Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1603,2,Mod(1,1603)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1603, 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("1603.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1603 = 7 \cdot 229 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1603.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: | \(12.8000194440\) |
Analytic rank: | \(1\) |
Dimension: | \(23\) |
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 | −2.76760 | −2.82712 | 5.65960 | −2.67108 | 7.82434 | −1.00000 | −10.1283 | 4.99263 | 7.39249 | ||||||||||||||||||
1.2 | −2.65943 | −0.121745 | 5.07259 | 2.20734 | 0.323773 | −1.00000 | −8.17134 | −2.98518 | −5.87026 | ||||||||||||||||||
1.3 | −2.46810 | 0.378895 | 4.09151 | 3.19560 | −0.935150 | −1.00000 | −5.16206 | −2.85644 | −7.88706 | ||||||||||||||||||
1.4 | −2.21674 | 2.74556 | 2.91394 | −2.71914 | −6.08619 | −1.00000 | −2.02597 | 4.53810 | 6.02762 | ||||||||||||||||||
1.5 | −2.11537 | −2.44257 | 2.47478 | 0.0175122 | 5.16694 | −1.00000 | −1.00434 | 2.96616 | −0.0370447 | ||||||||||||||||||
1.6 | −1.76477 | −2.21290 | 1.11441 | 3.10488 | 3.90525 | −1.00000 | 1.56287 | 1.89691 | −5.47940 | ||||||||||||||||||
1.7 | −1.62221 | −1.03642 | 0.631569 | −1.37974 | 1.68129 | −1.00000 | 2.21988 | −1.92584 | 2.23823 | ||||||||||||||||||
1.8 | −1.28438 | 2.12869 | −0.350357 | 0.183940 | −2.73406 | −1.00000 | 3.01876 | 1.53132 | −0.236249 | ||||||||||||||||||
1.9 | −1.17790 | 2.06555 | −0.612555 | −0.860179 | −2.43301 | −1.00000 | 3.07732 | 1.26651 | 1.01320 | ||||||||||||||||||
1.10 | −1.08887 | −2.25071 | −0.814361 | −3.88554 | 2.45073 | −1.00000 | 3.06447 | 2.06569 | 4.23085 | ||||||||||||||||||
1.11 | −0.804527 | 2.68478 | −1.35274 | 1.43476 | −2.15998 | −1.00000 | 2.69737 | 4.20805 | −1.15430 | ||||||||||||||||||
1.12 | −0.505895 | −1.90057 | −1.74407 | −1.46969 | 0.961490 | −1.00000 | 1.89411 | 0.612177 | 0.743509 | ||||||||||||||||||
1.13 | −0.437554 | 0.0510985 | −1.80855 | 0.814986 | −0.0223584 | −1.00000 | 1.66645 | −2.99739 | −0.356601 | ||||||||||||||||||
1.14 | −0.105554 | −1.28811 | −1.98886 | −3.68177 | 0.135965 | −1.00000 | 0.421041 | −1.34078 | 0.388627 | ||||||||||||||||||
1.15 | 0.268576 | 2.24747 | −1.92787 | −1.58319 | 0.603618 | −1.00000 | −1.05493 | 2.05114 | −0.425207 | ||||||||||||||||||
1.16 | 0.971214 | −3.28281 | −1.05674 | −0.336425 | −3.18831 | −1.00000 | −2.96875 | 7.77685 | −0.326741 | ||||||||||||||||||
1.17 | 1.20935 | −1.52471 | −0.537465 | 1.63811 | −1.84392 | −1.00000 | −3.06869 | −0.675248 | 1.98105 | ||||||||||||||||||
1.18 | 1.39614 | 1.38130 | −0.0507911 | −1.76142 | 1.92849 | −1.00000 | −2.86319 | −1.09201 | −2.45919 | ||||||||||||||||||
1.19 | 1.41538 | 2.79946 | 0.00330323 | −3.84231 | 3.96231 | −1.00000 | −2.82609 | 4.83700 | −5.43834 | ||||||||||||||||||
1.20 | 1.97463 | 0.0675157 | 1.89915 | −0.410631 | 0.133318 | −1.00000 | −0.199140 | −2.99544 | −0.810842 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(1\) |
\(229\) | \(1\) |
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 | 1603.2.a.b | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1603.2.a.b | ✓ | 23 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + 7 T_{2}^{22} - 9 T_{2}^{21} - 164 T_{2}^{20} - 146 T_{2}^{19} + 1533 T_{2}^{18} + \cdots - 107 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1603))\).