# Properties

 Label 3744.2.m.d Level $3744$ Weight $2$ Character orbit 3744.m Analytic conductor $29.896$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$29.8959905168$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{5}+O(q^{10})$$ q + 2 * q^5 $$q + 2 q^{5} - 4 q^{11} + ( - \beta + 3) q^{13} + 6 q^{17} + 6 q^{19} - q^{25} + 3 \beta q^{29} - 6 q^{37} - \beta q^{41} - 6 \beta q^{43} + 4 \beta q^{47} + 7 q^{49} + 3 \beta q^{53} - 8 q^{55} + 4 q^{59} - 4 \beta q^{61} + ( - 2 \beta + 6) q^{65} + 6 q^{67} + 2 \beta q^{71} - 6 \beta q^{73} - 6 q^{79} + 8 q^{83} + 12 q^{85} + \beta q^{89} + 12 q^{95} + 6 \beta q^{97} +O(q^{100})$$ q + 2 * q^5 - 4 * q^11 + (-b + 3) * q^13 + 6 * q^17 + 6 * q^19 - q^25 + 3*b * q^29 - 6 * q^37 - b * q^41 - 6*b * q^43 + 4*b * q^47 + 7 * q^49 + 3*b * q^53 - 8 * q^55 + 4 * q^59 - 4*b * q^61 + (-2*b + 6) * q^65 + 6 * q^67 + 2*b * q^71 - 6*b * q^73 - 6 * q^79 + 8 * q^83 + 12 * q^85 + b * q^89 + 12 * q^95 + 6*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} - 8 q^{11} + 6 q^{13} + 12 q^{17} + 12 q^{19} - 2 q^{25} - 12 q^{37} + 14 q^{49} - 16 q^{55} + 8 q^{59} + 12 q^{65} + 12 q^{67} - 12 q^{79} + 16 q^{83} + 24 q^{85} + 24 q^{95}+O(q^{100})$$ 2 * q + 4 * q^5 - 8 * q^11 + 6 * q^13 + 12 * q^17 + 12 * q^19 - 2 * q^25 - 12 * q^37 + 14 * q^49 - 16 * q^55 + 8 * q^59 + 12 * q^65 + 12 * q^67 - 12 * q^79 + 16 * q^83 + 24 * q^85 + 24 * q^95

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1585.1
 1.00000i − 1.00000i
0 0 0 2.00000 0 0 0 0 0
1585.2 0 0 0 2.00000 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
104.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.m.d 2
3.b odd 2 1 1248.2.m.a 2
4.b odd 2 1 936.2.m.d 2
8.b even 2 1 3744.2.m.a 2
8.d odd 2 1 936.2.m.a 2
12.b even 2 1 312.2.m.a 2
13.b even 2 1 3744.2.m.a 2
24.f even 2 1 312.2.m.b yes 2
24.h odd 2 1 1248.2.m.b 2
39.d odd 2 1 1248.2.m.b 2
52.b odd 2 1 936.2.m.a 2
104.e even 2 1 inner 3744.2.m.d 2
104.h odd 2 1 936.2.m.d 2
156.h even 2 1 312.2.m.b yes 2
312.b odd 2 1 1248.2.m.a 2
312.h even 2 1 312.2.m.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.m.a 2 12.b even 2 1
312.2.m.a 2 312.h even 2 1
312.2.m.b yes 2 24.f even 2 1
312.2.m.b yes 2 156.h even 2 1
936.2.m.a 2 8.d odd 2 1
936.2.m.a 2 52.b odd 2 1
936.2.m.d 2 4.b odd 2 1
936.2.m.d 2 104.h odd 2 1
1248.2.m.a 2 3.b odd 2 1
1248.2.m.a 2 312.b odd 2 1
1248.2.m.b 2 24.h odd 2 1
1248.2.m.b 2 39.d odd 2 1
3744.2.m.a 2 8.b even 2 1
3744.2.m.a 2 13.b even 2 1
3744.2.m.d 2 1.a even 1 1 trivial
3744.2.m.d 2 104.e even 2 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 4$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 64$$
$67$ $$(T - 6)^{2}$$
$71$ $$T^{2} + 16$$
$73$ $$T^{2} + 144$$
$79$ $$(T + 6)^{2}$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} + 4$$
$97$ $$T^{2} + 144$$