# Properties

 Label 3200.2.a.br Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, 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("3200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.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: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 640) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 + (b2 + 1) * q^7 + (b2 - b1 + 1) * q^9 $$q + (\beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + 2 \beta_1 q^{11} + (\beta_{2} + \beta_1 + 2) q^{13} + ( - 2 \beta_1 + 2) q^{17} + 2 \beta_{2} q^{19} + ( - \beta_{2} + 3 \beta_1 - 2) q^{21} + ( - \beta_{2} + 3) q^{23} + ( - 2 \beta_{2} - 2) q^{27} - 2 q^{29} + (2 \beta_{2} + 2 \beta_1 - 4) q^{31} + (2 \beta_{2} + 6) q^{33} + ( - \beta_{2} - 3 \beta_1 + 4) q^{37} + 4 \beta_1 q^{39} + ( - \beta_{2} - 3 \beta_1 + 2) q^{41} + (2 \beta_{2} - \beta_1 - 5) q^{43} + ( - \beta_{2} + 2 \beta_1 + 1) q^{47} + (\beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{51} + ( - \beta_{2} - \beta_1 + 6) q^{53} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{57} - 2 \beta_{2} q^{59} + ( - \beta_{2} + 3 \beta_1 - 4) q^{61} + (\beta_{2} - 4 \beta_1 + 9) q^{63} + (2 \beta_{2} - 3 \beta_1 - 3) q^{67} + (\beta_{2} + \beta_1 - 2) q^{69} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + 4 \beta_1 + 6) q^{73} + (6 \beta_1 - 2) q^{77} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + ( - 2 \beta_{2} - \beta_1 - 9) q^{83} + ( - 2 \beta_1 + 2) q^{87} + ( - 4 \beta_{2} - 2) q^{89} + (2 \beta_{2} + 2 \beta_1 + 8) q^{91} + 8 q^{93} + ( - 2 \beta_1 + 10) q^{97} + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 + (b2 + 1) * q^7 + (b2 - b1 + 1) * q^9 + 2*b1 * q^11 + (b2 + b1 + 2) * q^13 + (-2*b1 + 2) * q^17 + 2*b2 * q^19 + (-b2 + 3*b1 - 2) * q^21 + (-b2 + 3) * q^23 + (-2*b2 - 2) * q^27 - 2 * q^29 + (2*b2 + 2*b1 - 4) * q^31 + (2*b2 + 6) * q^33 + (-b2 - 3*b1 + 4) * q^37 + 4*b1 * q^39 + (-b2 - 3*b1 + 2) * q^41 + (2*b2 - b1 - 5) * q^43 + (-b2 + 2*b1 + 1) * q^47 + (b2 - b1 + 1) * q^49 + (-2*b2 + 2*b1 - 8) * q^51 + (-b2 - b1 + 6) * q^53 + (-2*b2 + 4*b1 - 2) * q^57 - 2*b2 * q^59 + (-b2 + 3*b1 - 4) * q^61 + (b2 - 4*b1 + 9) * q^63 + (2*b2 - 3*b1 - 3) * q^67 + (b2 + b1 - 2) * q^69 + (-2*b2 + 2*b1 + 4) * q^71 + (-2*b2 + 4*b1 + 6) * q^73 + (6*b1 - 2) * q^77 + (-2*b2 - 2*b1 - 4) * q^79 + (-b2 - 3*b1 + 1) * q^81 + (-2*b2 - b1 - 9) * q^83 + (-2*b1 + 2) * q^87 + (-4*b2 - 2) * q^89 + (2*b2 + 2*b1 + 8) * q^91 + 8 * q^93 + (-2*b1 + 10) * q^97 + (-2*b2 + 4*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 4 * q^7 + 3 * q^9 $$3 q - 2 q^{3} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 8 q^{13} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 8 q^{23} - 8 q^{27} - 6 q^{29} - 8 q^{31} + 20 q^{33} + 8 q^{37} + 4 q^{39} + 2 q^{41} - 14 q^{43} + 4 q^{47} + 3 q^{49} - 24 q^{51} + 16 q^{53} - 4 q^{57} - 2 q^{59} - 10 q^{61} + 24 q^{63} - 10 q^{67} - 4 q^{69} + 12 q^{71} + 20 q^{73} - 16 q^{79} - q^{81} - 30 q^{83} + 4 q^{87} - 10 q^{89} + 28 q^{91} + 24 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 8 * q^13 + 4 * q^17 + 2 * q^19 - 4 * q^21 + 8 * q^23 - 8 * q^27 - 6 * q^29 - 8 * q^31 + 20 * q^33 + 8 * q^37 + 4 * q^39 + 2 * q^41 - 14 * q^43 + 4 * q^47 + 3 * q^49 - 24 * q^51 + 16 * q^53 - 4 * q^57 - 2 * q^59 - 10 * q^61 + 24 * q^63 - 10 * q^67 - 4 * q^69 + 12 * q^71 + 20 * q^73 - 16 * q^79 - q^81 - 30 * q^83 + 4 * q^87 - 10 * q^89 + 28 * q^91 + 24 * q^93 + 28 * q^97 - 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

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

## 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
 0.311108 −1.48119 2.17009
0 −2.90321 0 0 0 3.52543 0 5.42864 0
1.2 0 −0.806063 0 0 0 −2.15633 0 −2.35026 0
1.3 0 1.70928 0 0 0 2.63090 0 −0.0783777 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
$$2$$ $$1$$
$$5$$ $$-1$$

## 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 3200.2.a.br 3
4.b odd 2 1 3200.2.a.bs 3
5.b even 2 1 3200.2.a.bt 3
5.c odd 4 2 640.2.c.b yes 6
8.b even 2 1 3200.2.a.bu 3
8.d odd 2 1 3200.2.a.bp 3
20.d odd 2 1 3200.2.a.bq 3
20.e even 4 2 640.2.c.a 6
40.e odd 2 1 3200.2.a.bv 3
40.f even 2 1 3200.2.a.bo 3
40.i odd 4 2 640.2.c.c yes 6
40.k even 4 2 640.2.c.d yes 6
80.i odd 4 2 1280.2.f.k 6
80.j even 4 2 1280.2.f.l 6
80.s even 4 2 1280.2.f.i 6
80.t odd 4 2 1280.2.f.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.c.a 6 20.e even 4 2
640.2.c.b yes 6 5.c odd 4 2
640.2.c.c yes 6 40.i odd 4 2
640.2.c.d yes 6 40.k even 4 2
1280.2.f.i 6 80.s even 4 2
1280.2.f.j 6 80.t odd 4 2
1280.2.f.k 6 80.i odd 4 2
1280.2.f.l 6 80.j even 4 2
3200.2.a.bo 3 40.f even 2 1
3200.2.a.bp 3 8.d odd 2 1
3200.2.a.bq 3 20.d odd 2 1
3200.2.a.br 3 1.a even 1 1 trivial
3200.2.a.bs 3 4.b odd 2 1
3200.2.a.bt 3 5.b even 2 1
3200.2.a.bu 3 8.b even 2 1
3200.2.a.bv 3 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3200))$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4$$ T3^3 + 2*T3^2 - 4*T3 - 4 $$T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 20$$ T7^3 - 4*T7^2 - 4*T7 + 20 $$T_{11}^{3} - 2T_{11}^{2} - 20T_{11} + 8$$ T11^3 - 2*T11^2 - 20*T11 + 8 $$T_{13}^{3} - 8T_{13}^{2} + 8T_{13} + 16$$ T13^3 - 8*T13^2 + 8*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 2 T^{2} - 4 T - 4$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 4 T^{2} - 4 T + 20$$
$11$ $$T^{3} - 2 T^{2} - 20 T + 8$$
$13$ $$T^{3} - 8 T^{2} + 8 T + 16$$
$17$ $$T^{3} - 4 T^{2} - 16 T + 32$$
$19$ $$T^{3} - 2 T^{2} - 36 T + 104$$
$23$ $$T^{3} - 8 T^{2} + 12 T - 4$$
$29$ $$(T + 2)^{3}$$
$31$ $$T^{3} + 8 T^{2} - 32 T - 128$$
$37$ $$T^{3} - 8 T^{2} - 32 T + 272$$
$41$ $$T^{3} - 2 T^{2} - 52 T + 184$$
$43$ $$T^{3} + 14 T^{2} + 20 T - 100$$
$47$ $$T^{3} - 4 T^{2} - 28 T + 116$$
$53$ $$T^{3} - 16 T^{2} + 72 T - 80$$
$59$ $$T^{3} + 2 T^{2} - 36 T - 104$$
$61$ $$T^{3} + 10 T^{2} - 28 T - 8$$
$67$ $$T^{3} + 10 T^{2} - 60 T - 604$$
$71$ $$T^{3} - 12 T^{2} - 16 T + 320$$
$73$ $$T^{3} - 20T^{2} + 1184$$
$79$ $$T^{3} + 16 T^{2} + 32 T - 128$$
$83$ $$T^{3} + 30 T^{2} + 260 T + 524$$
$89$ $$T^{3} + 10 T^{2} - 116 T - 1096$$
$97$ $$T^{3} - 28 T^{2} + 240 T - 608$$
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