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Properties

Label 375.3.k.b
Level $375$
Weight $3$
Character orbit 375.k
Analytic conductor $10.218$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [375,3,Mod(7,375)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("375.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 17])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.k (of order \(20\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80,-4,0,0,0,0,4,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2180099135\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} + 4 q^{7} + 72 q^{8} + 24 q^{12} - 32 q^{13} + 80 q^{16} - 40 q^{17} + 48 q^{18} + 100 q^{19} - 280 q^{22} - 264 q^{23} - 40 q^{26} + 44 q^{28} - 200 q^{29} - 636 q^{32} - 36 q^{33} - 100 q^{34}+ \cdots - 828 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.89947 + 1.47736i −1.71073 + 0.270952i 3.87323 − 5.33104i 0 4.55991 − 3.31297i 8.41509 + 8.41509i −1.31823 + 8.32297i 2.85317 − 0.927051i 0
7.2 −2.22224 + 1.13229i 1.71073 − 0.270952i 1.30514 − 1.79637i 0 −3.49485 + 2.53916i −3.10232 − 3.10232i 0.694313 − 4.38372i 2.85317 − 0.927051i 0
7.3 −2.10631 + 1.07322i −1.71073 + 0.270952i 0.933591 − 1.28498i 0 3.31252 − 2.40669i −9.10511 − 9.10511i 0.891852 − 5.63093i 2.85317 − 0.927051i 0
7.4 −1.52253 + 0.775767i 1.71073 − 0.270952i −0.634862 + 0.873813i 0 −2.39443 + 1.73966i 3.94736 + 3.94736i 1.35797 − 8.57385i 2.85317 − 0.927051i 0
7.5 0.706305 − 0.359880i 1.71073 − 0.270952i −1.98179 + 2.72770i 0 1.11078 − 0.807032i −8.25978 − 8.25978i −0.914128 + 5.77157i 2.85317 − 0.927051i 0
7.6 0.827404 − 0.421583i −1.71073 + 0.270952i −1.84428 + 2.53843i 0 −1.30123 + 0.945401i 4.18270 + 4.18270i −1.03687 + 6.54656i 2.85317 − 0.927051i 0
7.7 1.31554 − 0.670302i 1.71073 − 0.270952i −1.06979 + 1.47245i 0 2.06891 − 1.50315i −0.393993 − 0.393993i −1.34426 + 8.48731i 2.85317 − 0.927051i 0
7.8 2.03626 − 1.03753i −1.71073 + 0.270952i 0.718759 − 0.989286i 0 −3.20237 + 2.32666i −0.410561 − 0.410561i −0.992861 + 6.26868i 2.85317 − 0.927051i 0
7.9 2.98300 − 1.51991i 1.71073 − 0.270952i 4.23700 − 5.83172i 0 4.69127 − 3.40841i 3.58606 + 3.58606i 1.68033 − 10.6092i 2.85317 − 0.927051i 0
7.10 3.40219 − 1.73350i −1.71073 + 0.270952i 6.21871 − 8.55932i 0 −5.35052 + 3.88738i −7.30480 − 7.30480i 3.93033 − 24.8151i 2.85317 − 0.927051i 0
43.1 −3.83332 + 0.607139i −0.786335 + 1.54327i 10.5215 − 3.41865i 0 2.07730 − 6.39326i 2.30555 − 2.30555i −24.4245 + 12.4449i −1.76336 − 2.42705i 0
43.2 −3.12901 + 0.495586i 0.786335 − 1.54327i 5.74087 − 1.86532i 0 −1.69563 + 5.21860i −1.18776 + 1.18776i −5.74792 + 2.92871i −1.76336 − 2.42705i 0
43.3 −2.35878 + 0.373594i 0.786335 − 1.54327i 1.62005 − 0.526386i 0 −1.27823 + 3.93400i 5.77573 − 5.77573i 4.88686 − 2.48998i −1.76336 − 2.42705i 0
43.4 −1.44598 + 0.229020i −0.786335 + 1.54327i −1.76583 + 0.573753i 0 0.783582 − 2.41162i −8.32198 + 8.32198i 7.63968 − 3.89261i −1.76336 − 2.42705i 0
43.5 −1.05885 + 0.167706i −0.786335 + 1.54327i −2.71118 + 0.880917i 0 0.573797 − 1.76597i 2.58808 − 2.58808i 6.54382 − 3.33424i −1.76336 − 2.42705i 0
43.6 −0.213109 + 0.0337531i 0.786335 − 1.54327i −3.75995 + 1.22168i 0 −0.115485 + 0.355426i −2.53517 + 2.53517i 1.52904 − 0.779083i −1.76336 − 2.42705i 0
43.7 1.27601 − 0.202100i 0.786335 − 1.54327i −2.21687 + 0.720305i 0 0.691476 − 2.12814i −1.27981 + 1.27981i −7.28759 + 3.71321i −1.76336 − 2.42705i 0
43.8 1.41726 − 0.224472i −0.786335 + 1.54327i −1.84598 + 0.599796i 0 −0.768021 + 2.36373i 1.57196 − 1.57196i −7.59573 + 3.87022i −1.76336 − 2.42705i 0
43.9 3.02809 − 0.479602i 0.786335 − 1.54327i 5.13507 − 1.66849i 0 1.64094 − 5.05028i 5.93757 − 5.93757i 3.82253 − 1.94768i −1.76336 − 2.42705i 0
43.10 3.52409 − 0.558161i −0.786335 + 1.54327i 8.30342 − 2.69795i 0 −1.90972 + 5.87751i 8.56695 − 8.56695i 15.0396 − 7.66306i −1.76336 − 2.42705i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.3.k.b 80
5.b even 2 1 375.3.k.c 80
5.c odd 4 1 75.3.k.a ✓ 80
5.c odd 4 1 375.3.k.a 80
15.e even 4 1 225.3.r.b 80
25.d even 5 1 375.3.k.a 80
25.e even 10 1 75.3.k.a ✓ 80
25.f odd 20 1 inner 375.3.k.b 80
25.f odd 20 1 375.3.k.c 80
75.h odd 10 1 225.3.r.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.k.a ✓ 80 5.c odd 4 1
75.3.k.a ✓ 80 25.e even 10 1
225.3.r.b 80 15.e even 4 1
225.3.r.b 80 75.h odd 10 1
375.3.k.a 80 5.c odd 4 1
375.3.k.a 80 25.d even 5 1
375.3.k.b 80 1.a even 1 1 trivial
375.3.k.b 80 25.f odd 20 1 inner
375.3.k.c 80 5.b even 2 1
375.3.k.c 80 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 4 T_{2}^{79} + 8 T_{2}^{78} - 24 T_{2}^{77} - 428 T_{2}^{76} - 740 T_{2}^{75} + \cdots + 87\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display