Properties

Label 5929.2.a.y
Level $5929$
Weight $2$
Character orbit 5929.a
Self dual yes
Analytic conductor $47.343$
Analytic rank $0$
Dimension $3$
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] = [5929,2,Mod(1,5929)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5929, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5929.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.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: \(47.3433033584\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + \beta_1 q^{3} + (\beta_{2} + 3) q^{4} + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1 - 4) q^{6} + (2 \beta_{2} - \beta_1 + 3) q^{8} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + \beta_1 q^{3} + (\beta_{2} + 3) q^{4} + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1 - 4) q^{6} + (2 \beta_{2} - \beta_1 + 3) q^{8} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{2} + 2) q^{10} + ( - \beta_{2} + 3 \beta_1 - 2) q^{12} + (\beta_{2} - \beta_1 - 2) q^{13} + ( - \beta_{2} - 2) q^{15} + (3 \beta_{2} - 2 \beta_1 + 5) q^{16} + (\beta_{2} - \beta_1 - 2) q^{17} + ( - 3 \beta_1 - 5) q^{18} + (2 \beta_{2} + 2 \beta_1) q^{19} + (2 \beta_{2} - 2 \beta_1 + 6) q^{20} + ( - \beta_{2} + 2) q^{23} + ( - 3 \beta_{2} + \beta_1 - 8) q^{24} + ( - \beta_{2} - 2 \beta_1 + 1) q^{25} + (3 \beta_{2} + 3 \beta_1 + 4) q^{26} + (\beta_{2} + 2 \beta_1 + 6) q^{27} + (2 \beta_{2} + 2 \beta_1) q^{29} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{30} + (\beta_1 + 4) q^{31} + (4 \beta_{2} - \beta_1 + 13) q^{32} + (3 \beta_{2} + 3 \beta_1 + 4) q^{34} + (\beta_{2} + 4 \beta_1 + 5) q^{36} + (\beta_{2} + 2 \beta_1 - 6) q^{37} + (2 \beta_{2} - 2 \beta_1 - 4) q^{38} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{39} + (2 \beta_{2} - 4 \beta_1 + 14) q^{40} + ( - \beta_{2} + \beta_1 - 6) q^{41} + (2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{45} + ( - 2 \beta_{2} - 2 \beta_1) q^{46} + ( - \beta_{2} + \beta_1 - 2) q^{47} + ( - 5 \beta_{2} + \beta_1 - 14) q^{48} + (\beta_1 + 7) q^{50} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{51} + (\beta_{2} - 5 \beta_1 + 2) q^{52} + ( - 2 \beta_{2} - 4) q^{53} - 8 \beta_1 q^{54} + (4 \beta_1 + 4) q^{57} + (2 \beta_{2} - 2 \beta_1 - 4) q^{58} - \beta_1 q^{59} + ( - 4 \beta_{2} + 2 \beta_1 - 12) q^{60} + (\beta_{2} - \beta_1 + 6) q^{61} + ( - \beta_{2} - 5 \beta_1) q^{62} + (3 \beta_{2} - 8 \beta_1 + 15) q^{64} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{65} + (\beta_{2} + 4 \beta_1 - 2) q^{67} + (\beta_{2} - 5 \beta_1 + 2) q^{68} + (\beta_{2} + 2 \beta_1 + 2) q^{69} + (5 \beta_{2} + 4 \beta_1 + 2) q^{71} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{72} + ( - \beta_{2} + \beta_1 - 6) q^{73} + (4 \beta_1 - 12) q^{74} + ( - \beta_{2} - 3 \beta_1 - 6) q^{75} + (2 \beta_{2} + 2 \beta_1 + 8) q^{76} + (10 \beta_1 + 6) q^{78} + (2 \beta_{2} + 10) q^{79} + (4 \beta_{2} - 6 \beta_1 + 22) q^{80} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + ( - 3 \beta_{2} + 5 \beta_1 - 12) q^{82} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{85} + (4 \beta_{2} - 2 \beta_1 + 6) q^{86} + (4 \beta_1 + 4) q^{87} + (\beta_{2} - 2 \beta_1 + 8) q^{89} + ( - 2 \beta_{2} + 6) q^{90} + 2 \beta_1 q^{92} + (\beta_{2} + 6 \beta_1 + 4) q^{93} + ( - 3 \beta_{2} + \beta_1 - 8) q^{94} + ( - 4 \beta_{2} - 4 \beta_1 + 8) q^{95} + ( - 5 \beta_{2} + 11 \beta_1 - 12) q^{96} + (3 \beta_{2} + 2 \beta_1 + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + q^{3} + 8 q^{4} - q^{5} - 12 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + q^{3} + 8 q^{4} - q^{5} - 12 q^{6} + 6 q^{8} + 4 q^{9} + 4 q^{10} - 2 q^{12} - 8 q^{13} - 5 q^{15} + 10 q^{16} - 8 q^{17} - 18 q^{18} + 14 q^{20} + 7 q^{23} - 20 q^{24} + 2 q^{25} + 12 q^{26} + 19 q^{27} - 8 q^{30} + 13 q^{31} + 34 q^{32} + 12 q^{34} + 18 q^{36} - 17 q^{37} - 16 q^{38} - 20 q^{39} + 36 q^{40} - 16 q^{41} + 4 q^{43} + 6 q^{45} - 4 q^{47} - 36 q^{48} + 22 q^{50} - 20 q^{51} - 10 q^{53} - 8 q^{54} + 16 q^{57} - 16 q^{58} - q^{59} - 30 q^{60} + 16 q^{61} - 4 q^{62} + 34 q^{64} + 24 q^{65} - 3 q^{67} + 7 q^{69} + 5 q^{71} + 2 q^{72} - 16 q^{73} - 32 q^{74} - 20 q^{75} + 24 q^{76} + 28 q^{78} + 28 q^{79} + 56 q^{80} + 15 q^{81} - 28 q^{82} - 8 q^{83} + 24 q^{85} + 12 q^{86} + 16 q^{87} + 21 q^{89} + 20 q^{90} + 2 q^{92} + 17 q^{93} - 20 q^{94} + 24 q^{95} - 20 q^{96} + 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
3.12489
−0.363328
−1.76156
−2.12489 3.12489 2.51514 −0.484862 −6.64002 0 −1.09461 6.76491 1.03028
1.2 1.36333 −0.363328 −0.141336 −3.14134 −0.495336 0 −2.91934 −2.86799 −4.28267
1.3 2.76156 −1.76156 5.62620 2.62620 −4.86464 0 10.0140 0.103084 7.25240
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)

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 5929.2.a.y 3
7.b odd 2 1 847.2.a.j yes 3
11.b odd 2 1 5929.2.a.t 3
21.c even 2 1 7623.2.a.bz 3
77.b even 2 1 847.2.a.i 3
77.j odd 10 4 847.2.f.t 12
77.l even 10 4 847.2.f.u 12
231.h odd 2 1 7623.2.a.ce 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i 3 77.b even 2 1
847.2.a.j yes 3 7.b odd 2 1
847.2.f.t 12 77.j odd 10 4
847.2.f.u 12 77.l even 10 4
5929.2.a.t 3 11.b odd 2 1
5929.2.a.y 3 1.a even 1 1 trivial
7623.2.a.bz 3 21.c even 2 1
7623.2.a.ce 3 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5929))\):

\( T_{2}^{3} - 2T_{2}^{2} - 5T_{2} + 8 \) Copy content Toggle raw display
\( T_{3}^{3} - T_{3}^{2} - 6T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 8T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} - 5 T + 8 \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 8T - 4 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 2 T - 64 \) Copy content Toggle raw display
$17$ \( T^{3} + 8 T^{2} + 2 T - 64 \) Copy content Toggle raw display
$19$ \( T^{3} - 40T + 64 \) Copy content Toggle raw display
$23$ \( T^{3} - 7 T^{2} + 8 T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} - 40T + 64 \) Copy content Toggle raw display
$31$ \( T^{3} - 13 T^{2} + 50 T - 58 \) Copy content Toggle raw display
$37$ \( T^{3} + 17 T^{2} + 72 T + 16 \) Copy content Toggle raw display
$41$ \( T^{3} + 16 T^{2} + 66 T + 80 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} - 28 T + 32 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 14 T + 8 \) Copy content Toggle raw display
$53$ \( T^{3} + 10T^{2} - 64 \) Copy content Toggle raw display
$59$ \( T^{3} + T^{2} - 6T + 2 \) Copy content Toggle raw display
$61$ \( T^{3} - 16 T^{2} + 66 T - 80 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} - 88 T - 424 \) Copy content Toggle raw display
$71$ \( T^{3} - 5 T^{2} - 208 T + 1480 \) Copy content Toggle raw display
$73$ \( T^{3} + 16 T^{2} + 66 T + 80 \) Copy content Toggle raw display
$79$ \( T^{3} - 28 T^{2} + 228 T - 512 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 56 T + 64 \) Copy content Toggle raw display
$89$ \( T^{3} - 21 T^{2} + 104 T - 100 \) Copy content Toggle raw display
$97$ \( T^{3} - 11 T^{2} - 32 T + 452 \) Copy content Toggle raw display
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