# Properties

 Label 4025.2.a.j Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 161) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{6} + q^{7} + (\beta_{2} + 3 \beta_1 + 2) q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + (b2 + b1) * q^2 + (b1 - 1) * q^3 + (2*b1 + 1) * q^4 + (b1 + 1) * q^6 + q^7 + (b2 + 3*b1 + 2) * q^8 + (b2 - b1) * q^9 $$q + (\beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{6} + q^{7} + (\beta_{2} + 3 \beta_1 + 2) q^{8} + (\beta_{2} - \beta_1) q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} + (2 \beta_{2} + \beta_1 + 3) q^{12} - 2 \beta_1 q^{13} + (\beta_{2} + \beta_1) q^{14} + (4 \beta_{2} + 4 \beta_1 + 3) q^{16} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{18} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_1 - 1) q^{21} + (\beta_{2} + 3 \beta_1 + 3) q^{22} - q^{23} + (2 \beta_{2} + 3 \beta_1 + 3) q^{24} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{26} + ( - 2 \beta_{2} - 2 \beta_1) q^{27} + (2 \beta_1 + 1) q^{28} + ( - \beta_{2} + \beta_1 - 1) q^{29} + (4 \beta_{2} + \beta_1 + 5) q^{31} + (\beta_{2} + 5 \beta_1 + 8) q^{32} + 2 \beta_1 q^{33} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{34} + ( - \beta_{2} - \beta_1 - 6) q^{36} + (2 \beta_{2} + 2) q^{37} + (6 \beta_{2} + 6 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 4) q^{39} + (4 \beta_1 - 6) q^{41} + (\beta_1 + 1) q^{42} + (4 \beta_{2} + 2 \beta_1 + 2) q^{43} + (3 \beta_{2} + 7 \beta_1 + 3) q^{44} + ( - \beta_{2} - \beta_1) q^{46} + (2 \beta_{2} - \beta_1 - 5) q^{47} + (7 \beta_1 + 1) q^{48} + q^{49} + ( - \beta_{2} - \beta_1 - 1) q^{51} + ( - 4 \beta_{2} - 6 \beta_1 - 8) q^{52} + ( - 4 \beta_{2} - 2) q^{53} + ( - 4 \beta_1 - 6) q^{54} + (\beta_{2} + 3 \beta_1 + 2) q^{56} + (4 \beta_{2} + 4) q^{57} + (\beta_{2} + \beta_1 - 1) q^{58} + ( - 6 \beta_{2} - \beta_1 - 3) q^{59} + ( - 2 \beta_{2} - 5 \beta_1 + 5) q^{61} + (2 \beta_{2} + 7 \beta_1 + 9) q^{62} + (\beta_{2} - \beta_1) q^{63} + (4 \beta_{2} + 10 \beta_1 + 1) q^{64} + (2 \beta_{2} + 4 \beta_1 + 2) q^{66} + ( - 5 \beta_{2} - \beta_1 - 5) q^{67} + ( - 2 \beta_{2} - 5 \beta_1 - 5) q^{68} + ( - \beta_1 + 1) q^{69} + (4 \beta_{2} + 4 \beta_1) q^{71} + ( - 2 \beta_{2} - 4 \beta_1 - 5) q^{72} + 6 \beta_1 q^{73} + (2 \beta_1 + 4) q^{74} + (2 \beta_{2} + 6 \beta_1 + 14) q^{76} + (\beta_{2} + \beta_1 + 1) q^{77} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{78} + ( - \beta_{2} - 5 \beta_1 + 7) q^{79} + ( - 3 \beta_{2} + \beta_1 - 2) q^{81} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + ( - 8 \beta_{2} - 2 \beta_1 - 6) q^{83} + (2 \beta_{2} + \beta_1 + 3) q^{84} + (6 \beta_1 + 10) q^{86} + (2 \beta_{2} - 2 \beta_1 + 4) q^{87} + (5 \beta_{2} + 11 \beta_1 + 7) q^{88} + (2 \beta_{2} - 5 \beta_1 + 3) q^{89} - 2 \beta_1 q^{91} + ( - 2 \beta_1 - 1) q^{92} + ( - 3 \beta_{2} + 9 \beta_1 - 7) q^{93} + ( - 8 \beta_{2} - 7 \beta_1 + 3) q^{94} + (4 \beta_{2} + 9 \beta_1 + 1) q^{96} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{97} + (\beta_{2} + \beta_1) q^{98} + ( - \beta_{2} - 3 \beta_1 + 1) q^{99}+O(q^{100})$$ q + (b2 + b1) * q^2 + (b1 - 1) * q^3 + (2*b1 + 1) * q^4 + (b1 + 1) * q^6 + q^7 + (b2 + 3*b1 + 2) * q^8 + (b2 - b1) * q^9 + (b2 + b1 + 1) * q^11 + (2*b2 + b1 + 3) * q^12 - 2*b1 * q^13 + (b2 + b1) * q^14 + (4*b2 + 4*b1 + 3) * q^16 + (-b1 - 1) * q^17 + (-2*b2 - 2*b1 + 1) * q^18 + (-2*b2 + 2*b1 + 2) * q^19 + (b1 - 1) * q^21 + (b2 + 3*b1 + 3) * q^22 - q^23 + (2*b2 + 3*b1 + 3) * q^24 + (-2*b2 - 4*b1 - 2) * q^26 + (-2*b2 - 2*b1) * q^27 + (2*b1 + 1) * q^28 + (-b2 + b1 - 1) * q^29 + (4*b2 + b1 + 5) * q^31 + (b2 + 5*b1 + 8) * q^32 + 2*b1 * q^33 + (-2*b2 - 3*b1 - 1) * q^34 + (-b2 - b1 - 6) * q^36 + (2*b2 + 2) * q^37 + (6*b2 + 6*b1 - 2) * q^38 + (-2*b2 - 4) * q^39 + (4*b1 - 6) * q^41 + (b1 + 1) * q^42 + (4*b2 + 2*b1 + 2) * q^43 + (3*b2 + 7*b1 + 3) * q^44 + (-b2 - b1) * q^46 + (2*b2 - b1 - 5) * q^47 + (7*b1 + 1) * q^48 + q^49 + (-b2 - b1 - 1) * q^51 + (-4*b2 - 6*b1 - 8) * q^52 + (-4*b2 - 2) * q^53 + (-4*b1 - 6) * q^54 + (b2 + 3*b1 + 2) * q^56 + (4*b2 + 4) * q^57 + (b2 + b1 - 1) * q^58 + (-6*b2 - b1 - 3) * q^59 + (-2*b2 - 5*b1 + 5) * q^61 + (2*b2 + 7*b1 + 9) * q^62 + (b2 - b1) * q^63 + (4*b2 + 10*b1 + 1) * q^64 + (2*b2 + 4*b1 + 2) * q^66 + (-5*b2 - b1 - 5) * q^67 + (-2*b2 - 5*b1 - 5) * q^68 + (-b1 + 1) * q^69 + (4*b2 + 4*b1) * q^71 + (-2*b2 - 4*b1 - 5) * q^72 + 6*b1 * q^73 + (2*b1 + 4) * q^74 + (2*b2 + 6*b1 + 14) * q^76 + (b2 + b1 + 1) * q^77 + (-2*b2 - 4*b1 - 4) * q^78 + (-b2 - 5*b1 + 7) * q^79 + (-3*b2 + b1 - 2) * q^81 + (-2*b2 + 2*b1 + 4) * q^82 + (-8*b2 - 2*b1 - 6) * q^83 + (2*b2 + b1 + 3) * q^84 + (6*b1 + 10) * q^86 + (2*b2 - 2*b1 + 4) * q^87 + (5*b2 + 11*b1 + 7) * q^88 + (2*b2 - 5*b1 + 3) * q^89 - 2*b1 * q^91 + (-2*b1 - 1) * q^92 + (-3*b2 + 9*b1 - 7) * q^93 + (-8*b2 - 7*b1 + 3) * q^94 + (4*b2 + 9*b1 + 1) * q^96 + (-2*b2 - 3*b1 - 1) * q^97 + (b2 + b1) * q^98 + (-b2 - 3*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9}+O(q^{10})$$ 3 * q + q^2 - 2 * q^3 + 5 * q^4 + 4 * q^6 + 3 * q^7 + 9 * q^8 - q^9 $$3 q + q^{2} - 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} + 4 q^{11} + 10 q^{12} - 2 q^{13} + q^{14} + 13 q^{16} - 4 q^{17} + q^{18} + 8 q^{19} - 2 q^{21} + 12 q^{22} - 3 q^{23} + 12 q^{24} - 10 q^{26} - 2 q^{27} + 5 q^{28} - 2 q^{29} + 16 q^{31} + 29 q^{32} + 2 q^{33} - 6 q^{34} - 19 q^{36} + 6 q^{37} - 12 q^{39} - 14 q^{41} + 4 q^{42} + 8 q^{43} + 16 q^{44} - q^{46} - 16 q^{47} + 10 q^{48} + 3 q^{49} - 4 q^{51} - 30 q^{52} - 6 q^{53} - 22 q^{54} + 9 q^{56} + 12 q^{57} - 2 q^{58} - 10 q^{59} + 10 q^{61} + 34 q^{62} - q^{63} + 13 q^{64} + 10 q^{66} - 16 q^{67} - 20 q^{68} + 2 q^{69} + 4 q^{71} - 19 q^{72} + 6 q^{73} + 14 q^{74} + 48 q^{76} + 4 q^{77} - 16 q^{78} + 16 q^{79} - 5 q^{81} + 14 q^{82} - 20 q^{83} + 10 q^{84} + 36 q^{86} + 10 q^{87} + 32 q^{88} + 4 q^{89} - 2 q^{91} - 5 q^{92} - 12 q^{93} + 2 q^{94} + 12 q^{96} - 6 q^{97} + q^{98}+O(q^{100})$$ 3 * q + q^2 - 2 * q^3 + 5 * q^4 + 4 * q^6 + 3 * q^7 + 9 * q^8 - q^9 + 4 * q^11 + 10 * q^12 - 2 * q^13 + q^14 + 13 * q^16 - 4 * q^17 + q^18 + 8 * q^19 - 2 * q^21 + 12 * q^22 - 3 * q^23 + 12 * q^24 - 10 * q^26 - 2 * q^27 + 5 * q^28 - 2 * q^29 + 16 * q^31 + 29 * q^32 + 2 * q^33 - 6 * q^34 - 19 * q^36 + 6 * q^37 - 12 * q^39 - 14 * q^41 + 4 * q^42 + 8 * q^43 + 16 * q^44 - q^46 - 16 * q^47 + 10 * q^48 + 3 * q^49 - 4 * q^51 - 30 * q^52 - 6 * q^53 - 22 * q^54 + 9 * q^56 + 12 * q^57 - 2 * q^58 - 10 * q^59 + 10 * q^61 + 34 * q^62 - q^63 + 13 * q^64 + 10 * q^66 - 16 * q^67 - 20 * q^68 + 2 * q^69 + 4 * q^71 - 19 * q^72 + 6 * q^73 + 14 * q^74 + 48 * q^76 + 4 * q^77 - 16 * q^78 + 16 * q^79 - 5 * q^81 + 14 * q^82 - 20 * q^83 + 10 * q^84 + 36 * q^86 + 10 * q^87 + 32 * q^88 + 4 * q^89 - 2 * q^91 - 5 * q^92 - 12 * q^93 + 2 * q^94 + 12 * q^96 - 6 * q^97 + q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + 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
 0.311108 −1.48119 2.17009
−1.90321 −0.688892 1.62222 0 1.31111 1.00000 0.719004 −2.52543 0
1.2 0.193937 −2.48119 −1.96239 0 −0.481194 1.00000 −0.768452 3.15633 0
1.3 2.70928 1.17009 5.34017 0 3.17009 1.00000 9.04945 −1.63090 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
$$5$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## 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 4025.2.a.j 3
5.b even 2 1 161.2.a.c 3
15.d odd 2 1 1449.2.a.m 3
20.d odd 2 1 2576.2.a.v 3
35.c odd 2 1 1127.2.a.f 3
115.c odd 2 1 3703.2.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.c 3 5.b even 2 1
1127.2.a.f 3 35.c odd 2 1
1449.2.a.m 3 15.d odd 2 1
2576.2.a.v 3 20.d odd 2 1
3703.2.a.c 3 115.c odd 2 1
4025.2.a.j 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1$$ T2^3 - T2^2 - 5*T2 + 1 $$T_{3}^{3} + 2T_{3}^{2} - 2T_{3} - 2$$ T3^3 + 2*T3^2 - 2*T3 - 2 $$T_{11}^{3} - 4T_{11}^{2} + 4$$ T11^3 - 4*T11^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 5T + 1$$
$3$ $$T^{3} + 2 T^{2} - 2 T - 2$$
$5$ $$T^{3}$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3} - 4T^{2} + 4$$
$13$ $$T^{3} + 2 T^{2} - 12 T - 8$$
$17$ $$T^{3} + 4 T^{2} + 2 T - 2$$
$19$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 2 T^{2} - 8 T + 4$$
$31$ $$T^{3} - 16 T^{2} + 26 T + 338$$
$37$ $$T^{3} - 6 T^{2} - 4 T + 40$$
$41$ $$T^{3} + 14 T^{2} + 12 T - 152$$
$43$ $$T^{3} - 8 T^{2} - 40 T + 304$$
$47$ $$T^{3} + 16 T^{2} + 62 T + 10$$
$53$ $$T^{3} + 6 T^{2} - 52 T - 248$$
$59$ $$T^{3} + 10 T^{2} - 102 T - 970$$
$61$ $$T^{3} - 10 T^{2} - 46 T + 494$$
$67$ $$T^{3} + 16 T^{2} - 8 T - 676$$
$71$ $$T^{3} - 4 T^{2} - 80 T + 64$$
$73$ $$T^{3} - 6 T^{2} - 108 T + 216$$
$79$ $$T^{3} - 16 T^{2} + 8 T + 428$$
$83$ $$T^{3} + 20 T^{2} - 104 T - 2672$$
$89$ $$T^{3} - 4 T^{2} - 114 T - 278$$
$97$ $$T^{3} + 6 T^{2} - 22 T + 2$$