# Properties

 Label 9702.2.a.ck Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9702.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$77.4708600410$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3234) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( - \beta - 1) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (-b - 1) * q^5 - q^8 $$q - q^{2} + q^{4} + ( - \beta - 1) q^{5} - q^{8} + (\beta + 1) q^{10} + q^{11} + q^{16} + (\beta - 3) q^{17} + ( - 3 \beta + 1) q^{19} + ( - \beta - 1) q^{20} - q^{22} + ( - 2 \beta - 2) q^{23} + (2 \beta + 1) q^{25} + 2 \beta q^{29} + (\beta + 5) q^{31} - q^{32} + ( - \beta + 3) q^{34} + (4 \beta - 2) q^{37} + (3 \beta - 1) q^{38} + (\beta + 1) q^{40} + (\beta - 3) q^{41} + ( - 2 \beta + 2) q^{43} + q^{44} + (2 \beta + 2) q^{46} + ( - 3 \beta - 3) q^{47} + ( - 2 \beta - 1) q^{50} + 6 q^{53} + ( - \beta - 1) q^{55} - 2 \beta q^{58} + ( - 2 \beta - 6) q^{59} + ( - \beta - 5) q^{62} + q^{64} + ( - 6 \beta + 2) q^{67} + (\beta - 3) q^{68} + ( - 2 \beta - 2) q^{71} + ( - 3 \beta + 9) q^{73} + ( - 4 \beta + 2) q^{74} + ( - 3 \beta + 1) q^{76} + ( - 6 \beta - 2) q^{79} + ( - \beta - 1) q^{80} + ( - \beta + 3) q^{82} + ( - \beta - 1) q^{83} + (2 \beta - 2) q^{85} + (2 \beta - 2) q^{86} - q^{88} + (2 \beta - 2) q^{89} + ( - 2 \beta - 2) q^{92} + (3 \beta + 3) q^{94} + (2 \beta + 14) q^{95} +O(q^{100})$$ q - q^2 + q^4 + (-b - 1) * q^5 - q^8 + (b + 1) * q^10 + q^11 + q^16 + (b - 3) * q^17 + (-3*b + 1) * q^19 + (-b - 1) * q^20 - q^22 + (-2*b - 2) * q^23 + (2*b + 1) * q^25 + 2*b * q^29 + (b + 5) * q^31 - q^32 + (-b + 3) * q^34 + (4*b - 2) * q^37 + (3*b - 1) * q^38 + (b + 1) * q^40 + (b - 3) * q^41 + (-2*b + 2) * q^43 + q^44 + (2*b + 2) * q^46 + (-3*b - 3) * q^47 + (-2*b - 1) * q^50 + 6 * q^53 + (-b - 1) * q^55 - 2*b * q^58 + (-2*b - 6) * q^59 + (-b - 5) * q^62 + q^64 + (-6*b + 2) * q^67 + (b - 3) * q^68 + (-2*b - 2) * q^71 + (-3*b + 9) * q^73 + (-4*b + 2) * q^74 + (-3*b + 1) * q^76 + (-6*b - 2) * q^79 + (-b - 1) * q^80 + (-b + 3) * q^82 + (-b - 1) * q^83 + (2*b - 2) * q^85 + (2*b - 2) * q^86 - q^88 + (2*b - 2) * q^89 + (-2*b - 2) * q^92 + (3*b + 3) * q^94 + (2*b + 14) * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} + 2 q^{11} + 2 q^{16} - 6 q^{17} + 2 q^{19} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} + 10 q^{31} - 2 q^{32} + 6 q^{34} - 4 q^{37} - 2 q^{38} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 2 q^{44} + 4 q^{46} - 6 q^{47} - 2 q^{50} + 12 q^{53} - 2 q^{55} - 12 q^{59} - 10 q^{62} + 2 q^{64} + 4 q^{67} - 6 q^{68} - 4 q^{71} + 18 q^{73} + 4 q^{74} + 2 q^{76} - 4 q^{79} - 2 q^{80} + 6 q^{82} - 2 q^{83} - 4 q^{85} - 4 q^{86} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 6 q^{94} + 28 q^{95}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 + 2 * q^11 + 2 * q^16 - 6 * q^17 + 2 * q^19 - 2 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^25 + 10 * q^31 - 2 * q^32 + 6 * q^34 - 4 * q^37 - 2 * q^38 + 2 * q^40 - 6 * q^41 + 4 * q^43 + 2 * q^44 + 4 * q^46 - 6 * q^47 - 2 * q^50 + 12 * q^53 - 2 * q^55 - 12 * q^59 - 10 * q^62 + 2 * q^64 + 4 * q^67 - 6 * q^68 - 4 * q^71 + 18 * q^73 + 4 * q^74 + 2 * q^76 - 4 * q^79 - 2 * q^80 + 6 * q^82 - 2 * q^83 - 4 * q^85 - 4 * q^86 - 2 * q^88 - 4 * q^89 - 4 * q^92 + 6 * q^94 + 28 * q^95

## 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
 1.61803 −0.618034
−1.00000 0 1.00000 −3.23607 0 0 −1.00000 0 3.23607
1.2 −1.00000 0 1.00000 1.23607 0 0 −1.00000 0 −1.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 9702.2.a.ck 2
3.b odd 2 1 3234.2.a.be yes 2
7.b odd 2 1 9702.2.a.cw 2
21.c even 2 1 3234.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bb 2 21.c even 2 1
3234.2.a.be yes 2 3.b odd 2 1
9702.2.a.ck 2 1.a even 1 1 trivial
9702.2.a.cw 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} + 2T_{5} - 4$$ T5^2 + 2*T5 - 4 $$T_{13}$$ T13 $$T_{17}^{2} + 6T_{17} + 4$$ T17^2 + 6*T17 + 4 $$T_{19}^{2} - 2T_{19} - 44$$ T19^2 - 2*T19 - 44 $$T_{23}^{2} + 4T_{23} - 16$$ T23^2 + 4*T23 - 16 $$T_{29}^{2} - 20$$ T29^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T - 4$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 6T + 4$$
$19$ $$T^{2} - 2T - 44$$
$23$ $$T^{2} + 4T - 16$$
$29$ $$T^{2} - 20$$
$31$ $$T^{2} - 10T + 20$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} + 6T + 4$$
$43$ $$T^{2} - 4T - 16$$
$47$ $$T^{2} + 6T - 36$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12T + 16$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 4T - 176$$
$71$ $$T^{2} + 4T - 16$$
$73$ $$T^{2} - 18T + 36$$
$79$ $$T^{2} + 4T - 176$$
$83$ $$T^{2} + 2T - 4$$
$89$ $$T^{2} + 4T - 16$$
$97$ $$T^{2}$$