# Properties

 Label 9702.2.a.cj 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: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1386) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 + \beta ) q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 + \beta ) q^{5} - q^{8} + ( 1 - \beta ) q^{10} - q^{11} + ( -4 + 2 \beta ) q^{13} + q^{16} + ( 5 - 2 \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} + ( -1 + \beta ) q^{20} + q^{22} + ( -1 + 3 \beta ) q^{23} + ( -2 - 2 \beta ) q^{25} + ( 4 - 2 \beta ) q^{26} -2 \beta q^{29} + ( 2 - 6 \beta ) q^{31} - q^{32} + ( -5 + 2 \beta ) q^{34} + ( 4 + 4 \beta ) q^{37} + ( 2 - 2 \beta ) q^{38} + ( 1 - \beta ) q^{40} + ( 1 - 4 \beta ) q^{41} -2 \beta q^{43} - q^{44} + ( 1 - 3 \beta ) q^{46} + ( 5 + 3 \beta ) q^{47} + ( 2 + 2 \beta ) q^{50} + ( -4 + 2 \beta ) q^{52} + ( 8 - 2 \beta ) q^{53} + ( 1 - \beta ) q^{55} + 2 \beta q^{58} + ( -2 + 4 \beta ) q^{59} + ( -3 + \beta ) q^{61} + ( -2 + 6 \beta ) q^{62} + q^{64} + ( 8 - 6 \beta ) q^{65} + ( -5 - 6 \beta ) q^{67} + ( 5 - 2 \beta ) q^{68} + ( -2 - 8 \beta ) q^{71} + ( -2 - 2 \beta ) q^{73} + ( -4 - 4 \beta ) q^{74} + ( -2 + 2 \beta ) q^{76} + ( 9 - 3 \beta ) q^{79} + ( -1 + \beta ) q^{80} + ( -1 + 4 \beta ) q^{82} + ( 7 + 2 \beta ) q^{83} + ( -9 + 7 \beta ) q^{85} + 2 \beta q^{86} + q^{88} + ( -4 - 6 \beta ) q^{89} + ( -1 + 3 \beta ) q^{92} + ( -5 - 3 \beta ) q^{94} + ( 6 - 4 \beta ) q^{95} + ( -13 + 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + 2q^{10} - 2q^{11} - 8q^{13} + 2q^{16} + 10q^{17} - 4q^{19} - 2q^{20} + 2q^{22} - 2q^{23} - 4q^{25} + 8q^{26} + 4q^{31} - 2q^{32} - 10q^{34} + 8q^{37} + 4q^{38} + 2q^{40} + 2q^{41} - 2q^{44} + 2q^{46} + 10q^{47} + 4q^{50} - 8q^{52} + 16q^{53} + 2q^{55} - 4q^{59} - 6q^{61} - 4q^{62} + 2q^{64} + 16q^{65} - 10q^{67} + 10q^{68} - 4q^{71} - 4q^{73} - 8q^{74} - 4q^{76} + 18q^{79} - 2q^{80} - 2q^{82} + 14q^{83} - 18q^{85} + 2q^{88} - 8q^{89} - 2q^{92} - 10q^{94} + 12q^{95} - 26q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
−1.00000 0 1.00000 −2.41421 0 0 −1.00000 0 2.41421
1.2 −1.00000 0 1.00000 0.414214 0 0 −1.00000 0 −0.414214
 $$n$$: 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.cj 2
3.b odd 2 1 9702.2.a.ds 2
7.b odd 2 1 9702.2.a.cv 2
7.c even 3 2 1386.2.k.u yes 4
21.c even 2 1 9702.2.a.da 2
21.h odd 6 2 1386.2.k.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.q 4 21.h odd 6 2
1386.2.k.u yes 4 7.c even 3 2
9702.2.a.cj 2 1.a even 1 1 trivial
9702.2.a.cv 2 7.b odd 2 1
9702.2.a.da 2 21.c even 2 1
9702.2.a.ds 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} + 2 T_{5} - 1$$ $$T_{13}^{2} + 8 T_{13} + 8$$ $$T_{17}^{2} - 10 T_{17} + 17$$ $$T_{19}^{2} + 4 T_{19} - 4$$ $$T_{23}^{2} + 2 T_{23} - 17$$ $$T_{29}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-1 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$8 + 8 T + T^{2}$$
$17$ $$17 - 10 T + T^{2}$$
$19$ $$-4 + 4 T + T^{2}$$
$23$ $$-17 + 2 T + T^{2}$$
$29$ $$-8 + T^{2}$$
$31$ $$-68 - 4 T + T^{2}$$
$37$ $$-16 - 8 T + T^{2}$$
$41$ $$-31 - 2 T + T^{2}$$
$43$ $$-8 + T^{2}$$
$47$ $$7 - 10 T + T^{2}$$
$53$ $$56 - 16 T + T^{2}$$
$59$ $$-28 + 4 T + T^{2}$$
$61$ $$7 + 6 T + T^{2}$$
$67$ $$-47 + 10 T + T^{2}$$
$71$ $$-124 + 4 T + T^{2}$$
$73$ $$-4 + 4 T + T^{2}$$
$79$ $$63 - 18 T + T^{2}$$
$83$ $$41 - 14 T + T^{2}$$
$89$ $$-56 + 8 T + T^{2}$$
$97$ $$161 + 26 T + T^{2}$$