# Properties

 Label 3744.1.bd.b Level $3744$ Weight $1$ Character orbit 3744.bd Analytic conductor $1.868$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3744.bd (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 416) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.35152.1 Artin image: $C_2\times C_4\wr C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{5} +O(q^{10})$$ $$q + ( -1 - i ) q^{5} + i q^{13} + 2 i q^{17} + i q^{25} + ( 1 - i ) q^{37} + ( 1 + i ) q^{41} + i q^{49} + 2 q^{53} -2 q^{61} + ( 1 - i ) q^{65} + ( 1 - i ) q^{73} + ( 2 - 2 i ) q^{85} + ( 1 - i ) q^{89} + ( 1 + i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + O(q^{10})$$ $$2 q - 2 q^{5} + 2 q^{37} + 2 q^{41} + 4 q^{53} - 4 q^{61} + 2 q^{65} + 2 q^{73} + 4 q^{85} + 2 q^{89} + 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$1$$ $$i$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 0 0 −1.00000 + 1.00000i 0 0 0 0 0
2881.1 0 0 0 −1.00000 1.00000i 0 0 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
13.d odd 4 1 inner
52.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.bd.b 2
3.b odd 2 1 416.1.t.a 2
4.b odd 2 1 CM 3744.1.bd.b 2
12.b even 2 1 416.1.t.a 2
13.d odd 4 1 inner 3744.1.bd.b 2
24.f even 2 1 832.1.t.a 2
24.h odd 2 1 832.1.t.a 2
39.f even 4 1 416.1.t.a 2
48.i odd 4 1 3328.1.j.a 2
48.i odd 4 1 3328.1.j.b 2
48.k even 4 1 3328.1.j.a 2
48.k even 4 1 3328.1.j.b 2
52.f even 4 1 inner 3744.1.bd.b 2
156.l odd 4 1 416.1.t.a 2
312.w odd 4 1 832.1.t.a 2
312.y even 4 1 832.1.t.a 2
624.s odd 4 1 3328.1.j.b 2
624.u even 4 1 3328.1.j.b 2
624.bm even 4 1 3328.1.j.a 2
624.bo odd 4 1 3328.1.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.t.a 2 3.b odd 2 1
416.1.t.a 2 12.b even 2 1
416.1.t.a 2 39.f even 4 1
416.1.t.a 2 156.l odd 4 1
832.1.t.a 2 24.f even 2 1
832.1.t.a 2 24.h odd 2 1
832.1.t.a 2 312.w odd 4 1
832.1.t.a 2 312.y even 4 1
3328.1.j.a 2 48.i odd 4 1
3328.1.j.a 2 48.k even 4 1
3328.1.j.a 2 624.bm even 4 1
3328.1.j.a 2 624.bo odd 4 1
3328.1.j.b 2 48.i odd 4 1
3328.1.j.b 2 48.k even 4 1
3328.1.j.b 2 624.s odd 4 1
3328.1.j.b 2 624.u even 4 1
3744.1.bd.b 2 1.a even 1 1 trivial
3744.1.bd.b 2 4.b odd 2 1 CM
3744.1.bd.b 2 13.d odd 4 1 inner
3744.1.bd.b 2 52.f even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3744, [\chi])$$:

 $$T_{5}^{2} + 2 T_{5} + 2$$ $$T_{11}$$ $$T_{17}^{2} + 4$$

## Hecke characteristic polynomials

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