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Properties

Label 1728.4.d.k
Level $1728$
Weight $4$
Character orbit 1728.d
Analytic conductor $101.955$
Analytic rank $0$
Dimension $16$
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] = [1728,4,Mod(865,1728)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.865");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.d (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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 170x^{12} + 7609x^{8} + 59868x^{4} + 104976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} - \beta_{7} q^{7} + ( - \beta_{12} + 6 \beta_{4}) q^{11} + ( - \beta_{13} - \beta_{10} - \beta_{8}) q^{13} + (\beta_{3} - \beta_{2} + 15) q^{17} + (\beta_{15} - \beta_{12} + \beta_{4}) q^{19} + (\beta_{14} - 2 \beta_{11} + \cdots + 2 \beta_{7}) q^{23}+ \cdots + (2 \beta_{3} - 8 \beta_1 - 161) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 240 q^{17} - 304 q^{25} - 1008 q^{41} + 1616 q^{49} - 2736 q^{65} + 128 q^{73} - 5856 q^{89} - 2576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 170x^{12} + 7609x^{8} + 59868x^{4} + 104976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -66\nu^{12} - 12426\nu^{8} - 607650\nu^{4} - 3320223 ) / 44587 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -44\nu^{12} - 8284\nu^{8} - 315926\nu^{4} + 1576413 ) / 44587 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -932\nu^{12} - 151150\nu^{8} - 6010922\nu^{4} - 20854872 ) / 133761 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -763\nu^{14} - 131492\nu^{10} - 6141169\nu^{6} - 69308928\nu^{2} ) / 86677128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 425\nu^{14} + 92176\nu^{10} + 9611927\nu^{6} + 339571296\nu^{2} ) / 9630792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14282 \nu^{15} - 23274 \nu^{13} - 2409229 \nu^{11} - 4053537 \nu^{9} - 105148967 \nu^{7} + \cdots - 841086180 \nu ) / 260031384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14282 \nu^{15} - 23274 \nu^{13} + 2409229 \nu^{11} - 4053537 \nu^{9} + 105148967 \nu^{7} + \cdots - 841086180 \nu ) / 260031384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11056 \nu^{15} - 32400 \nu^{13} - 1777541 \nu^{11} - 5115069 \nu^{9} - 68865109 \nu^{7} + \cdots - 391495356 \nu ) / 86677128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11056 \nu^{15} + 32400 \nu^{13} - 1777541 \nu^{11} + 5115069 \nu^{9} - 68865109 \nu^{7} + \cdots + 391495356 \nu ) / 86677128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9704 \nu^{15} + 17928 \nu^{13} + 1620277 \nu^{11} + 3047031 \nu^{9} + 68301953 \nu^{7} + \cdots + 561313476 \nu ) / 32503923 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9704 \nu^{15} + 17928 \nu^{13} - 1620277 \nu^{11} + 3047031 \nu^{9} - 68301953 \nu^{7} + \cdots + 561313476 \nu ) / 32503923 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -32297\nu^{14} - 5509120\nu^{10} - 241375655\nu^{6} - 1337461920\nu^{2} ) / 28892376 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28774 \nu^{15} - 42498 \nu^{13} - 4821515 \nu^{11} - 7016247 \nu^{9} - 208376617 \nu^{7} + \cdots - 2017865196 \nu ) / 28892376 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28774 \nu^{15} + 42498 \nu^{13} - 4821515 \nu^{11} + 7016247 \nu^{9} - 208376617 \nu^{7} + \cdots + 2017865196 \nu ) / 28892376 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -11143\nu^{14} - 1830401\nu^{10} - 74724700\nu^{6} - 290478564\nu^{2} ) / 3611547 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{14} - 2\beta_{13} - \beta_{11} - \beta_{10} - 4\beta_{9} + 4\beta_{8} + 10\beta_{7} + 10\beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{5} - 132\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{14} - 4\beta_{13} - 7\beta_{11} + 7\beta_{10} + 8\beta_{9} + 8\beta_{8} - 92\beta_{7} + 92\beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{2} - 2\beta _1 - 255 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4 \beta_{14} - 4 \beta_{13} - 191 \beta_{11} - 191 \beta_{10} + 64 \beta_{9} - 64 \beta_{8} + \cdots - 844 \beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6\beta_{15} - 49\beta_{12} + 125\beta_{5} + 8952\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 236 \beta_{14} - 236 \beta_{13} + 2701 \beta_{11} - 2701 \beta_{10} - 1112 \beta_{9} + \cdots - 7820 \beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 33\beta_{3} - 317\beta_{2} + 56\beta _1 + 20523 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3604 \beta_{14} + 3604 \beta_{13} + 32573 \beta_{11} + 32573 \beta_{10} - 16336 \beta_{9} + \cdots + 73276 \beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1962\beta_{15} + 127\beta_{12} - 12771\beta_{5} - 767832\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 45140 \beta_{14} + 45140 \beta_{13} - 367063 \beta_{11} + 367063 \beta_{10} + 207944 \beta_{9} + \cdots + 693812 \beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -6213\beta_{3} + 32062\beta_{2} + 3817\beta _1 - 1818021 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 520828 \beta_{14} - 520828 \beta_{13} - 3985343 \beta_{11} - 3985343 \beta_{10} + \cdots - 6629716 \beta_{6} ) / 144 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -289830\beta_{15} + 281663\beta_{12} + 1285645\beta_{5} + 69536760\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 5743580 \beta_{14} - 5743580 \beta_{13} + 42267877 \beta_{11} - 42267877 \beta_{10} + \cdots - 63848540 \beta_{6} ) / 144 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
1.14496 + 1.14496i
−1.14496 + 1.14496i
1.97781 − 1.97781i
−1.97781 − 1.97781i
2.23276 + 2.23276i
−2.23276 + 2.23276i
−0.890014 + 0.890014i
0.890014 + 0.890014i
−0.890014 − 0.890014i
0.890014 − 0.890014i
2.23276 − 2.23276i
−2.23276 − 2.23276i
1.97781 + 1.97781i
−1.97781 + 1.97781i
1.14496 − 1.14496i
−1.14496 − 1.14496i
0 0 0 − 18.8216i 0 −3.15603 0 0 0
865.2 0 0 0 − 18.8216i 0 3.15603 0 0 0
865.3 0 0 0 − 10.8148i 0 −24.7677 0 0 0
865.4 0 0 0 − 10.8148i 0 24.7677 0 0 0
865.5 0 0 0 − 10.1512i 0 −33.8599 0 0 0
865.6 0 0 0 − 10.1512i 0 33.8599 0 0 0
865.7 0 0 0 − 1.31976i 0 −2.47210 0 0 0
865.8 0 0 0 − 1.31976i 0 2.47210 0 0 0
865.9 0 0 0 1.31976i 0 −2.47210 0 0 0
865.10 0 0 0 1.31976i 0 2.47210 0 0 0
865.11 0 0 0 10.1512i 0 −33.8599 0 0 0
865.12 0 0 0 10.1512i 0 33.8599 0 0 0
865.13 0 0 0 10.8148i 0 −24.7677 0 0 0
865.14 0 0 0 10.8148i 0 24.7677 0 0 0
865.15 0 0 0 18.8216i 0 −3.15603 0 0 0
865.16 0 0 0 18.8216i 0 3.15603 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.16
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 1728.4.d.k yes 16
3.b odd 2 1 1728.4.d.i ✓ 16
4.b odd 2 1 inner 1728.4.d.k yes 16
8.b even 2 1 inner 1728.4.d.k yes 16
8.d odd 2 1 inner 1728.4.d.k yes 16
12.b even 2 1 1728.4.d.i ✓ 16
24.f even 2 1 1728.4.d.i ✓ 16
24.h odd 2 1 1728.4.d.i ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.d.i ✓ 16 3.b odd 2 1
1728.4.d.i ✓ 16 12.b even 2 1
1728.4.d.i ✓ 16 24.f even 2 1
1728.4.d.i ✓ 16 24.h odd 2 1
1728.4.d.k yes 16 1.a even 1 1 trivial
1728.4.d.k yes 16 4.b odd 2 1 inner
1728.4.d.k yes 16 8.b even 2 1 inner
1728.4.d.k yes 16 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{8} + 576T_{5}^{6} + 90990T_{5}^{4} + 4426272T_{5}^{2} + 7436529 \) Copy content Toggle raw display
\( T_{7}^{8} - 1776T_{7}^{6} + 731646T_{7}^{4} - 11410416T_{7}^{2} + 42810849 \) Copy content Toggle raw display
\( T_{17}^{4} - 60T_{17}^{3} - 11772T_{17}^{2} + 228096T_{17} + 11943936 \) Copy content Toggle raw display
\( T_{23}^{8} - 18792T_{23}^{6} + 108111888T_{23}^{4} - 210946235904T_{23}^{2} + 108609079578624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 576 T^{6} + \cdots + 7436529)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 1776 T^{6} + \cdots + 42810849)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 5364 T^{6} + \cdots + 139858796529)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 8916100448256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 60 T^{3} + \cdots + 11943936)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 18472 T^{6} + \cdots + 437333561344)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 108609079578624)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 17\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 47\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 252 T^{3} + \cdots + 2875362624)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 26\!\cdots\!01)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 30\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 37\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 66\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 32 T^{3} + \cdots + 132320439073)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 22\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 23\!\cdots\!49)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1464 T^{3} + \cdots - 31131122688)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 644 T^{3} + \cdots - 12464063)^{4} \) Copy content Toggle raw display
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