# Properties

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

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## 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1078) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + q^{8} + q^{11} -\beta q^{13} + q^{16} -\beta q^{17} + 2 \beta q^{19} + q^{22} -8 q^{23} -5 q^{25} -\beta q^{26} -2 q^{29} + 3 \beta q^{31} + q^{32} -\beta q^{34} + 2 q^{37} + 2 \beta q^{38} -\beta q^{41} -4 q^{43} + q^{44} -8 q^{46} -\beta q^{47} -5 q^{50} -\beta q^{52} -14 q^{53} -2 q^{58} + 3 \beta q^{59} -3 \beta q^{61} + 3 \beta q^{62} + q^{64} + 4 q^{67} -\beta q^{68} -5 \beta q^{73} + 2 q^{74} + 2 \beta q^{76} -16 q^{79} -\beta q^{82} + 6 \beta q^{83} -4 q^{86} + q^{88} + 4 \beta q^{89} -8 q^{92} -\beta q^{94} -6 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} - 16 q^{23} - 10 q^{25} - 4 q^{29} + 2 q^{32} + 4 q^{37} - 8 q^{43} + 2 q^{44} - 16 q^{46} - 10 q^{50} - 28 q^{53} - 4 q^{58} + 2 q^{64} + 8 q^{67} + 4 q^{74} - 32 q^{79} - 8 q^{86} + 2 q^{88} - 16 q^{92} + O(q^{100})$$

## 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 0 0 0 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 0 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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.dn 2
3.b odd 2 1 1078.2.a.r 2
7.b odd 2 1 inner 9702.2.a.dn 2
12.b even 2 1 8624.2.a.by 2
21.c even 2 1 1078.2.a.r 2
21.g even 6 2 1078.2.e.r 4
21.h odd 6 2 1078.2.e.r 4
84.h odd 2 1 8624.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.r 2 3.b odd 2 1
1078.2.a.r 2 21.c even 2 1
1078.2.e.r 4 21.g even 6 2
1078.2.e.r 4 21.h odd 6 2
8624.2.a.by 2 12.b even 2 1
8624.2.a.by 2 84.h odd 2 1
9702.2.a.dn 2 1.a even 1 1 trivial
9702.2.a.dn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}$$ $$T_{13}^{2} - 8$$ $$T_{17}^{2} - 8$$ $$T_{19}^{2} - 32$$ $$T_{23} + 8$$ $$T_{29} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-8 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( 14 + T )^{2}$$
$59$ $$-72 + T^{2}$$
$61$ $$-72 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$-200 + T^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$-288 + T^{2}$$
$89$ $$-128 + T^{2}$$
$97$ $$-288 + T^{2}$$
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