# Properties

 Label 9702.2.a.cz 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: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.64575 2.64575
1.00000 0 1.00000 −3.64575 0 0 1.00000 0 −3.64575
1.2 1.00000 0 1.00000 1.64575 0 0 1.00000 0 1.64575
 $$n$$: 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.cz 2
3.b odd 2 1 1078.2.a.s 2
7.b odd 2 1 9702.2.a.dr 2
7.c even 3 2 1386.2.k.s 4
12.b even 2 1 8624.2.a.ca 2
21.c even 2 1 1078.2.a.n 2
21.g even 6 2 1078.2.e.v 4
21.h odd 6 2 154.2.e.f 4
84.h odd 2 1 8624.2.a.bk 2
84.n even 6 2 1232.2.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 21.h odd 6 2
1078.2.a.n 2 21.c even 2 1
1078.2.a.s 2 3.b odd 2 1
1078.2.e.v 4 21.g even 6 2
1232.2.q.g 4 84.n even 6 2
1386.2.k.s 4 7.c even 3 2
8624.2.a.bk 2 84.h odd 2 1
8624.2.a.ca 2 12.b even 2 1
9702.2.a.cz 2 1.a even 1 1 trivial
9702.2.a.dr 2 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} + 2 T_{5} - 6$$ $$T_{13} - 5$$ $$T_{17} + 6$$ $$T_{19}^{2} + 6 T_{19} + 2$$ $$T_{23}^{2} - 2 T_{23} - 6$$ $$T_{29}^{2} + 2 T_{29} - 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-6 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$2 + 6 T + T^{2}$$
$23$ $$-6 - 2 T + T^{2}$$
$29$ $$-27 + 2 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$-6 - 2 T + T^{2}$$
$41$ $$-54 + 6 T + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$36 + 16 T + T^{2}$$
$53$ $$-6 - 2 T + T^{2}$$
$59$ $$-3 + 4 T + T^{2}$$
$61$ $$53 - 18 T + T^{2}$$
$67$ $$-47 + 8 T + T^{2}$$
$71$ $$42 + 14 T + T^{2}$$
$73$ $$2 - 6 T + T^{2}$$
$79$ $$-7 + T^{2}$$
$83$ $$36 + 16 T + T^{2}$$
$89$ $$-96 - 8 T + T^{2}$$
$97$ $$93 + 22 T + T^{2}$$