# Properties

 Label 9200.2.a.cq Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.5744.1 Defining polynomial: $$x^{4} - 5x^{2} - 2x + 1$$ x^4 - 5*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 2) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + (b2 + 2) * q^7 + (b3 - b2 - b1 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 2) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_1 - 3) q^{13} + ( - \beta_{3} + \beta_{2} - 3) q^{17} + (\beta_{3} + 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{21} - q^{23} + (\beta_{3} - 3 \beta_{2} + 2) q^{27} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{29} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{31} + (\beta_{2} + 2 \beta_1 - 4) q^{33} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{37} + (\beta_{3} - \beta_{2} + 3 \beta_1) q^{39} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{41} + ( - \beta_{3} + 4 \beta_1 - 1) q^{43} + ( - 3 \beta_{3} + 3 \beta_{2} + 2) q^{47} + (4 \beta_{2} - 2 \beta_1 + 3) q^{49} + (2 \beta_{2} + 5 \beta_1 - 4) q^{51} + (2 \beta_{3} + 2 \beta_{2}) q^{53} + ( - \beta_{3} - 2 \beta_1 + 1) q^{57} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{59} + ( - 3 \beta_{3} + 2 \beta_1 - 3) q^{61} + (2 \beta_1 - 4) q^{63} + \beta_{2} q^{67} + (\beta_1 - 1) q^{69} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{71} + ( - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{73} + ( - 2 \beta_{2} - 2) q^{77} + (\beta_{3} - 2 \beta_{2} - 7) q^{79} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{81} + (\beta_{3} + 3 \beta_{2} - 6 \beta_1 + 3) q^{83} + (\beta_{3} - \beta_{2} - 2) q^{87} + (3 \beta_{3} - 2 \beta_{2} - 3) q^{89} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 7) q^{91} + (\beta_{3} - 5 \beta_{2} - 2 \beta_1 - 2) q^{93} + (3 \beta_{2} - 2 \beta_1 - 6) q^{97} + (4 \beta_{2} + 2 \beta_1 - 8) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + (b2 + 2) * q^7 + (b3 - b2 - b1 + 1) * q^9 + (-b3 + b1 - 1) * q^11 + (-b1 - 3) * q^13 + (-b3 + b2 - 3) * q^17 + (b3 + 1) * q^19 + (-b3 + 2*b2 - b1 + 1) * q^21 - q^23 + (b3 - 3*b2 + 2) * q^27 + (-b3 - b2 + b1) * q^29 + (b3 - 3*b2 + b1 - 2) * q^31 + (b2 + 2*b1 - 4) * q^33 + (2*b3 - b2 + 2*b1 - 2) * q^37 + (b3 - b2 + 3*b1) * q^39 + (-2*b2 + 4*b1 - 5) * q^41 + (-b3 + 4*b1 - 1) * q^43 + (-3*b3 + 3*b2 + 2) * q^47 + (4*b2 - 2*b1 + 3) * q^49 + (2*b2 + 5*b1 - 4) * q^51 + (2*b3 + 2*b2) * q^53 + (-b3 - 2*b1 + 1) * q^57 + (b3 - b2 + b1 - 1) * q^59 + (-3*b3 + 2*b1 - 3) * q^61 + (2*b1 - 4) * q^63 + b2 * q^67 + (b1 - 1) * q^69 + (2*b3 - 4*b2 + 2*b1 + 3) * q^71 + (-3*b3 - b2 + 2*b1 - 6) * q^73 + (-2*b2 - 2) * q^77 + (b3 - 2*b2 - 7) * q^79 + (-b3 - 3*b2 - 3*b1 + 2) * q^81 + (b3 + 3*b2 - 6*b1 + 3) * q^83 + (b3 - b2 - 2) * q^87 + (3*b3 - 2*b2 - 3) * q^89 + (-b3 - 2*b2 - b1 - 7) * q^91 + (b3 - 5*b2 - 2*b1 - 2) * q^93 + (3*b2 - 2*b1 - 6) * q^97 + (4*b2 + 2*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 6 * q^7 + 4 * q^9 $$4 q + 2 q^{3} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 14 q^{13} - 14 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} + 14 q^{27} + 4 q^{29} - 14 q^{33} - 2 q^{37} + 8 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} - 10 q^{51} - 4 q^{53} - 8 q^{61} - 12 q^{63} - 2 q^{67} - 2 q^{69} + 24 q^{71} - 18 q^{73} - 4 q^{77} - 24 q^{79} + 8 q^{81} - 6 q^{83} - 6 q^{87} - 8 q^{89} - 26 q^{91} - 2 q^{93} - 34 q^{97} - 36 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 14 * q^13 - 14 * q^17 + 4 * q^19 - 2 * q^21 - 4 * q^23 + 14 * q^27 + 4 * q^29 - 14 * q^33 - 2 * q^37 + 8 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 - 10 * q^51 - 4 * q^53 - 8 * q^61 - 12 * q^63 - 2 * q^67 - 2 * q^69 + 24 * q^71 - 18 * q^73 - 4 * q^77 - 24 * q^79 + 8 * q^81 - 6 * q^83 - 6 * q^87 - 8 * q^89 - 26 * q^91 - 2 * q^93 - 34 * q^97 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} - 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5\nu - 1$$ -v^3 + v^2 + 5*v - 1
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2$$ (b3 - b2 + b1 - 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 2$$ (b3 + b2 + b1 + 5) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{3} - 2\beta_{2} + 3\beta _1 - 1$$ 2*b3 - 2*b2 + 3*b1 - 1

## 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
 2.37988 −0.751024 −1.92022 0.291367
0 −1.95969 0 0 0 2.28394 0 0.840379 0
1.2 0 −0.580491 0 0 0 0.315061 0 −2.66303 0
1.3 0 1.39945 0 0 0 4.60747 0 −1.04155 0
1.4 0 3.14073 0 0 0 −1.20647 0 6.86420 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.cq 4
4.b odd 2 1 575.2.a.i 4
5.b even 2 1 9200.2.a.ck 4
5.c odd 4 2 1840.2.e.d 8
12.b even 2 1 5175.2.a.bv 4
20.d odd 2 1 575.2.a.j 4
20.e even 4 2 115.2.b.b 8
60.h even 2 1 5175.2.a.bw 4
60.l odd 4 2 1035.2.b.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 20.e even 4 2
575.2.a.i 4 4.b odd 2 1
575.2.a.j 4 20.d odd 2 1
1035.2.b.e 8 60.l odd 4 2
1840.2.e.d 8 5.c odd 4 2
5175.2.a.bv 4 12.b even 2 1
5175.2.a.bw 4 60.h even 2 1
9200.2.a.ck 4 5.b even 2 1
9200.2.a.cq 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 6T_{3} + 5$$ T3^4 - 2*T3^3 - 6*T3^2 + 6*T3 + 5 $$T_{7}^{4} - 6T_{7}^{3} + 4T_{7}^{2} + 12T_{7} - 4$$ T7^4 - 6*T7^3 + 4*T7^2 + 12*T7 - 4 $$T_{11}^{4} + 2T_{11}^{3} - 16T_{11}^{2} - 44T_{11} - 28$$ T11^4 + 2*T11^3 - 16*T11^2 - 44*T11 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} - 6 T^{2} + 6 T + 5$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 6 T^{3} + 4 T^{2} + 12 T - 4$$
$11$ $$T^{4} + 2 T^{3} - 16 T^{2} - 44 T - 28$$
$13$ $$T^{4} + 14 T^{3} + 66 T^{2} + 118 T + 61$$
$17$ $$T^{4} + 14 T^{3} + 58 T^{2} + 64 T + 20$$
$19$ $$T^{4} - 4 T^{3} - 6 T^{2} + 28 T - 20$$
$23$ $$(T + 1)^{4}$$
$29$ $$T^{4} - 4 T^{3} - 22 T^{2} + 4 T + 5$$
$31$ $$T^{4} - 74 T^{2} + 256 T - 167$$
$37$ $$T^{4} + 2 T^{3} - 72 T^{2} - 380 T - 476$$
$41$ $$T^{4} + 8 T^{3} - 94 T^{2} + \cdots + 2485$$
$43$ $$T^{4} - 4 T^{3} - 118 T^{2} + \cdots + 1964$$
$47$ $$T^{4} - 2 T^{3} - 138 T^{2} + \cdots + 4513$$
$53$ $$T^{4} + 4 T^{3} - 104 T^{2} + \cdots + 2192$$
$59$ $$T^{4} - 20 T^{2} - 16 T + 16$$
$61$ $$T^{4} + 8 T^{3} - 102 T^{2} + \cdots - 2756$$
$67$ $$T^{4} + 2 T^{3} - 8 T^{2} - 12 T + 4$$
$71$ $$T^{4} - 24 T^{3} + 66 T^{2} + \cdots - 7435$$
$73$ $$T^{4} + 18 T^{3} - 22 T^{2} + \cdots - 8339$$
$79$ $$T^{4} + 24 T^{3} + 178 T^{2} + \cdots + 28$$
$83$ $$T^{4} + 6 T^{3} - 270 T^{2} + \cdots + 14948$$
$89$ $$T^{4} + 8 T^{3} - 86 T^{2} + \cdots + 2380$$
$97$ $$T^{4} + 34 T^{3} + 348 T^{2} + \cdots - 4676$$