# Properties

 Label 8036.2.a.i Level $8036$ Weight $2$ Character orbit 8036.a Self dual yes Analytic conductor $64.168$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

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## 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: $$4$$ Coefficient field: 4.4.25808.1 Defining polynomial: $$x^{4} - 10 x^{2} - 6 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 164) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{3} + ( -2 + \beta_{2} - \beta_{3} ) q^{5} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{3} + ( -2 + \beta_{2} - \beta_{3} ) q^{5} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( -1 - 2 \beta_{1} + 3 \beta_{3} ) q^{15} -2 \beta_{3} q^{17} + ( -2 + \beta_{1} + \beta_{2} ) q^{19} + ( -2 - 2 \beta_{2} ) q^{23} + ( 3 - 2 \beta_{1} ) q^{25} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 8 + 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{33} + ( 4 - \beta_{2} - \beta_{3} ) q^{37} + ( -6 + 2 \beta_{1} ) q^{39} + q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -6 - 2 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{45} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{47} + ( 4 + 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -1 - 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{55} + ( -4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{57} + ( -2 - 2 \beta_{2} ) q^{59} + ( -6 - 2 \beta_{1} ) q^{61} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 7 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( 6 - 2 \beta_{3} ) q^{69} + ( -4 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -4 + \beta_{2} - 3 \beta_{3} ) q^{73} + ( -6 + 5 \beta_{1} - 3 \beta_{2} ) q^{75} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{79} + ( 7 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{81} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -12 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -2 - 4 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 7 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{95} + ( -2 + 4 \beta_{3} ) q^{97} + ( 16 + 7 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} - 4q^{5} + 12q^{9} + O(q^{10})$$ $$4q - 2q^{3} - 4q^{5} + 12q^{9} + 4q^{11} - 10q^{15} + 4q^{17} - 6q^{19} - 12q^{23} + 12q^{25} + 10q^{27} - 4q^{29} + 8q^{31} + 20q^{33} + 16q^{37} - 24q^{39} + 4q^{41} + 4q^{43} - 4q^{45} + 6q^{47} - 4q^{51} - 16q^{53} + 2q^{55} + 4q^{57} - 12q^{59} - 24q^{61} + 4q^{65} + 28q^{67} + 28q^{69} - 2q^{71} - 8q^{73} - 30q^{75} - 18q^{79} + 28q^{81} + 12q^{83} + 32q^{85} - 44q^{87} - 4q^{89} - 28q^{93} + 14q^{95} - 16q^{97} + 58q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 10 x^{2} - 6 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 7 \nu - 3$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu^{2} + 7 \nu - 12$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + 7 \beta_{1} + 3$$

## 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
 −1.55466 −2.46810 3.31526 0.707500
0 −2.92968 0 3.33307 0 0 0 5.58303 0
1.2 0 −2.21551 0 −3.59669 0 0 0 1.90849 0
1.3 0 −0.0950939 0 −1.17025 0 0 0 −2.99096 0
1.4 0 3.24028 0 −2.56613 0 0 0 7.49944 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.i 4
7.b odd 2 1 164.2.a.a 4
21.c even 2 1 1476.2.a.g 4
28.d even 2 1 656.2.a.i 4
35.c odd 2 1 4100.2.a.c 4
35.f even 4 2 4100.2.d.c 8
56.e even 2 1 2624.2.a.y 4
56.h odd 2 1 2624.2.a.v 4
84.h odd 2 1 5904.2.a.bp 4
287.d odd 2 1 6724.2.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 7.b odd 2 1
656.2.a.i 4 28.d even 2 1
1476.2.a.g 4 21.c even 2 1
2624.2.a.v 4 56.h odd 2 1
2624.2.a.y 4 56.e even 2 1
4100.2.a.c 4 35.c odd 2 1
4100.2.d.c 8 35.f even 4 2
5904.2.a.bp 4 84.h odd 2 1
6724.2.a.c 4 287.d odd 2 1
8036.2.a.i 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{4} + 2 T_{3}^{3} - 10 T_{3}^{2} - 22 T_{3} - 2$$ $$T_{5}^{4} + 4 T_{5}^{3} - 8 T_{5}^{2} - 44 T_{5} - 36$$ $$T_{11}^{4} - 4 T_{11}^{3} - 18 T_{11}^{2} + 18 T_{11} + 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$-2 - 22 T - 10 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$-36 - 44 T - 8 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$54 + 18 T - 18 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$144 + 48 T - 40 T^{2} + T^{4}$$
$17$ $$432 + 80 T - 48 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$-186 - 134 T - 14 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$-192 - 128 T + 16 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$144 - 40 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$64 + 32 T - 32 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$-324 + 36 T + 64 T^{2} - 16 T^{3} + T^{4}$$
$41$ $$( -1 + T )^{4}$$
$43$ $$-288 + 272 T - 48 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$1182 + 206 T - 62 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$-1296 - 720 T + 16 T^{3} + T^{4}$$
$59$ $$-192 - 128 T + 16 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$288 + 432 T + 176 T^{2} + 24 T^{3} + T^{4}$$
$67$ $$1094 - 1010 T + 270 T^{2} - 28 T^{3} + T^{4}$$
$71$ $$-426 - 694 T - 186 T^{2} + 2 T^{3} + T^{4}$$
$73$ $$-404 - 692 T - 80 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$-18 - 42 T + 50 T^{2} + 18 T^{3} + T^{4}$$
$83$ $$-3456 + 1344 T - 80 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$4272 - 272 T - 128 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$4944 - 1280 T - 120 T^{2} + 16 T^{3} + T^{4}$$
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