# Properties

 Label 4025.2.a.o Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2777.1 Defining polynomial: $$x^{4} - x^{3} - 4x^{2} + x + 2$$ x^4 - x^3 - 4*x^2 + x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 805) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 1) q^{2} + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} + 1) q^{6} - q^{7} + ( - 2 \beta_{2} + \beta_1 + 2) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (-b3 + 1) * q^2 + (b1 + 1) * q^3 + (-b3 - b2 + 1) * q^4 + (-b3 - b2 + 1) * q^6 - q^7 + (-2*b2 + b1 + 2) * q^8 + (b2 + 3*b1) * q^9 $$q + ( - \beta_{3} + 1) q^{2} + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} + 1) q^{6} - q^{7} + ( - 2 \beta_{2} + \beta_1 + 2) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{3} + 2 \beta_1 - 2) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{12} + (2 \beta_{3} - \beta_{2} + 1) q^{13} + (\beta_{3} - 1) q^{14} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{16} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{17} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{18} + (3 \beta_{3} - 3 \beta_{2} - 1) q^{19} + ( - \beta_1 - 1) q^{21} + (2 \beta_{3} - 3 \beta_{2}) q^{22} + q^{23} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 6) q^{24} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{26} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{27} + (\beta_{3} + \beta_{2} - 1) q^{28} + ( - \beta_{3} + \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{31} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{32} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{33} + ( - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{34} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 1) q^{36} + (2 \beta_{2} - \beta_1 + 2) q^{37} + ( - 2 \beta_{3} + 3 \beta_1 - 4) q^{38} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{39} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{41} + (\beta_{3} + \beta_{2} - 1) q^{42} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 1) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{44} + ( - \beta_{3} + 1) q^{46} + (\beta_{2} + \beta_1 + 5) q^{47} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 7) q^{48} + q^{49} + ( - 3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{51} + ( - \beta_{3} - \beta_1 - 1) q^{52} + ( - 4 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 4) q^{53} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 - 4) q^{54} + (2 \beta_{2} - \beta_1 - 2) q^{56} + ( - \beta_1 + 2) q^{57} + ( - 2 \beta_{2} + 2) q^{58} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 9) q^{59} + ( - \beta_{2} + \beta_1) q^{61} + (4 \beta_{2} - \beta_1 + 2) q^{62} + ( - \beta_{2} - 3 \beta_1) q^{63} + ( - \beta_{2} - 3 \beta_1 + 6) q^{64} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{66} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 9) q^{67} + ( - 7 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 13) q^{68} + (\beta_1 + 1) q^{69} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{71} + ( - 3 \beta_{3} + 5 \beta_{2} + 4 \beta_1 + 5) q^{72} + (4 \beta_{3} - 3 \beta_{2} - \beta_1 - 8) q^{73} + (3 \beta_{2} - 2 \beta_1) q^{74} + ( - 2 \beta_{3} + \beta_{2} + 2) q^{76} + (\beta_{3} - 2 \beta_1 + 2) q^{77} + ( - \beta_{3} - \beta_1 - 1) q^{78} + (5 \beta_{3} + \beta_{2} + 3 \beta_1 - 8) q^{79} + (5 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 6) q^{81} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{82} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{83} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{84} + (3 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{86} + ( - \beta_{3} + \beta_1 + 2) q^{87} + ( - 8 \beta_{3} + 5 \beta_{2} + \beta_1 + 6) q^{88} + (4 \beta_{3} + 2 \beta_{2} - \beta_1) q^{89} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{91} + ( - \beta_{3} - \beta_{2} + 1) q^{92} + ( - 4 \beta_{2} - 7 \beta_1 - 8) q^{93} + ( - 4 \beta_{3} - \beta_1 + 4) q^{94} + ( - 4 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 2) q^{96} + (4 \beta_{2} + 6 \beta_1 - 1) q^{97} + ( - \beta_{3} + 1) q^{98} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 + 9) q^{99}+O(q^{100})$$ q + (-b3 + 1) * q^2 + (b1 + 1) * q^3 + (-b3 - b2 + 1) * q^4 + (-b3 - b2 + 1) * q^6 - q^7 + (-2*b2 + b1 + 2) * q^8 + (b2 + 3*b1) * q^9 + (-b3 + 2*b1 - 2) * q^11 + (-2*b3 - 2*b2 - b1 + 2) * q^12 + (2*b3 - b2 + 1) * q^13 + (b3 - 1) * q^14 + (-2*b3 - b2 + 2*b1 + 2) * q^16 + (-b3 - 2*b2 + 2) * q^17 + (b3 - 2*b2 - b1 - 1) * q^18 + (3*b3 - 3*b2 - 1) * q^19 + (-b1 - 1) * q^21 + (2*b3 - 3*b2) * q^22 + q^23 + (-2*b3 - b2 + 2*b1 + 6) * q^24 + (-2*b3 + b2 + b1 - 2) * q^26 + (b3 + 4*b2 + 4*b1 + 2) * q^27 + (b3 + b2 - 1) * q^28 + (-b3 + b1) * q^29 + (-b3 + b2 - 4*b1 + 1) * q^31 + (-3*b3 - b2 - b1 + 3) * q^32 + (-b3 + b2 + b1 + 2) * q^33 + (-4*b3 - 3*b2 + 2*b1 + 6) * q^34 + (-b3 - 2*b2 - 4*b1 - 1) * q^36 + (2*b2 - b1 + 2) * q^37 + (-2*b3 + 3*b1 - 4) * q^38 + (b3 + b2 + 2*b1 + 2) * q^39 + (-3*b3 + 2*b2 + b1 - 1) * q^41 + (b3 + b2 - 1) * q^42 + (-b3 + 4*b2 + 2*b1 + 1) * q^43 + (-b3 - b2 - b1 + 3) * q^44 + (-b3 + 1) * q^46 + (b2 + b1 + 5) * q^47 + (-3*b3 - b2 + 3*b1 + 7) * q^48 + q^49 + (-3*b3 - 3*b2 - b1 + 4) * q^51 + (-b3 - b1 - 1) * q^52 + (-4*b3 + 5*b2 + 3*b1 + 4) * q^53 + (2*b3 + b2 - 4*b1 - 4) * q^54 + (2*b2 - b1 - 2) * q^56 + (-b1 + 2) * q^57 + (-2*b2 + 2) * q^58 + (-b3 + 3*b2 + b1 - 9) * q^59 + (-b2 + b1) * q^61 + (4*b2 - b1 + 2) * q^62 + (-b2 - 3*b1) * q^63 + (-b2 - 3*b1 + 6) * q^64 + (-b3 - b2 - b1 + 3) * q^66 + (-b3 - b2 - 3*b1 + 9) * q^67 + (-7*b3 - 5*b2 + 3*b1 + 13) * q^68 + (b1 + 1) * q^69 + (b3 - b2 + 2*b1 - 3) * q^71 + (-3*b3 + 5*b2 + 4*b1 + 5) * q^72 + (4*b3 - 3*b2 - b1 - 8) * q^73 + (3*b2 - 2*b1) * q^74 + (-2*b3 + b2 + 2) * q^76 + (b3 - 2*b1 + 2) * q^77 + (-b3 - b1 - 1) * q^78 + (5*b3 + b2 + 3*b1 - 8) * q^79 + (5*b3 + 6*b2 + 6*b1 + 6) * q^81 + (3*b3 - 2*b2 - 2*b1 + 3) * q^82 + (-3*b3 - b2 + b1 + 3) * q^83 + (2*b3 + 2*b2 + b1 - 2) * q^84 + (3*b3 + b2 - 4*b1 - 1) * q^86 + (-b3 + b1 + 2) * q^87 + (-8*b3 + 5*b2 + b1 + 6) * q^88 + (4*b3 + 2*b2 - b1) * q^89 + (-2*b3 + b2 - 1) * q^91 + (-b3 - b2 + 1) * q^92 + (-4*b2 - 7*b1 - 8) * q^93 + (-4*b3 - b1 + 4) * q^94 + (-4*b3 - 5*b2 - 3*b1 + 2) * q^96 + (4*b2 + 6*b1 - 1) * q^97 + (-b3 + 1) * q^98 + (3*b3 + b2 - 2*b1 + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 5 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 9 * q^8 + 3 * q^9 $$4 q + 2 q^{2} + 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 3 q^{9} - 8 q^{11} + 3 q^{12} + 8 q^{13} - 2 q^{14} + 6 q^{16} + 6 q^{17} - 3 q^{18} + 2 q^{19} - 5 q^{21} + 4 q^{22} + 4 q^{23} + 22 q^{24} - 11 q^{26} + 14 q^{27} - 2 q^{28} - q^{29} - 2 q^{31} + 5 q^{32} + 7 q^{33} + 18 q^{34} - 10 q^{36} + 7 q^{37} - 17 q^{38} + 12 q^{39} - 9 q^{41} - 2 q^{42} + 4 q^{43} + 9 q^{44} + 2 q^{46} + 21 q^{47} + 25 q^{48} + 4 q^{49} + 9 q^{51} - 7 q^{52} + 11 q^{53} - 16 q^{54} - 9 q^{56} + 7 q^{57} + 8 q^{58} - 37 q^{59} + q^{61} + 7 q^{62} - 3 q^{63} + 21 q^{64} + 9 q^{66} + 31 q^{67} + 41 q^{68} + 5 q^{69} - 8 q^{71} + 18 q^{72} - 25 q^{73} - 2 q^{74} + 4 q^{76} + 8 q^{77} - 7 q^{78} - 19 q^{79} + 40 q^{81} + 16 q^{82} + 7 q^{83} - 3 q^{84} - 2 q^{86} + 7 q^{87} + 9 q^{88} + 7 q^{89} - 8 q^{91} + 2 q^{92} - 39 q^{93} + 7 q^{94} - 3 q^{96} + 2 q^{97} + 2 q^{98} + 40 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 5 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 9 * q^8 + 3 * q^9 - 8 * q^11 + 3 * q^12 + 8 * q^13 - 2 * q^14 + 6 * q^16 + 6 * q^17 - 3 * q^18 + 2 * q^19 - 5 * q^21 + 4 * q^22 + 4 * q^23 + 22 * q^24 - 11 * q^26 + 14 * q^27 - 2 * q^28 - q^29 - 2 * q^31 + 5 * q^32 + 7 * q^33 + 18 * q^34 - 10 * q^36 + 7 * q^37 - 17 * q^38 + 12 * q^39 - 9 * q^41 - 2 * q^42 + 4 * q^43 + 9 * q^44 + 2 * q^46 + 21 * q^47 + 25 * q^48 + 4 * q^49 + 9 * q^51 - 7 * q^52 + 11 * q^53 - 16 * q^54 - 9 * q^56 + 7 * q^57 + 8 * q^58 - 37 * q^59 + q^61 + 7 * q^62 - 3 * q^63 + 21 * q^64 + 9 * q^66 + 31 * q^67 + 41 * q^68 + 5 * q^69 - 8 * q^71 + 18 * q^72 - 25 * q^73 - 2 * q^74 + 4 * q^76 + 8 * q^77 - 7 * q^78 - 19 * q^79 + 40 * q^81 + 16 * q^82 + 7 * q^83 - 3 * q^84 - 2 * q^86 + 7 * q^87 + 9 * q^88 + 7 * q^89 - 8 * q^91 + 2 * q^92 - 39 * q^93 + 7 * q^94 - 3 * q^96 + 2 * q^97 + 2 * q^98 + 40 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1

## 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
 −0.679643 2.36234 −1.50848 0.825785
−1.26308 0.320357 −0.404635 0 −0.404635 −1.00000 3.03724 −2.89737 0
1.2 −0.515722 3.36234 −1.73403 0 −1.73403 −1.00000 1.92572 8.30533 0
1.3 1.18264 −0.508481 −0.601352 0 −0.601352 −1.00000 −3.07647 −2.74145 0
1.4 2.59615 1.82578 4.74002 0 4.74002 −1.00000 7.11351 0.333489 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.o 4
5.b even 2 1 805.2.a.g 4
15.d odd 2 1 7245.2.a.bf 4
35.c odd 2 1 5635.2.a.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.g 4 5.b even 2 1
4025.2.a.o 4 1.a even 1 1 trivial
5635.2.a.s 4 35.c odd 2 1
7245.2.a.bf 4 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 3T_{2} + 2$$ T2^4 - 2*T2^3 - 3*T2^2 + 3*T2 + 2 $$T_{3}^{4} - 5T_{3}^{3} + 5T_{3}^{2} + 2T_{3} - 1$$ T3^4 - 5*T3^3 + 5*T3^2 + 2*T3 - 1 $$T_{11}^{4} + 8T_{11}^{3} + 3T_{11}^{2} - 51T_{11} + 41$$ T11^4 + 8*T11^3 + 3*T11^2 - 51*T11 + 41

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 3 T^{2} + 3 T + 2$$
$3$ $$T^{4} - 5 T^{3} + 5 T^{2} + 2 T - 1$$
$5$ $$T^{4}$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} + 8 T^{3} + 3 T^{2} - 51 T + 41$$
$13$ $$T^{4} - 8 T^{3} + 7 T^{2} + 21 T + 1$$
$17$ $$T^{4} - 6 T^{3} - 17 T^{2} + 13 T + 31$$
$19$ $$T^{4} - 2 T^{3} - 57 T^{2} + 31 T + 4$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} + T^{3} - 8 T^{2} - 4 T + 8$$
$31$ $$T^{4} + 2 T^{3} - 81 T^{2} - 177 T + 134$$
$37$ $$T^{4} - 7 T^{3} - 8 T^{2} + 59 T - 44$$
$41$ $$T^{4} + 9 T^{3} - 13 T^{2} - 9 T + 4$$
$43$ $$T^{4} - 4 T^{3} - 75 T^{2} + \cdots + 1262$$
$47$ $$T^{4} - 21 T^{3} + 157 T^{2} + \cdots + 577$$
$53$ $$T^{4} - 11 T^{3} - 110 T^{2} + \cdots - 2474$$
$59$ $$T^{4} + 37 T^{3} + 472 T^{2} + \cdots + 4226$$
$61$ $$T^{4} - T^{3} - 10 T^{2} + 13 T - 2$$
$67$ $$T^{4} - 31 T^{3} + 310 T^{2} + \cdots - 968$$
$71$ $$T^{4} + 8 T^{3} - 3 T^{2} - 27 T - 16$$
$73$ $$T^{4} + 25 T^{3} + 154 T^{2} + \cdots - 3782$$
$79$ $$T^{4} + 19 T^{3} - 41 T^{2} + \cdots + 601$$
$83$ $$T^{4} - 7 T^{3} - 40 T^{2} - 5 T + 4$$
$89$ $$T^{4} - 7 T^{3} - 102 T^{2} + \cdots - 2458$$
$97$ $$T^{4} - 2 T^{3} - 212 T^{2} + \cdots - 2069$$