Properties

Label 207.5.d.a
Level $207$
Weight $5$
Character orbit 207.d
Self dual yes
Analytic conductor $21.398$
Analytic rank $0$
Dimension $3$
CM discriminant -23
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,5,Mod(91,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.91");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 207.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.3975823584\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 23)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{2} - 3 \beta_1 + 16) q^{4} + ( - 16 \beta_{2} + 16 \beta_1 + 79) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{2} - 3 \beta_1 + 16) q^{4} + ( - 16 \beta_{2} + 16 \beta_1 + 79) q^{8} + ( - 35 \beta_{2} - 52 \beta_1) q^{13} + ( - 79 \beta_{2} + 79 \beta_1 + 256) q^{16} - 529 q^{23} + 625 q^{25} + ( - 140 \beta_{2} + 243 \beta_1 + 511) q^{26} + (51 \beta_{2} + 404 \beta_1) q^{29} + (189 \beta_{2} + 316 \beta_1) q^{31} + ( - 316 \beta_{2} - 237 \beta_1 + 1264) q^{32} + (259 \beta_{2} - 732 \beta_1) q^{41} + (529 \beta_{2} - 529 \beta_1) q^{46} + (595 \beta_{2} - 348 \beta_1) q^{47} + 2401 q^{49} + ( - 625 \beta_{2} + 625 \beta_1) q^{50} + ( - 511 \beta_{2} + 511 \beta_1 + 5201) q^{52} + (204 \beta_{2} - 1667 \beta_1 + 1553) q^{58} + 6286 q^{59} + (756 \beta_{2} - 1453 \beta_1 - 2513) q^{62} + ( - 1264 \beta_{2} + 1264 \beta_1 + 2145) q^{64} + (1395 \beta_{2} + 68 \beta_1) q^{71} + (861 \beta_{2} - 2276 \beta_1) q^{73} + (1036 \beta_{2} + 2669 \beta_1 - 11599) q^{82} + (2116 \beta_{2} + 1587 \beta_1 - 8464) q^{92} + (2380 \beta_{2} + 797 \beta_1 - 17311) q^{94} + ( - 2401 \beta_{2} + 2401 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 48 q^{4} + 237 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 48 q^{4} + 237 q^{8} + 768 q^{16} - 1587 q^{23} + 1875 q^{25} + 1533 q^{26} + 3792 q^{32} + 7203 q^{49} + 15603 q^{52} + 4659 q^{58} + 18858 q^{59} - 7539 q^{62} + 6435 q^{64} - 34797 q^{82} - 25392 q^{92} - 51933 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 3\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.523976
2.66908
−2.14510
−5.87897 0 18.5623 0 0 0 −15.0635 0 0
91.2 −1.75927 0 −12.9050 0 0 0 50.8517 0 0
91.3 7.63824 0 42.3427 0 0 0 201.212 0 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.5.d.a 3
3.b odd 2 1 23.5.b.a 3
12.b even 2 1 368.5.f.a 3
23.b odd 2 1 CM 207.5.d.a 3
69.c even 2 1 23.5.b.a 3
276.h odd 2 1 368.5.f.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.5.b.a 3 3.b odd 2 1
23.5.b.a 3 69.c even 2 1
207.5.d.a 3 1.a even 1 1 trivial
207.5.d.a 3 23.b odd 2 1 CM
368.5.f.a 3 12.b even 2 1
368.5.f.a 3 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 48T_{2} - 79 \) acting on \(S_{5}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 48T - 79 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 85683 T + 8482894 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T + 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 2121843 T - 244330126 \) Copy content Toggle raw display
$31$ \( T^{3} - 2770563 T - 1677025154 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 8477283 T - 7596282526 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 14639043 T + 20606906306 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( (T - 6286)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 76235043 T + 188893891874 \) Copy content Toggle raw display
$73$ \( T^{3} - 85194723 T - 223017449186 \) Copy content Toggle raw display
$79$ \( T^{3} \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
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