LMFDB - Newform orbit 912.6.a.s

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,6,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("912.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$912 = 2^{4} \cdot 3 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	912.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:	$146.270043669$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 4327x^{2} + 78705x - 258666$ x^4 - x^3 - 4327x^2 + 78705x - 258666
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 3$
Twist minimal:	no (minimal twist has level 228)
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

4.29948
55.1334
−73.3165
14.8836

9.00000

−31.7936

136.490

81.0000

1.2

9.00000

−21.2328

−102.241

81.0000

1.3

9.00000

74.6969

−227.637

81.0000

1.4

9.00000

95.3295

226.388

81.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$19$	$-1$

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	912.6.a.s		4
4.b	odd	2	1		228.6.a.c	✓	4
12.b	even	2	1		684.6.a.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.6.a.c	✓	4	4.b	odd	2	1
684.6.a.c		4	12.b	even	2	1
912.6.a.s		4	1.a	even	1	1	trivial

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{4} - 117T_{5}^{3} - 1220T_{5}^{2} + 262812T_{5} + 4807024$ T5^4 - 117*T5^3 - 1220*T5^2 + 262812*T5 + 4807024 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(912))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$(T - 9)^{4}$ (T - 9)^4
$5$	$T^{4} - 117 T^{3} + \cdots + 4807024$ T^4 - 117T^3 - 1220T^2 + 262812*T + 4807024
$7$	$T^{4} - 33 T^{3} + \cdots + 719152128$ T^4 - 33T^3 - 65532T^2 + 1747584*T + 719152128
$11$	$T^{4} + \cdots - 2947580480$ T^4 - 129T^3 - 297842T^2 - 58981176*T - 2947580480
$13$	$T^{4} + \cdots + 34302938368$ T^4 - 638T^3 - 440916T^2 + 77326360*T + 34302938368
$17$	$T^{4} + \cdots + 2211826294584$ T^4 - 285T^3 - 3036258T^2 + 476874324*T + 2211826294584
$19$	$(T - 361)^{4}$ (T - 361)^4
$23$	$T^{4} + \cdots + 6466156229824$ T^4 - 2340T^3 - 4661276T^2 + 6901100736*T + 6466156229824
$29$	$T^{4} + \cdots + 41401237109728$ T^4 - 438T^3 - 22071404T^2 + 2034320376*T + 41401237109728
$31$	$T^{4} + \cdots - 1260880553600$ T^4 + 160T^3 - 8454012T^2 - 8235269552*T - 1260880553600
$37$	$T^{4} + \cdots - 39\!\cdots\!04$ T^4 - 15520T^3 - 64089348T^2 + 1579978343936*T - 3965088310575104
$41$	$T^{4} + \cdots - 54\!\cdots\!32$ T^4 - 14664T^3 - 42766052T^2 + 1446370337568*T - 5419292258757632
$43$	$T^{4} + \cdots + 28\!\cdots\!56$ T^4 + 22519T^3 - 1533216T^2 - 1127134473872*T + 2863097248253056
$47$	$T^{4} + \cdots + 63\!\cdots\!28$ T^4 + 16395T^3 - 455644016T^2 - 3911731527708*T + 63661868207378128
$53$	$T^{4} + \cdots + 52\!\cdots\!16$ T^4 - 61266T^3 + 1395982756T^2 - 14025416158968*T + 52433138938107616
$59$	$T^{4} + \cdots - 16\!\cdots\!56$ T^4 + 67512T^3 + 1120782192T^2 + 3402483707520*T - 16894233059475456
$61$	$T^{4} + \cdots + 23\!\cdots\!12$ T^4 - 104285T^3 + 3720087678T^2 - 51950793978956*T + 238588030499195512
$67$	$T^{4} + \cdots - 30\!\cdots\!64$ T^4 + 40584T^3 - 548264784T^2 - 34354140916608*T - 308320429311166464
$71$	$T^{4} + \cdots - 20\!\cdots\!80$ T^4 + 98856T^3 - 66280784T^2 - 154898717803392*T - 2024404454247495680
$73$	$T^{4} + \cdots + 51\!\cdots\!84$ T^4 - 46323T^3 - 5504539422T^2 + 149932050454188*T + 5158733995962143784
$79$	$T^{4} + \cdots + 33\!\cdots\!80$ T^4 - 25126T^3 - 3627837960T^2 + 45159564393344*T + 3381863652163809280
$83$	$T^{4} + \cdots - 96\!\cdots\!36$ T^4 + 34026T^3 - 11392884792T^2 - 734492452986144*T - 9613831860676158336
$89$	$T^{4} + \cdots + 80\!\cdots\!36$ T^4 - 132606T^3 + 1021854156T^2 + 181941515249688*T + 801709249679531136
$97$	$T^{4} + \cdots + 39\!\cdots\!44$ T^4 + 31892T^3 - 7739385216T^2 - 79423323574480*T + 3955186033163720944
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + 9 q^{3} + ( - \beta_{2} + 29) q^{5} + ( - \beta_1 + 8) q^{7} + 81 q^{9} + ( - \beta_{3} + 4 \beta_{2} + \cdots + 33) q^{11} + ( - 9 \beta_{2} - \beta_1 + 157) q^{13} + ( - 9 \beta_{2} + 261) q^{15}+ \cdots + ( - 81 \beta_{3} + 324 \beta_{2} + \cdots + 2673) q^{99}+O(q^{100})$ q + 9 * q^3 + (-b2 + 29) * q^5 + (-b1 + 8) * q^7 + 81 * q^9 + (-b3 + 4b2 - b1 + 33) q^11 + (-9b2 - b1 + 157) q^13 + (-9b2 + 261) q^15 + (-6b3 + 3b1 + 72) * q^17 + 361 * q^19 + (-9b1 + 72) q^21 + (-9b3 + b2 + 3b1 + 586) * q^23 + (-60b2 - 5b1 + 891) * q^25 + 729 * q^27 + (9b3 + 52b2 - 6b1 + 121) q^29 + (-6b3 - 27b2 + 7b1 - 45) q^31 + (-9b3 + 36b2 - 9b1 + 297) q^33 + (40b3 - 42b2 - 71b1 + 574) q^35 + (30b3 + 39b2 - 43b1 + 3879) q^37 + (-81b2 - 9b1 + 1413) * q^39 + (-11b3 + 82b2 + 34b1 + 3695) q^41 + (42b3 + 30b2 - 39b1 - 5632) q^43 + (-81b2 + 2349) q^45 + (-81b3 - 23b2 + 42b1 - 4094) q^47 + (-24b3 - 330b2 + 9b1 + 16151) q^49 + (-54b3 + 27b1 + 648) * q^51 + (-7b3 + 32b2 + 2b1 + 15325) q^53 + (54b3 + 54b2 - 21b1 - 12472) q^55 + 3249 * q^57 + (-22b3 + 84b2 + 92b1 - 16834) q^59 + (6b3 - 234b2 + 5b1 + 26014) q^61 + (-81b1 + 648) q^63 + (40b3 - 470b2 - 116b1 + 33470) q^65 + (60b3 - 294b2 - 74b1 - 10238) q^67 + (-81b3 + 9b2 + 27b1 + 5274) q^69 + (162b3 + 100b2 + 108b1 - 24662) q^71 + (258b3 + 300b2 - 175b1 + 11612) q^73 + (-540b2 - 45b1 + 8019) * q^75 + (-184b3 + 282b2 + 131b1 + 41282) q^77 + (-198b3 + 3b2 + 155b1 + 6321) q^79 + 6561 * q^81 + (-231b3 - 432b2 + 390b1 - 8517) q^83 + (-36b3 + 12b2 + 393b1 - 5364) q^85 + (81b3 + 468b2 - 54b1 + 1089) q^87 + (-31b3 - 864b2 + 134b1 + 32969) q^89 + (336b3 - 708b2 - 518b1 + 37228) q^91 + (-54b3 - 243b2 + 63b1 - 405) q^93 + (-361b2 + 10469) q^95 + (-210b3 + 114b2 + 314b1 - 7866) q^97 + (-81b3 + 324b2 - 81b1 + 2673) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 36 q^{3} + 117 q^{5} + 33 q^{7} + 324 q^{9} + 129 q^{11} + 638 q^{13} + 1053 q^{15} + 285 q^{17} + 1444 q^{19} + 297 q^{21} + 2340 q^{23} + 3629 q^{25} + 2916 q^{27} + 438 q^{29} - 160 q^{31} + 1161 q^{33}+ \cdots + 10449 q^{99}+O(q^{100})$ 4 * q + 36 * q^3 + 117 * q^5 + 33 * q^7 + 324 * q^9 + 129 * q^11 + 638 * q^13 + 1053 * q^15 + 285 * q^17 + 1444 * q^19 + 297 * q^21 + 2340 * q^23 + 3629 * q^25 + 2916 * q^27 + 438 * q^29 - 160 * q^31 + 1161 * q^33 + 2409 * q^35 + 15520 * q^37 + 5742 * q^39 + 14664 * q^41 - 22519 * q^43 + 9477 * q^45 - 16395 * q^47 + 64925 * q^49 + 2565 * q^51 + 61266 * q^53 - 49921 * q^55 + 12996 * q^57 - 67512 * q^59 + 104285 * q^61 + 2673 * q^63 + 134466 * q^65 - 40584 * q^67 + 21060 * q^69 - 98856 * q^71 + 46323 * q^73 + 32661 * q^75 + 164715 * q^77 + 25126 * q^79 + 26244 * q^81 - 34026 * q^83 - 21861 * q^85 + 3942 * q^87 + 132606 * q^89 + 150138 * q^91 - 1440 * q^93 + 42237 * q^95 - 31892 * q^97 + 10449 * q^99

$\beta_{1}$	$=$	$( 5\nu^{3} + 196\nu^{2} - 19991\nu - 140614 ) / 1732$ (5v^3 + 196v^2 - 19991*v - 140614) / 1732
$\beta_{2}$	$=$	$( 3\nu^{3} + 31\nu^{2} - 11908\nu + 103034 ) / 866$ (3v^3 + 31v^2 - 11908*v + 103034) / 866
$\beta_{3}$	$=$	$( -13\nu^{3} + 10\nu^{2} + 61849\nu - 762282 ) / 1732$ (-13v^3 + 10v^2 + 61849*v - 762282) / 1732

Introduction

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Database

Properties

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Newspace parameters

Newform invariants

$q$ -expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	912.6.a.s

Level	$912$
Weight	$6$
Character orbit	912.a
Self dual	yes
Analytic conductor	$146.270$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$

$\nu$	$=$	$( \beta_{3} + 3\beta_{2} - \beta _1 + 2 ) / 6$ (b3 + 3*b2 - b1 + 2) / 6
$\nu^{2}$	$=$	$( \beta_{3} - 57\beta_{2} + 71\beta _1 + 12986 ) / 6$ (b3 - 57b2 + 71b1 + 12986) / 6
$\nu^{3}$	$=$	$( 3959\beta_{3} + 14229\beta_{2} - 4703\beta _1 - 332318 ) / 6$ (3959b3 + 14229b2 - 4703*b1 - 332318) / 6

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format: