Properties

Label 99.4.a.c
Level $99$
Weight $4$
Character orbit 99.a
Self dual yes
Analytic conductor $5.841$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

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: yes
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
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: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + (4 \beta + 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 9 \beta + 25) q^{10} + 11 q^{11} + (20 \beta + 40) q^{13} + (6 \beta + 2) q^{14}+ \cdots + ( - 275 \beta + 435) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28}+ \cdots + 870 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−2.73205 0 −0.535898 −14.8564 0 3.07180 23.3205 0 40.5885
1.2 0.732051 0 −7.46410 12.8564 0 16.9282 −11.3205 0 9.41154
\(n\): e.g. 2-40 or 990-1000
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 99.4.a.c 2
3.b odd 2 1 11.4.a.a 2
4.b odd 2 1 1584.4.a.bc 2
5.b even 2 1 2475.4.a.q 2
11.b odd 2 1 1089.4.a.v 2
12.b even 2 1 176.4.a.i 2
15.d odd 2 1 275.4.a.b 2
15.e even 4 2 275.4.b.c 4
21.c even 2 1 539.4.a.e 2
24.f even 2 1 704.4.a.n 2
24.h odd 2 1 704.4.a.p 2
33.d even 2 1 121.4.a.c 2
33.f even 10 4 121.4.c.f 8
33.h odd 10 4 121.4.c.c 8
39.d odd 2 1 1859.4.a.a 2
132.d odd 2 1 1936.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 3.b odd 2 1
99.4.a.c 2 1.a even 1 1 trivial
121.4.a.c 2 33.d even 2 1
121.4.c.c 8 33.h odd 10 4
121.4.c.f 8 33.f even 10 4
176.4.a.i 2 12.b even 2 1
275.4.a.b 2 15.d odd 2 1
275.4.b.c 4 15.e even 4 2
539.4.a.e 2 21.c even 2 1
704.4.a.n 2 24.f even 2 1
704.4.a.p 2 24.h odd 2 1
1089.4.a.v 2 11.b odd 2 1
1584.4.a.bc 2 4.b odd 2 1
1859.4.a.a 2 39.d odd 2 1
1936.4.a.w 2 132.d odd 2 1
2475.4.a.q 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 52 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 80T + 400 \) Copy content Toggle raw display
$17$ \( T^{2} - 124T + 3412 \) Copy content Toggle raw display
$19$ \( T^{2} - 72T - 9504 \) Copy content Toggle raw display
$23$ \( T^{2} - 98T - 1487 \) Copy content Toggle raw display
$29$ \( T^{2} + 144T - 4224 \) Copy content Toggle raw display
$31$ \( T^{2} + 34T - 2063 \) Copy content Toggle raw display
$37$ \( T^{2} - 54T + 537 \) Copy content Toggle raw display
$41$ \( T^{2} + 536T + 71776 \) Copy content Toggle raw display
$43$ \( T^{2} + 60T + 132 \) Copy content Toggle raw display
$47$ \( T^{2} - 272T - 24704 \) Copy content Toggle raw display
$53$ \( T^{2} - 492T + 51108 \) Copy content Toggle raw display
$59$ \( T^{2} + 634T + 48217 \) Copy content Toggle raw display
$61$ \( T^{2} - 840T + 74832 \) Copy content Toggle raw display
$67$ \( T^{2} - 754T + 140929 \) Copy content Toggle raw display
$71$ \( T^{2} - 678T + 97593 \) Copy content Toggle raw display
$73$ \( T^{2} + 400T - 617072 \) Copy content Toggle raw display
$79$ \( T^{2} - 316 T - 1266044 \) Copy content Toggle raw display
$83$ \( T^{2} + 468T + 11556 \) Copy content Toggle raw display
$89$ \( T^{2} - 1842 T + 525489 \) Copy content Toggle raw display
$97$ \( T^{2} - 2194 T + 1141201 \) Copy content Toggle raw display
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