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Properties

Label 1008.3.f.g
Level $1008$
Weight $3$
Character orbit 1008.f
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,3,Mod(433,1008)] 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("1008.433"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-8,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{7} + ( - \beta_{3} - 6) q^{11} + (8 \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{2} + 11 \beta_1) q^{17} + (2 \beta_{2} - 8 \beta_1) q^{19}+ \cdots + (44 \beta_{2} + 10 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} - 24 q^{11} + 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} - 80 q^{37} + 128 q^{43} - 20 q^{49} + 216 q^{53} - 240 q^{65} - 176 q^{67} - 120 q^{71} - 24 q^{77} - 128 q^{79} + 216 q^{85}+ \cdots - 240 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 − 1.22474i
0 0 0 − 5.91359i 0 −6.24264 − 3.16693i 0 0 0
433.2 0 0 0 − 1.01461i 0 2.24264 + 6.63103i 0 0 0
433.3 0 0 0 1.01461i 0 2.24264 − 6.63103i 0 0 0
433.4 0 0 0 5.91359i 0 −6.24264 + 3.16693i 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.f.g 4
3.b odd 2 1 336.3.f.c 4
4.b odd 2 1 126.3.c.b 4
7.b odd 2 1 inner 1008.3.f.g 4
12.b even 2 1 42.3.c.a ✓ 4
21.c even 2 1 336.3.f.c 4
24.f even 2 1 1344.3.f.f 4
24.h odd 2 1 1344.3.f.e 4
28.d even 2 1 126.3.c.b 4
28.f even 6 1 882.3.n.a 4
28.f even 6 1 882.3.n.d 4
28.g odd 6 1 882.3.n.a 4
28.g odd 6 1 882.3.n.d 4
60.h even 2 1 1050.3.f.a 4
60.l odd 4 2 1050.3.h.a 8
84.h odd 2 1 42.3.c.a ✓ 4
84.j odd 6 1 294.3.g.b 4
84.j odd 6 1 294.3.g.c 4
84.n even 6 1 294.3.g.b 4
84.n even 6 1 294.3.g.c 4
168.e odd 2 1 1344.3.f.f 4
168.i even 2 1 1344.3.f.e 4
420.o odd 2 1 1050.3.f.a 4
420.w even 4 2 1050.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a ✓ 4 12.b even 2 1
42.3.c.a ✓ 4 84.h odd 2 1
126.3.c.b 4 4.b odd 2 1
126.3.c.b 4 28.d even 2 1
294.3.g.b 4 84.j odd 6 1
294.3.g.b 4 84.n even 6 1
294.3.g.c 4 84.j odd 6 1
294.3.g.c 4 84.n even 6 1
336.3.f.c 4 3.b odd 2 1
336.3.f.c 4 21.c even 2 1
882.3.n.a 4 28.f even 6 1
882.3.n.a 4 28.g odd 6 1
882.3.n.d 4 28.f even 6 1
882.3.n.d 4 28.g odd 6 1
1008.3.f.g 4 1.a even 1 1 trivial
1008.3.f.g 4 7.b odd 2 1 inner
1050.3.f.a 4 60.h even 2 1
1050.3.f.a 4 420.o odd 2 1
1050.3.h.a 8 60.l odd 4 2
1050.3.h.a 8 420.w even 4 2
1344.3.f.e 4 24.h odd 2 1
1344.3.f.e 4 168.i even 2 1
1344.3.f.f 4 24.f even 2 1
1344.3.f.f 4 168.e odd 2 1

Hecke kernels

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

\( T_{5}^{4} + 36T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} + 12T_{11} + 18 \) 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
$5$ \( T^{4} + 36T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$17$ \( T^{4} + 1476 T^{2} + 509796 \) Copy content Toggle raw display
$19$ \( T^{4} + 792 T^{2} + 138384 \) Copy content Toggle raw display
$23$ \( (T^{2} - 12 T - 126)^{2} \) Copy content Toggle raw display
$29$ \( (T + 30)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2592 T^{2} + 186624 \) Copy content Toggle raw display
$37$ \( (T^{2} + 40 T - 2192)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 1764 T^{2} + 86436 \) Copy content Toggle raw display
$43$ \( (T^{2} - 64 T - 776)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4896 T^{2} + 5089536 \) Copy content Toggle raw display
$53$ \( (T^{2} - 108 T + 2628)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 9504 T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 288T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( (T^{2} + 88 T + 1648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 60 T - 5598)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8208 T^{2} + 16064064 \) Copy content Toggle raw display
$79$ \( (T^{2} + 64 T - 9344)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 10944 T^{2} + 451584 \) Copy content Toggle raw display
$89$ \( T^{4} + 22788 T^{2} + 37234404 \) Copy content Toggle raw display
$97$ \( T^{4} + 12816 T^{2} + 27123264 \) Copy content Toggle raw display
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