↑

Properties

Label 400.3.bg.e
Level $400$
Weight $3$
Character orbit 400.bg
Analytic conductor $10.899$
Analytic rank $0$
Dimension $56$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,3,Mod(17,400)] 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("400.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 0, 13])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.bg (of order \(20\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 200)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 10 q^{5} + 4 q^{7} + 40 q^{9} + 16 q^{11} + 14 q^{13} + 10 q^{15} + 22 q^{17} - 50 q^{19} + 100 q^{21} + 48 q^{23} + 150 q^{25} + 210 q^{27} + 108 q^{31} - 140 q^{33} - 70 q^{35} + 236 q^{37} - 80 q^{39}+ \cdots + 386 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −0.706680 + 4.46180i 0 −4.37575 − 2.41926i 0 0.0526437 + 0.0526437i 0 −10.8488 − 3.52499i 0
17.2 0 −0.466746 + 2.94692i 0 4.43224 − 2.31415i 0 −6.13972 − 6.13972i 0 0.0930182 + 0.0302234i 0
17.3 0 −0.267752 + 1.69052i 0 −1.42022 + 4.79406i 0 −1.33100 − 1.33100i 0 5.77334 + 1.87587i 0
17.4 0 −0.140767 + 0.888769i 0 4.81120 − 1.36101i 0 9.28221 + 9.28221i 0 7.78941 + 2.53093i 0
17.5 0 0.453949 − 2.86612i 0 0.0427179 − 4.99982i 0 −5.66477 − 5.66477i 0 0.550933 + 0.179009i 0
17.6 0 0.489530 − 3.09077i 0 −4.99979 + 0.0458106i 0 5.04313 + 5.04313i 0 −0.753717 − 0.244897i 0
17.7 0 0.638468 − 4.03113i 0 2.77827 + 4.15707i 0 −0.800018 − 0.800018i 0 −7.28283 − 2.36633i 0
33.1 0 −4.37171 − 0.692411i 0 −3.19425 + 3.84666i 0 6.50707 − 6.50707i 0 10.0729 + 3.27289i 0
33.2 0 −3.98022 − 0.630404i 0 4.90048 − 0.992634i 0 −1.90476 + 1.90476i 0 6.88520 + 2.23714i 0
33.3 0 −2.11884 − 0.335591i 0 −4.81583 − 1.34455i 0 −8.52465 + 8.52465i 0 −4.18265 − 1.35902i 0
33.4 0 0.304092 + 0.0481634i 0 −1.19185 − 4.85587i 0 5.20977 − 5.20977i 0 −8.46936 − 2.75186i 0
33.5 0 1.80371 + 0.285680i 0 −0.621686 + 4.96120i 0 −3.65826 + 3.65826i 0 −5.38774 − 1.75058i 0
33.6 0 2.88863 + 0.457514i 0 4.99800 + 0.141281i 0 0.412482 − 0.412482i 0 −0.424657 − 0.137979i 0
33.7 0 5.47433 + 0.867049i 0 −4.96157 + 0.618740i 0 4.75195 − 4.75195i 0 20.6571 + 6.71188i 0
97.1 0 −4.37171 + 0.692411i 0 −3.19425 − 3.84666i 0 6.50707 + 6.50707i 0 10.0729 − 3.27289i 0
97.2 0 −3.98022 + 0.630404i 0 4.90048 + 0.992634i 0 −1.90476 − 1.90476i 0 6.88520 − 2.23714i 0
97.3 0 −2.11884 + 0.335591i 0 −4.81583 + 1.34455i 0 −8.52465 − 8.52465i 0 −4.18265 + 1.35902i 0
97.4 0 0.304092 − 0.0481634i 0 −1.19185 + 4.85587i 0 5.20977 + 5.20977i 0 −8.46936 + 2.75186i 0
97.5 0 1.80371 − 0.285680i 0 −0.621686 − 4.96120i 0 −3.65826 − 3.65826i 0 −5.38774 + 1.75058i 0
97.6 0 2.88863 − 0.457514i 0 4.99800 − 0.141281i 0 0.412482 + 0.412482i 0 −0.424657 + 0.137979i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.7
Significant digits:
Format:

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 400.3.bg.e 56
4.b odd 2 1 200.3.u.a ✓ 56
25.f odd 20 1 inner 400.3.bg.e 56
100.l even 20 1 200.3.u.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.u.a ✓ 56 4.b odd 2 1
200.3.u.a ✓ 56 100.l even 20 1
400.3.bg.e 56 1.a even 1 1 trivial
400.3.bg.e 56 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 20 T_{3}^{54} - 70 T_{3}^{53} - 315 T_{3}^{52} + 1348 T_{3}^{51} + 5230 T_{3}^{50} + \cdots + 35\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display