Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,4,Mod(1,1859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1859.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: | \(109.684550701\) |
Analytic rank: | \(0\) |
Dimension: | \(51\) |
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 | −5.57916 | 9.06032 | 23.1271 | −7.74998 | −50.5490 | 23.9301 | −84.3964 | 55.0894 | 43.2384 | ||||||||||||||||||
1.2 | −5.41392 | −5.45021 | 21.3105 | −17.8801 | 29.5070 | −29.5931 | −72.0621 | 2.70474 | 96.8014 | ||||||||||||||||||
1.3 | −5.33739 | −5.24916 | 20.4878 | 17.7171 | 28.0168 | −32.5164 | −66.6520 | 0.553673 | −94.5632 | ||||||||||||||||||
1.4 | −5.05441 | 0.801365 | 17.5470 | −17.5680 | −4.05042 | 1.71020 | −48.2547 | −26.3578 | 88.7960 | ||||||||||||||||||
1.5 | −4.95360 | 8.81730 | 16.5381 | 19.6953 | −43.6773 | 6.57864 | −42.2943 | 50.7448 | −97.5628 | ||||||||||||||||||
1.6 | −4.84162 | 9.88636 | 15.4413 | −0.786780 | −47.8660 | −26.4479 | −36.0278 | 70.7402 | 3.80929 | ||||||||||||||||||
1.7 | −4.54788 | −9.03091 | 12.6832 | 5.41468 | 41.0715 | 26.3763 | −21.2987 | 54.5573 | −24.6253 | ||||||||||||||||||
1.8 | −4.52468 | 7.83415 | 12.4727 | −15.8043 | −35.4471 | −30.6432 | −20.2377 | 34.3740 | 71.5092 | ||||||||||||||||||
1.9 | −4.24016 | −4.67403 | 9.97893 | −7.09125 | 19.8186 | −21.9298 | −8.39097 | −5.15349 | 30.0680 | ||||||||||||||||||
1.10 | −4.08285 | −3.72122 | 8.66965 | 2.40264 | 15.1932 | 5.44341 | −2.73408 | −13.1525 | −9.80961 | ||||||||||||||||||
1.11 | −3.69645 | 1.78714 | 5.66374 | 8.08739 | −6.60608 | 3.93004 | 8.63587 | −23.8061 | −29.8946 | ||||||||||||||||||
1.12 | −3.57359 | −4.93113 | 4.77054 | 6.17910 | 17.6218 | 19.8272 | 11.5408 | −2.68401 | −22.0816 | ||||||||||||||||||
1.13 | −3.39599 | 6.27073 | 3.53273 | −11.8750 | −21.2953 | −17.2790 | 15.1708 | 12.3221 | 40.3274 | ||||||||||||||||||
1.14 | −3.26686 | 5.88118 | 2.67240 | 20.5347 | −19.2130 | 32.9664 | 17.4046 | 7.58824 | −67.0842 | ||||||||||||||||||
1.15 | −2.96025 | 1.91558 | 0.763092 | 5.98319 | −5.67060 | 31.3583 | 21.4231 | −23.3305 | −17.7117 | ||||||||||||||||||
1.16 | −2.92218 | −6.70384 | 0.539146 | 12.5727 | 19.5898 | −15.7135 | 21.8020 | 17.9414 | −36.7396 | ||||||||||||||||||
1.17 | −2.41499 | 1.53640 | −2.16780 | 15.9482 | −3.71040 | −30.4066 | 24.5552 | −24.6395 | −38.5149 | ||||||||||||||||||
1.18 | −1.90937 | −0.199217 | −4.35431 | −18.1265 | 0.380379 | 33.8392 | 23.5889 | −26.9603 | 34.6102 | ||||||||||||||||||
1.19 | −1.77976 | −6.74491 | −4.83246 | −21.4141 | 12.0043 | −5.98846 | 22.8387 | 18.4938 | 38.1119 | ||||||||||||||||||
1.20 | −1.66790 | 7.00149 | −5.21812 | −13.7691 | −11.6778 | 14.0493 | 22.0465 | 22.0209 | 22.9654 | ||||||||||||||||||
See all 51 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(-1\) |
\(13\) | \(-1\) |
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 | 1859.4.a.q | yes | 51 |
13.b | even | 2 | 1 | 1859.4.a.p | ✓ | 51 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1859.4.a.p | ✓ | 51 | 13.b | even | 2 | 1 | |
1859.4.a.q | yes | 51 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{51} - 321 T_{2}^{49} + 7 T_{2}^{48} + 48173 T_{2}^{47} - 2338 T_{2}^{46} - 4492292 T_{2}^{45} + \cdots + 10\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).