# 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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$58$$ Relative dimension: $$29$$ over $$\Q(\zeta_{3})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$58q - 6q^{2} + 3q^{3} + 54q^{4} - q^{5} + 12q^{6} + 7q^{7} - 12q^{8} - 22q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$58q - 6q^{2} + 3q^{3} + 54q^{4} - q^{5} + 12q^{6} + 7q^{7} - 12q^{8} - 22q^{9} + 4q^{10} + 16q^{11} + 12q^{12} + 2q^{13} - 11q^{14} + 7q^{15} + 30q^{16} + 29q^{17} + 8q^{18} + 8q^{19} - 33q^{20} - 26q^{21} - 22q^{22} - 5q^{23} + 12q^{24} - 36q^{25} - 12q^{27} + 15q^{28} + 2q^{29} + 11q^{30} + 3q^{31} - 40q^{32} + 17q^{33} - 3q^{34} + 38q^{35} - 7q^{36} + 2q^{37} + q^{38} - 54q^{39} + 5q^{40} + 14q^{41} - 112q^{42} + 31q^{43} - 24q^{44} - 46q^{45} - 13q^{46} - 28q^{47} - 28q^{49} - 13q^{50} + 6q^{51} + 85q^{52} - 10q^{53} + 34q^{54} + 36q^{55} - 54q^{56} - 23q^{57} + 3q^{58} + 12q^{59} + 2q^{60} - q^{61} - q^{62} - 14q^{63} + 28q^{64} + 80q^{65} - 74q^{66} + 11q^{67} + 27q^{68} - 11q^{69} + 2q^{70} + 16q^{71} + 21q^{72} + 14q^{73} + 21q^{74} - 54q^{75} + 44q^{76} + 25q^{77} + 88q^{78} - 4q^{79} - 112q^{80} + 11q^{81} - 176q^{82} - 3q^{83} + 100q^{84} - 2q^{85} + 44q^{86} + 8q^{87} - 106q^{88} + 82q^{89} + 54q^{90} - 15q^{91} + 42q^{92} + 88q^{94} + 29q^{95} + 20q^{96} + 20q^{97} + 44q^{98} - 54q^{99} + O(q^{100})$$

## 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.

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 681.29 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{29} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database