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Properties

Label 731.2.bj.a
Level $731$
Weight $2$
Character orbit 731.bj
Analytic conductor $5.837$
Analytic rank $0$
Dimension $3072$
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] = [731,2,Mod(22,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(112))
 
chi = DirichletCharacter(H, H._module([35, 40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.bj (of order \(112\), degree \(48\), 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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(3072\)
Relative dimension: \(64\) over \(\Q(\zeta_{112})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{112}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3072 q - 56 q^{2} - 56 q^{3} - 40 q^{4} - 56 q^{5} - 96 q^{6} - 56 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3072 q - 56 q^{2} - 56 q^{3} - 40 q^{4} - 56 q^{5} - 96 q^{6} - 56 q^{8} - 40 q^{9} - 40 q^{10} - 56 q^{11} - 56 q^{12} - 48 q^{13} - 40 q^{14} - 40 q^{15} - 40 q^{17} - 112 q^{18} - 56 q^{19} - 56 q^{20} - 40 q^{21} - 56 q^{22} - 32 q^{23} - 160 q^{24} - 56 q^{25} - 56 q^{26} - 56 q^{27} - 56 q^{28} - 56 q^{29} - 56 q^{30} - 104 q^{31} - 56 q^{32} - 56 q^{34} - 16 q^{35} - 128 q^{36} - 24 q^{38} - 56 q^{39} - 40 q^{40} - 40 q^{41} - 120 q^{43} - 32 q^{44} + 168 q^{45} + 168 q^{46} + 40 q^{47} - 56 q^{48} - 96 q^{49} - 56 q^{51} - 208 q^{52} - 40 q^{53} - 200 q^{54} - 56 q^{55} - 40 q^{56} - 8 q^{57} + 8 q^{58} - 8 q^{59} + 168 q^{60} - 56 q^{61} - 56 q^{62} - 56 q^{63} - 8 q^{64} - 56 q^{65} + 216 q^{66} - 40 q^{68} - 112 q^{69} - 56 q^{70} - 56 q^{71} + 672 q^{72} + 224 q^{73} - 280 q^{74} - 56 q^{75} + 336 q^{76} - 56 q^{77} - 120 q^{78} - 352 q^{79} - 152 q^{81} - 56 q^{82} + 208 q^{83} - 208 q^{86} - 96 q^{87} - 56 q^{88} - 56 q^{89} - 120 q^{90} - 56 q^{91} - 480 q^{92} - 728 q^{94} + 40 q^{95} + 200 q^{96} + 24 q^{97} - 56 q^{98} + 352 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −2.78025 − 0.156135i 0.285872 − 0.484623i 5.71798 + 0.644261i −0.0688676 − 2.45453i −0.870461 + 1.30274i −1.65739 + 0.329675i −10.3062 − 1.75110i 1.29802 + 2.34859i −0.191771 + 6.83497i
22.2 −2.71053 − 0.152220i −1.45195 + 2.46141i 5.33635 + 0.601262i 0.0365495 + 1.30267i 4.31022 − 6.45070i −0.745437 + 0.148277i −9.01992 − 1.53255i −2.49923 − 4.52202i 0.0992243 − 3.53649i
22.3 −2.60649 − 0.146377i −0.304024 + 0.515395i 4.78492 + 0.539131i 0.0135060 + 0.481371i 0.867876 − 1.29887i 4.93075 − 0.980788i −7.24551 − 1.23106i 1.27796 + 2.31229i 0.0352586 − 1.25666i
22.4 −2.59483 − 0.145722i 0.575733 − 0.976009i 4.72448 + 0.532321i 0.0913480 + 3.25577i −1.63615 + 2.44868i 0.226007 − 0.0449555i −7.05724 − 1.19907i 0.830032 + 1.50183i 0.237406 − 8.46147i
22.5 −2.41679 − 0.135724i 1.39523 − 2.36526i 3.83503 + 0.432104i −0.0158836 − 0.566113i −3.69300 + 5.52696i −4.07166 + 0.809904i −4.43701 − 0.753879i −2.19662 − 3.97448i −0.0384478 + 1.37033i
22.6 −2.33533 − 0.131149i 1.38003 − 2.33948i 3.44914 + 0.388625i −0.117447 − 4.18597i −3.52964 + 5.28248i 4.30943 − 0.857199i −3.39199 − 0.576322i −2.11755 − 3.83142i −0.274709 + 9.79101i
22.7 −2.30362 − 0.129368i 0.135076 − 0.228987i 3.30249 + 0.372101i 0.109747 + 3.91152i −0.340787 + 0.510024i −3.67839 + 0.731677i −3.01024 − 0.511460i 1.41697 + 2.56381i 0.253213 − 9.02485i
22.8 −2.29854 − 0.129083i −0.973987 + 1.65115i 3.27919 + 0.369476i −0.0268989 − 0.958715i 2.45188 − 3.66950i 0.592911 − 0.117937i −2.95039 − 0.501292i −0.326481 − 0.590723i −0.0619257 + 2.20712i
22.9 −2.22229 − 0.124801i −1.23180 + 2.08821i 2.93558 + 0.330760i −0.0969845 − 3.45666i 2.99803 − 4.48687i 3.43548 − 0.683360i −2.09374 − 0.355740i −1.39212 − 2.51885i −0.215867 + 7.69380i
22.10 −2.18353 − 0.122624i −0.306057 + 0.518842i 2.76535 + 0.311580i −0.0690731 − 2.46186i 0.731908 − 1.09538i −3.13875 + 0.624337i −1.68787 − 0.286781i 1.27563 + 2.30808i −0.151061 + 5.38402i
22.11 −2.12958 − 0.119594i 1.33637 − 2.26548i 2.53337 + 0.285442i 0.0355757 + 1.26797i −3.11684 + 4.66468i 0.618541 − 0.123035i −1.15528 − 0.196290i −1.89534 − 3.42936i 0.0758807 − 2.70449i
22.12 −2.05612 − 0.115469i 0.208025 − 0.352653i 2.22688 + 0.250909i 0.0498617 + 1.77714i −0.468445 + 0.701077i 2.53057 − 0.503362i −0.489238 − 0.0831248i 1.37007 + 2.47895i 0.102683 − 3.65978i
22.13 −1.98160 − 0.111284i −1.12012 + 1.89887i 1.92692 + 0.217111i 0.0401335 + 1.43041i 2.43093 − 3.63815i −1.99298 + 0.396427i 0.119142 + 0.0202431i −0.899896 − 1.62824i 0.0796537 − 2.83897i
22.14 −1.82092 − 0.102261i −1.49639 + 2.53675i 1.31788 + 0.148489i 0.117477 + 4.18705i 2.98422 − 4.46620i 1.63082 − 0.324389i 1.21148 + 0.205839i −2.74475 − 4.96624i 0.214254 − 7.63631i
22.15 −1.70659 − 0.0958400i 0.0386797 − 0.0655716i 0.915836 + 0.103190i −0.0888877 − 3.16808i −0.0722948 + 0.108197i −0.745802 + 0.148349i 1.81719 + 0.308753i 1.44835 + 2.62060i −0.151934 + 5.41512i
22.16 −1.61554 − 0.0907268i 1.27339 − 2.15871i 0.614316 + 0.0692168i −0.0395108 − 1.40822i −2.25307 + 3.37195i −0.0343223 + 0.00682714i 2.20428 + 0.374522i −1.58735 − 2.87209i −0.0639319 + 2.27862i
22.17 −1.49751 − 0.0840984i −0.399239 + 0.676808i 0.248041 + 0.0279476i 0.0330786 + 1.17897i 0.654783 − 0.979952i −3.04840 + 0.606365i 2.58826 + 0.439764i 1.15248 + 2.08525i 0.0496137 − 1.76830i
22.18 −1.44014 − 0.0808763i 0.635718 − 1.07770i 0.0800273 + 0.00901691i −0.0181415 − 0.646587i −1.00268 + 1.50062i 0.663159 − 0.131911i 2.72953 + 0.463766i 0.693859 + 1.25544i −0.0261674 + 0.932641i
22.19 −1.37635 − 0.0772944i 1.36620 − 2.31605i −0.0990493 − 0.0111602i 0.104308 + 3.71769i −2.05940 + 3.08210i 3.73039 − 0.742022i 2.85355 + 0.484838i −2.04642 − 3.70272i 0.143791 − 5.12492i
22.20 −1.32882 − 0.0746250i −0.509080 + 0.863016i −0.227228 − 0.0256025i 0.0579141 + 2.06414i 0.740879 − 1.10880i 4.35710 − 0.866681i 2.92426 + 0.496851i 0.965523 + 1.74698i 0.0770787 − 2.74719i
See next 80 embeddings (of 3072 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner
43.f odd 14 1 inner
731.bj even 112 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.bj.a ✓ 3072
17.e odd 16 1 inner 731.2.bj.a ✓ 3072
43.f odd 14 1 inner 731.2.bj.a ✓ 3072
731.bj even 112 1 inner 731.2.bj.a ✓ 3072
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.bj.a ✓ 3072 1.a even 1 1 trivial
731.2.bj.a ✓ 3072 17.e odd 16 1 inner
731.2.bj.a ✓ 3072 43.f odd 14 1 inner
731.2.bj.a ✓ 3072 731.bj even 112 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(731, [\chi])\).