Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6041,2,Mod(1,6041)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6041, 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("6041.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6041 = 7 \cdot 863 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6041.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: | \(48.2376278611\) |
Analytic rank: | \(0\) |
Dimension: | \(132\) |
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 | −2.81389 | 0.225620 | 5.91799 | −1.24518 | −0.634869 | 1.00000 | −11.0248 | −2.94910 | 3.50380 | ||||||||||||||||||
1.2 | −2.78355 | 3.33951 | 5.74813 | 2.68755 | −9.29568 | 1.00000 | −10.4331 | 8.15233 | −7.48091 | ||||||||||||||||||
1.3 | −2.77966 | −1.97929 | 5.72652 | −3.34292 | 5.50177 | 1.00000 | −10.3585 | 0.917605 | 9.29218 | ||||||||||||||||||
1.4 | −2.75252 | −2.26942 | 5.57639 | 4.03697 | 6.24664 | 1.00000 | −9.84409 | 2.15028 | −11.1119 | ||||||||||||||||||
1.5 | −2.71866 | −0.898623 | 5.39111 | 3.58669 | 2.44305 | 1.00000 | −9.21926 | −2.19248 | −9.75100 | ||||||||||||||||||
1.6 | −2.65744 | 2.58260 | 5.06196 | 1.09780 | −6.86308 | 1.00000 | −8.13696 | 3.66980 | −2.91732 | ||||||||||||||||||
1.7 | −2.62362 | −1.27143 | 4.88340 | 1.41870 | 3.33576 | 1.00000 | −7.56495 | −1.38346 | −3.72213 | ||||||||||||||||||
1.8 | −2.60157 | 0.915852 | 4.76817 | −4.25711 | −2.38265 | 1.00000 | −7.20159 | −2.16122 | 11.0752 | ||||||||||||||||||
1.9 | −2.58774 | −2.55272 | 4.69637 | −0.926438 | 6.60577 | 1.00000 | −6.97750 | 3.51640 | 2.39738 | ||||||||||||||||||
1.10 | −2.58389 | 1.40748 | 4.67649 | −2.16822 | −3.63678 | 1.00000 | −6.91577 | −1.01899 | 5.60245 | ||||||||||||||||||
1.11 | −2.57785 | −0.937633 | 4.64529 | −3.87896 | 2.41707 | 1.00000 | −6.81916 | −2.12084 | 9.99936 | ||||||||||||||||||
1.12 | −2.56875 | −3.19690 | 4.59849 | −1.91049 | 8.21205 | 1.00000 | −6.67489 | 7.22017 | 4.90758 | ||||||||||||||||||
1.13 | −2.49223 | 0.173328 | 4.21120 | 2.29751 | −0.431973 | 1.00000 | −5.51083 | −2.96996 | −5.72592 | ||||||||||||||||||
1.14 | −2.47641 | 1.36965 | 4.13259 | −1.36117 | −3.39181 | 1.00000 | −5.28115 | −1.12406 | 3.37081 | ||||||||||||||||||
1.15 | −2.46575 | 3.35396 | 4.07990 | −4.25898 | −8.27001 | 1.00000 | −5.12851 | 8.24904 | 10.5016 | ||||||||||||||||||
1.16 | −2.32322 | 2.41590 | 3.39733 | 0.249961 | −5.61265 | 1.00000 | −3.24629 | 2.83657 | −0.580712 | ||||||||||||||||||
1.17 | −2.26422 | −1.83534 | 3.12671 | 1.84374 | 4.15563 | 1.00000 | −2.55113 | 0.368489 | −4.17465 | ||||||||||||||||||
1.18 | −2.25429 | −3.03067 | 3.08183 | 3.72305 | 6.83200 | 1.00000 | −2.43875 | 6.18494 | −8.39284 | ||||||||||||||||||
1.19 | −2.23793 | 1.90424 | 3.00834 | 4.23440 | −4.26157 | 1.00000 | −2.25659 | 0.626144 | −9.47631 | ||||||||||||||||||
1.20 | −2.22730 | 1.19588 | 2.96085 | 2.18062 | −2.66358 | 1.00000 | −2.14009 | −1.56986 | −4.85688 | ||||||||||||||||||
See next 80 embeddings (of 132 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(-1\) |
\(863\) | \(1\) |
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 | 6041.2.a.f | ✓ | 132 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6041.2.a.f | ✓ | 132 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} - 8 T_{2}^{131} - 187 T_{2}^{130} + 1646 T_{2}^{129} + 16720 T_{2}^{128} - 164947 T_{2}^{127} + \cdots + 57534003 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).