# Properties

 Label 5625.2.a.x 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: 8.8.14884000000.2 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.

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

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
 −2.30927 −2.08529 −1.13370 −0.183172 0.183172 1.13370 2.08529 2.30927
−2.30927 0 3.33275 0 0 3.03582 −3.07768 0 0
1.2 −2.08529 0 2.34841 0 0 −0.992398 −0.726543 0 0
1.3 −1.13370 0 −0.714715 0 0 0.407162 3.07768 0 0
1.4 −0.183172 0 −1.96645 0 0 3.26086 0.726543 0 0
1.5 0.183172 0 −1.96645 0 0 −3.26086 −0.726543 0 0
1.6 1.13370 0 −0.714715 0 0 −0.407162 −3.07768 0 0
1.7 2.08529 0 2.34841 0 0 0.992398 0.726543 0 0
1.8 2.30927 0 3.33275 0 0 −3.03582 3.07768 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
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 5625.2.a.x 8
3.b odd 2 1 625.2.a.f 8
5.b even 2 1 inner 5625.2.a.x 8
12.b even 2 1 10000.2.a.bj 8
15.d odd 2 1 625.2.a.f 8
15.e even 4 2 625.2.b.c 8
25.f odd 20 2 225.2.m.a 8
60.h even 2 1 10000.2.a.bj 8
75.h odd 10 2 125.2.d.b 16
75.h odd 10 2 625.2.d.o 16
75.j odd 10 2 125.2.d.b 16
75.j odd 10 2 625.2.d.o 16
75.l even 20 2 25.2.e.a 8
75.l even 20 2 125.2.e.b 8
75.l even 20 2 625.2.e.a 8
75.l even 20 2 625.2.e.i 8
300.u odd 20 2 400.2.y.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 75.l even 20 2
125.2.d.b 16 75.h odd 10 2
125.2.d.b 16 75.j odd 10 2
125.2.e.b 8 75.l even 20 2
225.2.m.a 8 25.f odd 20 2
400.2.y.c 8 300.u odd 20 2
625.2.a.f 8 3.b odd 2 1
625.2.a.f 8 15.d odd 2 1
625.2.b.c 8 15.e even 4 2
625.2.d.o 16 75.h odd 10 2
625.2.d.o 16 75.j odd 10 2
625.2.e.a 8 75.l even 20 2
625.2.e.i 8 75.l even 20 2
5625.2.a.x 8 1.a even 1 1 trivial
5625.2.a.x 8 5.b even 2 1 inner
10000.2.a.bj 8 12.b even 2 1
10000.2.a.bj 8 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{8} - 11T_{2}^{6} + 36T_{2}^{4} - 31T_{2}^{2} + 1$$ T2^8 - 11*T2^6 + 36*T2^4 - 31*T2^2 + 1 $$T_{7}^{8} - 21T_{7}^{6} + 121T_{7}^{4} - 116T_{7}^{2} + 16$$ T7^8 - 21*T7^6 + 121*T7^4 - 116*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 11 T^{6} + 36 T^{4} - 31 T^{2} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 21 T^{6} + 121 T^{4} + \cdots + 16$$
$11$ $$(T + 2)^{8}$$
$13$ $$T^{8} - 14 T^{6} + 31 T^{4} - 14 T^{2} + \cdots + 1$$
$17$ $$T^{8} - 41 T^{6} + 441 T^{4} + \cdots + 1936$$
$19$ $$(T^{4} - 5 T^{3} - 5 T^{2} + 30 T - 20)^{2}$$
$23$ $$T^{8} - 34 T^{6} + 301 T^{4} + \cdots + 256$$
$29$ $$(T^{4} + 10 T^{3} - 5 T^{2} - 290 T - 695)^{2}$$
$31$ $$(T^{4} - 8 T^{3} - 41 T^{2} + 328 T - 44)^{2}$$
$37$ $$T^{8} - 111 T^{6} + 3556 T^{4} + \cdots + 116281$$
$41$ $$(T^{4} + 13 T^{3} + 19 T^{2} - 148 T + 116)^{2}$$
$43$ $$T^{8} - 129 T^{6} + 4421 T^{4} + \cdots + 246016$$
$47$ $$T^{8} - 141 T^{6} + 4661 T^{4} + \cdots + 65536$$
$53$ $$T^{8} - 239 T^{6} + 20356 T^{4} + \cdots + 8755681$$
$59$ $$(T^{4} + 15 T^{3} + 5 T^{2} - 630 T - 2020)^{2}$$
$61$ $$(T^{4} - 3 T^{3} - 146 T^{2} - 237 T + 341)^{2}$$
$67$ $$T^{8} - 176 T^{6} + 7776 T^{4} + \cdots + 246016$$
$71$ $$(T^{4} + 23 T^{3} + 99 T^{2} - 798 T - 4924)^{2}$$
$73$ $$T^{8} - 79 T^{6} + 76 T^{4} - 19 T^{2} + \cdots + 1$$
$79$ $$(T^{4} - 5 T^{3} - 195 T^{2} + 5780)^{2}$$
$83$ $$T^{8} - 374 T^{6} + 35061 T^{4} + \cdots + 99856$$
$89$ $$(T^{4} + 15 T^{3} - 35 T^{2} - 530 T + 1180)^{2}$$
$97$ $$T^{8} - 666 T^{6} + \cdots + 301334881$$