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Properties

Label 648.2.l.d
Level $648$
Weight $2$
Character orbit 648.l
Analytic conductor $5.174$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(107,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.l (of order \(6\), degree \(2\), 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 216)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{2} + ( - \beta_{7} - \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + (\beta_{6} + \beta_{5} + \cdots + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{2} + ( - \beta_{7} - \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - 2 \beta_{7} - 6 \beta_{6} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - q^{4} + 6 q^{10} - 15 q^{14} - q^{16} - 8 q^{19} - 21 q^{20} - 11 q^{22} - 4 q^{25} - 18 q^{28} - 33 q^{32} - 8 q^{34} + 36 q^{38} + 21 q^{40} - 36 q^{41} + 16 q^{43} - 24 q^{46} - 4 q^{49} + 36 q^{50} + 12 q^{52} + 27 q^{56} + 18 q^{58} + 26 q^{64} - 60 q^{65} + 16 q^{67} + 36 q^{68} + 3 q^{70} + 16 q^{73} - 36 q^{74} - 32 q^{76} + 40 q^{82} + 48 q^{83} - 12 q^{86} - 11 q^{88} + 24 q^{91} - 48 q^{92} - 36 q^{94} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{6} + \nu^{5} + 2\nu^{4} + 2\nu^{3} - 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{5} + 2\nu^{4} + 4\nu^{2} - 4\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 5\nu^{5} + 2\nu^{4} - 2\nu^{3} + 8\nu^{2} - 20\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 4\nu^{5} - 3\nu^{4} + 4\nu^{3} - 10\nu^{2} + 12\nu - 16 ) / 2 \) Copy content Toggle raw display

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.774115 − 1.18353i
1.41203 − 0.0786378i
−1.02187 − 0.977642i
0.335728 + 1.37379i
0.774115 + 1.18353i
1.41203 + 0.0786378i
−1.02187 + 0.977642i
0.335728 − 1.37379i
−1.41203 − 0.0786378i 0 1.98763 + 0.222077i −1.71352 − 2.96790i 0 −0.437696 − 0.252704i −2.78912 − 0.469882i 0 2.18614 + 4.32550i
107.2 −0.774115 − 1.18353i 0 −0.801492 + 1.83238i 1.71352 + 2.96790i 0 0.437696 + 0.252704i 2.78912 − 0.469882i 0 2.18614 − 4.32550i
107.3 −0.335728 + 1.37379i 0 −1.77457 − 0.922437i 0.252704 + 0.437696i 0 2.96790 + 1.71352i 1.86301 − 2.12819i 0 −0.686141 + 0.200214i
107.4 1.02187 − 0.977642i 0 0.0884324 − 1.99804i −0.252704 − 0.437696i 0 −2.96790 − 1.71352i −1.86301 − 2.12819i 0 −0.686141 − 0.200214i
539.1 −1.41203 + 0.0786378i 0 1.98763 − 0.222077i −1.71352 + 2.96790i 0 −0.437696 + 0.252704i −2.78912 + 0.469882i 0 2.18614 − 4.32550i
539.2 −0.774115 + 1.18353i 0 −0.801492 − 1.83238i 1.71352 − 2.96790i 0 0.437696 − 0.252704i 2.78912 + 0.469882i 0 2.18614 + 4.32550i
539.3 −0.335728 − 1.37379i 0 −1.77457 + 0.922437i 0.252704 − 0.437696i 0 2.96790 − 1.71352i 1.86301 + 2.12819i 0 −0.686141 − 0.200214i
539.4 1.02187 + 0.977642i 0 0.0884324 + 1.99804i −0.252704 + 0.437696i 0 −2.96790 + 1.71352i −1.86301 + 2.12819i 0 −0.686141 + 0.200214i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.l.d 8
3.b odd 2 1 648.2.l.e 8
4.b odd 2 1 2592.2.p.d 8
8.b even 2 1 2592.2.p.d 8
8.d odd 2 1 inner 648.2.l.d 8
9.c even 3 1 216.2.f.b ✓ 8
9.c even 3 1 648.2.l.e 8
9.d odd 6 1 216.2.f.b ✓ 8
9.d odd 6 1 inner 648.2.l.d 8
12.b even 2 1 2592.2.p.e 8
24.f even 2 1 648.2.l.e 8
24.h odd 2 1 2592.2.p.e 8
36.f odd 6 1 864.2.f.b 8
36.f odd 6 1 2592.2.p.e 8
36.h even 6 1 864.2.f.b 8
36.h even 6 1 2592.2.p.d 8
72.j odd 6 1 864.2.f.b 8
72.j odd 6 1 2592.2.p.d 8
72.l even 6 1 216.2.f.b ✓ 8
72.l even 6 1 inner 648.2.l.d 8
72.n even 6 1 864.2.f.b 8
72.n even 6 1 2592.2.p.e 8
72.p odd 6 1 216.2.f.b ✓ 8
72.p odd 6 1 648.2.l.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.f.b ✓ 8 9.c even 3 1
216.2.f.b ✓ 8 9.d odd 6 1
216.2.f.b ✓ 8 72.l even 6 1
216.2.f.b ✓ 8 72.p odd 6 1
648.2.l.d 8 1.a even 1 1 trivial
648.2.l.d 8 8.d odd 2 1 inner
648.2.l.d 8 9.d odd 6 1 inner
648.2.l.d 8 72.l even 6 1 inner
648.2.l.e 8 3.b odd 2 1
648.2.l.e 8 9.c even 3 1
648.2.l.e 8 24.f even 2 1
648.2.l.e 8 72.p odd 6 1
864.2.f.b 8 36.f odd 6 1
864.2.f.b 8 36.h even 6 1
864.2.f.b 8 72.j odd 6 1
864.2.f.b 8 72.n even 6 1
2592.2.p.d 8 4.b odd 2 1
2592.2.p.d 8 8.b even 2 1
2592.2.p.d 8 36.h even 6 1
2592.2.p.d 8 72.j odd 6 1
2592.2.p.e 8 12.b even 2 1
2592.2.p.e 8 24.h odd 2 1
2592.2.p.e 8 36.f odd 6 1
2592.2.p.e 8 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{8} + 12T_{5}^{6} + 141T_{5}^{4} + 36T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{8} - 12T_{7}^{6} + 141T_{7}^{4} - 36T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{41}^{4} + 18T_{41}^{3} + 124T_{41}^{2} + 288T_{41} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{6} + \cdots + 36864 \) Copy content Toggle raw display
$17$ \( (T^{4} + 28 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 32)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 60 T^{6} + \cdots + 589824 \) Copy content Toggle raw display
$29$ \( T^{8} + 36 T^{6} + \cdots + 36864 \) Copy content Toggle raw display
$31$ \( T^{8} - 60 T^{6} + \cdots + 751689 \) Copy content Toggle raw display
$37$ \( (T^{4} + 180 T^{2} + 6912)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 18 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 144 T^{6} + \cdots + 9437184 \) Copy content Toggle raw display
$53$ \( (T^{4} - 36 T^{2} + 27)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 44 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 144 T^{6} + \cdots + 9437184 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 108 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 29)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} - 60 T^{6} + \cdots + 589824 \) Copy content Toggle raw display
$83$ \( (T^{4} - 24 T^{3} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 304 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
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