# Properties

 Label 1776.1.n.a Level $1776$ Weight $1$ Character orbit 1776.n Self dual yes Analytic conductor $0.886$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -111, 37 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1776 = 2^{4} \cdot 3 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1776.n (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.886339462436$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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: $D_4$ Artin field: Galois closure of 4.0.5328.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 2q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + 2q^{7} + q^{9} - 2q^{21} - q^{25} - q^{27} + q^{37} + 3q^{49} + 2q^{63} + 2q^{67} - 2q^{73} + q^{75} + q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times$$.

 $$n$$ $$223$$ $$593$$ $$1297$$ $$1333$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1553.1
 0
0 −1.00000 0 0 0 2.00000 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 $$\Q(\sqrt{-111})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1776.1.n.a 1
3.b odd 2 1 CM 1776.1.n.a 1
4.b odd 2 1 111.1.d.a 1
12.b even 2 1 111.1.d.a 1
20.d odd 2 1 2775.1.h.a 1
20.e even 4 2 2775.1.b.a 2
36.f odd 6 2 2997.1.n.b 2
36.h even 6 2 2997.1.n.b 2
37.b even 2 1 RM 1776.1.n.a 1
60.h even 2 1 2775.1.h.a 1
60.l odd 4 2 2775.1.b.a 2
111.d odd 2 1 CM 1776.1.n.a 1
148.b odd 2 1 111.1.d.a 1
444.g even 2 1 111.1.d.a 1
740.g odd 2 1 2775.1.h.a 1
740.m even 4 2 2775.1.b.a 2
1332.z even 6 2 2997.1.n.b 2
1332.bl odd 6 2 2997.1.n.b 2
2220.p even 2 1 2775.1.h.a 1
2220.bf odd 4 2 2775.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.a 1 4.b odd 2 1
111.1.d.a 1 12.b even 2 1
111.1.d.a 1 148.b odd 2 1
111.1.d.a 1 444.g even 2 1
1776.1.n.a 1 1.a even 1 1 trivial
1776.1.n.a 1 3.b odd 2 1 CM
1776.1.n.a 1 37.b even 2 1 RM
1776.1.n.a 1 111.d odd 2 1 CM
2775.1.b.a 2 20.e even 4 2
2775.1.b.a 2 60.l odd 4 2
2775.1.b.a 2 740.m even 4 2
2775.1.b.a 2 2220.bf odd 4 2
2775.1.h.a 1 20.d odd 2 1
2775.1.h.a 1 60.h even 2 1
2775.1.h.a 1 740.g odd 2 1
2775.1.h.a 1 2220.p even 2 1
2997.1.n.b 2 36.f odd 6 2
2997.1.n.b 2 36.h even 6 2
2997.1.n.b 2 1332.z even 6 2
2997.1.n.b 2 1332.bl odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1776, [\chi])$$:

 $$T_{5}$$ $$T_{7} - 2$$

## Hecke characteristic polynomials

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