# Properties

 Label 5929.2.a.p Level $5929$ Weight $2$ Character orbit 5929.a Self dual yes Analytic conductor $47.343$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5929 = 7^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5929.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.3433033584$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 847) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + \beta q^{3} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} + (\beta + 3) q^{6} + 3 q^{8} + \beta q^{9}+O(q^{10})$$ q + b * q^2 + b * q^3 + (b + 1) * q^4 + (-2*b + 1) * q^5 + (b + 3) * q^6 + 3 * q^8 + b * q^9 $$q + \beta q^{2} + \beta q^{3} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} + (\beta + 3) q^{6} + 3 q^{8} + \beta q^{9} + ( - \beta - 6) q^{10} + (2 \beta + 3) q^{12} + ( - 2 \beta - 2) q^{13} + ( - \beta - 6) q^{15} + (\beta - 2) q^{16} + (\beta - 5) q^{17} + (\beta + 3) q^{18} + 3 q^{19} + ( - 3 \beta - 5) q^{20} + (\beta - 5) q^{23} + 3 \beta q^{24} + 8 q^{25} + ( - 4 \beta - 6) q^{26} + ( - 2 \beta + 3) q^{27} + ( - \beta + 7) q^{29} + ( - 7 \beta - 3) q^{30} - q^{31} + ( - \beta - 3) q^{32} + ( - 4 \beta + 3) q^{34} + (2 \beta + 3) q^{36} + ( - 4 \beta + 4) q^{37} + 3 \beta q^{38} + ( - 4 \beta - 6) q^{39} + ( - 6 \beta + 3) q^{40} - 7 q^{41} + ( - \beta + 4) q^{43} + ( - \beta - 6) q^{45} + ( - 4 \beta + 3) q^{46} + (3 \beta - 5) q^{47} + ( - \beta + 3) q^{48} + 8 \beta q^{50} + ( - 4 \beta + 3) q^{51} + ( - 6 \beta - 8) q^{52} + ( - 3 \beta - 6) q^{53} + (\beta - 6) q^{54} + 3 \beta q^{57} + (6 \beta - 3) q^{58} + (\beta - 9) q^{59} + ( - 8 \beta - 9) q^{60} + (\beta + 2) q^{61} - \beta q^{62} + ( - 6 \beta + 1) q^{64} + (6 \beta + 10) q^{65} + (5 \beta - 3) q^{67} + ( - 3 \beta - 2) q^{68} + ( - 4 \beta + 3) q^{69} + ( - \beta - 2) q^{71} + 3 \beta q^{72} + 5 q^{73} - 12 q^{74} + 8 \beta q^{75} + (3 \beta + 3) q^{76} + ( - 10 \beta - 12) q^{78} + ( - \beta - 6) q^{79} + (3 \beta - 8) q^{80} + ( - 2 \beta - 6) q^{81} - 7 \beta q^{82} - 3 q^{83} + (9 \beta - 11) q^{85} + (3 \beta - 3) q^{86} + (6 \beta - 3) q^{87} + (9 \beta - 6) q^{89} + ( - 7 \beta - 3) q^{90} + ( - 3 \beta - 2) q^{92} - \beta q^{93} + ( - 2 \beta + 9) q^{94} + ( - 6 \beta + 3) q^{95} + ( - 4 \beta - 3) q^{96} + (2 \beta - 1) q^{97} +O(q^{100})$$ q + b * q^2 + b * q^3 + (b + 1) * q^4 + (-2*b + 1) * q^5 + (b + 3) * q^6 + 3 * q^8 + b * q^9 + (-b - 6) * q^10 + (2*b + 3) * q^12 + (-2*b - 2) * q^13 + (-b - 6) * q^15 + (b - 2) * q^16 + (b - 5) * q^17 + (b + 3) * q^18 + 3 * q^19 + (-3*b - 5) * q^20 + (b - 5) * q^23 + 3*b * q^24 + 8 * q^25 + (-4*b - 6) * q^26 + (-2*b + 3) * q^27 + (-b + 7) * q^29 + (-7*b - 3) * q^30 - q^31 + (-b - 3) * q^32 + (-4*b + 3) * q^34 + (2*b + 3) * q^36 + (-4*b + 4) * q^37 + 3*b * q^38 + (-4*b - 6) * q^39 + (-6*b + 3) * q^40 - 7 * q^41 + (-b + 4) * q^43 + (-b - 6) * q^45 + (-4*b + 3) * q^46 + (3*b - 5) * q^47 + (-b + 3) * q^48 + 8*b * q^50 + (-4*b + 3) * q^51 + (-6*b - 8) * q^52 + (-3*b - 6) * q^53 + (b - 6) * q^54 + 3*b * q^57 + (6*b - 3) * q^58 + (b - 9) * q^59 + (-8*b - 9) * q^60 + (b + 2) * q^61 - b * q^62 + (-6*b + 1) * q^64 + (6*b + 10) * q^65 + (5*b - 3) * q^67 + (-3*b - 2) * q^68 + (-4*b + 3) * q^69 + (-b - 2) * q^71 + 3*b * q^72 + 5 * q^73 - 12 * q^74 + 8*b * q^75 + (3*b + 3) * q^76 + (-10*b - 12) * q^78 + (-b - 6) * q^79 + (3*b - 8) * q^80 + (-2*b - 6) * q^81 - 7*b * q^82 - 3 * q^83 + (9*b - 11) * q^85 + (3*b - 3) * q^86 + (6*b - 3) * q^87 + (9*b - 6) * q^89 + (-7*b - 3) * q^90 + (-3*b - 2) * q^92 - b * q^93 + (-2*b + 9) * q^94 + (-6*b + 3) * q^95 + (-4*b - 3) * q^96 + (2*b - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} + 3 q^{4} + 7 q^{6} + 6 q^{8} + q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 + 3 * q^4 + 7 * q^6 + 6 * q^8 + q^9 $$2 q + q^{2} + q^{3} + 3 q^{4} + 7 q^{6} + 6 q^{8} + q^{9} - 13 q^{10} + 8 q^{12} - 6 q^{13} - 13 q^{15} - 3 q^{16} - 9 q^{17} + 7 q^{18} + 6 q^{19} - 13 q^{20} - 9 q^{23} + 3 q^{24} + 16 q^{25} - 16 q^{26} + 4 q^{27} + 13 q^{29} - 13 q^{30} - 2 q^{31} - 7 q^{32} + 2 q^{34} + 8 q^{36} + 4 q^{37} + 3 q^{38} - 16 q^{39} - 14 q^{41} + 7 q^{43} - 13 q^{45} + 2 q^{46} - 7 q^{47} + 5 q^{48} + 8 q^{50} + 2 q^{51} - 22 q^{52} - 15 q^{53} - 11 q^{54} + 3 q^{57} - 17 q^{59} - 26 q^{60} + 5 q^{61} - q^{62} - 4 q^{64} + 26 q^{65} - q^{67} - 7 q^{68} + 2 q^{69} - 5 q^{71} + 3 q^{72} + 10 q^{73} - 24 q^{74} + 8 q^{75} + 9 q^{76} - 34 q^{78} - 13 q^{79} - 13 q^{80} - 14 q^{81} - 7 q^{82} - 6 q^{83} - 13 q^{85} - 3 q^{86} - 3 q^{89} - 13 q^{90} - 7 q^{92} - q^{93} + 16 q^{94} - 10 q^{96}+O(q^{100})$$ 2 * q + q^2 + q^3 + 3 * q^4 + 7 * q^6 + 6 * q^8 + q^9 - 13 * q^10 + 8 * q^12 - 6 * q^13 - 13 * q^15 - 3 * q^16 - 9 * q^17 + 7 * q^18 + 6 * q^19 - 13 * q^20 - 9 * q^23 + 3 * q^24 + 16 * q^25 - 16 * q^26 + 4 * q^27 + 13 * q^29 - 13 * q^30 - 2 * q^31 - 7 * q^32 + 2 * q^34 + 8 * q^36 + 4 * q^37 + 3 * q^38 - 16 * q^39 - 14 * q^41 + 7 * q^43 - 13 * q^45 + 2 * q^46 - 7 * q^47 + 5 * q^48 + 8 * q^50 + 2 * q^51 - 22 * q^52 - 15 * q^53 - 11 * q^54 + 3 * q^57 - 17 * q^59 - 26 * q^60 + 5 * q^61 - q^62 - 4 * q^64 + 26 * q^65 - q^67 - 7 * q^68 + 2 * q^69 - 5 * q^71 + 3 * q^72 + 10 * q^73 - 24 * q^74 + 8 * q^75 + 9 * q^76 - 34 * q^78 - 13 * q^79 - 13 * q^80 - 14 * q^81 - 7 * q^82 - 6 * q^83 - 13 * q^85 - 3 * q^86 - 3 * q^89 - 13 * q^90 - 7 * q^92 - q^93 + 16 * q^94 - 10 * q^96

## 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
 −1.30278 2.30278
−1.30278 −1.30278 −0.302776 3.60555 1.69722 0 3.00000 −1.30278 −4.69722
1.2 2.30278 2.30278 3.30278 −3.60555 5.30278 0 3.00000 2.30278 −8.30278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 5929.2.a.p 2
7.b odd 2 1 847.2.a.g yes 2
11.b odd 2 1 5929.2.a.k 2
21.c even 2 1 7623.2.a.bc 2
77.b even 2 1 847.2.a.e 2
77.j odd 10 4 847.2.f.o 8
77.l even 10 4 847.2.f.r 8
231.h odd 2 1 7623.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 77.b even 2 1
847.2.a.g yes 2 7.b odd 2 1
847.2.f.o 8 77.j odd 10 4
847.2.f.r 8 77.l even 10 4
5929.2.a.k 2 11.b odd 2 1
5929.2.a.p 2 1.a even 1 1 trivial
7623.2.a.bc 2 21.c even 2 1
7623.2.a.bs 2 231.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5929))$$:

 $$T_{2}^{2} - T_{2} - 3$$ T2^2 - T2 - 3 $$T_{3}^{2} - T_{3} - 3$$ T3^2 - T3 - 3 $$T_{5}^{2} - 13$$ T5^2 - 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 3$$
$3$ $$T^{2} - T - 3$$
$5$ $$T^{2} - 13$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T - 4$$
$17$ $$T^{2} + 9T + 17$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} + 9T + 17$$
$29$ $$T^{2} - 13T + 39$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} - 4T - 48$$
$41$ $$(T + 7)^{2}$$
$43$ $$T^{2} - 7T + 9$$
$47$ $$T^{2} + 7T - 17$$
$53$ $$T^{2} + 15T + 27$$
$59$ $$T^{2} + 17T + 69$$
$61$ $$T^{2} - 5T + 3$$
$67$ $$T^{2} + T - 81$$
$71$ $$T^{2} + 5T + 3$$
$73$ $$(T - 5)^{2}$$
$79$ $$T^{2} + 13T + 39$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} + 3T - 261$$
$97$ $$T^{2} - 13$$