Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4007,2,Mod(1,4007)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4007, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4007.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4007 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4007.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: | \(31.9960560899\) |
Analytic rank: | \(1\) |
Dimension: | \(139\) |
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 | −2.81570 | −0.263068 | 5.92814 | 2.38290 | 0.740719 | −2.49985 | −11.0604 | −2.93080 | −6.70951 | ||||||||||||||||||
1.2 | −2.77109 | 1.88840 | 5.67894 | −0.563279 | −5.23292 | 0.553316 | −10.1947 | 0.566050 | 1.56090 | ||||||||||||||||||
1.3 | −2.72706 | −2.54460 | 5.43687 | −1.96557 | 6.93928 | −3.80223 | −9.37255 | 3.47499 | 5.36024 | ||||||||||||||||||
1.4 | −2.70090 | −3.24744 | 5.29488 | −0.141100 | 8.77102 | 1.56207 | −8.89915 | 7.54587 | 0.381096 | ||||||||||||||||||
1.5 | −2.64078 | 1.79793 | 4.97369 | −0.236310 | −4.74793 | −1.11879 | −7.85286 | 0.232555 | 0.624041 | ||||||||||||||||||
1.6 | −2.62057 | −0.101978 | 4.86739 | 0.441388 | 0.267240 | 0.132029 | −7.51419 | −2.98960 | −1.15669 | ||||||||||||||||||
1.7 | −2.60765 | 2.41443 | 4.79982 | 2.62983 | −6.29599 | −4.53793 | −7.30095 | 2.82948 | −6.85767 | ||||||||||||||||||
1.8 | −2.53714 | −2.64168 | 4.43706 | 4.10471 | 6.70230 | −0.283767 | −6.18314 | 3.97846 | −10.4142 | ||||||||||||||||||
1.9 | −2.52380 | −1.27675 | 4.36956 | 2.82369 | 3.22226 | 3.70991 | −5.98028 | −1.36991 | −7.12642 | ||||||||||||||||||
1.10 | −2.51027 | −1.47560 | 4.30145 | −0.0269498 | 3.70415 | 2.66833 | −5.77725 | −0.822611 | 0.0676513 | ||||||||||||||||||
1.11 | −2.46958 | 0.828101 | 4.09882 | −3.19925 | −2.04506 | −2.79800 | −5.18321 | −2.31425 | 7.90080 | ||||||||||||||||||
1.12 | −2.42814 | −1.84708 | 3.89588 | −2.88367 | 4.48497 | −1.14516 | −4.60348 | 0.411691 | 7.00198 | ||||||||||||||||||
1.13 | −2.41992 | 2.90998 | 3.85601 | −2.04833 | −7.04192 | 0.763772 | −4.49140 | 5.46800 | 4.95680 | ||||||||||||||||||
1.14 | −2.39846 | −2.50987 | 3.75261 | 0.204658 | 6.01982 | −4.29057 | −4.20358 | 3.29943 | −0.490864 | ||||||||||||||||||
1.15 | −2.38393 | −0.401819 | 3.68311 | −0.901205 | 0.957907 | −4.24761 | −4.01240 | −2.83854 | 2.14841 | ||||||||||||||||||
1.16 | −2.37198 | 1.06086 | 3.62628 | 2.75167 | −2.51634 | 1.78055 | −3.85749 | −1.87457 | −6.52689 | ||||||||||||||||||
1.17 | −2.24444 | −1.81943 | 3.03751 | 1.76577 | 4.08359 | 0.689140 | −2.32862 | 0.310309 | −3.96317 | ||||||||||||||||||
1.18 | −2.23150 | 1.82610 | 2.97957 | 2.21560 | −4.07492 | 3.73360 | −2.18591 | 0.334626 | −4.94411 | ||||||||||||||||||
1.19 | −2.19482 | 2.00637 | 2.81722 | −1.97221 | −4.40361 | 2.24566 | −1.79364 | 1.02552 | 4.32863 | ||||||||||||||||||
1.20 | −2.15727 | 1.62818 | 2.65379 | 2.01382 | −3.51242 | −1.64574 | −1.41041 | −0.349027 | −4.34434 | ||||||||||||||||||
See next 80 embeddings (of 139 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(4007\) | \(1\) |
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 | 4007.2.a.a | ✓ | 139 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4007.2.a.a | ✓ | 139 | 1.a | even | 1 | 1 | trivial |