Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,4,Mod(16,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([20, 6]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.16");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.q (of order \(15\), degree \(8\), 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: | \(704\) |
Relative dimension: | \(88\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −4.99262 | + | 2.22286i | −0.539608 | − | 5.16806i | 14.6321 | − | 16.2506i | −11.1000 | + | 1.33789i | 14.1819 | + | 24.6027i | 8.61806 | − | 14.9269i | −23.4194 | + | 72.0775i | −26.4176 | + | 5.57745i | 52.4442 | − | 31.3533i |
16.2 | −4.95886 | + | 2.20783i | 5.16428 | + | 0.574673i | 14.3627 | − | 15.9514i | 5.23221 | − | 9.88049i | −26.8777 | + | 8.55210i | 11.6110 | − | 20.1109i | −22.5857 | + | 69.5116i | 26.3395 | + | 5.93554i | −4.13139 | + | 60.5477i |
16.3 | −4.88581 | + | 2.17530i | 3.34361 | + | 3.97747i | 13.7862 | − | 15.3111i | −5.63199 | + | 9.65819i | −24.9884 | − | 12.1598i | −3.78510 | + | 6.55599i | −20.8289 | + | 64.1048i | −4.64059 | + | 26.5982i | 6.50737 | − | 59.4394i |
16.4 | −4.86733 | + | 2.16708i | −2.70456 | + | 4.43682i | 13.6417 | − | 15.1506i | 9.93766 | − | 5.12279i | 3.54905 | − | 27.4565i | −12.9967 | + | 22.5110i | −20.3946 | + | 62.7682i | −12.3708 | − | 23.9993i | −37.2684 | + | 46.4700i |
16.5 | −4.73105 | + | 2.10640i | −4.59183 | − | 2.43210i | 12.5929 | − | 13.9858i | 6.40846 | + | 9.16142i | 26.8472 | + | 1.83415i | −0.678500 | + | 1.17520i | −17.3152 | + | 53.2908i | 15.1698 | + | 22.3356i | −49.6164 | − | 29.8444i |
16.6 | −4.55459 | + | 2.02783i | −2.55636 | + | 4.52383i | 11.2791 | − | 12.5267i | 2.67963 | + | 10.8545i | 2.46961 | − | 25.7880i | 15.1279 | − | 26.2024i | −13.6445 | + | 41.9935i | −13.9300 | − | 23.1291i | −34.2157 | − | 44.0038i |
16.7 | −4.46544 | + | 1.98814i | −5.18899 | + | 0.272686i | 10.6344 | − | 11.8107i | −11.1764 | + | 0.297118i | 22.6290 | − | 11.5341i | −12.8663 | + | 22.2852i | −11.9221 | + | 36.6924i | 26.8513 | − | 2.82993i | 49.3168 | − | 23.5470i |
16.8 | −4.41308 | + | 1.96483i | 3.87578 | − | 3.46097i | 10.2617 | − | 11.3968i | −6.88435 | − | 8.80941i | −10.3039 | + | 22.8888i | −13.7602 | + | 23.8334i | −10.9508 | + | 33.7031i | 3.04337 | − | 26.8279i | 47.6902 | + | 25.3501i |
16.9 | −4.40428 | + | 1.96091i | −4.10868 | + | 3.18099i | 10.1995 | − | 11.3277i | −7.46697 | − | 8.32132i | 11.8582 | − | 22.0668i | 13.0363 | − | 22.5795i | −10.7905 | + | 33.2096i | 6.76255 | − | 26.1394i | 49.2040 | + | 22.0074i |
16.10 | −4.39884 | + | 1.95849i | 0.262179 | − | 5.18953i | 10.1611 | − | 11.2850i | 9.88111 | − | 5.23102i | 9.01036 | + | 23.3414i | −5.56077 | + | 9.63153i | −10.6917 | + | 32.9056i | −26.8625 | − | 2.72118i | −33.2205 | + | 42.3625i |
16.11 | −4.21154 | + | 1.87510i | 1.87188 | + | 4.84727i | 8.86800 | − | 9.84891i | −8.20714 | − | 7.59229i | −16.9726 | − | 16.9045i | −4.19737 | + | 7.27006i | −7.48344 | + | 23.0317i | −19.9921 | + | 18.1470i | 48.8009 | + | 16.5860i |
16.12 | −4.11599 | + | 1.83256i | 5.19523 | + | 0.0977072i | 8.23008 | − | 9.14043i | 9.68777 | + | 5.58096i | −21.5626 | + | 9.11840i | −10.5565 | + | 18.2844i | −5.98634 | + | 18.4240i | 26.9809 | + | 1.01522i | −50.1022 | − | 5.21779i |
16.13 | −4.10662 | + | 1.82839i | −3.62152 | − | 3.72620i | 8.16829 | − | 9.07181i | 2.22738 | − | 10.9562i | 21.6852 | + | 8.68055i | 5.37596 | − | 9.31143i | −5.84444 | + | 17.9873i | −0.769138 | + | 26.9890i | 10.8852 | + | 49.0655i |
16.14 | −3.94276 | + | 1.75543i | 4.68747 | − | 2.24224i | 7.11079 | − | 7.89733i | −7.42625 | + | 8.35768i | −14.5455 | + | 17.0691i | 3.16503 | − | 5.48200i | −3.50346 | + | 10.7825i | 16.9447 | − | 21.0208i | 14.6086 | − | 45.9886i |
16.15 | −3.79552 | + | 1.68988i | 2.86836 | − | 4.33273i | 6.19727 | − | 6.88277i | 9.82087 | + | 5.34326i | −3.56515 | + | 21.2921i | 17.2551 | − | 29.8867i | −1.61984 | + | 4.98535i | −10.5450 | − | 24.8556i | −46.3048 | − | 3.68441i |
16.16 | −3.51417 | + | 1.56461i | 3.51869 | + | 3.82345i | 4.54836 | − | 5.05146i | 9.92601 | + | 5.14531i | −18.3475 | − | 7.93089i | 3.48252 | − | 6.03190i | 1.42952 | − | 4.39961i | −2.23760 | + | 26.9071i | −42.9321 | − | 2.55117i |
16.17 | −3.44830 | + | 1.53528i | 0.918757 | + | 5.11428i | 4.18063 | − | 4.64306i | 6.75551 | − | 8.90859i | −11.0200 | − | 16.2250i | 5.43165 | − | 9.40790i | 2.04374 | − | 6.28999i | −25.3118 | + | 9.39757i | −9.61783 | + | 41.0911i |
16.18 | −3.38912 | + | 1.50893i | −0.512070 | − | 5.17086i | 3.85619 | − | 4.28273i | −0.951580 | + | 11.1398i | 9.53794 | + | 16.7520i | −8.94160 | + | 15.4873i | 2.56453 | − | 7.89280i | −26.4756 | + | 5.29569i | −13.5841 | − | 39.1899i |
16.19 | −3.33000 | + | 1.48261i | −5.18559 | + | 0.331159i | 3.53770 | − | 3.92901i | 10.7249 | − | 3.15860i | 16.7770 | − | 8.79097i | 5.87729 | − | 10.1798i | 3.05594 | − | 9.40521i | 26.7807 | − | 3.43451i | −31.0309 | + | 26.4190i |
16.20 | −3.24605 | + | 1.44523i | −4.88286 | + | 1.77700i | 3.09509 | − | 3.43745i | −4.43805 | + | 10.2618i | 13.2818 | − | 12.8251i | −4.89407 | + | 8.47677i | 3.70520 | − | 11.4034i | 20.6845 | − | 17.3537i | −0.424499 | − | 39.7242i |
See next 80 embeddings (of 704 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
25.d | even | 5 | 1 | inner |
225.q | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.4.q.a | ✓ | 704 |
9.c | even | 3 | 1 | inner | 225.4.q.a | ✓ | 704 |
25.d | even | 5 | 1 | inner | 225.4.q.a | ✓ | 704 |
225.q | even | 15 | 1 | inner | 225.4.q.a | ✓ | 704 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.4.q.a | ✓ | 704 | 1.a | even | 1 | 1 | trivial |
225.4.q.a | ✓ | 704 | 9.c | even | 3 | 1 | inner |
225.4.q.a | ✓ | 704 | 25.d | even | 5 | 1 | inner |
225.4.q.a | ✓ | 704 | 225.q | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(225, [\chi])\).