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Properties

Label 1728.4.d.h
Level $1728$
Weight $4$
Character orbit 1728.d
Analytic conductor $101.955$
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] = [1728,4,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: \(8\)
Coefficient field: 8.0.2261390379264.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} + (\beta_{6} + \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} + (\beta_{6} + \beta_{4}) q^{7} + ( - \beta_{2} - 12 \beta_1) q^{11} + (\beta_{7} + 2 \beta_{5}) q^{13} + (\beta_{3} + 21) q^{17} + (3 \beta_{2} + 43 \beta_1) q^{19} + (9 \beta_{6} + 2 \beta_{4}) q^{23} + ( - 3 \beta_{3} - 37) q^{25} + (9 \beta_{7} - 10 \beta_{5}) q^{29} + ( - 8 \beta_{6} + 13 \beta_{4}) q^{31} + (6 \beta_{2} + 135 \beta_1) q^{35} + ( - 7 \beta_{7} + 22 \beta_{5}) q^{37} + ( - 5 \beta_{3} + 3) q^{41} + ( - 6 \beta_{2} - 106 \beta_1) q^{43} + 20 \beta_{4} q^{47} + (3 \beta_{3} + 35) q^{49} + (9 \beta_{7} - 11 \beta_{5}) q^{53} + (21 \beta_{6} + 45 \beta_{4}) q^{55} + ( - 10 \beta_{2} + 78 \beta_1) q^{59} + (12 \beta_{7} + 48 \beta_{5}) q^{61} + (3 \beta_{3} + 351) q^{65} + 20 \beta_1 q^{67} + (9 \beta_{6} + 22 \beta_{4}) q^{71} + (9 \beta_{3} + 38) q^{73} + 69 \beta_{5} q^{77} + ( - 7 \beta_{6} + 8 \beta_{4}) q^{79} + ( - 3 \beta_{2} + 468 \beta_1) q^{83} + (21 \beta_{7} - 54 \beta_{5}) q^{85} + (18 \beta_{3} + 774) q^{89} + ( - 15 \beta_{2} - 27 \beta_1) q^{91} + ( - 63 \beta_{6} - 142 \beta_{4}) q^{95} + (12 \beta_{3} + 403) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 168 q^{17} - 296 q^{25} + 24 q^{41} + 280 q^{49} + 2808 q^{65} + 304 q^{73} + 6192 q^{89} + 3224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 15\nu^{4} + 40\nu^{3} + 18\nu^{2} - 39\nu + 35 ) / 639 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 7\nu - 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 12\nu^{5} + 30\nu^{4} - 80\nu^{3} + 390\nu^{2} - 348\nu - 1561 ) / 71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -50\nu^{7} + 175\nu^{6} - 2043\nu^{5} + 4670\nu^{4} + 36047\nu^{3} - 58653\nu^{2} + 85648\nu - 32897 ) / 51333 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -58\nu^{7} + 203\nu^{6} - 1001\nu^{5} + 1995\nu^{4} + 2117\nu^{3} - 5069\nu^{2} + 67183\nu - 32685 ) / 17111 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -740\nu^{7} + 2590\nu^{6} + 10830\nu^{5} - 33550\nu^{4} - 41434\nu^{3} + 96996\nu^{2} - 5468\nu - 14612 ) / 51333 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 280\nu^{7} - 980\nu^{6} - 2248\nu^{5} + 8070\nu^{4} - 10220\nu^{3} + 6770\nu^{2} + 150056\nu - 75864 ) / 17111 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{4} + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{4} + 2\beta_{3} + 36\beta _1 + 48 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - 10\beta_{5} + 24\beta_{4} + 3\beta_{3} + 54\beta _1 + 69 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 5\beta_{6} - 20\beta_{5} + 46\beta_{4} + 18\beta_{3} + 12\beta_{2} + 324\beta _1 + 660 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43\beta_{7} + 77\beta_{6} - 190\beta_{5} + 244\beta_{4} + 40\beta_{3} + 30\beta_{2} + 720\beta _1 + 1536 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 122\beta_{7} + 219\beta_{6} - 520\beta_{5} + 618\beta_{4} + 177\beta_{3} + 180\beta_{2} + 7020\beta _1 + 7653 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 932 \beta_{7} + 790 \beta_{6} - 3134 \beta_{5} + 2552 \beta_{4} + 483 \beta_{3} + 525 \beta_{2} + \cdots + 21489 ) / 12 \) 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
−2.75792 − 0.460416i
3.75792 − 0.460416i
−0.297464 + 1.88096i
1.29746 + 1.88096i
−0.297464 − 1.88096i
1.29746 − 1.88096i
−2.75792 + 0.460416i
3.75792 + 0.460416i
0 0 0 − 16.7850i 0 −22.3100 0 0 0
865.2 0 0 0 − 16.7850i 0 22.3100 0 0 0
865.3 0 0 0 − 6.50098i 0 −16.0706 0 0 0
865.4 0 0 0 − 6.50098i 0 16.0706 0 0 0
865.5 0 0 0 6.50098i 0 −16.0706 0 0 0
865.6 0 0 0 6.50098i 0 16.0706 0 0 0
865.7 0 0 0 16.7850i 0 −22.3100 0 0 0
865.8 0 0 0 16.7850i 0 22.3100 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.8
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.h yes 8
3.b odd 2 1 1728.4.d.e ✓ 8
4.b odd 2 1 inner 1728.4.d.h yes 8
8.b even 2 1 inner 1728.4.d.h yes 8
8.d odd 2 1 inner 1728.4.d.h yes 8
12.b even 2 1 1728.4.d.e ✓ 8
24.f even 2 1 1728.4.d.e ✓ 8
24.h odd 2 1 1728.4.d.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.d.e ✓ 8 3.b odd 2 1
1728.4.d.e ✓ 8 12.b even 2 1
1728.4.d.e ✓ 8 24.f even 2 1
1728.4.d.e ✓ 8 24.h odd 2 1
1728.4.d.h yes 8 1.a even 1 1 trivial
1728.4.d.h yes 8 4.b odd 2 1 inner
1728.4.d.h yes 8 8.b even 2 1 inner
1728.4.d.h yes 8 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 324T_{5}^{2} + 11907 \) Copy content Toggle raw display
\( T_{7}^{4} - 756T_{7}^{2} + 128547 \) Copy content Toggle raw display
\( T_{17}^{2} - 42T_{17} - 1152 \) Copy content Toggle raw display
\( T_{23}^{4} - 43092T_{23}^{2} + 250838208 \) 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
$5$ \( (T^{4} + 324 T^{2} + 11907)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 756 T^{2} + 128547)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3474 T^{2} + 2099601)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2052 T^{2} + 995328)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 42 T - 1152)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 32372 T^{2} + 155950144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 43092 T^{2} + 250838208)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 66420 T^{2} + 903275712)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 100548 T^{2} + 2127630483)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 166644 T^{2} + 37340352)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 39816)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 137168 T^{2} + 2126316544)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 129600 T^{2} + 1905120000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 72252 T^{2} + 951695163)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 330768 T^{2} + 23475142656)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 886464 T^{2} + 185752092672)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 400)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 179172 T^{2} + 634583808)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 76 T - 127589)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 53244 T^{2} + 403123392)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 466722 T^{2} + 41896767969)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 1548 T + 82944)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 806 T - 66983)^{4} \) Copy content Toggle raw display
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