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Properties

Label 300.4.o.a
Level $300$
Weight $4$
Character orbit 300.o
Analytic conductor $17.701$
Analytic rank $0$
Dimension $56$
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] = [300,4,Mod(109,300)] 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("300.109"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.o (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.7005730017\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 20 q^{5} + 126 q^{9} - 84 q^{11} + 30 q^{15} + 320 q^{17} - 40 q^{19} + 84 q^{21} - 580 q^{23} - 390 q^{25} + 224 q^{29} + 414 q^{31} + 480 q^{33} + 1080 q^{35} - 700 q^{37} + 400 q^{41} - 180 q^{45}+ \cdots - 504 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 0 −2.85317 + 0.927051i 0 4.19386 − 10.3640i 0 31.2639i 0 7.28115 − 5.29007i 0
109.2 0 −2.85317 + 0.927051i 0 −1.84316 − 11.0274i 0 − 23.4790i 0 7.28115 − 5.29007i 0
109.3 0 −2.85317 + 0.927051i 0 1.06958 + 11.1291i 0 17.6905i 0 7.28115 − 5.29007i 0
109.4 0 −2.85317 + 0.927051i 0 10.3579 + 4.20878i 0 − 8.38073i 0 7.28115 − 5.29007i 0
109.5 0 −2.85317 + 0.927051i 0 11.0474 − 1.71884i 0 − 4.14369i 0 7.28115 − 5.29007i 0
109.6 0 −2.85317 + 0.927051i 0 −8.72117 + 6.99580i 0 − 21.9741i 0 7.28115 − 5.29007i 0
109.7 0 −2.85317 + 0.927051i 0 −10.6080 − 3.53141i 0 26.3784i 0 7.28115 − 5.29007i 0
109.8 0 2.85317 − 0.927051i 0 −7.47544 + 8.31371i 0 − 31.0451i 0 7.28115 − 5.29007i 0
109.9 0 2.85317 − 0.927051i 0 7.90711 − 7.90428i 0 − 18.2216i 0 7.28115 − 5.29007i 0
109.10 0 2.85317 − 0.927051i 0 5.60335 + 9.67484i 0 − 5.98575i 0 7.28115 − 5.29007i 0
109.11 0 2.85317 − 0.927051i 0 1.59182 − 11.0664i 0 9.56491i 0 7.28115 − 5.29007i 0
109.12 0 2.85317 − 0.927051i 0 −7.59490 + 8.20472i 0 9.77901i 0 7.28115 − 5.29007i 0
109.13 0 2.85317 − 0.927051i 0 −10.2182 − 4.53738i 0 15.3693i 0 7.28115 − 5.29007i 0
109.14 0 2.85317 − 0.927051i 0 7.45376 + 8.33315i 0 9.89455i 0 7.28115 − 5.29007i 0
169.1 0 −1.76336 − 2.42705i 0 9.97041 + 5.05875i 0 24.5869i 0 −2.78115 + 8.55951i 0
169.2 0 −1.76336 − 2.42705i 0 −8.55562 + 7.19732i 0 26.1619i 0 −2.78115 + 8.55951i 0
169.3 0 −1.76336 − 2.42705i 0 −9.49233 − 5.90726i 0 − 18.7477i 0 −2.78115 + 8.55951i 0
169.4 0 −1.76336 − 2.42705i 0 −0.506784 − 11.1688i 0 17.0205i 0 −2.78115 + 8.55951i 0
169.5 0 −1.76336 − 2.42705i 0 3.85044 + 10.4964i 0 − 9.05088i 0 −2.78115 + 8.55951i 0
169.6 0 −1.76336 − 2.42705i 0 8.97456 − 6.66762i 0 − 1.49943i 0 −2.78115 + 8.55951i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.14
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 300.4.o.a ✓ 56
25.e even 10 1 inner 300.4.o.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.4.o.a ✓ 56 1.a even 1 1 trivial
300.4.o.a ✓ 56 25.e even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(300, [\chi])\).