Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2015,2,Mod(1,2015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, 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("2015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2015 = 5 \cdot 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2015.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: | \(16.0898560073\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
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 | −2.80211 | −2.07248 | 5.85184 | 1.00000 | 5.80733 | −0.584705 | −10.7933 | 1.29518 | −2.80211 | ||||||||||||||||||
1.2 | −2.76992 | 1.79224 | 5.67245 | 1.00000 | −4.96436 | −5.11218 | −10.1724 | 0.212127 | −2.76992 | ||||||||||||||||||
1.3 | −2.12740 | 0.161318 | 2.52584 | 1.00000 | −0.343187 | −0.411943 | −1.11868 | −2.97398 | −2.12740 | ||||||||||||||||||
1.4 | −2.04109 | 3.04107 | 2.16603 | 1.00000 | −6.20708 | 4.34888 | −0.338882 | 6.24810 | −2.04109 | ||||||||||||||||||
1.5 | −1.77849 | −0.614278 | 1.16304 | 1.00000 | 1.09249 | 2.73573 | 1.48853 | −2.62266 | −1.77849 | ||||||||||||||||||
1.6 | −1.51285 | 2.86334 | 0.288701 | 1.00000 | −4.33179 | −4.20074 | 2.58893 | 5.19870 | −1.51285 | ||||||||||||||||||
1.7 | −1.07130 | −2.90938 | −0.852317 | 1.00000 | 3.11682 | −0.609202 | 3.05569 | 5.46452 | −1.07130 | ||||||||||||||||||
1.8 | −0.802845 | 2.11595 | −1.35544 | 1.00000 | −1.69878 | 3.29759 | 2.69390 | 1.47723 | −0.802845 | ||||||||||||||||||
1.9 | −0.414001 | −0.0596484 | −1.82860 | 1.00000 | 0.0246945 | 1.37645 | 1.58505 | −2.99644 | −0.414001 | ||||||||||||||||||
1.10 | 0.0156266 | −0.642037 | −1.99976 | 1.00000 | −0.0100329 | −4.79628 | −0.0625027 | −2.58779 | 0.0156266 | ||||||||||||||||||
1.11 | 0.149185 | −0.122146 | −1.97774 | 1.00000 | −0.0182224 | −2.03839 | −0.593420 | −2.98508 | 0.149185 | ||||||||||||||||||
1.12 | 0.524389 | 2.55681 | −1.72502 | 1.00000 | 1.34076 | 4.00960 | −1.95336 | 3.53726 | 0.524389 | ||||||||||||||||||
1.13 | 0.763611 | −3.39028 | −1.41690 | 1.00000 | −2.58885 | 3.22243 | −2.60918 | 8.49398 | 0.763611 | ||||||||||||||||||
1.14 | 1.14097 | 3.32499 | −0.698177 | 1.00000 | 3.79373 | −2.66542 | −3.07855 | 8.05557 | 1.14097 | ||||||||||||||||||
1.15 | 1.36079 | −2.47539 | −0.148263 | 1.00000 | −3.36847 | −4.84366 | −2.92333 | 3.12754 | 1.36079 | ||||||||||||||||||
1.16 | 1.56300 | −2.37379 | 0.442981 | 1.00000 | −3.71025 | 4.14002 | −2.43363 | 2.63489 | 1.56300 | ||||||||||||||||||
1.17 | 2.18281 | 1.66528 | 2.76464 | 1.00000 | 3.63498 | 1.10265 | 1.66907 | −0.226842 | 2.18281 | ||||||||||||||||||
1.18 | 2.25541 | 3.24132 | 3.08686 | 1.00000 | 7.31049 | 2.29814 | 2.45132 | 7.50614 | 2.25541 | ||||||||||||||||||
1.19 | 2.26081 | −0.979957 | 3.11127 | 1.00000 | −2.21550 | 1.84324 | 2.51236 | −2.03968 | 2.26081 | ||||||||||||||||||
1.20 | 2.53893 | 1.96468 | 4.44617 | 1.00000 | 4.98819 | −1.49124 | 6.21065 | 0.859981 | 2.53893 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(13\) | \(1\) |
\(31\) | \(1\) |
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 | 2015.2.a.l | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2015.2.a.l | ✓ | 22 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):
\( T_{2}^{22} - 5 T_{2}^{21} - 25 T_{2}^{20} + 157 T_{2}^{19} + 203 T_{2}^{18} - 2026 T_{2}^{17} + \cdots - 16 \) |
\( T_{3}^{22} - 4 T_{3}^{21} - 46 T_{3}^{20} + 195 T_{3}^{19} + 858 T_{3}^{18} - 3954 T_{3}^{17} + \cdots - 64 \) |