# Properties

 Label 8036.2.a.l Level 8036 Weight 2 Character orbit 8036.a Self dual yes Analytic conductor 64.168 Analytic rank 1 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.470117.1 Defining polynomial: $$x^{5} - 2 x^{4} - 6 x^{3} + 8 x^{2} + 7 x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{4} ) q^{13} + ( -2 - \beta_{3} + \beta_{4} ) q^{15} + ( -2 \beta_{2} - \beta_{3} ) q^{17} + ( 2 \beta_{3} + \beta_{4} ) q^{19} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} + ( -1 + \beta_{3} - \beta_{4} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{27} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( 1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} ) q^{33} + ( -4 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} - q^{41} + ( -6 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{43} + ( \beta_{3} + \beta_{4} ) q^{45} + ( 2 + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{51} + ( -5 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{53} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{55} + ( -3 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{57} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{61} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{65} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{67} + ( -1 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{69} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{73} + ( -3 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{81} + ( -3 + 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{83} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{85} + ( 7 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{87} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{89} + ( 4 + 3 \beta_{3} - 3 \beta_{4} ) q^{93} + ( 2 + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{95} + ( -2 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( -7 + \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 2q^{3} + q^{5} + q^{9} + O(q^{10})$$ $$5q + 2q^{3} + q^{5} + q^{9} - 2q^{11} + q^{13} - 9q^{15} + 3q^{17} + 4q^{19} - 8q^{23} - 6q^{25} + 8q^{27} - 9q^{29} + 11q^{31} - 5q^{33} - 11q^{37} - 17q^{39} - 5q^{41} - 27q^{43} + 3q^{45} + 3q^{47} - 3q^{51} - 19q^{53} + 13q^{55} - 11q^{57} + 15q^{59} + 7q^{65} - 21q^{67} - 14q^{69} - 16q^{71} + 10q^{73} - 12q^{75} - 14q^{79} - 7q^{81} + 2q^{83} - 21q^{85} + 36q^{87} - 6q^{89} + 17q^{93} + 9q^{95} - 20q^{97} - 17q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 6 x^{3} + 8 x^{2} + 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 8 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 9 \nu - 5$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 3 \nu^{3} + 5 \nu^{2} - 13 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} + 5 \beta_{3} + 8 \beta_{2} + 7 \beta_{1} + 17$$

## 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
 −2.05768 −0.475832 −0.189142 1.94177 2.78088
0 −2.05768 0 2.77770 0 0 0 1.23404 0
1.2 0 −0.475832 0 −1.19617 0 0 0 −2.77358 0
1.3 0 −0.189142 0 1.51194 0 0 0 −2.96423 0
1.4 0 1.94177 0 −2.68629 0 0 0 0.770473 0
1.5 0 2.78088 0 0.592821 0 0 0 4.73330 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.l 5
7.b odd 2 1 1148.2.a.c 5
28.d even 2 1 4592.2.a.be 5

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

 $$T_{3}^{5} - 2 T_{3}^{4} - 6 T_{3}^{3} + 8 T_{3}^{2} + 7 T_{3} + 1$$ $$T_{5}^{5} - T_{5}^{4} - 9 T_{5}^{3} + 8 T_{5}^{2} + 12 T_{5} - 8$$ $$T_{11}^{5} + 2 T_{11}^{4} - 29 T_{11}^{3} - 98 T_{11}^{2} - 88 T_{11} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 9 T^{2} - 16 T^{3} + 43 T^{4} - 59 T^{5} + 129 T^{6} - 144 T^{7} + 243 T^{8} - 162 T^{9} + 243 T^{10}$$
$5$ $$1 - T + 16 T^{2} - 12 T^{3} + 127 T^{4} - 78 T^{5} + 635 T^{6} - 300 T^{7} + 2000 T^{8} - 625 T^{9} + 3125 T^{10}$$
$7$ 1
$11$ $$1 + 2 T + 26 T^{2} - 10 T^{3} + 165 T^{4} - 728 T^{5} + 1815 T^{6} - 1210 T^{7} + 34606 T^{8} + 29282 T^{9} + 161051 T^{10}$$
$13$ $$1 - T + 42 T^{2} - 4 T^{3} + 835 T^{4} + 131 T^{5} + 10855 T^{6} - 676 T^{7} + 92274 T^{8} - 28561 T^{9} + 371293 T^{10}$$
$17$ $$1 - 3 T + 56 T^{2} - 126 T^{3} + 1591 T^{4} - 3013 T^{5} + 27047 T^{6} - 36414 T^{7} + 275128 T^{8} - 250563 T^{9} + 1419857 T^{10}$$
$19$ $$1 - 4 T + 39 T^{2} - 48 T^{3} + 441 T^{4} + 365 T^{5} + 8379 T^{6} - 17328 T^{7} + 267501 T^{8} - 521284 T^{9} + 2476099 T^{10}$$
$23$ $$1 + 8 T + 83 T^{2} + 574 T^{3} + 3355 T^{4} + 18011 T^{5} + 77165 T^{6} + 303646 T^{7} + 1009861 T^{8} + 2238728 T^{9} + 6436343 T^{10}$$
$29$ $$1 + 9 T + 68 T^{2} + 24 T^{3} - 1301 T^{4} - 16106 T^{5} - 37729 T^{6} + 20184 T^{7} + 1658452 T^{8} + 6365529 T^{9} + 20511149 T^{10}$$
$31$ $$1 - 11 T + 112 T^{2} - 656 T^{3} + 5003 T^{4} - 25610 T^{5} + 155093 T^{6} - 630416 T^{7} + 3336592 T^{8} - 10158731 T^{9} + 28629151 T^{10}$$
$37$ $$1 + 11 T + 145 T^{2} + 663 T^{3} + 4956 T^{4} + 12852 T^{5} + 183372 T^{6} + 907647 T^{7} + 7344685 T^{8} + 20615771 T^{9} + 69343957 T^{10}$$
$41$ $$( 1 + T )^{5}$$
$43$ $$1 + 27 T + 382 T^{2} + 3986 T^{3} + 34437 T^{4} + 248301 T^{5} + 1480791 T^{6} + 7370114 T^{7} + 30371674 T^{8} + 92307627 T^{9} + 147008443 T^{10}$$
$47$ $$1 - 3 T + 167 T^{2} - 461 T^{3} + 13280 T^{4} - 29348 T^{5} + 624160 T^{6} - 1018349 T^{7} + 17338441 T^{8} - 14639043 T^{9} + 229345007 T^{10}$$
$53$ $$1 + 19 T + 298 T^{2} + 2774 T^{3} + 25329 T^{4} + 175326 T^{5} + 1342437 T^{6} + 7792166 T^{7} + 44365346 T^{8} + 149919139 T^{9} + 418195493 T^{10}$$
$59$ $$1 - 15 T + 316 T^{2} - 3050 T^{3} + 36379 T^{4} - 253030 T^{5} + 2146361 T^{6} - 10617050 T^{7} + 64899764 T^{8} - 181760415 T^{9} + 714924299 T^{10}$$
$61$ $$1 + 190 T^{2} + 30 T^{3} + 18957 T^{4} + 5364 T^{5} + 1156377 T^{6} + 111630 T^{7} + 43126390 T^{8} + 844596301 T^{10}$$
$67$ $$1 + 21 T + 310 T^{2} + 3174 T^{3} + 31197 T^{4} + 258962 T^{5} + 2090199 T^{6} + 14248086 T^{7} + 93236530 T^{8} + 423173541 T^{9} + 1350125107 T^{10}$$
$71$ $$1 + 16 T + 338 T^{2} + 4312 T^{3} + 46713 T^{4} + 451448 T^{5} + 3316623 T^{6} + 21736792 T^{7} + 120973918 T^{8} + 406586896 T^{9} + 1804229351 T^{10}$$
$73$ $$1 - 10 T + 222 T^{2} - 2492 T^{3} + 25941 T^{4} - 257572 T^{5} + 1893693 T^{6} - 13279868 T^{7} + 86361774 T^{8} - 283982410 T^{9} + 2073071593 T^{10}$$
$79$ $$1 + 14 T + 315 T^{2} + 3176 T^{3} + 45082 T^{4} + 346676 T^{5} + 3561478 T^{6} + 19821416 T^{7} + 155307285 T^{8} + 545301134 T^{9} + 3077056399 T^{10}$$
$83$ $$1 - 2 T + 114 T^{2} - 338 T^{3} + 12089 T^{4} - 78912 T^{5} + 1003387 T^{6} - 2328482 T^{7} + 65183718 T^{8} - 94916642 T^{9} + 3939040643 T^{10}$$
$89$ $$1 + 6 T + 305 T^{2} + 1284 T^{3} + 45117 T^{4} + 153431 T^{5} + 4015413 T^{6} + 10170564 T^{7} + 215015545 T^{8} + 376453446 T^{9} + 5584059449 T^{10}$$
$97$ $$1 + 20 T + 207 T^{2} + 2236 T^{3} + 33747 T^{4} + 423407 T^{5} + 3273459 T^{6} + 21038524 T^{7} + 188923311 T^{8} + 1770585620 T^{9} + 8587340257 T^{10}$$