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Properties

Label 3564.2.i.g
Level $3564$
Weight $2$
Character orbit 3564.i
Analytic conductor $28.459$
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] = [3564,2,Mod(1189,3564)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3564, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) 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("3564.1189"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3564.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,-2,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.4586832804\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + (\zeta_{6} - 1) q^{11} - 6 \zeta_{6} q^{13} + 4 q^{17} - 2 q^{19} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 4 q^{35} - 6 q^{37} + (10 \zeta_{6} - 10) q^{43} + \cdots + ( - 2 \zeta_{6} + 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} - q^{11} - 6 q^{13} + 8 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} - 8 q^{35} - 12 q^{37} - 10 q^{43} + 3 q^{49} - 28 q^{53} - 4 q^{55} - 12 q^{59} + 14 q^{61} + 12 q^{65} - 4 q^{67}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3564\mathbb{Z}\right)^\times\).

\(n\) \(1541\) \(1783\) \(2917\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 1.00000 + 1.73205i 0 −1.00000 + 1.73205i 0 0 0
2377.1 0 0 0 1.00000 − 1.73205i 0 −1.00000 − 1.73205i 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 3564.2.i.g 2
3.b odd 2 1 3564.2.i.b 2
9.c even 3 1 396.2.a.b 1
9.c even 3 1 inner 3564.2.i.g 2
9.d odd 6 1 132.2.a.a ✓ 1
9.d odd 6 1 3564.2.i.b 2
36.f odd 6 1 1584.2.a.c 1
36.h even 6 1 528.2.a.h 1
45.h odd 6 1 3300.2.a.k 1
45.j even 6 1 9900.2.a.g 1
45.k odd 12 2 9900.2.c.k 2
45.l even 12 2 3300.2.c.c 2
63.o even 6 1 6468.2.a.l 1
72.j odd 6 1 2112.2.a.t 1
72.l even 6 1 2112.2.a.b 1
72.n even 6 1 6336.2.a.cf 1
72.p odd 6 1 6336.2.a.bz 1
99.g even 6 1 1452.2.a.c 1
99.h odd 6 1 4356.2.a.c 1
99.n odd 30 4 1452.2.i.k 4
99.p even 30 4 1452.2.i.l 4
396.o odd 6 1 5808.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.a.a ✓ 1 9.d odd 6 1
396.2.a.b 1 9.c even 3 1
528.2.a.h 1 36.h even 6 1
1452.2.a.c 1 99.g even 6 1
1452.2.i.k 4 99.n odd 30 4
1452.2.i.l 4 99.p even 30 4
1584.2.a.c 1 36.f odd 6 1
2112.2.a.b 1 72.l even 6 1
2112.2.a.t 1 72.j odd 6 1
3300.2.a.k 1 45.h odd 6 1
3300.2.c.c 2 45.l even 12 2
3564.2.i.b 2 3.b odd 2 1
3564.2.i.b 2 9.d odd 6 1
3564.2.i.g 2 1.a even 1 1 trivial
3564.2.i.g 2 9.c even 3 1 inner
4356.2.a.c 1 99.h odd 6 1
5808.2.a.bf 1 396.o odd 6 1
6336.2.a.bz 1 72.p odd 6 1
6336.2.a.cf 1 72.n even 6 1
6468.2.a.l 1 63.o even 6 1
9900.2.a.g 1 45.j even 6 1
9900.2.c.k 2 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3564, [\chi])\):

\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{17} - 4 \) 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
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 14)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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