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: | \(112\) |
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 | −2.81373 | −3.44120 | 5.91707 | 1.30649 | 9.68259 | −1.00000 | −11.0216 | 8.84183 | −3.67611 | ||||||||||||||||||
1.2 | −2.80869 | 2.13107 | 5.88877 | −3.52544 | −5.98552 | −1.00000 | −10.9224 | 1.54145 | 9.90189 | ||||||||||||||||||
1.3 | −2.77254 | −2.14289 | 5.68701 | −0.156494 | 5.94125 | −1.00000 | −10.2224 | 1.59196 | 0.433887 | ||||||||||||||||||
1.4 | −2.76309 | 2.60057 | 5.63466 | −1.29315 | −7.18561 | −1.00000 | −10.0429 | 3.76297 | 3.57310 | ||||||||||||||||||
1.5 | −2.71246 | −0.244067 | 5.35744 | −2.10613 | 0.662022 | −1.00000 | −9.10692 | −2.94043 | 5.71281 | ||||||||||||||||||
1.6 | −2.71151 | 1.27956 | 5.35230 | 1.05589 | −3.46954 | −1.00000 | −9.08979 | −1.36273 | −2.86306 | ||||||||||||||||||
1.7 | −2.67889 | −2.12545 | 5.17645 | −3.74850 | 5.69384 | −1.00000 | −8.50937 | 1.51753 | 10.0418 | ||||||||||||||||||
1.8 | −2.57911 | −0.702301 | 4.65183 | 2.85376 | 1.81131 | −1.00000 | −6.83937 | −2.50677 | −7.36016 | ||||||||||||||||||
1.9 | −2.52776 | 3.15958 | 4.38957 | 3.22941 | −7.98667 | −1.00000 | −6.04026 | 6.98297 | −8.16316 | ||||||||||||||||||
1.10 | −2.46248 | −0.0194781 | 4.06380 | −1.44141 | 0.0479644 | −1.00000 | −5.08207 | −2.99962 | 3.54944 | ||||||||||||||||||
1.11 | −2.45673 | 3.01946 | 4.03553 | −1.39061 | −7.41801 | −1.00000 | −5.00075 | 6.11715 | 3.41635 | ||||||||||||||||||
1.12 | −2.43414 | −1.84866 | 3.92503 | 3.83434 | 4.49989 | −1.00000 | −4.68578 | 0.417546 | −9.33330 | ||||||||||||||||||
1.13 | −2.41077 | 2.26439 | 3.81183 | 4.38346 | −5.45893 | −1.00000 | −4.36791 | 2.12745 | −10.5675 | ||||||||||||||||||
1.14 | −2.37296 | 0.810716 | 3.63095 | 0.528258 | −1.92380 | −1.00000 | −3.87017 | −2.34274 | −1.25354 | ||||||||||||||||||
1.15 | −2.27713 | −2.93009 | 3.18534 | −2.32769 | 6.67221 | −1.00000 | −2.69918 | 5.58544 | 5.30045 | ||||||||||||||||||
1.16 | −2.25493 | 0.275101 | 3.08473 | 2.79647 | −0.620335 | −1.00000 | −2.44598 | −2.92432 | −6.30585 | ||||||||||||||||||
1.17 | −2.22590 | −2.92856 | 2.95465 | 1.01796 | 6.51870 | −1.00000 | −2.12496 | 5.57647 | −2.26588 | ||||||||||||||||||
1.18 | −2.13001 | −1.53283 | 2.53693 | −2.08639 | 3.26494 | −1.00000 | −1.14367 | −0.650428 | 4.44403 | ||||||||||||||||||
1.19 | −2.10602 | −0.719393 | 2.43533 | 1.34496 | 1.51506 | −1.00000 | −0.916817 | −2.48247 | −2.83252 | ||||||||||||||||||
1.20 | −2.06364 | −0.217047 | 2.25861 | 3.48472 | 0.447908 | −1.00000 | −0.533675 | −2.95289 | −7.19121 | ||||||||||||||||||
See next 80 embeddings (of 112 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.e | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6041.2.a.e | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} + 3 T_{2}^{111} - 173 T_{2}^{110} - 521 T_{2}^{109} + 14516 T_{2}^{108} + 43895 T_{2}^{107} + \cdots + 24980734323 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).