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Properties

Label 731.2.bm.a
Level $731$
Weight $2$
Character orbit 731.bm
Analytic conductor $5.837$
Analytic rank $0$
Dimension $6144$
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(3,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(336))
 
chi = DirichletCharacter(H, H._module([21, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.bm (of order \(336\), degree \(96\), 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: \(6144\)
Relative dimension: \(64\) over \(\Q(\zeta_{336})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{336}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 6144 q - 112 q^{2} - 88 q^{3} - 80 q^{4} - 88 q^{5} - 48 q^{6} - 144 q^{7} - 112 q^{8} - 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6144 q - 112 q^{2} - 88 q^{3} - 80 q^{4} - 88 q^{5} - 48 q^{6} - 144 q^{7} - 112 q^{8} - 104 q^{9} - 104 q^{10} - 64 q^{11} - 88 q^{12} - 96 q^{13} - 104 q^{14} - 104 q^{15} - 104 q^{17} - 176 q^{18} - 88 q^{19} - 88 q^{20} - 80 q^{21} - 112 q^{22} - 112 q^{23} + 16 q^{24} - 88 q^{25} - 88 q^{26} - 112 q^{27} - 88 q^{28} - 88 q^{29} - 88 q^{30} - 136 q^{31} - 112 q^{32} - 88 q^{34} - 32 q^{35} - 64 q^{36} - 144 q^{37} - 96 q^{38} - 112 q^{39} - 104 q^{40} - 80 q^{41} - 24 q^{43} - 256 q^{44} - 336 q^{45} + 264 q^{46} - 160 q^{47} - 160 q^{48} - 48 q^{49} - 112 q^{51} - 80 q^{52} - 104 q^{53} - 256 q^{54} - 184 q^{55} - 104 q^{56} - 88 q^{57} + 16 q^{58} - 208 q^{59} - 136 q^{61} + 80 q^{62} - 88 q^{63} - 304 q^{64} - 112 q^{65} + 24 q^{66} - 104 q^{68} - 128 q^{69} - 112 q^{70} - 88 q^{71} - 408 q^{72} - 368 q^{73} + 136 q^{74} - 112 q^{75} + 528 q^{76} - 88 q^{77} + 384 q^{78} + 208 q^{79} - 432 q^{80} + 8 q^{81} - 112 q^{82} - 376 q^{83} - 176 q^{86} - 192 q^{87} - 112 q^{88} - 88 q^{89} - 88 q^{91} - 240 q^{92} + 1008 q^{93} + 560 q^{94} - 184 q^{95} - 488 q^{96} + 48 q^{97} + 320 q^{98} - 400 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.156865 + 2.79323i 0.267317 + 0.0743131i −5.79012 − 0.652390i 0.113790 − 0.0339378i −0.249506 + 0.735022i −0.279348 + 4.26203i 1.79330 − 10.5546i −2.50364 − 1.50859i 0.0769465 + 0.323166i
3.2 −0.150391 + 2.67795i 1.86923 + 0.519639i −5.16138 − 0.581548i −1.77483 + 0.529341i −1.67268 + 4.92756i 0.231555 − 3.53284i 1.43502 − 8.44591i 0.654429 + 0.394332i −1.15063 − 4.83252i
3.3 −0.149160 + 2.65604i −1.79989 − 0.500363i −5.04489 − 0.568422i 2.00210 − 0.597122i 1.59746 − 4.70595i 0.110823 − 1.69083i 1.37104 − 8.06936i 0.419675 + 0.252879i 1.28735 + 5.40672i
3.4 −0.143269 + 2.55114i 1.61076 + 0.447785i −4.50036 − 0.507068i 3.44136 − 1.02638i −1.37313 + 4.04512i 0.195447 − 2.98195i 1.08235 − 6.37028i −0.175531 − 0.105768i 2.12540 + 8.92643i
3.5 −0.136402 + 2.42887i −0.519679 − 0.144469i −3.89336 − 0.438676i −2.97658 + 0.887760i 0.421780 − 1.24253i 0.222665 − 3.39721i 0.781566 − 4.59997i −2.32038 − 1.39816i −1.75024 − 7.35080i
3.6 −0.131534 + 2.34218i −2.82542 − 0.785456i −3.48110 − 0.392226i 0.213187 − 0.0635828i 2.21132 − 6.51434i −0.289512 + 4.41709i 0.590651 − 3.47632i 4.79650 + 2.89017i 0.120881 + 0.507687i
3.7 −0.127458 + 2.26959i 0.484904 + 0.134801i −3.14738 − 0.354624i −2.86214 + 0.853630i −0.367749 + 1.08335i −0.184225 + 2.81073i 0.444469 − 2.61596i −2.35261 − 1.41759i −1.57259 − 6.60469i
3.8 −0.126007 + 2.24375i −1.77486 − 0.493405i −3.03113 − 0.341526i −2.14551 + 0.639895i 1.33072 − 3.92019i 0.0233145 − 0.355711i 0.395373 − 2.32700i 0.337119 + 0.203134i −1.16542 − 4.89463i
3.9 −0.125640 + 2.23723i 2.33558 + 0.649282i −3.00198 − 0.338241i 0.799277 − 0.238383i −1.74603 + 5.14365i −0.0317705 + 0.484724i 0.383211 − 2.25542i 2.46380 + 1.48459i 0.432896 + 1.81811i
3.10 −0.123196 + 2.19370i −2.12773 − 0.591500i −2.80974 − 0.316581i 3.31407 − 0.988418i 1.55970 − 4.59474i −0.0369461 + 0.563689i 0.304558 − 1.79250i 1.60779 + 0.968786i 1.76002 + 7.39187i
3.11 −0.121276 + 2.15952i 3.02862 + 0.841943i −2.66141 − 0.299869i −3.65250 + 1.08935i −2.18549 + 6.43826i −0.191575 + 2.92287i 0.245733 − 1.44628i 5.89407 + 3.55152i −1.90952 − 8.01977i
3.12 −0.118949 + 2.11808i 1.78091 + 0.495087i −2.48467 − 0.279955i 2.82526 − 0.842630i −1.26047 + 3.71322i −0.237309 + 3.62064i 0.177815 − 1.04655i 0.356968 + 0.215094i 1.44869 + 6.08434i
3.13 −0.112439 + 2.00217i −0.984592 − 0.273712i −2.00860 − 0.226315i 1.34100 − 0.399952i 0.658724 − 1.94054i 0.223439 − 3.40902i 0.00715764 − 0.0421269i −1.67507 − 1.00933i 0.649988 + 2.72988i
3.14 −0.111887 + 1.99234i 0.201359 + 0.0559769i −1.96946 − 0.221905i −0.0447030 + 0.0133326i −0.134054 + 0.394911i −0.0381184 + 0.581575i −0.00604176 + 0.0355593i −2.53216 − 1.52578i −0.0215614 − 0.0905552i
3.15 −0.0866269 + 1.54253i 3.15778 + 0.877851i −0.384483 − 0.0433208i 2.87377 − 0.857099i −1.62766 + 4.79494i 0.140784 − 2.14795i −0.417452 + 2.45695i 6.63139 + 3.99580i 1.07316 + 4.50714i
3.16 −0.0865416 + 1.54102i −3.01053 − 0.836914i −0.379818 − 0.0427952i −1.76044 + 0.525049i 1.55024 − 4.56685i −0.0377505 + 0.575962i −0.418255 + 2.46167i 5.79328 + 3.49079i −0.656758 − 2.75831i
3.17 −0.0844139 + 1.50313i −2.55867 − 0.711301i −0.264845 − 0.0298409i −1.66292 + 0.495964i 1.28516 − 3.78597i 0.256638 − 3.91553i −0.437149 + 2.57287i 3.47129 + 2.09166i −0.605124 − 2.54145i
3.18 −0.0835198 + 1.48721i −1.38205 − 0.384204i −0.217387 − 0.0244936i 0.891315 − 0.265833i 0.686819 − 2.02330i −0.154100 + 2.35111i −0.444435 + 2.61576i −0.807130 − 0.486343i 0.320907 + 1.34777i
3.19 −0.0785544 + 1.39879i 0.219726 + 0.0610829i 0.0369822 + 0.00416690i 2.77686 − 0.828196i −0.102703 + 0.302552i 0.307292 − 4.68837i −0.478084 + 2.81380i −2.52503 − 1.52148i 0.940337 + 3.94931i
3.20 −0.0742301 + 1.32179i 1.64473 + 0.457230i 0.245807 + 0.0276958i −2.87165 + 0.856465i −0.726450 + 2.14005i −0.0389237 + 0.593861i −0.498368 + 2.93318i −0.0734806 − 0.0442763i −0.918904 − 3.85929i
See next 80 embeddings (of 6144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner
43.h odd 42 1 inner
731.bm even 336 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.bm.a ✓ 6144
17.e odd 16 1 inner 731.2.bm.a ✓ 6144
43.h odd 42 1 inner 731.2.bm.a ✓ 6144
731.bm even 336 1 inner 731.2.bm.a ✓ 6144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.bm.a ✓ 6144 1.a even 1 1 trivial
731.2.bm.a ✓ 6144 17.e odd 16 1 inner
731.2.bm.a ✓ 6144 43.h odd 42 1 inner
731.2.bm.a ✓ 6144 731.bm even 336 1 inner

Hecke kernels

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