# Properties

 Label 2700.1.b.b Level $2700$ Weight $1$ Character orbit 2700.b Analytic conductor $1.347$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34747553411$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.108.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.265720500000000.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{7} +O(q^{10})$$ $$q -i q^{7} + i q^{13} + q^{19} + 2 q^{31} -i q^{37} -2 i q^{43} - q^{61} -i q^{67} + i q^{73} + q^{79} + q^{91} -i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{19} + 4q^{31} - 2q^{61} + 2q^{79} + 2q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1349.1
 1.00000i − 1.00000i
0 0 0 0 0 1.00000i 0 0 0
1349.2 0 0 0 0 0 1.00000i 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.b.b 2
3.b odd 2 1 CM 2700.1.b.b 2
5.b even 2 1 inner 2700.1.b.b 2
5.c odd 4 1 108.1.c.a 1
5.c odd 4 1 2700.1.g.b 1
15.d odd 2 1 inner 2700.1.b.b 2
15.e even 4 1 108.1.c.a 1
15.e even 4 1 2700.1.g.b 1
20.e even 4 1 432.1.e.a 1
40.i odd 4 1 1728.1.e.a 1
40.k even 4 1 1728.1.e.b 1
45.k odd 12 2 324.1.g.a 2
45.l even 12 2 324.1.g.a 2
60.l odd 4 1 432.1.e.a 1
120.q odd 4 1 1728.1.e.b 1
120.w even 4 1 1728.1.e.a 1
135.q even 36 6 2916.1.k.c 6
135.r odd 36 6 2916.1.k.c 6
180.v odd 12 2 1296.1.q.a 2
180.x even 12 2 1296.1.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 5.c odd 4 1
108.1.c.a 1 15.e even 4 1
324.1.g.a 2 45.k odd 12 2
324.1.g.a 2 45.l even 12 2
432.1.e.a 1 20.e even 4 1
432.1.e.a 1 60.l odd 4 1
1296.1.q.a 2 180.v odd 12 2
1296.1.q.a 2 180.x even 12 2
1728.1.e.a 1 40.i odd 4 1
1728.1.e.a 1 120.w even 4 1
1728.1.e.b 1 40.k even 4 1
1728.1.e.b 1 120.q odd 4 1
2700.1.b.b 2 1.a even 1 1 trivial
2700.1.b.b 2 3.b odd 2 1 CM
2700.1.b.b 2 5.b even 2 1 inner
2700.1.b.b 2 15.d odd 2 1 inner
2700.1.g.b 1 5.c odd 4 1
2700.1.g.b 1 15.e even 4 1
2916.1.k.c 6 135.q even 36 6
2916.1.k.c 6 135.r odd 36 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2700, [\chi])$$.

## Hecke characteristic polynomials

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