# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.73205 −1.00000 1.00000 1.00000 1.73205 0 1.73205 1.00000 −1.73205
1.2 1.73205 −1.00000 1.00000 1.00000 −1.73205 0 −1.73205 1.00000 1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 8085.2.a.bd 2
7.b odd 2 1 165.2.a.b 2
21.c even 2 1 495.2.a.c 2
28.d even 2 1 2640.2.a.x 2
35.c odd 2 1 825.2.a.e 2
35.f even 4 2 825.2.c.c 4
77.b even 2 1 1815.2.a.i 2
84.h odd 2 1 7920.2.a.bz 2
105.g even 2 1 2475.2.a.r 2
105.k odd 4 2 2475.2.c.n 4
231.h odd 2 1 5445.2.a.s 2
385.h even 2 1 9075.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 7.b odd 2 1
495.2.a.c 2 21.c even 2 1
825.2.a.e 2 35.c odd 2 1
825.2.c.c 4 35.f even 4 2
1815.2.a.i 2 77.b even 2 1
2475.2.a.r 2 105.g even 2 1
2475.2.c.n 4 105.k odd 4 2
2640.2.a.x 2 28.d even 2 1
5445.2.a.s 2 231.h odd 2 1
7920.2.a.bz 2 84.h odd 2 1
8085.2.a.bd 2 1.a even 1 1 trivial
9075.2.a.bh 2 385.h even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ $$T_{13}^{2} + 4 T_{13} - 8$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-8 + 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-8 + 4 T + T^{2}$$
$23$ $$-48 + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$-32 - 8 T + T^{2}$$
$37$ $$-44 - 4 T + T^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-44 - 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 + 12 T + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$-192 + T^{2}$$
$73$ $$-104 + 4 T + T^{2}$$
$79$ $$88 + 20 T + T^{2}$$
$83$ $$132 + 24 T + T^{2}$$
$89$ $$-12 - 12 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$
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