# Properties

 Label 8048.2.a.m Level 8048 Weight 2 Character orbit 8048.a Self dual yes Analytic conductor 64.264 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8048 = 2^{4} \cdot 503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8048.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2636035467$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 503) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{7} + \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{7} + \beta_{2} q^{9} + ( -4 + \beta_{1} ) q^{11} + ( 2 - \beta_{1} - \beta_{2} ) q^{13} + ( 4 + 2 \beta_{2} ) q^{17} -4 q^{19} + ( 3 - \beta_{1} ) q^{21} + 4 q^{23} -5 q^{25} + ( 2 \beta_{1} - \beta_{2} ) q^{27} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{33} + ( 2 - 4 \beta_{1} ) q^{37} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{39} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{41} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{47} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{49} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{53} + 4 \beta_{1} q^{57} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{59} + 2 q^{61} + ( 3 - 2 \beta_{2} ) q^{63} + ( -11 - 2 \beta_{1} + \beta_{2} ) q^{67} -4 \beta_{1} q^{69} + ( -4 + 2 \beta_{2} ) q^{71} + ( 12 - \beta_{1} + 2 \beta_{2} ) q^{73} + 5 \beta_{1} q^{75} + ( -3 + 5 \beta_{1} - 4 \beta_{2} ) q^{77} + ( 4 - 5 \beta_{1} - 3 \beta_{2} ) q^{79} + ( -6 + \beta_{1} - 4 \beta_{2} ) q^{81} + ( \beta_{1} - \beta_{2} ) q^{83} + ( 6 + 2 \beta_{1} + 4 \beta_{2} ) q^{87} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{91} + ( 6 - 6 \beta_{1} ) q^{93} + ( -6 - 5 \beta_{1} + 3 \beta_{2} ) q^{97} + ( \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{3} - q^{7} + O(q^{10})$$ $$3q - q^{3} - q^{7} - 11q^{11} + 5q^{13} + 12q^{17} - 12q^{19} + 8q^{21} + 12q^{23} - 15q^{25} + 2q^{27} - 2q^{29} + 10q^{31} - 5q^{33} + 2q^{37} + 8q^{39} + 2q^{41} - 5q^{43} - 19q^{47} - 4q^{49} - 6q^{51} + 14q^{53} + 4q^{57} + 5q^{59} + 6q^{61} + 9q^{63} - 35q^{67} - 4q^{69} - 12q^{71} + 35q^{73} + 5q^{75} - 4q^{77} + 7q^{79} - 17q^{81} + q^{83} + 20q^{87} + 4q^{89} - 3q^{91} + 12q^{93} - 23q^{97} + q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.19869 0.713538 −1.91223
0 −2.19869 0 0 0 −0.364448 0 1.83424 0
1.2 0 −0.713538 0 0 0 −3.20440 0 −2.49086 0
1.3 0 1.91223 0 0 0 2.56885 0 0.656620 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$503$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8048.2.a.m 3
4.b odd 2 1 503.2.a.d 3
12.b even 2 1 4527.2.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.d 3 4.b odd 2 1
4527.2.a.j 3 12.b even 2 1
8048.2.a.m 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8048))$$:

 $$T_{3}^{3} + T_{3}^{2} - 4 T_{3} - 3$$ $$T_{5}$$ $$T_{7}^{3} + T_{7}^{2} - 8 T_{7} - 3$$ $$T_{13}^{3} - 5 T_{13}^{2} - 2 T_{13} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T + 5 T^{2} + 3 T^{3} + 15 T^{4} + 9 T^{5} + 27 T^{6}$$
$5$ $$( 1 + 5 T^{2} )^{3}$$
$7$ $$1 + T + 13 T^{2} + 11 T^{3} + 91 T^{4} + 49 T^{5} + 343 T^{6}$$
$11$ $$1 + 11 T + 69 T^{2} + 277 T^{3} + 759 T^{4} + 1331 T^{5} + 1331 T^{6}$$
$13$ $$1 - 5 T + 37 T^{2} - 105 T^{3} + 481 T^{4} - 845 T^{5} + 2197 T^{6}$$
$17$ $$1 - 12 T + 79 T^{2} - 368 T^{3} + 1343 T^{4} - 3468 T^{5} + 4913 T^{6}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{3}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{3}$$
$29$ $$1 + 2 T + 47 T^{2} + 188 T^{3} + 1363 T^{4} + 1682 T^{5} + 24389 T^{6}$$
$31$ $$1 - 10 T + 93 T^{2} - 548 T^{3} + 2883 T^{4} - 9610 T^{5} + 29791 T^{6}$$
$37$ $$1 - 2 T + 43 T^{2} - 204 T^{3} + 1591 T^{4} - 2738 T^{5} + 50653 T^{6}$$
$41$ $$1 - 2 T + 19 T^{2} - 284 T^{3} + 779 T^{4} - 3362 T^{5} + 68921 T^{6}$$
$43$ $$1 + 5 T + 81 T^{2} + 435 T^{3} + 3483 T^{4} + 9245 T^{5} + 79507 T^{6}$$
$47$ $$1 + 19 T + 237 T^{2} + 1849 T^{3} + 11139 T^{4} + 41971 T^{5} + 103823 T^{6}$$
$53$ $$1 - 14 T + 127 T^{2} - 836 T^{3} + 6731 T^{4} - 39326 T^{5} + 148877 T^{6}$$
$59$ $$1 - 5 T + 161 T^{2} - 591 T^{3} + 9499 T^{4} - 17405 T^{5} + 205379 T^{6}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{3}$$
$67$ $$1 + 35 T + 589 T^{2} + 6009 T^{3} + 39463 T^{4} + 157115 T^{5} + 300763 T^{6}$$
$71$ $$1 + 12 T + 241 T^{2} + 1712 T^{3} + 17111 T^{4} + 60492 T^{5} + 357911 T^{6}$$
$73$ $$1 - 35 T + 605 T^{2} - 6403 T^{3} + 44165 T^{4} - 186515 T^{5} + 389017 T^{6}$$
$79$ $$1 - 7 T + 85 T^{2} + 39 T^{3} + 6715 T^{4} - 43687 T^{5} + 493039 T^{6}$$
$83$ $$1 - T + 241 T^{2} - 163 T^{3} + 20003 T^{4} - 6889 T^{5} + 571787 T^{6}$$
$89$ $$1 - 4 T + 175 T^{2} - 880 T^{3} + 15575 T^{4} - 31684 T^{5} + 704969 T^{6}$$
$97$ $$1 + 23 T + 329 T^{2} + 3379 T^{3} + 31913 T^{4} + 216407 T^{5} + 912673 T^{6}$$