Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8043,2,Mod(1,8043)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8043, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8043.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8043 = 3 \cdot 7 \cdot 383 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8043.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: | \(64.2236783457\) |
Analytic rank: | \(1\) |
Dimension: | \(46\) |
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 | −2.74370 | −1.00000 | 5.52791 | −1.87586 | 2.74370 | −1.00000 | −9.67954 | 1.00000 | 5.14679 | ||||||||||||||||||
1.2 | −2.68548 | −1.00000 | 5.21178 | 2.43353 | 2.68548 | −1.00000 | −8.62517 | 1.00000 | −6.53519 | ||||||||||||||||||
1.3 | −2.54493 | −1.00000 | 4.47665 | 0.653154 | 2.54493 | −1.00000 | −6.30289 | 1.00000 | −1.66223 | ||||||||||||||||||
1.4 | −2.41337 | −1.00000 | 3.82437 | 1.40118 | 2.41337 | −1.00000 | −4.40289 | 1.00000 | −3.38157 | ||||||||||||||||||
1.5 | −2.37441 | −1.00000 | 3.63784 | −1.15289 | 2.37441 | −1.00000 | −3.88892 | 1.00000 | 2.73743 | ||||||||||||||||||
1.6 | −2.17399 | −1.00000 | 2.72624 | −2.42046 | 2.17399 | −1.00000 | −1.57885 | 1.00000 | 5.26206 | ||||||||||||||||||
1.7 | −2.13482 | −1.00000 | 2.55746 | −3.00499 | 2.13482 | −1.00000 | −1.19007 | 1.00000 | 6.41512 | ||||||||||||||||||
1.8 | −1.90871 | −1.00000 | 1.64317 | 2.98743 | 1.90871 | −1.00000 | 0.681087 | 1.00000 | −5.70214 | ||||||||||||||||||
1.9 | −1.80468 | −1.00000 | 1.25687 | −3.36288 | 1.80468 | −1.00000 | 1.34111 | 1.00000 | 6.06893 | ||||||||||||||||||
1.10 | −1.77170 | −1.00000 | 1.13894 | 3.36587 | 1.77170 | −1.00000 | 1.52555 | 1.00000 | −5.96332 | ||||||||||||||||||
1.11 | −1.68230 | −1.00000 | 0.830121 | −0.636152 | 1.68230 | −1.00000 | 1.96808 | 1.00000 | 1.07020 | ||||||||||||||||||
1.12 | −1.59029 | −1.00000 | 0.529036 | 3.24933 | 1.59029 | −1.00000 | 2.33927 | 1.00000 | −5.16740 | ||||||||||||||||||
1.13 | −1.51659 | −1.00000 | 0.300048 | −2.76194 | 1.51659 | −1.00000 | 2.57813 | 1.00000 | 4.18873 | ||||||||||||||||||
1.14 | −1.34694 | −1.00000 | −0.185749 | −2.41165 | 1.34694 | −1.00000 | 2.94408 | 1.00000 | 3.24834 | ||||||||||||||||||
1.15 | −0.944841 | −1.00000 | −1.10728 | 3.73621 | 0.944841 | −1.00000 | 2.93588 | 1.00000 | −3.53012 | ||||||||||||||||||
1.16 | −0.861946 | −1.00000 | −1.25705 | 1.66275 | 0.861946 | −1.00000 | 2.80740 | 1.00000 | −1.43320 | ||||||||||||||||||
1.17 | −0.696464 | −1.00000 | −1.51494 | 1.11267 | 0.696464 | −1.00000 | 2.44803 | 1.00000 | −0.774934 | ||||||||||||||||||
1.18 | −0.661222 | −1.00000 | −1.56279 | −1.64966 | 0.661222 | −1.00000 | 2.35579 | 1.00000 | 1.09079 | ||||||||||||||||||
1.19 | −0.522616 | −1.00000 | −1.72687 | −3.91798 | 0.522616 | −1.00000 | 1.94772 | 1.00000 | 2.04760 | ||||||||||||||||||
1.20 | −0.490976 | −1.00000 | −1.75894 | 0.513476 | 0.490976 | −1.00000 | 1.84555 | 1.00000 | −0.252105 | ||||||||||||||||||
See all 46 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(1\) |
\(383\) | \(1\) |
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 | 8043.2.a.r | ✓ | 46 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8043.2.a.r | ✓ | 46 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):
\( T_{2}^{46} - 3 T_{2}^{45} - 64 T_{2}^{44} + 195 T_{2}^{43} + 1889 T_{2}^{42} - 5860 T_{2}^{41} + \cdots - 12176 \) |
\( T_{5}^{46} + 9 T_{5}^{45} - 92 T_{5}^{44} - 1024 T_{5}^{43} + 3435 T_{5}^{42} + 53382 T_{5}^{41} + \cdots - 1464900474868 \) |
\( T_{11}^{46} - 31 T_{11}^{45} + 223 T_{11}^{44} + 2679 T_{11}^{43} - 44538 T_{11}^{42} + \cdots + 22295759075792 \) |