Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,4,Mod(32,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.32");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.p (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: | \(13.2754297513\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 45) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 | −4.88245 | − | 1.30825i | 1.83715 | + | 4.86054i | 15.1986 | + | 8.77490i | 0 | −2.61101 | − | 26.1348i | 7.62587 | − | 28.4601i | −34.1329 | − | 34.1329i | −20.2497 | + | 17.8591i | 0 | ||||
32.2 | −4.29910 | − | 1.15194i | 4.54171 | − | 2.52446i | 10.2271 | + | 5.90461i | 0 | −22.4333 | + | 5.62113i | −1.91190 | + | 7.13532i | −11.9883 | − | 11.9883i | 14.2542 | − | 22.9307i | 0 | ||||
32.3 | −4.14355 | − | 1.11026i | −1.80338 | − | 4.87317i | 9.00816 | + | 5.20086i | 0 | 2.06189 | + | 22.1945i | −3.90791 | + | 14.5845i | −7.28515 | − | 7.28515i | −20.4957 | + | 17.5763i | 0 | ||||
32.4 | −3.56967 | − | 0.956489i | −4.00002 | + | 3.31660i | 4.89944 | + | 2.82870i | 0 | 17.4510 | − | 8.01320i | −1.11867 | + | 4.17492i | 6.12165 | + | 6.12165i | 5.00028 | − | 26.5329i | 0 | ||||
32.5 | −2.15214 | − | 0.576664i | −4.75638 | − | 2.09209i | −2.62904 | − | 1.51788i | 0 | 9.02996 | + | 7.24530i | 3.70051 | − | 13.8105i | 17.3866 | + | 17.3866i | 18.2463 | + | 19.9016i | 0 | ||||
32.6 | −1.43923 | − | 0.385641i | 2.38802 | + | 4.61491i | −5.00553 | − | 2.88994i | 0 | −1.65722 | − | 7.56285i | −6.62758 | + | 24.7344i | 14.5184 | + | 14.5184i | −15.5947 | + | 22.0410i | 0 | ||||
32.7 | −1.25351 | − | 0.335877i | 5.15697 | + | 0.636882i | −5.46973 | − | 3.15795i | 0 | −6.25041 | − | 2.53045i | 0.803840 | − | 2.99997i | 13.1367 | + | 13.1367i | 26.1888 | + | 6.56877i | 0 | ||||
32.8 | −0.145161 | − | 0.0388957i | 0.490448 | − | 5.17295i | −6.90864 | − | 3.98871i | 0 | −0.272400 | + | 0.731834i | 4.62436 | − | 17.2584i | 1.69784 | + | 1.69784i | −26.5189 | − | 5.07413i | 0 | ||||
32.9 | 0.471331 | + | 0.126293i | −3.69630 | + | 3.65204i | −6.72200 | − | 3.88095i | 0 | −2.20341 | + | 1.25450i | 1.53228 | − | 5.71856i | −5.43846 | − | 5.43846i | 0.325220 | − | 26.9980i | 0 | ||||
32.10 | 1.60134 | + | 0.429078i | 1.67949 | − | 4.91725i | −4.54802 | − | 2.62580i | 0 | 4.79931 | − | 7.15356i | −8.15082 | + | 30.4193i | −15.5344 | − | 15.5344i | −21.3587 | − | 16.5169i | 0 | ||||
32.11 | 2.57169 | + | 0.689083i | 0.974542 | + | 5.10395i | −0.789441 | − | 0.455784i | 0 | −1.01082 | + | 13.7973i | 0.400135 | − | 1.49332i | −16.7770 | − | 16.7770i | −25.1005 | + | 9.94802i | 0 | ||||
32.12 | 2.79258 | + | 0.748271i | −5.18419 | − | 0.352314i | 0.310415 | + | 0.179218i | 0 | −14.2137 | − | 4.86305i | −5.03374 | + | 18.7862i | −15.6218 | − | 15.6218i | 26.7517 | + | 3.65293i | 0 | ||||
32.13 | 2.87860 | + | 0.771317i | 5.17956 | + | 0.414910i | 0.763180 | + | 0.440622i | 0 | 14.5898 | + | 5.18944i | 8.39055 | − | 31.3140i | −15.0012 | − | 15.0012i | 26.6557 | + | 4.29810i | 0 | ||||
32.14 | 3.92041 | + | 1.05047i | −3.82410 | − | 3.51799i | 7.33795 | + | 4.23656i | 0 | −11.2965 | − | 17.8091i | 6.13411 | − | 22.8928i | 1.35786 | + | 1.35786i | 2.24743 | + | 26.9063i | 0 | ||||
32.15 | 4.73973 | + | 1.27001i | 4.68975 | − | 2.23746i | 13.9239 | + | 8.03899i | 0 | 25.0698 | − | 4.64896i | −5.69629 | + | 21.2589i | 28.0284 | + | 28.0284i | 16.9875 | − | 20.9863i | 0 | ||||
32.16 | 5.27515 | + | 1.41347i | −1.94123 | + | 4.81992i | 18.9011 | + | 10.9125i | 0 | −17.0531 | + | 22.6819i | −1.13077 | + | 4.22008i | 53.3880 | + | 53.3880i | −19.4633 | − | 18.7131i | 0 | ||||
68.1 | −1.30825 | + | 4.88245i | −4.86054 | + | 1.83715i | −15.1986 | − | 8.77490i | 0 | −2.61101 | − | 26.1348i | −28.4601 | − | 7.62587i | 34.1329 | − | 34.1329i | 20.2497 | − | 17.8591i | 0 | ||||
68.2 | −1.15194 | + | 4.29910i | 2.52446 | + | 4.54171i | −10.2271 | − | 5.90461i | 0 | −22.4333 | + | 5.62113i | 7.13532 | + | 1.91190i | 11.9883 | − | 11.9883i | −14.2542 | + | 22.9307i | 0 | ||||
68.3 | −1.11026 | + | 4.14355i | 4.87317 | − | 1.80338i | −9.00816 | − | 5.20086i | 0 | 2.06189 | + | 22.1945i | 14.5845 | + | 3.90791i | 7.28515 | − | 7.28515i | 20.4957 | − | 17.5763i | 0 | ||||
68.4 | −0.956489 | + | 3.56967i | −3.31660 | − | 4.00002i | −4.89944 | − | 2.82870i | 0 | 17.4510 | − | 8.01320i | 4.17492 | + | 1.11867i | −6.12165 | + | 6.12165i | −5.00028 | + | 26.5329i | 0 | ||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.4.p.b | 64 | |
5.b | even | 2 | 1 | 45.4.l.a | ✓ | 64 | |
5.c | odd | 4 | 1 | 45.4.l.a | ✓ | 64 | |
5.c | odd | 4 | 1 | inner | 225.4.p.b | 64 | |
9.d | odd | 6 | 1 | inner | 225.4.p.b | 64 | |
15.d | odd | 2 | 1 | 135.4.m.a | 64 | ||
15.e | even | 4 | 1 | 135.4.m.a | 64 | ||
45.h | odd | 6 | 1 | 45.4.l.a | ✓ | 64 | |
45.j | even | 6 | 1 | 135.4.m.a | 64 | ||
45.k | odd | 12 | 1 | 135.4.m.a | 64 | ||
45.l | even | 12 | 1 | 45.4.l.a | ✓ | 64 | |
45.l | even | 12 | 1 | inner | 225.4.p.b | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.4.l.a | ✓ | 64 | 5.b | even | 2 | 1 | |
45.4.l.a | ✓ | 64 | 5.c | odd | 4 | 1 | |
45.4.l.a | ✓ | 64 | 45.h | odd | 6 | 1 | |
45.4.l.a | ✓ | 64 | 45.l | even | 12 | 1 | |
135.4.m.a | 64 | 15.d | odd | 2 | 1 | ||
135.4.m.a | 64 | 15.e | even | 4 | 1 | ||
135.4.m.a | 64 | 45.j | even | 6 | 1 | ||
135.4.m.a | 64 | 45.k | odd | 12 | 1 | ||
225.4.p.b | 64 | 1.a | even | 1 | 1 | trivial | |
225.4.p.b | 64 | 5.c | odd | 4 | 1 | inner | |
225.4.p.b | 64 | 9.d | odd | 6 | 1 | inner | |
225.4.p.b | 64 | 45.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 6 T_{2}^{63} + 18 T_{2}^{62} - 36 T_{2}^{61} - 1721 T_{2}^{60} + 9540 T_{2}^{59} + \cdots + 47\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).