Properties

Label 1728.3.q.h
Level $1728$
Weight $3$
Character orbit 1728.q
Analytic conductor $47.085$
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] = [1728,3,Mod(449,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, 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("1728.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1 + 3) q^{5} + (\beta_{3} - 2 \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1 + 3) q^{5} + (\beta_{3} - 2 \beta_1 + 1) q^{7} + (2 \beta_{3} - 5 \beta_{2} + 7) q^{11} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} + 15 \beta_{2} + \beta_1 + 7) q^{17} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{19} + (17 \beta_{2} + \beta_1 + 33) q^{23} + (3 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 8) q^{25} + (7 \beta_{3} + 14 \beta_{2} - 7) q^{29} + ( - 6 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 3) q^{31} + (5 \beta_{3} + 51 \beta_{2} - 5 \beta_1 + 28) q^{35} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 18) q^{37} + ( - \beta_{2} - 8 \beta_1 + 6) q^{41} + ( - 23 \beta_{2} - 23) q^{43} + ( - 3 \beta_{3} + 12 \beta_{2} - 15) q^{47} + (2 \beta_{3} + 26 \beta_{2} - \beta_1 + 1) q^{49} + (8 \beta_{3} + 24 \beta_{2} - 8 \beta_1 + 16) q^{53} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 24) q^{55} + ( - 17 \beta_{2} + 8 \beta_1 - 42) q^{59} + (3 \beta_{3} - 22 \beta_{2} - 6 \beta_1 - 19) q^{61} + (7 \beta_{3} + 32 \beta_{2} - 25) q^{65} + ( - 12 \beta_{3} - 61 \beta_{2} + 6 \beta_1 - 6) q^{67} + ( - 2 \beta_{3} + 30 \beta_{2} + 2 \beta_1 + 14) q^{71} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 23) q^{73} + ( - 55 \beta_{2} - 17 \beta_1 - 93) q^{77} + (5 \beta_{3} - 44 \beta_{2} - 10 \beta_1 - 39) q^{79} + (3 \beta_{3} - 12 \beta_{2} + 15) q^{83} + (12 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 6) q^{85} + (8 \beta_{3} - 120 \beta_{2} - 8 \beta_1 - 56) q^{89} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 74) q^{91} + ( - 22 \beta_{2} + 4 \beta_1 - 48) q^{95} + ( - 2 \beta_{3} - 97 \beta_{2} + 4 \beta_1 - 99) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 9 q^{5} + q^{7} + 36 q^{11} - 5 q^{13} + 2 q^{19} + 99 q^{23} + 13 q^{25} - 63 q^{29} + 7 q^{31} + 64 q^{37} + 18 q^{41} - 46 q^{43} - 81 q^{47} - 51 q^{49} - 90 q^{55} - 126 q^{59} - 41 q^{61} - 171 q^{65} + 116 q^{67} + 86 q^{73} - 279 q^{77} - 83 q^{79} + 81 q^{83} - 18 q^{85} - 302 q^{91} - 144 q^{95} - 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 16\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + \nu^{2} + 8\nu + 12 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 8\beta_{2} + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.68614 + 0.396143i
−1.18614 1.26217i
1.68614 0.396143i
−1.18614 + 1.26217i
0 0 0 −2.05842 1.18843i 0 −4.05842 7.02939i 0 0 0
449.2 0 0 0 6.55842 + 3.78651i 0 4.55842 + 7.89542i 0 0 0
1601.1 0 0 0 −2.05842 + 1.18843i 0 −4.05842 + 7.02939i 0 0 0
1601.2 0 0 0 6.55842 3.78651i 0 4.55842 7.89542i 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
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 1728.3.q.h 4
3.b odd 2 1 576.3.q.g 4
4.b odd 2 1 1728.3.q.g 4
8.b even 2 1 432.3.q.b 4
8.d odd 2 1 108.3.g.a 4
9.c even 3 1 576.3.q.g 4
9.d odd 6 1 inner 1728.3.q.h 4
12.b even 2 1 576.3.q.d 4
24.f even 2 1 36.3.g.a 4
24.h odd 2 1 144.3.q.b 4
36.f odd 6 1 576.3.q.d 4
36.h even 6 1 1728.3.q.g 4
40.e odd 2 1 2700.3.p.b 4
40.k even 4 2 2700.3.u.b 8
72.j odd 6 1 432.3.q.b 4
72.j odd 6 1 1296.3.e.e 4
72.l even 6 1 108.3.g.a 4
72.l even 6 1 324.3.c.b 4
72.n even 6 1 144.3.q.b 4
72.n even 6 1 1296.3.e.e 4
72.p odd 6 1 36.3.g.a 4
72.p odd 6 1 324.3.c.b 4
120.m even 2 1 900.3.p.a 4
120.q odd 4 2 900.3.u.a 8
360.z odd 6 1 900.3.p.a 4
360.bd even 6 1 2700.3.p.b 4
360.bo even 12 2 900.3.u.a 8
360.bt odd 12 2 2700.3.u.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 24.f even 2 1
36.3.g.a 4 72.p odd 6 1
108.3.g.a 4 8.d odd 2 1
108.3.g.a 4 72.l even 6 1
144.3.q.b 4 24.h odd 2 1
144.3.q.b 4 72.n even 6 1
324.3.c.b 4 72.l even 6 1
324.3.c.b 4 72.p odd 6 1
432.3.q.b 4 8.b even 2 1
432.3.q.b 4 72.j odd 6 1
576.3.q.d 4 12.b even 2 1
576.3.q.d 4 36.f odd 6 1
576.3.q.g 4 3.b odd 2 1
576.3.q.g 4 9.c even 3 1
900.3.p.a 4 120.m even 2 1
900.3.p.a 4 360.z odd 6 1
900.3.u.a 8 120.q odd 4 2
900.3.u.a 8 360.bo even 12 2
1296.3.e.e 4 72.j odd 6 1
1296.3.e.e 4 72.n even 6 1
1728.3.q.g 4 4.b odd 2 1
1728.3.q.g 4 36.h even 6 1
1728.3.q.h 4 1.a even 1 1 trivial
1728.3.q.h 4 9.d odd 6 1 inner
2700.3.p.b 4 40.e odd 2 1
2700.3.p.b 4 360.bd even 6 1
2700.3.u.b 8 40.k even 4 2
2700.3.u.b 8 360.bt odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} - 9T_{5}^{3} + 9T_{5}^{2} + 162T_{5} + 324 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} + 75T_{7}^{2} + 74T_{7} + 5476 \) 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
$5$ \( T^{4} - 9 T^{3} + 9 T^{2} + 162 T + 324 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + 75 T^{2} + 74 T + 5476 \) Copy content Toggle raw display
$11$ \( T^{4} - 36 T^{3} + 441 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + 93 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( T^{4} + 387 T^{2} + 20736 \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 99 T^{3} + 4059 T^{2} + \cdots + 627264 \) Copy content Toggle raw display
$29$ \( T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924 \) Copy content Toggle raw display
$31$ \( T^{4} - 7 T^{3} + 705 T^{2} + \cdots + 430336 \) Copy content Toggle raw display
$37$ \( (T^{2} - 32 T - 932)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} - 1449 T^{2} + \cdots + 2424249 \) Copy content Toggle raw display
$43$ \( (T^{2} + 23 T + 529)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$53$ \( T^{4} + 4032 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121 \) Copy content Toggle raw display
$61$ \( T^{4} + 41 T^{3} + 1929 T^{2} + \cdots + 61504 \) Copy content Toggle raw display
$67$ \( T^{4} - 116 T^{3} + 12765 T^{2} + \cdots + 477481 \) Copy content Toggle raw display
$71$ \( T^{4} + 1548 T^{2} + 331776 \) Copy content Toggle raw display
$73$ \( (T^{2} - 43 T - 206)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 83 T^{3} + 7023 T^{2} + \cdots + 17956 \) Copy content Toggle raw display
$83$ \( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( T^{4} + 24768 T^{2} + \cdots + 84934656 \) Copy content Toggle raw display
$97$ \( T^{4} + 196 T^{3} + \cdots + 86620249 \) Copy content Toggle raw display
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