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: | \(0\) |
Dimension: | \(37\) |
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 | −2.70375 | −1.27224 | 5.31028 | 1.00000 | 3.43983 | −0.968765 | −8.95019 | −1.38140 | −2.70375 | ||||||||||||||||||
1.2 | −2.58511 | −0.179292 | 4.68277 | 1.00000 | 0.463488 | 2.09214 | −6.93525 | −2.96785 | −2.58511 | ||||||||||||||||||
1.3 | −2.53170 | 2.11924 | 4.40953 | 1.00000 | −5.36530 | 2.19751 | −6.10021 | 1.49120 | −2.53170 | ||||||||||||||||||
1.4 | −2.41139 | −3.18109 | 3.81483 | 1.00000 | 7.67086 | −3.73754 | −4.37626 | 7.11932 | −2.41139 | ||||||||||||||||||
1.5 | −2.14348 | 2.01146 | 2.59449 | 1.00000 | −4.31152 | −3.47503 | −1.27427 | 1.04598 | −2.14348 | ||||||||||||||||||
1.6 | −1.99859 | 2.62046 | 1.99434 | 1.00000 | −5.23721 | 2.62346 | 0.0113035 | 3.86680 | −1.99859 | ||||||||||||||||||
1.7 | −1.93175 | −1.55022 | 1.73167 | 1.00000 | 2.99464 | 5.03826 | 0.518349 | −0.596820 | −1.93175 | ||||||||||||||||||
1.8 | −1.86942 | −2.63227 | 1.49473 | 1.00000 | 4.92082 | −2.31866 | 0.944554 | 3.92884 | −1.86942 | ||||||||||||||||||
1.9 | −1.82541 | 0.572536 | 1.33213 | 1.00000 | −1.04512 | −3.97729 | 1.21914 | −2.67220 | −1.82541 | ||||||||||||||||||
1.10 | −1.58931 | −0.358295 | 0.525917 | 1.00000 | 0.569443 | 2.77249 | 2.34278 | −2.87162 | −1.58931 | ||||||||||||||||||
1.11 | −1.16251 | −1.28072 | −0.648575 | 1.00000 | 1.48884 | −0.641997 | 3.07899 | −1.35976 | −1.16251 | ||||||||||||||||||
1.12 | −1.11064 | 3.24263 | −0.766474 | 1.00000 | −3.60140 | 2.31774 | 3.07256 | 7.51466 | −1.11064 | ||||||||||||||||||
1.13 | −0.745458 | 3.05905 | −1.44429 | 1.00000 | −2.28039 | −3.66055 | 2.56757 | 6.35779 | −0.745458 | ||||||||||||||||||
1.14 | −0.742421 | 0.782455 | −1.44881 | 1.00000 | −0.580911 | 0.262449 | 2.56047 | −2.38776 | −0.742421 | ||||||||||||||||||
1.15 | −0.446590 | 1.08482 | −1.80056 | 1.00000 | −0.484472 | 4.20380 | 1.69729 | −1.82316 | −0.446590 | ||||||||||||||||||
1.16 | −0.421939 | −2.05594 | −1.82197 | 1.00000 | 0.867482 | −0.0580639 | 1.61264 | 1.22690 | −0.421939 | ||||||||||||||||||
1.17 | −0.327392 | −2.36457 | −1.89281 | 1.00000 | 0.774141 | 2.46288 | 1.27448 | 2.59119 | −0.327392 | ||||||||||||||||||
1.18 | −0.199490 | −3.23393 | −1.96020 | 1.00000 | 0.645135 | −5.08849 | 0.790020 | 7.45829 | −0.199490 | ||||||||||||||||||
1.19 | 0.132711 | 1.97030 | −1.98239 | 1.00000 | 0.261480 | 2.54454 | −0.528506 | 0.882080 | 0.132711 | ||||||||||||||||||
1.20 | 0.503541 | 0.0692183 | −1.74645 | 1.00000 | 0.0348542 | −2.92313 | −1.88649 | −2.99521 | 0.503541 | ||||||||||||||||||
See all 37 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.h | ✓ | 37 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4015.2.a.h | ✓ | 37 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 5 T_{2}^{36} - 46 T_{2}^{35} + 261 T_{2}^{34} + 910 T_{2}^{33} - 6168 T_{2}^{32} + \cdots - 7872 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).