# Properties

 Label 3744.2.m.f Level $3744$ Weight $2$ Character orbit 3744.m Analytic conductor $29.896$ Analytic rank $0$ Dimension $8$ CM discriminant -39 Inner twists $8$

# 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: $$8$$ Coefficient field: 8.0.151613669376.21 Defining polynomial: $$x^{8} + 5x^{4} + 16$$ x^8 + 5*x^4 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 936) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5}+O(q^{10})$$ q + b2 * q^5 $$q + \beta_{2} q^{5} + (\beta_{6} - \beta_{2}) q^{11} - \beta_1 q^{13} + ( - 2 \beta_{3} + 5) q^{25} + ( - 2 \beta_{5} - \beta_{4}) q^{41} + \beta_{7} q^{43} + (\beta_{5} - 3 \beta_{4}) q^{47} + 7 q^{49} + ( - \beta_{3} - 8) q^{55} + (\beta_{6} + 3 \beta_{2}) q^{59} + 2 \beta_1 q^{61} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{65} + ( - 3 \beta_{5} + \beta_{4}) q^{71} - 3 \beta_{3} q^{79} + (3 \beta_{6} + \beta_{2}) q^{83} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{89}+O(q^{100})$$ q + b2 * q^5 + (b6 - b2) * q^11 - b1 * q^13 + (-2*b3 + 5) * q^25 + (-2*b5 - b4) * q^41 + b7 * q^43 + (b5 - 3*b4) * q^47 + 7 * q^49 + (-b3 - 8) * q^55 + (b6 + 3*b2) * q^59 + 2*b1 * q^61 + (-2*b5 + 3*b4) * q^65 + (-3*b5 + b4) * q^71 - 3*b3 * q^79 + (3*b6 + b2) * q^83 + (-2*b5 - 3*b4) * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 40 q^{25} + 56 q^{49} - 64 q^{55}+O(q^{100})$$ 8 * q + 40 * q^25 + 56 * q^49 - 64 * q^55

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5x^{4} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 9\nu^{2} ) / 4$$ (v^6 + 9*v^2) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 4\nu^{5} - 3\nu^{3} + 12\nu ) / 8$$ (v^7 + 4*v^5 - 3*v^3 + 12*v) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} - \nu^{2} ) / 2$$ (-v^6 - v^2) / 2 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{5} + 3\nu^{3} + 12\nu ) / 8$$ (-v^7 + 4*v^5 + 3*v^3 + 12*v) / 8 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 5\nu^{3} + 8\nu ) / 4$$ (v^7 + 5*v^3 + 8*v) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} - 5\nu^{3} + 8\nu ) / 4$$ (-v^7 - 5*v^3 + 8*v) / 4 $$\beta_{7}$$ $$=$$ $$4\nu^{4} + 10$$ 4*v^4 + 10
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{5} ) / 4$$ (b6 + b5) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 4$$ (b3 + 2*b1) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{2} ) / 4$$ (-b6 + b5 + 2*b4 - 2*b2) / 4 $$\nu^{4}$$ $$=$$ $$( \beta_{7} - 10 ) / 4$$ (b7 - 10) / 4 $$\nu^{5}$$ $$=$$ $$( -3\beta_{6} - 3\beta_{5} + 4\beta_{4} + 4\beta_{2} ) / 4$$ (-3*b6 - 3*b5 + 4*b4 + 4*b2) / 4 $$\nu^{6}$$ $$=$$ $$( -9\beta_{3} - 2\beta_1 ) / 4$$ (-9*b3 - 2*b1) / 4 $$\nu^{7}$$ $$=$$ $$( -3\beta_{6} + 3\beta_{5} - 10\beta_{4} + 10\beta_{2} ) / 4$$ (-3*b6 + 3*b5 - 10*b4 + 10*b2) / 4

## 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
 −0.752986 − 1.19709i −0.752986 + 1.19709i 1.19709 + 0.752986i 1.19709 − 0.752986i −1.19709 − 0.752986i −1.19709 + 0.752986i 0.752986 + 1.19709i 0.752986 − 1.19709i
0 0 0 −4.11439 0 0 0 0 0
1585.2 0 0 0 −4.11439 0 0 0 0 0
1585.3 0 0 0 −1.75265 0 0 0 0 0
1585.4 0 0 0 −1.75265 0 0 0 0 0
1585.5 0 0 0 1.75265 0 0 0 0 0
1585.6 0 0 0 1.75265 0 0 0 0 0
1585.7 0 0 0 4.11439 0 0 0 0 0
1585.8 0 0 0 4.11439 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1585.8 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.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
104.e even 2 1 inner
312.b odd 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.f 8
3.b odd 2 1 inner 3744.2.m.f 8
4.b odd 2 1 936.2.m.g 8
8.b even 2 1 inner 3744.2.m.f 8
8.d odd 2 1 936.2.m.g 8
12.b even 2 1 936.2.m.g 8
13.b even 2 1 inner 3744.2.m.f 8
24.f even 2 1 936.2.m.g 8
24.h odd 2 1 inner 3744.2.m.f 8
39.d odd 2 1 CM 3744.2.m.f 8
52.b odd 2 1 936.2.m.g 8
104.e even 2 1 inner 3744.2.m.f 8
104.h odd 2 1 936.2.m.g 8
156.h even 2 1 936.2.m.g 8
312.b odd 2 1 inner 3744.2.m.f 8
312.h even 2 1 936.2.m.g 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 20T_{5}^{2} + 52$$ acting on $$S_{2}^{\mathrm{new}}(3744, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 20 T^{2} + 52)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 44 T^{2} + 52)^{2}$$
$13$ $$(T^{2} + 13)^{4}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} + 164 T^{2} + 6292)^{2}$$
$43$ $$(T^{2} + 156)^{4}$$
$47$ $$(T^{4} + 188 T^{2} + 8788)^{2}$$
$53$ $$T^{8}$$
$59$ $$(T^{4} - 236 T^{2} + 52)^{2}$$
$61$ $$(T^{2} + 52)^{4}$$
$67$ $$T^{8}$$
$71$ $$(T^{4} + 284 T^{2} + 6292)^{2}$$
$73$ $$T^{8}$$
$79$ $$(T^{2} - 108)^{4}$$
$83$ $$(T^{4} - 332 T^{2} + 27508)^{2}$$
$89$ $$(T^{4} + 356 T^{2} + 6292)^{2}$$
$97$ $$T^{8}$$