# Properties

 Label 4025.2.a.i Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $1$ 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: $$1$$ 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 161) 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.

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

## 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
 −0.618034 1.61803
−0.618034 1.00000 −1.61803 0 −0.618034 1.00000 2.23607 −2.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 1.00000 −2.23607 −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.i 2
5.b even 2 1 161.2.a.b 2
15.d odd 2 1 1449.2.a.i 2
20.d odd 2 1 2576.2.a.s 2
35.c odd 2 1 1127.2.a.d 2
115.c odd 2 1 3703.2.a.b 2

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

 $$T_{2}^{2} - T_{2} - 1$$ T2^2 - T2 - 1 $$T_{3} - 1$$ T3 - 1 $$T_{11}^{2} - 20$$ T11^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 20$$
$13$ $$T^{2} - 4T - 1$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 10T + 20$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} - 6T - 11$$
$31$ $$(T + 9)^{2}$$
$37$ $$T^{2} + 2T - 44$$
$41$ $$T^{2} - 5$$
$43$ $$T^{2} - 4T - 16$$
$47$ $$T^{2} + 2T - 19$$
$53$ $$T^{2} + 18T + 76$$
$59$ $$T^{2} + 12T + 16$$
$61$ $$T^{2} - 180$$
$67$ $$T^{2} - 2T - 124$$
$71$ $$T^{2} + 16T + 59$$
$73$ $$T^{2} - 45$$
$79$ $$T^{2} + 10T + 20$$
$83$ $$T^{2} + 4T - 16$$
$89$ $$T^{2} - 80$$
$97$ $$T^{2} - 6T - 36$$