# Properties

 Label 6498.2.a.s Level $6498$ Weight $2$ Character orbit 6498.a Self dual yes Analytic conductor $51.887$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 4q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} - 4q^{7} + q^{8} - 3q^{11} + 2q^{13} - 4q^{14} + q^{16} + 6q^{17} - 3q^{22} + 6q^{23} - 5q^{25} + 2q^{26} - 4q^{28} + 2q^{31} + q^{32} + 6q^{34} - 10q^{37} - 9q^{41} - 4q^{43} - 3q^{44} + 6q^{46} + 9q^{49} - 5q^{50} + 2q^{52} - 6q^{53} - 4q^{56} + 9q^{59} - 4q^{61} + 2q^{62} + q^{64} - 7q^{67} + 6q^{68} + 6q^{71} - q^{73} - 10q^{74} + 12q^{77} - 4q^{79} - 9q^{82} - 3q^{83} - 4q^{86} - 3q^{88} - 6q^{89} - 8q^{91} + 6q^{92} + 17q^{97} + 9q^{98} + O(q^{100})$$

## 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 0 0 −4.00000 1.00000 0 0
 $$n$$: 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.s 1
3.b odd 2 1 722.2.a.c 1
12.b even 2 1 5776.2.a.g 1
19.b odd 2 1 6498.2.a.e 1
19.c even 3 2 342.2.g.b 2
57.d even 2 1 722.2.a.d 1
57.f even 6 2 722.2.c.b 2
57.h odd 6 2 38.2.c.a 2
57.j even 18 6 722.2.e.i 6
57.l odd 18 6 722.2.e.j 6
76.g odd 6 2 2736.2.s.m 2
228.b odd 2 1 5776.2.a.n 1
228.m even 6 2 304.2.i.c 2
285.n odd 6 2 950.2.e.d 2
285.v even 12 4 950.2.j.e 4
456.u even 6 2 1216.2.i.d 2
456.x odd 6 2 1216.2.i.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 57.h odd 6 2
304.2.i.c 2 228.m even 6 2
342.2.g.b 2 19.c even 3 2
722.2.a.c 1 3.b odd 2 1
722.2.a.d 1 57.d even 2 1
722.2.c.b 2 57.f even 6 2
722.2.e.i 6 57.j even 18 6
722.2.e.j 6 57.l odd 18 6
950.2.e.d 2 285.n odd 6 2
950.2.j.e 4 285.v even 12 4
1216.2.i.d 2 456.u even 6 2
1216.2.i.h 2 456.x odd 6 2
2736.2.s.m 2 76.g odd 6 2
5776.2.a.g 1 12.b even 2 1
5776.2.a.n 1 228.b odd 2 1
6498.2.a.e 1 19.b odd 2 1
6498.2.a.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6498))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$3 + T$$
$13$ $$-2 + T$$
$17$ $$-6 + T$$
$19$ $$T$$
$23$ $$-6 + T$$
$29$ $$T$$
$31$ $$-2 + T$$
$37$ $$10 + T$$
$41$ $$9 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$-9 + T$$
$61$ $$4 + T$$
$67$ $$7 + T$$
$71$ $$-6 + T$$
$73$ $$1 + T$$
$79$ $$4 + T$$
$83$ $$3 + T$$
$89$ $$6 + T$$
$97$ $$-17 + T$$