Properties

Label 648.3.m.f
Level $648$
Weight $3$
Character orbit 648.m
Analytic conductor $17.657$
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] = [648,3,Mod(377,648)]
 
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([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.377");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.m (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: \(17.6567211305\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} - 3 \beta_{2} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} - 3 \beta_{2} + 3) q^{7} + ( - 2 \beta_{5} + 3 \beta_1) q^{11} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{13} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3}) q^{17} + (2 \beta_{7} + 8) q^{19} + ( - 10 \beta_{4} + 4 \beta_{3} - 4 \beta_1) q^{23} + ( - 2 \beta_{7} + 2 \beta_{6} + 8 \beta_{2} - 8) q^{25} - 14 \beta_1 q^{29} + ( - \beta_{6} + \beta_{2}) q^{31} + (12 \beta_{5} - 12 \beta_{4} - 11 \beta_{3}) q^{35} + (2 \beta_{7} - 36) q^{37} + ( - 18 \beta_{4} - 4 \beta_{3} + 4 \beta_1) q^{41} + (2 \beta_{7} - 2 \beta_{6} - 14 \beta_{2} + 14) q^{43} + ( - 12 \beta_{5} + 14 \beta_1) q^{47} + (6 \beta_{6} - 32 \beta_{2}) q^{49} + (17 \beta_{5} - 17 \beta_{4} + 17 \beta_{3}) q^{53} + ( - 5 \beta_{7} + 43) q^{55} + ( - 28 \beta_{4} + 6 \beta_{3} - 6 \beta_1) q^{59} + (8 \beta_{7} - 8 \beta_{6} + 52 \beta_{2} - 52) q^{61} + ( - 20 \beta_{5} + 18 \beta_1) q^{65} + (2 \beta_{6} + 58 \beta_{2}) q^{67} + (10 \beta_{5} - 10 \beta_{4} + 6 \beta_{3}) q^{71} + ( - 6 \beta_{7} - 71) q^{73} + ( - 33 \beta_{4} - 25 \beta_{3} + 25 \beta_1) q^{77} + (4 \beta_{7} - 4 \beta_{6} - 74 \beta_{2} + 74) q^{79} + (14 \beta_{5} - 23 \beta_1) q^{83} + (2 \beta_{6} - 14 \beta_{2}) q^{85} + (6 \beta_{5} - 6 \beta_{4} - 30 \beta_{3}) q^{89} + ( - 8 \beta_{7} + 150) q^{91} + (10 \beta_{4} + 8 \beta_{3} - 8 \beta_1) q^{95} + ( - 8 \beta_{7} + 8 \beta_{6} + 11 \beta_{2} - 11) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{7} + 8 q^{13} + 64 q^{19} - 32 q^{25} + 4 q^{31} - 288 q^{37} + 56 q^{43} - 128 q^{49} + 344 q^{55} - 208 q^{61} + 232 q^{67} - 568 q^{73} + 296 q^{79} - 56 q^{85} + 1200 q^{91} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 6\zeta_{24}^{7} + 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} + 3\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + 3\beta_{5} - 3\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 3\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{6} - 3\beta_{4} ) / 12 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0 0 0 −5.04757 2.91421i 0 5.74264 + 9.94655i 0 0 0
377.2 0 0 0 −0.148586 0.0857864i 0 −2.74264 4.75039i 0 0 0
377.3 0 0 0 0.148586 + 0.0857864i 0 −2.74264 4.75039i 0 0 0
377.4 0 0 0 5.04757 + 2.91421i 0 5.74264 + 9.94655i 0 0 0
593.1 0 0 0 −5.04757 + 2.91421i 0 5.74264 9.94655i 0 0 0
593.2 0 0 0 −0.148586 + 0.0857864i 0 −2.74264 + 4.75039i 0 0 0
593.3 0 0 0 0.148586 0.0857864i 0 −2.74264 + 4.75039i 0 0 0
593.4 0 0 0 5.04757 2.91421i 0 5.74264 9.94655i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.3.m.f 8
3.b odd 2 1 inner 648.3.m.f 8
4.b odd 2 1 1296.3.q.l 8
9.c even 3 1 216.3.e.c 4
9.c even 3 1 inner 648.3.m.f 8
9.d odd 6 1 216.3.e.c 4
9.d odd 6 1 inner 648.3.m.f 8
12.b even 2 1 1296.3.q.l 8
36.f odd 6 1 432.3.e.h 4
36.f odd 6 1 1296.3.q.l 8
36.h even 6 1 432.3.e.h 4
36.h even 6 1 1296.3.q.l 8
72.j odd 6 1 1728.3.e.p 4
72.l even 6 1 1728.3.e.s 4
72.n even 6 1 1728.3.e.p 4
72.p odd 6 1 1728.3.e.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.c 4 9.c even 3 1
216.3.e.c 4 9.d odd 6 1
432.3.e.h 4 36.f odd 6 1
432.3.e.h 4 36.h even 6 1
648.3.m.f 8 1.a even 1 1 trivial
648.3.m.f 8 3.b odd 2 1 inner
648.3.m.f 8 9.c even 3 1 inner
648.3.m.f 8 9.d odd 6 1 inner
1296.3.q.l 8 4.b odd 2 1
1296.3.q.l 8 12.b even 2 1
1296.3.q.l 8 36.f odd 6 1
1296.3.q.l 8 36.h even 6 1
1728.3.e.p 4 72.j odd 6 1
1728.3.e.p 4 72.n even 6 1
1728.3.e.s 4 72.l even 6 1
1728.3.e.s 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{8} - 34T_{5}^{6} + 1155T_{5}^{4} - 34T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 99T_{7}^{2} + 378T_{7} + 3969 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 34 T^{6} + 1155 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} + 99 T^{2} + 378 T + 3969)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 226 T^{6} + 48675 T^{4} + \cdots + 5764801 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + 300 T^{2} + 1136 T + 80656)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 328 T^{2} + 8464)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T - 224)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 1888 T^{6} + \cdots + 185189072896 \) Copy content Toggle raw display
$29$ \( (T^{4} - 1764 T^{2} + 3111696)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + 75 T^{2} + 142 T + 5041)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72 T + 1008)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 5472 T^{6} + \cdots + 35912501231616 \) Copy content Toggle raw display
$43$ \( (T^{4} - 28 T^{3} + 876 T^{2} + 2576 T + 8464)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 5832 T^{6} + \cdots + 140283207936 \) Copy content Toggle raw display
$53$ \( (T^{4} + 9826 T^{2} + 83521)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 13192 T^{6} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 104 T^{3} + 12720 T^{2} + \cdots + 3625216)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 116 T^{3} + 10380 T^{2} + \cdots + 9461776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2248 T^{2} + 226576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 142 T + 2449)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 148 T^{3} + 17580 T^{2} + \cdots + 18696976)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 103943102172001 \) Copy content Toggle raw display
$89$ \( (T^{4} + 16776 T^{2} + 61027344)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 22 T^{3} + 4971 T^{2} + \cdots + 20133169)^{2} \) Copy content Toggle raw display
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