# Properties

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

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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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1386) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 3q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + 3q^{5} - q^{8} - 3q^{10} + q^{11} - 2q^{13} + q^{16} - 3q^{17} - 2q^{19} + 3q^{20} - q^{22} - 3q^{23} + 4q^{25} + 2q^{26} - 2q^{31} - q^{32} + 3q^{34} + 8q^{37} + 2q^{38} - 3q^{40} - 9q^{41} - 4q^{43} + q^{44} + 3q^{46} + 3q^{47} - 4q^{50} - 2q^{52} - 6q^{53} + 3q^{55} + 6q^{59} - 5q^{61} + 2q^{62} + q^{64} - 6q^{65} + 11q^{67} - 3q^{68} - 2q^{73} - 8q^{74} - 2q^{76} - 13q^{79} + 3q^{80} + 9q^{82} + 9q^{83} - 9q^{85} + 4q^{86} - q^{88} + 12q^{89} - 3q^{92} - 3q^{94} - 6q^{95} - 5q^{97} + O(q^{100})$$

## 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
 0
−1.00000 0 1.00000 3.00000 0 0 −1.00000 0 −3.00000
 $$n$$: 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.z 1
3.b odd 2 1 9702.2.a.bc 1
7.b odd 2 1 9702.2.a.c 1
7.d odd 6 2 1386.2.k.p yes 2
21.c even 2 1 9702.2.a.cd 1
21.g even 6 2 1386.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.b 2 21.g even 6 2
1386.2.k.p yes 2 7.d odd 6 2
9702.2.a.c 1 7.b odd 2 1
9702.2.a.z 1 1.a even 1 1 trivial
9702.2.a.bc 1 3.b odd 2 1
9702.2.a.cd 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5} - 3$$ $$T_{13} + 2$$ $$T_{17} + 3$$ $$T_{19} + 2$$ $$T_{23} + 3$$ $$T_{29}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$-3 + T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$2 + T$$
$17$ $$3 + T$$
$19$ $$2 + T$$
$23$ $$3 + T$$
$29$ $$T$$
$31$ $$2 + T$$
$37$ $$-8 + T$$
$41$ $$9 + T$$
$43$ $$4 + T$$
$47$ $$-3 + T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$5 + T$$
$67$ $$-11 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$13 + T$$
$83$ $$-9 + T$$
$89$ $$-12 + T$$
$97$ $$5 + T$$
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