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Properties

Label 875.2.n.c
Level $875$
Weight $2$
Character orbit 875.n
Analytic conductor $6.987$
Analytic rank $0$
Dimension $56$
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] = [875,2,Mod(99,875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.n (of order \(10\), degree \(4\), 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: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{4} + 6 q^{9} + 8 q^{11} + 40 q^{12} - 4 q^{14} - 32 q^{16} + 12 q^{19} + 4 q^{21} + 30 q^{22} - 10 q^{23} - 28 q^{24} + 12 q^{26} - 30 q^{27} - 2 q^{29} + 12 q^{31} - 20 q^{33} - 14 q^{36} + 70 q^{37} + 70 q^{38} - 4 q^{39} + 4 q^{41} - 50 q^{42} + 22 q^{44} - 4 q^{46} + 10 q^{47} - 30 q^{48} - 56 q^{49} - 44 q^{51} + 20 q^{53} + 54 q^{54} + 12 q^{56} - 10 q^{58} - 6 q^{59} - 4 q^{61} - 50 q^{62} - 20 q^{63} + 24 q^{64} - 74 q^{66} - 10 q^{67} - 78 q^{69} - 8 q^{71} - 140 q^{72} - 40 q^{73} + 60 q^{74} + 52 q^{76} + 20 q^{77} + 90 q^{78} - 72 q^{81} + 30 q^{83} - 12 q^{84} - 20 q^{86} - 30 q^{87} - 140 q^{88} + 38 q^{89} + 8 q^{91} - 80 q^{92} + 88 q^{94} - 28 q^{96} + 30 q^{97} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −2.44207 + 0.793475i 1.88310 + 2.59187i 3.71604 − 2.69986i 0 −6.65525 − 4.83532i − 1.00000i −3.91399 + 5.38714i −2.24466 + 6.90836i 0
99.2 −2.11404 + 0.686892i −0.515049 − 0.708904i 2.37929 − 1.72866i 0 1.57577 + 1.14486i 1.00000i −1.22941 + 1.69214i 0.689782 − 2.12293i 0
99.3 −1.99559 + 0.648406i 0.448050 + 0.616687i 1.94391 − 1.41233i 0 −1.29399 − 0.940136i − 1.00000i −0.496791 + 0.683774i 0.747496 − 2.30056i 0
99.4 −1.19597 + 0.388594i −0.374566 − 0.515546i −0.338697 + 0.246078i 0 0.648308 + 0.471023i − 1.00000i 1.78775 − 2.46062i 0.801563 − 2.46696i 0
99.5 −1.09195 + 0.354796i −1.51811 − 2.08949i −0.551558 + 0.400730i 0 2.39904 + 1.74301i 1.00000i 1.80982 − 2.49101i −1.13429 + 3.49098i 0
99.6 −1.03051 + 0.334833i 1.58321 + 2.17910i −0.668196 + 0.485473i 0 −2.36115 − 1.71547i 1.00000i 1.79981 − 2.47723i −1.31487 + 4.04676i 0
99.7 −0.225611 + 0.0733055i −1.21485 − 1.67209i −1.57251 + 1.14249i 0 0.396656 + 0.288188i − 1.00000i 0.549895 − 0.756865i −0.392990 + 1.20950i 0
99.8 0.387846 − 0.126019i 0.531527 + 0.731584i −1.48349 + 1.07782i 0 0.298344 + 0.216760i − 1.00000i −0.918945 + 1.26482i 0.674357 − 2.07546i 0
99.9 0.798956 − 0.259597i 0.142668 + 0.196366i −1.04709 + 0.760758i 0 0.164962 + 0.119852i 1.00000i −1.62666 + 2.23890i 0.908846 − 2.79714i 0
99.10 0.977435 − 0.317588i −1.68681 − 2.32169i −0.763516 + 0.554727i 0 −2.38608 − 1.73359i 1.00000i −1.77829 + 2.44761i −1.61787 + 4.97930i 0
99.11 1.51762 − 0.493105i 1.52541 + 2.09954i 0.441990 − 0.321125i 0 3.35029 + 2.43413i − 1.00000i −1.36346 + 1.87664i −1.15416 + 3.55215i 0
99.12 1.82157 − 0.591863i 1.68900 + 2.32470i 1.34977 − 0.980668i 0 4.45253 + 3.23495i 1.00000i −0.373298 + 0.513801i −1.62449 + 4.99966i 0
99.13 2.05165 − 0.666622i −0.505071 − 0.695171i 2.14685 − 1.55978i 0 −1.49965 − 1.08956i − 1.00000i 0.828832 − 1.14079i 0.698885 − 2.15095i 0
99.14 2.54065 − 0.825507i 0.247551 + 0.340724i 4.15540 − 3.01908i 0 0.910210 + 0.661306i 1.00000i 4.92473 − 6.77832i 0.872239 − 2.68448i 0
274.1 −2.44207 − 0.793475i 1.88310 − 2.59187i 3.71604 + 2.69986i 0 −6.65525 + 4.83532i 1.00000i −3.91399 − 5.38714i −2.24466 − 6.90836i 0
274.2 −2.11404 − 0.686892i −0.515049 + 0.708904i 2.37929 + 1.72866i 0 1.57577 − 1.14486i − 1.00000i −1.22941 − 1.69214i 0.689782 + 2.12293i 0
274.3 −1.99559 − 0.648406i 0.448050 − 0.616687i 1.94391 + 1.41233i 0 −1.29399 + 0.940136i 1.00000i −0.496791 − 0.683774i 0.747496 + 2.30056i 0
274.4 −1.19597 − 0.388594i −0.374566 + 0.515546i −0.338697 − 0.246078i 0 0.648308 − 0.471023i 1.00000i 1.78775 + 2.46062i 0.801563 + 2.46696i 0
274.5 −1.09195 − 0.354796i −1.51811 + 2.08949i −0.551558 − 0.400730i 0 2.39904 − 1.74301i − 1.00000i 1.80982 + 2.49101i −1.13429 − 3.49098i 0
274.6 −1.03051 − 0.334833i 1.58321 − 2.17910i −0.668196 − 0.485473i 0 −2.36115 + 1.71547i − 1.00000i 1.79981 + 2.47723i −1.31487 − 4.04676i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c 56
5.b even 2 1 175.2.n.a ✓ 56
5.c odd 4 1 875.2.h.d 56
5.c odd 4 1 875.2.h.e 56
25.d even 5 1 175.2.n.a ✓ 56
25.e even 10 1 inner 875.2.n.c 56
25.f odd 20 1 875.2.h.d 56
25.f odd 20 1 875.2.h.e 56
25.f odd 20 1 4375.2.a.o 28
25.f odd 20 1 4375.2.a.p 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.n.a ✓ 56 5.b even 2 1
175.2.n.a ✓ 56 25.d even 5 1
875.2.h.d 56 5.c odd 4 1
875.2.h.d 56 25.f odd 20 1
875.2.h.e 56 5.c odd 4 1
875.2.h.e 56 25.f odd 20 1
875.2.n.c 56 1.a even 1 1 trivial
875.2.n.c 56 25.e even 10 1 inner
4375.2.a.o 28 25.f odd 20 1
4375.2.a.p 28 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 20 T_{2}^{54} + 254 T_{2}^{52} - 2636 T_{2}^{50} - 90 T_{2}^{49} + 24025 T_{2}^{48} + \cdots + 42025 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display