Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,8,Mod(2,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.2");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 23 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 23.c (of order \(11\), degree \(10\), 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: | \(7.18485558613\) |
Analytic rank: | \(0\) |
Dimension: | \(130\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | −18.0654 | − | 11.6099i | −7.91347 | − | 55.0394i | 138.396 | + | 303.044i | −448.477 | − | 131.685i | −496.044 | + | 1086.19i | −507.891 | + | 586.137i | 626.966 | − | 4360.65i | −868.303 | + | 254.957i | 6573.08 | + | 7585.74i |
2.2 | −15.1360 | − | 9.72735i | 9.41538 | + | 65.4854i | 81.3055 | + | 178.034i | −34.9931 | − | 10.2749i | 494.488 | − | 1082.78i | 215.101 | − | 248.240i | 173.405 | − | 1206.06i | −2101.28 | + | 616.990i | 429.710 | + | 495.912i |
2.3 | −14.3153 | − | 9.19986i | −4.38914 | − | 30.5271i | 67.1162 | + | 146.964i | 504.542 | + | 148.147i | −218.014 | + | 477.383i | 166.777 | − | 192.471i | 81.2827 | − | 565.334i | 1185.77 | − | 348.173i | −5859.71 | − | 6762.47i |
2.4 | −9.64791 | − | 6.20034i | 2.52225 | + | 17.5426i | 1.46493 | + | 3.20774i | −111.583 | − | 32.7636i | 84.4358 | − | 184.889i | −461.777 | + | 532.919i | −203.158 | + | 1412.99i | 1797.03 | − | 527.655i | 873.395 | + | 1007.95i |
2.5 | −6.77816 | − | 4.35606i | −7.10944 | − | 49.4473i | −26.2049 | − | 57.3807i | −177.765 | − | 52.1965i | −167.206 | + | 366.131i | 653.761 | − | 754.481i | −219.106 | + | 1523.91i | −296.076 | + | 86.9357i | 977.549 | + | 1128.15i |
2.6 | −1.43748 | − | 0.923809i | 6.16653 | + | 42.8892i | −51.9602 | − | 113.777i | −101.289 | − | 29.7411i | 30.7572 | − | 67.3488i | 679.427 | − | 784.101i | −61.5435 | + | 428.044i | 296.957 | − | 87.1944i | 118.125 | + | 136.324i |
2.7 | −0.160791 | − | 0.103334i | 9.57404 | + | 66.5889i | −53.1579 | − | 116.400i | 401.331 | + | 117.841i | 5.34148 | − | 11.6962i | −786.238 | + | 907.366i | −6.96246 | + | 48.4250i | −2244.01 | + | 658.900i | −52.3533 | − | 60.4189i |
2.8 | −0.0532626 | − | 0.0342298i | −10.0505 | − | 69.9027i | −53.1715 | − | 116.429i | 133.885 | + | 39.3123i | −1.85744 | + | 4.06723i | −916.475 | + | 1057.67i | −2.30664 | + | 16.0430i | −2686.96 | + | 788.963i | −5.78544 | − | 6.67675i |
2.9 | 7.29123 | + | 4.68579i | 2.75598 | + | 19.1683i | −21.9677 | − | 48.1025i | −520.166 | − | 152.735i | −69.7240 | + | 152.674i | −684.009 | + | 789.388i | 223.109 | − | 1551.76i | 1738.58 | − | 510.494i | −3076.97 | − | 3551.01i |
2.10 | 7.91581 | + | 5.08718i | −2.63237 | − | 18.3086i | −16.3925 | − | 35.8946i | 233.598 | + | 68.5907i | 72.3016 | − | 158.318i | 430.086 | − | 496.346i | 224.249 | − | 1559.69i | 1770.14 | − | 519.759i | 1500.19 | + | 1731.31i |
2.11 | 13.5156 | + | 8.68593i | 11.4363 | + | 79.5415i | 54.0522 | + | 118.358i | −29.6143 | − | 8.69556i | −536.323 | + | 1174.38i | 538.265 | − | 621.191i | −4.83843 | + | 33.6520i | −4097.64 | + | 1203.18i | −324.726 | − | 374.754i |
2.12 | 13.7906 | + | 8.86268i | −11.8811 | − | 82.6349i | 58.4604 | + | 128.010i | −324.875 | − | 95.3920i | 568.519 | − | 1244.88i | 482.425 | − | 556.748i | −29.6934 | + | 206.522i | −4588.96 | + | 1347.44i | −3634.80 | − | 4194.78i |
2.13 | 16.7731 | + | 10.7794i | −0.358102 | − | 2.49065i | 111.969 | + | 245.177i | 118.801 | + | 34.8831i | 20.8414 | − | 45.6362i | −507.110 | + | 585.236i | −401.607 | + | 2793.23i | 2092.34 | − | 614.365i | 1616.64 | + | 1865.71i |
3.1 | −2.99956 | − | 20.8624i | −29.5319 | + | 64.6658i | −303.428 | + | 89.0945i | 64.3692 | − | 74.2860i | 1437.67 | + | 422.138i | 967.886 | − | 622.023i | 1648.15 | + | 3608.95i | −1877.36 | − | 2166.59i | −1742.86 | − | 1120.07i |
3.2 | −2.78932 | − | 19.4002i | 32.2381 | − | 70.5916i | −245.771 | + | 72.1648i | −273.530 | + | 315.670i | −1459.41 | − | 428.522i | 1047.63 | − | 673.270i | 1043.37 | + | 2284.66i | −2511.70 | − | 2898.66i | 6887.02 | + | 4426.02i |
3.3 | −2.33422 | − | 16.2349i | −1.38221 | + | 3.02662i | −135.307 | + | 39.7298i | −80.9070 | + | 93.3716i | 52.3631 | + | 15.3752i | −1369.85 | + | 880.347i | 88.7099 | + | 194.247i | 1424.93 | + | 1644.46i | 1704.73 | + | 1095.56i |
3.4 | −2.00769 | − | 13.9638i | 16.2316 | − | 35.5423i | −68.1408 | + | 20.0079i | 299.281 | − | 345.388i | −528.892 | − | 155.297i | 217.245 | − | 139.615i | −333.941 | − | 731.228i | 432.393 | + | 499.008i | −5423.78 | − | 3485.65i |
3.5 | −1.38091 | − | 9.60444i | −13.2015 | + | 28.9072i | 32.4767 | − | 9.53603i | −1.45769 | + | 1.68227i | 295.868 | + | 86.8746i | 778.129 | − | 500.073i | −652.385 | − | 1428.52i | 770.833 | + | 889.589i | 18.1702 | + | 11.6773i |
3.6 | −0.505895 | − | 3.51858i | −37.0467 | + | 81.1210i | 110.691 | − | 32.5017i | 33.4805 | − | 38.6385i | 304.172 | + | 89.3130i | −1023.97 | + | 658.067i | −359.375 | − | 786.922i | −3775.97 | − | 4357.70i | −152.890 | − | 98.2566i |
3.7 | −0.266253 | − | 1.85183i | −1.93441 | + | 4.23577i | 119.457 | − | 35.0757i | −325.785 | + | 375.976i | 8.35896 | + | 2.45441i | 537.138 | − | 345.198i | −196.240 | − | 429.705i | 1417.98 | + | 1636.44i | 782.983 | + | 503.193i |
See next 80 embeddings (of 130 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 23.8.c.a | ✓ | 130 |
23.c | even | 11 | 1 | inner | 23.8.c.a | ✓ | 130 |
23.c | even | 11 | 1 | 529.8.a.k | 65 | ||
23.d | odd | 22 | 1 | 529.8.a.j | 65 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.8.c.a | ✓ | 130 | 1.a | even | 1 | 1 | trivial |
23.8.c.a | ✓ | 130 | 23.c | even | 11 | 1 | inner |
529.8.a.j | 65 | 23.d | odd | 22 | 1 | ||
529.8.a.k | 65 | 23.c | even | 11 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(23, [\chi])\).