# Properties

 Label 10000.2.a.r 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: 4.4.108625.1 Defining polynomial: $$x^{4} - x^{3} - 34x^{2} + 9x + 261$$ x^4 - x^3 - 34*x^2 + 9*x + 261 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5000) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + (-b2 + b1) * q^7 + (b2 - 1) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{2} - \beta_1) q^{11} + (\beta_{3} + \beta_{2} + 1) q^{13} + (2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{19} + ( - \beta_{3} - \beta_1 + 1) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} + (4 \beta_{2} + 3) q^{27} + (2 \beta_{2} - 2) q^{29} + (4 \beta_{2} + 2) q^{31} + (\beta_{3} + \beta_1 + 1) q^{33} + ( - \beta_{3} + 3 \beta_{2} + 3) q^{37} + ( - \beta_{2} - \beta_1 - 2) q^{39} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{41} + ( - 6 \beta_{2} + \beta_1 - 3) q^{43} + (\beta_{3} + \beta_1 - 3) q^{47} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 12) q^{49} + (\beta_{3} + \beta_1 - 2) q^{51} + ( - 4 \beta_{2} + 2) q^{53} + (4 \beta_{2} + \beta_1 + 6) q^{57} + ( - \beta_1 + 6) q^{59} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{61} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{63} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{67} + ( - \beta_{3} - \beta_1 + 3) q^{69} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{71} + ( - \beta_{3} + 6 \beta_{2} + 3) q^{73} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 - 17) q^{77} + ( - \beta_{3} - \beta_{2} - 1) q^{79} + ( - 6 \beta_{2} - 4) q^{81} - 8 q^{83} + 2 \beta_{2} q^{87} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{89} + (2 \beta_{3} + 15 \beta_{2} + \beta_1 + 2) q^{91} + ( - 2 \beta_{2} - 6) q^{93} + (\beta_{3} + 2 \beta_1 - 7) q^{97} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + (-b2 + b1) * q^7 + (b2 - 1) * q^9 + (-b2 - b1) * q^11 + (b3 + b2 + 1) * q^13 + (2*b2 - b1) * q^17 + (-b3 - 2*b2 - 4) * q^19 + (-b3 - b1 + 1) * q^21 + (-3*b2 + b1) * q^23 + (4*b2 + 3) * q^27 + (2*b2 - 2) * q^29 + (4*b2 + 2) * q^31 + (b3 + b1 + 1) * q^33 + (-b3 + 3*b2 + 3) * q^37 + (-b2 - b1 - 2) * q^39 + (b3 - b2 + b1 - 3) * q^41 + (-6*b2 + b1 - 3) * q^43 + (b3 + b1 - 3) * q^47 + (-b3 + 2*b2 + b1 + 12) * q^49 + (b3 + b1 - 2) * q^51 + (-4*b2 + 2) * q^53 + (4*b2 + b1 + 6) * q^57 + (-b1 + 6) * q^59 + (b3 + 2*b2 - b1 + 7) * q^61 + (b3 + 2*b2 - b1 - 1) * q^63 + (-b3 - b2 - b1 - 2) * q^67 + (-b3 - b1 + 3) * q^69 + (-b3 + 4*b2 - b1 + 1) * q^71 + (-b3 + 6*b2 + 3) * q^73 + (-b3 - 4*b2 - b1 - 17) * q^77 + (-b3 - b2 - 1) * q^79 + (-6*b2 - 4) * q^81 - 8 * q^83 + 2*b2 * q^87 + (-b3 + 2*b2 - 2*b1 - 2) * q^89 + (2*b3 + 15*b2 + b1 + 2) * q^91 + (-2*b2 - 6) * q^93 + (b3 + 2*b1 - 7) * q^97 + (-b3 + 2*b2 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 3 q^{7} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9 $$4 q - 2 q^{3} + 3 q^{7} - 6 q^{9} + q^{11} + 4 q^{13} - 5 q^{17} - 14 q^{19} + q^{21} + 7 q^{23} + 4 q^{27} - 12 q^{29} + 7 q^{33} + 4 q^{37} - 7 q^{39} - 7 q^{41} + q^{43} - 9 q^{47} + 43 q^{49} - 5 q^{51} + 16 q^{53} + 17 q^{57} + 23 q^{59} + 25 q^{61} - 7 q^{63} - 9 q^{67} + 9 q^{69} - 7 q^{71} - 2 q^{73} - 63 q^{77} - 4 q^{79} - 4 q^{81} - 32 q^{83} - 4 q^{87} - 16 q^{89} - 17 q^{91} - 20 q^{93} - 24 q^{97} - 9 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9 + q^11 + 4 * q^13 - 5 * q^17 - 14 * q^19 + q^21 + 7 * q^23 + 4 * q^27 - 12 * q^29 + 7 * q^33 + 4 * q^37 - 7 * q^39 - 7 * q^41 + q^43 - 9 * q^47 + 43 * q^49 - 5 * q^51 + 16 * q^53 + 17 * q^57 + 23 * q^59 + 25 * q^61 - 7 * q^63 - 9 * q^67 + 9 * q^69 - 7 * q^71 - 2 * q^73 - 63 * q^77 - 4 * q^79 - 4 * q^81 - 32 * q^83 - 4 * q^87 - 16 * q^89 - 17 * q^91 - 20 * q^93 - 24 * q^97 - 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 34x^{2} + 9x + 261$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 4\nu^{2} - 16\nu + 51 ) / 6$$ (v^3 - 4*v^2 - 16*v + 51) / 6 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 6\nu^{2} + 14\nu - 87 ) / 2$$ (-v^3 + 6*v^2 + 14*v - 87) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2} + \beta _1 + 18$$ b3 + 3*b2 + b1 + 18 $$\nu^{3}$$ $$=$$ $$4\beta_{3} + 18\beta_{2} + 20\beta _1 + 21$$ 4*b3 + 18*b2 + 20*b1 + 21

## 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
 −3.71963 5.33766 −3.94789 3.32986
0 −1.61803 0 0 0 −4.33766 0 −0.381966 0
1.2 0 −1.61803 0 0 0 4.71963 0 −0.381966 0
1.3 0 0.618034 0 0 0 −2.32986 0 −2.61803 0
1.4 0 0.618034 0 0 0 4.94789 0 −2.61803 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.r 4
4.b odd 2 1 5000.2.a.h yes 4
5.b even 2 1 10000.2.a.y 4
20.d odd 2 1 5000.2.a.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5000.2.a.g 4 20.d odd 2 1
5000.2.a.h yes 4 4.b odd 2 1
10000.2.a.r 4 1.a even 1 1 trivial
10000.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{7}^{4} - 3T_{7}^{3} - 31T_{7}^{2} + 58T_{7} + 236$$ T7^4 - 3*T7^3 - 31*T7^2 + 58*T7 + 236 $$T_{11}^{4} - T_{11}^{3} - 39T_{11}^{2} + 44T_{11} + 176$$ T11^4 - T11^3 - 39*T11^2 + 44*T11 + 176

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + T - 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 3 T^{3} - 31 T^{2} + 58 T + 236$$
$11$ $$T^{4} - T^{3} - 39 T^{2} + 44 T + 176$$
$13$ $$T^{4} - 4 T^{3} - 39 T^{2} + 146 T + 116$$
$17$ $$T^{4} + 5 T^{3} - 30 T^{2} - 115 T + 95$$
$19$ $$T^{4} + 14 T^{3} + 21 T^{2} + \cdots - 829$$
$23$ $$T^{4} - 7 T^{3} - 31 T^{2} + 192 T - 144$$
$29$ $$(T^{2} + 6 T + 4)^{2}$$
$31$ $$(T^{2} - 20)^{2}$$
$37$ $$T^{4} - 4 T^{3} - 59 T^{2} + 306 T - 324$$
$41$ $$T^{4} + 7 T^{3} - 41 T^{2} - 132 T + 176$$
$43$ $$T^{4} - T^{3} - 109 T^{2} + 144 T + 396$$
$47$ $$T^{4} + 9 T^{3} - 29 T^{2} - 276 T - 144$$
$53$ $$(T^{2} - 8 T - 4)^{2}$$
$59$ $$T^{4} - 23 T^{3} + 164 T^{2} + \cdots + 171$$
$61$ $$T^{4} - 25 T^{3} + 135 T^{2} + \cdots - 4180$$
$67$ $$T^{4} + 9 T^{3} - 34 T^{2} - 81 T + 181$$
$71$ $$T^{4} + 7 T^{3} - 71 T^{2} + \cdots - 1324$$
$73$ $$T^{4} + 2 T^{3} - 131 T^{2} + \cdots + 531$$
$79$ $$T^{4} + 4 T^{3} - 39 T^{2} - 146 T + 116$$
$83$ $$(T + 8)^{4}$$
$89$ $$T^{4} + 16 T^{3} - 49 T^{2} + \cdots - 3209$$
$97$ $$T^{4} + 24 T^{3} + 71 T^{2} + \cdots - 5689$$