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Properties

Label 1728.3.o.d
Level $1728$
Weight $3$
Character orbit 1728.o
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
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(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: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4} - 2 \beta_{2}) q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4} - 2 \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - \beta_{7} + \beta_{3}) q^{11} + ( - 3 \beta_{6} + 3 \beta_{4} + 4 \beta_{2}) q^{13} + (\beta_{4} - 7) q^{17} + (3 \beta_{7} - 3 \beta_{5} + 2 \beta_1) q^{19} + (4 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{23} + ( - 3 \beta_{6} + 5 \beta_{2} - 5) q^{25} + (\beta_{6} + 16 \beta_{2} - 16) q^{29} + (6 \beta_{5} - \beta_{3} + \beta_1) q^{31} + (2 \beta_{7} - 2 \beta_{5} + 7 \beta_1) q^{35} + ( - 6 \beta_{4} - 8) q^{37} + (6 \beta_{6} - 6 \beta_{4} + 33 \beta_{2}) q^{41} + (3 \beta_{7} + 7 \beta_{3}) q^{43} + (8 \beta_{7} + \beta_{3}) q^{47} + ( - 9 \beta_{6} + 9 \beta_{4} + 5 \beta_{2}) q^{49} + (6 \beta_{4} + 48) q^{53} + ( - 6 \beta_{7} + 6 \beta_{5} - 9 \beta_1) q^{55} + ( - 5 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{59} + ( - 3 \beta_{6} - 2 \beta_{2} + 2) q^{61} + (7 \beta_{6} - 86 \beta_{2} + 86) q^{65} + (3 \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{67} + (4 \beta_{7} - 4 \beta_{5} - 4 \beta_1) q^{71} + ( - 15 \beta_{4} - 35) q^{73} + (9 \beta_{6} - 9 \beta_{4} - 72 \beta_{2}) q^{77} + (6 \beta_{7} - \beta_{3}) q^{79} + (10 \beta_{7} - \beta_{3}) q^{83} + ( - 6 \beta_{6} + 6 \beta_{4} - 12 \beta_{2}) q^{85} + (8 \beta_{4} - 110) q^{89} + ( - 6 \beta_{7} + 6 \beta_{5} - 19 \beta_1) q^{91} + ( - 8 \beta_{5} + 8 \beta_{3} - 8 \beta_1) q^{95} + (18 \beta_{6} - 59 \beta_{2} + 59) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{7} + 93\nu^{6} - 244\nu^{5} - 547\nu^{4} - 659\nu^{3} + 3622\nu^{2} + 1884\nu - 1240 ) / 408 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\nu^{7} - 111\nu^{6} - 74\nu^{5} - 173\nu^{4} + 1517\nu^{3} - 1036\nu^{2} - 768\nu + 1480 ) / 816 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{7} - 15\nu^{6} - 248\nu^{5} - 311\nu^{4} + 953\nu^{3} + 2546\nu^{2} - 600\nu - 2384 ) / 408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -65\nu^{7} + 69\nu^{6} + 284\nu^{5} + 533\nu^{4} - 1691\nu^{3} - 1226\nu^{2} + 2556\nu - 376 ) / 408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -199\nu^{7} + 489\nu^{6} + 190\nu^{5} + 1387\nu^{4} - 6955\nu^{3} + 4700\nu^{2} - 24\nu - 1352 ) / 816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 169\nu^{7} - 261\nu^{6} - 412\nu^{5} - 1345\nu^{4} + 4315\nu^{3} - 566\nu^{2} + 168\nu + 2528 ) / 408 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 427\nu^{7} - 753\nu^{6} - 1114\nu^{5} - 2767\nu^{4} + 11515\nu^{3} - 1520\nu^{2} - 6864\nu + 10040 ) / 816 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + 2\beta_{5} + 3\beta_{4} + 2\beta_{3} + 3\beta_{2} - \beta _1 + 3 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} - 13\beta_{2} - \beta _1 + 14 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{7} + 12\beta_{6} + \beta_{5} - 6\beta_{4} + 10\beta_{3} - 6\beta_{2} - 11\beta _1 + 93 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 9\beta_{6} + 10\beta_{5} + 9\beta_{4} + 10\beta_{3} - 15\beta_{2} - 11\beta _1 + 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 44\beta_{7} - 15\beta_{6} - 55\beta_{5} + 30\beta_{4} + 44\beta_{3} - 537\beta_{2} - 55\beta _1 + 552 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -57\beta_{7} + 80\beta_{6} + 33\beta_{5} - 40\beta_{4} + 24\beta_{3} - 40\beta_{2} - 57\beta _1 + 263 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 169\beta_{7} + 273\beta_{6} + 220\beta_{5} + 273\beta_{4} + 220\beta_{3} - 1833\beta_{2} - 389\beta _1 + 273 ) / 18 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
2.25820 − 0.369600i
0.553538 − 0.676408i
−1.44918 + 1.77086i
−0.862555 + 0.141174i
2.25820 + 0.369600i
0.553538 + 0.676408i
−1.44918 − 1.77086i
−0.862555 − 0.141174i
0 0 0 −3.31174 − 5.73610i 0 −8.46808 − 4.88905i 0 0 0
127.2 0 0 0 −3.31174 − 5.73610i 0 8.46808 + 4.88905i 0 0 0
127.3 0 0 0 1.81174 + 3.13802i 0 −1.59422 − 0.920424i 0 0 0
127.4 0 0 0 1.81174 + 3.13802i 0 1.59422 + 0.920424i 0 0 0
1279.1 0 0 0 −3.31174 + 5.73610i 0 −8.46808 + 4.88905i 0 0 0
1279.2 0 0 0 −3.31174 + 5.73610i 0 8.46808 − 4.88905i 0 0 0
1279.3 0 0 0 1.81174 − 3.13802i 0 −1.59422 + 0.920424i 0 0 0
1279.4 0 0 0 1.81174 − 3.13802i 0 1.59422 − 0.920424i 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
4.b odd 2 1 inner
9.c even 3 1 inner
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.d 8
3.b odd 2 1 576.3.o.e 8
4.b odd 2 1 inner 1728.3.o.d 8
8.b even 2 1 432.3.o.c 8
8.d odd 2 1 432.3.o.c 8
9.c even 3 1 inner 1728.3.o.d 8
9.d odd 6 1 576.3.o.e 8
12.b even 2 1 576.3.o.e 8
24.f even 2 1 144.3.o.b ✓ 8
24.h odd 2 1 144.3.o.b ✓ 8
36.f odd 6 1 inner 1728.3.o.d 8
36.h even 6 1 576.3.o.e 8
72.j odd 6 1 144.3.o.b ✓ 8
72.j odd 6 1 1296.3.g.h 4
72.l even 6 1 144.3.o.b ✓ 8
72.l even 6 1 1296.3.g.h 4
72.n even 6 1 432.3.o.c 8
72.n even 6 1 1296.3.g.d 4
72.p odd 6 1 432.3.o.c 8
72.p odd 6 1 1296.3.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.b ✓ 8 24.f even 2 1
144.3.o.b ✓ 8 24.h odd 2 1
144.3.o.b ✓ 8 72.j odd 6 1
144.3.o.b ✓ 8 72.l even 6 1
432.3.o.c 8 8.b even 2 1
432.3.o.c 8 8.d odd 2 1
432.3.o.c 8 72.n even 6 1
432.3.o.c 8 72.p odd 6 1
576.3.o.e 8 3.b odd 2 1
576.3.o.e 8 9.d odd 6 1
576.3.o.e 8 12.b even 2 1
576.3.o.e 8 36.h even 6 1
1296.3.g.d 4 72.n even 6 1
1296.3.g.d 4 72.p odd 6 1
1296.3.g.h 4 72.j odd 6 1
1296.3.g.h 4 72.l even 6 1
1728.3.o.d 8 1.a even 1 1 trivial
1728.3.o.d 8 4.b odd 2 1 inner
1728.3.o.d 8 9.c even 3 1 inner
1728.3.o.d 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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 3T_{5}^{3} + 33T_{5}^{2} - 72T_{5} + 576 \) Copy content Toggle raw display
\( T_{7}^{8} - 99T_{7}^{6} + 9477T_{7}^{4} - 32076T_{7}^{2} + 104976 \) 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
$5$ \( (T^{4} + 3 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 99 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$11$ \( (T^{4} - 135 T^{2} + 18225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} + \cdots + 52900)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 15 T + 30)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 855 T^{2} + 129600)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 2379503694096 \) Copy content Toggle raw display
$29$ \( (T^{4} + 33 T^{3} + \cdots + 60516)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 2520473760000 \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T - 920)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 72 T^{3} + \cdots + 123201)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1147523000625 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20415837456 \) Copy content Toggle raw display
$53$ \( (T^{2} - 90 T + 1080)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 3375 T^{2} + 11390625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 7 T^{3} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 567 T^{2} + 321489)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2160)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 55 T - 5150)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 2520473760000 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 62171080298496 \) Copy content Toggle raw display
$89$ \( (T^{2} + 228 T + 11316)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 100 T^{3} + \cdots + 36060025)^{2} \) Copy content Toggle raw display
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