↑

Properties

Label 672.2.q.l
Level $672$
Weight $2$
Character orbit 672.q
Analytic conductor $5.366$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(193,672)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.q (of order \(3\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{4} - \beta_1) q^{5} + (\beta_{5} - \beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{4} - \beta_1) q^{5} + (\beta_{5} - \beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 1) q^{9} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{5} + \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 3 q^{7} - 3 q^{9} - 6 q^{13} - 6 q^{17} + 3 q^{19} - 6 q^{23} - 3 q^{25} - 6 q^{27} + 24 q^{29} + 3 q^{31} + 12 q^{35} - 3 q^{37} - 3 q^{39} - 12 q^{41} - 30 q^{43} + 12 q^{47} + 9 q^{49} + 6 q^{51} - 6 q^{53} + 24 q^{55} + 6 q^{57} + 12 q^{59} + 18 q^{61} - 3 q^{63} - 24 q^{65} + 9 q^{67} - 12 q^{69} - 33 q^{73} + 3 q^{75} - 12 q^{77} - 27 q^{79} - 3 q^{81} - 36 q^{83} + 72 q^{85} + 12 q^{87} + 12 q^{89} + 51 q^{91} - 3 q^{93} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{5} + 7\nu^{4} - 31\nu^{3} + 37\nu^{2} - 56\nu + 22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{5} + 8\nu^{4} - 33\nu^{3} + 44\nu^{2} - 62\nu + 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{5} + 8\beta_{4} - 3\beta_{3} + 6\beta_{2} - \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} + 18\beta_{4} - 9\beta_{3} + 12\beta_{2} - 17\beta _1 + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{5} - 28\beta_{4} + 3\beta_{3} - 34\beta_{2} - 11\beta _1 + 49 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.0585812i
0.500000 + 1.51496i
0.500000 − 2.43956i
0.500000 − 0.0585812i
0.500000 − 1.51496i
0.500000 + 2.43956i
0 0.500000 − 0.866025i 0 −1.37328 − 2.37860i 0 2.64510 + 0.0585812i 0 −0.500000 − 0.866025i 0
193.2 0 0.500000 − 0.866025i 0 −0.227452 − 0.393958i 0 −2.16908 + 1.51496i 0 −0.500000 − 0.866025i 0
193.3 0 0.500000 − 0.866025i 0 1.60074 + 2.77256i 0 1.02398 − 2.43956i 0 −0.500000 − 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −1.37328 + 2.37860i 0 2.64510 − 0.0585812i 0 −0.500000 + 0.866025i 0
289.2 0 0.500000 + 0.866025i 0 −0.227452 + 0.393958i 0 −2.16908 − 1.51496i 0 −0.500000 + 0.866025i 0
289.3 0 0.500000 + 0.866025i 0 1.60074 − 2.77256i 0 1.02398 + 2.43956i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.q.l yes 6
3.b odd 2 1 2016.2.s.v 6
4.b odd 2 1 672.2.q.k ✓ 6
7.c even 3 1 inner 672.2.q.l yes 6
7.c even 3 1 4704.2.a.bs 3
7.d odd 6 1 4704.2.a.bv 3
8.b even 2 1 1344.2.q.y 6
8.d odd 2 1 1344.2.q.z 6
12.b even 2 1 2016.2.s.u 6
21.h odd 6 1 2016.2.s.v 6
28.f even 6 1 4704.2.a.bt 3
28.g odd 6 1 672.2.q.k ✓ 6
28.g odd 6 1 4704.2.a.bu 3
56.j odd 6 1 9408.2.a.eg 3
56.k odd 6 1 1344.2.q.z 6
56.k odd 6 1 9408.2.a.eh 3
56.m even 6 1 9408.2.a.ei 3
56.p even 6 1 1344.2.q.y 6
56.p even 6 1 9408.2.a.ej 3
84.n even 6 1 2016.2.s.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.k ✓ 6 4.b odd 2 1
672.2.q.k ✓ 6 28.g odd 6 1
672.2.q.l yes 6 1.a even 1 1 trivial
672.2.q.l yes 6 7.c even 3 1 inner
1344.2.q.y 6 8.b even 2 1
1344.2.q.y 6 56.p even 6 1
1344.2.q.z 6 8.d odd 2 1
1344.2.q.z 6 56.k odd 6 1
2016.2.s.u 6 12.b even 2 1
2016.2.s.u 6 84.n even 6 1
2016.2.s.v 6 3.b odd 2 1
2016.2.s.v 6 21.h odd 6 1
4704.2.a.bs 3 7.c even 3 1
4704.2.a.bt 3 28.f even 6 1
4704.2.a.bu 3 28.g odd 6 1
4704.2.a.bv 3 7.d odd 6 1
9408.2.a.eg 3 56.j odd 6 1
9408.2.a.eh 3 56.k odd 6 1
9408.2.a.ei 3 56.m even 6 1
9408.2.a.ej 3 56.p even 6 1

Hecke kernels

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

\( T_{5}^{6} + 9T_{5}^{4} + 8T_{5}^{3} + 81T_{5}^{2} + 36T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 27T_{11}^{4} - 76T_{11}^{3} + 729T_{11}^{2} - 1026T_{11} + 1444 \) Copy content Toggle raw display
\( T_{13}^{3} + 3T_{13}^{2} - 36T_{13} - 112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 9 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 27 T^{4} + \cdots + 1444 \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} + \cdots - 112)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots + 12544 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( (T^{3} - 12 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{5} + \cdots + 135424 \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} + \cdots - 512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 15 T^{2} + \cdots + 28)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 12 T^{5} + \cdots + 12544 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + \cdots + 27556 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 6724 \) Copy content Toggle raw display
$61$ \( T^{6} - 18 T^{5} + \cdots + 304704 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} + \cdots + 26896 \) Copy content Toggle raw display
$71$ \( (T^{3} - 36 T + 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 33 T^{5} + \cdots + 992016 \) Copy content Toggle raw display
$79$ \( T^{6} + 27 T^{5} + \cdots + 124609 \) Copy content Toggle raw display
$83$ \( (T^{3} + 18 T^{2} + \cdots - 948)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 12 T^{5} + \cdots + 12544 \) Copy content Toggle raw display
$97$ \( (T^{3} - 27 T + 38)^{2} \) Copy content Toggle raw display
show more
show less