Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [16,15,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, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16.3");
S:= CuspForms(chi, 15);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 15 \) |
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: | \(19.8926349043\) |
Analytic rank: | \(0\) |
Dimension: | \(54\) |
Relative dimension: | \(27\) 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 | −127.662 | + | 9.30177i | 1995.33 | − | 1995.33i | 16211.0 | − | 2374.96i | −5949.31 | + | 5949.31i | −236166. | + | 273287.i | −842293. | −2.04742e6 | + | 453982.i | − | 3.17969e6i | 704159. | − | 814837.i | |||
3.2 | −123.321 | − | 34.2916i | −39.1610 | + | 39.1610i | 14032.2 | + | 8457.74i | −4748.26 | + | 4748.26i | 6172.27 | − | 3486.48i | 903713. | −1.44043e6 | − | 1.52420e6i | 4.77990e6i | 748385. | − | 422735.i | ||||
3.3 | −122.284 | − | 37.8242i | −2920.51 | + | 2920.51i | 13522.7 | + | 9250.58i | −497.809 | + | 497.809i | 467597. | − | 246665.i | −436996. | −1.30371e6 | − | 1.64268e6i | − | 1.22758e7i | 79703.3 | − | 42044.8i | |||
3.4 | −118.943 | + | 47.2917i | −1108.18 | + | 1108.18i | 11911.0 | − | 11250.1i | −55052.8 | + | 55052.8i | 79403.1 | − | 184219.i | 519580. | −884698. | + | 1.90141e6i | 2.32682e6i | 3.94462e6 | − | 9.15171e6i | ||||
3.5 | −108.596 | + | 67.7560i | −1164.54 | + | 1164.54i | 7202.24 | − | 14716.1i | 98063.2 | − | 98063.2i | 47560.0 | − | 205370.i | −431458. | 214968. | + | 2.08611e6i | 2.07065e6i | −4.00491e6 | + | 1.72937e7i | ||||
3.6 | −99.8787 | − | 80.0515i | 223.330 | − | 223.330i | 3567.51 | + | 15990.9i | 87742.2 | − | 87742.2i | −40183.8 | + | 4428.01i | −415149. | 923776. | − | 1.88273e6i | 4.68322e6i | −1.57875e7 | + | 1.73968e6i | ||||
3.7 | −85.4809 | + | 95.2734i | 2412.96 | − | 2412.96i | −1770.02 | − | 16288.1i | 56427.7 | − | 56427.7i | 23628.8 | + | 436153.i | 1.31174e6 | 1.70313e6 | + | 1.22369e6i | − | 6.86182e6i | 552565. | + | 1.01996e7i | |||
3.8 | −84.2589 | − | 96.3558i | 2495.14 | − | 2495.14i | −2184.86 | + | 16237.7i | −43850.1 | + | 43850.1i | −450659. | − | 30183.2i | 1.15759e6 | 1.74869e6 | − | 1.15764e6i | − | 7.66847e6i | 7.91997e6 | + | 530446.i | |||
3.9 | −79.3579 | − | 100.431i | −417.666 | + | 417.666i | −3788.66 | + | 15939.9i | −101107. | + | 101107.i | 75091.6 | + | 8801.42i | −1.56037e6 | 1.90152e6 | − | 884462.i | 4.43408e6i | 1.81780e7 | + | 2.13062e6i | ||||
3.10 | −72.6989 | + | 105.351i | 692.853 | − | 692.853i | −5813.74 | − | 15317.8i | −62322.7 | + | 62322.7i | 22623.2 | + | 123363.i | −300008. | 2.03640e6 | + | 501105.i | 3.82288e6i | −2.03498e6 | − | 1.10966e7i | ||||
3.11 | −40.7689 | + | 121.334i | −2416.37 | + | 2416.37i | −13059.8 | − | 9893.30i | −20189.0 | + | 20189.0i | −194674. | − | 391700.i | −480275. | 1.73283e6 | − | 1.18125e6i | − | 6.89468e6i | −1.62653e6 | − | 3.27270e6i | |||
3.12 | −36.7331 | − | 122.616i | −1658.66 | + | 1658.66i | −13685.4 | + | 9008.14i | 7093.51 | − | 7093.51i | 264306. | + | 142450.i | 747019. | 1.60725e6 | + | 1.34715e6i | − | 719331.i | −1.13034e6 | − | 609211.i | |||
3.13 | −11.9550 | − | 127.440i | 1873.37 | − | 1873.37i | −16098.2 | + | 3047.10i | 56067.7 | − | 56067.7i | −261139. | − | 216347.i | −548166. | 580777. | + | 2.01513e6i | − | 2.23606e6i | −7.81558e6 | − | 6.47501e6i | |||
3.14 | −6.21757 | + | 127.849i | 1390.71 | − | 1390.71i | −16306.7 | − | 1589.82i | 31054.7 | − | 31054.7i | 169154. | + | 186448.i | −1.17718e6 | 304644. | − | 2.07491e6i | 914816.i | 3.77722e6 | + | 4.16339e6i | ||||
3.15 | 4.45002 | + | 127.923i | −837.187 | + | 837.187i | −16344.4 | + | 1138.52i | 37060.2 | − | 37060.2i | −110821. | − | 103370.i | 1.50447e6 | −218375. | − | 2.08575e6i | 3.38120e6i | 4.90575e6 | + | 4.57592e6i | ||||
3.16 | 42.3109 | − | 120.805i | 859.991 | − | 859.991i | −12803.6 | − | 10222.7i | −55370.8 | + | 55370.8i | −67504.0 | − | 140278.i | 175518. | −1.77668e6 | + | 1.11420e6i | 3.30380e6i | 4.34627e6 | + | 9.03184e6i | ||||
3.17 | 47.5205 | + | 118.852i | 2009.03 | − | 2009.03i | −11867.6 | + | 11295.8i | −90795.6 | + | 90795.6i | 334247. | + | 143307.i | 656977. | −1.90649e6 | − | 873704.i | − | 3.28943e6i | −1.51059e7 | − | 6.47658e6i | |||
3.18 | 57.2769 | − | 114.470i | −1878.56 | + | 1878.56i | −9822.72 | − | 13113.0i | 35992.2 | − | 35992.2i | 107440. | + | 322636.i | −784164. | −2.06365e6 | + | 373337.i | − | 2.27499e6i | −2.05850e6 | − | 6.18154e6i | |||
3.19 | 74.1318 | + | 104.348i | −498.544 | + | 498.544i | −5392.95 | + | 15471.0i | 40758.2 | − | 40758.2i | −88979.9 | − | 15064.0i | −589264. | −2.01415e6 | + | 584149.i | 4.28588e6i | 7.27451e6 | + | 1.23155e6i | ||||
3.20 | 83.7629 | + | 96.7872i | −2050.98 | + | 2050.98i | −2351.54 | + | 16214.4i | −70340.6 | + | 70340.6i | −370305. | − | 26712.6i | −283091. | −1.76632e6 | + | 1.13056e6i | − | 3.63010e6i | −1.27000e7 | − | 916137.i | |||
See all 54 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 | 16.15.f.a | ✓ | 54 |
4.b | odd | 2 | 1 | 64.15.f.a | 54 | ||
16.e | even | 4 | 1 | 64.15.f.a | 54 | ||
16.f | odd | 4 | 1 | inner | 16.15.f.a | ✓ | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.15.f.a | ✓ | 54 | 1.a | even | 1 | 1 | trivial |
16.15.f.a | ✓ | 54 | 16.f | odd | 4 | 1 | inner |
64.15.f.a | 54 | 4.b | odd | 2 | 1 | ||
64.15.f.a | 54 | 16.e | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(16, [\chi])\).