# Properties

 Label 1805.2.a.e Level $1805$ Weight $2$ Character orbit 1805.a Self dual yes Analytic conductor $14.413$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4129975648$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + 3 \beta q^{4} - q^{5} + (3 \beta + 2) q^{6} + (2 \beta - 3) q^{7} + (4 \beta + 1) q^{8} + (3 \beta - 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + (b + 1) * q^3 + 3*b * q^4 - q^5 + (3*b + 2) * q^6 + (2*b - 3) * q^7 + (4*b + 1) * q^8 + (3*b - 1) * q^9 $$q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + 3 \beta q^{4} - q^{5} + (3 \beta + 2) q^{6} + (2 \beta - 3) q^{7} + (4 \beta + 1) q^{8} + (3 \beta - 1) q^{9} + ( - \beta - 1) q^{10} + ( - \beta - 3) q^{11} + (6 \beta + 3) q^{12} + 5 q^{13} + (\beta - 1) q^{14} + ( - \beta - 1) q^{15} + (3 \beta + 5) q^{16} - 6 q^{17} + (5 \beta + 2) q^{18} - 3 \beta q^{20} + (\beta - 1) q^{21} + ( - 5 \beta - 4) q^{22} - \beta q^{23} + (9 \beta + 5) q^{24} + q^{25} + (5 \beta + 5) q^{26} + (2 \beta - 1) q^{27} + ( - 3 \beta + 6) q^{28} + ( - 3 \beta + 3) q^{29} + ( - 3 \beta - 2) q^{30} + ( - 3 \beta + 9) q^{31} + (3 \beta + 6) q^{32} + ( - 5 \beta - 4) q^{33} + ( - 6 \beta - 6) q^{34} + ( - 2 \beta + 3) q^{35} + (6 \beta + 9) q^{36} + ( - 3 \beta + 3) q^{37} + (5 \beta + 5) q^{39} + ( - 4 \beta - 1) q^{40} + (10 \beta - 5) q^{41} + \beta q^{42} + ( - 3 \beta + 2) q^{43} + ( - 12 \beta - 3) q^{44} + ( - 3 \beta + 1) q^{45} + ( - 2 \beta - 1) q^{46} + ( - 2 \beta - 7) q^{47} + (11 \beta + 8) q^{48} + ( - 8 \beta + 6) q^{49} + (\beta + 1) q^{50} + ( - 6 \beta - 6) q^{51} + 15 \beta q^{52} + (\beta + 4) q^{53} + (3 \beta + 1) q^{54} + (\beta + 3) q^{55} + ( - 2 \beta + 5) q^{56} - 3 \beta q^{58} + (5 \beta - 8) q^{59} + ( - 6 \beta - 3) q^{60} - 3 q^{61} + (3 \beta + 6) q^{62} + ( - 5 \beta + 9) q^{63} + (6 \beta - 1) q^{64} - 5 q^{65} + ( - 14 \beta - 9) q^{66} + ( - 6 \beta + 5) q^{67} - 18 \beta q^{68} + ( - 2 \beta - 1) q^{69} + ( - \beta + 1) q^{70} + ( - 2 \beta + 7) q^{71} + (11 \beta + 11) q^{72} - q^{73} - 3 \beta q^{74} + (\beta + 1) q^{75} + ( - 5 \beta + 7) q^{77} + (15 \beta + 10) q^{78} + 8 q^{79} + ( - 3 \beta - 5) q^{80} + ( - 6 \beta + 4) q^{81} + (15 \beta + 5) q^{82} + (4 \beta - 2) q^{83} + 3 q^{84} + 6 q^{85} + ( - 4 \beta - 1) q^{86} - 3 \beta q^{87} + ( - 17 \beta - 7) q^{88} + ( - 6 \beta + 3) q^{89} + ( - 5 \beta - 2) q^{90} + (10 \beta - 15) q^{91} + ( - 3 \beta - 3) q^{92} + (3 \beta + 6) q^{93} + ( - 11 \beta - 9) q^{94} + (12 \beta + 9) q^{96} + (5 \beta - 2) q^{97} + ( - 10 \beta - 2) q^{98} - 11 \beta q^{99} +O(q^{100})$$ q + (b + 1) * q^2 + (b + 1) * q^3 + 3*b * q^4 - q^5 + (3*b + 2) * q^6 + (2*b - 3) * q^7 + (4*b + 1) * q^8 + (3*b - 1) * q^9 + (-b - 1) * q^10 + (-b - 3) * q^11 + (6*b + 3) * q^12 + 5 * q^13 + (b - 1) * q^14 + (-b - 1) * q^15 + (3*b + 5) * q^16 - 6 * q^17 + (5*b + 2) * q^18 - 3*b * q^20 + (b - 1) * q^21 + (-5*b - 4) * q^22 - b * q^23 + (9*b + 5) * q^24 + q^25 + (5*b + 5) * q^26 + (2*b - 1) * q^27 + (-3*b + 6) * q^28 + (-3*b + 3) * q^29 + (-3*b - 2) * q^30 + (-3*b + 9) * q^31 + (3*b + 6) * q^32 + (-5*b - 4) * q^33 + (-6*b - 6) * q^34 + (-2*b + 3) * q^35 + (6*b + 9) * q^36 + (-3*b + 3) * q^37 + (5*b + 5) * q^39 + (-4*b - 1) * q^40 + (10*b - 5) * q^41 + b * q^42 + (-3*b + 2) * q^43 + (-12*b - 3) * q^44 + (-3*b + 1) * q^45 + (-2*b - 1) * q^46 + (-2*b - 7) * q^47 + (11*b + 8) * q^48 + (-8*b + 6) * q^49 + (b + 1) * q^50 + (-6*b - 6) * q^51 + 15*b * q^52 + (b + 4) * q^53 + (3*b + 1) * q^54 + (b + 3) * q^55 + (-2*b + 5) * q^56 - 3*b * q^58 + (5*b - 8) * q^59 + (-6*b - 3) * q^60 - 3 * q^61 + (3*b + 6) * q^62 + (-5*b + 9) * q^63 + (6*b - 1) * q^64 - 5 * q^65 + (-14*b - 9) * q^66 + (-6*b + 5) * q^67 - 18*b * q^68 + (-2*b - 1) * q^69 + (-b + 1) * q^70 + (-2*b + 7) * q^71 + (11*b + 11) * q^72 - q^73 - 3*b * q^74 + (b + 1) * q^75 + (-5*b + 7) * q^77 + (15*b + 10) * q^78 + 8 * q^79 + (-3*b - 5) * q^80 + (-6*b + 4) * q^81 + (15*b + 5) * q^82 + (4*b - 2) * q^83 + 3 * q^84 + 6 * q^85 + (-4*b - 1) * q^86 - 3*b * q^87 + (-17*b - 7) * q^88 + (-6*b + 3) * q^89 + (-5*b - 2) * q^90 + (10*b - 15) * q^91 + (-3*b - 3) * q^92 + (3*b + 6) * q^93 + (-11*b - 9) * q^94 + (12*b + 9) * q^96 + (5*b - 2) * q^97 + (-10*b - 2) * q^98 - 11*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} + 7 q^{6} - 4 q^{7} + 6 q^{8} + q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 2 * q^5 + 7 * q^6 - 4 * q^7 + 6 * q^8 + q^9 $$2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} + 7 q^{6} - 4 q^{7} + 6 q^{8} + q^{9} - 3 q^{10} - 7 q^{11} + 12 q^{12} + 10 q^{13} - q^{14} - 3 q^{15} + 13 q^{16} - 12 q^{17} + 9 q^{18} - 3 q^{20} - q^{21} - 13 q^{22} - q^{23} + 19 q^{24} + 2 q^{25} + 15 q^{26} + 9 q^{28} + 3 q^{29} - 7 q^{30} + 15 q^{31} + 15 q^{32} - 13 q^{33} - 18 q^{34} + 4 q^{35} + 24 q^{36} + 3 q^{37} + 15 q^{39} - 6 q^{40} + q^{42} + q^{43} - 18 q^{44} - q^{45} - 4 q^{46} - 16 q^{47} + 27 q^{48} + 4 q^{49} + 3 q^{50} - 18 q^{51} + 15 q^{52} + 9 q^{53} + 5 q^{54} + 7 q^{55} + 8 q^{56} - 3 q^{58} - 11 q^{59} - 12 q^{60} - 6 q^{61} + 15 q^{62} + 13 q^{63} + 4 q^{64} - 10 q^{65} - 32 q^{66} + 4 q^{67} - 18 q^{68} - 4 q^{69} + q^{70} + 12 q^{71} + 33 q^{72} - 2 q^{73} - 3 q^{74} + 3 q^{75} + 9 q^{77} + 35 q^{78} + 16 q^{79} - 13 q^{80} + 2 q^{81} + 25 q^{82} + 6 q^{84} + 12 q^{85} - 6 q^{86} - 3 q^{87} - 31 q^{88} - 9 q^{90} - 20 q^{91} - 9 q^{92} + 15 q^{93} - 29 q^{94} + 30 q^{96} + q^{97} - 14 q^{98} - 11 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 2 * q^5 + 7 * q^6 - 4 * q^7 + 6 * q^8 + q^9 - 3 * q^10 - 7 * q^11 + 12 * q^12 + 10 * q^13 - q^14 - 3 * q^15 + 13 * q^16 - 12 * q^17 + 9 * q^18 - 3 * q^20 - q^21 - 13 * q^22 - q^23 + 19 * q^24 + 2 * q^25 + 15 * q^26 + 9 * q^28 + 3 * q^29 - 7 * q^30 + 15 * q^31 + 15 * q^32 - 13 * q^33 - 18 * q^34 + 4 * q^35 + 24 * q^36 + 3 * q^37 + 15 * q^39 - 6 * q^40 + q^42 + q^43 - 18 * q^44 - q^45 - 4 * q^46 - 16 * q^47 + 27 * q^48 + 4 * q^49 + 3 * q^50 - 18 * q^51 + 15 * q^52 + 9 * q^53 + 5 * q^54 + 7 * q^55 + 8 * q^56 - 3 * q^58 - 11 * q^59 - 12 * q^60 - 6 * q^61 + 15 * q^62 + 13 * q^63 + 4 * q^64 - 10 * q^65 - 32 * q^66 + 4 * q^67 - 18 * q^68 - 4 * q^69 + q^70 + 12 * q^71 + 33 * q^72 - 2 * q^73 - 3 * q^74 + 3 * q^75 + 9 * q^77 + 35 * q^78 + 16 * q^79 - 13 * q^80 + 2 * q^81 + 25 * q^82 + 6 * q^84 + 12 * q^85 - 6 * q^86 - 3 * q^87 - 31 * q^88 - 9 * q^90 - 20 * q^91 - 9 * q^92 + 15 * q^93 - 29 * q^94 + 30 * q^96 + q^97 - 14 * q^98 - 11 * 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
 −0.618034 1.61803
0.381966 0.381966 −1.85410 −1.00000 0.145898 −4.23607 −1.47214 −2.85410 −0.381966
1.2 2.61803 2.61803 4.85410 −1.00000 6.85410 0.236068 7.47214 3.85410 −2.61803
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 1805.2.a.e yes 2
5.b even 2 1 9025.2.a.k 2
19.b odd 2 1 1805.2.a.c 2
95.d odd 2 1 9025.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.c 2 19.b odd 2 1
1805.2.a.e yes 2 1.a even 1 1 trivial
9025.2.a.k 2 5.b even 2 1
9025.2.a.v 2 95.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1805))$$:

 $$T_{2}^{2} - 3T_{2} + 1$$ T2^2 - 3*T2 + 1 $$T_{3}^{2} - 3T_{3} + 1$$ T3^2 - 3*T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 1$$
$3$ $$T^{2} - 3T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T - 1$$
$11$ $$T^{2} + 7T + 11$$
$13$ $$(T - 5)^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + T - 1$$
$29$ $$T^{2} - 3T - 9$$
$31$ $$T^{2} - 15T + 45$$
$37$ $$T^{2} - 3T - 9$$
$41$ $$T^{2} - 125$$
$43$ $$T^{2} - T - 11$$
$47$ $$T^{2} + 16T + 59$$
$53$ $$T^{2} - 9T + 19$$
$59$ $$T^{2} + 11T - 1$$
$61$ $$(T + 3)^{2}$$
$67$ $$T^{2} - 4T - 41$$
$71$ $$T^{2} - 12T + 31$$
$73$ $$(T + 1)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 20$$
$89$ $$T^{2} - 45$$
$97$ $$T^{2} - T - 31$$