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Properties

Label 1620.4.i.h
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
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] = [1620,4,Mod(541,1620)] 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("1620.541"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,5,0,-17] 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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{5} + (17 \zeta_{6} - 17) q^{7} + (30 \zeta_{6} - 30) q^{11} + 61 \zeta_{6} q^{13} + 120 q^{17} - 43 q^{19} + 90 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (90 \zeta_{6} - 90) q^{29}+ \cdots + ( - 997 \zeta_{6} + 997) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} - 17 q^{7} - 30 q^{11} + 61 q^{13} + 240 q^{17} - 86 q^{19} + 90 q^{23} - 25 q^{25} - 90 q^{29} - 8 q^{31} - 170 q^{35} + 634 q^{37} - 30 q^{41} + 220 q^{43} - 180 q^{47} + 54 q^{49} + 1260 q^{53}+ \cdots + 997 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 2.50000 + 4.33013i 0 −8.50000 + 14.7224i 0 0 0
1081.1 0 0 0 2.50000 − 4.33013i 0 −8.50000 − 14.7224i 0 0 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.i.h 2
3.b odd 2 1 1620.4.i.b 2
9.c even 3 1 540.4.a.b ✓ 1
9.c even 3 1 inner 1620.4.i.h 2
9.d odd 6 1 540.4.a.d yes 1
9.d odd 6 1 1620.4.i.b 2
36.f odd 6 1 2160.4.a.a 1
36.h even 6 1 2160.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.b ✓ 1 9.c even 3 1
540.4.a.d yes 1 9.d odd 6 1
1620.4.i.b 2 3.b odd 2 1
1620.4.i.b 2 9.d odd 6 1
1620.4.i.h 2 1.a even 1 1 trivial
1620.4.i.h 2 9.c even 3 1 inner
2160.4.a.a 1 36.f odd 6 1
2160.4.a.k 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{2} + 17T_{7} + 289 \) Copy content Toggle raw display
\( T_{11}^{2} + 30T_{11} + 900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$11$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 61T + 3721 \) Copy content Toggle raw display
$17$ \( (T - 120)^{2} \) Copy content Toggle raw display
$19$ \( (T + 43)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 90T + 8100 \) Copy content Toggle raw display
$29$ \( T^{2} + 90T + 8100 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( (T - 317)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$43$ \( T^{2} - 220T + 48400 \) Copy content Toggle raw display
$47$ \( T^{2} + 180T + 32400 \) Copy content Toggle raw display
$53$ \( (T - 630)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 840T + 705600 \) Copy content Toggle raw display
$61$ \( T^{2} + 599T + 358801 \) Copy content Toggle raw display
$67$ \( T^{2} + 107T + 11449 \) Copy content Toggle raw display
$71$ \( (T - 210)^{2} \) Copy content Toggle raw display
$73$ \( (T + 421)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 353T + 124609 \) Copy content Toggle raw display
$83$ \( T^{2} + 1350 T + 1822500 \) Copy content Toggle raw display
$89$ \( (T + 1020)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 997T + 994009 \) Copy content Toggle raw display
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