Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,8,Mod(7,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.7");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.f (of order \(9\), degree \(6\), 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: | \(132\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −20.8388 | + | 7.58471i | −2.65397 | − | 0.965964i | 278.675 | − | 233.836i | 66.0192 | + | 374.414i | 62.6321 | −150.560 | − | 853.866i | −2614.40 | + | 4528.27i | −1669.23 | − | 1400.65i | −4215.58 | − | 7301.60i | ||
7.2 | −18.9650 | + | 6.90268i | −58.5849 | − | 21.3231i | 213.969 | − | 179.541i | −54.5768 | − | 309.520i | 1258.25 | 222.299 | + | 1260.72i | −1526.94 | + | 2644.75i | 1302.17 | + | 1092.65i | 3171.57 | + | 5493.32i | ||
7.3 | −17.7108 | + | 6.44620i | 49.8626 | + | 18.1485i | 174.065 | − | 146.058i | −63.8672 | − | 362.209i | −1000.10 | 43.2399 | + | 245.226i | −935.075 | + | 1619.60i | 481.575 | + | 404.089i | 3466.01 | + | 6003.30i | ||
7.4 | −14.2518 | + | 5.18723i | 85.5870 | + | 31.1511i | 78.1529 | − | 65.5781i | 80.5878 | + | 457.036i | −1381.36 | 115.337 | + | 654.111i | 197.001 | − | 341.217i | 4679.40 | + | 3926.48i | −3519.28 | − | 6095.56i | ||
7.5 | −13.3313 | + | 4.85219i | 11.3243 | + | 4.12171i | 56.1255 | − | 47.0949i | 4.98573 | + | 28.2755i | −170.967 | 7.15684 | + | 40.5884i | 388.247 | − | 672.464i | −1564.09 | − | 1312.43i | −203.664 | − | 352.756i | ||
7.6 | −13.2275 | + | 4.81442i | −57.0958 | − | 20.7812i | 53.7348 | − | 45.0888i | −7.87815 | − | 44.6792i | 855.285 | −278.049 | − | 1576.89i | 407.190 | − | 705.274i | 1152.74 | + | 967.261i | 319.313 | + | 553.066i | ||
7.7 | −12.0215 | + | 4.37547i | −68.2052 | − | 24.8247i | 27.3180 | − | 22.9225i | 75.7434 | + | 429.562i | 928.548 | 226.548 | + | 1284.82i | 590.646 | − | 1023.03i | 2360.34 | + | 1980.56i | −2790.09 | − | 4832.57i | ||
7.8 | −5.31992 | + | 1.93629i | 16.2628 | + | 5.91916i | −73.5013 | + | 61.6750i | 5.57004 | + | 31.5893i | −97.9779 | 125.391 | + | 711.126i | 633.927 | − | 1097.99i | −1445.90 | − | 1213.25i | −90.7982 | − | 157.267i | ||
7.9 | −4.76396 | + | 1.73394i | 65.1424 | + | 23.7099i | −78.3649 | + | 65.7560i | −43.0745 | − | 244.287i | −351.448 | −294.271 | − | 1668.89i | 583.772 | − | 1011.12i | 2006.04 | + | 1683.27i | 628.785 | + | 1089.09i | ||
7.10 | −4.25018 | + | 1.54694i | −41.1038 | − | 14.9605i | −82.3827 | + | 69.1273i | −92.7474 | − | 525.997i | 197.841 | 9.66574 | + | 54.8172i | 532.674 | − | 922.618i | −209.638 | − | 175.907i | 1207.88 | + | 2092.10i | ||
7.11 | −0.910359 | + | 0.331344i | 19.9506 | + | 7.26143i | −97.3347 | + | 81.6735i | 96.6240 | + | 547.982i | −20.5683 | −202.895 | − | 1150.68i | 123.550 | − | 213.994i | −1330.04 | − | 1116.04i | −269.533 | − | 466.845i | ||
7.12 | −0.323158 | + | 0.117620i | −38.0853 | − | 13.8619i | −97.9631 | + | 82.2008i | 13.0043 | + | 73.7511i | 13.9380 | 126.462 | + | 717.201i | 43.9985 | − | 76.2077i | −417.005 | − | 349.909i | −12.8770 | − | 22.3037i | ||
7.13 | 2.24172 | − | 0.815920i | 67.7355 | + | 24.6537i | −93.6941 | + | 78.6187i | −36.3337 | − | 206.059i | 171.960 | 278.285 | + | 1578.23i | −298.567 | + | 517.134i | 2304.96 | + | 1934.09i | −249.578 | − | 432.281i | ||
7.14 | 3.79145 | − | 1.37998i | −69.9883 | − | 25.4737i | −85.5829 | + | 71.8126i | 27.7384 | + | 157.312i | −300.510 | −147.454 | − | 836.251i | −483.610 | + | 837.637i | 2574.12 | + | 2159.94i | 322.256 | + | 558.163i | ||
7.15 | 7.19462 | − | 2.61863i | 53.5140 | + | 19.4775i | −53.1484 | + | 44.5968i | 33.7756 | + | 191.551i | 436.017 | 41.4542 | + | 235.098i | −755.606 | + | 1308.75i | 809.034 | + | 678.860i | 744.604 | + | 1289.69i | ||
7.16 | 10.1961 | − | 3.71107i | 11.1844 | + | 4.07079i | −7.86580 | + | 6.60019i | −64.6668 | − | 366.744i | 129.144 | −94.8119 | − | 537.705i | −750.134 | + | 1299.27i | −1566.82 | − | 1314.72i | −2020.36 | − | 3499.36i | ||
7.17 | 10.2566 | − | 3.73308i | −19.4495 | − | 7.07902i | −6.79263 | + | 5.69969i | 7.78053 | + | 44.1256i | −225.911 | −27.0245 | − | 153.264i | −746.939 | + | 1293.74i | −1347.17 | − | 1130.41i | 244.526 | + | 423.531i | ||
7.18 | 14.2134 | − | 5.17327i | −80.6447 | − | 29.3523i | 77.2054 | − | 64.7830i | −48.8700 | − | 277.155i | −1298.08 | 203.583 | + | 1154.58i | −205.826 | + | 356.501i | 3966.67 | + | 3328.43i | −2128.41 | − | 3686.51i | ||
7.19 | 15.5384 | − | 5.65552i | −16.3965 | − | 5.96785i | 111.404 | − | 93.4791i | 86.0271 | + | 487.884i | −288.528 | 270.212 | + | 1532.45i | 144.088 | − | 249.568i | −1442.11 | − | 1210.07i | 4095.96 | + | 7094.42i | ||
7.20 | 16.4964 | − | 6.00419i | 64.3192 | + | 23.4103i | 138.026 | − | 115.818i | 23.4987 | + | 133.267i | 1201.59 | −96.3778 | − | 546.586i | 458.016 | − | 793.306i | 1913.58 | + | 1605.68i | 1187.81 | + | 2057.34i | ||
See next 80 embeddings (of 132 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.f | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.8.f.a | ✓ | 132 |
37.f | even | 9 | 1 | inner | 37.8.f.a | ✓ | 132 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.8.f.a | ✓ | 132 | 1.a | even | 1 | 1 | trivial |
37.8.f.a | ✓ | 132 | 37.f | even | 9 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).