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Properties

Label 864.4.i.c
Level $864$
Weight $4$
Character orbit 864.i
Analytic conductor $50.978$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [864,4,Mod(289,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 864.i (of order \(3\), 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: \(50.9776502450\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 11x^{14} + 37x^{12} - 702x^{10} - 15606x^{8} - 56862x^{6} + 242757x^{4} + 5845851x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{2} + 1) q^{5} - \beta_{14} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_{2} + 1) q^{5} - \beta_{14} q^{7} - \beta_{3} q^{11} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \cdots + 10) q^{13}+ \cdots + ( - 10 \beta_{9} + 27 \beta_{7} + \cdots + 26 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{5} + 78 q^{13} + 188 q^{17} - 238 q^{25} + 314 q^{29} - 320 q^{37} - 368 q^{41} + 14 q^{49} - 112 q^{53} + 482 q^{61} - 846 q^{65} - 3204 q^{73} - 822 q^{77} + 3248 q^{85} + 3832 q^{89} + 3864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 11x^{14} + 37x^{12} - 702x^{10} - 15606x^{8} - 56862x^{6} + 242757x^{4} + 5845851x^{2} + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11 \nu^{14} + 21100 \nu^{12} + 145883 \nu^{10} - 4244103 \nu^{8} - 23014881 \nu^{6} + \cdots + 23445582597 ) / 433655856 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 215 \nu^{14} - 988 \nu^{12} - 38735 \nu^{10} + 228123 \nu^{8} + 2815101 \nu^{6} + \cdots - 2891570481 ) / 867311712 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1453 \nu^{15} - 275212 \nu^{13} + 1029973 \nu^{11} + 2823687 \nu^{9} + 91916721 \nu^{7} + \cdots + 31832784459 \nu ) / 7805805408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58 \nu^{14} + 557 \nu^{12} - 932 \nu^{10} - 67770 \nu^{8} - 752058 \nu^{6} + 1981422 \nu^{4} + \cdots + 280069407 ) / 18068994 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1693 \nu^{15} + 11252 \nu^{13} + 696709 \nu^{11} + 559575 \nu^{9} - 35175519 \nu^{7} + \cdots + 26603405019 \nu ) / 2601935136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 76 \nu^{14} + 1727 \nu^{12} - 509 \nu^{10} - 164727 \nu^{8} - 1234170 \nu^{6} - 3168963 \nu^{4} + \cdots + 632414790 ) / 9034497 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7307 \nu^{14} - 46388 \nu^{12} + 102851 \nu^{10} - 3232575 \nu^{8} - 58301721 \nu^{6} + \cdots + 33253857693 ) / 867311712 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3203 \nu^{15} + 42604 \nu^{13} - 515557 \nu^{11} - 3996567 \nu^{9} - 41231457 \nu^{7} + \cdots + 2017881477 \nu ) / 2601935136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8621 \nu^{14} + 4516 \nu^{12} - 64315 \nu^{10} - 5693193 \nu^{8} - 40767327 \nu^{6} + \cdots + 21203964459 ) / 867311712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9601 \nu^{14} + 151700 \nu^{12} - 76007 \nu^{10} - 7737741 \nu^{8} - 115573851 \nu^{6} + \cdots + 24084374679 ) / 867311712 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 257 \nu^{15} + 964 \nu^{13} + 21821 \nu^{11} - 419931 \nu^{9} - 2552013 \nu^{7} + \cdots + 3216812373 \nu ) / 162620946 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7541 \nu^{15} + 54020 \nu^{13} - 222317 \nu^{11} + 4509297 \nu^{9} + 55038231 \nu^{7} + \cdots - 43447958955 \nu ) / 3902902704 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7315 \nu^{15} - 58028 \nu^{13} + 166421 \nu^{11} + 10715463 \nu^{9} + 82063665 \nu^{7} + \cdots - 30069463221 \nu ) / 2601935136 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15491 \nu^{15} + 86404 \nu^{13} - 121165 \nu^{11} - 14113791 \nu^{9} - 112643001 \nu^{7} + \cdots + 48572113077 \nu ) / 3902902704 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4175 \nu^{15} + 808 \nu^{13} + 45287 \nu^{11} - 2999295 \nu^{9} - 24474663 \nu^{7} + \cdots + 15017991219 \nu ) / 487862838 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{11} + \beta_{8} ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{7} + 2\beta_{6} - 2\beta_{4} - 9\beta_{2} - 2\beta _1 - 17 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{15} + 12\beta_{14} + 2\beta_{13} - 6\beta_{12} - \beta_{11} - \beta_{8} + 3\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{10} + \beta_{9} - 10\beta_{7} - 2\beta_{6} + 47\beta_{4} - 117\beta_{2} - 7\beta _1 - 19 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{15} - 12\beta_{14} - 8\beta_{13} - 48\beta_{12} - 2\beta_{11} - 68\beta_{8} - 33\beta_{5} - 57\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -46\beta_{10} + 152\beta_{9} - 98\beta_{7} + 65\beta_{6} - 272\beta_{4} - 612\beta_{2} - 56\beta _1 + 1972 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 129 \beta_{15} + 228 \beta_{14} + 437 \beta_{13} + 534 \beta_{12} + 356 \beta_{11} - 106 \beta_{8} + \cdots + 57 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 397\beta_{10} - 737\beta_{9} + 152\beta_{7} - 236\beta_{6} + 686\beta_{4} - 5868\beta_{2} - 826\beta _1 + 32147 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1479 \beta_{15} + 4416 \beta_{14} + 6586 \beta_{13} - 3450 \beta_{12} + 370 \beta_{11} + \cdots - 543 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7120 \beta_{10} - 2215 \beta_{9} - 3311 \beta_{7} + 6671 \beta_{6} + 655 \beta_{4} - 285570 \beta_{2} + \cdots - 449342 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16692 \beta_{15} + 38730 \beta_{14} - 30214 \beta_{13} - 21660 \beta_{12} - 25504 \beta_{11} + \cdots - 6828 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 39898 \beta_{10} + 51472 \beta_{9} - 130636 \beta_{7} - 61166 \beta_{6} + 166592 \beta_{4} + \cdots + 1814345 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 258504 \beta_{15} - 288588 \beta_{14} + 141565 \beta_{13} + 359106 \beta_{12} + 185935 \beta_{11} + \cdots - 406596 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 236123 \beta_{10} + 1138679 \beta_{9} - 143117 \beta_{7} + 144038 \beta_{6} - 1930538 \beta_{4} + \cdots + 5758327 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 546690 \beta_{15} + 4804230 \beta_{14} + 4217636 \beta_{13} + 1490124 \beta_{12} + \cdots + 1264143 \beta_{3} ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.86932 + 2.34641i
1.86932 − 2.34641i
−1.68504 − 2.48207i
1.68504 + 2.48207i
2.98521 − 0.297481i
−2.98521 + 0.297481i
−0.0690752 + 2.99920i
0.0690752 − 2.99920i
−1.86932 − 2.34641i
1.86932 + 2.34641i
−1.68504 + 2.48207i
1.68504 − 2.48207i
2.98521 + 0.297481i
−2.98521 − 0.297481i
−0.0690752 − 2.99920i
0.0690752 + 2.99920i
0 0 0 −7.51347 − 13.0137i 0 −4.38125 + 7.58855i 0 0 0
289.2 0 0 0 −7.51347 − 13.0137i 0 4.38125 − 7.58855i 0 0 0
289.3 0 0 0 −2.25676 − 3.90883i 0 −10.1548 + 17.5886i 0 0 0
289.4 0 0 0 −2.25676 − 3.90883i 0 10.1548 − 17.5886i 0 0 0
289.5 0 0 0 3.14134 + 5.44096i 0 −6.02458 + 10.4349i 0 0 0
289.6 0 0 0 3.14134 + 5.44096i 0 6.02458 − 10.4349i 0 0 0
289.7 0 0 0 9.12890 + 15.8117i 0 −13.5144 + 23.4077i 0 0 0
289.8 0 0 0 9.12890 + 15.8117i 0 13.5144 − 23.4077i 0 0 0
577.1 0 0 0 −7.51347 + 13.0137i 0 −4.38125 − 7.58855i 0 0 0
577.2 0 0 0 −7.51347 + 13.0137i 0 4.38125 + 7.58855i 0 0 0
577.3 0 0 0 −2.25676 + 3.90883i 0 −10.1548 − 17.5886i 0 0 0
577.4 0 0 0 −2.25676 + 3.90883i 0 10.1548 + 17.5886i 0 0 0
577.5 0 0 0 3.14134 − 5.44096i 0 −6.02458 − 10.4349i 0 0 0
577.6 0 0 0 3.14134 − 5.44096i 0 6.02458 + 10.4349i 0 0 0
577.7 0 0 0 9.12890 − 15.8117i 0 −13.5144 − 23.4077i 0 0 0
577.8 0 0 0 9.12890 − 15.8117i 0 13.5144 + 23.4077i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
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 864.4.i.c 16
3.b odd 2 1 288.4.i.c ✓ 16
4.b odd 2 1 inner 864.4.i.c 16
9.c even 3 1 inner 864.4.i.c 16
9.d odd 6 1 288.4.i.c ✓ 16
12.b even 2 1 288.4.i.c ✓ 16
36.f odd 6 1 inner 864.4.i.c 16
36.h even 6 1 288.4.i.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.4.i.c ✓ 16 3.b odd 2 1
288.4.i.c ✓ 16 9.d odd 6 1
288.4.i.c ✓ 16 12.b even 2 1
288.4.i.c ✓ 16 36.h even 6 1
864.4.i.c 16 1.a even 1 1 trivial
864.4.i.c 16 4.b odd 2 1 inner
864.4.i.c 16 9.c even 3 1 inner
864.4.i.c 16 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 \(S_{4}^{\mathrm{new}}(864, [\chi])\):

\( T_{5}^{8} - 5T_{5}^{7} + 322T_{5}^{6} + 331T_{5}^{5} + 83314T_{5}^{4} - 93569T_{5}^{3} + 2643589T_{5}^{2} + 4489060T_{5} + 60528400 \) Copy content Toggle raw display
\( T_{7}^{16} + 1365 T_{7}^{14} + 1297026 T_{7}^{12} + 613605429 T_{7}^{10} + 208529830674 T_{7}^{8} + \cdots + 11\!\cdots\!24 \) 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
$5$ \( (T^{8} - 5 T^{7} + \cdots + 60528400)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( (T^{8} - 39 T^{7} + \cdots + 449304771204)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 47 T^{3} + \cdots + 727936)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 12060748689408)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 28\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{4} + 80 T^{3} + \cdots + 2490348544)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 14\!\cdots\!81)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{4} + 28 T^{3} + \cdots + 18688520320)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 63\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 94\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 801 T^{3} + \cdots + 3441493440)^{4} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} - 958 T^{3} + \cdots - 138725768192)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 26\!\cdots\!25)^{2} \) Copy content Toggle raw display
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