# Properties

 Label 9702.2.a.dg Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta q^{5} + q^{8} + \beta q^{10} - q^{11} -4 \beta q^{13} + q^{16} -\beta q^{17} + 3 \beta q^{19} + \beta q^{20} - q^{22} -4 q^{23} -3 q^{25} -4 \beta q^{26} + 3 \beta q^{31} + q^{32} -\beta q^{34} -6 q^{37} + 3 \beta q^{38} + \beta q^{40} -3 \beta q^{41} -4 q^{43} - q^{44} -4 q^{46} + \beta q^{47} -3 q^{50} -4 \beta q^{52} + 6 q^{53} -\beta q^{55} + 2 \beta q^{59} + 4 \beta q^{61} + 3 \beta q^{62} + q^{64} -8 q^{65} -10 q^{67} -\beta q^{68} + 3 \beta q^{73} -6 q^{74} + 3 \beta q^{76} -6 q^{79} + \beta q^{80} -3 \beta q^{82} -11 \beta q^{83} -2 q^{85} -4 q^{86} - q^{88} -10 \beta q^{89} -4 q^{92} + \beta q^{94} + 6 q^{95} + 4 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} - 8 q^{23} - 6 q^{25} + 2 q^{32} - 12 q^{37} - 8 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{50} + 12 q^{53} + 2 q^{64} - 16 q^{65} - 20 q^{67} - 12 q^{74} - 12 q^{79} - 4 q^{85} - 8 q^{86} - 2 q^{88} - 8 q^{92} + 12 q^{95} + O(q^{100})$$

## 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
 −1.41421 1.41421
1.00000 0 1.00000 −1.41421 0 0 1.00000 0 −1.41421
1.2 1.00000 0 1.00000 1.41421 0 0 1.00000 0 1.41421
 $$n$$: 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.dg yes 2
3.b odd 2 1 9702.2.a.ct 2
7.b odd 2 1 inner 9702.2.a.dg yes 2
21.c even 2 1 9702.2.a.ct 2

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

## Hecke kernels

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

 $$T_{5}^{2} - 2$$ $$T_{13}^{2} - 32$$ $$T_{17}^{2} - 2$$ $$T_{19}^{2} - 18$$ $$T_{23} + 4$$ $$T_{29}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-18 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$-18 + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$-18 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-2 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-8 + T^{2}$$
$61$ $$-32 + T^{2}$$
$67$ $$( 10 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$-18 + T^{2}$$
$79$ $$( 6 + T )^{2}$$
$83$ $$-242 + T^{2}$$
$89$ $$-200 + T^{2}$$
$97$ $$-32 + T^{2}$$