# Properties

 Label 9800.2.a.bt Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9800 = 2^{3} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$78.2533939809$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1400) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 + (b + 1) * q^9 $$q - \beta q^{3} + (\beta + 1) q^{9} + (2 \beta - 3) q^{11} - 2 q^{13} + \beta q^{17} + (\beta - 2) q^{19} + (\beta + 3) q^{23} + (\beta - 4) q^{27} + (\beta + 5) q^{29} + ( - 2 \beta + 6) q^{31} + (\beta - 8) q^{33} + ( - 5 \beta + 1) q^{37} + 2 \beta q^{39} + (\beta + 4) q^{41} + ( - \beta + 5) q^{43} + ( - 4 \beta + 2) q^{47} + ( - \beta - 4) q^{51} + (2 \beta + 2) q^{53} + (\beta - 4) q^{57} + (6 \beta - 2) q^{59} + (2 \beta - 8) q^{61} + (2 \beta + 11) q^{67} + ( - 4 \beta - 4) q^{69} + ( - 3 \beta - 3) q^{71} + ( - \beta - 8) q^{73} + (3 \beta - 1) q^{79} - 7 q^{81} + (3 \beta + 6) q^{83} + ( - 6 \beta - 4) q^{87} + (5 \beta - 2) q^{89} + ( - 4 \beta + 8) q^{93} - 6 q^{97} + (\beta + 5) q^{99} +O(q^{100})$$ q - b * q^3 + (b + 1) * q^9 + (2*b - 3) * q^11 - 2 * q^13 + b * q^17 + (b - 2) * q^19 + (b + 3) * q^23 + (b - 4) * q^27 + (b + 5) * q^29 + (-2*b + 6) * q^31 + (b - 8) * q^33 + (-5*b + 1) * q^37 + 2*b * q^39 + (b + 4) * q^41 + (-b + 5) * q^43 + (-4*b + 2) * q^47 + (-b - 4) * q^51 + (2*b + 2) * q^53 + (b - 4) * q^57 + (6*b - 2) * q^59 + (2*b - 8) * q^61 + (2*b + 11) * q^67 + (-4*b - 4) * q^69 + (-3*b - 3) * q^71 + (-b - 8) * q^73 + (3*b - 1) * q^79 - 7 * q^81 + (3*b + 6) * q^83 + (-6*b - 4) * q^87 + (5*b - 2) * q^89 + (-4*b + 8) * q^93 - 6 * q^97 + (b + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + 3 * q^9 $$2 q - q^{3} + 3 q^{9} - 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + 7 q^{23} - 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} - 3 q^{37} + 2 q^{39} + 9 q^{41} + 9 q^{43} - 9 q^{51} + 6 q^{53} - 7 q^{57} + 2 q^{59} - 14 q^{61} + 24 q^{67} - 12 q^{69} - 9 q^{71} - 17 q^{73} + q^{79} - 14 q^{81} + 15 q^{83} - 14 q^{87} + q^{89} + 12 q^{93} - 12 q^{97} + 11 q^{99}+O(q^{100})$$ 2 * q - q^3 + 3 * q^9 - 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + 7 * q^23 - 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 - 3 * q^37 + 2 * q^39 + 9 * q^41 + 9 * q^43 - 9 * q^51 + 6 * q^53 - 7 * q^57 + 2 * q^59 - 14 * q^61 + 24 * q^67 - 12 * q^69 - 9 * q^71 - 17 * q^73 + q^79 - 14 * q^81 + 15 * q^83 - 14 * q^87 + q^89 + 12 * q^93 - 12 * q^97 + 11 * q^99

## 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
 2.56155 −1.56155
0 −2.56155 0 0 0 0 0 3.56155 0
1.2 0 1.56155 0 0 0 0 0 −0.561553 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 9800.2.a.bt 2
5.b even 2 1 9800.2.a.bx 2
7.b odd 2 1 1400.2.a.q yes 2
28.d even 2 1 2800.2.a.bj 2
35.c odd 2 1 1400.2.a.o 2
35.f even 4 2 1400.2.g.j 4
140.c even 2 1 2800.2.a.bo 2
140.j odd 4 2 2800.2.g.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.o 2 35.c odd 2 1
1400.2.a.q yes 2 7.b odd 2 1
1400.2.g.j 4 35.f even 4 2
2800.2.a.bj 2 28.d even 2 1
2800.2.a.bo 2 140.c even 2 1
2800.2.g.v 4 140.j odd 4 2
9800.2.a.bt 2 1.a even 1 1 trivial
9800.2.a.bx 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{11}^{2} + 4T_{11} - 13$$ T11^2 + 4*T11 - 13 $$T_{13} + 2$$ T13 + 2 $$T_{19}^{2} + 3T_{19} - 2$$ T19^2 + 3*T19 - 2 $$T_{23}^{2} - 7T_{23} + 8$$ T23^2 - 7*T23 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T - 13$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} + 3T - 2$$
$23$ $$T^{2} - 7T + 8$$
$29$ $$T^{2} - 11T + 26$$
$31$ $$T^{2} - 10T + 8$$
$37$ $$T^{2} + 3T - 104$$
$41$ $$T^{2} - 9T + 16$$
$43$ $$T^{2} - 9T + 16$$
$47$ $$T^{2} - 68$$
$53$ $$T^{2} - 6T - 8$$
$59$ $$T^{2} - 2T - 152$$
$61$ $$T^{2} + 14T + 32$$
$67$ $$T^{2} - 24T + 127$$
$71$ $$T^{2} + 9T - 18$$
$73$ $$T^{2} + 17T + 68$$
$79$ $$T^{2} - T - 38$$
$83$ $$T^{2} - 15T + 18$$
$89$ $$T^{2} - T - 106$$
$97$ $$(T + 6)^{2}$$