Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,8,Mod(10,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.10");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.c (of order \(3\), 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: | \(11.5582459429\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
Relative dimension: | \(21\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −11.0007 | + | 19.0538i | 11.3160 | + | 19.5999i | −178.031 | − | 308.359i | 50.3125 | + | 87.1438i | −497.937 | −276.316 | − | 478.593i | 5017.70 | 837.396 | − | 1450.41i | −2213.89 | ||||||
10.2 | −9.34255 | + | 16.1818i | −43.2405 | − | 74.8948i | −110.566 | − | 191.507i | −171.836 | − | 297.629i | 1615.91 | −666.202 | − | 1153.90i | 1740.20 | −2645.99 | + | 4582.98i | 6421.56 | ||||||
10.3 | −8.90423 | + | 15.4226i | 0.464514 | + | 0.804563i | −94.5706 | − | 163.801i | −251.123 | − | 434.958i | −16.5446 | 772.514 | + | 1338.03i | 1088.83 | 1093.07 | − | 1893.25i | 8944.22 | ||||||
10.4 | −8.43866 | + | 14.6162i | −26.6641 | − | 46.1836i | −78.4219 | − | 135.831i | 196.505 | + | 340.357i | 900.038 | 448.397 | + | 776.647i | 486.805 | −328.451 | + | 568.894i | −6632.96 | ||||||
10.5 | −7.57692 | + | 13.1236i | 12.8050 | + | 22.1789i | −50.8193 | − | 88.0217i | 31.2763 | + | 54.1721i | −388.090 | −403.425 | − | 698.752i | −399.475 | 765.563 | − | 1325.99i | −947.911 | ||||||
10.6 | −7.43537 | + | 12.8784i | 41.2618 | + | 71.4676i | −46.5696 | − | 80.6609i | 62.4377 | + | 108.145i | −1227.19 | 308.286 | + | 533.967i | −518.407 | −2311.58 | + | 4003.77i | −1856.99 | ||||||
10.7 | −5.02707 | + | 8.70714i | −17.3307 | − | 30.0177i | 13.4571 | + | 23.3084i | 14.1829 | + | 24.5655i | 348.491 | −329.971 | − | 571.526i | −1557.53 | 492.792 | − | 853.541i | −285.194 | ||||||
10.8 | −3.42160 | + | 5.92638i | 28.0949 | + | 48.6617i | 40.5853 | + | 70.2959i | −242.165 | − | 419.442i | −384.517 | −642.579 | − | 1112.98i | −1431.40 | −485.141 | + | 840.289i | 3314.36 | ||||||
10.9 | −2.85295 | + | 4.94146i | −22.7033 | − | 39.3233i | 47.7213 | + | 82.6558i | −48.1581 | − | 83.4123i | 259.086 | 330.602 | + | 572.619i | −1274.94 | 62.6168 | − | 108.455i | 549.571 | ||||||
10.10 | −1.26811 | + | 2.19642i | 13.5559 | + | 23.4795i | 60.7838 | + | 105.281i | 253.342 | + | 438.801i | −68.7612 | −185.616 | − | 321.497i | −632.957 | 725.976 | − | 1257.43i | −1285.06 | ||||||
10.11 | −1.18492 | + | 2.05234i | 20.9206 | + | 36.2356i | 61.1919 | + | 105.988i | −19.7131 | − | 34.1440i | −99.1570 | 666.527 | + | 1154.46i | −593.369 | 218.155 | − | 377.855i | 93.4336 | ||||||
10.12 | 0.746546 | − | 1.29305i | −28.6079 | − | 49.5504i | 62.8853 | + | 108.921i | −130.514 | − | 226.056i | −85.4285 | −159.585 | − | 276.409i | 378.903 | −543.330 | + | 941.074i | −389.737 | ||||||
10.13 | 2.32071 | − | 4.01959i | −46.0982 | − | 79.8444i | 53.2286 | + | 92.1946i | 217.309 | + | 376.390i | −427.923 | 19.1096 | + | 33.0988i | 1088.22 | −3156.59 | + | 5467.37i | 2017.24 | ||||||
10.14 | 3.77156 | − | 6.53253i | 9.73006 | + | 16.8530i | 35.5507 | + | 61.5757i | −147.558 | − | 255.578i | 146.790 | 91.4021 | + | 158.313i | 1501.84 | 904.152 | − | 1566.04i | −2226.09 | ||||||
10.15 | 4.09845 | − | 7.09873i | 41.0784 | + | 71.1498i | 30.4054 | + | 52.6637i | 21.4152 | + | 37.0921i | 673.431 | −128.692 | − | 222.901i | 1547.66 | −2281.37 | + | 3951.44i | 351.076 | ||||||
10.16 | 4.93583 | − | 8.54911i | −6.61210 | − | 11.4525i | 15.2752 | + | 26.4574i | 56.3382 | + | 97.5806i | −130.545 | −783.646 | − | 1357.32i | 1565.16 | 1006.06 | − | 1742.55i | 1112.30 | ||||||
10.17 | 6.29543 | − | 10.9040i | −9.44168 | − | 16.3535i | −15.2650 | − | 26.4397i | 125.405 | + | 217.208i | −237.758 | 817.371 | + | 1415.73i | 1227.23 | 915.209 | − | 1585.19i | 3157.91 | ||||||
10.18 | 7.43768 | − | 12.8824i | −34.7930 | − | 60.2632i | −46.6382 | − | 80.7798i | −238.285 | − | 412.723i | −1035.12 | 249.349 | + | 431.886i | 516.525 | −1327.60 | + | 2299.47i | −7089.17 | ||||||
10.19 | 9.33278 | − | 16.1648i | 28.0828 | + | 48.6408i | −110.202 | − | 190.875i | 166.579 | + | 288.523i | 1048.36 | 1.55453 | + | 2.69252i | −1724.76 | −483.783 | + | 837.937i | 6218.56 | ||||||
10.20 | 9.53293 | − | 16.5115i | 19.4337 | + | 33.6601i | −117.754 | − | 203.955i | −179.660 | − | 311.180i | 741.040 | 35.6214 | + | 61.6981i | −2049.72 | 338.163 | − | 585.716i | −6850.74 | ||||||
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.8.c.a | ✓ | 42 |
37.c | even | 3 | 1 | inner | 37.8.c.a | ✓ | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.8.c.a | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
37.8.c.a | ✓ | 42 | 37.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).