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Properties

Label 99.2.p.a
Level $99$
Weight $2$
Character orbit 99.p
Analytic conductor $0.791$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 15 q^{2} - 3 q^{3} + 5 q^{4} - 6 q^{5} - 15 q^{6} - 5 q^{7} - q^{9} - 3 q^{11} - 54 q^{12} - 5 q^{13} - 9 q^{14} + 5 q^{16} - 50 q^{19} - 3 q^{20} - 11 q^{22} - 42 q^{23} - 5 q^{24} - 2 q^{25} + 3 q^{27}+ \cdots + 35 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.66735 − 1.85178i 1.13023 + 1.31247i −0.439972 + 4.18606i 1.05074 + 0.946093i 0.545914 − 4.28128i 1.27299 + 2.85917i 4.45339 − 3.23558i −0.445156 + 2.96679i − 3.52320i
2.2 −1.65440 − 1.83740i −0.642641 − 1.60842i −0.429934 + 4.09055i −1.51955 − 1.36821i −1.89212 + 3.84176i −0.441031 − 0.990572i 4.22672 − 3.07089i −2.17403 + 2.06727i 5.05558i
2.3 −1.02662 − 1.14018i −0.926210 + 1.46360i −0.0369992 + 0.352024i −2.07594 − 1.86918i 2.61964 − 0.446523i −1.07503 − 2.41456i −2.04313 + 1.48442i −1.28427 − 2.71121i 4.28588i
2.4 −0.906431 − 1.00669i 1.04844 − 1.37868i 0.0172421 − 0.164047i 2.90247 + 2.61340i −2.33825 + 0.194222i −0.686294 − 1.54144i −2.37263 + 1.72381i −0.801540 − 2.89094i − 5.29077i
2.5 −0.536887 − 0.596273i 1.73203 + 0.00818115i 0.141763 − 1.34878i −2.14661 − 1.93281i −0.925027 − 1.03716i 0.542079 + 1.21753i −2.17861 + 1.58285i 2.99987 + 0.0283400i 2.31767i
2.6 0.0350855 + 0.0389664i −0.778272 + 1.54735i 0.208770 − 1.98631i 1.98818 + 1.79017i −0.0876008 + 0.0239631i 1.00434 + 2.25577i 0.169565 − 0.123196i −1.78858 − 2.40852i 0.140281i
2.7 0.148911 + 0.165382i −1.37407 − 1.05449i 0.203880 − 1.93979i 0.397470 + 0.357884i −0.0302197 − 0.384271i −1.30165 − 2.92356i 0.711251 − 0.516754i 0.776111 + 2.89787i 0.119027i
2.8 0.780898 + 0.867275i 0.189274 − 1.72168i 0.0666926 − 0.634537i −1.15400 − 1.03907i 1.64097 − 1.18030i 1.76474 + 3.96367i 2.49070 − 1.80960i −2.92835 − 0.651738i − 1.81224i
2.9 1.13824 + 1.26415i 0.775984 + 1.54850i −0.0934131 + 0.888766i −1.30752 − 1.17729i −1.07427 + 2.74352i −1.14161 − 2.56410i 1.52254 − 1.10619i −1.79570 + 2.40322i − 2.99294i
2.10 1.48406 + 1.64822i −1.61485 + 0.626310i −0.305125 + 2.90307i −0.217939 − 0.196233i −3.42883 − 1.73214i 0.539627 + 1.21202i −1.64909 + 1.19814i 2.21547 − 2.02279i − 0.650432i
29.1 −2.46099 + 1.09570i −1.72861 + 0.109172i 3.51765 − 3.90675i 0.588502 − 1.32180i 4.13447 − 2.16271i −0.300148 + 1.41209i −2.71136 + 8.34470i 2.97616 − 0.377431i 3.89775i
29.2 −1.79897 + 0.800953i 0.666085 + 1.59885i 1.25651 − 1.39549i −0.804185 + 1.80623i −2.47887 − 2.34279i 0.0659410 − 0.310228i 0.0743474 − 0.228818i −2.11266 + 2.12994i − 3.89347i
29.3 −1.65764 + 0.738027i −0.00460448 − 1.73204i 0.864813 − 0.960472i −0.488756 + 1.09776i 1.28593 + 2.86770i 1.00654 − 4.73540i 0.396737 − 1.22103i −2.99996 + 0.0159503i − 2.18041i
29.4 −1.31398 + 0.585020i 1.60559 − 0.649686i 0.0460231 − 0.0511138i 1.45099 − 3.25897i −1.72962 + 1.79297i −0.449955 + 2.11687i 0.858363 − 2.64177i 2.15582 − 2.08625i 5.13106i
29.5 −0.571041 + 0.254244i −1.42001 − 0.991749i −1.07681 + 1.19592i −0.518875 + 1.16541i 1.06303 + 0.205300i −1.00684 + 4.73680i 0.697171 − 2.14567i 1.03287 + 2.81659i − 0.797419i
29.6 0.337994 − 0.150485i −1.17234 + 1.27500i −1.24667 + 1.38456i −0.966595 + 2.17101i −0.204377 + 0.607361i 0.327889 − 1.54259i −0.441671 + 1.35932i −0.251238 − 2.98946i 0.879247i
29.7 0.471493 − 0.209922i 1.16254 + 1.28394i −1.16002 + 1.28834i 0.661751 − 1.48632i 0.817657 + 0.361323i 0.171735 − 0.807949i −0.595467 + 1.83266i −0.296983 + 2.98526i − 0.839703i
29.8 1.05233 − 0.468527i 1.16424 − 1.28240i −0.450383 + 0.500201i −0.381354 + 0.856535i 0.624322 − 1.89498i 0.180225 − 0.847891i −0.951517 + 2.92847i −0.289100 − 2.98604i 1.08003i
29.9 1.93906 − 0.863327i −1.27072 + 1.17698i 1.67637 − 1.86180i 1.21148 − 2.72104i −1.44788 + 3.37928i −0.664357 + 3.12555i 0.331430 − 1.02004i 0.229440 − 2.99121i − 6.32217i
29.10 2.14162 − 0.953512i −1.25398 − 1.19479i 2.33911 − 2.59784i −1.18835 + 2.66909i −3.82481 − 1.36311i 0.273495 − 1.28669i 1.08356 − 3.33484i 0.144932 + 2.99650i 6.84930i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.10
Significant digits:
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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 99.2.p.a ✓ 80
3.b odd 2 1 297.2.t.a 80
9.c even 3 1 297.2.t.a 80
9.c even 3 1 891.2.k.a 80
9.d odd 6 1 inner 99.2.p.a ✓ 80
9.d odd 6 1 891.2.k.a 80
11.d odd 10 1 inner 99.2.p.a ✓ 80
33.f even 10 1 297.2.t.a 80
99.o odd 30 1 297.2.t.a 80
99.o odd 30 1 891.2.k.a 80
99.p even 30 1 inner 99.2.p.a ✓ 80
99.p even 30 1 891.2.k.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.p.a ✓ 80 1.a even 1 1 trivial
99.2.p.a ✓ 80 9.d odd 6 1 inner
99.2.p.a ✓ 80 11.d odd 10 1 inner
99.2.p.a ✓ 80 99.p even 30 1 inner
297.2.t.a 80 3.b odd 2 1
297.2.t.a 80 9.c even 3 1
297.2.t.a 80 33.f even 10 1
297.2.t.a 80 99.o odd 30 1
891.2.k.a 80 9.c even 3 1
891.2.k.a 80 9.d odd 6 1
891.2.k.a 80 99.o odd 30 1
891.2.k.a 80 99.p even 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(99, [\chi])\).