Properties

Label 592.2.s.a
Level $592$
Weight $2$
Character orbit 592.s
Analytic conductor $4.727$
Analytic rank $0$
Dimension $148$
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(339,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.339");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.s (of order \(4\), degree \(2\), 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: \(148\)
Relative dimension: \(74\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 148 q - 4 q^{2} - 4 q^{5} + 2 q^{6} - 8 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 148 q - 4 q^{2} - 4 q^{5} + 2 q^{6} - 8 q^{7} - 4 q^{8} - 4 q^{10} + 8 q^{11} - 4 q^{12} - 14 q^{14} - 12 q^{15} - 24 q^{16} - 4 q^{17} + 30 q^{18} - 4 q^{19} - 24 q^{20} + 6 q^{22} - 12 q^{23} - 28 q^{24} + 132 q^{25} - 4 q^{26} + 4 q^{28} + 12 q^{29} - 4 q^{30} - 24 q^{32} - 8 q^{33} + 12 q^{34} - 24 q^{35} - 32 q^{36} + 6 q^{37} + 52 q^{38} - 4 q^{39} - 20 q^{40} - 14 q^{42} - 4 q^{44} - 4 q^{46} + 28 q^{48} + 116 q^{49} - 44 q^{50} + 16 q^{52} - 4 q^{53} - 10 q^{54} - 4 q^{55} + 40 q^{56} - 12 q^{57} - 36 q^{58} - 16 q^{60} - 4 q^{61} - 28 q^{62} - 24 q^{64} + 62 q^{66} + 32 q^{68} - 32 q^{70} - 8 q^{71} - 44 q^{72} - 16 q^{73} - 28 q^{74} - 16 q^{75} - 12 q^{76} + 24 q^{78} + 40 q^{79} - 20 q^{80} - 124 q^{81} - 18 q^{82} - 4 q^{83} + 24 q^{84} - 20 q^{85} + 28 q^{86} + 52 q^{87} + 20 q^{88} - 8 q^{89} + 32 q^{90} - 48 q^{92} - 38 q^{94} - 40 q^{95} + 4 q^{96} - 4 q^{97} - 62 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1 −1.40895 0.121888i 1.82659 + 1.82659i 1.97029 + 0.343469i 2.73488 −2.35094 2.79622i 0.0550223 −2.73417 0.724086i 3.67289i −3.85331 0.333349i
339.2 −1.40004 0.199708i −1.00580 1.00580i 1.92023 + 0.559198i 3.89641 1.20729 + 1.60902i −2.11242 −2.57673 1.16639i 0.976747i −5.45514 0.778143i
339.3 −1.39780 + 0.214867i −1.10115 1.10115i 1.90766 0.600681i −1.71699 1.77579 + 1.30259i −0.420453 −2.53746 + 1.24952i 0.574923i 2.40000 0.368925i
339.4 −1.39081 0.256242i −2.29968 2.29968i 1.86868 + 0.712764i −2.15315 2.60914 + 3.78768i −4.08815 −2.41633 1.47015i 7.57707i 2.99461 + 0.551726i
339.5 −1.38151 + 0.302357i −0.641946 0.641946i 1.81716 0.835419i 1.23635 1.08095 + 0.692761i 3.44294 −2.25784 + 1.70357i 2.17581i −1.70804 + 0.373819i
339.6 −1.38027 0.307994i 2.14376 + 2.14376i 1.81028 + 0.850229i −3.28719 −2.29869 3.61922i 4.29351 −2.23680 1.73110i 6.19139i 4.53720 + 1.01244i
339.7 −1.37121 0.346101i −0.415690 0.415690i 1.76043 + 0.949154i −1.23281 0.426127 + 0.713868i 0.515574 −2.08541 1.91077i 2.65440i 1.69045 + 0.426678i
339.8 −1.36814 + 0.358024i 0.814535 + 0.814535i 1.74364 0.979658i −3.64825 −1.40603 0.822778i −2.22907 −2.03481 + 1.96458i 1.67306i 4.99133 1.30616i
339.9 −1.35541 + 0.403572i 1.14471 + 1.14471i 1.67426 1.09401i 1.84909 −2.01352 1.08957i 3.26353 −1.82779 + 2.15851i 0.379294i −2.50627 + 0.746241i
339.10 −1.30913 + 0.534969i 0.704988 + 0.704988i 1.42762 1.40068i 0.949460 −1.30006 0.545771i −3.64867 −1.11961 + 2.59740i 2.00598i −1.24296 + 0.507931i
339.11 −1.30119 0.553996i 0.972092 + 0.972092i 1.38618 + 1.44171i −1.78988 −0.726338 1.80341i −2.26792 −1.00497 2.64387i 1.11008i 2.32897 + 0.991587i
339.12 −1.19953 0.749083i −1.62819 1.62819i 0.877750 + 1.79710i 1.61926 0.733417 + 3.17272i 0.443847 0.293285 2.81318i 2.30203i −1.94236 1.21296i
339.13 −1.18524 + 0.771493i −2.05999 2.05999i 0.809597 1.82881i 3.07499 4.03085 + 0.852317i −1.16594 0.451348 + 2.79218i 5.48709i −3.64461 + 2.37234i
339.14 −1.17336 + 0.789447i −1.74500 1.74500i 0.753548 1.85261i −3.24438 3.42509 + 0.669927i 2.87009 0.578353 + 2.76867i 3.09002i 3.80682 2.56126i
339.15 −1.13524 0.843348i −1.40169 1.40169i 0.577529 + 1.91480i −4.05599 0.409139 + 2.77336i 4.34286 0.959209 2.66081i 0.929469i 4.60452 + 3.42061i
339.16 −1.13093 0.849111i 1.98293 + 1.98293i 0.558020 + 1.92058i 2.31249 −0.558833 3.92628i −3.56433 0.999699 2.64587i 4.86399i −2.61527 1.96356i
339.17 −1.09621 + 0.893492i 2.36762 + 2.36762i 0.403344 1.95891i −1.01012 −4.71086 0.479956i −2.40965 1.30812 + 2.50775i 8.21129i 1.10731 0.902538i
339.18 −1.08578 0.906132i 0.326280 + 0.326280i 0.357848 + 1.96773i 3.56659 −0.0586163 0.649922i 3.97694 1.39447 2.46078i 2.78708i −3.87254 3.23181i
339.19 −1.08040 + 0.912541i 1.23308 + 1.23308i 0.334537 1.97182i −1.94551 −2.45746 0.206986i 4.34289 1.43794 + 2.43564i 0.0409839i 2.10193 1.77536i
339.20 −1.03953 + 0.958838i 0.444118 + 0.444118i 0.161261 1.99349i 3.03578 −0.887513 0.0358386i −2.92719 1.74380 + 2.22692i 2.60552i −3.15580 + 2.91082i
See next 80 embeddings (of 148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
592.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.s.a yes 148
16.f odd 4 1 592.2.m.a 148
37.d odd 4 1 592.2.m.a 148
592.s even 4 1 inner 592.2.s.a yes 148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
592.2.m.a 148 16.f odd 4 1
592.2.m.a 148 37.d odd 4 1
592.2.s.a yes 148 1.a even 1 1 trivial
592.2.s.a yes 148 592.s even 4 1 inner

Hecke kernels

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