# Properties

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

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 392) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{9}+O(q^{10})$$ q + b * q^3 - q^9 $$q + \beta q^{3} - q^{9} + 6 q^{11} - 4 \beta q^{13} + \beta q^{17} + 3 \beta q^{19} - 4 q^{23} - 4 \beta q^{27} - 6 q^{29} - 2 \beta q^{31} + 6 \beta q^{33} - 2 q^{37} - 8 q^{39} + \beta q^{41} - 10 q^{43} - 2 \beta q^{47} + 2 q^{51} + 2 q^{53} + 6 q^{57} - \beta q^{59} + 6 \beta q^{61} - 4 q^{67} - 4 \beta q^{69} - 12 q^{71} - 7 \beta q^{73} - 4 q^{79} - 5 q^{81} + \beta q^{83} - 6 \beta q^{87} + 3 \beta q^{89} - 4 q^{93} + 9 \beta q^{97} - 6 q^{99} +O(q^{100})$$ q + b * q^3 - q^9 + 6 * q^11 - 4*b * q^13 + b * q^17 + 3*b * q^19 - 4 * q^23 - 4*b * q^27 - 6 * q^29 - 2*b * q^31 + 6*b * q^33 - 2 * q^37 - 8 * q^39 + b * q^41 - 10 * q^43 - 2*b * q^47 + 2 * q^51 + 2 * q^53 + 6 * q^57 - b * q^59 + 6*b * q^61 - 4 * q^67 - 4*b * q^69 - 12 * q^71 - 7*b * q^73 - 4 * q^79 - 5 * q^81 + b * q^83 - 6*b * q^87 + 3*b * q^89 - 4 * q^93 + 9*b * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 12 q^{11} - 8 q^{23} - 12 q^{29} - 4 q^{37} - 16 q^{39} - 20 q^{43} + 4 q^{51} + 4 q^{53} + 12 q^{57} - 8 q^{67} - 24 q^{71} - 8 q^{79} - 10 q^{81} - 8 q^{93} - 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 12 * q^11 - 8 * q^23 - 12 * q^29 - 4 * q^37 - 16 * q^39 - 20 * q^43 + 4 * q^51 + 4 * q^53 + 12 * q^57 - 8 * q^67 - 24 * q^71 - 8 * q^79 - 10 * q^81 - 8 * q^93 - 12 * 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
 −1.41421 1.41421
0 −1.41421 0 0 0 0 0 −1.00000 0
1.2 0 1.41421 0 0 0 0 0 −1.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.bv 2
5.b even 2 1 392.2.a.g 2
7.b odd 2 1 inner 9800.2.a.bv 2
15.d odd 2 1 3528.2.a.be 2
20.d odd 2 1 784.2.a.k 2
35.c odd 2 1 392.2.a.g 2
35.i odd 6 2 392.2.i.h 4
35.j even 6 2 392.2.i.h 4
40.e odd 2 1 3136.2.a.bp 2
40.f even 2 1 3136.2.a.bk 2
60.h even 2 1 7056.2.a.ct 2
105.g even 2 1 3528.2.a.be 2
105.o odd 6 2 3528.2.s.bj 4
105.p even 6 2 3528.2.s.bj 4
140.c even 2 1 784.2.a.k 2
140.p odd 6 2 784.2.i.n 4
140.s even 6 2 784.2.i.n 4
280.c odd 2 1 3136.2.a.bk 2
280.n even 2 1 3136.2.a.bp 2
420.o odd 2 1 7056.2.a.ct 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 5.b even 2 1
392.2.a.g 2 35.c odd 2 1
392.2.i.h 4 35.i odd 6 2
392.2.i.h 4 35.j even 6 2
784.2.a.k 2 20.d odd 2 1
784.2.a.k 2 140.c even 2 1
784.2.i.n 4 140.p odd 6 2
784.2.i.n 4 140.s even 6 2
3136.2.a.bk 2 40.f even 2 1
3136.2.a.bk 2 280.c odd 2 1
3136.2.a.bp 2 40.e odd 2 1
3136.2.a.bp 2 280.n even 2 1
3528.2.a.be 2 15.d odd 2 1
3528.2.a.be 2 105.g even 2 1
3528.2.s.bj 4 105.o odd 6 2
3528.2.s.bj 4 105.p even 6 2
7056.2.a.ct 2 60.h even 2 1
7056.2.a.ct 2 420.o odd 2 1
9800.2.a.bv 2 1.a even 1 1 trivial
9800.2.a.bv 2 7.b odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{11} - 6$$ T11 - 6 $$T_{13}^{2} - 32$$ T13^2 - 32 $$T_{19}^{2} - 18$$ T19^2 - 18 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} - 32$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} - 18$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 8$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 2$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 2$$
$61$ $$T^{2} - 72$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 98$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 2$$
$89$ $$T^{2} - 18$$
$97$ $$T^{2} - 162$$