Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6013,2,Mod(1,6013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, 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("6013.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6013 = 7 \cdot 859 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6013.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: | \(48.0140467354\) |
Analytic rank: | \(1\) |
Dimension: | \(104\) |
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 | −2.80449 | −2.76312 | 5.86515 | 0.655993 | 7.74914 | −1.00000 | −10.8398 | 4.63484 | −1.83973 | ||||||||||||||||||
1.2 | −2.79954 | −0.206703 | 5.83741 | 2.43879 | 0.578672 | −1.00000 | −10.7430 | −2.95727 | −6.82749 | ||||||||||||||||||
1.3 | −2.79135 | −2.87879 | 5.79164 | −2.47936 | 8.03571 | −1.00000 | −10.5838 | 5.28744 | 6.92075 | ||||||||||||||||||
1.4 | −2.76981 | 0.446127 | 5.67184 | −3.40223 | −1.23569 | −1.00000 | −10.1703 | −2.80097 | 9.42353 | ||||||||||||||||||
1.5 | −2.68972 | 2.91852 | 5.23459 | 0.347371 | −7.85000 | −1.00000 | −8.70012 | 5.51776 | −0.934329 | ||||||||||||||||||
1.6 | −2.66314 | −0.352926 | 5.09230 | −3.56407 | 0.939890 | −1.00000 | −8.23520 | −2.87544 | 9.49161 | ||||||||||||||||||
1.7 | −2.57881 | 1.45096 | 4.65028 | 4.21886 | −3.74175 | −1.00000 | −6.83459 | −0.894725 | −10.8797 | ||||||||||||||||||
1.8 | −2.50501 | −2.73412 | 4.27508 | −0.209181 | 6.84900 | −1.00000 | −5.69911 | 4.47541 | 0.524002 | ||||||||||||||||||
1.9 | −2.49724 | 2.33621 | 4.23620 | −3.93107 | −5.83407 | −1.00000 | −5.58433 | 2.45787 | 9.81682 | ||||||||||||||||||
1.10 | −2.47703 | −3.45031 | 4.13569 | 3.27478 | 8.54653 | −1.00000 | −5.29018 | 8.90465 | −8.11174 | ||||||||||||||||||
1.11 | −2.46187 | 0.832905 | 4.06078 | 1.64347 | −2.05050 | −1.00000 | −5.07337 | −2.30627 | −4.04600 | ||||||||||||||||||
1.12 | −2.45278 | −0.285776 | 4.01611 | 1.53542 | 0.700944 | −1.00000 | −4.94507 | −2.91833 | −3.76604 | ||||||||||||||||||
1.13 | −2.42052 | −1.81832 | 3.85894 | −2.20687 | 4.40128 | −1.00000 | −4.49961 | 0.306272 | 5.34179 | ||||||||||||||||||
1.14 | −2.41334 | 3.15271 | 3.82421 | −0.650667 | −7.60855 | −1.00000 | −4.40244 | 6.93956 | 1.57028 | ||||||||||||||||||
1.15 | −2.37810 | 0.188335 | 3.65538 | 0.902654 | −0.447879 | −1.00000 | −3.93667 | −2.96453 | −2.14661 | ||||||||||||||||||
1.16 | −2.33681 | 0.433467 | 3.46066 | −2.70266 | −1.01293 | −1.00000 | −3.41328 | −2.81211 | 6.31560 | ||||||||||||||||||
1.17 | −2.32315 | 1.73935 | 3.39700 | 3.06159 | −4.04076 | −1.00000 | −3.24544 | 0.0253336 | −7.11252 | ||||||||||||||||||
1.18 | −2.13208 | −1.43746 | 2.54578 | 0.297217 | 3.06479 | −1.00000 | −1.16366 | −0.933703 | −0.633692 | ||||||||||||||||||
1.19 | −2.03131 | −2.52319 | 2.12623 | 0.0590478 | 5.12539 | −1.00000 | −0.256407 | 3.36649 | −0.119944 | ||||||||||||||||||
1.20 | −2.02779 | 2.00075 | 2.11193 | −3.78073 | −4.05710 | −1.00000 | −0.226966 | 1.00300 | 7.66652 | ||||||||||||||||||
See next 80 embeddings (of 104 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(1\) |
\(859\) | \(1\) |
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 | 6013.2.a.c | ✓ | 104 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6013.2.a.c | ✓ | 104 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{104} + 19 T_{2}^{103} + 27 T_{2}^{102} - 1730 T_{2}^{101} - 10052 T_{2}^{100} + 63142 T_{2}^{99} + \cdots + 86336 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).