# Properties

 Label 10000.2.a.bb Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5x^{2} + 5$$ x^4 - 5*x^2 + 5 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1 + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + (-b3 - b1 + 2) * q^7 + (b2 + 2*b1 + 1) * q^9 $$q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1 + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} - \beta_{2} - 4) q^{13} + (\beta_{3} - 2) q^{17} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{19} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{21} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + (\beta_{3} + 3 \beta_{2} + 4) q^{27} + (3 \beta_{2} - \beta_1 - 1) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} - 1) q^{31} + (3 \beta_{3} + 4 \beta_{2}) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{37} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 5) q^{39} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 5) q^{41} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 2) q^{43} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 4) q^{47} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{49} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{51} + (2 \beta_{2} - 2 \beta_1) q^{53} + ( - 3 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{57} - 4 \beta_{3} q^{59} + ( - 3 \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{61} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 6) q^{63} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{69} + ( - 6 \beta_{2} + 2 \beta_1 - 5) q^{71} + (4 \beta_{3} - 6) q^{73} + (3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{77} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{79} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{81} + (5 \beta_{3} + 5 \beta_{2} + 6) q^{83} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{87} + ( - 6 \beta_{2} - 8) q^{89} + (2 \beta_{3} - \beta_{2} + 5 \beta_1 - 5) q^{91} + ( - 2 \beta_{2} - \beta_1 - 3) q^{93} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 6) q^{97} + (\beta_{3} + 7 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 + (-b3 - b1 + 2) * q^7 + (b2 + 2*b1 + 1) * q^9 + (2*b3 + b2 - b1 + 1) * q^11 + (-b3 - b2 - 4) * q^13 + (b3 - 2) * q^17 + (-b3 - 2*b2 - 1) * q^19 + (-b3 - 3*b2 + b1 - 2) * q^21 + (b3 - 2*b2 - b1) * q^23 + (b3 + 3*b2 + 4) * q^27 + (3*b2 - b1 - 1) * q^29 + (-2*b3 + 2*b2 - 1) * q^31 + (3*b3 + 4*b2) * q^33 + (-b3 - b2 - b1 - 5) * q^37 + (-2*b3 - 3*b2 - 4*b1 - 5) * q^39 + (4*b3 - 4*b2 - 2*b1 - 5) * q^41 + (3*b3 - 3*b2 + 2*b1 + 2) * q^43 + (-3*b3 + 4*b2 + 3*b1 + 4) * q^47 + (-4*b3 + 4*b2 - 4*b1 + 4) * q^49 + (b3 + 2*b2 - 2*b1 - 1) * q^51 + (2*b2 - 2*b1) * q^53 + (-3*b3 - 4*b2 - b1 - 2) * q^57 - 4*b3 * q^59 + (-3*b3 - 3*b2 - b1 - 2) * q^61 + (-b3 - 4*b2 + 2*b1 - 6) * q^63 + (-b3 + b2 - 2*b1) * q^67 + (-b3 - b2 - b1 - 2) * q^69 + (-6*b2 + 2*b1 - 5) * q^71 + (4*b3 - 6) * q^73 + (3*b3 + 3*b2 - 4*b1) * q^77 + (2*b3 + 2*b2 + 4*b1 - 4) * q^79 + (4*b3 + 2*b2 - 2*b1 + 2) * q^81 + (5*b3 + 5*b2 + 6) * q^83 + (3*b3 + 2*b2 - 2*b1 - 4) * q^87 + (-6*b2 - 8) * q^89 + (2*b3 - b2 + 5*b1 - 5) * q^91 + (-2*b2 - b1 - 3) * q^93 + (2*b3 + 2*b2 + b1 - 6) * q^97 + (b3 + 7*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9 $$4 q + 4 q^{3} + 8 q^{7} + 2 q^{9} + 2 q^{11} - 14 q^{13} - 8 q^{17} - 2 q^{21} + 4 q^{23} + 10 q^{27} - 10 q^{29} - 8 q^{31} - 8 q^{33} - 18 q^{37} - 14 q^{39} - 12 q^{41} + 14 q^{43} + 8 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 2 q^{61} - 16 q^{63} - 2 q^{67} - 6 q^{69} - 8 q^{71} - 24 q^{73} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 14 q^{83} - 20 q^{87} - 20 q^{89} - 18 q^{91} - 8 q^{93} - 28 q^{97} - 14 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9 + 2 * q^11 - 14 * q^13 - 8 * q^17 - 2 * q^21 + 4 * q^23 + 10 * q^27 - 10 * q^29 - 8 * q^31 - 8 * q^33 - 18 * q^37 - 14 * q^39 - 12 * q^41 + 14 * q^43 + 8 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 2 * q^61 - 16 * q^63 - 2 * q^67 - 6 * q^69 - 8 * q^71 - 24 * q^73 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 14 * q^83 - 20 * q^87 - 20 * q^89 - 18 * q^91 - 8 * q^93 - 28 * q^97 - 14 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{20} + \zeta_{20}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## 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
 −1.90211 −1.17557 1.17557 1.90211
0 −0.902113 0 0 0 5.07768 0 −2.18619 0
1.2 0 −0.175571 0 0 0 1.27346 0 −2.96917 0
1.3 0 2.17557 0 0 0 2.72654 0 1.73311 0
1.4 0 2.90211 0 0 0 −1.07768 0 5.42226 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.bb 4
4.b odd 2 1 1250.2.a.i 4
5.b even 2 1 10000.2.a.o 4
20.d odd 2 1 1250.2.a.h 4
20.e even 4 2 1250.2.b.c 8
25.f odd 20 2 400.2.y.a 8
100.h odd 10 2 250.2.d.c 8
100.j odd 10 2 250.2.d.b 8
100.l even 20 2 50.2.e.a 8
100.l even 20 2 250.2.e.a 8
300.u odd 20 2 450.2.l.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.e.a 8 100.l even 20 2
250.2.d.b 8 100.j odd 10 2
250.2.d.c 8 100.h odd 10 2
250.2.e.a 8 100.l even 20 2
400.2.y.a 8 25.f odd 20 2
450.2.l.b 8 300.u odd 20 2
1250.2.a.h 4 20.d odd 2 1
1250.2.a.i 4 4.b odd 2 1
1250.2.b.c 8 20.e even 4 2
10000.2.a.o 4 5.b even 2 1
10000.2.a.bb 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{4} - 4T_{3}^{3} + T_{3}^{2} + 6T_{3} + 1$$ T3^4 - 4*T3^3 + T3^2 + 6*T3 + 1 $$T_{7}^{4} - 8T_{7}^{3} + 14T_{7}^{2} + 8T_{7} - 19$$ T7^4 - 8*T7^3 + 14*T7^2 + 8*T7 - 19 $$T_{11}^{4} - 2T_{11}^{3} - 26T_{11}^{2} + 82T_{11} - 59$$ T11^4 - 2*T11^3 - 26*T11^2 + 82*T11 - 59

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4 T^{3} + T^{2} + 6 T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 8 T^{3} + 14 T^{2} + 8 T - 19$$
$11$ $$T^{4} - 2 T^{3} - 26 T^{2} + 82 T - 59$$
$13$ $$T^{4} + 14 T^{3} + 66 T^{2} + 114 T + 41$$
$17$ $$T^{4} + 8 T^{3} + 19 T^{2} + 12 T + 1$$
$19$ $$T^{4} - 15 T^{2} - 10 T + 5$$
$23$ $$T^{4} - 4 T^{3} - 14 T^{2} - 4 T + 1$$
$29$ $$T^{4} + 10 T^{3} + 10 T^{2} - 90 T - 95$$
$31$ $$T^{4} + 8 T^{3} - 6 T^{2} - 48 T - 19$$
$37$ $$T^{4} + 18 T^{3} + 109 T^{2} + \cdots + 241$$
$41$ $$T^{4} + 12 T^{3} - 86 T^{2} + \cdots - 5339$$
$43$ $$T^{4} - 14 T^{3} - 14 T^{2} + \cdots - 1919$$
$47$ $$T^{4} - 8 T^{3} - 106 T^{2} + \cdots - 2939$$
$53$ $$T^{4} + 4 T^{3} - 24 T^{2} - 96 T - 64$$
$59$ $$T^{4} - 80T^{2} + 1280$$
$61$ $$T^{4} + 2 T^{3} - 71 T^{2} - 12 T + 181$$
$67$ $$T^{4} + 2 T^{3} - 26 T^{2} - 82 T - 59$$
$71$ $$T^{4} + 8 T^{3} - 86 T^{2} + \cdots + 1021$$
$73$ $$T^{4} + 24 T^{3} + 136 T^{2} + \cdots - 304$$
$79$ $$T^{4} + 20 T^{3} + 40 T^{2} + \cdots - 4720$$
$83$ $$T^{4} - 14 T^{3} - 114 T^{2} + \cdots - 4139$$
$89$ $$(T^{2} + 10 T - 20)^{2}$$
$97$ $$T^{4} + 28 T^{3} + 259 T^{2} + \cdots + 361$$