# Properties

 Label 731.2.m.a Level 731 Weight 2 Character orbit 731.m Analytic conductor 5.837 Analytic rank 1 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.m (of order $$8$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( -1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{4} + ( -2 + 2 \zeta_{8} ) q^{5} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 1 + 3 \zeta_{8} + \zeta_{8}^{2} ) q^{8} + ( -2 + \zeta_{8} - 2 \zeta_{8}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( -1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{4} + ( -2 + 2 \zeta_{8} ) q^{5} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 1 + 3 \zeta_{8} + \zeta_{8}^{2} ) q^{8} + ( -2 + \zeta_{8} - 2 \zeta_{8}^{2} ) q^{9} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{10} + ( 1 - \zeta_{8}^{3} ) q^{11} + ( -3 + 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{12} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{14} + ( 4 - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{15} -3 q^{16} + ( 2 - 3 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{17} + ( 3 + \zeta_{8} - \zeta_{8}^{3} ) q^{18} + ( -2 + 2 \zeta_{8}^{2} ) q^{19} + ( 4 + 4 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{20} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{21} + ( -1 + \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{22} + ( -2 - \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{23} + ( 1 - 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{24} + ( 4 - 3 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{25} + ( -6 - 8 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{26} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{27} + ( 1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{28} + 4 \zeta_{8}^{2} q^{30} + ( -3 - 3 \zeta_{8} ) q^{31} + ( 1 - \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{32} -2 q^{33} + ( 3 + 5 \zeta_{8} + 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{34} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{35} + 7 \zeta_{8}^{3} q^{36} + ( -2 - 2 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{37} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{38} + ( -4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{39} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{40} + ( -4 - \zeta_{8} - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{41} + ( 2 + 2 \zeta_{8}^{2} ) q^{42} -\zeta_{8} q^{43} + ( -2 - 3 \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{44} + ( 4 - 6 \zeta_{8} + 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{45} + ( 2 - 2 \zeta_{8} - 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{46} + ( -3 \zeta_{8} - 4 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{47} + ( 3 + 3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{48} + ( 2 - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{49} + ( -5 - \zeta_{8} + \zeta_{8}^{3} ) q^{50} + ( -3 - \zeta_{8} + 7 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{51} + ( 12 + 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{52} + ( 3 - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{53} + ( -6 - 6 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{54} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{55} + ( 1 - \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{56} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{57} + ( 1 - 12 \zeta_{8} + \zeta_{8}^{2} ) q^{59} + ( 4 - 12 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{60} + ( 2 + 2 \zeta_{8}^{3} ) q^{61} + ( 6 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{62} + ( 5 - 5 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{63} + ( -2 \zeta_{8} - 7 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{64} + ( -4 - 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{65} + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{66} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{67} + ( -8 - 8 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{68} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{69} -4 \zeta_{8}^{3} q^{70} + ( -1 - \zeta_{8} - 9 \zeta_{8}^{2} + 9 \zeta_{8}^{3} ) q^{71} + ( -5 \zeta_{8} - \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{72} + ( -8 + 8 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{74} + ( -11 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{75} + ( 2 + 8 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{76} + 2 \zeta_{8} q^{77} + ( 4 + 8 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{78} + ( 6 - \zeta_{8} + \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{79} + ( 6 - 6 \zeta_{8} ) q^{80} + ( 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{81} + ( 6 + 6 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{82} + ( -4 + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{83} + ( 4 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{84} + ( -4 + 10 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{85} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{86} + ( 4 + 4 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{88} + ( -2 \zeta_{8} - 12 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{89} + ( -4 + 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{90} + ( -4 + 4 \zeta_{8}^{3} ) q^{91} + ( 3 + 8 \zeta_{8} + 8 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{92} + 6 \zeta_{8} q^{93} + ( 7 + 10 \zeta_{8} + 7 \zeta_{8}^{2} ) q^{94} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{95} + ( 3 - 5 \zeta_{8} + 5 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{96} + ( -4 + 4 \zeta_{8} - 7 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{97} + ( 5 \zeta_{8} + 7 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{98} + ( -1 - \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} - 8q^{5} + 4q^{6} - 4q^{7} + 4q^{8} - 8q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} - 8q^{5} + 4q^{6} - 4q^{7} + 4q^{8} - 8q^{9} + 4q^{11} - 12q^{12} - 4q^{14} + 16q^{15} - 12q^{16} + 8q^{17} + 12q^{18} - 8q^{19} + 16q^{20} - 4q^{22} - 8q^{23} + 4q^{24} + 16q^{25} - 24q^{26} + 8q^{27} + 4q^{28} - 12q^{31} + 4q^{32} - 8q^{33} + 12q^{34} + 16q^{35} - 8q^{37} - 8q^{40} - 16q^{41} + 8q^{42} - 8q^{44} + 16q^{45} + 8q^{46} + 12q^{48} + 8q^{49} - 20q^{50} - 12q^{51} + 48q^{52} + 12q^{53} - 24q^{54} + 4q^{56} + 4q^{59} + 16q^{60} + 8q^{61} + 24q^{62} + 20q^{63} - 16q^{65} + 8q^{66} - 32q^{68} + 8q^{69} - 4q^{71} - 32q^{73} + 32q^{74} - 44q^{75} + 8q^{76} + 16q^{78} + 24q^{79} + 24q^{80} + 24q^{82} - 16q^{83} + 16q^{84} - 16q^{85} + 4q^{86} + 16q^{88} - 16q^{90} - 16q^{91} + 12q^{92} + 28q^{94} + 16q^{95} + 12q^{96} - 16q^{97} - 4q^{99} + O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$$.

 $$n$$ $$173$$ $$562$$ $$\chi(n)$$ $$\zeta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
87.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−1.70711 1.70711i −1.00000 + 0.414214i 3.82843i −0.585786 1.41421i 2.41421 + 1.00000i 0.414214 1.00000i 3.12132 3.12132i −1.29289 + 1.29289i −1.41421 + 3.41421i
474.1 −0.292893 0.292893i −1.00000 2.41421i 1.82843i −3.41421 + 1.41421i −0.414214 + 1.00000i −2.41421 1.00000i −1.12132 + 1.12132i −2.70711 + 2.70711i 1.41421 + 0.585786i
603.1 −0.292893 + 0.292893i −1.00000 + 2.41421i 1.82843i −3.41421 1.41421i −0.414214 1.00000i −2.41421 + 1.00000i −1.12132 1.12132i −2.70711 2.70711i 1.41421 0.585786i
689.1 −1.70711 + 1.70711i −1.00000 0.414214i 3.82843i −0.585786 + 1.41421i 2.41421 1.00000i 0.414214 + 1.00000i 3.12132 + 3.12132i −1.29289 1.29289i −1.41421 3.41421i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.d Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4} + 4 T_{2}^{3} + 8 T_{2}^{2} + 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.