# Properties

 Label 9800.2.a.by 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: $$x^{2} - x - 4$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( 1 + \beta ) q^{9} -\beta q^{11} + ( 2 - 3 \beta ) q^{13} + ( 6 - \beta ) q^{17} + ( 4 - 2 \beta ) q^{19} + 2 \beta q^{23} + ( 4 - \beta ) q^{27} + ( 2 + \beta ) q^{29} + 4 \beta q^{31} + ( -4 - \beta ) q^{33} + ( 2 - 4 \beta ) q^{37} + ( -12 - \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( 4 + \beta ) q^{47} + ( -4 + 5 \beta ) q^{51} + ( 10 - 2 \beta ) q^{53} + ( -8 + 2 \beta ) q^{57} + 4 q^{59} + ( 10 + 2 \beta ) q^{61} + ( 4 + 4 \beta ) q^{67} + ( 8 + 2 \beta ) q^{69} + ( 2 + 4 \beta ) q^{73} + ( -4 - 3 \beta ) q^{79} -7 q^{81} + 12 q^{83} + ( 4 + 3 \beta ) q^{87} + ( -2 + 2 \beta ) q^{89} + ( 16 + 4 \beta ) q^{93} + ( 6 + 3 \beta ) q^{97} + ( -4 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 3q^{9} + O(q^{10})$$ $$2q + q^{3} + 3q^{9} - q^{11} + q^{13} + 11q^{17} + 6q^{19} + 2q^{23} + 7q^{27} + 5q^{29} + 4q^{31} - 9q^{33} - 25q^{39} - 6q^{41} + 6q^{43} + 9q^{47} - 3q^{51} + 18q^{53} - 14q^{57} + 8q^{59} + 22q^{61} + 12q^{67} + 18q^{69} + 8q^{73} - 11q^{79} - 14q^{81} + 24q^{83} + 11q^{87} - 2q^{89} + 36q^{93} + 15q^{97} - 10q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
0 −1.56155 0 0 0 0 0 −0.561553 0
1.2 0 2.56155 0 0 0 0 0 3.56155 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.by 2
5.b even 2 1 1960.2.a.r 2
7.b odd 2 1 1400.2.a.p 2
20.d odd 2 1 3920.2.a.bu 2
28.d even 2 1 2800.2.a.bn 2
35.c odd 2 1 280.2.a.d 2
35.f even 4 2 1400.2.g.k 4
35.i odd 6 2 1960.2.q.s 4
35.j even 6 2 1960.2.q.u 4
105.g even 2 1 2520.2.a.w 2
140.c even 2 1 560.2.a.g 2
140.j odd 4 2 2800.2.g.u 4
280.c odd 2 1 2240.2.a.be 2
280.n even 2 1 2240.2.a.bi 2
420.o odd 2 1 5040.2.a.bq 2

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

 $$T_{3}^{2} - T_{3} - 4$$ $$T_{11}^{2} + T_{11} - 4$$ $$T_{13}^{2} - T_{13} - 38$$ $$T_{19}^{2} - 6 T_{19} - 8$$ $$T_{23}^{2} - 2 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 + T + T^{2}$$
$13$ $$-38 - T + T^{2}$$
$17$ $$26 - 11 T + T^{2}$$
$19$ $$-8 - 6 T + T^{2}$$
$23$ $$-16 - 2 T + T^{2}$$
$29$ $$2 - 5 T + T^{2}$$
$31$ $$-64 - 4 T + T^{2}$$
$37$ $$-68 + T^{2}$$
$41$ $$-8 + 6 T + T^{2}$$
$43$ $$-8 - 6 T + T^{2}$$
$47$ $$16 - 9 T + T^{2}$$
$53$ $$64 - 18 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$104 - 22 T + T^{2}$$
$67$ $$-32 - 12 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$-52 - 8 T + T^{2}$$
$79$ $$-8 + 11 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$-16 + 2 T + T^{2}$$
$97$ $$18 - 15 T + T^{2}$$