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Properties

Label 1008.3.cd.p
Level $1008$
Weight $3$
Character orbit 1008.cd
Analytic conductor $27.466$
Analytic rank $0$
Dimension $8$
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(415,1008)] 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([3, 0, 0, 2])) 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("1008.415"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cd (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 39x^{6} + 593x^{4} - 4512x^{2} + 20164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} + ( - 3 \beta_{4} - \beta_{3} + 5 \beta_1 + 1) q^{7} - \beta_{7} q^{11} + ( - 3 \beta_{4} + 3 \beta_{3} - 1) q^{13} + (\beta_{7} + \beta_{5} + 2 \beta_{2}) q^{17} + (5 \beta_{3} - 3 \beta_1 + 2) q^{19}+ \cdots + ( - 31 \beta_{4} + 31 \beta_{3} + 16) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{7} - 20 q^{13} - 6 q^{19} - 78 q^{25} + 156 q^{31} + 70 q^{37} - 20 q^{49} + 276 q^{61} + 114 q^{67} - 262 q^{73} + 576 q^{79} - 392 q^{85} + 282 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 39x^{6} + 593x^{4} - 4512x^{2} + 20164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 181\nu^{5} - 3007\nu^{3} + 12322\nu + 23288 ) / 46576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} - 276\nu^{4} + 3734\nu^{2} - 21108 ) / 697 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -79\nu^{7} + 426\nu^{6} + 2655\nu^{5} - 13064\nu^{4} - 33783\nu^{3} + 110760\nu^{2} + 146714\nu - 405268 ) / 197948 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -79\nu^{7} - 426\nu^{6} + 2655\nu^{5} + 13064\nu^{4} - 33783\nu^{3} - 110760\nu^{2} + 146714\nu + 405268 ) / 197948 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -79\nu^{7} + 781\nu^{6} + 2655\nu^{5} - 7455\nu^{4} - 33783\nu^{3} - 44375\nu^{2} + 344662\nu + 378714 ) / 197948 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -79\nu^{7} - 781\nu^{6} + 2655\nu^{5} + 7455\nu^{4} - 33783\nu^{3} + 44375\nu^{2} + 344662\nu - 378714 ) / 197948 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 299 \nu^{7} - 639 \nu^{6} + 7543 \nu^{5} + 19596 \nu^{4} - 34526 \nu^{3} - 265114 \nu^{2} + \cdots + 1498668 ) / 98974 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{4} - 3\beta_{3} + \beta_{2} + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 5\beta_{6} + 5\beta_{5} - 13\beta_{4} - 13\beta_{3} + \beta_{2} + 16\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{6} + 12\beta_{5} + 67\beta_{4} - 67\beta_{3} + 15\beta_{2} + 134 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 70\beta_{7} + 104\beta_{6} + 104\beta_{5} - 401\beta_{4} - 401\beta_{3} + 35\beta_{2} + 1192\beta _1 - 596 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -184\beta_{6} + 184\beta_{5} + 405\beta_{4} - 405\beta_{3} + 100\beta_{2} + 666 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 642 \beta_{7} + 1076 \beta_{6} + 1076 \beta_{5} - 6721 \beta_{4} - 6721 \beta_{3} + 321 \beta_{2} + \cdots - 13188 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
3.98041 + 0.656712i
2.53108 − 1.52274i
−2.53108 − 1.52274i
−3.98041 + 0.656712i
3.98041 − 0.656712i
2.53108 + 1.52274i
−2.53108 + 1.52274i
−3.98041 − 0.656712i
0 0 0 −3.98041 + 6.89427i 0 −0.774917 + 6.95698i 0 0 0
415.2 0 0 0 −2.53108 + 4.38395i 0 6.77492 − 1.76082i 0 0 0
415.3 0 0 0 2.53108 − 4.38395i 0 6.77492 − 1.76082i 0 0 0
415.4 0 0 0 3.98041 − 6.89427i 0 −0.774917 + 6.95698i 0 0 0
991.1 0 0 0 −3.98041 − 6.89427i 0 −0.774917 − 6.95698i 0 0 0
991.2 0 0 0 −2.53108 − 4.38395i 0 6.77492 + 1.76082i 0 0 0
991.3 0 0 0 2.53108 + 4.38395i 0 6.77492 + 1.76082i 0 0 0
991.4 0 0 0 3.98041 + 6.89427i 0 −0.774917 − 6.95698i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n 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.cd.p yes 8
3.b odd 2 1 inner 1008.3.cd.p yes 8
4.b odd 2 1 1008.3.cd.n ✓ 8
7.c even 3 1 1008.3.cd.n ✓ 8
12.b even 2 1 1008.3.cd.n ✓ 8
21.h odd 6 1 1008.3.cd.n ✓ 8
28.g odd 6 1 inner 1008.3.cd.p yes 8
84.n even 6 1 inner 1008.3.cd.p yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.3.cd.n ✓ 8 4.b odd 2 1
1008.3.cd.n ✓ 8 7.c even 3 1
1008.3.cd.n ✓ 8 12.b even 2 1
1008.3.cd.n ✓ 8 21.h odd 6 1
1008.3.cd.p yes 8 1.a even 1 1 trivial
1008.3.cd.p yes 8 3.b odd 2 1 inner
1008.3.cd.p yes 8 28.g odd 6 1 inner
1008.3.cd.p yes 8 84.n 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 \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{8} + 89T_{5}^{6} + 6297T_{5}^{4} + 144536T_{5}^{2} + 2637376 \) Copy content Toggle raw display
\( T_{11}^{8} - 347T_{11}^{6} + 94425T_{11}^{4} - 9016448T_{11}^{2} + 675168256 \) Copy content Toggle raw display
\( T_{13}^{2} + 5T_{13} - 122 \) Copy content Toggle raw display
\( T_{19}^{4} + 3T_{19}^{3} - 115T_{19}^{2} - 354T_{19} + 13924 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 89 T^{6} + \cdots + 2637376 \) Copy content Toggle raw display
$7$ \( (T^{4} - 12 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 347 T^{6} + \cdots + 675168256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T - 122)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 101317436416 \) Copy content Toggle raw display
$19$ \( (T^{4} + 3 T^{3} + \cdots + 13924)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 596 T^{6} + \cdots + 42198016 \) Copy content Toggle raw display
$29$ \( (T^{4} - 1361 T^{2} + 6496)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 78 T^{3} + \cdots + 31329)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 35 T^{3} + \cdots + 2010724)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4448 T^{2} + 1662976)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1619 T^{2} + 633616)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 4429778927616 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20810165217856 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} - 138 T^{3} + \cdots + 11128896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 57 T^{3} + \cdots + 12996)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 32456 T^{2} + 249550336)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 131 T^{3} + \cdots + 3541924)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 288 T^{3} + \cdots + 45441081)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8243 T^{2} + 1365784)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 356 T^{6} + \cdots + 675168256 \) Copy content Toggle raw display
$97$ \( (T^{2} - T - 13694)^{4} \) Copy content Toggle raw display
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