# Properties

 Label 845.2.a.b Level $845$ Weight $2$ Character orbit 845.a Self dual yes Analytic conductor $6.747$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.74735897080$$ 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 65) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + ( - 2 \beta + 1) q^{3} + (\beta - 1) q^{4} - q^{5} + (\beta + 2) q^{6} + (2 \beta - 3) q^{7} + (2 \beta - 1) q^{8} + 2 q^{9} +O(q^{10})$$ q - b * q^2 + (-2*b + 1) * q^3 + (b - 1) * q^4 - q^5 + (b + 2) * q^6 + (2*b - 3) * q^7 + (2*b - 1) * q^8 + 2 * q^9 $$q - \beta q^{2} + ( - 2 \beta + 1) q^{3} + (\beta - 1) q^{4} - q^{5} + (\beta + 2) q^{6} + (2 \beta - 3) q^{7} + (2 \beta - 1) q^{8} + 2 q^{9} + \beta q^{10} + ( - 2 \beta - 1) q^{11} + (\beta - 3) q^{12} + (\beta - 2) q^{14} + (2 \beta - 1) q^{15} - 3 \beta q^{16} + (4 \beta - 1) q^{17} - 2 \beta q^{18} + (2 \beta - 3) q^{19} + ( - \beta + 1) q^{20} + (4 \beta - 7) q^{21} + (3 \beta + 2) q^{22} + (2 \beta + 5) q^{23} - 5 q^{24} + q^{25} + (2 \beta - 1) q^{27} + ( - 3 \beta + 5) q^{28} + (4 \beta - 5) q^{29} + ( - \beta - 2) q^{30} + ( - \beta + 5) q^{32} + (4 \beta + 3) q^{33} + ( - 3 \beta - 4) q^{34} + ( - 2 \beta + 3) q^{35} + (2 \beta - 2) q^{36} - 3 q^{37} + (\beta - 2) q^{38} + ( - 2 \beta + 1) q^{40} + (8 \beta - 7) q^{41} + (3 \beta - 4) q^{42} + (2 \beta - 5) q^{43} + ( - \beta - 1) q^{44} - 2 q^{45} + ( - 7 \beta - 2) q^{46} - 8 \beta q^{47} + (3 \beta + 6) q^{48} + ( - 8 \beta + 6) q^{49} - \beta q^{50} + ( - 2 \beta - 9) q^{51} + 6 q^{53} + ( - \beta - 2) q^{54} + (2 \beta + 1) q^{55} + ( - 4 \beta + 7) q^{56} + (4 \beta - 7) q^{57} + (\beta - 4) q^{58} + ( - 6 \beta - 3) q^{59} + ( - \beta + 3) q^{60} + ( - 12 \beta + 7) q^{61} + (4 \beta - 6) q^{63} + (2 \beta + 1) q^{64} + ( - 7 \beta - 4) q^{66} + ( - 6 \beta - 1) q^{67} + ( - \beta + 5) q^{68} + ( - 12 \beta + 1) q^{69} + ( - \beta + 2) q^{70} + ( - 2 \beta + 5) q^{71} + (4 \beta - 2) q^{72} + 6 q^{73} + 3 \beta q^{74} + ( - 2 \beta + 1) q^{75} + ( - 3 \beta + 5) q^{76} - q^{77} + 3 \beta q^{80} - 11 q^{81} + ( - \beta - 8) q^{82} + (8 \beta - 4) q^{83} + ( - 7 \beta + 11) q^{84} + ( - 4 \beta + 1) q^{85} + (3 \beta - 2) q^{86} + (6 \beta - 13) q^{87} + ( - 4 \beta - 3) q^{88} + 9 q^{89} + 2 \beta q^{90} + (5 \beta - 3) q^{92} + (8 \beta + 8) q^{94} + ( - 2 \beta + 3) q^{95} + ( - 9 \beta + 7) q^{96} + ( - 4 \beta + 1) q^{97} + (2 \beta + 8) q^{98} + ( - 4 \beta - 2) q^{99} +O(q^{100})$$ q - b * q^2 + (-2*b + 1) * q^3 + (b - 1) * q^4 - q^5 + (b + 2) * q^6 + (2*b - 3) * q^7 + (2*b - 1) * q^8 + 2 * q^9 + b * q^10 + (-2*b - 1) * q^11 + (b - 3) * q^12 + (b - 2) * q^14 + (2*b - 1) * q^15 - 3*b * q^16 + (4*b - 1) * q^17 - 2*b * q^18 + (2*b - 3) * q^19 + (-b + 1) * q^20 + (4*b - 7) * q^21 + (3*b + 2) * q^22 + (2*b + 5) * q^23 - 5 * q^24 + q^25 + (2*b - 1) * q^27 + (-3*b + 5) * q^28 + (4*b - 5) * q^29 + (-b - 2) * q^30 + (-b + 5) * q^32 + (4*b + 3) * q^33 + (-3*b - 4) * q^34 + (-2*b + 3) * q^35 + (2*b - 2) * q^36 - 3 * q^37 + (b - 2) * q^38 + (-2*b + 1) * q^40 + (8*b - 7) * q^41 + (3*b - 4) * q^42 + (2*b - 5) * q^43 + (-b - 1) * q^44 - 2 * q^45 + (-7*b - 2) * q^46 - 8*b * q^47 + (3*b + 6) * q^48 + (-8*b + 6) * q^49 - b * q^50 + (-2*b - 9) * q^51 + 6 * q^53 + (-b - 2) * q^54 + (2*b + 1) * q^55 + (-4*b + 7) * q^56 + (4*b - 7) * q^57 + (b - 4) * q^58 + (-6*b - 3) * q^59 + (-b + 3) * q^60 + (-12*b + 7) * q^61 + (4*b - 6) * q^63 + (2*b + 1) * q^64 + (-7*b - 4) * q^66 + (-6*b - 1) * q^67 + (-b + 5) * q^68 + (-12*b + 1) * q^69 + (-b + 2) * q^70 + (-2*b + 5) * q^71 + (4*b - 2) * q^72 + 6 * q^73 + 3*b * q^74 + (-2*b + 1) * q^75 + (-3*b + 5) * q^76 - q^77 + 3*b * q^80 - 11 * q^81 + (-b - 8) * q^82 + (8*b - 4) * q^83 + (-7*b + 11) * q^84 + (-4*b + 1) * q^85 + (3*b - 2) * q^86 + (6*b - 13) * q^87 + (-4*b - 3) * q^88 + 9 * q^89 + 2*b * q^90 + (5*b - 3) * q^92 + (8*b + 8) * q^94 + (-2*b + 3) * q^95 + (-9*b + 7) * q^96 + (-4*b + 1) * q^97 + (2*b + 8) * q^98 + (-4*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 5 * q^6 - 4 * q^7 + 4 * q^9 $$2 q - q^{2} - q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{7} + 4 q^{9} + q^{10} - 4 q^{11} - 5 q^{12} - 3 q^{14} - 3 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} + 7 q^{22} + 12 q^{23} - 10 q^{24} + 2 q^{25} + 7 q^{28} - 6 q^{29} - 5 q^{30} + 9 q^{32} + 10 q^{33} - 11 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{37} - 3 q^{38} - 6 q^{41} - 5 q^{42} - 8 q^{43} - 3 q^{44} - 4 q^{45} - 11 q^{46} - 8 q^{47} + 15 q^{48} + 4 q^{49} - q^{50} - 20 q^{51} + 12 q^{53} - 5 q^{54} + 4 q^{55} + 10 q^{56} - 10 q^{57} - 7 q^{58} - 12 q^{59} + 5 q^{60} + 2 q^{61} - 8 q^{63} + 4 q^{64} - 15 q^{66} - 8 q^{67} + 9 q^{68} - 10 q^{69} + 3 q^{70} + 8 q^{71} + 12 q^{73} + 3 q^{74} + 7 q^{76} - 2 q^{77} + 3 q^{80} - 22 q^{81} - 17 q^{82} + 15 q^{84} - 2 q^{85} - q^{86} - 20 q^{87} - 10 q^{88} + 18 q^{89} + 2 q^{90} - q^{92} + 24 q^{94} + 4 q^{95} + 5 q^{96} - 2 q^{97} + 18 q^{98} - 8 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 5 * q^6 - 4 * q^7 + 4 * q^9 + q^10 - 4 * q^11 - 5 * q^12 - 3 * q^14 - 3 * q^16 + 2 * q^17 - 2 * q^18 - 4 * q^19 + q^20 - 10 * q^21 + 7 * q^22 + 12 * q^23 - 10 * q^24 + 2 * q^25 + 7 * q^28 - 6 * q^29 - 5 * q^30 + 9 * q^32 + 10 * q^33 - 11 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^37 - 3 * q^38 - 6 * q^41 - 5 * q^42 - 8 * q^43 - 3 * q^44 - 4 * q^45 - 11 * q^46 - 8 * q^47 + 15 * q^48 + 4 * q^49 - q^50 - 20 * q^51 + 12 * q^53 - 5 * q^54 + 4 * q^55 + 10 * q^56 - 10 * q^57 - 7 * q^58 - 12 * q^59 + 5 * q^60 + 2 * q^61 - 8 * q^63 + 4 * q^64 - 15 * q^66 - 8 * q^67 + 9 * q^68 - 10 * q^69 + 3 * q^70 + 8 * q^71 + 12 * q^73 + 3 * q^74 + 7 * q^76 - 2 * q^77 + 3 * q^80 - 22 * q^81 - 17 * q^82 + 15 * q^84 - 2 * q^85 - q^86 - 20 * q^87 - 10 * q^88 + 18 * q^89 + 2 * q^90 - q^92 + 24 * q^94 + 4 * q^95 + 5 * q^96 - 2 * q^97 + 18 * q^98 - 8 * 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
−1.61803 −2.23607 0.618034 −1.00000 3.61803 0.236068 2.23607 2.00000 1.61803
1.2 0.618034 2.23607 −1.61803 −1.00000 1.38197 −4.23607 −2.23607 2.00000 −0.618034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$13$$ $$1$$

## 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 845.2.a.b 2
3.b odd 2 1 7605.2.a.bf 2
5.b even 2 1 4225.2.a.y 2
13.b even 2 1 845.2.a.e 2
13.c even 3 2 845.2.e.g 4
13.d odd 4 2 845.2.c.c 4
13.e even 6 2 65.2.e.a 4
13.f odd 12 4 845.2.m.e 8
39.d odd 2 1 7605.2.a.ba 2
39.h odd 6 2 585.2.j.e 4
52.i odd 6 2 1040.2.q.n 4
65.d even 2 1 4225.2.a.u 2
65.l even 6 2 325.2.e.b 4
65.r odd 12 4 325.2.o.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 13.e even 6 2
325.2.e.b 4 65.l even 6 2
325.2.o.a 8 65.r odd 12 4
585.2.j.e 4 39.h odd 6 2
845.2.a.b 2 1.a even 1 1 trivial
845.2.a.e 2 13.b even 2 1
845.2.c.c 4 13.d odd 4 2
845.2.e.g 4 13.c even 3 2
845.2.m.e 8 13.f odd 12 4
1040.2.q.n 4 52.i odd 6 2
4225.2.a.u 2 65.d even 2 1
4225.2.a.y 2 5.b even 2 1
7605.2.a.ba 2 39.d odd 2 1
7605.2.a.bf 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(845))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2} - 5$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T - 1$$
$11$ $$T^{2} + 4T - 1$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 2T - 19$$
$19$ $$T^{2} + 4T - 1$$
$23$ $$T^{2} - 12T + 31$$
$29$ $$T^{2} + 6T - 11$$
$31$ $$T^{2}$$
$37$ $$(T + 3)^{2}$$
$41$ $$T^{2} + 6T - 71$$
$43$ $$T^{2} + 8T + 11$$
$47$ $$T^{2} + 8T - 64$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12T - 9$$
$61$ $$T^{2} - 2T - 179$$
$67$ $$T^{2} + 8T - 29$$
$71$ $$T^{2} - 8T + 11$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 80$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} + 2T - 19$$