Properties

Label 1134.2.h.h
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: 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 4 q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 4 q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + q^{8} + (4 \zeta_{6} - 4) q^{10} + 2 q^{11} + (6 \zeta_{6} - 6) q^{13} + (3 \zeta_{6} - 1) q^{14} + (\zeta_{6} - 1) q^{16} + (2 \zeta_{6} - 2) q^{17} + 4 \zeta_{6} q^{19} - 4 \zeta_{6} q^{20} + (2 \zeta_{6} - 2) q^{22} - q^{23} + 11 q^{25} - 6 \zeta_{6} q^{26} + ( - \zeta_{6} - 2) q^{28} - 4 \zeta_{6} q^{29} + 9 \zeta_{6} q^{31} - \zeta_{6} q^{32} - 2 \zeta_{6} q^{34} + ( - 8 \zeta_{6} + 12) q^{35} - 8 \zeta_{6} q^{37} - 4 q^{38} + 4 q^{40} + (3 \zeta_{6} - 3) q^{41} - 2 \zeta_{6} q^{43} - 2 \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{46} + ( - 9 \zeta_{6} + 9) q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + (11 \zeta_{6} - 11) q^{50} + 6 q^{52} + ( - 12 \zeta_{6} + 12) q^{53} + 8 q^{55} + ( - 2 \zeta_{6} + 3) q^{56} + 4 q^{58} - 4 \zeta_{6} q^{59} + (6 \zeta_{6} - 6) q^{61} - 9 q^{62} + q^{64} + (24 \zeta_{6} - 24) q^{65} + 14 \zeta_{6} q^{67} + 2 q^{68} + (12 \zeta_{6} - 4) q^{70} - q^{71} + (7 \zeta_{6} - 7) q^{73} + 8 q^{74} + ( - 4 \zeta_{6} + 4) q^{76} + ( - 4 \zeta_{6} + 6) q^{77} + ( - 3 \zeta_{6} + 3) q^{79} + (4 \zeta_{6} - 4) q^{80} - 3 \zeta_{6} q^{82} - 14 \zeta_{6} q^{83} + (8 \zeta_{6} - 8) q^{85} + 2 q^{86} + 2 q^{88} + 3 \zeta_{6} q^{89} + (18 \zeta_{6} - 6) q^{91} + \zeta_{6} q^{92} + 9 \zeta_{6} q^{94} + 16 \zeta_{6} q^{95} - 10 \zeta_{6} q^{97} + (5 \zeta_{6} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 8 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 8 q^{5} + 4 q^{7} + 2 q^{8} - 4 q^{10} + 4 q^{11} - 6 q^{13} + q^{14} - q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{23} + 22 q^{25} - 6 q^{26} - 5 q^{28} - 4 q^{29} + 9 q^{31} - q^{32} - 2 q^{34} + 16 q^{35} - 8 q^{37} - 8 q^{38} + 8 q^{40} - 3 q^{41} - 2 q^{43} - 2 q^{44} + q^{46} + 9 q^{47} + 2 q^{49} - 11 q^{50} + 12 q^{52} + 12 q^{53} + 16 q^{55} + 4 q^{56} + 8 q^{58} - 4 q^{59} - 6 q^{61} - 18 q^{62} + 2 q^{64} - 24 q^{65} + 14 q^{67} + 4 q^{68} + 4 q^{70} - 2 q^{71} - 7 q^{73} + 16 q^{74} + 4 q^{76} + 8 q^{77} + 3 q^{79} - 4 q^{80} - 3 q^{82} - 14 q^{83} - 8 q^{85} + 4 q^{86} + 4 q^{88} + 3 q^{89} + 6 q^{91} + q^{92} + 9 q^{94} + 16 q^{95} - 10 q^{97} + 11 q^{98}+O(q^{100}) \) 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 4.00000 0 2.00000 1.73205i 1.00000 0 −2.00000 + 3.46410i
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 4.00000 0 2.00000 + 1.73205i 1.00000 0 −2.00000 3.46410i
\(n\): 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.h 2
3.b odd 2 1 1134.2.h.i 2
7.c even 3 1 1134.2.e.i 2
9.c even 3 1 1134.2.e.i 2
9.c even 3 1 1134.2.g.a 2
9.d odd 6 1 1134.2.e.h 2
9.d odd 6 1 1134.2.g.h yes 2
21.h odd 6 1 1134.2.e.h 2
63.g even 3 1 inner 1134.2.h.h 2
63.g even 3 1 7938.2.a.bf 1
63.h even 3 1 1134.2.g.a 2
63.j odd 6 1 1134.2.g.h yes 2
63.k odd 6 1 7938.2.a.q 1
63.n odd 6 1 1134.2.h.i 2
63.n odd 6 1 7938.2.a.a 1
63.s even 6 1 7938.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.h 2 9.d odd 6 1
1134.2.e.h 2 21.h odd 6 1
1134.2.e.i 2 7.c even 3 1
1134.2.e.i 2 9.c even 3 1
1134.2.g.a 2 9.c even 3 1
1134.2.g.a 2 63.h even 3 1
1134.2.g.h yes 2 9.d odd 6 1
1134.2.g.h yes 2 63.j odd 6 1
1134.2.h.h 2 1.a even 1 1 trivial
1134.2.h.h 2 63.g even 3 1 inner
1134.2.h.i 2 3.b odd 2 1
1134.2.h.i 2 63.n odd 6 1
7938.2.a.a 1 63.n odd 6 1
7938.2.a.p 1 63.s even 6 1
7938.2.a.q 1 63.k odd 6 1
7938.2.a.bf 1 63.g even 3 1

Hecke kernels

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

\( T_{5} - 4 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 4 \) Copy content Toggle raw display
\( T_{23} + 1 \) 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
$5$ \( (T - 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$31$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$71$ \( (T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
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