LMFDB - Newform orbit 280.4.q.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,4,Mod(81,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, 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("280.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$280 = 2^{3} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	280.q (of order $3$ , degree $2$ , 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:	$16.5205348016$
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:	$x^{12} - x^{11} + 97 x^{10} + 136 x^{9} + 6932 x^{8} + 7120 x^{7} + 190192 x^{6} + 97856 x^{5} + \cdots + 199148544$ x^12 - x^11 + 97x^10 + 136x^9 + 6932x^8 + 7120x^7 + 190192x^6 + 97856x^5 + 3784384x^4 + 956928x^3 + 32401152x^2 - 9483264x + 199148544
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{14}$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + ( - \beta_{5} + \beta_{3} - 1) q^{3} + 5 \beta_{3} q^{5} + (\beta_{10} + 2 \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 4 \beta_1) q^{9} + ( - \beta_{10} - \beta_{9} + \beta_{6} + \cdots + 9) q^{11}+ \cdots + (25 \beta_{11} - 25 \beta_{10} + \cdots - 498) q^{99}+O(q^{100})$ q + (-b5 + b3 - 1) * q^3 + 5b3 q^5 + (b10 + 2b3 + b2 - 1) q^7 + (-b11 - b10 - 6b3 - 4b1) * q^9 + (-b10 - b9 + b6 - b4 - 8b3 - b1 + 9) q^11 + (b11 - b10 - b9 - 2b8 - 3b7 - 3b5 - 2b4 + b3 + 2b2 - 3b1) * q^13 + (-5b5 - 5b1 - 5) * q^15 + (b11 + b10 - 2b9 + 2b6 + b5 + b4 + b3 + b2 + b1 - 1) * q^17 + (2b11 + b10 - 4b8 - b7 - 2b3 + 3b2 + 5b1 + 4) q^19 + (-3b11 - 3b10 + 3b9 - 2b8 + 2b7 - 2b6 + 2b5 + b4 - b3 - 2b2 - 8b1 + 10) q^21 + (6b10 + 5b8 + 6b7 + 3b6 + 8b3 + b2 + 5b1 - 5) * q^23 + (25b3 - 25) q^25 + (-3b11 + 3b10 + b9 + 5b8 + 14b7 + 11b5 + 5b4 - 3b3 - 11b2 + 11b1 + 76) q^27 + (-2b11 + 2b10 - 6b9 - b8 - 3b7 - 3b5 - b4 - 2b3 + 5b2 - 3b1 - 16) * q^29 + (-3b11 - 9b10 + 5b9 + 4b7 - 5b6 - 7b5 - 5b4 + 23b3 - 3b2 - 5b1 - 21) * q^31 + (-2b11 + 4b10 + 4b8 + 6b7 + 18b3 + 2b2 + 26b1 - 4) q^33 + (5b3 + 5b2 - 10) * q^35 + (-7b11 - 4b10 - b8 + 3b7 - b6 - 89b3 + 4b2 - 9b1 + 1) * q^37 + (-11b11 - 13b10 + 5b9 + 14b7 - 5b6 + 15b5 + b4 - 59b3 - 11b2 + b1 + 47) * q^39 + (-7b11 + 7b10 + 3b9 + 3b8 + 14b7 + 14b5 + 3b4 - 7b3 - 7b2 + 14b1 + 62) * q^41 + (-2b11 + 2b10 + 8b9 + 8b7 - 33b5 - 2b3 - 6b2 - 33b1 - 101) * q^43 + (-5b11 - 5b10 + 5b7 + 20b5 - 30b3 - 5b2 + 25) * q^45 + (-b11 + 7b10 + 8b7 - 8b6 + 8b2 + 40b1) q^47 + (-14b11 - 7b9 + 7b8 + 7b7 + 28b5 + 7b4 + 54b3 - 2b2 + 18) * q^49 + (5b11 + 11b10 - 5b8 + 6b7 + b6 - b3 + 11b2 + 10b1 + 5) * q^51 + (-21b11 - 6b10 + 9b9 + 20b7 - 9b6 + b5 + 14b4 - 177b3 - 21b2 + 14b1 + 142) q^53 + (5b11 - 5b10 - 5b9 - 5b8 - 5b7 - 5b4 + 5b3 + 50) q^55 + (2b11 - 2b10 + 10b9 - 9b8 + 6b7 - 23b5 - 9b4 + 2b3 - 8b2 - 23b1 - 84) * q^57 + (3b11 + 5b10 - 17b9 + 17b6 + 49b5 + 5b4 + 23b3 + 3b2 + 5b1 - 25) q^59 + (-4b11 + b10 + 12b8 + 5b7 + 11b6 + 9b3 - 7b2 + 45b1 - 12) q^61 + (4b11 + 11b10 + 3b9 + 12b8 + 23b7 + 5b6 + 44b5 + b4 + 74b3 - 11b2 + 69b1 + 182) * q^63 + (-5b10 - 10b8 - 5b7 - 5b6 - 20b3 + 5b2 - 5b1 + 10) q^65 + (19b11 - 4b10 - 15b9 - 3b7 + 15b6 - 15b5 - 7b4 + 55b3 + 19b2 - 7b1 - 29) * q^67 + (-7b11 + 7b10 + b9 - 6b8 - 27b7 - 59b5 - 6b4 - 7b3 + 34b2 - 59b1 - 111) q^69 + (-14b11 + 14b10 - 3b9 + 23b8 + 22b7 - 65b5 + 23b4 - 14b3 - 8b2 - 65b1 - 330) * q^71 + (b11 + 9b10 + 10b9 - 10b7 - 10b6 + 43b5 - b4 - 49b3 + b2 - b1 + 51) * q^73 + (-25b3 - 25b1) * q^75 + (24b11 - 3b10 - 3b9 - 12b8 - 30b7 + 9b6 + 12b5 - 29b4 + 334b3 + 12b2 - 13b1 - 99) q^77 + (7b11 - 15b10 - 21b8 - 22b7 - 19b6 + 253b3 - b2 - 76b1 + 21) q^79 + (39b11 + 20b10 - 15b9 - 53b7 + 15b6 - 160b5 - 33b4 + 77b3 + 39b2 - 33b1 - 5) * q^81 + (23b11 - 23b10 - 9b9 - 33b8 - 54b7 - 81b5 - 33b4 + 23b3 + 31b2 - 81b1 + 222) * q^83 + (-10b9 + 5b8 + 5b5 + 5b4 + 5b1 - 10) q^85 + (-10b11 - 21b10 + 22b9 + 39b7 - 22b6 + 42b5 + 18b4 - 178b3 - 10b2 + 18b1 + 150) * q^87 + (-21b11 - 50b10 - 25b8 - 29b7 - 5b6 - 166b3 - 4b2 + 31b1 + 25) * q^89 + (16b11 - 16b10 - 16b9 - 29b8 - 27b7 - b6 + 99b5 - 10b4 + 60b3 + 11b2 + 45b1 + 201) * q^91 + (-7b11 - 5b10 + 39b8 + 2b7 + 18b6 - 71b3 - 37b2 + 2b1 - 39) * q^93 + (5b11 + 10b10 + 10b7 - 5b5 + 20b4 - 15b3 + 5b2 + 20b1) * q^95 + (2b11 - 2b10 + 4b9 + 26b8 + 8b7 - 120b5 + 26b4 + 2b3 - 10b2 - 120b1 - 8) * q^97 + (25b11 - 25b10 - 17b9 - 23b8 - 59b7 - 116b5 - 23b4 + 25b3 + 34b2 - 116b1 - 498) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 7 q^{3} + 30 q^{5} - q^{7} - 39 q^{9} + 51 q^{11} - 54 q^{13} - 70 q^{15} - 6 q^{17} - 9 q^{19} + 123 q^{21} + 72 q^{23} - 150 q^{25} + 1118 q^{27} - 242 q^{29} - 132 q^{31} + 156 q^{33} - 115 q^{35}+ \cdots - 6978 q^{99}+O(q^{100})$ 12 * q - 7 * q^3 + 30 * q^5 - q^7 - 39 * q^9 + 51 * q^11 - 54 * q^13 - 70 * q^15 - 6 * q^17 - 9 * q^19 + 123 * q^21 + 72 * q^23 - 150 * q^25 + 1118 * q^27 - 242 * q^29 - 132 * q^31 + 156 * q^33 - 115 * q^35 - 523 * q^37 + 358 * q^39 + 928 * q^41 - 1202 * q^43 + 195 * q^45 + 73 * q^47 + 753 * q^49 + 28 * q^51 + 999 * q^53 + 510 * q^55 - 1080 * q^57 - 98 * q^59 + 111 * q^61 + 3011 * q^63 - 135 * q^65 - 289 * q^67 - 1794 * q^69 - 3668 * q^71 + 332 * q^73 - 175 * q^75 + 285 * q^77 + 1368 * q^79 - 498 * q^81 + 1678 * q^83 - 60 * q^85 + 1075 * q^87 - 1035 * q^89 + 2368 * q^91 - 448 * q^93 + 45 * q^95 + 8 * q^97 - 6978 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - x^{11} + 97 x^{10} + 136 x^{9} + 6932 x^{8} + 7120 x^{7} + 190192 x^{6} + 97856 x^{5} + \cdots + 199148544$ x^12 - x^11 + 97*x^10 + 136*x^9 + 6932*x^8 + 7120*x^7 + 190192*x^6 + 97856*x^5 + 3784384*x^4 + 956928*x^3 + 32401152*x^2 - 9483264*x + 199148544 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 5144955608 \nu^{11} - 63399912935 \nu^{10} - 17165331791883 \nu^{9} + 45225604010075 \nu^{8} + \cdots + 34\!\cdots\!96 ) / 22\!\cdots\!84$ (5144955608v^11 - 63399912935v^10 - 17165331791883v^9 + 45225604010075v^8 - 1536241955811706v^7 - 994345747634300v^6 - 93714149455167552v^5 + 21750348854897088v^4 - 1789685105494217536v^3 + 979791893036651072v^2 - 20915547228016280448*v + 34722865993703037696) / 2240079821969117184
$\beta_{3}$	$=$	$( 23266726295 \nu^{11} + 288099243676 \nu^{10} + 1745352572186 \nu^{9} + \cdots + 91\!\cdots\!28 ) / 67\!\cdots\!52$ (23266726295v^11 + 288099243676v^10 + 1745352572186v^9 + 31001770600517v^8 + 188675684145538v^7 + 2108127797159588v^6 + 5016026918789504v^5 + 47111134798498816v^4 + 82866435050821760v^3 + 1097487474321782208v^2 + 476551843940850048*v + 9156072410258473728) / 6720239465907351552
$\beta_{4}$	$=$	$( 79243202735 \nu^{11} - 1106092690847 \nu^{10} + 13009368288059 \nu^{9} + \cdots + 58\!\cdots\!40 ) / 67\!\cdots\!52$ (79243202735v^11 - 1106092690847v^10 + 13009368288059v^9 - 69013894724812v^8 + 1473606612684256v^7 - 7785312392893192v^6 + 106301032574528576v^5 - 69699089622155072v^4 + 3915586195129337408v^3 - 2930464679593699584v^2 + 59638489235470301184*v + 58206196255989680640) / 6720239465907351552
$\beta_{5}$	$=$	$( - 14826950951 \nu^{11} + 24358089449 \nu^{10} - 1325595039257 \nu^{9} + \cdots + 22\!\cdots\!80 ) / 32\!\cdots\!12$ (-14826950951v^11 + 24358089449v^10 - 1325595039257v^9 - 1304320831838v^8 - 92498509806628v^7 - 28137224347184v^6 - 2134968858579776v^5 + 246847222493120v^4 - 51201090117226688v^3 + 13205566251706752v^2 - 446510329434747648*v + 220644507871226880) / 320011403138445312
$\beta_{6}$	$=$	$( 18305245152 \nu^{11} + 603360023261 \nu^{10} + 1436579804809 \nu^{9} + \cdots + 25\!\cdots\!92 ) / 37\!\cdots\!64$ (18305245152v^11 + 603360023261v^10 + 1436579804809v^9 + 78501769400335v^8 + 204359394582654v^7 + 5655349080791380v^6 + 11936959644431552v^5 + 190984410792418496v^4 + 211829003819899840v^3 + 3004536948144486720v^2 + 2893955699891553920*v + 25062346543689729792) / 373346636994852864
$\beta_{7}$	$=$	$( 279440238408 \nu^{11} + 383921255709 \nu^{10} + 6378010240201 \nu^{9} + \cdots - 17\!\cdots\!20 ) / 22\!\cdots\!84$ (279440238408v^11 + 383921255709v^10 + 6378010240201v^9 + 135459494955735v^8 + 300751846448638v^7 + 4194565570075988v^6 - 51940312091771456v^5 + 52662988553650880v^4 - 881112608453626944v^3 + 1032213699688791872v^2 - 18584305936053148032*v - 17139271740579313920) / 2240079821969117184
$\beta_{8}$	$=$	$( - 868684120625 \nu^{11} + 1942888905761 \nu^{10} - 90244855812413 \nu^{9} + \cdots + 61\!\cdots\!12 ) / 33\!\cdots\!76$ (-868684120625v^11 + 1942888905761v^10 - 90244855812413v^9 - 36375997655204v^8 - 5905042285127368v^7 - 2343375350886296v^6 - 172458649034088128v^5 + 49752151811688896v^4 - 2923836273381308864v^3 + 2212832545225650432v^2 - 36892825538681452032*v + 61570036252917222912) / 3360119732953675776
$\beta_{9}$	$=$	$( - 699311292110 \nu^{11} - 3038539372501 \nu^{10} - 56929262300333 \nu^{9} + \cdots + 64\!\cdots\!24 ) / 16\!\cdots\!88$ (-699311292110v^11 - 3038539372501v^10 - 56929262300333v^9 - 369783551657945v^8 - 4949289432335746v^7 - 20650335091264652v^6 - 111315744276130304v^5 - 153808705319591488v^4 - 1829262461067045824v^3 - 760870727483186880v^2 - 5924519881889664384*v + 64419065619259002624) / 1680059866476837888
$\beta_{10}$	$=$	$( 2977161987743 \nu^{11} - 10713100087955 \nu^{10} + 314020590874775 \nu^{9} + \cdots - 15\!\cdots\!56 ) / 67\!\cdots\!52$ (2977161987743v^11 - 10713100087955v^10 + 314020590874775v^9 - 316384659066136v^8 + 20132411756112808v^7 - 22855817271541528v^6 + 542404391834850368v^5 - 851995476845013056v^4 + 8435539053393769280v^3 - 18380038469972046336v^2 + 45817856185204551168*v - 155393318744802464256) / 6720239465907351552
$\beta_{11}$	$=$	$( - 1173847773575 \nu^{11} + 1286309184371 \nu^{10} - 118305612912423 \nu^{9} + \cdots - 25\!\cdots\!72 ) / 22\!\cdots\!84$ (-1173847773575v^11 + 1286309184371v^10 - 118305612912423v^9 - 153250600750208v^8 - 8181329884622456v^7 - 10073100569005384v^6 - 222421477735425600v^5 - 184151778088658880v^4 - 3503997756273938240v^3 - 3102474013296508928v^2 - 24275741717227252224*v - 25197639604198774272) / 2240079821969117184

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{11} + \beta_{10} - \beta_{7} - 2\beta_{5} + 32\beta_{3} + \beta_{2} - 31$ b11 + b10 - b7 - 2b5 + 32b3 + b2 - 31
$\nu^{3}$	$=$	$3 \beta_{11} - 3 \beta_{10} - \beta_{9} - 5 \beta_{8} - 11 \beta_{7} - 56 \beta_{5} - 5 \beta_{4} + \cdots - 36$ 3b11 - 3b10 - b9 - 5b8 - 11b7 - 56b5 - 5b4 + 3b3 + 8b2 - 56*b1 - 36
$\nu^{4}$	$=$	$-83\beta_{11} - 90\beta_{10} - 13\beta_{8} - 7\beta_{7} - 11\beta_{6} - 1613\beta_{3} + 6\beta_{2} - 231\beta _1 + 13$ -83b11 - 90b10 - 13b8 - 7b7 - 11b6 - 1613b3 + 6b2 - 231b1 + 13
$\nu^{5}$	$=$	$- 602 \beta_{11} - 391 \beta_{10} + 107 \beta_{9} + 800 \beta_{7} - 107 \beta_{6} + 3732 \beta_{5} + \cdots + 5569$ -602b11 - 391b10 + 107b9 + 800b7 - 107b6 + 3732b5 + 409b4 - 6580b3 - 602b2 + 409b1 + 5569
$\nu^{6}$	$=$	$- 943 \beta_{11} + 943 \beta_{10} + 1183 \beta_{9} + 1757 \beta_{8} + 8855 \beta_{7} + 23076 \beta_{5} + \cdots + 94192$ -943b11 + 943b10 + 1183b9 + 1757b8 + 8855b7 + 23076b5 + 1757b4 - 943b3 - 7912b2 + 23076b1 + 94192
$\nu^{7}$	$=$	$37887 \beta_{11} + 52434 \beta_{10} + 29961 \beta_{8} + 14547 \beta_{7} + 9411 \beta_{6} + 615105 \beta_{3} + \cdots - 29961$ 37887b11 + 52434b10 + 29961b8 + 14547b7 + 9411b6 + 615105b3 - 15414b2 + 242251b1 - 29961
$\nu^{8}$	$=$	$552202 \beta_{11} + 460771 \beta_{10} - 99543 \beta_{9} - 634792 \beta_{7} + 99543 \beta_{6} + \cdots - 6845173$ 552202b11 + 460771b10 - 99543b9 - 634792b7 + 99543b6 - 2000324b5 - 174021b4 + 7571396b3 + 552202b2 - 174021b1 - 6845173
$\nu^{9}$	$=$	$1066731 \beta_{11} - 1066731 \beta_{10} - 791131 \beta_{9} - 2221265 \beta_{8} - 6610355 \beta_{7} + \cdots - 46030512$ 1066731b11 - 1066731b10 - 791131b9 - 2221265b8 - 6610355b7 - 20771252b5 - 2221265b4 + 1066731b3 + 5543624b2 - 20771252b1 - 46030512
$\nu^{10}$	$=$	$- 35287739 \beta_{11} - 43244826 \beta_{10} - 15457261 \beta_{8} - 7957087 \beta_{7} - 7940495 \beta_{6} + \cdots + 15457261$ -35287739b11 - 43244826b10 - 15457261b8 - 7957087b7 - 7940495b6 - 567515525b3 + 7500174b2 - 151480695b1 + 15457261
$\nu^{11}$	$=$	$- 360656498 \beta_{11} - 279055087 \beta_{10} + 65288435 \beta_{9} + 447993608 \beta_{7} + \cdots + 4060966729$ -360656498b11 - 279055087b10 + 65288435b9 + 447993608b7 - 65288435b6 + 1621855572b5 + 168938521b4 - 4590561748b3 - 360656498b2 + 168938521b1 + 4060966729

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$ .

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$1$	$1$	$1$	$-\beta_{3}$

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

4.47849	+	7.75697i
2.25227	+	3.90105i
1.45114	+	2.51344i
−1.73749	−	3.00942i
−2.56740	−	4.44687i
−3.37701	−	5.84915i
4.47849	−	7.75697i
2.25227	−	3.90105i
1.45114	−	2.51344i
−1.73749	+	3.00942i
−2.56740	+	4.44687i
−3.37701	+	5.84915i

−4.97849

8.62300i

2.50000

4.33013i

−14.6186

11.3709i

−36.0707

−

62.4763i

81.2

−2.75227

4.76708i

2.50000

4.33013i

18.4919

1.02408i

−1.65001

−

2.85789i

81.3

−1.95114

3.37947i

2.50000

4.33013i

−5.33627

−

17.7348i

5.88613

10.1951i

81.4

1.23749

−

2.14339i

2.50000

4.33013i

17.8048

5.09810i

10.4372

18.0778i

81.5

2.06740

−

3.58085i

2.50000

4.33013i

−17.7560

−

5.26551i

4.95169

8.57658i

81.6

2.87701

−

4.98312i

2.50000

4.33013i

0.914156

18.4977i

−3.05435

−

5.29029i

121.1

−4.97849

−

8.62300i

2.50000

−

4.33013i

−14.6186

−

11.3709i

−36.0707

62.4763i

121.2

−2.75227

−

4.76708i

2.50000

−

4.33013i

18.4919

−

1.02408i

−1.65001

2.85789i

121.3

−1.95114

−

3.37947i

2.50000

−

4.33013i

−5.33627

17.7348i

5.88613

−

10.1951i

121.4

1.23749

2.14339i

2.50000

−

4.33013i

17.8048

−

5.09810i

10.4372

−

18.0778i

121.5

2.06740

3.58085i

2.50000

−

4.33013i

−17.7560

5.26551i

4.95169

−

8.57658i

121.6

2.87701

4.98312i

2.50000

−

4.33013i

0.914156

−

18.4977i

−3.05435

5.29029i

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	280.4.q.c	✓	12
4.b	odd	2	1		560.4.q.r		12
7.c	even	3	1	inner	280.4.q.c	✓	12
7.c	even	3	1		1960.4.a.ba		6
7.d	odd	6	1		1960.4.a.x		6
28.g	odd	6	1		560.4.q.r		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.4.q.c	✓	12	1.a	even	1	1	trivial
280.4.q.c	✓	12	7.c	even	3	1	inner
560.4.q.r		12	4.b	odd	2	1
560.4.q.r		12	28.g	odd	6	1
1960.4.a.x		6	7.d	odd	6	1
1960.4.a.ba		6	7.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{12} + 7 T_{3}^{11} + 125 T_{3}^{10} + 136 T_{3}^{9} + 6341 T_{3}^{8} + 4105 T_{3}^{7} + \cdots + 158608836$ T3^12 + 7*T3^11 + 125*T3^10 + 136*T3^9 + 6341*T3^8 + 4105*T3^7 + 196315*T3^6 - 141804*T3^5 + 3369747*T3^4 - 2131053*T3^3 + 34882011*T3^2 - 44620542*T3 + 158608836 acting on $S_{4}^{\mathrm{new}}(280, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12} + \cdots + 158608836$ T^12 + 7T^11 + 125T^10 + 136T^9 + 6341T^8 + 4105T^7 + 196315T^6 - 141804T^5 + 3369747T^4 - 2131053T^3 + 34882011T^2 - 44620542*T + 158608836
$5$	$(T^{2} - 5 T + 25)^{6}$ (T^2 - 5*T + 25)^6
$7$	$T^{12} + \cdots + 16\!\cdots\!49$ T^12 + T^11 - 376T^10 - 8971T^9 - 74004T^8 + 2080785T^7 + 91731234T^6 + 713709255T^5 - 8706496596T^4 - 362012208397T^3 - 5204323987576T^2 + 4747561509943T + 1628413597910449
$11$	$T^{12} + \cdots + 16\!\cdots\!76$ T^12 - 51T^11 + 5003T^10 - 225674T^9 + 15716956T^8 - 612825472T^7 + 25944967600T^6 - 663124342880T^5 + 19142467725632T^4 - 356473684468224T^3 + 8709900782303488T^2 - 110199269288391168*T + 1655709545074738176
$13$	$(T^{6} + 27 T^{5} + \cdots + 664762112)^{2}$ (T^6 + 27T^5 - 5328T^4 - 133140T^3 + 6439488T^2 + 139299840*T + 664762112)^2
$17$	$T^{12} + \cdots + 12\!\cdots\!56$ T^12 + 6T^11 + 11792T^10 + 300264T^9 + 117639616T^8 + 1800536480T^7 + 287682212032T^6 - 6821297208064T^5 + 434701771319296T^4 - 2991202313932800T^3 + 38321995498209280T^2 + 132657652958494720*T + 1244592776314617856
$19$	$T^{12} + \cdots + 25\!\cdots\!00$ T^12 + 9T^11 + 28607T^10 - 43862T^9 + 639947872T^8 - 1425397912T^7 + 5011090928320T^6 - 93677599140896T^5 + 30440644020987200T^4 - 229659094320440832T^3 + 1452393842637684736T^2 - 2109409197305856000*T + 2568403685376000000
$23$	$T^{12} + \cdots + 28\!\cdots\!89$ T^12 - 72T^11 + 52199T^10 + 3740344T^9 + 1578539830T^8 + 137009559032T^7 + 33052472367379T^6 + 3566685114870904T^5 + 415434206314507622T^4 + 24650300747425988856T^3 + 1229852351785886538007T^2 + 21139564610515307273016*T + 286263578937571325653689
$29$	$(T^{6} + 121 T^{5} + \cdots - 7144947798)^{2}$ (T^6 + 121T^5 - 76456T^4 - 7045378T^3 + 651289085T^2 + 41125983625*T - 7144947798)^2
$31$	$T^{12} + \cdots + 11\!\cdots\!96$ T^12 + 132T^11 + 150060T^10 + 13763792T^9 + 14913416592T^8 + 1240178311296T^7 + 751067135711232T^6 + 32394197759920128T^5 + 24576420399803891712T^4 + 932856244543857754112T^3 + 334548667683710272339968T^2 - 14069601181375198460903424*T + 1130757787976608963882909696
$37$	$T^{12} + \cdots + 20\!\cdots\!00$ T^12 + 523T^11 + 245085T^10 + 38148884T^9 + 7801894560T^8 + 673849304128T^7 + 140910473815552T^6 + 9727268439917568T^5 + 1346753031966216192T^4 + 54285841582105165824T^3 + 7243192703984210804736T^2 + 253385451881333123973120*T + 20414770576811872380518400
$41$	$(T^{6} - 464 T^{5} + \cdots + 445954812315)^{2}$ (T^6 - 464T^5 - 26479T^4 + 27014792T^3 - 201078765T^2 - 412067622648*T + 445954812315)^2
$43$	$(T^{6} + \cdots + 62569573723766)^{2}$ (T^6 + 601T^5 - 100452T^4 - 97371186T^3 - 9633605155T^2 + 985181811657*T + 62569573723766)^2
$47$	$T^{12} + \cdots + 52\!\cdots\!64$ T^12 - 73T^11 + 397601T^10 - 11631184T^9 + 127802716944T^8 - 972426638080T^7 + 11566719697912064T^6 + 2827539015882829824T^5 + 728566434660988289024T^4 + 77077193797837233848320T^3 + 6442675132044276504985600T^2 + 209803862757180020222853120*T + 5219866796842034489525796864
$53$	$T^{12} + \cdots + 25\!\cdots\!36$ T^12 - 999T^11 + 1070279T^10 - 410291870T^9 + 259531255840T^8 - 57343594431416T^7 + 44236654978040704T^6 - 5294676453945325728T^5 + 3405172826234142855744T^4 + 100711056581653841843712T^3 + 179908065875832782251815936T^2 - 2117590054983895518623342592*T + 25220978654704518325805531136
$59$	$T^{12} + \cdots + 19\!\cdots\!96$ T^12 + 98T^11 + 874136T^10 - 92399576T^9 + 520783491936T^8 - 63348866046432T^7 + 169448900622782016T^6 - 27609535801390823424T^5 + 39274530176743521779712T^4 - 3442456153180816819945472T^3 + 3380620511378093633960935424T^2 + 217301877905525543490912518144*T + 191940073631264821142312867332096
$61$	$T^{12} + \cdots + 13\!\cdots\!16$ T^12 - 111T^11 + 618167T^10 - 68222642T^9 + 281450342077T^8 - 25017919477045T^7 + 53066226351087745T^6 + 1081714442086224934T^5 + 6738226891746289904411T^4 + 69390882278241935058537T^3 + 367134748252109130769220437T^2 + 17262221112962498866734723372*T + 13765801910945945299474941142416
$67$	$T^{12} + \cdots + 56\!\cdots\!96$ T^12 + 289T^11 + 967991T^10 + 488474878T^9 + 909388512549T^8 + 340490597487379T^7 + 121066840518479129T^6 + 7482729512369687118T^5 + 425631214451880551459T^4 - 1629556435978656013423T^3 + 27156979491065702226277T^2 + 83739135482772655707180*T + 561933756320839431178896
$71$	$(T^{6} + \cdots + 16\!\cdots\!68)^{2}$ (T^6 + 1834T^5 - 50188T^4 - 1632778824T^3 - 547129326240T^2 + 362865223574784*T + 163888738205165568)^2
$73$	$T^{12} + \cdots + 44\!\cdots\!44$ T^12 - 332T^11 + 757852T^10 - 464860656T^9 + 546521080848T^8 - 240526659067392T^7 + 104040583700884992T^6 - 19204302718891763712T^5 + 4345482266406526795776T^4 - 296983797981302351757312T^3 + 128696828144723409558896640T^2 - 7266654903163992507486830592*T + 442915431072695751901597138944
$79$	$T^{12} + \cdots + 96\!\cdots\!04$ T^12 - 1368T^11 + 2614328T^10 - 1024107648T^9 + 1672412509360T^8 - 162179182960896T^7 + 968404679508477312T^6 + 89793048014722795008T^5 + 212855818063939860052224T^4 + 20049368439920807117426688T^3 + 33183089764688308759787999232T^2 + 4867307337359846108326200016896*T + 960904838164819551418764904955904
$83$	$(T^{6} + \cdots + 44\!\cdots\!56)^{2}$ (T^6 - 839T^5 - 1312678T^4 + 1156517670T^3 + 259352934409T^2 - 325796442341647*T + 44536977571818956)^2
$89$	$T^{12} + \cdots + 14\!\cdots\!16$ T^12 + 1035T^11 + 3577041T^10 + 3824978188T^9 + 9040506124317T^8 + 8497421904476745T^7 + 10800021984282151363T^6 + 6694910873576638249656T^5 + 6460610653899339677721639T^4 + 3371998331742556935444039831T^3 + 2393570291747795695648482404715T^2 + 620865833536955782689955376770314*T + 147753797308602858281793038865958116
$97$	$(T^{6} + \cdots - 25\!\cdots\!52)^{2}$ (T^6 - 4T^5 - 3303396T^4 - 504792544T^3 + 2565527496688T^2 + 387542093654976*T - 255416978923157952)^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 81.6
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	280.4.q.c

Level	$280$
Weight	$4$
Character orbit	280.q
Analytic conductor	$16.521$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$