# Properties

 Label 9025.2.a.bt Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.0649878242$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4227136.1 Defining polynomial: $$x^{6} - 6x^{4} + 7x^{2} - 1$$ x^6 - 6*x^4 + 7*x^2 - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{3} - \beta_{2}) q^{4} + ( - \beta_{2} + 2) q^{6} + ( - 2 \beta_{5} - \beta_1) q^{7} + (2 \beta_{5} + \beta_{4}) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (b5 + b1) * q^2 + (b5 - b4 + b1) * q^3 + (b3 - b2) * q^4 + (-b2 + 2) * q^6 + (-2*b5 - b1) * q^7 + (2*b5 + b4) * q^8 + (-2*b3 - b2 + 2) * q^9 $$q + (\beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{3} - \beta_{2}) q^{4} + ( - \beta_{2} + 2) q^{6} + ( - 2 \beta_{5} - \beta_1) q^{7} + (2 \beta_{5} + \beta_{4}) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9} + \beta_{3} q^{11} + (2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{12} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2} - 3) q^{14} + (\beta_{3} + 2) q^{16} + ( - \beta_{5} + 2 \beta_{4} - \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_1) q^{18} + ( - \beta_{3} + \beta_{2} - 3) q^{21} + (2 \beta_{5} + \beta_{4} + \beta_1) q^{22} + (\beta_{4} - \beta_1) q^{23} + (4 \beta_{3} - 1) q^{24} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{26} + ( - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_1) q^{27} + ( - 7 \beta_{5} - 2 \beta_{4} - 5 \beta_1) q^{28} + ( - 3 \beta_{3} + 3) q^{29} + (\beta_{2} - 5) q^{31} + ( - \beta_{4} + 3 \beta_1) q^{32} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{33} + (\beta_{3} + \beta_{2} - 2) q^{34} + (2 \beta_{3} + 2 \beta_{2} - 3) q^{36} + ( - 2 \beta_{4} - 3 \beta_1) q^{37} + ( - \beta_{3} + 2 \beta_{2}) q^{39} + (3 \beta_{3} + 2 \beta_{2} - 3) q^{41} + ( - 7 \beta_{5} - \beta_{4} - 5 \beta_1) q^{42} + ( - \beta_{5} + 4 \beta_1) q^{43} + (\beta_{3} - 2 \beta_{2} + 3) q^{44} + (\beta_{3} - 1) q^{46} + ( - 4 \beta_{5} + 4 \beta_{4} - \beta_1) q^{47} + (3 \beta_{5} + \beta_1) q^{48} + (4 \beta_{3} - 3 \beta_{2} - 1) q^{49} + (4 \beta_{3} + \beta_{2} - 8) q^{51} + ( - 7 \beta_{5} - 2 \beta_1) q^{52} + (\beta_{5} + 2 \beta_{4} - 3 \beta_1) q^{53} + ( - 5 \beta_{3} + 2 \beta_{2} + 1) q^{54} + ( - 5 \beta_{3} + 3 \beta_{2} - 6) q^{56} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{58} + ( - \beta_{3} + 4 \beta_{2} - 3) q^{59} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{61} + ( - 7 \beta_{5} - 6 \beta_1) q^{62} + (\beta_{5} + \beta_{4} - \beta_1) q^{63} + ( - 3 \beta_{3} - 1) q^{64} + (3 \beta_{3} - \beta_{2}) q^{66} + (2 \beta_{5} + 4 \beta_1) q^{67} - 3 \beta_{4} q^{68} + (3 \beta_{3} + \beta_{2} - 4) q^{69} + (\beta_{3} + 5 \beta_{2}) q^{71} + ( - 3 \beta_{5} + 6 \beta_{4} - 5 \beta_1) q^{72} + (2 \beta_{5} + \beta_{4} - 5 \beta_1) q^{73} + ( - 2 \beta_{3} - 3) q^{74} + ( - 4 \beta_{5} - \beta_{4} - 2 \beta_1) q^{77} + ( - 6 \beta_{5} - \beta_{4} - 3 \beta_1) q^{78} + (4 \beta_{2} - 4) q^{79} + ( - 5 \beta_{3} + 4) q^{81} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{82} + (5 \beta_{5} + 2 \beta_{4} + 9 \beta_1) q^{83} + ( - 6 \beta_{3} + 5 \beta_{2} - 6) q^{84} + ( - \beta_{3} + \beta_{2} + 3) q^{86} + ( - 9 \beta_{4} + 6 \beta_1) q^{87} + (5 \beta_{5} - \beta_{4} + 4 \beta_1) q^{88} + ( - 4 \beta_{2} - 6) q^{89} + (3 \beta_{3} - 4 \beta_{2} + 3) q^{91} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{92} + ( - 6 \beta_{5} + 5 \beta_{4} - 7 \beta_1) q^{93} + (4 \beta_{2} - 5) q^{94} + ( - 5 \beta_{3} - 3 \beta_{2} + 6) q^{96} + (5 \beta_{5} - 7 \beta_{4} + \beta_1) q^{97} + (13 \beta_{5} + 4 \beta_{4} + 6 \beta_1) q^{98} + (3 \beta_{3} + \beta_{2} - 6) q^{99}+O(q^{100})$$ q + (b5 + b1) * q^2 + (b5 - b4 + b1) * q^3 + (b3 - b2) * q^4 + (-b2 + 2) * q^6 + (-2*b5 - b1) * q^7 + (2*b5 + b4) * q^8 + (-2*b3 - b2 + 2) * q^9 + b3 * q^11 + (2*b5 + 2*b4 + b1) * q^12 + (-b5 - b4 - 2*b1) * q^13 + (-2*b3 + 2*b2 - 3) * q^14 + (b3 + 2) * q^16 + (-b5 + 2*b4 - b1) * q^17 + (-2*b4 + b1) * q^18 + (-b3 + b2 - 3) * q^21 + (2*b5 + b4 + b1) * q^22 + (b4 - b1) * q^23 + (4*b3 - 1) * q^24 + (-2*b3 + b2 - 3) * q^26 + (-2*b5 - 3*b4 + 3*b1) * q^27 + (-7*b5 - 2*b4 - 5*b1) * q^28 + (-3*b3 + 3) * q^29 + (b2 - 5) * q^31 + (-b4 + 3*b1) * q^32 + (b5 + 2*b4 - b1) * q^33 + (b3 + b2 - 2) * q^34 + (2*b3 + 2*b2 - 3) * q^36 + (-2*b4 - 3*b1) * q^37 + (-b3 + 2*b2) * q^39 + (3*b3 + 2*b2 - 3) * q^41 + (-7*b5 - b4 - 5*b1) * q^42 + (-b5 + 4*b1) * q^43 + (b3 - 2*b2 + 3) * q^44 + (b3 - 1) * q^46 + (-4*b5 + 4*b4 - b1) * q^47 + (3*b5 + b1) * q^48 + (4*b3 - 3*b2 - 1) * q^49 + (4*b3 + b2 - 8) * q^51 + (-7*b5 - 2*b1) * q^52 + (b5 + 2*b4 - 3*b1) * q^53 + (-5*b3 + 2*b2 + 1) * q^54 + (-5*b3 + 3*b2 - 6) * q^56 + (-3*b5 - 3*b4) * q^58 + (-b3 + 4*b2 - 3) * q^59 + (2*b3 - 2*b2 - 1) * q^61 + (-7*b5 - 6*b1) * q^62 + (b5 + b4 - b1) * q^63 + (-3*b3 - 1) * q^64 + (3*b3 - b2) * q^66 + (2*b5 + 4*b1) * q^67 - 3*b4 * q^68 + (3*b3 + b2 - 4) * q^69 + (b3 + 5*b2) * q^71 + (-3*b5 + 6*b4 - 5*b1) * q^72 + (2*b5 + b4 - 5*b1) * q^73 + (-2*b3 - 3) * q^74 + (-4*b5 - b4 - 2*b1) * q^77 + (-6*b5 - b4 - 3*b1) * q^78 + (4*b2 - 4) * q^79 + (-5*b3 + 4) * q^81 + (-b5 + 3*b4 - 2*b1) * q^82 + (5*b5 + 2*b4 + 9*b1) * q^83 + (-6*b3 + 5*b2 - 6) * q^84 + (-b3 + b2 + 3) * q^86 + (-9*b4 + 6*b1) * q^87 + (5*b5 - b4 + 4*b1) * q^88 + (-4*b2 - 6) * q^89 + (3*b3 - 4*b2 + 3) * q^91 + (b5 - b4 + 2*b1) * q^92 + (-6*b5 + 5*b4 - 7*b1) * q^93 + (4*b2 - 5) * q^94 + (-5*b3 - 3*b2 + 6) * q^96 + (5*b5 - 7*b4 + b1) * q^97 + (13*b5 + 4*b4 + 6*b1) * q^98 + (3*b3 + b2 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10})$$ 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9 $$6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} - 22 q^{14} + 14 q^{16} - 20 q^{21} + 2 q^{24} - 22 q^{26} + 12 q^{29} - 30 q^{31} - 10 q^{34} - 14 q^{36} - 2 q^{39} - 12 q^{41} + 20 q^{44} - 4 q^{46} + 2 q^{49} - 40 q^{51} - 4 q^{54} - 46 q^{56} - 20 q^{59} - 2 q^{61} - 12 q^{64} + 6 q^{66} - 18 q^{69} + 2 q^{71} - 22 q^{74} - 24 q^{79} + 14 q^{81} - 48 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} - 30 q^{94} + 26 q^{96} - 30 q^{99}+O(q^{100})$$ 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9 + 2 * q^11 - 22 * q^14 + 14 * q^16 - 20 * q^21 + 2 * q^24 - 22 * q^26 + 12 * q^29 - 30 * q^31 - 10 * q^34 - 14 * q^36 - 2 * q^39 - 12 * q^41 + 20 * q^44 - 4 * q^46 + 2 * q^49 - 40 * q^51 - 4 * q^54 - 46 * q^56 - 20 * q^59 - 2 * q^61 - 12 * q^64 + 6 * q^66 - 18 * q^69 + 2 * q^71 - 22 * q^74 - 24 * q^79 + 14 * q^81 - 48 * q^84 + 16 * q^86 - 36 * q^89 + 24 * q^91 - 30 * q^94 + 26 * q^96 - 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 6x^{4} + 7x^{2} - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 3$$ v^4 - 5*v^2 + 3 $$\beta_{4}$$ $$=$$ $$\nu^{5} - 5\nu^{3} + 3\nu$$ v^5 - 5*v^3 + 3*v $$\beta_{5}$$ $$=$$ $$\nu^{5} - 6\nu^{3} + 6\nu$$ v^5 - 6*v^3 + 6*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + 3\beta_1$$ -b5 + b4 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5\beta_{2} + 7$$ b3 + 5*b2 + 7 $$\nu^{5}$$ $$=$$ $$-5\beta_{5} + 6\beta_{4} + 12\beta_1$$ -5*b5 + 6*b4 + 12*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
 −0.407132 −1.15904 −2.11917 2.11917 1.15904 0.407132
−2.45620 −1.56104 4.03293 0 3.83424 4.50527 −4.99330 −0.563139 0
1.2 −0.862781 −3.07914 −1.25561 0 2.65662 0.566520 2.80888 6.48108 0
1.3 −0.471884 1.04022 −1.77733 0 −0.490864 −1.17540 1.78246 −1.91794 0
1.4 0.471884 −1.04022 −1.77733 0 −0.490864 1.17540 −1.78246 −1.91794 0
1.5 0.862781 3.07914 −1.25561 0 2.65662 −0.566520 −2.80888 6.48108 0
1.6 2.45620 1.56104 4.03293 0 3.83424 −4.50527 4.99330 −0.563139 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
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 9025.2.a.bt 6
5.b even 2 1 inner 9025.2.a.bt 6
5.c odd 4 2 1805.2.b.g 6
19.b odd 2 1 9025.2.a.bu 6
19.d odd 6 2 475.2.e.g 12
95.d odd 2 1 9025.2.a.bu 6
95.g even 4 2 1805.2.b.f 6
95.h odd 6 2 475.2.e.g 12
95.l even 12 4 95.2.i.b 12
285.w odd 12 4 855.2.be.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.b 12 95.l even 12 4
475.2.e.g 12 19.d odd 6 2
475.2.e.g 12 95.h odd 6 2
855.2.be.d 12 285.w odd 12 4
1805.2.b.f 6 95.g even 4 2
1805.2.b.g 6 5.c odd 4 2
9025.2.a.bt 6 1.a even 1 1 trivial
9025.2.a.bt 6 5.b even 2 1 inner
9025.2.a.bu 6 19.b odd 2 1
9025.2.a.bu 6 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}^{6} - 7T_{2}^{4} + 6T_{2}^{2} - 1$$ T2^6 - 7*T2^4 + 6*T2^2 - 1 $$T_{3}^{6} - 13T_{3}^{4} + 36T_{3}^{2} - 25$$ T3^6 - 13*T3^4 + 36*T3^2 - 25 $$T_{7}^{6} - 22T_{7}^{4} + 35T_{7}^{2} - 9$$ T7^6 - 22*T7^4 + 35*T7^2 - 9 $$T_{11}^{3} - T_{11}^{2} - 4T_{11} + 3$$ T11^3 - T11^2 - 4*T11 + 3 $$T_{29}^{3} - 6T_{29}^{2} - 27T_{29} + 27$$ T29^3 - 6*T29^2 - 27*T29 + 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 7 T^{4} + 6 T^{2} - 1$$
$3$ $$T^{6} - 13 T^{4} + 36 T^{2} - 25$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 22 T^{4} + 35 T^{2} - 9$$
$11$ $$(T^{3} - T^{2} - 4 T + 3)^{2}$$
$13$ $$T^{6} - 31 T^{4} + 239 T^{2} - 9$$
$17$ $$T^{6} - 35 T^{4} + 198 T^{2} + \cdots - 81$$
$19$ $$T^{6}$$
$23$ $$T^{6} - 12 T^{4} + 7 T^{2} - 1$$
$29$ $$(T^{3} - 6 T^{2} - 27 T + 27)^{2}$$
$31$ $$(T^{3} + 15 T^{2} + 70 T + 97)^{2}$$
$37$ $$T^{6} - 98 T^{4} + 887 T^{2} + \cdots - 729$$
$41$ $$(T^{3} + 6 T^{2} - 41 T - 3)^{2}$$
$43$ $$T^{6} - 127 T^{4} + 2516 T^{2} + \cdots - 441$$
$47$ $$T^{6} - 214 T^{4} + 13407 T^{2} + \cdots - 218089$$
$53$ $$T^{6} - 99 T^{4} + 2302 T^{2} + \cdots - 11449$$
$59$ $$(T^{3} + 10 T^{2} - 55 T - 291)^{2}$$
$61$ $$(T^{3} + T^{2} - 41 T - 113)^{2}$$
$67$ $$T^{6} - 76 T^{4} + 1856 T^{2} + \cdots - 14400$$
$71$ $$(T^{3} - T^{2} - 124 T - 477)^{2}$$
$73$ $$T^{6} - 236 T^{4} + 13331 T^{2} + \cdots - 99225$$
$79$ $$(T^{3} + 12 T^{2} - 32 T - 448)^{2}$$
$83$ $$T^{6} - 459 T^{4} + 56302 T^{2} + \cdots - 966289$$
$89$ $$(T^{3} + 18 T^{2} + 28 T - 72)^{2}$$
$97$ $$T^{6} - 529 T^{4} + 74192 T^{2} + \cdots - 1238769$$