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Properties

Label 592.2.ca.a
Level $592$
Weight $2$
Character orbit 592.ca
Analytic conductor $4.727$
Analytic rank $0$
Dimension $888$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [592,2,Mod(19,592)]
 
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([18, 27, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.ca (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.

\(\operatorname{Tr}(f)(q) = \) \( 888 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 888 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 24 q^{8} - 6 q^{10} - 18 q^{11} + 18 q^{12} - 12 q^{13} - 12 q^{14} + 18 q^{16} - 24 q^{17} - 120 q^{18} - 12 q^{19} - 12 q^{20} - 12 q^{21} - 42 q^{22} - 24 q^{23} - 24 q^{24} - 6 q^{26} - 18 q^{27} - 42 q^{28} - 6 q^{29} - 78 q^{30} + 24 q^{32} - 24 q^{33} - 12 q^{34} - 12 q^{35} + 36 q^{37} - 108 q^{38} - 24 q^{39} + 18 q^{40} - 30 q^{42} - 96 q^{44} - 18 q^{45} + 48 q^{46} - 18 q^{48} - 24 q^{49} + 30 q^{50} - 198 q^{51} - 12 q^{52} - 36 q^{53} + 18 q^{54} - 24 q^{55} + 156 q^{56} - 84 q^{58} - 12 q^{59} + 144 q^{60} - 12 q^{61} - 12 q^{62} - 144 q^{64} + 24 q^{66} - 12 q^{67} + 138 q^{68} + 6 q^{69} + 60 q^{70} - 24 q^{71} - 180 q^{72} + 48 q^{73} - 54 q^{74} - 24 q^{75} + 96 q^{76} - 30 q^{77} + 60 q^{78} - 144 q^{80} - 24 q^{81} - 36 q^{82} + 48 q^{83} - 12 q^{84} - 18 q^{85} + 12 q^{86} - 192 q^{87} - 120 q^{88} - 48 q^{89} - 78 q^{90} - 12 q^{91} + 336 q^{92} + 24 q^{93} - 60 q^{94} - 126 q^{96} - 24 q^{97} + 42 q^{98} - 210 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41119 + 0.0924507i −1.50299 − 0.131495i 1.98291 − 0.260931i 0.504969 + 0.183794i 2.13316 + 0.0466112i 1.34600 + 0.489902i −2.77413 + 0.551544i −0.712725 − 0.125673i −0.729598 − 0.212683i
19.2 −1.41032 − 0.104857i −1.73679 − 0.151950i 1.97801 + 0.295765i 2.08413 + 0.758560i 2.43350 + 0.396413i −0.289795 − 0.105477i −2.75862 − 0.624533i 0.0389394 + 0.00686607i −2.85975 − 1.28835i
19.3 −1.40723 + 0.140356i 2.58092 + 0.225801i 1.96060 − 0.395026i 3.86831 + 1.40795i −3.66364 + 0.0444926i −0.927106 − 0.337439i −2.70357 + 0.831075i 3.65573 + 0.644604i −5.64122 − 1.43837i
19.4 −1.39566 − 0.228297i −3.32903 − 0.291252i 1.89576 + 0.637253i −2.93393 − 1.06786i 4.57972 + 1.16650i 0.0360796 + 0.0131319i −2.50036 − 1.32219i 8.04320 + 1.41823i 3.85099 + 2.16019i
19.5 −1.39326 + 0.242541i 1.15967 + 0.101458i 1.88235 − 0.675846i −1.63459 − 0.594943i −1.64033 + 0.139910i 1.92757 + 0.701578i −2.45868 + 1.39818i −1.61989 − 0.285630i 2.42171 + 0.432455i
19.6 −1.36435 − 0.372207i 0.918006 + 0.0803151i 1.72292 + 1.01564i 0.530596 + 0.193121i −1.22259 − 0.451266i −2.83146 − 1.03057i −1.97265 − 2.02698i −2.11814 − 0.373485i −0.652040 − 0.460977i
19.7 −1.36347 + 0.375424i 1.30331 + 0.114025i 1.71811 − 1.02376i −3.66238 − 1.33300i −1.81984 + 0.333825i −0.291349 − 0.106042i −1.95826 + 2.04089i −1.26880 − 0.223724i 5.49400 + 0.442559i
19.8 −1.33235 − 0.474170i 2.80961 + 0.245809i 1.55033 + 1.26352i −2.11647 − 0.770330i −3.62683 − 1.65974i −2.80361 − 1.02043i −1.46645 − 2.41858i 4.87904 + 0.860307i 2.45461 + 2.02992i
19.9 −1.31292 − 0.525584i 1.65775 + 0.145034i 1.44752 + 1.38010i 1.44863 + 0.527259i −2.10027 − 1.06171i 1.93226 + 0.703287i −1.17512 − 2.57276i −0.227324 − 0.0400833i −1.62482 − 1.45363i
19.10 −1.30391 + 0.547554i 0.512931 + 0.0448756i 1.40037 − 1.42792i 0.968889 + 0.352647i −0.693388 + 0.222343i −3.22160 − 1.17257i −1.04409 + 2.62866i −2.69334 − 0.474908i −1.45644 + 0.0706986i
19.11 −1.29195 + 0.575218i 3.29420 + 0.288205i 1.33825 − 1.48630i −1.26639 − 0.460928i −4.42171 + 1.52254i 1.72948 + 0.629480i −0.873997 + 2.69001i 7.81427 + 1.37787i 1.90124 − 0.132956i
19.12 −1.28906 − 0.581647i −0.376009 − 0.0328965i 1.32337 + 1.49956i −2.69350 − 0.980352i 0.465565 + 0.261110i 3.92251 + 1.42768i −0.833698 − 2.70277i −2.81412 − 0.496206i 2.90187 + 2.83040i
19.13 −1.26263 + 0.636998i −1.96684 − 0.172076i 1.18847 − 1.60859i −2.44115 − 0.888506i 2.59301 − 1.03561i −4.30587 − 1.56721i −0.475927 + 2.78810i 0.884435 + 0.155950i 3.64825 − 0.433155i
19.14 −1.21820 − 0.718325i −2.33875 − 0.204615i 0.968019 + 1.75013i 3.21839 + 1.17140i 2.70209 + 1.92925i −4.33243 − 1.57687i 0.0779178 − 2.82735i 2.47348 + 0.436142i −3.07919 − 3.73884i
19.15 −1.18327 + 0.774514i −3.38481 − 0.296132i 0.800256 − 1.83292i 3.18032 + 1.15754i 4.23450 − 2.27118i 2.53070 + 0.921099i 0.472702 + 2.78865i 8.41479 + 1.48376i −4.65971 + 1.09352i
19.16 −1.15469 + 0.816508i 0.749115 + 0.0655391i 0.666628 − 1.88563i 1.69290 + 0.616165i −0.918511 + 0.535981i 4.76400 + 1.73396i 0.769884 + 2.72163i −2.39754 − 0.422752i −2.45788 + 0.670786i
19.17 −1.08326 − 0.909142i 0.455757 + 0.0398735i 0.346920 + 1.96968i 3.31320 + 1.20590i −0.457454 − 0.457541i 1.76089 + 0.640910i 1.41492 − 2.44908i −2.74830 − 0.484599i −2.49272 − 4.31848i
19.18 −1.06830 + 0.926676i −2.18057 − 0.190775i 0.282544 − 1.97994i −0.814616 − 0.296496i 2.50630 − 1.81688i 0.370051 + 0.134688i 1.53292 + 2.37700i 1.76407 + 0.311054i 1.14501 − 0.438137i
19.19 −0.999384 − 1.00062i 3.16305 + 0.276731i −0.00246231 + 2.00000i 0.453683 + 0.165127i −2.88420 − 3.44156i 3.58354 + 1.30430i 2.00369 − 1.99630i 6.97389 + 1.22969i −0.288175 − 0.618988i
19.20 −0.955667 − 1.04245i −1.28513 − 0.112435i −0.173402 + 1.99247i −2.47931 − 0.902394i 1.11095 + 1.44714i −2.23545 − 0.813638i 2.24276 − 1.72337i −1.31550 − 0.231958i 1.42869 + 3.44694i
See next 80 embeddings (of 888 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
592.ca 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.ca.a ✓ 888
16.f odd 4 1 592.2.cd.a yes 888
37.i odd 36 1 592.2.cd.a yes 888
592.ca even 36 1 inner 592.2.ca.a ✓ 888
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
592.2.ca.a ✓ 888 1.a even 1 1 trivial
592.2.ca.a ✓ 888 592.ca even 36 1 inner
592.2.cd.a yes 888 16.f odd 4 1
592.2.cd.a yes 888 37.i odd 36 1

Hecke kernels

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