Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,4,Mod(43,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.43");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 240.bc (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: | \(14.1604584014\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(36\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −2.80743 | + | 0.344004i | − | 3.00000i | 7.76332 | − | 1.93153i | −8.98271 | + | 6.65664i | 1.03201 | + | 8.42229i | −25.3381 | + | 25.3381i | −21.1305 | + | 8.09326i | −9.00000 | 22.9284 | − | 21.7781i | |||
43.2 | −2.77701 | + | 0.536838i | − | 3.00000i | 7.42361 | − | 2.98161i | 8.09867 | − | 7.70789i | 1.61051 | + | 8.33104i | −10.5146 | + | 10.5146i | −19.0148 | + | 12.2653i | −9.00000 | −18.3522 | + | 25.7526i | |||
43.3 | −2.75906 | − | 0.622574i | − | 3.00000i | 7.22480 | + | 3.43543i | −5.62604 | + | 9.66166i | −1.86772 | + | 8.27717i | 17.9728 | − | 17.9728i | −17.7948 | − | 13.9765i | −9.00000 | 21.5377 | − | 23.1544i | |||
43.4 | −2.73499 | − | 0.720989i | − | 3.00000i | 6.96035 | + | 3.94380i | 10.6538 | + | 3.39051i | −2.16297 | + | 8.20497i | 4.24019 | − | 4.24019i | −16.1931 | − | 15.8046i | −9.00000 | −26.6937 | − | 16.9543i | |||
43.5 | −2.72114 | + | 0.771611i | − | 3.00000i | 6.80923 | − | 4.19933i | −7.55101 | − | 8.24514i | 2.31483 | + | 8.16343i | 21.3060 | − | 21.3060i | −15.2886 | + | 16.6810i | −9.00000 | 26.9094 | + | 16.6098i | |||
43.6 | −2.56229 | − | 1.19778i | − | 3.00000i | 5.13063 | + | 6.13813i | −6.29244 | − | 9.24149i | −3.59335 | + | 7.68686i | −4.87036 | + | 4.87036i | −5.79401 | − | 21.8730i | −9.00000 | 5.05375 | + | 31.2163i | |||
43.7 | −2.40790 | + | 1.48391i | − | 3.00000i | 3.59601 | − | 7.14624i | 5.93185 | + | 9.47698i | 4.45174 | + | 7.22371i | 2.77829 | − | 2.77829i | 1.94555 | + | 22.5436i | −9.00000 | −28.3463 | − | 14.0173i | |||
43.8 | −2.09540 | − | 1.89981i | − | 3.00000i | 0.781412 | + | 7.96175i | −5.08323 | + | 9.95795i | −5.69944 | + | 6.28620i | 4.30573 | − | 4.30573i | 13.4885 | − | 18.1676i | −9.00000 | 29.5697 | − | 11.2087i | |||
43.9 | −2.03968 | + | 1.95951i | − | 3.00000i | 0.320620 | − | 7.99357i | −10.5032 | + | 3.83193i | 5.87854 | + | 6.11905i | −0.263728 | + | 0.263728i | 15.0095 | + | 16.9326i | −9.00000 | 13.9144 | − | 28.3970i | |||
43.10 | −1.99616 | − | 2.00384i | − | 3.00000i | −0.0307262 | + | 7.99994i | 11.1550 | − | 0.751921i | −6.01151 | + | 5.98847i | −15.9673 | + | 15.9673i | 16.0919 | − | 15.9076i | −9.00000 | −23.7739 | − | 20.8519i | |||
43.11 | −1.55489 | + | 2.36270i | − | 3.00000i | −3.16466 | − | 7.34744i | 9.88881 | − | 5.21646i | 7.08809 | + | 4.66466i | 14.5115 | − | 14.5115i | 22.2805 | + | 3.94729i | −9.00000 | −3.05106 | + | 31.4752i | |||
43.12 | −1.22495 | − | 2.54941i | − | 3.00000i | −4.99900 | + | 6.24580i | −11.1782 | + | 0.219756i | −7.64824 | + | 3.67485i | −2.70903 | + | 2.70903i | 22.0466 | + | 5.09373i | −9.00000 | 14.2529 | + | 28.2286i | |||
43.13 | −1.09484 | − | 2.60794i | − | 3.00000i | −5.60265 | + | 5.71054i | 4.50871 | − | 10.2309i | −7.82381 | + | 3.28452i | 13.9744 | − | 13.9744i | 21.0267 | + | 8.35923i | −9.00000 | −31.6179 | − | 0.557207i | |||
43.14 | −0.858261 | − | 2.69507i | − | 3.00000i | −6.52678 | + | 4.62614i | 8.57292 | + | 7.17671i | −8.08520 | + | 2.57478i | 10.0043 | − | 10.0043i | 18.0694 | + | 13.6197i | −9.00000 | 11.9839 | − | 29.2641i | |||
43.15 | −0.840991 | + | 2.70051i | − | 3.00000i | −6.58547 | − | 4.54220i | 2.05633 | + | 10.9896i | 8.10152 | + | 2.52297i | −9.67693 | + | 9.67693i | 17.8046 | − | 13.9642i | −9.00000 | −31.4069 | − | 3.68901i | |||
43.16 | −0.704829 | + | 2.73920i | − | 3.00000i | −7.00643 | − | 3.86134i | −5.93312 | − | 9.47618i | 8.21760 | + | 2.11449i | −18.7256 | + | 18.7256i | 15.5153 | − | 16.4704i | −9.00000 | 30.1390 | − | 9.57292i | |||
43.17 | −0.548362 | + | 2.77476i | − | 3.00000i | −7.39860 | − | 3.04315i | −11.0689 | − | 1.57442i | 8.32428 | + | 1.64509i | 21.0997 | − | 21.0997i | 12.5011 | − | 18.8606i | −9.00000 | 10.4384 | − | 29.8503i | |||
43.18 | −0.0874600 | + | 2.82707i | − | 3.00000i | −7.98470 | − | 0.494512i | 10.8481 | − | 2.70533i | 8.48122 | + | 0.262380i | −4.78389 | + | 4.78389i | 2.09637 | − | 22.5301i | −9.00000 | 6.69940 | + | 30.9050i | |||
43.19 | 0.148945 | − | 2.82450i | − | 3.00000i | −7.95563 | − | 0.841393i | 2.45210 | + | 10.9081i | −8.47351 | − | 0.446836i | −14.3673 | + | 14.3673i | −3.56147 | + | 22.3454i | −9.00000 | 31.1753 | − | 5.30125i | |||
43.20 | 0.305329 | − | 2.81190i | − | 3.00000i | −7.81355 | − | 1.71711i | 9.00133 | − | 6.63144i | −8.43570 | − | 0.915987i | −20.8231 | + | 20.8231i | −7.21404 | + | 21.4466i | −9.00000 | −15.8986 | − | 27.3356i | |||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
80.j | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 240.4.bc.b | yes | 72 |
5.c | odd | 4 | 1 | 240.4.y.a | ✓ | 72 | |
16.f | odd | 4 | 1 | 240.4.y.a | ✓ | 72 | |
80.j | even | 4 | 1 | inner | 240.4.bc.b | yes | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.4.y.a | ✓ | 72 | 5.c | odd | 4 | 1 | |
240.4.y.a | ✓ | 72 | 16.f | odd | 4 | 1 | |
240.4.bc.b | yes | 72 | 1.a | even | 1 | 1 | trivial |
240.4.bc.b | yes | 72 | 80.j | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{72} - 10888 T_{7}^{69} + 5505904 T_{7}^{68} - 13062816 T_{7}^{67} + 59274272 T_{7}^{66} - 44819447360 T_{7}^{65} + 12407660711680 T_{7}^{64} - 54447132823488 T_{7}^{63} + \cdots + 92\!\cdots\!00 \)
acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\).