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Properties

Label 882.4.g.d
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,4,Mod(361,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.361"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,-4,-7,0,0,16,0,-14,35,0,-132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 7 \zeta_{6} q^{5} + 8 q^{8} + (14 \zeta_{6} - 14) q^{10} + ( - 35 \zeta_{6} + 35) q^{11} - 66 q^{13} - 16 \zeta_{6} q^{16} + (59 \zeta_{6} - 59) q^{17} + \cdots + 290 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8} - 14 q^{10} + 35 q^{11} - 132 q^{13} - 16 q^{16} - 59 q^{17} + 137 q^{19} + 56 q^{20} - 140 q^{22} - 7 q^{23} + 76 q^{25} + 132 q^{26} - 212 q^{29} + 75 q^{31}+ \cdots + 580 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.00000 − 1.73205i 0 −2.00000 + 3.46410i −3.50000 − 6.06218i 0 0 8.00000 0 −7.00000 + 12.1244i
667.1 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i −3.50000 + 6.06218i 0 0 8.00000 0 −7.00000 − 12.1244i
\(n\): 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 882.4.g.d 2
3.b odd 2 1 98.4.c.e 2
7.b odd 2 1 126.4.g.c 2
7.c even 3 1 882.4.a.p 1
7.c even 3 1 inner 882.4.g.d 2
7.d odd 6 1 126.4.g.c 2
7.d odd 6 1 882.4.a.k 1
21.c even 2 1 14.4.c.b ✓ 2
21.g even 6 1 14.4.c.b ✓ 2
21.g even 6 1 98.4.a.b 1
21.h odd 6 1 98.4.a.c 1
21.h odd 6 1 98.4.c.e 2
84.h odd 2 1 112.4.i.b 2
84.j odd 6 1 112.4.i.b 2
84.j odd 6 1 784.4.a.l 1
84.n even 6 1 784.4.a.j 1
105.g even 2 1 350.4.e.b 2
105.k odd 4 2 350.4.j.d 4
105.o odd 6 1 2450.4.a.bf 1
105.p even 6 1 350.4.e.b 2
105.p even 6 1 2450.4.a.bh 1
105.w odd 12 2 350.4.j.d 4
168.e odd 2 1 448.4.i.d 2
168.i even 2 1 448.4.i.c 2
168.ba even 6 1 448.4.i.c 2
168.be odd 6 1 448.4.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b ✓ 2 21.c even 2 1
14.4.c.b ✓ 2 21.g even 6 1
98.4.a.b 1 21.g even 6 1
98.4.a.c 1 21.h odd 6 1
98.4.c.e 2 3.b odd 2 1
98.4.c.e 2 21.h odd 6 1
112.4.i.b 2 84.h odd 2 1
112.4.i.b 2 84.j odd 6 1
126.4.g.c 2 7.b odd 2 1
126.4.g.c 2 7.d odd 6 1
350.4.e.b 2 105.g even 2 1
350.4.e.b 2 105.p even 6 1
350.4.j.d 4 105.k odd 4 2
350.4.j.d 4 105.w odd 12 2
448.4.i.c 2 168.i even 2 1
448.4.i.c 2 168.ba even 6 1
448.4.i.d 2 168.e odd 2 1
448.4.i.d 2 168.be odd 6 1
784.4.a.j 1 84.n even 6 1
784.4.a.l 1 84.j odd 6 1
882.4.a.k 1 7.d odd 6 1
882.4.a.p 1 7.c even 3 1
882.4.g.d 2 1.a even 1 1 trivial
882.4.g.d 2 7.c even 3 1 inner
2450.4.a.bf 1 105.o odd 6 1
2450.4.a.bh 1 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 7T_{5} + 49 \) Copy content Toggle raw display
\( T_{11}^{2} - 35T_{11} + 1225 \) Copy content Toggle raw display
\( T_{13} + 66 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T + 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} - 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( (T + 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 498)^{2} \) Copy content Toggle raw display
$43$ \( (T - 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} + 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} + 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T - 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T - 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T - 290)^{2} \) Copy content Toggle raw display
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