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Properties

Label 731.2.m.b
Level $731$
Weight $2$
Character orbit 731.m
Analytic conductor $5.837$
Analytic rank $0$
Dimension $116$
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] = [731,2,Mod(87,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.87");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.m (of order \(8\), 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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(29\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q - 8 q^{6} - 8 q^{10} + 8 q^{14} + 4 q^{15} - 68 q^{16} - 4 q^{17} - 44 q^{18} + 12 q^{19} + 8 q^{20} - 16 q^{22} - 28 q^{23} - 12 q^{24} - 4 q^{25} - 8 q^{26} + 24 q^{28} + 80 q^{33} + 32 q^{34} - 112 q^{35} + 160 q^{36} - 20 q^{37} + 8 q^{39} - 112 q^{40} + 8 q^{41} + 4 q^{42} + 32 q^{44} - 52 q^{45} - 40 q^{46} + 40 q^{48} + 8 q^{49} + 100 q^{50} - 32 q^{51} - 152 q^{52} + 28 q^{53} - 36 q^{54} + 124 q^{56} - 104 q^{57} - 32 q^{58} - 36 q^{59} - 24 q^{60} + 52 q^{61} - 68 q^{62} + 20 q^{63} + 20 q^{65} - 60 q^{66} + 64 q^{67} - 128 q^{69} + 188 q^{70} + 52 q^{73} - 104 q^{74} + 36 q^{75} - 112 q^{76} + 28 q^{77} + 56 q^{78} - 108 q^{79} - 44 q^{80} + 52 q^{82} - 52 q^{83} + 120 q^{84} + 12 q^{85} - 20 q^{86} + 56 q^{87} + 36 q^{88} + 144 q^{90} - 16 q^{92} - 176 q^{93} - 8 q^{94} + 164 q^{95} - 164 q^{96} - 8 q^{97} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1 −1.95693 − 1.95693i −2.36371 + 0.979079i 5.65912i 0.741144 + 1.78928i 6.54158 + 2.70961i −1.36877 + 3.30450i 7.16062 − 7.16062i 2.50719 − 2.50719i 2.05112 − 4.95185i
87.2 −1.81709 − 1.81709i −2.81259 + 1.16501i 4.60365i −1.37650 − 3.32317i 7.22768 + 2.99380i 0.758828 − 1.83197i 4.73108 − 4.73108i 4.43209 − 4.43209i −3.53728 + 8.53975i
87.3 −1.75194 − 1.75194i 1.70725 − 0.707167i 4.13857i −1.12861 − 2.72471i −4.22991 − 1.75209i −1.06160 + 2.56293i 3.74664 − 3.74664i 0.293302 − 0.293302i −2.79626 + 6.75077i
87.4 −1.65901 − 1.65901i 1.85997 − 0.770425i 3.50461i −1.10929 − 2.67805i −4.36385 − 1.80756i 1.56474 − 3.77763i 2.49616 − 2.49616i 0.744616 − 0.744616i −2.60260 + 6.28322i
87.5 −1.60811 − 1.60811i −0.921811 + 0.381827i 3.17206i 0.244536 + 0.590361i 2.09640 + 0.868356i −1.44759 + 3.49480i 1.88481 − 1.88481i −1.41738 + 1.41738i 0.556127 − 1.34261i
87.6 −1.52705 − 1.52705i 0.458286 − 0.189828i 2.66378i 1.62594 + 3.92537i −0.989705 − 0.409949i −0.150783 + 0.364023i 1.01362 − 1.01362i −1.94733 + 1.94733i 3.51135 − 8.47714i
87.7 −1.35151 − 1.35151i 2.57265 − 1.06563i 1.65316i 1.19002 + 2.87295i −4.91716 − 2.03675i 1.13248 − 2.73404i −0.468760 + 0.468760i 3.36163 − 3.36163i 2.27450 − 5.49114i
87.8 −1.09522 − 1.09522i 2.12589 − 0.880571i 0.399034i −0.733376 − 1.77053i −3.29275 − 1.36390i −0.642781 + 1.55181i −1.75342 + 1.75342i 1.62267 − 1.62267i −1.13591 + 2.74234i
87.9 −1.00590 − 1.00590i 0.0816782 − 0.0338322i 0.0236750i 0.261308 + 0.630853i −0.116192 − 0.0481283i 0.939495 − 2.26814i −1.98799 + 1.98799i −2.11579 + 2.11579i 0.371726 − 0.897426i
87.10 −1.00313 − 1.00313i −1.37786 + 0.570727i 0.0125460i −0.979244 − 2.36410i 1.95469 + 0.809657i 1.38293 − 3.33870i −1.99368 + 1.99368i −0.548561 + 0.548561i −1.38920 + 3.35382i
87.11 −0.993125 − 0.993125i −1.41917 + 0.587838i − 0.0274062i 0.967808 + 2.33649i 1.99321 + 0.825613i 0.386865 − 0.933975i −2.01347 + 2.01347i −0.452840 + 0.452840i 1.35928 − 3.28158i
87.12 −0.740178 − 0.740178i 2.21650 − 0.918106i − 0.904273i 0.764492 + 1.84565i −2.32017 − 0.961046i −2.00667 + 4.84454i −2.14968 + 2.14968i 1.94865 − 1.94865i 0.800248 − 1.93197i
87.13 −0.482959 − 0.482959i −1.45483 + 0.602612i − 1.53350i 1.15531 + 2.78915i 0.993661 + 0.411588i −1.39214 + 3.36091i −1.70653 + 1.70653i −0.367918 + 0.367918i 0.789081 − 1.90501i
87.14 −0.292872 − 0.292872i 2.84462 − 1.17828i − 1.82845i 0.0154981 + 0.0374157i −1.17820 − 0.488025i −0.341796 + 0.825169i −1.12125 + 1.12125i 4.58219 − 4.58219i 0.00641907 − 0.0154970i
87.15 −0.0607158 − 0.0607158i −1.28452 + 0.532068i − 1.99263i −0.430997 − 1.04052i 0.110296 + 0.0456861i −0.993514 + 2.39855i −0.242416 + 0.242416i −0.754412 + 0.754412i −0.0370077 + 0.0893444i
87.16 −0.0400748 − 0.0400748i 0.683480 − 0.283106i − 1.99679i −0.594750 − 1.43585i −0.0387357 − 0.0160449i 1.79398 − 4.33104i −0.160170 + 0.160170i −1.73433 + 1.73433i −0.0337070 + 0.0813759i
87.17 0.0756963 + 0.0756963i 0.293333 − 0.121503i − 1.98854i −0.559792 − 1.35146i 0.0314015 + 0.0130069i −1.02421 + 2.47265i 0.301918 − 0.301918i −2.05004 + 2.05004i 0.0599261 − 0.144674i
87.18 0.544996 + 0.544996i −1.90994 + 0.791125i − 1.40596i 0.670095 + 1.61775i −1.47207 − 0.609752i 1.29182 − 3.11873i 1.85623 − 1.85623i 0.900688 − 0.900688i −0.516470 + 1.24687i
87.19 0.598059 + 0.598059i −1.51219 + 0.626370i − 1.28465i −1.53826 − 3.71369i −1.27899 − 0.529773i −0.409939 + 0.989680i 1.96441 − 1.96441i −0.226937 + 0.226937i 1.30103 − 3.14098i
87.20 0.629885 + 0.629885i −0.623684 + 0.258339i − 1.20649i 1.24172 + 2.99777i −0.555573 − 0.230126i 1.05484 − 2.54660i 2.01972 − 2.01972i −1.79908 + 1.79908i −1.10611 + 2.67039i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.m.b ✓ 116
17.d even 8 1 inner 731.2.m.b ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.m.b ✓ 116 1.a even 1 1 trivial
731.2.m.b ✓ 116 17.d even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} + 399 T_{2}^{112} + 72795 T_{2}^{108} - 36 T_{2}^{107} + 584 T_{2}^{105} + 8060769 T_{2}^{104} + \cdots + 29073664 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\). Copy content Toggle raw display