Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2057,4,Mod(1,2057)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2057.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2057 = 11^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2057.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: | \(121.366928882\) |
Analytic rank: | \(1\) |
Dimension: | \(44\) |
Twist minimal: | no (minimal twist has level 187) |
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 | −5.35848 | −7.08785 | 20.7133 | 2.40559 | 37.9801 | 1.79015 | −68.1240 | 23.2377 | −12.8903 | ||||||||||||||||||
1.2 | −5.17591 | 1.82863 | 18.7901 | −2.37867 | −9.46484 | 16.1706 | −55.8485 | −23.6561 | 12.3118 | ||||||||||||||||||
1.3 | −5.00754 | −0.478835 | 17.0755 | −10.2728 | 2.39778 | −33.8426 | −45.4457 | −26.7707 | 51.4414 | ||||||||||||||||||
1.4 | −4.74167 | −8.40276 | 14.4835 | −4.48116 | 39.8432 | 6.84168 | −30.7424 | 43.6065 | 21.2482 | ||||||||||||||||||
1.5 | −4.59510 | −5.20799 | 13.1150 | 19.0975 | 23.9312 | 19.5522 | −23.5038 | 0.123151 | −87.7548 | ||||||||||||||||||
1.6 | −4.48873 | 7.40887 | 12.1487 | −14.9848 | −33.2565 | 10.8452 | −18.6226 | 27.8914 | 67.2626 | ||||||||||||||||||
1.7 | −4.17775 | 8.39425 | 9.45359 | −8.30638 | −35.0691 | 10.0314 | −6.07275 | 43.4635 | 34.7020 | ||||||||||||||||||
1.8 | −4.15815 | −2.77821 | 9.29018 | 17.2462 | 11.5522 | −3.81447 | −5.36476 | −19.2815 | −71.7123 | ||||||||||||||||||
1.9 | −3.96050 | 4.65525 | 7.68556 | 8.14307 | −18.4371 | 9.05024 | 1.24534 | −5.32862 | −32.2506 | ||||||||||||||||||
1.10 | −3.71299 | 1.08053 | 5.78632 | −1.92298 | −4.01199 | −27.0782 | 8.21937 | −25.8325 | 7.14001 | ||||||||||||||||||
1.11 | −2.84199 | −5.34906 | 0.0768922 | −17.1347 | 15.2020 | 7.83596 | 22.5174 | 1.61241 | 48.6966 | ||||||||||||||||||
1.12 | −2.61578 | −10.1873 | −1.15767 | −1.97479 | 26.6478 | −29.0117 | 23.9545 | 76.7814 | 5.16562 | ||||||||||||||||||
1.13 | −2.47148 | 4.16424 | −1.89179 | 6.24963 | −10.2918 | −8.57958 | 24.4474 | −9.65907 | −15.4458 | ||||||||||||||||||
1.14 | −2.27024 | −5.58818 | −2.84602 | 4.32678 | 12.6865 | −4.76061 | 24.6230 | 4.22772 | −9.82282 | ||||||||||||||||||
1.15 | −1.97752 | −8.52550 | −4.08942 | −7.64484 | 16.8593 | 34.0014 | 23.9071 | 45.6842 | 15.1178 | ||||||||||||||||||
1.16 | −1.92546 | 7.26305 | −4.29259 | 8.62699 | −13.9847 | −8.70813 | 23.6689 | 25.7519 | −16.6110 | ||||||||||||||||||
1.17 | −1.86890 | 3.47806 | −4.50721 | 16.9463 | −6.50016 | 27.1967 | 23.3747 | −14.9031 | −31.6709 | ||||||||||||||||||
1.18 | −1.28604 | 5.35520 | −6.34610 | −17.4555 | −6.88701 | −12.0664 | 18.4497 | 1.67815 | 22.4485 | ||||||||||||||||||
1.19 | −0.556409 | 0.761089 | −7.69041 | −17.6266 | −0.423477 | 25.0433 | 8.73028 | −26.4207 | 9.80761 | ||||||||||||||||||
1.20 | −0.515458 | −3.56912 | −7.73430 | 2.09077 | 1.83973 | 17.6614 | 8.11038 | −14.2614 | −1.07771 | ||||||||||||||||||
See all 44 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(1\) |
\(17\) | \(-1\) |
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 | 2057.4.a.t | 44 | |
11.b | odd | 2 | 1 | 2057.4.a.s | 44 | ||
11.c | even | 5 | 2 | 187.4.g.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
187.4.g.a | ✓ | 88 | 11.c | even | 5 | 2 | |
2057.4.a.s | 44 | 11.b | odd | 2 | 1 | ||
2057.4.a.t | 44 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):
\( T_{2}^{44} - T_{2}^{43} - 245 T_{2}^{42} + 261 T_{2}^{41} + 27677 T_{2}^{40} - 31257 T_{2}^{39} - 1912633 T_{2}^{38} + 2280678 T_{2}^{37} + 90457195 T_{2}^{36} - 113516565 T_{2}^{35} + \cdots - 49\!\cdots\!00 \) |
\( T_{3}^{44} + 28 T_{3}^{43} - 380 T_{3}^{42} - 17274 T_{3}^{41} + 16330 T_{3}^{40} + 4770576 T_{3}^{39} + 17790024 T_{3}^{38} - 776750576 T_{3}^{37} - 5171824726 T_{3}^{36} + \cdots + 87\!\cdots\!09 \) |
\( T_{5}^{44} + 24 T_{5}^{43} - 2691 T_{5}^{42} - 68106 T_{5}^{41} + 3224114 T_{5}^{40} + 87105226 T_{5}^{39} - 2265590643 T_{5}^{38} - 66465918588 T_{5}^{37} + 1034498025966 T_{5}^{36} + \cdots - 15\!\cdots\!20 \) |