Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(59,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 6, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.cn (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: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(320\) |
Relative dimension: | \(80\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 | −1.94625 | − | 1.94625i | −1.64127 | + | 0.553396i | 5.57582i | −2.22473 | + | 0.224911i | 4.27137 | + | 2.11727i | 0.325483 | − | 0.0872129i | 6.95945 | − | 6.95945i | 2.38751 | − | 1.81654i | 4.76762 | + | 3.89215i | ||
59.2 | −1.93326 | − | 1.93326i | 1.04015 | + | 1.38495i | 5.47498i | 1.64977 | − | 1.50940i | 0.666590 | − | 4.68835i | 0.772264 | − | 0.206928i | 6.71804 | − | 6.71804i | −0.836178 | + | 2.88111i | −6.10748 | − | 0.271372i | ||
59.3 | −1.91331 | − | 1.91331i | −0.327571 | − | 1.70079i | 5.32153i | −0.814026 | − | 2.08263i | −2.62740 | + | 3.88089i | 2.46797 | − | 0.661291i | 6.35511 | − | 6.35511i | −2.78539 | + | 1.11426i | −2.42724 | + | 5.54221i | ||
59.4 | −1.85091 | − | 1.85091i | 0.745675 | − | 1.56332i | 4.85176i | 1.86108 | + | 1.23951i | −4.27375 | + | 1.51339i | −3.75625 | + | 1.00648i | 5.27836 | − | 5.27836i | −1.88794 | − | 2.33146i | −1.15046 | − | 5.73892i | ||
59.5 | −1.81323 | − | 1.81323i | 1.73151 | + | 0.0430874i | 4.57563i | −1.44588 | − | 1.70571i | −3.06151 | − | 3.21777i | −4.81725 | + | 1.29078i | 4.67022 | − | 4.67022i | 2.99629 | + | 0.149213i | −0.471139 | + | 5.71456i | ||
59.6 | −1.76245 | − | 1.76245i | −0.729865 | + | 1.57076i | 4.21247i | 1.51779 | + | 1.64205i | 4.05474 | − | 1.48204i | 3.72057 | − | 0.996925i | 3.89937 | − | 3.89937i | −1.93459 | − | 2.29289i | 0.218995 | − | 5.56906i | ||
59.7 | −1.70431 | − | 1.70431i | 0.0523905 | + | 1.73126i | 3.80934i | −0.388734 | + | 2.20202i | 2.86131 | − | 3.03989i | −3.51649 | + | 0.942241i | 3.08367 | − | 3.08367i | −2.99451 | + | 0.181403i | 4.41544 | − | 3.09040i | ||
59.8 | −1.68369 | − | 1.68369i | 1.08191 | − | 1.35258i | 3.66960i | −1.81752 | + | 1.30255i | −4.09892 | + | 0.455709i | 2.08936 | − | 0.559844i | 2.81109 | − | 2.81109i | −0.658923 | − | 2.92674i | 5.25321 | + | 0.867047i | ||
59.9 | −1.65542 | − | 1.65542i | −1.70938 | − | 0.279330i | 3.48083i | 1.89655 | + | 1.18452i | 2.36733 | + | 3.29215i | 1.66783 | − | 0.446893i | 2.45140 | − | 2.45140i | 2.84395 | + | 0.954963i | −1.17871 | − | 5.10047i | ||
59.10 | −1.62238 | − | 1.62238i | −1.49998 | + | 0.866052i | 3.26427i | 0.420150 | − | 2.19624i | 3.83862 | + | 1.02848i | −0.840913 | + | 0.225322i | 2.05113 | − | 2.05113i | 1.49991 | − | 2.59813i | −4.24479 | + | 2.88150i | ||
59.11 | −1.60167 | − | 1.60167i | 1.52777 | − | 0.816043i | 3.13069i | 2.18696 | − | 0.466059i | −3.75401 | − | 1.13995i | 2.59318 | − | 0.694839i | 1.81099 | − | 1.81099i | 1.66815 | − | 2.49345i | −4.24926 | − | 2.75631i | ||
59.12 | −1.54486 | − | 1.54486i | −0.679716 | − | 1.59311i | 2.77316i | −0.737910 | + | 2.11080i | −1.41106 | + | 3.51118i | 0.537039 | − | 0.143899i | 1.19441 | − | 1.19441i | −2.07597 | + | 2.16572i | 4.40085 | − | 2.12092i | ||
59.13 | −1.53587 | − | 1.53587i | 1.37280 | + | 1.05614i | 2.71780i | −1.91376 | + | 1.15651i | −0.486345 | − | 3.73053i | 0.538136 | − | 0.144193i | 1.10246 | − | 1.10246i | 0.769140 | + | 2.89973i | 4.71554 | + | 1.16304i | ||
59.14 | −1.41644 | − | 1.41644i | −0.980704 | − | 1.42766i | 2.01259i | −1.92422 | − | 1.13903i | −0.633088 | + | 3.41130i | −3.47546 | + | 0.931245i | 0.0178351 | − | 0.0178351i | −1.07644 | + | 2.80023i | 1.11216 | + | 4.33890i | ||
59.15 | −1.38071 | − | 1.38071i | 1.72508 | − | 0.155257i | 1.81274i | −1.23211 | − | 1.86598i | −2.59620 | − | 2.16747i | 2.69953 | − | 0.723336i | −0.258553 | + | 0.258553i | 2.95179 | − | 0.535662i | −0.875194 | + | 4.27759i | ||
59.16 | −1.34121 | − | 1.34121i | −0.277240 | + | 1.70972i | 1.59769i | −0.210743 | − | 2.22611i | 2.66493 | − | 1.92125i | −0.170239 | + | 0.0456153i | −0.539585 | + | 0.539585i | −2.84628 | − | 0.948004i | −2.70304 | + | 3.26834i | ||
59.17 | −1.28176 | − | 1.28176i | 1.12928 | + | 1.31329i | 1.28584i | 2.16084 | − | 0.575117i | 0.235850 | − | 3.13080i | −0.980219 | + | 0.262649i | −0.915390 | + | 0.915390i | −0.449443 | + | 2.96614i | −3.50685 | − | 2.03253i | ||
59.18 | −1.20447 | − | 1.20447i | −0.649954 | − | 1.60548i | 0.901508i | 2.10551 | − | 0.752877i | −1.15090 | + | 2.71661i | 4.16747 | − | 1.11667i | −1.32310 | + | 1.32310i | −2.15512 | + | 2.08697i | −3.44285 | − | 1.62921i | ||
59.19 | −1.20043 | − | 1.20043i | −1.41832 | + | 0.994165i | 0.882060i | 2.15454 | + | 0.598279i | 2.89602 | + | 0.509170i | −4.33313 | + | 1.16106i | −1.34201 | + | 1.34201i | 1.02327 | − | 2.82009i | −1.86819 | − | 3.30457i | ||
59.20 | −1.12962 | − | 1.12962i | 1.72615 | − | 0.142821i | 0.552084i | 1.05878 | + | 1.96951i | −2.11123 | − | 1.78856i | −0.591700 | + | 0.158546i | −1.63560 | + | 1.63560i | 2.95920 | − | 0.493061i | 1.02878 | − | 3.42082i | ||
See next 80 embeddings (of 320 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
117.x | even | 12 | 1 | inner |
585.cn | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.cn.a | ✓ | 320 |
5.b | even | 2 | 1 | inner | 585.2.cn.a | ✓ | 320 |
9.d | odd | 6 | 1 | 585.2.de.a | yes | 320 | |
13.f | odd | 12 | 1 | 585.2.de.a | yes | 320 | |
45.h | odd | 6 | 1 | 585.2.de.a | yes | 320 | |
65.s | odd | 12 | 1 | 585.2.de.a | yes | 320 | |
117.x | even | 12 | 1 | inner | 585.2.cn.a | ✓ | 320 |
585.cn | even | 12 | 1 | inner | 585.2.cn.a | ✓ | 320 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.cn.a | ✓ | 320 | 1.a | even | 1 | 1 | trivial |
585.2.cn.a | ✓ | 320 | 5.b | even | 2 | 1 | inner |
585.2.cn.a | ✓ | 320 | 117.x | even | 12 | 1 | inner |
585.2.cn.a | ✓ | 320 | 585.cn | even | 12 | 1 | inner |
585.2.de.a | yes | 320 | 9.d | odd | 6 | 1 | |
585.2.de.a | yes | 320 | 13.f | odd | 12 | 1 | |
585.2.de.a | yes | 320 | 45.h | odd | 6 | 1 | |
585.2.de.a | yes | 320 | 65.s | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).