Properties

Label 672.2.i.b
Level $672$
Weight $2$
Character orbit 672.i
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(209,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.i (of order \(2\), degree \(1\), not 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + 2 \beta_{2} q^{5} + (\beta_{3} - 1) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + 2 \beta_{2} q^{5} + (\beta_{3} - 1) q^{7} - 3 q^{9} - 4 \beta_1 q^{11} - 6 q^{15} + ( - \beta_{2} + 3 \beta_1) q^{21} - 7 q^{25} - 3 \beta_{2} q^{27} + 2 \beta_1 q^{29} - 2 \beta_{3} q^{31} + 4 \beta_{3} q^{33} + ( - 2 \beta_{2} + 6 \beta_1) q^{35} - 6 \beta_{2} q^{45} + ( - 2 \beta_{3} - 5) q^{49} - 10 \beta_1 q^{53} + 8 \beta_{3} q^{55} + 6 \beta_{2} q^{59} + ( - 3 \beta_{3} + 3) q^{63} - 4 \beta_{3} q^{73} - 7 \beta_{2} q^{75} + (8 \beta_{2} + 4 \beta_1) q^{77} - 10 q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} - 2 \beta_{3} q^{87} - 6 \beta_1 q^{93} + 8 \beta_{3} q^{97} + 12 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 12 q^{9} - 24 q^{15} - 28 q^{25} - 20 q^{49} + 12 q^{63} - 40 q^{79} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.707107 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 1.73205i 0 3.46410i 0 −1.00000 2.44949i 0 −3.00000 0
209.2 0 1.73205i 0 3.46410i 0 −1.00000 + 2.44949i 0 −3.00000 0
209.3 0 1.73205i 0 3.46410i 0 −1.00000 2.44949i 0 −3.00000 0
209.4 0 1.73205i 0 3.46410i 0 −1.00000 + 2.44949i 0 −3.00000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.i.b 4
3.b odd 2 1 inner 672.2.i.b 4
4.b odd 2 1 168.2.i.b 4
7.b odd 2 1 inner 672.2.i.b 4
8.b even 2 1 inner 672.2.i.b 4
8.d odd 2 1 168.2.i.b 4
12.b even 2 1 168.2.i.b 4
21.c even 2 1 inner 672.2.i.b 4
24.f even 2 1 168.2.i.b 4
24.h odd 2 1 CM 672.2.i.b 4
28.d even 2 1 168.2.i.b 4
56.e even 2 1 168.2.i.b 4
56.h odd 2 1 inner 672.2.i.b 4
84.h odd 2 1 168.2.i.b 4
168.e odd 2 1 168.2.i.b 4
168.i even 2 1 inner 672.2.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.b 4 4.b odd 2 1
168.2.i.b 4 8.d odd 2 1
168.2.i.b 4 12.b even 2 1
168.2.i.b 4 24.f even 2 1
168.2.i.b 4 28.d even 2 1
168.2.i.b 4 56.e even 2 1
168.2.i.b 4 84.h odd 2 1
168.2.i.b 4 168.e odd 2 1
672.2.i.b 4 1.a even 1 1 trivial
672.2.i.b 4 3.b odd 2 1 inner
672.2.i.b 4 7.b odd 2 1 inner
672.2.i.b 4 8.b even 2 1 inner
672.2.i.b 4 21.c even 2 1 inner
672.2.i.b 4 24.h odd 2 1 CM
672.2.i.b 4 56.h odd 2 1 inner
672.2.i.b 4 168.i even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 32 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 384)^{2} \) Copy content Toggle raw display
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