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Properties

Label 99.4.g.a
Level $99$
Weight $4$
Character orbit 99.g
Analytic conductor $5.841$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,Mod(32,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.32");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.g (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 4 \beta_{2}) q^{3} + 8 \beta_{2} q^{4} + (2 \beta_{3} - 9 \beta_{2} + 18) q^{5} + (5 \beta_{2} - 8 \beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 4 \beta_{2}) q^{3} + 8 \beta_{2} q^{4} + (2 \beta_{3} - 9 \beta_{2} + 18) q^{5} + (5 \beta_{2} - 8 \beta_1 - 5) q^{9} + 11 \beta_1 q^{11} + (32 \beta_{2} - 8 \beta_1 - 32) q^{12} + ( - 18 \beta_{3} + 58 \beta_{2} + \cdots + 14) q^{15}+ \cdots + ( - 55 \beta_{3} - 968 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{3} + 16 q^{4} + 54 q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{3} + 16 q^{4} + 54 q^{5} - 10 q^{9} - 64 q^{12} + 172 q^{15} - 128 q^{16} + 432 q^{20} + 324 q^{25} + 272 q^{27} - 340 q^{31} - 484 q^{33} - 160 q^{36} - 868 q^{37} - 704 q^{45} + 108 q^{47} - 1024 q^{48} - 686 q^{49} + 968 q^{55} + 2160 q^{59} - 704 q^{60} - 2048 q^{64} + 416 q^{67} + 1276 q^{69} + 2592 q^{75} + 1358 q^{81} + 766 q^{93} - 34 q^{97} - 1936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 10\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{3} + 2\nu^{2} + 10\nu + 15 ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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.

comment: embeddings in the coefficient field
 
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
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
0 −0.872281 − 5.12241i 4.00000 − 6.92820i 19.2446 + 11.1109i 0 0 0 −25.4783 + 8.93637i 0
32.2 0 4.87228 − 1.80579i 4.00000 − 6.92820i 7.75544 + 4.47760i 0 0 0 20.4783 − 17.5966i 0
65.1 0 −0.872281 + 5.12241i 4.00000 + 6.92820i 19.2446 − 11.1109i 0 0 0 −25.4783 − 8.93637i 0
65.2 0 4.87228 + 1.80579i 4.00000 + 6.92820i 7.75544 − 4.47760i 0 0 0 20.4783 + 17.5966i 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 \(\Q(\sqrt{-11}) \)
9.d odd 6 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.g.a ✓ 4
3.b odd 2 1 297.4.g.a 4
9.c even 3 1 297.4.g.a 4
9.d odd 6 1 inner 99.4.g.a ✓ 4
11.b odd 2 1 CM 99.4.g.a ✓ 4
33.d even 2 1 297.4.g.a 4
99.g even 6 1 inner 99.4.g.a ✓ 4
99.h odd 6 1 297.4.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.g.a ✓ 4 1.a even 1 1 trivial
99.4.g.a ✓ 4 9.d odd 6 1 inner
99.4.g.a ✓ 4 11.b odd 2 1 CM
99.4.g.a ✓ 4 99.g even 6 1 inner
297.4.g.a 4 3.b odd 2 1
297.4.g.a 4 9.c even 3 1
297.4.g.a 4 33.d even 2 1
297.4.g.a 4 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 8 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{4} - 54 T^{3} + \cdots + 39601 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 1331 T^{2} + 1771561 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 1369296016 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 340 T^{3} + \cdots + 687855529 \) Copy content Toggle raw display
$37$ \( (T^{2} + 434 T + 36397)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 10511785729 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 156631518289 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 97982146441 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 531780768289 \) Copy content Toggle raw display
$71$ \( T^{4} + 1090366 T^{2} + 276656689 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 17600)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 7490419080769 \) Copy content Toggle raw display
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