# Properties

 Label 8036.2.a.n Level 8036 Weight 2 Character orbit 8036.a Self dual yes Analytic conductor 64.168 Analytic rank 1 Dimension 8 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: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 14 x^{6} - 4 x^{5} + 60 x^{4} + 31 x^{3} - 75 x^{2} - 60 x - 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{2} ) q^{9} + ( -1 - \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{4} + \beta_{5} ) q^{13} + ( -\beta_{3} - \beta_{4} ) q^{15} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{17} + ( -1 - \beta_{2} - \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{27} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{29} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{31} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{33} + ( -4 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{39} - q^{41} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{45} + ( -3 - \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{47} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{51} + ( -\beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{57} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{61} + ( -3 + \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{65} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{69} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{75} + ( 2 + 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -4 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{81} + ( 5 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{85} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{89} + ( -4 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{95} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( -1 - 2 \beta_{1} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{9} + O(q^{10})$$ $$8q + 4q^{9} - 8q^{11} + 7q^{13} - q^{15} + q^{17} - 4q^{19} - 3q^{23} - 4q^{25} + 12q^{27} - 4q^{29} - 4q^{31} - 23q^{33} - 31q^{37} + 5q^{39} - 8q^{41} - 8q^{43} - q^{45} - 24q^{47} - 23q^{51} - q^{53} - 2q^{55} - 15q^{57} - 4q^{59} + 4q^{61} - 24q^{65} + 21q^{69} + 8q^{71} - 11q^{73} + 15q^{75} + 14q^{79} - 28q^{81} + 42q^{83} - 20q^{85} - 25q^{87} + 11q^{89} - 27q^{93} - 15q^{95} + 16q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 14 x^{6} - 4 x^{5} + 60 x^{4} + 31 x^{3} - 75 x^{2} - 60 x - 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 13 \nu^{4} - 34 \nu^{3} + 10 \nu^{2} + 46 \nu + 19$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 4 \nu^{6} - 17 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} - 29 \nu^{2} - 26 \nu - 8$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - 16 \nu^{5} - 11 \nu^{4} + 73 \nu^{3} + 35 \nu^{2} - 94 \nu - 40$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} - 4 \nu^{6} - 20 \nu^{5} + 29 \nu^{4} + 65 \nu^{3} - 41 \nu^{2} - 80 \nu - 17$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} + 26 \nu^{5} - 5 \nu^{4} - 101 \nu^{3} - 7 \nu^{2} + 113 \nu + 44$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + 8 \beta_{2} + \beta_{1} + 23$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} + \beta_{4} + 12 \beta_{3} + 13 \beta_{2} + 28 \beta_{1} + 22$$ $$\nu^{6}$$ $$=$$ $$15 \beta_{7} + 23 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 36 \beta_{3} + 62 \beta_{2} + 15 \beta_{1} + 149$$ $$\nu^{7}$$ $$=$$ $$83 \beta_{7} + 86 \beta_{6} + 85 \beta_{5} + 14 \beta_{4} + 116 \beta_{3} + 126 \beta_{2} + 173 \beta_{1} + 210$$

## 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
 −2.28881 −2.24904 −1.33338 −0.503623 −0.308117 1.71291 2.10300 2.86706
0 −2.28881 0 −0.350035 0 0 0 2.23863 0
1.2 0 −2.24904 0 1.47097 0 0 0 2.05817 0
1.3 0 −1.33338 0 −1.76687 0 0 0 −1.22209 0
1.4 0 −0.503623 0 −2.23729 0 0 0 −2.74636 0
1.5 0 −0.308117 0 3.30124 0 0 0 −2.90506 0
1.6 0 1.71291 0 −2.15120 0 0 0 −0.0659453 0
1.7 0 2.10300 0 2.93524 0 0 0 1.42260 0
1.8 0 2.86706 0 −1.20206 0 0 0 5.22005 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.n 8
7.b odd 2 1 8036.2.a.m 8
7.d odd 6 2 1148.2.i.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.d 16 7.d odd 6 2
8036.2.a.m 8 7.b odd 2 1
8036.2.a.n 8 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{8} - 14 T_{3}^{6} - 4 T_{3}^{5} + 60 T_{3}^{4} + 31 T_{3}^{3} - 75 T_{3}^{2} - 60 T_{3} - 11$$ $$T_{5}^{8} - 18 T_{5}^{6} - 12 T_{5}^{5} + 98 T_{5}^{4} + 117 T_{5}^{3} - 115 T_{5}^{2} - 196 T_{5} - 51$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 10 T^{2} - 4 T^{3} + 60 T^{4} - 29 T^{5} + 267 T^{6} - 141 T^{7} + 889 T^{8} - 423 T^{9} + 2403 T^{10} - 783 T^{11} + 4860 T^{12} - 972 T^{13} + 7290 T^{14} + 6561 T^{16}$$
$5$ $$1 + 22 T^{2} - 12 T^{3} + 258 T^{4} - 183 T^{5} + 2095 T^{6} - 1441 T^{7} + 12249 T^{8} - 7205 T^{9} + 52375 T^{10} - 22875 T^{11} + 161250 T^{12} - 37500 T^{13} + 343750 T^{14} + 390625 T^{16}$$
$7$ 1
$11$ $$1 + 8 T + 89 T^{2} + 522 T^{3} + 3323 T^{4} + 15312 T^{5} + 70852 T^{6} + 264099 T^{7} + 962211 T^{8} + 2905089 T^{9} + 8573092 T^{10} + 20380272 T^{11} + 48652043 T^{12} + 84068622 T^{13} + 157668929 T^{14} + 155897368 T^{15} + 214358881 T^{16}$$
$13$ $$1 - 7 T + 87 T^{2} - 414 T^{3} + 2973 T^{4} - 10570 T^{5} + 58450 T^{6} - 170152 T^{7} + 839348 T^{8} - 2211976 T^{9} + 9878050 T^{10} - 23222290 T^{11} + 84911853 T^{12} - 153715302 T^{13} + 419932383 T^{14} - 439239619 T^{15} + 815730721 T^{16}$$
$17$ $$1 - T + 87 T^{2} - 61 T^{3} + 3759 T^{4} - 1751 T^{5} + 105134 T^{6} - 34539 T^{7} + 2092941 T^{8} - 587163 T^{9} + 30383726 T^{10} - 8602663 T^{11} + 313955439 T^{12} - 86611277 T^{13} + 2099968503 T^{14} - 410338673 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 4 T + 118 T^{2} + 415 T^{3} + 6581 T^{4} + 20104 T^{5} + 225525 T^{6} + 588263 T^{7} + 5177111 T^{8} + 11176997 T^{9} + 81414525 T^{10} + 137893336 T^{11} + 857642501 T^{12} + 1027581085 T^{13} + 5551413958 T^{14} + 3575486956 T^{15} + 16983563041 T^{16}$$
$23$ $$1 + 3 T + 137 T^{2} + 183 T^{3} + 7715 T^{4} - 1513 T^{5} + 252455 T^{6} - 332692 T^{7} + 6224545 T^{8} - 7651916 T^{9} + 133548695 T^{10} - 18408671 T^{11} + 2158973315 T^{12} + 1177850769 T^{13} + 20280916793 T^{14} + 10214476341 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 4 T + 138 T^{2} + 516 T^{3} + 9467 T^{4} + 33071 T^{5} + 440608 T^{6} + 1382028 T^{7} + 14949058 T^{8} + 40078812 T^{9} + 370551328 T^{10} + 806568619 T^{11} + 6695829227 T^{12} + 10583752884 T^{13} + 82085618298 T^{14} + 68999505236 T^{15} + 500246412961 T^{16}$$
$31$ $$1 + 4 T + 136 T^{2} + 326 T^{3} + 9390 T^{4} + 15989 T^{5} + 452251 T^{6} + 616687 T^{7} + 16188363 T^{8} + 19117297 T^{9} + 434613211 T^{10} + 476328299 T^{11} + 8671862190 T^{12} + 9333103226 T^{13} + 120700500616 T^{14} + 110050456444 T^{15} + 852891037441 T^{16}$$
$37$ $$1 + 31 T + 643 T^{2} + 9644 T^{3} + 117278 T^{4} + 1183925 T^{5} + 10234846 T^{6} + 76402804 T^{7} + 497661632 T^{8} + 2826903748 T^{9} + 14011504174 T^{10} + 59969353025 T^{11} + 219797853758 T^{12} + 668753121308 T^{13} + 1649762080987 T^{14} + 2942888191123 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 + T )^{8}$$
$43$ $$1 + 8 T + 288 T^{2} + 2036 T^{3} + 38055 T^{4} + 236033 T^{5} + 3024756 T^{6} + 16054528 T^{7} + 158339468 T^{8} + 690344704 T^{9} + 5592773844 T^{10} + 18766275731 T^{11} + 130102472055 T^{12} + 299309189948 T^{13} + 1820552558112 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 + 24 T + 521 T^{2} + 7118 T^{3} + 90467 T^{4} + 890901 T^{5} + 8363341 T^{6} + 64948086 T^{7} + 485629743 T^{8} + 3052560042 T^{9} + 18474620269 T^{10} + 92496014523 T^{11} + 441450101027 T^{12} + 1632477759826 T^{13} + 5615971186409 T^{14} + 12158954891112 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + T + 280 T^{2} + 513 T^{3} + 36898 T^{4} + 94390 T^{5} + 3091933 T^{6} + 8855605 T^{7} + 188115325 T^{8} + 469347065 T^{9} + 8685239797 T^{10} + 14052500030 T^{11} + 291142967938 T^{12} + 214534287909 T^{13} + 6206021116120 T^{14} + 1174711139837 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 4 T + 341 T^{2} + 1058 T^{3} + 53761 T^{4} + 131677 T^{5} + 5303445 T^{6} + 10633338 T^{7} + 367344925 T^{8} + 627366942 T^{9} + 18461292045 T^{10} + 27043690583 T^{11} + 651441444721 T^{12} + 756389908342 T^{13} + 14383561971581 T^{14} + 9954605939276 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 4 T + 355 T^{2} - 1441 T^{3} + 61684 T^{4} - 231523 T^{5} + 6694222 T^{6} - 22066564 T^{7} + 491179868 T^{8} - 1346060404 T^{9} + 24909200062 T^{10} - 52551322063 T^{11} + 854066856244 T^{12} - 1217063269741 T^{13} + 18289732898155 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 340 T^{2} + 68 T^{3} + 59697 T^{4} + 16147 T^{5} + 6764659 T^{6} + 1849089 T^{7} + 536944362 T^{8} + 123888963 T^{9} + 30366554251 T^{10} + 4856420161 T^{11} + 1202961470337 T^{12} + 91808507276 T^{13} + 30755849937460 T^{14} + 406067677556641 T^{16}$$
$71$ $$1 - 8 T + 341 T^{2} - 2537 T^{3} + 60540 T^{4} - 400543 T^{5} + 7099493 T^{6} - 41272935 T^{7} + 591932642 T^{8} - 2930378385 T^{9} + 35788544213 T^{10} - 143358745673 T^{11} + 1538423167740 T^{12} - 4577329863487 T^{13} + 43682196817061 T^{14} - 72760961267128 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 11 T + 474 T^{2} + 4040 T^{3} + 94652 T^{4} + 643511 T^{5} + 11124356 T^{6} + 63241059 T^{7} + 927349325 T^{8} + 4616597307 T^{9} + 59281693124 T^{10} + 250336718687 T^{11} + 2687950307132 T^{12} + 8375209235720 T^{13} + 71732423260986 T^{14} + 121521383710067 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 14 T + 483 T^{2} - 5250 T^{3} + 107571 T^{4} - 967249 T^{5} + 14893663 T^{6} - 112979804 T^{7} + 1409336805 T^{8} - 8925404516 T^{9} + 92951350783 T^{10} - 476891479711 T^{11} + 4189899163251 T^{12} - 16154546094750 T^{13} + 117411241016643 T^{14} - 268854725806226 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 42 T + 1200 T^{2} - 24924 T^{3} + 427571 T^{4} - 6160059 T^{5} + 77219520 T^{6} - 845604102 T^{7} + 8200816788 T^{8} - 70185140466 T^{9} + 531965273280 T^{10} - 3522241655433 T^{11} + 20291801768291 T^{12} - 98176648986132 T^{13} + 392328448042800 T^{14} - 1139714141564334 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 - 11 T + 348 T^{2} - 2729 T^{3} + 52037 T^{4} - 403057 T^{5} + 6261873 T^{6} - 54404070 T^{7} + 655489911 T^{8} - 4841962230 T^{9} + 49600296033 T^{10} - 284142690233 T^{11} + 3264917994917 T^{12} - 15238898236321 T^{13} + 172949489254428 T^{14} - 486544683850819 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 - 16 T + 693 T^{2} - 8103 T^{3} + 198454 T^{4} - 1794697 T^{5} + 32962752 T^{6} - 243374175 T^{7} + 3748103050 T^{8} - 23607294975 T^{9} + 310146533568 T^{10} - 1637971495081 T^{11} + 17568989931574 T^{12} - 69583218102471 T^{13} + 577249599415797 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16}$$