Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [300,8,Mod(1,300)] 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("300.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-54,0,0,0,906,0,1458,0,6342] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1129}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 282 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 60)
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 \(\beta = 5\sqrt{1129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 27 q^{3} + ( - \beta + 453) q^{7} + 729 q^{9} + (17 \beta + 3171) q^{11} + (15 \beta + 3345) q^{13} + (87 \beta + 15653) q^{17} + ( - 132 \beta + 2820) q^{19} + (27 \beta - 12231) q^{21} + ( - 204 \beta - 17488) q^{23}+ \cdots + (12393 \beta + 2311659) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} + 906 q^{7} + 1458 q^{9} + 6342 q^{11} + 6690 q^{13} + 31306 q^{17} + 5640 q^{19} - 24462 q^{21} - 34976 q^{23} - 39366 q^{27} - 79758 q^{29} + 152708 q^{31} - 171234 q^{33} + 168318 q^{37}+ \cdots + 4623318 q^{99}+O(q^{100}) \) 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
17.3003
−16.3003
0 −27.0000 0 0 0 284.997 0 729.000 0
1.2 0 −27.0000 0 0 0 621.003 0 729.000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)

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 300.8.a.j 2
5.b even 2 1 300.8.a.k 2
5.c odd 4 2 60.8.d.b ✓ 4
15.e even 4 2 180.8.d.c 4
20.e even 4 2 240.8.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.d.b ✓ 4 5.c odd 4 2
180.8.d.c 4 15.e even 4 2
240.8.f.c 4 20.e even 4 2
300.8.a.j 2 1.a even 1 1 trivial
300.8.a.k 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(300))\):

\( T_{7}^{2} - 906T_{7} + 176984 \) Copy content Toggle raw display
\( T_{11}^{2} - 6342T_{11} + 1898216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 906T + 176984 \) Copy content Toggle raw display
$11$ \( T^{2} - 6342 T + 1898216 \) Copy content Toggle raw display
$13$ \( T^{2} - 6690 T + 4838400 \) Copy content Toggle raw display
$17$ \( T^{2} - 31306 T + 31381384 \) Copy content Toggle raw display
$19$ \( T^{2} - 5640 T - 483840000 \) Copy content Toggle raw display
$23$ \( T^{2} + 34976 T - 868781456 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 23936383584 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 52210714784 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 6997356656 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 132110549900 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 5472599536 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 70271336704 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 756658769976 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 1410030476104 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 65990395684 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 13206551822176 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 10803266611200 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 17372893691104 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 2846846202624 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 6944201174576 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 27181827906500 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 200488661648576 \) Copy content Toggle raw display
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