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: | \(32\) |
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 | −2.77670 | −1.00000 | 5.71004 | −0.322981 | 2.77670 | −2.70395 | −10.3017 | 1.00000 | 0.896820 | ||||||||||||||||||
1.2 | −2.65510 | −1.00000 | 5.04957 | −3.36488 | 2.65510 | 2.92058 | −8.09693 | 1.00000 | 8.93412 | ||||||||||||||||||
1.3 | −2.57711 | −1.00000 | 4.64151 | 2.80201 | 2.57711 | −2.75409 | −6.80748 | 1.00000 | −7.22109 | ||||||||||||||||||
1.4 | −2.50856 | −1.00000 | 4.29286 | −2.69434 | 2.50856 | 3.74400 | −5.75176 | 1.00000 | 6.75889 | ||||||||||||||||||
1.5 | −2.27866 | −1.00000 | 3.19228 | −3.13196 | 2.27866 | −0.0802156 | −2.71679 | 1.00000 | 7.13667 | ||||||||||||||||||
1.6 | −2.16757 | −1.00000 | 2.69838 | 2.47527 | 2.16757 | 5.12027 | −1.51379 | 1.00000 | −5.36532 | ||||||||||||||||||
1.7 | −2.03180 | −1.00000 | 2.12821 | 1.66964 | 2.03180 | −1.52463 | −0.260503 | 1.00000 | −3.39237 | ||||||||||||||||||
1.8 | −1.67650 | −1.00000 | 0.810659 | −3.83953 | 1.67650 | −4.72769 | 1.99393 | 1.00000 | 6.43697 | ||||||||||||||||||
1.9 | −1.61915 | −1.00000 | 0.621643 | 1.62143 | 1.61915 | 3.15810 | 2.23177 | 1.00000 | −2.62534 | ||||||||||||||||||
1.10 | −1.60231 | −1.00000 | 0.567397 | −0.489899 | 1.60231 | −0.574101 | 2.29547 | 1.00000 | 0.784971 | ||||||||||||||||||
1.11 | −1.34067 | −1.00000 | −0.202611 | 1.16053 | 1.34067 | −4.05917 | 2.95297 | 1.00000 | −1.55588 | ||||||||||||||||||
1.12 | −1.17408 | −1.00000 | −0.621542 | 3.86856 | 1.17408 | 3.06255 | 3.07789 | 1.00000 | −4.54199 | ||||||||||||||||||
1.13 | −0.857766 | −1.00000 | −1.26424 | −4.41756 | 0.857766 | 0.00991750 | 2.79995 | 1.00000 | 3.78923 | ||||||||||||||||||
1.14 | −0.510062 | −1.00000 | −1.73984 | −1.60726 | 0.510062 | −0.772958 | 1.90755 | 1.00000 | 0.819801 | ||||||||||||||||||
1.15 | −0.249798 | −1.00000 | −1.93760 | 0.601573 | 0.249798 | −2.69175 | 0.983604 | 1.00000 | −0.150272 | ||||||||||||||||||
1.16 | −0.0772262 | −1.00000 | −1.99404 | 3.97384 | 0.0772262 | 0.969209 | 0.308444 | 1.00000 | −0.306884 | ||||||||||||||||||
1.17 | −0.0717307 | −1.00000 | −1.99485 | −1.92367 | 0.0717307 | 3.89464 | 0.286554 | 1.00000 | 0.137987 | ||||||||||||||||||
1.18 | 0.403182 | −1.00000 | −1.83744 | 2.23661 | −0.403182 | −0.778796 | −1.54719 | 1.00000 | 0.901761 | ||||||||||||||||||
1.19 | 0.437930 | −1.00000 | −1.80822 | −1.50297 | −0.437930 | −4.33772 | −1.66773 | 1.00000 | −0.658196 | ||||||||||||||||||
1.20 | 0.937640 | −1.00000 | −1.12083 | 0.347083 | −0.937640 | 4.09662 | −2.92622 | 1.00000 | 0.325439 | ||||||||||||||||||
See all 32 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.l | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4029.2.a.l | ✓ | 32 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):
\( T_{2}^{32} + T_{2}^{31} - 52 T_{2}^{30} - 50 T_{2}^{29} + 1215 T_{2}^{28} + 1115 T_{2}^{27} - 16869 T_{2}^{26} + \cdots - 544 \) |
\( T_{5}^{32} + T_{5}^{31} - 111 T_{5}^{30} - 107 T_{5}^{29} + 5499 T_{5}^{28} + 5015 T_{5}^{27} + \cdots - 143498044 \) |