Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4013,2,Mod(1,4013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4013, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4013.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4013 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4013.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: | \(32.0439663311\) |
Analytic rank: | \(1\) |
Dimension: | \(157\) |
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 | −2.81982 | 1.03159 | 5.95140 | 1.63220 | −2.90891 | 3.00127 | −11.1422 | −1.93581 | −4.60252 | ||||||||||||||||||
1.2 | −2.78653 | −0.812844 | 5.76476 | −1.14246 | 2.26502 | −4.07959 | −10.4906 | −2.33929 | 3.18350 | ||||||||||||||||||
1.3 | −2.78090 | −3.27207 | 5.73341 | 3.80445 | 9.09929 | 2.37189 | −10.3822 | 7.70642 | −10.5798 | ||||||||||||||||||
1.4 | −2.68953 | 2.61392 | 5.23356 | 1.08630 | −7.03022 | −3.47806 | −8.69675 | 3.83259 | −2.92162 | ||||||||||||||||||
1.5 | −2.65659 | 1.81764 | 5.05746 | 3.63888 | −4.82872 | −3.82226 | −8.12240 | 0.303819 | −9.66700 | ||||||||||||||||||
1.6 | −2.65066 | −3.08601 | 5.02601 | −3.86295 | 8.17996 | −0.380070 | −8.02093 | 6.52343 | 10.2394 | ||||||||||||||||||
1.7 | −2.61349 | −2.30050 | 4.83033 | −0.685238 | 6.01232 | −4.02398 | −7.39704 | 2.29228 | 1.79086 | ||||||||||||||||||
1.8 | −2.60727 | 1.83175 | 4.79788 | −1.15472 | −4.77588 | −1.01890 | −7.29484 | 0.355310 | 3.01066 | ||||||||||||||||||
1.9 | −2.54071 | 0.214630 | 4.45522 | 0.391559 | −0.545314 | 2.42547 | −6.23801 | −2.95393 | −0.994840 | ||||||||||||||||||
1.10 | −2.53912 | −1.94061 | 4.44715 | 3.91571 | 4.92744 | −2.75166 | −6.21361 | 0.765960 | −9.94248 | ||||||||||||||||||
1.11 | −2.52958 | −1.28339 | 4.39880 | −1.06771 | 3.24644 | 0.845401 | −6.06796 | −1.35291 | 2.70087 | ||||||||||||||||||
1.12 | −2.51870 | −2.55292 | 4.34385 | 0.976255 | 6.43005 | 1.70978 | −5.90347 | 3.51742 | −2.45890 | ||||||||||||||||||
1.13 | −2.51500 | −3.29902 | 4.32520 | −0.150822 | 8.29703 | −3.52495 | −5.84787 | 7.88356 | 0.379318 | ||||||||||||||||||
1.14 | −2.47196 | 3.17720 | 4.11056 | 1.49907 | −7.85391 | 0.578951 | −5.21722 | 7.09463 | −3.70563 | ||||||||||||||||||
1.15 | −2.42617 | 1.73131 | 3.88629 | −0.987697 | −4.20045 | 1.01261 | −4.57644 | −0.00256059 | 2.39632 | ||||||||||||||||||
1.16 | −2.39096 | −0.986021 | 3.71671 | −2.27233 | 2.35754 | −0.314319 | −4.10458 | −2.02776 | 5.43305 | ||||||||||||||||||
1.17 | −2.35914 | −1.36085 | 3.56554 | 2.69250 | 3.21043 | 1.19672 | −3.69333 | −1.14810 | −6.35199 | ||||||||||||||||||
1.18 | −2.34257 | −1.48585 | 3.48762 | 0.440014 | 3.48071 | 1.63946 | −3.48485 | −0.792243 | −1.03076 | ||||||||||||||||||
1.19 | −2.28387 | 2.83481 | 3.21604 | −3.24199 | −6.47432 | 0.379335 | −2.77728 | 5.03614 | 7.40427 | ||||||||||||||||||
1.20 | −2.28352 | −2.81571 | 3.21448 | 2.96925 | 6.42974 | −4.34900 | −2.77330 | 4.92822 | −6.78035 | ||||||||||||||||||
See next 80 embeddings (of 157 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(4013\) | \(1\) |
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 | 4013.2.a.b | ✓ | 157 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4013.2.a.b | ✓ | 157 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{157} + 15 T_{2}^{156} - 113 T_{2}^{155} - 2786 T_{2}^{154} + 2130 T_{2}^{153} + 248978 T_{2}^{152} + \cdots + 608498 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4013))\).