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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,18,Mod(24,25)] 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("25.24"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(45.8055218361\)
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 1)
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.

\(f(q)\) \(=\) \( q + 264 \beta q^{2} - 2142 \beta q^{3} - 147712 q^{4} + 2261952 q^{6} - 1612996 \beta q^{7} - 4392960 \beta q^{8} + 110787507 q^{9} - 753618228 q^{11} + 316399104 \beta q^{12} + 1270532263 \beta q^{13} + \cdots - 83\!\cdots\!96 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 295424 q^{4} + 4523904 q^{6} + 221575014 q^{9} - 1507236456 q^{11} + 3406647552 q^{14} - 29443883008 q^{16} - 2974999720 q^{19} - 27640299456 q^{21} - 75277762560 q^{24} - 2683364139456 q^{26} - 4866821205180 q^{29}+ \cdots - 16\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 1.00000i
1.00000i
− 528.000i 4284.00i −147712. 0 2.26195e6 3.22599e6i 8.78592e6i 1.10788e8 0
24.2 528.000i − 4284.00i −147712. 0 2.26195e6 − 3.22599e6i − 8.78592e6i 1.10788e8 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 25.18.b.a 2
5.b even 2 1 inner 25.18.b.a 2
5.c odd 4 1 1.18.a.a ✓ 1
5.c odd 4 1 25.18.a.a 1
15.e even 4 1 9.18.a.b 1
20.e even 4 1 16.18.a.b 1
35.f even 4 1 49.18.a.a 1
40.i odd 4 1 64.18.a.d 1
40.k even 4 1 64.18.a.b 1
55.e even 4 1 121.18.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.18.a.a ✓ 1 5.c odd 4 1
9.18.a.b 1 15.e even 4 1
16.18.a.b 1 20.e even 4 1
25.18.a.a 1 5.c odd 4 1
25.18.b.a 2 1.a even 1 1 trivial
25.18.b.a 2 5.b even 2 1 inner
49.18.a.a 1 35.f even 4 1
64.18.a.b 1 40.k even 4 1
64.18.a.d 1 40.i odd 4 1
121.18.a.b 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 278784 \) acting on \(S_{18}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 278784 \) Copy content Toggle raw display
$3$ \( T^{2} + 18352656 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 10407024384064 \) Copy content Toggle raw display
$11$ \( (T + 753618228)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 64\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} + 29\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T + 1487499860)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T + 2433410602590)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8849722053088)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T - 48864151002282)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 82\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + 24\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + 52\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T + 32695090729980)^{2} \) Copy content Toggle raw display
$61$ \( (T + 13\!\cdots\!78)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 26\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T + 37\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T + 23\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 88\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T + 29\!\cdots\!70)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 40\!\cdots\!04 \) Copy content Toggle raw display
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