# Properties

 Label 9800.2.a.cf Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1944.1 Defining polynomial: $$x^{3} - 9 x - 6$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} ) q^{13} + 2 q^{17} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{23} + ( -6 - 3 \beta_{1} ) q^{27} + ( 4 - \beta_{1} - \beta_{2} ) q^{29} + ( -4 + 2 \beta_{1} ) q^{31} + ( -6 - 4 \beta_{1} ) q^{33} + ( -3 - \beta_{1} + \beta_{2} ) q^{37} + ( -6 - 2 \beta_{2} ) q^{39} + ( -3 - 2 \beta_{2} ) q^{41} + ( -4 + \beta_{1} ) q^{43} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{47} -2 \beta_{1} q^{51} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{53} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{57} -8 q^{59} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{61} + ( -2 - 3 \beta_{1} ) q^{67} + ( -\beta_{1} + \beta_{2} ) q^{69} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 9 + 6 \beta_{1} ) q^{81} + ( -10 + \beta_{1} ) q^{83} + ( 6 - \beta_{1} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{89} + ( -12 + 2 \beta_{1} - 2 \beta_{2} ) q^{93} -2 q^{97} + ( 21 + 7 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 9q^{9} + O(q^{10})$$ $$3q + 9q^{9} + 3q^{11} + 3q^{13} + 6q^{17} - 3q^{19} - 3q^{23} - 18q^{27} + 12q^{29} - 12q^{31} - 18q^{33} - 9q^{37} - 18q^{39} - 9q^{41} - 12q^{43} - 15q^{47} - 9q^{53} - 18q^{57} - 24q^{59} + 6q^{61} - 6q^{67} - 18q^{79} + 27q^{81} - 30q^{83} + 18q^{87} - 36q^{93} - 6q^{97} + 63q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 9 x - 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 6$$

## 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
 3.28995 −0.705720 −2.58423
0 −3.28995 0 0 0 0 0 7.82374 0
1.2 0 0.705720 0 0 0 0 0 −2.50196 0
1.3 0 2.58423 0 0 0 0 0 3.67822 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.cf 3
5.b even 2 1 1960.2.a.v 3
7.b odd 2 1 9800.2.a.ce 3
7.d odd 6 2 1400.2.q.j 6
20.d odd 2 1 3920.2.a.cb 3
35.c odd 2 1 1960.2.a.w 3
35.i odd 6 2 280.2.q.e 6
35.j even 6 2 1960.2.q.w 6
35.k even 12 4 1400.2.bh.i 12
105.p even 6 2 2520.2.bi.q 6
140.c even 2 1 3920.2.a.cc 3
140.s even 6 2 560.2.q.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 35.i odd 6 2
560.2.q.l 6 140.s even 6 2
1400.2.q.j 6 7.d odd 6 2
1400.2.bh.i 12 35.k even 12 4
1960.2.a.v 3 5.b even 2 1
1960.2.a.w 3 35.c odd 2 1
1960.2.q.w 6 35.j even 6 2
2520.2.bi.q 6 105.p even 6 2
3920.2.a.cb 3 20.d odd 2 1
3920.2.a.cc 3 140.c even 2 1
9800.2.a.ce 3 7.b odd 2 1
9800.2.a.cf 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{3} - 9 T_{3} + 6$$ $$T_{11}^{3} - 3 T_{11}^{2} - 24 T_{11} + 44$$ $$T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 68$$ $$T_{19}^{3} + 3 T_{19}^{2} - 24 T_{19} + 16$$ $$T_{23}^{3} + 3 T_{23}^{2} - 15 T_{23} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$6 - 9 T + T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$44 - 24 T - 3 T^{2} + T^{3}$$
$13$ $$68 - 24 T - 3 T^{2} + T^{3}$$
$17$ $$( -2 + T )^{3}$$
$19$ $$16 - 24 T + 3 T^{2} + T^{3}$$
$23$ $$7 - 15 T + 3 T^{2} + T^{3}$$
$29$ $$26 + 21 T - 12 T^{2} + T^{3}$$
$31$ $$-128 + 12 T + 12 T^{2} + T^{3}$$
$37$ $$-96 + 9 T^{2} + T^{3}$$
$41$ $$-381 - 45 T + 9 T^{2} + T^{3}$$
$43$ $$22 + 39 T + 12 T^{2} + T^{3}$$
$47$ $$-1588 - 96 T + 15 T^{2} + T^{3}$$
$53$ $$-624 - 72 T + 9 T^{2} + T^{3}$$
$59$ $$( 8 + T )^{3}$$
$61$ $$544 - 87 T - 6 T^{2} + T^{3}$$
$67$ $$8 - 69 T + 6 T^{2} + T^{3}$$
$71$ $$T^{3}$$
$73$ $$336 - 108 T + T^{3}$$
$79$ $$-768 + 18 T^{2} + T^{3}$$
$83$ $$904 + 291 T + 30 T^{2} + T^{3}$$
$89$ $$-42 - 27 T + T^{3}$$
$97$ $$( 2 + T )^{3}$$