# Properties

 Label 10000.2.a.n Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + 3 q^{7} + ( -1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + 3 q^{7} + ( -1 + 3 \beta ) q^{9} + ( -3 + 2 \beta ) q^{11} + q^{13} + ( 3 + 3 \beta ) q^{17} + ( -4 + 3 \beta ) q^{19} + ( 3 + 3 \beta ) q^{21} + ( 3 + 2 \beta ) q^{23} + ( -1 + 2 \beta ) q^{27} + ( -7 + 4 \beta ) q^{29} + ( -1 - 2 \beta ) q^{31} + ( -1 + \beta ) q^{33} + ( -4 + 7 \beta ) q^{37} + ( 1 + \beta ) q^{39} + ( -1 - 4 \beta ) q^{41} + ( 5 - 2 \beta ) q^{43} + ( 7 - 8 \beta ) q^{47} + 2 q^{49} + ( 6 + 9 \beta ) q^{51} + ( 8 - 4 \beta ) q^{53} + ( -1 + 2 \beta ) q^{57} + ( 2 - 4 \beta ) q^{59} + ( -7 + 3 \beta ) q^{61} + ( -3 + 9 \beta ) q^{63} + ( 9 - 2 \beta ) q^{67} + ( 5 + 7 \beta ) q^{69} + 3 q^{71} + ( 4 - 6 \beta ) q^{73} + ( -9 + 6 \beta ) q^{77} + ( 6 - 2 \beta ) q^{79} + ( 4 - 6 \beta ) q^{81} + ( 7 + 4 \beta ) q^{83} + ( -3 + \beta ) q^{87} + ( 2 - 4 \beta ) q^{89} + 3 q^{91} + ( -3 - 5 \beta ) q^{93} + ( 4 - 9 \beta ) q^{97} + ( 9 - 5 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 6q^{7} + q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 6q^{7} + q^{9} - 4q^{11} + 2q^{13} + 9q^{17} - 5q^{19} + 9q^{21} + 8q^{23} - 10q^{29} - 4q^{31} - q^{33} - q^{37} + 3q^{39} - 6q^{41} + 8q^{43} + 6q^{47} + 4q^{49} + 21q^{51} + 12q^{53} - 11q^{61} + 3q^{63} + 16q^{67} + 17q^{69} + 6q^{71} + 2q^{73} - 12q^{77} + 10q^{79} + 2q^{81} + 18q^{83} - 5q^{87} + 6q^{91} - 11q^{93} - q^{97} + 13q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 0.381966 0 0 0 3.00000 0 −2.85410 0
1.2 0 2.61803 0 0 0 3.00000 0 3.85410 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.n 2
4.b odd 2 1 1250.2.a.a 2
5.b even 2 1 10000.2.a.a 2
20.d odd 2 1 1250.2.a.d 2
20.e even 4 2 1250.2.b.b 4
25.d even 5 2 400.2.u.c 4
100.h odd 10 2 250.2.d.a 4
100.j odd 10 2 50.2.d.a 4
100.l even 20 4 250.2.e.b 8
300.n 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.j odd 10 2
250.2.d.a 4 100.h odd 10 2
250.2.e.b 8 100.l even 20 4
400.2.u.c 4 25.d even 5 2
450.2.h.a 4 300.n even 10 2
1250.2.a.a 2 4.b odd 2 1
1250.2.a.d 2 20.d odd 2 1
1250.2.b.b 4 20.e even 4 2
10000.2.a.a 2 5.b even 2 1
10000.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{2} - 3 T_{3} + 1$$ $$T_{7} - 3$$ $$T_{11}^{2} + 4 T_{11} - 1$$

## Hecke characteristic polynomials

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