Properties

Label 1134.2.h.j
Level $1134$
Weight $2$
Character orbit 1134.h
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(109,1134)]
 
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([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.h (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 126)
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.

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i −3.00000 0 −2.00000 + 1.73205i −1.00000 0 −1.50000 + 2.59808i
541.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.00000 0 −2.00000 1.73205i −1.00000 0 −1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g 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.h.j 2
3.b odd 2 1 1134.2.h.f 2
7.c even 3 1 1134.2.e.g 2
9.c even 3 1 126.2.g.d yes 2
9.c even 3 1 1134.2.e.g 2
9.d odd 6 1 126.2.g.a 2
9.d odd 6 1 1134.2.e.k 2
21.h odd 6 1 1134.2.e.k 2
36.f odd 6 1 1008.2.s.o 2
36.h even 6 1 1008.2.s.b 2
63.g even 3 1 882.2.a.a 1
63.g even 3 1 inner 1134.2.h.j 2
63.h even 3 1 126.2.g.d yes 2
63.i even 6 1 882.2.g.e 2
63.j odd 6 1 126.2.g.a 2
63.k odd 6 1 882.2.a.e 1
63.l odd 6 1 882.2.g.g 2
63.n odd 6 1 882.2.a.j 1
63.n odd 6 1 1134.2.h.f 2
63.o even 6 1 882.2.g.e 2
63.s even 6 1 882.2.a.h 1
63.t odd 6 1 882.2.g.g 2
252.n even 6 1 7056.2.a.bx 1
252.o even 6 1 7056.2.a.by 1
252.u odd 6 1 1008.2.s.o 2
252.bb even 6 1 1008.2.s.b 2
252.bl odd 6 1 7056.2.a.e 1
252.bn odd 6 1 7056.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 9.d odd 6 1
126.2.g.a 2 63.j odd 6 1
126.2.g.d yes 2 9.c even 3 1
126.2.g.d yes 2 63.h even 3 1
882.2.a.a 1 63.g even 3 1
882.2.a.e 1 63.k odd 6 1
882.2.a.h 1 63.s even 6 1
882.2.a.j 1 63.n odd 6 1
882.2.g.e 2 63.i even 6 1
882.2.g.e 2 63.o even 6 1
882.2.g.g 2 63.l odd 6 1
882.2.g.g 2 63.t odd 6 1
1008.2.s.b 2 36.h even 6 1
1008.2.s.b 2 252.bb even 6 1
1008.2.s.o 2 36.f odd 6 1
1008.2.s.o 2 252.u odd 6 1
1134.2.e.g 2 7.c even 3 1
1134.2.e.g 2 9.c even 3 1
1134.2.e.k 2 9.d odd 6 1
1134.2.e.k 2 21.h odd 6 1
1134.2.h.f 2 3.b odd 2 1
1134.2.h.f 2 63.n odd 6 1
1134.2.h.j 2 1.a even 1 1 trivial
1134.2.h.j 2 63.g even 3 1 inner
7056.2.a.e 1 252.bl odd 6 1
7056.2.a.h 1 252.bn odd 6 1
7056.2.a.bx 1 252.n even 6 1
7056.2.a.by 1 252.o even 6 1

Hecke kernels

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

\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) Copy content Toggle raw display
\( T_{23} - 6 \) 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
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
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