Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,2,Mod(112,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([5, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.112");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 605.m (of order \(20\), degree \(8\), 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: | \(4.83094932229\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 55) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
112.1 | −0.690094 | − | 1.35439i | −0.124714 | + | 0.787410i | −0.182561 | + | 0.251273i | 0.543872 | + | 2.16892i | 1.15252 | − | 0.374477i | 0.189114 | − | 0.0299527i | −2.53639 | − | 0.401725i | 2.24871 | + | 0.730650i | 2.56223 | − | 2.23337i |
112.2 | −0.221702 | − | 0.435114i | 0.178235 | − | 1.12533i | 1.03540 | − | 1.42510i | −0.207538 | − | 2.22642i | −0.529164 | + | 0.171936i | −2.88051 | + | 0.456227i | −1.81429 | − | 0.287355i | 1.61856 | + | 0.525902i | −0.922733 | + | 0.583903i |
112.3 | 0.562185 | + | 1.10335i | 0.397989 | − | 2.51281i | 0.274241 | − | 0.377461i | −0.841972 | + | 2.07149i | 2.99625 | − | 0.973540i | 3.42554 | − | 0.542552i | 3.01679 | + | 0.477813i | −3.30263 | − | 1.07309i | −2.75893 | + | 0.235572i |
112.4 | 1.10437 | + | 2.16745i | −0.0935509 | + | 0.590657i | −2.30265 | + | 3.16932i | 1.76571 | − | 1.37196i | −1.38354 | + | 0.449539i | 2.54993 | − | 0.403870i | −4.60707 | − | 0.729688i | 2.51305 | + | 0.816538i | 4.92366 | + | 2.31195i |
118.1 | −2.50409 | − | 0.396609i | 0.910200 | − | 1.78637i | 4.21106 | + | 1.36825i | −0.509700 | − | 2.17720i | −2.98771 | + | 4.11224i | 1.61096 | − | 0.820825i | −5.48426 | − | 2.79437i | −0.599293 | − | 0.824856i | 0.412838 | + | 5.65406i |
118.2 | −1.03195 | − | 0.163444i | −1.27853 | + | 2.50925i | −0.863913 | − | 0.280702i | 0.639384 | − | 2.14271i | 1.72949 | − | 2.38044i | 0.489900 | − | 0.249616i | 2.70750 | + | 1.37954i | −2.89835 | − | 3.98923i | −1.01002 | + | 2.10665i |
118.3 | −0.568744 | − | 0.0900803i | 0.349459 | − | 0.685851i | −1.58676 | − | 0.515569i | −1.84663 | + | 1.26093i | −0.260534 | + | 0.358595i | −1.06314 | + | 0.541696i | 1.88216 | + | 0.959009i | 1.41508 | + | 1.94770i | 1.16385 | − | 0.550801i |
118.4 | 1.44791 | + | 0.229326i | −0.377935 | + | 0.741739i | 0.141727 | + | 0.0460500i | 1.49572 | + | 1.66218i | −0.717314 | + | 0.987298i | 3.75588 | − | 1.91372i | −2.41770 | − | 1.23188i | 1.35601 | + | 1.86639i | 1.78448 | + | 2.74968i |
233.1 | −1.35439 | + | 0.690094i | −0.787410 | − | 0.124714i | 0.182561 | − | 0.251273i | 0.834856 | + | 2.07437i | 1.15252 | − | 0.374477i | −0.0299527 | − | 0.189114i | 0.401725 | − | 2.53639i | −2.24871 | − | 0.730650i | −2.56223 | − | 2.23337i |
233.2 | −0.435114 | + | 0.221702i | 1.12533 | + | 0.178235i | −1.03540 | + | 1.42510i | −1.14075 | − | 1.92320i | −0.529164 | + | 0.171936i | 0.456227 | + | 2.88051i | 0.287355 | − | 1.81429i | −1.61856 | − | 0.525902i | 0.922733 | + | 0.583903i |
233.3 | 1.10335 | − | 0.562185i | 2.51281 | + | 0.397989i | −0.274241 | + | 0.377461i | 1.89876 | + | 1.18097i | 2.99625 | − | 0.973540i | −0.542552 | − | 3.42554i | −0.477813 | + | 3.01679i | 3.30263 | + | 1.07309i | 2.75893 | + | 0.235572i |
233.4 | 2.16745 | − | 1.10437i | −0.590657 | − | 0.0935509i | 2.30265 | − | 3.16932i | −2.23491 | − | 0.0720754i | −1.38354 | + | 0.449539i | −0.403870 | − | 2.54993i | 0.729688 | − | 4.60707i | −2.51305 | − | 0.816538i | −4.92366 | + | 2.31195i |
282.1 | −2.50409 | + | 0.396609i | 0.910200 | + | 1.78637i | 4.21106 | − | 1.36825i | −0.509700 | + | 2.17720i | −2.98771 | − | 4.11224i | 1.61096 | + | 0.820825i | −5.48426 | + | 2.79437i | −0.599293 | + | 0.824856i | 0.412838 | − | 5.65406i |
282.2 | −1.03195 | + | 0.163444i | −1.27853 | − | 2.50925i | −0.863913 | + | 0.280702i | 0.639384 | + | 2.14271i | 1.72949 | + | 2.38044i | 0.489900 | + | 0.249616i | 2.70750 | − | 1.37954i | −2.89835 | + | 3.98923i | −1.01002 | − | 2.10665i |
282.3 | −0.568744 | + | 0.0900803i | 0.349459 | + | 0.685851i | −1.58676 | + | 0.515569i | −1.84663 | − | 1.26093i | −0.260534 | − | 0.358595i | −1.06314 | − | 0.541696i | 1.88216 | − | 0.959009i | 1.41508 | − | 1.94770i | 1.16385 | + | 0.550801i |
282.4 | 1.44791 | − | 0.229326i | −0.377935 | − | 0.741739i | 0.141727 | − | 0.0460500i | 1.49572 | − | 1.66218i | −0.717314 | − | 0.987298i | 3.75588 | + | 1.91372i | −2.41770 | + | 1.23188i | 1.35601 | − | 1.86639i | 1.78448 | − | 2.74968i |
403.1 | −0.229326 | − | 1.44791i | 0.741739 | − | 0.377935i | −0.141727 | + | 0.0460500i | 2.04303 | − | 0.908872i | −0.717314 | − | 0.987298i | 1.91372 | − | 3.75588i | −1.23188 | − | 2.41770i | −1.35601 | + | 1.86639i | −1.78448 | − | 2.74968i |
403.2 | 0.0900803 | + | 0.568744i | −0.685851 | + | 0.349459i | 1.58676 | − | 0.515569i | 0.628574 | + | 2.14590i | −0.260534 | − | 0.358595i | −0.541696 | + | 1.06314i | 0.959009 | + | 1.88216i | −1.41508 | + | 1.94770i | −1.16385 | + | 0.550801i |
403.3 | 0.163444 | + | 1.03195i | 2.50925 | − | 1.27853i | 0.863913 | − | 0.280702i | −1.84025 | − | 1.27022i | 1.72949 | + | 2.38044i | 0.249616 | − | 0.489900i | 1.37954 | + | 2.70750i | 2.89835 | − | 3.98923i | 1.01002 | − | 2.10665i |
403.4 | 0.396609 | + | 2.50409i | −1.78637 | + | 0.910200i | −4.21106 | + | 1.36825i | −2.22815 | − | 0.188039i | −2.98771 | − | 4.11224i | 0.820825 | − | 1.61096i | −2.79437 | − | 5.48426i | 0.599293 | − | 0.824856i | −0.412838 | − | 5.65406i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.d | odd | 10 | 1 | inner |
55.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 605.2.m.d | 32 | |
5.c | odd | 4 | 1 | inner | 605.2.m.d | 32 | |
11.b | odd | 2 | 1 | 605.2.m.c | 32 | ||
11.c | even | 5 | 1 | 55.2.l.a | ✓ | 32 | |
11.c | even | 5 | 1 | 605.2.e.b | 32 | ||
11.c | even | 5 | 1 | 605.2.m.c | 32 | ||
11.c | even | 5 | 1 | 605.2.m.e | 32 | ||
11.d | odd | 10 | 1 | 55.2.l.a | ✓ | 32 | |
11.d | odd | 10 | 1 | 605.2.e.b | 32 | ||
11.d | odd | 10 | 1 | inner | 605.2.m.d | 32 | |
11.d | odd | 10 | 1 | 605.2.m.e | 32 | ||
33.f | even | 10 | 1 | 495.2.bj.a | 32 | ||
33.h | odd | 10 | 1 | 495.2.bj.a | 32 | ||
44.g | even | 10 | 1 | 880.2.cm.a | 32 | ||
44.h | odd | 10 | 1 | 880.2.cm.a | 32 | ||
55.e | even | 4 | 1 | 605.2.m.c | 32 | ||
55.h | odd | 10 | 1 | 275.2.bm.b | 32 | ||
55.j | even | 10 | 1 | 275.2.bm.b | 32 | ||
55.k | odd | 20 | 1 | 55.2.l.a | ✓ | 32 | |
55.k | odd | 20 | 1 | 275.2.bm.b | 32 | ||
55.k | odd | 20 | 1 | 605.2.e.b | 32 | ||
55.k | odd | 20 | 1 | 605.2.m.c | 32 | ||
55.k | odd | 20 | 1 | 605.2.m.e | 32 | ||
55.l | even | 20 | 1 | 55.2.l.a | ✓ | 32 | |
55.l | even | 20 | 1 | 275.2.bm.b | 32 | ||
55.l | even | 20 | 1 | 605.2.e.b | 32 | ||
55.l | even | 20 | 1 | inner | 605.2.m.d | 32 | |
55.l | even | 20 | 1 | 605.2.m.e | 32 | ||
165.u | odd | 20 | 1 | 495.2.bj.a | 32 | ||
165.v | even | 20 | 1 | 495.2.bj.a | 32 | ||
220.v | even | 20 | 1 | 880.2.cm.a | 32 | ||
220.w | odd | 20 | 1 | 880.2.cm.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
55.2.l.a | ✓ | 32 | 11.c | even | 5 | 1 | |
55.2.l.a | ✓ | 32 | 11.d | odd | 10 | 1 | |
55.2.l.a | ✓ | 32 | 55.k | odd | 20 | 1 | |
55.2.l.a | ✓ | 32 | 55.l | even | 20 | 1 | |
275.2.bm.b | 32 | 55.h | odd | 10 | 1 | ||
275.2.bm.b | 32 | 55.j | even | 10 | 1 | ||
275.2.bm.b | 32 | 55.k | odd | 20 | 1 | ||
275.2.bm.b | 32 | 55.l | even | 20 | 1 | ||
495.2.bj.a | 32 | 33.f | even | 10 | 1 | ||
495.2.bj.a | 32 | 33.h | odd | 10 | 1 | ||
495.2.bj.a | 32 | 165.u | odd | 20 | 1 | ||
495.2.bj.a | 32 | 165.v | even | 20 | 1 | ||
605.2.e.b | 32 | 11.c | even | 5 | 1 | ||
605.2.e.b | 32 | 11.d | odd | 10 | 1 | ||
605.2.e.b | 32 | 55.k | odd | 20 | 1 | ||
605.2.e.b | 32 | 55.l | even | 20 | 1 | ||
605.2.m.c | 32 | 11.b | odd | 2 | 1 | ||
605.2.m.c | 32 | 11.c | even | 5 | 1 | ||
605.2.m.c | 32 | 55.e | even | 4 | 1 | ||
605.2.m.c | 32 | 55.k | odd | 20 | 1 | ||
605.2.m.d | 32 | 1.a | even | 1 | 1 | trivial | |
605.2.m.d | 32 | 5.c | odd | 4 | 1 | inner | |
605.2.m.d | 32 | 11.d | odd | 10 | 1 | inner | |
605.2.m.d | 32 | 55.l | even | 20 | 1 | inner | |
605.2.m.e | 32 | 11.c | even | 5 | 1 | ||
605.2.m.e | 32 | 11.d | odd | 10 | 1 | ||
605.2.m.e | 32 | 55.k | odd | 20 | 1 | ||
605.2.m.e | 32 | 55.l | even | 20 | 1 | ||
880.2.cm.a | 32 | 44.g | even | 10 | 1 | ||
880.2.cm.a | 32 | 44.h | odd | 10 | 1 | ||
880.2.cm.a | 32 | 220.v | even | 20 | 1 | ||
880.2.cm.a | 32 | 220.w | odd | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 20 T_{2}^{29} - 25 T_{2}^{28} + 200 T_{2}^{26} - 130 T_{2}^{25} + 555 T_{2}^{24} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).