Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(2,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bx (of order \(12\), degree \(4\), 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: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(176\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −2.69307 | − | 0.721605i | 1.50080 | − | 0.864638i | 4.99984 | + | 2.88666i | 1.13456 | + | 1.92686i | −4.66568 | + | 1.24554i | 1.58627 | − | 2.11749i | −7.43896 | − | 7.43896i | 1.50480 | − | 2.59530i | −1.66500 | − | 6.00786i |
2.2 | −2.59467 | − | 0.695240i | −0.295616 | + | 1.70664i | 4.51691 | + | 2.60784i | 1.82199 | − | 1.29628i | 1.95355 | − | 4.22264i | −2.60972 | − | 0.435150i | −6.10795 | − | 6.10795i | −2.82522 | − | 1.00902i | −5.62870 | + | 2.09669i |
2.3 | −2.48085 | − | 0.664741i | −0.0323957 | − | 1.73175i | 3.98067 | + | 2.29824i | −0.385521 | − | 2.20258i | −1.07079 | + | 4.31774i | −0.368703 | + | 2.61993i | −4.71549 | − | 4.71549i | −2.99790 | + | 0.112202i | −0.507729 | + | 5.72054i |
2.4 | −2.41637 | − | 0.647465i | −1.67924 | − | 0.424464i | 3.68760 | + | 2.12903i | −2.17829 | − | 0.505032i | 3.78283 | + | 2.11291i | 0.757887 | − | 2.53488i | −3.99432 | − | 3.99432i | 2.63966 | + | 1.42555i | 4.93657 | + | 2.63071i |
2.5 | −2.27659 | − | 0.610010i | −1.55968 | + | 0.753259i | 3.07869 | + | 1.77748i | −0.123575 | + | 2.23265i | 4.01024 | − | 0.763441i | 0.379075 | + | 2.61845i | −2.59147 | − | 2.59147i | 1.86520 | − | 2.34969i | 1.64327 | − | 5.00744i |
2.6 | −2.11283 | − | 0.566131i | 1.70063 | − | 0.328428i | 2.41150 | + | 1.39228i | −2.23606 | + | 0.00531389i | −3.77907 | − | 0.268867i | −1.66975 | + | 2.05230i | −1.21347 | − | 1.21347i | 2.78427 | − | 1.11707i | 4.72743 | + | 1.25468i |
2.7 | −2.07435 | − | 0.555820i | 1.52422 | + | 0.822645i | 2.26194 | + | 1.30593i | 0.0290922 | − | 2.23588i | −2.70453 | − | 2.55365i | 2.57272 | − | 0.617326i | −0.929129 | − | 0.929129i | 1.64651 | + | 2.50779i | −1.30309 | + | 4.62182i |
2.8 | −1.95085 | − | 0.522728i | −1.45630 | − | 0.937654i | 1.80051 | + | 1.03953i | 2.23387 | + | 0.0991923i | 2.35088 | + | 2.59047i | −1.87306 | − | 1.86860i | −0.112896 | − | 0.112896i | 1.24161 | + | 2.73101i | −4.30608 | − | 1.36121i |
2.9 | −1.93868 | − | 0.519467i | 0.966001 | + | 1.43765i | 1.75657 | + | 1.01416i | −1.10599 | + | 1.94340i | −1.12595 | − | 3.28895i | −1.57987 | − | 2.12227i | −0.0401862 | − | 0.0401862i | −1.13368 | + | 2.77755i | 3.15369 | − | 3.19309i |
2.10 | −1.62572 | − | 0.435610i | 0.558438 | − | 1.63956i | 0.721151 | + | 0.416356i | 1.01641 | + | 1.99171i | −1.62207 | + | 2.42220i | −2.57976 | + | 0.587246i | 1.38920 | + | 1.38920i | −2.37629 | − | 1.83118i | −0.784783 | − | 3.68072i |
2.11 | −1.58231 | − | 0.423978i | −0.361579 | − | 1.69389i | 0.591884 | + | 0.341725i | −1.51691 | + | 1.64286i | −0.146042 | + | 2.83355i | 2.64543 | − | 0.0411436i | 1.52500 | + | 1.52500i | −2.73852 | + | 1.22495i | 3.09675 | − | 1.95638i |
2.12 | −1.47675 | − | 0.395694i | −0.550393 | + | 1.64228i | 0.292161 | + | 0.168679i | 2.03723 | + | 0.921779i | 1.46263 | − | 2.20744i | 2.43080 | − | 1.04461i | 1.79741 | + | 1.79741i | −2.39414 | − | 1.80779i | −2.64374 | − | 2.16736i |
2.13 | −1.45905 | − | 0.390953i | −1.73094 | + | 0.0619698i | 0.243946 | + | 0.140842i | 1.00295 | − | 1.99852i | 2.54977 | + | 0.586299i | 0.279016 | + | 2.63100i | 1.83534 | + | 1.83534i | 2.99232 | − | 0.214532i | −2.24469 | + | 2.52385i |
2.14 | −1.17797 | − | 0.315636i | 1.34464 | − | 1.09176i | −0.444064 | − | 0.256381i | 2.21223 | − | 0.325607i | −1.92855 | + | 0.861638i | 2.06273 | + | 1.65685i | 2.16684 | + | 2.16684i | 0.616135 | − | 2.93605i | −2.70872 | − | 0.314706i |
2.15 | −0.983057 | − | 0.263409i | 1.53646 | − | 0.799565i | −0.835034 | − | 0.482107i | −1.82992 | − | 1.28507i | −1.72104 | + | 0.381301i | −0.244874 | − | 2.63439i | 2.13319 | + | 2.13319i | 1.72139 | − | 2.45699i | 1.46042 | + | 1.74531i |
2.16 | −0.904631 | − | 0.242395i | 0.776514 | + | 1.54823i | −0.972449 | − | 0.561443i | 0.304672 | − | 2.21521i | −0.327174 | − | 1.58880i | −1.94982 | + | 1.78835i | 2.06809 | + | 2.06809i | −1.79405 | + | 2.40445i | −0.812573 | + | 1.93010i |
2.17 | −0.839010 | − | 0.224812i | −0.882497 | − | 1.49037i | −1.07865 | − | 0.622761i | −1.58499 | − | 1.57728i | 0.405371 | + | 1.44883i | −2.60439 | − | 0.466002i | 1.99339 | + | 1.99339i | −1.44240 | + | 2.63049i | 0.975229 | + | 1.67968i |
2.18 | −0.816718 | − | 0.218839i | 0.0813481 | + | 1.73014i | −1.11291 | − | 0.642541i | −2.21384 | + | 0.314524i | 0.312183 | − | 1.43084i | 1.44664 | + | 2.21523i | 1.96408 | + | 1.96408i | −2.98676 | + | 0.281487i | 1.87691 | + | 0.227596i |
2.19 | −0.580282 | − | 0.155486i | −1.43439 | + | 0.970830i | −1.41950 | − | 0.819548i | −1.41555 | + | 1.73096i | 0.983304 | − | 0.340327i | −1.77488 | − | 1.96209i | 1.54587 | + | 1.54587i | 1.11498 | − | 2.78511i | 1.09056 | − | 0.784343i |
2.20 | −0.534455 | − | 0.143207i | 1.61596 | + | 0.623443i | −1.46692 | − | 0.846925i | 2.22680 | + | 0.203348i | −0.774376 | − | 0.564618i | −1.18425 | − | 2.36592i | 1.44521 | + | 1.44521i | 2.22264 | + | 2.01491i | −1.16101 | − | 0.427574i |
See next 80 embeddings (of 176 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
63.n | odd | 6 | 1 | inner |
315.bx | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.bx.a | yes | 176 |
3.b | odd | 2 | 1 | 945.2.ca.a | 176 | ||
5.c | odd | 4 | 1 | inner | 315.2.bx.a | yes | 176 |
7.c | even | 3 | 1 | 315.2.bv.a | ✓ | 176 | |
9.c | even | 3 | 1 | 945.2.by.a | 176 | ||
9.d | odd | 6 | 1 | 315.2.bv.a | ✓ | 176 | |
15.e | even | 4 | 1 | 945.2.ca.a | 176 | ||
21.h | odd | 6 | 1 | 945.2.by.a | 176 | ||
35.l | odd | 12 | 1 | 315.2.bv.a | ✓ | 176 | |
45.k | odd | 12 | 1 | 945.2.by.a | 176 | ||
45.l | even | 12 | 1 | 315.2.bv.a | ✓ | 176 | |
63.g | even | 3 | 1 | 945.2.ca.a | 176 | ||
63.n | odd | 6 | 1 | inner | 315.2.bx.a | yes | 176 |
105.x | even | 12 | 1 | 945.2.by.a | 176 | ||
315.bx | even | 12 | 1 | inner | 315.2.bx.a | yes | 176 |
315.ch | odd | 12 | 1 | 945.2.ca.a | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bv.a | ✓ | 176 | 7.c | even | 3 | 1 | |
315.2.bv.a | ✓ | 176 | 9.d | odd | 6 | 1 | |
315.2.bv.a | ✓ | 176 | 35.l | odd | 12 | 1 | |
315.2.bv.a | ✓ | 176 | 45.l | even | 12 | 1 | |
315.2.bx.a | yes | 176 | 1.a | even | 1 | 1 | trivial |
315.2.bx.a | yes | 176 | 5.c | odd | 4 | 1 | inner |
315.2.bx.a | yes | 176 | 63.n | odd | 6 | 1 | inner |
315.2.bx.a | yes | 176 | 315.bx | even | 12 | 1 | inner |
945.2.by.a | 176 | 9.c | even | 3 | 1 | ||
945.2.by.a | 176 | 21.h | odd | 6 | 1 | ||
945.2.by.a | 176 | 45.k | odd | 12 | 1 | ||
945.2.by.a | 176 | 105.x | even | 12 | 1 | ||
945.2.ca.a | 176 | 3.b | odd | 2 | 1 | ||
945.2.ca.a | 176 | 15.e | even | 4 | 1 | ||
945.2.ca.a | 176 | 63.g | even | 3 | 1 | ||
945.2.ca.a | 176 | 315.ch | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).