Properties

Label 1232.2.e.c
Level $1232$
Weight $2$
Character orbit 1232.e
Analytic conductor $9.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,2,Mod(769,1232)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.e (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{3} q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{3} q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} - 2 q^{9} + ( - \beta_{2} + 3) q^{11} + ( - 2 \beta_{2} + 4 \beta_1) q^{13} - 5 q^{15} + ( - \beta_{2} + 2 \beta_1) q^{19} + (\beta_{2} + 3 \beta_1) q^{21} + 3 q^{23} + \beta_{3} q^{27} - \beta_{2} q^{29} - 3 \beta_{3} q^{31} + (3 \beta_{3} - \beta_{2} + 2 \beta_1) q^{33} + (\beta_{2} + 3 \beta_1) q^{35} - q^{37} + 10 \beta_{2} q^{39} + (3 \beta_{2} - 6 \beta_1) q^{41} - 3 \beta_{2} q^{43} - 2 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( - 3 \beta_{3} - 2) q^{49} + (3 \beta_{3} - \beta_{2} + 2 \beta_1) q^{55} + 5 \beta_{2} q^{57} - \beta_{3} q^{59} + (\beta_{2} - 2 \beta_1) q^{61} + (4 \beta_{2} - 2 \beta_1) q^{63} + 10 \beta_{2} q^{65} - 11 q^{67} + 3 \beta_{3} q^{69} - 9 q^{71} + (\beta_{2} - 2 \beta_1) q^{73} + ( - \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{77} + 6 \beta_{2} q^{79} - 11 q^{81} + (3 \beta_{2} - 6 \beta_1) q^{83} + ( - \beta_{2} + 2 \beta_1) q^{87} + \beta_{3} q^{89} + ( - 6 \beta_{3} + 10) q^{91} + 15 q^{93} + 5 \beta_{2} q^{95} - 3 \beta_{3} q^{97} + (2 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} + 12 q^{11} - 20 q^{15} + 12 q^{23} - 4 q^{37} - 8 q^{49} - 44 q^{67} - 36 q^{71} - 12 q^{77} - 44 q^{81} + 40 q^{91} + 60 q^{93} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
−1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
1.58114 + 0.707107i
0 2.23607i 0 2.23607i 0 −1.58114 2.12132i 0 −2.00000 0
769.2 0 2.23607i 0 2.23607i 0 1.58114 + 2.12132i 0 −2.00000 0
769.3 0 2.23607i 0 2.23607i 0 −1.58114 + 2.12132i 0 −2.00000 0
769.4 0 2.23607i 0 2.23607i 0 1.58114 2.12132i 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.e.c 4
4.b odd 2 1 77.2.b.b 4
7.b odd 2 1 inner 1232.2.e.c 4
11.b odd 2 1 inner 1232.2.e.c 4
12.b even 2 1 693.2.c.b 4
28.d even 2 1 77.2.b.b 4
28.f even 6 2 539.2.i.b 8
28.g odd 6 2 539.2.i.b 8
44.c even 2 1 77.2.b.b 4
44.g even 10 4 847.2.l.g 16
44.h odd 10 4 847.2.l.g 16
77.b even 2 1 inner 1232.2.e.c 4
84.h odd 2 1 693.2.c.b 4
132.d odd 2 1 693.2.c.b 4
308.g odd 2 1 77.2.b.b 4
308.m odd 6 2 539.2.i.b 8
308.n even 6 2 539.2.i.b 8
308.s odd 10 4 847.2.l.g 16
308.t even 10 4 847.2.l.g 16
924.n even 2 1 693.2.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.b 4 4.b odd 2 1
77.2.b.b 4 28.d even 2 1
77.2.b.b 4 44.c even 2 1
77.2.b.b 4 308.g odd 2 1
539.2.i.b 8 28.f even 6 2
539.2.i.b 8 28.g odd 6 2
539.2.i.b 8 308.m odd 6 2
539.2.i.b 8 308.n even 6 2
693.2.c.b 4 12.b even 2 1
693.2.c.b 4 84.h odd 2 1
693.2.c.b 4 132.d odd 2 1
693.2.c.b 4 924.n even 2 1
847.2.l.g 16 44.g even 10 4
847.2.l.g 16 44.h odd 10 4
847.2.l.g 16 308.s odd 10 4
847.2.l.g 16 308.t even 10 4
1232.2.e.c 4 1.a even 1 1 trivial
1232.2.e.c 4 7.b odd 2 1 inner
1232.2.e.c 4 11.b odd 2 1 inner
1232.2.e.c 4 77.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{13}^{2} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 4T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 11)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$23$ \( (T - 3)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$67$ \( (T + 11)^{4} \) Copy content Toggle raw display
$71$ \( (T + 9)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
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