# Properties

 Label 6084.2.b.f Level $6084$ Weight $2$ Character orbit 6084.b Analytic conductor $48.581$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6084 = 2^{2} \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6084.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.5809845897$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 i q^{7} +O(q^{10})$$ $$q -4 i q^{7} -8 i q^{19} + 5 q^{25} + 4 i q^{31} -10 i q^{37} -8 q^{43} -9 q^{49} + 14 q^{61} + 16 i q^{67} -10 i q^{73} -4 q^{79} -14 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 10 q^{25} - 16 q^{43} - 18 q^{49} + 28 q^{61} - 8 q^{79} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6084\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$3043$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4393.1
 1.00000i − 1.00000i
0 0 0 0 0 4.00000i 0 0 0
4393.2 0 0 0 0 0 4.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 inner
39.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6084.2.b.f 2
3.b odd 2 1 CM 6084.2.b.f 2
13.b even 2 1 inner 6084.2.b.f 2
13.d odd 4 1 36.2.a.a 1
13.d odd 4 1 6084.2.a.i 1
39.d odd 2 1 inner 6084.2.b.f 2
39.f even 4 1 36.2.a.a 1
39.f even 4 1 6084.2.a.i 1
52.f even 4 1 144.2.a.a 1
65.f even 4 1 900.2.d.b 2
65.g odd 4 1 900.2.a.g 1
65.k even 4 1 900.2.d.b 2
91.i even 4 1 1764.2.a.e 1
91.z odd 12 2 1764.2.k.h 2
91.bb even 12 2 1764.2.k.g 2
104.j odd 4 1 576.2.a.e 1
104.m even 4 1 576.2.a.f 1
117.y odd 12 2 324.2.e.c 2
117.z even 12 2 324.2.e.c 2
143.g even 4 1 4356.2.a.g 1
156.l odd 4 1 144.2.a.a 1
195.j odd 4 1 900.2.d.b 2
195.n even 4 1 900.2.a.g 1
195.u odd 4 1 900.2.d.b 2
208.l even 4 1 2304.2.d.a 2
208.m odd 4 1 2304.2.d.q 2
208.r odd 4 1 2304.2.d.q 2
208.s even 4 1 2304.2.d.a 2
260.l odd 4 1 3600.2.f.m 2
260.s odd 4 1 3600.2.f.m 2
260.u even 4 1 3600.2.a.e 1
273.o odd 4 1 1764.2.a.e 1
273.cb odd 12 2 1764.2.k.g 2
273.cd even 12 2 1764.2.k.h 2
312.w odd 4 1 576.2.a.f 1
312.y even 4 1 576.2.a.e 1
364.p odd 4 1 7056.2.a.bb 1
429.l odd 4 1 4356.2.a.g 1
468.bs even 12 2 1296.2.i.h 2
468.ch odd 12 2 1296.2.i.h 2
624.s odd 4 1 2304.2.d.a 2
624.u even 4 1 2304.2.d.q 2
624.bm even 4 1 2304.2.d.q 2
624.bo odd 4 1 2304.2.d.a 2
780.u even 4 1 3600.2.f.m 2
780.bb odd 4 1 3600.2.a.e 1
780.bn even 4 1 3600.2.f.m 2
1092.u even 4 1 7056.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 13.d odd 4 1
36.2.a.a 1 39.f even 4 1
144.2.a.a 1 52.f even 4 1
144.2.a.a 1 156.l odd 4 1
324.2.e.c 2 117.y odd 12 2
324.2.e.c 2 117.z even 12 2
576.2.a.e 1 104.j odd 4 1
576.2.a.e 1 312.y even 4 1
576.2.a.f 1 104.m even 4 1
576.2.a.f 1 312.w odd 4 1
900.2.a.g 1 65.g odd 4 1
900.2.a.g 1 195.n even 4 1
900.2.d.b 2 65.f even 4 1
900.2.d.b 2 65.k even 4 1
900.2.d.b 2 195.j odd 4 1
900.2.d.b 2 195.u odd 4 1
1296.2.i.h 2 468.bs even 12 2
1296.2.i.h 2 468.ch odd 12 2
1764.2.a.e 1 91.i even 4 1
1764.2.a.e 1 273.o odd 4 1
1764.2.k.g 2 91.bb even 12 2
1764.2.k.g 2 273.cb odd 12 2
1764.2.k.h 2 91.z odd 12 2
1764.2.k.h 2 273.cd even 12 2
2304.2.d.a 2 208.l even 4 1
2304.2.d.a 2 208.s even 4 1
2304.2.d.a 2 624.s odd 4 1
2304.2.d.a 2 624.bo odd 4 1
2304.2.d.q 2 208.m odd 4 1
2304.2.d.q 2 208.r odd 4 1
2304.2.d.q 2 624.u even 4 1
2304.2.d.q 2 624.bm even 4 1
3600.2.a.e 1 260.u even 4 1
3600.2.a.e 1 780.bb odd 4 1
3600.2.f.m 2 260.l odd 4 1
3600.2.f.m 2 260.s odd 4 1
3600.2.f.m 2 780.u even 4 1
3600.2.f.m 2 780.bn even 4 1
4356.2.a.g 1 143.g even 4 1
4356.2.a.g 1 429.l odd 4 1
6084.2.a.i 1 13.d odd 4 1
6084.2.a.i 1 39.f even 4 1
6084.2.b.f 2 1.a even 1 1 trivial
6084.2.b.f 2 3.b odd 2 1 CM
6084.2.b.f 2 13.b even 2 1 inner
6084.2.b.f 2 39.d odd 2 1 inner
7056.2.a.bb 1 364.p odd 4 1
7056.2.a.bb 1 1092.u even 4 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 16$$ $$T_{11}$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$256 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$196 + T^{2}$$