Properties

 Label 9800.2.a.cg Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( 2 - \beta_{2} ) q^{11} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + ( -2 + \beta_{1} - \beta_{2} ) q^{19} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{27} -3 \beta_{2} q^{29} + ( -4 + 2 \beta_{1} ) q^{31} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{33} + 6 q^{37} + ( -4 \beta_{1} + \beta_{2} ) q^{39} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 2 - 3 \beta_{2} ) q^{47} + ( 6 + 4 \beta_{1} + \beta_{2} ) q^{51} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 6 + 2 \beta_{2} ) q^{57} + ( -2 + 3 \beta_{1} + 3 \beta_{2} ) q^{59} + ( -8 + \beta_{1} + \beta_{2} ) q^{61} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -4 - 6 \beta_{1} ) q^{69} + ( 2 - 2 \beta_{1} ) q^{71} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 4 + 4 \beta_{1} - \beta_{2} ) q^{79} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( 6 - 5 \beta_{1} + \beta_{2} ) q^{83} + ( 6 + 3 \beta_{2} ) q^{87} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 8 + 2 \beta_{2} ) q^{93} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{97} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} + 4q^{9} + O(q^{10})$$ $$3q + q^{3} + 4q^{9} + 7q^{11} - 5q^{13} + q^{17} - 4q^{19} - 6q^{23} + 19q^{27} + 3q^{29} - 10q^{31} + 7q^{33} + 18q^{37} - 5q^{39} - 18q^{41} + 9q^{47} + 21q^{51} + 10q^{53} + 16q^{57} - 6q^{59} - 24q^{61} - 10q^{67} - 18q^{69} + 4q^{71} + 6q^{73} + 17q^{79} + 15q^{81} + 12q^{83} + 15q^{87} + 22q^{93} + 9q^{97} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

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
 −1.76156 −0.363328 3.12489
0 −1.76156 0 0 0 0 0 0.103084 0
1.2 0 −0.363328 0 0 0 0 0 −2.86799 0
1.3 0 3.12489 0 0 0 0 0 6.76491 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.cg 3
5.b even 2 1 9800.2.a.cd 3
5.c odd 4 2 1960.2.g.c 6
7.b odd 2 1 1400.2.a.s 3
28.d even 2 1 2800.2.a.br 3
35.c odd 2 1 1400.2.a.t 3
35.f even 4 2 280.2.g.b 6
105.k odd 4 2 2520.2.t.g 6
140.c even 2 1 2800.2.a.bq 3
140.j odd 4 2 560.2.g.f 6
280.s even 4 2 2240.2.g.l 6
280.y odd 4 2 2240.2.g.m 6
420.w even 4 2 5040.2.t.y 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 35.f even 4 2
560.2.g.f 6 140.j odd 4 2
1400.2.a.s 3 7.b odd 2 1
1400.2.a.t 3 35.c odd 2 1
1960.2.g.c 6 5.c odd 4 2
2240.2.g.l 6 280.s even 4 2
2240.2.g.m 6 280.y odd 4 2
2520.2.t.g 6 105.k odd 4 2
2800.2.a.bq 3 140.c even 2 1
2800.2.a.br 3 28.d even 2 1
5040.2.t.y 6 420.w even 4 2
9800.2.a.cd 3 5.b even 2 1
9800.2.a.cg 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{3} - T_{3}^{2} - 6 T_{3} - 2$$ $$T_{11}^{3} - 7 T_{11}^{2} + 8 T_{11} + 8$$ $$T_{13}^{3} + 5 T_{13}^{2} - 22 T_{13} - 106$$ $$T_{19}^{3} + 4 T_{19}^{2} - 14 T_{19} + 8$$ $$T_{23}^{3} + 6 T_{23}^{2} - 28 T_{23} - 136$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-2 - 6 T - T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$8 + 8 T - 7 T^{2} + T^{3}$$
$13$ $$-106 - 22 T + 5 T^{2} + T^{3}$$
$17$ $$-20 - 24 T - T^{2} + T^{3}$$
$19$ $$8 - 14 T + 4 T^{2} + T^{3}$$
$23$ $$-136 - 28 T + 6 T^{2} + T^{3}$$
$29$ $$108 - 72 T - 3 T^{2} + T^{3}$$
$31$ $$-80 + 8 T + 10 T^{2} + T^{3}$$
$37$ $$( -6 + T )^{3}$$
$41$ $$-88 + 68 T + 18 T^{2} + T^{3}$$
$43$ $$64 - 40 T + T^{3}$$
$47$ $$232 - 48 T - 9 T^{2} + T^{3}$$
$53$ $$472 - 44 T - 10 T^{2} + T^{3}$$
$59$ $$44 - 78 T + 6 T^{2} + T^{3}$$
$61$ $$440 + 182 T + 24 T^{2} + T^{3}$$
$67$ $$-512 - 64 T + 10 T^{2} + T^{3}$$
$71$ $$64 - 20 T - 4 T^{2} + T^{3}$$
$73$ $$136 - 28 T - 6 T^{2} + T^{3}$$
$79$ $$548 - 32 T - 17 T^{2} + T^{3}$$
$83$ $$824 - 142 T - 12 T^{2} + T^{3}$$
$89$ $$464 - 172 T + T^{3}$$
$97$ $$44 - 16 T - 9 T^{2} + T^{3}$$