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Properties

Label 672.2.bc.c
Level $672$
Weight $2$
Character orbit 672.bc
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(257,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bc (of order \(6\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 2 \beta_{2}) q^{3} + \beta_{7} q^{5} + \beta_1 q^{7} + ( - 3 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 2 \beta_{2}) q^{3} + \beta_{7} q^{5} + \beta_1 q^{7} + ( - 3 \beta_{3} - 3) q^{9} - \beta_{4} q^{11} + (4 \beta_{3} + 2) q^{13} + ( - \beta_{6} + \beta_1) q^{15} + 2 \beta_{5} q^{17} - 2 \beta_{6} q^{19} + (\beta_{7} - 2 \beta_{5}) q^{21} - 2 \beta_{2} q^{23} + 2 \beta_{3} q^{25} + (3 \beta_{4} - 3 \beta_{2}) q^{27} + ( - \beta_{7} - \beta_{5}) q^{29} + 3 \beta_1 q^{31} + ( - \beta_{3} + 1) q^{33} + ( - 7 \beta_{4} - 7 \beta_{2}) q^{35} + ( - 6 \beta_{3} - 6) q^{37} - 6 \beta_{4} q^{39} + ( - 4 \beta_{7} + 4 \beta_{5}) q^{41} - 3 \beta_{5} q^{45} + (8 \beta_{4} + 4 \beta_{2}) q^{47} + (7 \beta_{3} + 7) q^{49} + (2 \beta_{6} + 4 \beta_1) q^{51} + (2 \beta_{7} - \beta_{5}) q^{53} + (\beta_{6} + \beta_1) q^{55} + ( - 2 \beta_{7} - 2 \beta_{5}) q^{57} + (7 \beta_{4} + 14 \beta_{2}) q^{59} + ( - 8 \beta_{3} + 8) q^{61} + ( - 3 \beta_{6} - 3 \beta_1) q^{63} + ( - 2 \beta_{7} + 4 \beta_{5}) q^{65} + ( - 4 \beta_{6} - 2 \beta_1) q^{67} + (4 \beta_{3} + 2) q^{69} + (4 \beta_{4} + 4 \beta_{2}) q^{71} + ( - 4 \beta_{3} - 8) q^{73} + ( - 4 \beta_{4} - 2 \beta_{2}) q^{75} + \beta_{7} q^{77} + ( - \beta_{6} - 2 \beta_1) q^{79} + 9 \beta_{3} q^{81} + (\beta_{4} - \beta_{2}) q^{83} - 14 q^{85} - 3 \beta_1 q^{87} + 2 \beta_{7} q^{89} + (4 \beta_{6} + 2 \beta_1) q^{91} + (3 \beta_{7} - 6 \beta_{5}) q^{93} - 14 \beta_{4} q^{95} + ( - 10 \beta_{3} - 5) q^{97} + (3 \beta_{4} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{9} - 8 q^{25} + 12 q^{33} - 24 q^{37} + 28 q^{49} + 96 q^{61} - 48 q^{73} - 36 q^{81} - 112 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 7\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 13\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 3\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} + 5\beta_{4} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 11\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} - 5\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{5} - 13\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.228425 + 1.39564i
−1.09445 − 0.895644i
1.09445 + 0.895644i
−0.228425 − 1.39564i
0.228425 − 1.39564i
−1.09445 + 0.895644i
1.09445 − 0.895644i
−0.228425 + 1.39564i
0 −0.866025 − 1.50000i 0 −1.32288 + 2.29129i 0 2.29129 − 1.32288i 0 −1.50000 + 2.59808i 0
257.2 0 −0.866025 − 1.50000i 0 1.32288 − 2.29129i 0 −2.29129 + 1.32288i 0 −1.50000 + 2.59808i 0
257.3 0 0.866025 + 1.50000i 0 −1.32288 + 2.29129i 0 −2.29129 + 1.32288i 0 −1.50000 + 2.59808i 0
257.4 0 0.866025 + 1.50000i 0 1.32288 − 2.29129i 0 2.29129 − 1.32288i 0 −1.50000 + 2.59808i 0
353.1 0 −0.866025 + 1.50000i 0 −1.32288 − 2.29129i 0 2.29129 + 1.32288i 0 −1.50000 − 2.59808i 0
353.2 0 −0.866025 + 1.50000i 0 1.32288 + 2.29129i 0 −2.29129 − 1.32288i 0 −1.50000 − 2.59808i 0
353.3 0 0.866025 − 1.50000i 0 −1.32288 − 2.29129i 0 −2.29129 − 1.32288i 0 −1.50000 − 2.59808i 0
353.4 0 0.866025 − 1.50000i 0 1.32288 + 2.29129i 0 2.29129 + 1.32288i 0 −1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.g even 6 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bc.c ✓ 8
3.b odd 2 1 inner 672.2.bc.c ✓ 8
4.b odd 2 1 inner 672.2.bc.c ✓ 8
7.d odd 6 1 inner 672.2.bc.c ✓ 8
12.b even 2 1 inner 672.2.bc.c ✓ 8
21.g even 6 1 inner 672.2.bc.c ✓ 8
28.f even 6 1 inner 672.2.bc.c ✓ 8
84.j odd 6 1 inner 672.2.bc.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bc.c ✓ 8 1.a even 1 1 trivial
672.2.bc.c ✓ 8 3.b odd 2 1 inner
672.2.bc.c ✓ 8 4.b odd 2 1 inner
672.2.bc.c ✓ 8 7.d odd 6 1 inner
672.2.bc.c ✓ 8 12.b even 2 1 inner
672.2.bc.c ✓ 8 21.g even 6 1 inner
672.2.bc.c ✓ 8 28.f even 6 1 inner
672.2.bc.c ✓ 8 84.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 7T_{5}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 63 T^{2} + 3969)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 112)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 48 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 147 T^{2} + 21609)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 24 T + 192)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 84 T^{2} + 7056)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 48)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 75)^{4} \) Copy content Toggle raw display
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