# Properties

 Label 3744.1.o.c Level $3744$ Weight $1$ Character orbit 3744.o Analytic conductor $1.868$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -39 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3744.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 936) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.32448.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} + \zeta_{8}^{2} q^{13} + q^{25} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} -2 q^{43} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} - q^{49} + 2 \zeta_{8}^{2} q^{55} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{59} -2 \zeta_{8}^{2} q^{61} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{65} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{71} -2 \zeta_{8}^{2} q^{79} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{25} - 8 q^{43} - 4 q^{49} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2287.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −1.41421 0 0 0 0 0
2287.2 0 0 0 −1.41421 0 0 0 0 0
2287.3 0 0 0 1.41421 0 0 0 0 0
2287.4 0 0 0 1.41421 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
24.f even 2 1 inner
104.h odd 2 1 inner
312.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.o.c 4
3.b odd 2 1 inner 3744.1.o.c 4
4.b odd 2 1 936.1.o.c 4
8.b even 2 1 936.1.o.c 4
8.d odd 2 1 inner 3744.1.o.c 4
12.b even 2 1 936.1.o.c 4
13.b even 2 1 inner 3744.1.o.c 4
24.f even 2 1 inner 3744.1.o.c 4
24.h odd 2 1 936.1.o.c 4
39.d odd 2 1 CM 3744.1.o.c 4
52.b odd 2 1 936.1.o.c 4
104.e even 2 1 936.1.o.c 4
104.h odd 2 1 inner 3744.1.o.c 4
156.h even 2 1 936.1.o.c 4
312.b odd 2 1 936.1.o.c 4
312.h even 2 1 inner 3744.1.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.1.o.c 4 4.b odd 2 1
936.1.o.c 4 8.b even 2 1
936.1.o.c 4 12.b even 2 1
936.1.o.c 4 24.h odd 2 1
936.1.o.c 4 52.b odd 2 1
936.1.o.c 4 104.e even 2 1
936.1.o.c 4 156.h even 2 1
936.1.o.c 4 312.b odd 2 1
3744.1.o.c 4 1.a even 1 1 trivial
3744.1.o.c 4 3.b odd 2 1 inner
3744.1.o.c 4 8.d odd 2 1 inner
3744.1.o.c 4 13.b even 2 1 inner
3744.1.o.c 4 24.f even 2 1 inner
3744.1.o.c 4 39.d odd 2 1 CM
3744.1.o.c 4 104.h odd 2 1 inner
3744.1.o.c 4 312.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(3744, [\chi])$$.

## Hecke characteristic polynomials

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