# Properties

 Label 10000.2.a.c 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{7} -2 q^{9} +O(q^{10})$$ $$q - q^{3} + \beta q^{7} -2 q^{9} + ( 4 - 2 \beta ) q^{11} -3 \beta q^{13} + ( -4 + 2 \beta ) q^{17} + ( 1 + 3 \beta ) q^{19} -\beta q^{21} + ( -5 - 2 \beta ) q^{23} + 5 q^{27} + ( -3 + \beta ) q^{29} + 3 q^{31} + ( -4 + 2 \beta ) q^{33} + ( 1 + 2 \beta ) q^{37} + 3 \beta q^{39} + ( -2 - 2 \beta ) q^{41} + ( 3 - 3 \beta ) q^{43} + \beta q^{47} + ( -6 + \beta ) q^{49} + ( 4 - 2 \beta ) q^{51} + ( -1 + 4 \beta ) q^{53} + ( -1 - 3 \beta ) q^{57} + ( 9 - 3 \beta ) q^{59} + ( 5 - 6 \beta ) q^{61} -2 \beta q^{63} + ( -6 - 2 \beta ) q^{67} + ( 5 + 2 \beta ) q^{69} + ( 6 - \beta ) q^{71} -9 q^{73} + ( -2 + 2 \beta ) q^{77} + ( 5 - 5 \beta ) q^{79} + q^{81} + ( 5 - 2 \beta ) q^{83} + ( 3 - \beta ) q^{87} + ( -4 + 8 \beta ) q^{89} + ( -3 - 3 \beta ) q^{91} -3 q^{93} + ( -2 + 3 \beta ) q^{97} + ( -8 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + q^{7} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{3} + q^{7} - 4q^{9} + 6q^{11} - 3q^{13} - 6q^{17} + 5q^{19} - q^{21} - 12q^{23} + 10q^{27} - 5q^{29} + 6q^{31} - 6q^{33} + 4q^{37} + 3q^{39} - 6q^{41} + 3q^{43} + q^{47} - 11q^{49} + 6q^{51} + 2q^{53} - 5q^{57} + 15q^{59} + 4q^{61} - 2q^{63} - 14q^{67} + 12q^{69} + 11q^{71} - 18q^{73} - 2q^{77} + 5q^{79} + 2q^{81} + 8q^{83} + 5q^{87} - 9q^{91} - 6q^{93} - q^{97} - 12q^{99} + O(q^{100})$$

## 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 −1.00000 0 0 0 −0.618034 0 −2.00000 0
1.2 0 −1.00000 0 0 0 1.61803 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.c 2
4.b odd 2 1 625.2.a.b 2
5.b even 2 1 10000.2.a.l 2
12.b even 2 1 5625.2.a.f 2
20.d odd 2 1 625.2.a.c 2
20.e even 4 2 625.2.b.a 4
25.d even 5 2 400.2.u.b 4
60.h even 2 1 5625.2.a.d 2
100.h odd 10 2 125.2.d.a 4
100.h odd 10 2 625.2.d.b 4
100.j odd 10 2 25.2.d.a 4
100.j odd 10 2 625.2.d.h 4
100.l even 20 4 125.2.e.a 8
100.l even 20 4 625.2.e.c 8
300.n even 10 2 225.2.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 100.j odd 10 2
125.2.d.a 4 100.h odd 10 2
125.2.e.a 8 100.l even 20 4
225.2.h.b 4 300.n even 10 2
400.2.u.b 4 25.d even 5 2
625.2.a.b 2 4.b odd 2 1
625.2.a.c 2 20.d odd 2 1
625.2.b.a 4 20.e even 4 2
625.2.d.b 4 100.h odd 10 2
625.2.d.h 4 100.j odd 10 2
625.2.e.c 8 100.l even 20 4
5625.2.a.d 2 60.h even 2 1
5625.2.a.f 2 12.b even 2 1
10000.2.a.c 2 1.a even 1 1 trivial
10000.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3} + 1$$ $$T_{7}^{2} - T_{7} - 1$$ $$T_{11}^{2} - 6 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 - T + T^{2}$$
$11$ $$4 - 6 T + T^{2}$$
$13$ $$-9 + 3 T + T^{2}$$
$17$ $$4 + 6 T + T^{2}$$
$19$ $$-5 - 5 T + T^{2}$$
$23$ $$31 + 12 T + T^{2}$$
$29$ $$5 + 5 T + T^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$-1 - 4 T + T^{2}$$
$41$ $$4 + 6 T + T^{2}$$
$43$ $$-9 - 3 T + T^{2}$$
$47$ $$-1 - T + T^{2}$$
$53$ $$-19 - 2 T + T^{2}$$
$59$ $$45 - 15 T + T^{2}$$
$61$ $$-41 - 4 T + T^{2}$$
$67$ $$44 + 14 T + T^{2}$$
$71$ $$29 - 11 T + T^{2}$$
$73$ $$( 9 + T )^{2}$$
$79$ $$-25 - 5 T + T^{2}$$
$83$ $$11 - 8 T + T^{2}$$
$89$ $$-80 + T^{2}$$
$97$ $$-11 + T + T^{2}$$