Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,9,Mod(5,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 2]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.5");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 48.i (of order \(4\), degree \(2\), 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: | \(19.5541732829\) |
Analytic rank: | \(0\) |
Dimension: | \(124\) |
Relative dimension: | \(62\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −16.0000 | + | 0.0181473i | −69.5660 | + | 41.4919i | 255.999 | − | 0.580715i | −76.5595 | + | 76.5595i | 1112.30 | − | 665.132i | 2902.85i | −4095.98 | + | 13.9371i | 3117.85 | − | 5772.85i | 1223.56 | − | 1226.34i | ||
5.2 | −15.9624 | − | 1.09650i | 69.1317 | − | 42.2115i | 253.595 | + | 35.0054i | −755.337 | + | 755.337i | −1149.79 | + | 597.993i | − | 255.661i | −4009.60 | − | 836.837i | 2997.38 | − | 5836.30i | 12885.2 | − | 11228.8i | |
5.3 | −15.8366 | − | 2.28109i | 55.2937 | + | 59.1913i | 245.593 | + | 72.2492i | −126.448 | + | 126.448i | −740.641 | − | 1063.52i | 2580.10i | −3724.55 | − | 1704.40i | −446.215 | + | 6545.81i | 2290.94 | − | 1714.06i | ||
5.4 | −15.8350 | − | 2.29186i | 73.5550 | − | 33.9214i | 245.495 | + | 72.5831i | 608.521 | − | 608.521i | −1242.49 | + | 368.568i | − | 3930.72i | −3721.06 | − | 1711.99i | 4259.68 | − | 4990.18i | −11030.6 | + | 8241.28i | |
5.5 | −15.5809 | − | 3.63791i | −70.8508 | − | 39.2576i | 229.531 | + | 113.364i | 396.522 | − | 396.522i | 961.107 | + | 869.419i | − | 243.614i | −3163.91 | − | 2601.33i | 3478.68 | + | 5562.87i | −7620.70 | + | 4735.68i | |
5.6 | −15.5700 | + | 3.68426i | 16.7996 | + | 79.2387i | 228.852 | − | 114.728i | 356.180 | − | 356.180i | −553.506 | − | 1171.86i | − | 2259.12i | −3140.55 | + | 2629.47i | −5996.55 | + | 2662.36i | −4233.48 | + | 6858.00i | |
5.7 | −15.1821 | + | 5.05022i | −2.05641 | − | 80.9739i | 204.991 | − | 153.346i | −38.4932 | + | 38.4932i | 440.156 | + | 1218.97i | 592.015i | −2337.75 | + | 3363.35i | −6552.54 | + | 333.032i | 390.007 | − | 778.805i | ||
5.8 | −14.4416 | − | 6.88759i | −47.3668 | + | 65.7068i | 161.122 | + | 198.936i | −414.223 | + | 414.223i | 1136.62 | − | 622.672i | − | 3504.66i | −956.682 | − | 3982.71i | −2073.78 | − | 6224.64i | 8835.05 | − | 3129.06i | |
5.9 | −14.1385 | + | 7.49024i | −77.7033 | − | 22.8735i | 143.792 | − | 211.801i | −492.059 | + | 492.059i | 1269.93 | − | 258.620i | − | 3803.67i | −446.563 | + | 4071.58i | 5514.60 | + | 3554.70i | 3271.32 | − | 10642.6i | |
5.10 | −13.8367 | + | 8.03403i | 79.4910 | − | 15.5621i | 126.909 | − | 222.329i | 632.583 | − | 632.583i | −974.868 | + | 853.960i | 4516.63i | 30.1960 | + | 4095.89i | 6076.65 | − | 2474.09i | −3670.67 | + | 13835.0i | ||
5.11 | −13.1212 | − | 9.15614i | −51.1569 | − | 62.8011i | 88.3302 | + | 240.279i | −771.584 | + | 771.584i | 96.2230 | + | 1292.42i | 1052.69i | 1041.03 | − | 3961.50i | −1326.94 | + | 6425.41i | 17188.8 | − | 3059.35i | ||
5.12 | −13.0661 | + | 9.23453i | −70.3483 | + | 40.1511i | 85.4470 | − | 241.319i | 651.651 | − | 651.651i | 548.403 | − | 1174.25i | − | 290.192i | 1112.00 | + | 3942.16i | 3336.77 | − | 5649.13i | −2496.86 | + | 14532.2i | |
5.13 | −12.9672 | − | 9.37299i | 15.1978 | − | 79.5615i | 80.2941 | + | 243.082i | 295.006 | − | 295.006i | −942.801 | + | 889.237i | 1083.57i | 1237.22 | − | 3904.68i | −6099.06 | − | 2418.31i | −6590.48 | + | 1060.30i | ||
5.14 | −12.8620 | + | 9.51679i | −13.6278 | + | 79.8454i | 74.8613 | − | 244.810i | −792.195 | + | 792.195i | −584.591 | − | 1156.66i | 2103.96i | 1366.94 | + | 3861.18i | −6189.56 | − | 2176.24i | 2650.04 | − | 17728.3i | ||
5.15 | −12.2522 | + | 10.2899i | 74.9445 | + | 30.7298i | 44.2340 | − | 252.149i | −204.371 | + | 204.371i | −1234.44 | + | 394.667i | − | 2167.67i | 2052.64 | + | 3544.56i | 4672.36 | + | 4606.06i | 401.033 | − | 4606.97i | |
5.16 | −11.8041 | − | 10.8011i | −12.1951 | + | 80.0767i | 22.6731 | + | 254.994i | 756.913 | − | 756.913i | 1008.87 | − | 813.513i | 1455.04i | 2486.58 | − | 3254.87i | −6263.56 | − | 1953.08i | −17110.2 | + | 759.188i | ||
5.17 | −11.6216 | − | 10.9972i | 63.5896 | + | 50.1733i | 14.1218 | + | 255.610i | −284.826 | + | 284.826i | −187.244 | − | 1282.40i | − | 2397.49i | 2646.89 | − | 3125.89i | 1526.28 | + | 6381.00i | 6442.42 | − | 177.828i | |
5.18 | −10.8590 | − | 11.7508i | 80.1414 | − | 11.7622i | −20.1629 | + | 255.205i | 106.550 | − | 106.550i | −1008.47 | − | 814.001i | 2233.15i | 3217.81 | − | 2534.35i | 6284.30 | − | 1885.27i | −2409.09 | − | 95.0190i | ||
5.19 | −8.62811 | + | 13.4743i | −68.9923 | − | 42.4389i | −107.111 | − | 232.515i | −120.715 | + | 120.715i | 1167.11 | − | 563.453i | 4118.69i | 4057.13 | + | 562.915i | 2958.88 | + | 5855.92i | −585.002 | − | 2668.08i | ||
5.20 | −7.87771 | + | 13.9263i | −23.6488 | − | 77.4709i | −131.884 | − | 219.415i | 784.467 | − | 784.467i | 1265.18 | + | 280.953i | − | 3281.28i | 4094.57 | − | 108.166i | −5442.47 | + | 3664.18i | 4744.92 | + | 17104.5i | |
See next 80 embeddings (of 124 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 48.9.i.a | ✓ | 124 |
3.b | odd | 2 | 1 | inner | 48.9.i.a | ✓ | 124 |
16.e | even | 4 | 1 | inner | 48.9.i.a | ✓ | 124 |
48.i | odd | 4 | 1 | inner | 48.9.i.a | ✓ | 124 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.9.i.a | ✓ | 124 | 1.a | even | 1 | 1 | trivial |
48.9.i.a | ✓ | 124 | 3.b | odd | 2 | 1 | inner |
48.9.i.a | ✓ | 124 | 16.e | even | 4 | 1 | inner |
48.9.i.a | ✓ | 124 | 48.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(48, [\chi])\).