# Properties

 Label 9025.2.a.x Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + 2) q^{6} + \beta_1 q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + ( - 3 \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - 1) * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 + 2) * q^6 + b1 * q^7 + (-3*b2 + 2*b1 - 2) * q^8 + (-3*b2 + b1) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + 2) q^{6} + \beta_1 q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + ( - 3 \beta_{2} + \beta_1) q^{9} + (\beta_{2} + \beta_1) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_{2} - 2 \beta_1) q^{13} + (\beta_{2} - \beta_1 + 2) q^{14} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + ( - \beta_{2} + \beta_1 - 2) q^{17} + (4 \beta_{2} - 4 \beta_1 - 1) q^{18} + q^{21} + 3 q^{22} + ( - 2 \beta_1 + 2) q^{23} + (4 \beta_{2} - 3 \beta_1 - 2) q^{24} + ( - \beta_{2} + \beta_1 - 5) q^{26} + (3 \beta_{2} - 3 \beta_1 - 2) q^{27} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{28} + ( - \beta_{2} + 5) q^{29} + ( - 2 \beta_{2} - \beta_1 + 3) q^{31} + 3 \beta_1 q^{32} + ( - 2 \beta_{2} + \beta_1 + 3) q^{33} + (2 \beta_{2} - 4 \beta_1 + 3) q^{34} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{36} + ( - 3 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{2} - \beta_1 - 4) q^{39} + (4 \beta_{2} - 3 \beta_1 + 4) q^{41} + (\beta_1 - 1) q^{42} + (3 \beta_{2} - 5 \beta_1) q^{43} + ( - 2 \beta_{2} + \beta_1 - 3) q^{44} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{46} + ( - 3 \beta_{2} + \beta_1 + 2) q^{47} + ( - 5 \beta_{2} + 3 \beta_1 + 2) q^{48} + (\beta_{2} - 5) q^{49} + ( - \beta_1 + 1) q^{51} + (4 \beta_{2} - 3 \beta_1 + 6) q^{52} + (\beta_{2} - 3 \beta_1 - 2) q^{53} + ( - 6 \beta_{2} + 4 \beta_1 - 1) q^{54} + (2 \beta_{2} - 5 \beta_1 + 1) q^{56} + (\beta_{2} + 4 \beta_1 - 6) q^{58} + (2 \beta_1 + 7) q^{59} + ( - 4 \beta_{2} + 4 \beta_1 + 3) q^{61} + (\beta_{2} + 2 \beta_1 - 7) q^{62} + (\beta_{2} - 3 \beta_1 - 1) q^{63} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + (3 \beta_{2} - 3) q^{66} + ( - 2 \beta_{2} + 6 \beta_1 + 6) q^{67} + ( - 4 \beta_{2} + 7 \beta_1 - 5) q^{68} + (2 \beta_{2} - 4) q^{69} + (2 \beta_1 + 10) q^{71} + ( - \beta_{2} - 2 \beta_1 + 13) q^{72} + 4 \beta_1 q^{73} + (5 \beta_{2} - 5 \beta_1 + 1) q^{74} + (\beta_{2} + \beta_1 + 3) q^{77} + ( - 3 \beta_{2} - \beta_1 + 4) q^{78} + ( - 6 \beta_{2} + 7 \beta_1 + 3) q^{79} + (\beta_{2} + 5) q^{81} + ( - 7 \beta_{2} + 11 \beta_1 - 6) q^{82} + ( - 9 \beta_{2} + 6 \beta_1) q^{83} + (\beta_{2} - 2 \beta_1 + 1) q^{84} + ( - 8 \beta_{2} + 8 \beta_1 - 7) q^{86} + (7 \beta_{2} - \beta_1 - 7) q^{87} + (3 \beta_{2} - 6 \beta_1 - 3) q^{88} + ( - 3 \beta_{2} + \beta_1 + 5) q^{89} + ( - 2 \beta_{2} - \beta_1 - 5) q^{91} + (6 \beta_{2} - 8 \beta_1 + 8) q^{92} + (7 \beta_{2} - 2 \beta_1 - 8) q^{93} + (4 \beta_{2} - 2 \beta_1 - 3) q^{94} + 3 q^{96} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{97} + ( - \beta_{2} - 4 \beta_1 + 6) q^{98} + (4 \beta_{2} - 5 \beta_1 - 6) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - 1) * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 + 2) * q^6 + b1 * q^7 + (-3*b2 + 2*b1 - 2) * q^8 + (-3*b2 + b1) * q^9 + (b2 + b1) * q^11 + (-b2 + b1 - 1) * q^12 + (-b2 - 2*b1) * q^13 + (b2 - b1 + 2) * q^14 + (3*b2 - 3*b1 + 1) * q^16 + (-b2 + b1 - 2) * q^17 + (4*b2 - 4*b1 - 1) * q^18 + q^21 + 3 * q^22 + (-2*b1 + 2) * q^23 + (4*b2 - 3*b1 - 2) * q^24 + (-b2 + b1 - 5) * q^26 + (3*b2 - 3*b1 - 2) * q^27 + (-2*b2 + 2*b1 - 3) * q^28 + (-b2 + 5) * q^29 + (-2*b2 - b1 + 3) * q^31 + 3*b1 * q^32 + (-2*b2 + b1 + 3) * q^33 + (2*b2 - 4*b1 + 3) * q^34 + (-2*b2 + 5*b1 - 3) * q^36 + (-3*b2 + 2*b1) * q^37 + (2*b2 - b1 - 4) * q^39 + (4*b2 - 3*b1 + 4) * q^41 + (b1 - 1) * q^42 + (3*b2 - 5*b1) * q^43 + (-2*b2 + b1 - 3) * q^44 + (-2*b2 + 4*b1 - 6) * q^46 + (-3*b2 + b1 + 2) * q^47 + (-5*b2 + 3*b1 + 2) * q^48 + (b2 - 5) * q^49 + (-b1 + 1) * q^51 + (4*b2 - 3*b1 + 6) * q^52 + (b2 - 3*b1 - 2) * q^53 + (-6*b2 + 4*b1 - 1) * q^54 + (2*b2 - 5*b1 + 1) * q^56 + (b2 + 4*b1 - 6) * q^58 + (2*b1 + 7) * q^59 + (-4*b2 + 4*b1 + 3) * q^61 + (b2 + 2*b1 - 7) * q^62 + (b2 - 3*b1 - 1) * q^63 + (-3*b2 + 3*b1 + 4) * q^64 + (3*b2 - 3) * q^66 + (-2*b2 + 6*b1 + 6) * q^67 + (-4*b2 + 7*b1 - 5) * q^68 + (2*b2 - 4) * q^69 + (2*b1 + 10) * q^71 + (-b2 - 2*b1 + 13) * q^72 + 4*b1 * q^73 + (5*b2 - 5*b1 + 1) * q^74 + (b2 + b1 + 3) * q^77 + (-3*b2 - b1 + 4) * q^78 + (-6*b2 + 7*b1 + 3) * q^79 + (b2 + 5) * q^81 + (-7*b2 + 11*b1 - 6) * q^82 + (-9*b2 + 6*b1) * q^83 + (b2 - 2*b1 + 1) * q^84 + (-8*b2 + 8*b1 - 7) * q^86 + (7*b2 - b1 - 7) * q^87 + (3*b2 - 6*b1 - 3) * q^88 + (-3*b2 + b1 + 5) * q^89 + (-2*b2 - b1 - 5) * q^91 + (6*b2 - 8*b1 + 8) * q^92 + (7*b2 - 2*b1 - 8) * q^93 + (4*b2 - 2*b1 - 3) * q^94 + 3 * q^96 + (-2*b2 - 2*b1 + 5) * q^97 + (-b2 - 4*b1 + 6) * q^98 + (4*b2 - 5*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 6 q^{6} - 6 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 6 * q^6 - 6 * q^8 $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 6 q^{6} - 6 q^{8} - 3 q^{12} + 6 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{18} + 3 q^{21} + 9 q^{22} + 6 q^{23} - 6 q^{24} - 15 q^{26} - 6 q^{27} - 9 q^{28} + 15 q^{29} + 9 q^{31} + 9 q^{33} + 9 q^{34} - 9 q^{36} - 12 q^{39} + 12 q^{41} - 3 q^{42} - 9 q^{44} - 18 q^{46} + 6 q^{47} + 6 q^{48} - 15 q^{49} + 3 q^{51} + 18 q^{52} - 6 q^{53} - 3 q^{54} + 3 q^{56} - 18 q^{58} + 21 q^{59} + 9 q^{61} - 21 q^{62} - 3 q^{63} + 12 q^{64} - 9 q^{66} + 18 q^{67} - 15 q^{68} - 12 q^{69} + 30 q^{71} + 39 q^{72} + 3 q^{74} + 9 q^{77} + 12 q^{78} + 9 q^{79} + 15 q^{81} - 18 q^{82} + 3 q^{84} - 21 q^{86} - 21 q^{87} - 9 q^{88} + 15 q^{89} - 15 q^{91} + 24 q^{92} - 24 q^{93} - 9 q^{94} + 9 q^{96} + 15 q^{97} + 18 q^{98} - 18 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 6 * q^6 - 6 * q^8 - 3 * q^12 + 6 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^18 + 3 * q^21 + 9 * q^22 + 6 * q^23 - 6 * q^24 - 15 * q^26 - 6 * q^27 - 9 * q^28 + 15 * q^29 + 9 * q^31 + 9 * q^33 + 9 * q^34 - 9 * q^36 - 12 * q^39 + 12 * q^41 - 3 * q^42 - 9 * q^44 - 18 * q^46 + 6 * q^47 + 6 * q^48 - 15 * q^49 + 3 * q^51 + 18 * q^52 - 6 * q^53 - 3 * q^54 + 3 * q^56 - 18 * q^58 + 21 * q^59 + 9 * q^61 - 21 * q^62 - 3 * q^63 + 12 * q^64 - 9 * q^66 + 18 * q^67 - 15 * q^68 - 12 * q^69 + 30 * q^71 + 39 * q^72 + 3 * q^74 + 9 * q^77 + 12 * q^78 + 9 * q^79 + 15 * q^81 - 18 * q^82 + 3 * q^84 - 21 * q^86 - 21 * q^87 - 9 * q^88 + 15 * q^89 - 15 * q^91 + 24 * q^92 - 24 * q^93 - 9 * q^94 + 9 * q^96 + 15 * q^97 + 18 * q^98 - 18 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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
 −1.53209 −0.347296 1.87939
−2.53209 −0.652704 4.41147 0 1.65270 −1.53209 −6.10607 −2.57398 0
1.2 −1.34730 −2.87939 −0.184793 0 3.87939 −0.347296 2.94356 5.29086 0
1.3 0.879385 0.532089 −1.22668 0 0.467911 1.87939 −2.83750 −2.71688 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 9025.2.a.x 3
5.b even 2 1 361.2.a.h 3
15.d odd 2 1 3249.2.a.s 3
19.b odd 2 1 9025.2.a.bd 3
19.f odd 18 2 475.2.l.a 6
20.d odd 2 1 5776.2.a.bi 3
95.d odd 2 1 361.2.a.g 3
95.h odd 6 2 361.2.c.i 6
95.i even 6 2 361.2.c.h 6
95.o odd 18 2 19.2.e.a 6
95.o odd 18 2 361.2.e.f 6
95.o odd 18 2 361.2.e.g 6
95.p even 18 2 361.2.e.a 6
95.p even 18 2 361.2.e.b 6
95.p even 18 2 361.2.e.h 6
95.r even 36 4 475.2.u.a 12
285.b even 2 1 3249.2.a.z 3
285.bf even 18 2 171.2.u.c 6
380.d even 2 1 5776.2.a.br 3
380.bb even 18 2 304.2.u.b 6
665.cl odd 18 2 931.2.v.b 6
665.co even 18 2 931.2.v.a 6
665.cp even 18 2 931.2.w.a 6
665.cr even 18 2 931.2.x.b 6
665.ct odd 18 2 931.2.x.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 95.o odd 18 2
171.2.u.c 6 285.bf even 18 2
304.2.u.b 6 380.bb even 18 2
361.2.a.g 3 95.d odd 2 1
361.2.a.h 3 5.b even 2 1
361.2.c.h 6 95.i even 6 2
361.2.c.i 6 95.h odd 6 2
361.2.e.a 6 95.p even 18 2
361.2.e.b 6 95.p even 18 2
361.2.e.f 6 95.o odd 18 2
361.2.e.g 6 95.o odd 18 2
361.2.e.h 6 95.p even 18 2
475.2.l.a 6 19.f odd 18 2
475.2.u.a 12 95.r even 36 4
931.2.v.a 6 665.co even 18 2
931.2.v.b 6 665.cl odd 18 2
931.2.w.a 6 665.cp even 18 2
931.2.x.a 6 665.ct odd 18 2
931.2.x.b 6 665.cr even 18 2
3249.2.a.s 3 15.d odd 2 1
3249.2.a.z 3 285.b even 2 1
5776.2.a.bi 3 20.d odd 2 1
5776.2.a.br 3 380.d even 2 1
9025.2.a.x 3 1.a even 1 1 trivial
9025.2.a.bd 3 19.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} + 3T_{2}^{2} - 3$$ T2^3 + 3*T2^2 - 3 $$T_{3}^{3} + 3T_{3}^{2} - 1$$ T3^3 + 3*T3^2 - 1 $$T_{7}^{3} - 3T_{7} - 1$$ T7^3 - 3*T7 - 1 $$T_{11}^{3} - 9T_{11} - 9$$ T11^3 - 9*T11 - 9 $$T_{29}^{3} - 15T_{29}^{2} + 72T_{29} - 111$$ T29^3 - 15*T29^2 + 72*T29 - 111

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3T^{2} - 3$$
$3$ $$T^{3} + 3T^{2} - 1$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 3T - 1$$
$11$ $$T^{3} - 9T - 9$$
$13$ $$T^{3} - 21T + 37$$
$17$ $$T^{3} + 6 T^{2} + 9 T + 3$$
$19$ $$T^{3}$$
$23$ $$T^{3} - 6T^{2} + 24$$
$29$ $$T^{3} - 15 T^{2} + 72 T - 111$$
$31$ $$T^{3} - 9 T^{2} + 6 T + 53$$
$37$ $$T^{3} - 21T - 17$$
$41$ $$T^{3} - 12 T^{2} + 9 T + 111$$
$43$ $$T^{3} - 57T - 163$$
$47$ $$T^{3} - 6 T^{2} - 9 T - 3$$
$53$ $$T^{3} + 6 T^{2} - 9 T - 51$$
$59$ $$T^{3} - 21 T^{2} + 135 T - 267$$
$61$ $$T^{3} - 9 T^{2} - 21 T + 181$$
$67$ $$T^{3} - 18 T^{2} + 24 T + 424$$
$71$ $$T^{3} - 30 T^{2} + 288 T - 888$$
$73$ $$T^{3} - 48T - 64$$
$79$ $$T^{3} - 9 T^{2} - 102 T + 809$$
$83$ $$T^{3} - 189T - 459$$
$89$ $$T^{3} - 15 T^{2} + 54 T - 57$$
$97$ $$T^{3} - 15 T^{2} + 39 T + 127$$
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