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Properties

Label 600.2.be.a
Level $600$
Weight $2$
Character orbit 600.be
Analytic conductor $4.791$
Analytic rank $0$
Dimension $120$
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(109,600)] 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("600.109"); 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, 5, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.be (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 30 q^{3} + 10 q^{4} - 15 q^{8} - 30 q^{9} + 9 q^{10} + 8 q^{13} + 6 q^{14} + 6 q^{16} + 13 q^{20} + 31 q^{22} - 2 q^{25} + 10 q^{26} - 30 q^{27} + 8 q^{28} + 9 q^{30} + 12 q^{31} - 10 q^{32} + 32 q^{34}+ \cdots - 31 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41414 − 0.0145696i 0.309017 + 0.951057i 1.99958 + 0.0412067i 0.432940 + 2.19376i −0.423136 − 1.34943i − 0.643164i −2.82708 − 0.0874049i −0.809017 + 0.587785i −0.580275 − 3.10858i
109.2 −1.40750 − 0.137619i 0.309017 + 0.951057i 1.96212 + 0.387399i 1.96621 + 1.06491i −0.304058 − 1.38114i 1.46554i −2.70838 − 0.815291i −0.809017 + 0.587785i −2.62089 − 1.76944i
109.3 −1.33241 + 0.474014i 0.309017 + 0.951057i 1.55062 − 1.26316i 1.59403 − 1.56814i −0.862551 − 1.12072i − 1.85305i −1.46730 + 2.41806i −0.809017 + 0.587785i −1.38058 + 2.84499i
109.4 −1.33227 − 0.474398i 0.309017 + 0.951057i 1.54989 + 1.26405i −2.08356 − 0.811661i 0.0394846 − 1.41366i − 4.66350i −1.46521 − 2.41933i −0.809017 + 0.587785i 2.39081 + 2.06979i
109.5 −1.31688 − 0.515578i 0.309017 + 0.951057i 1.46836 + 1.35791i 1.16726 − 1.90723i 0.0834045 − 1.41175i − 1.91925i −1.23355 − 2.54526i −0.809017 + 0.587785i −2.52046 + 1.90978i
109.6 −1.31138 + 0.529425i 0.309017 + 0.951057i 1.43942 − 1.38855i −1.06976 − 1.96357i −0.908750 − 1.08359i 3.44588i −1.15249 + 2.58298i −0.809017 + 0.587785i 2.44242 + 2.00863i
109.7 −1.30099 − 0.554466i 0.309017 + 0.951057i 1.38514 + 1.44271i −2.13436 + 0.666722i 0.125301 − 1.40865i 5.07654i −1.00211 − 2.64495i −0.809017 + 0.587785i 3.14645 + 0.316032i
109.8 −1.15683 + 0.813479i 0.309017 + 0.951057i 0.676504 − 1.88211i −2.09775 + 0.774239i −1.13114 − 0.848831i − 2.82006i 0.748458 + 2.72760i −0.809017 + 0.587785i 1.79691 − 2.60214i
109.9 −0.846671 + 1.13276i 0.309017 + 0.951057i −0.566296 − 1.91815i −0.00633452 + 2.23606i −1.33896 − 0.455190i 4.12547i 2.65228 + 0.982566i −0.809017 + 0.587785i −2.52756 − 1.90038i
109.10 −0.822953 − 1.15011i 0.309017 + 0.951057i −0.645497 + 1.89297i −1.36330 − 1.77240i 0.839511 − 1.13808i 0.996218i 2.70833 − 0.815433i −0.809017 + 0.587785i −0.916527 + 3.02655i
109.11 −0.571475 − 1.29361i 0.309017 + 0.951057i −1.34683 + 1.47853i −1.14312 + 1.92179i 1.05370 − 0.943251i − 0.646785i 2.68231 + 0.897331i −0.809017 + 0.587785i 3.13930 + 0.380487i
109.12 −0.447604 + 1.34151i 0.309017 + 0.951057i −1.59930 − 1.20093i 2.18469 − 0.476576i −1.41417 − 0.0111470i − 1.14258i 2.32691 − 1.60794i −0.809017 + 0.587785i −0.338544 + 3.14410i
109.13 −0.387062 + 1.36021i 0.309017 + 0.951057i −1.70037 − 1.05297i −2.15299 − 0.603864i −1.41325 + 0.0522115i 0.170277i 2.09042 − 1.90530i −0.809017 + 0.587785i 1.65472 − 2.69479i
109.14 −0.164617 − 1.40460i 0.309017 + 0.951057i −1.94580 + 0.462442i 1.43821 − 1.71218i 1.28498 − 0.590605i 3.38230i 0.969859 + 2.65695i −0.809017 + 0.587785i −2.64168 − 1.73826i
109.15 −0.142266 + 1.40704i 0.309017 + 0.951057i −1.95952 − 0.400348i −0.665214 + 2.13483i −1.38214 + 0.299496i − 2.27192i 0.842080 − 2.70017i −0.809017 + 0.587785i −2.90915 − 1.23970i
109.16 −0.0755657 − 1.41219i 0.309017 + 0.951057i −1.98858 + 0.213427i 2.18080 + 0.494088i 1.31972 − 0.508259i − 3.46121i 0.451668 + 2.79213i −0.809017 + 0.587785i 0.532954 − 3.11704i
109.17 0.495441 + 1.32459i 0.309017 + 0.951057i −1.50908 + 1.31251i 0.137607 − 2.23183i −1.10666 + 0.880513i − 5.06961i −2.48620 − 1.34863i −0.809017 + 0.587785i 3.02443 − 0.923468i
109.18 0.522022 − 1.31434i 0.309017 + 0.951057i −1.45499 − 1.37223i −1.71415 − 1.43586i 1.41133 + 0.0903187i − 1.87161i −2.56311 + 1.19601i −0.809017 + 0.587785i −2.78204 + 1.50342i
109.19 0.539646 + 1.30720i 0.309017 + 0.951057i −1.41757 + 1.41085i 2.17022 + 0.538631i −1.07647 + 0.917182i 2.20134i −2.60926 − 1.09169i −0.809017 + 0.587785i 0.467051 + 3.12760i
109.20 0.696059 − 1.23106i 0.309017 + 0.951057i −1.03100 − 1.71378i 0.311583 + 2.21425i 1.38590 + 0.281574i 1.39209i −2.82740 + 0.0763343i −0.809017 + 0.587785i 2.94275 + 1.15767i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
200.o 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.be.a ✓ 120
8.b even 2 1 600.2.be.b yes 120
25.e even 10 1 600.2.be.b yes 120
200.o even 10 1 inner 600.2.be.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.be.a ✓ 120 1.a even 1 1 trivial
600.2.be.a ✓ 120 200.o even 10 1 inner
600.2.be.b yes 120 8.b even 2 1
600.2.be.b yes 120 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{120} - 186 T_{11}^{118} + 18969 T_{11}^{116} - 1407040 T_{11}^{114} - 68880 T_{11}^{113} + \cdots + 20\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display