↑

Properties

Label 1176.2.q.d
Level $1176$
Weight $2$
Character orbit 1176.q
Analytic conductor $9.390$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1176,2,Mod(361,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.q (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: \(9.39040727770\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} - 6 q^{13} - 2 q^{15} + (2 \zeta_{6} - 2) q^{17} + 4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + q^{27} - 10 q^{29} + (8 \zeta_{6} - 8) q^{31} - 6 \zeta_{6} q^{37} + ( - 6 \zeta_{6} + 6) q^{39} + 2 q^{41} - 4 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 8 \zeta_{6} q^{47} - 2 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{53} - 4 q^{57} + ( - 12 \zeta_{6} + 12) q^{59} - 2 \zeta_{6} q^{61} - 12 \zeta_{6} q^{65} + (12 \zeta_{6} - 12) q^{67} - 4 q^{69} - 12 q^{71} + (14 \zeta_{6} - 14) q^{73} + \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} - 4 q^{85} + ( - 10 \zeta_{6} + 10) q^{87} - 2 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + (8 \zeta_{6} - 8) q^{95} - 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{5} - q^{9} - 12 q^{13} - 4 q^{15} - 2 q^{17} + 4 q^{19} + 4 q^{23} + q^{25} + 2 q^{27} - 20 q^{29} - 8 q^{31} - 6 q^{37} + 6 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 8 q^{47} - 2 q^{51} + 10 q^{53} - 8 q^{57} + 12 q^{59} - 2 q^{61} - 12 q^{65} - 12 q^{67} - 8 q^{69} - 24 q^{71} - 14 q^{73} + q^{75} + 8 q^{79} - q^{81} - 24 q^{83} - 8 q^{85} + 10 q^{87} - 2 q^{89} - 8 q^{93} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −0.500000 + 0.866025i 0 1.00000 + 1.73205i 0 0 0 −0.500000 − 0.866025i 0
961.1 0 −0.500000 − 0.866025i 0 1.00000 − 1.73205i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.q.d 2
3.b odd 2 1 3528.2.s.g 2
4.b odd 2 1 2352.2.q.w 2
7.b odd 2 1 1176.2.q.f 2
7.c even 3 1 1176.2.a.f 1
7.c even 3 1 inner 1176.2.q.d 2
7.d odd 6 1 168.2.a.a ✓ 1
7.d odd 6 1 1176.2.q.f 2
21.c even 2 1 3528.2.s.w 2
21.g even 6 1 504.2.a.e 1
21.g even 6 1 3528.2.s.w 2
21.h odd 6 1 3528.2.a.v 1
21.h odd 6 1 3528.2.s.g 2
28.d even 2 1 2352.2.q.d 2
28.f even 6 1 336.2.a.e 1
28.f even 6 1 2352.2.q.d 2
28.g odd 6 1 2352.2.a.c 1
28.g odd 6 1 2352.2.q.w 2
35.i odd 6 1 4200.2.a.t 1
35.k even 12 2 4200.2.t.j 2
56.j odd 6 1 1344.2.a.m 1
56.k odd 6 1 9408.2.a.da 1
56.m even 6 1 1344.2.a.b 1
56.p even 6 1 9408.2.a.be 1
84.j odd 6 1 1008.2.a.b 1
84.n even 6 1 7056.2.a.bq 1
112.v even 12 2 5376.2.c.bb 2
112.x odd 12 2 5376.2.c.d 2
140.s even 6 1 8400.2.a.y 1
168.ba even 6 1 4032.2.a.bh 1
168.be odd 6 1 4032.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a ✓ 1 7.d odd 6 1
336.2.a.e 1 28.f even 6 1
504.2.a.e 1 21.g even 6 1
1008.2.a.b 1 84.j odd 6 1
1176.2.a.f 1 7.c even 3 1
1176.2.q.d 2 1.a even 1 1 trivial
1176.2.q.d 2 7.c even 3 1 inner
1176.2.q.f 2 7.b odd 2 1
1176.2.q.f 2 7.d odd 6 1
1344.2.a.b 1 56.m even 6 1
1344.2.a.m 1 56.j odd 6 1
2352.2.a.c 1 28.g odd 6 1
2352.2.q.d 2 28.d even 2 1
2352.2.q.d 2 28.f even 6 1
2352.2.q.w 2 4.b odd 2 1
2352.2.q.w 2 28.g odd 6 1
3528.2.a.v 1 21.h odd 6 1
3528.2.s.g 2 3.b odd 2 1
3528.2.s.g 2 21.h odd 6 1
3528.2.s.w 2 21.c even 2 1
3528.2.s.w 2 21.g even 6 1
4032.2.a.bc 1 168.be odd 6 1
4032.2.a.bh 1 168.ba even 6 1
4200.2.a.t 1 35.i odd 6 1
4200.2.t.j 2 35.k even 12 2
5376.2.c.d 2 112.x odd 12 2
5376.2.c.bb 2 112.v even 12 2
7056.2.a.bq 1 84.n even 6 1
8400.2.a.y 1 140.s even 6 1
9408.2.a.be 1 56.p even 6 1
9408.2.a.da 1 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1176, [\chi])\):

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

Hecke characteristic polynomials

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