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Properties

Label 1008.2.t.g
Level $1008$
Weight $2$
Character orbit 1008.t
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(193,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-4,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{3} + (\beta_{3} + \beta_{2} + 2) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} + \cdots - 2) q^{11}+ \cdots + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + 10 q^{5} + 2 q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} + 3 q^{19} - 7 q^{21} - 14 q^{23} + 4 q^{25} - 7 q^{27} - 5 q^{29} + 14 q^{31} - 4 q^{33} + 19 q^{35} - 9 q^{37}+ \cdots + 41 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 − 2.05195i
0.500000 + 1.41036i
0.500000 − 0.224437i
0.500000 + 2.05195i
0.500000 − 1.41036i
0.500000 + 0.224437i
0 −1.73025 + 0.0789082i 0 −0.460505 0 −2.25729 − 1.38008i 0 2.98755 − 0.273062i 0
193.2 0 −0.619562 + 1.61745i 0 1.76088 0 1.85185 + 1.88962i 0 −2.23229 − 2.00422i 0
193.3 0 0.349814 − 1.69636i 0 3.69963 0 1.40545 − 2.24159i 0 −2.75526 − 1.18682i 0
961.1 0 −1.73025 − 0.0789082i 0 −0.460505 0 −2.25729 + 1.38008i 0 2.98755 + 0.273062i 0
961.2 0 −0.619562 − 1.61745i 0 1.76088 0 1.85185 − 1.88962i 0 −2.23229 + 2.00422i 0
961.3 0 0.349814 + 1.69636i 0 3.69963 0 1.40545 + 2.24159i 0 −2.75526 + 1.18682i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.t.g 6
3.b odd 2 1 3024.2.t.g 6
4.b odd 2 1 126.2.h.c yes 6
7.c even 3 1 1008.2.q.h 6
9.c even 3 1 1008.2.q.h 6
9.d odd 6 1 3024.2.q.h 6
12.b even 2 1 378.2.h.d 6
21.h odd 6 1 3024.2.q.h 6
28.d even 2 1 882.2.h.o 6
28.f even 6 1 882.2.e.p 6
28.f even 6 1 882.2.f.m 6
28.g odd 6 1 126.2.e.d ✓ 6
28.g odd 6 1 882.2.f.l 6
36.f odd 6 1 126.2.e.d ✓ 6
36.f odd 6 1 1134.2.g.k 6
36.h even 6 1 378.2.e.c 6
36.h even 6 1 1134.2.g.n 6
63.g even 3 1 inner 1008.2.t.g 6
63.n odd 6 1 3024.2.t.g 6
84.h odd 2 1 2646.2.h.p 6
84.j odd 6 1 2646.2.e.o 6
84.j odd 6 1 2646.2.f.n 6
84.n even 6 1 378.2.e.c 6
84.n even 6 1 2646.2.f.o 6
252.n even 6 1 882.2.h.o 6
252.n even 6 1 7938.2.a.by 3
252.o even 6 1 378.2.h.d 6
252.o even 6 1 7938.2.a.bu 3
252.r odd 6 1 2646.2.f.n 6
252.s odd 6 1 2646.2.e.o 6
252.u odd 6 1 882.2.f.l 6
252.u odd 6 1 1134.2.g.k 6
252.bb even 6 1 1134.2.g.n 6
252.bb even 6 1 2646.2.f.o 6
252.bi even 6 1 882.2.e.p 6
252.bj even 6 1 882.2.f.m 6
252.bl odd 6 1 126.2.h.c yes 6
252.bl odd 6 1 7938.2.a.cb 3
252.bn odd 6 1 2646.2.h.p 6
252.bn odd 6 1 7938.2.a.bx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d ✓ 6 28.g odd 6 1
126.2.e.d ✓ 6 36.f odd 6 1
126.2.h.c yes 6 4.b odd 2 1
126.2.h.c yes 6 252.bl odd 6 1
378.2.e.c 6 36.h even 6 1
378.2.e.c 6 84.n even 6 1
378.2.h.d 6 12.b even 2 1
378.2.h.d 6 252.o even 6 1
882.2.e.p 6 28.f even 6 1
882.2.e.p 6 252.bi even 6 1
882.2.f.l 6 28.g odd 6 1
882.2.f.l 6 252.u odd 6 1
882.2.f.m 6 28.f even 6 1
882.2.f.m 6 252.bj even 6 1
882.2.h.o 6 28.d even 2 1
882.2.h.o 6 252.n even 6 1
1008.2.q.h 6 7.c even 3 1
1008.2.q.h 6 9.c even 3 1
1008.2.t.g 6 1.a even 1 1 trivial
1008.2.t.g 6 63.g even 3 1 inner
1134.2.g.k 6 36.f odd 6 1
1134.2.g.k 6 252.u odd 6 1
1134.2.g.n 6 36.h even 6 1
1134.2.g.n 6 252.bb even 6 1
2646.2.e.o 6 84.j odd 6 1
2646.2.e.o 6 252.s odd 6 1
2646.2.f.n 6 84.j odd 6 1
2646.2.f.n 6 252.r odd 6 1
2646.2.f.o 6 84.n even 6 1
2646.2.f.o 6 252.bb even 6 1
2646.2.h.p 6 84.h odd 2 1
2646.2.h.p 6 252.bn odd 6 1
3024.2.q.h 6 9.d odd 6 1
3024.2.q.h 6 21.h odd 6 1
3024.2.t.g 6 3.b odd 2 1
3024.2.t.g 6 63.n odd 6 1
7938.2.a.bu 3 252.o even 6 1
7938.2.a.bx 3 252.bn odd 6 1
7938.2.a.by 3 252.n even 6 1
7938.2.a.cb 3 252.bl odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 26T_{11} + 33 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{3} - 5 T^{2} + 4 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 26 T + 33)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 28224 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( (T^{3} + 7 T^{2} + 4 T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 5 T^{5} + \cdots + 1089 \) Copy content Toggle raw display
$31$ \( T^{6} - 14 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( T^{6} + 18 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} + \cdots + 31329 \) Copy content Toggle raw display
$61$ \( T^{6} - 4 T^{5} + \cdots + 514089 \) Copy content Toggle raw display
$67$ \( T^{6} + 5 T^{5} + \cdots + 22201 \) Copy content Toggle raw display
$71$ \( (T^{3} + 7 T^{2} - 50 T - 99)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 25 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{6} + 7 T^{5} + \cdots + 594441 \) Copy content Toggle raw display
$83$ \( T^{6} + 8 T^{5} + \cdots + 8649 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$97$ \( T^{6} + 28 T^{5} + \cdots + 287296 \) Copy content Toggle raw display
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