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Properties

Label 3600.2.x.h
Level $3600$
Weight $2$
Character orbit 3600.x
Analytic conductor $28.746$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3600,2,Mod(2143,3600)] 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("3600.2143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,-48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-80,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(101)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.7461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 1200)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{7} - \beta_{6} q^{11} - \beta_{5} q^{13} - 2 \beta_{2} q^{17} - \beta_{7} q^{19} - \beta_{6} q^{31} + 3 \beta_{2} q^{37} - 6 q^{41} - 4 \beta_1 q^{43} - 6 \beta_{4} q^{47} + 3 \beta_{3} q^{49}+ \cdots + 2 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 48 q^{41} - 80 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\zeta_{24}^{7} - 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{6} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{5} + \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2143.1
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 − 0.965926i
0 0 0 0 0 −1.41421 − 1.41421i 0 0 0
2143.2 0 0 0 0 0 −1.41421 − 1.41421i 0 0 0
2143.3 0 0 0 0 0 1.41421 + 1.41421i 0 0 0
2143.4 0 0 0 0 0 1.41421 + 1.41421i 0 0 0
3007.1 0 0 0 0 0 −1.41421 + 1.41421i 0 0 0
3007.2 0 0 0 0 0 −1.41421 + 1.41421i 0 0 0
3007.3 0 0 0 0 0 1.41421 − 1.41421i 0 0 0
3007.4 0 0 0 0 0 1.41421 − 1.41421i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2143.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.x.h 8
3.b odd 2 1 1200.2.w.c ✓ 8
4.b odd 2 1 inner 3600.2.x.h 8
5.b even 2 1 inner 3600.2.x.h 8
5.c odd 4 2 inner 3600.2.x.h 8
12.b even 2 1 1200.2.w.c ✓ 8
15.d odd 2 1 1200.2.w.c ✓ 8
15.e even 4 2 1200.2.w.c ✓ 8
20.d odd 2 1 inner 3600.2.x.h 8
20.e even 4 2 inner 3600.2.x.h 8
60.h even 2 1 1200.2.w.c ✓ 8
60.l odd 4 2 1200.2.w.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.w.c ✓ 8 3.b odd 2 1
1200.2.w.c ✓ 8 12.b even 2 1
1200.2.w.c ✓ 8 15.d odd 2 1
1200.2.w.c ✓ 8 15.e even 4 2
1200.2.w.c ✓ 8 60.h even 2 1
1200.2.w.c ✓ 8 60.l odd 4 2
3600.2.x.h 8 1.a even 1 1 trivial
3600.2.x.h 8 4.b odd 2 1 inner
3600.2.x.h 8 5.b even 2 1 inner
3600.2.x.h 8 5.c odd 4 2 inner
3600.2.x.h 8 20.d odd 2 1 inner
3600.2.x.h 8 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{4} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display
\( T_{13}^{4} + 144 \) Copy content Toggle raw display
\( T_{17}^{4} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 11664)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$61$ \( (T + 10)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 192)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 324)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
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