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: | \(1\) |
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.
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 | −11.0807 | 0 | 90.7814 | −11.4823 | 0 | 125.641 | −651.337 | 0 | 127.232 | ||||||||||||||||||
1.2 | −10.4480 | 0 | 77.1612 | 105.061 | 0 | −81.7882 | −471.845 | 0 | −1097.68 | ||||||||||||||||||
1.3 | −9.86624 | 0 | 65.3426 | −108.240 | 0 | 90.6662 | −328.966 | 0 | 1067.92 | ||||||||||||||||||
1.4 | −9.66139 | 0 | 61.3424 | −39.6926 | 0 | −119.507 | −283.488 | 0 | 383.485 | ||||||||||||||||||
1.5 | −7.90769 | 0 | 30.5316 | −60.1089 | 0 | 159.350 | 11.6117 | 0 | 475.323 | ||||||||||||||||||
1.6 | −7.26893 | 0 | 20.8374 | 35.8720 | 0 | 166.523 | 81.1403 | 0 | −260.751 | ||||||||||||||||||
1.7 | −6.25162 | 0 | 7.08273 | 42.9125 | 0 | −166.751 | 155.773 | 0 | −268.273 | ||||||||||||||||||
1.8 | −5.83028 | 0 | 1.99222 | 62.8474 | 0 | −155.625 | 174.954 | 0 | −366.418 | ||||||||||||||||||
1.9 | −4.75667 | 0 | −9.37413 | −105.105 | 0 | −231.297 | 196.803 | 0 | 499.948 | ||||||||||||||||||
1.10 | −4.37837 | 0 | −12.8299 | −19.4294 | 0 | 60.4821 | 196.282 | 0 | 85.0690 | ||||||||||||||||||
1.11 | −2.04262 | 0 | −27.8277 | −13.8772 | 0 | 247.966 | 122.205 | 0 | 28.3458 | ||||||||||||||||||
1.12 | −1.91492 | 0 | −28.3331 | 73.3501 | 0 | 98.0282 | 115.533 | 0 | −140.460 | ||||||||||||||||||
1.13 | −0.175920 | 0 | −31.9691 | −61.3303 | 0 | −222.143 | 11.2535 | 0 | 10.7892 | ||||||||||||||||||
1.14 | 0.0218368 | 0 | −31.9995 | −99.4365 | 0 | 17.4784 | −1.39754 | 0 | −2.17137 | ||||||||||||||||||
1.15 | 0.841476 | 0 | −31.2919 | 0.517435 | 0 | −123.064 | −53.2587 | 0 | 0.435409 | ||||||||||||||||||
1.16 | 2.75452 | 0 | −24.4126 | 13.7477 | 0 | −2.95197 | −155.390 | 0 | 37.8685 | ||||||||||||||||||
1.17 | 3.44711 | 0 | −20.1174 | −85.6439 | 0 | 80.0154 | −179.655 | 0 | −295.224 | ||||||||||||||||||
1.18 | 3.88748 | 0 | −16.8875 | 74.8246 | 0 | 160.771 | −190.049 | 0 | 290.879 | ||||||||||||||||||
1.19 | 5.67155 | 0 | 0.166494 | 50.5345 | 0 | −40.8775 | −180.545 | 0 | 286.609 | ||||||||||||||||||
1.20 | 8.03586 | 0 | 32.5750 | 53.4308 | 0 | 1.28687 | 4.62034 | 0 | 429.362 | ||||||||||||||||||
See all 25 embeddings |
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.g | ✓ | 25 |
3.b | odd | 2 | 1 | 531.6.a.h | yes | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
531.6.a.g | ✓ | 25 | 1.a | even | 1 | 1 | trivial |
531.6.a.h | yes | 25 | 3.b | odd | 2 | 1 |
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\).