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Properties

Label 99.3.h.b
Level $99$
Weight $3$
Character orbit 99.h
Analytic conductor $2.698$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [99,3,Mod(43,99)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("99.43"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 99.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.69755461717\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{3} + 46 q^{4} + 2 q^{5} - 6 q^{9} - 23 q^{11} + 42 q^{12} - 18 q^{14} - 48 q^{15} - 26 q^{16} + 50 q^{20} - 21 q^{22} + 32 q^{23} + 30 q^{25} - 216 q^{26} - 90 q^{27} + 50 q^{31} - 51 q^{33}+ \cdots + 768 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −3.30671 − 1.90913i 0.253226 − 2.98929i 5.28955 + 9.16177i −2.91356 − 5.04644i −6.54430 + 9.40128i −5.33694 − 3.08128i − 25.1207i −8.87175 − 1.51394i 22.2495i
43.2 −3.03057 − 1.74970i −2.16106 + 2.08083i 4.12292 + 7.14111i −0.198992 − 0.344665i 10.1901 − 2.52489i 2.03014 + 1.17210i − 14.8579i 0.340331 − 8.99356i 1.39271i
43.3 −2.67675 − 1.54542i 2.53282 + 1.60774i 2.77667 + 4.80934i −1.89480 − 3.28189i −4.29510 − 8.21780i 5.27354 + 3.04468i − 4.80116i 3.83037 + 8.14422i 11.7131i
43.4 −2.41819 − 1.39614i 1.93703 − 2.29083i 1.89841 + 3.28815i 3.95412 + 6.84874i −7.88243 + 2.83528i 11.3316 + 6.54229i 0.567308i −1.49581 − 8.87483i − 22.0820i
43.5 −2.38735 − 1.37834i −2.62560 − 1.45128i 1.79962 + 3.11703i 3.89274 + 6.74242i 4.26788 + 7.08367i −6.72208 − 3.88100i 1.10477i 4.78758 + 7.62097i − 21.4620i
43.6 −1.81513 − 1.04797i 0.432421 + 2.96867i 0.196474 + 0.340304i 1.28105 + 2.21884i 2.32617 − 5.84170i −4.60933 − 2.66120i 7.56015i −8.62602 + 2.56743i − 5.37000i
43.7 −1.64469 − 0.949563i −2.45615 − 1.72260i −0.196660 − 0.340626i −3.05607 − 5.29326i 2.40389 + 5.16541i 4.13935 + 2.38985i 8.34347i 3.06532 + 8.46190i 11.6077i
43.8 −1.10231 − 0.636421i 2.90522 − 0.748144i −1.18994 − 2.06103i −1.84774 − 3.20038i −3.67859 − 1.02425i −6.26503 − 3.61711i 8.12057i 7.88056 − 4.34704i 4.70376i
43.9 −0.981412 − 0.566618i 0.439224 − 2.96767i −1.35789 − 2.35193i 0.0827703 + 0.143362i −2.11260 + 2.66364i −3.82724 − 2.20966i 7.61056i −8.61416 − 2.60695i − 0.187597i
43.10 −0.491166 − 0.283575i −2.75714 + 1.18246i −1.83917 − 3.18554i 1.20048 + 2.07930i 1.68953 + 0.201072i 10.1718 + 5.87270i 4.35477i 6.20359 − 6.52039i − 1.36171i
43.11 0.491166 + 0.283575i −2.75714 + 1.18246i −1.83917 − 3.18554i 1.20048 + 2.07930i −1.68953 − 0.201072i −10.1718 − 5.87270i − 4.35477i 6.20359 − 6.52039i 1.36171i
43.12 0.981412 + 0.566618i 0.439224 − 2.96767i −1.35789 − 2.35193i 0.0827703 + 0.143362i 2.11260 − 2.66364i 3.82724 + 2.20966i − 7.61056i −8.61416 − 2.60695i 0.187597i
43.13 1.10231 + 0.636421i 2.90522 − 0.748144i −1.18994 − 2.06103i −1.84774 − 3.20038i 3.67859 + 1.02425i 6.26503 + 3.61711i − 8.12057i 7.88056 − 4.34704i − 4.70376i
43.14 1.64469 + 0.949563i −2.45615 − 1.72260i −0.196660 − 0.340626i −3.05607 − 5.29326i −2.40389 − 5.16541i −4.13935 − 2.38985i − 8.34347i 3.06532 + 8.46190i − 11.6077i
43.15 1.81513 + 1.04797i 0.432421 + 2.96867i 0.196474 + 0.340304i 1.28105 + 2.21884i −2.32617 + 5.84170i 4.60933 + 2.66120i − 7.56015i −8.62602 + 2.56743i 5.37000i
43.16 2.38735 + 1.37834i −2.62560 − 1.45128i 1.79962 + 3.11703i 3.89274 + 6.74242i −4.26788 − 7.08367i 6.72208 + 3.88100i − 1.10477i 4.78758 + 7.62097i 21.4620i
43.17 2.41819 + 1.39614i 1.93703 − 2.29083i 1.89841 + 3.28815i 3.95412 + 6.84874i 7.88243 − 2.83528i −11.3316 − 6.54229i − 0.567308i −1.49581 − 8.87483i 22.0820i
43.18 2.67675 + 1.54542i 2.53282 + 1.60774i 2.77667 + 4.80934i −1.89480 − 3.28189i 4.29510 + 8.21780i −5.27354 − 3.04468i 4.80116i 3.83037 + 8.14422i − 11.7131i
43.19 3.03057 + 1.74970i −2.16106 + 2.08083i 4.12292 + 7.14111i −0.198992 − 0.344665i −10.1901 + 2.52489i −2.03014 − 1.17210i 14.8579i 0.340331 − 8.99356i − 1.39271i
43.20 3.30671 + 1.90913i 0.253226 − 2.98929i 5.28955 + 9.16177i −2.91356 − 5.04644i 6.54430 − 9.40128i 5.33694 + 3.08128i 25.1207i −8.87175 − 1.51394i − 22.2495i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.b odd 2 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.h.b ✓ 40
3.b odd 2 1 297.3.h.b 40
9.c even 3 1 inner 99.3.h.b ✓ 40
9.c even 3 1 891.3.c.e 20
9.d odd 6 1 297.3.h.b 40
9.d odd 6 1 891.3.c.f 20
11.b odd 2 1 inner 99.3.h.b ✓ 40
33.d even 2 1 297.3.h.b 40
99.g even 6 1 297.3.h.b 40
99.g even 6 1 891.3.c.f 20
99.h odd 6 1 inner 99.3.h.b ✓ 40
99.h odd 6 1 891.3.c.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.h.b ✓ 40 1.a even 1 1 trivial
99.3.h.b ✓ 40 9.c even 3 1 inner
99.3.h.b ✓ 40 11.b odd 2 1 inner
99.3.h.b ✓ 40 99.h odd 6 1 inner
297.3.h.b 40 3.b odd 2 1
297.3.h.b 40 9.d odd 6 1
297.3.h.b 40 33.d even 2 1
297.3.h.b 40 99.g even 6 1
891.3.c.e 20 9.c even 3 1
891.3.c.e 20 99.h odd 6 1
891.3.c.f 20 9.d odd 6 1
891.3.c.f 20 99.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 63 T_{2}^{38} + 2289 T_{2}^{36} - 56286 T_{2}^{34} + 1039872 T_{2}^{32} + \cdots + 1148217259401 \) acting on \(S_{3}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display