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Properties

Label 1008.3.cg.k
Level $1008$
Weight $3$
Character orbit 1008.cg
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,3,Mod(145,1008)] 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, 0, 5])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.145"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2}) q^{5} + ( - 3 \beta_1 + 8) q^{7} + \beta_{2} q^{11} + (12 \beta_1 - 6) q^{13} + (8 \beta_{3} + 4 \beta_{2}) q^{17} + (8 \beta_1 + 8) q^{19} + (2 \beta_{3} + 2 \beta_{2}) q^{23}+ \cdots + (42 \beta_1 - 21) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{7} + 48 q^{19} + 28 q^{25} + 102 q^{31} - 16 q^{37} + 16 q^{43} + 142 q^{49} + 132 q^{61} + 208 q^{67} + 168 q^{73} + 22 q^{79} + 624 q^{85} + 108 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 5 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.15139 + 1.99426i
−0.651388 − 1.12824i
1.15139 − 1.99426i
−0.651388 + 1.12824i
0 0 0 −5.40833 − 3.12250i 0 6.50000 − 2.59808i 0 0 0
145.2 0 0 0 5.40833 + 3.12250i 0 6.50000 − 2.59808i 0 0 0
577.1 0 0 0 −5.40833 + 3.12250i 0 6.50000 + 2.59808i 0 0 0
577.2 0 0 0 5.40833 − 3.12250i 0 6.50000 + 2.59808i 0 0 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.k 4
3.b odd 2 1 inner 1008.3.cg.k 4
4.b odd 2 1 63.3.m.e ✓ 4
7.d odd 6 1 inner 1008.3.cg.k 4
12.b even 2 1 63.3.m.e ✓ 4
21.g even 6 1 inner 1008.3.cg.k 4
28.d even 2 1 441.3.m.h 4
28.f even 6 1 63.3.m.e ✓ 4
28.f even 6 1 441.3.d.f 4
28.g odd 6 1 441.3.d.f 4
28.g odd 6 1 441.3.m.h 4
84.h odd 2 1 441.3.m.h 4
84.j odd 6 1 63.3.m.e ✓ 4
84.j odd 6 1 441.3.d.f 4
84.n even 6 1 441.3.d.f 4
84.n even 6 1 441.3.m.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.e ✓ 4 4.b odd 2 1
63.3.m.e ✓ 4 12.b even 2 1
63.3.m.e ✓ 4 28.f even 6 1
63.3.m.e ✓ 4 84.j odd 6 1
441.3.d.f 4 28.f even 6 1
441.3.d.f 4 28.g odd 6 1
441.3.d.f 4 84.j odd 6 1
441.3.d.f 4 84.n even 6 1
441.3.m.h 4 28.d even 2 1
441.3.m.h 4 28.g odd 6 1
441.3.m.h 4 84.h odd 2 1
441.3.m.h 4 84.n even 6 1
1008.3.cg.k 4 1.a even 1 1 trivial
1008.3.cg.k 4 3.b odd 2 1 inner
1008.3.cg.k 4 7.d odd 6 1 inner
1008.3.cg.k 4 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} - 39T_{5}^{2} + 1521 \) Copy content Toggle raw display
\( T_{11}^{4} + 13T_{11}^{2} + 169 \) Copy content Toggle raw display
\( T_{13}^{2} + 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 39T^{2} + 1521 \) Copy content Toggle raw display
$7$ \( (T^{2} - 13 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$13$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 624 T^{2} + 389376 \) Copy content Toggle raw display
$19$ \( (T^{2} - 24 T + 192)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$29$ \( (T^{2} - 637)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 51 T + 867)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1404)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 156 T^{2} + 24336 \) Copy content Toggle raw display
$53$ \( T^{4} + 6877 T^{2} + 47293129 \) Copy content Toggle raw display
$59$ \( T^{4} - 6591 T^{2} + 43441281 \) Copy content Toggle raw display
$61$ \( (T^{2} - 66 T + 1452)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 104 T + 10816)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8788)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 84 T + 2352)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 17199)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 18876 T^{2} + 356303376 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1323)^{2} \) Copy content Toggle raw display
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