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Properties

Label 731.2.m.c
Level $731$
Weight $2$
Character orbit 731.m
Analytic conductor $5.837$
Analytic rank $0$
Dimension $128$
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: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{2} + 4 q^{3} + 8 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{2} + 4 q^{3} + 8 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{8} + 8 q^{9} - 8 q^{10} - 4 q^{11} + 12 q^{12} + 12 q^{14} - 12 q^{15} - 144 q^{16} - 12 q^{17} + 64 q^{18} - 28 q^{19} - 8 q^{20} - 12 q^{22} + 16 q^{23} - 16 q^{24} - 20 q^{25} + 16 q^{26} - 8 q^{27} + 20 q^{28} + 12 q^{31} - 4 q^{32} - 104 q^{33} + 20 q^{34} + 32 q^{35} - 96 q^{36} - 12 q^{37} + 8 q^{39} + 216 q^{40} + 24 q^{41} - 4 q^{42} + 24 q^{44} - 28 q^{45} - 48 q^{46} + 28 q^{48} - 80 q^{50} - 20 q^{51} + 56 q^{52} - 36 q^{53} - 12 q^{54} - 8 q^{56} + 72 q^{57} - 32 q^{58} + 48 q^{59} - 40 q^{60} - 76 q^{61} - 44 q^{62} + 36 q^{65} - 68 q^{66} - 48 q^{67} + 32 q^{68} + 216 q^{69} - 196 q^{70} + 4 q^{71} + 20 q^{73} + 88 q^{74} + 80 q^{75} + 72 q^{76} + 28 q^{77} - 120 q^{78} + 68 q^{79} - 68 q^{80} + 28 q^{82} - 36 q^{83} - 152 q^{84} + 28 q^{85} - 24 q^{86} - 56 q^{87} + 20 q^{88} - 112 q^{90} + 96 q^{91} - 28 q^{92} + 24 q^{93} - 36 q^{94} - 108 q^{95} + 272 q^{96} + 8 q^{97} - 20 q^{99}+O(q^{100}) \) 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 −1.95137 − 1.95137i 1.54767 − 0.641064i 5.61567i 0.779040 + 1.88077i −4.27102 − 1.76911i −0.326309 + 0.787780i 7.05550 − 7.05550i −0.137012 + 0.137012i 2.14988 − 5.19027i
87.2 −1.80058 − 1.80058i −0.368420 + 0.152604i 4.48416i 0.0622629 + 0.150316i 0.938144 + 0.388592i 1.14823 − 2.77206i 4.47291 − 4.47291i −2.00888 + 2.00888i 0.158546 − 0.382765i
87.3 −1.73249 − 1.73249i 2.93607 − 1.21616i 4.00305i 0.137451 + 0.331835i −7.19370 − 2.97973i −0.639449 + 1.54377i 3.47026 − 3.47026i 5.02014 − 5.02014i 0.336769 − 0.813033i
87.4 −1.56129 − 1.56129i −2.36310 + 0.978828i 2.87526i 1.21177 + 2.92548i 5.21772 + 2.16125i 1.72340 − 4.16065i 1.36654 − 1.36654i 2.50481 − 2.50481i 2.67560 − 6.45946i
87.5 −1.33120 − 1.33120i −1.44462 + 0.598380i 1.54419i −1.33546 − 3.22408i 2.71964 + 1.12651i −1.37427 + 3.31778i −0.606774 + 0.606774i −0.392459 + 0.392459i −2.51414 + 6.06966i
87.6 −1.15376 − 1.15376i 0.645285 − 0.267286i 0.662305i 0.125261 + 0.302406i −1.05288 − 0.436119i 0.205763 − 0.496757i −1.54337 + 1.54337i −1.77637 + 1.77637i 0.204383 − 0.493423i
87.7 −1.14159 − 1.14159i −2.60557 + 1.07926i 0.606454i 0.611497 + 1.47629i 4.20657 + 1.74242i −0.771748 + 1.86317i −1.59086 + 1.59086i 3.50287 − 3.50287i 0.987233 − 2.38339i
87.8 −1.01254 − 1.01254i 1.18533 − 0.490982i 0.0504804i 0.326712 + 0.788752i −1.69734 − 0.703061i −1.15418 + 2.78645i −1.97397 + 1.97397i −0.957365 + 0.957365i 0.467835 − 1.12945i
87.9 −0.768358 − 0.768358i −1.04116 + 0.431262i − 0.819252i −0.778759 − 1.88009i 1.13135 + 0.468619i 0.510833 − 1.23326i −2.16619 + 2.16619i −1.22330 + 1.22330i −0.846217 + 2.04295i
87.10 −0.692528 − 0.692528i 0.747939 − 0.309806i − 1.04081i −1.63001 − 3.93520i −0.732518 − 0.303419i −0.263818 + 0.636914i −2.10585 + 2.10585i −1.65789 + 1.65789i −1.59641 + 3.85406i
87.11 −0.624813 − 0.624813i 2.88236 − 1.19391i − 1.21922i −1.21824 − 2.94109i −2.54691 − 1.05496i 0.920964 − 2.22340i −2.01141 + 2.01141i 4.76125 − 4.76125i −1.07646 + 2.59880i
87.12 −0.349181 − 0.349181i 2.07415 − 0.859139i − 1.75615i 0.898278 + 2.16864i −1.02425 − 0.424257i 1.36146 − 3.28687i −1.31157 + 1.31157i 1.44264 − 1.44264i 0.443585 − 1.07091i
87.13 −0.286708 − 0.286708i −2.99617 + 1.24105i − 1.83560i −0.918391 − 2.21719i 1.21484 + 0.503205i −1.74237 + 4.20646i −1.09970 + 1.09970i 5.31549 − 5.31549i −0.372376 + 0.898996i
87.14 −0.177484 − 0.177484i 0.604214 − 0.250274i − 1.93700i 1.47313 + 3.55644i −0.151658 − 0.0628188i −0.253865 + 0.612883i −0.698755 + 0.698755i −1.81888 + 1.81888i 0.369755 − 0.892669i
87.15 −0.150441 − 0.150441i 0.303692 − 0.125793i − 1.95474i 0.0508915 + 0.122863i −0.0646121 − 0.0267632i −0.398942 + 0.963131i −0.594953 + 0.594953i −2.04492 + 2.04492i 0.0108274 − 0.0261397i
87.16 −0.0668491 − 0.0668491i −2.64125 + 1.09404i − 1.99106i −0.846330 − 2.04322i 0.249701 + 0.103430i 1.54141 − 3.72129i −0.266799 + 0.266799i 3.65797 − 3.65797i −0.0800112 + 0.193164i
87.17 0.413388 + 0.413388i 2.13234 − 0.883246i − 1.65822i 0.291415 + 0.703539i 1.24661 + 0.516363i 0.379677 − 0.916620i 1.51227 − 1.51227i 1.64545 − 1.64545i −0.170367 + 0.411303i
87.18 0.493410 + 0.493410i 2.20257 − 0.912334i − 1.51309i −1.26013 − 3.04222i 1.53692 + 0.636615i −1.60257 + 3.86896i 1.73339 − 1.73339i 1.89764 − 1.89764i 0.879301 − 2.12282i
87.19 0.639626 + 0.639626i −0.916759 + 0.379734i − 1.18176i 0.199818 + 0.482403i −0.829270 − 0.343495i 0.462611 − 1.11684i 2.03513 − 2.03513i −1.42507 + 1.42507i −0.180749 + 0.436366i
87.20 0.810350 + 0.810350i 3.13521 − 1.29865i − 0.686666i 1.23003 + 2.96956i 3.59298 + 1.48826i −0.604922 + 1.46041i 2.17714 − 2.17714i 6.02174 − 6.02174i −1.40963 + 3.40314i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.32
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.c ✓ 128
17.d even 8 1 inner 731.2.m.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.m.c ✓ 128 1.a even 1 1 trivial
731.2.m.c ✓ 128 17.d even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 4 T_{2}^{127} + 8 T_{2}^{126} - 4 T_{2}^{125} + 480 T_{2}^{124} - 1916 T_{2}^{123} + \cdots + 129777664 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\). Copy content Toggle raw display