Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,2,Mod(2,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([10, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.w (of order \(60\), degree \(16\), 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: | \(1.79663404548\) |
Analytic rank: | \(0\) |
Dimension: | \(448\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{60})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | −2.08185 | − | 1.68585i | 0.798183 | + | 1.53717i | 1.07618 | + | 5.06304i | −2.23432 | + | 0.0883769i | 0.929742 | − | 4.54577i | 0.0946799 | − | 0.0253694i | 3.86273 | − | 7.58103i | −1.72581 | + | 2.45389i | 4.80050 | + | 3.58273i |
2.2 | −2.00482 | − | 1.62347i | −0.535601 | − | 1.64716i | 0.967816 | + | 4.55321i | −0.251379 | + | 2.22189i | −1.60033 | + | 4.17178i | 3.21084 | − | 0.860341i | 3.10937 | − | 6.10249i | −2.42626 | + | 1.76444i | 4.11114 | − | 4.04638i |
2.3 | −1.92972 | − | 1.56265i | 1.72141 | − | 0.191662i | 0.866095 | + | 4.07466i | 2.21571 | − | 0.301067i | −3.62134 | − | 2.32012i | −3.08775 | + | 0.827360i | 2.44137 | − | 4.79145i | 2.92653 | − | 0.659860i | −4.74615 | − | 2.88141i |
2.4 | −1.88812 | − | 1.52897i | −1.53403 | − | 0.804204i | 0.811429 | + | 3.81747i | −0.154565 | − | 2.23072i | 1.66684 | + | 3.86393i | −3.12915 | + | 0.838454i | 2.09874 | − | 4.11900i | 1.70651 | + | 2.46735i | −3.11887 | + | 4.44820i |
2.5 | −1.54331 | − | 1.24975i | 1.24822 | − | 1.20081i | 0.404110 | + | 1.90119i | −0.779714 | − | 2.09572i | −3.42709 | + | 0.293265i | 3.96539 | − | 1.06252i | −0.0507944 | + | 0.0996896i | 0.116102 | − | 2.99775i | −1.41578 | + | 4.20878i |
2.6 | −1.44398 | − | 1.16931i | 0.219166 | + | 1.71813i | 0.301968 | + | 1.42065i | 2.00481 | + | 0.990313i | 1.69256 | − | 2.73722i | 0.739070 | − | 0.198033i | −0.461936 | + | 0.906601i | −2.90393 | + | 0.753111i | −1.73693 | − | 3.77425i |
2.7 | −1.32373 | − | 1.07193i | −1.37118 | + | 1.05824i | 0.187388 | + | 0.881593i | −2.23256 | + | 0.125228i | 2.94942 | + | 0.0689920i | 0.893133 | − | 0.239314i | −0.849622 | + | 1.66748i | 0.760261 | − | 2.90207i | 3.08953 | + | 2.22738i |
2.8 | −1.26732 | − | 1.02626i | 0.979850 | − | 1.42825i | 0.137078 | + | 0.644900i | −1.08873 | + | 1.95312i | −2.70753 | + | 0.804473i | −3.51917 | + | 0.942959i | −0.992568 | + | 1.94802i | −1.07979 | − | 2.79894i | 3.38417 | − | 1.35792i |
2.9 | −1.14980 | − | 0.931089i | −1.68761 | − | 0.389856i | 0.0392881 | + | 0.184836i | 2.21919 | − | 0.274221i | 1.57742 | + | 2.01957i | 3.95765 | − | 1.06045i | −1.21645 | + | 2.38741i | 2.69602 | + | 1.31585i | −2.80695 | − | 1.75096i |
2.10 | −0.870681 | − | 0.705063i | −0.448846 | + | 1.67288i | −0.154853 | − | 0.728525i | 0.686598 | − | 2.12805i | 1.57029 | − | 1.14008i | −3.87586 | + | 1.03853i | −1.39609 | + | 2.73998i | −2.59708 | − | 1.50173i | −2.09822 | + | 1.36875i |
2.11 | −0.863435 | − | 0.699196i | 1.54952 | + | 0.773953i | −0.159178 | − | 0.748875i | −0.594209 | + | 2.15567i | −0.796761 | − | 1.75167i | 1.93702 | − | 0.519023i | −1.39497 | + | 2.73778i | 1.80200 | + | 2.39850i | 2.02030 | − | 1.44581i |
2.12 | −0.476183 | − | 0.385605i | −1.66967 | − | 0.460650i | −0.337765 | − | 1.58906i | 0.528402 | + | 2.17274i | 0.617440 | + | 0.863188i | −3.94730 | + | 1.05768i | −1.00826 | + | 1.97882i | 2.57560 | + | 1.53827i | 0.586203 | − | 1.23838i |
2.13 | −0.350664 | − | 0.283962i | −0.107492 | − | 1.72871i | −0.373493 | − | 1.75714i | 0.997580 | − | 2.00121i | −0.453195 | + | 0.636721i | −0.764485 | + | 0.204843i | −0.777692 | + | 1.52631i | −2.97689 | + | 0.371647i | −0.918083 | + | 0.418477i |
2.14 | −0.235350 | − | 0.190582i | −1.06897 | − | 1.36283i | −0.396756 | − | 1.86659i | −2.22710 | + | 0.200096i | −0.00815054 | + | 0.524468i | 2.13576 | − | 0.572276i | −0.537334 | + | 1.05458i | −0.714618 | + | 2.91364i | 0.562281 | + | 0.377353i |
2.15 | −0.123603 | − | 0.100092i | 1.69052 | + | 0.376997i | −0.410564 | − | 1.93155i | 1.70202 | − | 1.45021i | −0.171220 | − | 0.215806i | −0.310258 | + | 0.0831334i | −0.286998 | + | 0.563265i | 2.71575 | + | 1.27465i | −0.355530 | + | 0.00889246i |
2.16 | 0.209703 | + | 0.169814i | 0.333716 | + | 1.69960i | −0.400685 | − | 1.88507i | −1.21183 | − | 1.87922i | −0.218634 | + | 0.413080i | 4.98040 | − | 1.33450i | 0.481094 | − | 0.944201i | −2.77727 | + | 1.13437i | 0.0649950 | − | 0.599864i |
2.17 | 0.319818 | + | 0.258983i | 1.43991 | − | 0.962634i | −0.380612 | − | 1.79064i | −2.05377 | − | 0.884313i | 0.709815 | + | 0.0650446i | −2.53527 | + | 0.679325i | 0.715680 | − | 1.40460i | 1.14667 | − | 2.77221i | −0.427811 | − | 0.814713i |
2.18 | 0.614967 | + | 0.497991i | −1.48472 | + | 0.891965i | −0.285633 | − | 1.34380i | 2.12267 | − | 0.703044i | −1.35725 | − | 0.190848i | 0.864298 | − | 0.231588i | 1.21204 | − | 2.37877i | 1.40880 | − | 2.64864i | 1.65548 | + | 0.624721i |
2.19 | 0.700361 | + | 0.567141i | 0.0516917 | − | 1.73128i | −0.246967 | − | 1.16189i | 1.17286 | + | 1.90378i | 1.01808 | − | 1.18320i | 1.15542 | − | 0.309594i | 1.30426 | − | 2.55975i | −2.99466 | − | 0.178986i | −0.258288 | + | 1.99851i |
2.20 | 0.802972 | + | 0.650234i | −1.64915 | + | 0.529426i | −0.193864 | − | 0.912056i | −1.97227 | − | 1.05364i | −1.66847 | − | 0.647221i | −2.82373 | + | 0.756615i | 1.37554 | − | 2.69964i | 2.43942 | − | 1.74621i | −0.898562 | − | 2.12848i |
See next 80 embeddings (of 448 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
25.f | odd | 20 | 1 | inner |
225.w | even | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.2.w.a | ✓ | 448 |
3.b | odd | 2 | 1 | 675.2.bd.a | 448 | ||
9.c | even | 3 | 1 | 675.2.bd.a | 448 | ||
9.d | odd | 6 | 1 | inner | 225.2.w.a | ✓ | 448 |
25.f | odd | 20 | 1 | inner | 225.2.w.a | ✓ | 448 |
75.l | even | 20 | 1 | 675.2.bd.a | 448 | ||
225.w | even | 60 | 1 | inner | 225.2.w.a | ✓ | 448 |
225.x | odd | 60 | 1 | 675.2.bd.a | 448 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.2.w.a | ✓ | 448 | 1.a | even | 1 | 1 | trivial |
225.2.w.a | ✓ | 448 | 9.d | odd | 6 | 1 | inner |
225.2.w.a | ✓ | 448 | 25.f | odd | 20 | 1 | inner |
225.2.w.a | ✓ | 448 | 225.w | even | 60 | 1 | inner |
675.2.bd.a | 448 | 3.b | odd | 2 | 1 | ||
675.2.bd.a | 448 | 9.c | even | 3 | 1 | ||
675.2.bd.a | 448 | 75.l | even | 20 | 1 | ||
675.2.bd.a | 448 | 225.x | odd | 60 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(225, [\chi])\).