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Properties

Label 5929.2.a.cf
Level $5929$
Weight $2$
Character orbit 5929.a
Self dual yes
Analytic conductor $47.343$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $4$

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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: \(1\)
Dimension: \(16\)
Coefficient field: 16.16.367787275866480643670016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 28 x^{14} - 12 x^{13} + 272 x^{12} + 192 x^{11} - 1122 x^{10} - 816 x^{9} + 2315 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} - \beta_{5} q^{3} + (\beta_{13} + 1) q^{4} + (\beta_{11} + \beta_{4} - \beta_{3}) q^{5} + (\beta_{6} - \beta_1) q^{6} - \beta_{9} q^{8} + ( - \beta_{10} + \beta_{8}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} - \beta_{5} q^{3} + (\beta_{13} + 1) q^{4} + (\beta_{11} + \beta_{4} - \beta_{3}) q^{5} + (\beta_{6} - \beta_1) q^{6} - \beta_{9} q^{8} + ( - \beta_{10} + \beta_{8}) q^{9} + (\beta_{7} + \beta_1) q^{10} + (\beta_{11} - \beta_{4} + \beta_{3}) q^{12} + ( - \beta_{6} + 2 \beta_1) q^{13} + ( - \beta_{13} - \beta_{8} - 1) q^{15} + ( - \beta_{13} - 2) q^{16} + (\beta_{14} - 2 \beta_{7} + \cdots - 2 \beta_1) q^{17}+ \cdots + (\beta_{11} + 2 \beta_{5} + \cdots + 5 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 8 q^{15} - 24 q^{16} - 56 q^{23} - 8 q^{25} + 8 q^{37} - 24 q^{53} - 56 q^{58} - 56 q^{60} - 48 q^{64} - 16 q^{67} - 144 q^{71} - 168 q^{78} - 56 q^{81} - 88 q^{86} + 24 q^{92} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 28 x^{14} - 12 x^{13} + 272 x^{12} + 192 x^{11} - 1122 x^{10} - 816 x^{9} + 2315 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 64330 \nu^{15} - 14957 \nu^{14} + 1796025 \nu^{13} + 1162602 \nu^{12} - 17166119 \nu^{11} + \cdots - 1768603 ) / 655479 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3591160 \nu^{15} - 1860368 \nu^{14} + 101376726 \nu^{13} + 94065945 \nu^{12} - 975609488 \nu^{11} + \cdots + 46042793 ) / 19008891 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8910676 \nu^{15} - 4203554 \nu^{14} + 246634545 \nu^{13} + 221646483 \nu^{12} + \cdots + 15427694 ) / 19008891 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24870664 \nu^{15} + 17475824 \nu^{14} - 678716382 \nu^{13} - 777767103 \nu^{12} + 6078281114 \nu^{11} + \cdots - 13050044 ) / 19008891 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31208885 \nu^{15} - 27732604 \nu^{14} + 851040276 \nu^{13} + 1128117483 \nu^{12} + \cdots - 5925998 ) / 19008891 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1480405 \nu^{15} - 1121957 \nu^{14} + 40513902 \nu^{13} + 48366996 \nu^{12} - 363571493 \nu^{11} + \cdots + 1459154 ) / 655479 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 142504 \nu^{15} + 114980 \nu^{14} - 3888834 \nu^{13} - 4844202 \nu^{12} + 34616117 \nu^{11} + \cdots + 31033 ) / 59589 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1639618 \nu^{15} + 1426787 \nu^{14} - 44877018 \nu^{13} - 58459653 \nu^{12} + 400490096 \nu^{11} + \cdots - 1962185 ) / 655479 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 48033406 \nu^{15} + 38406413 \nu^{14} - 1312032942 \nu^{13} - 1623400500 \nu^{12} + \cdots - 23457251 ) / 19008891 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1924952 \nu^{15} + 1773268 \nu^{14} - 52384488 \nu^{13} - 71376318 \nu^{12} + 461020438 \nu^{11} + \cdots - 3094939 ) / 655479 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25658 \nu^{15} + 22723 \nu^{14} - 699333 \nu^{13} - 926802 \nu^{12} + 6185329 \nu^{11} + 10406531 \nu^{10} + \cdots - 42508 ) / 8613 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64184140 \nu^{15} + 52254788 \nu^{14} - 1751504832 \nu^{13} - 2196346800 \nu^{12} + \cdots - 69464165 ) / 19008891 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 87664 \nu^{15} - 78840 \nu^{14} + 2384748 \nu^{13} + 3200244 \nu^{12} - 20997112 \nu^{11} + \cdots + 34591 ) / 24277 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1052252 \nu^{15} + 969464 \nu^{14} - 28627148 \nu^{13} - 39016402 \nu^{12} + 251779142 \nu^{11} + \cdots - 1150857 ) / 218493 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 125901440 \nu^{15} + 106981081 \nu^{14} - 3427210380 \nu^{13} - 4428395562 \nu^{12} + \cdots - 181427359 ) / 19008891 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{12} - \beta_{10} + \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} + 2\beta_{11} - \beta_{9} - \beta_{8} - 2\beta_{7} - 2\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{14} + 3 \beta_{13} - 7 \beta_{12} - 10 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} + 4 \beta_{14} - 14 \beta_{12} + 14 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} - 8 \beta_{8} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{15} + 114 \beta_{14} + 50 \beta_{13} - 87 \beta_{12} + 19 \beta_{11} - 102 \beta_{10} + \cdots + 115 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 105 \beta_{15} + 170 \beta_{14} + 45 \beta_{13} - 379 \beta_{12} + 390 \beta_{11} - 77 \beta_{10} + \cdots + 694 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 154 \beta_{15} + 1412 \beta_{14} + 728 \beta_{13} - 1221 \beta_{12} + 537 \beta_{11} - 1109 \beta_{10} + \cdots + 1750 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 784 \beta_{15} + 1424 \beta_{14} + 609 \beta_{13} - 2588 \beta_{12} + 2712 \beta_{11} - 590 \beta_{10} + \cdots + 4423 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3213 \beta_{15} + 18280 \beta_{14} + 10434 \beta_{13} - 17390 \beta_{12} + 10661 \beta_{11} + \cdots + 25332 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 22695 \beta_{15} + 44284 \beta_{14} + 23625 \beta_{13} - 71465 \beta_{12} + 75740 \beta_{11} + \cdots + 117067 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 57123 \beta_{15} + 242928 \beta_{14} + 149424 \beta_{13} - 247883 \beta_{12} + 184266 \beta_{11} + \cdots + 362043 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 163053 \beta_{15} + 333098 \beta_{14} + 201366 \beta_{13} - 497529 \beta_{12} + 531664 \beta_{11} + \cdots + 790398 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 935623 \beta_{15} + 3286491 \beta_{14} + 2140775 \beta_{13} - 3532466 \beta_{12} + 2967771 \beta_{11} + \cdots + 5156671 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 4676672 \beta_{15} + 9842296 \beta_{14} + 6431880 \beta_{13} - 13941286 \beta_{12} + 14999236 \beta_{11} + \cdots + 21638995 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 14619855 \beta_{15} + 45052809 \beta_{14} + 30671432 \beta_{13} - 50334158 \beta_{12} + \cdots + 73387318 ) / 2 \) 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
1.47254
0.864778
3.77659
−1.96528
1.38242
0.233229
−3.20459
3.25895
−0.459309
−2.04454
0.867273
−0.0334283
0.349635
−1.27274
−1.69862
−1.52690
−2.07431 −2.21874 2.30278 2.28084 4.60237 0 −0.628052 1.92282 −4.73117
1.2 −2.07431 −1.03787 2.30278 −0.438435 2.15288 0 −0.628052 −1.92282 0.909452
1.3 −2.07431 1.03787 2.30278 0.438435 −2.15288 0 −0.628052 −1.92282 −0.909452
1.4 −2.07431 2.21874 2.30278 −2.28084 −4.60237 0 −0.628052 1.92282 4.73117
1.5 −0.835000 −2.38796 −1.30278 −0.282552 1.99395 0 2.75782 2.70236 0.235931
1.6 −0.835000 −0.545560 −1.30278 −3.53917 0.455542 0 2.75782 −2.70236 2.95520
1.7 −0.835000 0.545560 −1.30278 3.53917 −0.455542 0 2.75782 −2.70236 −2.95520
1.8 −0.835000 2.38796 −1.30278 0.282552 −1.99395 0 2.75782 2.70236 −0.235931
1.9 0.835000 −2.38796 −1.30278 −0.282552 −1.99395 0 −2.75782 2.70236 −0.235931
1.10 0.835000 −0.545560 −1.30278 −3.53917 −0.455542 0 −2.75782 −2.70236 −2.95520
1.11 0.835000 0.545560 −1.30278 3.53917 0.455542 0 −2.75782 −2.70236 2.95520
1.12 0.835000 2.38796 −1.30278 0.282552 1.99395 0 −2.75782 2.70236 0.235931
1.13 2.07431 −2.21874 2.30278 2.28084 −4.60237 0 0.628052 1.92282 4.73117
1.14 2.07431 −1.03787 2.30278 −0.438435 −2.15288 0 0.628052 −1.92282 −0.909452
1.15 2.07431 1.03787 2.30278 0.438435 2.15288 0 0.628052 −1.92282 0.909452
1.16 2.07431 2.21874 2.30278 −2.28084 4.60237 0 0.628052 1.92282 −4.73117
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5929.2.a.cf ✓ 16
7.b odd 2 1 inner 5929.2.a.cf ✓ 16
11.b odd 2 1 inner 5929.2.a.cf ✓ 16
77.b even 2 1 inner 5929.2.a.cf ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5929.2.a.cf ✓ 16 1.a even 1 1 trivial
5929.2.a.cf ✓ 16 7.b odd 2 1 inner
5929.2.a.cf ✓ 16 11.b odd 2 1 inner
5929.2.a.cf ✓ 16 77.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 5T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{8} - 12T_{3}^{6} + 43T_{3}^{4} - 42T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{8} - 18T_{5}^{6} + 70T_{5}^{4} - 18T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 5 T^{2} + 3)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} - 12 T^{6} + 43 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 18 T^{6} + 70 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 68 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 110 T^{6} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 62 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 14 T^{3} + \cdots - 731)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} - 202 T^{6} + \cdots + 751689)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 118 T^{6} + \cdots + 149769)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} - 90 T^{2} + \cdots + 52)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 198 T^{6} + \cdots + 2313441)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 130 T^{6} + \cdots + 25281)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 186 T^{6} + \cdots + 169)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} + \cdots + 117)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 306 T^{6} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 212 T^{6} + \cdots + 1734489)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 5391)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 36 T^{3} + \cdots + 3267)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 302 T^{6} + \cdots + 439569)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 226 T^{6} + \cdots + 6610041)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 494 T^{6} + \cdots + 131537961)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 476 T^{6} + \cdots + 88566921)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 338 T^{6} + \cdots + 279841)^{2} \) Copy content Toggle raw display
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