Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,4,Mod(8,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.8");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 57.f (of order \(6\), degree \(2\), minimal) |
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: | no |
Analytic conductor: | \(3.36310887033\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −2.74302 | + | 4.75104i | 0.850422 | − | 5.12609i | −11.0483 | − | 19.1362i | 8.62601 | + | 4.98023i | 22.0215 | + | 18.1013i | 16.5845 | 77.3340 | −25.5536 | − | 8.71867i | −47.3226 | + | 27.3217i | ||||
8.2 | −2.50340 | + | 4.33601i | 3.91185 | + | 3.42015i | −8.53398 | − | 14.7813i | −10.6454 | − | 6.14614i | −24.6227 | + | 8.39985i | −23.1478 | 45.4014 | 3.60518 | + | 26.7582i | 53.2994 | − | 30.7724i | ||||
8.3 | −2.16509 | + | 3.75004i | −5.01325 | + | 1.36650i | −5.37519 | − | 9.31010i | −4.00786 | − | 2.31394i | 5.72967 | − | 21.7585i | 2.99271 | 11.9096 | 23.2653 | − | 13.7013i | 17.3547 | − | 10.0197i | ||||
8.4 | −1.70071 | + | 2.94572i | 0.694828 | + | 5.14949i | −1.78484 | − | 3.09144i | 16.5498 | + | 9.55504i | −16.3506 | − | 6.71103i | 8.54097 | −15.0694 | −26.0344 | + | 7.15601i | −56.2930 | + | 32.5008i | ||||
8.5 | −1.44157 | + | 2.49688i | 5.08799 | − | 1.05468i | −0.156269 | − | 0.270666i | 1.32057 | + | 0.762434i | −4.70130 | + | 14.2245i | 22.8542 | −22.1641 | 24.7753 | − | 10.7324i | −3.80741 | + | 2.19821i | ||||
8.6 | −1.31744 | + | 2.28188i | 0.493663 | − | 5.17265i | 0.528681 | + | 0.915703i | −14.8435 | − | 8.56992i | 11.1530 | + | 7.94116i | −7.94772 | −23.8651 | −26.5126 | − | 5.10709i | 39.1111 | − | 22.5808i | ||||
8.7 | −1.25916 | + | 2.18093i | −3.68522 | − | 3.66322i | 0.829040 | + | 1.43594i | 13.8138 | + | 7.97539i | 12.6295 | − | 3.42461i | −28.8532 | −24.3221 | 0.161628 | + | 26.9995i | −34.7875 | + | 20.0846i | ||||
8.8 | −0.479152 | + | 0.829916i | −2.14066 | + | 4.73472i | 3.54083 | + | 6.13289i | −8.81261 | − | 5.08796i | −2.90372 | − | 4.04522i | −5.02367 | −14.4528 | −17.8352 | − | 20.2708i | 8.44517 | − | 4.87582i | ||||
8.9 | 0.479152 | − | 0.829916i | 3.03006 | − | 4.22122i | 3.54083 | + | 6.13289i | 8.81261 | + | 5.08796i | −2.05140 | − | 4.53731i | −5.02367 | 14.4528 | −8.63744 | − | 25.5811i | 8.44517 | − | 4.87582i | ||||
8.10 | 1.25916 | − | 2.18093i | −5.01505 | − | 1.35988i | 0.829040 | + | 1.43594i | −13.8138 | − | 7.97539i | −9.28054 | + | 9.22515i | −28.8532 | 24.3221 | 23.3015 | + | 13.6397i | −34.7875 | + | 20.0846i | ||||
8.11 | 1.31744 | − | 2.28188i | −4.23281 | + | 3.01385i | 0.528681 | + | 0.915703i | 14.8435 | + | 8.56992i | 1.30075 | + | 13.6294i | −7.94772 | 23.8651 | 8.83343 | − | 25.5141i | 39.1111 | − | 22.5808i | ||||
8.12 | 1.44157 | − | 2.49688i | 1.63061 | + | 4.93367i | −0.156269 | − | 0.270666i | −1.32057 | − | 0.762434i | 14.6694 | + | 3.04080i | 22.8542 | 22.1641 | −21.6822 | + | 16.0898i | −3.80741 | + | 2.19821i | ||||
8.13 | 1.70071 | − | 2.94572i | 4.80700 | − | 1.97300i | −1.78484 | − | 3.09144i | −16.5498 | − | 9.55504i | 2.36340 | − | 17.5156i | 8.54097 | 15.0694 | 19.2145 | − | 18.9685i | −56.2930 | + | 32.5008i | ||||
8.14 | 2.16509 | − | 3.75004i | −1.32320 | − | 5.02485i | −5.37519 | − | 9.31010i | 4.00786 | + | 2.31394i | −21.7082 | − | 5.91720i | 2.99271 | −11.9096 | −23.4983 | + | 13.2977i | 17.3547 | − | 10.0197i | ||||
8.15 | 2.50340 | − | 4.33601i | 4.91786 | + | 1.67769i | −8.53398 | − | 14.7813i | 10.6454 | + | 6.14614i | 19.5858 | − | 17.1240i | −23.1478 | −45.4014 | 21.3707 | + | 16.5013i | 53.2994 | − | 30.7724i | ||||
8.16 | 2.74302 | − | 4.75104i | −4.01411 | + | 3.29953i | −11.0483 | − | 19.1362i | −8.62601 | − | 4.98023i | 4.66544 | + | 28.1219i | 16.5845 | −77.3340 | 5.22619 | − | 26.4894i | −47.3226 | + | 27.3217i | ||||
50.1 | −2.74302 | − | 4.75104i | 0.850422 | + | 5.12609i | −11.0483 | + | 19.1362i | 8.62601 | − | 4.98023i | 22.0215 | − | 18.1013i | 16.5845 | 77.3340 | −25.5536 | + | 8.71867i | −47.3226 | − | 27.3217i | ||||
50.2 | −2.50340 | − | 4.33601i | 3.91185 | − | 3.42015i | −8.53398 | + | 14.7813i | −10.6454 | + | 6.14614i | −24.6227 | − | 8.39985i | −23.1478 | 45.4014 | 3.60518 | − | 26.7582i | 53.2994 | + | 30.7724i | ||||
50.3 | −2.16509 | − | 3.75004i | −5.01325 | − | 1.36650i | −5.37519 | + | 9.31010i | −4.00786 | + | 2.31394i | 5.72967 | + | 21.7585i | 2.99271 | 11.9096 | 23.2653 | + | 13.7013i | 17.3547 | + | 10.0197i | ||||
50.4 | −1.70071 | − | 2.94572i | 0.694828 | − | 5.14949i | −1.78484 | + | 3.09144i | 16.5498 | − | 9.55504i | −16.3506 | + | 6.71103i | 8.54097 | −15.0694 | −26.0344 | − | 7.15601i | −56.2930 | − | 32.5008i | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
57.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 57.4.f.c | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 57.4.f.c | ✓ | 32 |
19.d | odd | 6 | 1 | inner | 57.4.f.c | ✓ | 32 |
57.f | even | 6 | 1 | inner | 57.4.f.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
57.4.f.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
57.4.f.c | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
57.4.f.c | ✓ | 32 | 19.d | odd | 6 | 1 | inner |
57.4.f.c | ✓ | 32 | 57.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\):
\( T_{2}^{32} + 108 T_{2}^{30} + 6921 T_{2}^{28} + 292684 T_{2}^{26} + 9187074 T_{2}^{24} + 218025072 T_{2}^{22} + 4053581529 T_{2}^{20} + 59091427476 T_{2}^{18} + 685653653682 T_{2}^{16} + \cdots + 30\!\cdots\!36 \) |
\( T_{7}^{8} + 14 T_{7}^{7} - 1070 T_{7}^{6} - 9260 T_{7}^{5} + 327467 T_{7}^{4} + 1064026 T_{7}^{3} - 21464462 T_{7}^{2} - 39409468 T_{7} + 258350560 \) |