Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6034,2,Mod(1,6034)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6034, 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("6034.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6034 = 2 \cdot 7 \cdot 431 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6034.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: | \(48.1817325796\) |
Analytic rank: | \(1\) |
Dimension: | \(25\) |
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 | −1.00000 | −3.32592 | 1.00000 | 4.15520 | 3.32592 | −1.00000 | −1.00000 | 8.06177 | −4.15520 | ||||||||||||||||||
1.2 | −1.00000 | −3.01499 | 1.00000 | 0.160841 | 3.01499 | −1.00000 | −1.00000 | 6.09019 | −0.160841 | ||||||||||||||||||
1.3 | −1.00000 | −2.93664 | 1.00000 | 1.14962 | 2.93664 | −1.00000 | −1.00000 | 5.62386 | −1.14962 | ||||||||||||||||||
1.4 | −1.00000 | −2.89955 | 1.00000 | −3.35107 | 2.89955 | −1.00000 | −1.00000 | 5.40739 | 3.35107 | ||||||||||||||||||
1.5 | −1.00000 | −2.44573 | 1.00000 | 1.36313 | 2.44573 | −1.00000 | −1.00000 | 2.98159 | −1.36313 | ||||||||||||||||||
1.6 | −1.00000 | −2.25211 | 1.00000 | 2.47943 | 2.25211 | −1.00000 | −1.00000 | 2.07199 | −2.47943 | ||||||||||||||||||
1.7 | −1.00000 | −1.45476 | 1.00000 | 0.421306 | 1.45476 | −1.00000 | −1.00000 | −0.883670 | −0.421306 | ||||||||||||||||||
1.8 | −1.00000 | −1.28274 | 1.00000 | −1.76478 | 1.28274 | −1.00000 | −1.00000 | −1.35459 | 1.76478 | ||||||||||||||||||
1.9 | −1.00000 | −1.09451 | 1.00000 | −2.81071 | 1.09451 | −1.00000 | −1.00000 | −1.80204 | 2.81071 | ||||||||||||||||||
1.10 | −1.00000 | −0.804173 | 1.00000 | −3.18296 | 0.804173 | −1.00000 | −1.00000 | −2.35331 | 3.18296 | ||||||||||||||||||
1.11 | −1.00000 | −0.655704 | 1.00000 | −2.51498 | 0.655704 | −1.00000 | −1.00000 | −2.57005 | 2.51498 | ||||||||||||||||||
1.12 | −1.00000 | −0.642803 | 1.00000 | −0.120045 | 0.642803 | −1.00000 | −1.00000 | −2.58680 | 0.120045 | ||||||||||||||||||
1.13 | −1.00000 | −0.286451 | 1.00000 | 3.05880 | 0.286451 | −1.00000 | −1.00000 | −2.91795 | −3.05880 | ||||||||||||||||||
1.14 | −1.00000 | −0.221832 | 1.00000 | −0.411075 | 0.221832 | −1.00000 | −1.00000 | −2.95079 | 0.411075 | ||||||||||||||||||
1.15 | −1.00000 | 0.0257691 | 1.00000 | 3.06772 | −0.0257691 | −1.00000 | −1.00000 | −2.99934 | −3.06772 | ||||||||||||||||||
1.16 | −1.00000 | 0.684006 | 1.00000 | 2.60861 | −0.684006 | −1.00000 | −1.00000 | −2.53214 | −2.60861 | ||||||||||||||||||
1.17 | −1.00000 | 0.809571 | 1.00000 | 3.41286 | −0.809571 | −1.00000 | −1.00000 | −2.34460 | −3.41286 | ||||||||||||||||||
1.18 | −1.00000 | 0.817521 | 1.00000 | −4.27702 | −0.817521 | −1.00000 | −1.00000 | −2.33166 | 4.27702 | ||||||||||||||||||
1.19 | −1.00000 | 1.42126 | 1.00000 | −0.924054 | −1.42126 | −1.00000 | −1.00000 | −0.980018 | 0.924054 | ||||||||||||||||||
1.20 | −1.00000 | 1.89436 | 1.00000 | 0.229039 | −1.89436 | −1.00000 | −1.00000 | 0.588591 | −0.229039 | ||||||||||||||||||
See all 25 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(7\) | \(1\) |
\(431\) | \(1\) |
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 | 6034.2.a.o | ✓ | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6034.2.a.o | ✓ | 25 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):
\( T_{3}^{25} + 4 T_{3}^{24} - 42 T_{3}^{23} - 179 T_{3}^{22} + 718 T_{3}^{21} + 3378 T_{3}^{20} + \cdots - 96 \) |
\( T_{5}^{25} - 74 T_{5}^{23} + 8 T_{5}^{22} + 2323 T_{5}^{21} - 463 T_{5}^{20} - 40445 T_{5}^{19} + \cdots - 1792 \) |
\( T_{11}^{25} + 13 T_{11}^{24} - 54 T_{11}^{23} - 1317 T_{11}^{22} - 1060 T_{11}^{21} + 51515 T_{11}^{20} + \cdots + 314754048 \) |