LMFDB - Newform orbit 160.6.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$160 = 2^{5} \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	160.d (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776$ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{90}\cdot 3^{4}\cdot 5^{12}$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{4} + 10) q^{7} + (\beta_{5} - \beta_{4} - 81) q^{9} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{11} + ( - \beta_{9} + \beta_{2} + 7 \beta_1) q^{13} + (\beta_{11} - 45) q^{15}+ \cdots + ( - 14 \beta_{18} + 14 \beta_{17} + \cdots - 101 \beta_1) q^{99}+O(q^{100})$ q - b1 * q^3 + b2 * q^5 + (b4 + 10) * q^7 + (b5 - b4 - 81) * q^9 + (b3 - 2b2 + b1) q^11 + (-b9 + b2 + 7b1) q^13 + (b11 - 45) * q^15 - b7 * q^17 + (-b14 - b6 + b3 + 12b2 + 3b1) * q^19 + (b10 - b9 + b3 + 4b2 - 44b1) * q^21 + (-b19 + b15 + 4b5 + 3b4 + 235) * q^23 - 625 * q^25 + (-b17 + b16 - b10 - b9 + 2b6 + b3 - 18b2 + 69b1) q^27 + (-b18 + 2b17 + b16 - b14 + b9 + 2b6 + b3 + 3b2 + 53b1) * q^29 + (-b19 + 2b11 + b8 + b7 + 2b5 + 5b4 - 357) q^31 + (3b19 + b15 + 2b13 - 2b12 - b11 + b8 + b7 - 2b5 - 17b4 + 280) q^33 + (b18 - b17 - b16 - b14 + b10 - b9 + 10b2 - 12b1) * q^35 + (b18 - b16 - 5b14 - b10 - 6b9 + 6b6 - 4b3 - b2 + 21b1) q^37 + (b19 + 2b15 + 4b13 - 4b12 + 4b11 + b8 + b7 - 10b5 - 25b4 + 2239) * q^39 + (-4b19 + 3b15 + 5b13 - b12 + 10b11 - 2b7 - 18b5 - 12b4 + 575) q^41 + (3b18 + b17 + 7b16 + 3b14 + 5b10 + 11b9 + 3b3 - 104b2 + 117b1) * q^43 + (b18 - b17 + 4b16 + 4b14 + b10 + 4b9 - 80b2 + 68b1) q^45 + (-4b19 + 4b15 + 5b13 + 2b12 + 3b11 - 6b7 - 24b5 - 2214) q^47 + (-6b19 + 5b15 - b13 + 3b12 - 10b11 + 2b8 + 18b5 + 16b4 + 942) q^49 + (-4b18 + 8b17 - 6b16 - 17b14 - 6b10 - 18b9 + 17b6 + 2b3 + 142b2 + 58b1) * q^51 + (9b18 - 6b17 + 3b16 - 3b14 + 9b10 + 5b9 - 10b6 + 16b3 + 12b2 + 250b1) * q^53 + (4b19 + b15 - b13 - 2b11 + b8 - b7 + 4b5 - 5b4 + 1209) * q^55 + (-4b19 - 2b15 - 6b12 + 34b11 + 4b8 + b7 + 18b5 + 24b4 + 260) q^57 + (6b16 - 3b14 + 6b10 - 22b9 - b6 - 3b3 - 116b2 - 231b1) q^59 + (-9b18 + 7b16 - 7b14 - 11b10 - 14b9 + 6b6 + 22b3 + 69b2 + 153b1) q^61 + (13b19 + 5b15 + 5b13 - 10b12 - 3b11 + 2b8 + 4b7 + 60b5 - 76b4 - 12033) q^63 + (b19 + 4b15 + b13 - 5b12 - 9b11 - b8 + b7 + 11b5 - 10b4 + 1) q^65 + (-10b18 + b17 - b16 + 2b14 + 9b10 - 19b9 + 6b6 + 41b3 + 314b2 - 230b1) * q^67 + (-5b18 - 6b17 - 11b16 - 13b14 - 2b10 - 21b9 + 18b6 + 35b3 + 121b2 + 341b1) * q^69 + (-12b19 - 6b15 - 2b13 + 4b12 + 20b11 + 2b8 - 26b7 + 36b5 + 20b4 + 10026) q^71 + (5b19 + 15b15 + 2b12 + 43b11 - b8 + 3b7 + 88b5 + 21b4 - 5238) q^73 + 625b1 q^75 + (-3b18 + 22b17 - 3b16 - 23b14 - 5b10 - b9 - 14b6 + 32b3 - 6b2 - 780b1) q^77 + (-11b19 - 6b15 - 10b13 - 8b12 - 3b8 - 15b7 - 10b5 + 25b4 - 14099) * q^79 + (-14b19 - 6b15 + 10b13 + 8b12 - 68b11 - 2b8 - 8b7 - 87b5 + 127b4 + 3269) q^81 + (35b18 - 23b17 + 15b16 + 33b14 + 13b10 - 5b9 - 42b6 - 19b3 - 168b2 + 569b1) * q^83 + (6b18 - b17 + 9b16 - b14 + 11b10 + 19b9 - 30b6 - 35b3 - 24b2 - 257b1) * q^85 + (-11b19 + 9b15 + 7b13 + 26b12 - 15b11 - 2b8 + 28b7 - 28b5 - b4 + 16619) * q^87 + (22b19 - 13b15 - 11b13 + 21b12 + 86b11 - 2b8 - 8b7 - 139b5 - 11b4 - 191) q^89 + (-8b18 + 6b17 - 30b16 - 23b14 - 18b10 - 18b9 + 13b6 + 6b3 + 6b2 - 1348b1) * q^91 + (29b18 - 14b17 + 15b16 + 81b14 + 23b10 + 56b9 + 14b6 - 54b3 - 609b2 + 443b1) * q^93 + (5b19 + 5b15 - 10b13 + 10b12 - 21b11 + 20b7 + 70b5 - 45b4 - 7220) * q^95 + (-b19 - 23b15 - 4b13 - 30b12 - 127b11 - 3b8 + b7 + 26b5 - 195b4 + 7346) q^97 + (-14b18 + 14b17 + 28b16 + b14 - 32b10 + 160b9 - 3b6 - 121b3 - 392b2 - 101b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63}+ \cdots + 147376 q^{97}+O(q^{100})$ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776$ x^20 - 2*x^19 - 17*x^18 + 78*x^17 + 253*x^16 - 884*x^15 + 2396*x^14 + 19376*x^13 - 109104*x^12 - 96128*x^11 + 3580672*x^10 - 1538048*x^9 - 27930624*x^8 + 79364096*x^7 + 157024256*x^6 - 926941184*x^5 + 4244635648*x^4 + 20937965568*x^3 - 73014444032*x^2 - 137438953472*x + 1099511627776 :

$\beta_{1}$	$=$	$( - 79348087 \nu^{19} + 164467438 \nu^{18} + 2750967783 \nu^{17} + 2347892670 \nu^{16} + \cdots + 38\!\cdots\!76 ) / 71\!\cdots\!80$ (-79348087v^19 + 164467438v^18 + 2750967783v^17 + 2347892670v^16 - 17200470427v^15 + 69605863660v^14 - 24303499972v^13 - 719687080656v^12 + 16148446793808v^11 + 66989955357824v^10 - 209376218715392v^9 - 308378042677248v^8 + 2858199953264640v^7 + 4761986870738944v^6 + 7314399222824960v^5 + 210707338900799488v^4 + 19270179452092416v^3 - 2314359262031118336v^2 + 3379581800966782976*v + 38936417104430104576) / 713485464769658880
$\beta_{2}$	$=$	$( - 6673115 \nu^{19} - 30164170 \nu^{18} + 153706635 \nu^{17} + 266710470 \nu^{16} + \cdots + 15\!\cdots\!20 ) / 32\!\cdots\!56$ (-6673115v^19 - 30164170v^18 + 153706635v^17 + 266710470v^16 - 3862360175v^15 - 2106440260v^14 - 4737874100v^13 - 236780073360v^12 + 50299499280v^11 + 4828965765760v^10 - 12397823115520v^9 - 82271702046720v^8 + 93706914631680v^7 + 50248724971520v^6 - 1926810069893120v^5 + 7838825908797440v^4 + 6061205571502080v^3 - 180832148914176000v^2 - 275660321136312320*v + 1550266383906897920) / 32930098373984256
$\beta_{3}$	$=$	$( 392347319 \nu^{19} - 3248676206 \nu^{18} - 37138350375 \nu^{17} - 56596678206 \nu^{16} + \cdots - 46\!\cdots\!32 ) / 71\!\cdots\!80$ (392347319v^19 - 3248676206v^18 - 37138350375v^17 - 56596678206v^16 + 116030366171v^15 - 215726096876v^14 - 7354614456892v^13 - 28326961252656v^12 - 89620996541520v^11 - 695657667918976v^10 + 64356184347904v^9 + 4877709075462144v^8 - 35110862377537536v^7 - 176315402435624960v^6 - 252028719960162304v^5 - 172884124007137280v^4 - 5395871712027869184v^3 + 4882972880129753088v^2 - 18049481855350079488*v - 464189588174098399232) / 713485464769658880
$\beta_{4}$	$=$	$( - 13756221 \nu^{19} - 7338982 \nu^{18} + 200878797 \nu^{17} - 336652854 \nu^{16} + \cdots + 16\!\cdots\!56 ) / 13\!\cdots\!80$ (-13756221v^19 - 7338982v^18 + 200878797v^17 - 336652854v^16 - 2577873801v^15 + 10357630916v^14 - 40516598892v^13 - 198756032368v^12 + 1182900850544v^11 + 3523610475392v^10 - 19032642164480v^9 - 17076135725056v^8 + 214125830696960v^7 - 344410878705664v^6 - 66398979883008v^5 + 14093127287570432v^4 - 25707255580065792v^3 - 64272954002767872v^2 + 140142841543262208*v + 1695627043779117056) / 13989911073914880
$\beta_{5}$	$=$	$( 16634889 \nu^{19} + 110715086 \nu^{18} - 463943001 \nu^{17} - 1145810466 \nu^{16} + \cdots - 79\!\cdots\!76 ) / 13\!\cdots\!80$ (16634889v^19 + 110715086v^18 - 463943001v^17 - 1145810466v^16 + 6592520229v^15 + 24985916492v^14 - 39165383748v^13 + 407299798448v^12 - 444635751856v^11 - 16455359245696v^10 + 31229684808448v^9 + 254075443275776v^8 - 337690320547840v^7 - 1102678310453248v^6 + 1472660100612096v^5 - 18613792178962432v^4 - 49157013485125632v^3 + 692899524384915456v^2 - 81055567702917120*v - 7971915667421003776) / 13989911073914880
$\beta_{6}$	$=$	$( - 1009053331 \nu^{19} + 5955093094 \nu^{18} + 92664246339 \nu^{17} + 23483121654 \nu^{16} + \cdots + 17\!\cdots\!04 ) / 53\!\cdots\!60$ (-1009053331v^19 + 5955093094v^18 + 92664246339v^17 + 23483121654v^16 - 677089321159v^15 + 3386184621532v^14 + 8948119875116v^13 + 28585511805168v^12 + 422451709184400v^11 + 1842365967659648v^10 - 6803910244993280v^9 - 7117010603464704v^8 + 163360662163329024v^7 + 266180090303021056v^6 + 667307502036844544v^5 + 4427672373038153728v^4 - 1299921691522105344v^3 - 48226044564681523200v^2 + 165267730063935143936*v + 1789430898294368763904) / 535114098577244160
$\beta_{7}$	$=$	$( 836343 \nu^{19} + 4825138 \nu^{18} + 26902617 \nu^{17} - 67997406 \nu^{16} + \cdots + 32\!\cdots\!04 ) / 241205363343360$ (836343v^19 + 4825138v^18 + 26902617v^17 - 67997406v^16 - 250315557v^15 + 785814196v^14 + 4487363652v^13 + 26267975248v^12 + 131794625968v^11 - 71168402048v^10 - 2500521196288v^9 - 1606553298944v^8 + 47753989304320v^7 + 65189977194496v^6 + 147694426521600v^5 + 212935867629568v^4 - 4662885082988544v^3 - 2307979059658752v^2 + 67795045155078144*v + 326224344046895104) / 241205363343360
$\beta_{8}$	$=$	$( 3296223 \nu^{19} - 144154430 \nu^{18} - 510836175 \nu^{17} + 2513179506 \nu^{16} + \cdots - 26\!\cdots\!04 ) / 874369442119680$ (3296223v^19 - 144154430v^18 - 510836175v^17 + 2513179506v^16 - 302758557v^15 - 38622839180v^14 + 23498844708v^13 - 598187911088v^12 - 4937404206032v^11 + 6016066387840v^10 + 56099885206784v^9 - 258574748911616v^8 - 831571548753920v^7 + 138450305351680v^6 - 10419102624251904v^5 - 7726509879984128v^4 + 131950309923618816v^3 - 207790633207726080v^2 - 3088787054461452288*v - 2698545235025199104) / 874369442119680
$\beta_{9}$	$=$	$( - 4110998567 \nu^{19} + 2877435278 \nu^{18} + 134445630231 \nu^{17} + \cdots + 12\!\cdots\!36 ) / 10\!\cdots\!20$ (-4110998567v^19 + 2877435278v^18 + 134445630231v^17 + 31119529182v^16 - 1125456230411v^15 + 2395691636972v^14 - 1327532567300v^13 - 65684958912720v^12 + 749430500797776v^11 + 2601937693755520v^10 - 10940955354203392v^9 - 16243278503245824v^8 + 132142915823357952v^7 + 69003910881148928v^6 + 910441646450802688v^5 + 7670849992021508096v^4 - 9197198500901683200v^3 - 96771021572960944128v^2 + 283901422599109869568*v + 1261662170434487975936) / 1070228197154488320
$\beta_{10}$	$=$	$( 260801237 \nu^{19} - 1824079658 \nu^{18} - 2872664613 \nu^{17} + 20832970854 \nu^{16} + \cdots - 13\!\cdots\!60 ) / 54\!\cdots\!60$ (260801237v^19 - 1824079658v^18 - 2872664613v^17 + 20832970854v^16 - 17140065151v^15 - 40099565828v^14 - 527426412916v^13 - 6963291085968v^12 - 19077433527792v^11 + 68364900248192v^10 + 239411450138368v^9 - 2610189305567232v^8 - 2069946286534656v^7 - 23652572186869760v^6 - 84020616738635776v^5 + 108333149439655936v^4 - 87124716917096448v^3 - 8007115074579726336v^2 - 20198180721396809728*v - 13320290281937960960) / 54883497289973760
$\beta_{11}$	$=$	$( 5795805 \nu^{19} + 4381350 \nu^{18} - 62274925 \nu^{17} + 58776310 \nu^{16} + \cdots - 62\!\cdots\!00 ) / 932660738260992$ (5795805v^19 + 4381350v^18 - 62274925v^17 + 58776310v^16 + 1200456105v^15 + 95469500v^14 + 19123443180v^13 + 95638449520v^12 - 344796491120v^11 - 1303549488000v^10 + 9868738819840v^9 + 21718028912640v^8 - 8979288780800v^7 + 149102573977600v^6 + 60096778076160v^5 - 4584323499950080v^4 + 10719961295093760v^3 + 94815187999129600v^2 - 7978245349703680*v - 620428641750220800) / 932660738260992
$\beta_{12}$	$=$	$( - 1017903 \nu^{19} + 5922366 \nu^{18} + 40945759 \nu^{17} + 18273230 \nu^{16} + \cdots + 10\!\cdots\!80 ) / 160803575562240$ (-1017903v^19 + 5922366v^18 + 40945759v^17 + 18273230v^16 - 696329523v^15 + 1014965932v^14 + 12216570780v^13 - 10837650640v^12 + 233765807312v^11 + 1469673965184v^10 - 4388269305088v^9 - 10438841456640v^8 + 89504132157440v^7 + 232292172234752v^6 - 248550134120448v^5 + 3349064691220480v^4 + 5978665091334144v^3 - 59776185520881664v^2 + 93891988000079872*v + 1056826868398817280) / 160803575562240
$\beta_{13}$	$=$	$( - 6553647 \nu^{19} + 4110910 \nu^{18} + 234956895 \nu^{17} - 334260274 \nu^{16} + \cdots + 28\!\cdots\!96 ) / 777217281884160$ (-6553647v^19 + 4110910v^18 + 234956895v^17 - 334260274v^16 - 2157450547v^15 + 5693424300v^14 - 2662055012v^13 - 75569044688v^12 + 1156498253008v^11 + 4108959401600v^10 - 19669523737856v^9 - 30778824841216v^8 + 234445426708480v^7 + 143375976693760v^6 + 1313387938840576v^5 + 11563371375624192v^4 - 4002535035633664v^3 - 178325683846512640v^2 + 36119523908124672*v + 2851010354920554496) / 777217281884160
$\beta_{14}$	$=$	$( - 11729669627 \nu^{19} - 7756672522 \nu^{18} + 162936990891 \nu^{17} + \cdots + 24\!\cdots\!40 ) / 10\!\cdots\!20$ (-11729669627v^19 - 7756672522v^18 + 162936990891v^17 - 194149219962v^16 - 2398382102543v^15 - 3278410510660v^14 - 28704426242612v^13 - 254793725413776v^12 + 774486071124240v^11 + 4078553528162944v^10 - 21442484581097728v^9 - 60625577667385344v^8 + 112534678343651328v^7 - 26970025576431616v^6 - 581936411104772096v^5 + 10340798851136356352v^4 + 6747486828795789312v^3 - 293300664712524988416v^2 + 80374664847897395200*v + 2488771395149512048640) / 1070228197154488320
$\beta_{15}$	$=$	$( 57066063 \nu^{19} + 274798978 \nu^{18} - 936303743 \nu^{17} + 2504816114 \nu^{16} + \cdots - 51\!\cdots\!96 ) / 46\!\cdots\!60$ (57066063v^19 + 274798978v^18 - 936303743v^17 + 2504816114v^16 + 13924163283v^15 - 23482753324v^14 + 192206479332v^13 + 2782240740048v^12 + 1439053804848v^11 - 30347313231488v^10 + 88322490633472v^9 + 496240116422656v^8 - 152588912087040v^7 + 5925686790782976v^6 + 17277039999713280v^5 - 54335566891712512v^4 - 153856678847053824v^3 + 1497369574205554688v^2 + 4219199795111133184*v - 5100257240045060096) / 4663303691304960
$\beta_{16}$	$=$	$( 445476863 \nu^{19} + 755206978 \nu^{18} - 2379736431 \nu^{17} + 690187890 \nu^{16} + \cdots - 55\!\cdots\!20 ) / 31\!\cdots\!80$ (445476863v^19 + 755206978v^18 - 2379736431v^17 + 690187890v^16 + 41497627523v^15 + 424482480052v^14 + 2364490799780v^13 + 8491155454800v^12 - 2213726830032v^11 - 135462442775680v^10 + 443775654153472v^9 + 3113792848312320v^8 + 7601811276902400v^7 + 8147471708520448v^6 + 7364567046029312v^5 - 78951472682762240v^4 + 130657928091795456v^3 + 9577769516843139072v^2 + 24592412889225101312*v - 55918171747851960320) / 31477299916308480
$\beta_{17}$	$=$	$( - 279607999 \nu^{19} + 207980926 \nu^{18} + 4651875759 \nu^{17} - 5763844146 \nu^{16} + \cdots + 21\!\cdots\!60 ) / 18\!\cdots\!20$ (-279607999v^19 + 207980926v^18 + 4651875759v^17 - 5763844146v^16 - 69380146627v^15 - 15389892212v^14 - 180631027876v^13 - 5214217274448v^12 + 24807733624272v^11 + 58583395989632v^10 - 593749860183296v^9 - 880987800532992v^8 + 2653284081782784v^7 + 1255725749960704v^6 - 25940664486461440v^5 + 165537541947129856v^4 - 637830318060994560v^3 - 6394634195309690880v^2 + 7376531761657806848*v + 21477900337870274560) / 18294499096657920
$\beta_{18}$	$=$	$( - 34415252287 \nu^{19} + 44118579838 \nu^{18} + 614301056559 \nu^{17} + \cdots + 46\!\cdots\!88 ) / 21\!\cdots\!40$ (-34415252287v^19 + 44118579838v^18 + 614301056559v^17 - 769774935858v^16 - 4977821113411v^15 - 2337983947892v^14 - 48612162875044v^13 - 299854001953872v^12 + 2489242855970256v^11 + 9428986422343808v^10 - 59405241520221440v^9 - 80251333231441920v^8 + 266111322945933312v^7 - 85505550715781120v^6 + 1030837898012524544v^5 + 13514949335179067392v^4 - 21716798153781411840v^3 - 471425196044209618944v^2 - 274160655454336712704*v + 4667308348567283826688) / 2140456394308976640
$\beta_{19}$	$=$	$( 240970995 \nu^{19} + 955006394 \nu^{18} - 2146959459 \nu^{17} - 1890330006 \nu^{16} + \cdots - 22\!\cdots\!56 ) / 13\!\cdots\!80$ (240970995v^19 + 955006394v^18 - 2146959459v^17 - 1890330006v^16 + 60572738535v^15 + 136049634308v^14 + 605040164052v^13 + 7864865425808v^12 + 7262658201200v^11 - 95866524978304v^10 + 297809594185984v^9 + 2218657466710016v^8 + 1657499988684800v^7 + 6687192417763328v^6 + 58820173022625792v^5 - 45042636666437632v^4 - 419669485478215680v^3 + 4829801159772340224v^2 + 13555780229221318656*v - 22273251727592390656) / 13989911073914880

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

$\nu$	$=$	$( 4 \beta_{19} - 14 \beta_{18} - \beta_{17} - 6 \beta_{16} - 9 \beta_{15} - \beta_{14} - 21 \beta_{13} + \cdots + 1269 ) / 12800$ (4b19 - 14b18 - b17 - 6b16 - 9b15 - b14 - 21b13 + 5b12 - 28b11 - 4b10 + 14b9 - 4b8 + 4b7 - 10b6 - 31b5 - 15b4 - 45b3 + 2b2 - 52*b1 + 1269) / 12800
$\nu^{2}$	$=$	$( - 44 \beta_{19} + 14 \beta_{18} - 39 \beta_{17} + 6 \beta_{16} + 19 \beta_{15} - 19 \beta_{14} + \cdots + 24401 ) / 12800$ (-44b19 + 14b18 - 39b17 + 6b16 + 19b15 - 19b14 - 9b13 + 5b12 + 84b11 - 36b10 + 106b9 + 4b8 + 56b7 - 50b6 + b5 + 445b4 + 125b3 - 378b2 + 1252b1 + 24401) / 12800
$\nu^{3}$	$=$	$( - 84 \beta_{19} + 18 \beta_{18} - 108 \beta_{17} + 112 \beta_{16} + 39 \beta_{15} + 192 \beta_{14} + \cdots - 39819 ) / 6400$ (-84b19 + 18b18 - 108b17 + 112b16 + 39b15 + 192b14 + 41b13 - 5b12 - 126b11 + 28b10 - 68b9 - 16b8 - 34b7 - 110b6 + 251b5 - 985b4 - 220b3 + 212b2 + 2714*b1 - 39819) / 6400
$\nu^{4}$	$=$	$( 36 \beta_{19} - 10 \beta_{18} - \beta_{17} + 10 \beta_{16} - 17 \beta_{15} + 11 \beta_{14} + \cdots - 19739 ) / 512$ (36b19 - 10b18 - b17 + 10b16 - 17b15 + 11b14 + 7b13 - 11b12 - 88b11 - 36b10 - 34b9 + 12b8 + 28b7 - 2b6 - 71b5 - 75b4 + 43b3 - 30b2 + 1776b1 - 19739) / 512
$\nu^{5}$	$=$	$( 68 \beta_{19} - 654 \beta_{18} - 911 \beta_{17} + 134 \beta_{16} + 347 \beta_{15} + 1289 \beta_{14} + \cdots - 1730847 ) / 12800$ (68b19 - 654b18 - 911b17 + 134b16 + 347b15 + 1289b14 + 2843b13 - 515b12 - 3920b11 - 1044b10 + 9754b9 + 532b8 - 2232b7 + 3790b6 + 13073b5 - 35655b4 - 195b3 - 218b2 - 289672*b1 - 1730847) / 12800
$\nu^{6}$	$=$	$( - 3196 \beta_{19} - 10606 \beta_{18} + 7856 \beta_{17} - 5024 \beta_{16} + 1491 \beta_{15} + \cdots - 9927991 ) / 6400$ (-3196b19 - 10606b18 + 7856b17 - 5024b16 + 1491b15 - 1324b14 + 4689b13 - 2585b12 + 18014b11 - 7356b10 - 8124b9 + 296b8 - 2066b7 + 9850b6 + 14639b5 - 2185b4 + 6400b3 + 43172b2 + 285842*b1 - 9927991) / 6400
$\nu^{7}$	$=$	$( 16324 \beta_{19} + 49938 \beta_{18} - 14313 \beta_{17} + 19162 \beta_{16} + 14471 \beta_{15} + \cdots - 78838411 ) / 12800$ (16324b19 + 49938b18 - 14313b17 + 19162b16 + 14471b15 + 85487b14 - 15301b13 - 9795b12 + 106052b11 + 27388b10 + 5262b9 + 10476b8 + 8124b7 + 71150b6 + 16089b5 + 514385b4 + 84475b3 - 125070b2 - 2814756*b1 - 78838411) / 12800
$\nu^{8}$	$=$	$( 2676 \beta_{19} + 4086 \beta_{18} + 3473 \beta_{17} - 10 \beta_{16} + 1195 \beta_{15} - 1331 \beta_{14} + \cdots + 6757289 ) / 512$ (2676b19 + 4086b18 + 3473b17 - 10b16 + 1195b15 - 1331b14 + 3991b13 - 1923b12 + 21532b11 - 1412b10 - 13574b9 - 1532b8 - 14040b7 + 6094b6 - 7767b5 - 54467b4 + 28837b3 + 410198b2 + 227036*b1 + 6757289) / 512
$\nu^{9}$	$=$	$( 525340 \beta_{19} - 91758 \beta_{18} + 136748 \beta_{17} - 210672 \beta_{16} - 145665 \beta_{15} + \cdots + 480581725 ) / 6400$ (525340b19 - 91758b18 + 136748b17 - 210672b16 - 145665b15 + 233448b14 - 86535b13 + 41875b12 + 372282b11 + 194732b10 + 98508b9 + 16160b8 - 71810b7 - 347390b6 - 120285b5 + 4778375b4 - 1566980b3 + 8138228b2 - 22020534*b1 + 480581725) / 6400
$\nu^{10}$	$=$	$( 325380 \beta_{19} + 731134 \beta_{18} - 1210009 \beta_{17} - 3982214 \beta_{16} - 936425 \beta_{15} + \cdots - 2969570195 ) / 12800$ (325380b19 + 731134b18 - 1210009b17 - 3982214b16 - 936425b15 - 6039989b14 - 2324665b13 + 2845805b12 + 6414912b11 + 220284b10 + 3760686b9 - 1129940b8 - 1680900b7 - 2164850b6 - 9232575b5 + 15202165b4 + 5661475b3 + 63180562b2 - 25585688*b1 - 2969570195) / 12800
$\nu^{11}$	$=$	$( - 13704092 \beta_{19} + 5590458 \beta_{18} - 4960383 \beta_{17} + 5939142 \beta_{16} + \cdots + 13219713793 ) / 12800$ (-13704092b19 + 5590458b18 - 4960383b17 + 5939142b16 + 13609707b15 + 3599617b14 + 13910883b13 + 3487885b12 + 30064360b11 + 14815308b10 + 6984442b9 - 1260108b8 + 889608b7 - 30912850b6 - 2883887b5 + 146464945b4 + 764125b3 - 291693850b2 + 186641504*b1 + 13219713793) / 12800
$\nu^{12}$	$=$	$( 526228 \beta_{19} + 377730 \beta_{18} - 609080 \beta_{17} + 191424 \beta_{16} - 227613 \beta_{15} + \cdots - 1250502055 ) / 256$ (526228b19 + 377730b18 - 609080b17 + 191424b16 - 227613b15 - 677156b14 - 400887b13 + 325399b12 - 4508218b11 + 575860b10 - 3495116b9 - 200168b8 + 293886b7 - 897510b6 - 5893585b5 + 1560751b4 - 2596520b3 - 8567708b2 + 25492274*b1 - 1250502055) / 256
$\nu^{13}$	$=$	$( - 14642492 \beta_{19} + 103278602 \beta_{18} + 45947223 \beta_{17} + 146964698 \beta_{16} + \cdots + 272916930693 ) / 12800$ (-14642492b19 + 103278602b18 + 45947223b17 + 146964698b16 - 13739193b15 - 274655577b14 + 167747283b13 + 59506685b12 - 363730660b11 - 19081348b10 + 268349198b9 + 54022892b8 + 15173308b7 - 142812050b6 - 602015287b5 - 286535655b4 + 132428075b3 - 8925510030b2 - 8029473324*b1 + 272916930693) / 12800
$\nu^{14}$	$=$	$( - 610849420 \beta_{19} - 1191174514 \beta_{18} + 436423289 \beta_{17} - 95003706 \beta_{16} + \cdots - 341144190495 ) / 12800$ (-610849420b19 - 1191174514b18 + 436423289b17 - 95003706b16 - 268281725b15 - 994905731b14 + 79948935b13 + 103955205b12 - 2654362828b11 + 149850236b10 - 341290806b9 - 108402940b8 - 428176600b7 + 779741950b6 + 3793620225b5 - 4026719635b4 - 3556998275b3 - 21797731162b2 + 7334948*b1 - 341144190495) / 12800
$\nu^{15}$	$=$	$( - 871984500 \beta_{19} - 160118542 \beta_{18} + 880534852 \beta_{17} - 86269328 \beta_{16} + \cdots - 1625292214715 ) / 6400$ (-871984500b19 - 160118542b18 + 880534852b17 - 86269328b16 - 506109225b15 - 680664048b14 - 869833975b13 - 620805525b12 + 1940428082b11 - 653569732b10 + 179384092b9 + 437480400b8 + 1615280350b7 + 1956387090b6 - 4627899125b5 + 9640071975b4 + 2470103380b3 - 135376526828b2 + 99281887834*b1 - 1625292214715) / 6400
$\nu^{16}$	$=$	$( - 408459356 \beta_{19} + 447320870 \beta_{18} - 120078593 \beta_{17} + 597058826 \beta_{16} + \cdots + 326130296645 ) / 512$ (-408459356b19 + 447320870b18 - 120078593b17 + 597058826b16 + 373970095b15 + 404508155b14 - 145902025b13 - 287005067b12 - 513172456b11 + 377274204b10 + 399285342b9 + 62833164b8 - 308012196b7 + 640417470b6 + 2260962329b5 - 5024644539b4 + 90840331b3 + 2217999970b2 + 13561143584*b1 + 326130296645) / 512
$\nu^{17}$	$=$	$( 42668508804 \beta_{19} - 14442191358 \beta_{18} + 46349039633 \beta_{17} + 2523089158 \beta_{16} + \cdots + 54880897505569 ) / 12800$ (42668508804b19 - 14442191358b18 + 46349039633b17 + 2523089158b16 - 39595954309b15 + 18573143033b14 - 20870356821b13 - 55490564195b12 + 38806841632b11 - 36430765908b10 - 71473967142b9 + 1357837396b8 + 21933976904b7 + 18461223950b6 - 36070946031b5 - 412462979415b4 - 143906710275b3 + 1840354413990b2 + 3895031802696*b1 + 54880897505569) / 12800
$\nu^{18}$	$=$	$( 79647000164 \beta_{19} + 72314538066 \beta_{18} - 46315267616 \beta_{17} - 50923610336 \beta_{16} + \cdots + 31192842076569 ) / 6400$ (79647000164b19 + 72314538066b18 - 46315267616b17 - 50923610336b16 - 46613619069b15 + 68569046564b14 - 145785265551b13 + 42593445015b12 - 35420251826b11 - 4963779484b10 + 314785960164b9 + 15303372936b8 + 81780472494b7 - 3031464550b6 + 734392870399b5 - 459204576585b4 - 162479678800b3 + 2226298022628b2 - 4383724445262*b1 + 31192842076569) / 6400
$\nu^{19}$	$=$	$( - 417829414972 \beta_{19} - 400451474046 \beta_{18} - 181387347689 \beta_{17} + \cdots + 255280619285013 ) / 12800$ (-417829414972b19 - 400451474046b18 - 181387347689b17 - 512963191334b16 + 119021860487b15 - 54094601889b14 + 205611761003b13 - 62903746115b12 + 3357788221300b11 - 23728061956b10 + 971350946446b9 - 236613835028b8 + 381057629628b7 - 549069993490b6 + 2266265144633b5 - 9631575098655b4 + 2731604216795b3 + 41915195352178b2 + 56906488516172*b1 + 255280619285013) / 12800

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$ .

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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}

81.1

3.46430	+	1.99965i
−3.90102	−	0.884346i
0.236693	−	3.99299i
2.93366	+	2.71913i
3.18502	−	2.41984i
−3.80026	−	1.24819i
3.72553	−	1.45618i
−2.80358	+	2.85306i
0.593959	+	3.95566i
−2.63430	+	3.01006i
−2.63430	−	3.01006i
0.593959	−	3.95566i
−2.80358	−	2.85306i
3.72553	+	1.45618i
−3.80026	+	1.24819i
3.18502	+	2.41984i
2.93366	−	2.71913i
0.236693	+	3.99299i
−3.90102	+	0.884346i
3.46430	−	1.99965i

−

29.2080i

−

25.0000i

168.173

−610.110

81.2

−

25.4343i

25.0000i

56.4938

−403.904

81.3

−

25.0521i

−

25.0000i

−103.624

−384.607

81.4

−

18.7876i

25.0000i

107.536

−109.975

81.5

−

17.3148i

25.0000i

9.19080

−56.8021

81.6

−

11.5927i

−

25.0000i

231.529

108.609

81.7

−

10.8240i

−

25.0000i

−163.706

125.841

81.8

−

10.7455i

25.0000i

−198.733

127.535

81.9

−

6.93089i

−

25.0000i

−47.1406

194.963

81.10

−

6.67450i

−

25.0000i

38.2812

198.451

81.11

6.67450i

25.0000i

38.2812

198.451

81.12

6.93089i

25.0000i

−47.1406

194.963

81.13

10.7455i

−

25.0000i

−198.733

127.535

81.14

10.8240i

25.0000i

−163.706

125.841

81.15

11.5927i

25.0000i

231.529

108.609

81.16

17.3148i

−

25.0000i

9.19080

−56.8021

81.17

18.7876i

−

25.0000i

107.536

−109.975

81.18

25.0521i

25.0000i

−103.624

−384.607

81.19

25.4343i

−

25.0000i

56.4938

−403.904

81.20

29.2080i

25.0000i

168.173

−610.110

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	160.6.d.a		20
4.b	odd	2	1		40.6.d.a	✓	20
5.b	even	2	1		800.6.d.c		20
5.c	odd	4	1		800.6.f.b		20
5.c	odd	4	1		800.6.f.c		20
8.b	even	2	1	inner	160.6.d.a		20
8.d	odd	2	1		40.6.d.a	✓	20
12.b	even	2	1		360.6.k.b		20
20.d	odd	2	1		200.6.d.b		20
20.e	even	4	1		200.6.f.b		20
20.e	even	4	1		200.6.f.c		20
24.f	even	2	1		360.6.k.b		20
40.e	odd	2	1		200.6.d.b		20
40.f	even	2	1		800.6.d.c		20
40.i	odd	4	1		800.6.f.b		20
40.i	odd	4	1		800.6.f.c		20
40.k	even	4	1		200.6.f.b		20
40.k	even	4	1		200.6.f.c		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.6.d.a	✓	20	4.b	odd	2	1
40.6.d.a	✓	20	8.d	odd	2	1
160.6.d.a		20	1.a	even	1	1	trivial
160.6.d.a		20	8.b	even	2	1	inner
200.6.d.b		20	20.d	odd	2	1
200.6.d.b		20	40.e	odd	2	1
200.6.f.b		20	20.e	even	4	1
200.6.f.b		20	40.k	even	4	1
200.6.f.c		20	20.e	even	4	1
200.6.f.c		20	40.k	even	4	1
360.6.k.b		20	12.b	even	2	1
360.6.k.b		20	24.f	even	2	1
800.6.d.c		20	5.b	even	2	1
800.6.d.c		20	40.f	even	2	1
800.6.f.b		20	5.c	odd	4	1
800.6.f.b		20	40.i	odd	4	1
800.6.f.c		20	5.c	odd	4	1
800.6.f.c		20	40.i	odd	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{6}^{\mathrm{new}}(160, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20}$ T^20
$3$	$T^{20} + \cdots + 14\!\cdots\!84$ T^20 + 3240T^18 + 4346772T^16 + 3151305344T^14 + 1355603009184T^12 + 359218958085888T^10 + 59164106793818752T^8 + 5978764455444817920T^6 + 354916377079609533696T^4 + 11177338398394463004672*T^2 + 142602076715343595766784
$5$	$(T^{2} + 625)^{10}$ (T^2 + 625)^10
$7$	$(T^{10} + \cdots + 13\!\cdots\!08)^{2}$ (T^10 - 98T^9 - 83922T^8 + 6806560T^7 + 2129001128T^6 - 149256489744T^5 - 16715051341136T^4 + 1088925123904640T^3 + 22278527346308496T^2 - 1721950139896063008*T + 13226564728477155808)^2
$11$	$T^{20} + \cdots + 70\!\cdots\!00$ T^20 + 1711016T^18 + 1187234134224T^16 + 438539966696586752T^14 + 94879747424584980517376T^12 + 12478678172254103334566031360T^10 + 1004907526767812027210486637469696T^8 + 48442269228536219186486307527193395200T^6 + 1306753581309652020110605216191306670080000T^4 + 16880002061984162758418768553020776095744000000*T^2 + 70167852897926169794298630715984656885760000000000
$13$	$T^{20} + \cdots + 16\!\cdots\!00$ T^20 + 3905528T^18 + 6136454126416T^16 + 4988469520919693824T^14 + 2258733925223246490825216T^12 + 577651295500612071992616407040T^10 + 82106651575366167887526352582877184T^8 + 6137788881525358860884565374533307596800T^6 + 215711768162188164741601449589076573429760000T^4 + 3217057431922814635360338733419212496494592000000*T^2 + 16511112125491170401455720758406419700190822400000000
$17$	$(T^{10} + \cdots + 29\!\cdots\!68)^{2}$ (T^10 - 8750908T^8 + 1859072000T^7 + 23998151145888T^6 - 11063772518400000T^5 - 21446330507601376128T^4 + 14317463194873602048000T^3 + 1988905722934233183245568T^2 - 2313907995449384505999360000T + 294580496690248704818606650368)^2
$19$	$T^{20} + \cdots + 23\!\cdots\!56$ T^20 + 25883288T^18 + 285046332695760T^16 + 1746271114538976571904T^14 + 6516200664459955340153573888T^12 + 15202268167893997261028763450249216T^10 + 21804659081583891282267655790665406685184T^8 + 17974632097588711413964184354778560588432932864T^6 + 7196396079747636051136014188955800980746248402042880T^4 + 834234046827879548853650177744267006656537537814315139072*T^2 + 23751782553356803907066925574188970785243151884587053540704256
$23$	$(T^{10} + \cdots - 88\!\cdots\!16)^{2}$ (T^10 - 2338T^9 - 28365770T^8 + 90426486752T^7 + 117354068596520T^6 - 637751007312995728T^5 + 365256046547394812400T^4 + 854744129145344502855552T^3 - 1245351955498942534596098160T^2 + 573732979732993040095696684512*T - 88471012791702948854644084275616)^2
$29$	$T^{20} + \cdots + 42\!\cdots\!00$ T^20 + 195130080T^18 + 15351673563972864T^16 + 640582862784030126800896T^14 + 15646238138232348012837933809664T^12 + 232336863610611750902129383428694474752T^10 + 2095588350906987898603680361094488537247514624T^8 + 11020729012782863909635144395483876077710638632140800T^6 + 30510345197779619183407906041426326120480519560636661760000T^4 + 34783649959736879549990698432082433127279711185333682962432000000*T^2 + 4257741901690041830921675255673030673204495643468955751179878400000000
$31$	$(T^{10} + \cdots - 42\!\cdots\!88)^{2}$ (T^10 + 3580T^9 - 153869896T^8 - 437494937344T^7 + 7341181528205952T^6 + 15797753164246990336T^5 - 108500883515612631520256T^4 - 149396246874032910324449280T^3 + 346432850950705074038378958848T^2 + 271599668713170267985848132943872*T - 42506672219643903529913221101682688)^2
$37$	$T^{20} + \cdots + 17\!\cdots\!36$ T^20 + 821506472T^18 + 283168092338137296T^16 + 53614409791166010231193088T^14 + 6122137772446843167011525370335744T^12 + 435476238395264394888571662139150206824448T^10 + 19258147448360316285191821612057874042851730464768T^8 + 513926248767002937244179018545905586444147911666188550144T^6 + 7784155353020183914146654116937200373273586250173081078993846272T^4 + 59973502819027341913874854008893787249829232173475324215775608825184256*T^2 + 178725925096385086823221609054630064267136066709897011138677253330512181723136
$41$	$(T^{10} + \cdots - 21\!\cdots\!00)^{2}$ (T^10 - 5804T^9 - 577824776T^8 + 1875250039808T^7 + 112673734070960256T^6 - 80442400781905077760T^5 - 8724196432503550885413888T^4 - 8072080537977110753976320000T^3 + 232931407962635431304496499200000T^2 + 199866639445197069256822776907776000*T - 2129594087346616896515164046253117440000)^2
$43$	$T^{20} + \cdots + 24\!\cdots\!76$ T^20 + 1732042120T^18 + 1264805616873797588T^16 + 508983510551730053880975488T^14 + 123765439745221626490204363571884704T^12 + 18741109459618538200143459210026335609594624T^10 + 1752931352996502522619923709865477273421215088243328T^8 + 96489295973780224138283590103208494998079285542187504732160T^6 + 2776189875669765470744881687914064251111731104099509007677874693376T^4 + 30596627988727262560117702312678421906091402339625222547883720417390790656*T^2 + 24226051988890617857696446673164212398296144306700432902045548976473856364594176
$47$	$(T^{10} + \cdots + 16\!\cdots\!92)^{2}$ (T^10 + 22090T^9 - 744939994T^8 - 19816630242400T^7 + 65043919811739304T^6 + 4220725961073437022160T^5 + 18381802218896663308929136T^4 - 177736154425140630725117301120T^3 - 1290246234067258465992384824144496T^2 - 1088469430275725328209235292887111520*T + 1633295114563215923585010454685317693792)^2
$53$	$T^{20} + \cdots + 82\!\cdots\!44$ T^20 + 4360240504T^18 + 7547899229497442640T^16 + 6586760769321828854564055552T^14 + 3055113607573734123194975509915267584T^12 + 748448778205602133781496299733062999342764032T^10 + 99289215064468658604196106516391089018876921962831872T^8 + 7104624154343797858245194153211702542985561006376336839278592T^6 + 256940821319584926627329786737352507382807947685669409504974584217600T^4 + 3804504188681598899657666365308957629805554311077278765485913150344893300736*T^2 + 8218543534197564144428436599117103779959729611227226932849035850171679617758265344
$59$	$T^{20} + \cdots + 40\!\cdots\!76$ T^20 + 4301536152T^18 + 7893526636419653840T^16 + 8057909856982563696503468544T^14 + 4997240412317258284245273221006938624T^12 + 1926809239516110941759681445082404257229869056T^10 + 452552672078695105663606535205093469091837361806450688T^8 + 60333291889295619630686363579727304955206280899012722800328704T^6 + 3885689727393451818198156298396423768861904695585144732307083831869440T^4 + 81120499824905058930649873005614923911871573274805097068089259040402178572288*T^2 + 403540028902294593914841834126903142619544297303781223921385131870339566130116427776
$61$	$T^{20} + \cdots + 29\!\cdots\!00$ T^20 + 9739073672T^18 + 39049057139936353872T^16 + 83604204136394781079545099776T^14 + 103864876633929765741157010751854555648T^12 + 76497978730975905181579976859465337533970722816T^10 + 32940149491321999386327126552634621600395737162649051136T^8 + 7867646420793382063636770322422120013417044016640913399778508800T^6 + 932336525459732023596754647103098594070787814585152355478000006471680000T^4 + 44543458855540294227023108297954067311982092847836388949181729965520265216000000*T^2 + 299537713907462762131805512876854462668349802239887200099961261832556558428160000000000
$67$	$T^{20} + \cdots + 50\!\cdots\!04$ T^20 + 15173983176T^18 + 91092031416369225300T^16 + 277467605920414066270535337088T^14 + 461485195273952838210263922149980158624T^12 + 426166422723961317457240497185137134478478162688T^10 + 214225754003586296081280925050694334801430561241165587072T^8 + 55621794683804803098661024739985475701241961075651216203962697728T^6 + 6666929836164768286085153987487191803105409505566468283608631589071709440T^4 + 318839816024018478222547961436447167288934485116688345883375545267211411597871104*T^2 + 5089672139612270986738224343903797791662025920318980266106715498794674280744413391995904
$71$	$(T^{10} + \cdots + 19\!\cdots\!32)^{2}$ (T^10 - 100156T^9 - 4593845064T^8 + 668893729933056T^7 - 3305303406670534016T^6 - 1029998039283869641104896T^5 + 17117410470145166387347434496T^4 + 243496951842460889215838375100416T^3 - 4045029571199781234215837031742468096T^2 - 11429043178618743826173449632452362878976*T + 199215962933176171034180132498264362490232832)^2
$73$	$(T^{10} + \cdots - 24\!\cdots\!16)^{2}$ (T^10 + 52568T^9 - 11351373036T^8 - 606266359383680T^7 + 32586480428083645088T^6 + 1815124743592703150563584T^5 - 25534871464500172795398923648T^4 - 1472196464690865551891671779665920T^3 + 11296022259716543464884773703751623936T^2 + 340283614301808839867207685493974187726848*T - 2484162303900341550147901680105283004478422016)^2
$79$	$(T^{10} + \cdots + 28\!\cdots\!00)^{2}$ (T^10 + 141040T^9 - 262124464T^8 - 878655619892224T^7 - 38639251881508836864T^6 + 368804259967459976519680T^5 + 46432072671986912096531685376T^4 + 246422077195686913044334639841280T^3 - 18993047487205440986435864422388531200T^2 - 116990025584986458985990635244300664832000*T + 2898380858667191038277431903243203835985920000)^2
$83$	$T^{20} + \cdots + 23\!\cdots\!36$ T^20 + 44012159480T^18 + 727681127398744111668T^16 + 5681169449754670235756897534848T^14 + 21878543541004798036916296912839587999904T^12 + 41275199782849078422880207380152868984441202162944T^10 + 36334764354335363723945978723444427299335734590758150059648T^8 + 13998863075049807812435566178223805110575070833853972557663735592960T^6 + 2036771244426971092848443599539582042165494837732410886300739271290843284736T^4 + 119627794582621301277879879485562069254756029387002744588319042504128263242759796736*T^2 + 2389527001683006690330779300263178280690821049985093737529872372090877116613739578206299136
$89$	$(T^{10} + \cdots + 21\!\cdots\!00)^{2}$ (T^10 + 1580T^9 - 28442400444T^8 + 380635615133504T^7 + 252005360584542180896T^6 - 7563365388046180744772480T^5 - 671540242766408546810621844864T^4 + 34610702600086976220464891122242560T^3 - 377933131026737681673796074215287980800T^2 - 1425549484010168477903499693504159761536000*T + 2148613791468302474557427046244603353019520000)^2
$97$	$(T^{10} + \cdots + 22\!\cdots\!48)^{2}$ (T^10 - 73688T^9 - 42200485708T^8 + 2877738584271488T^7 + 564180396810943351968T^6 - 37013898413631660974129408T^5 - 2458922010489390687930848405888T^4 + 177124349795910544583780048030803968T^3 + 1182400804278885328761347209268667876608T^2 - 193385696003934905939679749756550287788136448*T + 2230363856141118897530997543954042514224613762048)^2
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 81.20
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

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	160.6.d.a

Level	$160$
Weight	$6$
Character orbit	160.d
Analytic conductor	$25.661$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$