Properties

Label 1134.2.f.r
Level $1134$
Weight $2$
Character orbit 1134.f
Analytic conductor $9.055$
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] = [1134,2,Mod(379,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.f (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: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{5} + \beta_1 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{5} + \beta_1 q^{7} + q^{8} + ( - \beta_{3} - 2) q^{10} + ( - 3 \beta_{2} + \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{13} + ( - \beta_1 + 1) q^{14} - \beta_1 q^{16} - 7 q^{17} + ( - \beta_{3} - 1) q^{19} + (\beta_{2} + 2 \beta_1) q^{20} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{22} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{23} + ( - 4 \beta_{2} - 2 \beta_1) q^{25} + (2 \beta_{3} - 3) q^{26} - q^{28} + (2 \beta_{2} + 5 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{31} + (\beta_1 - 1) q^{32} + 7 \beta_1 q^{34} + (\beta_{3} + 2) q^{35} + ( - 5 \beta_{3} + 2) q^{37} + (\beta_{2} + \beta_1) q^{38} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{40} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{43} + (3 \beta_{3} - 1) q^{44} + ( - 3 \beta_{3} - 1) q^{46} + ( - \beta_{2} + 3 \beta_1) q^{47} + (\beta_1 - 1) q^{49} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{50} + ( - 2 \beta_{2} + 3 \beta_1) q^{52} + ( - 2 \beta_{3} + 6) q^{53} + ( - 5 \beta_{3} - 7) q^{55} + \beta_1 q^{56} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 5) q^{58} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{59} + (4 \beta_{2} + 3 \beta_1) q^{61} + ( - 3 \beta_{3} + 3) q^{62} + q^{64} + \beta_{2} q^{65} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{67} + ( - 7 \beta_1 + 7) q^{68} + ( - \beta_{2} - 2 \beta_1) q^{70} + ( - 2 \beta_{3} - 10) q^{71} + (\beta_{3} + 10) q^{73} + (5 \beta_{2} - 2 \beta_1) q^{74} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{76} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{77} + ( - 7 \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{3} - 2) q^{80} + ( - 2 \beta_{3} - 6) q^{82} + ( - 9 \beta_{2} + \beta_1) q^{83} + ( - 7 \beta_{3} + 7 \beta_{2} + 14 \beta_1 - 14) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{86} + ( - 3 \beta_{2} + \beta_1) q^{88} + (4 \beta_{3} - 3) q^{89} + ( - 2 \beta_{3} + 3) q^{91} + (3 \beta_{2} + \beta_1) q^{92} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{94} + ( - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{95} + (4 \beta_{2} - 4 \beta_1) q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 2 q^{7} + 4 q^{8} - 8 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{14} - 2 q^{16} - 28 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{23} - 4 q^{25} - 12 q^{26} - 4 q^{28} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34} + 8 q^{35} + 8 q^{37} + 2 q^{38} + 4 q^{40} + 12 q^{41} - 4 q^{43} - 4 q^{44} - 4 q^{46} + 6 q^{47} - 2 q^{49} - 4 q^{50} + 6 q^{52} + 24 q^{53} - 28 q^{55} + 2 q^{56} + 10 q^{58} - 2 q^{59} + 6 q^{61} + 12 q^{62} + 4 q^{64} + 10 q^{67} + 14 q^{68} - 4 q^{70} - 40 q^{71} + 40 q^{73} - 4 q^{74} + 2 q^{76} - 2 q^{77} - 6 q^{79} - 8 q^{80} - 24 q^{82} + 2 q^{83} - 28 q^{85} - 4 q^{86} + 2 q^{88} - 12 q^{89} + 12 q^{91} + 2 q^{92} + 6 q^{94} - 10 q^{95} - 8 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0.133975 + 0.232051i 0 0.500000 0.866025i 1.00000 0 −0.267949
379.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.86603 + 3.23205i 0 0.500000 0.866025i 1.00000 0 −3.73205
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.133975 0.232051i 0 0.500000 + 0.866025i 1.00000 0 −0.267949
757.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.86603 3.23205i 0 0.500000 + 0.866025i 1.00000 0 −3.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.f.r 4
3.b odd 2 1 1134.2.f.s 4
9.c even 3 1 1134.2.a.m yes 2
9.c even 3 1 inner 1134.2.f.r 4
9.d odd 6 1 1134.2.a.l 2
9.d odd 6 1 1134.2.f.s 4
36.f odd 6 1 9072.2.a.y 2
36.h even 6 1 9072.2.a.bp 2
63.l odd 6 1 7938.2.a.bt 2
63.o even 6 1 7938.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 9.d odd 6 1
1134.2.a.m yes 2 9.c even 3 1
1134.2.f.r 4 1.a even 1 1 trivial
1134.2.f.r 4 9.c even 3 1 inner
1134.2.f.s 4 3.b odd 2 1
1134.2.f.s 4 9.d odd 6 1
7938.2.a.bg 2 63.o even 6 1
7938.2.a.bt 2 63.l odd 6 1
9072.2.a.y 2 36.f odd 6 1
9072.2.a.bp 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{4} - 4T_{5}^{3} + 15T_{5}^{2} - 4T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} + 30T_{11}^{2} + 52T_{11} + 676 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} + 39T_{13}^{2} + 18T_{13} + 9 \) Copy content Toggle raw display
\( T_{17} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$17$ \( (T + 7)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 87 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 71)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 24 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 75 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$71$ \( (T^{2} + 20 T + 88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 20 T + 97)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 174 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} + 246 T^{2} + \cdots + 58564 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 39)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + 96 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
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