# Properties

 Label 5625.2.a.p Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.44400625.1 Defining polynomial: $$x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1$$ x^6 - 11*x^4 - x^3 + 29*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} - \beta_{3} + 1) q^{4} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b4 - b3 + 1) * q^4 + (b1 - 1) * q^7 + (-b3 + b2 + 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{4} - \beta_{3} + 1) q^{4} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{8} + (\beta_{5} - \beta_1) q^{11} + (\beta_{5} + \beta_{3} + \beta_1) q^{13} + (\beta_{4} - \beta_{3} - \beta_1 + 3) q^{14} + (\beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{16} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{17} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{19} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{22} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 2) q^{23} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{26} + ( - \beta_{4} + \beta_{2} + 4 \beta_1 - 1) q^{28} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{29} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{31} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{32} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{34} + ( - \beta_{5} - \beta_{2} + \beta_1 - 4) q^{37} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{38} + (\beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{41} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{43} + ( - 5 \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{44} + (\beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 2) q^{46} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 4) q^{47} + (\beta_{4} - \beta_{3} - 2 \beta_1 - 3) q^{49} + ( - \beta_{5} - \beta_{4} - 5 \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{52} + ( - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 2) q^{53} + (\beta_{5} + 2 \beta_{4} - 4 \beta_{3} - \beta_1 + 4) q^{56} + ( - \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2) q^{58} + ( - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{59} + (\beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{61} + ( - 2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} - 1) q^{62} + (\beta_{5} + \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 6) q^{64} + (2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{67} + ( - \beta_{5} + 2 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{68} + (\beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{71} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{73} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{74} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 4) q^{76} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{77} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{79} + (3 \beta_{5} + \beta_{4} - 11 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{82} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{83} + ( - 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{86} + ( - \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{88} + ( - 2 \beta_{4} + 8 \beta_{3} - 3 \beta_{2} + \beta_1 + 5) q^{89} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{91} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 11) q^{92} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{94} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{97} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - 6) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b4 - b3 + 1) * q^4 + (b1 - 1) * q^7 + (-b3 + b2 + 2*b1) * q^8 + (b5 - b1) * q^11 + (b5 + b3 + b1) * q^13 + (b4 - b3 - b1 + 3) * q^14 + (b5 - 3*b3 + b2 + b1 + 2) * q^16 + (b5 + b4 + b3 - b2 + 3) * q^17 + (-b4 + 2*b3 + b2 + 3) * q^19 + (-b4 + b3 + 2*b2 - 2) * q^22 + (-b5 + b4 + b3 + b1 + 2) * q^23 + (b4 - b3 + b2 - b1 + 4) * q^26 + (-b4 + b2 + 4*b1 - 1) * q^28 + (-b5 - b2 - b1) * q^29 + (b5 + b4 + b3 - 2*b2 - 2*b1 - 1) * q^31 + (b5 + b4 - 2*b3 + 3*b2 + b1 + 2) * q^32 + (-b5 + 2*b3 + b2 + 4*b1 + 3) * q^34 + (-b5 - b2 + b1 - 4) * q^37 + (b5 - 2*b3 - 2*b2 - b1 - 2) * q^38 + (b5 + b4 + b3 + 3*b2 + b1 + 1) * q^41 + (b4 - b3 - 2*b2 + b1 - 1) * q^43 + (-5*b3 - b2 - 3*b1 - 4) * q^44 + (b4 - 2*b3 - 3*b2 + 3*b1 + 2) * q^46 + (-b4 + 3*b3 + 2*b1 + 4) * q^47 + (b4 - b3 - 2*b1 - 3) * q^49 + (-b5 - b4 - 5*b3 + b2 + 5*b1 - 5) * q^52 + (-b4 - 4*b3 - 2*b2 + 2) * q^53 + (b5 + 2*b4 - 4*b3 - b1 + 4) * q^56 + (-b5 - b4 + 4*b3 - 2*b2 - 2) * q^58 + (-2*b5 - 3*b3 - b2 + b1 - 4) * q^59 + (b5 - b4 + b3 - 3*b2 - b1 + 3) * q^61 + (-2*b5 - 2*b4 + 7*b3 + b2 - 1) * q^62 + (b5 + b4 - 5*b3 + 2*b2 + 4*b1 - 6) * q^64 + (2*b5 + b4 - b3 + 2*b2 + b1 - 1) * q^67 + (-b5 + 2*b4 - 9*b3 - 2*b2 + b1 + 3) * q^68 + (b5 + 2*b4 + 3*b3 - b2 + b1 + 3) * q^71 + (-b5 + 2*b4 + 2*b3 - b2 - 2) * q^73 + (-b5 + b4 + 2*b3 - 2*b2 - 4*b1 + 4) * q^74 + (-2*b5 + b4 + 3*b3 + 2*b2 - 4) * q^76 + (-b5 - b4 + b3 + 2*b2 + b1 - 2) * q^77 + (-2*b5 - 2*b4 + b3 + b2 + 2*b1 - 1) * q^79 + (3*b5 + b4 - 11*b3 + b2 + 2*b1 - 2) * q^82 + (b5 - 2*b4 + b3 - b2 - 2) * q^83 + (-2*b5 + b4 + 4*b3 + b2 + 2*b1 + 7) * q^86 + (-b5 - b4 + 4*b3 + b2 + b1 - 3) * q^88 + (-2*b4 + 8*b3 - 3*b2 + b1 + 5) * q^89 + (-b5 + b4 - 2*b3 + b2 - 2*b1 + 4) * q^91 + (-b5 + b4 + 3*b3 + 2*b2 + 4*b1 + 11) * q^92 + (2*b4 - b3 - 3*b2 - b1 + 6) * q^94 + (-2*b5 + b4 + 2*b2 + 3*b1) * q^97 + (-2*b4 + b3 + b2 - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10})$$ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 $$6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100})$$ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} - 1 ) / 4$$ (v^5 - v^4 - 6*v^3 + 5*v^2 - 1) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - \nu^{4} - 10\nu^{3} + 5\nu^{2} + 24\nu - 1 ) / 4$$ (v^5 - v^4 - 10*v^3 + 5*v^2 + 24*v - 1) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - \nu^{4} - 10\nu^{3} + 9\nu^{2} + 24\nu - 13 ) / 4$$ (v^5 - v^4 - 10*v^3 + 9*v^2 + 24*v - 13) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + \nu^{4} - 12\nu^{3} - 7\nu^{2} + 34\nu + 3 ) / 2$$ (v^5 + v^4 - 12*v^3 - 7*v^2 + 34*v + 3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + 3$$ b4 - b3 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 6\beta_1$$ -b3 + b2 + 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + 6\beta_{4} - 9\beta_{3} + \beta_{2} + \beta _1 + 16$$ b5 + 6*b4 - 9*b3 + b2 + b1 + 16 $$\nu^{5}$$ $$=$$ $$\beta_{5} + \beta_{4} - 10\beta_{3} + 11\beta_{2} + 37\beta _1 + 2$$ b5 + b4 - 10*b3 + 11*b2 + 37*b1 + 2

## 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
 −2.44028 −2.16056 −0.246759 0.141689 2.01887 2.68704
−2.44028 0 3.95498 0 0 −3.44028 −4.77071 0 0
1.2 −2.16056 0 2.66802 0 0 −3.16056 −1.44329 0 0
1.3 −0.246759 0 −1.93911 0 0 −1.24676 0.972011 0 0
1.4 0.141689 0 −1.97992 0 0 −0.858311 −0.563913 0 0
1.5 2.01887 0 2.07584 0 0 1.01887 0.153106 0 0
1.6 2.68704 0 5.22020 0 0 1.68704 8.65280 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.p 6
3.b odd 2 1 1875.2.a.j 6
5.b even 2 1 5625.2.a.q 6
15.d odd 2 1 1875.2.a.k 6
15.e even 4 2 1875.2.b.f 12
25.d even 5 2 225.2.h.d 12
75.h odd 10 2 375.2.g.c 12
75.j odd 10 2 75.2.g.c 12
75.l even 20 4 375.2.i.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 75.j odd 10 2
225.2.h.d 12 25.d even 5 2
375.2.g.c 12 75.h odd 10 2
375.2.i.d 24 75.l even 20 4
1875.2.a.j 6 3.b odd 2 1
1875.2.a.k 6 15.d odd 2 1
1875.2.b.f 12 15.e even 4 2
5625.2.a.p 6 1.a even 1 1 trivial
5625.2.a.q 6 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{6} - 11T_{2}^{4} - T_{2}^{3} + 29T_{2}^{2} + 3T_{2} - 1$$ T2^6 - 11*T2^4 - T2^3 + 29*T2^2 + 3*T2 - 1 $$T_{7}^{6} + 6T_{7}^{5} + 4T_{7}^{4} - 25T_{7}^{3} - 25T_{7}^{2} + 20T_{7} + 20$$ T7^6 + 6*T7^5 + 4*T7^4 - 25*T7^3 - 25*T7^2 + 20*T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 11 T^{4} - T^{3} + 29 T^{2} + \cdots - 1$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 6 T^{5} + 4 T^{4} - 25 T^{3} + \cdots + 20$$
$11$ $$T^{6} + 3 T^{5} - 24 T^{4} - 66 T^{3} + \cdots + 244$$
$13$ $$T^{6} + 6 T^{5} - 26 T^{4} - 173 T^{3} + \cdots + 101$$
$17$ $$T^{6} - 13 T^{5} + 15 T^{4} + \cdots + 4639$$
$19$ $$T^{6} - 11 T^{5} + 11 T^{4} + \cdots - 380$$
$23$ $$T^{6} - 13 T^{5} + 6 T^{4} + \cdots - 2020$$
$29$ $$T^{6} - 3 T^{5} - 41 T^{4} + \cdots + 2105$$
$31$ $$T^{6} + 11 T^{5} - 46 T^{4} + \cdots - 2900$$
$37$ $$T^{6} + 21 T^{5} + 139 T^{4} + \cdots - 6025$$
$41$ $$T^{6} - T^{5} - 181 T^{4} + \cdots - 82655$$
$43$ $$T^{6} + 2 T^{5} - 91 T^{4} + \cdots - 6284$$
$47$ $$T^{6} - 14 T^{5} - 4 T^{4} + \cdots + 2284$$
$53$ $$T^{6} - 23 T^{5} + 61 T^{4} + \cdots - 34495$$
$59$ $$T^{6} + 9 T^{5} - 89 T^{4} + \cdots + 3920$$
$61$ $$T^{6} - 11 T^{5} - 139 T^{4} + \cdots + 168269$$
$67$ $$T^{6} + 8 T^{5} - 155 T^{4} + \cdots + 14684$$
$71$ $$T^{6} - 8 T^{5} - 150 T^{4} + \cdots - 196$$
$73$ $$T^{6} + 13 T^{5} - 74 T^{4} + \cdots + 22205$$
$79$ $$T^{6} + 5 T^{5} - 160 T^{4} + \cdots - 8000$$
$83$ $$T^{6} + 20 T^{5} + 26 T^{4} + \cdots + 41036$$
$89$ $$T^{6} - 4 T^{5} - 464 T^{4} + \cdots - 377055$$
$97$ $$T^{6} - 7 T^{5} - 215 T^{4} + \cdots + 2399$$