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Properties

Label 1728.4.f.h
Level $1728$
Weight $4$
Character orbit 1728.f
Analytic conductor $101.955$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,4,Mod(863,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.863"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-384,0,304,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1248,0,-720] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 29x^{14} + 601x^{12} - 5608x^{10} + 37420x^{8} - 128832x^{6} + 318736x^{4} - 389376x^{2} + 331776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} + ( - \beta_{13} + 3 \beta_{5}) q^{7} + ( - \beta_{2} + 2 \beta_1) q^{11} + (\beta_{6} + \beta_{2} - 2 \beta_1) q^{13} + (3 \beta_{13} + \beta_{12} - 11 \beta_{5}) q^{17} + (\beta_{14} - 5 \beta_{9} + \cdots - 5 \beta_{3}) q^{19}+ \cdots + (11 \beta_{10} + 2 \beta_{8} + \cdots + 287) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 384 q^{23} + 304 q^{25} + 1248 q^{47} - 720 q^{49} + 5088 q^{71} - 128 q^{73} + 11712 q^{95} + 4592 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 29x^{14} + 601x^{12} - 5608x^{10} + 37420x^{8} - 128832x^{6} + 318736x^{4} - 389376x^{2} + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 806351179 \nu^{14} + 22302352511 \nu^{12} - 456609841939 \nu^{10} + \cdots + 178948978038144 ) / 85620594939264 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 544104999 \nu^{14} + 20632180035 \nu^{12} - 443664027575 \nu^{10} + \cdots + 159504613879680 ) / 9513399437696 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 89077 \nu^{15} - 1824353 \nu^{13} + 32701069 \nu^{11} - 77425960 \nu^{9} + \cdots + 55071991296 \nu ) / 9602578944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25577187959 \nu^{15} + 171785176133 \nu^{13} - 10123932679969 \nu^{11} + \cdots + 12\!\cdots\!20 \nu ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 240167 \nu^{15} + 6845323 \nu^{13} - 140148527 \nu^{11} + 1256253752 \nu^{9} + \cdots + 28645539840 \nu ) / 10353885696 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3137426613 \nu^{14} + 95147268057 \nu^{12} - 1980328191733 \nu^{10} + \cdots + 751199042272896 ) / 9513399437696 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 406209420 \nu^{14} - 10610867853 \nu^{12} + 211926224571 \nu^{10} - 1618656526659 \nu^{8} + \cdots + 59943850365648 ) / 1189174929712 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 69220260 \nu^{14} + 1819496907 \nu^{12} - 36113363313 \nu^{10} + 275827738377 \nu^{8} + \cdots + 3694452567120 ) / 169882132816 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 63159926977 \nu^{15} + 1731191184317 \nu^{13} - 35069627452345 \nu^{11} + \cdots + 31\!\cdots\!96 \nu ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 743179500 \nu^{14} + 19529958285 \nu^{12} - 387729131475 \nu^{10} + 2961409285275 \nu^{8} + \cdots - 50662256670576 ) / 1189174929712 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23998092127 \nu^{14} - 696561407195 \nu^{12} + 14388731131279 \nu^{10} + \cdots - 54\!\cdots\!12 ) / 28540198313088 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 230610021713 \nu^{15} - 6330796865485 \nu^{13} + 129614315560265 \nu^{11} + \cdots - 26\!\cdots\!00 \nu ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 400315956361 \nu^{15} - 11347260960965 \nu^{13} + 232319484290785 \nu^{11} + \cdots - 47\!\cdots\!00 \nu ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8646096297 \nu^{15} + 198390580133 \nu^{13} - 3747935587073 \nu^{11} + \cdots - 19\!\cdots\!88 \nu ) / 38053597750784 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2168619054487 \nu^{15} + 62444203045979 \nu^{13} + \cdots + 26\!\cdots\!20 \nu ) / 20\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 3\beta_{14} + 7\beta_{13} - 2\beta_{12} - 6\beta_{9} - 11\beta_{5} + 6\beta_{4} + 9\beta_{3} ) / 144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} - 2\beta_{8} - \beta_{7} + 3\beta_{6} - 6\beta_{2} - 174\beta _1 + 174 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{15} + 19\beta_{13} - 18\beta_{12} - 155\beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{11} - 71\beta_{10} + 118\beta_{8} + 15\beta_{7} + 213\beta_{6} - 354\beta_{2} - 6498\beta _1 - 6498 ) / 144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 251 \beta_{15} - 753 \beta_{14} + 653 \beta_{13} - 862 \beta_{12} + 2586 \beta_{9} + \cdots - 9183 \beta_{3} ) / 144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -469\beta_{10} + 690\beta_{8} - 35\beta_{7} - 33222 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4307 \beta_{15} - 12921 \beta_{14} - 9077 \beta_{13} + 13870 \beta_{12} + 41610 \beta_{9} + \cdots - 172743 \beta_{3} ) / 144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1539 \beta_{11} - 8597 \beta_{10} + 11922 \beta_{8} - 1539 \beta_{7} - 25791 \beta_{6} + \cdots - 547398 ) / 48 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -74227\beta_{15} - 141013\beta_{13} + 230702\beta_{12} + 3308981\beta_{5} ) / 72 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 100281 \beta_{11} + 456527 \beta_{10} - 616726 \beta_{8} + 100281 \beta_{7} - 1369581 \beta_{6} + \cdots + 27795762 ) / 144 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1279939 \beta_{15} + 3839817 \beta_{14} - 2319493 \beta_{13} + 3913166 \beta_{12} + \cdots + 53782743 \beta_{3} ) / 144 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 7959679\beta_{10} - 10632566\beta_{8} + 1879689\beta_{7} + 475581906 ) / 72 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 22070707 \beta_{15} + 66212121 \beta_{14} + 39182101 \beta_{13} - 66992942 \beta_{12} + \cdots + 932397447 \beta_{3} ) / 144 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 33504153 \beta_{11} + 137888495 \beta_{10} - 183315670 \beta_{8} + 33504153 \beta_{7} + \cdots + 8173607922 ) / 144 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 126853921\beta_{15} + 223233015\beta_{13} - 383866970\beta_{12} - 5754895063\beta_{5} ) / 24 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
1.48258 − 0.855971i
1.48258 + 0.855971i
1.08543 + 0.626671i
1.08543 − 0.626671i
−2.33286 + 1.34688i
−2.33286 − 1.34688i
−3.59605 + 2.07618i
−3.59605 − 2.07618i
3.59605 + 2.07618i
3.59605 − 2.07618i
2.33286 + 1.34688i
2.33286 − 1.34688i
−1.08543 + 0.626671i
−1.08543 − 0.626671i
−1.48258 − 0.855971i
−1.48258 + 0.855971i
0 0 0 −16.5042 0 − 0.800305i 0 0 0
863.2 0 0 0 −16.5042 0 0.800305i 0 0 0
863.3 0 0 0 −12.9089 0 − 31.5675i 0 0 0
863.4 0 0 0 −12.9089 0 31.5675i 0 0 0
863.5 0 0 0 −8.34090 0 − 21.0128i 0 0 0
863.6 0 0 0 −8.34090 0 21.0128i 0 0 0
863.7 0 0 0 −8.20969 0 − 10.6450i 0 0 0
863.8 0 0 0 −8.20969 0 10.6450i 0 0 0
863.9 0 0 0 8.20969 0 − 10.6450i 0 0 0
863.10 0 0 0 8.20969 0 10.6450i 0 0 0
863.11 0 0 0 8.34090 0 − 21.0128i 0 0 0
863.12 0 0 0 8.34090 0 21.0128i 0 0 0
863.13 0 0 0 12.9089 0 − 31.5675i 0 0 0
863.14 0 0 0 12.9089 0 31.5675i 0 0 0
863.15 0 0 0 16.5042 0 − 0.800305i 0 0 0
863.16 0 0 0 16.5042 0 0.800305i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.f.h ✓ 16
3.b odd 2 1 1728.4.f.j yes 16
4.b odd 2 1 1728.4.f.j yes 16
8.b even 2 1 inner 1728.4.f.h ✓ 16
8.d odd 2 1 1728.4.f.j yes 16
12.b even 2 1 inner 1728.4.f.h ✓ 16
24.f even 2 1 inner 1728.4.f.h ✓ 16
24.h odd 2 1 1728.4.f.j yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.f.h ✓ 16 1.a even 1 1 trivial
1728.4.f.h ✓ 16 8.b even 2 1 inner
1728.4.f.h ✓ 16 12.b even 2 1 inner
1728.4.f.h ✓ 16 24.f even 2 1 inner
1728.4.f.j yes 16 3.b odd 2 1
1728.4.f.j yes 16 4.b odd 2 1
1728.4.f.j yes 16 8.d odd 2 1
1728.4.f.j yes 16 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{8} - 576T_{5}^{6} + 110214T_{5}^{4} - 8275824T_{5}^{2} + 212838921 \) Copy content Toggle raw display
\( T_{7}^{8} + 1552T_{7}^{6} + 603942T_{7}^{4} + 50244736T_{7}^{2} + 31933801 \) Copy content Toggle raw display
\( T_{23}^{4} + 96T_{23}^{3} - 27684T_{23}^{2} - 881280T_{23} + 112928256 \) 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
$5$ \( (T^{8} - 576 T^{6} + \cdots + 212838921)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 1552 T^{6} + \cdots + 31933801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 6732 T^{6} + \cdots + 178860480561)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 9336 T^{6} + \cdots + 218754514944)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 6295362011136)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 135592423981056)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 96 T^{3} + \cdots + 112928256)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 12\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 12\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 57\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 312 T^{3} + \cdots - 6864694272)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 14\!\cdots\!41)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 38\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 1272 T^{3} + \cdots + 34769564928)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 32 T^{3} + \cdots - 3799632599)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 35\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 64\!\cdots\!89)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1148 T^{3} + \cdots - 388402318367)^{4} \) Copy content Toggle raw display
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