Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,9,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, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.8");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
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: | \(15.0730085723\) |
Analytic rank: | \(0\) |
Dimension: | \(100\) |
Relative dimension: | \(25\) 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 | −29.9817 | − | 8.03357i | 96.7293 | + | 55.8467i | 612.661 | + | 353.720i | −928.181 | + | 248.705i | −2451.46 | − | 2451.46i | −1962.55 | + | 3399.23i | −9908.26 | − | 9908.26i | 2957.20 | + | 5122.03i | 29826.4 | ||
8.2 | −29.0187 | − | 7.77554i | −54.3802 | − | 31.3964i | 559.924 | + | 323.272i | 923.655 | − | 247.493i | 1333.92 | + | 1333.92i | −635.915 | + | 1101.44i | −8296.40 | − | 8296.40i | −1309.03 | − | 2267.30i | −28727.7 | ||
8.3 | −26.9838 | − | 7.23030i | −71.9894 | − | 41.5631i | 454.148 | + | 262.202i | −763.969 | + | 204.705i | 1642.04 | + | 1642.04i | 1794.23 | − | 3107.71i | −5301.94 | − | 5301.94i | 174.479 | + | 302.206i | 22094.9 | ||
8.4 | −23.7708 | − | 6.36938i | 94.2638 | + | 54.4232i | 302.782 | + | 174.811i | 460.453 | − | 123.378i | −1894.09 | − | 1894.09i | 1918.35 | − | 3322.68i | −1629.17 | − | 1629.17i | 2643.27 | + | 4578.28i | −11731.2 | ||
8.5 | −22.4552 | − | 6.01684i | −2.68847 | − | 1.55219i | 246.329 | + | 142.218i | 152.334 | − | 40.8178i | 51.0307 | + | 51.0307i | −434.094 | + | 751.872i | −467.444 | − | 467.444i | −3275.68 | − | 5673.65i | −3666.28 | ||
8.6 | −19.4436 | − | 5.20990i | −78.9822 | − | 45.6004i | 129.209 | + | 74.5989i | −557.634 | + | 149.418i | 1298.13 | + | 1298.13i | −564.669 | + | 978.035i | 1520.19 | + | 1520.19i | 878.292 | + | 1521.25i | 11620.9 | ||
8.7 | −17.9701 | − | 4.81507i | 55.1301 | + | 31.8294i | 78.0367 | + | 45.0545i | −19.5258 | + | 5.23193i | −837.432 | − | 837.432i | −774.650 | + | 1341.73i | 2182.30 | + | 2182.30i | −1254.28 | − | 2172.48i | 376.073 | ||
8.8 | −14.5407 | − | 3.89616i | −127.969 | − | 73.8828i | −25.4520 | − | 14.6947i | 528.293 | − | 141.556i | 1572.89 | + | 1572.89i | −30.8038 | + | 53.3538i | 3037.83 | + | 3037.83i | 7636.83 | + | 13227.4i | −8233.25 | ||
8.9 | −9.75824 | − | 2.61471i | 46.3249 | + | 26.7457i | −133.316 | − | 76.9700i | −1105.12 | + | 296.117i | −382.117 | − | 382.117i | 938.117 | − | 1624.87i | 2928.42 | + | 2928.42i | −1849.84 | − | 3204.01i | 11558.3 | ||
8.10 | −6.85365 | − | 1.83643i | 29.3470 | + | 16.9435i | −178.102 | − | 102.827i | 1138.20 | − | 304.979i | −170.019 | − | 170.019i | −1165.96 | + | 2019.51i | 2316.23 | + | 2316.23i | −2706.34 | − | 4687.51i | −8360.89 | ||
8.11 | −6.32937 | − | 1.69595i | 134.573 | + | 77.6955i | −184.518 | − | 106.531i | 44.9983 | − | 12.0573i | −719.992 | − | 719.992i | −750.951 | + | 1300.69i | 2173.37 | + | 2173.37i | 8792.70 | + | 15229.4i | −305.259 | ||
8.12 | −5.47162 | − | 1.46612i | −38.0479 | − | 21.9670i | −193.913 | − | 111.956i | 326.464 | − | 87.4758i | 175.978 | + | 175.978i | 1762.78 | − | 3053.23i | 1922.29 | + | 1922.29i | −2315.40 | − | 4010.40i | −1914.54 | ||
8.13 | −3.23067 | − | 0.865656i | −66.6884 | − | 38.5025i | −212.015 | − | 122.407i | −678.195 | + | 181.722i | 182.118 | + | 182.118i | −2300.40 | + | 3984.40i | 1184.43 | + | 1184.43i | −315.607 | − | 546.648i | 2348.34 | ||
8.14 | 5.19180 | + | 1.39114i | −60.8932 | − | 35.1567i | −196.683 | − | 113.555i | −61.3978 | + | 16.4515i | −267.238 | − | 267.238i | −417.786 | + | 723.627i | −1836.14 | − | 1836.14i | −808.514 | − | 1400.39i | −341.651 | ||
8.15 | 7.07517 | + | 1.89579i | 65.9616 | + | 38.0829i | −175.238 | − | 101.174i | −317.418 | + | 85.0520i | 394.492 | + | 394.492i | 582.766 | − | 1009.38i | −2373.96 | − | 2373.96i | −379.880 | − | 657.971i | −2407.03 | ||
8.16 | 10.3785 | + | 2.78090i | −134.250 | − | 77.5093i | −121.724 | − | 70.2771i | −903.574 | + | 242.112i | −1177.76 | − | 1177.76i | 1653.64 | − | 2864.19i | −3012.84 | − | 3012.84i | 8734.88 | + | 15129.2i | −10051.0 | ||
8.17 | 11.5756 | + | 3.10167i | 46.5342 | + | 26.8665i | −97.3286 | − | 56.1927i | −129.992 | + | 34.8313i | 455.330 | + | 455.330i | −1422.98 | + | 2464.68i | −3121.67 | − | 3121.67i | −1836.88 | − | 3181.57i | −1612.77 | ||
8.18 | 13.2543 | + | 3.55149i | 86.8010 | + | 50.1146i | −58.6381 | − | 33.8547i | 964.131 | − | 258.338i | 972.509 | + | 972.509i | 1465.21 | − | 2537.82i | −3140.90 | − | 3140.90i | 1742.45 | + | 3018.00i | 13696.4 | ||
8.19 | 14.0121 | + | 3.75453i | −97.4491 | − | 56.2622i | −39.4602 | − | 22.7824i | 1028.99 | − | 275.718i | −1154.23 | − | 1154.23i | −1032.28 | + | 1787.96i | −3093.32 | − | 3093.32i | 3050.38 | + | 5283.41i | 15453.5 | ||
8.20 | 21.1757 | + | 5.67401i | −53.5012 | − | 30.8889i | 194.514 | + | 112.303i | 280.857 | − | 75.2555i | −957.661 | − | 957.661i | 727.522 | − | 1260.11i | −486.675 | − | 486.675i | −1372.25 | − | 2376.80i | 6374.36 | ||
See all 100 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 | 37.9.g.a | ✓ | 100 |
37.g | odd | 12 | 1 | inner | 37.9.g.a | ✓ | 100 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.9.g.a | ✓ | 100 | 1.a | even | 1 | 1 | trivial |
37.9.g.a | ✓ | 100 | 37.g | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(37, [\chi])\).