# Properties

 Label 1232.2.a.p Level $1232$ Weight $2$ Character orbit 1232.a Self dual yes Analytic conductor $9.838$ Analytic rank $0$ 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: $$0$$ 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 154) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + (\beta + 1) q^{5} - q^{7} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (b + 1) * q^5 - q^7 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} + (\beta + 1) q^{5} - q^{7} + (2 \beta + 3) q^{9} - q^{11} + (\beta - 1) q^{13} + (2 \beta + 6) q^{15} + ( - 2 \beta - 2) q^{17} + ( - \beta + 5) q^{19} + ( - \beta - 1) q^{21} - 4 q^{23} + (2 \beta + 1) q^{25} + (2 \beta + 10) q^{27} - 2 \beta q^{29} - 2 q^{31} + ( - \beta - 1) q^{33} + ( - \beta - 1) q^{35} + ( - 4 \beta - 2) q^{37} + 4 q^{39} + (2 \beta + 2) q^{41} + ( - 2 \beta + 6) q^{43} + (5 \beta + 13) q^{45} + 2 q^{47} + q^{49} + ( - 4 \beta - 12) q^{51} + ( - 2 \beta + 4) q^{53} + ( - \beta - 1) q^{55} + 4 \beta q^{57} + ( - \beta - 5) q^{59} + ( - \beta - 3) q^{61} + ( - 2 \beta - 3) q^{63} + 4 q^{65} + (6 \beta + 2) q^{67} + ( - 4 \beta - 4) q^{69} + (2 \beta - 2) q^{71} + ( - 4 \beta + 4) q^{73} + (3 \beta + 11) q^{75} + q^{77} + (6 \beta + 11) q^{81} + ( - 5 \beta + 1) q^{83} + ( - 4 \beta - 12) q^{85} + ( - 2 \beta - 10) q^{87} + 10 q^{89} + ( - \beta + 1) q^{91} + ( - 2 \beta - 2) q^{93} + 4 \beta q^{95} + ( - 2 \beta + 8) q^{97} + ( - 2 \beta - 3) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + (b + 1) * q^5 - q^7 + (2*b + 3) * q^9 - q^11 + (b - 1) * q^13 + (2*b + 6) * q^15 + (-2*b - 2) * q^17 + (-b + 5) * q^19 + (-b - 1) * q^21 - 4 * q^23 + (2*b + 1) * q^25 + (2*b + 10) * q^27 - 2*b * q^29 - 2 * q^31 + (-b - 1) * q^33 + (-b - 1) * q^35 + (-4*b - 2) * q^37 + 4 * q^39 + (2*b + 2) * q^41 + (-2*b + 6) * q^43 + (5*b + 13) * q^45 + 2 * q^47 + q^49 + (-4*b - 12) * q^51 + (-2*b + 4) * q^53 + (-b - 1) * q^55 + 4*b * q^57 + (-b - 5) * q^59 + (-b - 3) * q^61 + (-2*b - 3) * q^63 + 4 * q^65 + (6*b + 2) * q^67 + (-4*b - 4) * q^69 + (2*b - 2) * q^71 + (-4*b + 4) * q^73 + (3*b + 11) * q^75 + q^77 + (6*b + 11) * q^81 + (-5*b + 1) * q^83 + (-4*b - 12) * q^85 + (-2*b - 10) * q^87 + 10 * q^89 + (-b + 1) * q^91 + (-2*b - 2) * q^93 + 4*b * q^95 + (-2*b + 8) * q^97 + (-2*b - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 6 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{13} + 12 q^{15} - 4 q^{17} + 10 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 4 q^{37} + 8 q^{39} + 4 q^{41} + 12 q^{43} + 26 q^{45} + 4 q^{47} + 2 q^{49} - 24 q^{51} + 8 q^{53} - 2 q^{55} - 10 q^{59} - 6 q^{61} - 6 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 4 q^{71} + 8 q^{73} + 22 q^{75} + 2 q^{77} + 22 q^{81} + 2 q^{83} - 24 q^{85} - 20 q^{87} + 20 q^{89} + 2 q^{91} - 4 q^{93} + 16 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 6 * q^9 - 2 * q^11 - 2 * q^13 + 12 * q^15 - 4 * q^17 + 10 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 + 20 * q^27 - 4 * q^31 - 2 * q^33 - 2 * q^35 - 4 * q^37 + 8 * q^39 + 4 * q^41 + 12 * q^43 + 26 * q^45 + 4 * q^47 + 2 * q^49 - 24 * q^51 + 8 * q^53 - 2 * q^55 - 10 * q^59 - 6 * q^61 - 6 * q^63 + 8 * q^65 + 4 * q^67 - 8 * q^69 - 4 * q^71 + 8 * q^73 + 22 * q^75 + 2 * q^77 + 22 * q^81 + 2 * q^83 - 24 * q^85 - 20 * q^87 + 20 * q^89 + 2 * q^91 - 4 * q^93 + 16 * 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
 −0.618034 1.61803
0 −1.23607 0 −1.23607 0 −1.00000 0 −1.47214 0
1.2 0 3.23607 0 3.23607 0 −1.00000 0 7.47214 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.p 2
4.b odd 2 1 154.2.a.d 2
7.b odd 2 1 8624.2.a.bf 2
8.b even 2 1 4928.2.a.bk 2
8.d odd 2 1 4928.2.a.bt 2
12.b even 2 1 1386.2.a.m 2
20.d odd 2 1 3850.2.a.bj 2
20.e even 4 2 3850.2.c.q 4
28.d even 2 1 1078.2.a.w 2
28.f even 6 2 1078.2.e.n 4
28.g odd 6 2 1078.2.e.q 4
44.c even 2 1 1694.2.a.l 2
84.h odd 2 1 9702.2.a.cu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 4.b odd 2 1
1078.2.a.w 2 28.d even 2 1
1078.2.e.n 4 28.f even 6 2
1078.2.e.q 4 28.g odd 6 2
1232.2.a.p 2 1.a even 1 1 trivial
1386.2.a.m 2 12.b even 2 1
1694.2.a.l 2 44.c even 2 1
3850.2.a.bj 2 20.d odd 2 1
3850.2.c.q 4 20.e even 4 2
4928.2.a.bk 2 8.b even 2 1
4928.2.a.bt 2 8.d odd 2 1
8624.2.a.bf 2 7.b odd 2 1
9702.2.a.cu 2 84.h 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 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{13}^{2} + 2T_{13} - 4$$ T13^2 + 2*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} + 4T - 16$$
$19$ $$T^{2} - 10T + 20$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - 4T - 16$$
$43$ $$T^{2} - 12T + 16$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 10T + 20$$
$61$ $$T^{2} + 6T + 4$$
$67$ $$T^{2} - 4T - 176$$
$71$ $$T^{2} + 4T - 16$$
$73$ $$T^{2} - 8T - 64$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 2T - 124$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} - 16T + 44$$