# Properties

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

# Related objects

## 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 200) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta + 1) q^{3} + (\beta - 2) q^{7} + 2 q^{9}+O(q^{10})$$ q + (-2*b + 1) * q^3 + (b - 2) * q^7 + 2 * q^9 $$q + ( - 2 \beta + 1) q^{3} + (\beta - 2) q^{7} + 2 q^{9} + 2 \beta q^{11} + (\beta + 4) q^{13} + ( - 2 \beta + 4) q^{17} + (\beta - 7) q^{19} + (3 \beta - 4) q^{21} + (4 \beta - 3) q^{23} + (2 \beta - 1) q^{27} + (\beta + 5) q^{29} + ( - 6 \beta + 1) q^{31} + ( - 2 \beta - 4) q^{33} + (2 \beta - 7) q^{37} + ( - 9 \beta + 2) q^{39} + ( - 6 \beta + 2) q^{41} + ( - \beta - 5) q^{43} + ( - 3 \beta - 2) q^{47} + ( - 3 \beta - 2) q^{49} + ( - 6 \beta + 8) q^{51} + ( - 8 \beta + 3) q^{53} + (13 \beta - 9) q^{57} + ( - \beta - 3) q^{59} + (10 \beta - 7) q^{61} + (2 \beta - 4) q^{63} + (6 \beta - 10) q^{67} + (2 \beta - 11) q^{69} + ( - \beta + 8) q^{71} + 11 q^{73} + ( - 2 \beta + 2) q^{77} + ( - 7 \beta + 1) q^{79} - 11 q^{81} + (4 \beta + 7) q^{83} + ( - 11 \beta + 3) q^{87} + 4 q^{89} + (3 \beta - 7) q^{91} + (4 \beta + 13) q^{93} + (15 \beta - 6) q^{97} + 4 \beta q^{99}+O(q^{100})$$ q + (-2*b + 1) * q^3 + (b - 2) * q^7 + 2 * q^9 + 2*b * q^11 + (b + 4) * q^13 + (-2*b + 4) * q^17 + (b - 7) * q^19 + (3*b - 4) * q^21 + (4*b - 3) * q^23 + (2*b - 1) * q^27 + (b + 5) * q^29 + (-6*b + 1) * q^31 + (-2*b - 4) * q^33 + (2*b - 7) * q^37 + (-9*b + 2) * q^39 + (-6*b + 2) * q^41 + (-b - 5) * q^43 + (-3*b - 2) * q^47 + (-3*b - 2) * q^49 + (-6*b + 8) * q^51 + (-8*b + 3) * q^53 + (13*b - 9) * q^57 + (-b - 3) * q^59 + (10*b - 7) * q^61 + (2*b - 4) * q^63 + (6*b - 10) * q^67 + (2*b - 11) * q^69 + (-b + 8) * q^71 + 11 * q^73 + (-2*b + 2) * q^77 + (-7*b + 1) * q^79 - 11 * q^81 + (4*b + 7) * q^83 + (-11*b + 3) * q^87 + 4 * q^89 + (3*b - 7) * q^91 + (4*b + 13) * q^93 + (15*b - 6) * q^97 + 4*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - 3 * q^7 + 4 * q^9 $$2 q - 3 q^{7} + 4 q^{9} + 2 q^{11} + 9 q^{13} + 6 q^{17} - 13 q^{19} - 5 q^{21} - 2 q^{23} + 11 q^{29} - 4 q^{31} - 10 q^{33} - 12 q^{37} - 5 q^{39} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 7 q^{49} + 10 q^{51} - 2 q^{53} - 5 q^{57} - 7 q^{59} - 4 q^{61} - 6 q^{63} - 14 q^{67} - 20 q^{69} + 15 q^{71} + 22 q^{73} + 2 q^{77} - 5 q^{79} - 22 q^{81} + 18 q^{83} - 5 q^{87} + 8 q^{89} - 11 q^{91} + 30 q^{93} + 3 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 3 * q^7 + 4 * q^9 + 2 * q^11 + 9 * q^13 + 6 * q^17 - 13 * q^19 - 5 * q^21 - 2 * q^23 + 11 * q^29 - 4 * q^31 - 10 * q^33 - 12 * q^37 - 5 * q^39 - 2 * q^41 - 11 * q^43 - 7 * q^47 - 7 * q^49 + 10 * q^51 - 2 * q^53 - 5 * q^57 - 7 * q^59 - 4 * q^61 - 6 * q^63 - 14 * q^67 - 20 * q^69 + 15 * q^71 + 22 * q^73 + 2 * q^77 - 5 * q^79 - 22 * q^81 + 18 * q^83 - 5 * q^87 + 8 * q^89 - 11 * q^91 + 30 * q^93 + 3 * q^97 + 4 * 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 −2.23607 0 0 0 −0.381966 0 2.00000 0
1.2 0 2.23607 0 0 0 −2.61803 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.g 2
4.b odd 2 1 5000.2.a.c 2
5.b even 2 1 10000.2.a.i 2
20.d odd 2 1 5000.2.a.a 2
25.d even 5 2 400.2.u.a 4
100.h odd 10 2 1000.2.m.a 4
100.j odd 10 2 200.2.m.a 4
100.l even 20 4 1000.2.q.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.a 4 100.j odd 10 2
400.2.u.a 4 25.d even 5 2
1000.2.m.a 4 100.h odd 10 2
1000.2.q.a 8 100.l even 20 4
5000.2.a.a 2 20.d odd 2 1
5000.2.a.c 2 4.b odd 2 1
10000.2.a.g 2 1.a even 1 1 trivial
10000.2.a.i 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{2} - 5$$ T3^2 - 5 $$T_{7}^{2} + 3T_{7} + 1$$ T7^2 + 3*T7 + 1 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3T + 1$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} - 9T + 19$$
$17$ $$T^{2} - 6T + 4$$
$19$ $$T^{2} + 13T + 41$$
$23$ $$T^{2} + 2T - 19$$
$29$ $$T^{2} - 11T + 29$$
$31$ $$T^{2} + 4T - 41$$
$37$ $$T^{2} + 12T + 31$$
$41$ $$T^{2} + 2T - 44$$
$43$ $$T^{2} + 11T + 29$$
$47$ $$T^{2} + 7T + 1$$
$53$ $$T^{2} + 2T - 79$$
$59$ $$T^{2} + 7T + 11$$
$61$ $$T^{2} + 4T - 121$$
$67$ $$T^{2} + 14T + 4$$
$71$ $$T^{2} - 15T + 55$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} + 5T - 55$$
$83$ $$T^{2} - 18T + 61$$
$89$ $$(T - 4)^{2}$$
$97$ $$T^{2} - 3T - 279$$