# Properties

 Label 5625.2.a.be Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ 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: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.6152203125.1 Defining polynomial: $$x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1$$ x^8 - 3*x^7 - 8*x^6 + 20*x^5 + 26*x^4 - 35*x^3 - 27*x^2 + 16*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 625) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_1 - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{2} - \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 2) * q^4 + (-b7 + b5 - b1 - 1) * q^7 + (b7 - b6 + b2 - b1 + 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_1 - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{2} - \beta_1 + 2) q^{8} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{11} + (\beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{3} - 3) q^{13} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{14} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_1 + 2) q^{16} + (\beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{3} + \beta_{2} + 4) q^{17} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{19} + ( - \beta_{7} - 3 \beta_{6} + \beta_{4} - 6 \beta_{3} - \beta_1 - 3) q^{22} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 3) q^{23} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} - 4) q^{26} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{28} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{29} + (4 \beta_{7} + \beta_{6} - \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + 1) q^{31} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 6) q^{32} + (3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{34} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{37} + ( - 4 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - 7 \beta_{3} - \beta_{2} - 2) q^{38} + ( - \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{41} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{43} + ( - 6 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 8 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{44} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{46} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{47} + (4 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{2} + 3 \beta_1 - 1) q^{49} + ( - 5 \beta_{7} - \beta_{6} + 2 \beta_{5} - 5 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{52} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{53} + ( - 3 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 2) q^{56} + (4 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} + \cdots + 9) q^{58}+ \cdots + (4 \beta_{7} + 6 \beta_{6} - \beta_{5} + 3 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} - \beta_1) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 2) * q^4 + (-b7 + b5 - b1 - 1) * q^7 + (b7 - b6 + b2 - b1 + 2) * q^8 + (b7 - 2*b6 + 2*b4 - b3 - b1 - 2) * q^11 + (b7 - 2*b6 + b4 - b3 - 3) * q^13 + (-b7 - 2*b6 + b5 - b4 - b3 + b1) * q^14 + (b7 - b6 + b5 + b3 - 2*b1 + 2) * q^16 + (b7 + b6 - b5 + 3*b3 + b2 + 4) * q^17 + (-2*b7 - b6 + b4 - 2*b3 - b2 - 2*b1 - 2) * q^19 + (-b7 - 3*b6 + b4 - 6*b3 - b1 - 3) * q^22 + (b7 - b5 + b4 + b1 + 3) * q^23 + (-b7 - 2*b6 + b5 + b4 - 3*b3 - 4) * q^26 + (-b7 - 3*b6 + 2*b5 + b3 - b2 - b1) * q^28 + (b7 + 2*b6 - b5 - b4 + 3*b3 + b2 - b1 + 2) * q^29 + (4*b7 + b6 - b5 + 4*b3 + 2*b2 + 1) * q^31 + (-2*b7 + b6 + 2*b5 - 2*b4 + 2*b3 + b2 + 6) * q^32 + (3*b7 + 2*b6 - 2*b5 - 2*b4 + 4*b3 + b2 - b1 + 5) * q^34 + (2*b7 + b6 + b5 + b4 + 2*b3 + 2*b2 - b1 + 2) * q^37 + (-4*b7 - 3*b6 + 2*b4 - 7*b3 - b2 - 2) * q^38 + (-b7 + 4*b6 + 2*b5 - b4 + 3*b3 + b2 - 2*b1 + 4) * q^41 + (b7 + 2*b6 - 2*b5 - b4 + b3 + 2) * q^43 + (-6*b7 - b6 + 2*b5 + 2*b4 - 8*b3 - b2 - 4*b1 - 1) * q^44 + (b7 + b6 - 2*b5 + b4 - 2*b3 - b2 - 3*b1 - 1) * q^46 + (2*b7 - b6 + b5 + b4 - b3 - b1 + 2) * q^47 + (4*b7 - b6 - 3*b5 - b4 + b2 + 3*b1 - 1) * q^49 + (-5*b7 - b6 + 2*b5 - 5*b3 - 2*b2 - b1 - 3) * q^52 + (-3*b7 - 3*b6 - b5 + b4 - 2*b3 - b2 - 3*b1 + 1) * q^53 + (-3*b7 - b6 + 3*b5 - b4 + 2*b3 - 3*b1 + 2) * q^56 + (4*b7 + 4*b6 - 2*b5 - 2*b4 + 7*b3 + 3*b2 + 2*b1 + 9) * q^58 + (2*b7 + 2*b6 + b5 - b4 + 4*b3 + b2 + 3) * q^59 + (-b7 + 3*b5 + 2*b4 - b2 - b1 - 1) * q^61 + (7*b7 + 4*b6 - 2*b5 - 3*b4 + 8*b3 + 4*b2 + 2*b1 + 7) * q^62 + (-2*b7 + b5 - 4*b4 + 4*b3 + 3) * q^64 + (b7 + 4*b6 + b4 + b3 + b2 + 2*b1 - 1) * q^67 + (4*b7 + 6*b6 - 2*b4 + 7*b3 + 4*b2 + 11) * q^68 + (-2*b7 + b5 - b4 - 2*b3 - 2*b2 - 2*b1 + 1) * q^71 + (b6 - b5 + 3*b4 - 4*b3 - b2 - 3*b1 - 2) * q^73 + (5*b7 - b6 - b5 - 3*b4 + b3 + 2*b2 - b1 + 4) * q^74 + (-4*b7 - 6*b6 + b5 + 5*b4 - 13*b3 - 4*b2 - 3*b1 - 11) * q^76 + (-5*b7 + b6 - 3*b4 - 5*b3 + b2 + b1 + 1) * q^77 + (-4*b7 - b6 + b5 + 4*b3 - b2 - b1) * q^79 + (4*b7 + b6 - b5 - 5*b4 + 5*b3 + 2*b2 + 2*b1 + 10) * q^82 + (-2*b7 + 2*b6 - 3*b5 - b4 + b3 + 2*b2 - b1 + 9) * q^83 + (3*b7 + 6*b6 - 3*b5 + b4 + 5*b3 + 2*b2 + b1 + 7) * q^86 + (-6*b7 - 4*b6 + b5 + 4*b4 - 8*b3 - 4*b2 - 5*b1) * q^88 + (-3*b7 - b6 + 3*b5 + b4 - 3*b3 - 4*b2 - 3) * q^89 + (-2*b7 + 3*b6 - b5 - b4 - b3 + b2 - b1 + 3) * q^91 + (-b7 + 4*b6 - 2*b5 + 2*b4 - 4*b3 + 3*b2 - b1 + 2) * q^92 + (b7 - b6 + b5 - 2*b3 + 2*b2 - 4*b1 + 6) * q^94 + (b7 - b6 - 2*b5 + b4 + 4*b3 - 3*b2 + 1) * q^97 + (4*b7 + 6*b6 - b5 + 3*b4 + 7*b3 + 2*b2 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10})$$ 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8 $$8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8} - q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} + 15 q^{17} - 10 q^{19} + 5 q^{22} + 30 q^{23} - 11 q^{26} + 5 q^{28} - 10 q^{29} - 9 q^{31} + 30 q^{32} + 7 q^{34} + 10 q^{37} + 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} + 30 q^{47} - 4 q^{49} - 5 q^{52} + 10 q^{53} + 30 q^{58} + 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 10 q^{67} + 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} + 5 q^{77} - 20 q^{79} + 45 q^{82} + 40 q^{83} + 24 q^{86} + 40 q^{88} + 5 q^{89} + 6 q^{91} + 15 q^{92} + 47 q^{94} - 30 q^{98}+O(q^{100})$$ 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8 - q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 + 15 * q^17 - 10 * q^19 + 5 * q^22 + 30 * q^23 - 11 * q^26 + 5 * q^28 - 10 * q^29 - 9 * q^31 + 30 * q^32 + 7 * q^34 + 10 * q^37 + 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 + 30 * q^47 - 4 * q^49 - 5 * q^52 + 10 * q^53 + 30 * q^58 + 5 * q^59 + 6 * q^61 + 10 * q^62 - 9 * q^64 - 10 * q^67 + 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 + 5 * q^77 - 20 * q^79 + 45 * q^82 + 40 * q^83 + 24 * q^86 + 40 * q^88 + 5 * q^89 + 6 * q^91 + 15 * q^92 + 47 * q^94 - 30 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{6} - 4\nu^{5} - 2\nu^{4} + 17\nu^{3} - \nu^{2} - 15\nu + 1 ) / 3$$ (v^6 - 4*v^5 - 2*v^4 + 17*v^3 - v^2 - 15*v + 1) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 17\nu^{4} - \nu^{3} - 15\nu^{2} + 4\nu ) / 3$$ (v^7 - 4*v^6 - 2*v^5 + 17*v^4 - v^3 - 15*v^2 + 4*v) / 3 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 5\nu^{4} - 26\nu^{3} - 5\nu^{2} + 36\nu - 4 ) / 3$$ (-v^6 + 4*v^5 + 5*v^4 - 26*v^3 - 5*v^2 + 36*v - 4) / 3 $$\beta_{6}$$ $$=$$ $$( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 4\nu^{3} + 51\nu^{2} - 8\nu ) / 3$$ (-2*v^7 + 8*v^6 + 7*v^5 - 43*v^4 - 4*v^3 + 51*v^2 - 8*v) / 3 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 7\nu^{3} + 57\nu^{2} + \nu - 6 ) / 3$$ (-2*v^7 + 8*v^6 + 7*v^5 - 43*v^4 - 7*v^3 + 57*v^2 + v - 6) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} + 2\beta_{2} + 5\beta _1 + 4$$ -b7 + b6 + 2*b2 + 5*b1 + 4 $$\nu^{4}$$ $$=$$ $$-3\beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 19$$ -3*b7 + 3*b6 + b5 + b3 + 8*b2 + 10*b1 + 19 $$\nu^{5}$$ $$=$$ $$-11\beta_{7} + 12\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 21\beta_{2} + 33\beta _1 + 44$$ -11*b7 + 12*b6 + 3*b5 + 2*b4 + 3*b3 + 21*b2 + 33*b1 + 44 $$\nu^{6}$$ $$=$$ $$-33\beta_{7} + 37\beta_{6} + 14\beta_{5} + 8\beta_{4} + 17\beta_{3} + 67\beta_{2} + 83\beta _1 + 148$$ -33*b7 + 37*b6 + 14*b5 + 8*b4 + 17*b3 + 67*b2 + 83*b1 + 148 $$\nu^{7}$$ $$=$$ $$-104\beta_{7} + 122\beta_{6} + 45\beta_{5} + 39\beta_{4} + 57\beta_{3} + 191\beta_{2} + 244\beta _1 + 406$$ -104*b7 + 122*b6 + 45*b5 + 39*b4 + 57*b3 + 191*b2 + 244*b1 + 406

## 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
 3.01367 2.68341 1.32675 0.499011 −0.0573749 −1.32610 −1.47435 −1.66501
−2.01367 0 2.05487 0 0 −0.369971 −0.110485 0 0
1.2 −1.68341 0 0.833870 0 0 −4.59110 1.96307 0 0
1.3 −0.326751 0 −1.89323 0 0 −3.42409 1.27212 0 0
1.4 0.500989 0 −1.74901 0 0 −0.0237879 −1.87821 0 0
1.5 1.05737 0 −0.881958 0 0 −1.01199 −3.04731 0 0
1.6 2.32610 0 3.41075 0 0 −3.59425 3.28154 0 0
1.7 2.47435 0 4.12242 0 0 0.973070 5.25163 0 0
1.8 2.66501 0 5.10229 0 0 2.04213 8.26764 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.be 8
3.b odd 2 1 625.2.a.e 8
5.b even 2 1 5625.2.a.s 8
12.b even 2 1 10000.2.a.bn 8
15.d odd 2 1 625.2.a.g yes 8
15.e even 4 2 625.2.b.d 16
60.h even 2 1 10000.2.a.be 8
75.h odd 10 2 625.2.d.m 16
75.h odd 10 2 625.2.d.n 16
75.j odd 10 2 625.2.d.p 16
75.j odd 10 2 625.2.d.q 16
75.l even 20 4 625.2.e.j 32
75.l even 20 4 625.2.e.k 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e 8 3.b odd 2 1
625.2.a.g yes 8 15.d odd 2 1
625.2.b.d 16 15.e even 4 2
625.2.d.m 16 75.h odd 10 2
625.2.d.n 16 75.h odd 10 2
625.2.d.p 16 75.j odd 10 2
625.2.d.q 16 75.j odd 10 2
625.2.e.j 32 75.l even 20 4
625.2.e.k 32 75.l even 20 4
5625.2.a.s 8 5.b even 2 1
5625.2.a.be 8 1.a even 1 1 trivial
10000.2.a.be 8 60.h even 2 1
10000.2.a.bn 8 12.b even 2 1

## Hecke kernels

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

 $$T_{2}^{8} - 5T_{2}^{7} - T_{2}^{6} + 35T_{2}^{5} - 29T_{2}^{4} - 60T_{2}^{3} + 69T_{2}^{2} - 9$$ T2^8 - 5*T2^7 - T2^6 + 35*T2^5 - 29*T2^4 - 60*T2^3 + 69*T2^2 - 9 $$T_{7}^{8} + 10T_{7}^{7} + 24T_{7}^{6} - 35T_{7}^{5} - 154T_{7}^{4} - 25T_{7}^{3} + 124T_{7}^{2} + 45T_{7} + 1$$ T7^8 + 10*T7^7 + 24*T7^6 - 35*T7^5 - 154*T7^4 - 25*T7^3 + 124*T7^2 + 45*T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 5 T^{7} - T^{6} + 35 T^{5} + \cdots - 9$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 10 T^{7} + 24 T^{6} - 35 T^{5} + \cdots + 1$$
$11$ $$T^{8} + T^{7} - 58 T^{6} - 82 T^{5} + \cdots + 2421$$
$13$ $$T^{8} + 10 T^{7} + 11 T^{6} + \cdots + 361$$
$17$ $$T^{8} - 15 T^{7} + 59 T^{6} + \cdots + 1611$$
$19$ $$T^{8} + 10 T^{7} - 15 T^{6} + \cdots + 10525$$
$23$ $$T^{8} - 30 T^{7} + 351 T^{6} + \cdots - 46089$$
$29$ $$T^{8} + 10 T^{7} - 25 T^{6} + \cdots - 60975$$
$31$ $$T^{8} + 9 T^{7} - 83 T^{6} + \cdots + 24001$$
$37$ $$T^{8} - 10 T^{7} - 91 T^{6} + 985 T^{5} + \cdots + 81$$
$41$ $$T^{8} - 4 T^{7} - 193 T^{6} + \cdots - 487629$$
$43$ $$T^{8} - 99 T^{6} - 180 T^{5} + \cdots - 1949$$
$47$ $$T^{8} - 30 T^{7} + 314 T^{6} + \cdots + 56961$$
$53$ $$T^{8} - 10 T^{7} - 139 T^{6} + \cdots - 1899$$
$59$ $$T^{8} - 5 T^{7} - 100 T^{6} + \cdots - 225$$
$61$ $$T^{8} - 6 T^{7} - 173 T^{6} + \cdots - 103529$$
$67$ $$T^{8} + 10 T^{7} - 181 T^{6} + \cdots - 1746299$$
$71$ $$T^{8} - 9 T^{7} - 133 T^{6} + \cdots - 16749$$
$73$ $$T^{8} - 314 T^{6} - 185 T^{5} + \cdots + 237091$$
$79$ $$T^{8} + 20 T^{7} - 115 T^{6} + \cdots + 249525$$
$83$ $$T^{8} - 40 T^{7} + 431 T^{6} + \cdots + 12262851$$
$89$ $$T^{8} - 5 T^{7} - 295 T^{6} + \cdots + 1849275$$
$97$ $$T^{8} - 321 T^{6} - 1875 T^{5} + \cdots + 972421$$