Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,8,Mod(4,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.4");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.k (of order \(30\), degree \(8\), 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: | \(19.0554865545\) |
Analytic rank: | \(0\) |
Dimension: | \(280\) |
Relative dimension: | \(35\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −22.3704 | − | 2.35123i | 23.5452 | + | 17.1066i | 369.706 | + | 78.5834i | 13.2121 | − | 14.6735i | −486.495 | − | 438.042i | −299.515 | + | 672.721i | −5347.44 | − | 1737.49i | −414.080 | − | 1274.41i | −330.061 | + | 297.188i |
4.2 | −20.1873 | − | 2.12177i | −54.3152 | − | 39.4623i | 277.823 | + | 59.0530i | −258.320 | + | 286.893i | 1012.75 | + | 911.882i | −4.30841 | + | 9.67685i | −3012.15 | − | 978.708i | 717.048 | + | 2206.85i | 5823.50 | − | 5243.50i |
4.3 | −18.4887 | − | 1.94324i | −39.3769 | − | 28.6090i | 212.852 | + | 45.2430i | 102.648 | − | 114.002i | 672.431 | + | 605.460i | 586.619 | − | 1317.57i | −1584.31 | − | 514.772i | 56.2439 | + | 173.101i | −2119.36 | + | 1908.28i |
4.4 | −18.3230 | − | 1.92582i | −48.9231 | − | 35.5447i | 206.820 | + | 43.9610i | 255.825 | − | 284.122i | 827.964 | + | 745.502i | −376.576 | + | 845.803i | −1462.07 | − | 475.054i | 454.222 | + | 1397.95i | −5234.65 | + | 4713.30i |
4.5 | −18.1111 | − | 1.90355i | 28.7250 | + | 20.8700i | 199.186 | + | 42.3383i | 262.089 | − | 291.079i | −480.515 | − | 432.658i | 423.140 | − | 950.387i | −1309.98 | − | 425.638i | −286.248 | − | 880.981i | −5300.81 | + | 4772.87i |
4.6 | −17.2245 | − | 1.81037i | 35.7941 | + | 26.0059i | 168.203 | + | 35.7527i | −259.548 | + | 288.257i | −569.455 | − | 512.739i | 334.683 | − | 751.710i | −724.109 | − | 235.277i | −70.9123 | − | 218.246i | 4992.43 | − | 4495.21i |
4.7 | −16.3593 | − | 1.71943i | 71.4731 | + | 51.9282i | 139.468 | + | 29.6449i | 73.9058 | − | 82.0807i | −1079.96 | − | 972.404i | −296.425 | + | 665.782i | −228.156 | − | 74.1325i | 1736.04 | + | 5342.98i | −1350.18 | + | 1215.71i |
4.8 | −15.4812 | − | 1.62714i | −7.88263 | − | 5.72707i | 111.816 | + | 23.7672i | 23.1417 | − | 25.7015i | 112.714 | + | 101.488i | −251.425 | + | 564.710i | 202.615 | + | 65.8335i | −646.484 | − | 1989.67i | −400.080 | + | 360.234i |
4.9 | −13.1192 | − | 1.37888i | −14.6901 | − | 10.6729i | 45.0095 | + | 9.56707i | −238.059 | + | 264.391i | 178.005 | + | 160.277i | −519.837 | + | 1167.57i | 1028.57 | + | 334.203i | −573.934 | − | 1766.39i | 3487.71 | − | 3140.35i |
4.10 | −9.55874 | − | 1.00466i | 32.1225 | + | 23.3383i | −34.8428 | − | 7.40606i | 117.153 | − | 130.112i | −283.603 | − | 255.357i | −343.257 | + | 770.967i | 1495.66 | + | 485.968i | −188.646 | − | 580.592i | −1250.56 | + | 1126.01i |
4.11 | −8.43552 | − | 0.886609i | −24.8267 | − | 18.0377i | −54.8309 | − | 11.6547i | −86.8519 | + | 96.4588i | 193.434 | + | 174.169i | 489.842 | − | 1100.20i | 1484.75 | + | 482.425i | −384.811 | − | 1184.33i | 818.163 | − | 736.677i |
4.12 | −8.18508 | − | 0.860287i | 19.3379 | + | 14.0498i | −58.9474 | − | 12.5297i | 310.083 | − | 344.383i | −146.196 | − | 131.635i | 145.721 | − | 327.295i | 1473.61 | + | 478.806i | −499.263 | − | 1536.57i | −2834.33 | + | 2552.04i |
4.13 | −8.04495 | − | 0.845558i | −60.1212 | − | 43.6806i | −61.1967 | − | 13.0078i | 184.705 | − | 205.136i | 446.737 | + | 402.244i | −99.8404 | + | 224.245i | 1466.07 | + | 476.356i | 1030.74 | + | 3172.30i | −1659.40 | + | 1494.13i |
4.14 | −7.93518 | − | 0.834021i | −69.1989 | − | 50.2759i | −62.9314 | − | 13.3765i | −215.468 | + | 239.302i | 507.174 | + | 456.662i | 51.7877 | − | 116.317i | 1459.53 | + | 474.229i | 1585.00 | + | 4878.11i | 1909.36 | − | 1719.20i |
4.15 | −6.72757 | − | 0.707096i | 58.4048 | + | 42.4336i | −80.4427 | − | 17.0986i | −103.444 | + | 114.886i | −362.918 | − | 326.773i | 488.155 | − | 1096.41i | 1352.59 | + | 439.482i | 934.695 | + | 2876.70i | 777.161 | − | 699.759i |
4.16 | −2.81062 | − | 0.295408i | 44.2542 | + | 32.1526i | −117.391 | − | 24.9521i | −225.809 | + | 250.786i | −114.884 | − | 103.442i | −351.558 | + | 789.612i | 666.606 | + | 216.593i | 248.827 | + | 765.812i | 708.748 | − | 638.160i |
4.17 | 0.641807 | + | 0.0674566i | 2.06335 | + | 1.49911i | −124.796 | − | 26.5261i | −93.9757 | + | 104.371i | 1.22315 | + | 1.10133i | 266.020 | − | 597.492i | −156.866 | − | 50.9689i | −673.810 | − | 2073.77i | −67.3548 | + | 60.6465i |
4.18 | 0.882056 | + | 0.0927078i | −16.0556 | − | 11.6650i | −124.433 | − | 26.4491i | 221.610 | − | 246.122i | −13.0805 | − | 11.7777i | −163.494 | + | 367.213i | −215.274 | − | 69.9467i | −554.112 | − | 1705.38i | 218.289 | − | 196.549i |
4.19 | 1.62861 | + | 0.171174i | −39.5814 | − | 28.7576i | −122.580 | − | 26.0551i | 25.9408 | − | 28.8101i | −59.5400 | − | 53.6101i | −674.746 | + | 1515.50i | −394.526 | − | 128.189i | 63.8690 | + | 196.568i | 47.1789 | − | 42.4800i |
4.20 | 1.96131 | + | 0.206142i | 66.1438 | + | 48.0563i | −121.399 | − | 25.8041i | 222.448 | − | 247.054i | 119.822 | + | 107.888i | 456.781 | − | 1025.95i | −472.856 | − | 153.640i | 1389.78 | + | 4277.30i | 487.217 | − | 438.692i |
See next 80 embeddings (of 280 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.k | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 61.8.k.a | ✓ | 280 |
61.k | even | 30 | 1 | inner | 61.8.k.a | ✓ | 280 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.8.k.a | ✓ | 280 | 1.a | even | 1 | 1 | trivial |
61.8.k.a | ✓ | 280 | 61.k | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(61, [\chi])\).