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Properties

Label 1050.6.g.o
Level $1050$
Weight $6$
Character orbit 1050.g
Analytic conductor $168.403$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1050,6,Mod(799,1050)] 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("1050.799"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1050.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-32,0,72,0,0,-162,0,432] 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: \(168.403010804\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 i q^{2} + 9 i q^{3} - 16 q^{4} + 36 q^{6} + 49 i q^{7} + 64 i q^{8} - 81 q^{9} + 216 q^{11} - 144 i q^{12} - 998 i q^{13} + 196 q^{14} + 256 q^{16} + 1302 i q^{17} + 324 i q^{18} - 884 q^{19} - 441 q^{21} + \cdots - 17496 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 432 q^{11} + 392 q^{14} + 512 q^{16} - 1768 q^{19} - 882 q^{21} - 1152 q^{24} - 7984 q^{26} + 2964 q^{29} + 16720 q^{31} + 10416 q^{34} + 2592 q^{36} + 17964 q^{39}+ \cdots - 34992 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
− 1.00000i
− 4.00000i 9.00000i −16.0000 0 36.0000 49.0000i 64.0000i −81.0000 0
799.2 4.00000i − 9.00000i −16.0000 0 36.0000 − 49.0000i − 64.0000i −81.0000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.6.g.o 2
5.b even 2 1 inner 1050.6.g.o 2
5.c odd 4 1 42.6.a.a ✓ 1
5.c odd 4 1 1050.6.a.n 1
15.e even 4 1 126.6.a.k 1
20.e even 4 1 336.6.a.j 1
35.f even 4 1 294.6.a.h 1
35.k even 12 2 294.6.e.h 2
35.l odd 12 2 294.6.e.r 2
60.l odd 4 1 1008.6.a.x 1
105.k odd 4 1 882.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a ✓ 1 5.c odd 4 1
126.6.a.k 1 15.e even 4 1
294.6.a.h 1 35.f even 4 1
294.6.e.h 2 35.k even 12 2
294.6.e.r 2 35.l odd 12 2
336.6.a.j 1 20.e even 4 1
882.6.a.o 1 105.k odd 4 1
1008.6.a.x 1 60.l odd 4 1
1050.6.a.n 1 5.c odd 4 1
1050.6.g.o 2 1.a even 1 1 trivial
1050.6.g.o 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 216 \) acting on \(S_{6}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 216)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 996004 \) Copy content Toggle raw display
$17$ \( T^{2} + 1695204 \) Copy content Toggle raw display
$19$ \( (T + 884)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5143824 \) Copy content Toggle raw display
$29$ \( (T - 1482)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8360)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 22221796 \) Copy content Toggle raw display
$41$ \( (T + 9786)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 377758096 \) Copy content Toggle raw display
$47$ \( T^{2} + 492840000 \) Copy content Toggle raw display
$53$ \( T^{2} + 717704100 \) Copy content Toggle raw display
$59$ \( (T + 28092)^{2} \) Copy content Toggle raw display
$61$ \( (T + 38866)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 573506704 \) Copy content Toggle raw display
$71$ \( (T + 20628)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 84100 \) Copy content Toggle raw display
$79$ \( (T - 99544)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 372798864 \) Copy content Toggle raw display
$89$ \( (T + 36390)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6253330084 \) Copy content Toggle raw display
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