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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,18,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.4899878610\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{910}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 910 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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 \(\beta = 12\sqrt{910}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 32 q^{4} - 280 \beta q^{5} - 196420 q^{7} - 131104 \beta q^{8} - 36691200 q^{10} - 3057440 \beta q^{11} - 1308631870 q^{13} - 196420 \beta q^{14} - 17175673856 q^{16} + 96664848 \beta q^{17} + \cdots - 232591933170807 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 392840 q^{7} - 73382400 q^{10} - 2617263740 q^{13} - 34351347712 q^{16} - 222645082352 q^{19} - 801293875200 q^{22} - 1505331834250 q^{25} + 12570880 q^{28} + 8147437309816 q^{31} + 25333923363840 q^{34}+ \cdots - 10\!\cdots\!40 q^{97}+O(q^{100}) \) 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
−30.1662
30.1662
−361.994 0 −32.0000 101358. 0 −196420. 4.74589e7 0 −3.66912e7
1.2 361.994 0 −32.0000 −101358. 0 −196420. −4.74589e7 0 −3.66912e7
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

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 9.18.a.d ✓ 2
3.b odd 2 1 inner 9.18.a.d ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.18.a.d ✓ 2 1.a even 1 1 trivial
9.18.a.d ✓ 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 131040 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 131040 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10273536000 \) Copy content Toggle raw display
$7$ \( (T + 196420)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T + 1308631870)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T + 111322541176)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 11\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{2} - 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T - 4073718654908)^{2} \) Copy content Toggle raw display
$37$ \( (T - 17155445649590)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 23\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T + 60341472064120)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 38\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{2} - 91\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{2} - 88\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T - 702559953764942)^{2} \) Copy content Toggle raw display
$67$ \( (T + 21\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 90\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T + 10\!\cdots\!10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 95\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{2} - 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T + 54\!\cdots\!70)^{2} \) Copy content Toggle raw display
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