# Properties

 Label 1232.2.q.e Level $1232$ Weight $2$ Character orbit 1232.q Analytic conductor $9.838$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(177,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1232.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.q (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{3} - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (-3*z + 3) * q^3 - 2*z * q^5 + (z + 2) * q^7 - 6*z * q^9 $$q + ( - 3 \zeta_{6} + 3) q^{3} - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} - 6 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{11} - 7 q^{13} - 6 q^{15} + (2 \zeta_{6} - 2) q^{17} + ( - 6 \zeta_{6} + 9) q^{21} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 9 q^{27} - 5 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 3 \zeta_{6} q^{33} + ( - 6 \zeta_{6} + 2) q^{35} - 4 \zeta_{6} q^{37} + (21 \zeta_{6} - 21) q^{39} + 4 q^{41} + 8 q^{43} + (12 \zeta_{6} - 12) q^{45} + 2 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + 6 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} + 2 q^{55} + ( - 3 \zeta_{6} + 3) q^{59} - \zeta_{6} q^{61} + ( - 18 \zeta_{6} + 6) q^{63} + 14 \zeta_{6} q^{65} + ( - 9 \zeta_{6} + 9) q^{67} - 24 q^{69} + 2 q^{71} + (4 \zeta_{6} - 4) q^{73} - 3 \zeta_{6} q^{75} + (2 \zeta_{6} - 3) q^{77} + 9 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - 6 q^{83} + 4 q^{85} + (15 \zeta_{6} - 15) q^{87} - 6 \zeta_{6} q^{89} + ( - 7 \zeta_{6} - 14) q^{91} - 12 \zeta_{6} q^{93} + 7 q^{97} + 6 q^{99} +O(q^{100})$$ q + (-3*z + 3) * q^3 - 2*z * q^5 + (z + 2) * q^7 - 6*z * q^9 + (z - 1) * q^11 - 7 * q^13 - 6 * q^15 + (2*z - 2) * q^17 + (-6*z + 9) * q^21 - 8*z * q^23 + (-z + 1) * q^25 - 9 * q^27 - 5 * q^29 + (-4*z + 4) * q^31 + 3*z * q^33 + (-6*z + 2) * q^35 - 4*z * q^37 + (21*z - 21) * q^39 + 4 * q^41 + 8 * q^43 + (12*z - 12) * q^45 + 2*z * q^47 + (5*z + 3) * q^49 + 6*z * q^51 + (-6*z + 6) * q^53 + 2 * q^55 + (-3*z + 3) * q^59 - z * q^61 + (-18*z + 6) * q^63 + 14*z * q^65 + (-9*z + 9) * q^67 - 24 * q^69 + 2 * q^71 + (4*z - 4) * q^73 - 3*z * q^75 + (2*z - 3) * q^77 + 9*z * q^79 + (9*z - 9) * q^81 - 6 * q^83 + 4 * q^85 + (15*z - 15) * q^87 - 6*z * q^89 + (-7*z - 14) * q^91 - 12*z * q^93 + 7 * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 2 q^{5} + 5 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 2 * q^5 + 5 * q^7 - 6 * q^9 $$2 q + 3 q^{3} - 2 q^{5} + 5 q^{7} - 6 q^{9} - q^{11} - 14 q^{13} - 12 q^{15} - 2 q^{17} + 12 q^{21} - 8 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} + 4 q^{31} + 3 q^{33} - 2 q^{35} - 4 q^{37} - 21 q^{39} + 8 q^{41} + 16 q^{43} - 12 q^{45} + 2 q^{47} + 11 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} + 3 q^{59} - q^{61} - 6 q^{63} + 14 q^{65} + 9 q^{67} - 48 q^{69} + 4 q^{71} - 4 q^{73} - 3 q^{75} - 4 q^{77} + 9 q^{79} - 9 q^{81} - 12 q^{83} + 8 q^{85} - 15 q^{87} - 6 q^{89} - 35 q^{91} - 12 q^{93} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 2 * q^5 + 5 * q^7 - 6 * q^9 - q^11 - 14 * q^13 - 12 * q^15 - 2 * q^17 + 12 * q^21 - 8 * q^23 + q^25 - 18 * q^27 - 10 * q^29 + 4 * q^31 + 3 * q^33 - 2 * q^35 - 4 * q^37 - 21 * q^39 + 8 * q^41 + 16 * q^43 - 12 * q^45 + 2 * q^47 + 11 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^55 + 3 * q^59 - q^61 - 6 * q^63 + 14 * q^65 + 9 * q^67 - 48 * q^69 + 4 * q^71 - 4 * q^73 - 3 * q^75 - 4 * q^77 + 9 * q^79 - 9 * q^81 - 12 * q^83 + 8 * q^85 - 15 * q^87 - 6 * q^89 - 35 * q^91 - 12 * q^93 + 14 * q^97 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$$.

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 + 2.59808i 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 0 −3.00000 + 5.19615i 0
529.1 0 1.50000 2.59808i 0 −1.00000 1.73205i 0 2.50000 + 0.866025i 0 −3.00000 5.19615i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.e 2
4.b odd 2 1 154.2.e.a 2
7.c even 3 1 inner 1232.2.q.e 2
7.c even 3 1 8624.2.a.b 1
7.d odd 6 1 8624.2.a.be 1
12.b even 2 1 1386.2.k.o 2
28.d even 2 1 1078.2.e.f 2
28.f even 6 1 1078.2.a.g 1
28.f even 6 1 1078.2.e.f 2
28.g odd 6 1 154.2.e.a 2
28.g odd 6 1 1078.2.a.m 1
84.j odd 6 1 9702.2.a.y 1
84.n even 6 1 1386.2.k.o 2
84.n even 6 1 9702.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 4.b odd 2 1
154.2.e.a 2 28.g odd 6 1
1078.2.a.g 1 28.f even 6 1
1078.2.a.m 1 28.g odd 6 1
1078.2.e.f 2 28.d even 2 1
1078.2.e.f 2 28.f even 6 1
1232.2.q.e 2 1.a even 1 1 trivial
1232.2.q.e 2 7.c even 3 1 inner
1386.2.k.o 2 12.b even 2 1
1386.2.k.o 2 84.n even 6 1
8624.2.a.b 1 7.c even 3 1
8624.2.a.be 1 7.d odd 6 1
9702.2.a.i 1 84.n even 6 1
9702.2.a.y 1 84.j odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} - 3T_{3} + 9$$ T3^2 - 3*T3 + 9 $$T_{13} + 7$$ T13 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T + 7)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T - 4)^{2}$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} + 4T + 16$$
$79$ $$T^{2} - 9T + 81$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T - 7)^{2}$$