↑

Properties

Label 531.8.a.h
Level $531$
Weight $8$
Character orbit 531.a
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $33$
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] = [531,8,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 33 q + 24 q^{2} + 1776 q^{4} + 1000 q^{5} + 154 q^{7} + 4608 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q + 24 q^{2} + 1776 q^{4} + 1000 q^{5} + 154 q^{7} + 4608 q^{8} + 5042 q^{10} + 13310 q^{11} - 14172 q^{13} + 16464 q^{14} + 95772 q^{16} + 39304 q^{17} - 56302 q^{19} - 17936 q^{20} + 152764 q^{22} + 17988 q^{23} + 468523 q^{25} - 27624 q^{26} - 59896 q^{28} + 474564 q^{29} + 188186 q^{31} + 251068 q^{32} + 169888 q^{34} + 514500 q^{35} + 1148200 q^{37} + 446594 q^{38} + 501214 q^{40} + 1246152 q^{41} + 62268 q^{43} + 2555520 q^{44} - 1289942 q^{46} + 1485654 q^{47} + 3829555 q^{49} + 6430160 q^{50} - 2624804 q^{52} + 4086740 q^{53} - 1119118 q^{55} + 8448352 q^{56} + 2966706 q^{58} - 6777507 q^{59} + 2436146 q^{61} + 9005952 q^{62} + 11117562 q^{64} + 16730354 q^{65} - 2652248 q^{67} + 15929124 q^{68} + 3359254 q^{70} + 7356324 q^{71} + 1900454 q^{73} + 25386964 q^{74} - 16047360 q^{76} + 20774826 q^{77} - 5912712 q^{79} + 13568404 q^{80} + 1579434 q^{82} - 1052766 q^{83} + 18372730 q^{85} + 43499960 q^{86} + 18209214 q^{88} + 174788 q^{89} - 18891512 q^{91} + 46033270 q^{92} + 365448 q^{94} + 31505580 q^{95} + 14418540 q^{97} - 15964056 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −21.6578 0 341.059 −72.2224 0 −507.922 −4614.38 0 1564.18
1.2 −20.6786 0 299.604 201.025 0 −1518.16 −3548.54 0 −4156.91
1.3 −18.4202 0 211.304 223.120 0 836.763 −1534.48 0 −4109.92
1.4 −16.9368 0 158.854 −242.916 0 −217.217 −522.560 0 4114.21
1.5 −16.6761 0 150.092 −179.277 0 1623.32 −368.400 0 2989.64
1.6 −14.8457 0 92.3958 91.9070 0 −20.3654 528.571 0 −1364.43
1.7 −14.6544 0 86.7500 −491.700 0 −488.511 604.491 0 7205.54
1.8 −12.2720 0 22.6015 −21.4488 0 197.214 1293.45 0 263.220
1.9 −10.4698 0 −18.3823 512.219 0 1732.46 1532.60 0 −5362.85
1.10 −8.90712 0 −48.6632 513.910 0 −1280.82 1573.56 0 −4577.46
1.11 −7.37606 0 −73.5937 −217.499 0 −1252.90 1486.97 0 1604.29
1.12 −7.24734 0 −75.4761 1.17055 0 1097.81 1474.66 0 −8.48334
1.13 −5.32951 0 −99.5963 288.047 0 −224.025 1212.98 0 −1535.15
1.14 −3.44034 0 −116.164 −382.136 0 −641.295 840.007 0 1314.68
1.15 −1.65238 0 −125.270 −348.169 0 493.067 418.497 0 575.306
1.16 −1.54690 0 −125.607 282.598 0 −727.135 392.304 0 −437.150
1.17 −1.18257 0 −126.602 236.792 0 −950.759 301.084 0 −280.023
1.18 0.306188 0 −127.906 −108.774 0 1364.63 −78.3553 0 −33.3052
1.19 5.77426 0 −94.6579 −218.484 0 606.083 −1285.68 0 −1261.58
1.20 6.24354 0 −89.0182 531.463 0 445.980 −1354.96 0 3318.21
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(59\) \(1\)

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 531.8.a.h yes 33
3.b odd 2 1 531.8.a.g ✓ 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.8.a.g ✓ 33 3.b odd 2 1
531.8.a.h yes 33 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{33} - 24 T_{2}^{32} - 2712 T_{2}^{31} + 66112 T_{2}^{30} + 3271065 T_{2}^{29} + \cdots + 18\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display