Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [232,4,Mod(45,232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(232, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 7, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("232.45");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 232.s (of order \(14\), degree \(6\), 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: | \(13.6884431213\) |
Analytic rank: | \(0\) |
Dimension: | \(528\) |
Relative dimension: | \(88\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
45.1 | −2.82734 | + | 0.0784509i | 3.89696 | + | 3.10772i | 7.98769 | − | 0.443615i | −2.39521 | + | 4.97371i | −11.2618 | − | 8.48087i | 4.79075 | − | 6.00741i | −22.5491 | + | 1.88089i | −0.479694 | − | 2.10168i | 6.38189 | − | 14.2503i |
45.2 | −2.82725 | − | 0.0816801i | 5.31996 | + | 4.24253i | 7.98666 | + | 0.461860i | −1.20408 | + | 2.50030i | −14.6943 | − | 12.4292i | −2.55349 | + | 3.20198i | −22.5425 | − | 1.95814i | 4.29489 | + | 18.8171i | 3.60846 | − | 6.97062i |
45.3 | −2.82293 | + | 0.176241i | −2.18494 | − | 1.74244i | 7.93788 | − | 0.995030i | 5.84824 | − | 12.1440i | 6.47504 | + | 4.53370i | −7.03876 | + | 8.82633i | −22.2327 | + | 4.20788i | −4.27016 | − | 18.7088i | −14.3689 | + | 35.3124i |
45.4 | −2.80684 | − | 0.348789i | −3.62321 | − | 2.88941i | 7.75669 | + | 1.95799i | 3.00172 | − | 6.23314i | 9.16197 | + | 9.37385i | 15.7972 | − | 19.8091i | −21.0889 | − | 8.20120i | −1.22913 | − | 5.38517i | −10.5994 | + | 16.4485i |
45.5 | −2.76834 | − | 0.579913i | 0.136331 | + | 0.108720i | 7.32740 | + | 3.21079i | −3.15436 | + | 6.55010i | −0.314361 | − | 0.380034i | −17.1707 | + | 21.5314i | −18.4228 | − | 13.1378i | −6.00130 | − | 26.2934i | 12.5308 | − | 16.3036i |
45.6 | −2.74857 | + | 0.667358i | −6.45303 | − | 5.14612i | 7.10927 | − | 3.66856i | −2.02619 | + | 4.20744i | 21.1709 | + | 9.83798i | 9.78624 | − | 12.2716i | −17.0921 | + | 14.8277i | 9.15096 | + | 40.0930i | 2.76127 | − | 12.9166i |
45.7 | −2.73296 | − | 0.728638i | −6.28332 | − | 5.01078i | 6.93817 | + | 3.98268i | 7.75346 | − | 16.1002i | 13.5210 | + | 18.2725i | −10.1355 | + | 12.7095i | −16.0598 | − | 15.9399i | 8.36410 | + | 36.6455i | −32.9211 | + | 38.3518i |
45.8 | −2.71900 | + | 0.779112i | −4.55540 | − | 3.63281i | 6.78597 | − | 4.23682i | −6.30657 | + | 13.0957i | 15.2165 | + | 6.32847i | −20.2050 | + | 25.3362i | −15.1501 | + | 16.8069i | 1.54631 | + | 6.77482i | 6.94456 | − | 40.5209i |
45.9 | −2.65907 | + | 0.964034i | 3.90835 | + | 3.11681i | 6.14128 | − | 5.12686i | 9.06800 | − | 18.8299i | −13.3973 | − | 4.52001i | −12.7745 | + | 16.0187i | −11.3876 | + | 19.5531i | −0.447331 | − | 1.95988i | −5.95975 | + | 58.8118i |
45.10 | −2.62649 | + | 1.04956i | 1.06985 | + | 0.853177i | 5.79685 | − | 5.51331i | −9.50600 | + | 19.7394i | −3.70541 | − | 1.11799i | 9.71878 | − | 12.1870i | −9.43879 | + | 20.5648i | −5.59140 | − | 24.4975i | 4.24968 | − | 61.8224i |
45.11 | −2.59668 | − | 1.12126i | −2.56621 | − | 2.04649i | 5.48553 | + | 5.82314i | −6.45055 | + | 13.3947i | 4.36900 | + | 8.19149i | 10.5690 | − | 13.2531i | −7.71493 | − | 21.2716i | −3.61072 | − | 15.8196i | 31.7690 | − | 27.5491i |
45.12 | −2.57901 | − | 1.16135i | 7.45001 | + | 5.94119i | 5.30255 | + | 5.99024i | 4.54665 | − | 9.44122i | −12.3139 | − | 23.9744i | −7.61140 | + | 9.54439i | −6.71857 | − | 21.6070i | 14.1969 | + | 62.2008i | −22.6904 | + | 19.0687i |
45.13 | −2.53868 | + | 1.24705i | 0.541991 | + | 0.432224i | 4.88976 | − | 6.33169i | 3.55640 | − | 7.38494i | −1.91494 | − | 0.421388i | 17.4431 | − | 21.8729i | −4.51761 | + | 22.1719i | −5.90113 | − | 25.8545i | 0.180806 | + | 23.1830i |
45.14 | −2.52067 | + | 1.28304i | 7.63258 | + | 6.08678i | 4.70760 | − | 6.46827i | 4.15605 | − | 8.63012i | −27.0489 | − | 5.54985i | 19.9393 | − | 25.0030i | −3.56724 | + | 22.3445i | 15.1993 | + | 66.5927i | 0.596784 | + | 27.0861i |
45.15 | −2.51596 | − | 1.29226i | −7.45001 | − | 5.94119i | 4.66012 | + | 6.50256i | −4.54665 | + | 9.44122i | 11.0664 | + | 24.5752i | −7.61140 | + | 9.54439i | −3.32168 | − | 22.3823i | 14.1969 | + | 62.2008i | 23.6397 | − | 17.8783i |
45.16 | −2.49565 | − | 1.33107i | 2.56621 | + | 2.04649i | 4.45649 | + | 6.64377i | 6.45055 | − | 13.3947i | −3.68034 | − | 8.52313i | 10.5690 | − | 13.2531i | −2.27848 | − | 22.5124i | −3.61072 | − | 15.8196i | −33.9276 | + | 24.8423i |
45.17 | −2.45878 | + | 1.39801i | −1.85288 | − | 1.47762i | 4.09116 | − | 6.87477i | 0.379073 | − | 0.787153i | 6.62153 | + | 1.04281i | −7.38060 | + | 9.25499i | −0.448292 | + | 22.6230i | −4.75827 | − | 20.8474i | 0.168389 | + | 2.46538i |
45.18 | −2.30539 | + | 1.63865i | 7.18394 | + | 5.72900i | 2.62963 | − | 7.55546i | −8.29334 | + | 17.2213i | −25.9496 | − | 1.43559i | −7.58299 | + | 9.50877i | 6.31845 | + | 21.7273i | 12.7795 | + | 55.9906i | −9.10035 | − | 53.2917i |
45.19 | −2.27365 | − | 1.68242i | 6.28332 | + | 5.01078i | 2.33894 | + | 7.65045i | −7.75346 | + | 16.1002i | −5.85582 | − | 21.9639i | −10.1355 | + | 12.7095i | 7.55334 | − | 21.3295i | 8.36410 | + | 36.6455i | 44.7159 | − | 23.5617i |
45.20 | −2.21092 | + | 1.76404i | −7.04646 | − | 5.61936i | 1.77633 | − | 7.80030i | 5.26037 | − | 10.9233i | 25.4919 | − | 0.00627760i | 1.30089 | − | 1.63127i | 9.83273 | + | 20.3793i | 12.0673 | + | 52.8702i | 7.63884 | + | 33.4300i |
See next 80 embeddings (of 528 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
29.d | even | 7 | 1 | inner |
232.s | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 232.4.s.a | ✓ | 528 |
8.b | even | 2 | 1 | inner | 232.4.s.a | ✓ | 528 |
29.d | even | 7 | 1 | inner | 232.4.s.a | ✓ | 528 |
232.s | even | 14 | 1 | inner | 232.4.s.a | ✓ | 528 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
232.4.s.a | ✓ | 528 | 1.a | even | 1 | 1 | trivial |
232.4.s.a | ✓ | 528 | 8.b | even | 2 | 1 | inner |
232.4.s.a | ✓ | 528 | 29.d | even | 7 | 1 | inner |
232.4.s.a | ✓ | 528 | 232.s | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).