# Properties

 Label 4998.2.a.bv Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 714) 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.

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

## 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 −2.41421 −1.00000 0 −1.00000 1.00000 2.41421
1.2 −1.00000 1.00000 1.00000 0.414214 −1.00000 0 −1.00000 1.00000 −0.414214
 $$n$$: 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.bv 2
7.b odd 2 1 4998.2.a.bt 2
7.c even 3 2 714.2.i.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.i.n 4 7.c even 3 2
4998.2.a.bt 2 7.b odd 2 1
4998.2.a.bv 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5}^{2} + 2 T_{5} - 1$$ $$T_{11}^{2} - 4 T_{11} - 4$$ $$T_{13}^{2} - 32$$ $$T_{23}^{2} - 6 T_{23} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-1 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 - 4 T + T^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$( -3 + T )^{2}$$
$23$ $$7 - 6 T + T^{2}$$
$29$ $$8 + 8 T + T^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$7 + 6 T + T^{2}$$
$41$ $$-16 + 8 T + T^{2}$$
$43$ $$41 - 14 T + T^{2}$$
$47$ $$( -10 + T )^{2}$$
$53$ $$28 + 12 T + T^{2}$$
$59$ $$-7 - 10 T + T^{2}$$
$61$ $$-56 - 8 T + T^{2}$$
$67$ $$-31 + 2 T + T^{2}$$
$71$ $$-1 - 2 T + T^{2}$$
$73$ $$56 - 16 T + T^{2}$$
$79$ $$4 - 12 T + T^{2}$$
$83$ $$-16 - 8 T + T^{2}$$
$89$ $$-103 + 10 T + T^{2}$$
$97$ $$-16 + 8 T + T^{2}$$