# Properties

 Label 3744.1.bd.d Level $3744$ Weight $1$ Character orbit 3744.bd Analytic conductor $1.868$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ RM discriminant 12 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.1265472.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q + ( 1 - i ) q^{11} - q^{13} -2 i q^{23} -i q^{25} + ( -1 + i ) q^{37} + ( 1 - i ) q^{47} + i q^{49} + ( 1 - i ) q^{59} + ( 1 + i ) q^{71} + ( 1 - i ) q^{73} + ( -1 - i ) q^{83} + ( 1 + i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 2 q^{11} - 2 q^{13} - 2 q^{37} + 2 q^{47} + 2 q^{59} + 2 q^{71} + 2 q^{73} - 2 q^{83} + 2 q^{97} + O(q^{100})$$

## 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 0 0 0 0 0 0
2881.1 0 0 0 0 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
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
13.d odd 4 1 inner
156.l odd 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.d yes 2
3.b odd 2 1 3744.1.bd.c 2
4.b odd 2 1 3744.1.bd.c 2
12.b even 2 1 RM 3744.1.bd.d yes 2
13.d odd 4 1 inner 3744.1.bd.d yes 2
39.f even 4 1 3744.1.bd.c 2
52.f even 4 1 3744.1.bd.c 2
156.l odd 4 1 inner 3744.1.bd.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.bd.c 2 3.b odd 2 1
3744.1.bd.c 2 4.b odd 2 1
3744.1.bd.c 2 39.f even 4 1
3744.1.bd.c 2 52.f even 4 1
3744.1.bd.d yes 2 1.a even 1 1 trivial
3744.1.bd.d yes 2 12.b even 2 1 RM
3744.1.bd.d yes 2 13.d odd 4 1 inner
3744.1.bd.d yes 2 156.l odd 4 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

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