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Properties

Label 400.2.j.e
Level $400$
Weight $2$
Character orbit 400.j
Analytic conductor $3.194$
Analytic rank $0$
Dimension $24$
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] = [400,2,Mod(43,400)] 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("400.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} + 12 q^{6} - 40 q^{9} - 20 q^{11} + 44 q^{14} - 4 q^{16} - 12 q^{19} + 24 q^{24} + 32 q^{26} + 8 q^{29} + 32 q^{34} - 60 q^{36} - 44 q^{44} - 76 q^{46} + 20 q^{51} + 16 q^{54} - 28 q^{56}+ \cdots + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.33179 + 0.475759i − 2.35800i 1.54731 − 1.26722i 0 1.12184 + 3.14035i −2.66357 + 2.66357i −1.45779 + 2.42381i −2.56018 0
43.2 −1.27006 − 0.622050i 3.25766i 1.22611 + 1.58008i 0 2.02643 − 4.13743i −2.54012 + 2.54012i −0.574339 − 2.76950i −7.61238 0
43.3 −0.911865 + 1.08097i 0.619018i −0.337006 − 1.97140i 0 −0.669142 − 0.564461i −1.82373 + 1.82373i 2.43834 + 1.43336i 2.61682 0
43.4 −0.804554 − 1.16305i − 2.70780i −0.705387 + 1.87148i 0 −3.14932 + 2.17857i −1.60911 + 1.60911i 2.74415 − 0.685302i −4.33218 0
43.5 −0.359994 − 1.36763i 1.86755i −1.74081 + 0.984676i 0 2.55411 − 0.672307i −0.719989 + 0.719989i 1.97335 + 2.02630i −0.487737 0
43.6 −0.0699536 + 1.41248i − 0.790153i −1.99021 − 0.197616i 0 1.11608 + 0.0552740i −0.139907 + 0.139907i 0.418352 − 2.79732i 2.37566 0
43.7 0.0699536 − 1.41248i 0.790153i −1.99021 − 0.197616i 0 1.11608 + 0.0552740i 0.139907 − 0.139907i −0.418352 + 2.79732i 2.37566 0
43.8 0.359994 + 1.36763i − 1.86755i −1.74081 + 0.984676i 0 2.55411 − 0.672307i 0.719989 − 0.719989i −1.97335 − 2.02630i −0.487737 0
43.9 0.804554 + 1.16305i 2.70780i −0.705387 + 1.87148i 0 −3.14932 + 2.17857i 1.60911 − 1.60911i −2.74415 + 0.685302i −4.33218 0
43.10 0.911865 − 1.08097i − 0.619018i −0.337006 − 1.97140i 0 −0.669142 − 0.564461i 1.82373 − 1.82373i −2.43834 − 1.43336i 2.61682 0
43.11 1.27006 + 0.622050i − 3.25766i 1.22611 + 1.58008i 0 2.02643 − 4.13743i 2.54012 − 2.54012i 0.574339 + 2.76950i −7.61238 0
43.12 1.33179 − 0.475759i 2.35800i 1.54731 − 1.26722i 0 1.12184 + 3.14035i 2.66357 − 2.66357i 1.45779 − 2.42381i −2.56018 0
307.1 −1.33179 − 0.475759i 2.35800i 1.54731 + 1.26722i 0 1.12184 − 3.14035i −2.66357 − 2.66357i −1.45779 − 2.42381i −2.56018 0
307.2 −1.27006 + 0.622050i − 3.25766i 1.22611 − 1.58008i 0 2.02643 + 4.13743i −2.54012 − 2.54012i −0.574339 + 2.76950i −7.61238 0
307.3 −0.911865 − 1.08097i − 0.619018i −0.337006 + 1.97140i 0 −0.669142 + 0.564461i −1.82373 − 1.82373i 2.43834 − 1.43336i 2.61682 0
307.4 −0.804554 + 1.16305i 2.70780i −0.705387 − 1.87148i 0 −3.14932 − 2.17857i −1.60911 − 1.60911i 2.74415 + 0.685302i −4.33218 0
307.5 −0.359994 + 1.36763i − 1.86755i −1.74081 − 0.984676i 0 2.55411 + 0.672307i −0.719989 − 0.719989i 1.97335 − 2.02630i −0.487737 0
307.6 −0.0699536 − 1.41248i 0.790153i −1.99021 + 0.197616i 0 1.11608 − 0.0552740i −0.139907 − 0.139907i 0.418352 + 2.79732i 2.37566 0
307.7 0.0699536 + 1.41248i − 0.790153i −1.99021 + 0.197616i 0 1.11608 − 0.0552740i 0.139907 + 0.139907i −0.418352 − 2.79732i 2.37566 0
307.8 0.359994 − 1.36763i 1.86755i −1.74081 − 0.984676i 0 2.55411 + 0.672307i 0.719989 + 0.719989i −1.97335 + 2.02630i −0.487737 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
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
80.j even 4 1 inner
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.j.e ✓ 24
4.b odd 2 1 1600.2.j.e 24
5.b even 2 1 inner 400.2.j.e ✓ 24
5.c odd 4 2 400.2.s.e yes 24
16.e even 4 1 1600.2.s.e 24
16.f odd 4 1 400.2.s.e yes 24
20.d odd 2 1 1600.2.j.e 24
20.e even 4 2 1600.2.s.e 24
80.i odd 4 1 1600.2.j.e 24
80.j even 4 1 inner 400.2.j.e ✓ 24
80.k odd 4 1 400.2.s.e yes 24
80.q even 4 1 1600.2.s.e 24
80.s even 4 1 inner 400.2.j.e ✓ 24
80.t odd 4 1 1600.2.j.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.e ✓ 24 1.a even 1 1 trivial
400.2.j.e ✓ 24 5.b even 2 1 inner
400.2.j.e ✓ 24 80.j even 4 1 inner
400.2.j.e ✓ 24 80.s even 4 1 inner
400.2.s.e yes 24 5.c odd 4 2
400.2.s.e yes 24 16.f odd 4 1
400.2.s.e yes 24 80.k odd 4 1
1600.2.j.e 24 4.b odd 2 1
1600.2.j.e 24 20.d odd 2 1
1600.2.j.e 24 80.i odd 4 1
1600.2.j.e 24 80.t odd 4 1
1600.2.s.e 24 16.e even 4 1
1600.2.s.e 24 20.e even 4 2
1600.2.s.e 24 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 28T_{3}^{10} + 287T_{3}^{8} + 1320T_{3}^{6} + 2631T_{3}^{4} + 1772T_{3}^{2} + 361 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display