# Properties

 Label 10000.2.a.k Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$10000 = 2^{4} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 10000.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.8504020213$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1250) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta q^{7} - 2 q^{9} +O(q^{10})$$ q + q^3 - b * q^7 - 2 * q^9 $$q + q^{3} - \beta q^{7} - 2 q^{9} + 3 q^{11} + ( - 4 \beta + 1) q^{13} + (3 \beta - 6) q^{17} + (2 \beta + 4) q^{19} - \beta q^{21} + (3 \beta - 3) q^{23} - 5 q^{27} + (6 \beta - 3) q^{29} + ( - 2 \beta + 9) q^{31} + 3 q^{33} + (5 \beta - 2) q^{37} + ( - 4 \beta + 1) q^{39} - 6 \beta q^{41} + ( - 5 \beta - 4) q^{43} - 3 q^{47} + (\beta - 6) q^{49} + (3 \beta - 6) q^{51} + ( - 6 \beta - 3) q^{53} + (2 \beta + 4) q^{57} + (6 \beta - 3) q^{59} + (2 \beta - 9) q^{61} + 2 \beta q^{63} + ( - 4 \beta + 9) q^{67} + (3 \beta - 3) q^{69} + (9 \beta - 9) q^{71} - 2 \beta q^{73} - 3 \beta q^{77} + ( - 8 \beta + 9) q^{79} + q^{81} - 3 \beta q^{83} + (6 \beta - 3) q^{87} + (12 \beta - 6) q^{89} + (3 \beta + 4) q^{91} + ( - 2 \beta + 9) q^{93} + ( - 2 \beta + 9) q^{97} - 6 q^{99} +O(q^{100})$$ q + q^3 - b * q^7 - 2 * q^9 + 3 * q^11 + (-4*b + 1) * q^13 + (3*b - 6) * q^17 + (2*b + 4) * q^19 - b * q^21 + (3*b - 3) * q^23 - 5 * q^27 + (6*b - 3) * q^29 + (-2*b + 9) * q^31 + 3 * q^33 + (5*b - 2) * q^37 + (-4*b + 1) * q^39 - 6*b * q^41 + (-5*b - 4) * q^43 - 3 * q^47 + (b - 6) * q^49 + (3*b - 6) * q^51 + (-6*b - 3) * q^53 + (2*b + 4) * q^57 + (6*b - 3) * q^59 + (2*b - 9) * q^61 + 2*b * q^63 + (-4*b + 9) * q^67 + (3*b - 3) * q^69 + (9*b - 9) * q^71 - 2*b * q^73 - 3*b * q^77 + (-8*b + 9) * q^79 + q^81 - 3*b * q^83 + (6*b - 3) * q^87 + (12*b - 6) * q^89 + (3*b + 4) * q^91 + (-2*b + 9) * q^93 + (-2*b + 9) * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^7 - 4 * q^9 $$2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 9 q^{17} + 10 q^{19} - q^{21} - 3 q^{23} - 10 q^{27} + 16 q^{31} + 6 q^{33} + q^{37} - 2 q^{39} - 6 q^{41} - 13 q^{43} - 6 q^{47} - 11 q^{49} - 9 q^{51} - 12 q^{53} + 10 q^{57} - 16 q^{61} + 2 q^{63} + 14 q^{67} - 3 q^{69} - 9 q^{71} - 2 q^{73} - 3 q^{77} + 10 q^{79} + 2 q^{81} - 3 q^{83} + 11 q^{91} + 16 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 - 2 * q^13 - 9 * q^17 + 10 * q^19 - q^21 - 3 * q^23 - 10 * q^27 + 16 * q^31 + 6 * q^33 + q^37 - 2 * q^39 - 6 * q^41 - 13 * q^43 - 6 * q^47 - 11 * q^49 - 9 * q^51 - 12 * q^53 + 10 * q^57 - 16 * q^61 + 2 * q^63 + 14 * q^67 - 3 * q^69 - 9 * q^71 - 2 * q^73 - 3 * q^77 + 10 * q^79 + 2 * q^81 - 3 * q^83 + 11 * q^91 + 16 * q^93 + 16 * q^97 - 12 * q^99

## 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
 1.61803 −0.618034
0 1.00000 0 0 0 −1.61803 0 −2.00000 0
1.2 0 1.00000 0 0 0 0.618034 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## 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 10000.2.a.k 2
4.b odd 2 1 1250.2.a.c yes 2
5.b even 2 1 10000.2.a.d 2
20.d odd 2 1 1250.2.a.b 2
20.e even 4 2 1250.2.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.b 2 20.d odd 2 1
1250.2.a.c yes 2 4.b odd 2 1
1250.2.b.a 4 20.e even 4 2
10000.2.a.d 2 5.b even 2 1
10000.2.a.k 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7}^{2} + T_{7} - 1$$ T7^2 + T7 - 1 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 1$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 2T - 19$$
$17$ $$T^{2} + 9T + 9$$
$19$ $$T^{2} - 10T + 20$$
$23$ $$T^{2} + 3T - 9$$
$29$ $$T^{2} - 45$$
$31$ $$T^{2} - 16T + 59$$
$37$ $$T^{2} - T - 31$$
$41$ $$T^{2} + 6T - 36$$
$43$ $$T^{2} + 13T + 11$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2} + 12T - 9$$
$59$ $$T^{2} - 45$$
$61$ $$T^{2} + 16T + 59$$
$67$ $$T^{2} - 14T + 29$$
$71$ $$T^{2} + 9T - 81$$
$73$ $$T^{2} + 2T - 4$$
$79$ $$T^{2} - 10T - 55$$
$83$ $$T^{2} + 3T - 9$$
$89$ $$T^{2} - 180$$
$97$ $$T^{2} - 16T + 59$$
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