Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,10,Mod(3,37)]
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([13]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.3");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.h (of order \(18\), 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: | \(19.0563259381\) |
Analytic rank: | \(0\) |
Dimension: | \(168\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −27.9981 | + | 33.3669i | 65.8541 | − | 55.2582i | −240.545 | − | 1364.20i | −399.807 | + | 1098.46i | 3744.47i | 1891.41 | + | 688.417i | 32940.3 | + | 19018.1i | −2134.62 | + | 12106.0i | −25458.3 | − | 44095.1i | ||
3.2 | −26.0193 | + | 31.0086i | −128.286 | + | 107.645i | −195.621 | − | 1109.42i | 759.168 | − | 2085.80i | − | 6778.81i | −10701.4 | − | 3894.99i | 21542.9 | + | 12437.8i | 1452.00 | − | 8234.72i | 44924.6 | + | 77811.7i | |
3.3 | −23.4612 | + | 27.9600i | −156.305 | + | 131.155i | −142.425 | − | 807.730i | −731.487 | + | 2009.75i | − | 7447.34i | −678.538 | − | 246.968i | 9741.66 | + | 5624.35i | 3811.56 | − | 21616.4i | −39030.9 | − | 67603.5i | |
3.4 | −22.9919 | + | 27.4007i | 183.862 | − | 154.278i | −133.262 | − | 755.768i | 816.313 | − | 2242.80i | 8585.09i | 8119.51 | + | 2955.26i | 7912.34 | + | 4568.19i | 6585.41 | − | 37347.7i | 42685.7 | + | 73933.9i | ||
3.5 | −22.5902 | + | 26.9219i | −90.5435 | + | 75.9750i | −125.566 | − | 712.119i | 323.614 | − | 889.121i | − | 4153.89i | 8597.44 | + | 3129.21i | 6425.09 | + | 3709.53i | −991.995 | + | 5625.89i | 16626.4 | + | 28797.7i | |
3.6 | −22.1667 | + | 26.4172i | 114.438 | − | 96.0253i | −117.600 | − | 666.941i | 92.1806 | − | 253.264i | 5151.70i | −10964.7 | − | 3990.81i | 4934.58 | + | 2848.98i | 457.392 | − | 2594.00i | 4647.19 | + | 8049.17i | ||
3.7 | −17.6863 | + | 21.0777i | 49.4482 | − | 41.4920i | −42.5571 | − | 241.353i | 60.4467 | − | 166.076i | 1776.10i | 276.139 | + | 100.506i | −6360.44 | − | 3672.20i | −2694.37 | + | 15280.6i | 2431.42 | + | 4211.35i | ||
3.8 | −15.3360 | + | 18.2767i | −37.8456 | + | 31.7562i | −9.93794 | − | 56.3609i | −278.257 | + | 764.505i | − | 1178.71i | −396.292 | − | 144.238i | −9396.50 | − | 5425.07i | −2994.09 | + | 16980.3i | −9705.29 | − | 16810.1i | |
3.9 | −13.9806 | + | 16.6614i | 158.307 | − | 132.835i | 6.76230 | + | 38.3509i | −726.809 | + | 1996.89i | 4494.73i | 3057.98 | + | 1113.01i | −10377.5 | − | 5991.47i | 3997.96 | − | 22673.6i | −23109.8 | − | 40027.3i | ||
3.10 | −9.94057 | + | 11.8467i | −168.752 | + | 141.600i | 47.3783 | + | 268.696i | −172.317 | + | 473.436i | − | 3406.75i | −6406.43 | − | 2331.75i | −10511.3 | − | 6068.69i | 5008.88 | − | 28406.8i | −3895.74 | − | 6747.61i | |
3.11 | −8.27130 | + | 9.85735i | 1.31238 | − | 1.10122i | 60.1549 | + | 341.155i | 785.378 | − | 2157.81i | 22.0451i | −1325.67 | − | 482.503i | −9566.12 | − | 5523.00i | −3417.41 | + | 19381.1i | 14774.2 | + | 25589.6i | ||
3.12 | −7.09899 | + | 8.46024i | −176.851 | + | 148.396i | 67.7278 | + | 384.103i | 403.268 | − | 1107.97i | − | 2549.67i | 4613.96 | + | 1679.34i | −8627.40 | − | 4981.03i | 5837.15 | − | 33104.1i | 6510.90 | + | 11277.2i | |
3.13 | −3.30493 | + | 3.93866i | 198.043 | − | 166.177i | 84.3174 | + | 478.188i | 299.908 | − | 823.990i | 1329.23i | −9977.77 | − | 3631.61i | −4441.87 | − | 2564.52i | 8188.00 | − | 46436.5i | 2254.24 | + | 3904.46i | ||
3.14 | −1.46466 | + | 1.74551i | −23.1371 | + | 19.4143i | 88.0063 | + | 499.108i | −648.756 | + | 1782.44i | − | 68.8214i | −9128.06 | − | 3322.34i | −2010.44 | − | 1160.73i | −3259.51 | + | 18485.6i | −2161.07 | − | 3743.08i | |
3.15 | −1.15965 | + | 1.38201i | 114.860 | − | 96.3791i | 88.3427 | + | 501.016i | 184.041 | − | 505.649i | 270.504i | 6332.05 | + | 2304.68i | −1594.80 | − | 920.758i | 486.007 | − | 2756.29i | 485.390 | + | 840.720i | ||
3.16 | −0.0979000 | + | 0.116673i | −67.1156 | + | 56.3167i | 88.9038 | + | 504.199i | −720.391 | + | 1979.26i | − | 13.3440i | 10808.8 | + | 3934.07i | −135.063 | − | 77.9786i | −2084.98 | + | 11824.5i | −160.399 | − | 277.819i | |
3.17 | 6.70758 | − | 7.99378i | 25.8863 | − | 21.7212i | 69.9990 | + | 396.984i | 99.8626 | − | 274.370i | − | 352.626i | −3442.46 | − | 1252.95i | 8269.92 | + | 4774.64i | −3219.63 | + | 18259.4i | −1523.42 | − | 2638.64i | |
3.18 | 9.72159 | − | 11.5857i | −144.436 | + | 121.196i | 49.1878 | + | 278.958i | −28.8177 | + | 79.1760i | 2851.61i | 3077.98 | + | 1120.29i | 10416.2 | + | 6013.81i | 2755.31 | − | 15626.1i | 637.158 | + | 1103.59i | ||
3.19 | 10.9936 | − | 13.1017i | −104.954 | + | 88.0672i | 38.1135 | + | 216.153i | 606.182 | − | 1665.47i | 2343.25i | −6557.17 | − | 2386.61i | 10834.5 | + | 6255.32i | −158.323 | + | 897.896i | −15156.3 | − | 26251.5i | ||
3.20 | 12.3034 | − | 14.6627i | 154.895 | − | 129.973i | 25.2886 | + | 143.419i | −383.475 | + | 1053.59i | − | 3870.29i | 1863.39 | + | 678.219i | 10901.1 | + | 6293.78i | 3681.76 | − | 20880.3i | 10730.4 | + | 18585.5i | |
See next 80 embeddings (of 168 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.h | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.10.h.a | ✓ | 168 |
37.h | even | 18 | 1 | inner | 37.10.h.a | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.10.h.a | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
37.10.h.a | ✓ | 168 | 37.h | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).