# Properties

 Label 9200.2.a.cu Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.13955077.1 Defining polynomial: $$x^{5} - 14 x^{3} - x^{2} + 32 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 + \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 + \beta_{2} + \beta_{4} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + ( -1 - \beta_{2} + \beta_{4} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{21} - q^{23} + ( 1 + 3 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{27} + ( 1 - \beta_{2} + \beta_{3} ) q^{29} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{31} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{33} + ( -3 + \beta_{3} ) q^{37} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{39} + ( 5 - \beta_{1} + \beta_{3} ) q^{41} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{47} + ( 6 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{49} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{51} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + ( -9 - 3 \beta_{1} - \beta_{2} - \beta_{4} ) q^{57} + ( 1 + \beta_{3} - 2 \beta_{4} ) q^{59} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{61} + ( -8 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{63} + ( 1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{67} -\beta_{1} q^{69} + ( -1 + 3 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{71} + ( \beta_{2} + 2 \beta_{3} ) q^{73} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{77} + 2 \beta_{2} q^{79} + ( 10 - 3 \beta_{1} + 5 \beta_{2} + \beta_{4} ) q^{81} + ( -9 + \beta_{3} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{87} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{89} + ( -4 + 5 \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{91} + ( 7 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{93} + ( -6 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{97} + ( 11 - 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{7} + 13 q^{9} + O(q^{10})$$ $$5 q - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 4 q^{17} - 7 q^{19} + 6 q^{21} - 5 q^{23} + 3 q^{27} + 4 q^{29} - 19 q^{31} - 17 q^{33} - 15 q^{37} - 19 q^{39} + 25 q^{41} - 11 q^{47} + 25 q^{49} - 19 q^{51} - 3 q^{53} - 48 q^{57} + q^{59} - 5 q^{61} - 41 q^{63} + 9 q^{67} - q^{71} + q^{73} - 18 q^{77} + 2 q^{79} + 57 q^{81} - 45 q^{83} - 9 q^{87} + 6 q^{89} - 11 q^{91} + 39 q^{93} - 25 q^{97} + 65 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 14 x^{3} - x^{2} + 32 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 10 \nu^{2} + 3 \nu + 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} + 4 \nu^{3} + 14 \nu^{2} - 39 \nu - 28$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{4} + 14 \nu^{2} - 3 \nu - 24$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + 2 \beta_{3} + 9 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$10 \beta_{4} + 14 \beta_{2} - 3 \beta_{1} + 46$$

## 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
 −3.36002 −1.31091 −0.568386 1.93283 3.30649
0 −3.36002 0 0 0 −1.90754 0 8.28974 0
1.2 0 −1.31091 0 0 0 −4.66212 0 −1.28151 0
1.3 0 −0.568386 0 0 0 4.73770 0 −2.67694 0
1.4 0 1.93283 0 0 0 2.38236 0 0.735829 0
1.5 0 3.30649 0 0 0 −2.55040 0 7.93288 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.cu 5
4.b odd 2 1 4600.2.a.be 5
5.b even 2 1 1840.2.a.v 5
20.d odd 2 1 920.2.a.j 5
20.e even 4 2 4600.2.e.u 10
40.e odd 2 1 7360.2.a.co 5
40.f even 2 1 7360.2.a.cp 5
60.h even 2 1 8280.2.a.bs 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 20.d odd 2 1
1840.2.a.v 5 5.b even 2 1
4600.2.a.be 5 4.b odd 2 1
4600.2.e.u 10 20.e even 4 2
7360.2.a.co 5 40.e odd 2 1
7360.2.a.cp 5 40.f even 2 1
8280.2.a.bs 5 60.h even 2 1
9200.2.a.cu 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{5} - 14 T_{3}^{3} - T_{3}^{2} + 32 T_{3} + 16$$ $$T_{7}^{5} + 2 T_{7}^{4} - 28 T_{7}^{3} - 57 T_{7}^{2} + 128 T_{7} + 256$$ $$T_{11}^{5} - T_{11}^{4} - 35 T_{11}^{3} + 28 T_{11}^{2} + 172 T_{11} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$16 + 32 T - T^{2} - 14 T^{3} + T^{5}$$
$5$ $$T^{5}$$
$7$ $$256 + 128 T - 57 T^{2} - 28 T^{3} + 2 T^{4} + T^{5}$$
$11$ $$-64 + 172 T + 28 T^{2} - 35 T^{3} - T^{4} + T^{5}$$
$13$ $$-500 + 600 T - 75 T^{2} - 46 T^{3} + 4 T^{4} + T^{5}$$
$17$ $$-32 - 138 T - 157 T^{2} - 46 T^{3} + 4 T^{4} + T^{5}$$
$19$ $$-512 + 676 T - 180 T^{2} - 41 T^{3} + 7 T^{4} + T^{5}$$
$23$ $$( 1 + T )^{5}$$
$29$ $$8 + 64 T - 36 T^{2} - 41 T^{3} - 4 T^{4} + T^{5}$$
$31$ $$128 - 53 T - 183 T^{2} + 72 T^{3} + 19 T^{4} + T^{5}$$
$37$ $$-64 - 176 T + 36 T^{2} + 64 T^{3} + 15 T^{4} + T^{5}$$
$41$ $$2182 + 27 T - 653 T^{2} + 212 T^{3} - 25 T^{4} + T^{5}$$
$43$ $$T^{5}$$
$47$ $$512 - 800 T - 1116 T^{2} - 110 T^{3} + 11 T^{4} + T^{5}$$
$53$ $$20272 + 5808 T - 440 T^{2} - 160 T^{3} + 3 T^{4} + T^{5}$$
$59$ $$-13568 + 4560 T + 236 T^{2} - 142 T^{3} - T^{4} + T^{5}$$
$61$ $$7664 + 2092 T - 568 T^{2} - 115 T^{3} + 5 T^{4} + T^{5}$$
$67$ $$8192 + 4832 T + 432 T^{2} - 126 T^{3} - 9 T^{4} + T^{5}$$
$71$ $$3968 + 8139 T - 407 T^{2} - 214 T^{3} + T^{4} + T^{5}$$
$73$ $$1328 + 2072 T + 272 T^{2} - 158 T^{3} - T^{4} + T^{5}$$
$79$ $$1024 + 2432 T - 128 T^{2} - 128 T^{3} - 2 T^{4} + T^{5}$$
$83$ $$41216 + 26608 T + 6588 T^{2} + 784 T^{3} + 45 T^{4} + T^{5}$$
$89$ $$-8192 + 2560 T + 512 T^{2} - 136 T^{3} - 6 T^{4} + T^{5}$$
$97$ $$49616 - 16908 T - 4164 T^{2} - 39 T^{3} + 25 T^{4} + T^{5}$$