Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2303,4,Mod(1,2303)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2303, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2303.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2303 = 7^{2} \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2303.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: | \(135.881398743\) |
Analytic rank: | \(0\) |
Dimension: | \(35\) |
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 | −5.48152 | −2.93802 | 22.0470 | 19.1722 | 16.1048 | 0 | −76.9989 | −18.3680 | −105.093 | ||||||||||||||||||
1.2 | −5.33525 | 8.84301 | 20.4649 | 3.71171 | −47.1797 | 0 | −66.5034 | 51.1989 | −19.8029 | ||||||||||||||||||
1.3 | −4.92815 | −4.29114 | 16.2866 | 2.55035 | 21.1474 | 0 | −40.8377 | −8.58611 | −12.5685 | ||||||||||||||||||
1.4 | −4.60891 | −5.06004 | 13.2421 | −19.1036 | 23.3213 | 0 | −24.1603 | −1.39599 | 88.0469 | ||||||||||||||||||
1.5 | −4.14945 | 8.07223 | 9.21794 | 8.04659 | −33.4953 | 0 | −5.05376 | 38.1609 | −33.3889 | ||||||||||||||||||
1.6 | −3.90337 | 4.20489 | 7.23626 | −15.3320 | −16.4132 | 0 | 2.98114 | −9.31892 | 59.8464 | ||||||||||||||||||
1.7 | −3.81980 | −8.81241 | 6.59091 | 14.5625 | 33.6617 | 0 | 5.38246 | 50.6586 | −55.6259 | ||||||||||||||||||
1.8 | −3.77029 | 0.853932 | 6.21511 | −1.45940 | −3.21958 | 0 | 6.72955 | −26.2708 | 5.50238 | ||||||||||||||||||
1.9 | −3.31842 | −4.09865 | 3.01190 | −16.7525 | 13.6010 | 0 | 16.5526 | −10.2011 | 55.5919 | ||||||||||||||||||
1.10 | −2.62935 | 5.20657 | −1.08651 | −5.47417 | −13.6899 | 0 | 23.8916 | 0.108399 | 14.3935 | ||||||||||||||||||
1.11 | −2.23222 | 1.75933 | −3.01718 | 8.98350 | −3.92721 | 0 | 24.5928 | −23.9048 | −20.0532 | ||||||||||||||||||
1.12 | −2.07002 | 9.29307 | −3.71503 | 12.3025 | −19.2368 | 0 | 24.2503 | 59.3612 | −25.4663 | ||||||||||||||||||
1.13 | −1.67304 | −8.29020 | −5.20094 | 3.53349 | 13.8698 | 0 | 22.0857 | 41.7274 | −5.91167 | ||||||||||||||||||
1.14 | −1.17274 | 3.53925 | −6.62469 | −19.8836 | −4.15061 | 0 | 17.1509 | −14.4737 | 23.3182 | ||||||||||||||||||
1.15 | −0.936762 | 3.03943 | −7.12248 | 14.9851 | −2.84722 | 0 | 14.1662 | −17.7619 | −14.0375 | ||||||||||||||||||
1.16 | −0.479509 | −5.23227 | −7.77007 | 17.2964 | 2.50892 | 0 | 7.56190 | 0.376695 | −8.29377 | ||||||||||||||||||
1.17 | −0.230045 | −5.72475 | −7.94708 | −2.70751 | 1.31695 | 0 | 3.66854 | 5.77272 | 0.622850 | ||||||||||||||||||
1.18 | 0.482933 | 0.143091 | −7.76678 | −10.0508 | 0.0691036 | 0 | −7.61430 | −26.9795 | −4.85389 | ||||||||||||||||||
1.19 | 0.529428 | −9.49502 | −7.71971 | −20.1615 | −5.02693 | 0 | −8.32246 | 63.1555 | −10.6741 | ||||||||||||||||||
1.20 | 1.11403 | 0.352262 | −6.75894 | 4.08963 | 0.392430 | 0 | −16.4419 | −26.8759 | 4.55597 | ||||||||||||||||||
See all 35 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(-1\) |
\(47\) | \(-1\) |
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 | 2303.4.a.h | yes | 35 |
7.b | odd | 2 | 1 | 2303.4.a.g | ✓ | 35 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2303.4.a.g | ✓ | 35 | 7.b | odd | 2 | 1 | |
2303.4.a.h | yes | 35 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2303))\):
\( T_{2}^{35} - 5 T_{2}^{34} - 197 T_{2}^{33} + 987 T_{2}^{32} + 17436 T_{2}^{31} - 87638 T_{2}^{30} + \cdots + 18089615294464 \) |
\( T_{3}^{35} - 12 T_{3}^{34} - 552 T_{3}^{33} + 6760 T_{3}^{32} + 136694 T_{3}^{31} + \cdots - 63\!\cdots\!56 \) |