# Properties

 Label 5625.2.a.bc Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# 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: 8.8.13366265625.1 Defining polynomial: $$x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16$$ x^8 - x^7 - 12*x^6 + 10*x^5 + 41*x^4 - 20*x^3 - 48*x^2 + 8*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1875) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{4} - \beta_{2} - 1) q^{7} + (\beta_{6} + 3 \beta_{2}) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b5 + b4 + 1) * q^4 + (b4 - b2 - 1) * q^7 + (b6 + 3*b2) * q^8 $$q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{4} - \beta_{2} - 1) q^{7} + (\beta_{6} + 3 \beta_{2}) q^{8} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{11} + (\beta_{5} + \beta_{3} + \beta_1 - 2) q^{13} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 1) q^{14} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{16} + (2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{17} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 3) q^{19} + (\beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{22} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 1) q^{23} + ( - 2 \beta_{7} + 2 \beta_{5} + \beta_{4} - 2 \beta_1 + 3) q^{26} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 4) q^{28} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 + 1) q^{29} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{3} + \beta_1 + 2) q^{31} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_1 - 3) q^{32} + ( - 4 \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} - 6) q^{34} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 1) q^{37} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{38} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{41} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 2) q^{43} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 4) q^{44} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 6) q^{46} + (\beta_{6} - \beta_{4} - 4 \beta_{3} + \beta_{2} - 3) q^{47} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{49} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \cdots - 2) q^{52}+ \cdots + ( - 8 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b5 + b4 + 1) * q^4 + (b4 - b2 - 1) * q^7 + (b6 + 3*b2) * q^8 + (-2*b7 - b6 - b5 - b4 - 2*b3 - b2 - 3) * q^11 + (b5 + b3 + b1 - 2) * q^13 + (2*b7 - 2*b5 - b4 + 2*b2 - 1) * q^14 + (2*b5 + b4 - 2*b3 - b1 + 1) * q^16 + (2*b7 + b5 - b4 + 2*b3 - 2*b1 + 1) * q^17 + (b7 - b6 - b5 + b3 - b2 + 3) * q^19 + (b6 - b4 + 2*b3 - b2 - 3*b1 - 1) * q^22 + (-2*b7 - b6 - 2*b5 - b4 - 3*b2 + 3*b1 - 1) * q^23 + (-2*b7 + 2*b5 + b4 - 2*b1 + 3) * q^26 + (2*b7 - 2*b6 - b5 - 2*b2 - 2*b1 + 4) * q^28 + (2*b7 + 2*b6 + b5 - b4 - b1 + 1) * q^29 + (-b7 + b6 - 3*b5 - 2*b3 + b1 + 2) * q^31 + (-2*b7 + 2*b6 - b4 + 2*b1 - 3) * q^32 + (-4*b7 - b6 - 2*b5 - 2*b4 - 3*b2 - 6) * q^34 + (-b7 + b6 - b5 + b2 + 1) * q^37 + (2*b7 - 2*b6 - 4*b5 - b4 + 2*b3 - 2*b2 + 4*b1 - 1) * q^38 + (-2*b7 - b6 - b5 + b4 - b2 + 1) * q^41 + (b7 + 2*b6 + 2*b5 + b4 + b3 + 4*b2 - 4*b1 - 2) * q^43 + (2*b7 - 2*b4 + 2*b3 - 2*b2 - 3*b1 - 4) * q^44 + (2*b7 - 2*b6 + 2*b3 - 4*b2 - b1 + 6) * q^46 + (b6 - b4 - 4*b3 + b2 - 3) * q^47 + (2*b7 - b6 + 2*b5 - 2*b4 + b3 + 3*b2 - 3*b1 + 3) * q^49 + (-2*b7 + 2*b6 - b5 - 2*b4 - 2*b3 + 4*b2 + 2*b1 - 2) * q^52 + (-4*b7 - b6 + 2*b4 - b2 - b1 - 2) * q^53 + (-2*b7 - b6 - 5*b5 - 2*b4 + 4*b3 - 5*b2 + 6*b1 - 6) * q^56 + (-4*b7 + b6 + b5 - b4 - 4*b3 - b2 - 2*b1 - 3) * q^58 + (4*b7 - 2*b6 + b5 - b4 + 2*b3 + 2*b2 - b1 - 1) * q^59 + (b7 + 2*b6 + b5 - b4 + 2*b3 - b2 - 3*b1 + 3) * q^61 + (6*b7 - b6 + b4 - 2*b3 - b2 + b1 + 3) * q^62 + (-2*b7 + 3*b5 - 2*b2 - 4*b1 + 4) * q^64 + (3*b7 - 2*b6 - 2*b5 - 2*b3 + 3*b2 - b1 - 2) * q^67 + (-4*b7 - 2*b6 - 2*b5 - b4 - 2*b3 - 6*b2 - 3*b1 - 5) * q^68 + (-2*b7 + 2*b6 - 2*b3 + 4*b2 - 3*b1 - 4) * q^71 + (3*b7 + b5 + 4*b3 + 2*b2 + 4) * q^73 + (2*b7 - b6 + 2*b5 + b4 - 2*b3 - b2 + 1) * q^74 + (4*b7 - 4*b6 - 3*b5 + 2*b4 + 2*b3 - 6*b2 + 4) * q^76 + (8*b7 + 3*b6 + 6*b5 + 4*b3 + 3*b2 - 2*b1 + 2) * q^77 + (-b7 + 3*b6 + 4*b5 + b4 + 2*b3 + 3*b2 - 5*b1 + 1) * q^79 + (4*b7 - b6 - 2*b5 - b4 + 2*b3 + b2 + 3*b1 - 1) * q^82 + (-2*b5 - 2*b3 + 4*b2 - b1 - 4) * q^83 + (-2*b7 + b6 + 2*b5 - 4*b3 + 3*b2 - 3*b1 - 8) * q^86 + (-4*b7 - 4*b6 - 5*b5 - 3*b4 - 4*b3 - 4*b2 - 9) * q^88 + (6*b7 + 4*b5 + b4 + 4*b3 + 4*b2 - b1 + 3) * q^89 + (b7 - 5*b5 - 3*b4 - 4*b3 + 5*b2 - b1 - 2) * q^91 + (4*b7 - 5*b5 - 3*b4 + 4*b3 + 4*b2 + 2*b1 - 5) * q^92 + (-2*b7 + 4*b6 + 3*b5 + b4 - 2*b3 + 2*b2 - 5*b1 + 1) * q^94 + (3*b7 - 2*b4 - b2 + 2*b1 - 2) * q^97 + (-8*b7 + b6 + b5 + 2*b3 - 3*b2 + 2*b1 - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10})$$ 8 * q + q^2 + 9 * q^4 - 12 * q^7 + 3 * q^8 $$8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8} - 12 q^{11} - 14 q^{13} - 16 q^{14} + 15 q^{16} - q^{17} + 16 q^{19} - 18 q^{22} - 4 q^{23} + 34 q^{26} + 21 q^{28} - 2 q^{29} + 13 q^{31} - 18 q^{32} - 37 q^{34} + 8 q^{37} - 24 q^{38} + 12 q^{41} - 20 q^{43} - 47 q^{44} + 33 q^{46} - 15 q^{47} + 30 q^{49} + q^{52} - 4 q^{53} - 60 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} + 4 q^{62} + 41 q^{64} - 19 q^{67} - 33 q^{68} - 21 q^{71} + 19 q^{73} + 9 q^{74} - q^{76} - 11 q^{77} + 10 q^{79} - 24 q^{82} - 27 q^{83} - 42 q^{86} - 53 q^{88} + 9 q^{89} - 12 q^{91} - 63 q^{92} + 14 q^{94} - 24 q^{97} - 24 q^{98}+O(q^{100})$$ 8 * q + q^2 + 9 * q^4 - 12 * q^7 + 3 * q^8 - 12 * q^11 - 14 * q^13 - 16 * q^14 + 15 * q^16 - q^17 + 16 * q^19 - 18 * q^22 - 4 * q^23 + 34 * q^26 + 21 * q^28 - 2 * q^29 + 13 * q^31 - 18 * q^32 - 37 * q^34 + 8 * q^37 - 24 * q^38 + 12 * q^41 - 20 * q^43 - 47 * q^44 + 33 * q^46 - 15 * q^47 + 30 * q^49 + q^52 - 4 * q^53 - 60 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 + 4 * q^62 + 41 * q^64 - 19 * q^67 - 33 * q^68 - 21 * q^71 + 19 * q^73 + 9 * q^74 - q^76 - 11 * q^77 + 10 * q^79 - 24 * q^82 - 27 * q^83 - 42 * q^86 - 53 * q^88 + 9 * q^89 - 12 * q^91 - 63 * q^92 + 14 * q^94 - 24 * q^97 - 24 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} - 12\nu^{5} + 10\nu^{4} + 41\nu^{3} - 20\nu^{2} - 40\nu + 8 ) / 8$$ (v^7 - v^6 - 12*v^5 + 10*v^4 + 41*v^3 - 20*v^2 - 40*v + 8) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 34\nu^{4} - 21\nu^{3} + 94\nu^{2} + 8\nu - 56 ) / 8$$ (-v^7 + 3*v^6 + 10*v^5 - 34*v^4 - 21*v^3 + 94*v^2 + 8*v - 56) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} - 10\nu^{5} + 30\nu^{4} + 21\nu^{3} - 62\nu^{2} - 12\nu + 20 ) / 4$$ (v^7 - 3*v^6 - 10*v^5 + 30*v^4 + 21*v^3 - 62*v^2 - 12*v + 20) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 30\nu^{4} - 21\nu^{3} + 66\nu^{2} + 12\nu - 32 ) / 4$$ (-v^7 + 3*v^6 + 10*v^5 - 30*v^4 - 21*v^3 + 66*v^2 + 12*v - 32) / 4 $$\beta_{6}$$ $$=$$ $$( -3\nu^{7} + 3\nu^{6} + 36\nu^{5} - 30\nu^{4} - 115\nu^{3} + 60\nu^{2} + 88\nu - 24 ) / 8$$ (-3*v^7 + 3*v^6 + 36*v^5 - 30*v^4 - 115*v^3 + 60*v^2 + 88*v - 24) / 8 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 30\nu^{4} - 52\nu^{3} + 61\nu^{2} + 30\nu - 28 ) / 4$$ (-2*v^7 + 3*v^6 + 21*v^5 - 30*v^4 - 52*v^3 + 61*v^2 + 30*v - 28) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + 3$$ b5 + b4 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{6} + 3\beta_{2} + 4\beta_1$$ b6 + 3*b2 + 4*b1 $$\nu^{4}$$ $$=$$ $$8\beta_{5} + 7\beta_{4} - 2\beta_{3} - \beta _1 + 15$$ 8*b5 + 7*b4 - 2*b3 - b1 + 15 $$\nu^{5}$$ $$=$$ $$-2\beta_{7} + 10\beta_{6} - \beta_{4} + 24\beta_{2} + 22\beta _1 - 3$$ -2*b7 + 10*b6 - b4 + 24*b2 + 22*b1 - 3 $$\nu^{6}$$ $$=$$ $$-2\beta_{7} + 59\beta_{5} + 46\beta_{4} - 20\beta_{3} - 2\beta_{2} - 14\beta _1 + 90$$ -2*b7 + 59*b5 + 46*b4 - 20*b3 - 2*b2 - 14*b1 + 90 $$\nu^{7}$$ $$=$$ $$-26\beta_{7} + 79\beta_{6} - \beta_{5} - 16\beta_{4} + 171\beta_{2} + 136\beta _1 - 44$$ -26*b7 + 79*b6 - b5 - 16*b4 + 171*b2 + 136*b1 - 44

## 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
 −2.69767 −1.52260 −0.895394 −0.770071 0.741379 1.31354 2.23365 2.59716
−2.69767 0 5.27745 0 0 3.56649 −8.84149 0 0
1.2 −1.52260 0 0.318310 0 0 0.990985 2.56054 0 0
1.3 −0.895394 0 −1.19827 0 0 −5.08992 2.86371 0 0
1.4 −0.770071 0 −1.40699 0 0 −3.98808 2.62363 0 0
1.5 0.741379 0 −1.45036 0 0 −1.03586 −2.55802 0 0
1.6 1.31354 0 −0.274605 0 0 −4.19091 −2.98779 0 0
1.7 2.23365 0 2.98921 0 0 1.03143 2.20956 0 0
1.8 2.59716 0 4.74525 0 0 −3.28414 7.12986 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

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 5625.2.a.bc 8
3.b odd 2 1 1875.2.a.n 8
5.b even 2 1 5625.2.a.u 8
15.d odd 2 1 1875.2.a.o yes 8
15.e even 4 2 1875.2.b.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.n 8 3.b odd 2 1
1875.2.a.o yes 8 15.d odd 2 1
1875.2.b.g 16 15.e even 4 2
5625.2.a.u 8 5.b even 2 1
5625.2.a.bc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{8} - T_{2}^{7} - 12T_{2}^{6} + 10T_{2}^{5} + 41T_{2}^{4} - 20T_{2}^{3} - 48T_{2}^{2} + 8T_{2} + 16$$ T2^8 - T2^7 - 12*T2^6 + 10*T2^5 + 41*T2^4 - 20*T2^3 - 48*T2^2 + 8*T2 + 16 $$T_{7}^{8} + 12T_{7}^{7} + 29T_{7}^{6} - 142T_{7}^{5} - 654T_{7}^{4} - 160T_{7}^{3} + 1650T_{7}^{2} + 320T_{7} - 1055$$ T7^8 + 12*T7^7 + 29*T7^6 - 142*T7^5 - 654*T7^4 - 160*T7^3 + 1650*T7^2 + 320*T7 - 1055

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} - 12 T^{6} + 10 T^{5} + \cdots + 16$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 12 T^{7} + 29 T^{6} + \cdots - 1055$$
$11$ $$T^{8} + 12 T^{7} + 2 T^{6} - 455 T^{5} + \cdots + 16$$
$13$ $$T^{8} + 14 T^{7} + 43 T^{6} + \cdots + 8731$$
$17$ $$T^{8} + T^{7} - 88 T^{6} - 122 T^{5} + \cdots + 8656$$
$19$ $$T^{8} - 16 T^{7} + 51 T^{6} + \cdots - 14975$$
$23$ $$T^{8} + 4 T^{7} - 104 T^{6} + \cdots + 28720$$
$29$ $$T^{8} + 2 T^{7} - 126 T^{6} + \cdots + 57520$$
$31$ $$T^{8} - 13 T^{7} - 96 T^{6} + \cdots + 801025$$
$37$ $$T^{8} - 8 T^{7} - 31 T^{6} + 313 T^{5} + \cdots + 25$$
$41$ $$T^{8} - 12 T^{7} - 6 T^{6} + \cdots - 48080$$
$43$ $$T^{8} + 20 T^{7} + 6 T^{6} + \cdots - 74369$$
$47$ $$T^{8} + 15 T^{7} - 71 T^{6} + \cdots - 255824$$
$53$ $$T^{8} + 4 T^{7} - 159 T^{6} + \cdots + 28720$$
$59$ $$T^{8} + 14 T^{7} - 164 T^{6} + \cdots + 11920$$
$61$ $$T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 1093919$$
$67$ $$T^{8} + 19 T^{7} - 183 T^{6} + \cdots + 20204221$$
$71$ $$T^{8} + 21 T^{7} - 18 T^{6} + \cdots + 67696$$
$73$ $$T^{8} - 19 T^{7} + 21 T^{6} + \cdots - 379655$$
$79$ $$T^{8} - 10 T^{7} - 320 T^{6} + \cdots + 6951025$$
$83$ $$T^{8} + 27 T^{7} + 132 T^{6} + \cdots - 113744$$
$89$ $$T^{8} - 9 T^{7} - 294 T^{6} + \cdots - 12105680$$
$97$ $$T^{8} + 24 T^{7} + 87 T^{6} + \cdots - 401939$$