Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,16,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,5182520] 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: \(205.478647344\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{370}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 370 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 9)
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 \(\beta = 288\sqrt{370}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 29 \beta q^{5} + 2591260 q^{7} - 6820 \beta q^{11} - 77911990 q^{13} + 13578 \beta q^{17} + 976566184 q^{19} + 3761416 \beta q^{23} - 4707893645 q^{25} - 20546645 \beta q^{29} - 162227927108 q^{31} + \cdots - 10\!\cdots\!10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5182520 q^{7} - 155823980 q^{13} + 1953132368 q^{19} - 9415787290 q^{25} - 324455854216 q^{31} - 1213210694660 q^{37} - 3529665379120 q^{43} + 3934133755314 q^{49} - 12139451596800 q^{55} + 63956550008524 q^{61}+ \cdots - 20\!\cdots\!20 q^{97}+O(q^{100}) \) 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
−19.2354
19.2354
0 0 0 −160654. 0 2.59126e6 0 0 0
1.2 0 0 0 160654. 0 2.59126e6 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.16.a.t 2
3.b odd 2 1 inner 144.16.a.t 2
4.b odd 2 1 9.16.a.e ✓ 2
12.b even 2 1 9.16.a.e ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.16.a.e ✓ 2 4.b odd 2 1
9.16.a.e ✓ 2 12.b even 2 1
144.16.a.t 2 1.a even 1 1 trivial
144.16.a.t 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 25809684480 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(144))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 25809684480 \) Copy content Toggle raw display
$7$ \( (T - 2591260)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T + 77911990)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 56\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( (T - 976566184)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 43\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{2} - 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T + 162227927108)^{2} \) Copy content Toggle raw display
$37$ \( (T + 606605347330)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 51\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T + 1764832689560)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 14\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{2} - 34\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} - 75\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T - 31978275004262)^{2} \) Copy content Toggle raw display
$67$ \( (T + 126767977040)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T - 101136696626630)^{2} \) Copy content Toggle raw display
$79$ \( (T + 163797938010884)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 95\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{2} - 45\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T + 10\!\cdots\!10)^{2} \) Copy content Toggle raw display
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