# Properties

 Label 9800.2.a.ca Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1960) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + 2 \beta q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + 2 \beta q^{9} - q^{11} + ( 1 - \beta ) q^{13} + ( 1 + \beta ) q^{17} -2 q^{19} + ( -2 - 3 \beta ) q^{23} + ( 1 - \beta ) q^{27} + q^{29} + ( -6 - 3 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( -2 - 7 \beta ) q^{37} - q^{39} + ( -6 + \beta ) q^{41} + ( 6 + 4 \beta ) q^{43} + ( 9 - \beta ) q^{47} + ( 3 + 2 \beta ) q^{51} + ( -8 + \beta ) q^{53} + ( -2 - 2 \beta ) q^{57} + ( -6 + 3 \beta ) q^{59} + ( 4 + 2 \beta ) q^{61} + \beta q^{67} + ( -8 - 5 \beta ) q^{69} + ( 6 - 6 \beta ) q^{71} + ( -8 - 2 \beta ) q^{73} + ( -9 + 4 \beta ) q^{79} + ( -1 - 6 \beta ) q^{81} -8 \beta q^{83} + ( 1 + \beta ) q^{87} + ( -4 - 4 \beta ) q^{89} + ( -12 - 9 \beta ) q^{93} + ( 7 + 5 \beta ) q^{97} -2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{11} + 2q^{13} + 2q^{17} - 4q^{19} - 4q^{23} + 2q^{27} + 2q^{29} - 12q^{31} - 2q^{33} - 4q^{37} - 2q^{39} - 12q^{41} + 12q^{43} + 18q^{47} + 6q^{51} - 16q^{53} - 4q^{57} - 12q^{59} + 8q^{61} - 16q^{69} + 12q^{71} - 16q^{73} - 18q^{79} - 2q^{81} + 2q^{87} - 8q^{89} - 24q^{93} + 14q^{97} + O(q^{100})$$

## 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 −0.414214 0 0 0 0 0 −2.82843 0
1.2 0 2.41421 0 0 0 0 0 2.82843 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.ca 2
5.b even 2 1 1960.2.a.q 2
7.b odd 2 1 9800.2.a.bs 2
20.d odd 2 1 3920.2.a.by 2
35.c odd 2 1 1960.2.a.u yes 2
35.i odd 6 2 1960.2.q.p 4
35.j even 6 2 1960.2.q.v 4
140.c even 2 1 3920.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.q 2 5.b even 2 1
1960.2.a.u yes 2 35.c odd 2 1
1960.2.q.p 4 35.i odd 6 2
1960.2.q.v 4 35.j even 6 2
3920.2.a.bn 2 140.c even 2 1
3920.2.a.by 2 20.d odd 2 1
9800.2.a.bs 2 7.b odd 2 1
9800.2.a.ca 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{2} - 2 T_{3} - 1$$ $$T_{11} + 1$$ $$T_{13}^{2} - 2 T_{13} - 1$$ $$T_{19} + 2$$ $$T_{23}^{2} + 4 T_{23} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-1 - 2 T + T^{2}$$
$17$ $$-1 - 2 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-14 + 4 T + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$18 + 12 T + T^{2}$$
$37$ $$-94 + 4 T + T^{2}$$
$41$ $$34 + 12 T + T^{2}$$
$43$ $$4 - 12 T + T^{2}$$
$47$ $$79 - 18 T + T^{2}$$
$53$ $$62 + 16 T + T^{2}$$
$59$ $$18 + 12 T + T^{2}$$
$61$ $$8 - 8 T + T^{2}$$
$67$ $$-2 + T^{2}$$
$71$ $$-36 - 12 T + T^{2}$$
$73$ $$56 + 16 T + T^{2}$$
$79$ $$49 + 18 T + T^{2}$$
$83$ $$-128 + T^{2}$$
$89$ $$-16 + 8 T + T^{2}$$
$97$ $$-1 - 14 T + T^{2}$$