↑

Properties

Label 5929.2.a.bs
Level $5929$
Weight $2$
Character orbit 5929.a
Self dual yes
Analytic conductor $47.343$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{7} + \beta_{4} - 1) q^{3} + (\beta_{6} + \beta_{5} + 1) q^{4} + ( - \beta_{3} - \beta_{2} - 2) q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{7} - \beta_{6} + \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{7} + \beta_{4} - 1) q^{3} + (\beta_{6} + \beta_{5} + 1) q^{4} + ( - \beta_{3} - \beta_{2} - 2) q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{7} + \beta_{6} - 3 \beta_{5} + \cdots - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 4 q^{3} + 7 q^{4} - 10 q^{5} - q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - 4 q^{3} + 7 q^{4} - 10 q^{5} - q^{6} + 14 q^{9} + 6 q^{10} - 9 q^{12} - 6 q^{13} + 11 q^{15} + q^{16} - 5 q^{17} - 8 q^{18} - 13 q^{19} - 23 q^{20} + 16 q^{23} - 10 q^{24} + 16 q^{25} + 6 q^{26} - 10 q^{27} - 9 q^{29} + 36 q^{30} - 9 q^{31} - 16 q^{32} + 12 q^{34} - 14 q^{36} + 7 q^{37} + 10 q^{38} - 13 q^{39} + 5 q^{40} - 10 q^{41} + 4 q^{43} - 35 q^{45} - 4 q^{46} - 16 q^{47} + 3 q^{48} - 6 q^{50} - 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} - 2 q^{57} - 15 q^{58} - q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} + 9 q^{68} - 20 q^{69} + 13 q^{71} + 35 q^{72} - 25 q^{73} - 33 q^{74} + 13 q^{75} + 26 q^{76} - 29 q^{78} - 4 q^{80} + 8 q^{81} + 13 q^{82} - 25 q^{83} - 3 q^{85} + 4 q^{86} - 36 q^{87} - 37 q^{89} - 2 q^{90} + 35 q^{92} + 21 q^{93} - 42 q^{94} - 21 q^{95} - 6 q^{96} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 5\nu^{5} - 30\nu^{4} - 30\nu^{3} + 130\nu^{2} + 135\nu - 140 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} - 4\nu^{6} + 35\nu^{5} + 35\nu^{4} - 165\nu^{3} - 60\nu^{2} + 180\nu - 45 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + 6\nu^{6} - 40\nu^{5} - 65\nu^{4} + 160\nu^{3} + 165\nu^{2} - 170\nu - 20 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 7\nu^{6} + 95\nu^{5} - 55\nu^{4} - 280\nu^{3} + 105\nu^{2} + 210\nu - 90 ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 7\nu^{6} - 95\nu^{5} + 55\nu^{4} + 280\nu^{3} - 80\nu^{2} - 210\nu + 15 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\nu^{7} - \nu^{6} - 135\nu^{5} + 15\nu^{4} + 440\nu^{3} - 90\nu^{2} - 405\nu + 170 ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 6\beta_{6} + 7\beta_{5} - \beta_{4} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 7\beta_{6} + 8\beta_{5} + 8\beta_{4} + 10\beta_{3} - 7\beta_{2} + 28\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{7} + 35\beta_{6} + 47\beta_{5} - 7\beta_{4} + 2\beta_{3} + \beta_{2} + 10\beta _1 + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{7} + 45\beta_{6} + 56\beta_{5} + 54\beta_{4} + 76\beta_{3} - 42\beta_{2} + 165\beta _1 + 31 \) Copy content Toggle raw display

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

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
2.55194
1.98451
1.11447
0.669744
0.226211
−1.40927
−1.70716
−2.43045
−2.55194 0.410598 4.51241 −3.42898 −1.04782 0 −6.41153 −2.83141 8.75055
1.2 −1.98451 −2.78639 1.93830 0.0269243 5.52964 0 0.122446 4.76399 −0.0534317
1.3 −1.11447 2.85882 −0.757964 −3.45608 −3.18606 0 3.07366 5.17284 3.85168
1.4 −0.669744 −3.13977 −1.55144 −2.14378 2.10284 0 2.37856 6.85818 1.43578
1.5 −0.226211 0.219130 −1.94883 2.49552 −0.0495696 0 0.893270 −2.95198 −0.564516
1.6 1.40927 2.16338 −0.0139645 1.83139 3.04878 0 −2.83822 1.68022 2.58091
1.7 1.70716 −2.29155 0.914391 −4.06637 −3.91205 0 −1.85331 2.25122 −6.94194
1.8 2.43045 −1.43421 3.90710 −1.25863 −3.48577 0 4.63512 −0.943053 −3.05904
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bs 8
7.b odd 2 1 847.2.a.o 8
11.b odd 2 1 5929.2.a.bt 8
11.d odd 10 2 539.2.f.e 16
21.c even 2 1 7623.2.a.cw 8
77.b even 2 1 847.2.a.p 8
77.j odd 10 2 847.2.f.v 16
77.j odd 10 2 847.2.f.x 16
77.l even 10 2 77.2.f.b ✓ 16
77.l even 10 2 847.2.f.w 16
77.n even 30 4 539.2.q.g 32
77.o odd 30 4 539.2.q.f 32
231.h odd 2 1 7623.2.a.ct 8
231.r odd 10 2 693.2.m.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b ✓ 16 77.l even 10 2
539.2.f.e 16 11.d odd 10 2
539.2.q.f 32 77.o odd 30 4
539.2.q.g 32 77.n even 30 4
693.2.m.i 16 231.r odd 10 2
847.2.a.o 8 7.b odd 2 1
847.2.a.p 8 77.b even 2 1
847.2.f.v 16 77.j odd 10 2
847.2.f.w 16 77.l even 10 2
847.2.f.x 16 77.j odd 10 2
5929.2.a.bs 8 1.a even 1 1 trivial
5929.2.a.bt 8 11.b odd 2 1
7623.2.a.ct 8 231.h odd 2 1
7623.2.a.cw 8 21.c even 2 1

Hecke kernels

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

\( T_{2}^{8} + T_{2}^{7} - 11T_{2}^{6} - 10T_{2}^{5} + 35T_{2}^{4} + 30T_{2}^{3} - 30T_{2}^{2} - 30T_{2} - 5 \) Copy content Toggle raw display
\( T_{3}^{8} + 4T_{3}^{7} - 11T_{3}^{6} - 54T_{3}^{5} + 15T_{3}^{4} + 186T_{3}^{3} + 64T_{3}^{2} - 96T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} + 10T_{5}^{7} + 22T_{5}^{6} - 67T_{5}^{5} - 285T_{5}^{4} - 62T_{5}^{3} + 680T_{5}^{2} + 576T_{5} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 11 T^{6} + \cdots - 5 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} + 10 T^{7} + \cdots - 16 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots - 13744 \) Copy content Toggle raw display
$17$ \( T^{8} + 5 T^{7} + \cdots - 1616 \) Copy content Toggle raw display
$19$ \( T^{8} + 13 T^{7} + \cdots + 7920 \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + \cdots + 859 \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + \cdots - 495 \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + \cdots - 51280 \) Copy content Toggle raw display
$37$ \( T^{8} - 7 T^{7} + \cdots - 461 \) Copy content Toggle raw display
$41$ \( T^{8} + 10 T^{7} + \cdots + 3664 \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} + \cdots - 971 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + \cdots - 2312080 \) Copy content Toggle raw display
$53$ \( T^{8} - 37 T^{7} + \cdots + 557531 \) Copy content Toggle raw display
$59$ \( T^{8} + T^{7} + \cdots - 13680 \) Copy content Toggle raw display
$61$ \( T^{8} - 19 T^{7} + \cdots + 15920 \) Copy content Toggle raw display
$67$ \( T^{8} + 19 T^{7} + \cdots - 27395 \) Copy content Toggle raw display
$71$ \( T^{8} - 13 T^{7} + \cdots - 19471 \) Copy content Toggle raw display
$73$ \( T^{8} + 25 T^{7} + \cdots + 324144 \) Copy content Toggle raw display
$79$ \( T^{8} - 295 T^{6} + \cdots - 3982275 \) Copy content Toggle raw display
$83$ \( T^{8} + 25 T^{7} + \cdots + 27504 \) Copy content Toggle raw display
$89$ \( T^{8} + 37 T^{7} + \cdots + 952400 \) Copy content Toggle raw display
$97$ \( T^{8} + 15 T^{7} + \cdots - 2155696 \) Copy content Toggle raw display
show more
show less