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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,8,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-128,0,688] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.6192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 i q^{2} + 43 i q^{3} - 64 q^{4} + 344 q^{6} - 974 i q^{7} + 512 i q^{8} + 338 q^{9} + 87 q^{11} - 2752 i q^{12} + 14828 i q^{13} - 7792 q^{14} + 4096 q^{16} + 35571 i q^{17} - 2704 i q^{18} - 20615 q^{19} + \cdots + 29406 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} + 688 q^{6} + 676 q^{9} + 174 q^{11} - 15584 q^{14} + 8192 q^{16} - 41230 q^{19} + 83764 q^{21} - 44032 q^{24} + 237248 q^{26} + 11520 q^{29} + 605884 q^{31} + 569136 q^{34} - 43264 q^{36}+ \cdots + 58812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
− 1.00000i
− 8.00000i 43.0000i −64.0000 0 344.000 − 974.000i 512.000i 338.000 0
49.2 8.00000i − 43.0000i −64.0000 0 344.000 974.000i − 512.000i 338.000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.8.b.e 2
3.b odd 2 1 450.8.c.i 2
4.b odd 2 1 400.8.c.g 2
5.b even 2 1 inner 50.8.b.e 2
5.c odd 4 1 50.8.a.a ✓ 1
5.c odd 4 1 50.8.a.h yes 1
15.d odd 2 1 450.8.c.i 2
15.e even 4 1 450.8.a.j 1
15.e even 4 1 450.8.a.q 1
20.d odd 2 1 400.8.c.g 2
20.e even 4 1 400.8.a.f 1
20.e even 4 1 400.8.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.8.a.a ✓ 1 5.c odd 4 1
50.8.a.h yes 1 5.c odd 4 1
50.8.b.e 2 1.a even 1 1 trivial
50.8.b.e 2 5.b even 2 1 inner
400.8.a.f 1 20.e even 4 1
400.8.a.o 1 20.e even 4 1
400.8.c.g 2 4.b odd 2 1
400.8.c.g 2 20.d odd 2 1
450.8.a.j 1 15.e even 4 1
450.8.a.q 1 15.e even 4 1
450.8.c.i 2 3.b odd 2 1
450.8.c.i 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1849 \) acting on \(S_{8}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{2} + 1849 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 948676 \) Copy content Toggle raw display
$11$ \( (T - 87)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 219869584 \) Copy content Toggle raw display
$17$ \( T^{2} + 1265296041 \) Copy content Toggle raw display
$19$ \( (T + 20615)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 493639524 \) Copy content Toggle raw display
$29$ \( (T - 5760)^{2} \) Copy content Toggle raw display
$31$ \( (T - 302942)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 39746801956 \) Copy content Toggle raw display
$41$ \( (T + 668523)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 20509676944 \) Copy content Toggle raw display
$47$ \( T^{2} + 114457715856 \) Copy content Toggle raw display
$53$ \( T^{2} + 1197540639684 \) Copy content Toggle raw display
$59$ \( (T - 2135520)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1939318)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 11214133260001 \) Copy content Toggle raw display
$71$ \( (T - 3005652)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9292724269609 \) Copy content Toggle raw display
$79$ \( (T + 5485130)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 27101738400489 \) Copy content Toggle raw display
$89$ \( (T - 832665)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 28246610080516 \) Copy content Toggle raw display
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