Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [731,2,Mod(259,731)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("731.259");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 731 = 17 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 731.f (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: | \(5.83706438776\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
259.1 | − | 2.81279i | −1.57952 | + | 1.57952i | −5.91177 | −2.62660 | + | 2.62660i | 4.44284 | + | 4.44284i | 0.229862 | + | 0.229862i | 11.0030i | − | 1.98974i | 7.38806 | + | 7.38806i | ||||||
259.2 | − | 2.77460i | 1.70485 | − | 1.70485i | −5.69840 | 0.433295 | − | 0.433295i | −4.73027 | − | 4.73027i | −2.87350 | − | 2.87350i | 10.2616i | − | 2.81302i | −1.20222 | − | 1.20222i | ||||||
259.3 | − | 2.68403i | 0.675148 | − | 0.675148i | −5.20400 | −0.573416 | + | 0.573416i | −1.81211 | − | 1.81211i | 1.88065 | + | 1.88065i | 8.59962i | 2.08835i | 1.53907 | + | 1.53907i | |||||||
259.4 | − | 2.23190i | −0.335489 | + | 0.335489i | −2.98138 | −0.357194 | + | 0.357194i | 0.748779 | + | 0.748779i | 0.708232 | + | 0.708232i | 2.19035i | 2.77489i | 0.797222 | + | 0.797222i | |||||||
259.5 | − | 2.18921i | 0.566370 | − | 0.566370i | −2.79262 | 2.16421 | − | 2.16421i | −1.23990 | − | 1.23990i | 0.402819 | + | 0.402819i | 1.73522i | 2.35845i | −4.73791 | − | 4.73791i | |||||||
259.6 | − | 2.09057i | 2.10973 | − | 2.10973i | −2.37046 | 0.614601 | − | 0.614601i | −4.41052 | − | 4.41052i | 0.361953 | + | 0.361953i | 0.774475i | − | 5.90190i | −1.28486 | − | 1.28486i | ||||||
259.7 | − | 1.74414i | −0.697298 | + | 0.697298i | −1.04202 | 1.44943 | − | 1.44943i | 1.21618 | + | 1.21618i | 1.15181 | + | 1.15181i | − | 1.67086i | 2.02755i | −2.52800 | − | 2.52800i | ||||||
259.8 | − | 1.64399i | −2.17617 | + | 2.17617i | −0.702716 | −2.09084 | + | 2.09084i | 3.57761 | + | 3.57761i | −2.29672 | − | 2.29672i | − | 2.13273i | − | 6.47143i | 3.43732 | + | 3.43732i | |||||
259.9 | − | 1.52916i | −0.854970 | + | 0.854970i | −0.338329 | −2.11844 | + | 2.11844i | 1.30739 | + | 1.30739i | 0.0998235 | + | 0.0998235i | − | 2.54096i | 1.53805i | 3.23944 | + | 3.23944i | ||||||
259.10 | − | 1.47026i | −1.92339 | + | 1.92339i | −0.161659 | 2.97219 | − | 2.97219i | 2.82787 | + | 2.82787i | −2.28989 | − | 2.28989i | − | 2.70284i | − | 4.39883i | −4.36988 | − | 4.36988i | |||||
259.11 | − | 1.21542i | 0.483456 | − | 0.483456i | 0.522759 | 1.73874 | − | 1.73874i | −0.587601 | − | 0.587601i | 3.36984 | + | 3.36984i | − | 3.06621i | 2.53254i | −2.11330 | − | 2.11330i | ||||||
259.12 | − | 1.09376i | 0.497569 | − | 0.497569i | 0.803697 | −2.18307 | + | 2.18307i | −0.544219 | − | 0.544219i | 0.291940 | + | 0.291940i | − | 3.06656i | 2.50485i | 2.38774 | + | 2.38774i | ||||||
259.13 | − | 0.607091i | −1.55808 | + | 1.55808i | 1.63144 | 1.22244 | − | 1.22244i | 0.945900 | + | 0.945900i | 1.59838 | + | 1.59838i | − | 2.20462i | − | 1.85526i | −0.742133 | − | 0.742133i | |||||
259.14 | − | 0.600676i | 0.381351 | − | 0.381351i | 1.63919 | 0.0334344 | − | 0.0334344i | −0.229068 | − | 0.229068i | −3.21348 | − | 3.21348i | − | 2.18597i | 2.70914i | −0.0200832 | − | 0.0200832i | ||||||
259.15 | − | 0.395202i | −2.06164 | + | 2.06164i | 1.84382 | −2.29638 | + | 2.29638i | 0.814765 | + | 0.814765i | 2.83331 | + | 2.83331i | − | 1.51908i | − | 5.50076i | 0.907534 | + | 0.907534i | |||||
259.16 | 0.265604i | 0.626669 | − | 0.626669i | 1.92945 | −0.906788 | + | 0.906788i | 0.166446 | + | 0.166446i | −0.565427 | − | 0.565427i | 1.04368i | 2.21457i | −0.240847 | − | 0.240847i | ||||||||
259.17 | 0.401786i | 1.92935 | − | 1.92935i | 1.83857 | 1.46367 | − | 1.46367i | 0.775185 | + | 0.775185i | 0.274168 | + | 0.274168i | 1.54228i | − | 4.44476i | 0.588082 | + | 0.588082i | |||||||
259.18 | 0.478501i | −0.772701 | + | 0.772701i | 1.77104 | 2.77993 | − | 2.77993i | −0.369738 | − | 0.369738i | −2.40550 | − | 2.40550i | 1.80444i | 1.80587i | 1.33020 | + | 1.33020i | ||||||||
259.19 | 0.831038i | −0.711849 | + | 0.711849i | 1.30938 | 0.619206 | − | 0.619206i | −0.591574 | − | 0.591574i | 2.80838 | + | 2.80838i | 2.75022i | 1.98654i | 0.514584 | + | 0.514584i | ||||||||
259.20 | 1.30582i | 2.39223 | − | 2.39223i | 0.294832 | −1.70398 | + | 1.70398i | 3.12383 | + | 3.12383i | 2.61268 | + | 2.61268i | 2.99664i | − | 8.44554i | −2.22509 | − | 2.22509i | |||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.c | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 731.2.f.c | ✓ | 56 |
17.c | even | 4 | 1 | inner | 731.2.f.c | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
731.2.f.c | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
731.2.f.c | ✓ | 56 | 17.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\):
\( T_{2}^{56} + 86 T_{2}^{54} + 3485 T_{2}^{52} + 88532 T_{2}^{50} + 1582396 T_{2}^{48} + 21171884 T_{2}^{46} + 220263882 T_{2}^{44} + 1827107802 T_{2}^{42} + 12294280942 T_{2}^{40} + \cdots + 8573184 \) |
\( T_{3}^{56} + 4 T_{3}^{55} + 8 T_{3}^{54} + 4 T_{3}^{53} + 325 T_{3}^{52} + 1338 T_{3}^{51} + 2760 T_{3}^{50} + 1094 T_{3}^{49} + 39192 T_{3}^{48} + 166940 T_{3}^{47} + 356722 T_{3}^{46} + 115248 T_{3}^{45} + 2213376 T_{3}^{44} + \cdots + 39564100 \) |