# Properties

 Label 9800.2.a.da Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 18 x^{6} + 85 x^{4} - 38 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{9} + ( 1 + \beta_{3} + \beta_{6} ) q^{11} -\beta_{5} q^{13} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{17} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( 3 \beta_{1} - \beta_{4} + \beta_{5} ) q^{27} + ( 2 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{31} + ( \beta_{1} - 3 \beta_{4} + \beta_{7} ) q^{33} + ( \beta_{2} + \beta_{3} - \beta_{6} ) q^{37} + 2 q^{39} + ( -\beta_{1} - 2 \beta_{5} ) q^{41} + ( -2 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{43} + ( 4 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{47} + ( 7 + \beta_{2} + \beta_{3} ) q^{51} + ( 4 + 2 \beta_{3} + \beta_{6} ) q^{53} + ( 1 + \beta_{2} - 5 \beta_{3} - \beta_{6} ) q^{57} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{61} + ( -5 + 2 \beta_{3} ) q^{67} + ( -4 \beta_{1} - 4 \beta_{4} - \beta_{5} + \beta_{7} ) q^{69} + ( 2 + \beta_{2} - \beta_{3} + \beta_{6} ) q^{71} + ( \beta_{1} + 5 \beta_{4} - \beta_{7} ) q^{73} + ( 2 - \beta_{2} - 3 \beta_{3} ) q^{79} + ( 7 + \beta_{3} ) q^{81} + ( -\beta_{1} - \beta_{7} ) q^{83} + ( 6 \beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{7} ) q^{87} + ( -5 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{89} + ( 8 + 2 \beta_{2} - 2 \beta_{3} ) q^{93} + ( 5 \beta_{4} - \beta_{5} + \beta_{7} ) q^{97} + ( 4 + \beta_{2} + 4 \beta_{3} - 2 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{9} + O(q^{10})$$ $$8q + 12q^{9} + 4q^{11} + 4q^{23} + 8q^{29} + 16q^{39} - 16q^{43} + 52q^{51} + 28q^{53} + 8q^{57} - 40q^{67} + 8q^{71} + 20q^{79} + 56q^{81} + 56q^{93} + 36q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 18 x^{6} + 85 x^{4} - 38 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 9 \nu^{2} + 2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 18 \nu^{5} + 83 \nu^{3} - 20 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 18 \nu^{5} + 85 \nu^{3} - 38 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 18 \nu^{4} + 83 \nu^{2} - 20$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} - 88 \nu^{5} + 397 \nu^{3} - 96 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - \beta_{4} + 9 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 9 \beta_{2} + 43$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 9 \beta_{5} - 14 \beta_{4} + 79 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + 18 \beta_{3} + 79 \beta_{2} + 379$$ $$\nu^{7}$$ $$=$$ $$18 \beta_{7} + 79 \beta_{5} - 167 \beta_{4} + 695 \beta_{1}$$

## 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
 −3.04118 −2.87509 −0.568463 −0.402377 0.402377 0.568463 2.87509 3.04118
0 −3.04118 0 0 0 0 0 6.24878 0
1.2 0 −2.87509 0 0 0 0 0 5.26617 0
1.3 0 −0.568463 0 0 0 0 0 −2.67685 0
1.4 0 −0.402377 0 0 0 0 0 −2.83809 0
1.5 0 0.402377 0 0 0 0 0 −2.83809 0
1.6 0 0.568463 0 0 0 0 0 −2.67685 0
1.7 0 2.87509 0 0 0 0 0 5.26617 0
1.8 0 3.04118 0 0 0 0 0 6.24878 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.da yes 8
5.b even 2 1 9800.2.a.cz 8
7.b odd 2 1 inner 9800.2.a.da yes 8
35.c odd 2 1 9800.2.a.cz 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9800.2.a.cz 8 5.b even 2 1
9800.2.a.cz 8 35.c odd 2 1
9800.2.a.da yes 8 1.a even 1 1 trivial
9800.2.a.da yes 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{8} - 18 T_{3}^{6} + 85 T_{3}^{4} - 38 T_{3}^{2} + 4$$ $$T_{11}^{4} - 2 T_{11}^{3} - 38 T_{11}^{2} + 62 T_{11} + 257$$ $$T_{13}^{8} - 38 T_{13}^{6} + 340 T_{13}^{4} - 288 T_{13}^{2} + 64$$ $$T_{19}^{8} - 160 T_{19}^{6} + 7997 T_{19}^{4} - 119588 T_{19}^{2} + 84100$$ $$T_{23}^{4} - 2 T_{23}^{3} - 47 T_{23}^{2} + 156 T_{23} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 - 38 T^{2} + 85 T^{4} - 18 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 257 + 62 T - 38 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$13$ $$64 - 288 T^{2} + 340 T^{4} - 38 T^{6} + T^{8}$$
$17$ $$40000 - 16600 T^{2} + 2089 T^{4} - 84 T^{6} + T^{8}$$
$19$ $$84100 - 119588 T^{2} + 7997 T^{4} - 160 T^{6} + T^{8}$$
$23$ $$( -8 + 156 T - 47 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$29$ $$( -292 + 328 T - 85 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$31$ $$4096 - 21248 T^{2} + 3812 T^{4} - 118 T^{6} + T^{8}$$
$37$ $$( 488 + 92 T - 105 T^{2} + T^{4} )^{2}$$
$41$ $$24964 - 20590 T^{2} + 4501 T^{4} - 138 T^{6} + T^{8}$$
$43$ $$( -100 - 480 T - 81 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$47$ $$66716224 - 3457024 T^{2} + 58260 T^{4} - 404 T^{6} + T^{8}$$
$53$ $$( 512 + 320 T - 4 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$59$ $$2704 - 14952 T^{2} + 19760 T^{4} - 282 T^{6} + T^{8}$$
$61$ $$1000000 - 260000 T^{2} + 13300 T^{4} - 222 T^{6} + T^{8}$$
$67$ $$( -1207 - 252 T + 78 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$71$ $$( -340 + 444 T - 105 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$73$ $$676 - 23134 T^{2} + 21717 T^{4} - 314 T^{6} + T^{8}$$
$79$ $$( -7724 + 2552 T - 179 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$83$ $$532900 - 204902 T^{2} + 10293 T^{4} - 178 T^{6} + T^{8}$$
$89$ $$41860900 - 2808558 T^{2} + 59893 T^{4} - 442 T^{6} + T^{8}$$
$97$ $$38416 - 238984 T^{2} + 27456 T^{4} - 386 T^{6} + T^{8}$$