↑

Properties

Label 33.12.d.a
Level $33$
Weight $12$
Character orbit 33.d
Analytic conductor $25.355$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [33,12,Mod(32,33)] 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("33.32"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.3553249585\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 11 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(-1 + 11\sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (23 \beta + 45) q^{3} - 2048 q^{4} + ( - 178 \beta - 89) q^{5} + (1541 \beta - 174132) q^{9} + ( - 29282 \beta - 14641) q^{11} + ( - 47104 \beta - 92160) q^{12} + ( - 5963 \beta + 1359297) q^{15} + 4194304 q^{16}+ \cdots + (5121495005 \beta + 17575612758) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 67 q^{3} - 4096 q^{4} - 349805 q^{9} - 137216 q^{12} + 2724557 q^{15} + 8388608 q^{16} + 76570548 q^{25} - 35305784 q^{27} + 584788390 q^{31} + 448204933 q^{33} + 716400640 q^{36} + 750230698 q^{37}+ \cdots + 30029730511 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
0.500000 − 1.65831i
0.500000 + 1.65831i
0 33.5000 − 419.553i −2048.00 3246.98i 0 0 0 −174902. − 28110.1i 0
32.2 0 33.5000 + 419.553i −2048.00 − 3246.98i 0 0 0 −174902. + 28110.1i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.12.d.a ✓ 2
3.b odd 2 1 inner 33.12.d.a ✓ 2
11.b odd 2 1 CM 33.12.d.a ✓ 2
33.d even 2 1 inner 33.12.d.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.12.d.a ✓ 2 1.a even 1 1 trivial
33.12.d.a ✓ 2 3.b odd 2 1 inner
33.12.d.a ✓ 2 11.b odd 2 1 CM
33.12.d.a ✓ 2 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{12}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 67T + 177147 \) Copy content Toggle raw display
$5$ \( T^{2} + 10542851 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 285311670611 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17\!\cdots\!19 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 292394195)^{2} \) Copy content Toggle raw display
$37$ \( (T - 375115349)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 43\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{2} + 60\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{2} + 55\!\cdots\!11 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 13691120599)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 13\!\cdots\!75 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 52\!\cdots\!75 \) Copy content Toggle raw display
$97$ \( (T - 151692012401)^{2} \) Copy content Toggle raw display
show more
show less