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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 = [2,0,0,0,500] 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 60)
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.

\(f(q)\) \(=\) \( q - 27 i q^{3} + ( - 125 i + 250) q^{5} + 722 i q^{7} - 729 q^{9} + 3994 q^{11} - 3030 i q^{13} + ( - 6750 i - 3375) q^{15} - 20582 i q^{17} - 25320 q^{19} + 19494 q^{21} + 66652 i q^{23} + ( - 62500 i + 46875) q^{25} + \cdots - 2911626 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 500 q^{5} - 1458 q^{9} + 7988 q^{11} - 6750 q^{15} - 50640 q^{19} + 38988 q^{21} + 93750 q^{25} + 305328 q^{29} + 247552 q^{31} + 180500 q^{35} - 163620 q^{39} + 793060 q^{41} - 364500 q^{45} + 604518 q^{49}+ \cdots - 5823252 q^{99}+O(q^{100}) \) 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 0 250.000 − 125.000i 0 722.000i 0 −729.000 0
49.2 0 27.0000i 0 250.000 + 125.000i 0 − 722.000i 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.b 2
4.b odd 2 1 60.8.d.a ✓ 2
5.b even 2 1 inner 240.8.f.b 2
12.b even 2 1 180.8.d.a 2
20.d odd 2 1 60.8.d.a ✓ 2
20.e even 4 1 300.8.a.b 1
20.e even 4 1 300.8.a.f 1
60.h even 2 1 180.8.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.d.a ✓ 2 4.b odd 2 1
60.8.d.a ✓ 2 20.d odd 2 1
180.8.d.a 2 12.b even 2 1
180.8.d.a 2 60.h even 2 1
240.8.f.b 2 1.a even 1 1 trivial
240.8.f.b 2 5.b even 2 1 inner
300.8.a.b 1 20.e even 4 1
300.8.a.f 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 521284 \) 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
$5$ \( T^{2} - 500T + 78125 \) Copy content Toggle raw display
$7$ \( T^{2} + 521284 \) Copy content Toggle raw display
$11$ \( (T - 3994)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 9180900 \) Copy content Toggle raw display
$17$ \( T^{2} + 423618724 \) Copy content Toggle raw display
$19$ \( (T + 25320)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4442489104 \) Copy content Toggle raw display
$29$ \( (T - 152664)^{2} \) Copy content Toggle raw display
$31$ \( (T - 123776)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 114166948996 \) Copy content Toggle raw display
$41$ \( (T - 396530)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 196117893904 \) Copy content Toggle raw display
$47$ \( T^{2} + 29047066624 \) Copy content Toggle raw display
$53$ \( T^{2} + 1536176809476 \) Copy content Toggle raw display
$59$ \( (T + 302354)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2830198)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 13997116177984 \) Copy content Toggle raw display
$71$ \( (T - 1007580)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5782274292496 \) Copy content Toggle raw display
$79$ \( (T - 7517832)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 28086056938384 \) Copy content Toggle raw display
$89$ \( (T - 7650250)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 101122009731136 \) Copy content Toggle raw display
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