# 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 $2$

# Related objects

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: $$x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331$$ x^6 - 3*x^5 + 118*x^4 - 231*x^3 + 3700*x^2 - 3585*x + 32331 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - 18 \beta_1 + 18) q^{5} + (\beta_{4} + 44 \beta_1) q^{7}+O(q^{10})$$ q + (-b5 - 18*b1 + 18) * q^5 + (b4 + 44*b1) * q^7 $$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})$$ q + (-b5 - 18*b1 + 18) * q^5 + (b4 + 44*b1) * q^7 + (b4 - 5*b3 - 105*b1) * q^11 + (-11*b5 - 2*b4 - 2*b2 + 248*b1 - 248) * q^13 + (-9*b5 + 9*b3 - 4*b2 - 483) * q^17 + (5*b5 - 5*b3 - 6*b2 - 377) * q^19 + (-14*b5 + b4 + b2 + 1056*b1 - 1056) * q^23 + (-10*b4 - 5*b3 - 961*b1) * q^25 + (6*b4 - 59*b3 + 1716*b1) * q^29 + (44*b5 - 17*b4 - 17*b2 - 2870*b1 + 2870) * q^31 + (-136*b5 + 136*b3 - 49*b2 + 450) * q^35 + (40*b5 - 40*b3 + 10*b2 + 6656) * q^37 + (-80*b5 + 70*b4 + 70*b2 + 1683*b1 - 1683) * q^41 + (-39*b4 + 117*b3 + 10463*b1) * q^43 + (-49*b4 + 40*b3 + 4308*b1) * q^47 + (251*b5 - 4*b4 - 4*b2 + 17619*b1 - 17619) * q^49 + (-116*b5 + 116*b3 - 54*b2 + 16008) * q^53 + (-52*b5 + 52*b3 + b2 - 21042) * q^55 + (-255*b5 + 71*b4 + 71*b2 - 20985*b1 + 20985) * q^59 + (-82*b4 + 37*b3 - 25322*b1) * q^61 + (-208*b4 + 207*b3 - 36234*b1) * q^65 + (519*b5 + 15*b4 + 15*b2 - 10997*b1 + 10997) * q^67 + (354*b5 - 354*b3 + 130*b2 - 21612) * q^71 + (-901*b5 + 901*b3 - 68*b2 - 1411) * q^73 + (-429*b5 - 308*b4 - 308*b2 + 29580*b1 - 29580) * q^77 + (15*b4 - 724*b3 - 29734*b1) * q^79 + (-211*b4 - 476*b3 + 10878*b1) * q^83 + (232*b5 - 286*b4 - 286*b2 - 23796*b1 + 23796) * q^85 + (-294*b5 + 294*b3 + 98*b2 - 11022) * q^89 + (-1998*b5 + 1998*b3 + 3*b2 + 50306) * q^91 + (-240*b5 - 244*b4 - 244*b2 + 27648*b1 - 27648) * q^95 + (386*b4 - 16*b3 + 15415*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 54 q^{5} + 132 q^{7}+O(q^{10})$$ 6 * q + 54 * q^5 + 132 * q^7 $$6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47} - 52857 q^{49} + 96048 q^{53} - 126252 q^{55} + 62955 q^{59} - 75966 q^{61} - 108702 q^{65} + 32991 q^{67} - 129672 q^{71} - 8466 q^{73} - 88740 q^{77} - 89202 q^{79} + 32634 q^{83} + 71388 q^{85} - 66132 q^{89} + 301836 q^{91} - 82944 q^{95} + 46245 q^{97}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} + 176\nu^{3} - 269\nu^{2} + 16626\nu + 22035 ) / 60606$$ (-2*v^5 + 5*v^4 + 176*v^3 - 269*v^2 + 16626*v + 22035) / 60606 $$\beta_{2}$$ $$=$$ $$( -3\nu^{4} + 6\nu^{3} - 843\nu^{2} + 840\nu - 26145 ) / 74$$ (-3*v^4 + 6*v^3 - 843*v^2 + 840*v - 26145) / 74 $$\beta_{3}$$ $$=$$ $$( 331\nu^{5} + 128\nu^{4} + 29567\nu^{3} + 40288\nu^{2} + 435399\nu + 1104636 ) / 6734$$ (331*v^5 + 128*v^4 + 29567*v^3 + 40288*v^2 + 435399*v + 1104636) / 6734 $$\beta_{4}$$ $$=$$ $$( 333\nu^{5} - 696\nu^{4} + 31029\nu^{3} - 7764\nu^{2} + 648093\nu + 854100 ) / 6734$$ (333*v^5 - 696*v^4 + 31029*v^3 - 7764*v^2 + 648093*v + 854100) / 6734 $$\beta_{5}$$ $$=$$ $$( 331\nu^{5} - 1783\nu^{4} + 33389\nu^{3} - 133067\nu^{2} + 606843\nu - 1610349 ) / 6734$$ (331*v^5 - 1783*v^4 + 33389*v^3 - 133067*v^2 + 606843*v - 1610349) / 6734
 $$\nu$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 18\beta _1 + 18 ) / 54$$ (-b5 + 2*b4 - b3 + b2 + 18*b1 + 18) / 54 $$\nu^{2}$$ $$=$$ $$( \beta_{4} - \beta_{3} - 3\beta_{2} + 9\beta _1 - 1026 ) / 27$$ (b4 - b3 - 3*b2 + 9*b1 - 1026) / 27 $$\nu^{3}$$ $$=$$ $$( 19\beta_{5} - 35\beta_{4} + 18\beta_{3} - 21\beta_{2} + 2664\beta _1 - 2358 ) / 18$$ (19*b5 - 35*b4 + 18*b3 - 21*b2 + 2664*b1 - 2358) / 18 $$\nu^{4}$$ $$=$$ $$( -83\beta_{5} - 106\beta_{4} + 195\beta_{3} + 254\beta_{2} + 7983\beta _1 + 48447 ) / 27$$ (-83*b5 - 106*b4 + 195*b3 + 254*b2 + 7983*b1 + 48447) / 27 $$\nu^{5}$$ $$=$$ $$( -3712\beta_{5} + 6587\beta_{4} - 2317\beta_{3} + 4846\beta_{2} - 745938\beta _1 + 640296 ) / 54$$ (-3712*b5 + 6587*b4 - 2317*b3 + 4846*b2 - 745938*b1 + 640296) / 54

## 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
 0.5 + 5.23712i 0.5 − 8.40123i 0.5 + 4.03013i 0.5 − 5.23712i 0.5 + 8.40123i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{6}^{\mathrm{new}}(432, [\chi])$$.

## Hecke characteristic polynomials

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