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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,8,Mod(1,400)] 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("400.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-56,0,0,0,-408] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 10\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 28) q^{3} + ( - 7 \beta - 204) q^{7} + ( - 56 \beta + 1697) q^{9} + ( - 16 \beta - 4452) q^{11} + (78 \beta - 1752) q^{13} + (168 \beta + 21824) q^{17} + ( - 624 \beta + 15900) q^{19} + ( - 8 \beta - 15988) q^{21}+ \cdots + (222160 \beta - 4777444) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 56 q^{3} - 408 q^{7} + 3394 q^{9} - 8904 q^{11} - 3504 q^{13} + 43648 q^{17} + 31800 q^{19} - 31976 q^{21} - 71656 q^{23} - 319760 q^{27} - 84780 q^{29} + 197056 q^{31} + 150112 q^{33} + 195408 q^{37}+ \cdots - 9554888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.56776
5.56776
0 −83.6776 0 0 0 185.744 0 4814.95 0
1.2 0 27.6776 0 0 0 −593.744 0 −1420.95 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.8.a.t 2
4.b odd 2 1 50.8.a.j 2
5.b even 2 1 400.8.a.bf 2
5.c odd 4 2 80.8.c.d 4
12.b even 2 1 450.8.a.bd 2
20.d odd 2 1 50.8.a.i 2
20.e even 4 2 10.8.b.a ✓ 4
40.i odd 4 2 320.8.c.g 4
40.k even 4 2 320.8.c.f 4
60.h even 2 1 450.8.a.bi 2
60.l odd 4 2 90.8.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.8.b.a ✓ 4 20.e even 4 2
50.8.a.i 2 20.d odd 2 1
50.8.a.j 2 4.b odd 2 1
80.8.c.d 4 5.c odd 4 2
90.8.c.c 4 60.l odd 4 2
320.8.c.f 4 40.k even 4 2
320.8.c.g 4 40.i odd 4 2
400.8.a.t 2 1.a even 1 1 trivial
400.8.a.bf 2 5.b even 2 1
450.8.a.bd 2 12.b even 2 1
450.8.a.bi 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 56T_{3} - 2316 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 56T - 2316 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 408T - 110284 \) Copy content Toggle raw display
$11$ \( T^{2} + 8904 T + 19026704 \) Copy content Toggle raw display
$13$ \( T^{2} + 3504 T - 15790896 \) Copy content Toggle raw display
$17$ \( T^{2} - 43648 T + 388792576 \) Copy content Toggle raw display
$19$ \( T^{2} - 31800 T - 954255600 \) Copy content Toggle raw display
$23$ \( T^{2} + 71656 T - 950837516 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 19030326300 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 4879504384 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 51511044784 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 183681487516 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 17685717524 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 599256039404 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 504522481104 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 159475001200 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1716951910564 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 2422233726604 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 4934358737856 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 296239095104 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 3963839328000 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 36935334540236 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 66278240560100 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1192948931584 \) Copy content Toggle raw display
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