# Properties

 Label 8036.2.a.k Level $8036$ Weight $2$ Character orbit 8036.a Self dual yes Analytic conductor $64.168$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.287349.1 Defining polynomial: $$x^{5} - 6 x^{3} + 7 x - 1$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} + \beta_{4} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( 2 - \beta_{1} + \beta_{4} ) q^{15} + ( \beta_{3} + \beta_{4} ) q^{17} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{19} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{23} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{25} + ( -2 - \beta_{1} + \beta_{2} ) q^{27} + ( 2 - 3 \beta_{1} - \beta_{4} ) q^{29} + ( -3 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{33} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{37} + ( 4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{39} + q^{41} + ( 6 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{45} + ( -2 - 3 \beta_{4} ) q^{47} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{51} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{55} + ( 3 + 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{57} + ( -1 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{59} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{61} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( 3 - 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -3 + 8 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{69} + ( 3 + 7 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{71} + ( 3 - 5 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{73} + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{75} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{79} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{81} + ( 1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} + ( 3 - \beta_{2} - \beta_{3} ) q^{85} + ( -1 + 7 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{87} + ( 5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{89} + ( -2 + 9 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{95} + ( 6 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{97} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{3} + 3q^{5} + q^{9} + O(q^{10})$$ $$5q - 2q^{3} + 3q^{5} + q^{9} + 6q^{11} - 7q^{13} + 9q^{15} - q^{17} - 2q^{19} + 4q^{23} + 2q^{25} - 8q^{27} + 11q^{29} - 13q^{31} - 11q^{33} + 5q^{37} + 23q^{39} + 5q^{41} + 29q^{43} - 11q^{45} - 7q^{47} - 3q^{51} + 21q^{53} - 19q^{55} + 9q^{57} - 3q^{59} + 8q^{61} - 5q^{65} + 3q^{67} - 10q^{69} + 22q^{71} + 16q^{73} + 18q^{75} + 4q^{79} - 15q^{81} + 6q^{83} + 13q^{85} - 6q^{87} + 20q^{89} - 5q^{93} - 7q^{95} + 24q^{97} + 27q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 6 x^{3} + 7 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2} + 7$$

## 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
 2.13797 −2.05679 −1.36870 1.14203 0.145487
0 −2.57090 0 −0.350332 0 0 0 3.60954 0
1.2 0 −2.23037 0 −1.70418 0 0 0 1.97454 0
1.3 0 0.126667 0 4.03743 0 0 0 −2.98396 0
1.4 0 0.695770 0 −1.38288 0 0 0 −2.51590 0
1.5 0 1.97883 0 2.39996 0 0 0 0.915782 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.k 5
7.b odd 2 1 1148.2.a.d 5
28.d even 2 1 4592.2.a.bc 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.d 5 7.b odd 2 1
4592.2.a.bc 5 28.d even 2 1
8036.2.a.k 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{5} + 2 T_{3}^{4} - 6 T_{3}^{3} - 8 T_{3}^{2} + 9 T_{3} - 1$$ $$T_{5}^{5} - 3 T_{5}^{4} - 9 T_{5}^{3} + 12 T_{5}^{2} + 28 T_{5} + 8$$ $$T_{11}^{5} - 6 T_{11}^{4} - 7 T_{11}^{3} + 68 T_{11}^{2} - 84 T_{11} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 2 T + 9 T^{2} + 16 T^{3} + 45 T^{4} + 59 T^{5} + 135 T^{6} + 144 T^{7} + 243 T^{8} + 162 T^{9} + 243 T^{10}$$
$5$ $$1 - 3 T + 16 T^{2} - 48 T^{3} + 143 T^{4} - 322 T^{5} + 715 T^{6} - 1200 T^{7} + 2000 T^{8} - 1875 T^{9} + 3125 T^{10}$$
$7$ 1
$11$ $$1 - 6 T + 48 T^{2} - 196 T^{3} + 895 T^{4} - 2836 T^{5} + 9845 T^{6} - 23716 T^{7} + 63888 T^{8} - 87846 T^{9} + 161051 T^{10}$$
$13$ $$1 + 7 T + 60 T^{2} + 328 T^{3} + 1513 T^{4} + 6189 T^{5} + 19669 T^{6} + 55432 T^{7} + 131820 T^{8} + 199927 T^{9} + 371293 T^{10}$$
$17$ $$1 + T + 68 T^{2} + 82 T^{3} + 2041 T^{4} + 2209 T^{5} + 34697 T^{6} + 23698 T^{7} + 334084 T^{8} + 83521 T^{9} + 1419857 T^{10}$$
$19$ $$1 + 2 T + 27 T^{2} - 24 T^{3} + 901 T^{4} + 1287 T^{5} + 17119 T^{6} - 8664 T^{7} + 185193 T^{8} + 260642 T^{9} + 2476099 T^{10}$$
$23$ $$1 - 4 T + 23 T^{2} - 58 T^{3} + 1051 T^{4} - 4157 T^{5} + 24173 T^{6} - 30682 T^{7} + 279841 T^{8} - 1119364 T^{9} + 6436343 T^{10}$$
$29$ $$1 - 11 T + 114 T^{2} - 876 T^{3} + 6109 T^{4} - 32522 T^{5} + 177161 T^{6} - 736716 T^{7} + 2780346 T^{8} - 7780091 T^{9} + 20511149 T^{10}$$
$31$ $$1 + 13 T + 130 T^{2} + 830 T^{3} + 4765 T^{4} + 25298 T^{5} + 147715 T^{6} + 797630 T^{7} + 3872830 T^{8} + 12005773 T^{9} + 28629151 T^{10}$$
$37$ $$1 - 5 T + 117 T^{2} - 517 T^{3} + 6812 T^{4} - 26316 T^{5} + 252044 T^{6} - 707773 T^{7} + 5926401 T^{8} - 9370805 T^{9} + 69343957 T^{10}$$
$41$ $$( 1 - T )^{5}$$
$43$ $$1 - 29 T + 522 T^{2} - 6438 T^{3} + 61101 T^{4} - 448587 T^{5} + 2627343 T^{6} - 11903862 T^{7} + 41502654 T^{8} - 99145229 T^{9} + 147008443 T^{10}$$
$47$ $$1 + 7 T + 161 T^{2} + 1027 T^{3} + 12666 T^{4} + 67144 T^{5} + 595302 T^{6} + 2268643 T^{7} + 16715503 T^{8} + 34157767 T^{9} + 229345007 T^{10}$$
$53$ $$1 - 21 T + 354 T^{2} - 4212 T^{3} + 41957 T^{4} - 328646 T^{5} + 2223721 T^{6} - 11831508 T^{7} + 52702458 T^{8} - 165700101 T^{9} + 418195493 T^{10}$$
$59$ $$1 + 3 T + 32 T^{2} + 122 T^{3} + 2667 T^{4} + 48958 T^{5} + 157353 T^{6} + 424682 T^{7} + 6572128 T^{8} + 36352083 T^{9} + 714924299 T^{10}$$
$61$ $$1 - 8 T + 138 T^{2} - 1074 T^{3} + 13585 T^{4} - 83436 T^{5} + 828685 T^{6} - 3996354 T^{7} + 31323378 T^{8} - 110766728 T^{9} + 844596301 T^{10}$$
$67$ $$1 - 3 T - 10 T^{2} - 696 T^{3} + 4469 T^{4} - 15122 T^{5} + 299423 T^{6} - 3124344 T^{7} - 3007630 T^{8} - 60453363 T^{9} + 1350125107 T^{10}$$
$71$ $$1 - 22 T + 152 T^{2} - 652 T^{3} + 13507 T^{4} - 179324 T^{5} + 958997 T^{6} - 3286732 T^{7} + 54402472 T^{8} - 559056982 T^{9} + 1804229351 T^{10}$$
$73$ $$1 - 16 T + 338 T^{2} - 4158 T^{3} + 47185 T^{4} - 439204 T^{5} + 3444505 T^{6} - 22157982 T^{7} + 131487746 T^{8} - 454371856 T^{9} + 2073071593 T^{10}$$
$79$ $$1 - 4 T + 107 T^{2} + 144 T^{3} + 11770 T^{4} - 28216 T^{5} + 929830 T^{6} + 898704 T^{7} + 52755173 T^{8} - 155800324 T^{9} + 3077056399 T^{10}$$
$83$ $$1 - 6 T + 326 T^{2} - 1666 T^{3} + 48333 T^{4} - 199160 T^{5} + 4011639 T^{6} - 11477074 T^{7} + 186402562 T^{8} - 284749926 T^{9} + 3939040643 T^{10}$$
$89$ $$1 - 20 T + 491 T^{2} - 6302 T^{3} + 87253 T^{4} - 799619 T^{5} + 7765517 T^{6} - 49918142 T^{7} + 346139779 T^{8} - 1254844820 T^{9} + 5584059449 T^{10}$$
$97$ $$1 - 24 T + 363 T^{2} - 2360 T^{3} + 3237 T^{4} + 115965 T^{5} + 313989 T^{6} - 22205240 T^{7} + 331300299 T^{8} - 2124702744 T^{9} + 8587340257 T^{10}$$