Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [592,2,Mod(21,592)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([0, 9, 22]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("592.21");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 592.by (of order \(36\), degree \(12\), 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: | \(4.72714379966\) |
Analytic rank: | \(0\) |
Dimension: | \(888\) |
Relative dimension: | \(74\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
21.1 | −1.41362 | − | 0.0410164i | 0.325684 | − | 0.698432i | 1.99664 | + | 0.115963i | −1.77468 | − | 2.53451i | −0.489040 | + | 0.973958i | 3.01385 | + | 0.531424i | −2.81772 | − | 0.245822i | 1.54663 | + | 1.84320i | 2.40477 | + | 3.65562i |
21.2 | −1.41159 | − | 0.0860838i | 0.721348 | − | 1.54694i | 1.98518 | + | 0.243030i | 0.124449 | + | 0.177732i | −1.15141 | + | 2.12154i | −0.313377 | − | 0.0552568i | −2.78134 | − | 0.513951i | 0.0556948 | + | 0.0663745i | −0.160372 | − | 0.261598i |
21.3 | −1.40791 | + | 0.133387i | −0.914826 | + | 1.96185i | 1.96442 | − | 0.375592i | −0.747883 | − | 1.06809i | 1.02631 | − | 2.88413i | −5.03623 | − | 0.888024i | −2.71562 | + | 0.790827i | −1.08359 | − | 1.29137i | 1.19542 | + | 1.40401i |
21.4 | −1.40246 | + | 0.181942i | 1.12487 | − | 2.41229i | 1.93379 | − | 0.510333i | 2.49161 | + | 3.55839i | −1.13869 | + | 3.58780i | 0.998928 | + | 0.176138i | −2.61922 | + | 1.06756i | −2.62544 | − | 3.12888i | −4.14181 | − | 4.53717i |
21.5 | −1.40165 | − | 0.188059i | −0.430006 | + | 0.922152i | 1.92927 | + | 0.527186i | 0.141897 | + | 0.202651i | 0.776139 | − | 1.21167i | 4.65418 | + | 0.820658i | −2.60502 | − | 1.10175i | 1.26290 | + | 1.50507i | −0.160781 | − | 0.310731i |
21.6 | −1.37515 | − | 0.330112i | −0.186355 | + | 0.399639i | 1.78205 | + | 0.907904i | 1.60552 | + | 2.29292i | 0.388191 | − | 0.488044i | −2.35981 | − | 0.416098i | −2.15087 | − | 1.83678i | 1.80338 | + | 2.14918i | −1.45090 | − | 3.68310i |
21.7 | −1.37177 | + | 0.343878i | −1.06094 | + | 2.27519i | 1.76350 | − | 0.943442i | −1.23965 | − | 1.77040i | 0.672972 | − | 3.48586i | 1.93431 | + | 0.341070i | −2.09468 | + | 1.90061i | −2.12252 | − | 2.52952i | 2.30931 | + | 2.00229i |
21.8 | −1.34493 | + | 0.437211i | −1.20635 | + | 2.58703i | 1.61769 | − | 1.17604i | 1.75480 | + | 2.50612i | 0.491385 | − | 4.00682i | 2.24528 | + | 0.395904i | −1.66151 | + | 2.28897i | −3.30909 | − | 3.94362i | −3.45580 | − | 2.60334i |
21.9 | −1.32449 | − | 0.495701i | 0.800436 | − | 1.71654i | 1.50856 | + | 1.31310i | −1.99406 | − | 2.84781i | −1.91106 | + | 1.87677i | −3.27919 | − | 0.578209i | −1.34717 | − | 2.48699i | −0.377450 | − | 0.449828i | 1.22946 | + | 4.76037i |
21.10 | −1.29049 | + | 0.578487i | 1.11244 | − | 2.38565i | 1.33071 | − | 1.49306i | −0.520407 | − | 0.743219i | −0.0555303 | + | 3.72218i | −3.41953 | − | 0.602955i | −0.853545 | + | 2.69656i | −2.52541 | − | 3.00966i | 1.10152 | + | 0.658064i |
21.11 | −1.25320 | + | 0.655357i | 1.29134 | − | 2.76928i | 1.14101 | − | 1.64259i | −1.34239 | − | 1.91713i | 0.196566 | + | 4.31675i | 3.98945 | + | 0.703448i | −0.353439 | + | 2.80626i | −4.07301 | − | 4.85403i | 2.93868 | + | 1.52280i |
21.12 | −1.23971 | + | 0.680527i | −0.157088 | + | 0.336877i | 1.07377 | − | 1.68731i | 1.54171 | + | 2.20179i | −0.0345098 | − | 0.524533i | 0.337590 | + | 0.0595263i | −0.182898 | + | 2.82251i | 1.83955 | + | 2.19229i | −3.40966 | − | 1.68041i |
21.13 | −1.20829 | − | 0.734872i | −1.41627 | + | 3.03721i | 0.919926 | + | 1.77588i | 0.903800 | + | 1.29076i | 3.94323 | − | 2.62905i | −1.44697 | − | 0.255140i | 0.193504 | − | 2.82180i | −5.29045 | − | 6.30491i | −0.143508 | − | 2.22379i |
21.14 | −1.20801 | − | 0.735337i | 0.740018 | − | 1.58697i | 0.918559 | + | 1.77658i | 0.781659 | + | 1.11633i | −2.06091 | + | 1.37291i | 2.31880 | + | 0.408867i | 0.196762 | − | 2.82157i | −0.0424939 | − | 0.0506422i | −0.123374 | − | 1.92331i |
21.15 | −1.20289 | + | 0.743684i | 0.202563 | − | 0.434399i | 0.893867 | − | 1.78913i | −0.955196 | − | 1.36416i | 0.0793949 | + | 0.673175i | −0.195194 | − | 0.0344180i | 0.255332 | + | 2.81688i | 1.78069 | + | 2.12215i | 2.16350 | + | 0.930565i |
21.16 | −1.18439 | − | 0.772796i | −0.817536 | + | 1.75321i | 0.805573 | + | 1.83059i | −1.69346 | − | 2.41852i | 2.32316 | − | 1.44470i | 0.418842 | + | 0.0738532i | 0.460557 | − | 2.79068i | −0.477024 | − | 0.568495i | 0.136706 | + | 4.17317i |
21.17 | −1.11062 | − | 0.875519i | −0.371753 | + | 0.797226i | 0.466932 | + | 1.94473i | −1.16010 | − | 1.65679i | 1.11086 | − | 0.559935i | −0.822247 | − | 0.144984i | 1.18407 | − | 2.56865i | 1.43099 | + | 1.70539i | −0.162131 | + | 2.85575i |
21.18 | −1.05070 | + | 0.946588i | −0.539330 | + | 1.15660i | 0.207943 | − | 1.98916i | −2.17855 | − | 3.11129i | −0.528146 | − | 1.72576i | −1.06027 | − | 0.186955i | 1.66443 | + | 2.28685i | 0.881525 | + | 1.05056i | 5.23411 | + | 1.20685i |
21.19 | −0.983490 | − | 1.01624i | 1.15158 | − | 2.46958i | −0.0654955 | + | 1.99893i | 0.243204 | + | 0.347331i | −3.64226 | + | 1.25852i | 1.36630 | + | 0.240915i | 2.09581 | − | 1.89937i | −2.84432 | − | 3.38973i | 0.113784 | − | 0.588751i |
21.20 | −0.904166 | + | 1.08742i | −1.13200 | + | 2.42758i | −0.364967 | − | 1.96642i | 0.149951 | + | 0.214152i | −1.61629 | − | 3.42590i | 0.637028 | + | 0.112325i | 2.46831 | + | 1.38110i | −2.68336 | − | 3.19791i | −0.368454 | − | 0.0305693i |
See next 80 embeddings (of 888 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
37.h | even | 18 | 1 | inner |
592.by | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 592.2.by.a | ✓ | 888 |
16.e | even | 4 | 1 | inner | 592.2.by.a | ✓ | 888 |
37.h | even | 18 | 1 | inner | 592.2.by.a | ✓ | 888 |
592.by | even | 36 | 1 | inner | 592.2.by.a | ✓ | 888 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
592.2.by.a | ✓ | 888 | 1.a | even | 1 | 1 | trivial |
592.2.by.a | ✓ | 888 | 16.e | even | 4 | 1 | inner |
592.2.by.a | ✓ | 888 | 37.h | even | 18 | 1 | inner |
592.2.by.a | ✓ | 888 | 592.by | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).