LMFDB - Newform orbit 38.6.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,6,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$38 = 2 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	38.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:	$6.09458515289$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 454x + 3760$ x^3 - x^2 - 454*x + 3760
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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

−24.1916
10.7990
14.3926

4.00000

−20.1916

16.0000

57.7548

−80.7665

142.950

64.0000

164.701

231.019

1.2

4.00000

14.7990

16.0000

−19.0956

59.1959

212.287

64.0000

−23.9901

−76.3823

1.3

4.00000

18.3926

16.0000

42.3408

73.5705

−127.237

64.0000

95.2889

169.363

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$19$	$+1$

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	38.6.a.d	✓	3
3.b	odd	2	1		342.6.a.l		3
4.b	odd	2	1		304.6.a.h		3
5.b	even	2	1		950.6.a.f		3
19.b	odd	2	1		722.6.a.d		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.6.a.d	✓	3	1.a	even	1	1	trivial
304.6.a.h		3	4.b	odd	2	1
342.6.a.l		3	3.b	odd	2	1
722.6.a.d		3	19.b	odd	2	1
950.6.a.f		3	5.b	even	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{3} - 13T_{3}^{2} - 398T_{3} + 5496$ T3^3 - 13*T3^2 - 398*T3 + 5496 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(38))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 4)^{3}$ (T - 4)^3
$3$	$T^{3} - 13 T^{2} + \cdots + 5496$ T^3 - 13T^2 - 398T + 5496
$5$	$T^{3} - 81 T^{2} + \cdots + 46696$ T^3 - 81T^2 + 534T + 46696
$7$	$T^{3} - 228 T^{2} + \cdots + 3861216$ T^3 - 228T^2 - 14853T + 3861216
$11$	$T^{3} - 363 T^{2} + \cdots + 162880120$ T^3 - 363T^2 - 433794T + 162880120
$13$	$T^{3} - 501 T^{2} + \cdots + 93082696$ T^3 - 501T^2 - 763650T + 93082696
$17$	$T^{3} + 1206 T^{2} + \cdots - 963841518$ T^3 + 1206T^2 - 1488321T - 963841518
$19$	$(T + 361)^{3}$ (T + 361)^3
$23$	$T^{3} + \cdots - 18644491520$ T^3 + 1077T^2 - 12667656T - 18644491520
$29$	$T^{3} + \cdots - 14808989500$ T^3 + 8349T^2 + 13250232T - 14808989500
$31$	$T^{3} + \cdots - 164301107200$ T^3 + 7332T^2 - 24785856T - 164301107200
$37$	$T^{3} + \cdots - 19509523912$ T^3 + 1650T^2 - 42089988T - 19509523912
$41$	$T^{3} + \cdots + 164480699264$ T^3 - 10140T^2 - 22487436T + 164480699264
$43$	$T^{3} + \cdots - 2228453451472$ T^3 - 3777T^2 - 361789272T - 2228453451472
$47$	$T^{3} + \cdots - 1331645125760$ T^3 - 33231T^2 + 365742456T - 1331645125760
$53$	$T^{3} + \cdots + 5330002936312$ T^3 - 31029T^2 - 62214738T + 5330002936312
$59$	$T^{3} + \cdots + 56358292470552$ T^3 - 20409T^2 - 2487789426T + 56358292470552
$61$	$T^{3} + \cdots + 3097994159068$ T^3 - 17115T^2 - 281445804T + 3097994159068
$67$	$T^{3} + \cdots - 6192188856432$ T^3 + 789T^2 - 3448744536T - 6192188856432
$71$	$T^{3} + \cdots + 33641692455520$ T^3 - 19164T^2 - 2231133564T + 33641692455520
$73$	$T^{3} + \cdots - 23123416049802$ T^3 + 76260T^2 - 133871259T - 23123416049802
$79$	$T^{3} + \cdots + 2286937169920$ T^3 - 68358T^2 + 369119904T + 2286937169920
$83$	$T^{3} + \cdots + 183940117843104$ T^3 - 6762T^2 - 8479184976T + 183940117843104
$89$	$T^{3} + \cdots - 63263160328320$ T^3 + 85506T^2 - 8953343256T - 63263160328320
$97$	$T^{3} + \cdots + 11475767642528$ T^3 - 105024T^2 - 9580226580T + 11475767642528
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + 4 q^{2} + (\beta_1 + 4) q^{3} + 16 q^{4} + (\beta_{2} - \beta_1 + 27) q^{5} + (4 \beta_1 + 16) q^{6} + ( - 5 \beta_{2} - 4 \beta_1 + 79) q^{7} + 64 q^{8} + (2 \beta_{2} - 3 \beta_1 + 79) q^{9} + (4 \beta_{2} - 4 \beta_1 + 108) q^{10}+ \cdots + (429 \beta_{2} - 2063 \beta_1 + 53317) q^{99}+O(q^{100})$ q + 4 * q^2 + (b1 + 4) * q^3 + 16 * q^4 + (b2 - b1 + 27) * q^5 + (4b1 + 16) q^6 + (-5b2 - 4b1 + 79) * q^7 + 64 * q^8 + (2b2 - 3b1 + 79) * q^9 + (4b2 - 4b1 + 108) * q^10 + (17b2 - 17b1 + 121) * q^11 + (16b1 + 64) q^12 + (-28b2 - 5b1 + 178) * q^13 + (-20b2 - 16b1 + 316) * q^14 + (14b2 + 42b1 - 242) * q^15 + 256 * q^16 + (21b2 + 60b1 - 429) * q^17 + (8b2 - 12b1 + 316) * q^18 - 361 * q^19 + (16b2 - 16b1 + 432) * q^20 + (-88b2 + 67b1 - 688) * q^21 + (68b2 - 68b1 + 484) * q^22 + (34b2 - 157b1 - 318) * q^23 + (64b1 + 256) q^24 + (25b2 - 55b1 - 1284) * q^25 + (-112b2 - 20b1 + 712) * q^26 + (26b2 - 127b1 - 1662) * q^27 + (-80b2 - 64b1 + 1264) * q^28 + (62b2 + 121b1 - 2844) * q^29 + (56b2 + 168b1 - 968) * q^30 + (-196b2 + 28b1 - 2388) * q^31 + 1024 * q^32 + (238b2 + 376b1 - 5466) * q^33 + (84b2 + 240b1 - 1716) * q^34 + (-b2 - 353b1 - 277) q^35 + (32b2 - 48b1 + 1264) * q^36 + (116b2 - 236b1 - 510) * q^37 - 1444 * q^38 + (-458b2 - 11b1 + 414) * q^39 + (64b2 - 64b1 + 1728) * q^40 + (-228b2 - 66b1 + 3478) * q^41 + (-352b2 + 268b1 - 2752) * q^42 + (489b2 + 549b1 + 913) * q^43 + (272b2 - 272b1 + 1936) * q^44 + (65b2 - 181b1 + 4707) * q^45 + (136b2 - 628b1 - 1272) * q^46 + (-5b2 + 71b1 + 11055) * q^47 + (256b1 + 1024) q^48 + (-453b2 + 162b1 + 10520) * q^49 + (100b2 - 220b1 - 5136) * q^50 + (456b2 - 681b1 + 15720) * q^51 + (-448b2 - 80b1 + 2848) * q^52 + (-156b2 + 867b1 + 10106) * q^53 + (104b2 - 508b1 - 6648) * q^54 + (87b2 - 597b1 + 22171) * q^55 + (-320b2 - 256b1 + 5056) * q^56 + (-361b1 - 1444) q^57 + (248b2 + 484b1 - 11376) * q^58 + (244b2 + 2405b1 + 5920) * q^59 + (224b2 + 672b1 - 3872) * q^60 + (401b2 - 623b1 + 5779) * q^61 + (-784b2 + 112b1 - 9552) * q^62 + (-59b2 - 889b1 + 2425) * q^63 + 4096 * q^64 + (-96b2 - 912b1 - 14780) * q^65 + (952b2 + 1504b1 - 21864) * q^66 + (-1054b2 + 2095b1 - 610) * q^67 + (336b2 + 960b1 - 6864) * q^68 + (230b2 + 1053b1 - 50810) * q^69 + (-4b2 - 1412b1 - 1108) * q^70 + (-818b2 - 1996b1 + 7326) * q^71 + (128b2 - 192b1 + 5056) * q^72 + (-739b2 - 1898b1 - 24541) * q^73 + (464b2 - 944b1 - 2040) * q^74 + (290b2 - 699b1 - 23066) * q^75 - 5776 * q^76 + (1673b2 - 4649b1 - 31411) * q^77 + (-1832b2 - 44b1 + 1656) * q^78 + (964b2 + 758b1 + 22212) * q^79 + (256b2 - 256b1 + 6912) * q^80 + (-324b2 + 164b1 - 65851) * q^81 + (-912b2 - 264b1 + 13912) * q^82 + (684b2 + 4278b1 + 600) * q^83 + (-1408b2 + 1072b1 - 11008) * q^84 + (339b2 + 3567b1 - 16581) * q^85 + (1956b2 + 2196b1 + 3652) * q^86 + (1234b2 - 3195b1 + 22922) * q^87 + (1088b2 - 1088b1 + 7744) * q^88 + (-1354b2 + 4384b1 - 29512) * q^89 + (260b2 - 724b1 + 18828) * q^90 + (-2398b2 + 3409b1 + 114674) * q^91 + (544b2 - 2512b1 - 5088) * q^92 + (-3080b2 - 4152b1 + 7640) * q^93 + (-20b2 + 284b1 + 44220) * q^94 + (-361b2 + 361b1 - 9747) * q^95 + (1024b1 + 4096) q^96 + (3302b2 + 2218b1 + 33168) * q^97 + (-1812b2 + 648b1 + 42080) * q^98 + (429b2 - 2063b1 + 53317) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 12 q^{2} + 13 q^{3} + 48 q^{4} + 81 q^{5} + 52 q^{6} + 228 q^{7} + 192 q^{8} + 236 q^{9} + 324 q^{10} + 363 q^{11} + 208 q^{12} + 501 q^{13} + 912 q^{14} - 670 q^{15} + 768 q^{16} - 1206 q^{17} + 944 q^{18}+ \cdots + 158317 q^{99}+O(q^{100})$ 3 * q + 12 * q^2 + 13 * q^3 + 48 * q^4 + 81 * q^5 + 52 * q^6 + 228 * q^7 + 192 * q^8 + 236 * q^9 + 324 * q^10 + 363 * q^11 + 208 * q^12 + 501 * q^13 + 912 * q^14 - 670 * q^15 + 768 * q^16 - 1206 * q^17 + 944 * q^18 - 1083 * q^19 + 1296 * q^20 - 2085 * q^21 + 1452 * q^22 - 1077 * q^23 + 832 * q^24 - 3882 * q^25 + 2004 * q^26 - 5087 * q^27 + 3648 * q^28 - 8349 * q^29 - 2680 * q^30 - 7332 * q^31 + 3072 * q^32 - 15784 * q^33 - 4824 * q^34 - 1185 * q^35 + 3776 * q^36 - 1650 * q^37 - 4332 * q^38 + 773 * q^39 + 5184 * q^40 + 10140 * q^41 - 8340 * q^42 + 3777 * q^43 + 5808 * q^44 + 14005 * q^45 - 4308 * q^46 + 33231 * q^47 + 3328 * q^48 + 31269 * q^49 - 15528 * q^50 + 46935 * q^51 + 8016 * q^52 + 31029 * q^53 - 20348 * q^54 + 66003 * q^55 + 14592 * q^56 - 4693 * q^57 - 33396 * q^58 + 20409 * q^59 - 10720 * q^60 + 17115 * q^61 - 29328 * q^62 + 6327 * q^63 + 12288 * q^64 - 45348 * q^65 - 63136 * q^66 - 789 * q^67 - 19296 * q^68 - 151147 * q^69 - 4740 * q^70 + 19164 * q^71 + 15104 * q^72 - 76260 * q^73 - 6600 * q^74 - 69607 * q^75 - 17328 * q^76 - 97209 * q^77 + 3092 * q^78 + 68358 * q^79 + 20736 * q^80 - 197713 * q^81 + 40560 * q^82 + 6762 * q^83 - 33360 * q^84 - 45837 * q^85 + 15108 * q^86 + 66805 * q^87 + 23232 * q^88 - 85506 * q^89 + 56020 * q^90 + 345033 * q^91 - 17232 * q^92 + 15688 * q^93 + 132924 * q^94 - 29241 * q^95 + 13312 * q^96 + 105024 * q^97 + 125076 * q^98 + 158317 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} + 11\nu - 306 ) / 2$ (v^2 + 11*v - 306) / 2

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

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	38.6.a.d

Level	$38$
Weight	$6$
Character orbit	38.a
Self dual	yes
Analytic conductor	$6.095$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$2\beta_{2} - 11\beta _1 + 306$ 2b2 - 11b1 + 306

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