↑

Properties

Label 208.3.t.c
Level $208$
Weight $3$
Character orbit 208.t
Analytic conductor $5.668$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [208,3,Mod(161,208)] 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("208.161"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 208.t (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,8] 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: \(5.66758949869\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{3} - 2 \beta_{2} + 2) q^{5} + (3 \beta_{2} - \beta_1 + 3) q^{7} + (2 \beta_{3} - 2 \beta_1 + 2) q^{9} + (\beta_{2} + 4 \beta_1 + 1) q^{11} + ( - 3 \beta_{3} + 10 \beta_{2} + \cdots + 2) q^{13}+ \cdots + ( - 38 \beta_{2} + 4 \beta_1 - 38) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{5} + 12 q^{7} + 8 q^{9} + 4 q^{11} + 8 q^{13} + 28 q^{15} + 32 q^{21} + 52 q^{27} + 40 q^{29} - 40 q^{31} - 76 q^{33} + 68 q^{35} + 40 q^{37} - 92 q^{39} + 32 q^{41} + 56 q^{45} + 4 q^{47}+ \cdots - 152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.58114 + 1.58114i
−1.58114 − 1.58114i
1.58114 − 1.58114i
−1.58114 + 1.58114i
0 −2.16228 0 0.418861 − 0.418861i 0 1.41886 + 1.41886i 0 −4.32456 0
161.2 0 4.16228 0 3.58114 − 3.58114i 0 4.58114 + 4.58114i 0 8.32456 0
177.1 0 −2.16228 0 0.418861 + 0.418861i 0 1.41886 − 1.41886i 0 −4.32456 0
177.2 0 4.16228 0 3.58114 + 3.58114i 0 4.58114 − 4.58114i 0 8.32456 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
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.3.t.c 4
4.b odd 2 1 13.3.d.a ✓ 4
12.b even 2 1 117.3.j.a 4
13.d odd 4 1 inner 208.3.t.c 4
20.d odd 2 1 325.3.j.a 4
20.e even 4 1 325.3.g.a 4
20.e even 4 1 325.3.g.b 4
52.b odd 2 1 169.3.d.d 4
52.f even 4 1 13.3.d.a ✓ 4
52.f even 4 1 169.3.d.d 4
52.i odd 6 2 169.3.f.d 8
52.j odd 6 2 169.3.f.f 8
52.l even 12 2 169.3.f.d 8
52.l even 12 2 169.3.f.f 8
156.l odd 4 1 117.3.j.a 4
260.l odd 4 1 325.3.g.b 4
260.s odd 4 1 325.3.g.a 4
260.u even 4 1 325.3.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.d.a ✓ 4 4.b odd 2 1
13.3.d.a ✓ 4 52.f even 4 1
117.3.j.a 4 12.b even 2 1
117.3.j.a 4 156.l odd 4 1
169.3.d.d 4 52.b odd 2 1
169.3.d.d 4 52.f even 4 1
169.3.f.d 8 52.i odd 6 2
169.3.f.d 8 52.l even 12 2
169.3.f.f 8 52.j odd 6 2
169.3.f.f 8 52.l even 12 2
208.3.t.c 4 1.a even 1 1 trivial
208.3.t.c 4 13.d odd 4 1 inner
325.3.g.a 4 20.e even 4 1
325.3.g.a 4 260.s odd 4 1
325.3.g.b 4 20.e even 4 1
325.3.g.b 4 260.l odd 4 1
325.3.j.a 4 20.d odd 2 1
325.3.j.a 4 260.u even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 9 \) acting on \(S_{3}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{4} - 12 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} + 738 T^{2} + 123201 \) Copy content Toggle raw display
$19$ \( T^{4} + 400 \) Copy content Toggle raw display
$23$ \( T^{4} + 828 T^{2} + 54756 \) Copy content Toggle raw display
$29$ \( (T^{2} - 20 T - 150)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 40 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{4} - 40 T^{3} + \cdots + 42025 \) Copy content Toggle raw display
$41$ \( T^{4} - 32 T^{3} + \cdots + 11664 \) Copy content Toggle raw display
$43$ \( T^{4} + 1062 T^{2} + 123201 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} + \cdots + 6985449 \) Copy content Toggle raw display
$53$ \( (T^{2} + 40 T + 150)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 56 T^{3} + \cdots + 12446784 \) Copy content Toggle raw display
$61$ \( (T^{2} + 148 T + 5436)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 84 T^{3} + \cdots + 40170244 \) Copy content Toggle raw display
$71$ \( T^{4} + 284 T^{3} + \cdots + 85322169 \) Copy content Toggle raw display
$73$ \( T^{4} + 4000000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 32 T - 954)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 52 T^{3} + \cdots + 2762244 \) Copy content Toggle raw display
$89$ \( T^{4} - 200 T^{3} + \cdots + 2624400 \) Copy content Toggle raw display
$97$ \( T^{4} + 68 T^{3} + \cdots + 449524804 \) Copy content Toggle raw display
show more
show less