# Properties

 Label 1840.1.g.b Level $1840$ Weight $1$ Character orbit 1840.g Self dual yes Analytic conductor $0.918$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -115 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1840.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.918279623245$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.460.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.3385600.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{5} + q^{7} + q^{9} - q^{17} - q^{23} + q^{25} - q^{29} + q^{31} + q^{35} - q^{37} - q^{41} - 2q^{43} + q^{45} - q^{53} + q^{59} + q^{63} + q^{67} + q^{71} + q^{81} + q^{83} - q^{85} + 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
689.1
 0
0 0 0 1.00000 0 1.00000 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
115.c odd 2 1 CM by $$\Q(\sqrt{-115})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.1.g.b 1
4.b odd 2 1 460.1.d.b yes 1
5.b even 2 1 1840.1.g.a 1
20.d odd 2 1 460.1.d.a 1
20.e even 4 2 2300.1.f.a 2
23.b odd 2 1 1840.1.g.a 1
92.b even 2 1 460.1.d.a 1
115.c odd 2 1 CM 1840.1.g.b 1
460.g even 2 1 460.1.d.b yes 1
460.k odd 4 2 2300.1.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.1.d.a 1 20.d odd 2 1
460.1.d.a 1 92.b even 2 1
460.1.d.b yes 1 4.b odd 2 1
460.1.d.b yes 1 460.g even 2 1
1840.1.g.a 1 5.b even 2 1
1840.1.g.a 1 23.b odd 2 1
1840.1.g.b 1 1.a even 1 1 trivial
1840.1.g.b 1 115.c odd 2 1 CM
2300.1.f.a 2 20.e even 4 2
2300.1.f.a 2 460.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1840, [\chi])$$.

## Hecke characteristic polynomials

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