# Properties

 Label 3724.1.e.d Level $3724$ Weight $1$ Character orbit 3724.e Self dual yes Analytic conductor $1.859$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3724.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.85851810705$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.3724.2 Artin image: $D_6$ Artin field: Galois closure of 6.0.97077232.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{9} + O(q^{10})$$ $$q + q^{5} + q^{9} + 2q^{11} + q^{17} - q^{19} - q^{23} - q^{43} + q^{45} - 2q^{47} + 2q^{55} - 2q^{61} - 2q^{73} + q^{81} + q^{83} + q^{85} - q^{95} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times$$.

 $$n$$ $$1863$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1177.1
 0
0 0 0 1.00000 0 0 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.e.d 1
7.b odd 2 1 3724.1.e.b 1
7.c even 3 2 3724.1.bc.a 2
7.d odd 6 2 532.1.bc.b 2
19.b odd 2 1 CM 3724.1.e.d 1
28.f even 6 2 2128.1.cl.b 2
133.c even 2 1 3724.1.e.b 1
133.o even 6 2 532.1.bc.b 2
133.r odd 6 2 3724.1.bc.a 2
532.bh odd 6 2 2128.1.cl.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.1.bc.b 2 7.d odd 6 2
532.1.bc.b 2 133.o even 6 2
2128.1.cl.b 2 28.f even 6 2
2128.1.cl.b 2 532.bh odd 6 2
3724.1.e.b 1 7.b odd 2 1
3724.1.e.b 1 133.c even 2 1
3724.1.e.d 1 1.a even 1 1 trivial
3724.1.e.d 1 19.b odd 2 1 CM
3724.1.bc.a 2 7.c even 3 2
3724.1.bc.a 2 133.r odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3724, [\chi])$$:

 $$T_{5} - 1$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

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