# Properties

 Label 5625.2.a.ba Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.46980000000.2 Defining polynomial: $$x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145$$ x^8 - 15*x^6 + 80*x^4 - 180*x^2 + 145 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - 2 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (-2*b5 + b3) * q^7 + (b6 + b4 + b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - 2 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{8} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_1) q^{11} + (2 \beta_{5} + \beta_{3} + 2) q^{13} + (\beta_{7} - \beta_{6} + \beta_1) q^{14} + (\beta_{3} + 2 \beta_{2} - 2) q^{16} + (\beta_{7} + 2 \beta_1) q^{17} + ( - \beta_{3} - 2 \beta_{2}) q^{19} + (2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 5) q^{22} + ( - \beta_{7} + \beta_{4} - \beta_1) q^{23} + (\beta_{7} + 3 \beta_{6} + 3 \beta_1) q^{26} + ( - \beta_{5} + \beta_{2} + 1) q^{28} + (\beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} - 1) q^{31} + (\beta_{7} + \beta_{6} - \beta_1) q^{32} + ( - 2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 5) q^{34} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{37} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{4} - 3 \beta_1) q^{38} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_1) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 7) q^{43} + (2 \beta_{6} + 2 \beta_{4} + 5 \beta_1) q^{44} + ( - \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 3) q^{46} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{4} - 2 \beta_1) q^{47} + ( - 5 \beta_{5} + 4 \beta_{3} - 3 \beta_{2} - 5) q^{49} + (3 \beta_{5} + 4 \beta_{3} + 3 \beta_{2} + 5) q^{52} + ( - 2 \beta_{7} + \beta_{6} - \beta_{4}) q^{53} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{4}) q^{56} + ( - 8 \beta_{5} + 3 \beta_{3} + 9 \beta_{2} + 5) q^{58} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + \beta_1) q^{59} + ( - 3 \beta_{5} + 3 \beta_{2} + 5) q^{61} + (3 \beta_{7} - 3 \beta_{4} - \beta_1) q^{62} + (\beta_{5} + 2 \beta_{3} - 5 \beta_{2} - 3) q^{64} + (\beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 5) q^{67} + (\beta_{7} + 3 \beta_{6} + 2 \beta_{4} + 6 \beta_1) q^{68} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 3 \beta_1) q^{71} + (2 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 6) q^{73} + ( - 3 \beta_{7} - 6 \beta_{6} - 3 \beta_{4} - 6 \beta_1) q^{74} + ( - \beta_{5} - 4 \beta_{3} - 5 \beta_{2} - 5) q^{76} + ( - 4 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - \beta_1) q^{77} + (\beta_{5} + 4 \beta_{2}) q^{79} + (2 \beta_{5} - 4 \beta_{3} - 2 \beta_{2} + 9) q^{82} + (4 \beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{83} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4} + 6 \beta_1) q^{86} + ( - 4 \beta_{5} + 6 \beta_{3} + 3 \beta_{2} + 6) q^{88} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{89} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} - 2) q^{91} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{92} + ( - 3 \beta_{5} - 11 \beta_{3} + \beta_{2} - 1) q^{94} + (4 \beta_{5} - 5 \beta_{3} + 2 \beta_{2} + 10) q^{97} + (4 \beta_{7} - 4 \beta_{6} - 3 \beta_{4} - 4 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (-2*b5 + b3) * q^7 + (b6 + b4 + b1) * q^8 + (-b7 + b6 + b4 + b1) * q^11 + (2*b5 + b3 + 2) * q^13 + (b7 - b6 + b1) * q^14 + (b3 + 2*b2 - 2) * q^16 + (b7 + 2*b1) * q^17 + (-b3 - 2*b2) * q^19 + (2*b5 - 2*b3 + 4*b2 + 5) * q^22 + (-b7 + b4 - b1) * q^23 + (b7 + 3*b6 + 3*b1) * q^26 + (-b5 + b2 + 1) * q^28 + (b7 + 2*b4 + 3*b1) * q^29 + (3*b3 - 3*b2 - 1) * q^31 + (b7 + b6 - b1) * q^32 + (-2*b5 + 3*b3 + 2*b2 + 5) * q^34 + (-3*b3 - 3*b2) * q^37 + (-b7 - 3*b6 - 2*b4 - 3*b1) * q^38 + (-b7 - b6 - b4 + b1) * q^41 + (-2*b3 + b2 + 7) * q^43 + (2*b6 + 2*b4 + 5*b1) * q^44 + (-b5 - 3*b3 + 2*b2 - 3) * q^46 + (-3*b7 - 2*b6 + b4 - 2*b1) * q^47 + (-5*b5 + 4*b3 - 3*b2 - 5) * q^49 + (3*b5 + 4*b3 + 3*b2 + 5) * q^52 + (-2*b7 + b6 - b4) * q^53 + (-2*b7 + 2*b6 + b4) * q^56 + (-8*b5 + 3*b3 + 9*b2 + 5) * q^58 + (-b7 - 2*b6 - 2*b4 + b1) * q^59 + (-3*b5 + 3*b2 + 5) * q^61 + (3*b7 - 3*b4 - b1) * q^62 + (b5 + 2*b3 - 5*b2 - 3) * q^64 + (b5 - 2*b3 - 4*b2 + 5) * q^67 + (b7 + 3*b6 + 2*b4 + 6*b1) * q^68 + (4*b7 + 2*b6 - 2*b4 + 3*b1) * q^71 + (2*b5 + 2*b3 - 3*b2 + 6) * q^73 + (-3*b7 - 6*b6 - 3*b4 - 6*b1) * q^74 + (-b5 - 4*b3 - 5*b2 - 5) * q^76 + (-4*b7 + 3*b6 + 3*b4 - b1) * q^77 + (b5 + 4*b2) * q^79 + (2*b5 - 4*b3 - 2*b2 + 9) * q^82 + (4*b7 + 2*b4 + 3*b1) * q^83 + (-2*b7 - b6 + b4 + 6*b1) * q^86 + (-4*b5 + 6*b3 + 3*b2 + 6) * q^88 + (-b7 - 3*b6 + 2*b4 - 4*b1) * q^89 + (-b5 + 2*b3 + b2 - 2) * q^91 + (-b7 - 2*b6 - 2*b1) * q^92 + (-3*b5 - 11*b3 + b2 - 1) * q^94 + (4*b5 - 5*b3 + 2*b2 + 10) * q^97 + (4*b7 - 4*b6 - 3*b4 - 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 14 q^{4} + 10 q^{7}+O(q^{10})$$ 8 * q + 14 * q^4 + 10 * q^7 $$8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} - 18 q^{16} + 2 q^{19} + 20 q^{22} + 10 q^{28} + 4 q^{31} + 50 q^{34} + 50 q^{43} - 30 q^{46} - 6 q^{49} + 30 q^{52} + 60 q^{58} + 46 q^{61} - 14 q^{64} + 40 q^{67} + 50 q^{73} - 34 q^{76} - 12 q^{79} + 60 q^{82} + 70 q^{88} - 10 q^{91} - 20 q^{94} + 50 q^{97}+O(q^{100})$$ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 - 18 * q^16 + 2 * q^19 + 20 * q^22 + 10 * q^28 + 4 * q^31 + 50 * q^34 + 50 * q^43 - 30 * q^46 - 6 * q^49 + 30 * q^52 + 60 * q^58 + 46 * q^61 - 14 * q^64 + 40 * q^67 + 50 * q^73 - 34 * q^76 - 12 * q^79 + 60 * q^82 + 70 * q^88 - 10 * q^91 - 20 * q^94 + 50 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 8\nu^{2} + 14$$ v^4 - 8*v^2 + 14 $$\beta_{4}$$ $$=$$ $$-\nu^{7} + 12\nu^{5} - 44\nu^{3} + 48\nu$$ -v^7 + 12*v^5 - 44*v^3 + 48*v $$\beta_{5}$$ $$=$$ $$\nu^{6} - 12\nu^{4} + 45\nu^{2} - 53$$ v^6 - 12*v^4 + 45*v^2 - 53 $$\beta_{6}$$ $$=$$ $$\nu^{7} - 12\nu^{5} + 45\nu^{3} - 53\nu$$ v^7 - 12*v^5 + 45*v^3 - 53*v $$\beta_{7}$$ $$=$$ $$-\nu^{7} + 13\nu^{5} - 53\nu^{3} + 66\nu$$ -v^7 + 13*v^5 - 53*v^3 + 66*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} + 5\beta_1$$ b6 + b4 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 8\beta_{2} + 18$$ b3 + 8*b2 + 18 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 9\beta_{6} + 8\beta_{4} + 27\beta_1$$ b7 + 9*b6 + 8*b4 + 27*b1 $$\nu^{6}$$ $$=$$ $$\beta_{5} + 12\beta_{3} + 51\beta_{2} + 89$$ b5 + 12*b3 + 51*b2 + 89 $$\nu^{7}$$ $$=$$ $$12\beta_{7} + 64\beta_{6} + 51\beta_{4} + 152\beta_1$$ 12*b7 + 64*b6 + 51*b4 + 152*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
 −2.44055 −2.05160 −1.63148 −1.47408 1.47408 1.63148 2.05160 2.44055
−2.44055 0 3.95630 0 0 0.591023 −4.77444 0 0
1.2 −2.05160 0 2.20906 0 0 1.27977 −0.428901 0 0
1.3 −1.63148 0 0.661739 0 0 −1.44512 2.18335 0 0
1.4 −1.47408 0 0.172909 0 0 4.57433 2.69328 0 0
1.5 1.47408 0 0.172909 0 0 4.57433 −2.69328 0 0
1.6 1.63148 0 0.661739 0 0 −1.44512 −2.18335 0 0
1.7 2.05160 0 2.20906 0 0 1.27977 0.428901 0 0
1.8 2.44055 0 3.95630 0 0 0.591023 4.77444 0 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
$$3$$ $$1$$
$$5$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.ba yes 8
3.b odd 2 1 inner 5625.2.a.ba yes 8
5.b even 2 1 5625.2.a.y 8
15.d odd 2 1 5625.2.a.y 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5625.2.a.y 8 5.b even 2 1
5625.2.a.y 8 15.d odd 2 1
5625.2.a.ba yes 8 1.a even 1 1 trivial
5625.2.a.ba yes 8 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{8} - 15T_{2}^{6} + 80T_{2}^{4} - 180T_{2}^{2} + 145$$ T2^8 - 15*T2^6 + 80*T2^4 - 180*T2^2 + 145 $$T_{7}^{4} - 5T_{7}^{3} + 10T_{7} - 5$$ T7^4 - 5*T7^3 + 10*T7 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 15 T^{6} + 80 T^{4} + \cdots + 145$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 5 T^{3} + 10 T - 5)^{2}$$
$11$ $$T^{8} - 60 T^{6} + 1100 T^{4} + \cdots + 3625$$
$13$ $$(T^{4} - 5 T^{3} - 10 T^{2} + 50 T - 5)^{2}$$
$17$ $$T^{8} - 65 T^{6} + 1110 T^{4} + \cdots + 145$$
$19$ $$(T^{4} - T^{3} - 24 T^{2} + 74 T - 59)^{2}$$
$23$ $$T^{8} - 60 T^{6} + 990 T^{4} + \cdots + 11745$$
$29$ $$T^{8} - 210 T^{6} + 15050 T^{4} + \cdots + 3483625$$
$31$ $$(T^{4} - 2 T^{3} - 66 T^{2} + 67 T + 211)^{2}$$
$37$ $$(T^{4} - 90 T^{2} + 405 T - 405)^{2}$$
$41$ $$T^{8} - 120 T^{6} + 3800 T^{4} + \cdots + 3625$$
$43$ $$(T^{4} - 25 T^{3} + 215 T^{2} - 755 T + 895)^{2}$$
$47$ $$T^{8} - 305 T^{6} + 28935 T^{4} + \cdots + 1148545$$
$53$ $$T^{8} - 180 T^{6} + 5705 T^{4} + \cdots + 145$$
$59$ $$T^{8} - 270 T^{6} + 21950 T^{4} + \cdots + 3483625$$
$61$ $$(T^{4} - 23 T^{3} + 159 T^{2} - 257 T - 359)^{2}$$
$67$ $$(T^{4} - 20 T^{3} + 65 T^{2} + 680 T - 3455)^{2}$$
$71$ $$T^{8} - 575 T^{6} + \cdots + 118758625$$
$73$ $$(T^{4} - 25 T^{3} + 180 T^{2} - 190 T - 1055)^{2}$$
$79$ $$(T^{4} + 6 T^{3} - 69 T^{2} - 414 T - 279)^{2}$$
$83$ $$T^{8} - 555 T^{6} + \cdots + 10648945$$
$89$ $$T^{8} - 525 T^{6} + \cdots + 76215625$$
$97$ $$(T^{4} - 25 T^{3} + 110 T^{2} + 1150 T - 7355)^{2}$$