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Properties

Label 1008.4.a.bg
Level $1008$
Weight $4$
Character orbit 1008.a
Self dual yes
Analytic conductor $59.474$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [1008,4,Mod(1,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,6,0,14,0,0,0,26,0,96] 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: yes
Analytic conductor: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{337}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{337}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{5} + 7 q^{7} + (\beta + 13) q^{11} + (2 \beta + 48) q^{13} + (3 \beta - 39) q^{17} - 20 q^{19} + ( - 5 \beta + 11) q^{23} + (6 \beta + 221) q^{25} - 102 q^{29} + (16 \beta - 48) q^{31}+ \cdots + (70 \beta + 200) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 14 q^{7} + 26 q^{11} + 96 q^{13} - 78 q^{17} - 40 q^{19} + 22 q^{23} + 442 q^{25} - 204 q^{29} - 96 q^{31} + 42 q^{35} + 504 q^{37} + 102 q^{41} - 296 q^{43} - 780 q^{47} + 98 q^{49} - 192 q^{53}+ \cdots + 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.67878
9.67878
0 0 0 −15.3576 0 7.00000 0 0 0
1.2 0 0 0 21.3576 0 7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.a.bg 2
3.b odd 2 1 336.4.a.o 2
4.b odd 2 1 504.4.a.n 2
12.b even 2 1 168.4.a.g ✓ 2
21.c even 2 1 2352.4.a.br 2
24.f even 2 1 1344.4.a.bq 2
24.h odd 2 1 1344.4.a.bi 2
84.h odd 2 1 1176.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.g ✓ 2 12.b even 2 1
336.4.a.o 2 3.b odd 2 1
504.4.a.n 2 4.b odd 2 1
1008.4.a.bg 2 1.a even 1 1 trivial
1176.4.a.w 2 84.h odd 2 1
1344.4.a.bi 2 24.h odd 2 1
1344.4.a.bq 2 24.f even 2 1
2352.4.a.br 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 6T_{5} - 328 \) Copy content Toggle raw display
\( T_{11}^{2} - 26T_{11} - 168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 328 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 26T - 168 \) Copy content Toggle raw display
$13$ \( T^{2} - 96T + 956 \) Copy content Toggle raw display
$17$ \( T^{2} + 78T - 1512 \) Copy content Toggle raw display
$19$ \( (T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 22T - 8304 \) Copy content Toggle raw display
$29$ \( (T + 102)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 96T - 83968 \) Copy content Toggle raw display
$37$ \( T^{2} - 504T + 62156 \) Copy content Toggle raw display
$41$ \( T^{2} - 102T - 38176 \) Copy content Toggle raw display
$43$ \( T^{2} + 296T - 64368 \) Copy content Toggle raw display
$47$ \( T^{2} + 780T + 118400 \) Copy content Toggle raw display
$53$ \( T^{2} + 192T - 99972 \) Copy content Toggle raw display
$59$ \( T^{2} - 212T - 54816 \) Copy content Toggle raw display
$61$ \( T^{2} + 100T - 83772 \) Copy content Toggle raw display
$67$ \( T^{2} + 212T + 9888 \) Copy content Toggle raw display
$71$ \( T^{2} + 534T + 30512 \) Copy content Toggle raw display
$73$ \( T^{2} - 1128 T + 284396 \) Copy content Toggle raw display
$79$ \( T^{2} + 468 T - 1078912 \) Copy content Toggle raw display
$83$ \( T^{2} + 824T - 606704 \) Copy content Toggle raw display
$89$ \( T^{2} - 2118 T + 875808 \) Copy content Toggle raw display
$97$ \( T^{2} - 400 T - 1611300 \) Copy content Toggle raw display
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