Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [59,11,Mod(2,59)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(59, base_ring=CyclotomicField(58))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("59.2");
S:= CuspForms(chi, 11);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 59 \) |
Weight: | \( k \) | \(=\) | \( 11 \) |
Character orbit: | \([\chi]\) | \(=\) | 59.d (of order \(58\), degree \(28\), 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: | \(37.4860779077\) |
Analytic rank: | \(0\) |
Dimension: | \(1372\) |
Relative dimension: | \(49\) over \(\Q(\zeta_{58})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{58}]$ |
$q$-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 | −63.5072 | − | 3.44326i | 228.394 | − | 105.666i | 3003.31 | + | 326.630i | −2502.91 | − | 843.328i | −14868.5 | + | 5924.15i | −13636.9 | − | 20113.0i | −125338. | − | 20548.2i | 2770.88 | − | 3262.14i | 156049. | + | 62175.6i |
2.2 | −58.7326 | − | 3.18439i | −356.429 | + | 164.902i | 2421.38 | + | 263.341i | 953.655 | + | 321.324i | 21459.1 | − | 8550.10i | −7498.83 | − | 11059.9i | −81938.2 | − | 13433.1i | 61621.8 | − | 72546.8i | −54987.4 | − | 21909.0i |
2.3 | −58.5728 | − | 3.17572i | −210.180 | + | 97.2396i | 2402.69 | + | 261.308i | −3651.31 | − | 1230.27i | 12619.6 | − | 5028.12i | 7707.96 | + | 11368.4i | −80627.0 | − | 13218.1i | −3507.47 | + | 4129.31i | 209961. | + | 83656.0i |
2.4 | −57.0594 | − | 3.09367i | −43.1529 | + | 19.9647i | 2228.21 | + | 242.332i | 4330.52 | + | 1459.12i | 2524.04 | − | 1005.67i | 11516.2 | + | 16985.1i | −68646.8 | − | 11254.1i | −36763.9 | + | 43281.8i | −242583. | − | 96653.9i |
2.5 | −55.0395 | − | 2.98415i | 298.048 | − | 137.892i | 2002.44 | + | 217.779i | 2237.07 | + | 753.757i | −16815.9 | + | 6700.07i | 4424.83 | + | 6526.14i | −53863.9 | − | 8830.55i | 31590.9 | − | 37191.7i | −120878. | − | 48162.2i |
2.6 | −52.5638 | − | 2.84992i | −130.502 | + | 60.3767i | 1736.83 | + | 188.892i | 3331.84 | + | 1122.63i | 7031.74 | − | 2801.70i | −8390.68 | − | 12375.3i | −37561.7 | − | 6157.93i | −24842.1 | + | 29246.4i | −171934. | − | 68504.9i |
2.7 | −47.9598 | − | 2.60030i | −24.3936 | + | 11.2857i | 1275.38 | + | 138.706i | −3095.79 | − | 1043.09i | 1199.26 | − | 477.829i | −221.486 | − | 326.667i | −12271.3 | − | 2011.77i | −37759.8 | + | 44454.3i | 145761. | + | 58076.6i |
2.8 | −47.8443 | − | 2.59404i | 236.528 | − | 109.429i | 1264.35 | + | 137.507i | −3457.65 | − | 1165.02i | −11600.4 | + | 4622.01i | 13550.7 | + | 19985.9i | −11717.2 | − | 1920.94i | 5743.02 | − | 6761.21i | 162407. | + | 64708.7i |
2.9 | −44.3605 | − | 2.40516i | 33.8748 | − | 15.6722i | 944.075 | + | 102.674i | −347.715 | − | 117.159i | −1540.40 | + | 613.752i | −12957.4 | − | 19110.7i | 3259.91 | + | 534.435i | −37325.6 | + | 43943.1i | 15143.1 | + | 6033.55i |
2.10 | −41.5655 | − | 2.25361i | 364.487 | − | 168.629i | 704.611 | + | 76.6310i | 1994.16 | + | 671.909i | −15530.1 | + | 6187.75i | −6225.84 | − | 9182.43i | 12949.2 | + | 2122.92i | 66187.0 | − | 77921.4i | −81373.7 | − | 32422.3i |
2.11 | −39.7014 | − | 2.15255i | −365.544 | + | 169.118i | 553.573 | + | 60.2047i | −2.80624 | − | 0.945534i | 14876.6 | − | 5927.39i | 10190.2 | + | 15029.5i | 18329.6 | + | 3004.98i | 66793.5 | − | 78635.4i | 109.377 | + | 43.5796i |
2.12 | −36.1667 | − | 1.96090i | −322.067 | + | 149.004i | 286.188 | + | 31.1249i | −5356.12 | − | 1804.69i | 11940.3 | − | 4757.45i | −16562.6 | − | 24428.1i | 26311.1 | + | 4313.48i | 43297.6 | − | 50973.8i | 190174. | + | 75772.4i |
2.13 | −33.0583 | − | 1.79237i | −330.750 | + | 153.021i | 71.6382 | + | 7.79112i | 4318.18 | + | 1454.96i | 11208.3 | − | 4465.79i | −5648.75 | − | 8331.28i | 31100.5 | + | 5098.67i | 47752.5 | − | 56218.6i | −140144. | − | 55838.3i |
2.14 | −32.5255 | − | 1.76348i | 427.138 | − | 197.615i | 36.8041 | + | 4.00268i | −5102.57 | − | 1719.25i | −14241.4 | + | 5674.29i | −10663.2 | − | 15727.1i | 31725.7 | + | 5201.16i | 105168. | − | 123813.i | 162932. | + | 64918.0i |
2.15 | −31.8894 | − | 1.72899i | −104.626 | + | 48.4054i | −4.05577 | − | 0.441091i | −182.378 | − | 61.4504i | 3420.16 | − | 1362.72i | 6593.18 | + | 9724.21i | 32400.4 | + | 5311.78i | −29623.9 | + | 34875.9i | 5709.68 | + | 2274.94i |
2.16 | −29.8836 | − | 1.62024i | 201.453 | − | 93.2019i | −127.594 | − | 13.8767i | 4428.02 | + | 1491.98i | −6171.13 | + | 2458.80i | −6659.33 | − | 9821.78i | 34032.5 | + | 5579.35i | −6330.97 | + | 7453.40i | −129908. | − | 51760.0i |
2.17 | −23.8525 | − | 1.29324i | 123.988 | − | 57.3631i | −450.729 | − | 49.0198i | −2276.31 | − | 766.978i | −3031.61 | + | 1207.91i | −3127.24 | − | 4612.33i | 34826.2 | + | 5709.46i | −26144.9 | + | 30780.2i | 53303.7 | + | 21238.1i |
2.18 | −22.0553 | − | 1.19581i | −8.21651 | + | 3.80136i | −532.989 | − | 57.9661i | 4105.81 | + | 1383.41i | 185.764 | − | 74.0150i | 15487.3 | + | 22842.0i | 34005.8 | + | 5574.97i | −38174.5 | + | 44942.4i | −88900.7 | − | 35421.3i |
2.19 | −18.9233 | − | 1.02599i | 316.805 | − | 146.569i | −660.958 | − | 71.8835i | 463.180 | + | 156.064i | −6145.38 | + | 2448.54i | 14010.4 | + | 20663.8i | 31584.1 | + | 5177.94i | 40655.1 | − | 47862.9i | −8604.78 | − | 3428.46i |
2.20 | −16.0810 | − | 0.871886i | 113.519 | − | 52.5194i | −760.159 | − | 82.6723i | −3672.04 | − | 1237.25i | −1871.28 | + | 745.588i | −1377.94 | − | 2032.31i | 28425.9 | + | 4660.19i | −28099.3 | + | 33081.0i | 57971.3 | + | 23097.9i |
See next 80 embeddings (of 1372 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.d | odd | 58 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 59.11.d.a | ✓ | 1372 |
59.d | odd | 58 | 1 | inner | 59.11.d.a | ✓ | 1372 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
59.11.d.a | ✓ | 1372 | 1.a | even | 1 | 1 | trivial |
59.11.d.a | ✓ | 1372 | 59.d | odd | 58 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(59, [\chi])\).