# Properties

 Label 9200.2.a.ca Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( 2 - \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( 2 - \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} + ( 3 + \beta ) q^{11} + ( -2 + \beta ) q^{13} + ( -3 + 3 \beta ) q^{17} + ( -2 + 3 \beta ) q^{19} - q^{21} - q^{23} + ( 7 + 4 \beta ) q^{27} + 2 \beta q^{29} + ( 4 - 3 \beta ) q^{31} + ( 6 + 5 \beta ) q^{33} -8 q^{37} + q^{39} + ( -3 - 3 \beta ) q^{41} + ( -4 + 4 \beta ) q^{43} + 2 \beta q^{47} -3 \beta q^{49} + ( 6 + 3 \beta ) q^{51} + ( 6 - 4 \beta ) q^{53} + ( 7 + 4 \beta ) q^{57} + ( 6 + 2 \beta ) q^{59} + ( 5 - 5 \beta ) q^{61} + ( -7 + 2 \beta ) q^{63} -4 q^{67} + ( -1 - \beta ) q^{69} + ( 15 - \beta ) q^{71} + ( -2 - 6 \beta ) q^{73} + ( 3 - 2 \beta ) q^{77} + ( 4 - 8 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 6 - 4 \beta ) q^{83} + ( 6 + 4 \beta ) q^{87} + ( -7 + 3 \beta ) q^{91} + ( -5 - 2 \beta ) q^{93} + ( -5 + \beta ) q^{97} + ( 12 + 13 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 3q^{7} + 5q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 3q^{7} + 5q^{9} + 7q^{11} - 3q^{13} - 3q^{17} - q^{19} - 2q^{21} - 2q^{23} + 18q^{27} + 2q^{29} + 5q^{31} + 17q^{33} - 16q^{37} + 2q^{39} - 9q^{41} - 4q^{43} + 2q^{47} - 3q^{49} + 15q^{51} + 8q^{53} + 18q^{57} + 14q^{59} + 5q^{61} - 12q^{63} - 8q^{67} - 3q^{69} + 29q^{71} - 10q^{73} + 4q^{77} + 38q^{81} + 8q^{83} + 16q^{87} - 11q^{91} - 12q^{93} - 9q^{97} + 37q^{99} + O(q^{100})$$

## 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
 −1.30278 2.30278
0 −0.302776 0 0 0 3.30278 0 −2.90833 0
1.2 0 3.30278 0 0 0 −0.302776 0 7.90833 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.ca 2
4.b odd 2 1 1150.2.a.m 2
5.b even 2 1 1840.2.a.j 2
20.d odd 2 1 230.2.a.b 2
20.e even 4 2 1150.2.b.f 4
40.e odd 2 1 7360.2.a.bc 2
40.f even 2 1 7360.2.a.bu 2
60.h even 2 1 2070.2.a.w 2
460.g even 2 1 5290.2.a.j 2

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

 $$T_{3}^{2} - 3 T_{3} - 1$$ $$T_{7}^{2} - 3 T_{7} - 1$$ $$T_{11}^{2} - 7 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 - 3 T + T^{2}$$
$11$ $$9 - 7 T + T^{2}$$
$13$ $$-1 + 3 T + T^{2}$$
$17$ $$-27 + 3 T + T^{2}$$
$19$ $$-29 + T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-12 - 2 T + T^{2}$$
$31$ $$-23 - 5 T + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$-9 + 9 T + T^{2}$$
$43$ $$-48 + 4 T + T^{2}$$
$47$ $$-12 - 2 T + T^{2}$$
$53$ $$-36 - 8 T + T^{2}$$
$59$ $$36 - 14 T + T^{2}$$
$61$ $$-75 - 5 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$207 - 29 T + T^{2}$$
$73$ $$-92 + 10 T + T^{2}$$
$79$ $$-208 + T^{2}$$
$83$ $$-36 - 8 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$17 + 9 T + T^{2}$$