# Properties

 Label 9702.2.a.dl 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

## 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

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.dl 2
3.b odd 2 1 3234.2.a.z yes 2
7.b odd 2 1 9702.2.a.de 2
21.c even 2 1 3234.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.y 2 21.c even 2 1
3234.2.a.z yes 2 3.b odd 2 1
9702.2.a.de 2 7.b odd 2 1
9702.2.a.dl 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5}$$ $$T_{13}^{2} - 2$$ $$T_{17} - 4$$ $$T_{19}^{2} + 8 T_{19} + 14$$ $$T_{23}^{2} + 4 T_{23} - 28$$ $$T_{29}^{2} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-2 + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$14 + 8 T + T^{2}$$
$23$ $$-28 + 4 T + T^{2}$$
$29$ $$-32 + T^{2}$$
$31$ $$-34 + 8 T + T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$-16 - 8 T + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$-2 + T^{2}$$
$53$ $$-28 + 4 T + T^{2}$$
$59$ $$8 - 8 T + T^{2}$$
$61$ $$-2 + T^{2}$$
$67$ $$-128 + T^{2}$$
$71$ $$32 + 16 T + T^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$-16 + 8 T + T^{2}$$
$83$ $$14 + 16 T + T^{2}$$
$89$ $$14 - 8 T + T^{2}$$
$97$ $$-162 + T^{2}$$