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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,16,Mod(1,36)] 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("36.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-69660] 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: yes
Analytic conductor: \(51.3696618360\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{8017}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2004 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 12)
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 \(\beta = 2880\sqrt{8017}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 34830) q^{5} + ( - 9 \beta + 1245752) q^{7} + ( - 286 \beta + 7487964) q^{11} + ( - 1314 \beta + 5878790) q^{13} + (1198 \beta - 2128038642) q^{17} + (5382 \beta + 4620844700) q^{19} + (2002 \beta - 12127411368) q^{23}+ \cdots + (421648668 \beta - 907785730540126) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 69660 q^{5} + 2491504 q^{7} + 14975928 q^{11} + 11757580 q^{13} - 4256077284 q^{17} + 9241689400 q^{19} - 24254822736 q^{23} + 74383511150 q^{25} - 147290258412 q^{29} - 76677530432 q^{31} + 1110152602080 q^{35}+ \cdots - 18\!\cdots\!52 q^{97}+O(q^{100}) \) 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
45.2689
−44.2689
0 0 0 −292699. 0 −1.07507e6 0 0 0
1.2 0 0 0 223039. 0 3.56657e6 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

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 36.16.a.c 2
3.b odd 2 1 12.16.a.b ✓ 2
4.b odd 2 1 144.16.a.r 2
12.b even 2 1 48.16.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.16.a.b ✓ 2 3.b odd 2 1
36.16.a.c 2 1.a even 1 1 trivial
48.16.a.i 2 12.b even 2 1
144.16.a.r 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 69660T_{5} - 65283075900 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(36))\). 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
$5$ \( T^{2} + \cdots - 65283075900 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 3834294543296 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 53\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 50\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 73\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 78\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 28\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 18\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 55\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 81\!\cdots\!76 \) Copy content Toggle raw display
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