# Properties

 Label 1232.2.a.c Level $1232$ Weight $2$ Character orbit 1232.a Self dual yes Analytic conductor $9.838$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 2 q^{5} + q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + 2 * q^5 + q^7 + q^9 $$q - 2 q^{3} + 2 q^{5} + q^{7} + q^{9} - q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{19} - 2 q^{21} - 4 q^{23} - q^{25} + 4 q^{27} + 2 q^{29} + 10 q^{31} + 2 q^{33} + 2 q^{35} - 6 q^{37} + 8 q^{39} + 4 q^{43} + 2 q^{45} - 10 q^{47} + q^{49} - 14 q^{53} - 2 q^{55} + 8 q^{57} - 10 q^{59} - 8 q^{61} + q^{63} - 8 q^{65} - 8 q^{67} + 8 q^{69} + 4 q^{71} + 4 q^{73} + 2 q^{75} - q^{77} - 16 q^{79} - 11 q^{81} - 4 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{91} - 20 q^{93} - 8 q^{95} + 6 q^{97} - q^{99}+O(q^{100})$$ q - 2 * q^3 + 2 * q^5 + q^7 + q^9 - q^11 - 4 * q^13 - 4 * q^15 - 4 * q^19 - 2 * q^21 - 4 * q^23 - q^25 + 4 * q^27 + 2 * q^29 + 10 * q^31 + 2 * q^33 + 2 * q^35 - 6 * q^37 + 8 * q^39 + 4 * q^43 + 2 * q^45 - 10 * q^47 + q^49 - 14 * q^53 - 2 * q^55 + 8 * q^57 - 10 * q^59 - 8 * q^61 + q^63 - 8 * q^65 - 8 * q^67 + 8 * q^69 + 4 * q^71 + 4 * q^73 + 2 * q^75 - q^77 - 16 * q^79 - 11 * q^81 - 4 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^91 - 20 * q^93 - 8 * q^95 + 6 * q^97 - 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
0 −2.00000 0 2.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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.c 1
4.b odd 2 1 154.2.a.b 1
7.b odd 2 1 8624.2.a.z 1
8.b even 2 1 4928.2.a.bf 1
8.d odd 2 1 4928.2.a.d 1
12.b even 2 1 1386.2.a.f 1
20.d odd 2 1 3850.2.a.o 1
20.e even 4 2 3850.2.c.d 2
28.d even 2 1 1078.2.a.b 1
28.f even 6 2 1078.2.e.l 2
28.g odd 6 2 1078.2.e.h 2
44.c even 2 1 1694.2.a.i 1
84.h odd 2 1 9702.2.a.bz 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 4.b odd 2 1
1078.2.a.b 1 28.d even 2 1
1078.2.e.h 2 28.g odd 6 2
1078.2.e.l 2 28.f even 6 2
1232.2.a.c 1 1.a even 1 1 trivial
1386.2.a.f 1 12.b even 2 1
1694.2.a.i 1 44.c even 2 1
3850.2.a.o 1 20.d odd 2 1
3850.2.c.d 2 20.e even 4 2
4928.2.a.d 1 8.d odd 2 1
4928.2.a.bf 1 8.b even 2 1
8624.2.a.z 1 7.b odd 2 1
9702.2.a.bz 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1232))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 2$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T - 2$$
$31$ $$T - 10$$
$37$ $$T + 6$$
$41$ $$T$$
$43$ $$T - 4$$
$47$ $$T + 10$$
$53$ $$T + 14$$
$59$ $$T + 10$$
$61$ $$T + 8$$
$67$ $$T + 8$$
$71$ $$T - 4$$
$73$ $$T - 4$$
$79$ $$T + 16$$
$83$ $$T + 4$$
$89$ $$T - 10$$
$97$ $$T - 6$$
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