Properties

Label 1216.2.i.c
Level $1216$
Weight $2$
Character orbit 1216.i
Analytic conductor $9.710$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(577,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.i (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: \(9.70980888579\)
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{5} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{5} + 2 \zeta_{6} q^{9} + 4 q^{11} - \zeta_{6} q^{13} - \zeta_{6} q^{15} + (3 \zeta_{6} - 3) q^{17} + ( - 2 \zeta_{6} + 5) q^{19} - 5 \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} - 5 q^{27} + 7 \zeta_{6} q^{29} + 4 q^{31} + (4 \zeta_{6} - 4) q^{33} - 10 q^{37} + q^{39} + ( - 5 \zeta_{6} + 5) q^{41} + (5 \zeta_{6} - 5) q^{43} - 2 q^{45} + 7 \zeta_{6} q^{47} - 7 q^{49} - 3 \zeta_{6} q^{51} + 11 \zeta_{6} q^{53} + (4 \zeta_{6} - 4) q^{55} + (5 \zeta_{6} - 3) q^{57} + ( - 3 \zeta_{6} + 3) q^{59} + 11 \zeta_{6} q^{61} + q^{65} - 3 \zeta_{6} q^{67} + 5 q^{69} + (11 \zeta_{6} - 11) q^{71} + (15 \zeta_{6} - 15) q^{73} - 4 q^{75} + ( - 13 \zeta_{6} + 13) q^{79} + (\zeta_{6} - 1) q^{81} - 3 \zeta_{6} q^{85} - 7 q^{87} - 3 \zeta_{6} q^{89} + (4 \zeta_{6} - 4) q^{93} + (5 \zeta_{6} - 3) q^{95} + ( - 5 \zeta_{6} + 5) q^{97} + 8 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} + 2 q^{9} + 8 q^{11} - q^{13} - q^{15} - 3 q^{17} + 8 q^{19} - 5 q^{23} + 4 q^{25} - 10 q^{27} + 7 q^{29} + 8 q^{31} - 4 q^{33} - 20 q^{37} + 2 q^{39} + 5 q^{41} - 5 q^{43} - 4 q^{45} + 7 q^{47} - 14 q^{49} - 3 q^{51} + 11 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} + 11 q^{61} + 2 q^{65} - 3 q^{67} + 10 q^{69} - 11 q^{71} - 15 q^{73} - 8 q^{75} + 13 q^{79} - q^{81} - 3 q^{85} - 14 q^{87} - 3 q^{89} - 4 q^{93} - q^{95} + 5 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 1.73205i 0
961.1 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 + 1.73205i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.c 2
4.b odd 2 1 1216.2.i.g 2
8.b even 2 1 76.2.e.a 2
8.d odd 2 1 304.2.i.a 2
19.c even 3 1 inner 1216.2.i.c 2
24.f even 2 1 2736.2.s.g 2
24.h odd 2 1 684.2.k.b 2
40.f even 2 1 1900.2.i.a 2
40.i odd 4 2 1900.2.s.a 4
76.g odd 6 1 1216.2.i.g 2
152.g odd 2 1 1444.2.e.b 2
152.k odd 6 1 304.2.i.a 2
152.k odd 6 1 5776.2.a.k 1
152.l odd 6 1 1444.2.a.c 1
152.l odd 6 1 1444.2.e.b 2
152.o even 6 1 5776.2.a.f 1
152.p even 6 1 76.2.e.a 2
152.p even 6 1 1444.2.a.b 1
456.u even 6 1 2736.2.s.g 2
456.x odd 6 1 684.2.k.b 2
760.z even 6 1 1900.2.i.a 2
760.br odd 12 2 1900.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 8.b even 2 1
76.2.e.a 2 152.p even 6 1
304.2.i.a 2 8.d odd 2 1
304.2.i.a 2 152.k odd 6 1
684.2.k.b 2 24.h odd 2 1
684.2.k.b 2 456.x odd 6 1
1216.2.i.c 2 1.a even 1 1 trivial
1216.2.i.c 2 19.c even 3 1 inner
1216.2.i.g 2 4.b odd 2 1
1216.2.i.g 2 76.g odd 6 1
1444.2.a.b 1 152.p even 6 1
1444.2.a.c 1 152.l odd 6 1
1444.2.e.b 2 152.g odd 2 1
1444.2.e.b 2 152.l odd 6 1
1900.2.i.a 2 40.f even 2 1
1900.2.i.a 2 760.z even 6 1
1900.2.s.a 4 40.i odd 4 2
1900.2.s.a 4 760.br odd 12 2
2736.2.s.g 2 24.f even 2 1
2736.2.s.g 2 456.u even 6 1
5776.2.a.f 1 152.o even 6 1
5776.2.a.k 1 152.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$73$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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