# Properties

 Label 8036.2.a.h Level $8036$ Weight $2$ Character orbit 8036.a Self dual yes Analytic conductor $64.168$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8036 = 2^{2} \cdot 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8036.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1677830643$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1148) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta ) q^{3} + ( 4 + 2 \beta - 2 \beta^{2} ) q^{5} + ( -2 - 2 \beta + \beta^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta ) q^{3} + ( 4 + 2 \beta - 2 \beta^{2} ) q^{5} + ( -2 - 2 \beta + \beta^{2} ) q^{9} + ( 6 + 2 \beta - 4 \beta^{2} ) q^{11} + ( -5 - \beta + 3 \beta^{2} ) q^{13} + ( 6 + 4 \beta - 4 \beta^{2} ) q^{15} + ( 1 - \beta^{2} ) q^{17} + ( 5 - 2 \beta - \beta^{2} ) q^{19} + ( 5 - \beta - 2 \beta^{2} ) q^{23} + ( 3 - 4 \beta ) q^{25} + ( -6 + 3 \beta^{2} ) q^{27} + ( -4 + 2 \beta ) q^{29} + ( 12 + 4 \beta - 4 \beta^{2} ) q^{31} + ( 10 + 8 \beta - 6 \beta^{2} ) q^{33} + ( -9 - 3 \beta + \beta^{2} ) q^{37} + ( -8 - 5 \beta + 4 \beta^{2} ) q^{39} + q^{41} + ( 1 + 4 \beta - \beta^{2} ) q^{43} + ( -2 + 4 \beta - 2 \beta^{2} ) q^{45} + ( 3 - 5 \beta - \beta^{2} ) q^{47} + ( 2 + 2 \beta - \beta^{2} ) q^{51} + ( 2 + 2 \beta + 2 \beta^{2} ) q^{53} + ( 12 - 8 \beta ) q^{55} + ( 6 - 4 \beta + \beta^{2} ) q^{57} + ( 10 + 4 \beta - 4 \beta^{2} ) q^{59} + ( 10 - 2 \beta ) q^{61} + ( -12 + 4 \beta + 2 \beta^{2} ) q^{65} + ( -14 - 4 \beta + 4 \beta^{2} ) q^{67} + ( 7 - \beta^{2} ) q^{69} + ( 16 + 2 \beta - 6 \beta^{2} ) q^{71} + ( -4 - 4 \beta + 4 \beta^{2} ) q^{73} + ( 3 - 7 \beta + 4 \beta^{2} ) q^{75} + ( -8 - 6 \beta + 6 \beta^{2} ) q^{79} + ( -3 + 3 \beta ) q^{81} + ( 6 - 4 \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} + ( -4 + 6 \beta - 2 \beta^{2} ) q^{87} + ( -13 - 5 \beta + 7 \beta^{2} ) q^{89} + ( 16 + 4 \beta - 8 \beta^{2} ) q^{93} + ( 22 + 10 \beta - 12 \beta^{2} ) q^{95} + ( -9 - 5 \beta + 4 \beta^{2} ) q^{97} + ( -2 + 10 \beta - 2 \beta^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + O(q^{10})$$ $$3q + 3q^{3} - 6q^{11} + 3q^{13} - 6q^{15} - 3q^{17} + 9q^{19} + 3q^{23} + 9q^{25} - 12q^{29} + 12q^{31} - 6q^{33} - 21q^{37} + 3q^{41} - 3q^{43} - 18q^{45} + 3q^{47} + 18q^{53} + 36q^{55} + 24q^{57} + 6q^{59} + 30q^{61} - 24q^{65} - 18q^{67} + 15q^{69} + 12q^{71} + 12q^{73} + 33q^{75} + 12q^{79} - 9q^{81} + 18q^{83} + 6q^{85} - 24q^{87} + 3q^{89} - 6q^{95} - 3q^{97} - 18q^{99} + O(q^{100})$$

## 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
 1.87939 −0.347296 −1.53209
0 −0.879385 0 0.694593 0 0 0 −2.22668 0
1.2 0 1.34730 0 3.06418 0 0 0 −1.18479 0
1.3 0 2.53209 0 −3.75877 0 0 0 3.41147 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$-1$$

## 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 8036.2.a.h 3
7.b odd 2 1 1148.2.a.b 3
28.d even 2 1 4592.2.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.b 3 7.b odd 2 1
4592.2.a.v 3 28.d even 2 1
8036.2.a.h 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{3} - 3 T_{3}^{2} + 3$$ $$T_{5}^{3} - 12 T_{5} + 8$$ $$T_{11}^{3} + 6 T_{11}^{2} - 24 T_{11} - 136$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T + 9 T^{2} - 15 T^{3} + 27 T^{4} - 27 T^{5} + 27 T^{6}$$
$5$ $$1 + 3 T^{2} + 8 T^{3} + 15 T^{4} + 125 T^{6}$$
$7$ 1
$11$ $$1 + 6 T + 9 T^{2} - 4 T^{3} + 99 T^{4} + 726 T^{5} + 1331 T^{6}$$
$13$ $$1 - 3 T + 21 T^{2} - 21 T^{3} + 273 T^{4} - 507 T^{5} + 2197 T^{6}$$
$17$ $$1 + 3 T + 51 T^{2} + 99 T^{3} + 867 T^{4} + 867 T^{5} + 4913 T^{6}$$
$19$ $$1 - 9 T + 63 T^{2} - 269 T^{3} + 1197 T^{4} - 3249 T^{5} + 6859 T^{6}$$
$23$ $$1 - 3 T + 51 T^{2} - 101 T^{3} + 1173 T^{4} - 1587 T^{5} + 12167 T^{6}$$
$29$ $$1 + 12 T + 123 T^{2} + 704 T^{3} + 3567 T^{4} + 10092 T^{5} + 24389 T^{6}$$
$31$ $$1 - 12 T + 93 T^{2} - 552 T^{3} + 2883 T^{4} - 11532 T^{5} + 29791 T^{6}$$
$37$ $$1 + 21 T + 237 T^{2} + 1733 T^{3} + 8769 T^{4} + 28749 T^{5} + 50653 T^{6}$$
$41$ $$( 1 - T )^{3}$$
$43$ $$1 + 3 T + 93 T^{2} + 239 T^{3} + 3999 T^{4} + 5547 T^{5} + 79507 T^{6}$$
$47$ $$1 - 3 T + 51 T^{2} + 99 T^{3} + 2397 T^{4} - 6627 T^{5} + 103823 T^{6}$$
$53$ $$1 - 18 T + 231 T^{2} - 1980 T^{3} + 12243 T^{4} - 50562 T^{5} + 148877 T^{6}$$
$59$ $$1 - 6 T + 141 T^{2} - 556 T^{3} + 8319 T^{4} - 20886 T^{5} + 205379 T^{6}$$
$61$ $$1 - 30 T + 471 T^{2} - 4532 T^{3} + 28731 T^{4} - 111630 T^{5} + 226981 T^{6}$$
$67$ $$1 + 18 T + 261 T^{2} + 2276 T^{3} + 17487 T^{4} + 80802 T^{5} + 300763 T^{6}$$
$71$ $$1 - 12 T + 177 T^{2} - 1728 T^{3} + 12567 T^{4} - 60492 T^{5} + 357911 T^{6}$$
$73$ $$1 - 12 T + 219 T^{2} - 1688 T^{3} + 15987 T^{4} - 63948 T^{5} + 389017 T^{6}$$
$79$ $$1 - 12 T + 177 T^{2} - 1744 T^{3} + 13983 T^{4} - 74892 T^{5} + 493039 T^{6}$$
$83$ $$1 - 18 T + 309 T^{2} - 2852 T^{3} + 25647 T^{4} - 124002 T^{5} + 571787 T^{6}$$
$89$ $$1 - 3 T + 153 T^{2} - 265 T^{3} + 13617 T^{4} - 23763 T^{5} + 704969 T^{6}$$
$97$ $$1 + 3 T + 231 T^{2} + 349 T^{3} + 22407 T^{4} + 28227 T^{5} + 912673 T^{6}$$