# Properties

 Label 9200.2.a.ct Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.521397.1 Defining polynomial: $$x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + \beta_{4} q^{11} + ( 1 + \beta_{2} - \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{17} + \beta_{4} q^{19} + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + q^{23} + ( -2 + \beta_{1} - \beta_{3} ) q^{27} + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{29} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{31} + ( 1 - \beta_{3} + \beta_{4} ) q^{33} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{37} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{39} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{4} ) q^{43} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{47} + ( 3 + 2 \beta_{1} - \beta_{4} ) q^{49} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{53} + ( 1 - \beta_{3} + \beta_{4} ) q^{57} + ( -\beta_{3} - 2 \beta_{4} ) q^{59} + ( 1 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{61} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{63} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{67} -\beta_{1} q^{69} + ( -4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( -1 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{73} + ( 1 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{77} + ( -1 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{79} + ( -6 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{81} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} ) q^{83} + ( -9 - \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{87} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 8 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{91} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{93} + ( 1 - \beta_{3} + 3 \beta_{4} ) q^{97} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 4 q^{7} + 3 q^{9} + O(q^{10})$$ $$5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57} + q^{59} + 10 q^{61} - 8 q^{67} - 8 q^{71} + 6 q^{73} - 27 q^{81} + 2 q^{83} - 39 q^{87} + 14 q^{89} + 46 q^{91} - 3 q^{93} + 6 q^{97} + 6 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu - 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 7 \nu^{2} + 4 \nu + 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 21$$

## 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
 2.61696 1.83957 −0.794805 −1.36629 −2.29544
0 −2.61696 0 0 0 −3.83744 0 3.84849 0
1.2 0 −1.83957 0 0 0 3.97272 0 0.384010 0
1.3 0 0.794805 0 0 0 −2.47193 0 −2.36829 0
1.4 0 1.36629 0 0 0 −3.28093 0 −1.13327 0
1.5 0 2.29544 0 0 0 1.61758 0 2.26905 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.ct 5
4.b odd 2 1 4600.2.a.bf yes 5
5.b even 2 1 9200.2.a.cv 5
20.d odd 2 1 4600.2.a.bd 5
20.e even 4 2 4600.2.e.w 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bd 5 20.d odd 2 1
4600.2.a.bf yes 5 4.b odd 2 1
4600.2.e.w 10 20.e even 4 2
9200.2.a.ct 5 1.a even 1 1 trivial
9200.2.a.cv 5 5.b even 2 1

## Hecke kernels

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

 $$T_{3}^{5} - 9 T_{3}^{3} + 3 T_{3}^{2} + 18 T_{3} - 12$$ $$T_{7}^{5} + 4 T_{7}^{4} - 17 T_{7}^{3} - 76 T_{7}^{2} + 20 T_{7} + 200$$ $$T_{11}^{5} - 15 T_{11}^{3} + 6 T_{11}^{2} + 48 T_{11} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$-12 + 18 T + 3 T^{2} - 9 T^{3} + T^{5}$$
$5$ $$T^{5}$$
$7$ $$200 + 20 T - 76 T^{2} - 17 T^{3} + 4 T^{4} + T^{5}$$
$11$ $$-24 + 48 T + 6 T^{2} - 15 T^{3} + T^{5}$$
$13$ $$-347 + 287 T + 91 T^{2} - 32 T^{3} - 4 T^{4} + T^{5}$$
$17$ $$384 - 672 T + 360 T^{2} - 48 T^{3} - 6 T^{4} + T^{5}$$
$19$ $$-24 + 48 T + 6 T^{2} - 15 T^{3} + T^{5}$$
$23$ $$( -1 + T )^{5}$$
$29$ $$-2787 - 273 T + 339 T^{2} - 12 T^{4} + T^{5}$$
$31$ $$228 - 516 T - 45 T^{2} + 93 T^{3} - 18 T^{4} + T^{5}$$
$37$ $$-608 - 928 T + 496 T^{2} - 32 T^{3} - 10 T^{4} + T^{5}$$
$41$ $$453 + 1311 T - 237 T^{2} - 66 T^{3} + 6 T^{4} + T^{5}$$
$43$ $$1472 - 112 T - 352 T^{2} - 23 T^{3} + 10 T^{4} + T^{5}$$
$47$ $$-4 - 334 T - 301 T^{2} + 97 T^{3} + 22 T^{4} + T^{5}$$
$53$ $$256 + 704 T + 400 T^{2} - 92 T^{3} - 10 T^{4} + T^{5}$$
$59$ $$3004 + 1322 T - 107 T^{2} - 83 T^{3} - T^{4} + T^{5}$$
$61$ $$7648 + 12800 T + 1072 T^{2} - 200 T^{3} - 10 T^{4} + T^{5}$$
$67$ $$-9728 - 8704 T - 2408 T^{2} - 188 T^{3} + 8 T^{4} + T^{5}$$
$71$ $$-8084 + 5960 T - 689 T^{2} - 179 T^{3} + 8 T^{4} + T^{5}$$
$73$ $$28569 + 10407 T + 39 T^{2} - 222 T^{3} - 6 T^{4} + T^{5}$$
$79$ $$1728 + 5184 T + 36 T^{2} - 153 T^{3} + T^{5}$$
$83$ $$-2392 + 524 T + 1112 T^{2} - 191 T^{3} - 2 T^{4} + T^{5}$$
$89$ $$3104 - 4864 T + 1280 T^{2} - 44 T^{3} - 14 T^{4} + T^{5}$$
$97$ $$96 + 576 T + 816 T^{2} - 144 T^{3} - 6 T^{4} + T^{5}$$