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Properties

Label 775.2.ck.c
Level $775$
Weight $2$
Character orbit 775.ck
Analytic conductor $6.188$
Analytic rank $0$
Dimension $80$
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] = [775,2,Mod(49,775)] 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("775.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 26])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.ck (of order \(30\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{4} + 8 q^{6} - 4 q^{9} - 56 q^{11} + 12 q^{14} + 16 q^{16} + 26 q^{19} - 16 q^{21} + 164 q^{24} + 64 q^{26} - 84 q^{29} - 20 q^{31} - 64 q^{34} - 26 q^{36} - 74 q^{39} + 72 q^{41} + 112 q^{44}+ \cdots - 106 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.26139 − 1.73615i 0.138074 + 0.310120i −0.805084 + 2.47779i 0 0.364249 − 0.630898i −1.83635 + 1.65346i 1.23541 − 0.401408i 1.93028 − 2.14380i 0
49.2 −1.04097 − 1.43277i −0.449533 − 1.00967i −0.351182 + 1.08083i 0 −0.978673 + 1.69511i −1.31785 + 1.18660i −1.45450 + 0.472596i 1.19004 − 1.32168i 0
49.3 −0.866342 − 1.19242i −1.18190 − 2.65459i −0.0532763 + 0.163968i 0 −2.14145 + 3.70910i 2.57003 − 2.31407i −2.56187 + 0.832401i −3.64256 + 4.04548i 0
49.4 −0.276293 − 0.380285i 0.673286 + 1.51222i 0.549755 − 1.69197i 0 0.389052 − 0.673858i 3.07510 − 2.76883i −1.68943 + 0.548929i 0.173882 − 0.193116i 0
49.5 −0.0238549 − 0.0328335i −0.860841 − 1.93348i 0.617525 − 1.90055i 0 −0.0429476 + 0.0743875i −0.719636 + 0.647963i −0.154329 + 0.0501445i −0.989905 + 1.09940i 0
49.6 0.0238549 + 0.0328335i 0.860841 + 1.93348i 0.617525 − 1.90055i 0 −0.0429476 + 0.0743875i 0.719636 − 0.647963i 0.154329 − 0.0501445i −0.989905 + 1.09940i 0
49.7 0.276293 + 0.380285i −0.673286 − 1.51222i 0.549755 − 1.69197i 0 0.389052 − 0.673858i −3.07510 + 2.76883i 1.68943 − 0.548929i 0.173882 − 0.193116i 0
49.8 0.866342 + 1.19242i 1.18190 + 2.65459i −0.0532763 + 0.163968i 0 −2.14145 + 3.70910i −2.57003 + 2.31407i 2.56187 − 0.832401i −3.64256 + 4.04548i 0
49.9 1.04097 + 1.43277i 0.449533 + 1.00967i −0.351182 + 1.08083i 0 −0.978673 + 1.69511i 1.31785 − 1.18660i 1.45450 − 0.472596i 1.19004 − 1.32168i 0
49.10 1.26139 + 1.73615i −0.138074 − 0.310120i −0.805084 + 2.47779i 0 0.364249 − 0.630898i 1.83635 − 1.65346i −1.23541 + 0.401408i 1.93028 − 2.14380i 0
174.1 −1.26139 + 1.73615i 0.138074 − 0.310120i −0.805084 − 2.47779i 0 0.364249 + 0.630898i −1.83635 − 1.65346i 1.23541 + 0.401408i 1.93028 + 2.14380i 0
174.2 −1.04097 + 1.43277i −0.449533 + 1.00967i −0.351182 − 1.08083i 0 −0.978673 − 1.69511i −1.31785 − 1.18660i −1.45450 − 0.472596i 1.19004 + 1.32168i 0
174.3 −0.866342 + 1.19242i −1.18190 + 2.65459i −0.0532763 − 0.163968i 0 −2.14145 − 3.70910i 2.57003 + 2.31407i −2.56187 − 0.832401i −3.64256 − 4.04548i 0
174.4 −0.276293 + 0.380285i 0.673286 − 1.51222i 0.549755 + 1.69197i 0 0.389052 + 0.673858i 3.07510 + 2.76883i −1.68943 − 0.548929i 0.173882 + 0.193116i 0
174.5 −0.0238549 + 0.0328335i −0.860841 + 1.93348i 0.617525 + 1.90055i 0 −0.0429476 − 0.0743875i −0.719636 − 0.647963i −0.154329 − 0.0501445i −0.989905 − 1.09940i 0
174.6 0.0238549 − 0.0328335i 0.860841 − 1.93348i 0.617525 + 1.90055i 0 −0.0429476 − 0.0743875i 0.719636 + 0.647963i 0.154329 + 0.0501445i −0.989905 − 1.09940i 0
174.7 0.276293 − 0.380285i −0.673286 + 1.51222i 0.549755 + 1.69197i 0 0.389052 + 0.673858i −3.07510 − 2.76883i 1.68943 + 0.548929i 0.173882 + 0.193116i 0
174.8 0.866342 − 1.19242i 1.18190 − 2.65459i −0.0532763 − 0.163968i 0 −2.14145 − 3.70910i −2.57003 − 2.31407i 2.56187 + 0.832401i −3.64256 − 4.04548i 0
174.9 1.04097 − 1.43277i 0.449533 − 1.00967i −0.351182 − 1.08083i 0 −0.978673 − 1.69511i 1.31785 + 1.18660i 1.45450 + 0.472596i 1.19004 + 1.32168i 0
174.10 1.26139 − 1.73615i −0.138074 + 0.310120i −0.805084 − 2.47779i 0 0.364249 + 0.630898i 1.83635 + 1.65346i −1.23541 − 0.401408i 1.93028 + 2.14380i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
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
31.g even 15 1 inner
155.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.ck.c 80
5.b even 2 1 inner 775.2.ck.c 80
5.c odd 4 1 155.2.q.a ✓ 40
5.c odd 4 1 775.2.bl.c 40
31.g even 15 1 inner 775.2.ck.c 80
155.u even 30 1 inner 775.2.ck.c 80
155.w odd 60 1 155.2.q.a ✓ 40
155.w odd 60 1 775.2.bl.c 40
155.w odd 60 1 4805.2.a.x 20
155.x even 60 1 4805.2.a.y 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.q.a ✓ 40 5.c odd 4 1
155.2.q.a ✓ 40 155.w odd 60 1
775.2.bl.c 40 5.c odd 4 1
775.2.bl.c 40 155.w odd 60 1
775.2.ck.c 80 1.a even 1 1 trivial
775.2.ck.c 80 5.b even 2 1 inner
775.2.ck.c 80 31.g even 15 1 inner
775.2.ck.c 80 155.u even 30 1 inner
4805.2.a.x 20 155.w odd 60 1
4805.2.a.y 20 155.x even 60 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 30 T_{2}^{78} + 501 T_{2}^{76} - 6191 T_{2}^{74} + 64256 T_{2}^{72} - 573961 T_{2}^{70} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display