Properties

Label 585.2.cn.a
Level $585$
Weight $2$
Character orbit 585.cn
Analytic conductor $4.671$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 6 q^{5} - 24 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 6 q^{5} - 24 q^{6} - 4 q^{9} - 12 q^{10} - 12 q^{11} - 24 q^{14} + 10 q^{15} - 280 q^{16} - 16 q^{19} + 6 q^{20} + 8 q^{21} + 44 q^{24} - 48 q^{26} - 6 q^{30} - 4 q^{31} - 20 q^{34} + 66 q^{35} + 36 q^{36} + 20 q^{39} - 12 q^{40} + 12 q^{41} + 72 q^{44} + 4 q^{45} - 8 q^{46} - 12 q^{49} + 48 q^{50} + 84 q^{54} - 4 q^{55} - 12 q^{56} - 12 q^{59} - 76 q^{60} + 4 q^{61} - 84 q^{65} - 20 q^{66} - 12 q^{69} - 28 q^{70} - 48 q^{71} + 36 q^{74} + 12 q^{75} - 4 q^{76} - 20 q^{79} - 6 q^{80} - 48 q^{81} - 96 q^{84} - 16 q^{85} - 12 q^{86} - 60 q^{89} + 114 q^{90} - 16 q^{91} + 52 q^{94} - 12 q^{95} + 180 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.80
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(585, [\chi])\).