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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,4,Mod(101,425)] 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("425.101"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,72,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 72 q^{4} - 172 q^{9} - 64 q^{16} + 156 q^{19} - 120 q^{21} + 524 q^{26} + 132 q^{34} - 196 q^{36} - 160 q^{49} + 36 q^{51} + 532 q^{59} - 632 q^{64} - 1976 q^{66} + 1072 q^{69} + 3116 q^{76} - 24 q^{81}+ \cdots + 1564 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −5.07030 − 3.15076i 17.7079 0 15.9753i 2.84475i −49.2221 17.0727 0
101.2 −5.07030 3.15076i 17.7079 0 − 15.9753i − 2.84475i −49.2221 17.0727 0
101.3 −4.28591 − 8.78504i 10.3690 0 37.6519i − 13.5594i −10.1535 −50.1770 0
101.4 −4.28591 8.78504i 10.3690 0 − 37.6519i 13.5594i −10.1535 −50.1770 0
101.5 −3.30829 − 0.644147i 2.94478 0 2.13103i − 24.9231i 16.7241 26.5851 0
101.6 −3.30829 0.644147i 2.94478 0 − 2.13103i 24.9231i 16.7241 26.5851 0
101.7 −2.69457 − 5.79684i −0.739315 0 15.6200i 34.0883i 23.5487 −6.60337 0
101.8 −2.69457 5.79684i −0.739315 0 − 15.6200i − 34.0883i 23.5487 −6.60337 0
101.9 −1.90493 − 8.33875i −4.37124 0 15.8847i − 10.3098i 23.5663 −42.5348 0
101.10 −1.90493 8.33875i −4.37124 0 − 15.8847i 10.3098i 23.5663 −42.5348 0
101.11 −0.298039 − 3.78716i −7.91117 0 1.12872i − 4.07231i 4.74215 12.6574 0
101.12 −0.298039 3.78716i −7.91117 0 − 1.12872i 4.07231i 4.74215 12.6574 0
101.13 0.298039 − 3.78716i −7.91117 0 − 1.12872i − 4.07231i −4.74215 12.6574 0
101.14 0.298039 3.78716i −7.91117 0 1.12872i 4.07231i −4.74215 12.6574 0
101.15 1.90493 − 8.33875i −4.37124 0 − 15.8847i − 10.3098i −23.5663 −42.5348 0
101.16 1.90493 8.33875i −4.37124 0 15.8847i 10.3098i −23.5663 −42.5348 0
101.17 2.69457 − 5.79684i −0.739315 0 − 15.6200i 34.0883i −23.5487 −6.60337 0
101.18 2.69457 5.79684i −0.739315 0 15.6200i − 34.0883i −23.5487 −6.60337 0
101.19 3.30829 − 0.644147i 2.94478 0 − 2.13103i − 24.9231i −16.7241 26.5851 0
101.20 3.30829 0.644147i 2.94478 0 2.13103i 24.9231i −16.7241 26.5851 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 101.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.4.d.h 24
5.b even 2 1 inner 425.4.d.h 24
5.c odd 4 2 85.4.c.a ✓ 24
17.b even 2 1 inner 425.4.d.h 24
85.c even 2 1 inner 425.4.d.h 24
85.g odd 4 2 85.4.c.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.c.a ✓ 24 5.c odd 4 2
85.4.c.a ✓ 24 85.g odd 4 2
425.4.d.h 24 1.a even 1 1 trivial
425.4.d.h 24 5.b even 2 1 inner
425.4.d.h 24 17.b even 2 1 inner
425.4.d.h 24 85.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{12} - 66T_{2}^{10} + 1586T_{2}^{8} - 17154T_{2}^{6} + 82945T_{2}^{4} - 143408T_{2}^{2} + 12096 \) Copy content Toggle raw display
\( T_{3}^{12} + 205T_{3}^{10} + 14900T_{3}^{8} + 462048T_{3}^{6} + 6031852T_{3}^{4} + 28100516T_{3}^{2} + 10653728 \) Copy content Toggle raw display