Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,7,Mod(8,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.8");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.g (of order \(12\), degree \(4\), 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: | \(8.51200109393\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | −13.9637 | − | 3.74156i | 7.40363 | + | 4.27449i | 125.560 | + | 72.4919i | −72.2350 | + | 19.3553i | −87.3887 | − | 87.3887i | 180.588 | − | 312.787i | −827.827 | − | 827.827i | −327.958 | − | 568.039i | 1081.09 | ||
8.2 | −13.9349 | − | 3.73385i | −44.2547 | − | 25.5505i | 124.814 | + | 72.0615i | −58.7930 | + | 15.7535i | 521.284 | + | 521.284i | −230.098 | + | 398.542i | −817.340 | − | 817.340i | 941.155 | + | 1630.13i | 878.096 | ||
8.3 | −13.0056 | − | 3.48484i | 38.1387 | + | 22.0194i | 101.576 | + | 58.6446i | 226.086 | − | 60.5794i | −419.282 | − | 419.282i | −208.015 | + | 360.292i | −507.354 | − | 507.354i | 605.206 | + | 1048.25i | −3151.48 | ||
8.4 | −9.93212 | − | 2.66130i | −3.04848 | − | 1.76004i | 36.1388 | + | 20.8647i | −34.6811 | + | 9.29276i | 25.5938 | + | 25.5938i | −217.964 | + | 377.524i | 161.925 | + | 161.925i | −358.305 | − | 620.602i | 369.187 | ||
8.5 | −9.59622 | − | 2.57130i | −21.8555 | − | 12.6183i | 30.0502 | + | 17.3495i | 173.019 | − | 46.3602i | 177.285 | + | 177.285i | 188.845 | − | 327.090i | 205.837 | + | 205.837i | −46.0571 | − | 79.7733i | −1779.53 | ||
8.6 | −8.85799 | − | 2.37349i | 32.4379 | + | 18.7280i | 17.4050 | + | 10.0488i | −153.103 | + | 41.0238i | −242.884 | − | 242.884i | 76.7196 | − | 132.882i | 284.686 | + | 284.686i | 336.977 | + | 583.662i | 1453.55 | ||
8.7 | −4.02062 | − | 1.07732i | −24.6756 | − | 14.2465i | −40.4209 | − | 23.3370i | −157.005 | + | 42.0695i | 83.8633 | + | 83.8633i | 63.0530 | − | 109.211i | 325.747 | + | 325.747i | 41.4241 | + | 71.7487i | 676.582 | ||
8.8 | −3.54127 | − | 0.948881i | 11.5842 | + | 6.68812i | −43.7854 | − | 25.2795i | 59.9765 | − | 16.0707i | −34.6765 | − | 34.6765i | −141.297 | + | 244.734i | 296.982 | + | 296.982i | −275.038 | − | 476.380i | −227.643 | ||
8.9 | −2.34959 | − | 0.629570i | 32.2680 | + | 18.6299i | −50.3014 | − | 29.0415i | 85.5401 | − | 22.9204i | −64.0877 | − | 64.0877i | 207.449 | − | 359.312i | 209.985 | + | 209.985i | 329.649 | + | 570.970i | −215.414 | ||
8.10 | 1.46393 | + | 0.392259i | −38.8847 | − | 22.4501i | −53.4364 | − | 30.8515i | 78.2708 | − | 20.9726i | −48.1182 | − | 48.1182i | −50.3376 | + | 87.1872i | −134.712 | − | 134.712i | 643.512 | + | 1114.60i | 122.810 | ||
8.11 | 3.45422 | + | 0.925557i | 32.3417 | + | 18.6725i | −44.3506 | − | 25.6058i | −181.632 | + | 48.6682i | 94.4331 | + | 94.4331i | −292.762 | + | 507.079i | −291.332 | − | 291.332i | 332.825 | + | 576.470i | −672.444 | ||
8.12 | 3.89838 | + | 1.04457i | −2.90484 | − | 1.67711i | −41.3194 | − | 23.8558i | 124.161 | − | 33.2689i | −9.57232 | − | 9.57232i | −66.0477 | + | 114.398i | −318.804 | − | 318.804i | −358.875 | − | 621.589i | 518.779 | ||
8.13 | 5.90851 | + | 1.58318i | 1.50469 | + | 0.868733i | −23.0216 | − | 13.2915i | −121.426 | + | 32.5360i | 7.51511 | + | 7.51511i | 268.634 | − | 465.288i | −391.802 | − | 391.802i | −362.991 | − | 628.718i | −768.957 | ||
8.14 | 10.0194 | + | 2.68468i | 36.8050 | + | 21.2494i | 37.7546 | + | 21.7976i | 48.6688 | − | 13.0408i | 311.715 | + | 311.715i | 32.6865 | − | 56.6146i | −149.662 | − | 149.662i | 538.570 | + | 932.831i | 522.641 | ||
8.15 | 10.2697 | + | 2.75175i | −23.6115 | − | 13.6321i | 42.4685 | + | 24.5192i | −109.253 | + | 29.2742i | −204.970 | − | 204.970i | −217.179 | + | 376.165i | −112.480 | − | 112.480i | 7.16866 | + | 12.4165i | −1202.55 | ||
8.16 | 11.7663 | + | 3.15278i | 2.99680 | + | 1.73020i | 73.0812 | + | 42.1934i | 187.344 | − | 50.1987i | 29.8064 | + | 29.8064i | −71.1800 | + | 123.287i | 175.603 | + | 175.603i | −358.513 | − | 620.962i | 2362.62 | ||
8.17 | 12.6486 | + | 3.38919i | −37.5401 | − | 21.6738i | 93.0755 | + | 53.7372i | 72.3779 | − | 19.3936i | −401.374 | − | 401.374i | 296.773 | − | 514.026i | 402.548 | + | 402.548i | 575.006 | + | 995.940i | 981.210 | ||
8.18 | 14.9428 | + | 4.00391i | 13.6514 | + | 7.88162i | 151.830 | + | 87.6593i | −121.406 | + | 32.5306i | 172.432 | + | 172.432i | 6.42619 | − | 11.1305i | 1217.70 | + | 1217.70i | −240.260 | − | 416.143i | −1944.40 | ||
14.1 | −13.9637 | + | 3.74156i | 7.40363 | − | 4.27449i | 125.560 | − | 72.4919i | −72.2350 | − | 19.3553i | −87.3887 | + | 87.3887i | 180.588 | + | 312.787i | −827.827 | + | 827.827i | −327.958 | + | 568.039i | 1081.09 | ||
14.2 | −13.9349 | + | 3.73385i | −44.2547 | + | 25.5505i | 124.814 | − | 72.0615i | −58.7930 | − | 15.7535i | 521.284 | − | 521.284i | −230.098 | − | 398.542i | −817.340 | + | 817.340i | 941.155 | − | 1630.13i | 878.096 | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.g | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.7.g.a | ✓ | 72 |
37.g | odd | 12 | 1 | inner | 37.7.g.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.7.g.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
37.7.g.a | ✓ | 72 | 37.g | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(37, [\chi])\).