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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1089,4,Mod(1,1089)] 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("1089.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1089, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,10,10,0,-8] 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: \(64.2530799963\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 121)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 5) q^{4} + (\beta + 5) q^{5} + (7 \beta - 4) q^{7} + ( - \beta + 21) q^{8} + (6 \beta + 17) q^{10} + ( - \beta + 65) q^{13} + (3 \beta + 80) q^{14} + (4 \beta - 31) q^{16}+ \cdots + (205 \beta - 411) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 10 q^{4} + 10 q^{5} - 8 q^{7} + 42 q^{8} + 34 q^{10} + 130 q^{13} + 160 q^{14} - 62 q^{16} + 14 q^{17} - 48 q^{19} + 98 q^{20} + 128 q^{23} - 176 q^{25} + 106 q^{26} + 296 q^{28} + 30 q^{29}+ \cdots - 822 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−2.46410 0 −1.92820 1.53590 0 −28.2487 24.4641 0 −3.78461
1.2 4.46410 0 11.9282 8.46410 0 20.2487 17.5359 0 37.7846
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(11\) \( -1 \)

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 1089.4.a.x 2
3.b odd 2 1 121.4.a.b ✓ 2
11.b odd 2 1 1089.4.a.k 2
12.b even 2 1 1936.4.a.z 2
33.d even 2 1 121.4.a.e yes 2
33.f even 10 4 121.4.c.d 8
33.h odd 10 4 121.4.c.g 8
132.d odd 2 1 1936.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.b ✓ 2 3.b odd 2 1
121.4.a.e yes 2 33.d even 2 1
121.4.c.d 8 33.f even 10 4
121.4.c.g 8 33.h odd 10 4
1089.4.a.k 2 11.b odd 2 1
1089.4.a.x 2 1.a even 1 1 trivial
1936.4.a.y 2 132.d odd 2 1
1936.4.a.z 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{2} - 2T_{2} - 11 \) Copy content Toggle raw display
\( T_{5}^{2} - 10T_{5} + 13 \) Copy content Toggle raw display
\( T_{7}^{2} + 8T_{7} - 572 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T + 13 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T - 572 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 130T + 4213 \) Copy content Toggle raw display
$17$ \( T^{2} - 14T - 3839 \) Copy content Toggle raw display
$19$ \( T^{2} + 48T - 396 \) Copy content Toggle raw display
$23$ \( T^{2} - 128T - 8972 \) Copy content Toggle raw display
$29$ \( T^{2} - 30T - 16203 \) Copy content Toggle raw display
$31$ \( T^{2} + 184T - 18044 \) Copy content Toggle raw display
$37$ \( T^{2} - 126T - 70923 \) Copy content Toggle raw display
$41$ \( T^{2} + 370T + 34177 \) Copy content Toggle raw display
$43$ \( T^{2} + 264T + 11616 \) Copy content Toggle raw display
$47$ \( T^{2} + 256T - 131468 \) Copy content Toggle raw display
$53$ \( T^{2} - 162T - 80139 \) Copy content Toggle raw display
$59$ \( T^{2} - 1304 T + 403936 \) Copy content Toggle raw display
$61$ \( T^{2} + 300T - 79068 \) Copy content Toggle raw display
$67$ \( T^{2} + 656T + 106132 \) Copy content Toggle raw display
$71$ \( T^{2} - 1176 T + 308112 \) Copy content Toggle raw display
$73$ \( T^{2} + 668T - 277244 \) Copy content Toggle raw display
$79$ \( T^{2} - 416T - 99308 \) Copy content Toggle raw display
$83$ \( T^{2} - 960T + 199188 \) Copy content Toggle raw display
$89$ \( T^{2} - 1074T - 83343 \) Copy content Toggle raw display
$97$ \( T^{2} + 338T - 628511 \) Copy content Toggle raw display
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