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Properties

Label 667.2.f.a
Level $667$
Weight $2$
Character orbit 667.f
Analytic conductor $5.326$
Analytic rank $0$
Dimension $12$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [667,2,Mod(505,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.f (of order \(4\), degree \(2\), 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: no
Analytic conductor: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.322241908269256704.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{9} + \beta_1) q^{2} + (\beta_{8} - \beta_{3}) q^{3} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{9} + \beta_1) q^{2} + (\beta_{8} - \beta_{3}) q^{3} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{4}+ \cdots + ( - 7 \beta_{10} - 7 \beta_{9} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{8} - 30 q^{12} - 48 q^{16} - 42 q^{18} - 24 q^{24} - 60 q^{25} + 54 q^{26} + 24 q^{27} + 36 q^{32} + 60 q^{36} + 48 q^{39} + 18 q^{48} + 84 q^{49} - 36 q^{54} - 30 q^{58} - 30 q^{72} - 108 q^{81} + 96 q^{87} - 156 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{6} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + \nu^{2} + 12\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + \nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 25\nu^{5} ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 3\nu - 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 5\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} + 11\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} + 23\nu^{3} + 24\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} + 23\nu^{4} ) / 48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 7\nu^{5} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + 2\nu^{8} - \nu^{4} + 2\nu^{2} ) / 24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 8\nu^{8} - 21\nu^{5} - 8\nu^{2} ) / 96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{9} + 2\beta_{6} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - \beta_{9} + 2\beta_{7} + 2\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - 2\beta_{10} - \beta_{9} + 4\beta_{8} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} + \beta_{9} + 6\beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{11} - 3\beta_{9} + 6\beta_{4} - 3\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{11} + 5\beta_{9} + 12\beta_{5} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -11\beta_{11} - 11\beta_{9} - 2\beta_{6} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( \beta_{11} + \beta_{9} - 2\beta_{7} + 46\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -23\beta_{11} - 46\beta_{10} - 23\beta_{9} - 4\beta_{8} + 23\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 25\beta_{11} - 25\beta_{9} + 42\beta_{3} + 25\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/667\mathbb{Z}\right)^\times\).

\(n\) \(465\) \(553\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
0.921756 − 1.07255i
1.38973 − 0.261988i
−0.921756 − 1.07255i
0.467979 + 1.33454i
−1.38973 − 0.261988i
−0.467979 + 1.33454i
0.921756 + 1.07255i
1.38973 + 0.261988i
−0.921756 + 1.07255i
0.467979 − 1.33454i
−1.38973 + 0.261988i
−0.467979 − 1.33454i
−1.99431 − 1.99431i −0.317779 − 0.317779i 5.95452i 0 1.26750i 0 7.88653 − 7.88653i − 2.79803i 0
505.2 −1.65172 − 1.65172i 0.939190 + 0.939190i 3.45638i 0 − 3.10256i 0 2.40553 − 2.40553i − 1.23585i 0
505.3 −0.150796 − 0.150796i −2.42879 − 2.42879i − 1.95452i 0 0.732502i 0 −0.596325 + 0.596325i 8.79803i 0
505.4 0.866561 + 0.866561i −1.94450 − 1.94450i − 0.498145i 0 − 3.37006i 0 2.16479 − 2.16479i 4.56219i 0
505.5 1.12775 + 1.12775i 2.26228 + 2.26228i 0.543624i 0 5.10256i 0 1.64242 − 1.64242i 7.23585i 0
505.6 1.80252 + 1.80252i 1.48960 + 1.48960i 4.49815i 0 5.37006i 0 −4.50295 + 4.50295i 1.43781i 0
597.1 −1.99431 + 1.99431i −0.317779 + 0.317779i − 5.95452i 0 − 1.26750i 0 7.88653 + 7.88653i 2.79803i 0
597.2 −1.65172 + 1.65172i 0.939190 − 0.939190i − 3.45638i 0 3.10256i 0 2.40553 + 2.40553i 1.23585i 0
597.3 −0.150796 + 0.150796i −2.42879 + 2.42879i 1.95452i 0 − 0.732502i 0 −0.596325 − 0.596325i − 8.79803i 0
597.4 0.866561 − 0.866561i −1.94450 + 1.94450i 0.498145i 0 3.37006i 0 2.16479 + 2.16479i − 4.56219i 0
597.5 1.12775 − 1.12775i 2.26228 − 2.26228i − 0.543624i 0 − 5.10256i 0 1.64242 + 1.64242i − 7.23585i 0
597.6 1.80252 − 1.80252i 1.48960 − 1.48960i − 4.49815i 0 − 5.37006i 0 −4.50295 − 4.50295i − 1.43781i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
29.c odd 4 1 inner
667.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 667.2.f.a ✓ 12
23.b odd 2 1 CM 667.2.f.a ✓ 12
29.c odd 4 1 inner 667.2.f.a ✓ 12
667.f even 4 1 inner 667.2.f.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.f.a ✓ 12 1.a even 1 1 trivial
667.2.f.a ✓ 12 23.b odd 2 1 CM
667.2.f.a ✓ 12 29.c odd 4 1 inner
667.2.f.a ✓ 12 667.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 6 T_{2}^{9} + 72 T_{2}^{8} - 36 T_{2}^{7} + 18 T_{2}^{6} - 216 T_{2}^{5} + 1128 T_{2}^{4} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(667, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{12} - 8 T^{9} + \cdots + 1444 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + 78 T^{4} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{2} - 23)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 3452032516 \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1903925956 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 42165336964 \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{2} - 92)^{6} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( (T^{6} + 426 T^{4} + \cdots + 1382976)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 525685901764 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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