# Properties

 Label 9702.2.a.ea 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: $$x^{4} - 6x^{2} - 4x + 2$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta_{3} + 1) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b3 + 1) * q^5 - q^8 $$q - q^{2} + q^{4} + (\beta_{3} + 1) q^{5} - q^{8} + ( - \beta_{3} - 1) q^{10} - q^{11} + (\beta_{3} - \beta_{2} - 2) q^{13} + q^{16} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{17} + ( - \beta_{2} - 3) q^{19} + (\beta_{3} + 1) q^{20} + q^{22} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + ( - \beta_{3} + \beta_{2} + 2) q^{26} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{29} + (3 \beta_{3} + \beta_1 - 1) q^{31} - q^{32} + (\beta_{3} + 2 \beta_{2} - 1) q^{34} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{37} + (\beta_{2} + 3) q^{38} + ( - \beta_{3} - 1) q^{40} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 + 3) q^{41} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{43} - q^{44} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{46} + ( - \beta_{2} - 2 \beta_1 + 1) q^{47} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{50} + (\beta_{3} - \beta_{2} - 2) q^{52} + (3 \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{53} + ( - \beta_{3} - 1) q^{55} + (2 \beta_{3} - 2 \beta_1 - 2) q^{58} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{59} + ( - \beta_{3} - 3 \beta_{2} - 6) q^{61} + ( - 3 \beta_{3} - \beta_1 + 1) q^{62} + q^{64} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{65} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{67} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{68} + (4 \beta_{3} - 2 \beta_{2} - 2) q^{71} + ( - \beta_{3} - 4 \beta_1 - 1) q^{73} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{74} + ( - \beta_{2} - 3) q^{76} + (\beta_{3} + 3 \beta_{2} - 3 \beta_1 - 2) q^{79} + (\beta_{3} + 1) q^{80} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 3) q^{82} + (\beta_{3} + 5 \beta_1 + 1) q^{83} + ( - 2 \beta_{2} - 8 \beta_1 - 2) q^{85} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{86} + q^{88} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 6) q^{89} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{92} + (\beta_{2} + 2 \beta_1 - 1) q^{94} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{95} + (4 \beta_{3} + 5 \beta_1) q^{97}+O(q^{100})$$ q - q^2 + q^4 + (b3 + 1) * q^5 - q^8 + (-b3 - 1) * q^10 - q^11 + (b3 - b2 - 2) * q^13 + q^16 + (-b3 - 2*b2 + 1) * q^17 + (-b2 - 3) * q^19 + (b3 + 1) * q^20 + q^22 + (-b3 - b2 + b1 + 2) * q^23 + (2*b3 + 2*b1 + 1) * q^25 + (-b3 + b2 + 2) * q^26 + (-2*b3 + 2*b1 + 2) * q^29 + (3*b3 + b1 - 1) * q^31 - q^32 + (b3 + 2*b2 - 1) * q^34 + (-b3 - b2 + 3*b1 + 2) * q^37 + (b2 + 3) * q^38 + (-b3 - 1) * q^40 + (b3 - 2*b2 + 4*b1 + 3) * q^41 + (-b3 - b2 - 3*b1 - 2) * q^43 - q^44 + (b3 + b2 - b1 - 2) * q^46 + (-b2 - 2*b1 + 1) * q^47 + (-2*b3 - 2*b1 - 1) * q^50 + (b3 - b2 - 2) * q^52 + (3*b3 + 3*b2 - b1 + 2) * q^53 + (-b3 - 1) * q^55 + (2*b3 - 2*b1 - 2) * q^58 + (2*b3 + 2*b2 - 4*b1) * q^59 + (-b3 - 3*b2 - 6) * q^61 + (-3*b3 - b1 + 1) * q^62 + q^64 + (-b3 - b2 - b1 + 4) * q^65 + (-b3 + b2 + 3*b1 - 2) * q^67 + (-b3 - 2*b2 + 1) * q^68 + (4*b3 - 2*b2 - 2) * q^71 + (-b3 - 4*b1 - 1) * q^73 + (b3 + b2 - 3*b1 - 2) * q^74 + (-b2 - 3) * q^76 + (b3 + 3*b2 - 3*b1 - 2) * q^79 + (b3 + 1) * q^80 + (-b3 + 2*b2 - 4*b1 - 3) * q^82 + (b3 + 5*b1 + 1) * q^83 + (-2*b2 - 8*b1 - 2) * q^85 + (b3 + b2 + 3*b1 + 2) * q^86 + q^88 + (-b3 + b2 - 4*b1 + 6) * q^89 + (-b3 - b2 + b1 + 2) * q^92 + (b2 + 2*b1 - 1) * q^94 + (-3*b3 - b2 - 3*b1 - 2) * q^95 + (4*b3 + 5*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8 $$4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} + 4 q^{16} + 4 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} + 8 q^{26} + 8 q^{29} - 4 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{37} + 12 q^{38} - 4 q^{40} + 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{47} - 4 q^{50} - 8 q^{52} + 8 q^{53} - 4 q^{55} - 8 q^{58} - 24 q^{61} + 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} + 4 q^{68} - 8 q^{71} - 4 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} + 24 q^{89} + 8 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu$$ v^3 - v^2 - 4*v $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 3$$ -v^2 + 2*v + 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} + 4\nu - 3$$ -v^3 + 2*v^2 + 4*v - 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b3 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 3$$ b3 + b1 + 3 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + 2\beta_{2} + 4\beta _1 + 3$$ 3*b3 + 2*b2 + 4*b1 + 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
 −1.27133 0.334904 −1.74912 2.68554
−1.00000 0 1.00000 −1.79793 0 0 −1.00000 0 1.79793
1.2 −1.00000 0 1.00000 −0.473626 0 0 −1.00000 0 0.473626
1.3 −1.00000 0 1.00000 2.47363 0 0 −1.00000 0 −2.47363
1.4 −1.00000 0 1.00000 3.79793 0 0 −1.00000 0 −3.79793
 $$n$$: 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.ea 4
3.b odd 2 1 3234.2.a.bl 4
7.b odd 2 1 9702.2.a.dz 4
21.c even 2 1 3234.2.a.bm yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bl 4 3.b odd 2 1
3234.2.a.bm yes 4 21.c even 2 1
9702.2.a.dz 4 7.b odd 2 1
9702.2.a.ea 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{4} - 4T_{5}^{3} - 4T_{5}^{2} + 16T_{5} + 8$$ T5^4 - 4*T5^3 - 4*T5^2 + 16*T5 + 8 $$T_{13}^{4} + 8T_{13}^{3} - 4T_{13}^{2} - 80T_{13} - 28$$ T13^4 + 8*T13^3 - 4*T13^2 - 80*T13 - 28 $$T_{17}^{4} - 4T_{17}^{3} - 52T_{17}^{2} + 112T_{17} + 776$$ T17^4 - 4*T17^3 - 52*T17^2 + 112*T17 + 776 $$T_{19}^{4} + 12T_{19}^{3} + 40T_{19}^{2} + 24T_{19} - 28$$ T19^4 + 12*T19^3 + 40*T19^2 + 24*T19 - 28 $$T_{23}^{4} - 8T_{23}^{3} + 32T_{23} + 16$$ T23^4 - 8*T23^3 + 32*T23 + 16 $$T_{29}^{4} - 8T_{29}^{3} - 32T_{29}^{2} + 64T_{29} + 64$$ T29^4 - 8*T29^3 - 32*T29^2 + 64*T29 + 64

## Hecke characteristic polynomials

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