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Properties

Label 375.3.k.c
Level $375$
Weight $3$
Character orbit 375.k
Analytic conductor $10.218$
Analytic rank $0$
Dimension $80$
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] = [375,3,Mod(7,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.k (of order \(20\), degree \(8\), not 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: \(10.2180099135\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{2} - 4 q^{7} - 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{2} - 4 q^{7} - 72 q^{8} - 24 q^{12} + 32 q^{13} + 80 q^{16} + 40 q^{17} - 48 q^{18} + 100 q^{19} + 280 q^{22} + 264 q^{23} - 40 q^{26} - 44 q^{28} - 200 q^{29} + 636 q^{32} + 36 q^{33} - 100 q^{34} - 120 q^{36} - 124 q^{37} + 96 q^{38} + 160 q^{41} - 252 q^{42} - 472 q^{43} + 700 q^{44} - 128 q^{47} - 48 q^{48} - 620 q^{52} - 76 q^{53} + 72 q^{57} - 108 q^{58} - 800 q^{59} - 240 q^{61} - 328 q^{62} + 168 q^{63} - 100 q^{64} - 1240 q^{67} + 104 q^{68} + 216 q^{72} + 244 q^{73} - 648 q^{77} - 120 q^{78} - 200 q^{79} + 180 q^{81} - 168 q^{82} - 44 q^{83} + 1200 q^{84} + 84 q^{87} + 1008 q^{88} + 1500 q^{89} + 2236 q^{92} - 648 q^{93} + 200 q^{94} + 60 q^{96} + 200 q^{97} + 828 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −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
7.2 −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
7.3 −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
7.4 −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
7.5 −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
7.6 −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
7.7 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
7.8 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
7.9 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
7.10 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
43.1 −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
43.2 −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
43.3 −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
43.4 −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
43.5 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
43.6 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
43.7 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
43.8 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
43.9 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
43.10 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
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
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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.c 80
5.b even 2 1 375.3.k.b 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 75.3.k.a ✓ 80
25.e even 10 1 375.3.k.a 80
25.f odd 20 1 375.3.k.b 80
25.f odd 20 1 inner 375.3.k.c 80
75.j 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.d even 5 1
225.3.r.b 80 15.e even 4 1
225.3.r.b 80 75.j odd 10 1
375.3.k.a 80 5.c odd 4 1
375.3.k.a 80 25.e even 10 1
375.3.k.b 80 5.b even 2 1
375.3.k.b 80 25.f odd 20 1
375.3.k.c 80 1.a even 1 1 trivial
375.3.k.c 80 25.f odd 20 1 inner

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