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Properties

Label 648.3.e.b
Level $648$
Weight $3$
Character orbit 648.e
Analytic conductor $17.657$
Analytic rank $0$
Dimension $4$
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] = [648,3,Mod(161,648)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.e (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: \(17.6567211305\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7} + ( - 3 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} + 7) q^{13} + ( - 8 \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{3} + 2) q^{19} + ( - 5 \beta_{2} + \beta_1) q^{23} + (2 \beta_{3} - 10) q^{25} + (\beta_{2} - 3 \beta_1) q^{29} + (\beta_{3} - 37) q^{31} + 29 \beta_{2} q^{35} + (2 \beta_{3} - 30) q^{37} + 23 \beta_{2} q^{41} + (4 \beta_{3} - 5) q^{43} + (29 \beta_{2} + 3 \beta_1) q^{47} + (6 \beta_{3} + 56) q^{49} + ( - 32 \beta_{2} - 8 \beta_1) q^{53} + (\beta_{3} + 55) q^{55} + (3 \beta_{2} - 2 \beta_1) q^{59} + (\beta_{3} + 31) q^{61} + ( - 39 \beta_{2} + 10 \beta_1) q^{65} + (10 \beta_{3} + 11) q^{67} + (24 \beta_{2} + 2 \beta_1) q^{71} + (6 \beta_{3} + 10) q^{73} + ( - 73 \beta_{2} - 15 \beta_1) q^{77} + ( - 7 \beta_{3} + 43) q^{79} + ( - 11 \beta_{2} + 14 \beta_1) q^{83} + (10 \beta_{3} - 88) q^{85} + ( - 24 \beta_{2} + 6 \beta_1) q^{89} + (4 \beta_{3} - 75) q^{91} + (62 \beta_{2} - 4 \beta_1) q^{95} + (2 \beta_{3} - 121) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{7} + 28 q^{13} + 8 q^{19} - 40 q^{25} - 148 q^{31} - 120 q^{37} - 20 q^{43} + 224 q^{49} + 220 q^{55} + 124 q^{61} + 44 q^{67} + 40 q^{73} + 172 q^{79} - 352 q^{85} - 300 q^{91} - 484 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( 2\nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 8\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.22474 − 0.707107i
1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
0 0 0 − 7.38891i 0 −6.79796 0 0 0
161.2 0 0 0 − 3.92480i 0 12.7980 0 0 0
161.3 0 0 0 3.92480i 0 12.7980 0 0 0
161.4 0 0 0 7.38891i 0 −6.79796 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.3.e.b 4
3.b odd 2 1 inner 648.3.e.b 4
4.b odd 2 1 1296.3.e.c 4
9.c even 3 1 72.3.m.a ✓ 4
9.c even 3 1 216.3.m.a 4
9.d odd 6 1 72.3.m.a ✓ 4
9.d odd 6 1 216.3.m.a 4
12.b even 2 1 1296.3.e.c 4
36.f odd 6 1 144.3.q.d 4
36.f odd 6 1 432.3.q.c 4
36.h even 6 1 144.3.q.d 4
36.h even 6 1 432.3.q.c 4
72.j odd 6 1 576.3.q.h 4
72.j odd 6 1 1728.3.q.e 4
72.l even 6 1 576.3.q.c 4
72.l even 6 1 1728.3.q.f 4
72.n even 6 1 576.3.q.h 4
72.n even 6 1 1728.3.q.e 4
72.p odd 6 1 576.3.q.c 4
72.p odd 6 1 1728.3.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a ✓ 4 9.c even 3 1
72.3.m.a ✓ 4 9.d odd 6 1
144.3.q.d 4 36.f odd 6 1
144.3.q.d 4 36.h even 6 1
216.3.m.a 4 9.c even 3 1
216.3.m.a 4 9.d odd 6 1
432.3.q.c 4 36.f odd 6 1
432.3.q.c 4 36.h even 6 1
576.3.q.c 4 72.l even 6 1
576.3.q.c 4 72.p odd 6 1
576.3.q.h 4 72.j odd 6 1
576.3.q.h 4 72.n even 6 1
648.3.e.b 4 1.a even 1 1 trivial
648.3.e.b 4 3.b odd 2 1 inner
1296.3.e.c 4 4.b odd 2 1
1296.3.e.c 4 12.b even 2 1
1728.3.q.e 4 72.j odd 6 1
1728.3.q.e 4 72.n even 6 1
1728.3.q.f 4 72.l even 6 1
1728.3.q.f 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 70T_{5}^{2} + 841 \) acting on \(S_{3}^{\mathrm{new}}(648, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 70T^{2} + 841 \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T - 87)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 310 T^{2} + 10201 \) Copy content Toggle raw display
$13$ \( (T^{2} - 14 T - 47)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 640T^{2} + 4096 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 380)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 214T^{2} + 1849 \) Copy content Toggle raw display
$29$ \( T^{4} + 582 T^{2} + 81225 \) Copy content Toggle raw display
$31$ \( (T^{2} + 74 T + 1273)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 60 T + 516)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1587)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T - 1511)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 5622 T^{2} + \cdots + 4995225 \) Copy content Toggle raw display
$53$ \( T^{4} + 10240 T^{2} + \cdots + 1048576 \) Copy content Toggle raw display
$59$ \( T^{4} + 310 T^{2} + 10201 \) Copy content Toggle raw display
$61$ \( (T^{2} - 62 T + 865)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 22 T - 9479)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 3712 T^{2} + \cdots + 2560000 \) Copy content Toggle raw display
$73$ \( (T^{2} - 20 T - 3356)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 86 T - 2855)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 13270 T^{2} + \cdots + 34916281 \) Copy content Toggle raw display
$89$ \( T^{4} + 5760 T^{2} + 331776 \) Copy content Toggle raw display
$97$ \( (T^{2} + 242 T + 14257)^{2} \) Copy content Toggle raw display
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