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Properties

Label 288.4.i.c
Level $288$
Weight $4$
Character orbit 288.i
Analytic conductor $16.993$
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] = [288,4,Mod(97,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, 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("288.97");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.i (of order \(3\), degree \(2\), 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: \(16.9925500817\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{12} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} + ( - \beta_{12} - \beta_{5} + \beta_1 - 1) q^{5} + ( - \beta_{11} - \beta_{10}) q^{7} + ( - \beta_{13} - \beta_{7} + 5 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{3} + ( - \beta_{12} - \beta_{5} + \beta_1 - 1) q^{5} + ( - \beta_{11} - \beta_{10}) q^{7} + ( - \beta_{13} - \beta_{7} + 5 \beta_1 - 3) q^{9} + (\beta_{15} + \beta_{10} - \beta_{3}) q^{11} + (\beta_{12} + \beta_{7} + \beta_{6} + \cdots + 9) q^{13}+ \cdots + (24 \beta_{14} - 24 \beta_{11} + \cdots - 15 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{5} - 6 q^{9} + 78 q^{13} - 188 q^{17} + 66 q^{21} - 238 q^{25} - 314 q^{29} - 336 q^{33} - 320 q^{37} + 368 q^{41} + 402 q^{45} + 14 q^{49} + 112 q^{53} - 1146 q^{57} + 482 q^{61} + 846 q^{65} - 642 q^{69} - 3204 q^{73} + 822 q^{77} + 954 q^{81} + 3248 q^{85} - 3832 q^{89} - 1602 q^{93} + 3864 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 215 \nu^{14} - 988 \nu^{12} - 38735 \nu^{10} + 228123 \nu^{8} + 2815101 \nu^{6} + \cdots - 2024258769 ) / 867311712 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 215 \nu^{14} - 988 \nu^{12} - 38735 \nu^{10} + 228123 \nu^{8} + 2815101 \nu^{6} + \cdots - 723291201 ) / 433655856 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 181 \nu^{15} - 70436 \nu^{13} + 257363 \nu^{11} + 1872369 \nu^{9} + 26853255 \nu^{7} + \cdots + 3548431557 \nu ) / 1951451352 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2177 \nu^{15} - 6532 \nu^{13} - 521 \nu^{11} + 4665789 \nu^{9} + 15496299 \nu^{7} + \cdots - 17639058231 \nu ) / 7805805408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58 \nu^{14} + 557 \nu^{12} - 932 \nu^{10} - 67770 \nu^{8} - 752058 \nu^{6} + 1981422 \nu^{4} + \cdots + 280069407 ) / 18068994 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 191 \nu^{14} - 6080 \nu^{12} + 2369 \nu^{10} - 29997 \nu^{8} + 603747 \nu^{6} + 24518457 \nu^{4} + \cdots + 994326111 ) / 54206982 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2233 \nu^{14} + 25940 \nu^{12} + 51841 \nu^{10} - 1490373 \nu^{8} - 35388387 \nu^{6} + \cdots + 10118105199 ) / 433655856 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 215 \nu^{15} + 988 \nu^{13} + 38735 \nu^{11} - 228123 \nu^{9} - 2815101 \nu^{7} + \cdots + 1156947057 \nu ) / 289103904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 215 \nu^{15} - 988 \nu^{13} - 38735 \nu^{11} + 228123 \nu^{9} + 2815101 \nu^{7} + \cdots - 3758882193 \nu ) / 289103904 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7073 \nu^{15} + 60388 \nu^{13} + 181673 \nu^{11} - 8102781 \nu^{9} - 91903275 \nu^{7} + \cdots + 46260344727 \nu ) / 7805805408 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8009 \nu^{15} - 168428 \nu^{13} + 262961 \nu^{11} - 915813 \nu^{9} - 18173187 \nu^{7} + \cdots + 40635573183 \nu ) / 7805805408 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8621 \nu^{14} + 4516 \nu^{12} - 64315 \nu^{10} - 5693193 \nu^{8} - 40767327 \nu^{6} + \cdots + 21203964459 ) / 867311712 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 599 \nu^{14} + 3916 \nu^{12} + 3695 \nu^{10} - 930555 \nu^{8} - 6358365 \nu^{6} + \cdots + 4288906017 ) / 48183984 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28805 \nu^{15} - 166276 \nu^{13} + 242851 \nu^{11} + 23561793 \nu^{9} + \cdots - 79505167923 \nu ) / 7805805408 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28775 \nu^{15} - 26996 \nu^{13} + 271199 \nu^{11} - 17610075 \nu^{9} - 142097517 \nu^{7} + \cdots + 88194228273 \nu ) / 3902902704 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 2\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{15} - 12\beta_{14} + 6\beta_{11} - 9\beta_{10} - 2\beta_{9} + \beta_{8} + 3\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{13} - 6\beta_{7} + 18\beta_{5} - \beta_{2} - 40\beta _1 + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{15} + 12\beta_{14} + 48\beta_{11} - 126\beta_{10} - 25\beta_{9} + 35\beta_{8} - 54\beta_{4} - 57\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{13} + 54\beta_{12} - 48\beta_{7} - 39\beta_{6} - 72\beta_{5} + 25\beta_{2} - 218\beta _1 + 897 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 129 \beta_{15} - 228 \beta_{14} - 534 \beta_{11} - 2034 \beta_{10} - 605 \beta_{9} - 188 \beta_{8} + \cdots + 57 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -354\beta_{13} + 162\beta_{12} + 183\beta_{7} - 3\beta_{6} + 504\beta_{5} + 329\beta_{2} - 1630\beta _1 + 12267 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1479 \beta_{15} - 4416 \beta_{14} + 3450 \beta_{11} - 5040 \beta_{10} - 4651 \beta_{9} + \cdots - 543 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1482 \beta_{13} + 594 \beta_{12} - 3477 \beta_{7} - 954 \beta_{6} + 3924 \beta_{5} + 3556 \beta_{2} + \cdots - 54969 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16692 \beta_{15} - 38730 \beta_{14} + 21660 \beta_{11} - 144135 \beta_{10} - 27704 \beta_{9} + \cdots - 6828 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 15312 \beta_{13} + 25380 \beta_{12} - 30246 \beta_{7} - 36456 \beta_{6} + 50454 \beta_{5} + \cdots + 803397 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 258504 \beta_{15} + 288588 \beta_{14} - 359106 \beta_{11} - 893988 \beta_{10} - 386143 \beta_{9} + \cdots - 406596 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 149010 \beta_{13} + 655290 \beta_{12} + 31002 \beta_{7} - 174426 \beta_{6} - 446490 \beta_{5} + \cdots + 3530871 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 546690 \beta_{15} - 4804230 \beta_{14} - 1490124 \beta_{11} - 10744713 \beta_{10} + \cdots + 1264143 \beta_{3} ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
1.86932 + 2.34641i
2.98521 + 0.297481i
0.0690752 + 2.99920i
1.68504 − 2.48207i
−1.68504 + 2.48207i
−0.0690752 − 2.99920i
−2.98521 − 0.297481i
−1.86932 − 2.34641i
1.86932 − 2.34641i
2.98521 − 0.297481i
0.0690752 − 2.99920i
1.68504 + 2.48207i
−1.68504 − 2.48207i
−0.0690752 + 2.99920i
−2.98521 + 0.297481i
−1.86932 + 2.34641i
0 −4.83603 + 1.90073i 0 7.51347 + 13.0137i 0 4.38125 − 7.58855i 0 19.7745 − 18.3840i 0
97.2 0 −4.73545 − 2.13905i 0 −3.14134 − 5.44096i 0 −6.02458 + 10.4349i 0 17.8489 + 20.2587i 0
97.3 0 −2.70100 + 4.43899i 0 −9.12890 − 15.8117i 0 13.5144 − 23.4077i 0 −12.4092 − 23.9794i 0
97.4 0 −0.378024 − 5.18238i 0 2.25676 + 3.90883i 0 10.1548 − 17.5886i 0 −26.7142 + 3.91813i 0
97.5 0 0.378024 + 5.18238i 0 2.25676 + 3.90883i 0 −10.1548 + 17.5886i 0 −26.7142 + 3.91813i 0
97.6 0 2.70100 − 4.43899i 0 −9.12890 − 15.8117i 0 −13.5144 + 23.4077i 0 −12.4092 − 23.9794i 0
97.7 0 4.73545 + 2.13905i 0 −3.14134 − 5.44096i 0 6.02458 − 10.4349i 0 17.8489 + 20.2587i 0
97.8 0 4.83603 − 1.90073i 0 7.51347 + 13.0137i 0 −4.38125 + 7.58855i 0 19.7745 − 18.3840i 0
193.1 0 −4.83603 − 1.90073i 0 7.51347 − 13.0137i 0 4.38125 + 7.58855i 0 19.7745 + 18.3840i 0
193.2 0 −4.73545 + 2.13905i 0 −3.14134 + 5.44096i 0 −6.02458 − 10.4349i 0 17.8489 − 20.2587i 0
193.3 0 −2.70100 − 4.43899i 0 −9.12890 + 15.8117i 0 13.5144 + 23.4077i 0 −12.4092 + 23.9794i 0
193.4 0 −0.378024 + 5.18238i 0 2.25676 − 3.90883i 0 10.1548 + 17.5886i 0 −26.7142 − 3.91813i 0
193.5 0 0.378024 − 5.18238i 0 2.25676 − 3.90883i 0 −10.1548 − 17.5886i 0 −26.7142 − 3.91813i 0
193.6 0 2.70100 + 4.43899i 0 −9.12890 + 15.8117i 0 −13.5144 − 23.4077i 0 −12.4092 + 23.9794i 0
193.7 0 4.73545 − 2.13905i 0 −3.14134 + 5.44096i 0 6.02458 + 10.4349i 0 17.8489 − 20.2587i 0
193.8 0 4.83603 + 1.90073i 0 7.51347 − 13.0137i 0 −4.38125 − 7.58855i 0 19.7745 + 18.3840i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
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 288.4.i.c ✓ 16
3.b odd 2 1 864.4.i.c 16
4.b odd 2 1 inner 288.4.i.c ✓ 16
9.c even 3 1 inner 288.4.i.c ✓ 16
9.d odd 6 1 864.4.i.c 16
12.b even 2 1 864.4.i.c 16
36.f odd 6 1 inner 288.4.i.c ✓ 16
36.h even 6 1 864.4.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.4.i.c ✓ 16 1.a even 1 1 trivial
288.4.i.c ✓ 16 4.b odd 2 1 inner
288.4.i.c ✓ 16 9.c even 3 1 inner
288.4.i.c ✓ 16 36.f odd 6 1 inner
864.4.i.c 16 3.b odd 2 1
864.4.i.c 16 9.d odd 6 1
864.4.i.c 16 12.b even 2 1
864.4.i.c 16 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(288, [\chi])\):

\( T_{5}^{8} + 5T_{5}^{7} + 322T_{5}^{6} - 331T_{5}^{5} + 83314T_{5}^{4} + 93569T_{5}^{3} + 2643589T_{5}^{2} - 4489060T_{5} + 60528400 \) 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
$3$ \( T^{16} + \cdots + 282429536481 \) 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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