# Properties

 Label 10000.2.a.a Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $2$ 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: $$2$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} - 3 q^{7} + (3 \beta - 1) q^{9}+O(q^{10})$$ q + (-b - 1) * q^3 - 3 * q^7 + (3*b - 1) * q^9 $$q + ( - \beta - 1) q^{3} - 3 q^{7} + (3 \beta - 1) q^{9} + (2 \beta - 3) q^{11} - q^{13} + ( - 3 \beta - 3) q^{17} + (3 \beta - 4) q^{19} + (3 \beta + 3) q^{21} + ( - 2 \beta - 3) q^{23} + ( - 2 \beta + 1) q^{27} + (4 \beta - 7) q^{29} + ( - 2 \beta - 1) q^{31} + ( - \beta + 1) q^{33} + ( - 7 \beta + 4) q^{37} + (\beta + 1) q^{39} + ( - 4 \beta - 1) q^{41} + (2 \beta - 5) q^{43} + (8 \beta - 7) q^{47} + 2 q^{49} + (9 \beta + 6) q^{51} + (4 \beta - 8) q^{53} + ( - 2 \beta + 1) q^{57} + ( - 4 \beta + 2) q^{59} + (3 \beta - 7) q^{61} + ( - 9 \beta + 3) q^{63} + (2 \beta - 9) q^{67} + (7 \beta + 5) q^{69} + 3 q^{71} + (6 \beta - 4) q^{73} + ( - 6 \beta + 9) q^{77} + ( - 2 \beta + 6) q^{79} + ( - 6 \beta + 4) q^{81} + ( - 4 \beta - 7) q^{83} + ( - \beta + 3) q^{87} + ( - 4 \beta + 2) q^{89} + 3 q^{91} + (5 \beta + 3) q^{93} + (9 \beta - 4) q^{97} + ( - 5 \beta + 9) q^{99}+O(q^{100})$$ q + (-b - 1) * q^3 - 3 * q^7 + (3*b - 1) * q^9 + (2*b - 3) * q^11 - q^13 + (-3*b - 3) * q^17 + (3*b - 4) * q^19 + (3*b + 3) * q^21 + (-2*b - 3) * q^23 + (-2*b + 1) * q^27 + (4*b - 7) * q^29 + (-2*b - 1) * q^31 + (-b + 1) * q^33 + (-7*b + 4) * q^37 + (b + 1) * q^39 + (-4*b - 1) * q^41 + (2*b - 5) * q^43 + (8*b - 7) * q^47 + 2 * q^49 + (9*b + 6) * q^51 + (4*b - 8) * q^53 + (-2*b + 1) * q^57 + (-4*b + 2) * q^59 + (3*b - 7) * q^61 + (-9*b + 3) * q^63 + (2*b - 9) * q^67 + (7*b + 5) * q^69 + 3 * q^71 + (6*b - 4) * q^73 + (-6*b + 9) * q^77 + (-2*b + 6) * q^79 + (-6*b + 4) * q^81 + (-4*b - 7) * q^83 + (-b + 3) * q^87 + (-4*b + 2) * q^89 + 3 * q^91 + (5*b + 3) * q^93 + (9*b - 4) * q^97 + (-5*b + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 6 q^{7} + q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 6 * q^7 + q^9 $$2 q - 3 q^{3} - 6 q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - 9 q^{17} - 5 q^{19} + 9 q^{21} - 8 q^{23} - 10 q^{29} - 4 q^{31} + q^{33} + q^{37} + 3 q^{39} - 6 q^{41} - 8 q^{43} - 6 q^{47} + 4 q^{49} + 21 q^{51} - 12 q^{53} - 11 q^{61} - 3 q^{63} - 16 q^{67} + 17 q^{69} + 6 q^{71} - 2 q^{73} + 12 q^{77} + 10 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{87} + 6 q^{91} + 11 q^{93} + q^{97} + 13 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 6 * q^7 + q^9 - 4 * q^11 - 2 * q^13 - 9 * q^17 - 5 * q^19 + 9 * q^21 - 8 * q^23 - 10 * q^29 - 4 * q^31 + q^33 + q^37 + 3 * q^39 - 6 * q^41 - 8 * q^43 - 6 * q^47 + 4 * q^49 + 21 * q^51 - 12 * q^53 - 11 * q^61 - 3 * q^63 - 16 * q^67 + 17 * q^69 + 6 * q^71 - 2 * q^73 + 12 * q^77 + 10 * q^79 + 2 * q^81 - 18 * q^83 + 5 * q^87 + 6 * q^91 + 11 * q^93 + q^97 + 13 * 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 −2.61803 0 0 0 −3.00000 0 3.85410 0
1.2 0 −0.381966 0 0 0 −3.00000 0 −2.85410 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.a 2
4.b odd 2 1 1250.2.a.d 2
5.b even 2 1 10000.2.a.n 2
20.d odd 2 1 1250.2.a.a 2
20.e even 4 2 1250.2.b.b 4
25.e even 10 2 400.2.u.c 4
100.h odd 10 2 50.2.d.a 4
100.j odd 10 2 250.2.d.a 4
100.l even 20 4 250.2.e.b 8
300.r even 10 2 450.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 100.h odd 10 2
250.2.d.a 4 100.j odd 10 2
250.2.e.b 8 100.l even 20 4
400.2.u.c 4 25.e even 10 2
450.2.h.a 4 300.r even 10 2
1250.2.a.a 2 20.d odd 2 1
1250.2.a.d 2 4.b odd 2 1
1250.2.b.b 4 20.e even 4 2
10000.2.a.a 2 1.a even 1 1 trivial
10000.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 3)^{2}$$
$11$ $$T^{2} + 4T - 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 9T + 9$$
$19$ $$T^{2} + 5T - 5$$
$23$ $$T^{2} + 8T + 11$$
$29$ $$T^{2} + 10T + 5$$
$31$ $$T^{2} + 4T - 1$$
$37$ $$T^{2} - T - 61$$
$41$ $$T^{2} + 6T - 11$$
$43$ $$T^{2} + 8T + 11$$
$47$ $$T^{2} + 6T - 71$$
$53$ $$T^{2} + 12T + 16$$
$59$ $$T^{2} - 20$$
$61$ $$T^{2} + 11T + 19$$
$67$ $$T^{2} + 16T + 59$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 2T - 44$$
$79$ $$T^{2} - 10T + 20$$
$83$ $$T^{2} + 18T + 61$$
$89$ $$T^{2} - 20$$
$97$ $$T^{2} - T - 101$$