# Properties

 Label 10000.2.a.bj Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# 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: $$8$$ Coefficient field: 8.8.14884000000.2 Defining polynomial: $$x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1$$ x^8 - 11*x^6 + 36*x^4 - 31*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} + \beta_{5}) q^{3} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} - \beta_{3} q^{9}+O(q^{10})$$ q + (-b6 + b5) * q^3 + (b7 - b6 - b1) * q^7 - b3 * q^9 $$q + ( - \beta_{6} + \beta_{5}) q^{3} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} - \beta_{3} q^{9} - 2 q^{11} - \beta_{5} q^{13} + ( - 2 \beta_{7} + \beta_{6} + \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{2} - 2) q^{19} + (\beta_{3} - \beta_{2} + 1) q^{21} + (\beta_{6} + \beta_{5}) q^{23} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_1) q^{27} + (3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 4) q^{29} + (\beta_{3} + 2 \beta_{2} - 1) q^{31} + (2 \beta_{6} - 2 \beta_{5}) q^{33} + (4 \beta_{7} + \beta_{5} - \beta_1) q^{37} + (2 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{39} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{41} + (3 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_1) q^{43} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_1) q^{47} + (2 \beta_{3} - \beta_{2} - 1) q^{49} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{51} + ( - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_1) q^{53} + (\beta_{7} + 4 \beta_{6} - \beta_{5} - 3 \beta_1) q^{57} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 4) q^{59} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + 3) q^{61} + (2 \beta_{6} + 2 \beta_1) q^{63} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_1) q^{67} + ( - 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{69} + ( - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 7) q^{71} + (3 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_1) q^{73} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_1) q^{77} + (4 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 2) q^{79} + (4 \beta_{4} + 2 \beta_{3} - 1) q^{81} + ( - 4 \beta_{7} + 5 \beta_{6} - \beta_{5} + 4 \beta_1) q^{83} + (\beta_{7} - 6 \beta_{6} + \beta_{5} + 3 \beta_1) q^{87} + ( - 7 \beta_{4} - \beta_{3} + \beta_{2}) q^{89} + (\beta_{2} + 2) q^{91} + (7 \beta_{6} - 5 \beta_{5} - 4 \beta_1) q^{93} + ( - 5 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 4 \beta_1) q^{97} + 2 \beta_{3} q^{99}+O(q^{100})$$ q + (-b6 + b5) * q^3 + (b7 - b6 - b1) * q^7 - b3 * q^9 - 2 * q^11 - b5 * q^13 + (-2*b7 + b6 + b1) * q^17 + (-2*b4 + b2 - 2) * q^19 + (b3 - b2 + 1) * q^21 + (b6 + b5) * q^23 + (-2*b7 + b6 - b5 + 2*b1) * q^27 + (3*b4 + b3 - 2*b2 + 4) * q^29 + (b3 + 2*b2 - 1) * q^31 + (2*b6 - 2*b5) * q^33 + (4*b7 + b5 - b1) * q^37 + (2*b4 + b3 - b2 - 1) * q^39 + (-b4 - 2*b3 + b2 + 2) * q^41 + (3*b7 + 2*b6 - b5 - b1) * q^43 + (-b7 + b6 - 2*b5 + 3*b1) * q^47 + (2*b3 - b2 - 1) * q^49 + (-2*b3 + 2*b2) * q^51 + (-3*b7 + 2*b6 - 2*b5 - 3*b1) * q^53 + (b7 + 4*b6 - b5 - 3*b1) * q^57 + (2*b4 - 2*b3 - b2 - 4) * q^59 + (b4 + 3*b3 + b2 + 3) * q^61 + (2*b6 + 2*b1) * q^63 + (-2*b7 + 2*b6 + 4*b1) * q^67 + (-4*b4 - b3 + 2*b2 - 1) * q^69 + (-2*b4 + b3 - 3*b2 - 7) * q^71 + (3*b7 - 2*b6 + 2*b5 - b1) * q^73 + (-2*b7 + 2*b6 + 2*b1) * q^77 + (4*b4 + 4*b3 - 3*b2 + 2) * q^79 + (4*b4 + 2*b3 - 1) * q^81 + (-4*b7 + 5*b6 - b5 + 4*b1) * q^83 + (b7 - 6*b6 + b5 + 3*b1) * q^87 + (-7*b4 - b3 + b2) * q^89 + (b2 + 2) * q^91 + (7*b6 - 5*b5 - 4*b1) * q^93 + (-5*b7 - 4*b6 - 4*b5 + 4*b1) * q^97 + 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^9 $$8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89} + 14 q^{91} - 8 q^{99}+O(q^{100})$$ 8 * q + 4 * q^9 - 16 * q^11 - 10 * q^19 + 6 * q^21 + 20 * q^29 - 16 * q^31 - 18 * q^39 + 26 * q^41 - 14 * q^49 + 4 * q^51 - 30 * q^59 + 6 * q^61 + 8 * q^69 - 46 * q^71 - 10 * q^79 - 32 * q^81 + 30 * q^89 + 14 * q^91 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 1$$ v^4 - 5*v^2 + 1 $$\beta_{4}$$ $$=$$ $$( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 7 ) / 4$$ (v^6 - 8*v^4 + 16*v^2 - 7) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu ) / 4$$ (v^7 - 8*v^5 + 16*v^3 - 7*v) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 27\nu ) / 4$$ (-v^7 + 12*v^5 - 40*v^3 + 27*v) / 4 $$\beta_{7}$$ $$=$$ $$( \nu^{7} - 12\nu^{5} + 44\nu^{3} - 43\nu ) / 4$$ (v^7 - 12*v^5 + 44*v^3 - 43*v) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + 4\beta_1$$ b7 + b6 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5\beta_{2} + 14$$ b3 + 5*b2 + 14 $$\nu^{5}$$ $$=$$ $$6\beta_{7} + 7\beta_{6} + \beta_{5} + 19\beta_1$$ 6*b7 + 7*b6 + b5 + 19*b1 $$\nu^{6}$$ $$=$$ $$4\beta_{4} + 8\beta_{3} + 24\beta_{2} + 71$$ 4*b4 + 8*b3 + 24*b2 + 71 $$\nu^{7}$$ $$=$$ $$32\beta_{7} + 40\beta_{6} + 12\beta_{5} + 95\beta_1$$ 32*b7 + 40*b6 + 12*b5 + 95*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
 −1.13370 2.08529 0.183172 2.30927 −2.30927 −0.183172 −2.08529 1.13370
0 −2.60278 0 0 0 0.407162 0 3.77447 0
1.2 0 −2.19849 0 0 0 0.992398 0 1.83337 0
1.3 0 −1.47195 0 0 0 −3.26086 0 −0.833366 0
1.4 0 −0.474903 0 0 0 −3.03582 0 −2.77447 0
1.5 0 0.474903 0 0 0 3.03582 0 −2.77447 0
1.6 0 1.47195 0 0 0 3.26086 0 −0.833366 0
1.7 0 2.19849 0 0 0 −0.992398 0 1.83337 0
1.8 0 2.60278 0 0 0 −0.407162 0 3.77447 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10000.2.a.bj 8
4.b odd 2 1 625.2.a.f 8
5.b even 2 1 inner 10000.2.a.bj 8
12.b even 2 1 5625.2.a.x 8
20.d odd 2 1 625.2.a.f 8
20.e even 4 2 625.2.b.c 8
25.f odd 20 2 400.2.y.c 8
60.h even 2 1 5625.2.a.x 8
100.h odd 10 2 125.2.d.b 16
100.h odd 10 2 625.2.d.o 16
100.j odd 10 2 125.2.d.b 16
100.j odd 10 2 625.2.d.o 16
100.l even 20 2 25.2.e.a 8
100.l even 20 2 125.2.e.b 8
100.l even 20 2 625.2.e.a 8
100.l even 20 2 625.2.e.i 8
300.u odd 20 2 225.2.m.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 100.l even 20 2
125.2.d.b 16 100.h odd 10 2
125.2.d.b 16 100.j odd 10 2
125.2.e.b 8 100.l even 20 2
225.2.m.a 8 300.u odd 20 2
400.2.y.c 8 25.f odd 20 2
625.2.a.f 8 4.b odd 2 1
625.2.a.f 8 20.d odd 2 1
625.2.b.c 8 20.e even 4 2
625.2.d.o 16 100.h odd 10 2
625.2.d.o 16 100.j odd 10 2
625.2.e.a 8 100.l even 20 2
625.2.e.i 8 100.l even 20 2
5625.2.a.x 8 12.b even 2 1
5625.2.a.x 8 60.h even 2 1
10000.2.a.bj 8 1.a even 1 1 trivial
10000.2.a.bj 8 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{8} - 14T_{3}^{6} + 61T_{3}^{4} - 84T_{3}^{2} + 16$$ T3^8 - 14*T3^6 + 61*T3^4 - 84*T3^2 + 16 $$T_{7}^{8} - 21T_{7}^{6} + 121T_{7}^{4} - 116T_{7}^{2} + 16$$ T7^8 - 21*T7^6 + 121*T7^4 - 116*T7^2 + 16 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 14 T^{6} + 61 T^{4} - 84 T^{2} + \cdots + 16$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 21 T^{6} + 121 T^{4} + \cdots + 16$$
$11$ $$(T + 2)^{8}$$
$13$ $$T^{8} - 14 T^{6} + 31 T^{4} - 14 T^{2} + \cdots + 1$$
$17$ $$T^{8} - 41 T^{6} + 441 T^{4} + \cdots + 1936$$
$19$ $$(T^{4} + 5 T^{3} - 5 T^{2} - 30 T - 20)^{2}$$
$23$ $$T^{8} - 34 T^{6} + 301 T^{4} + \cdots + 256$$
$29$ $$(T^{4} - 10 T^{3} - 5 T^{2} + 290 T - 695)^{2}$$
$31$ $$(T^{4} + 8 T^{3} - 41 T^{2} - 328 T - 44)^{2}$$
$37$ $$T^{8} - 111 T^{6} + 3556 T^{4} + \cdots + 116281$$
$41$ $$(T^{4} - 13 T^{3} + 19 T^{2} + 148 T + 116)^{2}$$
$43$ $$T^{8} - 129 T^{6} + 4421 T^{4} + \cdots + 246016$$
$47$ $$T^{8} - 141 T^{6} + 4661 T^{4} + \cdots + 65536$$
$53$ $$T^{8} - 239 T^{6} + 20356 T^{4} + \cdots + 8755681$$
$59$ $$(T^{4} + 15 T^{3} + 5 T^{2} - 630 T - 2020)^{2}$$
$61$ $$(T^{4} - 3 T^{3} - 146 T^{2} - 237 T + 341)^{2}$$
$67$ $$T^{8} - 176 T^{6} + 7776 T^{4} + \cdots + 246016$$
$71$ $$(T^{4} + 23 T^{3} + 99 T^{2} - 798 T - 4924)^{2}$$
$73$ $$T^{8} - 79 T^{6} + 76 T^{4} - 19 T^{2} + \cdots + 1$$
$79$ $$(T^{4} + 5 T^{3} - 195 T^{2} + 5780)^{2}$$
$83$ $$T^{8} - 374 T^{6} + 35061 T^{4} + \cdots + 99856$$
$89$ $$(T^{4} - 15 T^{3} - 35 T^{2} + 530 T + 1180)^{2}$$
$97$ $$T^{8} - 666 T^{6} + \cdots + 301334881$$