Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(325,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.325");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.d (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.31435290000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{4} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{5} q^{4} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{6} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{5} q^{4} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{6} - \beta_1) q^{8} + (\beta_{7} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{10} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{11} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + (\beta_{5} - \beta_{4} - 2) q^{14} + ( - \beta_{7} + \beta_{3} + \beta_1 + 1) q^{16} + ( - \beta_{5} + \beta_{2} - 2) q^{17} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{20} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1) q^{22} + ( - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{23} + (\beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{25} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{2} + \beta_1 + 2) q^{26} + (\beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_1) q^{28} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{29} + (\beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{31} + (\beta_{7} + \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{32} + ( - \beta_{6} - 2 \beta_{4} - 3 \beta_1) q^{34} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - \beta_{7} - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{38} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{40} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{41} + \beta_{7} q^{43} + (3 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 1) q^{44} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{46} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 - 1) q^{47} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{49} + (\beta_{7} - 2 \beta_{5} - \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{50} + (3 \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 + 1) q^{52} + (2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1) q^{53} + (\beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{55} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} - 1) q^{56} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 3 \beta_1 + 2) q^{58} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{59} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{61} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{62} + ( - \beta_{7} - 2 \beta_{5} - \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 1) q^{64} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 1) q^{65} + ( - 3 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} - \beta_{3} - \beta_{2} + 5 \beta_1) q^{67} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3} + \beta_1 + 5) q^{68} + (2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{70} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 4) q^{71} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{73} + (3 \beta_{7} - \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 5) q^{74} + (\beta_{7} - 2 \beta_{6} - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{76} + (3 \beta_{7} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1) q^{77} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{79} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} - 6) q^{80} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 3) q^{82} + ( - \beta_{7} + \beta_{6} + 3 \beta_{4} - \beta_{3} + 3 \beta_1) q^{83} + (\beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1) q^{85} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{86} + ( - 4 \beta_{7} + \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \cdots - 4) q^{88}+ \cdots + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_1 + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + q^{4} - 6 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + q^{4} - 6 q^{7} - q^{8} - 8 q^{10} - 16 q^{14} + 9 q^{16} - 14 q^{17} + 8 q^{20} - q^{22} + 10 q^{23} - 2 q^{25} + 14 q^{26} + 2 q^{28} + 10 q^{31} - 11 q^{32} - q^{34} - 23 q^{38} - 6 q^{40} + 8 q^{41} + 9 q^{44} - 10 q^{46} - 6 q^{47} - 18 q^{49} + 23 q^{50} + 8 q^{52} - 2 q^{55} - 10 q^{56} + 14 q^{58} - 26 q^{62} + 13 q^{64} + 14 q^{65} + 39 q^{68} + 36 q^{71} - 22 q^{73} + 38 q^{74} - 5 q^{76} + 30 q^{79} - 48 q^{80} + 19 q^{82} - 7 q^{86} - 31 q^{88} + 32 q^{89} + 30 q^{92} + 12 q^{94} - 44 q^{95} + 33 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 6\nu^{3} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{3} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} - 2\nu^{5} - 2\nu^{3} - 4\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 2\nu^{5} + 4\nu^{4} + 2\nu^{3} + 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + \nu^{5} + 2\nu^{4} - 2\nu^{3} - 2\nu^{2} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - \beta_{6} - 2\beta_{5} - \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - \beta_{6} + 2\beta_{5} - 5\beta_{4} + 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
1.40782 0.134277i
1.40782 + 0.134277i
0.691926 1.23338i
0.691926 + 1.23338i
−0.295065 1.38309i
−0.295065 + 1.38309i
−1.30469 0.545705i
−1.30469 + 0.545705i
−1.40782 0.134277i 0 1.96394 + 0.378078i 2.28102i 0 1.81565 −2.71411 0.795980i 0 0.306290 3.21128i
325.2 −1.40782 + 0.134277i 0 1.96394 0.378078i 2.28102i 0 1.81565 −2.71411 + 0.795980i 0 0.306290 + 3.21128i
325.3 −0.691926 1.23338i 0 −1.04248 + 1.70682i 3.66342i 0 0.383852 2.82649 + 0.104780i 0 −4.51841 + 2.53482i
325.4 −0.691926 + 1.23338i 0 −1.04248 1.70682i 3.66342i 0 0.383852 2.82649 0.104780i 0 −4.51841 2.53482i
325.5 0.295065 1.38309i 0 −1.82587 0.816201i 0.696047i 0 −1.59013 −1.66763 + 2.28452i 0 0.962695 + 0.205379i
325.6 0.295065 + 1.38309i 0 −1.82587 + 0.816201i 0.696047i 0 −1.59013 −1.66763 2.28452i 0 0.962695 0.205379i
325.7 1.30469 0.545705i 0 1.40441 1.42395i 1.37542i 0 −3.60937 1.05526 2.62420i 0 −0.750573 1.79449i
325.8 1.30469 + 0.545705i 0 1.40441 + 1.42395i 1.37542i 0 −3.60937 1.05526 + 2.62420i 0 −0.750573 + 1.79449i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.d.j 8
3.b odd 2 1 648.2.d.k 8
4.b odd 2 1 2592.2.d.j 8
8.b even 2 1 inner 648.2.d.j 8
8.d odd 2 1 2592.2.d.j 8
9.c even 3 2 72.2.n.b 16
9.d odd 6 2 216.2.n.b 16
12.b even 2 1 2592.2.d.k 8
24.f even 2 1 2592.2.d.k 8
24.h odd 2 1 648.2.d.k 8
36.f odd 6 2 288.2.r.b 16
36.h even 6 2 864.2.r.b 16
72.j odd 6 2 216.2.n.b 16
72.l even 6 2 864.2.r.b 16
72.n even 6 2 72.2.n.b 16
72.p odd 6 2 288.2.r.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b 16 9.c even 3 2
72.2.n.b 16 72.n even 6 2
216.2.n.b 16 9.d odd 6 2
216.2.n.b 16 72.j odd 6 2
288.2.r.b 16 36.f odd 6 2
288.2.r.b 16 72.p odd 6 2
648.2.d.j 8 1.a even 1 1 trivial
648.2.d.j 8 8.b even 2 1 inner
648.2.d.k 8 3.b odd 2 1
648.2.d.k 8 24.h odd 2 1
864.2.r.b 16 36.h even 6 2
864.2.r.b 16 72.l even 6 2
2592.2.d.j 8 4.b odd 2 1
2592.2.d.j 8 8.d odd 2 1
2592.2.d.k 8 12.b even 2 1
2592.2.d.k 8 24.f even 2 1

Hecke kernels

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

\( T_{5}^{8} + 21T_{5}^{6} + 115T_{5}^{4} + 183T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{4} + 7T_{17}^{3} - 2T_{17}^{2} - 48T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 2 T^{4} + 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 21 T^{6} + 115 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{3} - 5 T^{2} - 9 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 40 T^{6} + 466 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$13$ \( T^{8} + 53 T^{6} + 715 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$17$ \( (T^{4} + 7 T^{3} - 2 T^{2} - 48 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 83 T^{6} + 1660 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} - 35 T^{2} + 225 T - 300)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 109 T^{6} + 2635 T^{4} + \cdots + 63504 \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{3} - 51 T^{2} + 205 T + 304)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + 8080 T^{4} + \cdots + 1327104 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 74 T^{2} + 456 T - 579)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 20 T^{6} + 106 T^{4} + 120 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$47$ \( (T^{4} + 3 T^{3} - 81 T^{2} - 153 T + 1476)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 160 T^{6} + 7744 T^{4} + \cdots + 451584 \) Copy content Toggle raw display
$59$ \( T^{8} + 108 T^{6} + 3130 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$61$ \( T^{8} + 165 T^{6} + 8379 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
$67$ \( T^{8} + 240 T^{6} + 18954 T^{4} + \cdots + 3143529 \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + 48 T^{2} + 252 T - 864)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11 T^{3} - 14 T^{2} - 84 T + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 15 T^{3} - 11 T^{2} + 615 T - 56)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 105 T^{6} + 3619 T^{4} + \cdots + 178084 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + 52 T^{2} + 156 T - 504)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 134 T^{2} + 780 T - 1151)^{2} \) Copy content Toggle raw display
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