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: | \(1\) |
Dimension: | \(39\) |
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 | −5.13931 | −6.39990 | 18.4125 | −3.19755 | 32.8911 | 11.5196 | −53.5133 | 13.9588 | 16.4332 | ||||||||||||||||||
1.2 | −5.01772 | 4.57830 | 17.1775 | 10.5311 | −22.9726 | 16.1445 | −46.0502 | −6.03920 | −52.8420 | ||||||||||||||||||
1.3 | −4.78473 | 6.91979 | 14.8936 | 9.47124 | −33.1093 | 13.6025 | −32.9841 | 20.8835 | −45.3173 | ||||||||||||||||||
1.4 | −4.54302 | 0.739868 | 12.6390 | 5.27775 | −3.36123 | −32.8528 | −21.0751 | −26.4526 | −23.9769 | ||||||||||||||||||
1.5 | −4.48614 | −3.10130 | 12.1255 | −9.82658 | 13.9129 | −8.62268 | −18.5074 | −17.3819 | 44.0834 | ||||||||||||||||||
1.6 | −4.05479 | −7.92582 | 8.44132 | −7.77507 | 32.1375 | −27.1534 | −1.78944 | 35.8186 | 31.5263 | ||||||||||||||||||
1.7 | −4.00081 | −8.62277 | 8.00649 | 18.4684 | 34.4981 | −15.5106 | −0.0259569 | 47.3521 | −73.8885 | ||||||||||||||||||
1.8 | −3.98336 | 6.46359 | 7.86718 | −1.70678 | −25.7468 | 18.7420 | 0.529060 | 14.7780 | 6.79871 | ||||||||||||||||||
1.9 | −3.79202 | 1.78265 | 6.37944 | −15.2422 | −6.75985 | 7.73177 | 6.14519 | −23.8222 | 57.7988 | ||||||||||||||||||
1.10 | −2.66272 | 7.39869 | −0.909937 | 10.9244 | −19.7006 | −23.1620 | 23.7246 | 27.7406 | −29.0887 | ||||||||||||||||||
1.11 | −2.51874 | 1.77429 | −1.65593 | −18.5552 | −4.46898 | 4.77905 | 24.3208 | −23.8519 | 46.7358 | ||||||||||||||||||
1.12 | −2.34716 | −6.88614 | −2.49082 | 4.99906 | 16.1629 | 27.5938 | 24.6237 | 20.4189 | −11.7336 | ||||||||||||||||||
1.13 | −2.30249 | 4.54839 | −2.69855 | −4.16672 | −10.4726 | −17.3608 | 24.6333 | −6.31212 | 9.59382 | ||||||||||||||||||
1.14 | −2.06825 | −3.79252 | −3.72233 | 11.8121 | 7.84389 | −6.57089 | 24.2447 | −12.6168 | −24.4303 | ||||||||||||||||||
1.15 | −1.49011 | −8.91494 | −5.77957 | −12.8908 | 13.2843 | 33.8928 | 20.5331 | 52.4761 | 19.2087 | ||||||||||||||||||
1.16 | −1.42955 | −6.49934 | −5.95638 | 0.801184 | 9.29115 | 16.7131 | 19.9514 | 15.2414 | −1.14534 | ||||||||||||||||||
1.17 | −1.40955 | 8.50422 | −6.01318 | −10.6119 | −11.9871 | 26.4105 | 19.7522 | 45.3218 | 14.9579 | ||||||||||||||||||
1.18 | −0.641292 | −1.91302 | −7.58874 | −0.642358 | 1.22681 | −23.2338 | 9.99694 | −23.3404 | 0.411939 | ||||||||||||||||||
1.19 | −0.150454 | 3.58277 | −7.97736 | 14.8852 | −0.539044 | 5.18902 | 2.40386 | −14.1637 | −2.23954 | ||||||||||||||||||
1.20 | 0.169295 | 6.82637 | −7.97134 | 1.80831 | 1.15567 | −16.1277 | −2.70387 | 19.5993 | 0.306138 | ||||||||||||||||||
See all 39 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.n | ✓ | 39 |
13.b | even | 2 | 1 | 1859.4.a.o | yes | 39 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1859.4.a.n | ✓ | 39 | 1.a | even | 1 | 1 | trivial |
1859.4.a.o | yes | 39 | 13.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{39} - 213 T_{2}^{37} - 7 T_{2}^{36} + 20695 T_{2}^{35} + 1070 T_{2}^{34} - 1215940 T_{2}^{33} + \cdots + 91431516471296 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).