# Properties

 Label 9800.2.a.ck 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: 4.4.43449.1 Defining polynomial: $$x^{4} - x^{3} - 7 x^{2} + 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1400) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} + ( -\beta_{2} + \beta_{3} ) q^{11} + ( 2 + \beta_{2} + \beta_{3} ) q^{13} + ( 2 + \beta_{1} + \beta_{3} ) q^{17} + ( 4 + \beta_{2} + 2 \beta_{3} ) q^{19} + ( 1 + 2 \beta_{1} + 3 \beta_{3} ) q^{23} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{27} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{29} + ( 3 - \beta_{2} ) q^{31} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( -4 + 4 \beta_{1} - \beta_{2} ) q^{39} + ( 3 - 2 \beta_{1} + 3 \beta_{3} ) q^{41} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{43} + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -2 + \beta_{3} ) q^{53} + ( -7 + 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{57} + ( 1 - 3 \beta_{1} ) q^{59} + ( 5 - \beta_{1} - 3 \beta_{3} ) q^{61} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{69} + ( -5 - 4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -1 - 3 \beta_{3} ) q^{79} + ( 7 - 8 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{81} + ( -6 - \beta_{2} + 2 \beta_{3} ) q^{83} + ( -5 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{87} + ( 4 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{2} - \beta_{3} ) q^{93} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} + ( -15 + 5 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{3} + 5q^{9} + O(q^{10})$$ $$4q - 3q^{3} + 5q^{9} + 4q^{13} + 7q^{17} + 10q^{19} - 12q^{27} + 2q^{29} + 14q^{31} - 2q^{33} - 2q^{37} - 10q^{39} + 4q^{41} + 15q^{43} + 15q^{47} + 5q^{51} - 10q^{53} - 19q^{57} + q^{59} + 25q^{61} - 4q^{67} + 16q^{69} - 20q^{71} + 2q^{73} + 2q^{79} + 24q^{81} - 26q^{83} - 13q^{87} + 19q^{89} - 8q^{93} + 6q^{97} - 51q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 7 x^{2} + 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 8 \beta_{1} + 3$$

## 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
 −2.31553 −0.265362 0.534166 3.04673
0 −3.31553 0 0 0 0 0 7.99276 0
1.2 0 −1.26536 0 0 0 0 0 −1.39886 0
1.3 0 −0.465834 0 0 0 0 0 −2.78300 0
1.4 0 2.04673 0 0 0 0 0 1.18910 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.ck 4
5.b even 2 1 9800.2.a.cu 4
7.b odd 2 1 9800.2.a.ct 4
7.d odd 6 2 1400.2.q.l 8
35.c odd 2 1 9800.2.a.cj 4
35.i odd 6 2 1400.2.q.m yes 8
35.k even 12 4 1400.2.bh.j 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.l 8 7.d odd 6 2
1400.2.q.m yes 8 35.i odd 6 2
1400.2.bh.j 16 35.k even 12 4
9800.2.a.cj 4 35.c odd 2 1
9800.2.a.ck 4 1.a even 1 1 trivial
9800.2.a.ct 4 7.b odd 2 1
9800.2.a.cu 4 5.b even 2 1

## Hecke kernels

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

 $$T_{3}^{4} + 3 T_{3}^{3} - 4 T_{3}^{2} - 11 T_{3} - 4$$ $$T_{11}^{4} - 37 T_{11}^{2} - 49 T_{11} + 134$$ $$T_{13}^{4} - 4 T_{13}^{3} - 23 T_{13}^{2} + 105 T_{13} - 84$$ $$T_{19}^{4} - 10 T_{19}^{3} - 13 T_{19}^{2} + 359 T_{19} - 824$$ $$T_{23}^{4} - 79 T_{23}^{2} - 91 T_{23} + 953$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$-4 - 11 T - 4 T^{2} + 3 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$134 - 49 T - 37 T^{2} + T^{4}$$
$13$ $$-84 + 105 T - 23 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$5 + 18 T + 7 T^{2} - 7 T^{3} + T^{4}$$
$19$ $$-824 + 359 T - 13 T^{2} - 10 T^{3} + T^{4}$$
$23$ $$953 - 91 T - 79 T^{2} + T^{4}$$
$29$ $$-222 + 219 T - 49 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$-25 + 3 T + 49 T^{2} - 14 T^{3} + T^{4}$$
$37$ $$-92 - 127 T - 43 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$3319 + 125 T - 127 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$-1956 + 621 T + 4 T^{2} - 15 T^{3} + T^{4}$$
$47$ $$-3015 + 816 T + T^{2} - 15 T^{3} + T^{4}$$
$53$ $$-32 + 13 T + 29 T^{2} + 10 T^{3} + T^{4}$$
$59$ $$70 + 77 T - 66 T^{2} - T^{3} + T^{4}$$
$61$ $$-2130 + 57 T + 164 T^{2} - 25 T^{3} + T^{4}$$
$67$ $$-1472 - 1016 T - 172 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$-1035 - 732 T + 34 T^{2} + 20 T^{3} + T^{4}$$
$73$ $$1152 + 384 T - 184 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$-149 + 265 T - 75 T^{2} - 2 T^{3} + T^{4}$$
$83$ $$-2826 - 15 T + 187 T^{2} + 26 T^{3} + T^{4}$$
$89$ $$-1437 + 474 T + 53 T^{2} - 19 T^{3} + T^{4}$$
$97$ $$-27 + 81 T - 57 T^{2} - 6 T^{3} + T^{4}$$