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Properties

Label 2900.2.s.d
Level $2900$
Weight $2$
Character orbit 2900.s
Analytic conductor $23.157$
Analytic rank $0$
Dimension $30$
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] = [2900,2,Mod(157,2900)] 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("2900.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,0,0] 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: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 580)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 38 q^{9} - 4 q^{11} + 6 q^{13} - 4 q^{21} - 12 q^{27} - 4 q^{31} - 4 q^{33} + 24 q^{37} - 12 q^{39} + 10 q^{41} + 16 q^{43} + 32 q^{47} + 18 q^{53} - 24 q^{57} - 22 q^{61} - 24 q^{63} - 16 q^{67}+ \cdots + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 0 −3.02258 0 0 0 −0.186244 − 0.186244i 0 6.13596 0
157.2 0 −2.99066 0 0 0 1.61250 + 1.61250i 0 5.94404 0
157.3 0 −2.68138 0 0 0 −3.30434 − 3.30434i 0 4.18982 0
157.4 0 −2.30110 0 0 0 2.56511 + 2.56511i 0 2.29505 0
157.5 0 −0.993874 0 0 0 −1.61651 − 1.61651i 0 −2.01221 0
157.6 0 −0.938623 0 0 0 −0.222978 − 0.222978i 0 −2.11899 0
157.7 0 −0.548138 0 0 0 −1.10894 − 1.10894i 0 −2.69955 0
157.8 0 0.121921 0 0 0 2.56056 + 2.56056i 0 −2.98514 0
157.9 0 0.357354 0 0 0 1.34106 + 1.34106i 0 −2.87230 0
157.10 0 1.57681 0 0 0 1.37454 + 1.37454i 0 −0.513683 0
157.11 0 1.59876 0 0 0 −2.73334 − 2.73334i 0 −0.443956 0
157.12 0 1.61432 0 0 0 −1.96009 − 1.96009i 0 −0.393986 0
157.13 0 1.94358 0 0 0 2.43648 + 2.43648i 0 0.777490 0
157.14 0 2.93055 0 0 0 1.87865 + 1.87865i 0 5.58815 0
157.15 0 3.33306 0 0 0 −2.63647 − 2.63647i 0 8.10929 0
1293.1 0 −3.02258 0 0 0 −0.186244 + 0.186244i 0 6.13596 0
1293.2 0 −2.99066 0 0 0 1.61250 − 1.61250i 0 5.94404 0
1293.3 0 −2.68138 0 0 0 −3.30434 + 3.30434i 0 4.18982 0
1293.4 0 −2.30110 0 0 0 2.56511 − 2.56511i 0 2.29505 0
1293.5 0 −0.993874 0 0 0 −1.61651 + 1.61651i 0 −2.01221 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2900.2.s.d 30
5.b even 2 1 580.2.s.a yes 30
5.c odd 4 1 580.2.j.a ✓ 30
5.c odd 4 1 2900.2.j.d 30
29.c odd 4 1 2900.2.j.d 30
145.e even 4 1 inner 2900.2.s.d 30
145.f odd 4 1 580.2.j.a ✓ 30
145.j even 4 1 580.2.s.a yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.j.a ✓ 30 5.c odd 4 1
580.2.j.a ✓ 30 145.f odd 4 1
580.2.s.a yes 30 5.b even 2 1
580.2.s.a yes 30 145.j even 4 1
2900.2.j.d 30 5.c odd 4 1
2900.2.j.d 30 29.c odd 4 1
2900.2.s.d 30 1.a even 1 1 trivial
2900.2.s.d 30 145.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 32 T_{3}^{13} + 2 T_{3}^{12} + 393 T_{3}^{11} - 62 T_{3}^{10} - 2336 T_{3}^{9} + 604 T_{3}^{8} + \cdots + 96 \) acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\). Copy content Toggle raw display