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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,4,Mod(49,600)] 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("600.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-36,0,-32,0,0,0,0,0,0,0,420] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(35.4011460034\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{109})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 55x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} - 9 q^{9} + ( - 3 \beta_{3} - 8) q^{11} + (2 \beta_{2} - 41 \beta_1) q^{13} + (5 \beta_{2} + 4 \beta_1) q^{17} + (\beta_{3} + 105) q^{19} + (3 \beta_{3} - 3) q^{21}+ \cdots + (27 \beta_{3} + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9} - 32 q^{11} + 420 q^{19} - 12 q^{21} + 480 q^{29} + 436 q^{31} + 492 q^{39} + 168 q^{41} - 376 q^{49} - 48 q^{51} + 2168 q^{59} + 1548 q^{61} + 240 q^{69} + 2216 q^{71} + 2656 q^{79} + 324 q^{81}+ \cdots + 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 55x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 28\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 164\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 110 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 110 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{2} + 41\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
5.72015i
− 4.72015i
4.72015i
− 5.72015i
0 − 3.00000i 0 0 0 − 21.8806i 0 −9.00000 0
49.2 0 − 3.00000i 0 0 0 19.8806i 0 −9.00000 0
49.3 0 3.00000i 0 0 0 − 19.8806i 0 −9.00000 0
49.4 0 3.00000i 0 0 0 21.8806i 0 −9.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.4.f.j 4
3.b odd 2 1 1800.4.f.z 4
4.b odd 2 1 1200.4.f.x 4
5.b even 2 1 inner 600.4.f.j 4
5.c odd 4 1 600.4.a.s ✓ 2
5.c odd 4 1 600.4.a.u yes 2
15.d odd 2 1 1800.4.f.z 4
15.e even 4 1 1800.4.a.bm 2
15.e even 4 1 1800.4.a.bo 2
20.d odd 2 1 1200.4.f.x 4
20.e even 4 1 1200.4.a.bp 2
20.e even 4 1 1200.4.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.s ✓ 2 5.c odd 4 1
600.4.a.u yes 2 5.c odd 4 1
600.4.f.j 4 1.a even 1 1 trivial
600.4.f.j 4 5.b even 2 1 inner
1200.4.a.bp 2 20.e even 4 1
1200.4.a.br 2 20.e even 4 1
1200.4.f.x 4 4.b odd 2 1
1200.4.f.x 4 20.d odd 2 1
1800.4.a.bm 2 15.e even 4 1
1800.4.a.bo 2 15.e even 4 1
1800.4.f.z 4 3.b odd 2 1
1800.4.f.z 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{4} + 874T_{7}^{2} + 189225 \) Copy content Toggle raw display
\( T_{11}^{2} + 16T_{11} - 3860 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 874 T^{2} + 189225 \) Copy content Toggle raw display
$11$ \( (T^{2} + 16 T - 3860)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 6850 T^{2} + 3969 \) Copy content Toggle raw display
$17$ \( T^{4} + 21832 T^{2} + 118461456 \) Copy content Toggle raw display
$19$ \( (T^{2} - 210 T + 10589)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1672 T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 240 T + 13964)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 218 T - 23435)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 153448 T^{2} + 109746576 \) Copy content Toggle raw display
$41$ \( (T^{2} - 84 T - 2160)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 59338 T^{2} + 61921161 \) Copy content Toggle raw display
$47$ \( T^{4} + 262856 T^{2} + 34339600 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 2057529600 \) Copy content Toggle raw display
$59$ \( (T^{2} - 1084 T + 265860)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 774 T + 142793)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 343230767881 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1108 T - 59760)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 2008832400 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1328 T + 99072)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 199155127824 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1424 T + 499968)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1670838026881 \) Copy content Toggle raw display
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