LMFDB - Newform orbit 464.4.a.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$464 = 2^{4} \cdot 29$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	464.a (trivial)

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:	yes
Analytic conductor:	$27.3768862427$
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:	$x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9$ x^6 - 40x^4 - 88x^3 - 8x^2 + 48x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{7}$
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

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}

1.1

0.488896
−1.90366
−4.70501
−1.32108
0.215317
7.22553

−7.88819

−14.0062

34.0826

35.2236

1.2

−2.88453

−3.33508

−0.0162797

−18.6795

1.3

−0.586661

15.4201

28.5777

−26.6558

1.4

0.116789

−14.4485

−30.0998

−26.9864

1.5

6.36242

8.72860

8.76166

13.4804

1.6

9.88017

2.64112

−3.30580

70.6177

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$29$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	464.4.a.n		6
4.b	odd	2	1		232.4.a.e	✓	6
8.b	even	2	1		1856.4.a.bc		6
8.d	odd	2	1		1856.4.a.bd		6
12.b	even	2	1		2088.4.a.l		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
232.4.a.e	✓	6	4.b	odd	2	1
464.4.a.n		6	1.a	even	1	1	trivial
1856.4.a.bc		6	8.b	even	2	1
1856.4.a.bd		6	8.d	odd	2	1
2088.4.a.l		6	12.b	even	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{6} - 5T_{3}^{5} - 92T_{3}^{4} + 266T_{3}^{3} + 1581T_{3}^{2} + 651T_{3} - 98$ T3^6 - 5*T3^5 - 92*T3^4 + 266*T3^3 + 1581*T3^2 + 651*T3 - 98 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(464))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 5 T^{5} + \cdots - 98$ T^6 - 5T^5 - 92T^4 + 266T^3 + 1581T^2 + 651*T - 98
$5$	$T^{6} + 5 T^{5} + \cdots - 239922$ T^6 + 5T^5 - 356T^4 - 1338T^3 + 29589T^2 + 28213*T - 239922
$7$	$T^{6} - 38 T^{5} + \cdots - 13824$ T^6 - 38T^5 - 764T^4 + 35224T^3 - 132960T^2 - 851328*T - 13824
$11$	$T^{6} + \cdots - 2827812238$ T^6 - 19T^5 - 5344T^4 + 61406T^3 + 7673789T^2 - 51413115*T - 2827812238
$13$	$T^{6} + \cdots - 7992521554$ T^6 - 13T^5 - 8340T^4 + 4154T^3 + 16168933T^2 + 2057683*T - 7992521554
$17$	$T^{6} + \cdots + 200391946752$ T^6 + 218T^5 - 4612T^4 - 3944840T^3 - 255383488T^2 - 488718592*T + 200391946752
$19$	$T^{6} + \cdots + 1256352636928$ T^6 - 290T^5 + 3712T^4 + 4805440T^3 - 261900032T^2 - 19018166784*T + 1256352636928
$23$	$T^{6} + \cdots + 4069096656896$ T^6 - 196T^5 - 42044T^4 + 10694016T^3 - 110795648T^2 - 85657167616*T + 4069096656896
$29$	$(T + 29)^{6}$ (T + 29)^6
$31$	$T^{6} + \cdots + 766872337410$ T^6 - 675T^5 + 142324T^4 - 7502686T^3 - 527106679T^2 + 23119116801*T + 766872337410
$37$	$T^{6} + \cdots - 38445998866432$ T^6 + 238T^5 - 103408T^4 - 14936320T^3 + 3556792320T^2 + 213735661568*T - 38445998866432
$41$	$T^{6} + \cdots + 64375103462400$ T^6 + 464T^5 - 96328T^4 - 58456480T^3 - 2270737392T^2 + 915304369536*T + 64375103462400
$43$	$T^{6} + \cdots - 905147314238$ T^6 - 579T^5 - 18928T^4 + 65222558T^3 - 12229664643T^2 + 565616130293*T - 905147314238
$47$	$T^{6} + \cdots - 154765125165962$ T^6 - 975T^5 + 297680T^4 - 4193966T^3 - 14779964871T^2 + 2757073381245*T - 154765125165962
$53$	$T^{6} + \cdots - 36699799804014$ T^6 - 515T^5 - 82464T^4 + 39108150T^3 + 5467943949T^2 - 226282260659*T - 36699799804014
$59$	$T^{6} + \cdots - 110093808390144$ T^6 - 108T^5 - 185404T^4 + 4854336T^3 + 8352643456T^2 - 51966529792*T - 110093808390144
$61$	$T^{6} + \cdots + 40\!\cdots\!72$ T^6 - 1158T^5 - 159652T^4 + 407518776T^3 - 26869577024T^2 - 28847810688512*T + 4093378478479872
$67$	$T^{6} + \cdots + 681378807496704$ T^6 - 80T^5 - 889504T^4 + 6793216T^3 + 138794423552T^2 - 24567547572224*T + 681378807496704
$71$	$T^{6} + \cdots - 16\!\cdots\!00$ T^6 + 438T^5 - 1753560T^4 - 622137648T^3 + 975903515408T^2 + 220943543379808*T - 166851083251168000
$73$	$T^{6} + \cdots + 21\!\cdots\!72$ T^6 - 262T^5 - 1253464T^4 - 245119136T^3 + 344293594752T^2 + 167554638915584*T + 21555325400809472
$79$	$T^{6} + \cdots - 10\!\cdots\!74$ T^6 - 237T^5 - 518592T^4 + 99483894T^3 + 60252645305T^2 - 3694182599273*T - 1096022985222874
$83$	$T^{6} + \cdots + 47\!\cdots\!44$ T^6 - 1288T^5 - 1112252T^4 + 1523011216T^3 - 128095368768T^2 - 223308345498624*T + 47348759682514944
$89$	$T^{6} + \cdots + 10\!\cdots\!00$ T^6 + 252T^5 - 1590016T^4 - 109713456T^3 + 646257064240T^2 - 74963689381888*T + 1024537299897600
$97$	$T^{6} + \cdots + 83\!\cdots\!28$ T^6 - 380T^5 - 3250128T^4 + 705512144T^3 + 1840854626480T^2 - 706922308534400*T + 8387409981948928
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Belyi maps

$f(q)$	$=$	$q + ( - \beta_{3} + 1) q^{3} + ( - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1 + 6) q^{7} + (2 \beta_{4} - 3 \beta_{3} - \beta_{2} + \cdots + 9) q^{9} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{2} + \cdots + 1) q^{11}+ \cdots + (15 \beta_{5} + 71 \beta_{4} + \cdots - 342) q^{99}+O(q^{100})$ q + (-b3 + 1) * q^3 + (-b4 - b3 - 1) * q^5 + (-b4 + b3 + b1 + 6) * q^7 + (2b4 - 3b3 - b2 + 2b1 + 9) q^9 + (b5 - 3b4 - 3b2 - 2b1 + 1) q^11 + (3b5 - 6b3 - b2 - 2b1 + 3) q^13 + (-b5 + 2b4 + b3 + b2 + 3b1 + 33) * q^15 + (-6b5 - 5b4 - 4b3 + 3b2 + 2b1 - 37) q^17 + (b5 - b4 - b3 + 11b2 + 52) q^19 + (-2b5 - 5b4 - 10b3 + b2 - 4b1 - 45) * q^21 + (-8b5 + 5b4 - 9b3 - 2b2 + b1 + 34) * q^23 + (5b5 - b4 + 6b3 + 6b2 - 2b1 - 1) * q^25 + (3b5 + 3b4 - 12b3 - 15b2 - 2b1 + 71) q^27 - 29 * q^29 + (4b5 - b4 - 10b3 + 10b2 - 5b1 + 117) * q^31 + (9b5 + 15b4 + 8b3 - 8b2 + 6b1 + 52) q^33 + (10b5 - 15b4 - 15b3 + 8b2 - 21b1 + 76) q^35 + (-2b5 + 14b4 + 2b3 - 10b2 - 6b1 - 40) q^37 + (8b5 + 21b4 - 6b3 - 24b2 + 9b1 + 219) q^39 + (-14b5 - 7b4 + 28b3 - 11b2 + 8b1 - 89) q^41 + (-13b5 - 7b4 - 12b3 - 9b2 + 12b1 + 93) q^43 + (-5b5 + 13b4 - 27b3 + b2 - 10b1 - 18) * q^45 + (-11b5 + 12b4 - b3 - 3b2 + 7b1 + 165) * q^47 + (24b5 + 10b4 + 44b3 + 22b2 - 4b1 + 159) q^49 + (-22b5 - 7b4 + 41b3 + 50b2 + 25b1 + 104) q^51 + (15b5 - 8b4 - 8b3 - 11b2 - 2b1 + 83) q^53 + (30b5 - 9b4 + 36b3 + 2b2 - 7b1 + 263) q^55 + (-33b5 - 31b4 - 9b3 + 39b2 - 58) * q^57 + (-6b5 - 13b4 + 33b3 - 16b2 + b1 + 2) * q^59 + (-24b5 + 13b4 + 6b3 + b2 + 38b1 + 199) * q^61 + (-6b5 + 56b4 + 46b3 + 24b2 + 16b1 + 210) q^63 + (10b5 - 8b4 + 37b3 - 17b2 + 14b1 + 196) q^65 + (10b5 - 4b4 - 6b3 - 52b2 + 4b1 - 2) q^67 + (2b5 + 21b4 - 70b3 + 19b2 + 34b1 + 459) q^69 + (8b5 - 24b4 + 70b3 - 16b2 - 56b1 - 106) q^71 + (-14b5 - 52b4 + 58b3 - 40b2 - 18b1 - 2) q^73 + (-12b5 - 24b4 + 50b3 + 6b2 - 20b1 - 314) q^75 + (40b5 - 77b4 - 34b3 - b2 - 70b1 + 7) * q^77 + (9b5 - 20b4 + 9b3 + 29b2 - 21b1 + 39) q^79 + (53b5 + 21b4 - 67b3 - 65b2 - 36b1 + 425) q^81 + (12b5 + 55b4 - 29b3 + 40b2 + 41b1 + 260) q^83 + (-16b5 + 67b4 + 22b3 + 67b2 + 24b1 + 561) q^85 + (29b3 - 29) q^87 + (-12b5 - 25b4 + 30b3 + 11b2 - 44b1 - 61) q^89 + (8b5 - 91b4 - 53b3 - 20b2 - 41b1 - 442) q^91 + (-22b5 + 5b4 - 59b3 + 18b2 + 24b1 + 373) q^93 + (-33b5 - 119b4 - 67b3 + b2 + 22b1 - 60) * q^95 + (-16b5 + 45b4 + 124b3 - 73b2 - 22b1 + 27) q^97 + (15b5 + 71b4 - 155b3 - 39b2 - 22b1 - 342) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 5 q^{3} - 5 q^{5} + 38 q^{7} + 47 q^{9} + 19 q^{11} + 13 q^{13} + 191 q^{15} - 218 q^{17} + 290 q^{19} - 266 q^{21} + 196 q^{23} - 13 q^{25} + 437 q^{27} - 174 q^{29} + 675 q^{31} + 291 q^{33} + 466 q^{35}+ \cdots - 2264 q^{99}+O(q^{100})$ 6 * q + 5 * q^3 - 5 * q^5 + 38 * q^7 + 47 * q^9 + 19 * q^11 + 13 * q^13 + 191 * q^15 - 218 * q^17 + 290 * q^19 - 266 * q^21 + 196 * q^23 - 13 * q^25 + 437 * q^27 - 174 * q^29 + 675 * q^31 + 291 * q^33 + 466 * q^35 - 238 * q^37 + 1297 * q^39 - 464 * q^41 + 579 * q^43 - 148 * q^45 + 975 * q^47 + 914 * q^49 + 576 * q^51 + 515 * q^53 + 1605 * q^55 - 340 * q^57 + 108 * q^59 + 1158 * q^61 + 1136 * q^63 + 1239 * q^65 + 80 * q^67 + 2568 * q^69 - 438 * q^71 + 262 * q^73 - 1766 * q^75 + 194 * q^77 + 237 * q^79 + 2554 * q^81 + 1288 * q^83 + 3112 * q^85 - 145 * q^87 - 252 * q^89 - 2450 * q^91 + 2131 * q^93 - 180 * q^95 + 380 * q^97 - 2264 * q^99

$\beta_{1}$	$=$	$( -\nu^{5} - 9\nu^{4} + 43\nu^{3} + 447\nu^{2} + 671\nu - 113 ) / 14$ (-v^5 - 9v^4 + 43v^3 + 447v^2 + 671v - 113) / 14
$\beta_{2}$	$=$	$( -\nu^{5} + 12\nu^{4} + 22\nu^{3} - 365\nu^{2} - 400\nu + 321 ) / 21$ (-v^5 + 12v^4 + 22v^3 - 365v^2 - 400v + 321) / 21
$\beta_{3}$	$=$	$( -5\nu^{5} - 3\nu^{4} + 201\nu^{3} + 555\nu^{2} + 261\nu - 159 ) / 14$ (-5v^5 - 3v^4 + 201v^3 + 555v^2 + 261*v - 159) / 14
$\beta_{4}$	$=$	$( -11\nu^{5} + 6\nu^{4} + 431\nu^{3} + 731\nu^{2} - 32\nu - 123 ) / 21$ (-11v^5 + 6v^4 + 431v^3 + 731v^2 - 32*v - 123) / 21
$\beta_{5}$	$=$	$( -41\nu^{5} + 9\nu^{4} + 1595\nu^{3} + 3347\nu^{2} + 1093\nu - 1035 ) / 42$ (-41v^5 + 9v^4 + 1595v^3 + 3347v^2 + 1093*v - 1035) / 42

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