Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6034,2,Mod(1,6034)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6034, 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("6034.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6034 = 2 \cdot 7 \cdot 431 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6034.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: | \(48.1817325796\) |
Analytic rank: | \(1\) |
Dimension: | \(21\) |
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 | 1.00000 | −3.34739 | 1.00000 | 1.24845 | −3.34739 | 1.00000 | 1.00000 | 8.20504 | 1.24845 | ||||||||||||||||||
1.2 | 1.00000 | −2.48749 | 1.00000 | 1.32783 | −2.48749 | 1.00000 | 1.00000 | 3.18763 | 1.32783 | ||||||||||||||||||
1.3 | 1.00000 | −2.30616 | 1.00000 | −2.85291 | −2.30616 | 1.00000 | 1.00000 | 2.31835 | −2.85291 | ||||||||||||||||||
1.4 | 1.00000 | −2.23818 | 1.00000 | 3.73239 | −2.23818 | 1.00000 | 1.00000 | 2.00943 | 3.73239 | ||||||||||||||||||
1.5 | 1.00000 | −2.23488 | 1.00000 | 0.812592 | −2.23488 | 1.00000 | 1.00000 | 1.99468 | 0.812592 | ||||||||||||||||||
1.6 | 1.00000 | −1.63595 | 1.00000 | −2.93756 | −1.63595 | 1.00000 | 1.00000 | −0.323656 | −2.93756 | ||||||||||||||||||
1.7 | 1.00000 | −1.49715 | 1.00000 | −0.207716 | −1.49715 | 1.00000 | 1.00000 | −0.758538 | −0.207716 | ||||||||||||||||||
1.8 | 1.00000 | −1.27918 | 1.00000 | −0.808180 | −1.27918 | 1.00000 | 1.00000 | −1.36370 | −0.808180 | ||||||||||||||||||
1.9 | 1.00000 | −1.16337 | 1.00000 | −3.30128 | −1.16337 | 1.00000 | 1.00000 | −1.64657 | −3.30128 | ||||||||||||||||||
1.10 | 1.00000 | −0.950934 | 1.00000 | −2.70877 | −0.950934 | 1.00000 | 1.00000 | −2.09572 | −2.70877 | ||||||||||||||||||
1.11 | 1.00000 | −0.235507 | 1.00000 | 1.96765 | −0.235507 | 1.00000 | 1.00000 | −2.94454 | 1.96765 | ||||||||||||||||||
1.12 | 1.00000 | −0.00804327 | 1.00000 | 3.67628 | −0.00804327 | 1.00000 | 1.00000 | −2.99994 | 3.67628 | ||||||||||||||||||
1.13 | 1.00000 | 0.402295 | 1.00000 | 1.29038 | 0.402295 | 1.00000 | 1.00000 | −2.83816 | 1.29038 | ||||||||||||||||||
1.14 | 1.00000 | 0.584190 | 1.00000 | 0.414785 | 0.584190 | 1.00000 | 1.00000 | −2.65872 | 0.414785 | ||||||||||||||||||
1.15 | 1.00000 | 0.800681 | 1.00000 | −2.16386 | 0.800681 | 1.00000 | 1.00000 | −2.35891 | −2.16386 | ||||||||||||||||||
1.16 | 1.00000 | 0.893081 | 1.00000 | 0.353208 | 0.893081 | 1.00000 | 1.00000 | −2.20241 | 0.353208 | ||||||||||||||||||
1.17 | 1.00000 | 1.34513 | 1.00000 | −2.31517 | 1.34513 | 1.00000 | 1.00000 | −1.19062 | −2.31517 | ||||||||||||||||||
1.18 | 1.00000 | 1.87377 | 1.00000 | −1.03478 | 1.87377 | 1.00000 | 1.00000 | 0.510997 | −1.03478 | ||||||||||||||||||
1.19 | 1.00000 | 2.14514 | 1.00000 | −0.309854 | 2.14514 | 1.00000 | 1.00000 | 1.60160 | −0.309854 | ||||||||||||||||||
1.20 | 1.00000 | 2.28514 | 1.00000 | −4.30862 | 2.28514 | 1.00000 | 1.00000 | 2.22188 | −4.30862 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(7\) | \(-1\) |
\(431\) | \(1\) |
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 | 6034.2.a.m | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6034.2.a.m | ✓ | 21 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):
\( T_{3}^{21} + 6 T_{3}^{20} - 16 T_{3}^{19} - 155 T_{3}^{18} - 5 T_{3}^{17} + 1546 T_{3}^{16} + 1479 T_{3}^{15} - 7657 T_{3}^{14} - 11444 T_{3}^{13} + 19951 T_{3}^{12} + 39484 T_{3}^{11} - 26535 T_{3}^{10} - 71161 T_{3}^{9} + 16140 T_{3}^{8} + \cdots - 4 \) |
\( T_{5}^{21} + 11 T_{5}^{20} + 5 T_{5}^{19} - 350 T_{5}^{18} - 1077 T_{5}^{17} + 3184 T_{5}^{16} + 17851 T_{5}^{15} - 1753 T_{5}^{14} - 113339 T_{5}^{13} - 86999 T_{5}^{12} + 353928 T_{5}^{11} + 407587 T_{5}^{10} - 607364 T_{5}^{9} + \cdots - 1721 \) |
\( T_{11}^{21} + 34 T_{11}^{20} + 462 T_{11}^{19} + 2886 T_{11}^{18} + 3617 T_{11}^{17} - 60918 T_{11}^{16} - 353290 T_{11}^{15} - 312972 T_{11}^{14} + 3485592 T_{11}^{13} + 11786108 T_{11}^{12} - 2134972 T_{11}^{11} + \cdots - 206981567 \) |