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Properties

Label 1050.6.a.g
Level $1050$
Weight $6$
Character orbit 1050.a
Self dual yes
Analytic conductor $168.403$
Analytic rank $0$
Dimension $1$
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] = [1050,6,Mod(1,1050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1050.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,4,-9,16,0,-36,-49,64,81,0,-414] 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: yes
Analytic conductor: \(168.403010804\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9} - 414 q^{11} - 144 q^{12} + 1054 q^{13} - 196 q^{14} + 256 q^{16} + 1848 q^{17} + 324 q^{18} + 236 q^{19} + 441 q^{21} - 1656 q^{22}+ \cdots - 33534 q^{99}+O(q^{100}) \) 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
0
4.00000 −9.00000 16.0000 0 −36.0000 −49.0000 64.0000 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 1050.6.a.g 1
5.b even 2 1 42.6.a.c ✓ 1
5.c odd 4 2 1050.6.g.b 2
15.d odd 2 1 126.6.a.l 1
20.d odd 2 1 336.6.a.b 1
35.c odd 2 1 294.6.a.c 1
35.i odd 6 2 294.6.e.n 2
35.j even 6 2 294.6.e.l 2
60.h even 2 1 1008.6.a.ba 1
105.g even 2 1 882.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.c ✓ 1 5.b even 2 1
126.6.a.l 1 15.d odd 2 1
294.6.a.c 1 35.c odd 2 1
294.6.e.l 2 35.j even 6 2
294.6.e.n 2 35.i odd 6 2
336.6.a.b 1 20.d odd 2 1
882.6.a.n 1 105.g even 2 1
1008.6.a.ba 1 60.h even 2 1
1050.6.a.g 1 1.a even 1 1 trivial
1050.6.g.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} + 414 \) Copy content Toggle raw display
\( T_{13} - 1054 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 414 \) Copy content Toggle raw display
$13$ \( T - 1054 \) Copy content Toggle raw display
$17$ \( T - 1848 \) Copy content Toggle raw display
$19$ \( T - 236 \) Copy content Toggle raw display
$23$ \( T + 2898 \) Copy content Toggle raw display
$29$ \( T + 6522 \) Copy content Toggle raw display
$31$ \( T - 6200 \) Copy content Toggle raw display
$37$ \( T + 9650 \) Copy content Toggle raw display
$41$ \( T - 8484 \) Copy content Toggle raw display
$43$ \( T - 10804 \) Copy content Toggle raw display
$47$ \( T + 60 \) Copy content Toggle raw display
$53$ \( T + 22506 \) Copy content Toggle raw display
$59$ \( T + 28176 \) Copy content Toggle raw display
$61$ \( T + 35194 \) Copy content Toggle raw display
$67$ \( T - 28216 \) Copy content Toggle raw display
$71$ \( T + 6642 \) Copy content Toggle raw display
$73$ \( T - 52090 \) Copy content Toggle raw display
$79$ \( T - 43340 \) Copy content Toggle raw display
$83$ \( T + 25716 \) Copy content Toggle raw display
$89$ \( T - 98724 \) Copy content Toggle raw display
$97$ \( T - 148954 \) Copy content Toggle raw display
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