# Properties

 Label 6534.2.a.bc Level $6534$ Weight $2$ Character orbit 6534.a Self dual yes Analytic conductor $52.174$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6534 = 2 \cdot 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6534.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$52.1742526807$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 3 q^{5} + q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + 3 q^{5} + q^{7} + q^{8} + 3 q^{10} + 4 q^{13} + q^{14} + q^{16} - 2 q^{19} + 3 q^{20} - 6 q^{23} + 4 q^{25} + 4 q^{26} + q^{28} - 6 q^{29} + 5 q^{31} + q^{32} + 3 q^{35} + 2 q^{37} - 2 q^{38} + 3 q^{40} + 6 q^{41} + 10 q^{43} - 6 q^{46} + 6 q^{47} - 6 q^{49} + 4 q^{50} + 4 q^{52} + 9 q^{53} + q^{56} - 6 q^{58} + 12 q^{59} - 8 q^{61} + 5 q^{62} + q^{64} + 12 q^{65} + 14 q^{67} + 3 q^{70} + 7 q^{73} + 2 q^{74} - 2 q^{76} - 8 q^{79} + 3 q^{80} + 6 q^{82} + 3 q^{83} + 10 q^{86} - 18 q^{89} + 4 q^{91} - 6 q^{92} + 6 q^{94} - 6 q^{95} - q^{97} - 6 q^{98} + O(q^{100})$$

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 6534.2.a.bc 1
3.b odd 2 1 6534.2.a.b 1
11.b odd 2 1 54.2.a.a 1
33.d even 2 1 54.2.a.b yes 1
44.c even 2 1 432.2.a.g 1
55.d odd 2 1 1350.2.a.r 1
55.e even 4 2 1350.2.c.b 2
77.b even 2 1 2646.2.a.a 1
88.b odd 2 1 1728.2.a.c 1
88.g even 2 1 1728.2.a.d 1
99.g even 6 2 162.2.c.b 2
99.h odd 6 2 162.2.c.c 2
132.d odd 2 1 432.2.a.b 1
143.d odd 2 1 9126.2.a.u 1
165.d even 2 1 1350.2.a.h 1
165.l odd 4 2 1350.2.c.k 2
231.h odd 2 1 2646.2.a.bd 1
264.m even 2 1 1728.2.a.y 1
264.p odd 2 1 1728.2.a.z 1
396.k even 6 2 1296.2.i.c 2
396.o odd 6 2 1296.2.i.o 2
429.e even 2 1 9126.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 11.b odd 2 1
54.2.a.b yes 1 33.d even 2 1
162.2.c.b 2 99.g even 6 2
162.2.c.c 2 99.h odd 6 2
432.2.a.b 1 132.d odd 2 1
432.2.a.g 1 44.c even 2 1
1296.2.i.c 2 396.k even 6 2
1296.2.i.o 2 396.o odd 6 2
1350.2.a.h 1 165.d even 2 1
1350.2.a.r 1 55.d odd 2 1
1350.2.c.b 2 55.e even 4 2
1350.2.c.k 2 165.l odd 4 2
1728.2.a.c 1 88.b odd 2 1
1728.2.a.d 1 88.g even 2 1
1728.2.a.y 1 264.m even 2 1
1728.2.a.z 1 264.p odd 2 1
2646.2.a.a 1 77.b even 2 1
2646.2.a.bd 1 231.h odd 2 1
6534.2.a.b 1 3.b odd 2 1
6534.2.a.bc 1 1.a even 1 1 trivial
9126.2.a.r 1 429.e even 2 1
9126.2.a.u 1 143.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6534))$$:

 $$T_{5} - 3$$ $$T_{7} - 1$$ $$T_{13} - 4$$ $$T_{17}$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$-3 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$-4 + T$$
$17$ $$T$$
$19$ $$2 + T$$
$23$ $$6 + T$$
$29$ $$6 + T$$
$31$ $$-5 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$-10 + T$$
$47$ $$-6 + T$$
$53$ $$-9 + T$$
$59$ $$-12 + T$$
$61$ $$8 + T$$
$67$ $$-14 + T$$
$71$ $$T$$
$73$ $$-7 + T$$
$79$ $$8 + T$$
$83$ $$-3 + T$$
$89$ $$18 + T$$
$97$ $$1 + T$$