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 $1$

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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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (\beta + 1) q^{5} - q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + (\beta + 1) q^{5} - q^{7} + (2 \beta + 3) q^{9} - q^{11} + (\beta - 1) q^{13} + (2 \beta + 6) q^{15} + ( - 2 \beta - 2) q^{17} + ( - \beta + 5) q^{19} + ( - \beta - 1) q^{21} - 4 q^{23} + (2 \beta + 1) q^{25} + (2 \beta + 10) q^{27} - 2 \beta q^{29} - 2 q^{31} + ( - \beta - 1) q^{33} + ( - \beta - 1) q^{35} + ( - 4 \beta - 2) q^{37} + 4 q^{39} + (2 \beta + 2) q^{41} + ( - 2 \beta + 6) q^{43} + (5 \beta + 13) q^{45} + 2 q^{47} + q^{49} + ( - 4 \beta - 12) q^{51} + ( - 2 \beta + 4) q^{53} + ( - \beta - 1) q^{55} + 4 \beta q^{57} + ( - \beta - 5) q^{59} + ( - \beta - 3) q^{61} + ( - 2 \beta - 3) q^{63} + 4 q^{65} + (6 \beta + 2) q^{67} + ( - 4 \beta - 4) q^{69} + (2 \beta - 2) q^{71} + ( - 4 \beta + 4) q^{73} + (3 \beta + 11) q^{75} + q^{77} + (6 \beta + 11) q^{81} + ( - 5 \beta + 1) q^{83} + ( - 4 \beta - 12) q^{85} + ( - 2 \beta - 10) q^{87} + 10 q^{89} + ( - \beta + 1) q^{91} + ( - 2 \beta - 2) q^{93} + 4 \beta q^{95} + ( - 2 \beta + 8) q^{97} + ( - 2 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{13} + 12 q^{15} - 4 q^{17} + 10 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 4 q^{37} + 8 q^{39} + 4 q^{41} + 12 q^{43} + 26 q^{45} + 4 q^{47} + 2 q^{49} - 24 q^{51} + 8 q^{53} - 2 q^{55} - 10 q^{59} - 6 q^{61} - 6 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 4 q^{71} + 8 q^{73} + 22 q^{75} + 2 q^{77} + 22 q^{81} + 2 q^{83} - 24 q^{85} - 20 q^{87} + 20 q^{89} + 2 q^{91} - 4 q^{93} + 16 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{2} - 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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