Properties

 Label 10000.2.a.j 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: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2500) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - \beta q^{7} + (\beta - 2) q^{9} +O(q^{10})$$ q + b * q^3 - b * q^7 + (b - 2) * q^9 $$q + \beta q^{3} - \beta q^{7} + (\beta - 2) q^{9} + ( - \beta - 2) q^{11} + 3 q^{13} + (4 \beta - 1) q^{17} + (\beta + 1) q^{19} + ( - \beta - 1) q^{21} + (\beta - 2) q^{23} + ( - 4 \beta + 1) q^{27} + ( - 6 \beta + 4) q^{29} + (4 \beta + 2) q^{31} + ( - 3 \beta - 1) q^{33} + ( - 4 \beta + 5) q^{37} + 3 \beta q^{39} + (4 \beta + 4) q^{41} - 10 q^{43} + (7 \beta - 7) q^{47} + (\beta - 6) q^{49} + (3 \beta + 4) q^{51} + ( - 4 \beta - 2) q^{53} + (2 \beta + 1) q^{57} + ( - 2 \beta + 9) q^{59} + (7 \beta + 1) q^{61} + (\beta - 1) q^{63} + ( - 10 \beta + 7) q^{67} + ( - \beta + 1) q^{69} + (5 \beta - 3) q^{71} + ( - \beta + 6) q^{73} + (3 \beta + 1) q^{77} - 13 q^{79} + ( - 6 \beta + 2) q^{81} + (8 \beta - 8) q^{83} + ( - 2 \beta - 6) q^{87} + ( - \beta + 7) q^{89} - 3 \beta q^{91} + (6 \beta + 4) q^{93} + (3 \beta + 14) q^{97} + ( - \beta + 3) q^{99} +O(q^{100})$$ q + b * q^3 - b * q^7 + (b - 2) * q^9 + (-b - 2) * q^11 + 3 * q^13 + (4*b - 1) * q^17 + (b + 1) * q^19 + (-b - 1) * q^21 + (b - 2) * q^23 + (-4*b + 1) * q^27 + (-6*b + 4) * q^29 + (4*b + 2) * q^31 + (-3*b - 1) * q^33 + (-4*b + 5) * q^37 + 3*b * q^39 + (4*b + 4) * q^41 - 10 * q^43 + (7*b - 7) * q^47 + (b - 6) * q^49 + (3*b + 4) * q^51 + (-4*b - 2) * q^53 + (2*b + 1) * q^57 + (-2*b + 9) * q^59 + (7*b + 1) * q^61 + (b - 1) * q^63 + (-10*b + 7) * q^67 + (-b + 1) * q^69 + (5*b - 3) * q^71 + (-b + 6) * q^73 + (3*b + 1) * q^77 - 13 * q^79 + (-6*b + 2) * q^81 + (8*b - 8) * q^83 + (-2*b - 6) * q^87 + (-b + 7) * q^89 - 3*b * q^91 + (6*b + 4) * q^93 + (3*b + 14) * q^97 + (-b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - q^7 - 3 * q^9 $$2 q + q^{3} - q^{7} - 3 q^{9} - 5 q^{11} + 6 q^{13} + 2 q^{17} + 3 q^{19} - 3 q^{21} - 3 q^{23} - 2 q^{27} + 2 q^{29} + 8 q^{31} - 5 q^{33} + 6 q^{37} + 3 q^{39} + 12 q^{41} - 20 q^{43} - 7 q^{47} - 11 q^{49} + 11 q^{51} - 8 q^{53} + 4 q^{57} + 16 q^{59} + 9 q^{61} - q^{63} + 4 q^{67} + q^{69} - q^{71} + 11 q^{73} + 5 q^{77} - 26 q^{79} - 2 q^{81} - 8 q^{83} - 14 q^{87} + 13 q^{89} - 3 q^{91} + 14 q^{93} + 31 q^{97} + 5 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^7 - 3 * q^9 - 5 * q^11 + 6 * q^13 + 2 * q^17 + 3 * q^19 - 3 * q^21 - 3 * q^23 - 2 * q^27 + 2 * q^29 + 8 * q^31 - 5 * q^33 + 6 * q^37 + 3 * q^39 + 12 * q^41 - 20 * q^43 - 7 * q^47 - 11 * q^49 + 11 * q^51 - 8 * q^53 + 4 * q^57 + 16 * q^59 + 9 * q^61 - q^63 + 4 * q^67 + q^69 - q^71 + 11 * q^73 + 5 * q^77 - 26 * q^79 - 2 * q^81 - 8 * q^83 - 14 * q^87 + 13 * q^89 - 3 * q^91 + 14 * q^93 + 31 * q^97 + 5 * 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
 −0.618034 1.61803
0 −0.618034 0 0 0 0.618034 0 −2.61803 0
1.2 0 1.61803 0 0 0 −1.61803 0 −0.381966 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.j 2
4.b odd 2 1 2500.2.a.a 2
5.b even 2 1 10000.2.a.e 2
20.d odd 2 1 2500.2.a.b yes 2
20.e even 4 2 2500.2.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2500.2.a.a 2 4.b odd 2 1
2500.2.a.b yes 2 20.d odd 2 1
2500.2.c.a 4 20.e even 4 2
10000.2.a.e 2 5.b even 2 1
10000.2.a.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 1$$
$11$ $$T^{2} + 5T + 5$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 2T - 19$$
$19$ $$T^{2} - 3T + 1$$
$23$ $$T^{2} + 3T + 1$$
$29$ $$T^{2} - 2T - 44$$
$31$ $$T^{2} - 8T - 4$$
$37$ $$T^{2} - 6T - 11$$
$41$ $$T^{2} - 12T + 16$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} + 7T - 49$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} - 16T + 59$$
$61$ $$T^{2} - 9T - 41$$
$67$ $$T^{2} - 4T - 121$$
$71$ $$T^{2} + T - 31$$
$73$ $$T^{2} - 11T + 29$$
$79$ $$(T + 13)^{2}$$
$83$ $$T^{2} + 8T - 64$$
$89$ $$T^{2} - 13T + 41$$
$97$ $$T^{2} - 31T + 229$$