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: | \(0\) |
Dimension: | \(52\) |
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 | −5.58278 | 1.54543 | 23.1674 | 18.5030 | −8.62782 | 4.56305 | −84.6764 | −24.6116 | −103.298 | ||||||||||||||||||
1.2 | −5.53450 | −6.28501 | 22.6307 | −12.2792 | 34.7844 | 31.7914 | −80.9739 | 12.5013 | 67.9590 | ||||||||||||||||||
1.3 | −5.49302 | −3.83066 | 22.1733 | 17.2968 | 21.0419 | −17.8053 | −77.8542 | −12.3261 | −95.0116 | ||||||||||||||||||
1.4 | −5.36660 | 9.52348 | 20.8004 | −15.8955 | −51.1087 | −21.2549 | −68.6944 | 63.6966 | 85.3050 | ||||||||||||||||||
1.5 | −4.78655 | 5.72685 | 14.9110 | 0.431714 | −27.4118 | −26.0630 | −33.0800 | 5.79678 | −2.06642 | ||||||||||||||||||
1.6 | −4.66855 | 9.98201 | 13.7954 | 20.2462 | −46.6016 | 10.9859 | −27.0561 | 72.6405 | −94.5205 | ||||||||||||||||||
1.7 | −4.59808 | −2.80460 | 13.1424 | 1.26573 | 12.8958 | −24.2666 | −23.6449 | −19.1342 | −5.81993 | ||||||||||||||||||
1.8 | −4.56499 | −9.42960 | 12.8392 | 6.53233 | 43.0460 | 20.4259 | −22.0907 | 61.9173 | −29.8200 | ||||||||||||||||||
1.9 | −4.20383 | 8.49638 | 9.67219 | 12.9101 | −35.7173 | −6.90657 | −7.02962 | 45.1885 | −54.2720 | ||||||||||||||||||
1.10 | −4.06732 | 0.0694426 | 8.54311 | −17.2565 | −0.282445 | 17.0158 | −2.20900 | −26.9952 | 70.1876 | ||||||||||||||||||
1.11 | −3.65613 | −8.25912 | 5.36730 | 3.13044 | 30.1964 | 14.4272 | 9.62551 | 41.2131 | −11.4453 | ||||||||||||||||||
1.12 | −3.43003 | −1.96781 | 3.76513 | 6.18310 | 6.74964 | 26.3490 | 14.5258 | −23.1277 | −21.2082 | ||||||||||||||||||
1.13 | −3.32824 | −1.32929 | 3.07716 | −9.90267 | 4.42418 | −6.00243 | 16.3844 | −25.2330 | 32.9584 | ||||||||||||||||||
1.14 | −3.11011 | 7.45275 | 1.67277 | −8.38904 | −23.1788 | 34.4760 | 19.6784 | 28.5434 | 26.0908 | ||||||||||||||||||
1.15 | −3.04850 | −6.77034 | 1.29337 | −1.02951 | 20.6394 | −3.20599 | 20.4452 | 18.8374 | 3.13845 | ||||||||||||||||||
1.16 | −2.97728 | 3.86255 | 0.864168 | −21.2217 | −11.4999 | 4.42148 | 21.2453 | −12.0807 | 63.1828 | ||||||||||||||||||
1.17 | −2.80404 | −0.374518 | −0.137383 | 21.1506 | 1.05016 | −7.42020 | 22.8175 | −26.8597 | −59.3071 | ||||||||||||||||||
1.18 | −2.36540 | 4.98259 | −2.40490 | −9.01519 | −11.7858 | 1.33216 | 24.6117 | −2.17379 | 21.3245 | ||||||||||||||||||
1.19 | −2.16938 | −6.51073 | −3.29381 | −10.9160 | 14.1242 | −25.2888 | 24.5005 | 15.3896 | 23.6809 | ||||||||||||||||||
1.20 | −1.69820 | 1.47404 | −5.11610 | 3.94546 | −2.50323 | −10.0206 | 22.2738 | −24.8272 | −6.70020 | ||||||||||||||||||
See all 52 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.u | 52 | |
11.b | odd | 2 | 1 | 2057.4.a.v | 52 | ||
11.d | odd | 10 | 2 | 187.4.g.b | ✓ | 104 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
187.4.g.b | ✓ | 104 | 11.d | odd | 10 | 2 | |
2057.4.a.u | 52 | 1.a | even | 1 | 1 | trivial | |
2057.4.a.v | 52 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):
\( T_{2}^{52} + T_{2}^{51} - 325 T_{2}^{50} - 305 T_{2}^{49} + 49453 T_{2}^{48} + 43381 T_{2}^{47} - 4683173 T_{2}^{46} - 3821674 T_{2}^{45} + 309481915 T_{2}^{44} + 233567301 T_{2}^{43} + \cdots + 19\!\cdots\!44 \) |
\( T_{3}^{52} - 20 T_{3}^{51} - 788 T_{3}^{50} + 17710 T_{3}^{49} + 279137 T_{3}^{48} - 7303284 T_{3}^{47} - 57851124 T_{3}^{46} + 1863625082 T_{3}^{45} + 7516015457 T_{3}^{44} + \cdots + 20\!\cdots\!09 \) |
\( T_{5}^{52} - 40 T_{5}^{51} - 3379 T_{5}^{50} + 150766 T_{5}^{49} + 5137542 T_{5}^{48} - 263620290 T_{5}^{47} - 4608509171 T_{5}^{46} + 284409116428 T_{5}^{45} + 2664627365542 T_{5}^{44} + \cdots + 23\!\cdots\!80 \) |