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Properties

Label 480.6.h.b
Level $480$
Weight $6$
Character orbit 480.h
Analytic conductor $76.984$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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] = [480,6,Mod(191,480)] 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("480.191"); S:= CuspForms(chi, 6); 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, 0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 480.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,0,0,0,0,44,0,1112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(76.9842335102\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 44 q^{9} + 1112 q^{11} + 232 q^{13} + 1100 q^{15} + 820 q^{21} - 14432 q^{23} - 25000 q^{25} - 11784 q^{27} + 13264 q^{33} - 2152 q^{37} + 10056 q^{39} + 5900 q^{45} - 3376 q^{47} - 110672 q^{49}+ \cdots + 170440 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −15.3326 − 2.81282i 0 25.0000i 0 144.063i 0 227.176 + 86.2555i 0
191.2 0 −15.3326 + 2.81282i 0 − 25.0000i 0 − 144.063i 0 227.176 − 86.2555i 0
191.3 0 −15.1062 − 3.84742i 0 − 25.0000i 0 95.3389i 0 213.395 + 116.240i 0
191.4 0 −15.1062 + 3.84742i 0 25.0000i 0 − 95.3389i 0 213.395 − 116.240i 0
191.5 0 −15.0458 − 4.07722i 0 25.0000i 0 − 232.663i 0 209.753 + 122.690i 0
191.6 0 −15.0458 + 4.07722i 0 − 25.0000i 0 232.663i 0 209.753 − 122.690i 0
191.7 0 −12.9206 − 8.72109i 0 25.0000i 0 163.490i 0 90.8850 + 225.364i 0
191.8 0 −12.9206 + 8.72109i 0 − 25.0000i 0 − 163.490i 0 90.8850 − 225.364i 0
191.9 0 −12.3372 − 9.52858i 0 − 25.0000i 0 − 103.877i 0 61.4124 + 235.112i 0
191.10 0 −12.3372 + 9.52858i 0 25.0000i 0 103.877i 0 61.4124 − 235.112i 0
191.11 0 −11.2333 − 10.8080i 0 − 25.0000i 0 74.7782i 0 9.37266 + 242.819i 0
191.12 0 −11.2333 + 10.8080i 0 25.0000i 0 − 74.7782i 0 9.37266 − 242.819i 0
191.13 0 −9.65614 − 12.2376i 0 25.0000i 0 56.9854i 0 −56.5180 + 236.336i 0
191.14 0 −9.65614 + 12.2376i 0 − 25.0000i 0 − 56.9854i 0 −56.5180 − 236.336i 0
191.15 0 −6.27490 − 14.2697i 0 25.0000i 0 − 128.112i 0 −164.251 + 179.083i 0
191.16 0 −6.27490 + 14.2697i 0 − 25.0000i 0 128.112i 0 −164.251 − 179.083i 0
191.17 0 −0.00614934 − 15.5885i 0 − 25.0000i 0 − 197.208i 0 −243.000 + 0.191717i 0
191.18 0 −0.00614934 + 15.5885i 0 25.0000i 0 197.208i 0 −243.000 − 0.191717i 0
191.19 0 0.797118 − 15.5681i 0 − 25.0000i 0 26.5383i 0 −241.729 − 24.8192i 0
191.20 0 0.797118 + 15.5681i 0 25.0000i 0 − 26.5383i 0 −241.729 + 24.8192i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.6.h.b yes 40
3.b odd 2 1 480.6.h.a ✓ 40
4.b odd 2 1 480.6.h.a ✓ 40
12.b even 2 1 inner 480.6.h.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.6.h.a ✓ 40 3.b odd 2 1
480.6.h.a ✓ 40 4.b odd 2 1
480.6.h.b yes 40 1.a even 1 1 trivial
480.6.h.b yes 40 12.b even 2 1 inner