# Properties

 Label 920.2.a.h Level $920$ Weight $2$ Character orbit 920.a Self dual yes Analytic conductor $7.346$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.34623698596$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2597.1 Defining polynomial: $$x^{3} - x^{2} - 9x + 8$$ x^3 - x^2 - 9*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + (\beta_{2} + 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + q^5 + (-b1 + 1) * q^7 + (b2 + 3) * q^9 $$q + \beta_1 q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + (\beta_{2} + 3) q^{9} + (\beta_1 + 2) q^{11} - \beta_{2} q^{13} + \beta_1 q^{15} + (\beta_{2} + 3) q^{17} + ( - \beta_{2} - 4) q^{19} + ( - \beta_{2} + \beta_1 - 6) q^{21} + q^{23} + q^{25} + (\beta_{2} + 3 \beta_1 - 2) q^{27} + ( - \beta_{2} - \beta_1 + 5) q^{29} + (\beta_1 - 3) q^{31} + (\beta_{2} + 2 \beta_1 + 6) q^{33} + ( - \beta_1 + 1) q^{35} + ( - \beta_{2} - 3 \beta_1 + 3) q^{37} + ( - \beta_{2} - 3 \beta_1 + 2) q^{39} + ( - \beta_{2} + 3) q^{41} + 8 q^{43} + (\beta_{2} + 3) q^{45} + 2 \beta_{2} q^{47} + (\beta_{2} - 2 \beta_1) q^{49} + (\beta_{2} + 6 \beta_1 - 2) q^{51} + (\beta_{2} + 3 \beta_1 - 1) q^{53} + (\beta_1 + 2) q^{55} + ( - \beta_{2} - 7 \beta_1 + 2) q^{57} + ( - \beta_{2} - \beta_1 - 5) q^{59} + (2 \beta_{2} - \beta_1 + 4) q^{61} + ( - 6 \beta_1 + 5) q^{63} - \beta_{2} q^{65} + ( - \beta_{2} - 3 \beta_1 + 3) q^{67} + \beta_1 q^{69} + (\beta_{2} - 2 \beta_1 - 7) q^{71} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{73} + \beta_1 q^{75} + ( - \beta_{2} - \beta_1 - 4) q^{77} - 4 \beta_1 q^{79} + (\beta_{2} + \beta_1 + 7) q^{81} + (\beta_{2} - \beta_1 - 5) q^{83} + (\beta_{2} + 3) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{87} + (4 \beta_1 - 4) q^{89} + (3 \beta_1 - 2) q^{91} + (\beta_{2} - 3 \beta_1 + 6) q^{93} + ( - \beta_{2} - 4) q^{95} + (3 \beta_1 - 2) q^{97} + (3 \beta_{2} + 6 \beta_1 + 4) q^{99}+O(q^{100})$$ q + b1 * q^3 + q^5 + (-b1 + 1) * q^7 + (b2 + 3) * q^9 + (b1 + 2) * q^11 - b2 * q^13 + b1 * q^15 + (b2 + 3) * q^17 + (-b2 - 4) * q^19 + (-b2 + b1 - 6) * q^21 + q^23 + q^25 + (b2 + 3*b1 - 2) * q^27 + (-b2 - b1 + 5) * q^29 + (b1 - 3) * q^31 + (b2 + 2*b1 + 6) * q^33 + (-b1 + 1) * q^35 + (-b2 - 3*b1 + 3) * q^37 + (-b2 - 3*b1 + 2) * q^39 + (-b2 + 3) * q^41 + 8 * q^43 + (b2 + 3) * q^45 + 2*b2 * q^47 + (b2 - 2*b1) * q^49 + (b2 + 6*b1 - 2) * q^51 + (b2 + 3*b1 - 1) * q^53 + (b1 + 2) * q^55 + (-b2 - 7*b1 + 2) * q^57 + (-b2 - b1 - 5) * q^59 + (2*b2 - b1 + 4) * q^61 + (-6*b1 + 5) * q^63 - b2 * q^65 + (-b2 - 3*b1 + 3) * q^67 + b1 * q^69 + (b2 - 2*b1 - 7) * q^71 + (-2*b2 - 4*b1 + 6) * q^73 + b1 * q^75 + (-b2 - b1 - 4) * q^77 - 4*b1 * q^79 + (b2 + b1 + 7) * q^81 + (b2 - b1 - 5) * q^83 + (b2 + 3) * q^85 + (-2*b2 + 2*b1 - 4) * q^87 + (4*b1 - 4) * q^89 + (3*b1 - 2) * q^91 + (b2 - 3*b1 + 6) * q^93 + (-b2 - 4) * q^95 + (3*b1 - 2) * q^97 + (3*b2 + 6*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 + 2 * q^7 + 10 * q^9 $$3 q + q^{3} + 3 q^{5} + 2 q^{7} + 10 q^{9} + 7 q^{11} - q^{13} + q^{15} + 10 q^{17} - 13 q^{19} - 18 q^{21} + 3 q^{23} + 3 q^{25} - 2 q^{27} + 13 q^{29} - 8 q^{31} + 21 q^{33} + 2 q^{35} + 5 q^{37} + 2 q^{39} + 8 q^{41} + 24 q^{43} + 10 q^{45} + 2 q^{47} - q^{49} + q^{51} + q^{53} + 7 q^{55} - 2 q^{57} - 17 q^{59} + 13 q^{61} + 9 q^{63} - q^{65} + 5 q^{67} + q^{69} - 22 q^{71} + 12 q^{73} + q^{75} - 14 q^{77} - 4 q^{79} + 23 q^{81} - 15 q^{83} + 10 q^{85} - 12 q^{87} - 8 q^{89} - 3 q^{91} + 16 q^{93} - 13 q^{95} - 3 q^{97} + 21 q^{99}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 + 2 * q^7 + 10 * q^9 + 7 * q^11 - q^13 + q^15 + 10 * q^17 - 13 * q^19 - 18 * q^21 + 3 * q^23 + 3 * q^25 - 2 * q^27 + 13 * q^29 - 8 * q^31 + 21 * q^33 + 2 * q^35 + 5 * q^37 + 2 * q^39 + 8 * q^41 + 24 * q^43 + 10 * q^45 + 2 * q^47 - q^49 + q^51 + q^53 + 7 * q^55 - 2 * q^57 - 17 * q^59 + 13 * q^61 + 9 * q^63 - q^65 + 5 * q^67 + q^69 - 22 * q^71 + 12 * q^73 + q^75 - 14 * q^77 - 4 * q^79 + 23 * q^81 - 15 * q^83 + 10 * q^85 - 12 * q^87 - 8 * q^89 - 3 * q^91 + 16 * q^93 - 13 * q^95 - 3 * q^97 + 21 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6$$ v^2 - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6$$ b2 + 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.95759 0.878468 3.07912
0 −2.95759 0 1.00000 0 3.95759 0 5.74732 0
1.2 0 0.878468 0 1.00000 0 0.121532 0 −2.22829 0
1.3 0 3.07912 0 1.00000 0 −2.07912 0 6.48097 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.2.a.h 3
3.b odd 2 1 8280.2.a.bj 3
4.b odd 2 1 1840.2.a.s 3
5.b even 2 1 4600.2.a.x 3
5.c odd 4 2 4600.2.e.p 6
8.b even 2 1 7360.2.a.by 3
8.d odd 2 1 7360.2.a.cc 3
20.d odd 2 1 9200.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 1.a even 1 1 trivial
1840.2.a.s 3 4.b odd 2 1
4600.2.a.x 3 5.b even 2 1
4600.2.e.p 6 5.c odd 4 2
7360.2.a.by 3 8.b even 2 1
7360.2.a.cc 3 8.d odd 2 1
8280.2.a.bj 3 3.b odd 2 1
9200.2.a.ce 3 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - T_{3}^{2} - 9T_{3} + 8$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(920))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 9T + 8$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 2 T^{2} - 8 T + 1$$
$11$ $$T^{3} - 7 T^{2} + 7 T + 14$$
$13$ $$T^{3} + T^{2} - 23 T - 50$$
$17$ $$T^{3} - 10 T^{2} + 10 T + 83$$
$19$ $$T^{3} + 13 T^{2} + 33 T - 62$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} - 13 T^{2} + 26 T + 76$$
$31$ $$T^{3} + 8 T^{2} + 12 T - 1$$
$37$ $$T^{3} - 5 T^{2} - 92 T + 496$$
$41$ $$T^{3} - 8 T^{2} - 2 T + 1$$
$43$ $$(T - 8)^{3}$$
$47$ $$T^{3} - 2 T^{2} - 92 T + 400$$
$53$ $$T^{3} - T^{2} - 100 T - 300$$
$59$ $$T^{3} + 17 T^{2} + 66 T + 36$$
$61$ $$T^{3} - 13 T^{2} - 51 T + 720$$
$67$ $$T^{3} - 5 T^{2} - 92 T + 496$$
$71$ $$T^{3} + 22 T^{2} + 96 T - 225$$
$73$ $$T^{3} - 12 T^{2} - 176 T + 2120$$
$79$ $$T^{3} + 4 T^{2} - 144 T - 512$$
$83$ $$T^{3} + 15 T^{2} + 40 T - 36$$
$89$ $$T^{3} + 8 T^{2} - 128 T - 64$$
$97$ $$T^{3} + 3 T^{2} - 81 T + 50$$