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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [725,2,Mod(157,725)] 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("725.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 145)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{3} - 22 q^{4} + 4 q^{7} + 10 q^{9} - 8 q^{11} + 8 q^{12} + 14 q^{13} + 4 q^{14} + 6 q^{16} + 16 q^{21} - 8 q^{22} + 4 q^{23} + 6 q^{26} + 4 q^{27} - 8 q^{28} + 8 q^{31} + 32 q^{34} - 22 q^{36}+ \cdots - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 − 2.26693i 2.65401 −3.13899 0 − 6.01647i 2.59753 + 2.59753i 2.58201i 4.04378 0
157.2 − 2.23019i −1.25170 −2.97373 0 2.79153i −0.483409 − 0.483409i 2.17160i −1.43324 0
157.3 − 1.41066i −2.58872 0.0100302 0 3.65181i 0.510628 + 0.510628i − 2.83547i 3.70148 0
157.4 − 1.26373i 0.913274 0.402981 0 − 1.15413i −2.03055 − 2.03055i − 3.03672i −2.16593 0
157.5 − 0.895351i 2.11245 1.19835 0 − 1.89138i −1.38465 − 1.38465i − 2.86364i 1.46244 0
157.6 − 0.222351i −1.02589 1.95056 0 0.228107i 2.35964 + 2.35964i − 0.878412i −1.94756 0
157.7 0.342532i 2.64611 1.88267 0 0.906377i 1.55474 + 1.55474i 1.32994i 4.00189 0
157.8 0.839004i −0.711801 1.29607 0 − 0.597203i −1.13987 − 1.13987i 2.76542i −2.49334 0
157.9 1.36192i 0.228160 0.145179 0 0.310736i −3.45046 − 3.45046i 2.92156i −2.94794 0
157.10 1.77873i 1.38965 −1.16389 0 2.47181i 3.41296 + 3.41296i 1.48721i −1.06888 0
157.11 1.82099i −2.59340 −1.31599 0 − 4.72256i 0.820621 + 0.820621i 1.24557i 3.72575 0
157.12 2.41122i 1.85973 −3.81399 0 4.48422i 0.291676 + 0.291676i − 4.37394i 0.458586 0
157.13 2.73482i −1.63186 −5.47925 0 − 4.46285i −1.05887 − 1.05887i − 9.51511i −0.337028 0
568.1 − 2.73482i −1.63186 −5.47925 0 4.46285i −1.05887 + 1.05887i 9.51511i −0.337028 0
568.2 − 2.41122i 1.85973 −3.81399 0 − 4.48422i 0.291676 − 0.291676i 4.37394i 0.458586 0
568.3 − 1.82099i −2.59340 −1.31599 0 4.72256i 0.820621 − 0.820621i − 1.24557i 3.72575 0
568.4 − 1.77873i 1.38965 −1.16389 0 − 2.47181i 3.41296 − 3.41296i − 1.48721i −1.06888 0
568.5 − 1.36192i 0.228160 0.145179 0 − 0.310736i −3.45046 + 3.45046i − 2.92156i −2.94794 0
568.6 − 0.839004i −0.711801 1.29607 0 0.597203i −1.13987 + 1.13987i − 2.76542i −2.49334 0
568.7 − 0.342532i 2.64611 1.88267 0 − 0.906377i 1.55474 − 1.55474i − 1.32994i 4.00189 0
See all 26 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 157.13
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 725.2.e.c 26
5.b even 2 1 145.2.e.a ✓ 26
5.c odd 4 1 145.2.j.a yes 26
5.c odd 4 1 725.2.j.c 26
29.c odd 4 1 725.2.j.c 26
145.e even 4 1 inner 725.2.e.c 26
145.f odd 4 1 145.2.j.a yes 26
145.j even 4 1 145.2.e.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.e.a ✓ 26 5.b even 2 1
145.2.e.a ✓ 26 145.j even 4 1
145.2.j.a yes 26 5.c odd 4 1
145.2.j.a yes 26 145.f odd 4 1
725.2.e.c 26 1.a even 1 1 trivial
725.2.e.c 26 145.e even 4 1 inner
725.2.j.c 26 5.c odd 4 1
725.2.j.c 26 29.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 37 T_{2}^{24} + 598 T_{2}^{22} + 5562 T_{2}^{20} + 33015 T_{2}^{18} + 131099 T_{2}^{16} + \cdots + 225 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display