Properties

Label 1216.2.s.b
Level $1216$
Weight $2$
Character orbit 1216.s
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
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(31,1216)]
 
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([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.s (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{2} - 2) q^{3} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{5} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{2} - 2) q^{3} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{5} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{7} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 1) q^{11} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 3) q^{15} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{17} + (5 \zeta_{12}^{2} - 3) q^{19} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{21} + (3 \zeta_{12}^{2} + \zeta_{12} + 3) q^{23} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{25} + ( - 6 \zeta_{12}^{2} + 3) q^{27} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{29} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 3) q^{31} + (6 \zeta_{12}^{3} - \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{33} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{35} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{37} + (6 \zeta_{12}^{3} - \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{41} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{43} + (5 \zeta_{12}^{2} - \zeta_{12} + 5) q^{47} + (6 \zeta_{12}^{3} - 12 \zeta_{12} - 5) q^{49} + ( - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{51} + ( - 4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{53} + ( - 7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 7 \zeta_{12} - 6) q^{55} + ( - 8 \zeta_{12}^{2} + 1) q^{57} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{59} + ( - 3 \zeta_{12}^{2} + 7 \zeta_{12} - 3) q^{61} + ( - \zeta_{12}^{2} - 6 \zeta_{12} - 1) q^{67} + (\zeta_{12}^{3} - 2 \zeta_{12} - 9) q^{69} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{71} + 5 \zeta_{12}^{2} q^{73} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{75} + (3 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 5) q^{77} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{79} + 9 \zeta_{12}^{2} q^{81} + (6 \zeta_{12}^{3} - 12 \zeta_{12} - 7) q^{83} + (4 \zeta_{12}^{2} - 8 \zeta_{12} + 4) q^{85} + (9 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{87} + ( - 4 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{89} + ( - 9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 9 \zeta_{12} + 6) q^{93} + ( - 3 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{95} + ( - 6 \zeta_{12}^{3} + \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 6 q^{5} - 4 q^{11} - 6 q^{15} + 4 q^{17} - 2 q^{19} - 6 q^{21} + 18 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 6 q^{33} + 12 q^{37} + 6 q^{41} + 30 q^{47} - 20 q^{49} - 12 q^{51} - 12 q^{53} - 18 q^{55} - 12 q^{57} - 18 q^{59} - 18 q^{61} - 6 q^{67} - 36 q^{69} + 12 q^{71} + 10 q^{73} + 12 q^{79} + 18 q^{81} - 28 q^{83} + 24 q^{85} - 24 q^{89} + 18 q^{93} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.50000 + 0.866025i 0 0.633975 0.366025i 0 4.73205i 0 0 0
31.2 0 −1.50000 + 0.866025i 0 2.36603 1.36603i 0 1.26795i 0 0 0
863.1 0 −1.50000 0.866025i 0 0.633975 + 0.366025i 0 4.73205i 0 0 0
863.2 0 −1.50000 0.866025i 0 2.36603 + 1.36603i 0 1.26795i 0 0 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
152.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.s.b yes 4
4.b odd 2 1 1216.2.s.f yes 4
8.b even 2 1 1216.2.s.e yes 4
8.d odd 2 1 1216.2.s.a 4
19.d odd 6 1 1216.2.s.a 4
76.f even 6 1 1216.2.s.e yes 4
152.l odd 6 1 1216.2.s.f yes 4
152.o even 6 1 inner 1216.2.s.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.s.a 4 8.d odd 2 1
1216.2.s.a 4 19.d odd 6 1
1216.2.s.b yes 4 1.a even 1 1 trivial
1216.2.s.b yes 4 152.o even 6 1 inner
1216.2.s.e yes 4 8.b even 2 1
1216.2.s.e yes 4 76.f even 6 1
1216.2.s.f yes 4 4.b odd 2 1
1216.2.s.f yes 4 152.l odd 6 1

Hecke kernels

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

\( T_{3}^{2} + 3T_{3} + 3 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{3} + 14T_{5}^{2} - 12T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 24 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + 134 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} - 21 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$43$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{4} - 30 T^{3} + 374 T^{2} + \cdots + 5476 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 86 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} - 21 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$73$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + 216 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$83$ \( (T^{2} + 14 T - 59)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} - 21 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
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