# Properties

 Label 1232.2.a.m Level $1232$ Weight $2$ Character orbit 1232.a Self dual yes Analytic conductor $9.838$ 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] = [1232,2,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1232.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.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: $$9.83756952902$$ 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: $$2$$ Twist minimal: no (minimal twist has level 77) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} - 2 q^{5} - q^{7} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (-b - 1) * q^3 - 2 * q^5 - q^7 + (2*b + 3) * q^9 $$q + ( - \beta - 1) q^{3} - 2 q^{5} - q^{7} + (2 \beta + 3) q^{9} + q^{11} + ( - \beta + 1) q^{13} + (2 \beta + 2) q^{15} + (\beta - 1) q^{17} + (2 \beta - 2) q^{19} + (\beta + 1) q^{21} + (2 \beta + 2) q^{23} - q^{25} + ( - 2 \beta - 10) q^{27} + ( - 2 \beta + 4) q^{29} + (\beta + 5) q^{31} + ( - \beta - 1) q^{33} + 2 q^{35} + (2 \beta - 4) q^{37} + 4 q^{39} + (\beta - 9) q^{41} - 8 q^{43} + ( - 4 \beta - 6) q^{45} + ( - \beta - 5) q^{47} + q^{49} - 4 q^{51} + (2 \beta + 4) q^{53} - 2 q^{55} - 8 q^{57} + ( - \beta - 1) q^{59} + (\beta - 5) q^{61} + ( - 2 \beta - 3) q^{63} + (2 \beta - 2) q^{65} + (2 \beta - 10) q^{67} + ( - 4 \beta - 12) q^{69} + ( - 2 \beta + 6) q^{71} + ( - \beta - 3) q^{73} + (\beta + 1) q^{75} - q^{77} - 4 \beta q^{79} + (6 \beta + 11) q^{81} + ( - 6 \beta - 2) q^{83} + ( - 2 \beta + 2) q^{85} + ( - 2 \beta + 6) q^{87} + 2 q^{89} + (\beta - 1) q^{91} + ( - 6 \beta - 10) q^{93} + ( - 4 \beta + 4) q^{95} + ( - 6 \beta + 4) q^{97} + (2 \beta + 3) q^{99} +O(q^{100})$$ q + (-b - 1) * q^3 - 2 * q^5 - q^7 + (2*b + 3) * q^9 + q^11 + (-b + 1) * q^13 + (2*b + 2) * q^15 + (b - 1) * q^17 + (2*b - 2) * q^19 + (b + 1) * q^21 + (2*b + 2) * q^23 - q^25 + (-2*b - 10) * q^27 + (-2*b + 4) * q^29 + (b + 5) * q^31 + (-b - 1) * q^33 + 2 * q^35 + (2*b - 4) * q^37 + 4 * q^39 + (b - 9) * q^41 - 8 * q^43 + (-4*b - 6) * q^45 + (-b - 5) * q^47 + q^49 - 4 * q^51 + (2*b + 4) * q^53 - 2 * q^55 - 8 * q^57 + (-b - 1) * q^59 + (b - 5) * q^61 + (-2*b - 3) * q^63 + (2*b - 2) * q^65 + (2*b - 10) * q^67 + (-4*b - 12) * q^69 + (-2*b + 6) * q^71 + (-b - 3) * q^73 + (b + 1) * q^75 - q^77 - 4*b * q^79 + (6*b + 11) * q^81 + (-6*b - 2) * q^83 + (-2*b + 2) * q^85 + (-2*b + 6) * q^87 + 2 * q^89 + (b - 1) * q^91 + (-6*b - 10) * q^93 + (-4*b + 4) * q^95 + (-6*b + 4) * q^97 + (2*b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9 $$2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 2 q^{13} + 4 q^{15} - 2 q^{17} - 4 q^{19} + 2 q^{21} + 4 q^{23} - 2 q^{25} - 20 q^{27} + 8 q^{29} + 10 q^{31} - 2 q^{33} + 4 q^{35} - 8 q^{37} + 8 q^{39} - 18 q^{41} - 16 q^{43} - 12 q^{45} - 10 q^{47} + 2 q^{49} - 8 q^{51} + 8 q^{53} - 4 q^{55} - 16 q^{57} - 2 q^{59} - 10 q^{61} - 6 q^{63} - 4 q^{65} - 20 q^{67} - 24 q^{69} + 12 q^{71} - 6 q^{73} + 2 q^{75} - 2 q^{77} + 22 q^{81} - 4 q^{83} + 4 q^{85} + 12 q^{87} + 4 q^{89} - 2 q^{91} - 20 q^{93} + 8 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9 + 2 * q^11 + 2 * q^13 + 4 * q^15 - 2 * q^17 - 4 * q^19 + 2 * q^21 + 4 * q^23 - 2 * q^25 - 20 * q^27 + 8 * q^29 + 10 * q^31 - 2 * q^33 + 4 * q^35 - 8 * q^37 + 8 * q^39 - 18 * q^41 - 16 * q^43 - 12 * q^45 - 10 * q^47 + 2 * q^49 - 8 * q^51 + 8 * q^53 - 4 * q^55 - 16 * q^57 - 2 * q^59 - 10 * q^61 - 6 * q^63 - 4 * q^65 - 20 * q^67 - 24 * q^69 + 12 * q^71 - 6 * q^73 + 2 * q^75 - 2 * q^77 + 22 * q^81 - 4 * q^83 + 4 * q^85 + 12 * q^87 + 4 * q^89 - 2 * q^91 - 20 * q^93 + 8 * q^95 + 8 * 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 −3.23607 0 −2.00000 0 −1.00000 0 7.47214 0
1.2 0 1.23607 0 −2.00000 0 −1.00000 0 −1.47214 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
$$2$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 1232.2.a.m 2
4.b odd 2 1 77.2.a.d 2
7.b odd 2 1 8624.2.a.ce 2
8.b even 2 1 4928.2.a.bv 2
8.d odd 2 1 4928.2.a.bm 2
12.b even 2 1 693.2.a.h 2
20.d odd 2 1 1925.2.a.r 2
20.e even 4 2 1925.2.b.h 4
28.d even 2 1 539.2.a.f 2
28.f even 6 2 539.2.e.j 4
28.g odd 6 2 539.2.e.i 4
44.c even 2 1 847.2.a.f 2
44.g even 10 2 847.2.f.b 4
44.g even 10 2 847.2.f.m 4
44.h odd 10 2 847.2.f.a 4
44.h odd 10 2 847.2.f.n 4
84.h odd 2 1 4851.2.a.y 2
132.d odd 2 1 7623.2.a.bl 2
308.g odd 2 1 5929.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 4.b odd 2 1
539.2.a.f 2 28.d even 2 1
539.2.e.i 4 28.g odd 6 2
539.2.e.j 4 28.f even 6 2
693.2.a.h 2 12.b even 2 1
847.2.a.f 2 44.c even 2 1
847.2.f.a 4 44.h odd 10 2
847.2.f.b 4 44.g even 10 2
847.2.f.m 4 44.g even 10 2
847.2.f.n 4 44.h odd 10 2
1232.2.a.m 2 1.a even 1 1 trivial
1925.2.a.r 2 20.d odd 2 1
1925.2.b.h 4 20.e even 4 2
4851.2.a.y 2 84.h odd 2 1
4928.2.a.bm 2 8.d odd 2 1
4928.2.a.bv 2 8.b even 2 1
5929.2.a.m 2 308.g odd 2 1
7623.2.a.bl 2 132.d odd 2 1
8624.2.a.ce 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1232))$$:

 $$T_{3}^{2} + 2T_{3} - 4$$ T3^2 + 2*T3 - 4 $$T_{5} + 2$$ T5 + 2 $$T_{13}^{2} - 2T_{13} - 4$$ T13^2 - 2*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$(T + 2)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} + 2T - 4$$
$19$ $$T^{2} + 4T - 16$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$T^{2} - 10T + 20$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} + 18T + 76$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 10T + 20$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 2T - 4$$
$61$ $$T^{2} + 10T + 20$$
$67$ $$T^{2} + 20T + 80$$
$71$ $$T^{2} - 12T + 16$$
$73$ $$T^{2} + 6T + 4$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} + 4T - 176$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} - 8T - 164$$