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Properties

Label 600.2.bc.b
Level $600$
Weight $2$
Character orbit 600.bc
Analytic conductor $4.791$
Analytic rank $0$
Dimension $24$
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] = [600,2,Mod(169,600)] 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("600.169"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bc (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-4] 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{5} + 6 q^{9} - 10 q^{11} - 20 q^{13} - 2 q^{15} - 10 q^{17} + 14 q^{19} - 12 q^{21} + 24 q^{25} + 28 q^{29} - 4 q^{31} + 30 q^{33} - 14 q^{35} - 10 q^{37} - 20 q^{39} - 6 q^{41} + 4 q^{45} + 40 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 0 −0.587785 − 0.809017i 0 −2.23321 + 0.113099i 0 1.02996i 0 −0.309017 + 0.951057i 0
169.2 0 −0.587785 − 0.809017i 0 −0.454498 − 2.18939i 0 − 4.18140i 0 −0.309017 + 0.951057i 0
169.3 0 −0.587785 − 0.809017i 0 2.18770 − 0.462550i 0 1.81748i 0 −0.309017 + 0.951057i 0
169.4 0 0.587785 + 0.809017i 0 −1.69051 − 1.46362i 0 0.184115i 0 −0.309017 + 0.951057i 0
169.5 0 0.587785 + 0.809017i 0 −0.696192 + 2.12493i 0 2.74003i 0 −0.309017 + 0.951057i 0
169.6 0 0.587785 + 0.809017i 0 1.88670 − 1.20015i 0 2.21404i 0 −0.309017 + 0.951057i 0
289.1 0 −0.951057 − 0.309017i 0 −2.14870 − 0.618935i 0 0.783294i 0 0.809017 + 0.587785i 0
289.2 0 −0.951057 − 0.309017i 0 −0.580331 + 2.15945i 0 − 4.32657i 0 0.809017 + 0.587785i 0
289.3 0 −0.951057 − 0.309017i 0 2.22903 − 0.177242i 0 3.48278i 0 0.809017 + 0.587785i 0
289.4 0 0.951057 + 0.309017i 0 −2.02922 + 0.939285i 0 0.466908i 0 0.809017 + 0.587785i 0
289.5 0 0.951057 + 0.309017i 0 −0.633377 − 2.14449i 0 − 1.68627i 0 0.809017 + 0.587785i 0
289.6 0 0.951057 + 0.309017i 0 2.16260 + 0.568475i 0 3.63100i 0 0.809017 + 0.587785i 0
409.1 0 −0.951057 + 0.309017i 0 −2.14870 + 0.618935i 0 − 0.783294i 0 0.809017 − 0.587785i 0
409.2 0 −0.951057 + 0.309017i 0 −0.580331 − 2.15945i 0 4.32657i 0 0.809017 − 0.587785i 0
409.3 0 −0.951057 + 0.309017i 0 2.22903 + 0.177242i 0 − 3.48278i 0 0.809017 − 0.587785i 0
409.4 0 0.951057 − 0.309017i 0 −2.02922 − 0.939285i 0 − 0.466908i 0 0.809017 − 0.587785i 0
409.5 0 0.951057 − 0.309017i 0 −0.633377 + 2.14449i 0 1.68627i 0 0.809017 − 0.587785i 0
409.6 0 0.951057 − 0.309017i 0 2.16260 − 0.568475i 0 − 3.63100i 0 0.809017 − 0.587785i 0
529.1 0 −0.587785 + 0.809017i 0 −2.23321 − 0.113099i 0 − 1.02996i 0 −0.309017 − 0.951057i 0
529.2 0 −0.587785 + 0.809017i 0 −0.454498 + 2.18939i 0 4.18140i 0 −0.309017 − 0.951057i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bc.b ✓ 24
25.e even 10 1 inner 600.2.bc.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bc.b ✓ 24 1.a even 1 1 trivial
600.2.bc.b ✓ 24 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 82 T_{7}^{22} + 2823 T_{7}^{20} + 53224 T_{7}^{18} + 602215 T_{7}^{16} + 4226114 T_{7}^{14} + \cdots + 87025 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display