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Properties

Label 5929.2.a.bx
Level $5929$
Weight $2$
Character orbit 5929.a
Self dual yes
Analytic conductor $47.343$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 14x^{8} + 14x^{7} + 61x^{6} - 57x^{5} - 84x^{4} + 63x^{3} + 20x^{2} - 15x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{8} q^{3} + ( - \beta_{8} + \beta_{4} + \beta_{3} + 1) q^{4} - \beta_{9} q^{5} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{8} q^{3} + ( - \beta_{8} + \beta_{4} + \beta_{3} + 1) q^{4} - \beta_{9} q^{5} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 3 q^{3} + 9 q^{4} + 2 q^{5} + 9 q^{6} + 3 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 3 q^{3} + 9 q^{4} + 2 q^{5} + 9 q^{6} + 3 q^{8} + 9 q^{9} + 7 q^{10} - 9 q^{12} + q^{13} - q^{15} + 15 q^{16} + 19 q^{17} - 17 q^{18} + 28 q^{19} + 5 q^{20} + 7 q^{23} + 19 q^{24} - 8 q^{25} + 5 q^{26} - 6 q^{27} + 15 q^{29} + 22 q^{30} + 14 q^{31} + 15 q^{32} - 12 q^{34} + 16 q^{36} + 13 q^{37} - 24 q^{38} - 4 q^{39} + 10 q^{40} + 35 q^{41} - 18 q^{43} - 8 q^{45} - 9 q^{46} - 16 q^{47} + 33 q^{48} - 6 q^{50} + 21 q^{51} - 4 q^{52} - 9 q^{53} + 17 q^{54} - 4 q^{57} - 9 q^{58} - 12 q^{59} - 21 q^{60} + 20 q^{61} + 38 q^{62} - 29 q^{64} + 20 q^{65} + 19 q^{67} + 56 q^{68} - 9 q^{69} + 15 q^{71} + 4 q^{72} + 3 q^{73} - 42 q^{74} + 27 q^{75} + 24 q^{76} - 25 q^{78} + 32 q^{79} + 6 q^{80} - 46 q^{81} + 18 q^{82} + 29 q^{83} + 23 q^{85} + 9 q^{86} + 24 q^{87} - 5 q^{89} + 12 q^{90} + 15 q^{92} - q^{93} - 19 q^{94} - q^{95} + 46 q^{96} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 14x^{8} + 14x^{7} + 61x^{6} - 57x^{5} - 84x^{4} + 63x^{3} + 20x^{2} - 15x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34 \nu^{9} - 59 \nu^{8} + 454 \nu^{7} + 726 \nu^{6} - 2159 \nu^{5} - 2952 \nu^{4} + 4578 \nu^{3} + \cdots - 832 ) / 677 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 95 \nu^{9} - 94 \nu^{8} - 1388 \nu^{7} + 1237 \nu^{6} + 6371 \nu^{5} - 4336 \nu^{4} - 9287 \nu^{3} + \cdots - 224 ) / 677 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 164 \nu^{9} - 34 \nu^{8} - 2389 \nu^{7} + 361 \nu^{6} + 11091 \nu^{5} - 416 \nu^{4} - 17144 \nu^{3} + \cdots - 686 ) / 677 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 224 \nu^{9} + 129 \nu^{8} + 3230 \nu^{7} - 1748 \nu^{6} - 14901 \nu^{5} + 6397 \nu^{4} + 23152 \nu^{3} + \cdots + 970 ) / 677 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 250 \nu^{9} + 283 \nu^{8} + 3617 \nu^{7} - 3861 \nu^{6} - 16552 \nu^{5} + 15330 \nu^{4} + \cdots + 3440 ) / 677 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 257 \nu^{9} - 12 \nu^{8} - 3591 \nu^{7} + 446 \nu^{6} + 15981 \nu^{5} - 2138 \nu^{4} - 23613 \nu^{3} + \cdots - 43 ) / 677 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 259 \nu^{9} - 128 \nu^{8} - 3777 \nu^{7} + 1598 \nu^{6} + 17462 \nu^{5} - 4752 \nu^{4} - 26431 \nu^{3} + \cdots + 1121 ) / 677 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 496 \nu^{9} + 334 \nu^{8} + 6862 \nu^{7} - 4741 \nu^{6} - 29465 \nu^{5} + 18662 \nu^{4} + \cdots + 2438 ) / 677 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} + 5\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{8} + \beta_{5} + 6\beta_{4} + 8\beta_{3} - \beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 8\beta_{6} - 18\beta_{5} - 9\beta_{4} - \beta_{3} + 6\beta_{2} + 28\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} - 36 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + 12 \beta_{5} + 35 \beta_{4} + 55 \beta_{3} + \cdots + 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{9} - 14\beta_{7} + 58\beta_{6} - 134\beta_{5} - 67\beta_{4} - 14\beta_{3} + 32\beta_{2} + 166\beta _1 - 126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{9} - 219 \beta_{8} + 43 \beta_{7} - 28 \beta_{6} + 114 \beta_{5} + 212 \beta_{4} + 364 \beta_{3} + \cdots + 564 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 43 \beta_{9} + 5 \beta_{8} - 134 \beta_{7} + 407 \beta_{6} - 944 \beta_{5} - 472 \beta_{4} + \cdots - 897 \) 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
2.44760
2.24539
1.84883
0.673964
0.437218
0.0760218
−0.585278
−1.27507
−2.23710
−2.63157
−2.44760 −2.91502 3.99072 0.331923 7.13479 0 −4.87248 5.49735 −0.812412
1.2 −2.24539 1.90908 3.04179 −2.46504 −4.28664 0 −2.33923 0.644603 5.53498
1.3 −1.84883 −1.18200 1.41817 2.36270 2.18532 0 1.07570 −1.60287 −4.36823
1.4 −0.673964 −2.15114 −1.54577 −1.68909 1.44979 0 2.38972 1.62738 1.13839
1.5 −0.437218 2.58353 −1.80884 1.21579 −1.12957 0 1.66530 3.67462 −0.531564
1.6 −0.0760218 2.35507 −1.99422 −2.57807 −0.179037 0 0.303648 2.54635 0.195990
1.7 0.585278 1.38265 −1.65745 3.96418 0.809236 0 −2.14063 −1.08827 2.32015
1.8 1.27507 0.139310 −0.374193 −1.18507 0.177630 0 −3.02727 −2.98059 −1.51105
1.9 2.23710 −1.33493 3.00462 0.866333 −2.98638 0 2.24744 −1.21795 1.93808
1.10 2.63157 2.21345 4.92517 1.17636 5.82486 0 7.69779 1.89937 3.09567
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bx 10
7.b odd 2 1 5929.2.a.bw 10
7.d odd 6 2 847.2.e.i 20
11.b odd 2 1 5929.2.a.bz 10
11.c even 5 2 539.2.f.g 20
77.b even 2 1 5929.2.a.by 10
77.i even 6 2 847.2.e.h 20
77.j odd 10 2 539.2.f.h 20
77.m even 15 4 539.2.q.h 40
77.n even 30 4 847.2.n.h 40
77.n even 30 4 847.2.n.j 40
77.p odd 30 4 77.2.m.b ✓ 40
77.p odd 30 4 847.2.n.i 40
231.bc even 30 4 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 77.p odd 30 4
539.2.f.g 20 11.c even 5 2
539.2.f.h 20 77.j odd 10 2
539.2.q.h 40 77.m even 15 4
693.2.by.b 40 231.bc even 30 4
847.2.e.h 20 77.i even 6 2
847.2.e.i 20 7.d odd 6 2
847.2.n.h 40 77.n even 30 4
847.2.n.i 40 77.p odd 30 4
847.2.n.j 40 77.n even 30 4
5929.2.a.bw 10 7.b odd 2 1
5929.2.a.bx 10 1.a even 1 1 trivial
5929.2.a.by 10 77.b even 2 1
5929.2.a.bz 10 11.b odd 2 1

Hecke kernels

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

\( T_{2}^{10} + T_{2}^{9} - 14T_{2}^{8} - 14T_{2}^{7} + 61T_{2}^{6} + 57T_{2}^{5} - 84T_{2}^{4} - 63T_{2}^{3} + 20T_{2}^{2} + 15T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{10} - 3T_{3}^{9} - 15T_{3}^{8} + 50T_{3}^{7} + 64T_{3}^{6} - 265T_{3}^{5} - 69T_{3}^{4} + 530T_{3}^{3} - 34T_{3}^{2} - 357T_{3} + 49 \) Copy content Toggle raw display
\( T_{5}^{10} - 2 T_{5}^{9} - 19 T_{5}^{8} + 27 T_{5}^{7} + 113 T_{5}^{6} - 134 T_{5}^{5} - 235 T_{5}^{4} + \cdots + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots - 49 \) Copy content Toggle raw display
$17$ \( T^{10} - 19 T^{9} + \cdots + 113141 \) Copy content Toggle raw display
$19$ \( T^{10} - 28 T^{9} + \cdots + 15631 \) Copy content Toggle raw display
$23$ \( T^{10} - 7 T^{9} + \cdots + 13711 \) Copy content Toggle raw display
$29$ \( T^{10} - 15 T^{9} + \cdots - 33129 \) Copy content Toggle raw display
$31$ \( T^{10} - 14 T^{9} + \cdots + 1421 \) Copy content Toggle raw display
$37$ \( T^{10} - 13 T^{9} + \cdots + 2238959 \) Copy content Toggle raw display
$41$ \( T^{10} - 35 T^{9} + \cdots - 49 \) Copy content Toggle raw display
$43$ \( T^{10} + 18 T^{9} + \cdots + 181456 \) Copy content Toggle raw display
$47$ \( T^{10} + 16 T^{9} + \cdots - 18712169 \) Copy content Toggle raw display
$53$ \( T^{10} + 9 T^{9} + \cdots - 1469471 \) Copy content Toggle raw display
$59$ \( T^{10} + 12 T^{9} + \cdots + 35231 \) Copy content Toggle raw display
$61$ \( T^{10} - 20 T^{9} + \cdots - 3483851 \) Copy content Toggle raw display
$67$ \( T^{10} - 19 T^{9} + \cdots + 2377069 \) Copy content Toggle raw display
$71$ \( T^{10} - 15 T^{9} + \cdots - 1017431 \) Copy content Toggle raw display
$73$ \( T^{10} - 3 T^{9} + \cdots - 54045775 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 128791951 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 326647279 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 112558831 \) Copy content Toggle raw display
$97$ \( T^{10} + 4 T^{9} + \cdots + 3426031 \) Copy content Toggle raw display
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