# Properties

 Label 9800.2.a.cw Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $1$ Dimension $6$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.239575536.1 Defining polynomial: $$x^{6} - x^{5} - 12 x^{4} + 8 x^{3} + 35 x^{2} - 15 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{5} ) q^{11} + ( -1 - \beta_{2} ) q^{13} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} ) q^{23} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{27} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{29} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{31} + ( -2 - 2 \beta_{2} + \beta_{4} ) q^{33} + ( -2 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{39} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{41} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{43} + ( -5 + \beta_{2} - \beta_{4} ) q^{47} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{51} + ( 1 + \beta_{2} + \beta_{4} ) q^{53} + ( 5 - 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{57} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{59} + ( -1 - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{61} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{67} + ( -3 + 4 \beta_{1} + \beta_{3} + \beta_{5} ) q^{69} + ( 1 - 3 \beta_{2} - \beta_{3} - \beta_{5} ) q^{71} + ( -3 - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{79} + ( -2 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{81} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{83} + ( -5 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{87} + ( -1 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{97} + ( 2 + 4 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{3} + 7q^{9} + O(q^{10})$$ $$6q - q^{3} + 7q^{9} + q^{11} - 7q^{13} - 4q^{17} + 5q^{19} - 6q^{23} - 7q^{27} - 3q^{29} + 2q^{31} - 16q^{33} - 9q^{37} + 10q^{39} - 6q^{41} + 3q^{43} - 27q^{47} + 5q^{53} + 26q^{57} + 24q^{59} - 9q^{61} - 17q^{67} - 15q^{69} + 4q^{71} - 18q^{73} - 22q^{79} - 6q^{81} - 9q^{83} - 39q^{87} - 15q^{89} - 10q^{93} - 12q^{97} + 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 12 x^{4} + 8 x^{3} + 35 x^{2} - 15 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 10 \nu^{3} - 4 \nu^{2} + 17 \nu + 8$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 14 \nu^{2} + 11 \nu - 16$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 14 \nu^{3} - 45 \nu + 4$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{3} + \beta_{2} + 7 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 10 \beta_{2} + 4 \beta_{1} + 25$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{5} + 14 \beta_{3} + 14 \beta_{2} + 53 \beta_{1} + 18$$

## 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
 3.04302 2.10821 0.751428 −0.319986 −2.03837 −2.54431
0 −3.04302 0 0 0 0 0 6.25996 0
1.2 0 −2.10821 0 0 0 0 0 1.44457 0
1.3 0 −0.751428 0 0 0 0 0 −2.43536 0
1.4 0 0.319986 0 0 0 0 0 −2.89761 0
1.5 0 2.03837 0 0 0 0 0 1.15495 0
1.6 0 2.54431 0 0 0 0 0 3.47349 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.cw 6
5.b even 2 1 9800.2.a.cy 6
5.c odd 4 2 1960.2.g.e 12
7.b odd 2 1 9800.2.a.cx 6
7.d odd 6 2 1400.2.q.n 12
35.c odd 2 1 9800.2.a.cv 6
35.f even 4 2 1960.2.g.f 12
35.i odd 6 2 1400.2.q.o 12
35.k even 12 4 280.2.bg.a 24
140.x odd 12 4 560.2.bw.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bg.a 24 35.k even 12 4
560.2.bw.f 24 140.x odd 12 4
1400.2.q.n 12 7.d odd 6 2
1400.2.q.o 12 35.i odd 6 2
1960.2.g.e 12 5.c odd 4 2
1960.2.g.f 12 35.f even 4 2
9800.2.a.cv 6 35.c odd 2 1
9800.2.a.cw 6 1.a even 1 1 trivial
9800.2.a.cx 6 7.b odd 2 1
9800.2.a.cy 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{6} + T_{3}^{5} - 12 T_{3}^{4} - 8 T_{3}^{3} + 35 T_{3}^{2} + 15 T_{3} - 8$$ $$T_{11}^{6} - T_{11}^{5} - 37 T_{11}^{4} + 69 T_{11}^{3} + 296 T_{11}^{2} - 832 T_{11} + 512$$ $$T_{13}^{6} + 7 T_{13}^{5} - 10 T_{13}^{4} - 100 T_{13}^{3} - 8 T_{13}^{2} + 320 T_{13} + 256$$ $$T_{19}^{6} - 5 T_{19}^{5} - 47 T_{19}^{4} + 301 T_{19}^{3} - 94 T_{19}^{2} - 1300 T_{19} + 568$$ $$T_{23}^{6} + 6 T_{23}^{5} - 19 T_{23}^{4} - 108 T_{23}^{3} + 123 T_{23}^{2} + 478 T_{23} - 337$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$-8 + 15 T + 35 T^{2} - 8 T^{3} - 12 T^{4} + T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$512 - 832 T + 296 T^{2} + 69 T^{3} - 37 T^{4} - T^{5} + T^{6}$$
$13$ $$256 + 320 T - 8 T^{2} - 100 T^{3} - 10 T^{4} + 7 T^{5} + T^{6}$$
$17$ $$-9552 + 3552 T + 1656 T^{2} - 244 T^{3} - 75 T^{4} + 4 T^{5} + T^{6}$$
$19$ $$568 - 1300 T - 94 T^{2} + 301 T^{3} - 47 T^{4} - 5 T^{5} + T^{6}$$
$23$ $$-337 + 478 T + 123 T^{2} - 108 T^{3} - 19 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$-7344 - 2700 T + 2652 T^{2} - 35 T^{3} - 105 T^{4} + 3 T^{5} + T^{6}$$
$31$ $$-1968 + 1728 T + 2016 T^{2} + 124 T^{3} - 115 T^{4} - 2 T^{5} + T^{6}$$
$37$ $$-3456 + 8064 T + 1392 T^{2} - 547 T^{3} - 75 T^{4} + 9 T^{5} + T^{6}$$
$41$ $$2764 - 5372 T + 2871 T^{2} - 266 T^{3} - 100 T^{4} + 6 T^{5} + T^{6}$$
$43$ $$-95736 - 2292 T + 6990 T^{2} + 203 T^{3} - 151 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$-1392 + 1320 T + 2604 T^{2} + 1271 T^{3} + 273 T^{4} + 27 T^{5} + T^{6}$$
$53$ $$344 - 1268 T + 1030 T^{2} + 267 T^{3} - 77 T^{4} - 5 T^{5} + T^{6}$$
$59$ $$-124704 + 93384 T - 23820 T^{2} + 2042 T^{3} + 93 T^{4} - 24 T^{5} + T^{6}$$
$61$ $$-105806 + 36181 T + 5289 T^{2} - 1550 T^{3} - 172 T^{4} + 9 T^{5} + T^{6}$$
$67$ $$3842 + 1883 T - 1201 T^{2} - 396 T^{3} + 38 T^{4} + 17 T^{5} + T^{6}$$
$71$ $$-183296 - 33536 T + 11984 T^{2} + 1280 T^{3} - 272 T^{4} - 4 T^{5} + T^{6}$$
$73$ $$-179072 - 130568 T - 35340 T^{2} - 3998 T^{3} - 79 T^{4} + 18 T^{5} + T^{6}$$
$79$ $$-272 + 1808 T - 1720 T^{2} - 872 T^{3} + 61 T^{4} + 22 T^{5} + T^{6}$$
$83$ $$-136 - 1556 T - 4230 T^{2} - 1389 T^{3} - 103 T^{4} + 9 T^{5} + T^{6}$$
$89$ $$115554 + 42867 T - 11367 T^{2} - 4498 T^{3} - 268 T^{4} + 15 T^{5} + T^{6}$$
$97$ $$-8192 + 14336 T + 13824 T^{2} - 1600 T^{3} - 208 T^{4} + 12 T^{5} + T^{6}$$