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Properties

Label 531.6.a.h
Level $531$
Weight $6$
Character orbit 531.a
Self dual yes
Analytic conductor $85.164$
Analytic rank $0$
Dimension $25$
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] = [531,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(85.1638083207\)
Analytic rank: \(0\)
Dimension: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 12 q^{2} + 400 q^{4} + 200 q^{5} - 38 q^{7} + 576 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 12 q^{2} + 400 q^{4} + 200 q^{5} - 38 q^{7} + 576 q^{8} + 370 q^{10} + 1210 q^{11} - 136 q^{13} + 1176 q^{14} + 5884 q^{16} + 2312 q^{17} + 1722 q^{19} + 944 q^{20} + 1700 q^{22} + 4476 q^{23} + 17547 q^{25} + 10320 q^{26} - 552 q^{28} + 30036 q^{29} - 822 q^{31} + 44684 q^{32} - 4240 q^{34} + 14700 q^{35} - 1220 q^{37} + 58378 q^{38} - 11890 q^{40} + 43656 q^{41} + 18476 q^{43} + 58080 q^{44} + 25322 q^{46} + 43962 q^{47} + 62015 q^{49} + 167260 q^{50} + 11740 q^{52} + 79780 q^{53} - 42830 q^{55} + 161888 q^{56} + 14274 q^{58} + 87025 q^{59} - 24390 q^{61} + 73224 q^{62} - 13574 q^{64} + 43378 q^{65} - 14168 q^{67} + 202548 q^{68} + 518 q^{70} + 27924 q^{71} + 44646 q^{73} + 186092 q^{74} + 56816 q^{76} + 207738 q^{77} + 21128 q^{79} + 150020 q^{80} + 84138 q^{82} + 84150 q^{83} + 7454 q^{85} + 376480 q^{86} - 54018 q^{88} + 227092 q^{89} + 21768 q^{91} + 470414 q^{92} - 83352 q^{94} + 230580 q^{95} + 177344 q^{97} + 299700 q^{98}+O(q^{100}) \) 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 −10.2122 0 72.2881 −8.99171 0 −195.581 −411.429 0 91.8247
1.2 −9.78286 0 63.7044 33.7701 0 54.0775 −310.160 0 −330.368
1.3 −8.44229 0 39.2723 60.3387 0 205.995 −61.3947 0 −509.397
1.4 −8.42706 0 39.0154 −43.8627 0 −144.906 −59.1192 0 369.634
1.5 −8.05913 0 32.9496 67.4986 0 −21.7889 −7.65325 0 −543.980
1.6 −8.03586 0 32.5750 −53.4308 0 1.28687 −4.62034 0 429.362
1.7 −5.67155 0 0.166494 −50.5345 0 −40.8775 180.545 0 286.609
1.8 −3.88748 0 −16.8875 −74.8246 0 160.771 190.049 0 290.879
1.9 −3.44711 0 −20.1174 85.6439 0 80.0154 179.655 0 −295.224
1.10 −2.75452 0 −24.4126 −13.7477 0 −2.95197 155.390 0 37.8685
1.11 −0.841476 0 −31.2919 −0.517435 0 −123.064 53.2587 0 0.435409
1.12 −0.0218368 0 −31.9995 99.4365 0 17.4784 1.39754 0 −2.17137
1.13 0.175920 0 −31.9691 61.3303 0 −222.143 −11.2535 0 10.7892
1.14 1.91492 0 −28.3331 −73.3501 0 98.0282 −115.533 0 −140.460
1.15 2.04262 0 −27.8277 13.8772 0 247.966 −122.205 0 28.3458
1.16 4.37837 0 −12.8299 19.4294 0 60.4821 −196.282 0 85.0690
1.17 4.75667 0 −9.37413 105.105 0 −231.297 −196.803 0 499.948
1.18 5.83028 0 1.99222 −62.8474 0 −155.625 −174.954 0 −366.418
1.19 6.25162 0 7.08273 −42.9125 0 −166.751 −155.773 0 −268.273
1.20 7.26893 0 20.8374 −35.8720 0 166.523 −81.1403 0 −260.751
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
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 531.6.a.h yes 25
3.b odd 2 1 531.6.a.g ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.6.a.g ✓ 25 3.b odd 2 1
531.6.a.h yes 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 12 T_{2}^{24} - 528 T_{2}^{23} + 6464 T_{2}^{22} + 119169 T_{2}^{21} + \cdots - 586688321126400 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display