# Properties

 Label 731.2.f.a Level 731 Weight 2 Character orbit 731.f Analytic conductor 5.837 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{2} + q^{4} + ( -1 - i ) q^{5} + ( -2 + 2 i ) q^{7} -3 i q^{8} -3 i q^{9} +O(q^{10})$$ $$q -i q^{2} + q^{4} + ( -1 - i ) q^{5} + ( -2 + 2 i ) q^{7} -3 i q^{8} -3 i q^{9} + ( -1 + i ) q^{10} + ( 2 - 2 i ) q^{11} + ( 2 + 2 i ) q^{14} - q^{16} + ( -1 + 4 i ) q^{17} -3 q^{18} -8 i q^{19} + ( -1 - i ) q^{20} + ( -2 - 2 i ) q^{22} + ( -4 + 4 i ) q^{23} -3 i q^{25} + ( -2 + 2 i ) q^{28} + ( 1 + i ) q^{29} + ( -4 - 4 i ) q^{31} -5 i q^{32} + ( 4 + i ) q^{34} + 4 q^{35} -3 i q^{36} + ( -5 - 5 i ) q^{37} -8 q^{38} + ( -3 + 3 i ) q^{40} + ( 1 - i ) q^{41} -i q^{43} + ( 2 - 2 i ) q^{44} + ( -3 + 3 i ) q^{45} + ( 4 + 4 i ) q^{46} + 12 q^{47} -i q^{49} -3 q^{50} -4 q^{55} + ( 6 + 6 i ) q^{56} + ( 1 - i ) q^{58} + ( 7 - 7 i ) q^{61} + ( -4 + 4 i ) q^{62} + ( 6 + 6 i ) q^{63} -7 q^{64} -4 q^{67} + ( -1 + 4 i ) q^{68} -4 i q^{70} + ( 10 + 10 i ) q^{71} -9 q^{72} + ( -9 - 9 i ) q^{73} + ( -5 + 5 i ) q^{74} -8 i q^{76} + 8 i q^{77} + ( -12 + 12 i ) q^{79} + ( 1 + i ) q^{80} -9 q^{81} + ( -1 - i ) q^{82} + 4 i q^{83} + ( 5 - 3 i ) q^{85} - q^{86} + ( -6 - 6 i ) q^{88} + 12 q^{89} + ( 3 + 3 i ) q^{90} + ( -4 + 4 i ) q^{92} -12 i q^{94} + ( -8 + 8 i ) q^{95} + ( 7 + 7 i ) q^{97} - q^{98} + ( -6 - 6 i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{5} - 4q^{7} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{5} - 4q^{7} - 2q^{10} + 4q^{11} + 4q^{14} - 2q^{16} - 2q^{17} - 6q^{18} - 2q^{20} - 4q^{22} - 8q^{23} - 4q^{28} + 2q^{29} - 8q^{31} + 8q^{34} + 8q^{35} - 10q^{37} - 16q^{38} - 6q^{40} + 2q^{41} + 4q^{44} - 6q^{45} + 8q^{46} + 24q^{47} - 6q^{50} - 8q^{55} + 12q^{56} + 2q^{58} + 14q^{61} - 8q^{62} + 12q^{63} - 14q^{64} - 8q^{67} - 2q^{68} + 20q^{71} - 18q^{72} - 18q^{73} - 10q^{74} - 24q^{79} + 2q^{80} - 18q^{81} - 2q^{82} + 10q^{85} - 2q^{86} - 12q^{88} + 24q^{89} + 6q^{90} - 8q^{92} - 16q^{95} + 14q^{97} - 2q^{98} - 12q^{99} + O(q^{100})$$

## Character Values

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

 $$n$$ $$173$$ $$562$$ $$\chi(n)$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
259.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 −1.00000 + 1.00000i 0 −2.00000 2.00000i 3.00000i 3.00000i −1.00000 1.00000i
302.1 1.00000i 0 1.00000 −1.00000 1.00000i 0 −2.00000 + 2.00000i 3.00000i 3.00000i −1.00000 + 1.00000i
 $$n$$: 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.c Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{3}$$