# Properties

 Label 4025.2.a.h Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.61803 2.23607 4.85410 0 −5.85410 1.00000 −7.47214 2.00000 0
1.2 −0.381966 −2.23607 −1.85410 0 0.854102 1.00000 1.47214 2.00000 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.h 2
5.b even 2 1 805.2.a.e 2
15.d odd 2 1 7245.2.a.v 2
35.c odd 2 1 5635.2.a.q 2

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

 $$T_{2}^{2} + 3T_{2} + 1$$ T2^2 + 3*T2 + 1 $$T_{3}^{2} - 5$$ T3^2 - 5 $$T_{11} - 6$$ T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 1$$
$3$ $$T^{2} - 5$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 6)^{2}$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} + 2T - 4$$
$19$ $$T^{2} - 8T - 4$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 6T - 11$$
$31$ $$T^{2} + 2T - 19$$
$37$ $$T^{2} - 10T + 20$$
$41$ $$T^{2} - 45$$
$43$ $$T^{2} - 6T - 36$$
$47$ $$T^{2} - 4T - 1$$
$53$ $$T^{2} - 22T + 116$$
$59$ $$T^{2} + 4T - 16$$
$61$ $$T^{2} - 10T - 20$$
$67$ $$T^{2} + 2T - 4$$
$71$ $$T^{2} + 4T - 1$$
$73$ $$(T + 9)^{2}$$
$79$ $$T^{2} - 180$$
$83$ $$T^{2} - 6T - 36$$
$89$ $$T^{2} + 14T + 44$$
$97$ $$T^{2} - 14T + 4$$