# Properties

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

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## 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: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + ( 5 + \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{3} + ( 5 + \beta ) q^{9} + ( 4 - \beta ) q^{11} + ( 2 - \beta ) q^{13} + ( 2 + \beta ) q^{17} -2 \beta q^{19} -2 \beta q^{23} + ( -8 - 3 \beta ) q^{27} + ( -2 + \beta ) q^{29} + 8 q^{31} + ( 8 - 3 \beta ) q^{33} + 2 q^{37} + ( 8 - \beta ) q^{39} + ( -2 + 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + 3 \beta q^{47} + ( -8 - 3 \beta ) q^{51} + ( -6 + 2 \beta ) q^{53} + ( 16 + 2 \beta ) q^{57} -8 q^{59} + ( -2 - 2 \beta ) q^{61} + 4 q^{67} + ( 16 + 2 \beta ) q^{69} + 8 q^{71} -6 q^{73} + ( 8 - 3 \beta ) q^{79} + ( 9 + 8 \beta ) q^{81} + 4 \beta q^{83} + ( -8 + \beta ) q^{87} + ( -10 + 2 \beta ) q^{89} -8 \beta q^{93} + ( 2 + 5 \beta ) q^{97} + ( 12 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 11q^{9} + O(q^{10})$$ $$2q - q^{3} + 11q^{9} + 7q^{11} + 3q^{13} + 5q^{17} - 2q^{19} - 2q^{23} - 19q^{27} - 3q^{29} + 16q^{31} + 13q^{33} + 4q^{37} + 15q^{39} - 2q^{41} + 6q^{43} + 3q^{47} - 19q^{51} - 10q^{53} + 34q^{57} - 16q^{59} - 6q^{61} + 8q^{67} + 34q^{69} + 16q^{71} - 12q^{73} + 13q^{79} + 26q^{81} + 4q^{83} - 15q^{87} - 18q^{89} - 8q^{93} + 9q^{97} + 22q^{99} + O(q^{100})$$

## 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
 3.37228 −2.37228
0 −3.37228 0 0 0 0 0 8.37228 0
1.2 0 2.37228 0 0 0 0 0 2.62772 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.bu 2
5.b even 2 1 1960.2.a.s 2
7.b odd 2 1 1400.2.a.r 2
20.d odd 2 1 3920.2.a.bt 2
28.d even 2 1 2800.2.a.bk 2
35.c odd 2 1 280.2.a.c 2
35.f even 4 2 1400.2.g.i 4
35.i odd 6 2 1960.2.q.t 4
35.j even 6 2 1960.2.q.r 4
105.g even 2 1 2520.2.a.x 2
140.c even 2 1 560.2.a.h 2
140.j odd 4 2 2800.2.g.r 4
280.c odd 2 1 2240.2.a.bk 2
280.n even 2 1 2240.2.a.bg 2
420.o odd 2 1 5040.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 35.c odd 2 1
560.2.a.h 2 140.c even 2 1
1400.2.a.r 2 7.b odd 2 1
1400.2.g.i 4 35.f even 4 2
1960.2.a.s 2 5.b even 2 1
1960.2.q.r 4 35.j even 6 2
1960.2.q.t 4 35.i odd 6 2
2240.2.a.bg 2 280.n even 2 1
2240.2.a.bk 2 280.c odd 2 1
2520.2.a.x 2 105.g even 2 1
2800.2.a.bk 2 28.d even 2 1
2800.2.g.r 4 140.j odd 4 2
3920.2.a.bt 2 20.d odd 2 1
5040.2.a.by 2 420.o odd 2 1
9800.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{2} + T_{3} - 8$$ $$T_{11}^{2} - 7 T_{11} + 4$$ $$T_{13}^{2} - 3 T_{13} - 6$$ $$T_{19}^{2} + 2 T_{19} - 32$$ $$T_{23}^{2} + 2 T_{23} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 - 7 T + T^{2}$$
$13$ $$-6 - 3 T + T^{2}$$
$17$ $$-2 - 5 T + T^{2}$$
$19$ $$-32 + 2 T + T^{2}$$
$23$ $$-32 + 2 T + T^{2}$$
$29$ $$-6 + 3 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-32 + 2 T + T^{2}$$
$43$ $$-24 - 6 T + T^{2}$$
$47$ $$-72 - 3 T + T^{2}$$
$53$ $$-8 + 10 T + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$-24 + 6 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$-32 - 13 T + T^{2}$$
$83$ $$-128 - 4 T + T^{2}$$
$89$ $$48 + 18 T + T^{2}$$
$97$ $$-186 - 9 T + T^{2}$$
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