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Properties

Label 1050.2.b.a
Level $1050$
Weight $2$
Character orbit 1050.b
Analytic conductor $8.384$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1050,2,Mod(251,1050)] 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([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1050.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,-4,0,2,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 210)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - 1) q^{3} - q^{4} + ( - \beta_{2} - \beta_1) q^{6} + (\beta_{3} - \beta_{2} - 2) q^{7} - \beta_{2} q^{8} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{11}+ \cdots + (4 \beta_{3} - 10 \beta_{2} - 4 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{4} + 2 q^{6} - 6 q^{7} + 2 q^{12} + 6 q^{14} + 4 q^{16} + 12 q^{17} + 8 q^{18} - 2 q^{24} + 4 q^{26} - 14 q^{27} + 6 q^{28} + 20 q^{33} - 12 q^{37} - 16 q^{38} - 12 q^{39} - 8 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
− 0.618034i
1.61803i
0.618034i
− 1.61803i
− 1.00000i −1.61803 − 0.618034i −1.00000 0 −0.618034 + 1.61803i −2.61803 + 0.381966i 1.00000i 2.23607 + 2.00000i 0
251.2 − 1.00000i 0.618034 + 1.61803i −1.00000 0 1.61803 − 0.618034i −0.381966 + 2.61803i 1.00000i −2.23607 + 2.00000i 0
251.3 1.00000i −1.61803 + 0.618034i −1.00000 0 −0.618034 − 1.61803i −2.61803 − 0.381966i − 1.00000i 2.23607 − 2.00000i 0
251.4 1.00000i 0.618034 − 1.61803i −1.00000 0 1.61803 + 0.618034i −0.381966 − 2.61803i − 1.00000i −2.23607 − 2.00000i 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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.b.a 4
3.b odd 2 1 1050.2.b.c 4
5.b even 2 1 210.2.b.b yes 4
5.c odd 4 1 1050.2.d.a 4
5.c odd 4 1 1050.2.d.f 4
7.b odd 2 1 1050.2.b.c 4
15.d odd 2 1 210.2.b.a ✓ 4
15.e even 4 1 1050.2.d.c 4
15.e even 4 1 1050.2.d.d 4
20.d odd 2 1 1680.2.f.e 4
21.c even 2 1 inner 1050.2.b.a 4
35.c odd 2 1 210.2.b.a ✓ 4
35.f even 4 1 1050.2.d.c 4
35.f even 4 1 1050.2.d.d 4
60.h even 2 1 1680.2.f.i 4
105.g even 2 1 210.2.b.b yes 4
105.k odd 4 1 1050.2.d.a 4
105.k odd 4 1 1050.2.d.f 4
140.c even 2 1 1680.2.f.i 4
420.o odd 2 1 1680.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.b.a ✓ 4 15.d odd 2 1
210.2.b.a ✓ 4 35.c odd 2 1
210.2.b.b yes 4 5.b even 2 1
210.2.b.b yes 4 105.g even 2 1
1050.2.b.a 4 1.a even 1 1 trivial
1050.2.b.a 4 21.c even 2 1 inner
1050.2.b.c 4 3.b odd 2 1
1050.2.b.c 4 7.b odd 2 1
1050.2.d.a 4 5.c odd 4 1
1050.2.d.a 4 105.k odd 4 1
1050.2.d.c 4 15.e even 4 1
1050.2.d.c 4 35.f even 4 1
1050.2.d.d 4 15.e even 4 1
1050.2.d.d 4 35.f even 4 1
1050.2.d.f 4 5.c odd 4 1
1050.2.d.f 4 105.k odd 4 1
1680.2.f.e 4 20.d odd 2 1
1680.2.f.e 4 420.o odd 2 1
1680.2.f.i 4 60.h even 2 1
1680.2.f.i 4 140.c even 2 1

Hecke kernels

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

\( T_{11}^{2} + 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 4 \) Copy content Toggle raw display
\( T_{37}^{2} + 6T_{37} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$67$ \( (T - 12)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$73$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 244)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 20 T + 80)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
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