$$ 24 * q - 3 * q^2 - 505 * q^4 + 72 * q^5 + 68 * q^7 + 4545 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 3 q^{2} - 505 q^{4} + 72 q^{5} + 68 q^{7} + 4545 q^{8} + 4240 q^{10} - 5952 q^{11} - 16564 q^{13} + 29544 q^{14} + 13279 q^{16} + 65592 q^{17} - 37804 q^{19} - 120702 q^{20} + 246233 q^{22} + 26304 q^{23} - 210646 q^{25} + 336384 q^{26} + 92334 q^{28} + 231048 q^{29} + 141004 q^{31} + 260892 q^{32} + 1512162 q^{34} + 342588 q^{35} - 836112 q^{37} + 2387040 q^{38} - 1906028 q^{40} - 1930344 q^{41} + 2877456 q^{43} - 5822808 q^{44} - 4951876 q^{46} + 2462532 q^{47} - 6581826 q^{49} + 2698797 q^{50} + 1655498 q^{52} - 6157836 q^{53} + 11721612 q^{55} - 11431080 q^{56} - 15150140 q^{58} + 2480268 q^{59} - 4195044 q^{61} + 6436524 q^{62} + 11301151 q^{64} - 11117640 q^{65} + 16682528 q^{67} + 14718306 q^{68} - 16625944 q^{70} + 20942628 q^{71} - 456824 q^{73} - 8747736 q^{74} + 18091206 q^{76} - 13904988 q^{77} - 12976692 q^{79} + 11893596 q^{80} - 32706099 q^{82} + 4471404 q^{83} + 25528128 q^{85} - 20374383 q^{86} + 25246815 q^{88} + 2477316 q^{89} - 18339604 q^{91} - 1026678 q^{92} - 4961738 q^{94} + 25328772 q^{95} - 34034202 q^{97} + 51534036 q^{98}+O(q^{100})$$ 24 * q - 3 * q^2 - 505 * q^4 + 72 * q^5 + 68 * q^7 + 4545 * q^8 + 4240 * q^10 - 5952 * q^11 - 16564 * q^13 + 29544 * q^14 + 13279 * q^16 + 65592 * q^17 - 37804 * q^19 - 120702 * q^20 + 246233 * q^22 + 26304 * q^23 - 210646 * q^25 + 336384 * q^26 + 92334 * q^28 + 231048 * q^29 + 141004 * q^31 + 260892 * q^32 + 1512162 * q^34 + 342588 * q^35 - 836112 * q^37 + 2387040 * q^38 - 1906028 * q^40 - 1930344 * q^41 + 2877456 * q^43 - 5822808 * q^44 - 4951876 * q^46 + 2462532 * q^47 - 6581826 * q^49 + 2698797 * q^50 + 1655498 * q^52 - 6157836 * q^53 + 11721612 * q^55 - 11431080 * q^56 - 15150140 * q^58 + 2480268 * q^59 - 4195044 * q^61 + 6436524 * q^62 + 11301151 * q^64 - 11117640 * q^65 + 16682528 * q^67 + 14718306 * q^68 - 16625944 * q^70 + 20942628 * q^71 - 456824 * q^73 - 8747736 * q^74 + 18091206 * q^76 - 13904988 * q^77 - 12976692 * q^79 + 11893596 * q^80 - 32706099 * q^82 + 4471404 * q^83 + 25528128 * q^85 - 20374383 * q^86 + 25246815 * q^88 + 2477316 * q^89 - 18339604 * q^91 - 1026678 * q^92 - 4961738 * q^94 + 25328772 * q^95 - 34034202 * q^97 + 51534036 * 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 −14.3633 10.4355i 0 57.8489 + 178.041i −53.3657 + 38.7724i 0 −143.396 441.326i 324.804 999.643i 0 1171.11
37.2 −7.83171 5.69007i 0 −10.5954 32.6093i −9.68480 + 7.03642i 0 520.681 + 1602.49i −485.474 + 1494.14i 0 115.886
37.3 −0.987911 0.717759i 0 −39.0934 120.317i 339.439 246.617i 0 312.595 + 962.069i −96.0385 + 295.576i 0 −512.347
37.4 1.19911 + 0.871205i 0 −38.8753 119.646i −171.086 + 124.301i 0 −463.381 1426.14i 116.247 357.771i 0 −313.442
37.5 12.7113 + 9.23528i 0 36.7318 + 113.049i 268.339 194.959i 0 −72.0370 221.707i 44.3449 136.480i 0 5211.43
37.6 13.5536 + 9.84730i 0 47.1778 + 145.198i −315.392 + 229.146i 0 356.708 + 1097.83i −127.721 + 393.085i 0 −6531.19
64.1 −5.78570 + 17.8065i 0 −180.045 130.810i −151.154 465.205i 0 −544.422 395.546i 1432.12 1040.50i 0 9158.23
64.2 −5.72800 + 17.6290i 0 −174.417 126.721i 66.8497 + 205.742i 0 −327.935 238.259i 1313.52 954.328i 0 −4009.94
64.3 −2.40997 + 7.41712i 0 54.3484 + 39.4864i 139.587 + 429.605i 0 40.1674 + 29.1833i −1231.45 + 894.704i 0 −3522.83
64.4 −0.419581 + 1.29134i 0 102.063 + 74.1529i −90.0532 277.155i 0 205.463 + 149.277i −279.185 + 202.840i 0 395.685
64.5 3.78266 11.6418i 0 −17.6695 12.8376i −31.6803 97.5020i 0 601.350 + 436.906i 1051.31 763.821i 0 −1254.94
64.6 4.77944 14.7096i 0 −89.9749 65.3706i 44.2018 + 136.039i 0 −451.793 328.247i 210.025 152.592i 0 2212.34
82.1 −5.78570 17.8065i 0 −180.045 + 130.810i −151.154 + 465.205i 0 −544.422 + 395.546i 1432.12 + 1040.50i 0 9158.23
82.2 −5.72800 17.6290i 0 −174.417 + 126.721i 66.8497 205.742i 0 −327.935 + 238.259i 1313.52 + 954.328i 0 −4009.94
82.3 −2.40997 7.41712i 0 54.3484 39.4864i 139.587 429.605i 0 40.1674 29.1833i −1231.45 894.704i 0 −3522.83
82.4 −0.419581 1.29134i 0 102.063 74.1529i −90.0532 + 277.155i 0 205.463 149.277i −279.185 202.840i 0 395.685
82.5 3.78266 + 11.6418i 0 −17.6695 + 12.8376i −31.6803 + 97.5020i 0 601.350 436.906i 1051.31 + 763.821i 0 −1254.94
82.6 4.77944 + 14.7096i 0 −89.9749 + 65.3706i 44.2018 136.039i 0 −451.793 + 328.247i 210.025 + 152.592i 0 2212.34
91.1 −14.3633 + 10.4355i 0 57.8489 178.041i −53.3657 38.7724i 0 −143.396 + 441.326i 324.804 + 999.643i 0 1171.11
91.2 −7.83171 + 5.69007i 0 −10.5954 + 32.6093i −9.68480 7.03642i 0 520.681 1602.49i −485.474 1494.14i 0 115.886
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.6 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.a 24
3.b odd 2 1 11.8.c.a 24
11.c even 5 1 inner 99.8.f.a 24
33.f even 10 1 121.8.a.g 12
33.h odd 10 1 11.8.c.a 24
33.h odd 10 1 121.8.a.i 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.c.a 24 3.b odd 2 1
11.8.c.a 24 33.h odd 10 1
99.8.f.a 24 1.a even 1 1 trivial
99.8.f.a 24 11.c even 5 1 inner
121.8.a.g 12 33.f even 10 1
121.8.a.i 12 33.h odd 10 1