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Properties

Label 1050.3.q.a
Level $1050$
Weight $3$
Character orbit 1050.q
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1050,3,Mod(199,1050)] 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("1050.199"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8,0,0,0,0,-12,0,-24,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - 2 \beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + (\beta_{7} - 2 \beta_{4}) q^{6} + ( - 4 \beta_{6} + 2 \beta_{5} + \cdots + 5 \beta_1) q^{7} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 12 q^{9} - 24 q^{11} - 48 q^{14} - 16 q^{16} + 84 q^{19} + 192 q^{26} + 204 q^{31} - 48 q^{36} - 12 q^{39} + 48 q^{44} - 144 q^{46} + 4 q^{49} - 96 q^{51} - 96 q^{56} + 48 q^{59} - 144 q^{61}+ \cdots + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 − 0.258819i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 − 1.73205i 0 − 2.44949i −1.88064 + 6.74264i 2.82843i −1.50000 − 2.59808i 0
199.2 −1.22474 + 0.707107i 0.866025 − 1.50000i 1.00000 − 1.73205i 0 2.44949i 6.77962 + 1.74264i 2.82843i −1.50000 − 2.59808i 0
199.3 1.22474 − 0.707107i −0.866025 + 1.50000i 1.00000 − 1.73205i 0 2.44949i −6.77962 − 1.74264i − 2.82843i −1.50000 − 2.59808i 0
199.4 1.22474 − 0.707107i 0.866025 − 1.50000i 1.00000 − 1.73205i 0 − 2.44949i 1.88064 − 6.74264i − 2.82843i −1.50000 − 2.59808i 0
649.1 −1.22474 − 0.707107i −0.866025 − 1.50000i 1.00000 + 1.73205i 0 2.44949i −1.88064 − 6.74264i − 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 − 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 − 2.44949i 6.77962 − 1.74264i − 2.82843i −1.50000 + 2.59808i 0
649.3 1.22474 + 0.707107i −0.866025 − 1.50000i 1.00000 + 1.73205i 0 − 2.44949i −6.77962 + 1.74264i 2.82843i −1.50000 + 2.59808i 0
649.4 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 1.88064 + 6.74264i 2.82843i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.a 8
5.b even 2 1 inner 1050.3.q.a 8
5.c odd 4 1 42.3.g.a ✓ 4
5.c odd 4 1 1050.3.p.a 4
7.d odd 6 1 inner 1050.3.q.a 8
15.e even 4 1 126.3.n.a 4
20.e even 4 1 336.3.bh.e 4
35.f even 4 1 294.3.g.a 4
35.i odd 6 1 inner 1050.3.q.a 8
35.k even 12 1 42.3.g.a ✓ 4
35.k even 12 1 294.3.c.a 4
35.k even 12 1 1050.3.p.a 4
35.l odd 12 1 294.3.c.a 4
35.l odd 12 1 294.3.g.a 4
60.l odd 4 1 1008.3.cg.h 4
105.k odd 4 1 882.3.n.e 4
105.w odd 12 1 126.3.n.a 4
105.w odd 12 1 882.3.c.b 4
105.x even 12 1 882.3.c.b 4
105.x even 12 1 882.3.n.e 4
140.w even 12 1 2352.3.f.e 4
140.x odd 12 1 336.3.bh.e 4
140.x odd 12 1 2352.3.f.e 4
420.br even 12 1 1008.3.cg.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a ✓ 4 5.c odd 4 1
42.3.g.a ✓ 4 35.k even 12 1
126.3.n.a 4 15.e even 4 1
126.3.n.a 4 105.w odd 12 1
294.3.c.a 4 35.k even 12 1
294.3.c.a 4 35.l odd 12 1
294.3.g.a 4 35.f even 4 1
294.3.g.a 4 35.l odd 12 1
336.3.bh.e 4 20.e even 4 1
336.3.bh.e 4 140.x odd 12 1
882.3.c.b 4 105.w odd 12 1
882.3.c.b 4 105.x even 12 1
882.3.n.e 4 105.k odd 4 1
882.3.n.e 4 105.x even 12 1
1008.3.cg.h 4 60.l odd 4 1
1008.3.cg.h 4 420.br even 12 1
1050.3.p.a 4 5.c odd 4 1
1050.3.p.a 4 35.k even 12 1
1050.3.q.a 8 1.a even 1 1 trivial
1050.3.q.a 8 5.b even 2 1 inner
1050.3.q.a 8 7.d odd 6 1 inner
1050.3.q.a 8 35.i odd 6 1 inner
2352.3.f.e 4 140.w even 12 1
2352.3.f.e 4 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 6T_{11} + 36 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 774 T^{2} + 145161)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 432 T^{6} + \cdots + 796594176 \) Copy content Toggle raw display
$19$ \( (T^{4} - 42 T^{3} + \cdots + 15129)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 64524128256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 102 T^{3} + \cdots + 423801)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 818 T^{6} + \cdots + 777796321 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4248 T^{2} + 3732624)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 242 T^{2} + 529)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 4158271385856 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 75939094204416 \) Copy content Toggle raw display
$59$ \( (T^{4} - 24 T^{3} + \cdots + 1272384)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 72 T^{3} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 64013554081 \) Copy content Toggle raw display
$71$ \( (T^{2} - 156 T + 2556)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 72\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + \cdots + 75463969)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 17928 T^{2} + 69956496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 36 T + 432)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1056 T^{2} + 112896)^{2} \) Copy content Toggle raw display
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