Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4017,2,Mod(1,4017)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4017, 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("4017.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4017 = 3 \cdot 13 \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4017.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: | \(32.0759064919\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | −2.64656 | −1.00000 | 5.00429 | 2.13528 | 2.64656 | 3.46294 | −7.95105 | 1.00000 | −5.65115 | ||||||||||||||||||
1.2 | −2.42509 | −1.00000 | 3.88107 | −0.736887 | 2.42509 | −2.40625 | −4.56178 | 1.00000 | 1.78702 | ||||||||||||||||||
1.3 | −2.29414 | −1.00000 | 3.26307 | −3.12687 | 2.29414 | −2.80124 | −2.89765 | 1.00000 | 7.17346 | ||||||||||||||||||
1.4 | −2.16909 | −1.00000 | 2.70494 | 0.657691 | 2.16909 | −0.817989 | −1.52908 | 1.00000 | −1.42659 | ||||||||||||||||||
1.5 | −1.94312 | −1.00000 | 1.77572 | −0.275874 | 1.94312 | 3.15290 | 0.435811 | 1.00000 | 0.536057 | ||||||||||||||||||
1.6 | −1.52294 | −1.00000 | 0.319360 | 3.31668 | 1.52294 | −2.60872 | 2.55952 | 1.00000 | −5.05112 | ||||||||||||||||||
1.7 | −1.32626 | −1.00000 | −0.241032 | −0.192408 | 1.32626 | 3.16535 | 2.97219 | 1.00000 | 0.255184 | ||||||||||||||||||
1.8 | −1.17908 | −1.00000 | −0.609765 | 4.22706 | 1.17908 | 3.50133 | 3.07713 | 1.00000 | −4.98405 | ||||||||||||||||||
1.9 | −0.607674 | −1.00000 | −1.63073 | −3.70104 | 0.607674 | 0.175250 | 2.20630 | 1.00000 | 2.24902 | ||||||||||||||||||
1.10 | −0.574597 | −1.00000 | −1.66984 | −2.72483 | 0.574597 | 0.829930 | 2.10868 | 1.00000 | 1.56568 | ||||||||||||||||||
1.11 | −0.389331 | −1.00000 | −1.84842 | 0.642702 | 0.389331 | −1.35418 | 1.49831 | 1.00000 | −0.250224 | ||||||||||||||||||
1.12 | 0.0298243 | −1.00000 | −1.99911 | 3.06004 | −0.0298243 | −1.69214 | −0.119271 | 1.00000 | 0.0912635 | ||||||||||||||||||
1.13 | 0.543206 | −1.00000 | −1.70493 | 2.45290 | −0.543206 | 0.874109 | −2.01254 | 1.00000 | 1.33243 | ||||||||||||||||||
1.14 | 0.671722 | −1.00000 | −1.54879 | −1.67775 | −0.671722 | 5.02977 | −2.38380 | 1.00000 | −1.12698 | ||||||||||||||||||
1.15 | 0.794802 | −1.00000 | −1.36829 | −2.06654 | −0.794802 | −0.419104 | −2.67712 | 1.00000 | −1.64249 | ||||||||||||||||||
1.16 | 1.05828 | −1.00000 | −0.880046 | −0.912505 | −1.05828 | −3.63309 | −3.04789 | 1.00000 | −0.965685 | ||||||||||||||||||
1.17 | 1.57828 | −1.00000 | 0.490964 | −1.83430 | −1.57828 | 4.32058 | −2.38168 | 1.00000 | −2.89504 | ||||||||||||||||||
1.18 | 1.68466 | −1.00000 | 0.838074 | 0.142092 | −1.68466 | −0.0282508 | −1.95745 | 1.00000 | 0.239376 | ||||||||||||||||||
1.19 | 1.73242 | −1.00000 | 1.00128 | 0.756937 | −1.73242 | −4.14115 | −1.73020 | 1.00000 | 1.31133 | ||||||||||||||||||
1.20 | 1.86289 | −1.00000 | 1.47035 | 4.02931 | −1.86289 | 3.22519 | −0.986675 | 1.00000 | 7.50616 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(13\) | \(1\) |
\(103\) | \(-1\) |
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 | 4017.2.a.g | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4017.2.a.g | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):
\( T_{2}^{24} - 3 T_{2}^{23} - 32 T_{2}^{22} + 99 T_{2}^{21} + 432 T_{2}^{20} - 1397 T_{2}^{19} - 3194 T_{2}^{18} + 11029 T_{2}^{17} + 14007 T_{2}^{16} - 53536 T_{2}^{15} - 36577 T_{2}^{14} + 165500 T_{2}^{13} + 52434 T_{2}^{12} + \cdots - 64 \) |
\( T_{23}^{24} - 41 T_{23}^{23} + 558 T_{23}^{22} - 493 T_{23}^{21} - 62179 T_{23}^{20} + 568371 T_{23}^{19} - 75174 T_{23}^{18} - 25272080 T_{23}^{17} + 115269488 T_{23}^{16} + 257472706 T_{23}^{15} + \cdots + 138468131840 \) |