# Properties

 Label 693.2.i.f Level $693$ Weight $2$ Character orbit 693.i Analytic conductor $5.534$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - 2 \beta_{2} - 2) q^{5} + (\beta_{3} + 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3) q^{8}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^2 + (-2*b3 + b2 - 2*b1) * q^4 + (-2*b2 - 2) * q^5 + (b3 + 2*b1 + 1) * q^7 + (b3 + 3) * q^8 $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - 2 \beta_{2} - 2) q^{5} + (\beta_{3} + 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3) q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{10} - \beta_{2} q^{11} + ( - 2 \beta_{3} + 2) q^{13} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{14} + ( - 3 \beta_{2} - 3) q^{16} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{17} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{19} + (4 \beta_{3} + 2) q^{20} + ( - \beta_{3} - 1) q^{22} + ( - 7 \beta_{2} - 7) q^{23} - \beta_{2} q^{25} + (2 \beta_{2} + 2) q^{26} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 8) q^{28} + ( - 3 \beta_{3} + 1) q^{29} + ( - 4 \beta_{3} - 4 \beta_1) q^{31} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{32} + (4 \beta_{3} + 5) q^{34} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{35} + (\beta_{2} - 6 \beta_1 + 1) q^{37} + ( - 6 \beta_{3} + 9 \beta_{2} - 6 \beta_1) q^{38} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{40} + ( - 2 \beta_{3} + 4) q^{41} + (3 \beta_{3} - 7) q^{43} + (\beta_{2} - 2 \beta_1 + 1) q^{44} + ( - 7 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{46} + (7 \beta_{2} - 2 \beta_1 + 7) q^{47} + (2 \beta_{3} + 4 \beta_1 - 5) q^{49} + ( - \beta_{3} - 1) q^{50} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{52} + (2 \beta_{3} + 10 \beta_{2} + 2 \beta_1) q^{53} - 2 q^{55} + (4 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 1) q^{56} + (5 \beta_{2} - 2 \beta_1 + 5) q^{58} + ( - 4 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{59} + (4 \beta_{2} + 4) q^{61} + (4 \beta_{3} + 8) q^{62} + ( - 2 \beta_{3} - 7) q^{64} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{65} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{67} + ( - 7 \beta_{2} + 7 \beta_1 - 7) q^{68} + (6 \beta_{2} - 4 \beta_1 + 8) q^{70} + ( - 2 \beta_{3} + 7) q^{71} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{73} + (7 \beta_{3} - 13 \beta_{2} + 7 \beta_1) q^{74} + (9 \beta_{3} + 15) q^{76} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{2} + 8 \beta_1 - 2) q^{79} + 6 \beta_{2} q^{80} + 2 \beta_1 q^{82} + 2 \beta_{3} q^{83} + (2 \beta_{3} + 6) q^{85} + (\beta_{2} - 4 \beta_1 + 1) q^{86} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{88} + 10 \beta_1 q^{89} + (8 \beta_{2} + 4 \beta_1 + 6) q^{91} + (14 \beta_{3} + 7) q^{92} + (9 \beta_{3} - 11 \beta_{2} + 9 \beta_1) q^{94} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{95} + ( - 6 \beta_{3} - 3) q^{97} + ( - 4 \beta_{3} + 9 \beta_{2} - 7 \beta_1 + 1) q^{98}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^2 + (-2*b3 + b2 - 2*b1) * q^4 + (-2*b2 - 2) * q^5 + (b3 + 2*b1 + 1) * q^7 + (b3 + 3) * q^8 + (-2*b3 + 2*b2 - 2*b1) * q^10 - b2 * q^11 + (-2*b3 + 2) * q^13 + (-2*b3 + b2 - 3) * q^14 + (-3*b2 - 3) * q^16 + (-b3 + 3*b2 - b1) * q^17 + (-3*b2 + 3*b1 - 3) * q^19 + (4*b3 + 2) * q^20 + (-b3 - 1) * q^22 + (-7*b2 - 7) * q^23 - b2 * q^25 + (2*b2 + 2) * q^26 + (-b3 + 5*b2 - 3*b1 + 8) * q^28 + (-3*b3 + 1) * q^29 + (-4*b3 - 4*b1) * q^31 + (-b3 - 3*b2 - b1) * q^32 + (4*b3 + 5) * q^34 + (-4*b3 - 2*b2 - 2*b1 - 2) * q^35 + (b2 - 6*b1 + 1) * q^37 + (-6*b3 + 9*b2 - 6*b1) * q^38 + (-6*b2 + 2*b1 - 6) * q^40 + (-2*b3 + 4) * q^41 + (3*b3 - 7) * q^43 + (b2 - 2*b1 + 1) * q^44 + (-7*b3 + 7*b2 - 7*b1) * q^46 + (7*b2 - 2*b1 + 7) * q^47 + (2*b3 + 4*b1 - 5) * q^49 + (-b3 - 1) * q^50 + (-2*b3 - 6*b2 - 2*b1) * q^52 + (2*b3 + 10*b2 + 2*b1) * q^53 - 2 * q^55 + (4*b3 - 4*b2 + 6*b1 + 1) * q^56 + (5*b2 - 2*b1 + 5) * q^58 + (-4*b3 + 3*b2 - 4*b1) * q^59 + (4*b2 + 4) * q^61 + (4*b3 + 8) * q^62 + (-2*b3 - 7) * q^64 + (-4*b2 - 4*b1 - 4) * q^65 + (-2*b3 - 6*b2 - 2*b1) * q^67 + (-7*b2 + 7*b1 - 7) * q^68 + (6*b2 - 4*b1 + 8) * q^70 + (-2*b3 + 7) * q^71 + (4*b3 + 6*b2 + 4*b1) * q^73 + (7*b3 - 13*b2 + 7*b1) * q^74 + (9*b3 + 15) * q^76 + (-b3 - b2 + b1) * q^77 + (-2*b2 + 8*b1 - 2) * q^79 + 6*b2 * q^80 + 2*b1 * q^82 + 2*b3 * q^83 + (2*b3 + 6) * q^85 + (b2 - 4*b1 + 1) * q^86 + (b3 - 3*b2 + b1) * q^88 + 10*b1 * q^89 + (8*b2 + 4*b1 + 6) * q^91 + (14*b3 + 7) * q^92 + (9*b3 - 11*b2 + 9*b1) * q^94 + (-6*b3 + 6*b2 - 6*b1) * q^95 + (-6*b3 - 3) * q^97 + (-4*b3 + 9*b2 - 7*b1 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^5 + 4 * q^7 + 12 * q^8 $$4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8} - 4 q^{10} + 2 q^{11} + 8 q^{13} - 14 q^{14} - 6 q^{16} - 6 q^{17} - 6 q^{19} + 8 q^{20} - 4 q^{22} - 14 q^{23} + 2 q^{25} + 4 q^{26} + 22 q^{28} + 4 q^{29} + 6 q^{32} + 20 q^{34} - 4 q^{35} + 2 q^{37} - 18 q^{38} - 12 q^{40} + 16 q^{41} - 28 q^{43} + 2 q^{44} - 14 q^{46} + 14 q^{47} - 20 q^{49} - 4 q^{50} + 12 q^{52} - 20 q^{53} - 8 q^{55} + 12 q^{56} + 10 q^{58} - 6 q^{59} + 8 q^{61} + 32 q^{62} - 28 q^{64} - 8 q^{65} + 12 q^{67} - 14 q^{68} + 20 q^{70} + 28 q^{71} - 12 q^{73} + 26 q^{74} + 60 q^{76} + 2 q^{77} - 4 q^{79} - 12 q^{80} + 24 q^{85} + 2 q^{86} + 6 q^{88} + 8 q^{91} + 28 q^{92} + 22 q^{94} - 12 q^{95} - 12 q^{97} - 14 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^5 + 4 * q^7 + 12 * q^8 - 4 * q^10 + 2 * q^11 + 8 * q^13 - 14 * q^14 - 6 * q^16 - 6 * q^17 - 6 * q^19 + 8 * q^20 - 4 * q^22 - 14 * q^23 + 2 * q^25 + 4 * q^26 + 22 * q^28 + 4 * q^29 + 6 * q^32 + 20 * q^34 - 4 * q^35 + 2 * q^37 - 18 * q^38 - 12 * q^40 + 16 * q^41 - 28 * q^43 + 2 * q^44 - 14 * q^46 + 14 * q^47 - 20 * q^49 - 4 * q^50 + 12 * q^52 - 20 * q^53 - 8 * q^55 + 12 * q^56 + 10 * q^58 - 6 * q^59 + 8 * q^61 + 32 * q^62 - 28 * q^64 - 8 * q^65 + 12 * q^67 - 14 * q^68 + 20 * q^70 + 28 * q^71 - 12 * q^73 + 26 * q^74 + 60 * q^76 + 2 * q^77 - 4 * q^79 - 12 * q^80 + 24 * q^85 + 2 * q^86 + 6 * q^88 + 8 * q^91 + 28 * q^92 + 22 * q^94 - 12 * q^95 - 12 * q^97 - 14 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−1.20711 + 2.09077i 0 −1.91421 3.31552i −1.00000 + 1.73205i 0 1.00000 + 2.44949i 4.41421 0 −2.41421 4.18154i
100.2 0.207107 0.358719i 0 0.914214 + 1.58346i −1.00000 + 1.73205i 0 1.00000 2.44949i 1.58579 0 0.414214 + 0.717439i
298.1 −1.20711 2.09077i 0 −1.91421 + 3.31552i −1.00000 1.73205i 0 1.00000 2.44949i 4.41421 0 −2.41421 + 4.18154i
298.2 0.207107 + 0.358719i 0 0.914214 1.58346i −1.00000 1.73205i 0 1.00000 + 2.44949i 1.58579 0 0.414214 0.717439i
 $$n$$: e.g. 2-40 or 990-1000 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.f 4
3.b odd 2 1 231.2.i.d 4
7.c even 3 1 inner 693.2.i.f 4
7.c even 3 1 4851.2.a.be 2
7.d odd 6 1 4851.2.a.bd 2
21.g even 6 1 1617.2.a.m 2
21.h odd 6 1 231.2.i.d 4
21.h odd 6 1 1617.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.d 4 3.b odd 2 1
231.2.i.d 4 21.h odd 6 1
693.2.i.f 4 1.a even 1 1 trivial
693.2.i.f 4 7.c even 3 1 inner
1617.2.a.m 2 21.g even 6 1
1617.2.a.n 2 21.h odd 6 1
4851.2.a.bd 2 7.d odd 6 1
4851.2.a.be 2 7.c even 3 1

## Hecke kernels

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

 $$T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1$$ T2^4 + 2*T2^3 + 5*T2^2 - 2*T2 + 1 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 2 T + 4)^{2}$$
$7$ $$(T^{2} - 2 T + 7)^{2}$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} - 4 T - 4)^{2}$$
$17$ $$T^{4} + 6 T^{3} + 29 T^{2} + 42 T + 49$$
$19$ $$T^{4} + 6 T^{3} + 45 T^{2} - 54 T + 81$$
$23$ $$(T^{2} + 7 T + 49)^{2}$$
$29$ $$(T^{2} - 2 T - 17)^{2}$$
$31$ $$T^{4} + 32T^{2} + 1024$$
$37$ $$T^{4} - 2 T^{3} + 75 T^{2} + \cdots + 5041$$
$41$ $$(T^{2} - 8 T + 8)^{2}$$
$43$ $$(T^{2} + 14 T + 31)^{2}$$
$47$ $$T^{4} - 14 T^{3} + 155 T^{2} + \cdots + 1681$$
$53$ $$T^{4} + 20 T^{3} + 308 T^{2} + \cdots + 8464$$
$59$ $$T^{4} + 6 T^{3} + 59 T^{2} - 138 T + 529$$
$61$ $$(T^{2} - 4 T + 16)^{2}$$
$67$ $$T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784$$
$71$ $$(T^{2} - 14 T + 41)^{2}$$
$73$ $$T^{4} + 12 T^{3} + 140 T^{2} + \cdots + 16$$
$79$ $$T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376$$
$83$ $$(T^{2} - 8)^{2}$$
$89$ $$T^{4} + 200 T^{2} + 40000$$
$97$ $$(T^{2} + 6 T - 63)^{2}$$