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Properties

Label 1050.2.o.g
Level $1050$
Weight $2$
Character orbit 1050.o
Analytic conductor $8.384$
Analytic rank $0$
Dimension $4$
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] = [1050,2,Mod(499,1050)] 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("1050.499"); S:= CuspForms(chi, 2); 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, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,-4,0,0,2,0,10,0,0,2] 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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} - q^{6} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( - 5 \zeta_{12}^{2} + 5) q^{11}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} + 10 q^{11} + 2 q^{14} - 2 q^{16} - 14 q^{19} - 10 q^{21} - 2 q^{24} + 10 q^{26} + 4 q^{31} + 16 q^{34} + 4 q^{36} + 10 q^{39} + 20 q^{41} - 10 q^{44} - 2 q^{46} - 4 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 0 −1.00000 −1.73205 − 2.00000i − 1.00000i 0.500000 − 0.866025i 0
499.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 1.73205 + 2.00000i 1.00000i 0.500000 − 0.866025i 0
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i 0 −1.00000 −1.73205 + 2.00000i 1.00000i 0.500000 + 0.866025i 0
949.2 0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 0 −1.00000 1.73205 − 2.00000i − 1.00000i 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.o.g 4
5.b even 2 1 inner 1050.2.o.g 4
5.c odd 4 1 210.2.i.b ✓ 2
5.c odd 4 1 1050.2.i.p 2
7.c even 3 1 inner 1050.2.o.g 4
15.e even 4 1 630.2.k.g 2
20.e even 4 1 1680.2.bg.d 2
35.f even 4 1 1470.2.i.e 2
35.j even 6 1 inner 1050.2.o.g 4
35.k even 12 1 1470.2.a.o 1
35.k even 12 1 1470.2.i.e 2
35.k even 12 1 7350.2.a.a 1
35.l odd 12 1 210.2.i.b ✓ 2
35.l odd 12 1 1050.2.i.p 2
35.l odd 12 1 1470.2.a.l 1
35.l odd 12 1 7350.2.a.u 1
105.w odd 12 1 4410.2.a.u 1
105.x even 12 1 630.2.k.g 2
105.x even 12 1 4410.2.a.j 1
140.w even 12 1 1680.2.bg.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.b ✓ 2 5.c odd 4 1
210.2.i.b ✓ 2 35.l odd 12 1
630.2.k.g 2 15.e even 4 1
630.2.k.g 2 105.x even 12 1
1050.2.i.p 2 5.c odd 4 1
1050.2.i.p 2 35.l odd 12 1
1050.2.o.g 4 1.a even 1 1 trivial
1050.2.o.g 4 5.b even 2 1 inner
1050.2.o.g 4 7.c even 3 1 inner
1050.2.o.g 4 35.j even 6 1 inner
1470.2.a.l 1 35.l odd 12 1
1470.2.a.o 1 35.k even 12 1
1470.2.i.e 2 35.f even 4 1
1470.2.i.e 2 35.k even 12 1
1680.2.bg.d 2 20.e even 4 1
1680.2.bg.d 2 140.w even 12 1
4410.2.a.j 1 105.x even 12 1
4410.2.a.u 1 105.w odd 12 1
7350.2.a.a 1 35.k even 12 1
7350.2.a.u 1 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T - 5)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$53$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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