Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,5,Mod(2,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.2");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.i (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: | \(3.82468863410\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −7.43299 | − | 0.650302i | −8.90517 | + | 10.6128i | 39.0695 | + | 6.88901i | −18.8246 | + | 40.3695i | 73.0935 | − | 73.0935i | −26.2057 | + | 9.53809i | −170.609 | − | 45.7146i | −19.2633 | − | 109.247i | 166.176 | − | 287.824i |
2.2 | −7.07168 | − | 0.618692i | 10.4867 | − | 12.4976i | 33.8689 | + | 5.97201i | −4.92870 | + | 10.5696i | −81.8911 | + | 81.8911i | 75.0961 | − | 27.3328i | −126.106 | − | 33.7901i | −32.1531 | − | 182.349i | 41.3935 | − | 71.6956i |
2.3 | −5.82370 | − | 0.509508i | 0.698255 | − | 0.832148i | 17.8989 | + | 3.15607i | 10.8573 | − | 23.2835i | −4.49041 | + | 4.49041i | −65.7606 | + | 23.9349i | −12.2821 | − | 3.29097i | 13.8606 | + | 78.6073i | −75.0928 | + | 130.064i |
2.4 | −4.38998 | − | 0.384074i | 0.247322 | − | 0.294747i | 3.36752 | + | 0.593785i | −2.39998 | + | 5.14676i | −1.19894 | + | 1.19894i | 28.5706 | − | 10.3989i | 53.5502 | + | 14.3487i | 14.0398 | + | 79.6236i | 12.5126 | − | 21.6724i |
2.5 | −2.93366 | − | 0.256662i | −9.35942 | + | 11.1541i | −7.21645 | − | 1.27246i | 9.43269 | − | 20.2285i | 30.3202 | − | 30.3202i | 39.1303 | − | 14.2423i | 66.3563 | + | 17.7801i | −22.7502 | − | 129.023i | −32.8641 | + | 56.9224i |
2.6 | −1.13246 | − | 0.0990776i | 9.68265 | − | 11.5393i | −14.4843 | − | 2.55397i | 6.08797 | − | 13.0557i | −12.1085 | + | 12.1085i | −66.6341 | + | 24.2528i | 33.7187 | + | 9.03489i | −25.3370 | − | 143.693i | −8.18792 | + | 14.1819i |
2.7 | −0.818396 | − | 0.0716004i | 2.52396 | − | 3.00793i | −15.0923 | − | 2.66118i | −18.6068 | + | 39.9023i | −2.28096 | + | 2.28096i | 8.45114 | − | 3.07597i | 24.8574 | + | 6.66052i | 11.3882 | + | 64.5856i | 18.0847 | − | 31.3237i |
2.8 | 2.03026 | + | 0.177625i | −6.00370 | + | 7.15494i | −11.6665 | − | 2.05712i | −1.02327 | + | 2.19440i | −13.4600 | + | 13.4600i | −47.7624 | + | 17.3841i | −54.8178 | − | 14.6884i | −1.08314 | − | 6.14280i | −2.46728 | + | 4.27345i |
2.9 | 2.08231 | + | 0.182179i | 2.85756 | − | 3.40550i | −11.4541 | − | 2.01967i | 15.1773 | − | 32.5477i | 6.57073 | − | 6.57073i | 66.1524 | − | 24.0775i | −55.7877 | − | 14.9483i | 10.6337 | + | 60.3066i | 37.5333 | − | 65.0095i |
2.10 | 5.32165 | + | 0.465584i | 8.08788 | − | 9.63876i | 12.3463 | + | 2.17699i | −8.12880 | + | 17.4323i | 47.5286 | − | 47.5286i | 2.51614 | − | 0.915801i | −17.8701 | − | 4.78828i | −13.4264 | − | 76.1449i | −51.3748 | + | 88.9838i |
2.11 | 5.93109 | + | 0.518903i | −7.67181 | + | 9.14291i | 19.1516 | + | 3.37695i | −13.8164 | + | 29.6295i | −50.2465 | + | 50.2465i | 80.3866 | − | 29.2583i | 19.8237 | + | 5.31175i | −10.6706 | − | 60.5159i | −97.3214 | + | 168.566i |
2.12 | 6.83559 | + | 0.598037i | −0.770402 | + | 0.918130i | 30.6107 | + | 5.39749i | 9.09446 | − | 19.5031i | −5.81523 | + | 5.81523i | −42.6753 | + | 15.5325i | 99.9681 | + | 26.7864i | 13.8161 | + | 78.3548i | 73.8296 | − | 127.877i |
5.1 | −3.29658 | + | 7.06954i | −3.57062 | + | 9.81021i | −28.8263 | − | 34.3539i | 0.215509 | − | 0.307778i | −57.5828 | − | 57.5828i | 11.8673 | + | 67.3028i | 217.341 | − | 58.2363i | −21.4413 | − | 17.9914i | 1.46541 | + | 2.53816i |
5.2 | −2.62972 | + | 5.63946i | 2.63021 | − | 7.22644i | −14.6034 | − | 17.4037i | 17.6934 | − | 25.2688i | 33.8365 | + | 33.8365i | −11.0584 | − | 62.7152i | 40.3836 | − | 10.8208i | 16.7462 | + | 14.0517i | 95.9735 | + | 166.231i |
5.3 | −2.37818 | + | 5.10001i | 4.82693 | − | 13.2619i | −10.0698 | − | 12.0008i | −25.5003 | + | 36.4182i | 56.1566 | + | 56.1566i | 13.1184 | + | 74.3980i | −1.81618 | + | 0.486643i | −90.5289 | − | 75.9628i | −125.089 | − | 216.661i |
5.4 | −1.86456 | + | 3.99856i | −1.37794 | + | 3.78585i | −2.22731 | − | 2.65440i | −13.4352 | + | 19.1875i | −12.5687 | − | 12.5687i | −8.06935 | − | 45.7636i | −53.4187 | + | 14.3135i | 49.6157 | + | 41.6325i | −51.6715 | − | 89.4976i |
5.5 | −1.15872 | + | 2.48488i | −5.54437 | + | 15.2330i | 5.45259 | + | 6.49814i | 6.68567 | − | 9.54813i | −31.4280 | − | 31.4280i | −2.18344 | − | 12.3829i | −64.8386 | + | 17.3735i | −139.256 | − | 116.849i | 15.9792 | + | 27.6767i |
5.6 | −1.02938 | + | 2.20752i | 0.471556 | − | 1.29559i | 6.47109 | + | 7.71195i | 15.7198 | − | 22.4501i | 2.37463 | + | 2.37463i | 13.1089 | + | 74.3445i | −61.3292 | + | 16.4331i | 60.5934 | + | 50.8439i | 33.3774 | + | 57.8114i |
5.7 | 0.240642 | − | 0.516058i | 4.70675 | − | 12.9317i | 10.0762 | + | 12.0083i | 1.54727 | − | 2.20973i | −5.54086 | − | 5.54086i | −8.44071 | − | 47.8697i | 17.4218 | − | 4.66816i | −83.0255 | − | 69.6667i | −0.768009 | − | 1.33023i |
5.8 | 0.479059 | − | 1.02734i | −0.611056 | + | 1.67886i | 9.45866 | + | 11.2724i | −16.2436 | + | 23.1983i | 1.43204 | + | 1.43204i | 0.371208 | + | 2.10523i | 33.6307 | − | 9.01132i | 59.6044 | + | 50.0140i | 16.0510 | + | 27.8012i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.i | odd | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.5.i.a | ✓ | 144 |
37.i | odd | 36 | 1 | inner | 37.5.i.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.5.i.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
37.5.i.a | ✓ | 144 | 37.i | odd | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(37, [\chi])\).