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Properties

Label 672.2.bc.a
Level $672$
Weight $2$
Character orbit 672.bc
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [672,2,Mod(257,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 5])) 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("672.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + 3 q^{9} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{13} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{15}+ \cdots + (15 \zeta_{12}^{3} - 15 \zeta_{12}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 12 q^{9} - 2 q^{17} - 12 q^{21} + 8 q^{25} + 30 q^{33} - 6 q^{37} - 32 q^{41} - 6 q^{45} - 4 q^{49} - 42 q^{53} - 6 q^{57} - 42 q^{61} + 12 q^{65} + 30 q^{69} + 30 q^{73} - 50 q^{77} + 36 q^{81}+ \cdots + 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1 - \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 −0.500000 + 0.866025i 0 1.73205 + 2.00000i 0 3.00000 0
257.2 0 1.73205 0 −0.500000 + 0.866025i 0 −1.73205 − 2.00000i 0 3.00000 0
353.1 0 −1.73205 0 −0.500000 − 0.866025i 0 1.73205 − 2.00000i 0 3.00000 0
353.2 0 1.73205 0 −0.500000 − 0.866025i 0 −1.73205 + 2.00000i 0 3.00000 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
4.b odd 2 1 inner
21.g even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bc.a ✓ 4
3.b odd 2 1 672.2.bc.b yes 4
4.b odd 2 1 inner 672.2.bc.a ✓ 4
7.d odd 6 1 672.2.bc.b yes 4
12.b even 2 1 672.2.bc.b yes 4
21.g even 6 1 inner 672.2.bc.a ✓ 4
28.f even 6 1 672.2.bc.b yes 4
84.j odd 6 1 inner 672.2.bc.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bc.a ✓ 4 1.a even 1 1 trivial
672.2.bc.a ✓ 4 4.b odd 2 1 inner
672.2.bc.a ✓ 4 21.g even 6 1 inner
672.2.bc.a ✓ 4 84.j odd 6 1 inner
672.2.bc.b yes 4 3.b odd 2 1
672.2.bc.b yes 4 7.d odd 6 1
672.2.bc.b yes 4 12.b even 2 1
672.2.bc.b yes 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T + 8)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$61$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
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