Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4015,2,Mod(1,4015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4015 = 5 \cdot 11 \cdot 73 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4015.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: | \(32.0599364115\) |
Analytic rank: | \(1\) |
Dimension: | \(23\) |
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 | −2.63381 | −0.811587 | 4.93698 | 1.00000 | 2.13757 | 5.26915 | −7.73545 | −2.34133 | −2.63381 | ||||||||||||||||||
1.2 | −2.41301 | 1.23805 | 3.82260 | 1.00000 | −2.98742 | −1.94006 | −4.39795 | −1.46724 | −2.41301 | ||||||||||||||||||
1.3 | −2.26107 | −2.17834 | 3.11244 | 1.00000 | 4.92537 | 1.01936 | −2.51531 | 1.74515 | −2.26107 | ||||||||||||||||||
1.4 | −2.15888 | 1.54599 | 2.66074 | 1.00000 | −3.33759 | 2.48210 | −1.42646 | −0.609922 | −2.15888 | ||||||||||||||||||
1.5 | −2.00725 | −2.63655 | 2.02904 | 1.00000 | 5.29220 | −1.13661 | −0.0582891 | 3.95138 | −2.00725 | ||||||||||||||||||
1.6 | −1.63550 | 2.32797 | 0.674867 | 1.00000 | −3.80740 | −0.601850 | 2.16726 | 2.41945 | −1.63550 | ||||||||||||||||||
1.7 | −1.22952 | −0.451066 | −0.488282 | 1.00000 | 0.554594 | −2.48868 | 3.05939 | −2.79654 | −1.22952 | ||||||||||||||||||
1.8 | −1.17870 | −1.42513 | −0.610675 | 1.00000 | 1.67979 | 3.29739 | 3.07719 | −0.969018 | −1.17870 | ||||||||||||||||||
1.9 | −0.975417 | 1.70884 | −1.04856 | 1.00000 | −1.66683 | 0.984932 | 2.97362 | −0.0798559 | −0.975417 | ||||||||||||||||||
1.10 | −0.760826 | 0.764917 | −1.42114 | 1.00000 | −0.581969 | −2.66383 | 2.60290 | −2.41490 | −0.760826 | ||||||||||||||||||
1.11 | −0.692372 | −2.63531 | −1.52062 | 1.00000 | 1.82461 | 0.337135 | 2.43758 | 3.94484 | −0.692372 | ||||||||||||||||||
1.12 | −0.197331 | 2.44280 | −1.96106 | 1.00000 | −0.482041 | −1.78119 | 0.781640 | 2.96728 | −0.197331 | ||||||||||||||||||
1.13 | 0.418197 | 2.15176 | −1.82511 | 1.00000 | 0.899862 | 0.524806 | −1.59965 | 1.63009 | 0.418197 | ||||||||||||||||||
1.14 | 0.422103 | −2.19245 | −1.82183 | 1.00000 | −0.925438 | −3.51396 | −1.61321 | 1.80682 | 0.422103 | ||||||||||||||||||
1.15 | 0.615983 | −0.785692 | −1.62056 | 1.00000 | −0.483973 | 2.63763 | −2.23021 | −2.38269 | 0.615983 | ||||||||||||||||||
1.16 | 0.661798 | −2.32231 | −1.56202 | 1.00000 | −1.53690 | −0.917277 | −2.35734 | 2.39311 | 0.661798 | ||||||||||||||||||
1.17 | 1.20516 | 1.14152 | −0.547593 | 1.00000 | 1.37572 | 3.42329 | −3.07025 | −1.69692 | 1.20516 | ||||||||||||||||||
1.18 | 1.30572 | 1.83317 | −0.295101 | 1.00000 | 2.39360 | −4.41824 | −2.99675 | 0.360505 | 1.30572 | ||||||||||||||||||
1.19 | 1.83759 | −0.530531 | 1.37674 | 1.00000 | −0.974898 | 1.03293 | −1.14530 | −2.71854 | 1.83759 | ||||||||||||||||||
1.20 | 1.92108 | −3.32354 | 1.69056 | 1.00000 | −6.38479 | 2.35268 | −0.594467 | 8.04590 | 1.92108 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(11\) | \(1\) |
\(73\) | \(-1\) |
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 | 4015.2.a.c | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4015.2.a.c | ✓ | 23 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + 3 T_{2}^{22} - 26 T_{2}^{21} - 81 T_{2}^{20} + 281 T_{2}^{19} + 922 T_{2}^{18} - 1638 T_{2}^{17} + \cdots - 67 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).