# Properties

 Label 9800.2.a.bw 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: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 5 q^{9} -4 q^{11} -\beta q^{13} -2 \beta q^{17} + \beta q^{19} + 2 \beta q^{27} + 2 q^{29} + 2 \beta q^{31} -4 \beta q^{33} -10 q^{37} -8 q^{39} -2 \beta q^{41} + 4 q^{43} + 2 \beta q^{47} -16 q^{51} -6 q^{53} + 8 q^{57} + \beta q^{59} -5 \beta q^{61} -12 q^{67} + 8 q^{79} + q^{81} -5 \beta q^{83} + 2 \beta q^{87} + 16 q^{93} + 2 \beta q^{97} -20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{9} + O(q^{10})$$ $$2q + 10q^{9} - 8q^{11} + 4q^{29} - 20q^{37} - 16q^{39} + 8q^{43} - 32q^{51} - 12q^{53} + 16q^{57} - 24q^{67} + 16q^{79} + 2q^{81} + 32q^{93} - 40q^{99} + O(q^{100})$$

## 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 −2.82843 0 0 0 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 0 5.00000 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.bw 2
5.b even 2 1 392.2.a.h 2
7.b odd 2 1 inner 9800.2.a.bw 2
15.d odd 2 1 3528.2.a.bj 2
20.d odd 2 1 784.2.a.n 2
35.c odd 2 1 392.2.a.h 2
35.i odd 6 2 392.2.i.g 4
35.j even 6 2 392.2.i.g 4
40.e odd 2 1 3136.2.a.bq 2
40.f even 2 1 3136.2.a.bt 2
60.h even 2 1 7056.2.a.cj 2
105.g even 2 1 3528.2.a.bj 2
105.o odd 6 2 3528.2.s.be 4
105.p even 6 2 3528.2.s.be 4
140.c even 2 1 784.2.a.n 2
140.p odd 6 2 784.2.i.k 4
140.s even 6 2 784.2.i.k 4
280.c odd 2 1 3136.2.a.bt 2
280.n even 2 1 3136.2.a.bq 2
420.o odd 2 1 7056.2.a.cj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.h 2 5.b even 2 1
392.2.a.h 2 35.c odd 2 1
392.2.i.g 4 35.i odd 6 2
392.2.i.g 4 35.j even 6 2
784.2.a.n 2 20.d odd 2 1
784.2.a.n 2 140.c even 2 1
784.2.i.k 4 140.p odd 6 2
784.2.i.k 4 140.s even 6 2
3136.2.a.bq 2 40.e odd 2 1
3136.2.a.bq 2 280.n even 2 1
3136.2.a.bt 2 40.f even 2 1
3136.2.a.bt 2 280.c odd 2 1
3528.2.a.bj 2 15.d odd 2 1
3528.2.a.bj 2 105.g even 2 1
3528.2.s.be 4 105.o odd 6 2
3528.2.s.be 4 105.p even 6 2
7056.2.a.cj 2 60.h even 2 1
7056.2.a.cj 2 420.o odd 2 1
9800.2.a.bw 2 1.a even 1 1 trivial
9800.2.a.bw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{2} - 8$$ $$T_{11} + 4$$ $$T_{13}^{2} - 8$$ $$T_{19}^{2} - 8$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$-32 + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$-32 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-32 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$-8 + T^{2}$$
$61$ $$-200 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-200 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$-32 + T^{2}$$