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Properties

Label 99.4.e.b
Level $99$
Weight $4$
Character orbit 99.e
Analytic conductor $5.841$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,Mod(34,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.34");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.e (of order \(3\), degree \(2\), 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: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 6 q^{3} - 32 q^{4} + 2 q^{5} + 57 q^{6} + 8 q^{7} - 6 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 6 q^{3} - 32 q^{4} + 2 q^{5} + 57 q^{6} + 8 q^{7} - 6 q^{8} - 54 q^{9} - 272 q^{10} - 132 q^{11} + 21 q^{12} + 62 q^{13} + 179 q^{14} - 24 q^{15} - 128 q^{16} - 160 q^{17} + 63 q^{18} - 128 q^{19} + 67 q^{20} - 48 q^{21} - 44 q^{22} + 280 q^{23} - 513 q^{24} + 112 q^{25} - 100 q^{26} + 756 q^{27} - 1158 q^{28} + 60 q^{29} + 618 q^{30} + 704 q^{31} - 755 q^{32} + 66 q^{33} + 778 q^{34} - 448 q^{35} + 1089 q^{36} - 1760 q^{37} - 171 q^{38} + 636 q^{39} + 1695 q^{40} + 374 q^{41} - 1041 q^{42} + 826 q^{43} + 704 q^{44} - 846 q^{45} - 1912 q^{46} + 214 q^{47} + 3009 q^{48} + 462 q^{49} - 1019 q^{50} + 372 q^{51} + 834 q^{52} - 756 q^{53} - 1026 q^{54} - 44 q^{55} + 750 q^{56} + 2574 q^{57} + 867 q^{58} + 748 q^{59} - 5061 q^{60} + 2662 q^{61} - 1258 q^{62} - 828 q^{63} - 306 q^{64} + 416 q^{65} + 330 q^{66} + 2430 q^{67} + 895 q^{68} - 1920 q^{69} + 136 q^{70} - 1224 q^{71} - 5166 q^{72} - 1696 q^{73} - 93 q^{74} + 4314 q^{75} + 934 q^{76} + 88 q^{77} - 2481 q^{78} + 1770 q^{79} - 4262 q^{80} - 1566 q^{81} - 3590 q^{82} - 156 q^{83} + 6114 q^{84} + 1414 q^{85} - 3576 q^{86} - 1980 q^{87} + 33 q^{88} - 504 q^{89} + 405 q^{90} - 5004 q^{91} + 3821 q^{92} + 7098 q^{93} + 4046 q^{94} + 510 q^{95} - 9939 q^{96} + 2436 q^{97} + 2448 q^{98} - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.70533 + 4.68576i −2.29336 − 4.66267i −10.6376 − 18.4248i 7.11703 + 12.3271i 28.0524 + 1.86788i 9.24748 − 16.0171i 71.8272 −16.4810 + 21.3864i −77.0155
34.2 −2.09032 + 3.62053i −3.47524 + 3.86299i −4.73885 − 8.20792i −2.12693 − 3.68395i −6.72172 − 20.6571i 10.7173 − 18.5629i 6.17770 −2.84535 − 26.8497i 17.7838
34.3 −1.67234 + 2.89657i 3.99103 + 3.32741i −1.59343 − 2.75990i −0.582904 − 1.00962i −16.3124 + 5.99575i −5.00112 + 8.66220i −16.0984 4.85663 + 26.5596i 3.89925
34.4 −1.55275 + 2.68944i −5.06696 − 1.15149i −0.822045 − 1.42382i 6.37846 + 11.0478i 10.9646 − 11.8393i −7.32878 + 12.6938i −19.7382 24.3482 + 11.6691i −39.6165
34.5 −0.692765 + 1.19990i 4.61150 − 2.39460i 3.04015 + 5.26570i −8.79243 − 15.2289i −0.321396 + 7.19225i 13.6589 − 23.6580i −19.5087 15.5318 − 22.0854i 24.3643
34.6 −0.316303 + 0.547853i −2.75190 − 4.40761i 3.79990 + 6.58163i −7.01119 − 12.1437i 3.28516 − 0.113494i −8.35675 + 14.4743i −9.86854 −11.8541 + 24.2586i 8.87065
34.7 −0.271684 + 0.470571i 1.12050 − 5.07390i 3.85238 + 6.67251i 7.19686 + 12.4653i 2.08321 + 1.90578i −5.62518 + 9.74309i −8.53347 −24.4889 − 11.3706i −7.82110
34.8 −0.0391684 + 0.0678417i 1.19695 + 5.05641i 3.99693 + 6.92289i 4.72166 + 8.17816i −0.389918 − 0.116849i 7.37292 − 12.7703i −1.25291 −24.1346 + 12.1045i −0.739761
34.9 1.26694 − 2.19441i 4.70254 − 2.21045i 0.789718 + 1.36783i 3.48893 + 6.04301i 1.10721 − 13.1198i 0.859077 − 1.48796i 24.2732 17.2278 − 20.7895i 17.6811
34.10 1.43205 − 2.48039i −0.831657 + 5.12917i −0.101546 − 0.175883i −1.38275 − 2.39500i 11.5313 + 9.40807i −16.4543 + 28.4996i 22.3312 −25.6167 − 8.53142i −7.92069
34.11 2.21996 − 3.84508i −3.20135 − 4.09284i −5.85642 − 10.1436i −2.57011 − 4.45156i −22.8442 + 3.22349i 4.51199 − 7.81499i −16.4847 −6.50276 + 26.2052i −22.8221
34.12 2.42170 − 4.19450i 4.99795 + 1.42143i −7.72923 − 13.3874i −5.43663 − 9.41652i 18.0657 − 17.5216i 0.398427 − 0.690096i −36.1242 22.9591 + 14.2085i −52.6635
67.1 −2.70533 − 4.68576i −2.29336 + 4.66267i −10.6376 + 18.4248i 7.11703 − 12.3271i 28.0524 − 1.86788i 9.24748 + 16.0171i 71.8272 −16.4810 − 21.3864i −77.0155
67.2 −2.09032 − 3.62053i −3.47524 − 3.86299i −4.73885 + 8.20792i −2.12693 + 3.68395i −6.72172 + 20.6571i 10.7173 + 18.5629i 6.17770 −2.84535 + 26.8497i 17.7838
67.3 −1.67234 − 2.89657i 3.99103 − 3.32741i −1.59343 + 2.75990i −0.582904 + 1.00962i −16.3124 − 5.99575i −5.00112 − 8.66220i −16.0984 4.85663 − 26.5596i 3.89925
67.4 −1.55275 − 2.68944i −5.06696 + 1.15149i −0.822045 + 1.42382i 6.37846 − 11.0478i 10.9646 + 11.8393i −7.32878 − 12.6938i −19.7382 24.3482 − 11.6691i −39.6165
67.5 −0.692765 − 1.19990i 4.61150 + 2.39460i 3.04015 − 5.26570i −8.79243 + 15.2289i −0.321396 − 7.19225i 13.6589 + 23.6580i −19.5087 15.5318 + 22.0854i 24.3643
67.6 −0.316303 − 0.547853i −2.75190 + 4.40761i 3.79990 − 6.58163i −7.01119 + 12.1437i 3.28516 + 0.113494i −8.35675 − 14.4743i −9.86854 −11.8541 − 24.2586i 8.87065
67.7 −0.271684 − 0.470571i 1.12050 + 5.07390i 3.85238 − 6.67251i 7.19686 − 12.4653i 2.08321 − 1.90578i −5.62518 − 9.74309i −8.53347 −24.4889 + 11.3706i −7.82110
67.8 −0.0391684 − 0.0678417i 1.19695 − 5.05641i 3.99693 − 6.92289i 4.72166 − 8.17816i −0.389918 + 0.116849i 7.37292 + 12.7703i −1.25291 −24.1346 − 12.1045i −0.739761
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.e.b ✓ 24
3.b odd 2 1 297.4.e.b 24
9.c even 3 1 inner 99.4.e.b ✓ 24
9.c even 3 1 891.4.a.l 12
9.d odd 6 1 297.4.e.b 24
9.d odd 6 1 891.4.a.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.e.b ✓ 24 1.a even 1 1 trivial
99.4.e.b ✓ 24 9.c even 3 1 inner
297.4.e.b 24 3.b odd 2 1
297.4.e.b 24 9.d odd 6 1
891.4.a.k 12 9.d odd 6 1
891.4.a.l 12 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 4 T_{2}^{23} + 72 T_{2}^{22} + 226 T_{2}^{21} + 3016 T_{2}^{20} + 8775 T_{2}^{19} + \cdots + 1871424 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display