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: | \(1\) |
Dimension: | \(88\) |
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 | −2.76833 | −0.546396 | 5.66368 | 1.00000 | 1.51261 | −1.48260 | −10.1423 | −2.70145 | −2.76833 | ||||||||||||||||||
1.2 | −2.70073 | −2.65354 | 5.29393 | 1.00000 | 7.16648 | −4.13839 | −8.89601 | 4.04126 | −2.70073 | ||||||||||||||||||
1.3 | −2.69844 | 0.716329 | 5.28159 | 1.00000 | −1.93297 | 1.67359 | −8.85517 | −2.48687 | −2.69844 | ||||||||||||||||||
1.4 | −2.69778 | −3.22929 | 5.27799 | 1.00000 | 8.71190 | 2.27707 | −8.84328 | 7.42831 | −2.69778 | ||||||||||||||||||
1.5 | −2.63715 | 2.33888 | 4.95457 | 1.00000 | −6.16798 | −0.829927 | −7.79166 | 2.47036 | −2.63715 | ||||||||||||||||||
1.6 | −2.56191 | 2.09483 | 4.56337 | 1.00000 | −5.36676 | −0.643691 | −6.56712 | 1.38832 | −2.56191 | ||||||||||||||||||
1.7 | −2.49547 | −2.51949 | 4.22735 | 1.00000 | 6.28731 | 2.97292 | −5.55828 | 3.34784 | −2.49547 | ||||||||||||||||||
1.8 | −2.45453 | −0.943055 | 4.02473 | 1.00000 | 2.31476 | −0.315218 | −4.96977 | −2.11065 | −2.45453 | ||||||||||||||||||
1.9 | −2.32346 | 1.00159 | 3.39848 | 1.00000 | −2.32717 | 3.48702 | −3.24933 | −1.99681 | −2.32346 | ||||||||||||||||||
1.10 | −2.28766 | −2.82727 | 3.23337 | 1.00000 | 6.46782 | −3.61501 | −2.82152 | 4.99347 | −2.28766 | ||||||||||||||||||
1.11 | −2.26517 | 2.21267 | 3.13101 | 1.00000 | −5.01209 | −3.04532 | −2.56193 | 1.89592 | −2.26517 | ||||||||||||||||||
1.12 | −2.25928 | −0.374349 | 3.10434 | 1.00000 | 0.845758 | −3.16273 | −2.49501 | −2.85986 | −2.25928 | ||||||||||||||||||
1.13 | −2.21323 | −3.30657 | 2.89841 | 1.00000 | 7.31821 | −1.75850 | −1.98838 | 7.93340 | −2.21323 | ||||||||||||||||||
1.14 | −2.21193 | 0.112500 | 2.89262 | 1.00000 | −0.248842 | −5.20479 | −1.97441 | −2.98734 | −2.21193 | ||||||||||||||||||
1.15 | −2.17289 | −1.65848 | 2.72145 | 1.00000 | 3.60369 | 5.14399 | −1.56762 | −0.249442 | −2.17289 | ||||||||||||||||||
1.16 | −2.07691 | −0.413168 | 2.31355 | 1.00000 | 0.858111 | 2.69878 | −0.651210 | −2.82929 | −2.07691 | ||||||||||||||||||
1.17 | −1.97194 | 2.67165 | 1.88854 | 1.00000 | −5.26832 | −0.758910 | 0.219801 | 4.13769 | −1.97194 | ||||||||||||||||||
1.18 | −1.97101 | −1.60435 | 1.88489 | 1.00000 | 3.16220 | 2.33314 | 0.226875 | −0.426059 | −1.97101 | ||||||||||||||||||
1.19 | −1.93565 | −2.84784 | 1.74674 | 1.00000 | 5.51242 | −0.215804 | 0.490231 | 5.11019 | −1.93565 | ||||||||||||||||||
1.20 | −1.89204 | 0.756696 | 1.57982 | 1.00000 | −1.43170 | 1.91014 | 0.794996 | −2.42741 | −1.89204 | ||||||||||||||||||
See all 88 embeddings |
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.e | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6005.2.a.e | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 14 T_{2}^{87} - 23 T_{2}^{86} - 1205 T_{2}^{85} - 2777 T_{2}^{84} + 47070 T_{2}^{83} + \cdots + 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).