# Properties

 Label 4998.2.a.ck Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4998.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.9092309302$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.10304.1 Defining polynomial: $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + ( -1 + \beta_{1} ) q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + ( -1 + \beta_{1} ) q^{5} + q^{6} - q^{8} + q^{9} + ( 1 - \beta_{1} ) q^{10} + ( -1 + \beta_{1} + \beta_{2} ) q^{11} - q^{12} + ( 2 + \beta_{3} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + q^{16} - q^{17} - q^{18} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{20} + ( 1 - \beta_{1} - \beta_{2} ) q^{22} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + q^{24} + ( -\beta_{1} + 2 \beta_{2} ) q^{25} + ( -2 - \beta_{3} ) q^{26} - q^{27} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{29} + ( -1 + \beta_{1} ) q^{30} + ( 3 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{31} - q^{32} + ( 1 - \beta_{1} - \beta_{2} ) q^{33} + q^{34} + q^{36} + ( -3 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{38} + ( -2 - \beta_{3} ) q^{39} + ( 1 - \beta_{1} ) q^{40} + ( 1 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{41} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( -1 + \beta_{1} + \beta_{2} ) q^{44} + ( -1 + \beta_{1} ) q^{45} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{46} + ( -1 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{47} - q^{48} + ( \beta_{1} - 2 \beta_{2} ) q^{50} + q^{51} + ( 2 + \beta_{3} ) q^{52} + ( -5 - \beta_{1} ) q^{53} + q^{54} + ( 4 + \beta_{2} + \beta_{3} ) q^{55} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{57} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{58} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 1 - \beta_{1} ) q^{60} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -3 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{62} + q^{64} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} ) q^{66} + ( -4 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{67} - q^{68} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( 6 + \beta_{2} + 2 \beta_{3} ) q^{71} - q^{72} + ( 3 - \beta_{1} + 4 \beta_{3} ) q^{73} + ( 3 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{74} + ( \beta_{1} - 2 \beta_{2} ) q^{75} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{76} + ( 2 + \beta_{3} ) q^{78} + ( 3 \beta_{2} - 5 \beta_{3} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + q^{81} + ( -1 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -2 + 3 \beta_{2} + \beta_{3} ) q^{83} + ( 1 - \beta_{1} ) q^{85} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{86} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{87} + ( 1 - \beta_{1} - \beta_{2} ) q^{88} + ( 1 - 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 1 - \beta_{1} ) q^{90} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{92} + ( -3 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{93} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{94} + ( -9 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{95} + q^{96} + ( 7 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} + 4q^{4} - 2q^{5} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} + 4q^{4} - 2q^{5} + 4q^{6} - 4q^{8} + 4q^{9} + 2q^{10} - 2q^{11} - 4q^{12} + 10q^{13} + 2q^{15} + 4q^{16} - 4q^{17} - 4q^{18} + 4q^{19} - 2q^{20} + 2q^{22} - 8q^{23} + 4q^{24} - 2q^{25} - 10q^{26} - 4q^{27} - 12q^{29} - 2q^{30} + 8q^{31} - 4q^{32} + 2q^{33} + 4q^{34} + 4q^{36} - 10q^{37} - 4q^{38} - 10q^{39} + 2q^{40} - 8q^{41} - 6q^{43} - 2q^{44} - 2q^{45} + 8q^{46} - 4q^{48} + 2q^{50} + 4q^{51} + 10q^{52} - 22q^{53} + 4q^{54} + 18q^{55} - 4q^{57} + 12q^{58} + 4q^{59} + 2q^{60} + 8q^{61} - 8q^{62} + 4q^{64} - 6q^{65} - 2q^{66} - 14q^{67} - 4q^{68} + 8q^{69} + 28q^{71} - 4q^{72} + 18q^{73} + 10q^{74} + 2q^{75} + 4q^{76} + 10q^{78} - 10q^{79} - 2q^{80} + 4q^{81} + 8q^{82} - 6q^{83} + 2q^{85} + 6q^{86} + 12q^{87} + 2q^{88} + 2q^{89} + 2q^{90} - 8q^{92} - 8q^{93} - 36q^{95} + 4q^{96} + 22q^{97} - 2q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 6 \nu + 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 2$$

## 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
 −2.16053 −0.692297 1.69230 3.16053
−1.00000 −1.00000 1.00000 −3.16053 1.00000 0 −1.00000 1.00000 3.16053
1.2 −1.00000 −1.00000 1.00000 −1.69230 1.00000 0 −1.00000 1.00000 1.69230
1.3 −1.00000 −1.00000 1.00000 0.692297 1.00000 0 −1.00000 1.00000 −0.692297
1.4 −1.00000 −1.00000 1.00000 2.16053 1.00000 0 −1.00000 1.00000 −2.16053
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$17$$ $$1$$

## 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 4998.2.a.ck 4
7.b odd 2 1 4998.2.a.cn yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.ck 4 1.a even 1 1 trivial
4998.2.a.cn yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5}^{4} + 2 T_{5}^{3} - 7 T_{5}^{2} - 8 T_{5} + 8$$ $$T_{11}^{4} + 2 T_{11}^{3} - 11 T_{11}^{2} - 28 T_{11} - 14$$ $$T_{13}^{4} - 10 T_{13}^{3} + 27 T_{13}^{2} - 46$$ $$T_{23}^{4} + 8 T_{23}^{3} - 2 T_{23}^{2} - 24 T_{23} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$8 - 8 T - 7 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$-14 - 28 T - 11 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$-46 + 27 T^{2} - 10 T^{3} + T^{4}$$
$17$ $$( 1 + T )^{4}$$
$19$ $$-4 + 136 T - 32 T^{2} - 4 T^{3} + T^{4}$$
$23$ $$-8 - 24 T - 2 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$-964 - 376 T + 12 T^{3} + T^{4}$$
$31$ $$-1600 + 800 T - 82 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$-2048 - 768 T - 47 T^{2} + 10 T^{3} + T^{4}$$
$41$ $$1096 - 1048 T - 138 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$-4 + 36 T - 37 T^{2} + 6 T^{3} + T^{4}$$
$47$ $$-16 + 80 T - 42 T^{2} + T^{4}$$
$53$ $$668 + 572 T + 173 T^{2} + 22 T^{3} + T^{4}$$
$59$ $$7232 + 480 T - 180 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$-2192 + 1280 T - 144 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$-388 - 548 T - 53 T^{2} + 14 T^{3} + T^{4}$$
$71$ $$-1444 - 456 T + 240 T^{2} - 28 T^{3} + T^{4}$$
$73$ $$-8036 + 2156 T - 59 T^{2} - 18 T^{3} + T^{4}$$
$79$ $$5344 - 880 T - 201 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$356 - 92 T - 45 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$-292 - 540 T - 219 T^{2} - 2 T^{3} + T^{4}$$
$97$ $$-100 - 140 T + 133 T^{2} - 22 T^{3} + T^{4}$$