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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [480,4,Mod(431,480)] 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("480.431"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.3209168028\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 120 q^{5} - 12 q^{19} + 4 q^{21} + 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} - 656 q^{39} - 924 q^{47} - 816 q^{49} + 700 q^{51} - 528 q^{53} - 172 q^{57} + 476 q^{63} - 1632 q^{67} - 980 q^{69}+ \cdots + 1328 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1 0 −5.02993 − 1.30378i 0 −5.00000 0 − 8.16056i 0 23.6003 + 13.1158i 0
431.2 0 −5.02993 + 1.30378i 0 −5.00000 0 8.16056i 0 23.6003 − 13.1158i 0
431.3 0 −4.99995 − 1.41438i 0 −5.00000 0 − 35.4366i 0 22.9991 + 14.1437i 0
431.4 0 −4.99995 + 1.41438i 0 −5.00000 0 35.4366i 0 22.9991 − 14.1437i 0
431.5 0 −3.91264 − 3.41925i 0 −5.00000 0 2.12151i 0 3.61744 + 26.7566i 0
431.6 0 −3.91264 + 3.41925i 0 −5.00000 0 − 2.12151i 0 3.61744 − 26.7566i 0
431.7 0 −3.31357 − 4.00253i 0 −5.00000 0 30.8728i 0 −5.04045 + 26.5253i 0
431.8 0 −3.31357 + 4.00253i 0 −5.00000 0 − 30.8728i 0 −5.04045 − 26.5253i 0
431.9 0 −2.58903 − 4.50521i 0 −5.00000 0 − 6.99225i 0 −13.5939 + 23.3282i 0
431.10 0 −2.58903 + 4.50521i 0 −5.00000 0 6.99225i 0 −13.5939 − 23.3282i 0
431.11 0 0.403912 − 5.18043i 0 −5.00000 0 − 23.0707i 0 −26.6737 − 4.18488i 0
431.12 0 0.403912 + 5.18043i 0 −5.00000 0 23.0707i 0 −26.6737 + 4.18488i 0
431.13 0 0.899959 − 5.11762i 0 −5.00000 0 23.1184i 0 −25.3801 − 9.21130i 0
431.14 0 0.899959 + 5.11762i 0 −5.00000 0 − 23.1184i 0 −25.3801 + 9.21130i 0
431.15 0 1.49310 − 4.97701i 0 −5.00000 0 − 14.5956i 0 −22.5413 − 14.8623i 0
431.16 0 1.49310 + 4.97701i 0 −5.00000 0 14.5956i 0 −22.5413 + 14.8623i 0
431.17 0 3.04181 − 4.21277i 0 −5.00000 0 − 13.6956i 0 −8.49478 − 25.6289i 0
431.18 0 3.04181 + 4.21277i 0 −5.00000 0 13.6956i 0 −8.49478 + 25.6289i 0
431.19 0 3.93281 − 3.39603i 0 −5.00000 0 20.9207i 0 3.93395 − 26.7119i 0
431.20 0 3.93281 + 3.39603i 0 −5.00000 0 − 20.9207i 0 3.93395 + 26.7119i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 431.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.4.b.a 24
3.b odd 2 1 480.4.b.b 24
4.b odd 2 1 120.4.b.b yes 24
8.b even 2 1 120.4.b.a ✓ 24
8.d odd 2 1 480.4.b.b 24
12.b even 2 1 120.4.b.a ✓ 24
24.f even 2 1 inner 480.4.b.a 24
24.h odd 2 1 120.4.b.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.b.a ✓ 24 8.b even 2 1
120.4.b.a ✓ 24 12.b even 2 1
120.4.b.b yes 24 4.b odd 2 1
120.4.b.b yes 24 24.h odd 2 1
480.4.b.a 24 1.a even 1 1 trivial
480.4.b.a 24 24.f even 2 1 inner
480.4.b.b 24 3.b odd 2 1
480.4.b.b 24 8.d odd 2 1