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Properties

Label 144.6.i.b
Level $144$
Weight $6$
Character orbit 144.i
Analytic conductor $23.095$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,6,Mod(49,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.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: \(23.0952700531\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 18)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 18 \beta_1 - 20) q^{5} + (3 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 37 \beta_1 + 5) q^{7} + (6 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 36 \beta_1 - 50) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 18 \beta_1 - 20) q^{5} + (3 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 37 \beta_1 + 5) q^{7} + (6 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 36 \beta_1 - 50) q^{9} + ( - 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 12 \beta_{2} + 107 \beta_1 - 10) q^{11} + ( - 13 \beta_{5} + 6 \beta_{4} - 31 \beta_{3} - 8 \beta_{2} + 260 \beta_1 - 228) q^{13} + ( - 9 \beta_{5} - 9 \beta_{4} - 6 \beta_{3} + 21 \beta_{2} + 558 \beta_1 - 663) q^{15} + (\beta_{5} - 13 \beta_{4} - 2 \beta_{3} + 38 \beta_{2} + 13 \beta_1 + 458) q^{17} + (17 \beta_{5} + \beta_{4} - 34 \beta_{3} - 20 \beta_{2} - \beta_1 - 358) q^{19} + (36 \beta_{5} + 9 \beta_{4} + 59 \beta_{3} + 6 \beta_{2} + 1037 \beta_1 - 2415) q^{21} + (13 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 35 \beta_{2} - 1050 \beta_1 + 1027) q^{23} + ( - 30 \beta_{5} + 15 \beta_{4} - 75 \beta_{3} + 120 \beta_{2} - 886 \beta_1 - 45) q^{25} + ( - 27 \beta_{5} + 18 \beta_{4} - 30 \beta_{3} + 93 \beta_{2} + \cdots - 4056) q^{27}+ \cdots + (504 \beta_{5} - 495 \beta_{4} + 3648 \beta_{3} - 2112 \beta_{2} + \cdots - 13191) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 54 q^{5} + 132 q^{7} - 177 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{3} - 54 q^{5} + 132 q^{7} - 177 q^{9} + 315 q^{11} - 744 q^{13} - 2286 q^{15} + 2898 q^{17} - 2262 q^{19} - 11076 q^{21} + 3168 q^{23} - 2883 q^{25} - 18144 q^{27} - 5148 q^{29} + 8610 q^{31} + 17469 q^{33} - 2700 q^{35} + 39936 q^{37} + 49026 q^{39} + 5049 q^{41} + 31389 q^{43} + 2538 q^{45} - 12924 q^{47} - 52857 q^{49} - 36837 q^{51} - 96048 q^{53} - 126252 q^{55} - 17469 q^{57} - 62955 q^{59} - 75966 q^{61} - 49578 q^{63} + 108702 q^{65} + 32991 q^{67} - 29250 q^{69} + 129672 q^{71} - 8466 q^{73} + 105483 q^{75} + 88740 q^{77} - 89202 q^{79} + 123435 q^{81} - 32634 q^{83} + 71388 q^{85} + 151524 q^{87} + 66132 q^{89} + 301836 q^{91} + 57678 q^{93} + 82944 q^{95} + 46245 q^{97} - 282168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 176\nu^{3} - 269\nu^{2} + 16626\nu + 22035 ) / 60606 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 331\nu^{5} - 2875\nu^{4} + 35573\nu^{3} - 258101\nu^{2} + 670179\nu - 4096833 ) / 60606 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 331\nu^{5} - 418\nu^{4} + 30659\nu^{3} - 22229\nu^{2} + 527673\nu - 168909 ) / 30303 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 991\nu^{5} + 3665\nu^{4} + 82325\nu^{3} + 495697\nu^{2} + 769179\nu + 11037819 ) / 60606 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1655\nu^{5} - 9461\nu^{4} + 168037\nu^{3} - 636943\nu^{2} + 3065883\nu - 5840445 ) / 60606 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 4\beta_{3} - 7\beta_{2} + 2\beta _1 + 9 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - 3\beta_{2} - 112 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -44\beta_{5} + 109\beta_{4} - 221\beta_{3} + 353\beta_{2} + 2870\beta _1 - 2505 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -101\beta_{5} + 17\beta_{4} + 98\beta_{3} + 264\beta_{2} + 976\beta _1 + 5208 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2119\beta_{5} - 6779\beta_{4} + 16081\beta_{3} - 20746\beta_{2} - 261628\beta _1 + 221196 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 + 5.23712i
0.500000 − 8.40123i
0.500000 + 4.03013i
0.500000 − 5.23712i
0.500000 + 8.40123i
0.500000 − 4.03013i
0 −15.5145 + 1.51695i 0 33.0434 − 57.2329i 0 57.0952 + 98.8918i 0 238.398 − 47.0695i 0
49.2 0 2.43381 − 15.3973i 0 −20.8014 + 36.0292i 0 −101.661 − 176.082i 0 −231.153 − 74.9483i 0
49.3 0 8.58066 + 13.0143i 0 −39.2420 + 67.9691i 0 110.566 + 191.505i 0 −95.7446 + 223.343i 0
97.1 0 −15.5145 − 1.51695i 0 33.0434 + 57.2329i 0 57.0952 − 98.8918i 0 238.398 + 47.0695i 0
97.2 0 2.43381 + 15.3973i 0 −20.8014 − 36.0292i 0 −101.661 + 176.082i 0 −231.153 + 74.9483i 0
97.3 0 8.58066 − 13.0143i 0 −39.2420 − 67.9691i 0 110.566 − 191.505i 0 −95.7446 − 223.343i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.6.i.b 6
3.b odd 2 1 432.6.i.b 6
4.b odd 2 1 18.6.c.b ✓ 6
9.c even 3 1 inner 144.6.i.b 6
9.d odd 6 1 432.6.i.b 6
12.b even 2 1 54.6.c.b 6
36.f odd 6 1 18.6.c.b ✓ 6
36.f odd 6 1 162.6.a.j 3
36.h even 6 1 54.6.c.b 6
36.h even 6 1 162.6.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.b ✓ 6 4.b odd 2 1
18.6.c.b ✓ 6 36.f odd 6 1
54.6.c.b 6 12.b even 2 1
54.6.c.b 6 36.h even 6 1
144.6.i.b 6 1.a even 1 1 trivial
144.6.i.b 6 9.c even 3 1 inner
162.6.a.i 3 36.h even 6 1
162.6.a.j 3 36.f odd 6 1
432.6.i.b 6 3.b odd 2 1
432.6.i.b 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 54T_{5}^{5} + 7587T_{5}^{4} + 179334T_{5}^{3} + 33470577T_{5}^{2} + 1007927064T_{5} + 46562734656 \) acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{5} + 129 T^{4} + \cdots + 14348907 \) Copy content Toggle raw display
$5$ \( T^{6} + 54 T^{5} + \cdots + 46562734656 \) Copy content Toggle raw display
$7$ \( T^{6} - 132 T^{5} + \cdots + 26358756910084 \) Copy content Toggle raw display
$11$ \( T^{6} - 315 T^{5} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$13$ \( T^{6} + 744 T^{5} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{3} - 1449 T^{2} - 500040 T + 930192444)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 1131 T^{2} - 1499928 T + 352455920)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 3168 T^{5} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{6} + 5148 T^{5} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{6} - 8610 T^{5} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{3} - 19968 T^{2} + \cdots - 188019064016)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 5049 T^{5} + \cdots + 46\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{6} - 31389 T^{5} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{6} + 12924 T^{5} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} + 48024 T^{2} + \cdots + 1764512817552)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 62955 T^{5} + \cdots + 33\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{6} + 75966 T^{5} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} - 32991 T^{5} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{3} - 64836 T^{2} + \cdots + 139951336896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 4233 T^{2} + \cdots + 14322358753732)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 89202 T^{5} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{6} + 32634 T^{5} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{3} - 33066 T^{2} + \cdots + 9104584153608)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 46245 T^{5} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
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