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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [100,11,Mod(51,100)] 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("100.51"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 100.b (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: \(63.5357252674\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q - 8 \beta q^{2} + 59 \beta q^{3} - 1024 q^{4} + 7552 q^{6} - 8341 \beta q^{7} + 8192 \beta q^{8} + 3353 q^{9} - 60416 \beta q^{12} - 1067648 q^{14} + 1048576 q^{16} - 26824 \beta q^{18} + 7873904 q^{21} + \cdots + 6645449976 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{4} + 15104 q^{6} + 6706 q^{9} - 2135296 q^{14} + 2097152 q^{16} + 15747808 q^{21} - 15466496 q^{24} + 76359404 q^{29} - 6866944 q^{36} - 422056196 q^{41} - 74852096 q^{46} - 1661362494 q^{49}+ \cdots + 15837691904 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.00000i
− 1.00000i
− 32.0000i 236.000i −1024.00 0 7552.00 − 33364.0i 32768.0i 3353.00 0
51.2 32.0000i − 236.000i −1024.00 0 7552.00 33364.0i − 32768.0i 3353.00 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
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 100.11.b.c 2
4.b odd 2 1 inner 100.11.b.c 2
5.b even 2 1 inner 100.11.b.c 2
5.c odd 4 1 20.11.d.a ✓ 1
5.c odd 4 1 20.11.d.b yes 1
20.d odd 2 1 CM 100.11.b.c 2
20.e even 4 1 20.11.d.a ✓ 1
20.e even 4 1 20.11.d.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.d.a ✓ 1 5.c odd 4 1
20.11.d.a ✓ 1 20.e even 4 1
20.11.d.b yes 1 5.c odd 4 1
20.11.d.b yes 1 20.e even 4 1
100.11.b.c 2 1.a even 1 1 trivial
100.11.b.c 2 4.b odd 2 1 inner
100.11.b.c 2 5.b even 2 1 inner
100.11.b.c 2 20.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{2} + 55696 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1024 \) Copy content Toggle raw display
$3$ \( T^{2} + 55696 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1113156496 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1367879950096 \) Copy content Toggle raw display
$29$ \( (T - 38179702)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 211028098)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 50\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} + 93\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 1041591898)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 54\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 29\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T + 11118190898)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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