# Properties

 Label 2368.2.a.u Level $2368$ Weight $2$ Character orbit 2368.a Self dual yes Analytic conductor $18.909$ 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] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2368 = 2^{6} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2368.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: $$18.9085751986$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + ( - 3 \beta + 1) q^{5} + 2 \beta q^{7} + (\beta - 2) q^{9} +O(q^{10})$$ q - b * q^3 + (-3*b + 1) * q^5 + 2*b * q^7 + (b - 2) * q^9 $$q - \beta q^{3} + ( - 3 \beta + 1) q^{5} + 2 \beta q^{7} + (\beta - 2) q^{9} + (\beta - 3) q^{11} + (3 \beta - 2) q^{13} + (2 \beta + 3) q^{15} + ( - 4 \beta + 2) q^{17} + (4 \beta - 2) q^{19} + ( - 2 \beta - 2) q^{21} + ( - 3 \beta + 2) q^{23} + (3 \beta + 5) q^{25} + (4 \beta - 1) q^{27} + (7 \beta - 2) q^{29} + (\beta - 9) q^{31} + (2 \beta - 1) q^{33} + ( - 4 \beta - 6) q^{35} + q^{37} + ( - \beta - 3) q^{39} + (\beta + 8) q^{41} + ( - 2 \beta - 2) q^{43} + (4 \beta - 5) q^{45} + (2 \beta - 2) q^{47} + (4 \beta - 3) q^{49} + (2 \beta + 4) q^{51} + ( - 4 \beta + 6) q^{53} + (7 \beta - 6) q^{55} + ( - 2 \beta - 4) q^{57} + (2 \beta - 8) q^{59} + ( - \beta - 9) q^{61} + ( - 2 \beta + 2) q^{63} - 11 q^{65} + (5 \beta - 7) q^{67} + (\beta + 3) q^{69} + ( - 8 \beta + 10) q^{71} + (5 \beta - 1) q^{73} + ( - 8 \beta - 3) q^{75} + ( - 4 \beta + 2) q^{77} + (9 \beta - 6) q^{79} + ( - 6 \beta + 2) q^{81} + ( - 4 \beta - 8) q^{83} + (2 \beta + 14) q^{85} + ( - 5 \beta - 7) q^{87} + (4 \beta - 8) q^{89} + (2 \beta + 6) q^{91} + (8 \beta - 1) q^{93} + ( - 2 \beta - 14) q^{95} + ( - 4 \beta + 6) q^{97} + ( - 4 \beta + 7) q^{99} +O(q^{100})$$ q - b * q^3 + (-3*b + 1) * q^5 + 2*b * q^7 + (b - 2) * q^9 + (b - 3) * q^11 + (3*b - 2) * q^13 + (2*b + 3) * q^15 + (-4*b + 2) * q^17 + (4*b - 2) * q^19 + (-2*b - 2) * q^21 + (-3*b + 2) * q^23 + (3*b + 5) * q^25 + (4*b - 1) * q^27 + (7*b - 2) * q^29 + (b - 9) * q^31 + (2*b - 1) * q^33 + (-4*b - 6) * q^35 + q^37 + (-b - 3) * q^39 + (b + 8) * q^41 + (-2*b - 2) * q^43 + (4*b - 5) * q^45 + (2*b - 2) * q^47 + (4*b - 3) * q^49 + (2*b + 4) * q^51 + (-4*b + 6) * q^53 + (7*b - 6) * q^55 + (-2*b - 4) * q^57 + (2*b - 8) * q^59 + (-b - 9) * q^61 + (-2*b + 2) * q^63 - 11 * q^65 + (5*b - 7) * q^67 + (b + 3) * q^69 + (-8*b + 10) * q^71 + (5*b - 1) * q^73 + (-8*b - 3) * q^75 + (-4*b + 2) * q^77 + (9*b - 6) * q^79 + (-6*b + 2) * q^81 + (-4*b - 8) * q^83 + (2*b + 14) * q^85 + (-5*b - 7) * q^87 + (4*b - 8) * q^89 + (2*b + 6) * q^91 + (8*b - 1) * q^93 + (-2*b - 14) * q^95 + (-4*b + 6) * q^97 + (-4*b + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - q^5 + 2 * q^7 - 3 * q^9 $$2 q - q^{3} - q^{5} + 2 q^{7} - 3 q^{9} - 5 q^{11} - q^{13} + 8 q^{15} - 6 q^{21} + q^{23} + 13 q^{25} + 2 q^{27} + 3 q^{29} - 17 q^{31} - 16 q^{35} + 2 q^{37} - 7 q^{39} + 17 q^{41} - 6 q^{43} - 6 q^{45} - 2 q^{47} - 2 q^{49} + 10 q^{51} + 8 q^{53} - 5 q^{55} - 10 q^{57} - 14 q^{59} - 19 q^{61} + 2 q^{63} - 22 q^{65} - 9 q^{67} + 7 q^{69} + 12 q^{71} + 3 q^{73} - 14 q^{75} - 3 q^{79} - 2 q^{81} - 20 q^{83} + 30 q^{85} - 19 q^{87} - 12 q^{89} + 14 q^{91} + 6 q^{93} - 30 q^{95} + 8 q^{97} + 10 q^{99}+O(q^{100})$$ 2 * q - q^3 - q^5 + 2 * q^7 - 3 * q^9 - 5 * q^11 - q^13 + 8 * q^15 - 6 * q^21 + q^23 + 13 * q^25 + 2 * q^27 + 3 * q^29 - 17 * q^31 - 16 * q^35 + 2 * q^37 - 7 * q^39 + 17 * q^41 - 6 * q^43 - 6 * q^45 - 2 * q^47 - 2 * q^49 + 10 * q^51 + 8 * q^53 - 5 * q^55 - 10 * q^57 - 14 * q^59 - 19 * q^61 + 2 * q^63 - 22 * q^65 - 9 * q^67 + 7 * q^69 + 12 * q^71 + 3 * q^73 - 14 * q^75 - 3 * q^79 - 2 * q^81 - 20 * q^83 + 30 * q^85 - 19 * q^87 - 12 * q^89 + 14 * q^91 + 6 * q^93 - 30 * q^95 + 8 * q^97 + 10 * 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 −1.61803 0 −3.85410 0 3.23607 0 −0.381966 0
1.2 0 0.618034 0 2.85410 0 −1.23607 0 −2.61803 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
$$2$$ $$-1$$
$$37$$ $$-1$$

## 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 2368.2.a.u 2
4.b odd 2 1 2368.2.a.y 2
8.b even 2 1 592.2.a.g 2
8.d odd 2 1 74.2.a.b 2
24.f even 2 1 666.2.a.i 2
24.h odd 2 1 5328.2.a.bc 2
40.e odd 2 1 1850.2.a.t 2
40.k even 4 2 1850.2.b.j 4
56.e even 2 1 3626.2.a.s 2
88.g even 2 1 8954.2.a.j 2
296.h odd 2 1 2738.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 8.d odd 2 1
592.2.a.g 2 8.b even 2 1
666.2.a.i 2 24.f even 2 1
1850.2.a.t 2 40.e odd 2 1
1850.2.b.j 4 40.k even 4 2
2368.2.a.u 2 1.a even 1 1 trivial
2368.2.a.y 2 4.b odd 2 1
2738.2.a.g 2 296.h odd 2 1
3626.2.a.s 2 56.e even 2 1
5328.2.a.bc 2 24.h odd 2 1
8954.2.a.j 2 88.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2368))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{5}^{2} + T_{5} - 11$$ T5^2 + T5 - 11 $$T_{7}^{2} - 2T_{7} - 4$$ T7^2 - 2*T7 - 4 $$T_{11}^{2} + 5T_{11} + 5$$ T11^2 + 5*T11 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2} + T - 11$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$T^{2} + 5T + 5$$
$13$ $$T^{2} + T - 11$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} - T - 11$$
$29$ $$T^{2} - 3T - 59$$
$31$ $$T^{2} + 17T + 71$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 17T + 71$$
$43$ $$T^{2} + 6T + 4$$
$47$ $$T^{2} + 2T - 4$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 14T + 44$$
$61$ $$T^{2} + 19T + 89$$
$67$ $$T^{2} + 9T - 11$$
$71$ $$T^{2} - 12T - 44$$
$73$ $$T^{2} - 3T - 29$$
$79$ $$T^{2} + 3T - 99$$
$83$ $$T^{2} + 20T + 80$$
$89$ $$T^{2} + 12T + 16$$
$97$ $$T^{2} - 8T - 4$$