# Properties

 Label 9200.2.a.ce Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ 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: no (minimal twist has level 920) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_1 + 1) q^{7} + (\beta_{2} + 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b1 + 1) * q^7 + (b2 + 3) * q^9 $$q + \beta_1 q^{3} + ( - \beta_1 + 1) q^{7} + (\beta_{2} + 3) q^{9} + ( - \beta_1 - 2) q^{11} + \beta_{2} q^{13} + ( - \beta_{2} - 3) q^{17} + (\beta_{2} + 4) q^{19} + ( - \beta_{2} + \beta_1 - 6) q^{21} + q^{23} + (\beta_{2} + 3 \beta_1 - 2) q^{27} + ( - \beta_{2} - \beta_1 + 5) q^{29} + ( - \beta_1 + 3) q^{31} + ( - \beta_{2} - 2 \beta_1 - 6) q^{33} + (\beta_{2} + 3 \beta_1 - 3) q^{37} + (\beta_{2} + 3 \beta_1 - 2) q^{39} + ( - \beta_{2} + 3) q^{41} + 8 q^{43} + 2 \beta_{2} q^{47} + (\beta_{2} - 2 \beta_1) q^{49} + ( - \beta_{2} - 6 \beta_1 + 2) q^{51} + ( - \beta_{2} - 3 \beta_1 + 1) q^{53} + (\beta_{2} + 7 \beta_1 - 2) q^{57} + (\beta_{2} + \beta_1 + 5) q^{59} + (2 \beta_{2} - \beta_1 + 4) q^{61} + ( - 6 \beta_1 + 5) q^{63} + ( - \beta_{2} - 3 \beta_1 + 3) q^{67} + \beta_1 q^{69} + ( - \beta_{2} + 2 \beta_1 + 7) q^{71} + (2 \beta_{2} + 4 \beta_1 - 6) q^{73} + (\beta_{2} + \beta_1 + 4) q^{77} + 4 \beta_1 q^{79} + (\beta_{2} + \beta_1 + 7) q^{81} + (\beta_{2} - \beta_1 - 5) q^{83} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{87} + (4 \beta_1 - 4) q^{89} + ( - 3 \beta_1 + 2) q^{91} + ( - \beta_{2} + 3 \beta_1 - 6) q^{93} + ( - 3 \beta_1 + 2) q^{97} + ( - 3 \beta_{2} - 6 \beta_1 - 4) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b1 + 1) * q^7 + (b2 + 3) * q^9 + (-b1 - 2) * q^11 + b2 * q^13 + (-b2 - 3) * q^17 + (b2 + 4) * q^19 + (-b2 + b1 - 6) * q^21 + q^23 + (b2 + 3*b1 - 2) * q^27 + (-b2 - b1 + 5) * q^29 + (-b1 + 3) * q^31 + (-b2 - 2*b1 - 6) * q^33 + (b2 + 3*b1 - 3) * q^37 + (b2 + 3*b1 - 2) * q^39 + (-b2 + 3) * q^41 + 8 * q^43 + 2*b2 * q^47 + (b2 - 2*b1) * q^49 + (-b2 - 6*b1 + 2) * q^51 + (-b2 - 3*b1 + 1) * q^53 + (b2 + 7*b1 - 2) * q^57 + (b2 + b1 + 5) * q^59 + (2*b2 - b1 + 4) * q^61 + (-6*b1 + 5) * q^63 + (-b2 - 3*b1 + 3) * q^67 + b1 * q^69 + (-b2 + 2*b1 + 7) * q^71 + (2*b2 + 4*b1 - 6) * q^73 + (b2 + b1 + 4) * q^77 + 4*b1 * q^79 + (b2 + b1 + 7) * q^81 + (b2 - b1 - 5) * q^83 + (-2*b2 + 2*b1 - 4) * q^87 + (4*b1 - 4) * q^89 + (-3*b1 + 2) * q^91 + (-b2 + 3*b1 - 6) * q^93 + (-3*b1 + 2) * q^97 + (-3*b2 - 6*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 2 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q + q^3 + 2 * q^7 + 10 * q^9 $$3 q + q^{3} + 2 q^{7} + 10 q^{9} - 7 q^{11} + q^{13} - 10 q^{17} + 13 q^{19} - 18 q^{21} + 3 q^{23} - 2 q^{27} + 13 q^{29} + 8 q^{31} - 21 q^{33} - 5 q^{37} - 2 q^{39} + 8 q^{41} + 24 q^{43} + 2 q^{47} - q^{49} - q^{51} - q^{53} + 2 q^{57} + 17 q^{59} + 13 q^{61} + 9 q^{63} + 5 q^{67} + q^{69} + 22 q^{71} - 12 q^{73} + 14 q^{77} + 4 q^{79} + 23 q^{81} - 15 q^{83} - 12 q^{87} - 8 q^{89} + 3 q^{91} - 16 q^{93} + 3 q^{97} - 21 q^{99}+O(q^{100})$$ 3 * q + q^3 + 2 * q^7 + 10 * q^9 - 7 * q^11 + q^13 - 10 * q^17 + 13 * q^19 - 18 * q^21 + 3 * q^23 - 2 * q^27 + 13 * q^29 + 8 * q^31 - 21 * q^33 - 5 * q^37 - 2 * q^39 + 8 * q^41 + 24 * q^43 + 2 * q^47 - q^49 - q^51 - q^53 + 2 * q^57 + 17 * q^59 + 13 * q^61 + 9 * q^63 + 5 * q^67 + q^69 + 22 * q^71 - 12 * q^73 + 14 * q^77 + 4 * q^79 + 23 * q^81 - 15 * q^83 - 12 * q^87 - 8 * q^89 + 3 * q^91 - 16 * q^93 + 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 0 0 3.95759 0 5.74732 0
1.2 0 0.878468 0 0 0 0.121532 0 −2.22829 0
1.3 0 3.07912 0 0 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 9200.2.a.ce 3
4.b odd 2 1 4600.2.a.x 3
5.b even 2 1 1840.2.a.s 3
20.d odd 2 1 920.2.a.h 3
20.e even 4 2 4600.2.e.p 6
40.e odd 2 1 7360.2.a.by 3
40.f even 2 1 7360.2.a.cc 3
60.h even 2 1 8280.2.a.bj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 20.d odd 2 1
1840.2.a.s 3 5.b even 2 1
4600.2.a.x 3 4.b odd 2 1
4600.2.e.p 6 20.e even 4 2
7360.2.a.by 3 40.e odd 2 1
7360.2.a.cc 3 40.f even 2 1
8280.2.a.bj 3 60.h even 2 1
9200.2.a.ce 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{3} - T_{3}^{2} - 9T_{3} + 8$$ T3^3 - T3^2 - 9*T3 + 8 $$T_{7}^{3} - 2T_{7}^{2} - 8T_{7} + 1$$ T7^3 - 2*T7^2 - 8*T7 + 1 $$T_{11}^{3} + 7T_{11}^{2} + 7T_{11} - 14$$ T11^3 + 7*T11^2 + 7*T11 - 14

## Hecke characteristic polynomials

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