LMFDB - Newform orbit 2088.4.a.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2088,4,Mod(1,2088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2088 = 2^{3} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2088.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:	$123.195988092$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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_{4} + \beta_{3} + 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 - 6) q^{7}+O(q^{10}) $ q + (b4 + b3 + 1) * q^5 + (b4 - b3 - b1 - 6) * q^7	$ q + (\beta_{4} + \beta_{3} + 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 - 6) q^{7} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{2} + \cdots + 1) q^{11}+ \cdots + ( - 16 \beta_{5} + 45 \beta_{4} + \cdots + 27) q^{97}+O(q^{100}) $ q + (b4 + b3 + 1) * q^5 + (b4 - b3 - b1 - 6) * q^7 + (b5 - 3b4 - 3b2 - 2b1 + 1) q^11 + (3b5 - 6b3 - b2 - 2b1 + 3) q^13 + (6b5 + 5b4 + 4b3 - 3b2 - 2b1 + 37) q^17 + (-b5 + b4 + b3 - 11b2 - 52) q^19 + (-8b5 + 5b4 - 9b3 - 2b2 + b1 + 34) * q^23 + (5b5 - b4 + 6b3 + 6b2 - 2b1 - 1) * q^25 + 29 * q^29 + (-4b5 + b4 + 10b3 - 10b2 + 5b1 - 117) * q^31 + (10b5 - 15b4 - 15b3 + 8b2 - 21b1 + 76) q^35 + (-2b5 + 14b4 + 2b3 - 10b2 - 6b1 - 40) q^37 + (14b5 + 7b4 - 28b3 + 11b2 - 8b1 + 89) q^41 + (13b5 + 7b4 + 12b3 + 9b2 - 12b1 - 93) q^43 + (-11b5 + 12b4 - b3 - 3b2 + 7b1 + 165) * q^47 + (24b5 + 10b4 + 44b3 + 22b2 - 4b1 + 159) q^49 + (-15b5 + 8b4 + 8b3 + 11b2 + 2b1 - 83) q^53 + (-30b5 + 9b4 - 36b3 - 2b2 + 7b1 - 263) q^55 + (-6b5 - 13b4 + 33b3 - 16b2 + b1 + 2) * q^59 + (-24b5 + 13b4 + 6b3 + b2 + 38b1 + 199) * q^61 + (-10b5 + 8b4 - 37b3 + 17b2 - 14b1 - 196) q^65 + (-10b5 + 4b4 + 6b3 + 52b2 - 4b1 + 2) q^67 + (8b5 - 24b4 + 70b3 - 16b2 - 56b1 - 106) q^71 + (-14b5 - 52b4 + 58b3 - 40b2 - 18b1 - 2) q^73 + (-40b5 + 77b4 + 34b3 + b2 + 70b1 - 7) * q^77 + (-9b5 + 20b4 - 9b3 - 29b2 + 21b1 - 39) q^79 + (12b5 + 55b4 - 29b3 + 40b2 + 41b1 + 260) q^83 + (-16b5 + 67b4 + 22b3 + 67b2 + 24b1 + 561) q^85 + (12b5 + 25b4 - 30b3 - 11b2 + 44b1 + 61) q^89 + (-8b5 + 91b4 + 53b3 + 20b2 + 41b1 + 442) q^91 + (-33b5 - 119b4 - 67b3 + b2 + 22b1 - 60) * q^95 + (-16b5 + 45b4 + 124b3 - 73b2 - 22b1 + 27) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{5} - 38 q^{7}+O(q^{10}) $ 6 * q + 5 * q^5 - 38 * q^7	$ 6 q + 5 q^{5} - 38 q^{7} + 19 q^{11} + 13 q^{13} + 218 q^{17} - 290 q^{19} + 196 q^{23} - 13 q^{25} + 174 q^{29} - 675 q^{31} + 466 q^{35} - 238 q^{37} + 464 q^{41} - 579 q^{43} + 975 q^{47} + 914 q^{49} - 515 q^{53} - 1605 q^{55} + 108 q^{59} + 1158 q^{61} - 1239 q^{65} - 80 q^{67} - 438 q^{71} + 262 q^{73} - 194 q^{77} - 237 q^{79} + 1288 q^{83} + 3112 q^{85} + 252 q^{89} + 2450 q^{91} - 180 q^{95} + 380 q^{97}+O(q^{100}) $ 6 * q + 5 * q^5 - 38 * q^7 + 19 * q^11 + 13 * q^13 + 218 * q^17 - 290 * q^19 + 196 * q^23 - 13 * q^25 + 174 * q^29 - 675 * q^31 + 466 * q^35 - 238 * q^37 + 464 * q^41 - 579 * q^43 + 975 * q^47 + 914 * q^49 - 515 * q^53 - 1605 * q^55 + 108 * q^59 + 1158 * q^61 - 1239 * q^65 - 80 * q^67 - 438 * q^71 + 262 * q^73 - 194 * q^77 - 237 * q^79 + 1288 * q^83 + 3112 * q^85 + 252 * q^89 + 2450 * q^91 - 180 * q^95 + 380 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 $ x^6 - 40*x^4 - 88*x^3 - 8*x^2 + 48*x - 9 :

$\beta_{1}$	$=$	$ ( -\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

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

$\nu$	$=$	$ ( 2\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 8 $ (2b4 - 3b3 - b2 + b1 + 1) / 8
$\nu^{2}$	$=$	$ ( 6\beta_{5} - 15\beta_{3} - 9\beta_{2} - \beta _1 + 107 ) / 8 $ (6b5 - 15b3 - 9*b2 - b1 + 107) / 8
$\nu^{3}$	$=$	$ ( 3\beta_{5} + 17\beta_{4} - 32\beta_{3} - 16\beta_{2} + 5\beta _1 + 95 ) / 2 $ (3b5 + 17b4 - 32b3 - 16b2 + 5*b1 + 95) / 2
$\nu^{4}$	$=$	$ ( 244\beta_{5} + 170\beta_{4} - 861\beta_{3} - 455\beta_{2} + 27\beta _1 + 4403 ) / 8 $ (244b5 + 170b4 - 861b3 - 455b2 + 27*b1 + 4403) / 8
$\nu^{5}$	$=$	$ ( 1002\beta_{5} + 2736\beta_{4} - 6473\beta_{3} - 3351\beta_{2} + 729\beta _1 + 24309 ) / 8 $ (1002b5 + 2736b4 - 6473b3 - 3351b2 + 729*b1 + 24309) / 8

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

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

−4.70501
0.215317
7.22553
−1.90366
0.488896
−1.32108

−15.4201

−28.5777

1.2

−8.72860

−8.76166

1.3

−2.64112

3.30580

1.4

3.33508

0.0162797

1.5

14.0062

−34.0826

1.6

14.4485

30.0998

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$3$	$-1$
$29$	$-1$

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	2088.4.a.l		6
3.b	odd	2	1		232.4.a.e	✓	6
12.b	even	2	1		464.4.a.n		6
24.f	even	2	1		1856.4.a.bc		6
24.h	odd	2	1		1856.4.a.bd		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
232.4.a.e	✓	6	3.b	odd	2	1
464.4.a.n		6	12.b	even	2	1
1856.4.a.bc		6	24.f	even	2	1
1856.4.a.bd		6	24.h	odd	2	1
2088.4.a.l		6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - 5T_{5}^{5} - 356T_{5}^{4} + 1338T_{5}^{3} + 29589T_{5}^{2} - 28213T_{5} - 239922 $ T5^6 - 5*T5^5 - 356*T5^4 + 1338*T5^3 + 29589*T5^2 - 28213*T5 - 239922 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2088))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ 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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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2088.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta_{4} + \beta_{3} + 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 - 6) q^{7}+O(q^{10}) \) q + (b4 + b3 + 1) * q^5 + (b4 - b3 - b1 - 6) * q^7	\( q + (\beta_{4} + \beta_{3} + 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 - 6) q^{7} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{2} + \cdots + 1) q^{11}+ \cdots + ( - 16 \beta_{5} + 45 \beta_{4} + \cdots + 27) q^{97}+O(q^{100}) \) q + (b4 + b3 + 1) * q^5 + (b4 - b3 - b1 - 6) * q^7 + (b5 - 3b4 - 3b2 - 2b1 + 1) q^11 + (3b5 - 6b3 - b2 - 2b1 + 3) q^13 + (6b5 + 5b4 + 4b3 - 3b2 - 2b1 + 37) q^17 + (-b5 + b4 + b3 - 11b2 - 52) q^19 + (-8b5 + 5b4 - 9b3 - 2b2 + b1 + 34) * q^23 + (5b5 - b4 + 6b3 + 6b2 - 2b1 - 1) * q^25 + 29 * q^29 + (-4b5 + b4 + 10b3 - 10b2 + 5b1 - 117) * q^31 + (10b5 - 15b4 - 15b3 + 8b2 - 21b1 + 76) q^35 + (-2b5 + 14b4 + 2b3 - 10b2 - 6b1 - 40) q^37 + (14b5 + 7b4 - 28b3 + 11b2 - 8b1 + 89) q^41 + (13b5 + 7b4 + 12b3 + 9b2 - 12b1 - 93) q^43 + (-11b5 + 12b4 - b3 - 3b2 + 7b1 + 165) * q^47 + (24b5 + 10b4 + 44b3 + 22b2 - 4b1 + 159) q^49 + (-15b5 + 8b4 + 8b3 + 11b2 + 2b1 - 83) q^53 + (-30b5 + 9b4 - 36b3 - 2b2 + 7b1 - 263) q^55 + (-6b5 - 13b4 + 33b3 - 16b2 + b1 + 2) * q^59 + (-24b5 + 13b4 + 6b3 + b2 + 38b1 + 199) * q^61 + (-10b5 + 8b4 - 37b3 + 17b2 - 14b1 - 196) q^65 + (-10b5 + 4b4 + 6b3 + 52b2 - 4b1 + 2) q^67 + (8b5 - 24b4 + 70b3 - 16b2 - 56b1 - 106) q^71 + (-14b5 - 52b4 + 58b3 - 40b2 - 18b1 - 2) q^73 + (-40b5 + 77b4 + 34b3 + b2 + 70b1 - 7) * q^77 + (-9b5 + 20b4 - 9b3 - 29b2 + 21b1 - 39) q^79 + (12b5 + 55b4 - 29b3 + 40b2 + 41b1 + 260) q^83 + (-16b5 + 67b4 + 22b3 + 67b2 + 24b1 + 561) q^85 + (12b5 + 25b4 - 30b3 - 11b2 + 44b1 + 61) q^89 + (-8b5 + 91b4 + 53b3 + 20b2 + 41b1 + 442) q^91 + (-33b5 - 119b4 - 67b3 + b2 + 22b1 - 60) * q^95 + (-16b5 + 45b4 + 124b3 - 73b2 - 22b1 + 27) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{5} - 38 q^{7}+O(q^{10}) \) 6 * q + 5 * q^5 - 38 * q^7	\( 6 q + 5 q^{5} - 38 q^{7} + 19 q^{11} + 13 q^{13} + 218 q^{17} - 290 q^{19} + 196 q^{23} - 13 q^{25} + 174 q^{29} - 675 q^{31} + 466 q^{35} - 238 q^{37} + 464 q^{41} - 579 q^{43} + 975 q^{47} + 914 q^{49} - 515 q^{53} - 1605 q^{55} + 108 q^{59} + 1158 q^{61} - 1239 q^{65} - 80 q^{67} - 438 q^{71} + 262 q^{73} - 194 q^{77} - 237 q^{79} + 1288 q^{83} + 3112 q^{85} + 252 q^{89} + 2450 q^{91} - 180 q^{95} + 380 q^{97}+O(q^{100}) \) 6 * q + 5 * q^5 - 38 * q^7 + 19 * q^11 + 13 * q^13 + 218 * q^17 - 290 * q^19 + 196 * q^23 - 13 * q^25 + 174 * q^29 - 675 * q^31 + 466 * q^35 - 238 * q^37 + 464 * q^41 - 579 * q^43 + 975 * q^47 + 914 * q^49 - 515 * q^53 - 1605 * q^55 + 108 * q^59 + 1158 * q^61 - 1239 * q^65 - 80 * q^67 - 438 * q^71 + 262 * q^73 - 194 * q^77 - 237 * q^79 + 1288 * q^83 + 3112 * q^85 + 252 * q^89 + 2450 * q^91 - 180 * q^95 + 380 * q^97

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