Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8015,2,Mod(1,8015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, 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("8015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8015 = 5 \cdot 7 \cdot 229 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8015.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: | \(64.0000972201\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
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 | −2.76919 | 2.35151 | 5.66840 | −1.00000 | −6.51177 | 1.00000 | −10.1585 | 2.52960 | 2.76919 | ||||||||||||||||||
1.2 | −2.73044 | 0.281096 | 5.45528 | −1.00000 | −0.767515 | 1.00000 | −9.43443 | −2.92099 | 2.73044 | ||||||||||||||||||
1.3 | −2.66015 | −2.46219 | 5.07642 | −1.00000 | 6.54979 | 1.00000 | −8.18374 | 3.06236 | 2.66015 | ||||||||||||||||||
1.4 | −2.55606 | −1.84104 | 4.53346 | −1.00000 | 4.70581 | 1.00000 | −6.47568 | 0.389420 | 2.55606 | ||||||||||||||||||
1.5 | −2.40811 | 3.06409 | 3.79900 | −1.00000 | −7.37867 | 1.00000 | −4.33220 | 6.38865 | 2.40811 | ||||||||||||||||||
1.6 | −2.40730 | 0.0150324 | 3.79510 | −1.00000 | −0.0361876 | 1.00000 | −4.32135 | −2.99977 | 2.40730 | ||||||||||||||||||
1.7 | −2.33888 | −1.00822 | 3.47038 | −1.00000 | 2.35811 | 1.00000 | −3.43905 | −1.98350 | 2.33888 | ||||||||||||||||||
1.8 | −2.33656 | 2.32181 | 3.45950 | −1.00000 | −5.42504 | 1.00000 | −3.41021 | 2.39080 | 2.33656 | ||||||||||||||||||
1.9 | −2.27050 | −2.16931 | 3.15517 | −1.00000 | 4.92541 | 1.00000 | −2.62281 | 1.70590 | 2.27050 | ||||||||||||||||||
1.10 | −2.17774 | 0.311923 | 2.74253 | −1.00000 | −0.679286 | 1.00000 | −1.61704 | −2.90270 | 2.17774 | ||||||||||||||||||
1.11 | −2.12688 | 0.933356 | 2.52363 | −1.00000 | −1.98514 | 1.00000 | −1.11369 | −2.12885 | 2.12688 | ||||||||||||||||||
1.12 | −2.04947 | −1.65528 | 2.20033 | −1.00000 | 3.39245 | 1.00000 | −0.410568 | −0.260041 | 2.04947 | ||||||||||||||||||
1.13 | −1.92788 | −3.19402 | 1.71671 | −1.00000 | 6.15768 | 1.00000 | 0.546142 | 7.20176 | 1.92788 | ||||||||||||||||||
1.14 | −1.76060 | 2.09671 | 1.09972 | −1.00000 | −3.69147 | 1.00000 | 1.58503 | 1.39619 | 1.76060 | ||||||||||||||||||
1.15 | −1.66934 | 1.36319 | 0.786696 | −1.00000 | −2.27563 | 1.00000 | 2.02542 | −1.14171 | 1.66934 | ||||||||||||||||||
1.16 | −1.64550 | 3.30855 | 0.707660 | −1.00000 | −5.44420 | 1.00000 | 2.12654 | 7.94648 | 1.64550 | ||||||||||||||||||
1.17 | −1.48933 | −2.36326 | 0.218100 | −1.00000 | 3.51967 | 1.00000 | 2.65384 | 2.58500 | 1.48933 | ||||||||||||||||||
1.18 | −1.39879 | −1.98679 | −0.0433739 | −1.00000 | 2.77911 | 1.00000 | 2.85826 | 0.947327 | 1.39879 | ||||||||||||||||||
1.19 | −1.37224 | 1.58108 | −0.116968 | −1.00000 | −2.16961 | 1.00000 | 2.90498 | −0.500189 | 1.37224 | ||||||||||||||||||
1.20 | −1.36878 | 0.541505 | −0.126449 | −1.00000 | −0.741199 | 1.00000 | 2.91064 | −2.70677 | 1.36878 | ||||||||||||||||||
See all 68 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(7\) | \(-1\) |
\(229\) | \(1\) |
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 | 8015.2.a.n | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.n | ✓ | 68 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{68} - 9 T_{2}^{67} - 69 T_{2}^{66} + 842 T_{2}^{65} + 1744 T_{2}^{64} - 37136 T_{2}^{63} + \cdots - 15565411 \) |
\( T_{3}^{68} - 145 T_{3}^{66} - 4 T_{3}^{65} + 9978 T_{3}^{64} + 547 T_{3}^{63} - 433678 T_{3}^{62} + \cdots - 504199168 \) |