# Properties

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

# 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 3 \beta q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 3*b * q^5 - q^8 $$q - q^{2} + q^{4} + 3 \beta q^{5} - q^{8} - 3 \beta q^{10} - q^{11} - 4 \beta q^{13} + q^{16} - 5 \beta q^{17} - \beta q^{19} + 3 \beta q^{20} + q^{22} + 8 q^{23} + 13 q^{25} + 4 \beta q^{26} + 8 q^{29} - 3 \beta q^{31} - q^{32} + 5 \beta q^{34} + 2 q^{37} + \beta q^{38} - 3 \beta q^{40} + \beta q^{41} + 8 q^{43} - q^{44} - 8 q^{46} + 7 \beta q^{47} - 13 q^{50} - 4 \beta q^{52} + 2 q^{53} - 3 \beta q^{55} - 8 q^{58} - 6 \beta q^{59} + 3 \beta q^{62} + q^{64} - 24 q^{65} - 2 q^{67} - 5 \beta q^{68} - 12 q^{71} - 5 \beta q^{73} - 2 q^{74} - \beta q^{76} + 14 q^{79} + 3 \beta q^{80} - \beta q^{82} + 9 \beta q^{83} - 30 q^{85} - 8 q^{86} + q^{88} + 2 \beta q^{89} + 8 q^{92} - 7 \beta q^{94} - 6 q^{95} +O(q^{100})$$ q - q^2 + q^4 + 3*b * q^5 - q^8 - 3*b * q^10 - q^11 - 4*b * q^13 + q^16 - 5*b * q^17 - b * q^19 + 3*b * q^20 + q^22 + 8 * q^23 + 13 * q^25 + 4*b * q^26 + 8 * q^29 - 3*b * q^31 - q^32 + 5*b * q^34 + 2 * q^37 + b * q^38 - 3*b * q^40 + b * q^41 + 8 * q^43 - q^44 - 8 * q^46 + 7*b * q^47 - 13 * q^50 - 4*b * q^52 + 2 * q^53 - 3*b * q^55 - 8 * q^58 - 6*b * q^59 + 3*b * q^62 + q^64 - 24 * q^65 - 2 * q^67 - 5*b * q^68 - 12 * q^71 - 5*b * q^73 - 2 * q^74 - b * q^76 + 14 * q^79 + 3*b * q^80 - b * q^82 + 9*b * q^83 - 30 * q^85 - 8 * q^86 + q^88 + 2*b * q^89 + 8 * q^92 - 7*b * q^94 - 6 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{11} + 2 q^{16} + 2 q^{22} + 16 q^{23} + 26 q^{25} + 16 q^{29} - 2 q^{32} + 4 q^{37} + 16 q^{43} - 2 q^{44} - 16 q^{46} - 26 q^{50} + 4 q^{53} - 16 q^{58} + 2 q^{64} - 48 q^{65} - 4 q^{67} - 24 q^{71} - 4 q^{74} + 28 q^{79} - 60 q^{85} - 16 q^{86} + 2 q^{88} + 16 q^{92} - 12 q^{95}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^11 + 2 * q^16 + 2 * q^22 + 16 * q^23 + 26 * q^25 + 16 * q^29 - 2 * q^32 + 4 * q^37 + 16 * q^43 - 2 * q^44 - 16 * q^46 - 26 * q^50 + 4 * q^53 - 16 * q^58 + 2 * q^64 - 48 * q^65 - 4 * q^67 - 24 * q^71 - 4 * q^74 + 28 * q^79 - 60 * q^85 - 16 * q^86 + 2 * q^88 + 16 * q^92 - 12 * 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
 −1.41421 1.41421
−1.00000 0 1.00000 −4.24264 0 0 −1.00000 0 4.24264
1.2 −1.00000 0 1.00000 4.24264 0 0 −1.00000 0 −4.24264
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9702.2.a.cm 2
3.b odd 2 1 9702.2.a.do yes 2
7.b odd 2 1 inner 9702.2.a.cm 2
21.c even 2 1 9702.2.a.do yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9702.2.a.cm 2 1.a even 1 1 trivial
9702.2.a.cm 2 7.b odd 2 1 inner
9702.2.a.do yes 2 3.b odd 2 1
9702.2.a.do yes 2 21.c even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 18$$ T5^2 - 18 $$T_{13}^{2} - 32$$ T13^2 - 32 $$T_{17}^{2} - 50$$ T17^2 - 50 $$T_{19}^{2} - 2$$ T19^2 - 2 $$T_{23} - 8$$ T23 - 8 $$T_{29} - 8$$ T29 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 18$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 32$$
$17$ $$T^{2} - 50$$
$19$ $$T^{2} - 2$$
$23$ $$(T - 8)^{2}$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} - 18$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 2$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} - 98$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 72$$
$61$ $$T^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 50$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} - 162$$
$89$ $$T^{2} - 8$$
$97$ $$T^{2}$$