# Properties

 Label 9702.2.a.ci Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $1$ 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: $$1$$ 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: 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2 q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 2 * q^5 - q^8 $$q - q^{2} + q^{4} - 2 q^{5} - q^{8} + 2 q^{10} + q^{11} + (\beta + 4) q^{13} + q^{16} - 2 q^{17} + ( - 3 \beta + 2) q^{19} - 2 q^{20} - q^{22} + (2 \beta + 2) q^{23} - q^{25} + ( - \beta - 4) q^{26} + ( - 4 \beta - 4) q^{29} + (3 \beta + 2) q^{31} - q^{32} + 2 q^{34} + (2 \beta - 2) q^{37} + (3 \beta - 2) q^{38} + 2 q^{40} + ( - 4 \beta - 6) q^{41} + ( - 2 \beta - 2) q^{43} + q^{44} + ( - 2 \beta - 2) q^{46} + (7 \beta - 2) q^{47} + q^{50} + (\beta + 4) q^{52} + (6 \beta + 2) q^{53} - 2 q^{55} + (4 \beta + 4) q^{58} + ( - 2 \beta - 4) q^{59} + ( - \beta + 4) q^{61} + ( - 3 \beta - 2) q^{62} + q^{64} + ( - 2 \beta - 8) q^{65} + (2 \beta - 4) q^{67} - 2 q^{68} - 4 \beta q^{71} + (4 \beta + 6) q^{73} + ( - 2 \beta + 2) q^{74} + ( - 3 \beta + 2) q^{76} + ( - 6 \beta - 8) q^{79} - 2 q^{80} + (4 \beta + 6) q^{82} + ( - 5 \beta + 2) q^{83} + 4 q^{85} + (2 \beta + 2) q^{86} - q^{88} + (3 \beta - 8) q^{89} + (2 \beta + 2) q^{92} + ( - 7 \beta + 2) q^{94} + (6 \beta - 4) q^{95} - 7 \beta q^{97} +O(q^{100})$$ q - q^2 + q^4 - 2 * q^5 - q^8 + 2 * q^10 + q^11 + (b + 4) * q^13 + q^16 - 2 * q^17 + (-3*b + 2) * q^19 - 2 * q^20 - q^22 + (2*b + 2) * q^23 - q^25 + (-b - 4) * q^26 + (-4*b - 4) * q^29 + (3*b + 2) * q^31 - q^32 + 2 * q^34 + (2*b - 2) * q^37 + (3*b - 2) * q^38 + 2 * q^40 + (-4*b - 6) * q^41 + (-2*b - 2) * q^43 + q^44 + (-2*b - 2) * q^46 + (7*b - 2) * q^47 + q^50 + (b + 4) * q^52 + (6*b + 2) * q^53 - 2 * q^55 + (4*b + 4) * q^58 + (-2*b - 4) * q^59 + (-b + 4) * q^61 + (-3*b - 2) * q^62 + q^64 + (-2*b - 8) * q^65 + (2*b - 4) * q^67 - 2 * q^68 - 4*b * q^71 + (4*b + 6) * q^73 + (-2*b + 2) * q^74 + (-3*b + 2) * q^76 + (-6*b - 8) * q^79 - 2 * q^80 + (4*b + 6) * q^82 + (-5*b + 2) * q^83 + 4 * q^85 + (2*b + 2) * q^86 - q^88 + (3*b - 8) * q^89 + (2*b + 2) * q^92 + (-7*b + 2) * q^94 + (6*b - 4) * q^95 - 7*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 8 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} - 8 q^{26} - 8 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{37} - 4 q^{38} + 4 q^{40} - 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} - 4 q^{47} + 2 q^{50} + 8 q^{52} + 4 q^{53} - 4 q^{55} + 8 q^{58} - 8 q^{59} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} - 4 q^{68} + 12 q^{73} + 4 q^{74} + 4 q^{76} - 16 q^{79} - 4 q^{80} + 12 q^{82} + 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} - 16 q^{89} + 4 q^{92} + 4 q^{94} - 8 q^{95}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 + 2 * q^11 + 8 * q^13 + 2 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^25 - 8 * q^26 - 8 * q^29 + 4 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^37 - 4 * q^38 + 4 * q^40 - 12 * q^41 - 4 * q^43 + 2 * q^44 - 4 * q^46 - 4 * q^47 + 2 * q^50 + 8 * q^52 + 4 * q^53 - 4 * q^55 + 8 * q^58 - 8 * q^59 + 8 * q^61 - 4 * q^62 + 2 * q^64 - 16 * q^65 - 8 * q^67 - 4 * q^68 + 12 * q^73 + 4 * q^74 + 4 * q^76 - 16 * q^79 - 4 * q^80 + 12 * q^82 + 4 * q^83 + 8 * q^85 + 4 * q^86 - 2 * q^88 - 16 * q^89 + 4 * q^92 + 4 * q^94 - 8 * 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 −2.00000 0 0 −1.00000 0 2.00000
1.2 −1.00000 0 1.00000 −2.00000 0 0 −1.00000 0 2.00000
 $$n$$: 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.ci 2
3.b odd 2 1 3234.2.a.bc 2
7.b odd 2 1 9702.2.a.cy 2
21.c even 2 1 3234.2.a.bd yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bc 2 3.b odd 2 1
3234.2.a.bd yes 2 21.c even 2 1
9702.2.a.ci 2 1.a even 1 1 trivial
9702.2.a.cy 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{13}^{2} - 8T_{13} + 14$$ T13^2 - 8*T13 + 14 $$T_{17} + 2$$ T17 + 2 $$T_{19}^{2} - 4T_{19} - 14$$ T19^2 - 4*T19 - 14 $$T_{23}^{2} - 4T_{23} - 4$$ T23^2 - 4*T23 - 4 $$T_{29}^{2} + 8T_{29} - 16$$ T29^2 + 8*T29 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 8T + 14$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 4T - 14$$
$23$ $$T^{2} - 4T - 4$$
$29$ $$T^{2} + 8T - 16$$
$31$ $$T^{2} - 4T - 14$$
$37$ $$T^{2} + 4T - 4$$
$41$ $$T^{2} + 12T + 4$$
$43$ $$T^{2} + 4T - 4$$
$47$ $$T^{2} + 4T - 94$$
$53$ $$T^{2} - 4T - 68$$
$59$ $$T^{2} + 8T + 8$$
$61$ $$T^{2} - 8T + 14$$
$67$ $$T^{2} + 8T + 8$$
$71$ $$T^{2} - 32$$
$73$ $$T^{2} - 12T + 4$$
$79$ $$T^{2} + 16T - 8$$
$83$ $$T^{2} - 4T - 46$$
$89$ $$T^{2} + 16T + 46$$
$97$ $$T^{2} - 98$$