# Properties

 Label 9200.2.a.bs Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + ( 1 - \beta ) q^{7} + ( 2 + \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{3} + ( 1 - \beta ) q^{7} + ( 2 + \beta ) q^{9} + ( -2 + \beta ) q^{11} + ( -3 - \beta ) q^{13} + ( 2 - \beta ) q^{17} + ( -3 - \beta ) q^{19} + 5 q^{21} + q^{23} -5 q^{27} + ( 2 + 2 \beta ) q^{29} + ( -5 + 3 \beta ) q^{31} + ( -5 + \beta ) q^{33} + 4 q^{37} + ( 5 + 4 \beta ) q^{39} + ( -4 - \beta ) q^{41} + 4 \beta q^{43} + ( -10 + 2 \beta ) q^{47} + ( -1 - \beta ) q^{49} + ( 5 - \beta ) q^{51} -6 q^{53} + ( 5 + 4 \beta ) q^{57} + ( 8 + 2 \beta ) q^{59} + ( 2 + 3 \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + 4 \beta q^{67} -\beta q^{69} -3 \beta q^{71} + ( 4 - 6 \beta ) q^{73} + ( -7 + 2 \beta ) q^{77} -8 q^{79} + ( -6 + 2 \beta ) q^{81} -6 q^{83} + ( -10 - 4 \beta ) q^{87} + ( 4 + 4 \beta ) q^{89} + ( 2 + 3 \beta ) q^{91} + ( -15 + 2 \beta ) q^{93} + ( -6 + 5 \beta ) q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + q^{7} + 5q^{9} + O(q^{10})$$ $$2q - q^{3} + q^{7} + 5q^{9} - 3q^{11} - 7q^{13} + 3q^{17} - 7q^{19} + 10q^{21} + 2q^{23} - 10q^{27} + 6q^{29} - 7q^{31} - 9q^{33} + 8q^{37} + 14q^{39} - 9q^{41} + 4q^{43} - 18q^{47} - 3q^{49} + 9q^{51} - 12q^{53} + 14q^{57} + 18q^{59} + 7q^{61} - 8q^{63} + 4q^{67} - q^{69} - 3q^{71} + 2q^{73} - 12q^{77} - 16q^{79} - 10q^{81} - 12q^{83} - 24q^{87} + 12q^{89} + 7q^{91} - 28q^{93} - 7q^{97} + 3q^{99} + O(q^{100})$$

## 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
 2.79129 −1.79129
0 −2.79129 0 0 0 −1.79129 0 4.79129 0
1.2 0 1.79129 0 0 0 2.79129 0 0.208712 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.bs 2
4.b odd 2 1 1150.2.a.o 2
5.b even 2 1 1840.2.a.n 2
20.d odd 2 1 230.2.a.a 2
20.e even 4 2 1150.2.b.g 4
40.e odd 2 1 7360.2.a.bq 2
40.f even 2 1 7360.2.a.bk 2
60.h even 2 1 2070.2.a.x 2
460.g even 2 1 5290.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 20.d odd 2 1
1150.2.a.o 2 4.b odd 2 1
1150.2.b.g 4 20.e even 4 2
1840.2.a.n 2 5.b even 2 1
2070.2.a.x 2 60.h even 2 1
5290.2.a.e 2 460.g even 2 1
7360.2.a.bk 2 40.f even 2 1
7360.2.a.bq 2 40.e odd 2 1
9200.2.a.bs 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{2} + T_{3} - 5$$ $$T_{7}^{2} - T_{7} - 5$$ $$T_{11}^{2} + 3 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-5 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-5 - T + T^{2}$$
$11$ $$-3 + 3 T + T^{2}$$
$13$ $$7 + 7 T + T^{2}$$
$17$ $$-3 - 3 T + T^{2}$$
$19$ $$7 + 7 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-12 - 6 T + T^{2}$$
$31$ $$-35 + 7 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$15 + 9 T + T^{2}$$
$43$ $$-80 - 4 T + T^{2}$$
$47$ $$60 + 18 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$60 - 18 T + T^{2}$$
$61$ $$-35 - 7 T + T^{2}$$
$67$ $$-80 - 4 T + T^{2}$$
$71$ $$-45 + 3 T + T^{2}$$
$73$ $$-188 - 2 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$-48 - 12 T + T^{2}$$
$97$ $$-119 + 7 T + T^{2}$$