Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,2,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, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
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: | \(14.8441897358\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
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 | −2.68791 | −2.90985 | 5.22486 | −3.09378 | 7.82142 | −1.70397 | −8.66813 | 5.46725 | 8.31580 | ||||||||||||||||||
1.2 | −2.56428 | 1.12676 | 4.57554 | −1.24792 | −2.88933 | −2.50384 | −6.60441 | −1.73041 | 3.20001 | ||||||||||||||||||
1.3 | −2.26490 | 3.17598 | 3.12978 | −2.32745 | −7.19329 | 4.39568 | −2.55884 | 7.08687 | 5.27144 | ||||||||||||||||||
1.4 | −2.23194 | 1.36188 | 2.98154 | −4.28736 | −3.03963 | −0.384100 | −2.19073 | −1.14528 | 9.56911 | ||||||||||||||||||
1.5 | −2.14442 | −1.72478 | 2.59856 | 3.54668 | 3.69866 | 2.57025 | −1.28356 | −0.0251423 | −7.60558 | ||||||||||||||||||
1.6 | −2.13563 | −2.20961 | 2.56090 | 1.38469 | 4.71890 | −5.12151 | −1.19787 | 1.88237 | −2.95718 | ||||||||||||||||||
1.7 | −1.43463 | 0.655312 | 0.0581570 | −1.45609 | −0.940129 | −3.21683 | 2.78582 | −2.57057 | 2.08894 | ||||||||||||||||||
1.8 | −1.17546 | −1.22926 | −0.618298 | 1.06938 | 1.44494 | 3.67782 | 3.07770 | −1.48892 | −1.25701 | ||||||||||||||||||
1.9 | −0.776244 | 3.01008 | −1.39745 | −0.0164567 | −2.33656 | −0.793722 | 2.63725 | 6.06060 | 0.0127744 | ||||||||||||||||||
1.10 | −0.149116 | −3.21882 | −1.97776 | −1.07607 | 0.479977 | 3.58386 | 0.593149 | 7.36077 | 0.160459 | ||||||||||||||||||
1.11 | 0.340522 | −1.21327 | −1.88404 | 3.18769 | −0.413146 | −4.05402 | −1.32260 | −1.52797 | 1.08548 | ||||||||||||||||||
1.12 | 0.392741 | 1.69948 | −1.84575 | −1.18432 | 0.667456 | −5.07600 | −1.51039 | −0.111771 | −0.465133 | ||||||||||||||||||
1.13 | 0.603243 | −0.154529 | −1.63610 | −4.17643 | −0.0932187 | −2.73997 | −2.19345 | −2.97612 | −2.51940 | ||||||||||||||||||
1.14 | 0.930026 | 2.53271 | −1.13505 | 3.38691 | 2.35548 | 3.38196 | −2.91568 | 3.41460 | 3.14991 | ||||||||||||||||||
1.15 | 1.50636 | −0.544988 | 0.269129 | −3.40659 | −0.820949 | 0.714897 | −2.60732 | −2.70299 | −5.13156 | ||||||||||||||||||
1.16 | 1.53678 | −3.28188 | 0.361702 | 1.21246 | −5.04353 | 1.98662 | −2.51771 | 7.77071 | 1.86328 | ||||||||||||||||||
1.17 | 1.77144 | 2.92005 | 1.13800 | 2.22672 | 5.17269 | −0.178675 | −1.52699 | 5.52669 | 3.94449 | ||||||||||||||||||
1.18 | 2.32475 | 2.00106 | 3.40448 | −3.04764 | 4.65197 | 5.08642 | 3.26506 | 1.00425 | −7.08500 | ||||||||||||||||||
1.19 | 2.68347 | 3.12668 | 5.20101 | −2.33476 | 8.39034 | −0.266736 | 8.58980 | 6.77612 | −6.26526 | ||||||||||||||||||
1.20 | 2.70607 | 0.960218 | 5.32282 | 3.57279 | 2.59842 | −3.95601 | 8.99178 | −2.07798 | 9.66821 | ||||||||||||||||||
See all 21 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 | 1859.2.a.s | ✓ | 21 |
13.b | even | 2 | 1 | 1859.2.a.t | yes | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1859.2.a.s | ✓ | 21 | 1.a | even | 1 | 1 | trivial |
1859.2.a.t | yes | 21 | 13.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):
\( T_{2}^{21} - 37 T_{2}^{19} - T_{2}^{18} + 582 T_{2}^{17} + 20 T_{2}^{16} - 5075 T_{2}^{15} - 99 T_{2}^{14} + \cdots - 448 \) |
\( T_{7}^{21} + T_{7}^{20} - 109 T_{7}^{19} - 94 T_{7}^{18} + 4999 T_{7}^{17} + 3763 T_{7}^{16} + \cdots - 2981888 \) |