# Properties

 Label 4025.2.a.p Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.2147108.1 Defining polynomial: $$x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27$$ x^5 - 2*x^4 - 9*x^3 + 17*x^2 + 16*x - 27 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{4} - \beta_{2}) q^{6} - q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{2} - \beta_1 + 3) q^{9}+O(q^{10})$$ q - b1 * q^2 - b3 * q^3 + (b2 + 2) * q^4 + (b4 - b2) * q^6 - q^7 + (-b4 + b3 - b2 - b1) * q^8 + (-b2 - b1 + 3) * q^9 $$q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{4} - \beta_{2}) q^{6} - q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{2} - \beta_1 + 3) q^{9} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{12} + ( - \beta_{4} - \beta_{3} + 1) q^{13} + \beta_1 q^{14} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + 1) q^{16} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{17} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{18} + ( - 2 \beta_1 + 2) q^{19} + \beta_{3} q^{21} + (3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{22} + q^{23} + ( - \beta_{3} + 2 \beta_1 - 8) q^{24} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{26} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{27} + ( - \beta_{2} - 2) q^{28} + ( - 3 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + 6) q^{31} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{32} + ( - 2 \beta_{4} + 4) q^{33} + (3 \beta_{4} + 4 \beta_{2} - 2 \beta_1 + 1) q^{34} + ( - 2 \beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{36} + ( - 2 \beta_{3} - 2 \beta_1) q^{37} + (2 \beta_{2} - 2 \beta_1 + 8) q^{38} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{39} + ( - \beta_{4} + \beta_{3} + 1) q^{41} + ( - \beta_{4} + \beta_{2}) q^{42} + ( - 2 \beta_{4} + 2) q^{43} + ( - 3 \beta_{4} - 3 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 5) q^{44} - \beta_1 q^{46} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{47} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 6) q^{48} + q^{49} + ( - \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{51} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} + 4) q^{52} + (2 \beta_{2} - 4) q^{53} + ( - 2 \beta_{3} - \beta_{2} + 7) q^{54} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{56} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{57} + (3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 4) q^{58} + ( - \beta_{4} - 2 \beta_1 + 5) q^{59} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 3) q^{61} + (\beta_{4} - \beta_{2} - 6 \beta_1) q^{62} + (\beta_{2} + \beta_1 - 3) q^{63} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 5) q^{64} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 2) q^{66} + (\beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_1 + 3) q^{67} + ( - 5 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{68} - \beta_{3} q^{69} + (\beta_{4} + 3 \beta_{3} + 1) q^{71} + (\beta_{4} + 5 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 6) q^{72} + (\beta_{4} + \beta_{3} + 4 \beta_{2} - 1) q^{73} + (2 \beta_{4} + 8) q^{74} + ( - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_1 + 4) q^{76} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (4 \beta_{4} + \beta_{2} - 2 \beta_1 - 7) q^{78} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 7) q^{79} + (3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{81} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{82} + ( - 2 \beta_{4} - 2) q^{83} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{84} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{86} + ( - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 3) q^{87} + (5 \beta_{4} - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 3) q^{88} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 5) q^{89} + (\beta_{4} + \beta_{3} - 1) q^{91} + (\beta_{2} + 2) q^{92} + ( - 6 \beta_{3} - \beta_{2} - \beta_1 + 6) q^{93} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 8) q^{94} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 1) q^{96} + (3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{97} - \beta_1 q^{98} + (\beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{99}+O(q^{100})$$ q - b1 * q^2 - b3 * q^3 + (b2 + 2) * q^4 + (b4 - b2) * q^6 - q^7 + (-b4 + b3 - b2 - b1) * q^8 + (-b2 - b1 + 3) * q^9 + (-b4 - b3 - b2 + b1 - 1) * q^11 + (-b3 - b2 + 2*b1 - 1) * q^12 + (-b4 - b3 + 1) * q^13 + b1 * q^14 + (b4 + b3 + 3*b2 + 1) * q^16 + (-b4 - 2*b2 + 3) * q^17 + (b4 - b3 + 2*b2 - 2*b1 + 4) * q^18 + (-2*b1 + 2) * q^19 + b3 * q^21 + (3*b4 + b3 + b2 + b1 - 3) * q^22 + q^23 + (-b3 + 2*b1 - 8) * q^24 + (2*b4 + 2*b3 + b2 - 2*b1 + 1) * q^26 + (b4 - b3 - 2*b1 + 1) * q^27 + (-b2 - 2) * q^28 + (-3*b2 + b1) * q^29 + (-b3 + 6) * q^31 + (-3*b4 - b3 - 2*b2 - b1 - 1) * q^32 + (-2*b4 + 4) * q^33 + (3*b4 + 4*b2 - 2*b1 + 1) * q^34 + (-2*b4 - b2 - 3*b1 + 1) * q^36 + (-2*b3 - 2*b1) * q^37 + (2*b2 - 2*b1 + 8) * q^38 + (-b4 - b3 - 2*b2 + 2*b1 + 3) * q^39 + (-b4 + b3 + 1) * q^41 + (-b4 + b2) * q^42 + (-2*b4 + 2) * q^43 + (-3*b4 - 3*b3 - 5*b2 + 3*b1 - 5) * q^44 - b1 * q^46 + (3*b3 - 2*b2 + 2*b1 - 2) * q^47 + (b4 + 2*b3 - b2 + 4*b1 - 6) * q^48 + q^49 + (-b4 - 5*b3 + b2 - b1 - 1) * q^51 + (-3*b4 - b3 - b2 + 4) * q^52 + (2*b2 - 4) * q^53 + (-2*b3 - b2 + 7) * q^54 + (b4 - b3 + b2 + b1) * q^56 + (2*b4 - 2*b3 - 2*b2) * q^57 + (3*b4 - 3*b3 + 2*b2 + 3*b1 - 4) * q^58 + (-b4 - 2*b1 + 5) * q^59 + (-b4 - 2*b3 - 2*b1 - 3) * q^61 + (b4 - b2 - 6*b1) * q^62 + (b2 + b1 - 3) * q^63 + (4*b4 + 2*b3 + 2*b2 + 5) * q^64 + (2*b4 + 4*b3 + 4*b2 - 6*b1 + 2) * q^66 + (b4 + b3 - b2 - 5*b1 + 3) * q^67 + (-5*b4 - 2*b3 - 4*b2 - 2*b1 - 1) * q^68 - b3 * q^69 + (b4 + 3*b3 + 1) * q^71 + (b4 + 5*b3 + 4*b2 + 2*b1 + 6) * q^72 + (b4 + b3 + 4*b2 - 1) * q^73 + (2*b4 + 8) * q^74 + (-2*b4 + 2*b3 - 6*b1 + 4) * q^76 + (b4 + b3 + b2 - b1 + 1) * q^77 + (4*b4 + b2 - 2*b1 - 7) * q^78 + (b4 + b3 - b2 - b1 + 7) * q^79 + (3*b4 - b3 + b2 - b1) * q^81 + (2*b3 + 3*b2 - 2*b1 + 1) * q^82 + (-2*b4 - 2) * q^83 + (b3 + b2 - 2*b1 + 1) * q^84 + (2*b4 + 4*b3 + 4*b2 - 4*b1 + 2) * q^86 + (-b4 - 3*b3 + 4*b2 - 6*b1 + 3) * q^87 + (5*b4 - b3 + 3*b2 + 5*b1 - 3) * q^88 + (-b4 - 2*b3 + 2*b1 - 5) * q^89 + (b4 + b3 - 1) * q^91 + (b2 + 2) * q^92 + (-6*b3 - b2 - b1 + 6) * q^93 + (-b4 - 2*b3 + 3*b2 + 4*b1 - 8) * q^94 + (-2*b4 - b3 - 3*b2 + 4*b1 - 1) * q^96 + (3*b4 + 2*b2 - 2*b1 + 3) * q^97 - b1 * q^98 + (b4 - b3 + b2 + 3*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} + 12 q^{4} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10})$$ 5 * q - 2 * q^2 + 12 * q^4 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 11 * q^9 $$5 q - 2 q^{2} + 12 q^{4} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 11 q^{9} - 4 q^{11} - 3 q^{12} + 6 q^{13} + 2 q^{14} + 10 q^{16} + 12 q^{17} + 19 q^{18} + 6 q^{19} - 14 q^{22} + 5 q^{23} - 36 q^{24} + q^{26} - 12 q^{28} - 4 q^{29} + 30 q^{31} - 8 q^{32} + 22 q^{33} + 6 q^{34} - q^{36} - 4 q^{37} + 40 q^{38} + 16 q^{39} + 6 q^{41} + 3 q^{42} + 12 q^{43} - 26 q^{44} - 2 q^{46} - 10 q^{47} - 25 q^{48} + 5 q^{49} - 4 q^{51} + 21 q^{52} - 16 q^{53} + 33 q^{54} + 3 q^{56} - 6 q^{57} - 13 q^{58} + 22 q^{59} - 18 q^{61} - 15 q^{62} - 11 q^{63} + 25 q^{64} + 4 q^{66} + 2 q^{67} - 12 q^{68} + 4 q^{71} + 41 q^{72} + 2 q^{73} + 38 q^{74} + 10 q^{76} + 4 q^{77} - 41 q^{78} + 30 q^{79} - 3 q^{81} + 7 q^{82} - 8 q^{83} + 3 q^{84} + 8 q^{86} + 12 q^{87} - 4 q^{88} - 20 q^{89} - 6 q^{91} + 12 q^{92} + 26 q^{93} - 25 q^{94} - q^{96} + 12 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100})$$ 5 * q - 2 * q^2 + 12 * q^4 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 11 * q^9 - 4 * q^11 - 3 * q^12 + 6 * q^13 + 2 * q^14 + 10 * q^16 + 12 * q^17 + 19 * q^18 + 6 * q^19 - 14 * q^22 + 5 * q^23 - 36 * q^24 + q^26 - 12 * q^28 - 4 * q^29 + 30 * q^31 - 8 * q^32 + 22 * q^33 + 6 * q^34 - q^36 - 4 * q^37 + 40 * q^38 + 16 * q^39 + 6 * q^41 + 3 * q^42 + 12 * q^43 - 26 * q^44 - 2 * q^46 - 10 * q^47 - 25 * q^48 + 5 * q^49 - 4 * q^51 + 21 * q^52 - 16 * q^53 + 33 * q^54 + 3 * q^56 - 6 * q^57 - 13 * q^58 + 22 * q^59 - 18 * q^61 - 15 * q^62 - 11 * q^63 + 25 * q^64 + 4 * q^66 + 2 * q^67 - 12 * q^68 + 4 * q^71 + 41 * q^72 + 2 * q^73 + 38 * q^74 + 10 * q^76 + 4 * q^77 - 41 * q^78 + 30 * q^79 - 3 * q^81 + 7 * q^82 - 8 * q^83 + 3 * q^84 + 8 * q^86 + 12 * q^87 - 4 * q^88 - 20 * q^89 - 6 * q^91 + 12 * q^92 + 26 * q^93 - 25 * q^94 - q^96 + 12 * q^97 - 2 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 11 ) / 2$$ (v^4 - v^3 - 8*v^2 + 5*v + 11) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 10\nu^{2} - 5\nu + 19 ) / 2$$ (v^4 + v^3 - 10*v^2 - 5*v + 19) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1$$ b4 - b3 + b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 9\beta_{2} + 21$$ b4 + b3 + 9*b2 + 21

## 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
 2.69017 2.11948 1.23828 −1.50216 −2.54577
−2.69017 0.269842 5.23702 0 −0.725921 −1.00000 −8.70812 −2.92719 0
1.2 −2.11948 1.84074 2.49221 0 −3.90141 −1.00000 −1.04322 0.388311 0
1.3 −1.23828 −2.68857 −0.466664 0 3.32920 −1.00000 3.05442 4.22838 0
1.4 1.50216 3.04067 0.256481 0 4.56757 −1.00000 −2.61904 6.24568 0
1.5 2.54577 −2.46268 4.48096 0 −6.26943 −1.00000 6.31597 3.06481 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.p 5
5.b even 2 1 161.2.a.d 5
15.d odd 2 1 1449.2.a.r 5
20.d odd 2 1 2576.2.a.bd 5
35.c odd 2 1 1127.2.a.h 5
115.c odd 2 1 3703.2.a.j 5

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

 $$T_{2}^{5} + 2T_{2}^{4} - 9T_{2}^{3} - 17T_{2}^{2} + 16T_{2} + 27$$ T2^5 + 2*T2^4 - 9*T2^3 - 17*T2^2 + 16*T2 + 27 $$T_{3}^{5} - 13T_{3}^{3} + 38T_{3} - 10$$ T3^5 - 13*T3^3 + 38*T3 - 10 $$T_{11}^{5} + 4T_{11}^{4} - 28T_{11}^{3} - 148T_{11}^{2} - 160T_{11} - 48$$ T11^5 + 4*T11^4 - 28*T11^3 - 148*T11^2 - 160*T11 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 2 T^{4} - 9 T^{3} - 17 T^{2} + \cdots + 27$$
$3$ $$T^{5} - 13 T^{3} + 38 T - 10$$
$5$ $$T^{5}$$
$7$ $$(T + 1)^{5}$$
$11$ $$T^{5} + 4 T^{4} - 28 T^{3} - 148 T^{2} + \cdots - 48$$
$13$ $$T^{5} - 6 T^{4} - 9 T^{3} + 46 T^{2} + \cdots - 56$$
$17$ $$T^{5} - 12 T^{4} + 6 T^{3} + \cdots + 1536$$
$19$ $$T^{5} - 6 T^{4} - 28 T^{3} + 96 T^{2} + \cdots + 128$$
$23$ $$(T - 1)^{5}$$
$29$ $$T^{5} + 4 T^{4} - 111 T^{3} + \cdots - 1452$$
$31$ $$T^{5} - 30 T^{4} + 347 T^{3} + \cdots - 5206$$
$37$ $$T^{5} + 4 T^{4} - 76 T^{3} - 376 T^{2} + \cdots + 32$$
$41$ $$T^{5} - 6 T^{4} - 29 T^{3} + 146 T^{2} + \cdots - 456$$
$43$ $$T^{5} - 12 T^{4} - 24 T^{3} + \cdots - 256$$
$47$ $$T^{5} + 10 T^{4} - 125 T^{3} + \cdots - 11142$$
$53$ $$T^{5} + 16 T^{4} + 52 T^{3} + \cdots + 480$$
$59$ $$T^{5} - 22 T^{4} + 118 T^{3} + \cdots + 1440$$
$61$ $$T^{5} + 18 T^{4} + 34 T^{3} - 438 T^{2} + \cdots + 56$$
$67$ $$T^{5} - 2 T^{4} - 300 T^{3} + \cdots - 61936$$
$71$ $$T^{5} - 4 T^{4} - 101 T^{3} + \cdots - 5184$$
$73$ $$T^{5} - 2 T^{4} - 197 T^{3} + \cdots + 27656$$
$79$ $$T^{5} - 30 T^{4} + 308 T^{3} + \cdots + 1936$$
$83$ $$T^{5} + 8 T^{4} - 56 T^{3} + \cdots + 5376$$
$89$ $$T^{5} + 20 T^{4} + 66 T^{3} + \cdots - 4704$$
$97$ $$T^{5} - 12 T^{4} - 114 T^{3} + \cdots - 4120$$