# Properties

 Label 10000.2.a.bn Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.8.6152203125.1 Defining polynomial: $$x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1$$ x^8 - 3*x^7 - 8*x^6 + 20*x^5 + 26*x^4 - 35*x^3 - 27*x^2 + 16*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 625) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{7} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{7} + (2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9}+O(q^{10})$$ q + (-b5 + 1) * q^3 + (-b7 + b5 - b3 - b1 + 1) * q^7 + (2*b6 - b5 - b4 + b3 + 3) * q^9 $$q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{7} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{7} + (2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9} + ( - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{11} + ( - \beta_{6} - \beta_{4} - 1) q^{13} + (\beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} - 2) q^{17} + ( - \beta_{6} + \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{19} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 2) q^{21} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 + 5) q^{23} + (4 \beta_{6} - \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 + 7) q^{27} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 3) q^{29} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 2) q^{31} + ( - 3 \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{33} + (\beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{37} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{39} + ( - 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{41} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2) q^{43} + (3 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 4) q^{47} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} - 5 \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{49} + (4 \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{51} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1) q^{53} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{57} + (3 \beta_{7} - \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 1) q^{59} + (2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{61} + (2 \beta_{7} + 2 \beta_{6} - \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{63} + (3 \beta_{7} - 4 \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{67} + ( - 4 \beta_{7} - 4 \beta_{5} - \beta_{2} - \beta_1 + 3) q^{69} + (4 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{71} + (5 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{73} + (4 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} + 9 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{77} + (\beta_{7} + \beta_{5} - 6 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{79} + (\beta_{6} - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 3) q^{81} + ( - 5 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{83} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 5) q^{87} + ( - 3 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 5 \beta_{3} - 4 \beta_{2} + \cdots - 6) q^{89}+ \cdots + ( - 7 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{2} + \beta_1) q^{99}+O(q^{100})$$ q + (-b5 + 1) * q^3 + (-b7 + b5 - b3 - b1 + 1) * q^7 + (2*b6 - b5 - b4 + b3 + 3) * q^9 + (-2*b4 + b3 + b1 + 1) * q^11 + (-b6 - b4 - 1) * q^13 + (b7 - b5 - b3 + b2 - 2) * q^17 + (-b6 + b4 - b2 - 2*b1 + 1) * q^19 + (-b7 - b6 - b5 + b4 - b3 - b1 - 2) * q^21 + (-2*b7 + b6 + b5 - b4 - b1 + 5) * q^23 + (4*b6 - b5 - b4 + 4*b3 + b2 - b1 + 7) * q^27 + (-b7 - b5 - b4 + b2 - b1 + 3) * q^29 + (-b7 + b6 - b5 - b3 + 2*b2 + 2) * q^31 + (-3*b7 - b6 - b4 - 2*b3 - b2 + 3*b1 - 2) * q^33 + (b7 - b5 - b4 + 2*b3 - 2*b2 + b1 + 2) * q^37 + (-2*b7 - b6 + 2*b5 - 3*b3 - b2 + 2*b1 - 4) * q^39 + (-3*b7 - 2*b6 + 2*b5 - b4 - 3*b3 + b2 - 2*b1 - 1) * q^41 + (-b6 - 2*b5 - b4 + 2*b3 + 2) * q^43 + (3*b7 - b5 - b4 + b3 + b1 + 4) * q^47 + (-2*b7 - 3*b6 + 3*b5 + b4 - 5*b3 - b2 - 3*b1 - 4) * q^49 + (4*b7 + 2*b6 + b5 - 2*b3 + 2*b2 + b1) * q^51 + (b7 + b6 - b5 + b4 + b3 - b2 - 3*b1) * q^53 + (-b7 + b6 - b5 + 2*b4 + 2*b3 - b2 - 3*b1 + 3) * q^57 + (3*b7 - b5 + b4 + 3*b3 - b2 + 1) * q^59 + (2*b7 + 3*b6 - 3*b5 - 2*b4 + 6*b3 + b2 + b1 + 6) * q^61 + (2*b7 + 2*b6 - b4 + 3*b3 - b2 + 2*b1 + 1) * q^63 + (3*b7 - 4*b6 + b4 - b3 + b2 + 2*b1 - 2) * q^67 + (-4*b7 - 4*b5 - b2 - b1 + 3) * q^69 + (4*b7 + b6 - b5 + b4 + 2*b2 + 2*b1) * q^71 + (5*b6 + b5 - 3*b4 - b3 + b2 + 3*b1 + 1) * q^73 + (4*b7 + 3*b6 - 3*b4 + 9*b3 + b2 + b1 + 5) * q^77 + (b7 + b5 - 6*b3 - b2 - b1 - 1) * q^79 + (b6 - 3*b5 + 3*b4 + 2*b3 - 3*b1 + 3) * q^81 + (-5*b7 - b6 + 3*b5 + b4 - 5*b3 - 2*b2 + b1 + 1) * q^83 + (-2*b7 + 3*b6 - b5 - b4 - 3*b3 - b2 + b1 + 5) * q^87 + (-3*b7 - 4*b6 + 3*b5 + b4 - 5*b3 - 4*b2 - 6) * q^89 + (2*b7 - b6 - b5 - b4 + 4*b3 + b2 - b1 + 2) * q^91 + (2*b7 + 3*b6 - b5 - 2*b4 - 3*b3 + b2 + b1 + 5) * q^93 + (b7 + 3*b6 + 2*b5 - b4 + 6*b3 + 3*b2 + 4) * q^97 + (-7*b7 - 4*b6 + 3*b5 - 3*b3 - 2*b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 5 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10})$$ 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9 $$8 q + 5 q^{3} + 10 q^{7} + 9 q^{9} - q^{11} - 10 q^{13} - 15 q^{17} + 10 q^{19} - 14 q^{21} + 30 q^{23} + 20 q^{27} + 10 q^{29} + 9 q^{31} - 5 q^{33} + 10 q^{37} - 8 q^{39} - 4 q^{41} + 30 q^{47} - 4 q^{49} + 14 q^{51} - 10 q^{53} + 10 q^{57} + 5 q^{59} + 6 q^{61} + 10 q^{67} + 3 q^{69} + 9 q^{71} - 5 q^{77} + 20 q^{79} + 8 q^{81} + 40 q^{83} + 40 q^{87} - 5 q^{89} - 6 q^{91} + 40 q^{93} + 22 q^{99}+O(q^{100})$$ 8 * q + 5 * q^3 + 10 * q^7 + 9 * q^9 - q^11 - 10 * q^13 - 15 * q^17 + 10 * q^19 - 14 * q^21 + 30 * q^23 + 20 * q^27 + 10 * q^29 + 9 * q^31 - 5 * q^33 + 10 * q^37 - 8 * q^39 - 4 * q^41 + 30 * q^47 - 4 * q^49 + 14 * q^51 - 10 * q^53 + 10 * q^57 + 5 * q^59 + 6 * q^61 + 10 * q^67 + 3 * q^69 + 9 * q^71 - 5 * q^77 + 20 * q^79 + 8 * q^81 + 40 * q^83 + 40 * q^87 - 5 * q^89 - 6 * q^91 + 40 * q^93 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{6} - 4\nu^{5} - 2\nu^{4} + 17\nu^{3} - \nu^{2} - 15\nu + 1 ) / 3$$ (v^6 - 4*v^5 - 2*v^4 + 17*v^3 - v^2 - 15*v + 1) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 17\nu^{4} - \nu^{3} - 15\nu^{2} + 4\nu ) / 3$$ (v^7 - 4*v^6 - 2*v^5 + 17*v^4 - v^3 - 15*v^2 + 4*v) / 3 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 5\nu^{4} - 26\nu^{3} - 5\nu^{2} + 36\nu - 4 ) / 3$$ (-v^6 + 4*v^5 + 5*v^4 - 26*v^3 - 5*v^2 + 36*v - 4) / 3 $$\beta_{6}$$ $$=$$ $$( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 4\nu^{3} + 51\nu^{2} - 8\nu ) / 3$$ (-2*v^7 + 8*v^6 + 7*v^5 - 43*v^4 - 4*v^3 + 51*v^2 - 8*v) / 3 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 7\nu^{3} + 57\nu^{2} + \nu - 6 ) / 3$$ (-2*v^7 + 8*v^6 + 7*v^5 - 43*v^4 - 7*v^3 + 57*v^2 + v - 6) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} + 2\beta_{2} + 5\beta _1 + 4$$ -b7 + b6 + 2*b2 + 5*b1 + 4 $$\nu^{4}$$ $$=$$ $$-3\beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 19$$ -3*b7 + 3*b6 + b5 + b3 + 8*b2 + 10*b1 + 19 $$\nu^{5}$$ $$=$$ $$-11\beta_{7} + 12\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 21\beta_{2} + 33\beta _1 + 44$$ -11*b7 + 12*b6 + 3*b5 + 2*b4 + 3*b3 + 21*b2 + 33*b1 + 44 $$\nu^{6}$$ $$=$$ $$-33\beta_{7} + 37\beta_{6} + 14\beta_{5} + 8\beta_{4} + 17\beta_{3} + 67\beta_{2} + 83\beta _1 + 148$$ -33*b7 + 37*b6 + 14*b5 + 8*b4 + 17*b3 + 67*b2 + 83*b1 + 148 $$\nu^{7}$$ $$=$$ $$-104\beta_{7} + 122\beta_{6} + 45\beta_{5} + 39\beta_{4} + 57\beta_{3} + 191\beta_{2} + 244\beta _1 + 406$$ -104*b7 + 122*b6 + 45*b5 + 39*b4 + 57*b3 + 191*b2 + 244*b1 + 406

## 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
 0.499011 −1.66501 3.01367 −1.47435 1.32675 2.68341 −0.0573749 −1.32610
0 −2.30231 0 0 0 3.59425 0 2.30065 0
1.2 0 −1.71538 0 0 0 3.42409 0 −0.0574791 0
1.3 0 −0.687404 0 0 0 1.01199 0 −2.52748 0
1.4 0 0.710340 0 0 0 4.59110 0 −2.49542 0
1.5 0 0.759083 0 0 0 −2.04213 0 −2.42379 0
1.6 0 2.11675 0 0 0 −0.973070 0 1.48063 0
1.7 0 3.02566 0 0 0 0.369971 0 6.15465 0
1.8 0 3.09326 0 0 0 0.0237879 0 6.56824 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

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.bn 8
4.b odd 2 1 625.2.a.e 8
5.b even 2 1 10000.2.a.be 8
12.b even 2 1 5625.2.a.be 8
20.d odd 2 1 625.2.a.g yes 8
20.e even 4 2 625.2.b.d 16
60.h even 2 1 5625.2.a.s 8
100.h odd 10 2 625.2.d.m 16
100.h odd 10 2 625.2.d.n 16
100.j odd 10 2 625.2.d.p 16
100.j odd 10 2 625.2.d.q 16
100.l even 20 4 625.2.e.j 32
100.l even 20 4 625.2.e.k 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e 8 4.b odd 2 1
625.2.a.g yes 8 20.d odd 2 1
625.2.b.d 16 20.e even 4 2
625.2.d.m 16 100.h odd 10 2
625.2.d.n 16 100.h odd 10 2
625.2.d.p 16 100.j odd 10 2
625.2.d.q 16 100.j odd 10 2
625.2.e.j 32 100.l even 20 4
625.2.e.k 32 100.l even 20 4
5625.2.a.s 8 60.h even 2 1
5625.2.a.be 8 12.b even 2 1
10000.2.a.be 8 5.b even 2 1
10000.2.a.bn 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{8} - 5T_{3}^{7} - 4T_{3}^{6} + 45T_{3}^{5} - 19T_{3}^{4} - 105T_{3}^{3} + 71T_{3}^{2} + 40T_{3} - 29$$ T3^8 - 5*T3^7 - 4*T3^6 + 45*T3^5 - 19*T3^4 - 105*T3^3 + 71*T3^2 + 40*T3 - 29 $$T_{7}^{8} - 10T_{7}^{7} + 24T_{7}^{6} + 35T_{7}^{5} - 154T_{7}^{4} + 25T_{7}^{3} + 124T_{7}^{2} - 45T_{7} + 1$$ T7^8 - 10*T7^7 + 24*T7^6 + 35*T7^5 - 154*T7^4 + 25*T7^3 + 124*T7^2 - 45*T7 + 1 $$T_{11}^{8} + T_{11}^{7} - 58T_{11}^{6} - 82T_{11}^{5} + 970T_{11}^{4} + 1617T_{11}^{3} - 3588T_{11}^{2} - 3591T_{11} + 2421$$ T11^8 + T11^7 - 58*T11^6 - 82*T11^5 + 970*T11^4 + 1617*T11^3 - 3588*T11^2 - 3591*T11 + 2421

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 5 T^{7} - 4 T^{6} + 45 T^{5} + \cdots - 29$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 10 T^{7} + 24 T^{6} + 35 T^{5} + \cdots + 1$$
$11$ $$T^{8} + T^{7} - 58 T^{6} - 82 T^{5} + \cdots + 2421$$
$13$ $$T^{8} + 10 T^{7} + 11 T^{6} + \cdots + 361$$
$17$ $$T^{8} + 15 T^{7} + 59 T^{6} + \cdots + 1611$$
$19$ $$T^{8} - 10 T^{7} - 15 T^{6} + \cdots + 10525$$
$23$ $$T^{8} - 30 T^{7} + 351 T^{6} + \cdots - 46089$$
$29$ $$T^{8} - 10 T^{7} - 25 T^{6} + \cdots - 60975$$
$31$ $$T^{8} - 9 T^{7} - 83 T^{6} + \cdots + 24001$$
$37$ $$T^{8} - 10 T^{7} - 91 T^{6} + 985 T^{5} + \cdots + 81$$
$41$ $$T^{8} + 4 T^{7} - 193 T^{6} + \cdots - 487629$$
$43$ $$T^{8} - 99 T^{6} + 180 T^{5} + \cdots - 1949$$
$47$ $$T^{8} - 30 T^{7} + 314 T^{6} + \cdots + 56961$$
$53$ $$T^{8} + 10 T^{7} - 139 T^{6} + \cdots - 1899$$
$59$ $$T^{8} - 5 T^{7} - 100 T^{6} + \cdots - 225$$
$61$ $$T^{8} - 6 T^{7} - 173 T^{6} + \cdots - 103529$$
$67$ $$T^{8} - 10 T^{7} - 181 T^{6} + \cdots - 1746299$$
$71$ $$T^{8} - 9 T^{7} - 133 T^{6} + \cdots - 16749$$
$73$ $$T^{8} - 314 T^{6} - 185 T^{5} + \cdots + 237091$$
$79$ $$T^{8} - 20 T^{7} - 115 T^{6} + \cdots + 249525$$
$83$ $$T^{8} - 40 T^{7} + 431 T^{6} + \cdots + 12262851$$
$89$ $$T^{8} + 5 T^{7} - 295 T^{6} + \cdots + 1849275$$
$97$ $$T^{8} - 321 T^{6} - 1875 T^{5} + \cdots + 972421$$