Properties

Label 1134.2.f.j
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 42)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} - 2 q^{10} + (4 \zeta_{6} - 4) q^{11} - 6 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 2 q^{17} - 4 q^{19} + (2 \zeta_{6} - 2) q^{20} + 4 \zeta_{6} q^{22} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 6 q^{26} - q^{28} + (2 \zeta_{6} - 2) q^{29} + \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{34} - 2 q^{35} - 10 q^{37} + (4 \zeta_{6} - 4) q^{38} + 2 \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + 4 q^{44} + 8 q^{46} - \zeta_{6} q^{49} - \zeta_{6} q^{50} + (6 \zeta_{6} - 6) q^{52} - 6 q^{53} + 8 q^{55} + (\zeta_{6} - 1) q^{56} + 2 \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} + (6 \zeta_{6} - 6) q^{61} + q^{64} + (12 \zeta_{6} - 12) q^{65} - 4 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + (2 \zeta_{6} - 2) q^{70} - 8 q^{71} + 10 q^{73} + (10 \zeta_{6} - 10) q^{74} + 4 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + 2 q^{80} - 6 q^{82} + (4 \zeta_{6} - 4) q^{83} + 4 \zeta_{6} q^{85} - 4 \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 4) q^{88} + 6 q^{89} - 6 q^{91} + ( - 8 \zeta_{6} + 8) q^{92} + 8 \zeta_{6} q^{95} + ( - 14 \zeta_{6} + 14) q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8} - 4 q^{10} - 4 q^{11} - 6 q^{13} - q^{14} - q^{16} - 4 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{22} + 8 q^{23} + q^{25} - 12 q^{26} - 2 q^{28} - 2 q^{29} + q^{32} - 2 q^{34} - 4 q^{35} - 20 q^{37} - 4 q^{38} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 8 q^{44} + 16 q^{46} - q^{49} - q^{50} - 6 q^{52} - 12 q^{53} + 16 q^{55} - q^{56} + 2 q^{58} + 4 q^{59} - 6 q^{61} + 2 q^{64} - 12 q^{65} - 4 q^{67} + 2 q^{68} - 2 q^{70} - 16 q^{71} + 20 q^{73} - 10 q^{74} + 4 q^{76} + 4 q^{77} + 4 q^{80} - 12 q^{82} - 4 q^{83} + 4 q^{85} - 4 q^{86} + 4 q^{88} + 12 q^{89} - 12 q^{91} + 8 q^{92} + 8 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) \) 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 −1.00000 1.73205i 0 0.500000 0.866025i −1.00000 0 −2.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 + 1.73205i 0 0.500000 + 0.866025i −1.00000 0 −2.00000
\(n\): 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.j 2
3.b odd 2 1 1134.2.f.g 2
9.c even 3 1 126.2.a.a 1
9.c even 3 1 inner 1134.2.f.j 2
9.d odd 6 1 42.2.a.a 1
9.d odd 6 1 1134.2.f.g 2
36.f odd 6 1 1008.2.a.j 1
36.h even 6 1 336.2.a.d 1
45.h odd 6 1 1050.2.a.i 1
45.j even 6 1 3150.2.a.bo 1
45.k odd 12 2 3150.2.g.r 2
45.l even 12 2 1050.2.g.a 2
63.g even 3 1 882.2.g.h 2
63.h even 3 1 882.2.g.h 2
63.i even 6 1 294.2.e.a 2
63.j odd 6 1 294.2.e.c 2
63.k odd 6 1 882.2.g.j 2
63.l odd 6 1 882.2.a.b 1
63.n odd 6 1 294.2.e.c 2
63.o even 6 1 294.2.a.g 1
63.s even 6 1 294.2.e.a 2
63.t odd 6 1 882.2.g.j 2
72.j odd 6 1 1344.2.a.q 1
72.l even 6 1 1344.2.a.i 1
72.n even 6 1 4032.2.a.e 1
72.p odd 6 1 4032.2.a.m 1
99.g even 6 1 5082.2.a.d 1
117.n odd 6 1 7098.2.a.f 1
144.u even 12 2 5376.2.c.e 2
144.w odd 12 2 5376.2.c.bc 2
180.n even 6 1 8400.2.a.k 1
252.o even 6 1 2352.2.q.i 2
252.r odd 6 1 2352.2.q.n 2
252.s odd 6 1 2352.2.a.l 1
252.bb even 6 1 2352.2.q.i 2
252.bi even 6 1 7056.2.a.k 1
252.bn odd 6 1 2352.2.q.n 2
315.z even 6 1 7350.2.a.f 1
504.cc even 6 1 9408.2.a.n 1
504.co odd 6 1 9408.2.a.bw 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 9.d odd 6 1
126.2.a.a 1 9.c even 3 1
294.2.a.g 1 63.o even 6 1
294.2.e.a 2 63.i even 6 1
294.2.e.a 2 63.s even 6 1
294.2.e.c 2 63.j odd 6 1
294.2.e.c 2 63.n odd 6 1
336.2.a.d 1 36.h even 6 1
882.2.a.b 1 63.l odd 6 1
882.2.g.h 2 63.g even 3 1
882.2.g.h 2 63.h even 3 1
882.2.g.j 2 63.k odd 6 1
882.2.g.j 2 63.t odd 6 1
1008.2.a.j 1 36.f odd 6 1
1050.2.a.i 1 45.h odd 6 1
1050.2.g.a 2 45.l even 12 2
1134.2.f.g 2 3.b odd 2 1
1134.2.f.g 2 9.d odd 6 1
1134.2.f.j 2 1.a even 1 1 trivial
1134.2.f.j 2 9.c even 3 1 inner
1344.2.a.i 1 72.l even 6 1
1344.2.a.q 1 72.j odd 6 1
2352.2.a.l 1 252.s odd 6 1
2352.2.q.i 2 252.o even 6 1
2352.2.q.i 2 252.bb even 6 1
2352.2.q.n 2 252.r odd 6 1
2352.2.q.n 2 252.bn odd 6 1
3150.2.a.bo 1 45.j even 6 1
3150.2.g.r 2 45.k odd 12 2
4032.2.a.e 1 72.n even 6 1
4032.2.a.m 1 72.p odd 6 1
5082.2.a.d 1 99.g even 6 1
5376.2.c.e 2 144.u even 12 2
5376.2.c.bc 2 144.w odd 12 2
7056.2.a.k 1 252.bi even 6 1
7098.2.a.f 1 117.n odd 6 1
7350.2.a.f 1 315.z even 6 1
8400.2.a.k 1 180.n even 6 1
9408.2.a.n 1 504.cc even 6 1
9408.2.a.bw 1 504.co odd 6 1

Hecke kernels

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

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