# Properties

 Label 9800.2.a.cc 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: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.53209 −0.347296 1.87939
0 −2.53209 0 0 0 0 0 3.41147 0
1.2 0 −1.34730 0 0 0 0 0 −1.18479 0
1.3 0 0.879385 0 0 0 0 0 −2.22668 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.cc 3
5.b even 2 1 9800.2.a.ci 3
7.b odd 2 1 9800.2.a.ch 3
7.c even 3 2 1400.2.q.k yes 6
35.c odd 2 1 9800.2.a.cb 3
35.j even 6 2 1400.2.q.i 6
35.l odd 12 4 1400.2.bh.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.i 6 35.j even 6 2
1400.2.q.k yes 6 7.c even 3 2
1400.2.bh.h 12 35.l odd 12 4
9800.2.a.cb 3 35.c odd 2 1
9800.2.a.cc 3 1.a even 1 1 trivial
9800.2.a.ch 3 7.b odd 2 1
9800.2.a.ci 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{3} + 3 T_{3}^{2} - 3$$ $$T_{11}^{3} + 6 T_{11}^{2} + 3 T_{11} - 19$$ $$T_{13}^{3} - 39 T_{13} - 19$$ $$T_{19}^{3} - 6 T_{19}^{2} - 9 T_{19} + 17$$ $$T_{23}^{3} + 3 T_{23}^{2} - 18 T_{23} + 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-3 + 3 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$-19 + 3 T + 6 T^{2} + T^{3}$$
$13$ $$-19 - 39 T + T^{3}$$
$17$ $$19 - 15 T - 6 T^{2} + T^{3}$$
$19$ $$17 - 9 T - 6 T^{2} + T^{3}$$
$23$ $$17 - 18 T + 3 T^{2} + T^{3}$$
$29$ $$-19 + 39 T - 12 T^{2} + T^{3}$$
$31$ $$-19 - 36 T + 3 T^{2} + T^{3}$$
$37$ $$51 - 27 T - 6 T^{2} + T^{3}$$
$41$ $$57 - 18 T - 3 T^{2} + T^{3}$$
$43$ $$179 - 12 T - 9 T^{2} + T^{3}$$
$47$ $$107 + 87 T + 18 T^{2} + T^{3}$$
$53$ $$-1077 - 171 T + 6 T^{2} + T^{3}$$
$59$ $$-53 - 24 T + 3 T^{2} + T^{3}$$
$61$ $$-37 - 66 T - 9 T^{2} + T^{3}$$
$67$ $$-584 - 60 T + 12 T^{2} + T^{3}$$
$71$ $$513 - 81 T - 9 T^{2} + T^{3}$$
$73$ $$192 - 12 T^{2} + T^{3}$$
$79$ $$-3 + 3 T^{2} + T^{3}$$
$83$ $$163 - 57 T + T^{3}$$
$89$ $$699 - 117 T - 6 T^{2} + T^{3}$$
$97$ $$467 + 66 T - 21 T^{2} + T^{3}$$