Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [16,13,Mod(3,16)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16.3");
S:= CuspForms(chi, 13);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.f (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: | \(14.6239010764\) |
Analytic rank: | \(0\) |
Dimension: | \(46\) |
Relative dimension: | \(23\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −63.9395 | + | 2.78179i | 398.082 | − | 398.082i | 4080.52 | − | 355.733i | 14857.0 | − | 14857.0i | −24345.8 | + | 26560.6i | 113515. | −259917. | + | 34096.6i | 214502.i | −908623. | + | 991281.i | ||||
3.2 | −62.9799 | + | 11.3812i | −429.462 | + | 429.462i | 3836.94 | − | 1433.57i | −2473.08 | + | 2473.08i | 22159.7 | − | 31935.2i | −198740. | −225334. | + | 133955.i | 162567.i | 127608. | − | 183901.i | ||||
3.3 | −59.3506 | − | 23.9480i | −234.930 | + | 234.930i | 2948.99 | + | 2842.66i | −18167.9 | + | 18167.9i | 19569.4 | − | 8317.14i | 112043. | −106948. | − | 239336.i | 421057.i | 1.51336e6 | − | 643191.i | ||||
3.4 | −55.0768 | + | 32.5967i | 769.939 | − | 769.939i | 1970.91 | − | 3590.64i | −17644.5 | + | 17644.5i | −17308.3 | + | 67503.3i | 15384.5 | 8491.78 | + | 262006.i | − | 654171.i | 396649. | − | 1.54695e6i | |||
3.5 | −50.3472 | − | 39.5115i | 739.301 | − | 739.301i | 973.684 | + | 3978.59i | −805.082 | + | 805.082i | −66432.6 | + | 8010.85i | −97582.7 | 108178. | − | 238782.i | − | 561690.i | 72343.6 | − | 8723.64i | |||
3.6 | −48.8546 | + | 41.3429i | −913.246 | + | 913.246i | 677.536 | − | 4039.57i | 70.8144 | − | 70.8144i | 6860.04 | − | 82372.5i | 225982. | 133907. | + | 225363.i | − | 1.13660e6i | −531.937 | + | 6387.27i | |||
3.7 | −48.1592 | − | 42.1508i | −640.742 | + | 640.742i | 542.622 | + | 4059.90i | 10737.8 | − | 10737.8i | 57865.4 | − | 3849.85i | 25233.9 | 144996. | − | 218394.i | − | 289658.i | −969728. | + | 64517.2i | |||
3.8 | −34.8144 | + | 53.7025i | 78.2777 | − | 78.2777i | −1671.91 | − | 3739.24i | 5406.83 | − | 5406.83i | 1478.51 | + | 6928.90i | −30061.3 | 259013. | + | 40393.8i | 519186.i | 102124. | + | 478596.i | ||||
3.9 | −18.3780 | − | 61.3046i | 37.8347 | − | 37.8347i | −3420.50 | + | 2253.31i | 5270.38 | − | 5270.38i | −3014.77 | − | 1624.12i | −115235. | 201000. | + | 168281.i | 528578.i | −419958. | − | 226239.i | ||||
3.10 | −11.7829 | − | 62.9060i | 252.451 | − | 252.451i | −3818.33 | + | 1482.43i | −9569.73 | + | 9569.73i | −18855.3 | − | 12906.1i | 168743. | 138244. | + | 222729.i | 403978.i | 714752. | + | 489234.i | ||||
3.11 | 2.53819 | + | 63.9496i | −322.288 | + | 322.288i | −4083.12 | + | 324.633i | −16463.9 | + | 16463.9i | −21428.3 | − | 19792.2i | 21051.9 | −31123.9 | − | 260290.i | 323701.i | −1.09465e6 | − | 1.01107e6i | ||||
3.12 | 3.36222 | + | 63.9116i | 837.232 | − | 837.232i | −4073.39 | + | 429.769i | 5671.68 | − | 5671.68i | 56323.8 | + | 50693.9i | −4585.41 | −41162.9 | − | 258892.i | − | 870475.i | 381556. | + | 343417.i | |||
3.13 | 4.15344 | − | 63.8651i | −980.484 | + | 980.484i | −4061.50 | − | 530.520i | −15901.7 | + | 15901.7i | 58546.3 | + | 66691.1i | −94838.7 | −50750.9 | + | 257184.i | − | 1.39126e6i | 949514. | + | 1.08161e6i | |||
3.14 | 14.0860 | + | 62.4306i | −832.058 | + | 832.058i | −3699.17 | + | 1758.79i | 17612.4 | − | 17612.4i | −63666.3 | − | 40225.6i | −164049. | −161909. | − | 206167.i | − | 853201.i | 1.34764e6 | + | 851466.i | |||
3.15 | 24.3843 | − | 59.1727i | 949.435 | − | 949.435i | −2906.81 | − | 2885.77i | 10755.1 | − | 10755.1i | −33029.3 | − | 79332.0i | 35819.9 | −241639. | + | 101636.i | − | 1.27141e6i | −374153. | − | 898665.i | |||
3.16 | 26.6058 | − | 58.2077i | −435.597 | + | 435.597i | −2680.26 | − | 3097.32i | 17337.6 | − | 17337.6i | 13765.7 | + | 36944.5i | 102408. | −251598. | + | 73605.0i | 151952.i | −547901. | − | 1.47046e6i | ||||
3.17 | 40.2856 | + | 49.7300i | 87.0633 | − | 87.0633i | −850.147 | + | 4006.80i | 8119.76 | − | 8119.76i | 7837.05 | + | 822.264i | 149756. | −233507. | + | 119138.i | 516281.i | 730905. | + | 76686.6i | ||||
3.18 | 41.7334 | − | 48.5213i | 193.300 | − | 193.300i | −612.641 | − | 4049.92i | −10355.8 | + | 10355.8i | −1312.10 | − | 17446.2i | −108466. | −222075. | − | 139291.i | 456712.i | 70294.5 | + | 934665.i | ||||
3.19 | 49.5161 | + | 40.5482i | 543.347 | − | 543.347i | 807.695 | + | 4015.58i | −12801.7 | + | 12801.7i | 48936.2 | − | 4872.73i | −188146. | −122830. | + | 231586.i | − | 59011.2i | −1.15298e6 | + | 114806.i | |||
3.20 | 58.9783 | − | 24.8508i | −513.579 | + | 513.579i | 2860.88 | − | 2931.31i | 208.882 | − | 208.882i | −17527.2 | + | 43052.8i | 42912.4 | 95884.1 | − | 243979.i | 3914.51i | 7128.62 | − | 17510.4i | ||||
See all 46 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 16.13.f.a | ✓ | 46 |
4.b | odd | 2 | 1 | 64.13.f.a | 46 | ||
8.b | even | 2 | 1 | 128.13.f.b | 46 | ||
8.d | odd | 2 | 1 | 128.13.f.a | 46 | ||
16.e | even | 4 | 1 | 64.13.f.a | 46 | ||
16.e | even | 4 | 1 | 128.13.f.a | 46 | ||
16.f | odd | 4 | 1 | inner | 16.13.f.a | ✓ | 46 |
16.f | odd | 4 | 1 | 128.13.f.b | 46 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.13.f.a | ✓ | 46 | 1.a | even | 1 | 1 | trivial |
16.13.f.a | ✓ | 46 | 16.f | odd | 4 | 1 | inner |
64.13.f.a | 46 | 4.b | odd | 2 | 1 | ||
64.13.f.a | 46 | 16.e | even | 4 | 1 | ||
128.13.f.a | 46 | 8.d | odd | 2 | 1 | ||
128.13.f.a | 46 | 16.e | even | 4 | 1 | ||
128.13.f.b | 46 | 8.b | even | 2 | 1 | ||
128.13.f.b | 46 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(16, [\chi])\).