Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [513,2,Mod(235,513)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(513, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("513.235");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 513 = 3^{3} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 513.h (of order \(3\), degree \(2\), not 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: | \(4.09632562369\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{3})\) |
Twist minimal: | no (minimal twist has level 171) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
235.1 | −2.46808 | 0 | 4.09140 | 0.957619 | − | 1.65865i | 0 | −0.324708 | + | 0.562412i | −5.16173 | 0 | −2.36348 | + | 4.09366i | ||||||||||||
235.2 | −2.09768 | 0 | 2.40028 | −1.44796 | + | 2.50795i | 0 | 0.116480 | − | 0.201749i | −0.839660 | 0 | 3.03737 | − | 5.26088i | ||||||||||||
235.3 | −1.95703 | 0 | 1.82996 | −0.0981173 | + | 0.169944i | 0 | −2.23368 | + | 3.86885i | 0.332766 | 0 | 0.192018 | − | 0.332586i | ||||||||||||
235.4 | −1.77797 | 0 | 1.16118 | −0.639786 | + | 1.10814i | 0 | 0.657761 | − | 1.13928i | 1.49140 | 0 | 1.13752 | − | 1.97024i | ||||||||||||
235.5 | −1.60662 | 0 | 0.581222 | 1.87940 | − | 3.25521i | 0 | 2.27973 | − | 3.94861i | 2.27943 | 0 | −3.01948 | + | 5.22989i | ||||||||||||
235.6 | −0.791858 | 0 | −1.37296 | 1.29546 | − | 2.24381i | 0 | −0.373088 | + | 0.646207i | 2.67091 | 0 | −1.02582 | + | 1.77678i | ||||||||||||
235.7 | −0.539090 | 0 | −1.70938 | −0.473662 | + | 0.820407i | 0 | 1.18430 | − | 2.05126i | 1.99969 | 0 | 0.255347 | − | 0.442274i | ||||||||||||
235.8 | −0.370889 | 0 | −1.86244 | −1.77761 | + | 3.07890i | 0 | −0.124876 | + | 0.216291i | 1.43254 | 0 | 0.659294 | − | 1.14193i | ||||||||||||
235.9 | −0.146534 | 0 | −1.97853 | 1.28502 | − | 2.22572i | 0 | −1.73898 | + | 3.01201i | 0.582990 | 0 | −0.188299 | + | 0.326143i | ||||||||||||
235.10 | 0.194693 | 0 | −1.96209 | −0.952817 | + | 1.65033i | 0 | 1.69446 | − | 2.93489i | −0.771394 | 0 | −0.185507 | + | 0.321308i | ||||||||||||
235.11 | 1.23359 | 0 | −0.478252 | −0.275772 | + | 0.477650i | 0 | −1.62156 | + | 2.80862i | −3.05715 | 0 | −0.340189 | + | 0.589225i | ||||||||||||
235.12 | 1.69423 | 0 | 0.870412 | 0.0441088 | − | 0.0763987i | 0 | 1.84695 | − | 3.19901i | −1.91378 | 0 | 0.0747304 | − | 0.129437i | ||||||||||||
235.13 | 2.02031 | 0 | 2.08166 | −2.09369 | + | 3.62638i | 0 | −0.976107 | + | 1.69067i | 0.164982 | 0 | −4.22991 | + | 7.32643i | ||||||||||||
235.14 | 2.39943 | 0 | 3.75726 | 0.359839 | − | 0.623259i | 0 | 1.65862 | − | 2.87282i | 4.21641 | 0 | 0.863408 | − | 1.49547i | ||||||||||||
235.15 | 2.60319 | 0 | 4.77661 | 1.43897 | − | 2.49237i | 0 | −1.80240 | + | 3.12185i | 7.22804 | 0 | 3.74592 | − | 6.48812i | ||||||||||||
235.16 | 2.61030 | 0 | 4.81368 | −1.00100 | + | 1.73379i | 0 | 0.257107 | − | 0.445323i | 7.34455 | 0 | −2.61292 | + | 4.52572i | ||||||||||||
334.1 | −2.46808 | 0 | 4.09140 | 0.957619 | + | 1.65865i | 0 | −0.324708 | − | 0.562412i | −5.16173 | 0 | −2.36348 | − | 4.09366i | ||||||||||||
334.2 | −2.09768 | 0 | 2.40028 | −1.44796 | − | 2.50795i | 0 | 0.116480 | + | 0.201749i | −0.839660 | 0 | 3.03737 | + | 5.26088i | ||||||||||||
334.3 | −1.95703 | 0 | 1.82996 | −0.0981173 | − | 0.169944i | 0 | −2.23368 | − | 3.86885i | 0.332766 | 0 | 0.192018 | + | 0.332586i | ||||||||||||
334.4 | −1.77797 | 0 | 1.16118 | −0.639786 | − | 1.10814i | 0 | 0.657761 | + | 1.13928i | 1.49140 | 0 | 1.13752 | + | 1.97024i | ||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 513.2.h.c | 32 | |
3.b | odd | 2 | 1 | 171.2.h.c | yes | 32 | |
9.c | even | 3 | 1 | 513.2.g.c | 32 | ||
9.d | odd | 6 | 1 | 171.2.g.c | ✓ | 32 | |
19.c | even | 3 | 1 | 513.2.g.c | 32 | ||
57.h | odd | 6 | 1 | 171.2.g.c | ✓ | 32 | |
171.h | even | 3 | 1 | inner | 513.2.h.c | 32 | |
171.j | odd | 6 | 1 | 171.2.h.c | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
171.2.g.c | ✓ | 32 | 9.d | odd | 6 | 1 | |
171.2.g.c | ✓ | 32 | 57.h | odd | 6 | 1 | |
171.2.h.c | yes | 32 | 3.b | odd | 2 | 1 | |
171.2.h.c | yes | 32 | 171.j | odd | 6 | 1 | |
513.2.g.c | 32 | 9.c | even | 3 | 1 | ||
513.2.g.c | 32 | 19.c | even | 3 | 1 | ||
513.2.h.c | 32 | 1.a | even | 1 | 1 | trivial | |
513.2.h.c | 32 | 171.h | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\):
\( T_{2}^{16} - T_{2}^{15} - 24 T_{2}^{14} + 17 T_{2}^{13} + 235 T_{2}^{12} - 96 T_{2}^{11} - 1193 T_{2}^{10} + \cdots - 9 \) |
\( T_{5}^{32} + 3 T_{5}^{31} + 49 T_{5}^{30} + 110 T_{5}^{29} + 1345 T_{5}^{28} + 2690 T_{5}^{27} + \cdots + 35721 \) |