Properties

Label 592.2.by.a
Level $592$
Weight $2$
Character orbit 592.by
Analytic conductor $4.727$
Analytic rank $0$
Dimension $888$
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] = [592,2,Mod(21,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([0, 9, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.by (of order \(36\), degree \(12\), 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(888\)
Relative dimension: \(74\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 888 q - 12 q^{2} - 12 q^{3} - 18 q^{4} - 12 q^{5} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 888 q - 12 q^{2} - 12 q^{3} - 18 q^{4} - 12 q^{5} - 18 q^{8} - 6 q^{10} - 6 q^{11} - 12 q^{12} - 12 q^{13} - 18 q^{14} - 24 q^{15} + 18 q^{16} - 24 q^{17} + 18 q^{18} - 12 q^{19} - 96 q^{20} - 12 q^{21} - 12 q^{22} + 18 q^{24} - 6 q^{26} - 6 q^{27} - 12 q^{28} - 18 q^{29} - 54 q^{30} + 18 q^{32} - 24 q^{33} - 42 q^{34} - 12 q^{35} - 24 q^{36} - 60 q^{37} - 108 q^{38} + 48 q^{40} - 84 q^{42} + 30 q^{44} - 18 q^{45} + 108 q^{47} + 42 q^{48} - 24 q^{49} + 30 q^{50} - 198 q^{51} - 12 q^{52} - 36 q^{53} - 84 q^{54} + 30 q^{56} + 30 q^{58} - 12 q^{59} - 144 q^{60} - 12 q^{61} + 12 q^{62} - 132 q^{63} + 36 q^{64} - 48 q^{65} - 18 q^{66} - 12 q^{67} + 6 q^{69} + 90 q^{70} + 204 q^{72} + 30 q^{74} - 24 q^{75} - 108 q^{76} - 30 q^{77} + 60 q^{78} - 24 q^{79} - 24 q^{81} - 198 q^{82} - 72 q^{83} - 90 q^{84} - 6 q^{85} - 42 q^{86} + 72 q^{88} + 6 q^{90} - 12 q^{91} + 114 q^{92} + 24 q^{93} - 24 q^{95} + 60 q^{96} - 36 q^{97} + 42 q^{98} + 186 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −1.41362 0.0410164i 0.325684 0.698432i 1.99664 + 0.115963i −1.77468 2.53451i −0.489040 + 0.973958i 3.01385 + 0.531424i −2.81772 0.245822i 1.54663 + 1.84320i 2.40477 + 3.65562i
21.2 −1.41159 0.0860838i 0.721348 1.54694i 1.98518 + 0.243030i 0.124449 + 0.177732i −1.15141 + 2.12154i −0.313377 0.0552568i −2.78134 0.513951i 0.0556948 + 0.0663745i −0.160372 0.261598i
21.3 −1.40791 + 0.133387i −0.914826 + 1.96185i 1.96442 0.375592i −0.747883 1.06809i 1.02631 2.88413i −5.03623 0.888024i −2.71562 + 0.790827i −1.08359 1.29137i 1.19542 + 1.40401i
21.4 −1.40246 + 0.181942i 1.12487 2.41229i 1.93379 0.510333i 2.49161 + 3.55839i −1.13869 + 3.58780i 0.998928 + 0.176138i −2.61922 + 1.06756i −2.62544 3.12888i −4.14181 4.53717i
21.5 −1.40165 0.188059i −0.430006 + 0.922152i 1.92927 + 0.527186i 0.141897 + 0.202651i 0.776139 1.21167i 4.65418 + 0.820658i −2.60502 1.10175i 1.26290 + 1.50507i −0.160781 0.310731i
21.6 −1.37515 0.330112i −0.186355 + 0.399639i 1.78205 + 0.907904i 1.60552 + 2.29292i 0.388191 0.488044i −2.35981 0.416098i −2.15087 1.83678i 1.80338 + 2.14918i −1.45090 3.68310i
21.7 −1.37177 + 0.343878i −1.06094 + 2.27519i 1.76350 0.943442i −1.23965 1.77040i 0.672972 3.48586i 1.93431 + 0.341070i −2.09468 + 1.90061i −2.12252 2.52952i 2.30931 + 2.00229i
21.8 −1.34493 + 0.437211i −1.20635 + 2.58703i 1.61769 1.17604i 1.75480 + 2.50612i 0.491385 4.00682i 2.24528 + 0.395904i −1.66151 + 2.28897i −3.30909 3.94362i −3.45580 2.60334i
21.9 −1.32449 0.495701i 0.800436 1.71654i 1.50856 + 1.31310i −1.99406 2.84781i −1.91106 + 1.87677i −3.27919 0.578209i −1.34717 2.48699i −0.377450 0.449828i 1.22946 + 4.76037i
21.10 −1.29049 + 0.578487i 1.11244 2.38565i 1.33071 1.49306i −0.520407 0.743219i −0.0555303 + 3.72218i −3.41953 0.602955i −0.853545 + 2.69656i −2.52541 3.00966i 1.10152 + 0.658064i
21.11 −1.25320 + 0.655357i 1.29134 2.76928i 1.14101 1.64259i −1.34239 1.91713i 0.196566 + 4.31675i 3.98945 + 0.703448i −0.353439 + 2.80626i −4.07301 4.85403i 2.93868 + 1.52280i
21.12 −1.23971 + 0.680527i −0.157088 + 0.336877i 1.07377 1.68731i 1.54171 + 2.20179i −0.0345098 0.524533i 0.337590 + 0.0595263i −0.182898 + 2.82251i 1.83955 + 2.19229i −3.40966 1.68041i
21.13 −1.20829 0.734872i −1.41627 + 3.03721i 0.919926 + 1.77588i 0.903800 + 1.29076i 3.94323 2.62905i −1.44697 0.255140i 0.193504 2.82180i −5.29045 6.30491i −0.143508 2.22379i
21.14 −1.20801 0.735337i 0.740018 1.58697i 0.918559 + 1.77658i 0.781659 + 1.11633i −2.06091 + 1.37291i 2.31880 + 0.408867i 0.196762 2.82157i −0.0424939 0.0506422i −0.123374 1.92331i
21.15 −1.20289 + 0.743684i 0.202563 0.434399i 0.893867 1.78913i −0.955196 1.36416i 0.0793949 + 0.673175i −0.195194 0.0344180i 0.255332 + 2.81688i 1.78069 + 2.12215i 2.16350 + 0.930565i
21.16 −1.18439 0.772796i −0.817536 + 1.75321i 0.805573 + 1.83059i −1.69346 2.41852i 2.32316 1.44470i 0.418842 + 0.0738532i 0.460557 2.79068i −0.477024 0.568495i 0.136706 + 4.17317i
21.17 −1.11062 0.875519i −0.371753 + 0.797226i 0.466932 + 1.94473i −1.16010 1.65679i 1.11086 0.559935i −0.822247 0.144984i 1.18407 2.56865i 1.43099 + 1.70539i −0.162131 + 2.85575i
21.18 −1.05070 + 0.946588i −0.539330 + 1.15660i 0.207943 1.98916i −2.17855 3.11129i −0.528146 1.72576i −1.06027 0.186955i 1.66443 + 2.28685i 0.881525 + 1.05056i 5.23411 + 1.20685i
21.19 −0.983490 1.01624i 1.15158 2.46958i −0.0654955 + 1.99893i 0.243204 + 0.347331i −3.64226 + 1.25852i 1.36630 + 0.240915i 2.09581 1.89937i −2.84432 3.38973i 0.113784 0.588751i
21.20 −0.904166 + 1.08742i −1.13200 + 2.42758i −0.364967 1.96642i 0.149951 + 0.214152i −1.61629 3.42590i 0.637028 + 0.112325i 2.46831 + 1.38110i −2.68336 3.19791i −0.368454 0.0305693i
See next 80 embeddings (of 888 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
37.h even 18 1 inner
592.by even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.by.a 888
16.e even 4 1 inner 592.2.by.a 888
37.h even 18 1 inner 592.2.by.a 888
592.by even 36 1 inner 592.2.by.a 888
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
592.2.by.a 888 1.a even 1 1 trivial
592.2.by.a 888 16.e even 4 1 inner
592.2.by.a 888 37.h even 18 1 inner
592.2.by.a 888 592.by even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).