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Properties

Label 1728.3.b.i
Level $1728$
Weight $3$
Character orbit 1728.b
Analytic conductor $47.085$
Analytic rank $0$
Dimension $12$
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] = [1728,3,Mod(1567,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1567");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.116304318664704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + \beta_{2} q^{7} + (\beta_{9} - 2 \beta_1) q^{11} + ( - \beta_{5} - 2 \beta_{4}) q^{13} + ( - \beta_{10} - 4) q^{17} + ( - \beta_{11} - \beta_{9} + 4 \beta_1) q^{19} + (\beta_{7} + 3 \beta_{3} + 3 \beta_{2}) q^{23} + ( - \beta_{10} - 2 \beta_{8} - 6) q^{25} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{29} + (3 \beta_{7} - 4 \beta_{2}) q^{31} + ( - 2 \beta_{9} - 5 \beta_1) q^{35} + ( - \beta_{6} + 3 \beta_{5} - 6 \beta_{4}) q^{37} + (2 \beta_{10} + 3 \beta_{8} - 28) q^{41} + ( - \beta_{11} + 2 \beta_{9} - 9 \beta_1) q^{43} + ( - 4 \beta_{7} + 9 \beta_{3}) q^{47} + (\beta_{10} + 2 \beta_{8} + 4) q^{49} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4}) q^{53} + (2 \beta_{3} + 5 \beta_{2}) q^{55} + (2 \beta_{11} + 2 \beta_{9} - 24 \beta_1) q^{59} + 4 \beta_{6} q^{61} + ( - \beta_{10} - 6 \beta_{8} - 76) q^{65} + (\beta_{11} + 4 \beta_{9} - \beta_1) q^{67} + (\beta_{7} + 18 \beta_{3} - 9 \beta_{2}) q^{71} + (\beta_{10} - 4 \beta_{8} + 5) q^{73} + ( - 10 \beta_{6} - 4 \beta_{5} - 7 \beta_{4}) q^{77} + ( - 3 \beta_{7} - 8 \beta_{3} - 3 \beta_{2}) q^{79} + ( - 4 \beta_{11} - 5 \beta_{9} - 22 \beta_1) q^{83} + ( - 2 \beta_{6} - 5 \beta_{5} + 14 \beta_{4}) q^{85} + (\beta_{10} - 3 \beta_{8} - 104) q^{89} + (2 \beta_{11} - 7 \beta_{9} + 31 \beta_1) q^{91} + (13 \beta_{7} + 27 \beta_{3} - 9 \beta_{2}) q^{95} + (2 \beta_{10} + 10 \beta_{8} - 17) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 10 \nu^{8} - 25 \nu^{7} + 30 \nu^{6} + 30 \nu^{5} - 68 \nu^{4} + \cdots - 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 8 \nu^{10} + 7 \nu^{9} + 4 \nu^{8} - 35 \nu^{7} + 16 \nu^{6} + 66 \nu^{5} - 52 \nu^{4} + \cdots - 248 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 10 \nu^{10} + 5 \nu^{9} + 14 \nu^{8} - 37 \nu^{7} - 10 \nu^{6} + 78 \nu^{5} - 20 \nu^{4} + \cdots - 256 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{11} - 30 \nu^{10} - 21 \nu^{9} + 122 \nu^{8} - 139 \nu^{7} - 158 \nu^{6} + 330 \nu^{5} + \cdots - 1408 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{11} + 11 \nu^{10} - 2 \nu^{9} - 23 \nu^{8} + 38 \nu^{7} + 27 \nu^{6} - 84 \nu^{5} - 16 \nu^{4} + \cdots + 256 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} + \nu^{9} - 15 \nu^{8} + 19 \nu^{7} + 19 \nu^{6} - 48 \nu^{5} - 18 \nu^{4} + \cdots + 168 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{11} - 30 \nu^{10} + 9 \nu^{9} + 66 \nu^{8} - 93 \nu^{7} - 78 \nu^{6} + 222 \nu^{5} + 60 \nu^{4} + \cdots - 552 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15 \nu^{11} - 12 \nu^{10} - 21 \nu^{9} + 72 \nu^{8} - 3 \nu^{7} - 132 \nu^{6} + 66 \nu^{5} + 252 \nu^{4} + \cdots - 144 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67 \nu^{11} + 86 \nu^{10} + 73 \nu^{9} - 386 \nu^{8} + 231 \nu^{7} + 630 \nu^{6} - 770 \nu^{5} + \cdots + 2176 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18 \nu^{11} - 15 \nu^{10} - 30 \nu^{9} + 87 \nu^{8} - 6 \nu^{7} - 171 \nu^{6} + 84 \nu^{5} + 300 \nu^{4} + \cdots - 184 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 157 \nu^{11} - 146 \nu^{10} - 247 \nu^{9} + 806 \nu^{8} - 153 \nu^{7} - 1554 \nu^{6} + 1022 \nu^{5} + \cdots - 2560 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 8 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 4 \beta_{3} + \cdots + 8 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} - 3\beta_{8} + 2\beta_{7} - 7\beta_{3} + 2\beta_{2} - 32 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} - 2 \beta_{5} + \cdots - 40 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - 7 \beta_{8} - 6 \beta_{7} - 14 \beta_{6} - 2 \beta_{5} + \cdots + 48 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{10} + 4\beta_{8} - \beta_{7} - 6\beta_{3} + 9\beta_{2} + 44 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + 12 \beta_{8} + 30 \beta_{7} + 3 \beta_{6} + 18 \beta_{5} + \cdots - 8 ) / 48 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -5\beta_{10} - \beta_{8} + 22\beta_{7} - 29\beta_{3} - 26\beta_{2} - 464 ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 43 \beta_{11} + 10 \beta_{10} - 82 \beta_{9} + 16 \beta_{8} - 30 \beta_{7} - 41 \beta_{6} + \cdots - 152 ) / 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 18 \beta_{11} - 33 \beta_{10} - 102 \beta_{9} - 5 \beta_{8} - 58 \beta_{7} + 122 \beta_{6} - 94 \beta_{5} + \cdots - 432 ) / 48 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−1.41362 − 0.0408194i
0.742163 + 1.20382i
1.33544 − 0.465413i
−1.07078 + 0.923815i
0.828615 + 1.14604i
0.578188 + 1.29062i
0.578188 − 1.29062i
0.828615 − 1.14604i
−1.07078 − 0.923815i
1.33544 + 0.465413i
0.742163 − 1.20382i
−1.41362 + 0.0408194i
0 0 0 − 8.36964i 0 − 2.43910i 0 0 0
1567.2 0 0 0 − 8.36964i 0 2.43910i 0 0 0
1567.3 0 0 0 − 3.99718i 0 − 7.74742i 0 0 0
1567.4 0 0 0 − 3.99718i 0 7.74742i 0 0 0
1567.5 0 0 0 − 2.64040i 0 − 8.30833i 0 0 0
1567.6 0 0 0 − 2.64040i 0 8.30833i 0 0 0
1567.7 0 0 0 2.64040i 0 − 8.30833i 0 0 0
1567.8 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.9 0 0 0 3.99718i 0 − 7.74742i 0 0 0
1567.10 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.11 0 0 0 8.36964i 0 − 2.43910i 0 0 0
1567.12 0 0 0 8.36964i 0 2.43910i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.i ✓ 12
3.b odd 2 1 1728.3.b.j yes 12
4.b odd 2 1 inner 1728.3.b.i ✓ 12
8.b even 2 1 inner 1728.3.b.i ✓ 12
8.d odd 2 1 inner 1728.3.b.i ✓ 12
12.b even 2 1 1728.3.b.j yes 12
24.f even 2 1 1728.3.b.j yes 12
24.h odd 2 1 1728.3.b.j yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.i ✓ 12 1.a even 1 1 trivial
1728.3.b.i ✓ 12 4.b odd 2 1 inner
1728.3.b.i ✓ 12 8.b even 2 1 inner
1728.3.b.i ✓ 12 8.d odd 2 1 inner
1728.3.b.j yes 12 3.b odd 2 1
1728.3.b.j yes 12 12.b even 2 1
1728.3.b.j yes 12 24.f even 2 1
1728.3.b.j yes 12 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{6} + 93T_{5}^{4} + 1719T_{5}^{2} + 7803 \) Copy content Toggle raw display
\( T_{7}^{6} + 135T_{7}^{4} + 4911T_{7}^{2} + 24649 \) Copy content Toggle raw display
\( T_{11}^{6} - 417T_{11}^{4} + 23211T_{11}^{2} - 121203 \) Copy content Toggle raw display
\( T_{17}^{3} + 12T_{17}^{2} - 540T_{17} - 6912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 93 T^{4} + \cdots + 7803)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 135 T^{4} + \cdots + 24649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 417 T^{4} + \cdots - 121203)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 936 T^{4} + \cdots + 442368)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 12 T^{2} + \cdots - 6912)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} - 1656 T^{4} + \cdots - 51121152)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 2520 T^{4} + \cdots + 394896384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 1512 T^{4} + \cdots + 20155392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 3447 T^{4} + \cdots + 262018969)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 5256 T^{4} + \cdots + 2674142208)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 84 T^{2} + \cdots - 113184)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} - 4320 T^{4} + \cdots - 322486272)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 7488 T^{4} + \cdots + 5780865024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 1101 T^{4} + \cdots + 8741547)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 11232 T^{4} + \cdots - 5159780352)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 768)^{6} \) Copy content Toggle raw display
$67$ \( (T^{6} - 6336 T^{4} + \cdots - 936050688)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 22392 T^{4} + \cdots + 4087812096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 15 T^{2} + \cdots - 120721)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8868 T^{4} + \cdots + 1846936576)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 30801 T^{4} + \cdots - 11178011043)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 312 T^{2} + \cdots + 680832)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + 51 T^{2} + \cdots - 1211279)^{4} \) Copy content Toggle raw display
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