# Properties

 Label 9800.2.a.cl Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.16448.2 Defining polynomial: $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( \beta_{1} - \beta_{2} ) q^{11} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -2 + \beta_{1} - 2 \beta_{3} ) q^{17} + ( -\beta_{2} - \beta_{3} ) q^{19} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{27} + ( -\beta_{1} - 3 \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{31} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -2 \beta_{2} - \beta_{3} ) q^{37} + ( 2 + 3 \beta_{1} - 3 \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + ( \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} + ( 2 + 4 \beta_{2} + 2 \beta_{3} ) q^{57} + ( 2 + 5 \beta_{2} ) q^{59} + ( 6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 2 + 6 \beta_{2} - \beta_{3} ) q^{67} + ( 6 - 2 \beta_{2} + \beta_{3} ) q^{69} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{71} + ( -2 - 4 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{73} + ( 6 - \beta_{1} + 5 \beta_{2} ) q^{79} + ( -5 + 2 \beta_{3} ) q^{81} + ( -8 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{83} + ( 4 + \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{87} + ( 10 - 3 \beta_{2} + \beta_{3} ) q^{89} + ( 6 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{93} + ( -6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 2 + 2 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 6q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 6q^{9} + 2q^{11} - 10q^{13} - 6q^{17} + 4q^{23} - 14q^{27} - 2q^{29} - 12q^{31} - 18q^{33} + 14q^{39} + 12q^{41} + 8q^{43} + 2q^{47} + 2q^{51} + 4q^{53} + 8q^{57} + 8q^{59} + 20q^{61} + 8q^{67} + 24q^{69} + 4q^{71} - 16q^{73} + 22q^{79} - 20q^{81} - 36q^{83} + 18q^{87} + 40q^{89} + 32q^{93} - 26q^{97} + 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1} + 4$$

## 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
 2.87996 2.18398 −1.18398 −1.87996
0 −2.87996 0 0 0 0 0 5.29417 0
1.2 0 −2.18398 0 0 0 0 0 1.76977 0
1.3 0 1.18398 0 0 0 0 0 −1.59819 0
1.4 0 1.87996 0 0 0 0 0 0.534253 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.cl 4
5.b even 2 1 1960.2.a.y yes 4
7.b odd 2 1 9800.2.a.cs 4
20.d odd 2 1 3920.2.a.cd 4
35.c odd 2 1 1960.2.a.x 4
35.i odd 6 2 1960.2.q.y 8
35.j even 6 2 1960.2.q.x 8
140.c even 2 1 3920.2.a.ce 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.x 4 35.c odd 2 1
1960.2.a.y yes 4 5.b even 2 1
1960.2.q.x 8 35.j even 6 2
1960.2.q.y 8 35.i odd 6 2
3920.2.a.cd 4 20.d odd 2 1
3920.2.a.ce 4 140.c even 2 1
9800.2.a.cl 4 1.a even 1 1 trivial
9800.2.a.cs 4 7.b odd 2 1

## Hecke kernels

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

 $$T_{3}^{4} + 2 T_{3}^{3} - 7 T_{3}^{2} - 8 T_{3} + 14$$ $$T_{11}^{4} - 2 T_{11}^{3} - 11 T_{11}^{2} + 20 T_{11} - 4$$ $$T_{13}^{4} + 10 T_{13}^{3} + 19 T_{13}^{2} - 52 T_{13} - 124$$ $$T_{19}^{4} - 26 T_{19}^{2} + 24 T_{19} + 8$$ $$T_{23}^{4} - 4 T_{23}^{3} - 38 T_{23}^{2} - 48 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$14 - 8 T - 7 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$-4 + 20 T - 11 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$-124 - 52 T + 19 T^{2} + 10 T^{3} + T^{4}$$
$17$ $$434 - 244 T - 51 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$8 + 24 T - 26 T^{2} + T^{4}$$
$23$ $$16 - 48 T - 38 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$188 - 20 T - 43 T^{2} + 2 T^{3} + T^{4}$$
$31$ $$-784 - 288 T + 10 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$8 + 40 T - 42 T^{2} + T^{4}$$
$41$ $$-4228 + 1528 T - 96 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$752 + 192 T - 48 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$-56 - 96 T - 41 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$-568 + 424 T - 70 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$( -46 - 4 T + T^{2} )^{2}$$
$61$ $$-5344 + 1088 T + 44 T^{2} - 20 T^{3} + T^{4}$$
$67$ $$2336 + 432 T - 114 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$10976 - 252 T^{2} - 4 T^{3} + T^{4}$$
$73$ $$-14192 - 3744 T - 170 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$-56 + 488 T + 73 T^{2} - 22 T^{3} + T^{4}$$
$83$ $$256 + 1600 T + 418 T^{2} + 36 T^{3} + T^{4}$$
$89$ $$5408 - 3120 T + 558 T^{2} - 40 T^{3} + T^{4}$$
$97$ $$-1022 + 36 T + 157 T^{2} + 26 T^{3} + T^{4}$$