# Properties

 Label 8085.2.a.bn Level 8085 Weight 2 Character orbit 8085.a Self dual yes Analytic conductor 64.559 Analytic rank 1 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8085.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.5590500342$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.13448.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1155) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} + \beta_{1} q^{10} - q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -2 - \beta_{1} - \beta_{3} ) q^{13} - q^{15} + ( 6 + \beta_{2} ) q^{16} + ( 1 + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( 2 + \beta_{2} ) q^{20} -\beta_{1} q^{22} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{23} + ( -2 \beta_{1} - \beta_{3} ) q^{24} + q^{25} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{26} - q^{27} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} -\beta_{1} q^{30} + ( -6 - \beta_{1} + \beta_{3} ) q^{31} + ( 4 \beta_{1} - \beta_{3} ) q^{32} + q^{33} + ( 3 \beta_{1} + \beta_{3} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 - 3 \beta_{1} + \beta_{3} ) q^{37} + ( -4 - 5 \beta_{1} - \beta_{3} ) q^{38} + ( 2 + \beta_{1} + \beta_{3} ) q^{39} + ( 2 \beta_{1} + \beta_{3} ) q^{40} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{41} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( -2 - \beta_{2} ) q^{44} + q^{45} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{46} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{47} + ( -6 - \beta_{2} ) q^{48} + \beta_{1} q^{50} + ( -1 - \beta_{2} ) q^{51} + ( -4 - 8 \beta_{1} - 2 \beta_{2} ) q^{52} + ( 3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{53} -\beta_{1} q^{54} - q^{55} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{57} + ( -6 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{58} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( -2 - \beta_{2} ) q^{60} + ( -1 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{61} + ( -2 - 6 \beta_{1} ) q^{62} + ( 2 + \beta_{2} ) q^{64} + ( -2 - \beta_{1} - \beta_{3} ) q^{65} + \beta_{1} q^{66} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{67} + ( 12 + 2 \beta_{2} ) q^{68} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{69} + ( -\beta_{1} - \beta_{3} ) q^{71} + ( 2 \beta_{1} + \beta_{3} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -10 + 2 \beta_{1} - 2 \beta_{2} ) q^{74} - q^{75} + ( -16 - 4 \beta_{2} - 4 \beta_{3} ) q^{76} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{78} + ( \beta_{1} + \beta_{3} ) q^{79} + ( 6 + \beta_{2} ) q^{80} + q^{81} + 6 q^{82} + ( -1 + \beta_{2} + 4 \beta_{3} ) q^{83} + ( 1 + \beta_{2} ) q^{85} + ( -10 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -2 \beta_{1} - \beta_{3} ) q^{88} + ( -5 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + \beta_{1} q^{90} + ( -12 - 8 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 6 + \beta_{1} - \beta_{3} ) q^{93} + ( 10 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} + ( -4 \beta_{1} + \beta_{3} ) q^{96} + ( 3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} - 4q^{11} - 6q^{12} - 8q^{13} - 4q^{15} + 22q^{16} + 2q^{17} - 10q^{19} + 6q^{20} - 2q^{23} + 4q^{25} - 20q^{26} - 4q^{27} - 2q^{29} - 24q^{31} + 4q^{33} + 6q^{36} + 8q^{37} - 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} + 4q^{45} - 12q^{46} - 4q^{47} - 22q^{48} - 2q^{51} - 12q^{52} + 14q^{53} - 4q^{55} + 10q^{57} - 20q^{58} + 2q^{59} - 6q^{60} - 6q^{61} - 8q^{62} + 6q^{64} - 8q^{65} - 8q^{67} + 44q^{68} + 2q^{69} - 4q^{73} - 36q^{74} - 4q^{75} - 56q^{76} + 20q^{78} + 22q^{80} + 4q^{81} + 24q^{82} - 6q^{83} + 2q^{85} - 36q^{86} + 2q^{87} - 18q^{89} - 44q^{92} + 24q^{93} + 36q^{94} - 10q^{95} + 6q^{97} - 4q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$

## 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
 −2.58874 −0.546295 0.546295 2.58874
−2.58874 −1.00000 4.70156 1.00000 2.58874 0 −6.99364 1.00000 −2.58874
1.2 −0.546295 −1.00000 −1.70156 1.00000 0.546295 0 2.02214 1.00000 −0.546295
1.3 0.546295 −1.00000 −1.70156 1.00000 −0.546295 0 −2.02214 1.00000 0.546295
1.4 2.58874 −1.00000 4.70156 1.00000 −2.58874 0 6.99364 1.00000 2.58874
 $$n$$: e.g. 2-40 or 990-1000 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 8085.2.a.bn 4
7.b odd 2 1 1155.2.a.u 4
21.c even 2 1 3465.2.a.bl 4
35.c odd 2 1 5775.2.a.bz 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.u 4 7.b odd 2 1
3465.2.a.bl 4 21.c even 2 1
5775.2.a.bz 4 35.c odd 2 1
8085.2.a.bn 4 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8085))$$:

 $$T_{2}^{4} - 7 T_{2}^{2} + 2$$ $$T_{13}^{4} + 8 T_{13}^{3} - 2 T_{13}^{2} - 72 T_{13} + 40$$ $$T_{17}^{2} - T_{17} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} - 2 T^{4} + 4 T^{6} + 16 T^{8}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( 1 - T )^{4}$$
$7$ 1
$11$ $$( 1 + T )^{4}$$
$13$ $$1 + 8 T + 50 T^{2} + 240 T^{3} + 1002 T^{4} + 3120 T^{5} + 8450 T^{6} + 17576 T^{7} + 28561 T^{8}$$
$17$ $$( 1 - T + 24 T^{2} - 17 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 + 10 T + 37 T^{2} - 78 T^{3} - 916 T^{4} - 1482 T^{5} + 13357 T^{6} + 68590 T^{7} + 130321 T^{8}$$
$23$ $$1 + 2 T + 21 T^{2} - 98 T^{3} - 108 T^{4} - 2254 T^{5} + 11109 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$1 + 2 T + 71 T^{2} + 210 T^{3} + 2432 T^{4} + 6090 T^{5} + 59711 T^{6} + 48778 T^{7} + 707281 T^{8}$$
$31$ $$1 + 24 T + 326 T^{2} + 2928 T^{3} + 19090 T^{4} + 90768 T^{5} + 313286 T^{6} + 714984 T^{7} + 923521 T^{8}$$
$37$ $$1 - 8 T + 114 T^{2} - 688 T^{3} + 6282 T^{4} - 25456 T^{5} + 156066 T^{6} - 405224 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 38 T^{2} + 402 T^{4} + 63878 T^{6} + 2825761 T^{8}$$
$43$ $$1 - 6 T + 107 T^{2} - 470 T^{3} + 5520 T^{4} - 20210 T^{5} + 197843 T^{6} - 477042 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 4 T + 54 T^{2} + 452 T^{3} + 786 T^{4} + 21244 T^{5} + 119286 T^{6} + 415292 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 14 T + 161 T^{2} - 1198 T^{3} + 10012 T^{4} - 63494 T^{5} + 452249 T^{6} - 2084278 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 2 T + 203 T^{2} - 238 T^{3} + 16912 T^{4} - 14042 T^{5} + 706643 T^{6} - 410758 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 6 T + 121 T^{2} + 866 T^{3} + 9380 T^{4} + 52826 T^{5} + 450241 T^{6} + 1361886 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 8 T + 176 T^{2} + 1176 T^{3} + 17358 T^{4} + 78792 T^{5} + 790064 T^{6} + 2406104 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 258 T^{2} + 26682 T^{4} + 1300578 T^{6} + 25411681 T^{8}$$
$73$ $$1 + 4 T + 112 T^{2} - 148 T^{3} + 4318 T^{4} - 10804 T^{5} + 596848 T^{6} + 1556068 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 290 T^{2} + 33466 T^{4} + 1809890 T^{6} + 38950081 T^{8}$$
$83$ $$1 + 6 T + 117 T^{2} + 1478 T^{3} + 12284 T^{4} + 122674 T^{5} + 806013 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 18 T + 255 T^{2} + 2594 T^{3} + 29096 T^{4} + 230866 T^{5} + 2019855 T^{6} + 12689442 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 6 T + 111 T^{2} + 834 T^{3} - 984 T^{4} + 80898 T^{5} + 1044399 T^{6} - 5476038 T^{7} + 88529281 T^{8}$$