Properties

Label 882.2.g.h
Level $882$
Weight $2$
Character orbit 882.g
Analytic conductor $7.043$
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] = [882,2,Mod(361,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("882.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.g (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: \(7.04280545828\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 1.73205i 0 0 −1.00000 0 1.00000 1.73205i
667.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0 −1.00000 0 1.00000 + 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
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 882.2.g.h 2
3.b odd 2 1 294.2.e.c 2
7.b odd 2 1 882.2.g.j 2
7.c even 3 1 126.2.a.a 1
7.c even 3 1 inner 882.2.g.h 2
7.d odd 6 1 882.2.a.b 1
7.d odd 6 1 882.2.g.j 2
12.b even 2 1 2352.2.q.i 2
21.c even 2 1 294.2.e.a 2
21.g even 6 1 294.2.a.g 1
21.g even 6 1 294.2.e.a 2
21.h odd 6 1 42.2.a.a 1
21.h odd 6 1 294.2.e.c 2
28.f even 6 1 7056.2.a.k 1
28.g odd 6 1 1008.2.a.j 1
35.j even 6 1 3150.2.a.bo 1
35.l odd 12 2 3150.2.g.r 2
56.k odd 6 1 4032.2.a.m 1
56.p even 6 1 4032.2.a.e 1
63.g even 3 1 1134.2.f.j 2
63.h even 3 1 1134.2.f.j 2
63.j odd 6 1 1134.2.f.g 2
63.n odd 6 1 1134.2.f.g 2
84.h odd 2 1 2352.2.q.n 2
84.j odd 6 1 2352.2.a.l 1
84.j odd 6 1 2352.2.q.n 2
84.n even 6 1 336.2.a.d 1
84.n even 6 1 2352.2.q.i 2
105.o odd 6 1 1050.2.a.i 1
105.p even 6 1 7350.2.a.f 1
105.x even 12 2 1050.2.g.a 2
168.s odd 6 1 1344.2.a.q 1
168.v even 6 1 1344.2.a.i 1
168.ba even 6 1 9408.2.a.n 1
168.be odd 6 1 9408.2.a.bw 1
231.l even 6 1 5082.2.a.d 1
273.w odd 6 1 7098.2.a.f 1
336.bt odd 12 2 5376.2.c.bc 2
336.bu even 12 2 5376.2.c.e 2
420.ba even 6 1 8400.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 21.h odd 6 1
126.2.a.a 1 7.c even 3 1
294.2.a.g 1 21.g even 6 1
294.2.e.a 2 21.c even 2 1
294.2.e.a 2 21.g even 6 1
294.2.e.c 2 3.b odd 2 1
294.2.e.c 2 21.h odd 6 1
336.2.a.d 1 84.n even 6 1
882.2.a.b 1 7.d odd 6 1
882.2.g.h 2 1.a even 1 1 trivial
882.2.g.h 2 7.c even 3 1 inner
882.2.g.j 2 7.b odd 2 1
882.2.g.j 2 7.d odd 6 1
1008.2.a.j 1 28.g odd 6 1
1050.2.a.i 1 105.o odd 6 1
1050.2.g.a 2 105.x even 12 2
1134.2.f.g 2 63.j odd 6 1
1134.2.f.g 2 63.n odd 6 1
1134.2.f.j 2 63.g even 3 1
1134.2.f.j 2 63.h even 3 1
1344.2.a.i 1 168.v even 6 1
1344.2.a.q 1 168.s odd 6 1
2352.2.a.l 1 84.j odd 6 1
2352.2.q.i 2 12.b even 2 1
2352.2.q.i 2 84.n even 6 1
2352.2.q.n 2 84.h odd 2 1
2352.2.q.n 2 84.j odd 6 1
3150.2.a.bo 1 35.j even 6 1
3150.2.g.r 2 35.l odd 12 2
4032.2.a.e 1 56.p even 6 1
4032.2.a.m 1 56.k odd 6 1
5082.2.a.d 1 231.l even 6 1
5376.2.c.e 2 336.bu even 12 2
5376.2.c.bc 2 336.bt odd 12 2
7056.2.a.k 1 28.f even 6 1
7098.2.a.f 1 273.w odd 6 1
7350.2.a.f 1 105.p even 6 1
8400.2.a.k 1 420.ba even 6 1
9408.2.a.n 1 168.ba even 6 1
9408.2.a.bw 1 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} - 6 \) 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
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) 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
$23$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) 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
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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