Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4029,2,Mod(1,4029)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4029, 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("4029.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4029 = 3 \cdot 17 \cdot 79 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4029.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: | \(32.1717269744\) |
Analytic rank: | \(0\) |
Dimension: | \(31\) |
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 | −2.61491 | 1.00000 | 4.83775 | −2.62328 | −2.61491 | −4.80522 | −7.42046 | 1.00000 | 6.85964 | ||||||||||||||||||
1.2 | −2.56879 | 1.00000 | 4.59868 | 3.14567 | −2.56879 | −0.137684 | −6.67546 | 1.00000 | −8.08058 | ||||||||||||||||||
1.3 | −2.41826 | 1.00000 | 3.84800 | 1.43707 | −2.41826 | 3.74160 | −4.46895 | 1.00000 | −3.47522 | ||||||||||||||||||
1.4 | −2.28064 | 1.00000 | 3.20133 | 2.67486 | −2.28064 | −3.60745 | −2.73981 | 1.00000 | −6.10039 | ||||||||||||||||||
1.5 | −2.22001 | 1.00000 | 2.92842 | −1.63428 | −2.22001 | −1.90265 | −2.06110 | 1.00000 | 3.62811 | ||||||||||||||||||
1.6 | −1.88694 | 1.00000 | 1.56054 | −0.846288 | −1.88694 | 3.01609 | 0.829236 | 1.00000 | 1.59689 | ||||||||||||||||||
1.7 | −1.73268 | 1.00000 | 1.00217 | 2.89501 | −1.73268 | 3.30216 | 1.72891 | 1.00000 | −5.01613 | ||||||||||||||||||
1.8 | −1.49453 | 1.00000 | 0.233609 | −1.69557 | −1.49453 | 0.268778 | 2.63992 | 1.00000 | 2.53407 | ||||||||||||||||||
1.9 | −1.27443 | 1.00000 | −0.375837 | −3.50336 | −1.27443 | −0.192219 | 3.02783 | 1.00000 | 4.46478 | ||||||||||||||||||
1.10 | −1.26690 | 1.00000 | −0.394972 | 3.60603 | −1.26690 | −2.46109 | 3.03418 | 1.00000 | −4.56847 | ||||||||||||||||||
1.11 | −0.719826 | 1.00000 | −1.48185 | 1.82945 | −0.719826 | 0.414755 | 2.50633 | 1.00000 | −1.31688 | ||||||||||||||||||
1.12 | −0.639237 | 1.00000 | −1.59138 | −1.53158 | −0.639237 | −2.74819 | 2.29574 | 1.00000 | 0.979040 | ||||||||||||||||||
1.13 | −0.626356 | 1.00000 | −1.60768 | −1.73582 | −0.626356 | −3.52892 | 2.25969 | 1.00000 | 1.08724 | ||||||||||||||||||
1.14 | −0.103363 | 1.00000 | −1.98932 | −2.25663 | −0.103363 | 4.11147 | 0.412347 | 1.00000 | 0.233252 | ||||||||||||||||||
1.15 | 0.0488355 | 1.00000 | −1.99762 | −0.188126 | 0.0488355 | 3.46836 | −0.195225 | 1.00000 | −0.00918720 | ||||||||||||||||||
1.16 | 0.0509119 | 1.00000 | −1.99741 | 3.95784 | 0.0509119 | 1.89285 | −0.203515 | 1.00000 | 0.201501 | ||||||||||||||||||
1.17 | 0.308270 | 1.00000 | −1.90497 | 2.48291 | 0.308270 | 3.17570 | −1.20379 | 1.00000 | 0.765408 | ||||||||||||||||||
1.18 | 0.617959 | 1.00000 | −1.61813 | 3.16624 | 0.617959 | −0.196167 | −2.23585 | 1.00000 | 1.95660 | ||||||||||||||||||
1.19 | 0.680303 | 1.00000 | −1.53719 | −1.16933 | 0.680303 | −3.22223 | −2.40636 | 1.00000 | −0.795496 | ||||||||||||||||||
1.20 | 0.959467 | 1.00000 | −1.07942 | −3.93035 | 0.959467 | −0.0159719 | −2.95460 | 1.00000 | −3.77104 | ||||||||||||||||||
See all 31 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(17\) | \(-1\) |
\(79\) | \(-1\) |
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 | 4029.2.a.k | ✓ | 31 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4029.2.a.k | ✓ | 31 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):
\( T_{2}^{31} - 4 T_{2}^{30} - 40 T_{2}^{29} + 172 T_{2}^{28} + 693 T_{2}^{27} - 3280 T_{2}^{26} - 6767 T_{2}^{25} + 36580 T_{2}^{24} + 40284 T_{2}^{23} - 264955 T_{2}^{22} - 144079 T_{2}^{21} + 1309226 T_{2}^{20} + 250363 T_{2}^{19} + \cdots - 16 \) |
\( T_{5}^{31} - 11 T_{5}^{30} - 33 T_{5}^{29} + 767 T_{5}^{28} - 751 T_{5}^{27} - 21949 T_{5}^{26} + 57042 T_{5}^{25} + 331924 T_{5}^{24} - 1299128 T_{5}^{23} - 2793266 T_{5}^{22} + 16257159 T_{5}^{21} + 11412047 T_{5}^{20} + \cdots - 245126 \) |