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Properties

Label 538.4.a.d
Level $538$
Weight $4$
Character orbit 538.a
Self dual yes
Analytic conductor $31.743$
Analytic rank $0$
Dimension $21$
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] = [538,4,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(31.7430275831\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 42 q^{2} + 6 q^{3} + 84 q^{4} + 54 q^{5} + 12 q^{6} + 52 q^{7} + 168 q^{8} + 309 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 42 q^{2} + 6 q^{3} + 84 q^{4} + 54 q^{5} + 12 q^{6} + 52 q^{7} + 168 q^{8} + 309 q^{9} + 108 q^{10} + 99 q^{11} + 24 q^{12} + 81 q^{13} + 104 q^{14} + 277 q^{15} + 336 q^{16} + 228 q^{17} + 618 q^{18} + 51 q^{19} + 216 q^{20} + 235 q^{21} + 198 q^{22} + 691 q^{23} + 48 q^{24} + 1035 q^{25} + 162 q^{26} + 291 q^{27} + 208 q^{28} + 540 q^{29} + 554 q^{30} + 312 q^{31} + 672 q^{32} + 513 q^{33} + 456 q^{34} + 410 q^{35} + 1236 q^{36} + 1120 q^{37} + 102 q^{38} + 904 q^{39} + 432 q^{40} + 731 q^{41} + 470 q^{42} + 788 q^{43} + 396 q^{44} + 1771 q^{45} + 1382 q^{46} + 1570 q^{47} + 96 q^{48} + 2337 q^{49} + 2070 q^{50} + 973 q^{51} + 324 q^{52} + 1801 q^{53} + 582 q^{54} + 1191 q^{55} + 416 q^{56} + 2122 q^{57} + 1080 q^{58} + 1352 q^{59} + 1108 q^{60} + 1868 q^{61} + 624 q^{62} + 2302 q^{63} + 1344 q^{64} + 2552 q^{65} + 1026 q^{66} + 1968 q^{67} + 912 q^{68} + 823 q^{69} + 820 q^{70} + 868 q^{71} + 2472 q^{72} + 1587 q^{73} + 2240 q^{74} - 1978 q^{75} + 204 q^{76} - 2020 q^{77} + 1808 q^{78} + 1168 q^{79} + 864 q^{80} + 3069 q^{81} + 1462 q^{82} - 1506 q^{83} + 940 q^{84} + 871 q^{85} + 1576 q^{86} - 6155 q^{87} + 792 q^{88} + 1471 q^{89} + 3542 q^{90} - 2157 q^{91} + 2764 q^{92} - 677 q^{93} + 3140 q^{94} + 544 q^{95} + 192 q^{96} - 801 q^{97} + 4674 q^{98} - 4470 q^{99}+O(q^{100}) \) 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 2.00000 −10.0400 4.00000 −10.3142 −20.0800 30.9953 8.00000 73.8018 −20.6284
1.2 2.00000 −8.94694 4.00000 17.4912 −17.8939 −29.4022 8.00000 53.0478 34.9825
1.3 2.00000 −8.51755 4.00000 16.7909 −17.0351 25.1792 8.00000 45.5487 33.5818
1.4 2.00000 −6.54123 4.00000 −13.1398 −13.0825 −12.7222 8.00000 15.7877 −26.2795
1.5 2.00000 −6.41803 4.00000 6.01917 −12.8361 26.2838 8.00000 14.1911 12.0383
1.6 2.00000 −5.47352 4.00000 −17.6842 −10.9470 −34.0172 8.00000 2.95944 −35.3683
1.7 2.00000 −4.26201 4.00000 6.83518 −8.52401 −31.0166 8.00000 −8.83529 13.6704
1.8 2.00000 −3.71008 4.00000 −7.25350 −7.42015 0.635451 8.00000 −13.2353 −14.5070
1.9 2.00000 −1.64279 4.00000 −16.1874 −3.28558 5.23738 8.00000 −24.3012 −32.3748
1.10 2.00000 −1.07079 4.00000 3.57975 −2.14158 21.8456 8.00000 −25.8534 7.15950
1.11 2.00000 −0.0924306 4.00000 22.1534 −0.184861 −8.74881 8.00000 −26.9915 44.3068
1.12 2.00000 0.291894 4.00000 10.4463 0.583789 19.7604 8.00000 −26.9148 20.8926
1.13 2.00000 2.97379 4.00000 −18.4835 5.94759 −11.2005 8.00000 −18.1566 −36.9670
1.14 2.00000 5.07974 4.00000 10.3107 10.1595 24.1687 8.00000 −1.19626 20.6214
1.15 2.00000 5.22913 4.00000 19.5728 10.4583 −1.73635 8.00000 0.343768 39.1455
1.16 2.00000 6.73834 4.00000 5.62452 13.4767 −24.9667 8.00000 18.4053 11.2490
1.17 2.00000 6.96110 4.00000 11.9308 13.9222 13.1346 8.00000 21.4569 23.8617
1.18 2.00000 7.70804 4.00000 −10.8854 15.4161 35.5612 8.00000 32.4139 −21.7708
1.19 2.00000 8.48473 4.00000 −6.56280 16.9695 3.72047 8.00000 44.9906 −13.1256
1.20 2.00000 9.24827 4.00000 15.1406 18.4965 12.8672 8.00000 58.5304 30.2812
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(269\) \( -1 \)

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 538.4.a.d ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.4.a.d ✓ 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{21} - 6 T_{3}^{20} - 420 T_{3}^{19} + 2387 T_{3}^{18} + 74643 T_{3}^{17} - 394163 T_{3}^{16} + \cdots + 2956968188160 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(538))\). Copy content Toggle raw display