# Properties

 Label 4998.2.a.cb Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 1.00000
 $$n$$: 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.cb 2
7.b odd 2 1 4998.2.a.cc yes 2

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

 $$T_{5} - 1$$ $$T_{11}^{2} + 6 T_{11} + 7$$ $$T_{13}^{2} + 2 T_{13} - 7$$ $$T_{23}^{2} + 12 T_{23} + 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$7 + 6 T + T^{2}$$
$13$ $$-7 + 2 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$-14 - 4 T + T^{2}$$
$23$ $$28 + 12 T + T^{2}$$
$29$ $$14 + 8 T + T^{2}$$
$31$ $$-2 - 8 T + T^{2}$$
$37$ $$-7 + 10 T + T^{2}$$
$41$ $$-14 - 4 T + T^{2}$$
$43$ $$23 + 10 T + T^{2}$$
$47$ $$46 + 16 T + T^{2}$$
$53$ $$-7 - 2 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$4 - 12 T + T^{2}$$
$67$ $$-73 + 10 T + T^{2}$$
$71$ $$-146 - 8 T + T^{2}$$
$73$ $$17 - 14 T + T^{2}$$
$79$ $$23 + 10 T + T^{2}$$
$83$ $$47 + 14 T + T^{2}$$
$89$ $$89 + 22 T + T^{2}$$
$97$ $$1 + 6 T + T^{2}$$