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Properties

Label 1050.3.l.e
Level $1050$
Weight $3$
Character orbit 1050.l
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [1050,3,Mod(43,1050)] 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("1050.43"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,0,0,0,0,0,-16,0,0,-32,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} - \beta_{6} q^{3} + 2 \beta_{3} q^{4} + ( - \beta_{6} - \beta_1) q^{6} - \beta_{2} q^{7} + (2 \beta_{3} - 2) q^{8} - 3 \beta_{3} q^{9} + ( - \beta_{7} + 3 \beta_{6} - \beta_{4} + \cdots - 4) q^{11}+ \cdots + ( - 3 \beta_{7} + 9 \beta_{6} + \cdots - 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{8} - 32 q^{11} + 40 q^{13} - 32 q^{16} - 40 q^{17} + 24 q^{18} - 32 q^{22} - 8 q^{23} + 80 q^{26} + 96 q^{31} - 32 q^{32} - 72 q^{33} + 48 q^{36} + 112 q^{37} - 24 q^{38} - 160 q^{41}+ \cdots - 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 19\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 29\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 24\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + 23 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{6} - 22\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} - 91\nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6\nu^{7} + 139\nu^{3} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{4} - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{2} + 29\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{5} - 55\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -91\beta_{7} - 139\beta_{6} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.323042 − 0.323042i
−1.54779 + 1.54779i
1.54779 − 1.54779i
−0.323042 + 0.323042i
0.323042 + 0.323042i
−1.54779 − 1.54779i
1.54779 + 1.54779i
−0.323042 − 0.323042i
1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i 0 −2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.2 1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i 0 −2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.3 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 0 2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0
43.4 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 0 2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i 0
757.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.3 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 0 2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0
757.4 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.l.e yes 8
5.b even 2 1 1050.3.l.a ✓ 8
5.c odd 4 1 1050.3.l.a ✓ 8
5.c odd 4 1 inner 1050.3.l.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.l.a ✓ 8 5.b even 2 1
1050.3.l.a ✓ 8 5.c odd 4 1
1050.3.l.e yes 8 1.a even 1 1 trivial
1050.3.l.e yes 8 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{4} + 16T_{11}^{3} - 82T_{11}^{2} - 160T_{11} + 625 \) Copy content Toggle raw display
\( T_{13}^{8} - 40 T_{13}^{7} + 800 T_{13}^{6} - 6720 T_{13}^{5} + 20648 T_{13}^{4} + 131040 T_{13}^{3} + \cdots + 1157776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 40 T^{7} + \cdots + 1157776 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 69482851216 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 145676568976 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots + 244140625 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1070879337889 \) Copy content Toggle raw display
$31$ \( (T^{4} - 48 T^{3} + \cdots - 379760)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 1163755500625 \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{3} + \cdots - 68864)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 64 T^{7} + \cdots + 223532401 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 6469392250000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 383805030400 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 125230513134736 \) Copy content Toggle raw display
$61$ \( (T^{4} + 64 T^{3} + \cdots - 22016000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 38759303135809 \) Copy content Toggle raw display
$71$ \( (T^{4} - 120 T^{3} + \cdots + 17125945)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 165058256250000 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 7023613249 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 66382491904 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 362929170490000 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 392594596000000 \) Copy content Toggle raw display
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