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Properties

Label 209.4.d.a
Level $209$
Weight $4$
Character orbit 209.d
Analytic conductor $12.331$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [209,4,Mod(208,209)] 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("209.208"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.3313991912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{4} - 14 q^{5} - 2 \beta q^{7} + 27 q^{9} + ( - 7 \beta - 20) q^{11} + 64 q^{16} + 32 \beta q^{17} + 19 \beta q^{19} + 112 q^{20} + 212 q^{23} + 71 q^{25} + 16 \beta q^{28} + 28 \beta q^{35} - 216 q^{36} + \cdots + ( - 189 \beta - 540) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 28 q^{5} + 54 q^{9} - 40 q^{11} + 128 q^{16} + 224 q^{20} + 424 q^{23} + 142 q^{25} - 432 q^{36} + 320 q^{44} - 756 q^{45} + 728 q^{47} + 534 q^{49} + 560 q^{55} - 1024 q^{64} - 532 q^{77}+ \cdots - 1080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
0.500000 + 2.17945i
0.500000 − 2.17945i
0 0 −8.00000 −14.0000 0 − 8.71780i 0 27.0000 0
208.2 0 0 −8.00000 −14.0000 0 8.71780i 0 27.0000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
11.b odd 2 1 inner
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.4.d.a ✓ 2
11.b odd 2 1 inner 209.4.d.a ✓ 2
19.b odd 2 1 CM 209.4.d.a ✓ 2
209.d even 2 1 inner 209.4.d.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.d.a ✓ 2 1.a even 1 1 trivial
209.4.d.a ✓ 2 11.b odd 2 1 inner
209.4.d.a ✓ 2 19.b odd 2 1 CM
209.4.d.a ✓ 2 209.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 14)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 76 \) Copy content Toggle raw display
$11$ \( T^{2} + 40T + 1331 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19456 \) Copy content Toggle raw display
$19$ \( T^{2} + 6859 \) Copy content Toggle raw display
$23$ \( (T - 212)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 301644 \) Copy content Toggle raw display
$47$ \( (T - 364)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 511024 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 393984 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2274604 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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