# Properties

 Label 3724.1.bc.d Level $3724$ Weight $1$ Character orbit 3724.bc Analytic conductor $1.859$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3724.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.85851810705$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.3724.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{17} + \zeta_{6} q^{19} + \zeta_{6} q^{23} + 3 \zeta_{6}^{2} q^{25} + 2 q^{43} -2 \zeta_{6}^{2} q^{45} -\zeta_{6} q^{47} + 2 q^{55} -\zeta_{6} q^{61} + \zeta_{6}^{2} q^{73} + \zeta_{6}^{2} q^{81} + q^{83} -2 q^{85} + 2 \zeta_{6}^{2} q^{95} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - q^{9} + O(q^{10})$$ $$2q + 2q^{5} - q^{9} + q^{11} - q^{17} + q^{19} + q^{23} - 3q^{25} + 4q^{43} + 2q^{45} - q^{47} + 4q^{55} - q^{61} - q^{73} - q^{81} + 2q^{83} - 4q^{85} - 2q^{95} - 2q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$1863$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.00000 1.73205i 0 0 0 −0.500000 + 0.866025i 0
949.1 0 0 0 1.00000 + 1.73205i 0 0 0 −0.500000 0.866025i 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 $$\Q(\sqrt{-19})$$
7.c even 3 1 inner
133.r odd 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{11}^{2} - T_{11} + 1$$

## Hecke characteristic polynomials

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