# Properties

 Label 9800.2.a.cp Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: 4.4.13448.1 Defining polynomial: $$x^{4} - 7 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} + ( 2 + \beta_{2} ) q^{11} -\beta_{1} q^{13} -\beta_{3} q^{17} + ( 3 \beta_{1} - \beta_{3} ) q^{19} + ( 2 + 2 \beta_{2} ) q^{23} + \beta_{3} q^{27} + ( -4 - \beta_{2} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( 4 \beta_{1} + \beta_{3} ) q^{33} + ( 2 + 2 \beta_{2} ) q^{37} + ( -4 - \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} + 4 q^{43} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{47} + ( -2 - \beta_{2} ) q^{51} + ( 6 + 2 \beta_{2} ) q^{53} + ( 10 + 2 \beta_{2} ) q^{57} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{59} + ( -\beta_{1} + 3 \beta_{3} ) q^{61} + 8 q^{67} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 2 + 2 \beta_{2} ) q^{71} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{73} -3 \beta_{2} q^{79} + ( -1 - 2 \beta_{2} ) q^{81} + ( \beta_{1} - 3 \beta_{3} ) q^{83} + ( -6 \beta_{1} - \beta_{3} ) q^{87} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{89} -4 q^{93} + ( 4 \beta_{1} - \beta_{3} ) q^{97} + ( 12 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} + 6q^{11} + 4q^{23} - 14q^{29} + 4q^{37} - 14q^{39} + 16q^{43} - 6q^{51} + 20q^{53} + 36q^{57} + 32q^{67} + 4q^{71} + 6q^{79} - 16q^{93} + 44q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$

## 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
 −2.58874 −0.546295 0.546295 2.58874
0 −2.58874 0 0 0 0 0 3.70156 0
1.2 0 −0.546295 0 0 0 0 0 −2.70156 0
1.3 0 0.546295 0 0 0 0 0 −2.70156 0
1.4 0 2.58874 0 0 0 0 0 3.70156 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.cp 4
5.b even 2 1 9800.2.a.co 4
5.c odd 4 2 1960.2.g.d 8
7.b odd 2 1 inner 9800.2.a.cp 4
35.c odd 2 1 9800.2.a.co 4
35.f even 4 2 1960.2.g.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.g.d 8 5.c odd 4 2
1960.2.g.d 8 35.f even 4 2
9800.2.a.co 4 5.b even 2 1
9800.2.a.co 4 35.c odd 2 1
9800.2.a.cp 4 1.a even 1 1 trivial
9800.2.a.cp 4 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{4} - 7 T_{3}^{2} + 2$$ $$T_{11}^{2} - 3 T_{11} - 8$$ $$T_{13}^{4} - 7 T_{13}^{2} + 2$$ $$T_{19}^{4} - 58 T_{19}^{2} + 800$$ $$T_{23}^{2} - 2 T_{23} - 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$2 - 7 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -8 - 3 T + T^{2} )^{2}$$
$13$ $$2 - 7 T^{2} + T^{4}$$
$17$ $$32 - 13 T^{2} + T^{4}$$
$19$ $$800 - 58 T^{2} + T^{4}$$
$23$ $$( -40 - 2 T + T^{2} )^{2}$$
$29$ $$( 2 + 7 T + T^{2} )^{2}$$
$31$ $$128 - 56 T^{2} + T^{4}$$
$37$ $$( -40 - 2 T + T^{2} )^{2}$$
$41$ $$128 - 56 T^{2} + T^{4}$$
$43$ $$( -4 + T )^{4}$$
$47$ $$7688 - 181 T^{2} + T^{4}$$
$53$ $$( -16 - 10 T + T^{2} )^{2}$$
$59$ $$10368 - 234 T^{2} + T^{4}$$
$61$ $$800 - 106 T^{2} + T^{4}$$
$67$ $$( -8 + T )^{4}$$
$71$ $$( -40 - 2 T + T^{2} )^{2}$$
$73$ $$20000 - 284 T^{2} + T^{4}$$
$79$ $$( -90 - 3 T + T^{2} )^{2}$$
$83$ $$800 - 106 T^{2} + T^{4}$$
$89$ $$51200 - 464 T^{2} + T^{4}$$
$97$ $$2048 - 101 T^{2} + T^{4}$$