# Properties

 Label 5625.2.a.w Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $1$ 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: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.33620000000.1 Defining polynomial: $$x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5$$ x^8 - 10*x^6 + 30*x^4 - 25*x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 225) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{2} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 1) * q^4 + (-b4 - b2 - 1) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{2} - 1) q^{7} + \beta_{3} q^{8} + (\beta_{7} + \beta_{5} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{2} + 2) q^{13} + ( - \beta_{5} - \beta_{3} - 2 \beta_1) q^{14} + (\beta_{4} - \beta_{2} - 1) q^{16} + (\beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{17} + (\beta_{6} - \beta_{2} - 3) q^{19} + (2 \beta_{6} + 4 \beta_{4} + \beta_{2} + 2) q^{22} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_1) q^{23} + ( - \beta_{7} - \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{26} + ( - \beta_{6} - 2 \beta_{4} - \beta_{2} - 5) q^{28} + ( - 3 \beta_{7} + 2 \beta_{3} - \beta_1) q^{29} + ( - 3 \beta_{6} - 2 \beta_{4} - \beta_{2} - 4) q^{31} + (\beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{32} + ( - 3 \beta_{4} - 2 \beta_{2} - 5) q^{34} + (2 \beta_{6} + 2 \beta_{4} + 3 \beta_{2} + 2) q^{37} + (\beta_{7} + \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{38} + ( - \beta_{7} - \beta_{5} - 3 \beta_{3} + 2 \beta_1) q^{41} + (2 \beta_{6} + \beta_{4} - 5) q^{43} + (4 \beta_{5} - \beta_{3} + 3 \beta_1) q^{44} + ( - \beta_{6} + \beta_{4} - 2 \beta_{2} - 4) q^{46} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{47} + (2 \beta_{6} + 2 \beta_{4} + \beta_{2} - 1) q^{49} + ( - 2 \beta_{4} + 2 \beta_{2} + 5) q^{52} + (\beta_{7} + 5 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{53} + ( - \beta_{7} - \beta_{5} + 2 \beta_{3} - 3 \beta_1) q^{56} + ( - 3 \beta_{6} - \beta_{4} + \beta_{2} + 2) q^{58} + (2 \beta_{7} - 6 \beta_{5} + \beta_{3} - 3 \beta_1) q^{59} + (2 \beta_{4} - 2 \beta_{2} - 3) q^{61} + ( - 3 \beta_{7} - 5 \beta_{5} + 2 \beta_{3} - 8 \beta_1) q^{62} + (\beta_{6} - 2 \beta_{4} - 3 \beta_{2} - 7) q^{64} + ( - 2 \beta_{6} + 5 \beta_{4} - 2 \beta_{2} - 3) q^{67} + ( - 2 \beta_{7} - \beta_{5} - 5 \beta_1) q^{68} + (2 \beta_{3} - 7 \beta_1) q^{71} + (2 \beta_{6} + 9 \beta_{4} - 2 \beta_{2} + 8) q^{73} + (2 \beta_{7} + 4 \beta_{5} + \beta_{3} + 7 \beta_1) q^{74} + (2 \beta_{4} - 3 \beta_{2} - 6) q^{76} + ( - 3 \beta_{5} - 4 \beta_1) q^{77} + (3 \beta_{6} - 2 \beta_{2} - 7) q^{79} + ( - 2 \beta_{6} - 7 \beta_{4} - \beta_{2} + 4) q^{82} + ( - \beta_{7} + \beta_{5} + 3 \beta_{3}) q^{83} + (2 \beta_{7} + 3 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{86} + (3 \beta_{4} + 4) q^{88} + (\beta_{7} - 2 \beta_{5} - 3 \beta_{3} - 5 \beta_1) q^{89} + ( - 2 \beta_{6} - \beta_{2} - 5) q^{91} + (3 \beta_{7} - 2 \beta_{5} - \beta_{3} - 3 \beta_1) q^{92} + ( - \beta_{6} - 3 \beta_{4} + \beta_{2} - 2) q^{94} + ( - 4 \beta_{6} - 4 \beta_{4} + 2 \beta_{2} - 5) q^{97} + (2 \beta_{7} + 4 \beta_{5} - \beta_{3} + 2 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 1) * q^4 + (-b4 - b2 - 1) * q^7 + b3 * q^8 + (b7 + b5 + b1) * q^11 + (-b6 + b2 + 2) * q^13 + (-b5 - b3 - 2*b1) * q^14 + (b4 - b2 - 1) * q^16 + (b7 - b5 - b3 - b1) * q^17 + (b6 - b2 - 3) * q^19 + (2*b6 + 4*b4 + b2 + 2) * q^22 + (-2*b7 + b5 - 2*b1) * q^23 + (-b7 - b5 + 2*b3 + 2*b1) * q^26 + (-b6 - 2*b4 - b2 - 5) * q^28 + (-3*b7 + 2*b3 - b1) * q^29 + (-3*b6 - 2*b4 - b2 - 4) * q^31 + (b5 - 3*b3 - 2*b1) * q^32 + (-3*b4 - 2*b2 - 5) * q^34 + (2*b6 + 2*b4 + 3*b2 + 2) * q^37 + (b7 + b5 - 2*b3 - 3*b1) * q^38 + (-b7 - b5 - 3*b3 + 2*b1) * q^41 + (2*b6 + b4 - 5) * q^43 + (4*b5 - b3 + 3*b1) * q^44 + (-b6 + b4 - 2*b2 - 4) * q^46 + (b7 - 2*b5 + 2*b3 - b1) * q^47 + (2*b6 + 2*b4 + b2 - 1) * q^49 + (-2*b4 + 2*b2 + 5) * q^52 + (b7 + 5*b5 - 3*b3 + 3*b1) * q^53 + (-b7 - b5 + 2*b3 - 3*b1) * q^56 + (-3*b6 - b4 + b2 + 2) * q^58 + (2*b7 - 6*b5 + b3 - 3*b1) * q^59 + (2*b4 - 2*b2 - 3) * q^61 + (-3*b7 - 5*b5 + 2*b3 - 8*b1) * q^62 + (b6 - 2*b4 - 3*b2 - 7) * q^64 + (-2*b6 + 5*b4 - 2*b2 - 3) * q^67 + (-2*b7 - b5 - 5*b1) * q^68 + (2*b3 - 7*b1) * q^71 + (2*b6 + 9*b4 - 2*b2 + 8) * q^73 + (2*b7 + 4*b5 + b3 + 7*b1) * q^74 + (2*b4 - 3*b2 - 6) * q^76 + (-3*b5 - 4*b1) * q^77 + (3*b6 - 2*b2 - 7) * q^79 + (-2*b6 - 7*b4 - b2 + 4) * q^82 + (-b7 + b5 + 3*b3) * q^83 + (2*b7 + 3*b5 - 2*b3 - 3*b1) * q^86 + (3*b4 + 4) * q^88 + (b7 - 2*b5 - 3*b3 - 5*b1) * q^89 + (-2*b6 - b2 - 5) * q^91 + (3*b7 - 2*b5 - b3 - 3*b1) * q^92 + (-b6 - 3*b4 + b2 - 2) * q^94 + (-4*b6 - 4*b4 + 2*b2 - 5) * q^97 + (2*b7 + 4*b5 - b3 + 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4}+O(q^{10})$$ 8 * q + 4 * q^4 $$8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100})$$ 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 2$$ v^4 - 5*v^2 + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5\nu^{3} + 2\nu$$ v^5 - 5*v^3 + 2*v $$\beta_{6}$$ $$=$$ $$\nu^{6} - 8\nu^{4} + 17\nu^{2} - 6$$ v^6 - 8*v^4 + 17*v^2 - 6 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 9\nu^{5} + 23\nu^{3} - 13\nu$$ v^7 - 9*v^5 + 23*v^3 - 13*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 13$$ b4 + 5*b2 + 13 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 5\beta_{3} + 18\beta_1$$ b5 + 5*b3 + 18*b1 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 8\beta_{4} + 23\beta_{2} + 59$$ b6 + 8*b4 + 23*b2 + 59 $$\nu^{7}$$ $$=$$ $$\beta_{7} + 9\beta_{5} + 22\beta_{3} + 83\beta_1$$ b7 + 9*b5 + 22*b3 + 83*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
 −2.16942 −2.03035 −0.936839 −0.541884 0.541884 0.936839 2.03035 2.16942
−2.16942 0 2.70636 0 0 −3.32440 −1.53239 0 0
1.2 −2.03035 0 2.12233 0 0 −0.504300 −0.248380 0 0
1.3 −0.936839 0 −1.12233 0 0 2.74037 2.92512 0 0
1.4 −0.541884 0 −1.70636 0 0 1.08833 2.00842 0 0
1.5 0.541884 0 −1.70636 0 0 1.08833 −2.00842 0 0
1.6 0.936839 0 −1.12233 0 0 2.74037 −2.92512 0 0
1.7 2.03035 0 2.12233 0 0 −0.504300 0.248380 0 0
1.8 2.16942 0 2.70636 0 0 −3.32440 1.53239 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.w 8
3.b odd 2 1 inner 5625.2.a.w 8
5.b even 2 1 5625.2.a.v 8
15.d odd 2 1 5625.2.a.v 8
25.d even 5 2 225.2.h.e 16
75.j odd 10 2 225.2.h.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.h.e 16 25.d even 5 2
225.2.h.e 16 75.j odd 10 2
5625.2.a.v 8 5.b even 2 1
5625.2.a.v 8 15.d odd 2 1
5625.2.a.w 8 1.a even 1 1 trivial
5625.2.a.w 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{8} - 10T_{2}^{6} + 30T_{2}^{4} - 25T_{2}^{2} + 5$$ T2^8 - 10*T2^6 + 30*T2^4 - 25*T2^2 + 5 $$T_{7}^{4} - 10T_{7}^{2} + 5T_{7} + 5$$ T7^4 - 10*T7^2 + 5*T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 10 T^{6} + 30 T^{4} - 25 T^{2} + \cdots + 5$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 10 T^{2} + 5 T + 5)^{2}$$
$11$ $$T^{8} - 35 T^{6} + 400 T^{4} + \cdots + 125$$
$13$ $$(T^{4} - 5 T^{3} - 10 T^{2} + 60 T - 45)^{2}$$
$17$ $$T^{8} - 50 T^{6} + 790 T^{4} + \cdots + 4205$$
$19$ $$(T^{4} + 9 T^{3} + 11 T^{2} - 61 T - 109)^{2}$$
$23$ $$T^{8} - 95 T^{6} + 2950 T^{4} + \cdots + 142805$$
$29$ $$T^{8} - 160 T^{6} + 7500 T^{4} + \cdots + 125$$
$31$ $$(T^{4} + 13 T^{3} - 16 T^{2} - 378 T + 801)^{2}$$
$37$ $$(T^{4} - 90 T^{2} - 195 T - 95)^{2}$$
$41$ $$T^{8} - 320 T^{6} + 33850 T^{4} + \cdots + 4005125$$
$43$ $$(T^{4} + 20 T^{3} + 115 T^{2} + 150 T - 45)^{2}$$
$47$ $$T^{8} - 100 T^{6} + 1620 T^{4} + \cdots + 12005$$
$53$ $$T^{8} - 310 T^{6} + \cdots + 10210205$$
$59$ $$T^{8} - 445 T^{6} + \cdots + 63190125$$
$61$ $$(T^{4} + 12 T^{3} + 14 T^{2} - 172 T - 319)^{2}$$
$67$ $$(T^{4} + 20 T^{3} + 5 T^{2} - 1580 T - 6395)^{2}$$
$71$ $$T^{8} - 550 T^{6} + \cdots + 259560125$$
$73$ $$(T^{4} - 20 T^{3} - 85 T^{2} + 3090 T - 10845)^{2}$$
$79$ $$(T^{4} + 21 T^{3} + 36 T^{2} - 1194 T - 5309)^{2}$$
$83$ $$T^{8} - 160 T^{6} + 4530 T^{4} + \cdots + 1805$$
$89$ $$T^{8} - 425 T^{6} + 37000 T^{4} + \cdots + 1128125$$
$97$ $$(T^{4} + 20 T^{3} - 50 T^{2} - 2340 T - 9295)^{2}$$