LMFDB - Newform orbit 560.4.q.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(81,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("560.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$560 = 2^{4} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	560.q (of order $3$ , degree $2$ , 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:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 2 x^{11} + 95 x^{10} - 258 x^{9} + 7289 x^{8} - 15564 x^{7} + 170984 x^{6} + 88720 x^{5} + \cdots + 6718464$ x^12 - 2x^11 + 95x^10 - 258x^9 + 7289x^8 - 15564x^7 + 170984x^6 + 88720x^5 + 2094592x^4 - 681024x^3 + 5329728x^2 + 3297024*x + 6718464
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{10}$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{3} - \beta_1 - 1) q^{3} + 5 \beta_{3} q^{5} + (\beta_{9} + 3 \beta_{3} + \beta_1) q^{7} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots + 1) q^{9} + ( - \beta_{8} + \beta_{5} + \beta_{4} + \cdots - 2) q^{11}+ \cdots + ( - 3 \beta_{11} - 12 \beta_{10} + \cdots - 21) q^{99}+O(q^{100})$ q + (b3 - b1 - 1) * q^3 + 5b3 q^5 + (b9 + 3b3 + b1) q^7 + (b9 + b8 + b7 - b6 + b5 + b4 - 6b3 + 1) q^9 + (-b8 + b5 + b4 + 2b3 - b2 + 4b1 - 2) * q^11 + (-b10 - 2b8 - 2b6 + 2b5 + 2b3 - 23) * q^13 + (5b4 - 5) q^15 + (-2b10 + b9 + 2b8 - 2b7 - 2b6 - 22b3 - b2 - 6b1 + 22) * q^17 + (-b9 - 2b8 - 4b7 + b6 - 2b5 + 4b4 + 3b3 + 6b1 - 2) * q^19 + (-b9 + b8 - 3b7 - b6 - 2b5 + 4b4 + 15b3 + b2 + 6b1 + 7) q^21 + (-3b11 + 2b8 - b5 - 4b4 + 2b3 - 3b1 + 2) q^23 + (25b3 - 25) q^25 + (b11 + 3b8 + 4b6 - 3b5 - 6b4 - 3b3 + b2 + 8) q^27 + (b11 + 3b10 - 6b9 - 5b8 - 4b6 - b5 + 13b4 + 5b3 + b2 + 35) * q^29 + (-4b10 + 5b9 - 4b7 + 4b6 - 4b5 - 4b4 + 14b3 - b2 - 10b1 - 14) * q^31 + (-4b11 + 6b9 + 6b8 - 4b7 - 6b6 + 2b5 + 2b4 + 96b3 + 6) * q^33 + (5b9 + 5b8 - 5b4 + 10b3 - 10) * q^35 + (-b11 + 6b9 + b8 + b7 - 6b6 + 24b4 + 64b3 + 24b1 + 1) q^37 + (4b10 + b9 + 4b7 + 2b6 - 2b5 - 2b4 + 32b3 + b2 + 42b1 - 32) q^39 + (-5b11 - 2b10 + 6b9 - 5b8 - 10b6 + 11b5 + 15b4 + 5b3 - 5b2 - 58) q^41 + (4b11 - 4b10 - 6b9 + 4b6 - 6b5 - 31b4 + 4b2 - 111) q^43 + (5b10 + 5b8 + 5b7 - 5b6 - 30b3 + 30) q^45 + (-4b11 - 12b9 - 7b8 + 12b6 - 11b5 + 35b4 + 56b3 + 46b1 - 7) * q^47 + (7b11 + 7b10 + 5b7 - 3b6 + b5 - 27b4 - 104b3 + 3b2 - 4b1 + 13) * q^49 + (3b11 - 7b8 + 12b7 - 4b5 - 18b4 - 135b3 - 14b1 - 7) q^51 + (-11b10 - 6b9 - 4b8 - 11b7 - 6b6 + 10b5 + 10b4 + 125b3 - 4b2 - 2b1 - 125) * q^53 + (-5b11 + 5b9 - 5b6 + 5b5 - 15b4 - 5b2 - 5) * q^55 + (b11 + 4b10 - 19b9 - 13b8 - 12b6 - 6b5 + 84b4 + 13b3 + b2 + 115) q^57 + (-8b10 + 11b9 + 12b8 - 8b7 + 4b6 - 16b5 - 16b4 + 128b3 + 5b2 - 40b1 - 128) * q^59 + (6b11 + 4b9 - 12b8 - 3b7 - 4b6 - 6b5 - 48b4 + 162b3 - 42b1 - 12) q^61 + (8b11 + 4b10 - 7b9 + 4b7 + 9b6 - 17b5 + 31b4 + 147b3 + b2 - 14b1 + 44) q^63 + (10b9 + 5b7 - 10b6 - 10b4 - 105b3 - 10b1) * q^65 + (-4b10 + 7b9 + 5b8 - 4b7 - 5b6 - 258b3 - 7b2 - 5b1 + 258) * q^67 + (8b11 - 5b10 - 16b9 + 2b8 + 10b6 - 18b5 + 8b4 - 2b3 + 8b2 - 62) q^69 + (-5b11 + 24b10 - 11b9 - 3b8 - 8b6 - 8b5 - 28b4 + 3b3 - 5b2 + 9) q^71 + (-16b10 + 7b9 + 12b8 - 16b7 + 4b6 - 16b5 - 16b4 + 22b3 + 9b2 + 10b1 - 22) * q^73 + (25b4 - 25b3 + 25b1) q^75 + (-8b11 + 3b10 + 15b9 + 20b8 + 27b7 - 22b6 - 2b5 - 14b4 - 183b3 - 9b2 + 36b1 + 145) q^77 + (7b11 + 16b9 - 9b8 - 20b7 - 16b6 - 2b5 + 78b4 + 75b3 + 80b1 - 9) q^79 + (16b10 + 3b9 + 13b8 + 16b7 - 15b6 + 2b5 + 2b4 - 44b3 - 5b2 + 8b1 + 44) * q^81 + (-b11 - 20b10 - 8b9 - 23b8 - 24b6 + 15b5 - 48b4 + 23b3 - b2 - 118) q^83 + (-5b11 - 10b10 - 5b9 + 5b8 - 10b5 + 20b4 - 5b3 - 5b2 + 105) * q^85 + (28b10 - 14b9 - 5b8 + 28b7 - 15b6 + 20b5 + 20b4 - 254b3 - 6b2 - 135b1 + 254) * q^87 + (-5b11 - 24b9 - b8 + 23b7 + 24b6 - 6b5 - 110b4 + 235b3 - 104b1 - 1) * q^89 + (3b11 + 12b10 - 20b9 + 12b8 + 28b7 - 19b6 + 11b5 + 17b4 + 508b3 + 14b2 - 70b1 - 255) q^91 + (-3b11 + 22b9 + 5b8 - 22b6 + 2b5 + 96b4 - 443b3 + 94b1 + 5) * q^93 + (-20b10 + 5b9 - 5b8 - 20b7 + 10b6 - 5b5 - 5b4 + 10b3 + 30b1 - 10) q^95 + (6b11 + 24b10 - 40b9 - 10b8 - 4b6 - 30b5 - 46b4 + 10b3 + 6b2 - 260) q^97 + (-3b11 - 12b10 - 43b9 - 12b8 - 15b6 - 31b5 + 35b4 + 12b3 - 3b2 - 21) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 8 q^{3} + 30 q^{5} + 14 q^{7} - 34 q^{9} - 240 q^{13} - 80 q^{15} + 122 q^{17} - 2 q^{19} + 170 q^{21} + 30 q^{23} - 150 q^{25} + 64 q^{27} + 476 q^{29} - 122 q^{31} + 584 q^{33} - 100 q^{35} + 342 q^{37}+ \cdots + 216 q^{99}+O(q^{100})$ 12 * q - 8 * q^3 + 30 * q^5 + 14 * q^7 - 34 * q^9 - 240 * q^13 - 80 * q^15 + 122 * q^17 - 2 * q^19 + 170 * q^21 + 30 * q^23 - 150 * q^25 + 64 * q^27 + 476 * q^29 - 122 * q^31 + 584 * q^33 - 100 * q^35 + 342 * q^37 - 126 * q^39 - 676 * q^41 - 1176 * q^43 + 170 * q^45 + 260 * q^47 - 368 * q^49 - 770 * q^51 - 680 * q^53 + 1380 * q^57 - 918 * q^59 + 1026 * q^61 + 1284 * q^63 - 600 * q^65 + 1532 * q^67 - 724 * q^69 + 300 * q^71 - 158 * q^73 - 200 * q^75 + 758 * q^77 + 222 * q^79 + 250 * q^81 - 704 * q^83 + 1220 * q^85 + 1306 * q^87 + 1684 * q^89 - 122 * q^91 - 2834 * q^93 + 10 * q^95 - 2600 * q^97 + 216 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 2 x^{11} + 95 x^{10} - 258 x^{9} + 7289 x^{8} - 15564 x^{7} + 170984 x^{6} + 88720 x^{5} + \cdots + 6718464$ x^12 - 2*x^11 + 95*x^10 - 258*x^9 + 7289*x^8 - 15564*x^7 + 170984*x^6 + 88720*x^5 + 2094592*x^4 - 681024*x^3 + 5329728*x^2 + 3297024*x + 6718464 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 267705222294445 \nu^{11} + \cdots + 82\!\cdots\!16 ) / 23\!\cdots\!76$ (-267705222294445v^11 + 18345339459436178v^10 - 104250581027502167v^9 + 1909135778072406762v^8 - 10432574427794419553v^7 + 148479202784471334108v^6 - 641773343274596300252v^5 + 3848504075500645661408v^4 - 5924971178025352770976v^3 + 30385784723642544948384v^2 - 100455519893376589860672*v + 82701721371070118615616) / 2314335974266486973376
$\beta_{3}$	$=$	$( 10\!\cdots\!79 \nu^{11} + \cdots + 20\!\cdots\!28 ) / 92\!\cdots\!04$ (1099737702803879v^11 - 3843000928119310v^10 + 105429395331439573v^9 - 433541768931860502v^8 + 8230001643151803667v^7 - 28419886218955489692v^6 + 197477850702440005972v^5 - 142585790609958806608v^4 + 1837005968461204372928v^3 - 4712553793049098316064v^2 + 4117322124117040342464*v + 2043101114661042793728) / 9257343897065947893504
$\beta_{4}$	$=$	$( 15217828912144 \nu^{11} - 8836236713621 \nu^{10} + \cdots + 68\!\cdots\!32 ) / 85\!\cdots\!88$ (15217828912144v^11 - 8836236713621v^10 + 1387124459337590v^9 - 1981606735317867v^8 + 104662672338110342v^7 - 87410178946495917v^6 + 2223652959284434736v^5 + 4319405795649797680v^4 + 36700055775322124296v^3 + 16147969473819185856v^2 + 14655189677668644096*v + 68412483016023704832) / 85716147195055073088
$\beta_{5}$	$=$	$( 12\!\cdots\!94 \nu^{11} + \cdots + 43\!\cdots\!52 ) / 23\!\cdots\!76$ (1290380775947894v^11 + 8217990220886249v^10 + 103790866302443290v^9 + 695994774241735875v^8 + 6707294877906992266v^7 + 56956268314878796185v^6 + 57564950446494337846v^5 + 1899392294684211962888v^4 + 3533227999223391885656v^3 + 17716533820499405383824v^2 + 5571754992786718116288*v + 43953110092068897125952) / 2314335974266486973376
$\beta_{6}$	$=$	$( - 59\!\cdots\!01 \nu^{11} + \cdots + 44\!\cdots\!24 ) / 92\!\cdots\!04$ (-5934406146733801v^11 + 63768427164990482v^10 - 665631961309809611v^9 + 6165385776280499946v^8 - 56726762536627967885v^7 + 442924272154750125828v^6 - 1776237035268319040156v^5 + 6567150663330637154288v^4 - 5830615511788440824320v^3 + 77250462748787507424096v^2 - 87548425163395999557696*v + 44247637271901847463424) / 9257343897065947893504
$\beta_{7}$	$=$	$( - 62\!\cdots\!81 \nu^{11} + \cdots - 56\!\cdots\!76 ) / 92\!\cdots\!04$ (-6251202919135781v^11 - 21923276966139638v^10 - 629883169962647287v^9 - 1680591398330758830v^8 - 45277280674422060097v^7 - 138394006703765156028v^6 - 1112037667902344577508v^5 - 5812321396881679238288v^4 - 24454044727833359321024v^3 - 78807563525225763034464v^2 - 74294805211745842476864*v - 56880675025847990136576) / 9257343897065947893504
$\beta_{8}$	$=$	$( 11\!\cdots\!63 \nu^{11} + \cdots - 11\!\cdots\!36 ) / 11\!\cdots\!88$ (1129583999685463v^11 - 1747483029584513v^10 + 111925758667853036v^9 - 248259888301084197v^8 + 8711843241252666968v^7 - 15420580529317780881v^6 + 227401226966467298465v^5 + 66014138881429556500v^4 + 3590297183431510066504v^3 - 1618256311410334261800v^2 + 13995494409704150055456*v - 11664882063556622084736) / 1157167987133243486688
$\beta_{9}$	$=$	$( 57\!\cdots\!95 \nu^{11} + \cdots + 20\!\cdots\!32 ) / 23\!\cdots\!76$ (5738433420840595v^11 - 14282683945156940v^10 + 544307941578691571v^9 - 1759818237295219212v^8 + 41672553472379605421v^7 - 110504702124813721254v^6 + 961444189299277235210v^5 + 39385074208882106752v^4 + 9628984191548149944928v^3 - 10415613823508600782608v^2 + 1086790262658792579648*v + 20536342093936301850432) / 2314335974266486973376
$\beta_{10}$	$=$	$( 335263802565144 \nu^{11} - 235819143081583 \nu^{10} + \cdots - 35\!\cdots\!28 ) / 12\!\cdots\!32$ (335263802565144v^11 - 235819143081583v^10 + 31007912951700402v^9 - 46738769496233585v^8 + 2306615178923364410v^7 - 2276269271876692855v^6 + 48822219413991862288v^5 + 93993375516054519888v^4 + 636416412649111092176v^3 + 352488040258069915200v^2 + 320577992544394393344*v - 352311571870543016928) / 128574220792582609632
$\beta_{11}$	$=$	$( 10\!\cdots\!93 \nu^{11} + \cdots + 19\!\cdots\!44 ) / 92\!\cdots\!04$ (103164954246750593v^11 - 234848018040038986v^10 + 9864180542445664147v^9 - 29459041213484789586v^8 + 760615468726927417189v^7 - 1812815322511020111468v^6 + 18224385917077818737356v^5 + 4144982116462562942576v^4 + 216843384622934160019328v^3 - 100080875026824846733536v^2 + 579644873115518369118528*v + 195301528296125220140544) / 9257343897065947893504

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 32\beta_{3} - 2\beta _1 + 1$ b9 + b8 + b7 - b6 + b5 - b4 - 32b3 - 2b1 + 1
$\nu^{3}$	$=$	$-\beta_{11} + 3\beta_{10} - 3\beta_{9} - 3\beta_{8} - 4\beta_{6} + 60\beta_{4} + 3\beta_{3} - \beta_{2} + 32$ -b11 + 3b10 - 3b9 - 3b8 - 4b6 + 60b4 + 3b3 - b2 + 32
$\nu^{4}$	$=$	$- 71 \beta_{10} + 15 \beta_{9} - 62 \beta_{8} - 71 \beta_{7} + 76 \beta_{6} - 14 \beta_{5} + \cdots - 1847$ -71b10 + 15b9 - 62b8 - 71b7 + 76b6 - 14b5 - 14b4 + 1847b3 - b2 + 240*b1 - 1847
$\nu^{5}$	$=$	$69 \beta_{11} + 84 \beta_{9} + 327 \beta_{8} + 351 \beta_{7} - 84 \beta_{6} + 396 \beta_{5} - 3664 \beta_{4} + \cdots + 327$ 69b11 + 84b9 + 327b8 + 351b7 - 84b6 + 396b5 - 3664b4 - 5754b3 - 4060*b1 + 327
$\nu^{6}$	$=$	$9 \beta_{11} + 4987 \beta_{10} - 5581 \beta_{9} - 1539 \beta_{8} - 1530 \beta_{6} - 4042 \beta_{5} + \cdots + 115942$ 9b11 + 4987b10 - 5581b9 - 1539b8 - 1530b6 - 4042b5 + 18742b4 + 1539b3 + 9*b2 + 115942
$\nu^{7}$	$=$	$- 32019 \beta_{10} + 18039 \beta_{9} - 11928 \beta_{8} - 32019 \beta_{7} + 34612 \beta_{6} + \cdots - 573971$ -32019b10 + 18039b9 - 11928b8 - 32019b7 + 34612b6 - 22684b5 - 22684b4 + 573971b3 + 4645b2 + 290820b1 - 573971
$\nu^{8}$	$=$	$5231 \beta_{11} + 276254 \beta_{9} + 408389 \beta_{8} + 364103 \beta_{7} - 276254 \beta_{6} + \cdots + 408389$ 5231b11 + 276254b9 + 408389b8 + 364103b7 - 276254b6 + 413620b5 - 1556468b4 - 8648726b3 - 1970088*b1 + 408389
$\nu^{9}$	$=$	$- 315309 \beta_{11} + 2697495 \beta_{10} - 2579607 \beta_{9} - 1356579 \beta_{8} - 1671888 \beta_{6} + \cdots + 49203216$ -315309b11 + 2697495b10 - 2579607b9 - 1356579b8 - 1671888b6 - 1223028b5 + 20350216b4 + 1356579b3 - 315309*b2 + 49203216
$\nu^{10}$	$=$	$- 27378739 \beta_{10} + 10805643 \beta_{9} - 19582930 \beta_{8} - 27378739 \beta_{7} + 31178404 \beta_{6} + \cdots - 621474251$ -27378739b10 + 10805643b9 - 19582930b8 - 27378739b7 + 31178404b6 - 11595474b5 - 11595474b4 + 621474251b3 + 789831b2 + 162942992b1 - 621474251
$\nu^{11}$	$=$	$21802933 \beta_{11} + 110900976 \beta_{9} + 214648479 \beta_{8} + 219532347 \beta_{7} - 110900976 \beta_{6} + \cdots + 214648479$ 21802933b11 + 110900976b9 + 214648479b8 + 219532347b7 - 110900976b6 + 236451412b5 - 1399335200b4 - 4484466698b3 - 1635786612*b1 + 214648479

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$ .

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-\beta_{3}$	$1$	$1$	$1$

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

3.54223	−	6.13533i
3.22348	−	5.58323i
0.889448	−	1.54057i
−0.531189	+	0.920046i
−1.69507	+	2.93594i
−4.42891	+	7.67109i
3.54223	+	6.13533i
3.22348	+	5.58323i
0.889448	+	1.54057i
−0.531189	−	0.920046i
−1.69507	−	2.93594i
−4.42891	−	7.67109i

−4.04223

7.00136i

2.50000

4.33013i

−18.4932

0.999858i

−19.1793

−

33.2196i

81.2

−3.72348

6.44926i

2.50000

4.33013i

15.4269

−

10.2474i

−14.2286

−

24.6447i

81.3

−1.38945

2.40659i

2.50000

4.33013i

−2.01227

18.4106i

9.63887

16.6950i

81.4

0.0311889

−

0.0540207i

2.50000

4.33013i

16.7481

7.90586i

13.4981

23.3793i

81.5

1.19507

−

2.06992i

2.50000

4.33013i

3.21331

−

18.2394i

10.6436

18.4353i

81.6

3.92891

−

6.80507i

2.50000

4.33013i

−7.88276

16.7589i

−17.3726

−

30.0903i

401.1

−4.04223

−

7.00136i

2.50000

−

4.33013i

−18.4932

−

0.999858i

−19.1793

33.2196i

401.2

−3.72348

−

6.44926i

2.50000

−

4.33013i

15.4269

10.2474i

−14.2286

24.6447i

401.3

−1.38945

−

2.40659i

2.50000

−

4.33013i

−2.01227

−

18.4106i

9.63887

−

16.6950i

401.4

0.0311889

0.0540207i

2.50000

−

4.33013i

16.7481

−

7.90586i

13.4981

−

23.3793i

401.5

1.19507

2.06992i

2.50000

−

4.33013i

3.21331

18.2394i

10.6436

−

18.4353i

401.6

3.92891

6.80507i

2.50000

−

4.33013i

−7.88276

−

16.7589i

−17.3726

30.0903i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	560.4.q.p		12
4.b	odd	2	1		280.4.q.e	✓	12
7.c	even	3	1	inner	560.4.q.p		12
28.f	even	6	1		1960.4.a.bb		6
28.g	odd	6	1		280.4.q.e	✓	12
28.g	odd	6	1		1960.4.a.w		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.4.q.e	✓	12	4.b	odd	2	1
280.4.q.e	✓	12	28.g	odd	6	1
560.4.q.p		12	1.a	even	1	1	trivial
560.4.q.p		12	7.c	even	3	1	inner
1960.4.a.w		6	28.g	odd	6	1
1960.4.a.bb		6	28.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(560, [\chi])$ :

T_{3}^{12} + 8 T_{3}^{11} + 130 T_{3}^{10} + 560 T_{3}^{9} + 8447 T_{3}^{8} + 34856 T_{3}^{7} + \cdots + 38416

T_{11}^{12} + 6101 T_{11}^{10} + 211064 T_{11}^{9} + 27970393 T_{11}^{8} + 961017372 T_{11}^{7} + \cdots + 68\!\cdots\!64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12} + 8 T^{11} + \cdots + 38416$ T^12 + 8T^11 + 130T^10 + 560T^9 + 8447T^8 + 34856T^7 + 287746T^6 + 269344T^5 + 1756817T^4 + 603160T^3 + 9835540T^2 - 613088*T + 38416
$5$	$(T^{2} - 5 T + 25)^{6}$ (T^2 - 5*T + 25)^6
$7$	$T^{12} + \cdots + 16\!\cdots\!49$ T^12 - 14T^11 + 282T^10 - 2444T^9 - 64138T^8 + 3833410T^7 - 45232194T^6 + 1314859630T^5 - 7545771562T^4 - 98624215508T^3 + 3903242990682T^2 - 66465861139202*T + 1628413597910449
$11$	$T^{12} + \cdots + 68\!\cdots\!64$ T^12 + 6101T^10 + 211064T^9 + 27970393T^8 + 961017372T^7 + 72818613648T^6 + 2893705539424T^5 + 135040605835152T^4 + 3486965236533632T^3 + 76371917562555136T^2 + 830394598552933120T + 6854788009115640064
$13$	$(T^{6} + 120 T^{5} + \cdots + 8623951872)^{2}$ (T^6 + 120T^5 - 1811T^4 - 603390T^3 - 14105480T^2 + 307760544*T + 8623951872)^2
$17$	$T^{12} + \cdots + 14\!\cdots\!76$ T^12 - 122T^11 + 33736T^10 - 2921000T^9 + 623314592T^8 - 50724314016T^7 + 6837125271616T^6 - 436451657017344T^5 + 45861161590548480T^4 - 2681310837476597760T^3 + 172596855450956267520T^2 - 5254909103885866500096*T + 143986418025895068696576
$19$	$T^{12} + \cdots + 10\!\cdots\!36$ T^12 + 2T^11 + 28893T^10 + 1867562T^9 + 663502361T^8 + 35991725414T^7 + 6108292656744T^6 + 346334615392664T^5 + 41381787339220064T^4 + 1729090622173862240T^3 + 61295678588906139456T^2 + 894352318691109615104*T + 10159270185460863963136
$23$	$T^{12} + \cdots + 13\!\cdots\!09$ T^12 - 30T^11 + 42341T^10 - 4097098T^9 + 1516472434T^8 - 113797376614T^7 + 18403286664949T^6 - 910993104937422T^5 + 137112109236395202T^4 - 6232256721436344354T^3 + 367907765637920346549T^2 - 2291485938121873559430*T + 13124056139065863340209
$29$	$(T^{6} - 238 T^{5} + \cdots + 931883943676)^{2}$ (T^6 - 238T^5 - 45708T^4 + 8770450T^3 + 66318319T^2 - 37433510532*T + 931883943676)^2
$31$	$T^{12} + \cdots + 26\!\cdots\!16$ T^12 + 122T^11 + 87876T^10 + 23424896T^9 + 8385954368T^8 + 1419054572032T^7 + 184862858816512T^6 + 13762730920452096T^5 + 749742374057525248T^4 + 22783841884839346176T^3 + 490423729807911813120T^2 + 4220300566724383604736*T + 26671736148957924950016
$37$	$T^{12} + \cdots + 46\!\cdots\!56$ T^12 - 342T^11 + 178811T^10 - 27306494T^9 + 11627655913T^8 - 1460042469316T^7 + 503356435314688T^6 - 22125042673756352T^5 + 6898648747531357184T^4 + 188066242634284778496T^3 + 88802502981848513769472T^2 + 1988554077854446479507456*T + 46242666296457368869011456
$41$	$(T^{6} + \cdots - 2860161422643)^{2}$ (T^6 + 338T^5 - 171509T^4 - 44862628T^3 - 358899801T^2 + 189089846994*T - 2860161422643)^2
$43$	$(T^{6} + \cdots - 44576110032948)^{2}$ (T^6 + 588T^5 - 108702T^4 - 126822756T^3 - 26332766719T^2 - 1992482500632*T - 44576110032948)^2
$47$	$T^{12} + \cdots + 63\!\cdots\!24$ T^12 - 260T^11 + 449845T^10 - 77687436T^9 + 143890721161T^8 - 26812569965264T^7 + 16407008667784528T^6 - 2490626185412191488T^5 + 1273256892606894354688T^4 - 198579699584040631758848T^3 + 30770832318587873184514048T^2 - 1540136853030465353482764288*T + 63852135622021316267237965824
$53$	$T^{12} + \cdots + 54\!\cdots\!04$ T^12 + 680T^11 + 716247T^10 + 395176516T^9 + 331291657245T^8 + 165055706341754T^7 + 66417740438044288T^6 + 17045681597974823640T^5 + 3291702623044709198976T^4 + 412820522366386222810080T^3 + 37345496803601511573358272T^2 + 1718341177500216903455587584*T + 54157135098710408057005999104
$59$	$T^{12} + \cdots + 99\!\cdots\!56$ T^12 + 918T^11 + 1109352T^10 + 388034200T^9 + 398010047136T^8 + 150566738229984T^7 + 91066823052129088T^6 + 11139914871435449856T^5 + 1094076534932439232512T^4 + 26493068341466591330304T^3 + 516935809946360189288448T^2 + 741722685332864020512768*T + 994316877528181040480256
$61$	$T^{12} + \cdots + 69\!\cdots\!84$ T^12 - 1026T^11 + 1303064T^10 - 701142956T^9 + 672172785841T^8 - 359270091197844T^7 + 204463162897116432T^6 - 55969316797636985986T^5 + 12784807105420486811865T^4 - 1128111699872879938871432T^3 + 105967230579281220480188464T^2 + 2338310313712695353109550976*T + 690367800517813034179371030784
$67$	$T^{12} + \cdots + 17\!\cdots\!36$ T^12 - 1532T^11 + 1668618T^10 - 1075895648T^9 + 570130668731T^8 - 219512850686672T^7 + 82295216446275162T^6 - 25204990248473204516T^5 + 7246246578282817930577T^4 - 1503854426951384732221040T^3 + 251910098322606900879230160T^2 - 24834254990221560877524487424*T + 1716994499715498278472056494336
$71$	$(T^{6} + \cdots - 15\!\cdots\!00)^{2}$ (T^6 - 150T^5 - 1259076T^4 - 37172968T^3 + 349067423040T^2 + 15829612444800*T - 15125099576832000)^2
$73$	$T^{12} + \cdots + 15\!\cdots\!96$ T^12 + 158T^11 + 1155780T^10 - 135201216T^9 + 971043716032T^8 - 93452067098112T^7 + 330226614007415808T^6 - 53330314603341479936T^5 + 82130345634503207501824T^4 - 6680207289078960144384000T^3 + 4308254736916039526973702144T^2 + 248316198858435087347623133184*T + 158715771710110394812149106999296
$79$	$T^{12} + \cdots + 61\!\cdots\!44$ T^12 - 222T^11 + 2082708T^10 + 588698304T^9 + 3097839427920T^8 + 854150398261344T^7 + 2090483202662724544T^6 + 996545842588776768000T^5 + 1011245916647753272274688T^4 + 263182194312524338831292928T^3 + 93512124502299909952652934144T^2 - 6493984960708140801473565978624*T + 619417525283588692781151735660544
$83$	$(T^{6} + \cdots - 28\!\cdots\!52)^{2}$ (T^6 + 352T^5 - 2106230T^4 - 1215836976T^3 + 724957075653T^2 + 427960246558608*T - 28187979507134352)^2
$89$	$T^{12} + \cdots + 25\!\cdots\!24$ T^12 - 1684T^11 + 4785938T^10 - 3623360824T^9 + 8827895272671T^8 - 5632290247329148T^7 + 10809493777178820386T^6 - 3330264351515510628144T^5 + 5985887584379797944076865T^4 - 1125208737878179339244731508T^3 + 2449373479160440643676307269100T^2 + 24718056566518612402513978345776*T + 250726214795092782887013018151824
$97$	$(T^{6} + \cdots + 32\!\cdots\!68)^{2}$ (T^6 + 1300T^5 - 1822516T^4 - 2732523040T^3 - 131112522384T^2 + 544285243941696*T + 32470471403189568)^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 $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 81.6
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	560.4.q.p

Level	$560$
Weight	$4$
Character orbit	560.q
Analytic conductor	$33.041$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$