# Properties

 Label 6592.2.a.t Level $6592$ Weight $2$ Character orbit 6592.a Self dual yes Analytic conductor $52.637$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6592,2,Mod(1,6592)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6592 = 2^{6} \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6592.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$52.6373850124$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( - \beta + 2) q^{5} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + (-b + 2) * q^5 - q^7 - 2 * q^9 $$q + q^{3} + ( - \beta + 2) q^{5} - q^{7} - 2 q^{9} + (\beta + 1) q^{11} + 3 \beta q^{13} + ( - \beta + 2) q^{15} + ( - \beta - 4) q^{17} + ( - 3 \beta - 1) q^{19} - q^{21} + (4 \beta - 2) q^{23} - 3 \beta q^{25} - 5 q^{27} + (2 \beta + 2) q^{29} + ( - 6 \beta + 3) q^{31} + (\beta + 1) q^{33} + (\beta - 2) q^{35} + ( - 6 \beta + 3) q^{37} + 3 \beta q^{39} + (8 \beta - 4) q^{41} + (6 \beta - 1) q^{43} + (2 \beta - 4) q^{45} + (5 \beta - 4) q^{47} - 6 q^{49} + ( - \beta - 4) q^{51} + ( - 5 \beta + 7) q^{53} + q^{55} + ( - 3 \beta - 1) q^{57} + (\beta - 8) q^{59} + (3 \beta - 9) q^{61} + 2 q^{63} + (3 \beta - 3) q^{65} + ( - 12 \beta + 5) q^{67} + (4 \beta - 2) q^{69} + ( - 5 \beta + 4) q^{71} + ( - 3 \beta - 6) q^{73} - 3 \beta q^{75} + ( - \beta - 1) q^{77} + ( - 9 \beta + 8) q^{79} + q^{81} + (7 \beta - 5) q^{83} + (3 \beta - 7) q^{85} + (2 \beta + 2) q^{87} + (6 \beta - 12) q^{89} - 3 \beta q^{91} + ( - 6 \beta + 3) q^{93} + ( - 2 \beta + 1) q^{95} + ( - 6 \beta + 8) q^{97} + ( - 2 \beta - 2) q^{99} +O(q^{100})$$ q + q^3 + (-b + 2) * q^5 - q^7 - 2 * q^9 + (b + 1) * q^11 + 3*b * q^13 + (-b + 2) * q^15 + (-b - 4) * q^17 + (-3*b - 1) * q^19 - q^21 + (4*b - 2) * q^23 - 3*b * q^25 - 5 * q^27 + (2*b + 2) * q^29 + (-6*b + 3) * q^31 + (b + 1) * q^33 + (b - 2) * q^35 + (-6*b + 3) * q^37 + 3*b * q^39 + (8*b - 4) * q^41 + (6*b - 1) * q^43 + (2*b - 4) * q^45 + (5*b - 4) * q^47 - 6 * q^49 + (-b - 4) * q^51 + (-5*b + 7) * q^53 + q^55 + (-3*b - 1) * q^57 + (b - 8) * q^59 + (3*b - 9) * q^61 + 2 * q^63 + (3*b - 3) * q^65 + (-12*b + 5) * q^67 + (4*b - 2) * q^69 + (-5*b + 4) * q^71 + (-3*b - 6) * q^73 - 3*b * q^75 + (-b - 1) * q^77 + (-9*b + 8) * q^79 + q^81 + (7*b - 5) * q^83 + (3*b - 7) * q^85 + (2*b + 2) * q^87 + (6*b - 12) * q^89 - 3*b * q^91 + (-6*b + 3) * q^93 + (-2*b + 1) * q^95 + (-6*b + 8) * q^97 + (-2*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 - 4 * q^9 $$2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} - 4 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{15} - 9 q^{17} - 5 q^{19} - 2 q^{21} - 3 q^{25} - 10 q^{27} + 6 q^{29} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 4 q^{43} - 6 q^{45} - 3 q^{47} - 12 q^{49} - 9 q^{51} + 9 q^{53} + 2 q^{55} - 5 q^{57} - 15 q^{59} - 15 q^{61} + 4 q^{63} - 3 q^{65} - 2 q^{67} + 3 q^{71} - 15 q^{73} - 3 q^{75} - 3 q^{77} + 7 q^{79} + 2 q^{81} - 3 q^{83} - 11 q^{85} + 6 q^{87} - 18 q^{89} - 3 q^{91} + 10 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 - 4 * q^9 + 3 * q^11 + 3 * q^13 + 3 * q^15 - 9 * q^17 - 5 * q^19 - 2 * q^21 - 3 * q^25 - 10 * q^27 + 6 * q^29 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 4 * q^43 - 6 * q^45 - 3 * q^47 - 12 * q^49 - 9 * q^51 + 9 * q^53 + 2 * q^55 - 5 * q^57 - 15 * q^59 - 15 * q^61 + 4 * q^63 - 3 * q^65 - 2 * q^67 + 3 * q^71 - 15 * q^73 - 3 * q^75 - 3 * q^77 + 7 * q^79 + 2 * q^81 - 3 * q^83 - 11 * q^85 + 6 * q^87 - 18 * q^89 - 3 * q^91 + 10 * q^97 - 6 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
0 1.00000 0 0.381966 0 −1.00000 0 −2.00000 0
1.2 0 1.00000 0 2.61803 0 −1.00000 0 −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
$$2$$ $$+1$$
$$103$$ $$+1$$

## 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 6592.2.a.t 2
4.b odd 2 1 6592.2.a.h 2
8.b even 2 1 103.2.a.a 2
8.d odd 2 1 1648.2.a.f 2
24.h odd 2 1 927.2.a.b 2
40.f even 2 1 2575.2.a.g 2
56.h odd 2 1 5047.2.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.2.a.a 2 8.b even 2 1
927.2.a.b 2 24.h odd 2 1
1648.2.a.f 2 8.d odd 2 1
2575.2.a.g 2 40.f even 2 1
5047.2.a.a 2 56.h odd 2 1
6592.2.a.h 2 4.b odd 2 1
6592.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6592))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5}^{2} - 3T_{5} + 1$$ T5^2 - 3*T5 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{11}^{2} - 3T_{11} + 1$$ T11^2 - 3*T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 3T + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 3T + 1$$
$13$ $$T^{2} - 3T - 9$$
$17$ $$T^{2} + 9T + 19$$
$19$ $$T^{2} + 5T - 5$$
$23$ $$T^{2} - 20$$
$29$ $$T^{2} - 6T + 4$$
$31$ $$T^{2} - 45$$
$37$ $$T^{2} - 45$$
$41$ $$T^{2} - 80$$
$43$ $$T^{2} - 4T - 41$$
$47$ $$T^{2} + 3T - 29$$
$53$ $$T^{2} - 9T - 11$$
$59$ $$T^{2} + 15T + 55$$
$61$ $$T^{2} + 15T + 45$$
$67$ $$T^{2} + 2T - 179$$
$71$ $$T^{2} - 3T - 29$$
$73$ $$T^{2} + 15T + 45$$
$79$ $$T^{2} - 7T - 89$$
$83$ $$T^{2} + 3T - 59$$
$89$ $$T^{2} + 18T + 36$$
$97$ $$T^{2} - 10T - 20$$