Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [625,8,Mod(1,625)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(625, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("625.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 625 = 5^{4} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 625.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: | \(195.240640928\) |
Analytic rank: | \(1\) |
Dimension: | \(22\) |
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.
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 | −21.3970 | −45.8462 | 329.831 | 0 | 980.970 | 104.481 | −4318.57 | −85.1275 | 0 | ||||||||||||||||||
1.2 | −19.9369 | 7.59174 | 269.481 | 0 | −151.356 | 250.277 | −2820.70 | −2129.37 | 0 | ||||||||||||||||||
1.3 | −17.3699 | −85.0611 | 173.712 | 0 | 1477.50 | −1277.84 | −794.018 | 5048.39 | 0 | ||||||||||||||||||
1.4 | −16.8316 | 60.3488 | 155.304 | 0 | −1015.77 | 1500.12 | −459.578 | 1454.98 | 0 | ||||||||||||||||||
1.5 | −16.3997 | 21.4355 | 140.949 | 0 | −351.535 | −1450.75 | −212.357 | −1727.52 | 0 | ||||||||||||||||||
1.6 | −14.6488 | 70.4760 | 86.5879 | 0 | −1032.39 | −98.3469 | 606.638 | 2779.87 | 0 | ||||||||||||||||||
1.7 | −9.65795 | −78.6437 | −34.7239 | 0 | 759.537 | 34.6342 | 1571.58 | 3997.83 | 0 | ||||||||||||||||||
1.8 | −9.55882 | −19.6493 | −36.6289 | 0 | 187.824 | 275.298 | 1573.66 | −1800.91 | 0 | ||||||||||||||||||
1.9 | −8.75606 | −29.2735 | −51.3314 | 0 | 256.321 | 528.656 | 1570.24 | −1330.06 | 0 | ||||||||||||||||||
1.10 | −3.59862 | 41.3836 | −115.050 | 0 | −148.924 | −1266.89 | 874.645 | −474.397 | 0 | ||||||||||||||||||
1.11 | −3.07018 | 68.4413 | −118.574 | 0 | −210.127 | −698.276 | 757.028 | 2497.20 | 0 | ||||||||||||||||||
1.12 | 2.72966 | −0.659241 | −120.549 | 0 | −1.79951 | 767.270 | −678.454 | −2186.57 | 0 | ||||||||||||||||||
1.13 | 3.20926 | 62.8158 | −117.701 | 0 | 201.593 | 1045.66 | −788.518 | 1758.83 | 0 | ||||||||||||||||||
1.14 | 5.27101 | −89.1473 | −100.216 | 0 | −469.896 | 1076.06 | −1202.93 | 5760.24 | 0 | ||||||||||||||||||
1.15 | 5.97357 | −39.4745 | −92.3164 | 0 | −235.804 | −1669.35 | −1316.08 | −628.764 | 0 | ||||||||||||||||||
1.16 | 7.52269 | −47.0815 | −71.4092 | 0 | −354.180 | −1676.07 | −1500.09 | 29.6718 | 0 | ||||||||||||||||||
1.17 | 9.79880 | 0.219481 | −31.9834 | 0 | 2.15065 | 700.879 | −1567.65 | −2186.95 | 0 | ||||||||||||||||||
1.18 | 13.4550 | −79.3155 | 53.0377 | 0 | −1067.19 | 1416.14 | −1008.62 | 4103.95 | 0 | ||||||||||||||||||
1.19 | 14.0235 | 58.7448 | 68.6591 | 0 | 823.809 | 136.950 | −832.168 | 1263.95 | 0 | ||||||||||||||||||
1.20 | 17.2747 | 56.9787 | 170.417 | 0 | 984.292 | −464.211 | 732.735 | 1059.57 | 0 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \( -1 \) |
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 | 625.8.a.a | ✓ | 22 |
5.b | even | 2 | 1 | 625.8.a.b | yes | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
625.8.a.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
625.8.a.b | yes | 22 | 5.b | even | 2 | 1 |