Properties

Label 1134.2.f.f
Level $1134$
Weight $2$
Character orbit 1134.f
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
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 14)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{7} + q^{8} + 4 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 6 q^{17} + 2 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} - 4 q^{26} + q^{28} + (6 \zeta_{6} - 6) q^{29} + 4 \zeta_{6} q^{31} - \zeta_{6} q^{32} + ( - 6 \zeta_{6} + 6) q^{34} + 2 q^{37} + (2 \zeta_{6} - 2) q^{38} + 6 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} + (12 \zeta_{6} - 12) q^{47} - \zeta_{6} q^{49} + 5 \zeta_{6} q^{50} + ( - 4 \zeta_{6} + 4) q^{52} - 6 q^{53} + (\zeta_{6} - 1) q^{56} - 6 \zeta_{6} q^{58} - 6 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} - 4 q^{62} + q^{64} + 4 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + 2 q^{73} + (2 \zeta_{6} - 2) q^{74} - 2 \zeta_{6} q^{76} + (8 \zeta_{6} - 8) q^{79} - 6 q^{82} + (6 \zeta_{6} - 6) q^{83} - 8 \zeta_{6} q^{86} + 6 q^{89} - 4 q^{91} - 12 \zeta_{6} q^{94} + ( - 10 \zeta_{6} + 10) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} + 4 q^{13} - q^{14} - q^{16} - 12 q^{17} + 4 q^{19} + 5 q^{25} - 8 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 6 q^{41} - 8 q^{43} - 12 q^{47} - q^{49} + 5 q^{50} + 4 q^{52} - 12 q^{53} - q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{67} + 6 q^{68} + 4 q^{73} - 2 q^{74} - 2 q^{76} - 8 q^{79} - 12 q^{82} - 6 q^{83} - 8 q^{86} + 12 q^{89} - 8 q^{91} - 12 q^{94} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 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 0 0 −0.500000 + 0.866025i 1.00000 0 0
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i 1.00000 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
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.f 2
3.b odd 2 1 1134.2.f.l 2
9.c even 3 1 126.2.a.b 1
9.c even 3 1 inner 1134.2.f.f 2
9.d odd 6 1 14.2.a.a 1
9.d odd 6 1 1134.2.f.l 2
36.f odd 6 1 1008.2.a.h 1
36.h even 6 1 112.2.a.c 1
45.h odd 6 1 350.2.a.f 1
45.j even 6 1 3150.2.a.i 1
45.k odd 12 2 3150.2.g.j 2
45.l even 12 2 350.2.c.d 2
63.g even 3 1 882.2.g.c 2
63.h even 3 1 882.2.g.c 2
63.i even 6 1 98.2.c.a 2
63.j odd 6 1 98.2.c.b 2
63.k odd 6 1 882.2.g.d 2
63.l odd 6 1 882.2.a.i 1
63.n odd 6 1 98.2.c.b 2
63.o even 6 1 98.2.a.a 1
63.s even 6 1 98.2.c.a 2
63.t odd 6 1 882.2.g.d 2
72.j odd 6 1 448.2.a.g 1
72.l even 6 1 448.2.a.a 1
72.n even 6 1 4032.2.a.w 1
72.p odd 6 1 4032.2.a.r 1
99.g even 6 1 1694.2.a.e 1
117.n odd 6 1 2366.2.a.j 1
117.z even 12 2 2366.2.d.b 2
144.u even 12 2 1792.2.b.g 2
144.w odd 12 2 1792.2.b.c 2
153.i odd 6 1 4046.2.a.f 1
171.l even 6 1 5054.2.a.c 1
180.n even 6 1 2800.2.a.g 1
180.v odd 12 2 2800.2.g.h 2
207.g even 6 1 7406.2.a.a 1
252.o even 6 1 784.2.i.c 2
252.r odd 6 1 784.2.i.i 2
252.s odd 6 1 784.2.a.b 1
252.bb even 6 1 784.2.i.c 2
252.bi even 6 1 7056.2.a.bd 1
252.bn odd 6 1 784.2.i.i 2
315.z even 6 1 2450.2.a.t 1
315.cf odd 12 2 2450.2.c.c 2
504.cc even 6 1 3136.2.a.e 1
504.co odd 6 1 3136.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 9.d odd 6 1
98.2.a.a 1 63.o even 6 1
98.2.c.a 2 63.i even 6 1
98.2.c.a 2 63.s even 6 1
98.2.c.b 2 63.j odd 6 1
98.2.c.b 2 63.n odd 6 1
112.2.a.c 1 36.h even 6 1
126.2.a.b 1 9.c even 3 1
350.2.a.f 1 45.h odd 6 1
350.2.c.d 2 45.l even 12 2
448.2.a.a 1 72.l even 6 1
448.2.a.g 1 72.j odd 6 1
784.2.a.b 1 252.s odd 6 1
784.2.i.c 2 252.o even 6 1
784.2.i.c 2 252.bb even 6 1
784.2.i.i 2 252.r odd 6 1
784.2.i.i 2 252.bn odd 6 1
882.2.a.i 1 63.l odd 6 1
882.2.g.c 2 63.g even 3 1
882.2.g.c 2 63.h even 3 1
882.2.g.d 2 63.k odd 6 1
882.2.g.d 2 63.t odd 6 1
1008.2.a.h 1 36.f odd 6 1
1134.2.f.f 2 1.a even 1 1 trivial
1134.2.f.f 2 9.c even 3 1 inner
1134.2.f.l 2 3.b odd 2 1
1134.2.f.l 2 9.d odd 6 1
1694.2.a.e 1 99.g even 6 1
1792.2.b.c 2 144.w odd 12 2
1792.2.b.g 2 144.u even 12 2
2366.2.a.j 1 117.n odd 6 1
2366.2.d.b 2 117.z even 12 2
2450.2.a.t 1 315.z even 6 1
2450.2.c.c 2 315.cf odd 12 2
2800.2.a.g 1 180.n even 6 1
2800.2.g.h 2 180.v odd 12 2
3136.2.a.e 1 504.cc even 6 1
3136.2.a.z 1 504.co odd 6 1
3150.2.a.i 1 45.j even 6 1
3150.2.g.j 2 45.k odd 12 2
4032.2.a.r 1 72.p odd 6 1
4032.2.a.w 1 72.n even 6 1
4046.2.a.f 1 153.i odd 6 1
5054.2.a.c 1 171.l even 6 1
7056.2.a.bd 1 252.bi even 6 1
7406.2.a.a 1 207.g even 6 1

Hecke kernels

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

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