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: | \(111\) |
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 | −2.74668 | 1.63661 | 5.54423 | 1.00000 | −4.49523 | −2.87647 | −9.73484 | −0.321512 | −2.74668 | ||||||||||||||||||
1.2 | −2.72268 | 3.38210 | 5.41301 | 1.00000 | −9.20838 | 2.13000 | −9.29255 | 8.43858 | −2.72268 | ||||||||||||||||||
1.3 | −2.71589 | 0.174668 | 5.37604 | 1.00000 | −0.474377 | 2.58754 | −9.16893 | −2.96949 | −2.71589 | ||||||||||||||||||
1.4 | −2.65700 | −0.218117 | 5.05966 | 1.00000 | 0.579538 | 4.50653 | −8.12951 | −2.95242 | −2.65700 | ||||||||||||||||||
1.5 | −2.63642 | 3.34656 | 4.95072 | 1.00000 | −8.82296 | −3.64864 | −7.77935 | 8.19949 | −2.63642 | ||||||||||||||||||
1.6 | −2.62342 | −2.19431 | 4.88231 | 1.00000 | 5.75659 | 1.89929 | −7.56149 | 1.81500 | −2.62342 | ||||||||||||||||||
1.7 | −2.61069 | −1.33217 | 4.81572 | 1.00000 | 3.47788 | −0.680814 | −7.35098 | −1.22533 | −2.61069 | ||||||||||||||||||
1.8 | −2.56895 | 1.63205 | 4.59952 | 1.00000 | −4.19265 | 3.45435 | −6.67805 | −0.336428 | −2.56895 | ||||||||||||||||||
1.9 | −2.47000 | −2.23996 | 4.10090 | 1.00000 | 5.53271 | 3.40548 | −5.18922 | 2.01743 | −2.47000 | ||||||||||||||||||
1.10 | −2.41916 | 1.95905 | 3.85233 | 1.00000 | −4.73925 | −1.88529 | −4.48108 | 0.837875 | −2.41916 | ||||||||||||||||||
1.11 | −2.34362 | −0.853258 | 3.49256 | 1.00000 | 1.99971 | −1.48080 | −3.49800 | −2.27195 | −2.34362 | ||||||||||||||||||
1.12 | −2.32829 | 0.726641 | 3.42095 | 1.00000 | −1.69183 | −3.13935 | −3.30840 | −2.47199 | −2.32829 | ||||||||||||||||||
1.13 | −2.32774 | 1.75930 | 3.41838 | 1.00000 | −4.09520 | 3.92990 | −3.30162 | 0.0951392 | −2.32774 | ||||||||||||||||||
1.14 | −2.32622 | −1.23055 | 3.41132 | 1.00000 | 2.86255 | −0.720355 | −3.28304 | −1.48574 | −2.32622 | ||||||||||||||||||
1.15 | −2.23901 | 3.10819 | 3.01316 | 1.00000 | −6.95927 | 3.97064 | −2.26846 | 6.66087 | −2.23901 | ||||||||||||||||||
1.16 | −2.17647 | 1.64482 | 2.73702 | 1.00000 | −3.57990 | −1.13164 | −1.60410 | −0.294573 | −2.17647 | ||||||||||||||||||
1.17 | −2.03891 | 2.79275 | 2.15716 | 1.00000 | −5.69418 | −0.0959578 | −0.320429 | 4.79947 | −2.03891 | ||||||||||||||||||
1.18 | −1.98616 | −2.45604 | 1.94482 | 1.00000 | 4.87807 | −0.764099 | 0.109603 | 3.03211 | −1.98616 | ||||||||||||||||||
1.19 | −1.98040 | 0.765234 | 1.92199 | 1.00000 | −1.51547 | −3.35858 | 0.154490 | −2.41442 | −1.98040 | ||||||||||||||||||
1.20 | −1.91654 | −2.98629 | 1.67314 | 1.00000 | 5.72336 | 0.938196 | 0.626446 | 5.91794 | −1.91654 | ||||||||||||||||||
See next 80 embeddings (of 111 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.f | ✓ | 111 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6005.2.a.f | ✓ | 111 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{111} - 20 T_{2}^{110} + 21 T_{2}^{109} + 2205 T_{2}^{108} - 12753 T_{2}^{107} - 99984 T_{2}^{106} + \cdots - 1711692 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).