LMFDB - Newform orbit 361.6.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [361,6,Mod(1,361)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("361.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$361 = 19^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	361.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.8985589525$
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:	$x^{4} - x^{3} - 84x^{2} - 154x + 484$ x^4 - x^3 - 84x^2 - 154x + 484
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 19)
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

−4.37522
−6.51487
10.2303
1.65978

−9.02832

−7.95930

49.5106

−73.8370

71.8591

41.7769

−158.091

−179.649

666.624

1.2

−7.37689

3.41393

22.4185

108.514

−25.1842

−80.6465

70.6815

−231.345

−800.499

1.3

−1.84953

−30.2395

−28.5793

10.2772

55.9287

19.2784

112.043

671.426

−19.0079

1.4

9.25474

28.7849

53.6502

45.0454

266.396

−170.409

200.367

585.568

416.883

Atkin-Lehner signs

$p$	Sign
$19$	$-1$

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	361.6.a.e		4
19.b	odd	2	1		19.6.a.d	✓	4
57.d	even	2	1		171.6.a.i		4
76.d	even	2	1		304.6.a.l		4
95.d	odd	2	1		475.6.a.e		4
133.c	even	2	1		931.6.a.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.6.a.d	✓	4	19.b	odd	2	1
171.6.a.i		4	57.d	even	2	1
304.6.a.l		4	76.d	even	2	1
361.6.a.e		4	1.a	even	1	1	trivial
475.6.a.e		4	95.d	odd	2	1
931.6.a.d		4	133.c	even	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{4} + 9T_{2}^{3} - 72T_{2}^{2} - 774T_{2} - 1140$ T2^4 + 9*T2^3 - 72*T2^2 - 774*T2 - 1140 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(361))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 9 T^{3} + \cdots - 1140$ T^4 + 9T^3 - 72T^2 - 774*T - 1140
$3$	$T^{4} + 6 T^{3} + \cdots + 23652$ T^4 + 6T^3 - 891T^2 - 3996*T + 23652
$5$	$T^{4} - 90 T^{3} + \cdots - 3709248$ T^4 - 90T^3 - 5631T^2 + 427212*T - 3709248
$7$	$T^{4} + 190 T^{3} + \cdots + 11068411$ T^4 + 190T^3 - 780T^2 - 636878*T + 11068411
$11$	$T^{4} + \cdots + 23023022796$ T^4 + 162T^3 - 422091T^2 - 81387504*T + 23023022796
$13$	$T^{4} + \cdots + 19147784368$ T^4 - 52T^3 - 421077T^2 + 43634984*T + 19147784368
$17$	$T^{4} + \cdots + 597258958449$ T^4 + 288T^3 - 2052402T^2 - 441826848*T + 597258958449
$19$	$T^{4}$ T^4
$23$	$T^{4} + \cdots - 6908113340160$ T^4 - 9900T^3 + 31520571T^2 - 29955652704*T - 6908113340160
$29$	$T^{4} + \cdots + 56401950031212$ T^4 + 4176T^3 - 10692777T^2 - 31339154700*T + 56401950031212
$31$	$T^{4} + \cdots - 5096621276672$ T^4 + 13580T^3 + 36785424T^2 - 1266177472*T - 5096621276672
$37$	$T^{4} + \cdots + 63\!\cdots\!52$ T^4 + 1172T^3 - 175373664T^2 - 17237560912*T + 6365660907920752
$41$	$T^{4} + \cdots + 813179406508032$ T^4 + 9540T^3 - 87034140T^2 - 468085180608*T + 813179406508032
$43$	$T^{4} + \cdots + 13\!\cdots\!16$ T^4 - 13370T^3 - 238114695T^2 + 1117530034132*T + 13224295964370016
$47$	$T^{4} + \cdots - 942340690593024$ T^4 - 28098T^3 + 16529481T^2 + 2125370457864*T - 942340690593024
$53$	$T^{4} + \cdots + 13\!\cdots\!52$ T^4 + 34740T^3 - 215994525T^2 - 2417877286776*T + 13514460982907952
$59$	$T^{4} + \cdots + 19\!\cdots\!36$ T^4 + 9702T^3 - 1487899683T^2 + 3184990747164*T + 197460228941644836
$61$	$T^{4} + \cdots - 10\!\cdots\!92$ T^4 + 37978T^3 - 31913283T^2 - 4643761518032*T - 10144006296432692
$67$	$T^{4} + \cdots + 19\!\cdots\!12$ T^4 - 2974T^3 - 3399843207T^2 + 8155288679000*T + 1992835358829060112
$71$	$T^{4} + \cdots + 16\!\cdots\!16$ T^4 + 32220T^3 - 718058208T^2 + 2958926025744*T + 1659943870844016
$73$	$T^{4} + \cdots + 12\!\cdots\!37$ T^4 + 86908T^3 + 2181828102T^2 + 16457736452476*T + 12195686687175937
$79$	$T^{4} + \cdots - 20\!\cdots\!56$ T^4 - 165736T^3 + 6058367052T^2 + 92232915500480*T - 2068098417951885056
$83$	$T^{4} + \cdots - 13\!\cdots\!16$ T^4 + 146448T^3 + 4137891228T^2 - 26558028132768*T - 1325549961450398016
$89$	$T^{4} + \cdots + 38\!\cdots\!72$ T^4 - 26604T^3 - 3031336176T^2 + 65630986054848*T + 38144875881948672
$97$	$T^{4} + \cdots + 42\!\cdots\!12$ T^4 + 313820T^3 + 30060887844T^2 + 896157150390848*T + 423617787303930112
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + ( - \beta_{2} + \beta_1 - 3) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + ( - \beta_{3} - 3 \beta_{2} + \cdots + 23) q^{4} + (4 \beta_{3} - 3 \beta_{2} + 22) q^{5} + ( - 3 \beta_{3} - 15 \beta_{2} + \cdots + 81) q^{6}+ \cdots + (3096 \beta_{3} - 17271 \beta_{2} + \cdots - 62406) q^{99}+O(q^{100})$ q + (-b2 + b1 - 3) * q^2 + (-3b2 - 3) q^3 + (-b3 - 3b2 - 2b1 + 23) * q^4 + (4b3 - 3b2 + 22) * q^5 + (-3b3 - 15b2 + 12b1 + 81) q^6 + (-2b3 + 12b2 - 4b1 - 41) q^7 + (5b3 - 13b2 + 16b1 + 47) q^8 + (-27b2 + 72b1 + 180) * q^9 + (-35b3 - 24b2 + 29b1 + 38) q^10 + (-28b3 - 17b2 - 16b1 - 52) q^11 + (9b3 - 87b2 + 114b1 + 453) q^12 + (-18b3 + 27b2 + 36b1 + 13) q^13 + (28b3 + 105b2 - 81b1 - 269) q^14 + (-12b3 - 78b2 + 444) * q^15 + (-21b3 - 9b2 + 102b1 - 173) q^16 + (16b3 - 138b2 - 137) * q^17 + (-27b3 - 342b2 + 27b1 + 1692) q^18 + (128b3 - 226b2 + 112b1 + 116) q^20 + (18b3 + 303b2 - 204b1 - 1317) q^21 + (207b3 - 162b2 + 153b1 - 12) q^22 + (-98b3 - 65b2 + 116b1 + 2389) q^23 + (-63b3 - 459b2 + 30b1 + 717) q^24 + (128b3 + 159b2 - 680b1 + 1997) q^25 + (171b3 + 77b2 - 230b1 - 39) q^26 + (-216b3 - 999b2 - 216b1 - 837) q^27 + (-55b3 + 627b2 - 398b1 - 1959) q^28 + (110b3 - 319b2 - 20b1 - 1171) q^29 + (18b3 - 960b2 + 858b1 + 444) q^30 + (236b3 + 24b2 + 256b1 - 3388) q^31 + (-b3 + 451b2 - 1006b1 + 1307) * q^32 + (132b3 - 24b2 + 1104b1 + 2886) q^33 + (-266b3 - 627b2 + 521b1 + 3851) q^34 + (-92b3 + 331b2 + 224b1 - 4742) q^35 + (-126b3 - 2988b2 + 1044b1 - 2250) q^36 + (-100b3 + 456b2 - 1568b1 + 302) q^37 + (-54b3 - 255b2 - 756b1 - 6573) q^39 + (-130b3 - 192b2 - 386b1 + 7348) q^40 + (-504b3 - 262b2 + 136b1 - 2676) q^41 + (159b3 + 3207b2 - 2052b1 - 7665) q^42 + (588b3 - 783b2 - 624b1 + 3254) q^43 + (-922b3 + 412b2 + 284b1 + 10610) q^44 + (-936b3 - 1611b2 + 2088b1 + 3798) q^45 + (719b3 - 3171b2 + 2446b1 - 3839) q^46 + (552b3 + 659b2 + 808b1 + 7290) q^47 + (-243b3 - 939b2 - 630b1 - 5475) q^48 + (-8b3 - 1824b2 + 848b1 - 8518) q^49 + (-865b3 - 531b2 + 3666b1 - 23743) q^50 + (-48b3 - 1149b2 + 3024b1 + 19839) q^51 + (-715b3 + 321b2 - 998b1 - 5839) q^52 + (1102b3 - 45b2 + 1068b1 - 8699) q^53 + (729b3 - 6021b2 + 5454b1 + 20007) q^54 + (-120b3 - 2457b2 + 4296b1 - 26094) q^55 + (171b3 + 2141b2 - 800b1 - 9759) q^56 + (-1199b3 - 303b2 + 284b1 + 11609) q^58 + (432b3 + 3377b2 - 4232b1 + 429) q^59 + (-720b3 - 3636b2 + 1776b1 + 26520) q^60 + (1020b3 - 873b2 + 240b1 - 9736) q^61 + (-1864b3 + 4476b2 - 5004b1 + 17108) q^62 + (1044b3 + 7227b2 - 4176b1 - 14004) q^63 + (1131b3 + 1683b2 - 186b1 - 31349) q^64 + (-956b3 - 804b2 + 4104b1 - 25256) q^65 + (-1080b3 - 2502b2 - 1674b1 + 17262) q^66 + (2088b3 + 4455b2 - 2304b1 + 4069) q^67 + (989b3 - 4261b2 + 5434b1 + 17213) q^68 + (-54b3 - 9927b2 + 1932b1 - 8205) q^69 + (1067b3 + 6360b2 - 7109b1 + 10474) q^70 + (-1052b3 - 956b2 + 3032b1 - 9554) q^71 + (-1116b3 - 5238b2 + 7902b1 + 46278) q^72 + (336b3 + 2304b2 - 2640b1 - 19831) q^73 + (1256b3 + 2034b2 + 4494b1 - 47146) q^74 + (1656b3 + 4845b2 + 2040b1 + 20019) q^75 + (304b3 + 2753b2 - 4376b1 + 9364) q^77 + (177b3 + 4827b2 - 2166b1 + 8775) q^78 + (2124b3 + 5418b2 - 1872b1 + 45142) q^79 + (-3248b3 - 1788b2 + 6528b1 - 30680) q^80 + (1296b3 - 1620b2 + 12960b1 + 105705) q^81 + (3770b3 - 912b2 - 902b1 + 13276) q^82 + (1172b3 + 5878b2 - 5704b1 - 31954) q^83 + (1359b3 + 17847b2 - 9282b1 - 55701) q^84 + (-388b3 - 2343b2 - 2720b1 + 36958) q^85 + (-5487b3 - 5600b2 + 8489b1 + 6) q^86 + (-270b3 + 585b2 + 5916b1 + 51495) q^87 + (1164b3 - 6642b2 + 4362b1 - 42462) q^88 + (-1148b3 + 144b2 - 5664b1 + 7852) q^89 + (5877b3 - 17208b2 + 5373b1 + 65718) q^90 + (784b3 - 663b2 - 1240b1 + 26395) q^91 + (-5787b3 - 10231b2 - 2918b1 + 70737) q^92 + (-1476b3 + 8796b2 - 7896b1 - 4380) q^93 + (-3757b3 - 1128b2 - 341b1 - 15494) q^94 + (3021b3 + 13557b2 + 1266b1 + 213) q^96 + (-3824b3 - 942b2 + 1280b1 - 80202) q^97 + (-1760b3 - 2458b2 - 2774b1 + 87922) q^98 + (3096b3 - 17271b2 - 11160b1 - 62406) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 9 q^{2} - 6 q^{3} + 97 q^{4} + 90 q^{5} + 369 q^{6} - 190 q^{7} + 225 q^{8} + 846 q^{9} + 264 q^{10} - 162 q^{11} + 2091 q^{12} + 52 q^{13} - 1395 q^{14} + 1944 q^{15} - 551 q^{16} - 288 q^{17} + 7506 q^{18}+ \cdots - 229338 q^{99}+O(q^{100})$ 4 * q - 9 * q^2 - 6 * q^3 + 97 * q^4 + 90 * q^5 + 369 * q^6 - 190 * q^7 + 225 * q^8 + 846 * q^9 + 264 * q^10 - 162 * q^11 + 2091 * q^12 + 52 * q^13 - 1395 * q^14 + 1944 * q^15 - 551 * q^16 - 288 * q^17 + 7506 * q^18 + 900 * q^20 - 6096 * q^21 + 222 * q^22 + 9900 * q^23 + 3879 * q^24 + 6862 * q^25 - 711 * q^26 - 1350 * q^27 - 9433 * q^28 - 4176 * q^29 + 4536 * q^30 - 13580 * q^31 + 3321 * q^32 + 12564 * q^33 + 17445 * q^34 - 19314 * q^35 - 1854 * q^36 - 1172 * q^37 - 26484 * q^39 + 29520 * q^40 - 9540 * q^41 - 39285 * q^42 + 13370 * q^43 + 42822 * q^44 + 21438 * q^45 - 7287 * q^46 + 28098 * q^47 - 20409 * q^48 - 29568 * q^49 - 89379 * q^50 + 84726 * q^51 - 24281 * q^52 - 34740 * q^53 + 96795 * q^54 - 95046 * q^55 - 44289 * q^56 + 48525 * q^58 - 9702 * q^59 + 115848 * q^60 - 37978 * q^61 + 56340 * q^62 - 75690 * q^63 - 130079 * q^64 - 94356 * q^65 + 73458 * q^66 + 2974 * q^67 + 81819 * q^68 - 10980 * q^69 + 21000 * q^70 - 32220 * q^71 + 204606 * q^72 - 86908 * q^73 - 189414 * q^74 + 70770 * q^75 + 27270 * q^77 + 23103 * q^78 + 165736 * q^79 - 109368 * q^80 + 437724 * q^81 + 50256 * q^82 - 146448 * q^83 - 269139 * q^84 + 150186 * q^85 + 25200 * q^86 + 210996 * q^87 - 153366 * q^88 + 26604 * q^89 + 296784 * q^90 + 104882 * q^91 + 306279 * q^92 - 41532 * q^93 - 56304 * q^94 - 28017 * q^96 - 313820 * q^97 + 355590 * q^98 - 229338 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{3} - \nu^{2} - 62\nu - 132 ) / 22$ (v^3 - v^2 - 62*v - 132) / 22
$\beta_{3}$	$=$	$( -2\nu^{3} + 13\nu^{2} + 102\nu - 220 ) / 11$ (-2v^3 + 13v^2 + 102*v - 220) / 11

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more