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: | \(1\) |
Dimension: | \(83\) |
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 | −2.73002 | −0.0142784 | 5.45300 | −0.484021 | 0.0389802 | 1.00000 | −9.42677 | −2.99980 | 1.32139 | ||||||||||||||||||
1.2 | −2.58676 | −0.792673 | 4.69133 | 0.0189688 | 2.05046 | 1.00000 | −6.96183 | −2.37167 | −0.0490676 | ||||||||||||||||||
1.3 | −2.50552 | 2.64911 | 4.27764 | −1.70463 | −6.63740 | 1.00000 | −5.70669 | 4.01777 | 4.27099 | ||||||||||||||||||
1.4 | −2.50484 | 1.86042 | 4.27422 | 2.20949 | −4.66004 | 1.00000 | −5.69655 | 0.461144 | −5.53442 | ||||||||||||||||||
1.5 | −2.49510 | −2.54767 | 4.22552 | 0.711578 | 6.35668 | 1.00000 | −5.55291 | 3.49060 | −1.77546 | ||||||||||||||||||
1.6 | −2.36693 | 2.60578 | 3.60235 | 0.153258 | −6.16771 | 1.00000 | −3.79266 | 3.79011 | −0.362751 | ||||||||||||||||||
1.7 | −2.28673 | 0.00653898 | 3.22911 | 3.50344 | −0.0149528 | 1.00000 | −2.81064 | −2.99996 | −8.01141 | ||||||||||||||||||
1.8 | −2.27758 | −1.23058 | 3.18737 | 1.22240 | 2.80274 | 1.00000 | −2.70432 | −1.48568 | −2.78412 | ||||||||||||||||||
1.9 | −2.25188 | −3.15195 | 3.07097 | 1.48598 | 7.09783 | 1.00000 | −2.41170 | 6.93481 | −3.34625 | ||||||||||||||||||
1.10 | −2.19298 | 0.436268 | 2.80917 | 2.07310 | −0.956728 | 1.00000 | −1.77450 | −2.80967 | −4.54628 | ||||||||||||||||||
1.11 | −2.16622 | −1.36958 | 2.69252 | −2.62137 | 2.96681 | 1.00000 | −1.50014 | −1.12425 | 5.67847 | ||||||||||||||||||
1.12 | −2.15056 | −0.709425 | 2.62493 | −2.42733 | 1.52566 | 1.00000 | −1.34395 | −2.49672 | 5.22013 | ||||||||||||||||||
1.13 | −2.14754 | −2.08287 | 2.61192 | −2.63815 | 4.47305 | 1.00000 | −1.31412 | 1.33837 | 5.66553 | ||||||||||||||||||
1.14 | −2.14224 | 0.933242 | 2.58920 | −2.33430 | −1.99923 | 1.00000 | −1.26222 | −2.12906 | 5.00064 | ||||||||||||||||||
1.15 | −1.96601 | 2.65332 | 1.86520 | 0.0208201 | −5.21646 | 1.00000 | 0.265017 | 4.04013 | −0.0409325 | ||||||||||||||||||
1.16 | −1.90567 | −1.35998 | 1.63156 | 3.50685 | 2.59167 | 1.00000 | 0.702124 | −1.15045 | −6.68289 | ||||||||||||||||||
1.17 | −1.79680 | 0.970719 | 1.22848 | −1.22425 | −1.74419 | 1.00000 | 1.38627 | −2.05770 | 2.19973 | ||||||||||||||||||
1.18 | −1.78752 | 2.42627 | 1.19524 | −3.07648 | −4.33701 | 1.00000 | 1.43852 | 2.88678 | 5.49928 | ||||||||||||||||||
1.19 | −1.77588 | −2.45572 | 1.15374 | 3.79745 | 4.36107 | 1.00000 | 1.50285 | 3.03058 | −6.74380 | ||||||||||||||||||
1.20 | −1.71609 | −2.62448 | 0.944964 | −3.91107 | 4.50383 | 1.00000 | 1.81054 | 3.88787 | 6.71175 | ||||||||||||||||||
See all 83 embeddings |
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.c | ✓ | 83 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6041.2.a.c | ✓ | 83 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{83} + 8 T_{2}^{82} - 75 T_{2}^{81} - 754 T_{2}^{80} + 2369 T_{2}^{79} + 33869 T_{2}^{78} + \cdots - 299 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).