# Properties

 Label 693.2.i.g Level $693$ Weight $2$ Character orbit 693.i Analytic conductor $5.534$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 693.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1783323.2 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{2} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( 3 - \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{2} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( 3 - \beta_{3} ) q^{8} + ( 3 - 3 \beta_{4} - \beta_{5} ) q^{10} + ( -1 + \beta_{4} ) q^{11} + ( -4 - \beta_{2} ) q^{13} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{14} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{16} + ( -1 + \beta_{4} - \beta_{5} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{19} + ( -5 - \beta_{2} + 2 \beta_{3} ) q^{20} -\beta_{3} q^{22} + ( -\beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{23} + ( -2 + 3 \beta_{1} + 2 \beta_{4} ) q^{25} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{26} + ( 3 - \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{28} + ( 2 - 3 \beta_{2} + \beta_{3} ) q^{29} + ( 2 - 3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{31} -3 \beta_{1} q^{32} + ( -3 + \beta_{2} ) q^{34} + ( -1 - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 3 - \beta_{1} - 3 \beta_{4} - 6 \beta_{5} ) q^{38} + ( -3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{40} + ( 2 + \beta_{2} - 4 \beta_{3} ) q^{41} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( 3 + \beta_{1} - 3 \beta_{4} - 5 \beta_{5} ) q^{46} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{47} + ( -3 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{49} + ( -3 \beta_{2} + \beta_{3} ) q^{50} + ( 1 + 4 \beta_{1} - \beta_{4} - 4 \beta_{5} ) q^{52} + ( 6 - \beta_{1} - 6 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 1 + \beta_{2} + \beta_{3} ) q^{55} + ( 3 - \beta_{1} + \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} ) q^{56} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{58} + ( 3 - \beta_{1} - 3 \beta_{4} + 6 \beta_{5} ) q^{59} + ( -2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{61} + ( -3 + 4 \beta_{2} ) q^{62} + ( -2 + \beta_{2} + \beta_{3} ) q^{64} + ( 6 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} ) q^{65} + ( 6 - 2 \beta_{1} - 6 \beta_{4} ) q^{67} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -3 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{70} + ( -1 + 4 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 5 + 5 \beta_{1} - 5 \beta_{4} ) q^{73} + ( 6 + 4 \beta_{1} - 6 \beta_{4} - 2 \beta_{5} ) q^{74} + ( -12 + 3 \beta_{2} + 6 \beta_{3} ) q^{76} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{77} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{79} + ( 2 + 3 \beta_{1} - 2 \beta_{4} - 3 \beta_{5} ) q^{80} + ( 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} - 5 \beta_{5} ) q^{82} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{83} + ( 4 + \beta_{2} ) q^{85} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{86} + ( -3 + 3 \beta_{4} + \beta_{5} ) q^{88} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{89} + ( -7 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -7 + 4 \beta_{2} + 7 \beta_{3} ) q^{92} + ( 3 + \beta_{1} - 3 \beta_{4} ) q^{94} + ( -6 + \beta_{1} + 6 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 4 + 3 \beta_{2} ) q^{97} + ( 6 + \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 4q^{4} - 2q^{5} + 2q^{7} + 18q^{8} + O(q^{10})$$ $$6q - 4q^{4} - 2q^{5} + 2q^{7} + 18q^{8} + 9q^{10} - 3q^{11} - 22q^{13} - 12q^{14} - 2q^{16} - 3q^{17} + 11q^{19} - 28q^{20} + 12q^{23} - 3q^{25} + q^{26} + 13q^{28} + 18q^{29} + 3q^{31} - 3q^{32} - 20q^{34} - 9q^{35} + 4q^{37} + 8q^{38} + 3q^{40} + 10q^{41} + 4q^{43} - 4q^{44} + 10q^{46} - 3q^{47} - 24q^{49} + 6q^{50} + 7q^{52} + 17q^{53} + 4q^{55} - 3q^{56} + 13q^{58} + 8q^{59} + 24q^{61} - 26q^{62} - 14q^{64} + 15q^{65} + 16q^{67} + 5q^{68} - 27q^{70} - 14q^{71} + 20q^{73} + 22q^{74} - 78q^{76} + 2q^{77} - 3q^{79} + 9q^{80} - 41q^{82} + 22q^{83} + 22q^{85} - 21q^{86} - 9q^{88} + q^{89} - 15q^{91} - 50q^{92} + 10q^{94} - 17q^{95} + 18q^{97} + 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 19 \nu^{2} + 12 \nu - 60$$$$)/83$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 48 \nu - 240$$$$)/83$$ $$\beta_{4}$$ $$=$$ $$($$$$-20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu + 204$$$$)/249$$ $$\beta_{5}$$ $$=$$ $$($$$$-16 \nu^{5} - 3 \nu^{4} - 68 \nu^{3} - 28 \nu^{2} - 275 \nu - 36$$$$)/83$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{5} + 12 \beta_{4} - \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$-6 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 17 \beta_{2} - 17 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 0.356769 + 0.617942i −0.956115 − 1.65604i 1.09935 + 1.90412i 0.356769 − 0.617942i −0.956115 + 1.65604i 1.09935 − 1.90412i
−1.24543 + 2.15715i 0 −2.10220 3.64112i 1.10220 1.90907i 0 −1.10220 + 2.40523i 5.49086 0 2.74543 + 4.75523i
100.2 0.328310 0.568650i 0 0.784425 + 1.35866i −1.78442 + 3.09071i 0 1.78442 + 1.95341i 2.34338 0 1.17169 + 2.02943i
100.3 0.917122 1.58850i 0 −0.682224 1.18165i −0.317776 + 0.550404i 0 0.317776 2.62660i 1.16576 0 0.582878 + 1.00958i
298.1 −1.24543 2.15715i 0 −2.10220 + 3.64112i 1.10220 + 1.90907i 0 −1.10220 2.40523i 5.49086 0 2.74543 4.75523i
298.2 0.328310 + 0.568650i 0 0.784425 1.35866i −1.78442 3.09071i 0 1.78442 1.95341i 2.34338 0 1.17169 2.02943i
298.3 0.917122 + 1.58850i 0 −0.682224 + 1.18165i −0.317776 0.550404i 0 0.317776 + 2.62660i 1.16576 0 0.582878 1.00958i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 298.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.i.g 6
3.b odd 2 1 77.2.e.b 6
7.c even 3 1 inner 693.2.i.g 6
7.c even 3 1 4851.2.a.bo 3
7.d odd 6 1 4851.2.a.bn 3
12.b even 2 1 1232.2.q.k 6
21.c even 2 1 539.2.e.l 6
21.g even 6 1 539.2.a.i 3
21.g even 6 1 539.2.e.l 6
21.h odd 6 1 77.2.e.b 6
21.h odd 6 1 539.2.a.h 3
33.d even 2 1 847.2.e.d 6
33.f even 10 4 847.2.n.d 24
33.h odd 10 4 847.2.n.e 24
84.j odd 6 1 8624.2.a.ck 3
84.n even 6 1 1232.2.q.k 6
84.n even 6 1 8624.2.a.cl 3
231.k odd 6 1 5929.2.a.w 3
231.l even 6 1 847.2.e.d 6
231.l even 6 1 5929.2.a.v 3
231.z odd 30 4 847.2.n.e 24
231.be even 30 4 847.2.n.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 3.b odd 2 1
77.2.e.b 6 21.h odd 6 1
539.2.a.h 3 21.h odd 6 1
539.2.a.i 3 21.g even 6 1
539.2.e.l 6 21.c even 2 1
539.2.e.l 6 21.g even 6 1
693.2.i.g 6 1.a even 1 1 trivial
693.2.i.g 6 7.c even 3 1 inner
847.2.e.d 6 33.d even 2 1
847.2.e.d 6 231.l even 6 1
847.2.n.d 24 33.f even 10 4
847.2.n.d 24 231.be even 30 4
847.2.n.e 24 33.h odd 10 4
847.2.n.e 24 231.z odd 30 4
1232.2.q.k 6 12.b even 2 1
1232.2.q.k 6 84.n even 6 1
4851.2.a.bn 3 7.d odd 6 1
4851.2.a.bo 3 7.c even 3 1
5929.2.a.v 3 231.l even 6 1
5929.2.a.w 3 231.k odd 6 1
8624.2.a.ck 3 84.j odd 6 1
8624.2.a.cl 3 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(693, [\chi])$$:

 $$T_{2}^{6} + 5 T_{2}^{4} - 6 T_{2}^{3} + 25 T_{2}^{2} - 15 T_{2} + 9$$ $$T_{5}^{6} + 2 T_{5}^{5} + 11 T_{5}^{4} - 4 T_{5}^{3} + 59 T_{5}^{2} + 35 T_{5} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 15 T + 25 T^{2} - 6 T^{3} + 5 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$25 + 35 T + 59 T^{2} - 4 T^{3} + 11 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$343 - 98 T + 98 T^{2} - 23 T^{3} + 14 T^{4} - 2 T^{5} + T^{6}$$
$11$ $$( 1 + T + T^{2} )^{3}$$
$13$ $$( 35 + 36 T + 11 T^{2} + T^{3} )^{2}$$
$17$ $$49 + 14 T + 25 T^{2} + 8 T^{3} + 11 T^{4} + 3 T^{5} + T^{6}$$
$19$ $$3249 + 1140 T + 1027 T^{2} - 334 T^{3} + 101 T^{4} - 11 T^{5} + T^{6}$$
$23$ $$2209 - 2021 T + 1285 T^{2} - 422 T^{3} + 101 T^{4} - 12 T^{5} + T^{6}$$
$29$ $$( 53 - 20 T - 9 T^{2} + T^{3} )^{2}$$
$31$ $$11449 - 4708 T + 2257 T^{2} - 82 T^{3} + 53 T^{4} - 3 T^{5} + T^{6}$$
$37$ $$23104 - 5472 T + 1904 T^{2} - 160 T^{3} + 52 T^{4} - 4 T^{5} + T^{6}$$
$41$ $$( 109 - 80 T - 5 T^{2} + T^{3} )^{2}$$
$43$ $$( 41 - 25 T - 2 T^{2} + T^{3} )^{2}$$
$47$ $$49 + 14 T + 25 T^{2} + 8 T^{3} + 11 T^{4} + 3 T^{5} + T^{6}$$
$53$ $$441 - 1554 T + 5119 T^{2} - 1216 T^{3} + 215 T^{4} - 17 T^{5} + T^{6}$$
$59$ $$1750329 - 207711 T + 35233 T^{2} - 1390 T^{3} + 221 T^{4} - 8 T^{5} + T^{6}$$
$61$ $$141376 - 64672 T + 20560 T^{2} - 3376 T^{3} + 404 T^{4} - 24 T^{5} + T^{6}$$
$67$ $$5184 - 4896 T + 3472 T^{2} - 944 T^{3} + 188 T^{4} - 16 T^{5} + T^{6}$$
$71$ $$( -419 - 86 T + 7 T^{2} + T^{3} )^{2}$$
$73$ $$390625 + 15625 T + 13125 T^{2} - 1750 T^{3} + 375 T^{4} - 20 T^{5} + T^{6}$$
$79$ $$19881 + 5358 T + 1867 T^{2} + 168 T^{3} + 47 T^{4} + 3 T^{5} + T^{6}$$
$83$ $$( 3 + 16 T - 11 T^{2} + T^{3} )^{2}$$
$89$ $$9 - 24 T + 67 T^{2} + 2 T^{3} + 9 T^{4} - T^{5} + T^{6}$$
$97$ $$( 47 - 12 T - 9 T^{2} + T^{3} )^{2}$$