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Properties

Label 1776.2.q.c
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,-1,0,0,0,-3,0,-1,0,4] 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 222)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{3} - 3 \zeta_{6} q^{7} + (\zeta_{6} - 1) q^{9} + 2 q^{11} + 5 \zeta_{6} q^{13} + (3 \zeta_{6} - 3) q^{21} + 4 q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + q^{27} + 2 q^{29} + 3 q^{31} - 2 \zeta_{6} q^{33} + \cdots + (2 \zeta_{6} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 3 q^{7} - q^{9} + 4 q^{11} + 5 q^{13} - 3 q^{21} + 8 q^{23} + 5 q^{25} + 2 q^{27} + 4 q^{29} + 6 q^{31} - 2 q^{33} + 10 q^{37} + 5 q^{39} - 2 q^{41} - 18 q^{43} - 12 q^{47} - 2 q^{49} + 10 q^{53}+ \cdots - 2 q^{99}+O(q^{100}) \) 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 0 −1.50000 + 2.59808i 0 −0.500000 − 0.866025i 0
1009.1 0 −0.500000 − 0.866025i 0 0 0 −1.50000 − 2.59808i 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.c 2
4.b odd 2 1 222.2.e.a ✓ 2
12.b even 2 1 666.2.f.e 2
37.c even 3 1 inner 1776.2.q.c 2
148.i odd 6 1 222.2.e.a ✓ 2
148.i odd 6 1 8214.2.a.h 1
148.j odd 6 1 8214.2.a.a 1
444.t even 6 1 666.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.e.a ✓ 2 4.b odd 2 1
222.2.e.a ✓ 2 148.i odd 6 1
666.2.f.e 2 12.b even 2 1
666.2.f.e 2 444.t even 6 1
1776.2.q.c 2 1.a even 1 1 trivial
1776.2.q.c 2 37.c even 3 1 inner
8214.2.a.a 1 148.j odd 6 1
8214.2.a.h 1 148.i odd 6 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 2 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( (T + 9)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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