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Properties

Label 731.2.e.a
Level $731$
Weight $2$
Character orbit 731.e
Analytic conductor $5.837$
Analytic rank $0$
Dimension $58$
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(307,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.e (of order \(3\), degree \(2\), 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: \(58\)
Relative dimension: \(29\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 58 q - 6 q^{2} + 3 q^{3} + 54 q^{4} - q^{5} + 12 q^{6} + 7 q^{7} - 12 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 58 q - 6 q^{2} + 3 q^{3} + 54 q^{4} - q^{5} + 12 q^{6} + 7 q^{7} - 12 q^{8} - 22 q^{9} + 4 q^{10} + 16 q^{11} + 12 q^{12} + 2 q^{13} - 11 q^{14} + 7 q^{15} + 30 q^{16} + 29 q^{17} + 8 q^{18} + 8 q^{19} - 33 q^{20} - 26 q^{21} - 22 q^{22} - 5 q^{23} + 12 q^{24} - 36 q^{25} - 12 q^{27} + 15 q^{28} + 2 q^{29} + 11 q^{30} + 3 q^{31} - 40 q^{32} + 17 q^{33} - 3 q^{34} + 38 q^{35} - 7 q^{36} + 2 q^{37} + q^{38} - 54 q^{39} + 5 q^{40} + 14 q^{41} - 112 q^{42} + 31 q^{43} - 24 q^{44} - 46 q^{45} - 13 q^{46} - 28 q^{47} - 28 q^{49} - 13 q^{50} + 6 q^{51} + 85 q^{52} - 10 q^{53} + 34 q^{54} + 36 q^{55} - 54 q^{56} - 23 q^{57} + 3 q^{58} + 12 q^{59} + 2 q^{60} - q^{61} - q^{62} - 14 q^{63} + 28 q^{64} + 80 q^{65} - 74 q^{66} + 11 q^{67} + 27 q^{68} - 11 q^{69} + 2 q^{70} + 16 q^{71} + 21 q^{72} + 14 q^{73} + 21 q^{74} - 54 q^{75} + 44 q^{76} + 25 q^{77} + 88 q^{78} - 4 q^{79} - 112 q^{80} + 11 q^{81} - 176 q^{82} - 3 q^{83} + 100 q^{84} - 2 q^{85} + 44 q^{86} + 8 q^{87} - 106 q^{88} + 82 q^{89} + 54 q^{90} - 15 q^{91} + 42 q^{92} + 88 q^{94} + 29 q^{95} + 20 q^{96} + 20 q^{97} + 44 q^{98} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −2.82259 0.178900 − 0.309865i 5.96703 −1.59268 + 2.75860i −0.504963 + 0.874622i 1.57025 + 2.71975i −11.1973 1.43599 + 2.48721i 4.49548 − 7.78640i
307.2 −2.66184 −1.22526 + 2.12222i 5.08540 −0.351592 + 0.608975i 3.26146 − 5.64901i −1.71463 − 2.96983i −8.21285 −1.50254 − 2.60247i 0.935881 − 1.62099i
307.3 −2.32507 1.39344 − 2.41351i 3.40595 1.48534 − 2.57269i −3.23985 + 5.61158i 1.79579 + 3.11040i −3.26893 −2.38336 − 4.12809i −3.45353 + 5.98168i
307.4 −2.13710 1.01873 − 1.76449i 2.56719 −1.08625 + 1.88145i −2.17712 + 3.77089i 0.371017 + 0.642620i −1.21215 −0.575612 − 0.996990i 2.32143 − 4.02084i
307.5 −1.99188 0.176804 − 0.306233i 1.96759 −1.32572 + 2.29622i −0.352172 + 0.609981i −1.66418 − 2.88245i 0.0645559 1.43748 + 2.48979i 2.64068 − 4.57379i
307.6 −1.94880 −0.712581 + 1.23423i 1.79780 2.11686 − 3.66650i 1.38867 − 2.40525i −0.444390 − 0.769706i 0.394042 0.484458 + 0.839105i −4.12532 + 7.14527i
307.7 −1.84257 −1.58757 + 2.74976i 1.39506 0.0845289 − 0.146408i 2.92522 − 5.06662i 1.30280 + 2.25652i 1.11464 −3.54079 − 6.13282i −0.155750 + 0.269768i
307.8 −1.79000 0.128663 − 0.222852i 1.20411 −1.10204 + 1.90879i −0.230308 + 0.398905i 1.71488 + 2.97026i 1.42464 1.46689 + 2.54073i 1.97265 − 3.41674i
307.9 −1.65452 −0.422449 + 0.731703i 0.737422 0.952877 − 1.65043i 0.698949 − 1.21061i 2.62410 + 4.54507i 2.08895 1.14307 + 1.97986i −1.57655 + 2.73067i
307.10 −1.31896 1.65592 − 2.86814i −0.260346 −0.0483968 + 0.0838257i −2.18409 + 3.78296i −1.22618 − 2.12381i 2.98130 −3.98416 − 6.90077i 0.0638334 − 0.110563i
307.11 −0.966012 −0.835198 + 1.44661i −1.06682 −0.811572 + 1.40568i 0.806812 − 1.39744i −1.19805 − 2.07509i 2.96259 0.104888 + 0.181671i 0.783988 − 1.35791i
307.12 −0.884748 −1.14431 + 1.98200i −1.21722 −1.47938 + 2.56236i 1.01242 − 1.75357i −0.890904 − 1.54309i 2.84643 −1.11888 − 1.93795i 1.30888 − 2.26704i
307.13 −0.546292 0.344032 − 0.595881i −1.70156 0.671732 − 1.16347i −0.187942 + 0.325525i 0.618902 + 1.07197i 2.02214 1.26328 + 2.18807i −0.366962 + 0.635597i
307.14 −0.410035 0.529064 − 0.916365i −1.83187 2.19207 − 3.79678i −0.216935 + 0.375742i −1.90032 − 3.29146i 1.57120 0.940183 + 1.62845i −0.898826 + 1.55681i
307.15 −0.317626 −0.924775 + 1.60176i −1.89911 1.30757 − 2.26478i 0.293732 − 0.508759i 0.730298 + 1.26491i 1.23846 −0.210416 − 0.364452i −0.415317 + 0.719351i
307.16 0.220069 1.00430 − 1.73949i −1.95157 −1.15229 + 1.99582i 0.221015 − 0.382808i −1.47386 − 2.55280i −0.869618 −0.517225 − 0.895859i −0.253582 + 0.439217i
307.17 0.469159 1.22299 − 2.11828i −1.77989 −1.20485 + 2.08687i 0.573775 − 0.993808i 1.21565 + 2.10557i −1.77337 −1.49139 − 2.58317i −0.565268 + 0.979072i
307.18 0.576537 0.0377307 − 0.0653514i −1.66761 0.551580 − 0.955365i 0.0217531 − 0.0376775i 0.781254 + 1.35317i −2.11451 1.49715 + 2.59314i 0.318006 − 0.550803i
307.19 0.715873 −1.27871 + 2.21478i −1.48753 0.380045 − 0.658258i −0.915390 + 1.58550i −1.25160 − 2.16783i −2.49662 −1.77018 − 3.06604i 0.272064 − 0.471229i
307.20 0.875460 −0.791784 + 1.37141i −1.23357 −1.30803 + 2.26557i −0.693175 + 1.20061i 1.95679 + 3.38927i −2.83086 0.246156 + 0.426354i −1.14512 + 1.98341i
See all 58 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.e.a ✓ 58
43.c even 3 1 inner 731.2.e.a ✓ 58
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.e.a ✓ 58 1.a even 1 1 trivial
731.2.e.a ✓ 58 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{29} + 3 T_{2}^{28} - 38 T_{2}^{27} - 117 T_{2}^{26} + 631 T_{2}^{25} + 2011 T_{2}^{24} + \cdots + 823 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\). Copy content Toggle raw display