# Properties

 Label 2700.1.g.b Level 2700 Weight 1 Character orbit 2700.g Self dual yes Analytic conductor 1.347 Analytic rank 0 Dimension 1 Projective image $$D_{3}$$ CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.34747553411$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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 $D_6$ Artin field Galois closure of 6.2.1458000.2

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 0
0 0 0 0 0 1.00000 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 $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.g.b 1
3.b odd 2 1 CM 2700.1.g.b 1
5.b even 2 1 108.1.c.a 1
5.c odd 4 2 2700.1.b.b 2
15.d odd 2 1 108.1.c.a 1
15.e even 4 2 2700.1.b.b 2
20.d odd 2 1 432.1.e.a 1
40.e odd 2 1 1728.1.e.b 1
40.f even 2 1 1728.1.e.a 1
45.h odd 6 2 324.1.g.a 2
45.j even 6 2 324.1.g.a 2
60.h even 2 1 432.1.e.a 1
120.i odd 2 1 1728.1.e.a 1
120.m even 2 1 1728.1.e.b 1
135.n odd 18 6 2916.1.k.c 6
135.p even 18 6 2916.1.k.c 6
180.n even 6 2 1296.1.q.a 2
180.p odd 6 2 1296.1.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 5.b even 2 1
108.1.c.a 1 15.d odd 2 1
324.1.g.a 2 45.h odd 6 2
324.1.g.a 2 45.j even 6 2
432.1.e.a 1 20.d odd 2 1
432.1.e.a 1 60.h even 2 1
1296.1.q.a 2 180.n even 6 2
1296.1.q.a 2 180.p odd 6 2
1728.1.e.a 1 40.f even 2 1
1728.1.e.a 1 120.i odd 2 1
1728.1.e.b 1 40.e odd 2 1
1728.1.e.b 1 120.m even 2 1
2700.1.b.b 2 5.c odd 4 2
2700.1.b.b 2 15.e even 4 2
2700.1.g.b 1 1.a even 1 1 trivial
2700.1.g.b 1 3.b odd 2 1 CM
2916.1.k.c 6 135.n odd 18 6
2916.1.k.c 6 135.p even 18 6

## Hecke kernels

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

## Hecke characteristic polynomials

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