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Properties

Label 432.6.i.b
Level $432$
Weight $6$
Character orbit 432.i
Analytic conductor $69.286$
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] = [432,6,Mod(145,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, 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("432.145");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(69.2858101592\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 18 \beta_1 + 18) q^{5} + (\beta_{4} + 44 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 18 \beta_1 + 18) q^{5} + (\beta_{4} + 44 \beta_1) q^{7} + (\beta_{4} - 5 \beta_{3} - 105 \beta_1) q^{11} + ( - 11 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 248 \beta_1 - 248) q^{13} + ( - 9 \beta_{5} + 9 \beta_{3} - 4 \beta_{2} - 483) q^{17} + (5 \beta_{5} - 5 \beta_{3} - 6 \beta_{2} - 377) q^{19} + ( - 14 \beta_{5} + \beta_{4} + \beta_{2} + 1056 \beta_1 - 1056) q^{23} + ( - 10 \beta_{4} - 5 \beta_{3} - 961 \beta_1) q^{25} + (6 \beta_{4} - 59 \beta_{3} + 1716 \beta_1) q^{29} + (44 \beta_{5} - 17 \beta_{4} - 17 \beta_{2} - 2870 \beta_1 + 2870) q^{31} + ( - 136 \beta_{5} + 136 \beta_{3} - 49 \beta_{2} + 450) q^{35} + (40 \beta_{5} - 40 \beta_{3} + 10 \beta_{2} + 6656) q^{37} + ( - 80 \beta_{5} + 70 \beta_{4} + 70 \beta_{2} + 1683 \beta_1 - 1683) q^{41} + ( - 39 \beta_{4} + 117 \beta_{3} + 10463 \beta_1) q^{43} + ( - 49 \beta_{4} + 40 \beta_{3} + 4308 \beta_1) q^{47} + (251 \beta_{5} - 4 \beta_{4} - 4 \beta_{2} + 17619 \beta_1 - 17619) q^{49} + ( - 116 \beta_{5} + 116 \beta_{3} - 54 \beta_{2} + 16008) q^{53} + ( - 52 \beta_{5} + 52 \beta_{3} + \beta_{2} - 21042) q^{55} + ( - 255 \beta_{5} + 71 \beta_{4} + 71 \beta_{2} - 20985 \beta_1 + 20985) q^{59} + ( - 82 \beta_{4} + 37 \beta_{3} - 25322 \beta_1) q^{61} + ( - 208 \beta_{4} + 207 \beta_{3} - 36234 \beta_1) q^{65} + (519 \beta_{5} + 15 \beta_{4} + 15 \beta_{2} - 10997 \beta_1 + 10997) q^{67} + (354 \beta_{5} - 354 \beta_{3} + 130 \beta_{2} - 21612) q^{71} + ( - 901 \beta_{5} + 901 \beta_{3} - 68 \beta_{2} - 1411) q^{73} + ( - 429 \beta_{5} - 308 \beta_{4} - 308 \beta_{2} + 29580 \beta_1 - 29580) q^{77} + (15 \beta_{4} - 724 \beta_{3} - 29734 \beta_1) q^{79} + ( - 211 \beta_{4} - 476 \beta_{3} + 10878 \beta_1) q^{83} + (232 \beta_{5} - 286 \beta_{4} - 286 \beta_{2} - 23796 \beta_1 + 23796) q^{85} + ( - 294 \beta_{5} + 294 \beta_{3} + 98 \beta_{2} - 11022) q^{89} + ( - 1998 \beta_{5} + 1998 \beta_{3} + 3 \beta_{2} + 50306) q^{91} + ( - 240 \beta_{5} - 244 \beta_{4} - 244 \beta_{2} + 27648 \beta_1 - 27648) q^{95} + (386 \beta_{4} - 16 \beta_{3} + 15415 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{5} + 132 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47} - 52857 q^{49} + 96048 q^{53} - 126252 q^{55} + 62955 q^{59} - 75966 q^{61} - 108702 q^{65} + 32991 q^{67} - 129672 q^{71} - 8466 q^{73} - 88740 q^{77} - 89202 q^{79} + 32634 q^{83} + 71388 q^{85} - 66132 q^{89} + 301836 q^{91} - 82944 q^{95} + 46245 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) : 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
\(\beta_{2}\)\(=\) \( ( -3\nu^{4} + 6\nu^{3} - 843\nu^{2} + 840\nu - 26145 ) / 74 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 331\nu^{5} + 128\nu^{4} + 29567\nu^{3} + 40288\nu^{2} + 435399\nu + 1104636 ) / 6734 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 333\nu^{5} - 696\nu^{4} + 31029\nu^{3} - 7764\nu^{2} + 648093\nu + 854100 ) / 6734 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 331\nu^{5} - 1783\nu^{4} + 33389\nu^{3} - 133067\nu^{2} + 606843\nu - 1610349 ) / 6734 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 18\beta _1 + 18 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} - 3\beta_{2} + 9\beta _1 - 1026 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 19\beta_{5} - 35\beta_{4} + 18\beta_{3} - 21\beta_{2} + 2664\beta _1 - 2358 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -83\beta_{5} - 106\beta_{4} + 195\beta_{3} + 254\beta_{2} + 7983\beta _1 + 48447 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3712\beta_{5} + 6587\beta_{4} - 2317\beta_{3} + 4846\beta_{2} - 745938\beta _1 + 640296 ) / 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
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 0 0 −33.0434 + 57.2329i 0 57.0952 + 98.8918i 0 0 0
145.2 0 0 0 20.8014 − 36.0292i 0 −101.661 − 176.082i 0 0 0
145.3 0 0 0 39.2420 − 67.9691i 0 110.566 + 191.505i 0 0 0
289.1 0 0 0 −33.0434 − 57.2329i 0 57.0952 − 98.8918i 0 0 0
289.2 0 0 0 20.8014 + 36.0292i 0 −101.661 + 176.082i 0 0 0
289.3 0 0 0 39.2420 + 67.9691i 0 110.566 − 191.505i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
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 432.6.i.b 6
3.b odd 2 1 144.6.i.b 6
4.b odd 2 1 54.6.c.b 6
9.c even 3 1 inner 432.6.i.b 6
9.d odd 6 1 144.6.i.b 6
12.b even 2 1 18.6.c.b ✓ 6
36.f odd 6 1 54.6.c.b 6
36.f odd 6 1 162.6.a.i 3
36.h even 6 1 18.6.c.b ✓ 6
36.h even 6 1 162.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.b ✓ 6 12.b even 2 1
18.6.c.b ✓ 6 36.h even 6 1
54.6.c.b 6 4.b odd 2 1
54.6.c.b 6 36.f odd 6 1
144.6.i.b 6 3.b odd 2 1
144.6.i.b 6 9.d odd 6 1
162.6.a.i 3 36.f odd 6 1
162.6.a.j 3 36.h even 6 1
432.6.i.b 6 1.a even 1 1 trivial
432.6.i.b 6 9.c even 3 1 inner

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 \(S_{6}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) 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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