Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [240,8,Mod(49,240)] 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("240.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(74.9724061162\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 27 i q^{3} + (275 i - 50) q^{5} + 1126 i q^{7} - 729 q^{9} + 5518 q^{11} - 12798 i q^{13} + ( - 1350 i - 7425) q^{15} - 32206 i q^{17} - 4440 q^{19} - 30402 q^{21} - 95452 i q^{23} + ( - 27500 i - 73125) q^{25} + \cdots - 4022622 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 100 q^{5} - 1458 q^{9} + 11036 q^{11} - 14850 q^{15} - 8880 q^{19} - 60804 q^{21} - 146250 q^{25} - 38880 q^{29} + 480496 q^{31} - 619300 q^{35} + 691092 q^{39} + 599044 q^{41} + 72900 q^{45} - 888666 q^{49}+ \cdots - 8045244 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 1.00000i
1.00000i
0 − 27.0000i 0 −50.0000 − 275.000i 0 − 1126.00i 0 −729.000 0
49.2 0 27.0000i 0 −50.0000 + 275.000i 0 1126.00i 0 −729.000 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 240.8.f.a 2
4.b odd 2 1 30.8.c.a ✓ 2
5.b even 2 1 inner 240.8.f.a 2
12.b even 2 1 90.8.c.a 2
20.d odd 2 1 30.8.c.a ✓ 2
20.e even 4 1 150.8.a.d 1
20.e even 4 1 150.8.a.m 1
60.h even 2 1 90.8.c.a 2
60.l odd 4 1 450.8.a.b 1
60.l odd 4 1 450.8.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.8.c.a ✓ 2 4.b odd 2 1
30.8.c.a ✓ 2 20.d odd 2 1
90.8.c.a 2 12.b even 2 1
90.8.c.a 2 60.h even 2 1
150.8.a.d 1 20.e even 4 1
150.8.a.m 1 20.e even 4 1
240.8.f.a 2 1.a even 1 1 trivial
240.8.f.a 2 5.b even 2 1 inner
450.8.a.b 1 60.l odd 4 1
450.8.a.y 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 1267876 \) acting on \(S_{8}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 729 \) Copy content Toggle raw display
$5$ \( T^{2} + 100T + 78125 \) Copy content Toggle raw display
$7$ \( T^{2} + 1267876 \) Copy content Toggle raw display
$11$ \( (T - 5518)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 163788804 \) Copy content Toggle raw display
$17$ \( T^{2} + 1037226436 \) Copy content Toggle raw display
$19$ \( (T + 4440)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9111084304 \) Copy content Toggle raw display
$29$ \( (T + 19440)^{2} \) Copy content Toggle raw display
$31$ \( (T - 240248)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6058131556 \) Copy content Toggle raw display
$41$ \( (T - 299522)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 173232428944 \) Copy content Toggle raw display
$47$ \( T^{2} + 104313496576 \) Copy content Toggle raw display
$53$ \( T^{2} + 775946050884 \) Copy content Toggle raw display
$59$ \( (T + 1845110)^{2} \) Copy content Toggle raw display
$61$ \( (T + 861718)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 454092690496 \) Copy content Toggle raw display
$71$ \( (T - 3426948)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 21890682847504 \) Copy content Toggle raw display
$79$ \( (T + 3137760)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 234383793424 \) Copy content Toggle raw display
$89$ \( (T + 6258710)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 74953622195776 \) Copy content Toggle raw display
show more
show less