# Properties

 Label 9702.2.a.dm Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + b * q^5 + q^8 $$q + q^{2} + q^{4} + \beta q^{5} + q^{8} + \beta q^{10} - q^{11} + 4 q^{13} + q^{16} - 3 q^{17} - 2 \beta q^{19} + \beta q^{20} - q^{22} + \beta q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{29} + 4 q^{31} + q^{32} - 3 q^{34} + (2 \beta - 4) q^{37} - 2 \beta q^{38} + \beta q^{40} + 9 q^{41} + (2 \beta + 4) q^{43} - q^{44} + \beta q^{46} + ( - 3 \beta + 4) q^{47} + 2 q^{50} + 4 q^{52} - 4 q^{53} - \beta q^{55} - 2 q^{58} + (4 \beta - 4) q^{59} + ( - 3 \beta - 4) q^{61} + 4 q^{62} + q^{64} + 4 \beta q^{65} + (4 \beta - 3) q^{67} - 3 q^{68} + ( - 2 \beta + 8) q^{71} + (2 \beta + 10) q^{73} + (2 \beta - 4) q^{74} - 2 \beta q^{76} + (\beta + 8) q^{79} + \beta q^{80} + 9 q^{82} + (4 \beta + 5) q^{83} - 3 \beta q^{85} + (2 \beta + 4) q^{86} - q^{88} + ( - 2 \beta + 8) q^{89} + \beta q^{92} + ( - 3 \beta + 4) q^{94} - 14 q^{95} + (4 \beta + 1) q^{97} +O(q^{100})$$ q + q^2 + q^4 + b * q^5 + q^8 + b * q^10 - q^11 + 4 * q^13 + q^16 - 3 * q^17 - 2*b * q^19 + b * q^20 - q^22 + b * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^29 + 4 * q^31 + q^32 - 3 * q^34 + (2*b - 4) * q^37 - 2*b * q^38 + b * q^40 + 9 * q^41 + (2*b + 4) * q^43 - q^44 + b * q^46 + (-3*b + 4) * q^47 + 2 * q^50 + 4 * q^52 - 4 * q^53 - b * q^55 - 2 * q^58 + (4*b - 4) * q^59 + (-3*b - 4) * q^61 + 4 * q^62 + q^64 + 4*b * q^65 + (4*b - 3) * q^67 - 3 * q^68 + (-2*b + 8) * q^71 + (2*b + 10) * q^73 + (2*b - 4) * q^74 - 2*b * q^76 + (b + 8) * q^79 + b * q^80 + 9 * q^82 + (4*b + 5) * q^83 - 3*b * q^85 + (2*b + 4) * q^86 - q^88 + (-2*b + 8) * q^89 + b * q^92 + (-3*b + 4) * q^94 - 14 * q^95 + (4*b + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{22} + 4 q^{25} + 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{34} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 2 q^{44} + 8 q^{47} + 4 q^{50} + 8 q^{52} - 8 q^{53} - 4 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{67} - 6 q^{68} + 16 q^{71} + 20 q^{73} - 8 q^{74} + 16 q^{79} + 18 q^{82} + 10 q^{83} + 8 q^{86} - 2 q^{88} + 16 q^{89} + 8 q^{94} - 28 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 8 * q^13 + 2 * q^16 - 6 * q^17 - 2 * q^22 + 4 * q^25 + 8 * q^26 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^34 - 8 * q^37 + 18 * q^41 + 8 * q^43 - 2 * q^44 + 8 * q^47 + 4 * q^50 + 8 * q^52 - 8 * q^53 - 4 * q^58 - 8 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 - 6 * q^67 - 6 * q^68 + 16 * q^71 + 20 * q^73 - 8 * q^74 + 16 * q^79 + 18 * q^82 + 10 * q^83 + 8 * q^86 - 2 * q^88 + 16 * q^89 + 8 * q^94 - 28 * q^95 + 2 * q^97

## 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 −2.64575 0 0 1.00000 0 −2.64575
1.2 1.00000 0 1.00000 2.64575 0 0 1.00000 0 2.64575
 $$n$$: 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.dm 2
3.b odd 2 1 3234.2.a.ba 2
7.b odd 2 1 9702.2.a.db 2
7.d odd 6 2 1386.2.k.r 4
21.c even 2 1 3234.2.a.w 2
21.g even 6 2 462.2.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 21.g even 6 2
1386.2.k.r 4 7.d odd 6 2
3234.2.a.w 2 21.c even 2 1
3234.2.a.ba 2 3.b odd 2 1
9702.2.a.db 2 7.b odd 2 1
9702.2.a.dm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} - 7$$ T5^2 - 7 $$T_{13} - 4$$ T13 - 4 $$T_{17} + 3$$ T17 + 3 $$T_{19}^{2} - 28$$ T19^2 - 28 $$T_{23}^{2} - 7$$ T23^2 - 7 $$T_{29} + 2$$ T29 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 7$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T - 4)^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} - 28$$
$23$ $$T^{2} - 7$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 8T - 12$$
$41$ $$(T - 9)^{2}$$
$43$ $$T^{2} - 8T - 12$$
$47$ $$T^{2} - 8T - 47$$
$53$ $$(T + 4)^{2}$$
$59$ $$T^{2} + 8T - 96$$
$61$ $$T^{2} + 8T - 47$$
$67$ $$T^{2} + 6T - 103$$
$71$ $$T^{2} - 16T + 36$$
$73$ $$T^{2} - 20T + 72$$
$79$ $$T^{2} - 16T + 57$$
$83$ $$T^{2} - 10T - 87$$
$89$ $$T^{2} - 16T + 36$$
$97$ $$T^{2} - 2T - 111$$