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Properties

Label 875.2.n.d
Level $875$
Weight $2$
Character orbit 875.n
Analytic conductor $6.987$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [875,2,Mod(99,875)] 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("875.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([9, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.n (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,18,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 18 q^{4} + 30 q^{6} + 12 q^{9} + 4 q^{11} + 6 q^{14} - 30 q^{16} + 16 q^{19} - 8 q^{21} + 92 q^{24} - 68 q^{26} - 26 q^{29} + 10 q^{31} - 50 q^{34} + 128 q^{36} + 36 q^{39} - 52 q^{41} + 74 q^{44}+ \cdots + 132 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −2.59530 + 0.843265i 0.886773 + 1.22054i 4.40646 − 3.20148i 0 −3.33068 − 2.41988i 1.00000i −5.52843 + 7.60923i 0.223704 − 0.688489i 0
99.2 −2.44884 + 0.795678i −1.34866 − 1.85627i 3.74570 − 2.72141i 0 4.77966 + 3.47263i − 1.00000i −3.98034 + 5.47846i −0.699813 + 2.15380i 0
99.3 −2.42712 + 0.788620i −1.80104 − 2.47892i 3.65097 − 2.65259i 0 6.32628 + 4.59631i 1.00000i −3.76938 + 5.18811i −1.97425 + 6.07612i 0
99.4 −1.16988 + 0.380117i 0.0547698 + 0.0753841i −0.393906 + 0.286190i 0 −0.0927287 − 0.0673714i 1.00000i 1.79809 − 2.47485i 0.924368 − 2.84491i 0
99.5 −1.13629 + 0.369205i 0.735989 + 1.01300i −0.463180 + 0.336520i 0 −1.21030 − 0.879338i − 1.00000i 1.80660 − 2.48657i 0.442558 − 1.36205i 0
99.6 −0.999270 + 0.324683i 0.772801 + 1.06367i −0.724912 + 0.526679i 0 −1.11759 − 0.811978i 1.00000i 1.78855 − 2.46172i 0.392880 − 1.20916i 0
99.7 −0.530468 + 0.172359i −1.94012 − 2.67034i −1.36635 + 0.992708i 0 1.48943 + 1.08213i − 1.00000i 1.20939 − 1.66459i −2.43961 + 7.50836i 0
99.8 −0.222797 + 0.0723910i 0.927248 + 1.27625i −1.57364 + 1.14331i 0 −0.298977 − 0.217219i − 1.00000i 0.543227 − 0.747688i 0.158032 − 0.486374i 0
99.9 0.222797 − 0.0723910i −0.927248 − 1.27625i −1.57364 + 1.14331i 0 −0.298977 − 0.217219i 1.00000i −0.543227 + 0.747688i 0.158032 − 0.486374i 0
99.10 0.530468 − 0.172359i 1.94012 + 2.67034i −1.36635 + 0.992708i 0 1.48943 + 1.08213i 1.00000i −1.20939 + 1.66459i −2.43961 + 7.50836i 0
99.11 0.999270 − 0.324683i −0.772801 − 1.06367i −0.724912 + 0.526679i 0 −1.11759 − 0.811978i − 1.00000i −1.78855 + 2.46172i 0.392880 − 1.20916i 0
99.12 1.13629 − 0.369205i −0.735989 − 1.01300i −0.463180 + 0.336520i 0 −1.21030 − 0.879338i 1.00000i −1.80660 + 2.48657i 0.442558 − 1.36205i 0
99.13 1.16988 − 0.380117i −0.0547698 − 0.0753841i −0.393906 + 0.286190i 0 −0.0927287 − 0.0673714i − 1.00000i −1.79809 + 2.47485i 0.924368 − 2.84491i 0
99.14 2.42712 − 0.788620i 1.80104 + 2.47892i 3.65097 − 2.65259i 0 6.32628 + 4.59631i − 1.00000i 3.76938 − 5.18811i −1.97425 + 6.07612i 0
99.15 2.44884 − 0.795678i 1.34866 + 1.85627i 3.74570 − 2.72141i 0 4.77966 + 3.47263i 1.00000i 3.98034 − 5.47846i −0.699813 + 2.15380i 0
99.16 2.59530 − 0.843265i −0.886773 − 1.22054i 4.40646 − 3.20148i 0 −3.33068 − 2.41988i − 1.00000i 5.52843 − 7.60923i 0.223704 − 0.688489i 0
274.1 −2.59530 − 0.843265i 0.886773 − 1.22054i 4.40646 + 3.20148i 0 −3.33068 + 2.41988i − 1.00000i −5.52843 − 7.60923i 0.223704 + 0.688489i 0
274.2 −2.44884 − 0.795678i −1.34866 + 1.85627i 3.74570 + 2.72141i 0 4.77966 − 3.47263i 1.00000i −3.98034 − 5.47846i −0.699813 − 2.15380i 0
274.3 −2.42712 − 0.788620i −1.80104 + 2.47892i 3.65097 + 2.65259i 0 6.32628 − 4.59631i − 1.00000i −3.76938 − 5.18811i −1.97425 − 6.07612i 0
274.4 −1.16988 − 0.380117i 0.0547698 − 0.0753841i −0.393906 − 0.286190i 0 −0.0927287 + 0.0673714i − 1.00000i 1.79809 + 2.47485i 0.924368 + 2.84491i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 875.2.n.d 64
5.b even 2 1 inner 875.2.n.d 64
5.c odd 4 1 175.2.h.c ✓ 32
5.c odd 4 1 875.2.h.c 32
25.d even 5 1 inner 875.2.n.d 64
25.e even 10 1 inner 875.2.n.d 64
25.f odd 20 1 175.2.h.c ✓ 32
25.f odd 20 1 875.2.h.c 32
25.f odd 20 1 4375.2.a.i 16
25.f odd 20 1 4375.2.a.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.h.c ✓ 32 5.c odd 4 1
175.2.h.c ✓ 32 25.f odd 20 1
875.2.h.c 32 5.c odd 4 1
875.2.h.c 32 25.f odd 20 1
875.2.n.d 64 1.a even 1 1 trivial
875.2.n.d 64 5.b even 2 1 inner
875.2.n.d 64 25.d even 5 1 inner
875.2.n.d 64 25.e even 10 1 inner
4375.2.a.i 16 25.f odd 20 1
4375.2.a.l 16 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 25 T_{2}^{62} + 379 T_{2}^{60} - 4531 T_{2}^{58} + 47740 T_{2}^{56} - 421543 T_{2}^{54} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display