# Properties

 Label 4368.2.a.br Level $4368$ Weight $2$ Character orbit 4368.a Self dual yes Analytic conductor $34.879$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.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: $$34.8786556029$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 273) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta_{2} - 1) q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b2 - 1) * q^5 - q^7 + q^9 $$q - q^{3} + ( - \beta_{2} - 1) q^{5} - q^{7} + q^{9} + \beta_1 q^{11} + q^{13} + (\beta_{2} + 1) q^{15} + ( - \beta_{3} - 1) q^{17} + (\beta_{2} - \beta_1 - 1) q^{19} + q^{21} + (\beta_{3} - \beta_{2} - \beta_1) q^{23} + (\beta_{2} + \beta_1 + 2) q^{25} - q^{27} + (\beta_{2} - \beta_1 + 1) q^{29} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{31} - \beta_1 q^{33} + (\beta_{2} + 1) q^{35} + (2 \beta_{2} - 2 \beta_1 + 4) q^{37} - q^{39} + (2 \beta_{2} + \beta_1 - 4) q^{41} + (\beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} - 1) q^{45} + ( - \beta_{3} - \beta_{2} - 2) q^{47} + q^{49} + (\beta_{3} + 1) q^{51} + (\beta_{2} + \beta_1 + 1) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{3} - \beta_1 + 5) q^{59} + (2 \beta_{3} + 4) q^{61} - q^{63} + ( - \beta_{2} - 1) q^{65} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{67} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{69} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{71} + ( - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{73} + ( - \beta_{2} - \beta_1 - 2) q^{75} - \beta_1 q^{77} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{79} + q^{81} + (\beta_{3} - \beta_{2}) q^{83} + (2 \beta_{3} + 2 \beta_1 + 2) q^{85} + ( - \beta_{2} + \beta_1 - 1) q^{87} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{89} - q^{91} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{93} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{95} + (2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{97} + \beta_1 q^{99}+O(q^{100})$$ q - q^3 + (-b2 - 1) * q^5 - q^7 + q^9 + b1 * q^11 + q^13 + (b2 + 1) * q^15 + (-b3 - 1) * q^17 + (b2 - b1 - 1) * q^19 + q^21 + (b3 - b2 - b1) * q^23 + (b2 + b1 + 2) * q^25 - q^27 + (b2 - b1 + 1) * q^29 + (2*b3 + b2 - b1 + 1) * q^31 - b1 * q^33 + (b2 + 1) * q^35 + (2*b2 - 2*b1 + 4) * q^37 - q^39 + (2*b2 + b1 - 4) * q^41 + (b2 + b1 - 1) * q^43 + (-b2 - 1) * q^45 + (-b3 - b2 - 2) * q^47 + q^49 + (b3 + 1) * q^51 + (b2 + b1 + 1) * q^53 + (-2*b3 - 2*b2 - 4) * q^55 + (-b2 + b1 + 1) * q^57 + (-b3 - b1 + 5) * q^59 + (2*b3 + 4) * q^61 - q^63 + (-b2 - 1) * q^65 + (-2*b3 + 2*b2 + 2*b1 + 4) * q^67 + (-b3 + b2 + b1) * q^69 + (-2*b2 + 3*b1 - 2) * q^71 + (-2*b3 + 3*b2 - b1 - 3) * q^73 + (-b2 - b1 - 2) * q^75 - b1 * q^77 + (-2*b3 + b2 - b1 - 3) * q^79 + q^81 + (b3 - b2) * q^83 + (2*b3 + 2*b1 + 2) * q^85 + (-b2 + b1 - 1) * q^87 + (-3*b2 + 2*b1 - 3) * q^89 - q^91 + (-2*b3 - b2 + b1 - 1) * q^93 + (2*b3 + 3*b2 - b1 - 1) * q^95 + (2*b3 - b2 - b1 - 3) * q^97 + b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 $$4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 - 7 * q^19 + 4 * q^21 - 3 * q^23 + 9 * q^25 - 4 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + 3 * q^35 + 10 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^43 - 3 * q^45 - 5 * q^47 + 4 * q^49 + 2 * q^51 + 5 * q^53 - 10 * q^55 + 7 * q^57 + 20 * q^59 + 12 * q^61 - 4 * q^63 - 3 * q^65 + 22 * q^67 + 3 * q^69 - 13 * q^73 - 9 * q^75 - 2 * q^77 - 11 * q^79 + 4 * q^81 - q^83 + 8 * q^85 - q^87 - 5 * q^89 - 4 * q^91 + 3 * q^93 - 13 * q^95 - 17 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 7$$ 2*v^2 - 7
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 7 ) / 2$$ (b3 + 7) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 1$$ b2 + 2*b1 + 1

## 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
 2.36865 −0.787711 −2.10710 1.52616
0 −1.00000 0 −3.81471 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 −2.66208 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 0.926817 0 −1.00000 0 1.00000 0
1.4 0 −1.00000 0 2.54997 0 −1.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$

## 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 4368.2.a.br 4
4.b odd 2 1 273.2.a.e 4
12.b even 2 1 819.2.a.k 4
20.d odd 2 1 6825.2.a.bg 4
28.d even 2 1 1911.2.a.s 4
52.b odd 2 1 3549.2.a.w 4
84.h odd 2 1 5733.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.e 4 4.b odd 2 1
819.2.a.k 4 12.b even 2 1
1911.2.a.s 4 28.d even 2 1
3549.2.a.w 4 52.b odd 2 1
4368.2.a.br 4 1.a even 1 1 trivial
5733.2.a.bf 4 84.h odd 2 1
6825.2.a.bg 4 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4368))$$:

 $$T_{5}^{4} + 3T_{5}^{3} - 10T_{5}^{2} - 20T_{5} + 24$$ T5^4 + 3*T5^3 - 10*T5^2 - 20*T5 + 24 $$T_{11}^{4} - 2T_{11}^{3} - 24T_{11}^{2} + 32T_{11} + 96$$ T11^4 - 2*T11^3 - 24*T11^2 + 32*T11 + 96 $$T_{17}^{4} + 2T_{17}^{3} - 28T_{17}^{2} - 40T_{17} + 96$$ T17^4 + 2*T17^3 - 28*T17^2 - 40*T17 + 96

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4} + 3 T^{3} - 10 T^{2} - 20 T + 24$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} - 2 T^{3} - 24 T^{2} + 32 T + 96$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} + 2 T^{3} - 28 T^{2} - 40 T + 96$$
$19$ $$T^{4} + 7 T^{3} - 12 T^{2} - 48 T + 64$$
$23$ $$T^{4} + 3 T^{3} - 52 T^{2} - 256 T - 288$$
$29$ $$T^{4} - T^{3} - 30 T^{2} + 52 T + 72$$
$31$ $$T^{4} + 3 T^{3} - 128 T^{2} + \cdots + 3968$$
$37$ $$T^{4} - 10 T^{3} - 84 T^{2} + \cdots - 128$$
$41$ $$T^{4} + 16 T^{3} - 688 T - 1392$$
$43$ $$T^{4} + 3 T^{3} - 44 T^{2} - 112 T - 64$$
$47$ $$T^{4} + 5 T^{3} - 40 T^{2} - 16 T + 144$$
$53$ $$T^{4} - 5 T^{3} - 38 T^{2} + 68 T - 24$$
$59$ $$T^{4} - 20 T^{3} + 80 T^{2} + \cdots - 1536$$
$61$ $$T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 496$$
$67$ $$T^{4} - 22 T^{3} - 40 T^{2} + \cdots - 15488$$
$71$ $$T^{4} - 232 T^{2} - 304 T + 10176$$
$73$ $$T^{4} + 13 T^{3} - 166 T^{2} + \cdots - 11672$$
$79$ $$T^{4} + 11 T^{3} - 120 T^{2} + \cdots - 3456$$
$83$ $$T^{4} + T^{3} - 36 T^{2} + 80 T - 48$$
$89$ $$T^{4} + 5 T^{3} - 162 T^{2} + \cdots - 1704$$
$97$ $$T^{4} + 17 T^{3} - 14 T^{2} + \cdots - 1528$$