# Properties

 Label 9702.2.a.dz Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 - \beta_{3} ) q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 - \beta_{3} ) q^{5} - q^{8} + ( 1 + \beta_{3} ) q^{10} - q^{11} + ( 2 + \beta_{2} - \beta_{3} ) q^{13} + q^{16} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{17} + ( 3 + \beta_{2} ) q^{19} + ( -1 - \beta_{3} ) q^{20} + q^{22} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( 1 + 2 \beta_{1} + 2 \beta_{3} ) q^{25} + ( -2 - \beta_{2} + \beta_{3} ) q^{26} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( 1 - \beta_{1} - 3 \beta_{3} ) q^{31} - q^{32} + ( 1 - 2 \beta_{2} - \beta_{3} ) q^{34} + ( 2 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( -3 - \beta_{2} ) q^{38} + ( 1 + \beta_{3} ) q^{40} + ( -3 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{41} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} - q^{44} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{46} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{47} + ( -1 - 2 \beta_{1} - 2 \beta_{3} ) q^{50} + ( 2 + \beta_{2} - \beta_{3} ) q^{52} + ( 2 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{53} + ( 1 + \beta_{3} ) q^{55} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 6 + 3 \beta_{2} + \beta_{3} ) q^{61} + ( -1 + \beta_{1} + 3 \beta_{3} ) q^{62} + q^{64} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{65} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{68} + ( -2 - 2 \beta_{2} + 4 \beta_{3} ) q^{71} + ( 1 + 4 \beta_{1} + \beta_{3} ) q^{73} + ( -2 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{74} + ( 3 + \beta_{2} ) q^{76} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{79} + ( -1 - \beta_{3} ) q^{80} + ( 3 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{82} + ( -1 - 5 \beta_{1} - \beta_{3} ) q^{83} + ( -2 - 8 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{86} + q^{88} + ( -6 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{92} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{94} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{95} + ( -5 \beta_{1} - 4 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + O(q^{10})$$ $$4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{16} - 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 8 q^{26} + 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{34} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} - 4 q^{47} - 4 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{55} - 8 q^{58} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} - 4 q^{68} - 8 q^{71} + 4 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} - 24 q^{89} + 8 q^{92} + 4 q^{94} - 8 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} + 4 \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 3$$

## 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
 2.68554 −1.74912 0.334904 −1.27133
−1.00000 0 1.00000 −3.79793 0 0 −1.00000 0 3.79793
1.2 −1.00000 0 1.00000 −2.47363 0 0 −1.00000 0 2.47363
1.3 −1.00000 0 1.00000 0.473626 0 0 −1.00000 0 −0.473626
1.4 −1.00000 0 1.00000 1.79793 0 0 −1.00000 0 −1.79793
 $$n$$: 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.dz 4
3.b odd 2 1 3234.2.a.bm yes 4
7.b odd 2 1 9702.2.a.ea 4
21.c even 2 1 3234.2.a.bl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bl 4 21.c even 2 1
3234.2.a.bm yes 4 3.b odd 2 1
9702.2.a.dz 4 1.a even 1 1 trivial
9702.2.a.ea 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{4} + 4 T_{5}^{3} - 4 T_{5}^{2} - 16 T_{5} + 8$$ $$T_{13}^{4} - 8 T_{13}^{3} - 4 T_{13}^{2} + 80 T_{13} - 28$$ $$T_{17}^{4} + 4 T_{17}^{3} - 52 T_{17}^{2} - 112 T_{17} + 776$$ $$T_{19}^{4} - 12 T_{19}^{3} + 40 T_{19}^{2} - 24 T_{19} - 28$$ $$T_{23}^{4} - 8 T_{23}^{3} + 32 T_{23} + 16$$ $$T_{29}^{4} - 8 T_{29}^{3} - 32 T_{29}^{2} + 64 T_{29} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$8 - 16 T - 4 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$-28 + 80 T - 4 T^{2} - 8 T^{3} + T^{4}$$
$17$ $$776 - 112 T - 52 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$-28 - 24 T + 40 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$16 + 32 T - 8 T^{3} + T^{4}$$
$29$ $$64 + 64 T - 32 T^{2} - 8 T^{3} + T^{4}$$
$31$ $$964 + 328 T - 88 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$16 + 96 T - 32 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$-4984 - 1552 T - 84 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$16 - 96 T - 32 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$4 + 8 T - 24 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$3856 + 992 T - 160 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$-448 - 512 T - 144 T^{2} + T^{4}$$
$61$ $$164 + 624 T + 92 T^{2} - 24 T^{3} + T^{4}$$
$67$ $$32 - 32 T - 40 T^{2} + 8 T^{3} + T^{4}$$
$71$ $$12352 - 960 T - 224 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$712 + 80 T - 68 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$32 - 992 T - 136 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$1988 - 136 T - 104 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$-284 + 16 T + 124 T^{2} + 24 T^{3} + T^{4}$$
$97$ $$-1148 + 1280 T - 260 T^{2} + T^{4}$$