Properties

Label 99.5.k.c
Level $99$
Weight $5$
Character orbit 99.k
Analytic conductor $10.234$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,5,Mod(19,99)]
 
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, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 99.k (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: \(10.2336263453\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 33)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 76 q^{4} - 36 q^{5} + 150 q^{7} - 480 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 76 q^{4} - 36 q^{5} + 150 q^{7} - 480 q^{8} + 246 q^{11} - 510 q^{13} + 1290 q^{14} - 232 q^{16} - 2490 q^{17} + 582 q^{20} - 510 q^{22} + 2196 q^{23} - 370 q^{25} + 5226 q^{26} + 4310 q^{28} - 960 q^{29} + 1658 q^{31} - 2320 q^{34} - 1920 q^{35} + 1374 q^{37} - 12054 q^{38} + 11070 q^{40} - 9360 q^{41} + 4350 q^{44} - 12950 q^{46} + 3450 q^{47} - 11838 q^{49} + 11550 q^{50} - 19250 q^{52} + 2790 q^{53} + 12356 q^{55} + 5604 q^{56} + 9486 q^{58} - 2682 q^{59} - 17190 q^{61} + 39360 q^{62} + 16248 q^{64} + 2796 q^{67} - 68160 q^{68} + 18188 q^{70} - 132 q^{71} - 21790 q^{73} + 2130 q^{74} - 4542 q^{77} + 12270 q^{79} - 32346 q^{80} + 29442 q^{82} - 35430 q^{83} - 11990 q^{85} + 49416 q^{86} + 1176 q^{88} + 38748 q^{89} - 51858 q^{91} + 25590 q^{92} - 34510 q^{94} + 71670 q^{95} + 30306 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.38915 4.66477i 0 −5.32944 + 16.4023i −4.81744 3.50008i 0 −13.8978 4.51567i 6.83522 2.22090i 0 34.3346i
19.2 −1.95429 2.68985i 0 1.52822 4.70338i −35.4362 25.7459i 0 8.04612 + 2.61434i −66.2318 + 21.5200i 0 145.633i
19.3 −1.01888 1.40237i 0 4.01574 12.3592i 32.0081 + 23.2552i 0 27.3116 + 8.87408i −47.8012 + 15.5316i 0 68.5817i
19.4 0.0886329 + 0.121993i 0 4.93725 15.1953i −11.7622 8.54577i 0 66.7007 + 21.6724i 4.58589 1.49005i 0 2.19234i
19.5 1.95593 + 2.69210i 0 1.52251 4.68579i −6.63359 4.81958i 0 −71.1959 23.1329i 66.2286 21.5190i 0 27.2851i
19.6 2.44632 + 3.36708i 0 −0.408428 + 1.25701i 26.0050 + 18.8937i 0 −15.8191 5.13994i 58.1002 18.8779i 0 133.781i
19.7 4.20830 + 5.79223i 0 −10.8958 + 33.5339i −33.2604 24.1651i 0 −32.2093 10.4654i −131.142 + 42.6108i 0 294.346i
19.8 4.37135 + 6.01665i 0 −12.1471 + 37.3849i 24.8968 + 18.0886i 0 40.6127 + 13.1959i −164.863 + 53.5673i 0 228.867i
28.1 −6.48369 + 2.10668i 0 24.6559 17.9136i −2.10789 + 6.48741i 0 12.4454 + 17.1296i −58.0090 + 79.8425i 0 46.5030i
28.2 −6.41780 + 2.08527i 0 23.8955 17.3611i 9.80980 30.1915i 0 −46.3989 63.8626i −53.6911 + 73.8995i 0 214.219i
28.3 −4.40557 + 1.43146i 0 4.41572 3.20821i −12.7557 + 39.2579i 0 37.7759 + 51.9941i 28.7033 39.5067i 0 191.213i
28.4 −0.743344 + 0.241527i 0 −12.4500 + 9.04549i 9.71709 29.9061i 0 16.8776 + 23.2301i 14.4205 19.8482i 0 24.5775i
28.5 −0.619915 + 0.201423i 0 −12.6005 + 9.15483i −12.5302 + 38.5641i 0 −44.8315 61.7053i 12.0973 16.6506i 0 26.4303i
28.6 0.173259 0.0562952i 0 −12.9174 + 9.38506i 6.40673 19.7179i 0 12.6762 + 17.4473i −3.42300 + 4.71136i 0 3.77697i
28.7 4.49285 1.45982i 0 5.11038 3.71291i −5.35863 + 16.4922i 0 48.2339 + 66.3883i −26.8877 + 37.0078i 0 81.9194i
28.8 7.29601 2.37062i 0 34.6676 25.1875i −2.18122 + 6.71310i 0 28.6722 + 39.4639i 121.078 166.650i 0 54.1497i
46.1 −6.48369 2.10668i 0 24.6559 + 17.9136i −2.10789 6.48741i 0 12.4454 17.1296i −58.0090 79.8425i 0 46.5030i
46.2 −6.41780 2.08527i 0 23.8955 + 17.3611i 9.80980 + 30.1915i 0 −46.3989 + 63.8626i −53.6911 73.8995i 0 214.219i
46.3 −4.40557 1.43146i 0 4.41572 + 3.20821i −12.7557 39.2579i 0 37.7759 51.9941i 28.7033 + 39.5067i 0 191.213i
46.4 −0.743344 0.241527i 0 −12.4500 9.04549i 9.71709 + 29.9061i 0 16.8776 23.2301i 14.4205 + 19.8482i 0 24.5775i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.5.k.c 32
3.b odd 2 1 33.5.g.a 32
11.d odd 10 1 inner 99.5.k.c 32
33.f even 10 1 33.5.g.a 32
33.f even 10 1 363.5.c.e 32
33.h odd 10 1 363.5.c.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.g.a 32 3.b odd 2 1
33.5.g.a 32 33.f even 10 1
99.5.k.c 32 1.a even 1 1 trivial
99.5.k.c 32 11.d odd 10 1 inner
363.5.c.e 32 33.f even 10 1
363.5.c.e 32 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 102 T_{2}^{30} + 480 T_{2}^{29} + 6836 T_{2}^{28} - 48960 T_{2}^{27} - 279108 T_{2}^{26} + 2770830 T_{2}^{25} + 10746133 T_{2}^{24} - 97459650 T_{2}^{23} - 216432384 T_{2}^{22} + \cdots + 7015378606336 \) acting on \(S_{5}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display