# Properties

 Label 2700.1.c.b Level $2700$ Weight $1$ Character orbit 2700.c Analytic conductor $1.347$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -15 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2700.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34747553411$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.1166400.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} - q^{8} +O(q^{10})$$ $$q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} - q^{8} + \zeta_{6}^{2} q^{16} - q^{17} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{19} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{23} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} + \zeta_{6} q^{32} + \zeta_{6}^{2} q^{34} + ( -1 - \zeta_{6} ) q^{38} + ( -1 - \zeta_{6} ) q^{46} + q^{49} - q^{53} + q^{61} + ( -1 - \zeta_{6} ) q^{62} + q^{64} + \zeta_{6} q^{68} + ( -1 + \zeta_{6}^{2} ) q^{76} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{79} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{83} + ( -1 + \zeta_{6}^{2} ) q^{92} -\zeta_{6}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} - 2q^{8} - q^{16} - 2q^{17} + q^{32} - q^{34} - 3q^{38} - 3q^{46} + 2q^{49} - 2q^{53} + 2q^{61} - 3q^{62} + 2q^{64} + q^{68} - 3q^{76} - 3q^{92} + q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 0 0
1351.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
4.b odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.c.b 2
3.b odd 2 1 2700.1.c.a 2
4.b odd 2 1 inner 2700.1.c.b 2
5.b even 2 1 2700.1.c.a 2
5.c odd 4 2 540.1.f.a 4
12.b even 2 1 2700.1.c.a 2
15.d odd 2 1 CM 2700.1.c.b 2
15.e even 4 2 540.1.f.a 4
20.d odd 2 1 2700.1.c.a 2
20.e even 4 2 540.1.f.a 4
45.k odd 12 2 1620.1.p.a 4
45.k odd 12 2 1620.1.p.d 4
45.l even 12 2 1620.1.p.a 4
45.l even 12 2 1620.1.p.d 4
60.h even 2 1 inner 2700.1.c.b 2
60.l odd 4 2 540.1.f.a 4
180.v odd 12 2 1620.1.p.a 4
180.v odd 12 2 1620.1.p.d 4
180.x even 12 2 1620.1.p.a 4
180.x even 12 2 1620.1.p.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.1.f.a 4 5.c odd 4 2
540.1.f.a 4 15.e even 4 2
540.1.f.a 4 20.e even 4 2
540.1.f.a 4 60.l odd 4 2
1620.1.p.a 4 45.k odd 12 2
1620.1.p.a 4 45.l even 12 2
1620.1.p.a 4 180.v odd 12 2
1620.1.p.a 4 180.x even 12 2
1620.1.p.d 4 45.k odd 12 2
1620.1.p.d 4 45.l even 12 2
1620.1.p.d 4 180.v odd 12 2
1620.1.p.d 4 180.x even 12 2
2700.1.c.a 2 3.b odd 2 1
2700.1.c.a 2 5.b even 2 1
2700.1.c.a 2 12.b even 2 1
2700.1.c.a 2 20.d odd 2 1
2700.1.c.b 2 1.a even 1 1 trivial
2700.1.c.b 2 4.b odd 2 1 inner
2700.1.c.b 2 15.d odd 2 1 CM
2700.1.c.b 2 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2700, [\chi])$$.

## Hecke characteristic polynomials

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