# Properties

 Label 2775.1.b.a Level $2775$ Weight $1$ Character orbit 2775.b Analytic conductor $1.385$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -111, 37 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2775 = 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2775.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.38490541006$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 111) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{37})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.1732640625.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{3} + q^{4} -2 i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + q^{4} -2 i q^{7} - q^{9} -i q^{12} + q^{16} -2 q^{21} + i q^{27} -2 i q^{28} - q^{36} + i q^{37} -i q^{48} -3 q^{49} + 2 i q^{63} + q^{64} -2 i q^{67} + 2 i q^{73} + q^{81} -2 q^{84} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{9} + 2q^{16} - 4q^{21} - 2q^{36} - 6q^{49} + 2q^{64} + 2q^{81} - 4q^{84} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times$$.

 $$n$$ $$76$$ $$926$$ $$1777$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2774.1
 1.00000i − 1.00000i
0 1.00000i 1.00000 0 0 2.00000i 0 −1.00000 0
2774.2 0 1.00000i 1.00000 0 0 2.00000i 0 −1.00000 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 $$\Q(\sqrt{-3})$$
37.b even 2 1 RM by $$\Q(\sqrt{37})$$
111.d odd 2 1 CM by $$\Q(\sqrt{-111})$$
5.b even 2 1 inner
15.d odd 2 1 inner
185.d even 2 1 inner
555.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.b.a 2
3.b odd 2 1 CM 2775.1.b.a 2
5.b even 2 1 inner 2775.1.b.a 2
5.c odd 4 1 111.1.d.a 1
5.c odd 4 1 2775.1.h.a 1
15.d odd 2 1 inner 2775.1.b.a 2
15.e even 4 1 111.1.d.a 1
15.e even 4 1 2775.1.h.a 1
20.e even 4 1 1776.1.n.a 1
37.b even 2 1 RM 2775.1.b.a 2
45.k odd 12 2 2997.1.n.b 2
45.l even 12 2 2997.1.n.b 2
60.l odd 4 1 1776.1.n.a 1
111.d odd 2 1 CM 2775.1.b.a 2
185.d even 2 1 inner 2775.1.b.a 2
185.h odd 4 1 111.1.d.a 1
185.h odd 4 1 2775.1.h.a 1
555.b odd 2 1 inner 2775.1.b.a 2
555.n even 4 1 111.1.d.a 1
555.n even 4 1 2775.1.h.a 1
740.m even 4 1 1776.1.n.a 1
1665.dc odd 12 2 2997.1.n.b 2
1665.di even 12 2 2997.1.n.b 2
2220.bf odd 4 1 1776.1.n.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.a 1 5.c odd 4 1
111.1.d.a 1 15.e even 4 1
111.1.d.a 1 185.h odd 4 1
111.1.d.a 1 555.n even 4 1
1776.1.n.a 1 20.e even 4 1
1776.1.n.a 1 60.l odd 4 1
1776.1.n.a 1 740.m even 4 1
1776.1.n.a 1 2220.bf odd 4 1
2775.1.b.a 2 1.a even 1 1 trivial
2775.1.b.a 2 3.b odd 2 1 CM
2775.1.b.a 2 5.b even 2 1 inner
2775.1.b.a 2 15.d odd 2 1 inner
2775.1.b.a 2 37.b even 2 1 RM
2775.1.b.a 2 111.d odd 2 1 CM
2775.1.b.a 2 185.d even 2 1 inner
2775.1.b.a 2 555.b odd 2 1 inner
2775.1.h.a 1 5.c odd 4 1
2775.1.h.a 1 15.e even 4 1
2775.1.h.a 1 185.h odd 4 1
2775.1.h.a 1 555.n even 4 1
2997.1.n.b 2 45.k odd 12 2
2997.1.n.b 2 45.l even 12 2
2997.1.n.b 2 1665.dc odd 12 2
2997.1.n.b 2 1665.di even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(2775, [\chi])$$.

## Hecke characteristic polynomials

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