Properties

Label 1134.2.f.p
Level $1134$
Weight $2$
Character orbit 1134.f
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
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: \(2\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} + 4 q^{10} + (4 \zeta_{6} - 4) q^{11} - 3 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 7 q^{17} + 2 q^{19} + ( - 4 \zeta_{6} + 4) q^{20} + 4 \zeta_{6} q^{22} - \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} - 3 q^{26} - q^{28} + ( - \zeta_{6} + 1) q^{29} + 9 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( - 7 \zeta_{6} + 7) q^{34} + 4 q^{35} + 2 q^{37} + ( - 2 \zeta_{6} + 2) q^{38} - 4 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + (11 \zeta_{6} - 11) q^{43} + 4 q^{44} - q^{46} + (6 \zeta_{6} - 6) q^{47} - \zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + (3 \zeta_{6} - 3) q^{52} + 9 q^{53} - 16 q^{55} + (\zeta_{6} - 1) q^{56} - \zeta_{6} q^{58} - 5 \zeta_{6} q^{59} + ( - 6 \zeta_{6} + 6) q^{61} + 9 q^{62} + q^{64} + ( - 12 \zeta_{6} + 12) q^{65} - 7 \zeta_{6} q^{67} - 7 \zeta_{6} q^{68} + ( - 4 \zeta_{6} + 4) q^{70} + 7 q^{71} - 14 q^{73} + ( - 2 \zeta_{6} + 2) q^{74} - 2 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + ( - 6 \zeta_{6} + 6) q^{79} - 4 q^{80} + 6 q^{82} + (4 \zeta_{6} - 4) q^{83} + 28 \zeta_{6} q^{85} + 11 \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 4) q^{88} + 3 q^{89} - 3 q^{91} + (\zeta_{6} - 1) q^{92} + 6 \zeta_{6} q^{94} + 8 \zeta_{6} q^{95} + ( - 8 \zeta_{6} + 8) q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 4 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 4 q^{5} + q^{7} - 2 q^{8} + 8 q^{10} - 4 q^{11} - 3 q^{13} - q^{14} - q^{16} + 14 q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{22} - q^{23} - 11 q^{25} - 6 q^{26} - 2 q^{28} + q^{29} + 9 q^{31} + q^{32} + 7 q^{34} + 8 q^{35} + 4 q^{37} + 2 q^{38} - 4 q^{40} + 6 q^{41} - 11 q^{43} + 8 q^{44} - 2 q^{46} - 6 q^{47} - q^{49} + 11 q^{50} - 3 q^{52} + 18 q^{53} - 32 q^{55} - q^{56} - q^{58} - 5 q^{59} + 6 q^{61} + 18 q^{62} + 2 q^{64} + 12 q^{65} - 7 q^{67} - 7 q^{68} + 4 q^{70} + 14 q^{71} - 28 q^{73} + 2 q^{74} - 2 q^{76} + 4 q^{77} + 6 q^{79} - 8 q^{80} + 12 q^{82} - 4 q^{83} + 28 q^{85} + 11 q^{86} + 4 q^{88} + 6 q^{89} - 6 q^{91} - q^{92} + 6 q^{94} + 8 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 2.00000 + 3.46410i 0 0.500000 0.866025i −1.00000 0 4.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 3.46410i 0 0.500000 + 0.866025i −1.00000 0 4.00000
\(n\): 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.p 2
3.b odd 2 1 1134.2.f.a 2
9.c even 3 1 378.2.a.a 1
9.c even 3 1 inner 1134.2.f.p 2
9.d odd 6 1 378.2.a.h yes 1
9.d odd 6 1 1134.2.f.a 2
36.f odd 6 1 3024.2.a.a 1
36.h even 6 1 3024.2.a.bd 1
45.h odd 6 1 9450.2.a.bc 1
45.j even 6 1 9450.2.a.dv 1
63.l odd 6 1 2646.2.a.o 1
63.o even 6 1 2646.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.a 1 9.c even 3 1
378.2.a.h yes 1 9.d odd 6 1
1134.2.f.a 2 3.b odd 2 1
1134.2.f.a 2 9.d odd 6 1
1134.2.f.p 2 1.a even 1 1 trivial
1134.2.f.p 2 9.c even 3 1 inner
2646.2.a.o 1 63.l odd 6 1
2646.2.a.p 1 63.o even 6 1
3024.2.a.a 1 36.f odd 6 1
3024.2.a.bd 1 36.h even 6 1
9450.2.a.bc 1 45.h odd 6 1
9450.2.a.dv 1 45.j even 6 1

Hecke kernels

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

\( T_{5}^{2} - 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} + 9 \) Copy content Toggle raw display
\( T_{17} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$17$ \( (T - 7)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$71$ \( (T - 7)^{2} \) Copy content Toggle raw display
$73$ \( (T + 14)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$89$ \( (T - 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
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