Properties

 Label 6084.2.a.b Level $6084$ Weight $2$ Character orbit 6084.a Self dual yes Analytic conductor $48.581$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6084 = 2^{2} \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6084.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$48.5809845897$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{5} + 2q^{7} + O(q^{10})$$ $$q - 4q^{5} + 2q^{7} - 4q^{11} - 2q^{17} + 2q^{19} + 11q^{25} + 6q^{29} + 10q^{31} - 8q^{35} - 10q^{37} + 8q^{41} + 4q^{43} - 4q^{47} - 3q^{49} + 10q^{53} + 16q^{55} - 8q^{59} - 14q^{61} - 2q^{67} + 16q^{71} + 10q^{73} - 8q^{77} - 16q^{79} + 8q^{85} - 4q^{89} - 8q^{95} + 2q^{97} + O(q^{100})$$

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
0 0 0 −4.00000 0 2.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$1$$

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 6084.2.a.b 1
3.b odd 2 1 2028.2.a.c 1
12.b even 2 1 8112.2.a.bi 1
13.b even 2 1 468.2.a.d 1
13.d odd 4 2 6084.2.b.j 2
39.d odd 2 1 156.2.a.a 1
39.f even 4 2 2028.2.b.a 2
39.h odd 6 2 2028.2.i.e 2
39.i odd 6 2 2028.2.i.g 2
39.k even 12 4 2028.2.q.h 4
52.b odd 2 1 1872.2.a.s 1
104.e even 2 1 7488.2.a.c 1
104.h odd 2 1 7488.2.a.d 1
117.n odd 6 2 4212.2.i.l 2
117.t even 6 2 4212.2.i.b 2
156.h even 2 1 624.2.a.e 1
195.e odd 2 1 3900.2.a.m 1
195.s even 4 2 3900.2.h.b 2
273.g even 2 1 7644.2.a.k 1
312.b odd 2 1 2496.2.a.bc 1
312.h even 2 1 2496.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 39.d odd 2 1
468.2.a.d 1 13.b even 2 1
624.2.a.e 1 156.h even 2 1
1872.2.a.s 1 52.b odd 2 1
2028.2.a.c 1 3.b odd 2 1
2028.2.b.a 2 39.f even 4 2
2028.2.i.e 2 39.h odd 6 2
2028.2.i.g 2 39.i odd 6 2
2028.2.q.h 4 39.k even 12 4
2496.2.a.o 1 312.h even 2 1
2496.2.a.bc 1 312.b odd 2 1
3900.2.a.m 1 195.e odd 2 1
3900.2.h.b 2 195.s even 4 2
4212.2.i.b 2 117.t even 6 2
4212.2.i.l 2 117.n odd 6 2
6084.2.a.b 1 1.a even 1 1 trivial
6084.2.b.j 2 13.d odd 4 2
7488.2.a.c 1 104.e even 2 1
7488.2.a.d 1 104.h odd 2 1
7644.2.a.k 1 273.g even 2 1
8112.2.a.bi 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6084))$$:

 $$T_{5} + 4$$ $$T_{7} - 2$$ $$T_{11} + 4$$

Hecke characteristic polynomials

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