Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4033,2,Mod(1,4033)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, 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("4033.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4033 = 37 \cdot 109 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4033.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: | \(32.2036671352\) |
Analytic rank: | \(0\) |
Dimension: | \(82\) |
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 | −2.75814 | 1.25704 | 5.60732 | 1.25507 | −3.46708 | 2.38515 | −9.94947 | −1.41986 | −3.46166 | ||||||||||||||||||
1.2 | −2.62838 | −1.37635 | 4.90837 | −2.91482 | 3.61757 | −1.06759 | −7.64429 | −1.10566 | 7.66124 | ||||||||||||||||||
1.3 | −2.58504 | −2.45542 | 4.68242 | 2.00515 | 6.34735 | −0.0308722 | −6.93416 | 3.02908 | −5.18339 | ||||||||||||||||||
1.4 | −2.58335 | 3.13276 | 4.67371 | 4.37342 | −8.09303 | 0.189689 | −6.90713 | 6.81421 | −11.2981 | ||||||||||||||||||
1.5 | −2.53999 | 0.468124 | 4.45153 | 3.58258 | −1.18903 | 0.512106 | −6.22686 | −2.78086 | −9.09972 | ||||||||||||||||||
1.6 | −2.52769 | 2.28955 | 4.38920 | −2.00544 | −5.78727 | −4.48140 | −6.03914 | 2.24205 | 5.06913 | ||||||||||||||||||
1.7 | −2.40882 | −0.455887 | 3.80243 | 1.02492 | 1.09815 | 3.87779 | −4.34175 | −2.79217 | −2.46884 | ||||||||||||||||||
1.8 | −2.37038 | 1.65069 | 3.61871 | −2.32287 | −3.91275 | −2.51278 | −3.83696 | −0.275237 | 5.50608 | ||||||||||||||||||
1.9 | −2.34943 | −2.83343 | 3.51983 | −1.24434 | 6.65694 | 2.59127 | −3.57074 | 5.02830 | 2.92348 | ||||||||||||||||||
1.10 | −2.33134 | −0.785356 | 3.43514 | 0.539311 | 1.83093 | −3.00025 | −3.34579 | −2.38322 | −1.25731 | ||||||||||||||||||
1.11 | −2.32285 | 2.68333 | 3.39564 | −2.18288 | −6.23298 | 1.55979 | −3.24187 | 4.20026 | 5.07050 | ||||||||||||||||||
1.12 | −2.23087 | −0.522576 | 2.97676 | 0.778391 | 1.16580 | −1.74225 | −2.17903 | −2.72691 | −1.73649 | ||||||||||||||||||
1.13 | −1.92046 | 1.69690 | 1.68818 | −0.668171 | −3.25884 | 1.48884 | 0.598834 | −0.120514 | 1.28320 | ||||||||||||||||||
1.14 | −1.75973 | 1.54621 | 1.09665 | 4.09824 | −2.72090 | −4.10916 | 1.58965 | −0.609248 | −7.21180 | ||||||||||||||||||
1.15 | −1.75361 | −2.25140 | 1.07516 | −3.95285 | 3.94809 | 2.44850 | 1.62182 | 2.06881 | 6.93177 | ||||||||||||||||||
1.16 | −1.67615 | 2.74461 | 0.809463 | 2.71622 | −4.60037 | 4.10502 | 1.99551 | 4.53291 | −4.55277 | ||||||||||||||||||
1.17 | −1.62784 | −2.32510 | 0.649851 | 2.52480 | 3.78489 | −3.47012 | 2.19782 | 2.40611 | −4.10996 | ||||||||||||||||||
1.18 | −1.58741 | −1.76709 | 0.519871 | 4.42213 | 2.80509 | 5.11364 | 2.34957 | 0.122598 | −7.01973 | ||||||||||||||||||
1.19 | −1.58089 | 0.557711 | 0.499209 | 2.89052 | −0.881678 | 1.64472 | 2.37258 | −2.68896 | −4.56959 | ||||||||||||||||||
1.20 | −1.52363 | −2.34833 | 0.321441 | 0.104108 | 3.57798 | −3.33397 | 2.55750 | 2.51466 | −0.158621 | ||||||||||||||||||
See all 82 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(37\) | \(1\) |
\(109\) | \(-1\) |
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 | 4033.2.a.e | ✓ | 82 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4033.2.a.e | ✓ | 82 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{82} - 10 T_{2}^{81} - 76 T_{2}^{80} + 1069 T_{2}^{79} + 1976 T_{2}^{78} - 54246 T_{2}^{77} + \cdots + 2256 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).