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Properties

Label 672.2.i.d
Level $672$
Weight $2$
Character orbit 672.i
Analytic conductor $5.366$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Coefficient field: 8.0.3317760000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{3} - \beta_{3} q^{5} + ( - \beta_{6} - 1) q^{7} + (\beta_{5} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{3} - \beta_{3} q^{5} + ( - \beta_{6} - 1) q^{7} + (\beta_{5} + 2) q^{9} + \beta_{4} q^{11} + ( - \beta_{3} - 2 \beta_{2}) q^{13} + (\beta_{5} - 1) q^{15} + (\beta_{3} + 2 \beta_{2}) q^{19} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{21} + 2 \beta_{5} q^{23} + 3 q^{25} + ( - 4 \beta_{3} - \beta_{2}) q^{27} + 2 \beta_{4} q^{29} + 2 \beta_{6} q^{31} + ( - \beta_{7} + \beta_{6}) q^{33} + (\beta_{4} + \beta_{3}) q^{35} + (\beta_{5} + 5) q^{39} + 2 \beta_{7} q^{41} + (\beta_{4} + 2 \beta_1) q^{43} + ( - \beta_{3} + 2 \beta_{2}) q^{45} + 2 \beta_{7} q^{47} + (2 \beta_{6} - 5) q^{49} + 2 \beta_{6} q^{55} + ( - \beta_{5} - 5) q^{57} + 7 \beta_{3} q^{59} + ( - \beta_{3} - 2 \beta_{2}) q^{61} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - 2) q^{63} + 2 \beta_{5} q^{65} + ( - \beta_{4} - 2 \beta_1) q^{67} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{69} - 4 \beta_{5} q^{71} - 6 \beta_{6} q^{73} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{75} + ( - \beta_{4} + 6 \beta_{3}) q^{77} + 10 q^{79} + (4 \beta_{5} - 1) q^{81} - 5 \beta_{3} q^{83} + ( - 2 \beta_{7} + 2 \beta_{6}) q^{87} + (\beta_{4} + \beta_{3} + \cdots + 2 \beta_1) q^{91}+ \cdots + (\beta_{4} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 16 q^{9} - 8 q^{15} + 24 q^{25} + 40 q^{39} - 40 q^{49} - 40 q^{57} - 16 q^{63} + 80 q^{79} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 10\nu^{4} + 71\nu^{2} - 6 ) / 24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 97\nu^{3} + 138\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 25\nu^{3} - 78\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 6\nu^{4} - 7\nu^{2} + 30 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 9\nu^{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 5\nu^{3} + 18\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 26\nu^{5} - 59\nu^{3} + 54\nu ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{3} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - 3\beta_{4} + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} - 2\beta_{3} - 14\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{5} - 15\beta_{4} - 6\beta _1 + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15\beta_{7} + 21\beta_{6} + 16\beta_{3} - 50\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -62\beta_{5} - 63\beta_{4} - 36\beta _1 + 164 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 57\beta_{7} + 141\beta_{6} + 142\beta_{3} - 194\beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
−2.15988 + 0.258819i
0.578737 − 0.965926i
0.578737 + 0.965926i
−2.15988 − 0.258819i
2.15988 + 0.258819i
−0.578737 − 0.965926i
−0.578737 + 0.965926i
2.15988 − 0.258819i
0 −1.58114 − 0.707107i 0 − 1.41421i 0 −1.00000 − 2.44949i 0 2.00000 + 2.23607i 0
209.2 0 −1.58114 − 0.707107i 0 − 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 + 2.23607i 0
209.3 0 −1.58114 + 0.707107i 0 1.41421i 0 −1.00000 − 2.44949i 0 2.00000 − 2.23607i 0
209.4 0 −1.58114 + 0.707107i 0 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 − 2.23607i 0
209.5 0 1.58114 − 0.707107i 0 − 1.41421i 0 −1.00000 − 2.44949i 0 2.00000 − 2.23607i 0
209.6 0 1.58114 − 0.707107i 0 − 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 − 2.23607i 0
209.7 0 1.58114 + 0.707107i 0 1.41421i 0 −1.00000 − 2.44949i 0 2.00000 + 2.23607i 0
209.8 0 1.58114 + 0.707107i 0 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 + 2.23607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 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.d 8
3.b odd 2 1 inner 672.2.i.d 8
4.b odd 2 1 168.2.i.e ✓ 8
7.b odd 2 1 inner 672.2.i.d 8
8.b even 2 1 inner 672.2.i.d 8
8.d odd 2 1 168.2.i.e ✓ 8
12.b even 2 1 168.2.i.e ✓ 8
21.c even 2 1 inner 672.2.i.d 8
24.f even 2 1 168.2.i.e ✓ 8
24.h odd 2 1 inner 672.2.i.d 8
28.d even 2 1 168.2.i.e ✓ 8
56.e even 2 1 168.2.i.e ✓ 8
56.h odd 2 1 inner 672.2.i.d 8
84.h odd 2 1 168.2.i.e ✓ 8
168.e odd 2 1 168.2.i.e ✓ 8
168.i even 2 1 inner 672.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.e ✓ 8 4.b odd 2 1
168.2.i.e ✓ 8 8.d odd 2 1
168.2.i.e ✓ 8 12.b even 2 1
168.2.i.e ✓ 8 24.f even 2 1
168.2.i.e ✓ 8 28.d even 2 1
168.2.i.e ✓ 8 56.e even 2 1
168.2.i.e ✓ 8 84.h odd 2 1
168.2.i.e ✓ 8 168.e odd 2 1
672.2.i.d 8 1.a even 1 1 trivial
672.2.i.d 8 3.b odd 2 1 inner
672.2.i.d 8 7.b odd 2 1 inner
672.2.i.d 8 8.b even 2 1 inner
672.2.i.d 8 21.c even 2 1 inner
672.2.i.d 8 24.h odd 2 1 inner
672.2.i.d 8 56.h odd 2 1 inner
672.2.i.d 8 168.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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 120)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 120)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 216)^{4} \) Copy content Toggle raw display
$79$ \( (T - 10)^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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