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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [20,11,Mod(19,20)] 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("20.19"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7071450535\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta q^{2} - 1024 q^{4} + (779 \beta - 237) q^{5} - 8192 \beta q^{8} - 59049 q^{9} + ( - 1896 \beta - 99712) q^{10} - 72834 \beta q^{13} + 1048576 q^{16} - 452884 \beta q^{17} - 472392 \beta q^{18} + \cdots - 2259801992 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{4} - 474 q^{5} - 118098 q^{9} - 199424 q^{10} + 2097152 q^{16} + 485376 q^{20} - 19306574 q^{25} + 18645504 q^{26} - 64639596 q^{29} + 115938304 q^{34} + 120932352 q^{36} + 204210176 q^{40}+ \cdots + 11775787776 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
− 1.00000i
1.00000i
− 32.0000i 0 −1024.00 −237.000 − 3116.00i 0 0 32768.0i −59049.0 −99712.0 + 7584.00i
19.2 32.0000i 0 −1024.00 −237.000 + 3116.00i 0 0 − 32768.0i −59049.0 −99712.0 − 7584.00i
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.11.d.c ✓ 2
4.b odd 2 1 CM 20.11.d.c ✓ 2
5.b even 2 1 inner 20.11.d.c ✓ 2
5.c odd 4 1 100.11.b.a 1
5.c odd 4 1 100.11.b.b 1
20.d odd 2 1 inner 20.11.d.c ✓ 2
20.e even 4 1 100.11.b.a 1
20.e even 4 1 100.11.b.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.d.c ✓ 2 1.a even 1 1 trivial
20.11.d.c ✓ 2 4.b odd 2 1 CM
20.11.d.c ✓ 2 5.b even 2 1 inner
20.11.d.c ✓ 2 20.d odd 2 1 inner
100.11.b.a 1 5.c odd 4 1
100.11.b.a 1 20.e even 4 1
100.11.b.b 1 5.c odd 4 1
100.11.b.b 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{11}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1024 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 474 T + 9765625 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 84876664896 \) Copy content Toggle raw display
$17$ \( T^{2} + 3281662679296 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 32319798)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 19\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T + 207190098)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 86\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 1330090102)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 32\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 8562016398)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 21\!\cdots\!96 \) Copy content Toggle raw display
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