# Properties

 Label 6498.2.a.x Level $6498$ Weight $2$ Character orbit 6498.a Self dual yes Analytic conductor $51.887$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6498 = 2 \cdot 3^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6498.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.8867912334$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 4q^{5} - 3q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + 4q^{5} - 3q^{7} + q^{8} + 4q^{10} - 2q^{11} + 7q^{13} - 3q^{14} + q^{16} + 4q^{20} - 2q^{22} + 4q^{23} + 11q^{25} + 7q^{26} - 3q^{28} + 4q^{29} - q^{31} + q^{32} - 12q^{35} - 7q^{37} + 4q^{40} + 4q^{41} + 7q^{43} - 2q^{44} + 4q^{46} - 2q^{47} + 2q^{49} + 11q^{50} + 7q^{52} - 4q^{53} - 8q^{55} - 3q^{56} + 4q^{58} - 6q^{59} - q^{61} - q^{62} + q^{64} + 28q^{65} - 3q^{67} - 12q^{70} + 2q^{71} - 3q^{73} - 7q^{74} + 6q^{77} - 5q^{79} + 4q^{80} + 4q^{82} + 12q^{83} + 7q^{86} - 2q^{88} + 18q^{89} - 21q^{91} + 4q^{92} - 2q^{94} - 10q^{97} + 2q^{98} + O(q^{100})$$

## 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
 0
1.00000 0 1.00000 4.00000 0 −3.00000 1.00000 0 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 6498.2.a.x 1
3.b odd 2 1 2166.2.a.c 1
19.b odd 2 1 6498.2.a.l 1
19.d odd 6 2 342.2.g.d 2
57.d even 2 1 2166.2.a.f 1
57.f even 6 2 114.2.e.a 2
76.f even 6 2 2736.2.s.c 2
228.n odd 6 2 912.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 57.f even 6 2
342.2.g.d 2 19.d odd 6 2
912.2.q.d 2 228.n odd 6 2
2166.2.a.c 1 3.b odd 2 1
2166.2.a.f 1 57.d even 2 1
2736.2.s.c 2 76.f even 6 2
6498.2.a.l 1 19.b odd 2 1
6498.2.a.x 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6498))$$:

 $$T_{5} - 4$$ $$T_{7} + 3$$ $$T_{11} + 2$$ $$T_{13} - 7$$ $$T_{29} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$3 + T$$
$11$ $$2 + T$$
$13$ $$-7 + T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$-4 + T$$
$29$ $$-4 + T$$
$31$ $$1 + T$$
$37$ $$7 + T$$
$41$ $$-4 + T$$
$43$ $$-7 + T$$
$47$ $$2 + T$$
$53$ $$4 + T$$
$59$ $$6 + T$$
$61$ $$1 + T$$
$67$ $$3 + T$$
$71$ $$-2 + T$$
$73$ $$3 + T$$
$79$ $$5 + T$$
$83$ $$-12 + T$$
$89$ $$-18 + T$$
$97$ $$10 + T$$