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Properties

Label 416.2.w.b
Level $416$
Weight $2$
Character orbit 416.w
Analytic conductor $3.322$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [416,2,Mod(225,416)] 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("416.225"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.w (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.32177672409\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 2 \beta_{2} - 1) q^{5} + ( - 3 \beta_{2} + 3) q^{9} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{13} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{17} + (2 \beta_{3} - 4 \beta_1 - 2) q^{25}+ \cdots + 4 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9} + 6 q^{13} - 2 q^{17} - 8 q^{25} + 10 q^{29} + 6 q^{37} + 30 q^{41} + 18 q^{45} - 14 q^{49} - 28 q^{53} - 10 q^{61} - 26 q^{65} - 18 q^{81} - 54 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0 0 0 − 3.73205i 0 0 0 1.50000 + 2.59808i 0
225.2 0 0 0 0.267949i 0 0 0 1.50000 + 2.59808i 0
257.1 0 0 0 − 0.267949i 0 0 0 1.50000 − 2.59808i 0
257.2 0 0 0 3.73205i 0 0 0 1.50000 − 2.59808i 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 CM by \(\Q(\sqrt{-1}) \)
13.e even 6 1 inner
52.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.w.b ✓ 4
4.b odd 2 1 CM 416.2.w.b ✓ 4
8.b even 2 1 832.2.w.f 4
8.d odd 2 1 832.2.w.f 4
13.e even 6 1 inner 416.2.w.b ✓ 4
13.f odd 12 1 5408.2.a.r 2
13.f odd 12 1 5408.2.a.bc 2
52.i odd 6 1 inner 416.2.w.b ✓ 4
52.l even 12 1 5408.2.a.r 2
52.l even 12 1 5408.2.a.bc 2
104.p odd 6 1 832.2.w.f 4
104.s even 6 1 832.2.w.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.b ✓ 4 1.a even 1 1 trivial
416.2.w.b ✓ 4 4.b odd 2 1 CM
416.2.w.b ✓ 4 13.e even 6 1 inner
416.2.w.b ✓ 4 52.i odd 6 1 inner
832.2.w.f 4 8.b even 2 1
832.2.w.f 4 8.d odd 2 1
832.2.w.f 4 104.p odd 6 1
832.2.w.f 4 104.s even 6 1
5408.2.a.r 2 13.f odd 12 1
5408.2.a.r 2 52.l even 12 1
5408.2.a.bc 2 13.f odd 12 1
5408.2.a.bc 2 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$41$ \( T^{4} - 30 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 14 T + 37)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + \cdots + 6889 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 182T^{2} + 1369 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$97$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
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