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Properties

Label 531.8.a.g
Level $531$
Weight $8$
Character orbit 531.a
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $33$
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] = [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: \(1\)
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 −21.9302 0 352.934 397.348 0 802.058 −4932.86 0 −8713.93
1.2 −21.2047 0 321.639 −386.781 0 166.779 −4106.06 0 8201.57
1.3 −19.9006 0 268.036 −368.296 0 −615.218 −2786.80 0 7329.32
1.4 −19.6384 0 257.669 −206.692 0 1442.78 −2546.49 0 4059.12
1.5 −16.4456 0 142.458 503.166 0 −880.461 −237.762 0 −8274.86
1.6 −16.1147 0 131.683 151.027 0 −1122.77 −59.3436 0 −2433.75
1.7 −15.7474 0 119.981 −232.890 0 −1600.47 126.277 0 3667.41
1.8 −14.1831 0 73.1601 347.413 0 522.395 777.799 0 −4927.40
1.9 −11.4478 0 3.05173 −248.349 0 1066.24 1430.38 0 2843.04
1.10 −10.5948 0 −15.7504 −27.5110 0 196.184 1523.01 0 291.473
1.11 −10.4428 0 −18.9469 −410.223 0 1249.19 1534.54 0 4283.89
1.12 −10.1400 0 −25.1797 176.104 0 −221.673 1553.25 0 −1785.70
1.13 −7.17931 0 −76.4574 −94.6886 0 −1419.24 1467.86 0 679.799
1.14 −6.24354 0 −89.0182 −531.463 0 445.980 1354.96 0 3318.21
1.15 −5.77426 0 −94.6579 218.484 0 606.083 1285.68 0 −1261.58
1.16 −0.306188 0 −127.906 108.774 0 1364.63 78.3553 0 −33.3052
1.17 1.18257 0 −126.602 −236.792 0 −950.759 −301.084 0 −280.023
1.18 1.54690 0 −125.607 −282.598 0 −727.135 −392.304 0 −437.150
1.19 1.65238 0 −125.270 348.169 0 493.067 −418.497 0 575.306
1.20 3.44034 0 −116.164 382.136 0 −641.295 −840.007 0 1314.68
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.g ✓ 33
3.b odd 2 1 531.8.a.h yes 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.8.a.g ✓ 33 1.a even 1 1 trivial
531.8.a.h yes 33 3.b odd 2 1

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