Properties

Label 592.2.bf.a
Level $592$
Weight $2$
Character orbit 592.bf
Analytic conductor $4.727$
Analytic rank $0$
Dimension $296$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [592,2,Mod(251,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bf (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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(296\)
Relative dimension: \(74\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 296 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{8} - 8 q^{10} - 8 q^{11} - 32 q^{12} - 6 q^{13} + 8 q^{14} + 12 q^{15} - 12 q^{16} - 8 q^{17} + 18 q^{18} - 2 q^{19} + 18 q^{20} - 6 q^{21} + 18 q^{22} + 34 q^{24} - 132 q^{25} - 8 q^{26} + 20 q^{28} - 24 q^{29} - 26 q^{30} - 18 q^{32} - 4 q^{33} - 18 q^{34} + 18 q^{35} + 32 q^{36} + 12 q^{37} + 20 q^{38} - 8 q^{39} - 16 q^{40} + 26 q^{42} - 44 q^{44} + 28 q^{46} - 28 q^{48} - 128 q^{49} - 4 q^{50} - 22 q^{52} + 22 q^{53} - 26 q^{54} - 8 q^{55} + 38 q^{56} + 12 q^{57} - 24 q^{58} - 6 q^{59} - 20 q^{60} - 2 q^{61} + 22 q^{62} + 24 q^{64} - 36 q^{65} + 4 q^{66} - 6 q^{67} - 80 q^{68} - 24 q^{69} - 46 q^{70} - 4 q^{71} - 46 q^{72} - 32 q^{73} + 64 q^{74} + 4 q^{75} + 42 q^{76} + 12 q^{77} + 6 q^{78} - 40 q^{79} - 52 q^{80} + 112 q^{81} + 36 q^{82} - 62 q^{83} + 132 q^{84} + 20 q^{85} - 22 q^{86} + 104 q^{87} - 8 q^{88} - 16 q^{89} + 28 q^{90} - 6 q^{91} - 36 q^{92} + 12 q^{93} - 82 q^{94} + 40 q^{95} + 104 q^{96} - 8 q^{97} + 2 q^{98} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −1.41416 + 0.0124895i 0.185341 + 0.691701i 1.99969 0.0353243i −0.445060 + 0.770867i −0.270740 0.975861i −0.429533 + 0.743973i −2.82743 + 0.0749293i 2.15398 1.24360i 0.619758 1.09569i
251.2 −1.40808 + 0.131531i 0.751687 + 2.80533i 1.96540 0.370413i 1.29725 2.24691i −1.42743 3.85127i 1.26308 2.18771i −2.71873 + 0.780083i −4.70679 + 2.71746i −1.53110 + 3.33446i
251.3 −1.40096 0.193131i 0.114417 + 0.427011i 1.92540 + 0.541140i −1.50227 + 2.60201i −0.0778251 0.620324i 2.30014 3.98397i −2.59291 1.12997i 2.42883 1.40229i 2.60715 3.35518i
251.4 −1.39710 0.219367i −0.618148 2.30696i 1.90376 + 0.612955i −0.349210 + 0.604850i 0.357540 + 3.35864i −0.422689 + 0.732118i −2.52527 1.27398i −2.34187 + 1.35208i 0.620564 0.768428i
251.5 −1.37078 + 0.347808i −0.659446 2.46109i 1.75806 0.953534i −0.530589 + 0.919007i 1.75994 + 3.14424i 0.701035 1.21423i −2.07826 + 1.91855i −3.02400 + 1.74591i 0.407681 1.44430i
251.6 −1.37025 + 0.349895i −0.403729 1.50674i 1.75515 0.958884i −1.73355 + 3.00260i 1.08041 + 1.92334i −2.23939 + 3.87874i −2.06947 + 1.92802i 0.490815 0.283372i 1.32480 4.72086i
251.7 −1.36182 + 0.381390i 0.0457446 + 0.170721i 1.70908 1.03876i 1.81094 3.13664i −0.127407 0.215044i 1.16260 2.01367i −1.93128 + 2.06643i 2.57102 1.48438i −1.26988 + 4.96220i
251.8 −1.34508 0.436771i −0.588609 2.19672i 1.61846 + 1.17498i 1.07481 1.86162i −0.167738 + 3.21184i 1.22576 2.12307i −1.66376 2.28734i −1.88103 + 1.08601i −2.25880 + 2.03457i
251.9 −1.33397 + 0.469606i 0.716356 + 2.67348i 1.55894 1.25288i −0.633595 + 1.09742i −2.21108 3.22993i −0.406156 + 0.703483i −1.49122 + 2.40339i −4.03624 + 2.33033i 0.329841 1.76146i
251.10 −1.32688 0.489282i 0.648209 + 2.41915i 1.52121 + 1.29844i 0.423601 0.733699i 0.323553 3.52707i −2.29063 + 3.96749i −1.38315 2.46716i −2.83403 + 1.63623i −0.921053 + 0.766268i
251.11 −1.32437 0.496022i −0.112260 0.418959i 1.50793 + 1.31383i 1.63460 2.83121i −0.0591389 + 0.610540i −2.06434 + 3.57553i −1.34536 2.48797i 2.43515 1.40594i −3.56916 + 2.93878i
251.12 −1.30534 + 0.544131i −0.729786 2.72360i 1.40784 1.42055i 1.96845 3.40945i 2.43461 + 3.15813i −2.04578 + 3.54340i −1.06475 + 2.62036i −4.28732 + 2.47528i −0.714313 + 5.52159i
251.13 −1.20413 + 0.741666i −0.257947 0.962671i 0.899863 1.78613i 0.293162 0.507772i 1.02458 + 0.967872i 2.60292 4.50838i 0.241156 + 2.81813i 1.73788 1.00336i 0.0235913 + 0.828852i
251.14 −1.19564 0.755273i 0.595709 + 2.22322i 0.859125 + 1.80607i −0.155650 + 0.269594i 0.966880 3.10810i 0.475920 0.824318i 0.336873 2.80829i −1.98974 + 1.14878i 0.389719 0.204780i
251.15 −1.18389 0.773561i −0.301998 1.12707i 0.803206 + 1.83163i −1.40057 + 2.42585i −0.514326 + 1.56795i −0.357830 + 0.619780i 0.465966 2.78978i 1.41899 0.819252i 3.53467 1.78853i
251.16 −1.14258 + 0.833371i 0.193662 + 0.722757i 0.610987 1.90439i 0.285864 0.495132i −0.823600 0.664417i −1.39274 + 2.41230i 0.888959 + 2.68510i 2.11320 1.22006i 0.0860048 + 0.803960i
251.17 −1.04334 0.954691i 0.149173 + 0.556722i 0.177128 + 1.99214i −0.354588 + 0.614164i 0.375859 0.723267i 0.432081 0.748387i 1.71707 2.24759i 2.31039 1.33390i 0.956294 0.302262i
251.18 −1.01537 0.984387i 0.278090 + 1.03785i 0.0619648 + 1.99904i 2.01095 3.48306i 0.739277 1.32755i 0.924173 1.60072i 1.90491 2.09077i 1.59829 0.922771i −5.47054 + 1.55706i
251.19 −0.946318 + 1.05094i 0.0103273 + 0.0385418i −0.208965 1.98905i −2.08235 + 3.60674i −0.0502782 0.0256195i 0.453175 0.784923i 2.28813 + 1.66267i 2.59670 1.49920i −1.81991 5.60156i
251.20 −0.892332 + 1.09715i 0.461372 + 1.72186i −0.407487 1.95805i 1.92757 3.33864i −2.30084 1.03028i −0.348836 + 0.604201i 2.51189 + 1.30015i −0.153875 + 0.0888400i 1.94297 + 5.09401i
See next 80 embeddings (of 296 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
592.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.bf.a 296
16.f odd 4 1 592.2.bl.a yes 296
37.g odd 12 1 592.2.bl.a yes 296
592.bf even 12 1 inner 592.2.bf.a 296
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
592.2.bf.a 296 1.a even 1 1 trivial
592.2.bf.a 296 592.bf even 12 1 inner
592.2.bl.a yes 296 16.f odd 4 1
592.2.bl.a yes 296 37.g odd 12 1

Hecke kernels

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