↑

Properties

Label 1134.2.m.b
Level $1134$
Weight $2$
Character orbit 1134.m
Analytic conductor $9.055$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1134,2,Mod(377,1134)]
 
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([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.m (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{5} + (\zeta_{12}^{2} - 3) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{5} + (\zeta_{12}^{2} - 3) q^{7} + \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{2} + 1) q^{10} + 3 \zeta_{12} q^{11} + ( - 2 \zeta_{12}^{2} + 4) q^{13} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{17} + (2 \zeta_{12}^{2} - 1) q^{19} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{20} + 3 \zeta_{12}^{2} q^{22} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{23} + ( - 2 \zeta_{12}^{2} + 2) q^{25} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{26} + ( - 2 \zeta_{12}^{2} - 1) q^{28} + 6 \zeta_{12} q^{29} + (3 \zeta_{12}^{2} - 6) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + (4 \zeta_{12}^{2} + 4) q^{34} + (\zeta_{12}^{3} + 4 \zeta_{12}) q^{35} + 7 q^{37} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{38} + ( - \zeta_{12}^{2} + 2) q^{40} + (7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{41} + ( - 2 \zeta_{12}^{2} + 2) q^{43} + 3 \zeta_{12}^{3} q^{44} - 3 q^{46} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{47} + ( - 5 \zeta_{12}^{2} + 8) q^{49} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{50} + (2 \zeta_{12}^{2} + 2) q^{52} + 12 \zeta_{12}^{3} q^{53} + ( - 6 \zeta_{12}^{2} + 3) q^{55} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{56} + 6 \zeta_{12}^{2} q^{58} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{59} + ( - 4 \zeta_{12}^{2} - 4) q^{61} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{62} - q^{64} - 6 \zeta_{12} q^{65} - 2 \zeta_{12}^{2} q^{67} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{68} + (5 \zeta_{12}^{2} - 1) q^{70} + 3 \zeta_{12}^{3} q^{71} + ( - 4 \zeta_{12}^{2} + 2) q^{73} + 7 \zeta_{12} q^{74} + (\zeta_{12}^{2} - 2) q^{76} + (3 \zeta_{12}^{3} - 9 \zeta_{12}) q^{77} + ( - 10 \zeta_{12}^{2} + 10) q^{79} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{80} + (14 \zeta_{12}^{2} - 7) q^{82} + ( - 20 \zeta_{12}^{3} + 10 \zeta_{12}) q^{83} - 12 \zeta_{12}^{2} q^{85} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{86} + (3 \zeta_{12}^{2} - 3) q^{88} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} + (8 \zeta_{12}^{2} - 10) q^{91} - 3 \zeta_{12} q^{92} + ( - 2 \zeta_{12}^{2} + 4) q^{94} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{95} + ( - 8 \zeta_{12}^{2} - 8) q^{97} + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 10 q^{7} + 12 q^{13} - 2 q^{16} + 6 q^{22} + 4 q^{25} - 8 q^{28} - 18 q^{31} + 24 q^{34} + 28 q^{37} + 6 q^{40} + 4 q^{43} - 12 q^{46} + 22 q^{49} + 12 q^{52} + 12 q^{58} - 24 q^{61} - 4 q^{64} - 4 q^{67} + 6 q^{70} - 6 q^{76} + 20 q^{79} - 24 q^{85} - 6 q^{88} - 24 q^{91} + 12 q^{94} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.866025 − 1.50000i 0 −2.50000 − 0.866025i 1.00000i 0 1.73205i
377.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.866025 + 1.50000i 0 −2.50000 − 0.866025i − 1.00000i 0 1.73205i
755.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.866025 + 1.50000i 0 −2.50000 + 0.866025i − 1.00000i 0 − 1.73205i
755.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.866025 − 1.50000i 0 −2.50000 + 0.866025i 1.00000i 0 − 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.m.b 4
3.b odd 2 1 inner 1134.2.m.b 4
7.b odd 2 1 1134.2.m.c 4
9.c even 3 1 378.2.d.b ✓ 4
9.c even 3 1 1134.2.m.c 4
9.d odd 6 1 378.2.d.b ✓ 4
9.d odd 6 1 1134.2.m.c 4
21.c even 2 1 1134.2.m.c 4
36.f odd 6 1 3024.2.k.e 4
36.h even 6 1 3024.2.k.e 4
63.l odd 6 1 378.2.d.b ✓ 4
63.l odd 6 1 inner 1134.2.m.b 4
63.o even 6 1 378.2.d.b ✓ 4
63.o even 6 1 inner 1134.2.m.b 4
252.s odd 6 1 3024.2.k.e 4
252.bi even 6 1 3024.2.k.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.b ✓ 4 9.c even 3 1
378.2.d.b ✓ 4 9.d odd 6 1
378.2.d.b ✓ 4 63.l odd 6 1
378.2.d.b ✓ 4 63.o even 6 1
1134.2.m.b 4 1.a even 1 1 trivial
1134.2.m.b 4 3.b odd 2 1 inner
1134.2.m.b 4 63.l odd 6 1 inner
1134.2.m.b 4 63.o even 6 1 inner
1134.2.m.c 4 7.b odd 2 1
1134.2.m.c 4 9.c even 3 1
1134.2.m.c 4 9.d odd 6 1
1134.2.m.c 4 21.c even 2 1
3024.2.k.e 4 36.f odd 6 1
3024.2.k.e 4 36.h even 6 1
3024.2.k.e 4 252.s odd 6 1
3024.2.k.e 4 252.bi even 6 1

Hecke kernels

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

\( T_{5}^{4} + 3T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} - 9T_{11}^{2} + 81 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} + 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$31$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T - 7)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 300 T^{2} + 90000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
show more
show less