Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8013,2,Mod(1,8013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8013, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8013.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8013 = 3 \cdot 2671 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8013.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: | \(63.9841271397\) |
Analytic rank: | \(1\) |
Dimension: | \(116\) |
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 | −2.78241 | −1.00000 | 5.74180 | −1.84035 | 2.78241 | −4.26418 | −10.4112 | 1.00000 | 5.12061 | ||||||||||||||||||
1.2 | −2.77241 | −1.00000 | 5.68625 | 1.80856 | 2.77241 | −1.74273 | −10.2198 | 1.00000 | −5.01408 | ||||||||||||||||||
1.3 | −2.74646 | −1.00000 | 5.54305 | −2.85945 | 2.74646 | 1.69114 | −9.73084 | 1.00000 | 7.85338 | ||||||||||||||||||
1.4 | −2.74034 | −1.00000 | 5.50946 | 3.71802 | 2.74034 | −2.61622 | −9.61710 | 1.00000 | −10.1886 | ||||||||||||||||||
1.5 | −2.72179 | −1.00000 | 5.40814 | 2.52515 | 2.72179 | 2.50598 | −9.27623 | 1.00000 | −6.87293 | ||||||||||||||||||
1.6 | −2.71385 | −1.00000 | 5.36498 | −0.752198 | 2.71385 | 4.51557 | −9.13205 | 1.00000 | 2.04135 | ||||||||||||||||||
1.7 | −2.66795 | −1.00000 | 5.11795 | −3.13798 | 2.66795 | 0.0813556 | −8.31854 | 1.00000 | 8.37196 | ||||||||||||||||||
1.8 | −2.59210 | −1.00000 | 4.71896 | −1.33515 | 2.59210 | −1.92470 | −7.04781 | 1.00000 | 3.46084 | ||||||||||||||||||
1.9 | −2.58282 | −1.00000 | 4.67097 | −1.48132 | 2.58282 | 0.817005 | −6.89864 | 1.00000 | 3.82598 | ||||||||||||||||||
1.10 | −2.57433 | −1.00000 | 4.62715 | −4.27877 | 2.57433 | −4.90523 | −6.76315 | 1.00000 | 11.0149 | ||||||||||||||||||
1.11 | −2.49870 | −1.00000 | 4.24352 | −4.37212 | 2.49870 | 1.33064 | −5.60588 | 1.00000 | 10.9246 | ||||||||||||||||||
1.12 | −2.45463 | −1.00000 | 4.02523 | 1.82739 | 2.45463 | −2.98046 | −4.97119 | 1.00000 | −4.48558 | ||||||||||||||||||
1.13 | −2.42944 | −1.00000 | 3.90217 | −3.31824 | 2.42944 | 4.69634 | −4.62122 | 1.00000 | 8.06146 | ||||||||||||||||||
1.14 | −2.39685 | −1.00000 | 3.74487 | 3.43258 | 2.39685 | −5.25948 | −4.18219 | 1.00000 | −8.22737 | ||||||||||||||||||
1.15 | −2.30332 | −1.00000 | 3.30530 | 0.401271 | 2.30332 | −3.38368 | −3.00653 | 1.00000 | −0.924257 | ||||||||||||||||||
1.16 | −2.24523 | −1.00000 | 3.04104 | −1.84042 | 2.24523 | 1.63491 | −2.33738 | 1.00000 | 4.13216 | ||||||||||||||||||
1.17 | −2.23895 | −1.00000 | 3.01291 | 1.55531 | 2.23895 | 1.34297 | −2.26786 | 1.00000 | −3.48228 | ||||||||||||||||||
1.18 | −2.20341 | −1.00000 | 2.85502 | −0.504029 | 2.20341 | −3.35275 | −1.88396 | 1.00000 | 1.11058 | ||||||||||||||||||
1.19 | −2.18030 | −1.00000 | 2.75372 | 1.60452 | 2.18030 | 4.21617 | −1.64333 | 1.00000 | −3.49835 | ||||||||||||||||||
1.20 | −2.15908 | −1.00000 | 2.66162 | −0.295164 | 2.15908 | 3.97405 | −1.42849 | 1.00000 | 0.637282 | ||||||||||||||||||
See next 80 embeddings (of 116 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(2671\) | \(1\) |
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 | 8013.2.a.c | ✓ | 116 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8013.2.a.c | ✓ | 116 | 1.a | even | 1 | 1 | trivial |