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Properties

Label 1728.3.o.f
Level $1728$
Weight $3$
Character orbit 1728.o
Analytic conductor $47.085$
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] = [1728,3,Mod(127,1728)]
 
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([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.o (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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 144)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 2) q^{11}+ \cdots + (8 \beta_{7} - 16 \beta_{6} + \cdots - 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{5} + 3 q^{7} + 18 q^{11} - 5 q^{13} - 6 q^{17} + 81 q^{23} - 23 q^{25} + 69 q^{29} + 45 q^{31} + 20 q^{37} - 54 q^{41} - 207 q^{47} + 41 q^{49} - 252 q^{53} - 306 q^{59} - 7 q^{61} - 93 q^{65} - 12 q^{67} + 74 q^{73} + 207 q^{77} + 33 q^{79} + 549 q^{83} + 30 q^{85} + 168 q^{89} + 684 q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - \nu^{3} + 14\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 6\nu^{4} + \nu^{3} + 24\nu^{2} - 14\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} + 8\nu^{4} - 4\nu^{3} + 14\nu^{2} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{5} + 8\nu^{4} + 4\nu^{3} + 14\nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 2\nu^{6} + 32\nu^{5} + 22\nu^{4} + 95\nu^{3} + 70\nu^{2} + 50\nu + 42 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} - 32\nu^{5} + 22\nu^{4} - 95\nu^{3} + 70\nu^{2} - 50\nu + 42 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 2\beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} - 24 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} - 2\beta_{4} + 3\beta_{2} - 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} + 4\beta_{5} + 4\beta_{4} + 14\beta_{3} + 7\beta_{2} + 105 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{5} + 9\beta_{4} - 16\beta_{2} + 16\beta _1 - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6\beta_{7} + 6\beta_{6} - 3\beta_{5} - 3\beta_{4} - 28\beta_{3} - 14\beta_{2} - 168 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 41\beta_{5} - 41\beta_{4} + 84\beta_{2} - 124\beta _1 + 62 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
− 1.07834i
− 0.385731i
− 2.33086i
2.06288i
1.07834i
0.385731i
2.33086i
− 2.06288i
0 0 0 −3.01729 − 5.22611i 0 10.2332 + 5.90815i 0 0 0
127.2 0 0 0 −0.454613 − 0.787412i 0 −6.10709 − 3.52593i 0 0 0
127.3 0 0 0 0.355304 + 0.615405i 0 2.70480 + 1.56162i 0 0 0
127.4 0 0 0 4.61660 + 7.99619i 0 −5.33093 − 3.07781i 0 0 0
1279.1 0 0 0 −3.01729 + 5.22611i 0 10.2332 − 5.90815i 0 0 0
1279.2 0 0 0 −0.454613 + 0.787412i 0 −6.10709 + 3.52593i 0 0 0
1279.3 0 0 0 0.355304 − 0.615405i 0 2.70480 − 1.56162i 0 0 0
1279.4 0 0 0 4.61660 − 7.99619i 0 −5.33093 + 3.07781i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f 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.o.f 8
3.b odd 2 1 576.3.o.f 8
4.b odd 2 1 1728.3.o.e 8
8.b even 2 1 432.3.o.b 8
8.d odd 2 1 432.3.o.a 8
9.c even 3 1 1728.3.o.e 8
9.d odd 6 1 576.3.o.d 8
12.b even 2 1 576.3.o.d 8
24.f even 2 1 144.3.o.c yes 8
24.h odd 2 1 144.3.o.a ✓ 8
36.f odd 6 1 inner 1728.3.o.f 8
36.h even 6 1 576.3.o.f 8
72.j odd 6 1 144.3.o.c yes 8
72.j odd 6 1 1296.3.g.j 8
72.l even 6 1 144.3.o.a ✓ 8
72.l even 6 1 1296.3.g.j 8
72.n even 6 1 432.3.o.a 8
72.n even 6 1 1296.3.g.k 8
72.p odd 6 1 432.3.o.b 8
72.p odd 6 1 1296.3.g.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a ✓ 8 24.h odd 2 1
144.3.o.a ✓ 8 72.l even 6 1
144.3.o.c yes 8 24.f even 2 1
144.3.o.c yes 8 72.j odd 6 1
432.3.o.a 8 8.d odd 2 1
432.3.o.a 8 72.n even 6 1
432.3.o.b 8 8.b even 2 1
432.3.o.b 8 72.p odd 6 1
576.3.o.d 8 9.d odd 6 1
576.3.o.d 8 12.b even 2 1
576.3.o.f 8 3.b odd 2 1
576.3.o.f 8 36.h even 6 1
1296.3.g.j 8 72.j odd 6 1
1296.3.g.j 8 72.l even 6 1
1296.3.g.k 8 72.n even 6 1
1296.3.g.k 8 72.p odd 6 1
1728.3.o.e 8 4.b odd 2 1
1728.3.o.e 8 9.c even 3 1
1728.3.o.f 8 1.a even 1 1 trivial
1728.3.o.f 8 36.f odd 6 1 inner

Hecke kernels

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

\( T_{5}^{8} - 3T_{5}^{7} + 66T_{5}^{6} + 189T_{5}^{5} + 3186T_{5}^{4} + 729T_{5}^{3} + 2133T_{5}^{2} - 324T_{5} + 1296 \) Copy content Toggle raw display
\( T_{7}^{8} - 3T_{7}^{7} - 114T_{7}^{6} + 351T_{7}^{5} + 12366T_{7}^{4} + 32643T_{7}^{3} - 161487T_{7}^{2} - 446958T_{7} + 2566404 \) 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
$5$ \( T^{8} - 3 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 2566404 \) Copy content Toggle raw display
$11$ \( T^{8} - 18 T^{7} + \cdots + 12131289 \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + \cdots + 10201636 \) Copy content Toggle raw display
$17$ \( (T^{4} + 3 T^{3} + \cdots + 84168)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2931572736 \) Copy content Toggle raw display
$23$ \( T^{8} - 81 T^{7} + \cdots + 19131876 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 4046639163876 \) Copy content Toggle raw display
$31$ \( T^{8} - 45 T^{7} + \cdots + 944784 \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots - 613568)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 5431756955769 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 29016737649 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 28643839776036 \) Copy content Toggle raw display
$53$ \( (T^{4} + 126 T^{3} + \cdots - 6508512)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 48359409452649 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 309954973696 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 68036119056801 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 726110197530624 \) Copy content Toggle raw display
$73$ \( (T^{4} - 37 T^{3} + \cdots + 416536)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 240627852449856 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} - 84 T^{3} + \cdots - 1161936)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 30429664983481 \) Copy content Toggle raw display
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