Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [731,2,Mod(9,731)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, base_ring=CyclotomicField(168))
chi = DirichletCharacter(H, H._module([21, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("731.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 731 = 17 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 731.bl (of order \(168\), degree \(48\), 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: | \(5.83706438776\) |
Analytic rank: | \(0\) |
Dimension: | \(3072\) |
Relative dimension: | \(64\) over \(\Q(\zeta_{168})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{168}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −2.72251 | − | 0.306753i | −0.620611 | − | 0.373954i | 5.36808 | + | 1.22523i | 0.404374 | − | 0.264759i | 1.57490 | + | 1.20847i | −5.03855 | − | 0.663338i | −9.06682 | − | 3.17262i | −1.15649 | − | 2.18819i | −1.18213 | + | 0.596764i |
9.2 | −2.63781 | − | 0.297210i | −1.45043 | − | 0.873969i | 4.91985 | + | 1.12292i | −0.178314 | + | 0.116749i | 3.56621 | + | 2.73645i | 3.75466 | + | 0.494311i | −7.63281 | − | 2.67084i | −0.0618808 | − | 0.117084i | 0.505058 | − | 0.254965i |
9.3 | −2.55439 | − | 0.287810i | 1.48660 | + | 0.895766i | 4.49220 | + | 1.02532i | −2.99252 | + | 1.95932i | −3.53955 | − | 2.71599i | 0.541511 | + | 0.0712913i | −6.32713 | − | 2.21396i | 0.00578988 | + | 0.0109550i | 8.20797 | − | 4.14357i |
9.4 | −2.54211 | − | 0.286426i | 2.35858 | + | 1.42118i | 4.43040 | + | 1.01121i | 1.38702 | − | 0.908135i | −5.58870 | − | 4.28836i | 3.61781 | + | 0.476294i | −6.14365 | − | 2.14976i | 2.14134 | + | 4.05160i | −3.78607 | + | 1.91129i |
9.5 | −2.52269 | − | 0.284239i | 0.0154490 | + | 0.00930894i | 4.33332 | + | 0.989053i | 0.766036 | − | 0.501553i | −0.0363272 | − | 0.0278748i | −0.223013 | − | 0.0293603i | −5.85813 | − | 2.04985i | −1.40165 | − | 2.65206i | −2.07503 | + | 1.04753i |
9.6 | −2.50024 | − | 0.281710i | 0.980817 | + | 0.591000i | 4.22200 | + | 0.963643i | 2.79856 | − | 1.83232i | −2.28579 | − | 1.75395i | −0.0420865 | − | 0.00554079i | −5.53481 | − | 1.93671i | −0.789084 | − | 1.49302i | −7.51326 | + | 3.79287i |
9.7 | −2.31943 | − | 0.261336i | −2.95396 | − | 1.77993i | 3.36158 | + | 0.767259i | −1.47851 | + | 0.968034i | 6.38632 | + | 4.90040i | −1.74587 | − | 0.229848i | −3.19018 | − | 1.11629i | 4.15589 | + | 7.86334i | 3.68227 | − | 1.85889i |
9.8 | −2.29109 | − | 0.258143i | −0.766885 | − | 0.462093i | 3.23258 | + | 0.737816i | −3.34001 | + | 2.18683i | 1.63771 | + | 1.25666i | −2.68123 | − | 0.352990i | −2.86326 | − | 1.00190i | −1.02722 | − | 1.94360i | 8.21676 | − | 4.14801i |
9.9 | −2.25200 | − | 0.253740i | 2.54761 | + | 1.53509i | 3.05728 | + | 0.697804i | 0.169346 | − | 0.110877i | −5.34772 | − | 4.10345i | −3.68545 | − | 0.485199i | −2.42979 | − | 0.850219i | 2.73203 | + | 5.16926i | −0.409502 | + | 0.206726i |
9.10 | −2.13253 | − | 0.240278i | −1.98754 | − | 1.19761i | 2.54008 | + | 0.579756i | −1.15205 | + | 0.754294i | 3.95072 | + | 3.03150i | 0.0673855 | + | 0.00887147i | −1.22629 | − | 0.429097i | 1.11425 | + | 2.10826i | 2.63803 | − | 1.33174i |
9.11 | −2.06687 | − | 0.232880i | −0.844141 | − | 0.508645i | 2.26786 | + | 0.517625i | 3.26327 | − | 2.13659i | 1.62628 | + | 1.24789i | 1.78259 | + | 0.234683i | −0.640380 | − | 0.224079i | −0.947951 | − | 1.79361i | −7.24233 | + | 3.65610i |
9.12 | −1.98773 | − | 0.223963i | 0.209805 | + | 0.126420i | 1.95104 | + | 0.445312i | −1.38577 | + | 0.907318i | −0.388722 | − | 0.298277i | 2.47032 | + | 0.325224i | −0.00229218 | 0.000802070i | −1.37377 | − | 2.59930i | 2.95774 | − | 1.49314i | |
9.13 | −1.75957 | − | 0.198256i | 1.94400 | + | 1.17137i | 1.10693 | + | 0.252650i | −1.51027 | + | 0.988831i | −3.18837 | − | 2.44652i | 0.127518 | + | 0.0167880i | 1.44504 | + | 0.505643i | 1.00521 | + | 1.90194i | 2.85347 | − | 1.44050i |
9.14 | −1.71083 | − | 0.192764i | 1.34544 | + | 0.810704i | 0.939930 | + | 0.214533i | 1.48366 | − | 0.971406i | −2.14554 | − | 1.64633i | −3.99522 | − | 0.525981i | 1.68338 | + | 0.589040i | −0.248850 | − | 0.470847i | −2.72554 | + | 1.37592i |
9.15 | −1.70400 | − | 0.191995i | −1.59121 | − | 0.958800i | 0.916901 | + | 0.209277i | 0.260745 | − | 0.170719i | 2.52735 | + | 1.93930i | −1.40514 | − | 0.184990i | 1.71489 | + | 0.600066i | 0.210860 | + | 0.398966i | −0.477086 | + | 0.240844i |
9.16 | −1.69079 | − | 0.190506i | −1.80830 | − | 1.08960i | 0.872615 | + | 0.199169i | 2.36534 | − | 1.54868i | 2.84987 | + | 2.18678i | −2.81774 | − | 0.370962i | 1.77455 | + | 0.620940i | 0.680893 | + | 1.28831i | −4.29432 | + | 2.16787i |
9.17 | −1.58124 | − | 0.178163i | 0.373386 | + | 0.224987i | 0.518729 | + | 0.118397i | −0.176659 | + | 0.115666i | −0.550330 | − | 0.422283i | 4.75303 | + | 0.625749i | 2.20476 | + | 0.771479i | −1.31301 | − | 2.48433i | 0.299949 | − | 0.151421i |
9.18 | −1.52146 | − | 0.171428i | −2.06462 | − | 1.24405i | 0.335610 | + | 0.0766007i | 1.34334 | − | 0.879534i | 2.92798 | + | 2.24672i | 4.03118 | + | 0.530715i | 2.39286 | + | 0.837297i | 1.31318 | + | 2.48465i | −2.19462 | + | 1.10789i |
9.19 | −1.36186 | − | 0.153444i | −0.627515 | − | 0.378115i | −0.118749 | − | 0.0271036i | −2.79353 | + | 1.82903i | 0.796566 | + | 0.611227i | −4.13974 | − | 0.545007i | 2.74469 | + | 0.960410i | −1.15100 | − | 2.17780i | 4.08504 | − | 2.06222i |
9.20 | −1.22085 | − | 0.137557i | 2.07165 | + | 1.24829i | −0.478291 | − | 0.109167i | 1.32412 | − | 0.866951i | −2.35747 | − | 1.80895i | 3.11502 | + | 0.410100i | 2.88818 | + | 1.01062i | 1.33169 | + | 2.51968i | −1.73581 | + | 0.876279i |
See next 80 embeddings (of 3072 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.d | even | 8 | 1 | inner |
43.g | even | 21 | 1 | inner |
731.bl | even | 168 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 731.2.bl.a | ✓ | 3072 |
17.d | even | 8 | 1 | inner | 731.2.bl.a | ✓ | 3072 |
43.g | even | 21 | 1 | inner | 731.2.bl.a | ✓ | 3072 |
731.bl | even | 168 | 1 | inner | 731.2.bl.a | ✓ | 3072 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
731.2.bl.a | ✓ | 3072 | 1.a | even | 1 | 1 | trivial |
731.2.bl.a | ✓ | 3072 | 17.d | even | 8 | 1 | inner |
731.2.bl.a | ✓ | 3072 | 43.g | even | 21 | 1 | inner |
731.2.bl.a | ✓ | 3072 | 731.bl | even | 168 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(731, [\chi])\).