# Properties

 Label 9200.2.a.cf 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: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} ) q^{13} + ( 2 + \beta_{1} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} ) q^{19} + ( -9 + 2 \beta_{1} - 3 \beta_{2} ) q^{21} - q^{23} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{27} + ( -2 + 2 \beta_{1} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} ) q^{31} + ( 3 - 3 \beta_{1} - 3 \beta_{2} ) q^{33} -2 \beta_{2} q^{37} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} ) q^{41} + 8 q^{43} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 11 - \beta_{1} + 2 \beta_{2} ) q^{49} + ( 7 + \beta_{1} + \beta_{2} ) q^{51} + 6 q^{53} + ( 9 - 2 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -4 - 2 \beta_{1} ) q^{59} + ( 2 - \beta_{1} + 4 \beta_{2} ) q^{61} + ( 5 - 8 \beta_{1} - \beta_{2} ) q^{63} + ( 4 + 4 \beta_{2} ) q^{67} -\beta_{1} q^{69} + ( -4 + \beta_{1} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 5 + 8 \beta_{1} - \beta_{2} ) q^{77} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{79} + ( 4 - 4 \beta_{1} + \beta_{2} ) q^{81} + ( 2 - 2 \beta_{2} ) q^{83} + ( 14 - 4 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 4 + 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 - 5 \beta_{1} + 2 \beta_{2} ) q^{91} + ( 5 - \beta_{2} ) q^{93} + ( 10 + 3 \beta_{1} ) q^{97} + ( -21 + 3 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} + 3q^{7} + 10q^{9} + O(q^{10})$$ $$3q + q^{3} + 3q^{7} + 10q^{9} - 3q^{11} + q^{13} + 7q^{17} - 3q^{19} - 22q^{21} - 3q^{23} - 14q^{27} - 4q^{29} + 5q^{31} + 9q^{33} + 2q^{37} - 14q^{39} + q^{41} + 24q^{43} - 14q^{47} + 30q^{49} + 21q^{51} + 18q^{53} + 22q^{57} - 14q^{59} + q^{61} + 8q^{63} + 8q^{67} - q^{69} - 11q^{71} + 8q^{73} + 24q^{77} + 4q^{79} + 7q^{81} + 8q^{83} + 36q^{87} + 18q^{89} - q^{91} + 16q^{93} + 33q^{97} - 57q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 7$$

## 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
 −3.11903 1.43163 2.68740
0 −3.11903 0 0 0 4.50973 0 6.72833 0
1.2 0 1.43163 0 0 0 3.08719 0 −0.950444 0
1.3 0 2.68740 0 0 0 −4.59692 0 4.22212 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.cf 3
4.b odd 2 1 1150.2.a.q 3
5.b even 2 1 1840.2.a.r 3
20.d odd 2 1 230.2.a.d 3
20.e even 4 2 1150.2.b.j 6
40.e odd 2 1 7360.2.a.bz 3
40.f even 2 1 7360.2.a.ce 3
60.h even 2 1 2070.2.a.z 3
460.g even 2 1 5290.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 20.d odd 2 1
1150.2.a.q 3 4.b odd 2 1
1150.2.b.j 6 20.e even 4 2
1840.2.a.r 3 5.b even 2 1
2070.2.a.z 3 60.h even 2 1
5290.2.a.r 3 460.g even 2 1
7360.2.a.bz 3 40.e odd 2 1
7360.2.a.ce 3 40.f even 2 1
9200.2.a.cf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{3} - T_{3}^{2} - 9 T_{3} + 12$$ $$T_{7}^{3} - 3 T_{7}^{2} - 21 T_{7} + 64$$ $$T_{11}^{3} + 3 T_{11}^{2} - 39 T_{11} - 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$12 - 9 T - T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$64 - 21 T - 3 T^{2} + T^{3}$$
$11$ $$-144 - 39 T + 3 T^{2} + T^{3}$$
$13$ $$18 - 15 T - T^{2} + T^{3}$$
$17$ $$18 + 7 T - 7 T^{2} + T^{3}$$
$19$ $$-64 - 21 T + 3 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$24 - 32 T + 4 T^{2} + T^{3}$$
$31$ $$8 - 7 T - 5 T^{2} + T^{3}$$
$37$ $$32 - 40 T - 2 T^{2} + T^{3}$$
$41$ $$186 - 59 T - T^{2} + T^{3}$$
$43$ $$( -8 + T )^{3}$$
$47$ $$-288 + 4 T + 14 T^{2} + T^{3}$$
$53$ $$( -6 + T )^{3}$$
$59$ $$-144 + 28 T + 14 T^{2} + T^{3}$$
$61$ $$526 - 157 T - T^{2} + T^{3}$$
$67$ $$384 - 144 T - 8 T^{2} + T^{3}$$
$71$ $$24 + 31 T + 11 T^{2} + T^{3}$$
$73$ $$248 - 40 T - 8 T^{2} + T^{3}$$
$79$ $$1152 - 240 T - 4 T^{2} + T^{3}$$
$83$ $$96 - 20 T - 8 T^{2} + T^{3}$$
$89$ $$1152 - 48 T - 18 T^{2} + T^{3}$$
$97$ $$-166 + 279 T - 33 T^{2} + T^{3}$$