Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [232,4,Mod(5,232)]
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, 11]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("232.5");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 232.o (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −2.82003 | + | 0.217821i | 1.61196 | + | 7.06244i | 7.90511 | − | 1.22852i | 3.64677 | − | 2.90821i | −6.08410 | − | 19.5652i | −4.02149 | − | 17.6193i | −22.0250 | + | 5.18635i | −22.9535 | + | 11.0538i | −9.65054 | + | 8.99556i |
5.2 | −2.81978 | + | 0.220999i | −0.700106 | − | 3.06736i | 7.90232 | − | 1.24634i | 5.00974 | − | 3.99514i | 2.65203 | + | 8.49457i | −0.0793850 | − | 0.347808i | −22.0074 | + | 5.26080i | 15.4076 | − | 7.41990i | −13.2435 | + | 12.3726i |
5.3 | −2.81099 | + | 0.313561i | −0.352368 | − | 1.54383i | 7.80336 | − | 1.76284i | 0.408717 | − | 0.325941i | 1.47459 | + | 4.22919i | 7.22862 | + | 31.6706i | −21.3824 | + | 7.40216i | 22.0669 | − | 10.6269i | −1.04670 | + | 1.04438i |
5.4 | −2.79039 | − | 0.462275i | −1.14212 | − | 5.00394i | 7.57260 | + | 2.57986i | −12.2351 | + | 9.75720i | 0.873762 | + | 14.4909i | −1.21693 | − | 5.33170i | −19.9380 | − | 10.6994i | 0.591207 | − | 0.284710i | 38.6514 | − | 21.5705i |
5.5 | −2.77642 | − | 0.539896i | −1.75561 | − | 7.69184i | 7.41702 | + | 2.99796i | 10.9392 | − | 8.72370i | 0.721524 | + | 22.3036i | 0.861400 | + | 3.77404i | −18.9742 | − | 12.3280i | −31.7560 | + | 15.2929i | −35.0816 | + | 18.3146i |
5.6 | −2.74634 | + | 0.676489i | 1.20561 | + | 5.28214i | 7.08473 | − | 3.71573i | −12.2064 | + | 9.73427i | −6.88432 | − | 13.6909i | −1.69935 | − | 7.44533i | −16.9434 | + | 14.9974i | −2.12130 | + | 1.02156i | 26.9377 | − | 34.9911i |
5.7 | −2.69280 | + | 0.865356i | −1.38368 | − | 6.06232i | 6.50232 | − | 4.66046i | −10.5510 | + | 8.41417i | 8.97204 | + | 15.1272i | 0.716771 | + | 3.14038i | −13.4765 | + | 18.1765i | −10.5109 | + | 5.06180i | 21.1305 | − | 31.7881i |
5.8 | −2.68679 | − | 0.883840i | 0.957403 | + | 4.19466i | 6.43765 | + | 4.74938i | −0.846064 | + | 0.674714i | 1.13507 | − | 12.1163i | 0.373905 | + | 1.63819i | −13.0989 | − | 18.4504i | 7.64763 | − | 3.68291i | 2.86953 | − | 1.06503i |
5.9 | −2.67889 | − | 0.907495i | 0.0922057 | + | 0.403980i | 6.35291 | + | 4.86216i | 9.57271 | − | 7.63398i | 0.119600 | − | 1.16589i | −5.17627 | − | 22.6787i | −12.6064 | − | 18.7904i | 24.1715 | − | 11.6404i | −32.5720 | + | 11.7634i |
5.10 | −2.64387 | − | 1.00496i | 0.884551 | + | 3.87547i | 5.98010 | + | 5.31398i | −13.4272 | + | 10.7078i | 1.55606 | − | 11.1352i | 5.93063 | + | 25.9838i | −10.4703 | − | 20.0592i | 10.0893 | − | 4.85876i | 46.2607 | − | 14.8163i |
5.11 | −2.63629 | − | 1.02467i | 2.16297 | + | 9.47659i | 5.90010 | + | 5.40267i | 5.65159 | − | 4.50699i | 4.00817 | − | 27.1994i | 3.79065 | + | 16.6079i | −10.0184 | − | 20.2887i | −60.8011 | + | 29.2803i | −19.5174 | + | 6.09074i |
5.12 | −2.63112 | + | 1.03788i | 1.25801 | + | 5.51171i | 5.84561 | − | 5.46158i | 15.8882 | − | 12.6704i | −9.03048 | − | 13.1963i | 3.23241 | + | 14.1621i | −9.71203 | + | 20.4371i | −4.47017 | + | 2.15272i | −28.6535 | + | 49.8276i |
5.13 | −2.61176 | + | 1.08569i | −1.99935 | − | 8.75970i | 5.64256 | − | 5.67111i | −2.62310 | + | 2.09185i | 14.7321 | + | 20.7076i | −7.04128 | − | 30.8499i | −8.57995 | + | 20.9376i | −48.4089 | + | 23.3125i | 4.57981 | − | 8.31129i |
5.14 | −2.53885 | + | 1.24668i | −0.295943 | − | 1.29661i | 4.89155 | − | 6.33030i | 9.62722 | − | 7.67745i | 2.36782 | + | 2.92296i | −7.01745 | − | 30.7455i | −4.52705 | + | 22.1699i | 22.7325 | − | 10.9474i | −14.8707 | + | 31.4940i |
5.15 | −2.42154 | − | 1.46155i | −0.539523 | − | 2.36380i | 3.72772 | + | 7.07843i | −9.56069 | + | 7.62440i | −2.14835 | + | 6.51259i | −5.30344 | − | 23.2359i | 1.31868 | − | 22.5890i | 19.0297 | − | 9.16421i | 34.2951 | − | 4.48932i |
5.16 | −2.36811 | − | 1.54663i | −1.97425 | − | 8.64975i | 3.21588 | + | 7.32517i | 0.213915 | − | 0.170592i | −8.70272 | + | 23.5370i | −1.55828 | − | 6.82725i | 3.71379 | − | 22.3206i | −46.5943 | + | 22.4387i | −0.770416 | + | 0.0731320i |
5.17 | −2.29689 | + | 1.65054i | 0.295031 | + | 1.29261i | 2.55142 | − | 7.58223i | −6.51562 | + | 5.19604i | −2.81117 | − | 2.48203i | 0.828590 | + | 3.63029i | 6.65448 | + | 21.6268i | 22.7424 | − | 10.9521i | 6.38940 | − | 22.6890i |
5.18 | −2.26638 | − | 1.69219i | 0.0472863 | + | 0.207175i | 2.27299 | + | 7.67030i | 16.2175 | − | 12.9330i | 0.243410 | − | 0.549555i | 5.54237 | + | 24.2827i | 7.82813 | − | 21.2302i | 24.2855 | − | 11.6953i | −58.6401 | + | 1.86812i |
5.19 | −2.21344 | + | 1.76088i | 2.00311 | + | 8.77622i | 1.79864 | − | 7.79519i | −6.00699 | + | 4.79041i | −19.8876 | − | 15.8984i | 5.84411 | + | 25.6047i | 9.74518 | + | 20.4213i | −48.6834 | + | 23.4447i | 4.86079 | − | 21.1809i |
5.20 | −2.20926 | + | 1.76611i | −2.00311 | − | 8.77622i | 1.76169 | − | 7.80362i | 6.00699 | − | 4.79041i | 19.9252 | + | 15.8513i | 5.84411 | + | 25.6047i | 9.89003 | + | 20.3516i | −48.6834 | + | 23.4447i | −4.81061 | + | 21.1923i |
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.e | even | 14 | 1 | inner |
232.o | 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.o.a | ✓ | 528 |
8.b | even | 2 | 1 | inner | 232.4.o.a | ✓ | 528 |
29.e | even | 14 | 1 | inner | 232.4.o.a | ✓ | 528 |
232.o | even | 14 | 1 | inner | 232.4.o.a | ✓ | 528 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
232.4.o.a | ✓ | 528 | 1.a | even | 1 | 1 | trivial |
232.4.o.a | ✓ | 528 | 8.b | even | 2 | 1 | inner |
232.4.o.a | ✓ | 528 | 29.e | even | 14 | 1 | inner |
232.4.o.a | ✓ | 528 | 232.o | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).