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Properties

Label 1776.2.q.d
Level $1776$
Weight $2$
Character orbit 1776.q
Analytic conductor $14.181$
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] = [1776,2,Mod(433,1776)] 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("1776.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,1,0,-4,0,-1,0,0] 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: \(14.1814313990\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{3} + \zeta_{6} q^{5} - 4 \zeta_{6} q^{7} + (\zeta_{6} - 1) q^{9} + 2 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{15} + (7 \zeta_{6} - 7) q^{17} + 8 \zeta_{6} q^{19} + (4 \zeta_{6} - 4) q^{21} + \cdots + 11 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} - 4 q^{7} - q^{9} + 2 q^{13} + q^{15} - 7 q^{17} + 8 q^{19} - 4 q^{21} - 8 q^{23} + 4 q^{25} + 2 q^{27} - 10 q^{29} + 4 q^{35} + 11 q^{37} + 2 q^{39} + 5 q^{41} - 2 q^{45} + 16 q^{47}+ \cdots + 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 −2.00000 + 3.46410i 0 −0.500000 − 0.866025i 0
1009.1 0 −0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 −2.00000 − 3.46410i 0 −0.500000 + 0.866025i 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
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1776.2.q.d 2
4.b odd 2 1 888.2.q.d ✓ 2
12.b even 2 1 2664.2.r.c 2
37.c even 3 1 inner 1776.2.q.d 2
148.i odd 6 1 888.2.q.d ✓ 2
444.t even 6 1 2664.2.r.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.q.d ✓ 2 4.b odd 2 1
888.2.q.d ✓ 2 148.i odd 6 1
1776.2.q.d 2 1.a even 1 1 trivial
1776.2.q.d 2 37.c even 3 1 inner
2664.2.r.c 2 12.b even 2 1
2664.2.r.c 2 444.t even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1776, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( (T - 11)^{2} \) Copy content Toggle raw display
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