# Properties

 Label 10000.2.a.u Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.18625.1 Defining polynomial: $$x^{4} - x^{3} - 14x^{2} + 9x + 41$$ x^4 - x^3 - 14*x^2 + 9*x + 41 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1250) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{7} + ( - \beta_{3} - \beta_{2} + 5) q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^3 + (-b3 - 2*b2 + 2) * q^7 + (-b3 - b2 + 5) * q^9 $$q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{7} + ( - \beta_{3} - \beta_{2} + 5) q^{9} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_1 - 1) q^{13} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{3} - \beta_{2} - 1) q^{19} + ( - \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 2) q^{21} + (2 \beta_{2} + \beta_1 + 1) q^{23} + ( - 8 \beta_{2} + 2 \beta_1 + 1) q^{27} + (\beta_{3} + \beta_1 - 3) q^{29} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{31} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 + 9) q^{33} + ( - \beta_{3} - 2 \beta_{2} - 2) q^{37} + (\beta_{3} - 6 \beta_{2} - \beta_1 + 7) q^{39} + ( - \beta_{2} + \beta_1 + 1) q^{41} + (3 \beta_{2} - 2 \beta_1 - 5) q^{43} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{47} + (2 \beta_{2} + 3 \beta_1 + 4) q^{49} + ( - 2 \beta_{3} + 6 \beta_{2} + \beta_1 - 6) q^{51} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{53} + ( - 2 \beta_{3} + 7 \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{3} - \beta_{2} - 8) q^{59} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{61} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 16) q^{63} + ( - 5 \beta_{2} + \beta_1 + 8) q^{67} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{71} + (\beta_{2} + 3 \beta_1 - 1) q^{73} + (\beta_{3} + 4 \beta_1 + 6) q^{77} + (3 \beta_{3} - \beta_1 - 3) q^{79} + ( - 5 \beta_{3} + \beta_1 + 7) q^{81} + ( - \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{83} + ( - \beta_{3} + 8 \beta_{2} - 3 \beta_1 + 7) q^{87} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{89} + ( - \beta_{3} + 2 \beta_1 + 4) q^{91} + (3 \beta_{3} - 5 \beta_1 - 9) q^{93} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{97} + ( - 2 \beta_{3} - 8 \beta_{2} + 6 \beta_1 + 2) q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^3 + (-b3 - 2*b2 + 2) * q^7 + (-b3 - b2 + 5) * q^9 + (-b3 - 2*b2 + b1 - 1) * q^11 + (-b3 + b1 - 1) * q^13 + (b3 - b2 - b1 + 1) * q^17 + (b3 - b2 - 1) * q^19 + (-b3 - 8*b2 + 2*b1 + 2) * q^21 + (2*b2 + b1 + 1) * q^23 + (-8*b2 + 2*b1 + 1) * q^27 + (b3 + b1 - 3) * q^29 + (-b3 + 2*b2 - b1 - 5) * q^31 + (-b3 - 6*b2 - b1 + 9) * q^33 + (-b3 - 2*b2 - 2) * q^37 + (b3 - 6*b2 - b1 + 7) * q^39 + (-b2 + b1 + 1) * q^41 + (3*b2 - 2*b1 - 5) * q^43 + (b3 + 2*b2 - b1 + 1) * q^47 + (2*b2 + 3*b1 + 4) * q^49 + (-2*b3 + 6*b2 + b1 - 6) * q^51 + (-b3 - 2*b2 + b1 + 5) * q^53 + (-2*b3 + 7*b2 - b1 + 1) * q^57 + (-b3 - b2 - 8) * q^59 + (-b3 + 2*b2 + b1 + 1) * q^61 + (-4*b3 - 4*b2 + 2*b1 + 16) * q^63 + (-5*b2 + b1 + 8) * q^67 + (2*b3 - 2*b2 + b1 + 5) * q^69 + (-2*b3 - 2*b2 + 3*b1 + 5) * q^71 + (b2 + 3*b1 - 1) * q^73 + (b3 + 4*b1 + 6) * q^77 + (3*b3 - b1 - 3) * q^79 + (-5*b3 + b1 + 7) * q^81 + (-b3 + 4*b2 - 2*b1 - 4) * q^83 + (-b3 + 8*b2 - 3*b1 + 7) * q^87 + (-b3 + b2 - 2*b1 - 1) * q^89 + (-b3 + 2*b1 + 4) * q^91 + (3*b3 - 5*b1 - 9) * q^93 + (b3 - 3*b2 + 2*b1) * q^97 + (-2*b3 - 8*b2 + 6*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 3 q^{7} + 17 q^{9}+O(q^{10})$$ 4 * q - q^3 + 3 * q^7 + 17 * q^9 $$4 q - q^{3} + 3 q^{7} + 17 q^{9} - 8 q^{11} - 4 q^{13} + 2 q^{17} - 5 q^{19} - 7 q^{21} + 9 q^{23} - 10 q^{27} - 10 q^{29} - 18 q^{31} + 22 q^{33} - 13 q^{37} + 16 q^{39} + 3 q^{41} - 16 q^{43} + 8 q^{47} + 23 q^{49} - 13 q^{51} + 16 q^{53} + 15 q^{57} - 35 q^{59} + 8 q^{61} + 54 q^{63} + 23 q^{67} + 19 q^{69} + 17 q^{71} + q^{73} + 29 q^{77} - 10 q^{79} + 24 q^{81} - 11 q^{83} + 40 q^{87} - 5 q^{89} + 17 q^{91} - 38 q^{93} - 3 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - q^3 + 3 * q^7 + 17 * q^9 - 8 * q^11 - 4 * q^13 + 2 * q^17 - 5 * q^19 - 7 * q^21 + 9 * q^23 - 10 * q^27 - 10 * q^29 - 18 * q^31 + 22 * q^33 - 13 * q^37 + 16 * q^39 + 3 * q^41 - 16 * q^43 + 8 * q^47 + 23 * q^49 - 13 * q^51 + 16 * q^53 + 15 * q^57 - 35 * q^59 + 8 * q^61 + 54 * q^63 + 23 * q^67 + 19 * q^69 + 17 * q^71 + q^73 + 29 * q^77 - 10 * q^79 + 24 * q^81 - 11 * q^83 + 40 * q^87 - 5 * q^89 + 17 * q^91 - 38 * q^93 - 3 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 14x^{2} + 9x + 41$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 8\nu + 1 ) / 6$$ (v^3 - 8*v + 1) / 6 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 7$$ v^2 - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 7$$ b3 + 7 $$\nu^{3}$$ $$=$$ $$6\beta_{2} + 8\beta _1 - 1$$ 6*b2 + 8*b1 - 1

## 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
 −1.64791 −3.08634 3.26594 2.46831
0 −3.26594 0 0 0 3.04834 0 7.66637 0
1.2 0 −2.46831 0 0 0 0.710571 0 3.09254 0
1.3 0 1.64791 0 0 0 −4.90244 0 −0.284403 0
1.4 0 3.08634 0 0 0 4.14353 0 6.52550 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.u 4
4.b odd 2 1 1250.2.a.j yes 4
5.b even 2 1 10000.2.a.v 4
20.d odd 2 1 1250.2.a.g 4
20.e even 4 2 1250.2.b.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.g 4 20.d odd 2 1
1250.2.a.j yes 4 4.b odd 2 1
1250.2.b.d 8 20.e even 4 2
10000.2.a.u 4 1.a even 1 1 trivial
10000.2.a.v 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{4} + T_{3}^{3} - 14T_{3}^{2} - 9T_{3} + 41$$ T3^4 + T3^3 - 14*T3^2 - 9*T3 + 41 $$T_{7}^{4} - 3T_{7}^{3} - 21T_{7}^{2} + 78T_{7} - 44$$ T7^4 - 3*T7^3 - 21*T7^2 + 78*T7 - 44 $$T_{11}^{4} + 8T_{11}^{3} - T_{11}^{2} - 108T_{11} - 144$$ T11^4 + 8*T11^3 - T11^2 - 108*T11 - 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + T^{3} - 14 T^{2} - 9 T + 41$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 3 T^{3} - 21 T^{2} + 78 T - 44$$
$11$ $$T^{4} + 8 T^{3} - T^{2} - 108 T - 144$$
$13$ $$T^{4} + 4 T^{3} - 19 T^{2} - 6 T + 36$$
$17$ $$T^{4} - 2 T^{3} - 31 T^{2} - 48 T - 9$$
$19$ $$T^{4} + 5 T^{3} - 15 T^{2} - 10 T + 20$$
$23$ $$T^{4} - 9 T^{3} + T^{2} + 96 T - 144$$
$29$ $$T^{4} + 10 T^{3} - 5 T^{2} - 150 T - 180$$
$31$ $$T^{4} + 18 T^{3} + 69 T^{2} + \cdots - 1604$$
$37$ $$T^{4} + 13 T^{3} + 39 T^{2} + 22 T - 4$$
$41$ $$T^{4} - 3 T^{3} - 11 T^{2} + 18 T + 36$$
$43$ $$T^{4} + 16 T^{3} + 31 T^{2} + \cdots - 169$$
$47$ $$T^{4} - 8 T^{3} - T^{2} + 108 T - 144$$
$53$ $$T^{4} - 16 T^{3} + 71 T^{2} - 96 T + 36$$
$59$ $$T^{4} + 35 T^{3} + 440 T^{2} + \cdots + 4545$$
$61$ $$T^{4} - 8 T^{3} - 21 T^{2} + 268 T - 484$$
$67$ $$T^{4} - 23 T^{3} + 134 T^{2} + \cdots - 599$$
$71$ $$T^{4} - 17 T^{3} - 31 T^{2} + \cdots - 2844$$
$73$ $$T^{4} - T^{3} - 139 T^{2} + 154 T + 2836$$
$79$ $$T^{4} + 10 T^{3} - 125 T^{2} + \cdots + 4220$$
$83$ $$T^{4} + 11 T^{3} - 79 T^{2} + \cdots - 2304$$
$89$ $$T^{4} + 5 T^{3} - 85 T^{2} - 300 T + 900$$
$97$ $$T^{4} + 3 T^{3} - 106 T^{2} + \cdots + 701$$