# Properties

 Label 731.2.s.a Level 731 Weight 2 Character orbit 731.s Analytic conductor 5.837 Analytic rank 0 Dimension 16 CM disc. -43 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.s (of order $$16$$ and degree $$8$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{16})$$ Coefficient field: 16.0.3289935900927224469054816256.1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{4} q^{4} -3 \beta_{5} q^{9} +O(q^{10})$$ $$q + 2 \beta_{4} q^{4} -3 \beta_{5} q^{9} + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{11} + ( -1 + \beta_{2} + \beta_{8} - \beta_{9} ) q^{13} + 4 \beta_{8} q^{16} + ( 3 \beta_{13} - \beta_{15} ) q^{17} + ( 3 - \beta_{2} - 4 \beta_{10} + \beta_{12} ) q^{23} + 5 \beta_{13} q^{25} + ( \beta_{1} - 4 \beta_{3} + 5 \beta_{4} + \beta_{6} ) q^{31} -6 \beta_{10} q^{36} + ( -6 \beta_{5} + \beta_{7} + 5 \beta_{11} + \beta_{14} ) q^{41} + ( \beta_{13} - 2 \beta_{15} ) q^{43} + ( -2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} ) q^{44} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{47} -7 \beta_{3} q^{49} + ( -4 \beta_{4} - 2 \beta_{6} + 2 \beta_{11} + 2 \beta_{14} ) q^{52} + ( 7 \beta_{8} + \beta_{9} - 6 \beta_{11} - \beta_{14} ) q^{53} + ( -3 + 2 \beta_{2} - 3 \beta_{11} - 2 \beta_{14} ) q^{59} + 8 \beta_{11} q^{64} + ( -\beta_{1} + 7 \beta_{3} + 7 \beta_{13} + \beta_{15} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{68} + ( 5 - 2 \beta_{2} + 7 \beta_{10} - 2 \beta_{12} ) q^{79} -9 \beta_{11} q^{81} + ( \beta_{10} - 2 \beta_{12} ) q^{83} + ( 8 \beta_{4} + 2 \beta_{6} - 8 \beta_{13} + 2 \beta_{15} ) q^{92} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{11} + 3 \beta_{14} ) q^{97} + ( 3 \beta_{12} - 3 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10319 x^{8} + 214358881$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{8} + 3070$$$$)/4179$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} + 3070 \nu$$$$)/45969$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{10} + 49039 \nu^{2}$$$$)/505659$$ $$\beta_{5}$$ $$=$$ $$($$$$-10 \nu^{11} + 15269 \nu^{3}$$$$)/5562249$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{10} - 3070 \nu^{2}$$$$)/45969$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} + 49039 \nu^{3}$$$$)/505659$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{12} + 24960 \nu^{4}$$$$)/2913559$$ $$\beta_{9}$$ $$=$$ $$($$$$-10 \nu^{12} + 15269 \nu^{4}$$$$)/5562249$$ $$\beta_{10}$$ $$=$$ $$($$$$-89 \nu^{13} + 692119 \nu^{5}$$$$)/ 673032129$$ $$\beta_{11}$$ $$=$$ $$($$$$320 \nu^{14} + 5073641 \nu^{6}$$$$)/ 7403353419$$ $$\beta_{12}$$ $$=$$ $$($$$$-320 \nu^{13} - 5073641 \nu^{5}$$$$)/ 673032129$$ $$\beta_{13}$$ $$=$$ $$($$$$-659 \nu^{15} + 12686950 \nu^{7}$$$$)/ 81436887609$$ $$\beta_{14}$$ $$=$$ $$($$$$89 \nu^{14} - 692119 \nu^{6}$$$$)/ 673032129$$ $$\beta_{15}$$ $$=$$ $$($$$$-\nu^{15} - 10319 \nu^{7}$$$$)/19487171$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 11 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{7} + 11 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$21 \beta_{9} + 110 \beta_{8}$$ $$\nu^{5}$$ $$=$$ $$-89 \beta_{12} + 320 \beta_{10}$$ $$\nu^{6}$$ $$=$$ $$-320 \beta_{14} + 979 \beta_{11}$$ $$\nu^{7}$$ $$=$$ $$-659 \beta_{15} + 4179 \beta_{13}$$ $$\nu^{8}$$ $$=$$ $$4179 \beta_{2} - 3070$$ $$\nu^{9}$$ $$=$$ $$45969 \beta_{3} - 3070 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-49039 \beta_{6} - 33770 \beta_{4}$$ $$\nu^{11}$$ $$=$$ $$15269 \beta_{7} - 539429 \beta_{5}$$ $$\nu^{12}$$ $$=$$ $$-524160 \beta_{9} + 167959 \beta_{8}$$ $$\nu^{13}$$ $$=$$ $$-692119 \beta_{12} - 5073641 \beta_{10}$$ $$\nu^{14}$$ $$=$$ $$5073641 \beta_{14} + 7613309 \beta_{11}$$ $$\nu^{15}$$ $$=$$ $$-12686950 \beta_{15} - 43123101 \beta_{13}$$

## Character Values

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

 $$n$$ $$173$$ $$562$$ $$\chi(n)$$ $$-\beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
214.1
 −0.792772 + 3.22048i 1.71665 − 2.83780i 3.22048 + 0.792772i −2.83780 − 1.71665i −3.22048 + 0.792772i 2.83780 − 1.71665i −0.792772 − 3.22048i 1.71665 + 2.83780i −1.71665 − 2.83780i 0.792772 + 3.22048i 3.22048 − 0.792772i −2.83780 + 1.71665i −3.22048 − 0.792772i 2.83780 + 1.71665i −1.71665 + 2.83780i 0.792772 − 3.22048i
0 0 −1.41421 1.41421i 0 0 0 0 1.14805 + 2.77164i 0
214.2 0 0 −1.41421 1.41421i 0 0 0 0 1.14805 + 2.77164i 0
300.1 0 0 1.41421 + 1.41421i 0 0 0 0 −2.77164 + 1.14805i 0
300.2 0 0 1.41421 + 1.41421i 0 0 0 0 −2.77164 + 1.14805i 0
343.1 0 0 1.41421 1.41421i 0 0 0 0 2.77164 + 1.14805i 0
343.2 0 0 1.41421 1.41421i 0 0 0 0 2.77164 + 1.14805i 0
386.1 0 0 −1.41421 + 1.41421i 0 0 0 0 1.14805 2.77164i 0
386.2 0 0 −1.41421 + 1.41421i 0 0 0 0 1.14805 2.77164i 0
515.1 0 0 −1.41421 + 1.41421i 0 0 0 0 −1.14805 + 2.77164i 0
515.2 0 0 −1.41421 + 1.41421i 0 0 0 0 −1.14805 + 2.77164i 0
558.1 0 0 1.41421 1.41421i 0 0 0 0 −2.77164 1.14805i 0
558.2 0 0 1.41421 1.41421i 0 0 0 0 −2.77164 1.14805i 0
601.1 0 0 1.41421 + 1.41421i 0 0 0 0 2.77164 1.14805i 0
601.2 0 0 1.41421 + 1.41421i 0 0 0 0 2.77164 1.14805i 0
687.1 0 0 −1.41421 1.41421i 0 0 0 0 −1.14805 2.77164i 0
687.2 0 0 −1.41421 1.41421i 0 0 0 0 −1.14805 2.77164i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 687.2 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
43.b Odd 1 CM by $$\Q(\sqrt{-43})$$ yes
17.e Odd 1 yes
731.s Even 1 yes

## Hecke kernels

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