Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [712,2,Mod(85,712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(712, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 11, 19]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("712.85");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 712 = 2^{3} \cdot 89 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 712.t (of order \(22\), degree \(10\), 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: | \(5.68534862392\) |
Analytic rank: | \(0\) |
Dimension: | \(880\) |
Relative dimension: | \(88\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
85.1 | −1.41408 | + | 0.0192097i | −0.669586 | − | 1.46619i | 1.99926 | − | 0.0543283i | 1.34067 | − | 0.192759i | 0.975016 | + | 2.06045i | 2.70826 | − | 0.389389i | −2.82608 | + | 0.115230i | 0.263216 | − | 0.303768i | −1.89212 | + | 0.298332i |
85.2 | −1.41352 | − | 0.0442608i | 1.27406 | + | 2.78980i | 1.99608 | + | 0.125127i | 3.44160 | − | 0.494827i | −1.67743 | − | 3.99983i | 5.17583 | − | 0.744172i | −2.81597 | − | 0.265218i | −4.19518 | + | 4.84150i | −4.88668 | + | 0.547121i |
85.3 | −1.40519 | + | 0.159466i | 0.485601 | + | 1.06332i | 1.94914 | − | 0.448161i | 1.57279 | − | 0.226133i | −0.851926 | − | 1.41673i | 0.0267271 | − | 0.00384277i | −2.66746 | + | 0.940575i | 1.06975 | − | 1.23455i | −2.17402 | + | 0.568567i |
85.4 | −1.40063 | − | 0.195543i | −1.41003 | − | 3.08753i | 1.92353 | + | 0.547767i | −1.40119 | + | 0.201460i | 1.37118 | + | 4.60021i | −3.81848 | + | 0.549015i | −2.58703 | − | 1.14335i | −5.58010 | + | 6.43978i | 2.00194 | − | 0.00817865i |
85.5 | −1.39418 | − | 0.237183i | 1.35569 | + | 2.96854i | 1.88749 | + | 0.661353i | −1.54432 | + | 0.222039i | −1.18599 | − | 4.46023i | −2.11521 | + | 0.304121i | −2.47464 | − | 1.36973i | −5.00976 | + | 5.78157i | 2.20572 | + | 0.0567224i |
85.6 | −1.38825 | + | 0.269764i | 0.0607019 | + | 0.132919i | 1.85445 | − | 0.748998i | −1.65789 | + | 0.238369i | −0.120126 | − | 0.168149i | −3.25832 | + | 0.468475i | −2.37239 | + | 1.54006i | 1.95060 | − | 2.25111i | 2.23726 | − | 0.778154i |
85.7 | −1.38616 | − | 0.280305i | 0.658002 | + | 1.44082i | 1.84286 | + | 0.777093i | −3.18682 | + | 0.458196i | −0.508223 | − | 2.18165i | 3.03966 | − | 0.437037i | −2.33667 | − | 1.59374i | 0.321578 | − | 0.371121i | 4.54587 | + | 0.258152i |
85.8 | −1.37553 | − | 0.328487i | −0.287478 | − | 0.629489i | 1.78419 | + | 0.903690i | 0.403160 | − | 0.0579657i | 0.188657 | + | 0.960317i | −1.18553 | + | 0.170454i | −2.15737 | − | 1.82914i | 1.65097 | − | 1.90532i | −0.573602 | − | 0.0526990i |
85.9 | −1.37498 | + | 0.330808i | −0.700665 | − | 1.53424i | 1.78113 | − | 0.909708i | 4.13654 | − | 0.594744i | 1.47094 | + | 1.87777i | −2.98673 | + | 0.429427i | −2.14808 | + | 1.84004i | 0.101613 | − | 0.117268i | −5.49090 | + | 2.18616i |
85.10 | −1.31714 | + | 0.514929i | 1.03166 | + | 2.25903i | 1.46970 | − | 1.35646i | −2.35114 | + | 0.338043i | −2.52208 | − | 2.44422i | −1.32338 | + | 0.190273i | −1.23731 | + | 2.54344i | −2.07430 | + | 2.39387i | 2.92270 | − | 1.65592i |
85.11 | −1.31126 | + | 0.529710i | −0.859677 | − | 1.88243i | 1.43881 | − | 1.38918i | −3.39203 | + | 0.487700i | 2.12440 | + | 2.01298i | 0.0843246 | − | 0.0121240i | −1.15080 | + | 2.58373i | −0.839913 | + | 0.969311i | 4.18950 | − | 2.43630i |
85.12 | −1.28844 | − | 0.583034i | 0.819041 | + | 1.79345i | 1.32014 | + | 1.50241i | 3.71925 | − | 0.534748i | −0.00964071 | − | 2.78828i | −4.70012 | + | 0.675775i | −0.824967 | − | 2.70544i | −0.581052 | + | 0.670570i | −5.10380 | − | 1.47946i |
85.13 | −1.28834 | + | 0.583249i | 0.149949 | + | 0.328342i | 1.31964 | − | 1.50285i | −1.97078 | + | 0.283355i | −0.384690 | − | 0.335558i | 4.21470 | − | 0.605983i | −0.823613 | + | 2.70586i | 1.87926 | − | 2.16878i | 2.37377 | − | 1.51451i |
85.14 | −1.22497 | − | 0.706718i | −0.860766 | − | 1.88481i | 1.00110 | + | 1.73142i | −2.43475 | + | 0.350064i | −0.277620 | + | 2.91716i | 2.81093 | − | 0.404150i | −0.00269505 | − | 2.82843i | −0.847023 | + | 0.977517i | 3.22989 | + | 1.29186i |
85.15 | −1.21317 | − | 0.726780i | −0.0387383 | − | 0.0848251i | 0.943582 | + | 1.76342i | 2.49889 | − | 0.359286i | −0.0146528 | + | 0.131062i | 2.76914 | − | 0.398142i | 0.136890 | − | 2.82511i | 1.95889 | − | 2.26068i | −3.29271 | − | 1.38026i |
85.16 | −1.20889 | − | 0.733883i | −0.442122 | − | 0.968112i | 0.922831 | + | 1.77437i | −1.09028 | + | 0.156758i | −0.176004 | + | 1.49481i | −3.31475 | + | 0.476589i | 0.186577 | − | 2.82227i | 1.22281 | − | 1.41120i | 1.43306 | + | 0.610631i |
85.17 | −1.19267 | + | 0.759957i | 0.872513 | + | 1.91054i | 0.844931 | − | 1.81276i | 0.846421 | − | 0.121697i | −2.49255 | − | 1.61557i | −1.44373 | + | 0.207577i | 0.369894 | + | 2.80414i | −0.924288 | + | 1.06669i | −0.917017 | + | 0.788388i |
85.18 | −1.18673 | + | 0.769195i | −0.872513 | − | 1.91054i | 0.816679 | − | 1.82566i | −0.846421 | + | 0.121697i | 2.50502 | + | 1.59617i | −1.44373 | + | 0.207577i | 0.435106 | + | 2.79476i | −0.924288 | + | 1.06669i | 0.910868 | − | 0.795484i |
85.19 | −1.15226 | − | 0.819936i | −1.23417 | − | 2.70245i | 0.655411 | + | 1.88956i | 4.10263 | − | 0.589869i | −0.793753 | + | 4.12587i | 1.32836 | − | 0.190989i | 0.794112 | − | 2.71466i | −3.81550 | + | 4.40333i | −5.21096 | − | 2.68421i |
85.20 | −1.06574 | + | 0.929625i | −0.149949 | − | 0.328342i | 0.271594 | − | 1.98147i | 1.97078 | − | 0.283355i | 0.465041 | + | 0.210530i | 4.21470 | − | 0.605983i | 1.55258 | + | 2.36421i | 1.87926 | − | 2.16878i | −1.83692 | + | 2.13407i |
See next 80 embeddings (of 880 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
89.f | even | 22 | 1 | inner |
712.t | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 712.2.t.a | ✓ | 880 |
8.b | even | 2 | 1 | inner | 712.2.t.a | ✓ | 880 |
89.f | even | 22 | 1 | inner | 712.2.t.a | ✓ | 880 |
712.t | even | 22 | 1 | inner | 712.2.t.a | ✓ | 880 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
712.2.t.a | ✓ | 880 | 1.a | even | 1 | 1 | trivial |
712.2.t.a | ✓ | 880 | 8.b | even | 2 | 1 | inner |
712.2.t.a | ✓ | 880 | 89.f | even | 22 | 1 | inner |
712.2.t.a | ✓ | 880 | 712.t | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(712, [\chi])\).