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Properties

Label 592.2.bl.a
Level $592$
Weight $2$
Character orbit 592.bl
Analytic conductor $4.727$
Analytic rank $0$
Dimension $296$
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(51,592)]
 
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, 9, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bl (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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 6 q^{3} - 12 q^{4} - 6 q^{5} - 4 q^{7} - 8 q^{10} + 8 q^{11} + 28 q^{12} - 2 q^{13} - 12 q^{14} - 12 q^{15} - 12 q^{16} - 8 q^{17} - 14 q^{18} - 6 q^{19} + 2 q^{20} - 6 q^{21} - 30 q^{22} + 30 q^{24}+ \cdots - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.41384 − 0.0324319i 0.129674 − 0.483951i 1.99790 + 0.0917070i 0.327392 − 0.189020i −0.199034 + 0.680024i −0.635574 − 1.10085i −2.82173 − 0.194455i 2.38068 + 1.37449i −0.469011 + 0.256627i
51.2 −1.41352 + 0.0443189i 0.0273586 − 0.102104i 1.99607 − 0.125291i 3.43045 − 1.98057i −0.0341468 + 0.145538i −1.61675 − 2.80030i −2.81593 + 0.265565i 2.58840 + 1.49441i −4.76123 + 2.95161i
51.3 −1.40679 − 0.144693i −0.552822 + 2.06316i 1.95813 + 0.407105i 2.54336 − 1.46841i 1.07623 − 2.82245i 2.37518 + 4.11393i −2.69577 − 0.856039i −1.35294 − 0.781122i −3.79044 + 1.69774i
51.4 −1.40654 + 0.147133i 0.286836 − 1.07049i 1.95670 − 0.413898i −3.56489 + 2.05819i −0.245942 + 1.54788i −0.00191601 − 0.00331863i −2.69128 + 0.870060i 1.53441 + 0.885891i 4.71132 − 3.41944i
51.5 −1.40488 − 0.162222i 0.663867 − 2.47758i 1.94737 + 0.455803i −1.07939 + 0.623185i −1.33457 + 3.37301i 2.26443 + 3.92211i −2.66188 − 0.956254i −3.09963 − 1.78957i 1.61750 − 0.700399i
51.6 −1.36124 − 0.383429i −0.655391 + 2.44595i 1.70596 + 1.04388i 1.22682 − 0.708307i 1.82999 − 3.07824i −0.990259 − 1.71518i −1.92198 − 2.07509i −2.95506 − 1.70611i −1.94159 + 0.493779i
51.7 −1.34319 + 0.442540i 0.751509 − 2.80467i 1.60832 − 1.18883i 1.61400 − 0.931844i 0.231760 + 4.09978i 0.0928006 + 0.160735i −1.63417 + 2.30857i −4.70333 − 2.71547i −1.75553 + 1.96590i
51.8 −1.33188 + 0.475498i −0.430251 + 1.60572i 1.54780 − 1.26661i −0.303976 + 0.175501i −0.190474 − 2.34320i −0.209310 − 0.362536i −1.45922 + 2.42295i 0.204864 + 0.118278i 0.321409 − 0.378286i
51.9 −1.28384 + 0.593089i −0.109465 + 0.408530i 1.29649 − 1.52286i 0.0413160 − 0.0238538i −0.101758 − 0.589409i 2.08451 + 3.61047i −0.761296 + 2.72405i 2.44316 + 1.41056i −0.0388957 + 0.0551285i
51.10 −1.28137 − 0.598400i 0.767320 − 2.86368i 1.28384 + 1.53355i −1.94098 + 1.12063i −2.69685 + 3.21028i −2.06845 − 3.58266i −0.727398 − 2.73329i −5.01379 − 2.89471i 3.15771 − 0.274458i
51.11 −1.27069 − 0.620773i 0.166554 − 0.621587i 1.22928 + 1.57761i 1.08801 − 0.628166i −0.597501 + 0.686449i 0.442359 + 0.766189i −0.582689 − 2.76776i 2.23945 + 1.29295i −1.77247 + 0.122790i
51.12 −1.26991 − 0.622361i −0.455855 + 1.70127i 1.22533 + 1.58068i −2.61123 + 1.50759i 1.63770 − 1.87675i −1.95644 − 3.38866i −0.572305 − 2.76992i −0.0884496 − 0.0510664i 4.25429 − 0.289378i
51.13 −1.25418 + 0.653476i −0.811153 + 3.02726i 1.14594 − 1.63915i −0.241412 + 0.139379i −0.960912 − 4.32681i −1.65877 − 2.87307i −0.366065 + 2.80464i −5.90829 − 3.41115i 0.211693 − 0.332563i
51.14 −1.22537 − 0.706025i −0.0976689 + 0.364505i 1.00306 + 1.73028i −1.29645 + 0.748507i 0.377030 − 0.377697i 1.29798 + 2.24817i −0.00749447 − 2.82842i 2.47475 + 1.42880i 2.11710 − 0.00186989i
51.15 −1.16062 − 0.808054i 0.679548 − 2.53611i 0.694098 + 1.87569i 2.93382 − 1.69384i −2.83801 + 2.39436i 0.314679 + 0.545040i 0.710074 − 2.73784i −3.37199 − 1.94682i −4.77378 − 0.404771i
51.16 −1.11179 + 0.874033i 0.260643 − 0.972734i 0.472134 − 1.94347i 2.77715 − 1.60339i 0.560422 + 1.30928i 0.627049 + 1.08608i 1.17375 + 2.57339i 1.71980 + 0.992927i −1.68618 + 4.20995i
51.17 −1.04406 + 0.953911i −0.521820 + 1.94746i 0.180107 − 1.99187i −3.57739 + 2.06541i −1.31289 − 2.53103i 1.63731 + 2.83590i 1.71203 + 2.25143i −0.922225 − 0.532447i 1.76478 − 5.56891i
51.18 −1.03842 + 0.960039i −0.0380764 + 0.142103i 0.156652 − 1.99386i −1.99008 + 1.14897i −0.0968850 − 0.184118i −2.39045 − 4.14038i 1.75151 + 2.22086i 2.57933 + 1.48918i 0.963487 − 3.10367i
51.19 −0.936454 + 1.05974i 0.510515 − 1.90527i −0.246107 − 1.98480i −1.97858 + 1.14233i 1.54102 + 2.32521i 0.307813 + 0.533148i 2.33384 + 1.59787i −0.771351 − 0.445339i 0.642271 − 3.16653i
51.20 −0.927755 − 1.06737i −0.469723 + 1.75303i −0.278541 + 1.98051i −2.20934 + 1.27556i 2.30691 − 1.12502i 1.57589 + 2.72952i 2.37235 − 1.54012i −0.254397 − 0.146876i 3.41122 + 1.17477i
See next 80 embeddings (of 296 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.74
Significant digits:
Format:

Inner twists

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

Hecke kernels

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