LMFDB - Newform orbit 528.4.b.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,4,Mod(65,528)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("528.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$528 = 2^{4} \cdot 3 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	528.b (of order $2$ , degree $1$ , 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:	$31.1530084830$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 21x^{4} + 114x^{3} - 567x^{2} - 729x + 19683$ x^6 - x^5 - 21x^4 + 114x^3 - 567x^2 - 729x + 19683
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}\cdot 3$
Twist minimal:	no (minimal twist has level 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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.

$f(q)$	$=$	$q - \beta_1 q^{3} - \beta_{5} q^{5} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{4} - \beta_{3} + 7) q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{11} + (4 \beta_{3} + 5 \beta_{2} + \beta_1) q^{13}+ \cdots + ( - 69 \beta_{5} + 8 \beta_{4} + \cdots - 280) q^{99}+O(q^{100})$ q - b1 * q^3 - b5 * q^5 + (-b5 - b3 - b2) * q^7 + (b4 - b3 + 7) * q^9 + (-b5 - b4 - b3 - b2 - 4b1 + 2) q^11 + (4b3 + 5b2 + b1) * q^13 + (-b4 - 5b3 - 3b2 - b1 + 2) * q^15 + (-b5 + 2b4 - b3 - 3b1 + 26) * q^17 + (-b5 + 7b3 - 7b1) * q^19 + (-6b5 - 4b3 - 5b2 - b1 + 6) q^21 + (-6b5 - 5b3 + 5b1) q^23 + (2b5 - 4b4 - b3 + 3b2 - 6b1 - 31) * q^25 + (-3b5 - b4 - b3 + 11b2 - 8b1 + 50) q^27 + (4b5 - 8b4 + 6b3 - 2b2 + 20b1 - 26) q^29 + (-b5 + 2b4 + 6b3 - 7b2 + 25b1 + 12) * q^31 + (-6b5 + 5b4 - 6b3 - 17b2 - 2b1 + 23) q^33 + (b5 - 2b4 + 7b3 - 6b2 + 27b1 - 104) * q^35 + (b5 - 2b4 - 2b3 + 3b2 - 9b1 - 6) * q^37 + (27b5 - 6b4 - 3b3 + 9b2 + 12) * q^39 + (4b5 - 8b4 - 2b3 + 6b2 - 12b1 - 62) q^41 + (-13b5 + 7b3 - 18b2 - 25b1) * q^43 + (-24b5 + 5b4 + 6b3 - 20b2 - b1 + 86) * q^45 + (-15b5 - 13b3 - 25b2 - 12b1) * q^47 + (3b5 - 6b4 + 13b3 - 10b2 + 49b1 + 91) q^49 + (-3b5 - 11b3 + 20b2 - 29b1 + 84) * q^51 + (-36b3 + 5b2 + 41b1) q^53 + (14b5 - 8b4 - 19b3 - 8b2 + 45b1 - 160) q^55 + (21b5 + 6b4 - 19b3 + 4b2 - b1 - 222) * q^57 + (14b5 - 13b3 - 15b2 - 2b1) * q^59 + (40b5 + 16b3 - 17b2 - 33b1) * q^61 + (-27b5 - 27b3 - 27b2 - 12b1) * q^63 + (3b5 - 6b4 - 35b3 + 38b2 - 143b1 - 240) q^65 + (7b5 - 14b4 - 12b3 + 19b2 - 55b1 + 8) q^67 + (-15b5 - 11b4 - 20b3 - 23b2 - 6b1 + 172) q^69 + (-4b5 + 15b3 + 32b2 + 17b1) * q^71 + (-8b5 + 12b3 - 52b2 - 64b1) * q^73 + (6b5 + 9b4 + 13b3 - 40b2 + 37b1 + 138) q^75 + (-8b5 - 8b4 + 14b3 - 19b2 + 67b1 - 314) q^77 + (55b5 + 59b3 + 3b2 - 56b1) * q^79 + (30b5 - 5b4 - 20b3 - 11b2 - 52b1 + 475) q^81 + (-8b5 + 16b4 - 28b3 + 20b2 - 104b1 - 212) q^83 + (-56b5 - 4b3 - 36b2 - 32b1) * q^85 + (12b5 - 6b4 + 50b3 - 80b2 + 38b1 - 540) q^87 + (21b5 + 52b3 + 7b2 - 45b1) * q^89 + (-9b5 + 18b4 - 27b3 + 18b2 - 99b1 + 648) q^91 + (-3b5 - 21b4 + 10b3 + 20b2 - 15b1 - 630) q^93 + (16b5 - 32b4 - 36b3 + 52b2 - 160b1 - 296) q^95 + (-8b5 + 16b4 - 21b3 + 13b2 - 76b1 + 222) q^97 + (-69b5 + 8b4 - 36b3 + 19b2 - 34b1 - 280) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - q^{3} + 43 q^{9} + 6 q^{11} + 7 q^{15} + 156 q^{17} + 36 q^{21} - 204 q^{25} + 278 q^{27} - 144 q^{29} + 114 q^{31} + 157 q^{33} - 588 q^{35} - 54 q^{37} + 48 q^{39} - 408 q^{41} + 551 q^{45} + 606 q^{49}+ \cdots - 1753 q^{99}+O(q^{100})$ 6 * q - q^3 + 43 * q^9 + 6 * q^11 + 7 * q^15 + 156 * q^17 + 36 * q^21 - 204 * q^25 + 278 * q^27 - 144 * q^29 + 114 * q^31 + 157 * q^33 - 588 * q^35 - 54 * q^37 + 48 * q^39 - 408 * q^41 + 551 * q^45 + 606 * q^49 + 444 * q^51 - 942 * q^55 - 1344 * q^57 - 12 * q^63 - 1668 * q^65 - 66 * q^67 + 1007 * q^69 + 936 * q^75 - 1800 * q^77 + 2779 * q^81 - 1392 * q^83 - 3084 * q^87 + 3780 * q^91 - 3847 * q^93 - 2088 * q^95 + 1254 * q^97 - 1753 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - x^{5} - 21x^{4} + 114x^{3} - 567x^{2} - 729x + 19683$ x^6 - x^5 - 21*x^4 + 114*x^3 - 567*x^2 - 729*x + 19683 :

$\beta_{1}$	$=$	$( -\nu^{5} + \nu^{4} + 21\nu^{3} - 114\nu^{2} + 567\nu + 729 ) / 729$ (-v^5 + v^4 + 21v^3 - 114v^2 + 567*v + 729) / 729
$\beta_{2}$	$=$	$( 7\nu^{5} - 34\nu^{4} + 123\nu^{3} + 393\nu^{2} - 7776\nu + 24057 ) / 1458$ (7v^5 - 34v^4 + 123v^3 + 393v^2 - 7776*v + 24057) / 1458
$\beta_{3}$	$=$	$( \nu^{5} - 4\nu^{4} + 9\nu^{3} + 69\nu^{2} - 504\nu + 2511 ) / 162$ (v^5 - 4v^4 + 9v^3 + 69v^2 - 504v + 2511) / 162
$\beta_{4}$	$=$	$( 7\nu^{5} - 88\nu^{4} + 177\nu^{3} + 1527\nu^{2} - 9558\nu + 44469 ) / 1458$ (7v^5 - 88v^4 + 177v^3 + 1527v^2 - 9558*v + 44469) / 1458
$\beta_{5}$	$=$	$( 11\nu^{5} - 92\nu^{4} + 93\nu^{3} + 525\nu^{2} - 10368\nu + 51759 ) / 1458$ (11v^5 - 92v^4 + 93v^3 + 525v^2 - 10368*v + 51759) / 1458

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_{3} - \beta_{2} + \beta_1 ) / 3$ (b3 - b2 + b1) / 3
$\nu^{2}$	$=$	$( -3\beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 5\beta _1 + 21 ) / 3$ (-3b5 + 3b4 + b3 - b2 - 5*b1 + 21) / 3
$\nu^{3}$	$=$	$( -12\beta_{5} + 3\beta_{4} + 16\beta_{3} + 11\beta_{2} + 55\beta _1 - 150 ) / 3$ (-12b5 + 3b4 + 16b3 + 11b2 + 55*b1 - 150) / 3
$\nu^{4}$	$=$	$( -75\beta_{5} - 15\beta_{4} + 4\beta_{3} + 104\beta_{2} - 83\beta _1 + 1425 ) / 3$ (-75b5 - 15b4 + 4b3 + 104b2 - 83*b1 + 1425) / 3
$\nu^{5}$	$=$	$( 15\beta_{5} - 294\beta_{4} + 793\beta_{3} - 118\beta_{2} + 22\beta _1 - 1932 ) / 3$ (15b5 - 294b4 + 793b3 - 118b2 + 22*b1 - 1932) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$ .

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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}

65.1

4.85873	−	1.84194i
4.85873	+	1.84194i
0.832570	−	5.12902i
0.832570	+	5.12902i
−5.19130	−	0.224541i
−5.19130	+	0.224541i

−4.85873

−

1.84194i

−

16.6397i

3.25743i

20.2145

17.8990i

65.2

−4.85873

1.84194i

16.6397i

−

3.25743i

20.2145

−

17.8990i

65.3

−0.832570

−

5.12902i

6.09929i

1.16981i

−25.6137

8.54053i

65.4

−0.832570

5.12902i

−

6.09929i

−

1.16981i

−25.6137

−

8.54053i

65.5

5.19130

−

0.224541i

12.7640i

26.7212i

26.8992

−

2.33132i

65.6

5.19130

0.224541i

−

12.7640i

−

26.7212i

26.8992

2.33132i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	528.4.b.d		6
3.b	odd	2	1		528.4.b.c		6
4.b	odd	2	1		66.4.b.a	✓	6
11.b	odd	2	1		528.4.b.c		6
12.b	even	2	1		66.4.b.b	yes	6
33.d	even	2	1	inner	528.4.b.d		6
44.c	even	2	1		66.4.b.b	yes	6
132.d	odd	2	1		66.4.b.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.4.b.a	✓	6	4.b	odd	2	1
66.4.b.a	✓	6	132.d	odd	2	1
66.4.b.b	yes	6	12.b	even	2	1
66.4.b.b	yes	6	44.c	even	2	1
528.4.b.c		6	3.b	odd	2	1
528.4.b.c		6	11.b	odd	2	1
528.4.b.d		6	1.a	even	1	1	trivial
528.4.b.d		6	33.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(528, [\chi])$ :

T_{5}^{6} + 477T_{5}^{4} + 61470T_{5}^{2} + 1678112

T_{17}^{3} - 78T_{17}^{2} - 1608T_{17} + 133056

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} + T^{5} + \cdots + 19683$ T^6 + T^5 - 21T^4 - 114T^3 - 567T^2 + 729T + 19683
$5$	$T^{6} + 477 T^{4} + \cdots + 1678112$ T^6 + 477T^4 + 61470T^2 + 1678112
$7$	$T^{6} + 726 T^{4} + \cdots + 10368$ T^6 + 726T^4 + 8568T^2 + 10368
$11$	$T^{6} + \cdots + 2357947691$ T^6 - 6T^5 - 495T^4 + 53724T^3 - 658845T^2 - 10629366*T + 2357947691
$13$	$T^{6} + \cdots + 38459847168$ T^6 + 12330T^4 + 42659136T^2 + 38459847168
$17$	$(T^{3} - 78 T^{2} + \cdots + 133056)^{2}$ (T^3 - 78T^2 - 1608T + 133056)^2
$19$	$T^{6} + \cdots + 184987680768$ T^6 + 24396T^4 + 148961088T^2 + 184987680768
$23$	$T^{6} + \cdots + 68188290632$ T^6 + 23937T^4 + 146924490T^2 + 68188290632
$29$	$(T^{3} + 72 T^{2} + \cdots - 2511360)^{2}$ (T^3 + 72T^2 - 71412T - 2511360)^2
$31$	$(T^{3} - 57 T^{2} + \cdots - 722848)^{2}$ (T^3 - 57T^2 - 37812T - 722848)^2
$37$	$(T^{3} + 27 T^{2} + \cdots + 188452)^{2}$ (T^3 + 27T^2 - 7452T + 188452)^2
$41$	$(T^{3} + 204 T^{2} + \cdots - 4045104)^{2}$ (T^3 + 204T^2 - 43620T - 4045104)^2
$43$	$T^{6} + \cdots + 338585981755392$ T^6 + 221604T^4 + 15339631968T^2 + 338585981755392
$47$	$T^{6} + \cdots + 27115367339552$ T^6 + 239970T^4 + 7933883616T^2 + 27115367339552
$53$	$T^{6} + \cdots + 52\!\cdots\!88$ T^6 + 653130T^4 + 123310706496T^2 + 5236187634610688
$59$	$T^{6} + \cdots + 2025875153408$ T^6 + 284991T^4 + 8655719028T^2 + 2025875153408
$61$	$T^{6} + \cdots + 38\!\cdots\!00$ T^6 + 1088538T^4 + 362872849248T^2 + 38252452331520000
$67$	$(T^{3} + 33 T^{2} + \cdots + 62825648)^{2}$ (T^3 + 33T^2 - 322752T + 62825648)^2
$71$	$T^{6} + \cdots + 22\!\cdots\!00$ T^6 + 425721T^4 + 56325658602T^2 + 2205235696752200
$73$	$T^{6} + \cdots + 965647255339008$ T^6 + 1192560T^4 + 131730287616T^2 + 965647255339008
$79$	$T^{6} + \cdots + 183824512639488$ T^6 + 2465286T^4 + 1499403679128T^2 + 183824512639488
$83$	$(T^{3} + 696 T^{2} + \cdots - 455107968)^{2}$ (T^3 + 696T^2 - 665232T - 455107968)^2
$89$	$T^{6} + \cdots + 54\!\cdots\!92$ T^6 + 1196355T^4 + 456720058656T^2 + 54320674642184192
$97$	$(T^{3} - 627 T^{2} + \cdots - 9054752)^{2}$ (T^3 - 627T^2 - 403950T - 9054752)^2
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