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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
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,-2097152,0,-146644992] 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: \(139.738672144\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 512 \beta q^{2} + 35802 \beta q^{3} - 1048576 q^{4} - 73322496 q^{6} + 426601196 \beta q^{7} - 536870912 \beta q^{8} + 5333220387 q^{9} + 86731179612 q^{11} - 37541117952 \beta q^{12} - 447661721393 \beta q^{13} + \cdots + 46\!\cdots\!44 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2097152 q^{4} - 146644992 q^{6} + 10666440774 q^{9} + 173462359224 q^{11} - 1747358498816 q^{14} + 2199023255552 q^{16} - 46064935288840 q^{19} - 122185408153536 q^{21} + 153768419131392 q^{24} + 18\!\cdots\!28 q^{26}+ \cdots + 92\!\cdots\!88 q^{99}+O(q^{100}) \) 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
− 1024.00i − 71604.0i −1.04858e6 0 −7.33225e7 − 8.53202e8i 1.07374e9i 5.33322e9 0
49.2 1024.00i 71604.0i −1.04858e6 0 −7.33225e7 8.53202e8i − 1.07374e9i 5.33322e9 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 50.22.b.a 2
5.b even 2 1 inner 50.22.b.a 2
5.c odd 4 1 2.22.a.a ✓ 1
5.c odd 4 1 50.22.a.c 1
15.e even 4 1 18.22.a.e 1
20.e even 4 1 16.22.a.a 1
40.i odd 4 1 64.22.a.b 1
40.k even 4 1 64.22.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.a ✓ 1 5.c odd 4 1
16.22.a.a 1 20.e even 4 1
18.22.a.e 1 15.e even 4 1
50.22.a.c 1 5.c odd 4 1
50.22.b.a 2 1.a even 1 1 trivial
50.22.b.a 2 5.b even 2 1 inner
64.22.a.b 1 40.i odd 4 1
64.22.a.f 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5127132816 \) acting on \(S_{22}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1048576 \) Copy content Toggle raw display
$3$ \( T^{2} + 5127132816 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 72\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( (T - 86731179612)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 80\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{2} + 10\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T + 23032467644420)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 21\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T - 734051633521170)^{2} \) Copy content Toggle raw display
$31$ \( (T + 31\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T - 45\!\cdots\!42)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 57\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + 20\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + 42\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T - 37\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( (T + 76\!\cdots\!38)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 35\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T + 45\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 65\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T + 99\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 87\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T + 11\!\cdots\!90)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 32\!\cdots\!04 \) Copy content Toggle raw display
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