# Properties

 Label 5625.2.a.bb 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: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45$$ x^8 - 15*x^6 + 70*x^4 - 105*x^2 + 45 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (-b4 + b2 + 1) * q^7 + (b3 + 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + \beta_{7} q^{11} + (\beta_{4} + \beta_{2} + 2) q^{13} + ( - \beta_{5} + \beta_{3} + 3 \beta_1) q^{14} + (\beta_{6} + 2 \beta_{2} + 3) q^{16} + (\beta_{7} - 2 \beta_1) q^{17} + ( - \beta_{4} + \beta_{2}) q^{19} + (2 \beta_{6} + \beta_{2} + 1) q^{22} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{23} + (\beta_{5} + \beta_{3} + 4 \beta_1) q^{26} + ( - \beta_{4} + 3 \beta_{2} + 9) q^{28} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{29} + ( - \beta_{6} + \beta_{4}) q^{31} + (\beta_{7} + 2 \beta_{5} + 3 \beta_1) q^{32} + (2 \beta_{6} - \beta_{2} - 7) q^{34} + ( - \beta_{6} + \beta_{4} + 7) q^{37} + ( - \beta_{5} + \beta_{3} + 2 \beta_1) q^{38} + ( - 4 \beta_{5} + \beta_{3} - \beta_1) q^{41} + ( - 2 \beta_{6} + \beta_{4} + 1) q^{43} + (4 \beta_{5} + \beta_{3} + 3 \beta_1) q^{44} + ( - 3 \beta_{6} - 6 \beta_{4} - 3 \beta_{2} - 3) q^{46} + ( - 2 \beta_{5} - 2 \beta_1) q^{47} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2} + 2) q^{49} + (2 \beta_{6} + \beta_{4} + 4 \beta_{2} + 11) q^{52} + (2 \beta_{5} - 2 \beta_1) q^{53} + (\beta_{5} + \beta_{3} + 9 \beta_1) q^{56} + ( - 3 \beta_{6} - 4 \beta_{2} - 4) q^{58} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_1) q^{59} + ( - \beta_{6} + \beta_{4} - 6) q^{61} + ( - \beta_{7} - \beta_{5}) q^{62} + (2 \beta_{6} + 6 \beta_{4} + 7) q^{64} + (3 \beta_{6} - \beta_{4}) q^{67} + (4 \beta_{5} - \beta_{3} - 5 \beta_1) q^{68} + (\beta_{7} - 4 \beta_{5} + \beta_{3} - \beta_1) q^{71} + ( - 2 \beta_{6} - \beta_{4} - 3 \beta_{2} + 3) q^{73} + ( - \beta_{7} - \beta_{5} + 7 \beta_1) q^{74} + ( - \beta_{4} + 2 \beta_{2} + 7) q^{76} + (4 \beta_{5} + 2 \beta_1) q^{77} + ( - 2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 4) q^{79} + ( - 3 \beta_{6} - 12 \beta_{4} + \beta_{2} - 5) q^{82} + (\beta_{7} - 2 \beta_{3} - 4 \beta_1) q^{83} + ( - 2 \beta_{7} - 3 \beta_{5} + \beta_1) q^{86} + (\beta_{6} + 12 \beta_{4} + 3 \beta_{2} + 9) q^{88} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{89} + (\beta_{6} + 3 \beta_{2} + 8) q^{91} + ( - 3 \beta_{7} - 8 \beta_{5} - \beta_{3} - 7 \beta_1) q^{92} + ( - 2 \beta_{6} - 6 \beta_{4} - 2 \beta_{2} - 8) q^{94} + (\beta_{6} - \beta_{4} - \beta_{2} - 1) q^{97} + ( - \beta_{7} - 3 \beta_{5} + 2 \beta_{3} + 6 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (-b4 + b2 + 1) * q^7 + (b3 + 2*b1) * q^8 + b7 * q^11 + (b4 + b2 + 2) * q^13 + (-b5 + b3 + 3*b1) * q^14 + (b6 + 2*b2 + 3) * q^16 + (b7 - 2*b1) * q^17 + (-b4 + b2) * q^19 + (2*b6 + b2 + 1) * q^22 + (-2*b5 - b3 - b1) * q^23 + (b5 + b3 + 4*b1) * q^26 + (-b4 + 3*b2 + 9) * q^28 + (-b7 - b3 - b1) * q^29 + (-b6 + b4) * q^31 + (b7 + 2*b5 + 3*b1) * q^32 + (2*b6 - b2 - 7) * q^34 + (-b6 + b4 + 7) * q^37 + (-b5 + b3 + 2*b1) * q^38 + (-4*b5 + b3 - b1) * q^41 + (-2*b6 + b4 + 1) * q^43 + (4*b5 + b3 + 3*b1) * q^44 + (-3*b6 - 6*b4 - 3*b2 - 3) * q^46 + (-2*b5 - 2*b1) * q^47 + (-b6 - b4 + 2*b2 + 2) * q^49 + (2*b6 + b4 + 4*b2 + 11) * q^52 + (2*b5 - 2*b1) * q^53 + (b5 + b3 + 9*b1) * q^56 + (-3*b6 - 4*b2 - 4) * q^58 + (-b7 + 4*b5 + 2*b1) * q^59 + (-b6 + b4 - 6) * q^61 + (-b7 - b5) * q^62 + (2*b6 + 6*b4 + 7) * q^64 + (3*b6 - b4) * q^67 + (4*b5 - b3 - 5*b1) * q^68 + (b7 - 4*b5 + b3 - b1) * q^71 + (-2*b6 - b4 - 3*b2 + 3) * q^73 + (-b7 - b5 + 7*b1) * q^74 + (-b4 + 2*b2 + 7) * q^76 + (4*b5 + 2*b1) * q^77 + (-2*b6 + b4 - 2*b2 + 4) * q^79 + (-3*b6 - 12*b4 + b2 - 5) * q^82 + (b7 - 2*b3 - 4*b1) * q^83 + (-2*b7 - 3*b5 + b1) * q^86 + (b6 + 12*b4 + 3*b2 + 9) * q^88 + (b7 + 2*b5 - 2*b3 + 4*b1) * q^89 + (b6 + 3*b2 + 8) * q^91 + (-3*b7 - 8*b5 - b3 - 7*b1) * q^92 + (-2*b6 - 6*b4 - 2*b2 - 8) * q^94 + (b6 - b4 - b2 - 1) * q^97 + (-b7 - 3*b5 + 2*b3 + 6*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 $$8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100})$$ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6\nu$$ v^3 - 6*v $$\beta_{4}$$ $$=$$ $$( \nu^{6} - 12\nu^{4} + 40\nu^{2} - 33 ) / 6$$ (v^6 - 12*v^4 + 40*v^2 - 33) / 6 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 12\nu^{5} + 40\nu^{3} - 33\nu ) / 6$$ (v^7 - 12*v^5 + 40*v^3 - 33*v) / 6 $$\beta_{6}$$ $$=$$ $$\nu^{4} - 8\nu^{2} + 9$$ v^4 - 8*v^2 + 9 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + 15\nu^{5} - 64\nu^{3} + 60\nu ) / 3$$ (-v^7 + 15*v^5 - 64*v^3 + 60*v) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6\beta_1$$ b3 + 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{6} + 8\beta_{2} + 23$$ b6 + 8*b2 + 23 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 2\beta_{5} + 8\beta_{3} + 39\beta_1$$ b7 + 2*b5 + 8*b3 + 39*b1 $$\nu^{6}$$ $$=$$ $$12\beta_{6} + 6\beta_{4} + 56\beta_{2} + 149$$ 12*b6 + 6*b4 + 56*b2 + 149 $$\nu^{7}$$ $$=$$ $$12\beta_{7} + 30\beta_{5} + 56\beta_{3} + 261\beta_1$$ 12*b7 + 30*b5 + 56*b3 + 261*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
 −2.66202 −2.37653 −1.23762 −0.856773 0.856773 1.23762 2.37653 2.66202
−2.66202 0 5.08634 0 0 3.46831 −8.21589 0 0
1.2 −2.37653 0 3.64791 0 0 4.26594 −3.91630 0 0
1.3 −1.23762 0 −0.468306 0 0 −2.08634 3.05482 0 0
1.4 −0.856773 0 −1.26594 0 0 −0.647907 2.79817 0 0
1.5 0.856773 0 −1.26594 0 0 −0.647907 −2.79817 0 0
1.6 1.23762 0 −0.468306 0 0 −2.08634 −3.05482 0 0
1.7 2.37653 0 3.64791 0 0 4.26594 3.91630 0 0
1.8 2.66202 0 5.08634 0 0 3.46831 8.21589 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.bb yes 8
3.b odd 2 1 inner 5625.2.a.bb yes 8
5.b even 2 1 5625.2.a.z 8
15.d odd 2 1 5625.2.a.z 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5625.2.a.z 8 5.b even 2 1
5625.2.a.z 8 15.d odd 2 1
5625.2.a.bb yes 8 1.a even 1 1 trivial
5625.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{8} - 15T_{2}^{6} + 70T_{2}^{4} - 105T_{2}^{2} + 45$$ T2^8 - 15*T2^6 + 70*T2^4 - 105*T2^2 + 45 $$T_{7}^{4} - 5T_{7}^{3} - 5T_{7}^{2} + 30T_{7} + 20$$ T7^4 - 5*T7^3 - 5*T7^2 + 30*T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 15 T^{6} + 70 T^{4} - 105 T^{2} + \cdots + 45$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 5 T^{3} - 5 T^{2} + 30 T + 20)^{2}$$
$11$ $$T^{8} - 60 T^{6} + 1075 T^{4} + \cdots + 18000$$
$13$ $$(T^{4} - 5 T^{3} - 10 T^{2} + 35 T - 5)^{2}$$
$17$ $$T^{8} - 100 T^{6} + 3115 T^{4} + \cdots + 87120$$
$19$ $$(T^{4} - T^{3} - 14 T^{2} + 9 T + 41)^{2}$$
$23$ $$T^{8} - 165 T^{6} + 8955 T^{4} + \cdots + 933120$$
$29$ $$T^{8} - 135 T^{6} + 4675 T^{4} + \cdots + 18000$$
$31$ $$(T^{4} + 3 T^{3} - 16 T^{2} - 13 T + 41)^{2}$$
$37$ $$(T^{4} - 25 T^{3} + 215 T^{2} - 720 T + 720)^{2}$$
$41$ $$T^{8} - 285 T^{6} + 24175 T^{4} + \cdots + 18000$$
$43$ $$(T^{4} - 75 T^{2} + 745)^{2}$$
$47$ $$T^{8} - 100 T^{6} + 2080 T^{4} + \cdots + 11520$$
$53$ $$T^{8} - 180 T^{6} + 4000 T^{4} + \cdots + 11520$$
$59$ $$T^{8} - 300 T^{6} + \cdots + 30258000$$
$61$ $$(T^{4} + 27 T^{3} + 254 T^{2} + 983 T + 1331)^{2}$$
$67$ $$(T^{4} - 5 T^{3} - 160 T^{2} + 545 T + 3295)^{2}$$
$71$ $$T^{8} - 295 T^{6} + 26275 T^{4} + \cdots + 4608000$$
$73$ $$(T^{4} - 15 T^{3} - 85 T^{2} + 2140 T - 8080)^{2}$$
$79$ $$(T^{4} - 14 T^{3} - 19 T^{2} + 636 T - 909)^{2}$$
$83$ $$T^{8} - 420 T^{6} + \cdots + 28512720$$
$89$ $$T^{8} - 600 T^{6} + 116875 T^{4} + \cdots + 7200000$$
$97$ $$(T^{4} - 45 T^{2} - 90 T - 5)^{2}$$