Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6005,2,Mod(1,6005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6005, 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("6005.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6005 = 5 \cdot 1201 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6005.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: | \(47.9501664138\) |
Analytic rank: | \(0\) |
Dimension: | \(113\) |
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 | −2.81643 | −3.03025 | 5.93229 | −1.00000 | 8.53449 | 1.00907 | −11.0750 | 6.18240 | 2.81643 | ||||||||||||||||||
1.2 | −2.80393 | 2.29330 | 5.86204 | −1.00000 | −6.43026 | 2.82627 | −10.8289 | 2.25923 | 2.80393 | ||||||||||||||||||
1.3 | −2.78685 | −1.57697 | 5.76652 | −1.00000 | 4.39478 | −0.289151 | −10.4967 | −0.513163 | 2.78685 | ||||||||||||||||||
1.4 | −2.77794 | 1.80495 | 5.71693 | −1.00000 | −5.01404 | −4.50639 | −10.3254 | 0.257853 | 2.77794 | ||||||||||||||||||
1.5 | −2.68650 | 2.64233 | 5.21726 | −1.00000 | −7.09862 | −4.41903 | −8.64317 | 3.98193 | 2.68650 | ||||||||||||||||||
1.6 | −2.67625 | 1.85244 | 5.16230 | −1.00000 | −4.95758 | 4.71146 | −8.46311 | 0.431523 | 2.67625 | ||||||||||||||||||
1.7 | −2.66122 | −2.12417 | 5.08210 | −1.00000 | 5.65288 | −3.49276 | −8.20215 | 1.51209 | 2.66122 | ||||||||||||||||||
1.8 | −2.62680 | −1.30643 | 4.90009 | −1.00000 | 3.43174 | −4.40647 | −7.61796 | −1.29324 | 2.62680 | ||||||||||||||||||
1.9 | −2.61532 | 0.118458 | 4.83989 | −1.00000 | −0.309805 | −0.279288 | −7.42721 | −2.98597 | 2.61532 | ||||||||||||||||||
1.10 | −2.57539 | −2.35580 | 4.63261 | −1.00000 | 6.06709 | 3.76299 | −6.78000 | 2.54978 | 2.57539 | ||||||||||||||||||
1.11 | −2.56745 | −0.556289 | 4.59182 | −1.00000 | 1.42824 | 3.15883 | −6.65437 | −2.69054 | 2.56745 | ||||||||||||||||||
1.12 | −2.49553 | −3.37079 | 4.22769 | −1.00000 | 8.41191 | −2.54255 | −5.55927 | 8.36220 | 2.49553 | ||||||||||||||||||
1.13 | −2.48400 | 0.156242 | 4.17024 | −1.00000 | −0.388105 | 1.49577 | −5.39087 | −2.97559 | 2.48400 | ||||||||||||||||||
1.14 | −2.47572 | 3.00665 | 4.12920 | −1.00000 | −7.44362 | 1.05496 | −5.27130 | 6.03992 | 2.47572 | ||||||||||||||||||
1.15 | −2.24487 | −2.47232 | 3.03945 | −1.00000 | 5.55004 | −1.67086 | −2.33343 | 3.11237 | 2.24487 | ||||||||||||||||||
1.16 | −2.21162 | 0.635696 | 2.89127 | −1.00000 | −1.40592 | −2.43179 | −1.97116 | −2.59589 | 2.21162 | ||||||||||||||||||
1.17 | −2.18171 | 2.09909 | 2.75987 | −1.00000 | −4.57961 | −2.68153 | −1.65781 | 1.40617 | 2.18171 | ||||||||||||||||||
1.18 | −2.16728 | 2.60896 | 2.69711 | −1.00000 | −5.65435 | 4.04826 | −1.51083 | 3.80667 | 2.16728 | ||||||||||||||||||
1.19 | −2.09533 | −3.35381 | 2.39043 | −1.00000 | 7.02736 | 3.72693 | −0.818074 | 8.24805 | 2.09533 | ||||||||||||||||||
1.20 | −2.08489 | −0.988264 | 2.34676 | −1.00000 | 2.06042 | 4.76657 | −0.722948 | −2.02333 | 2.08489 | ||||||||||||||||||
See next 80 embeddings (of 113 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(1201\) | \(-1\) |
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 | 6005.2.a.g | ✓ | 113 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6005.2.a.g | ✓ | 113 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{113} + 3 T_{2}^{112} - 179 T_{2}^{111} - 538 T_{2}^{110} + 15552 T_{2}^{109} + 46831 T_{2}^{108} + \cdots - 28158256848 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).