Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(52,315)]
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([8, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.52");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bs (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: | \(160\) |
Relative dimension: | \(40\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
52.1 | −1.89043 | − | 1.89043i | 0.147691 | − | 1.72574i | 5.14743i | 1.96858 | − | 1.06052i | −3.54159 | + | 2.98319i | 1.93775 | + | 1.80142i | 5.94998 | − | 5.94998i | −2.95637 | − | 0.509752i | −5.72629 | − | 1.71661i | ||
52.2 | −1.88273 | − | 1.88273i | −1.73169 | − | 0.0355078i | 5.08933i | −2.21562 | − | 0.301714i | 3.19344 | + | 3.32715i | 2.48041 | − | 0.920630i | 5.81636 | − | 5.81636i | 2.99748 | + | 0.122977i | 3.60336 | + | 4.73945i | ||
52.3 | −1.82176 | − | 1.82176i | −0.791575 | + | 1.54059i | 4.63759i | 1.54912 | − | 1.61252i | 4.24863 | − | 1.36452i | −2.63707 | − | 0.214125i | 4.80505 | − | 4.80505i | −1.74682 | − | 2.43898i | −5.75974 | + | 0.115499i | ||
52.4 | −1.69178 | − | 1.69178i | 1.41922 | + | 0.992886i | 3.72426i | −1.82828 | − | 1.28740i | −0.721260 | − | 4.08075i | 0.203370 | + | 2.63792i | 2.91707 | − | 2.91707i | 1.02836 | + | 2.81824i | 0.915043 | + | 5.27105i | ||
52.5 | −1.55445 | − | 1.55445i | −0.951641 | − | 1.44720i | 2.83266i | 1.24343 | + | 1.85846i | −0.770319 | + | 3.72889i | −1.65840 | − | 2.06148i | 1.29433 | − | 1.29433i | −1.18876 | + | 2.75442i | 0.956046 | − | 4.82175i | ||
52.6 | −1.53214 | − | 1.53214i | −1.70780 | + | 0.288849i | 2.69492i | 0.485164 | + | 2.18280i | 3.05914 | + | 2.17403i | −1.33658 | + | 2.28332i | 1.06472 | − | 1.06472i | 2.83313 | − | 0.986590i | 2.60102 | − | 4.08770i | ||
52.7 | −1.48808 | − | 1.48808i | 0.377509 | + | 1.69041i | 2.42876i | −1.63729 | + | 1.52292i | 1.95370 | − | 3.07723i | −1.04527 | − | 2.43052i | 0.638021 | − | 0.638021i | −2.71497 | + | 1.27629i | 4.70264 | + | 0.170180i | ||
52.8 | −1.45962 | − | 1.45962i | 1.30818 | + | 1.13520i | 2.26101i | 2.22626 | + | 0.209158i | −0.252491 | − | 3.56641i | 2.27033 | − | 1.35853i | 0.380972 | − | 0.380972i | 0.422664 | + | 2.97008i | −2.94422 | − | 3.55480i | ||
52.9 | −1.21497 | − | 1.21497i | −1.20986 | − | 1.23945i | 0.952318i | −0.123740 | − | 2.23264i | −0.0359577 | + | 2.97585i | 0.124742 | − | 2.64281i | −1.27291 | + | 1.27291i | −0.0724885 | + | 2.99912i | −2.56226 | + | 2.86294i | ||
52.10 | −1.07573 | − | 1.07573i | 1.65548 | − | 0.509290i | 0.314407i | 2.21353 | − | 0.316708i | −2.32872 | − | 1.23300i | −2.59482 | − | 0.516641i | −1.81325 | + | 1.81325i | 2.48125 | − | 1.68624i | −2.72186 | − | 2.04047i | ||
52.11 | −1.03040 | − | 1.03040i | −0.767189 | + | 1.55288i | 0.123432i | −0.154472 | − | 2.23073i | 2.39058 | − | 0.809568i | 2.47377 | + | 0.938328i | −1.93361 | + | 1.93361i | −1.82284 | − | 2.38270i | −2.13936 | + | 2.45770i | ||
52.12 | −0.899204 | − | 0.899204i | −0.810966 | − | 1.53047i | − | 0.382866i | −2.02048 | + | 0.957954i | −0.646980 | + | 2.10543i | 1.52253 | + | 2.16377i | −2.14268 | + | 2.14268i | −1.68467 | + | 2.48232i | 2.67822 | + | 0.955424i | |
52.13 | −0.690838 | − | 0.690838i | 0.0911190 | + | 1.72965i | − | 1.04549i | 1.77477 | + | 1.36022i | 1.13196 | − | 1.25786i | −0.704044 | + | 2.55036i | −2.10394 | + | 2.10394i | −2.98339 | + | 0.315208i | −0.286383 | − | 2.16577i | |
52.14 | −0.664590 | − | 0.664590i | −1.71542 | + | 0.239440i | − | 1.11664i | −2.09469 | − | 0.782473i | 1.29918 | + | 0.980921i | −2.43118 | + | 1.04372i | −2.07129 | + | 2.07129i | 2.88534 | − | 0.821482i | 0.872087 | + | 1.91213i | |
52.15 | −0.634954 | − | 0.634954i | −1.27037 | + | 1.17735i | − | 1.19367i | −0.623828 | + | 2.14729i | 1.55419 | + | 0.0590616i | 1.63520 | − | 2.07993i | −2.02783 | + | 2.02783i | 0.227680 | − | 2.99135i | 1.75953 | − | 0.967326i | |
52.16 | −0.565391 | − | 0.565391i | 1.73112 | − | 0.0568332i | − | 1.36066i | −2.17953 | − | 0.499648i | −1.01089 | − | 0.946626i | 1.69248 | − | 2.03359i | −1.90009 | + | 1.90009i | 2.99354 | − | 0.196770i | 0.949791 | + | 1.51478i | |
52.17 | −0.333535 | − | 0.333535i | 0.667654 | − | 1.59820i | − | 1.77751i | 1.47915 | − | 1.67694i | −0.755740 | + | 0.310369i | 2.25720 | − | 1.38023i | −1.25993 | + | 1.25993i | −2.10848 | − | 2.13409i | −1.05266 | + | 0.0659706i | |
52.18 | −0.309887 | − | 0.309887i | −1.71259 | + | 0.258881i | − | 1.80794i | 2.07755 | − | 0.826926i | 0.610935 | + | 0.450487i | −2.09180 | − | 1.61999i | −1.18003 | + | 1.18003i | 2.86596 | − | 0.886717i | −0.900059 | − | 0.387551i | |
52.19 | −0.141728 | − | 0.141728i | 0.505641 | − | 1.65660i | − | 1.95983i | −1.15314 | + | 1.91579i | −0.306450 | + | 0.163123i | −2.32441 | − | 1.26377i | −0.561218 | + | 0.561218i | −2.48865 | − | 1.67529i | 0.434954 | − | 0.108090i | |
52.20 | −0.108614 | − | 0.108614i | 1.61761 | + | 0.619140i | − | 1.97641i | 0.847505 | − | 2.06924i | −0.108448 | − | 0.242943i | 0.0705295 | + | 2.64481i | −0.431895 | + | 0.431895i | 2.23333 | + | 2.00306i | −0.316800 | + | 0.132697i | |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
63.t | odd | 6 | 1 | inner |
315.bs | 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.bs.e | ✓ | 160 |
3.b | odd | 2 | 1 | 945.2.bv.e | 160 | ||
5.c | odd | 4 | 1 | inner | 315.2.bs.e | ✓ | 160 |
7.d | odd | 6 | 1 | 315.2.cg.e | yes | 160 | |
9.c | even | 3 | 1 | 315.2.cg.e | yes | 160 | |
9.d | odd | 6 | 1 | 945.2.cj.e | 160 | ||
15.e | even | 4 | 1 | 945.2.bv.e | 160 | ||
21.g | even | 6 | 1 | 945.2.cj.e | 160 | ||
35.k | even | 12 | 1 | 315.2.cg.e | yes | 160 | |
45.k | odd | 12 | 1 | 315.2.cg.e | yes | 160 | |
45.l | even | 12 | 1 | 945.2.cj.e | 160 | ||
63.i | even | 6 | 1 | 945.2.bv.e | 160 | ||
63.t | odd | 6 | 1 | inner | 315.2.bs.e | ✓ | 160 |
105.w | odd | 12 | 1 | 945.2.cj.e | 160 | ||
315.bs | even | 12 | 1 | inner | 315.2.bs.e | ✓ | 160 |
315.bu | odd | 12 | 1 | 945.2.bv.e | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bs.e | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
315.2.bs.e | ✓ | 160 | 5.c | odd | 4 | 1 | inner |
315.2.bs.e | ✓ | 160 | 63.t | odd | 6 | 1 | inner |
315.2.bs.e | ✓ | 160 | 315.bs | even | 12 | 1 | inner |
315.2.cg.e | yes | 160 | 7.d | odd | 6 | 1 | |
315.2.cg.e | yes | 160 | 9.c | even | 3 | 1 | |
315.2.cg.e | yes | 160 | 35.k | even | 12 | 1 | |
315.2.cg.e | yes | 160 | 45.k | odd | 12 | 1 | |
945.2.bv.e | 160 | 3.b | odd | 2 | 1 | ||
945.2.bv.e | 160 | 15.e | even | 4 | 1 | ||
945.2.bv.e | 160 | 63.i | even | 6 | 1 | ||
945.2.bv.e | 160 | 315.bu | odd | 12 | 1 | ||
945.2.cj.e | 160 | 9.d | odd | 6 | 1 | ||
945.2.cj.e | 160 | 21.g | even | 6 | 1 | ||
945.2.cj.e | 160 | 45.l | even | 12 | 1 | ||
945.2.cj.e | 160 | 105.w | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):
\( T_{2}^{80} - 2 T_{2}^{79} + 2 T_{2}^{78} + 4 T_{2}^{77} + 289 T_{2}^{76} - 568 T_{2}^{75} + 566 T_{2}^{74} + \cdots + 17161 \) |
\( T_{11}^{80} + 8 T_{11}^{79} + 258 T_{11}^{78} + 1752 T_{11}^{77} + 34377 T_{11}^{76} + \cdots + 17\!\cdots\!16 \) |
\( T_{13}^{160} - 6473 T_{13}^{156} + 1314 T_{13}^{155} - 58698 T_{13}^{153} + 23984127 T_{13}^{152} + \cdots + 58\!\cdots\!81 \) |