LMFDB - Newform orbit 665.2.i.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [665,2,Mod(106,665)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(665, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("665.106");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$665 = 5 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	665.i (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:	$5.31005173442$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 3 x^{19} + 20 x^{18} - 43 x^{17} + 207 x^{16} - 401 x^{15} + 1351 x^{14} - 2135 x^{13} + \cdots + 1024$ x^20 - 3x^19 + 20x^18 - 43x^17 + 207x^16 - 401x^15 + 1351x^14 - 2135x^13 + 5714x^12 - 7983x^11 + 16723x^10 - 18718x^9 + 30867x^8 - 28726x^7 + 37857x^6 - 24848x^5 + 21360x^4 - 7488x^3 + 5888x^2 - 1536*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{12} - \beta_{8} + \beta_{6}) q^{4} + ( - \beta_{8} + 1) q^{5} + ( - \beta_{16} - \beta_{15} + \cdots - \beta_1) q^{6} - q^{7} + ( - \beta_{19} - \beta_{16} - \beta_{14} + \cdots + 1) q^{8}+ \cdots + ( - 3 \beta_{19} + 3 \beta_{18} + \cdots - 3 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^2 - b2 * q^3 + (-b12 - b8 + b6) * q^4 + (-b8 + 1) * q^5 + (-b16 - b15 + b14 - b13 + b9 - b8 - b7 - b2 - b1) * q^6 - q^7 + (-b19 - b16 - b14 + b11 + b7 + b6 - b5 + 1) * q^8 + (-b18 + b15 - b11 - b8 + b7 + b3 + b2 + b1) * q^9 + (b7 + b1) * q^10 + (b17 - b11 - b10 + b5) * q^11 + (-b19 - b17 - b16 - b15 + b11 + b10 - b5) * q^12 + (-b19 + b17 - b12 + b11 + b8 + b6 + b4) * q^13 - b1 * q^14 + (-b15 - b2) * q^15 + (-b18 + b12 + b10 - b9) * q^16 + (b13 - b9 + b5 - b4 + 2b1) q^17 + (b16 + b15 - b14 - 2b11 + b7 + b6 + 2b5 + b3 - 4) * q^18 + (b19 - b13 + b11 + b10 - b6 - b5 - b4 + 1) * q^19 + (b6 - 1) * q^20 + b2 * q^21 + (b18 - b13 - b10 + 2b9 + b8 + b5 + b4 - 1) q^22 + (-b19 - b18 + b15 + b12 - b6 + b4 + b3 + b2) * q^23 + (-b18 + b12 + 2b10 - b9 - b5 - b4) q^24 - b8 * q^25 + (b17 + b16 + b14 - b10 + b6 - b3 + 1) * q^26 + (-b16 - b15 + 2b14 - 2b11 - 3b7 - b6 + 2b5 + b3 - 1) * q^27 + (b12 + b8 - b6) * q^28 + (2b19 + b18 - b14 + b13 - b12 + b8 + b7 + b6 - 2b4 - b3 + b1) * q^29 + (-b16 - b15 + b14 - b7 - 1) * q^30 + (-b16 + b15 + 2b14 - b11 - b7 + b6 + b5) q^31 + (2b17 + b16 + b15 + b14 - b13 - 2b11 - b9 + b8 - b7 + b2 - b1) * q^32 + (b18 + 2b13 - 3b12 + b10 + 2b8 + 2b5 + 2b1 - 2) q^33 + (b19 + b18 + b17 + 2b15 + b14 - b13 - b12 - b11 - 2b8 + b7 + b6 - b4 - b3 + 2b2 + b1) q^34 + (b8 - 1) * q^35 + (-2b13 + b12 - b10 - b2 - 4b1) * q^36 + (b19 - b16 + 2b15 + 2b14 - b3 - 3) * q^37 + (-b18 + b17 + 2b16 - b15 - b14 + 2b13 - b12 - b9 - b8 + b7 + b5 + b4 - b2 + 3b1 + 1) q^38 + (-b19 - b17 + b16 + b15 + b11 + b10 - b5 - b3 + 2) * q^39 + (-b13 + b12 - b9 - b8 - b5 - b4 - b1 + 1) * q^40 + (-b13 + 2b12 + 2b9 + 3b8 + b5 - 2b4 - b2 - 3) * q^41 + (b16 + b15 - b14 + b13 - b9 + b8 + b7 + b2 + b1) * q^42 + (-b18 + b13 - b9 - b8 - 2b5 + b2 - b1 + 1) q^43 + (-2b19 + b18 - b17 - 2b16 - 3b15 - b14 + b13 - 2b12 + 2b11 + 2b9 - 2b8 - b7 + 2b6 + 2b4 - b3 - 3b2 - b1) * q^44 + (b15 - b11 + b7 + b5 + b3 - 1) * q^45 + (b19 + 2b16 + b15 + 2b14 - 3b11 - 3b7 - b6 + 3b5 - 2) q^46 + (-2b19 - b18 - b17 - 2b14 + 2b13 - b12 + 2b7 + b6 + 2b4 + b3 + 2b1) * q^47 + (b19 - b18 + b17 + 3b16 - b15 + b14 - b13 + 2b12 - 3b11 - 3b9 - b8 - 2b7 - 2b6 - b4 + b3 - b2 - 2b1) q^48 + q^49 + b7 * q^50 + (b18 - b17 - b16 - 2b15 + b14 - b13 + b12 + 2b11 + b9 + b8 - 3b7 - b6 - b3 - 2b2 - 3b1) q^51 + (-b13 - 2b12 + b9 - b5 + b4 - 2b1) * q^52 + (2b19 + 2b18 + b17 + b16 - 2b15 + 2b14 - 2b13 + b12 + b11 - b9 + 4b8 - b6 - 2b4 - 2b3 - 2b2) q^53 + (2b18 - 3b12 + b10 - b8 + b5 - b4 + 4b2 + 3b1 + 1) * q^54 + (-b10 + b5) * q^55 + (b19 + b16 + b14 - b11 - b7 - b6 + b5 - 1) * q^56 + (-b18 - b17 + b16 + b14 - 2b13 + 2b12 - b11 + b9 - 2b8 - 2b6 + b4 - b2 - b1 + 2) * q^57 + (-b16 + b15 - 4b14 + 2b11 + 4b7 + 2b6 - 2b5 + 2) q^58 + (-b12 - 2b10 - b9 + 2b4 - 2b1) q^59 + (b10 - b9 - b5 - b4 + b2) * q^60 + (3b19 - b18 - b16 + b15 + b12 - b11 + b9 + b8 + b7 - b6 - 3b4 + b3 + b2 + b1) * q^61 + (b18 - 4b13 + b12 + b10 - b8 - 3b5 - 3b4 + 2b2 - 3b1 + 1) q^62 + (b18 - b15 + b11 + b8 - b7 - b3 - b2 - b1) * q^63 + (b19 + b17 + 3b16 + b15 - 2b14 - b11 - b10 + b5 + 2) * q^64 + (-b19 + b17 + b11 - b10 + b6 - b5 + 1) * q^65 + (2b18 + b17 - b16 + 2b15 - 2b14 + 2b13 - 5b12 + 2b11 + b9 + 2b8 + 5b7 + 5b6 - 2b3 + 2b2 + 5b1) * q^66 + (-2b19 + b18 - b16 + b15 - b14 + b13 + b9 + 3b8 + 2b4 - b3 + b2) q^67 + (-3b19 + b17 + b16 + b15 - 4b14 + b11 - b10 + 4b7 + 2b6 - b5 + b3) * q^68 + (b19 + 2b17 + 2b16 - 2b15 - 2b11 - 2b10 - b7 + b6 + 2b5 + 2) * q^69 + (-b7 - b1) * q^70 + (b18 - b13 + b12 - b10 + b8 - b4 + 3b2 - 1) q^71 + (-2b19 - 2b18 - b17 - b15 + 3b12 - 2b11 - b7 - 3b6 + 2b4 + 2b3 - b2 - b1) q^72 + (3b13 + b10 + 2b5 + b1) * q^73 + (-5b13 + b12 + b10 + 2b9 - b8 - 4b5 - 2b4 + b2 - 7b1 + 1) q^74 - b15 * q^75 + (2b19 + b18 + 2b15 + 2b14 - b13 - 2b12 - 2b11 - b10 + b9 - 4b8 + b6 + 2b5 - 2b4 - b3 + b1 - 1) * q^76 + (-b17 + b11 + b10 - b5) * q^77 + (-b18 + b13 - b12 - b9 - 3b8 - 3b5 - b4 - 2b2 + b1 + 3) q^78 + (b12 - b10 + b9 + 3b4 + 3b1) * q^79 + (-b18 + b17 + b16 + b12 - b9 - b6 + b3) * q^80 + (b18 - b13 - 2b10 + 3b9 + 5b8 + 4b5 + b4 + b2 - 2b1 - 5) q^81 + (b19 + b18 - 2b17 + b16 - 4b15 - b11 - b9 - 3b8 - 6b7 - b4 - b3 - 4b2 - 6b1) * q^82 + (-2b19 - 2b15 - b14 + b3 - 3) * q^83 + (b19 + b17 + b16 + b15 - b11 - b10 + b5) * q^84 + (b19 + b16 - b14 + b13 + b11 - b9 + 2b7 - b4 + 2b1) * q^85 + (b19 - 2b18 + b17 + b16 + 3b15 - b14 + b13 + 2b12 - 3b11 - b9 + 3b8 + b7 - 2b6 - b4 + 2b3 + 3b2 + b1) * q^86 + (-b19 - 4b17 - 3b16 + b15 + b14 + 3b11 + 4b10 - 2b7 - 3b6 - 3b5 - b3 + 1) q^87 + (-b17 - 3b16 - 4b15 + 2b11 + b10 + 2b7 + 3b6 - 2b5) * q^88 + (b19 + b18 - 3b17 - 3b14 + 3b13 + b12 + 2b11 - b8 + 2b7 - b6 - b4 - b3 + 2b1) * q^89 + (b18 - b13 + b12 + b9 + 4b8 + 2b5 - b2 - b1 - 4) * q^90 + (b19 - b17 + b12 - b11 - b8 - b6 - b4) * q^91 + (b18 - 2b13 - 4b12 - 2b10 + 2b9 - 7b8 - b5 + b4 + b2 - 2b1 + 7) * q^92 + (b12 - b9 - 2b8 - b4 - 2b1 + 2) * q^93 + (-b17 - 3b16 + 2b15 + 2b14 - 2b11 + b10 - b7 + 3b6 + 2b5 - 6) * q^94 + (b19 + b17 + b14 - b13 - b12 - b8 - b5 + 1) * q^95 + (b16 + b15 + 3b14 - 2b11 - 4b7 - 5b6 + 2b5 + b3 + 2) q^96 + (-2b18 - b13 + 2b12 - 2b9 - 5b8 - 2b5 - 2b4 - b1 + 5) * q^97 + b1 * q^98 + (-3b19 + 3b18 - 3b17 - 3b16 + b15 + b14 - b13 + 2b12 + 3b11 + 3b9 + 4b8 - 3b7 - 2b6 + 3b4 - 3b3 + b2 - 3b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 3 q^{2} - q^{3} - 11 q^{4} + 10 q^{5} - 6 q^{6} - 20 q^{7} - 9 q^{9} - 3 q^{10} + 2 q^{11} - 4 q^{12} + 6 q^{13} - 3 q^{14} + q^{15} - 5 q^{16} + 8 q^{17} - 76 q^{18} + 17 q^{19} - 22 q^{20} + q^{21}+ \cdots + 34 q^{99}+O(q^{100})$ 20 * q + 3 * q^2 - q^3 - 11 * q^4 + 10 * q^5 - 6 * q^6 - 20 * q^7 - 9 * q^9 - 3 * q^10 + 2 * q^11 - 4 * q^12 + 6 * q^13 - 3 * q^14 + q^15 - 5 * q^16 + 8 * q^17 - 76 * q^18 + 17 * q^19 - 22 * q^20 + q^21 - 4 * q^22 + 3 * q^23 - 6 * q^24 - 10 * q^25 + 18 * q^26 + 20 * q^27 + 11 * q^28 + q^29 - 12 * q^30 + 10 * q^31 + 18 * q^32 + q^33 - 26 * q^34 - 10 * q^35 - 19 * q^36 - 66 * q^37 + 19 * q^38 + 34 * q^39 - 29 * q^41 + 6 * q^42 + q^43 - 28 * q^44 - 18 * q^45 + 6 * q^46 - 7 * q^47 + 15 * q^48 + 20 * q^49 - 6 * q^50 + 16 * q^51 - 6 * q^52 + 40 * q^53 + 35 * q^54 + q^55 + 24 * q^57 - 18 * q^58 - 9 * q^59 - 2 * q^60 + 10 * q^61 - 8 * q^62 + 9 * q^63 + 44 * q^64 + 12 * q^65 - 19 * q^66 + 22 * q^67 - 42 * q^68 + 60 * q^69 + 3 * q^70 - 8 * q^71 + 18 * q^72 + 14 * q^73 - 24 * q^74 + 2 * q^75 - 49 * q^76 - 2 * q^77 + 23 * q^78 + 9 * q^79 + 5 * q^80 - 42 * q^81 - 5 * q^82 - 54 * q^83 + 4 * q^84 - 8 * q^85 + 37 * q^86 + 18 * q^87 - 28 * q^88 - 25 * q^89 - 38 * q^90 - 6 * q^91 + 68 * q^92 + 11 * q^93 - 118 * q^94 + 10 * q^95 + 102 * q^96 + 29 * q^97 + 3 * q^98 + 34 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 3 x^{19} + 20 x^{18} - 43 x^{17} + 207 x^{16} - 401 x^{15} + 1351 x^{14} - 2135 x^{13} + \cdots + 1024$ x^20 - 3*x^19 + 20*x^18 - 43*x^17 + 207*x^16 - 401*x^15 + 1351*x^14 - 2135*x^13 + 5714*x^12 - 7983*x^11 + 16723*x^10 - 18718*x^9 + 30867*x^8 - 28726*x^7 + 37857*x^6 - 24848*x^5 + 21360*x^4 - 7488*x^3 + 5888*x^2 - 1536*x + 1024 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 19\!\cdots\!91 \nu^{19} + \cdots - 10\!\cdots\!68 ) / 37\!\cdots\!68$ (-19869127020279103661941691v^19 - 731036015982304764038698243v^18 + 1068267607767549839136865632v^17 - 12456746243219293441839723623v^16 + 13317138864266463209415233511v^15 - 122226941569345592195473864641v^14 + 126160848290505115724219304215v^13 - 719195516221250890555767000223v^12 + 574729347153592771765061004806v^11 - 2817468903192068490696016116819v^10 + 2031828555655060579061089068555v^9 - 7237470472177831669749310935506v^8 + 3703058054963194385794861315895v^7 - 11720818161755281605453135299978v^6 + 4993988703801509591480256406397v^5 - 11678746427881776219979714415652v^4 + 1788909675339773938516183757200v^3 - 4037765755594335138276063624096v^2 + 87092937628114915899580091392*v - 1019928662696777763686147335168) / 379959772356236311192938330368
$\beta_{3}$	$=$	$( - 33\!\cdots\!89 \nu^{19} + \cdots + 23\!\cdots\!16 ) / 47\!\cdots\!96$ (-3350788277842028339252289v^19 + 38462723761247523053945582v^18 - 149534437757662809077997817v^17 + 573276316800065624771967795v^16 - 1593449235471714025780098168v^15 + 5055876860573003255156370178v^14 - 12253085980729356507567069726v^13 + 25285281478357982003211814056v^12 - 48582311944785802335139648811v^11 + 80289189453968121256612387601v^10 - 145176126798923771899397546656v^9 + 149684576534580575545320586955v^8 - 163859640575877690977146091401v^7 + 134193687990614476778477006623v^6 - 119210439176460489842482486879v^5 + 80183009894358444566734312947v^4 + 289533091738025676411751555356v^3 + 21027242149846706533227056048v^2 - 6766508560668333169493806944*v + 234612658709522577377951431616) / 47494971544529538899117291296
$\beta_{4}$	$=$	$( - 70\!\cdots\!61 \nu^{19} + \cdots - 28\!\cdots\!64 ) / 75\!\cdots\!36$ (-70903227545747937657879961v^19 - 2251228917677159051391588413v^18 + 3946435230534642185069037444v^17 - 40308740200596374500054959533v^16 + 52786442715251042727502815233v^15 - 399518047167677831331274020975v^14 + 504263808901910947791661253017v^13 - 2416886682508939119860768504569v^12 + 2420189054681203563883982289142v^11 - 9562448214986893408435053094009v^10 + 8579383805435835958070585529085v^9 - 25239096383806554478618084049546v^8 + 16436259370746748634246283063957v^7 - 40847144239560296069318104459634v^6 + 20836465612822557855012366610455v^5 - 41477535119498175504551078991768v^4 + 7059485274107562320063664270256v^3 - 12005162940570048670285642902240v^2 - 3240934420828043581644218825728*v - 2879680941980920500966152715264) / 759919544712472622385876660736
$\beta_{5}$	$=$	$( - 20\!\cdots\!11 \nu^{19} + \cdots - 41\!\cdots\!76 ) / 18\!\cdots\!84$ (-20423406307088312444411611v^19 - 167970361064276084447882423v^18 - 208549849245469082734059108v^17 - 2571014535424813952274977623v^16 - 2536341145835998863142267549v^15 - 25063846452641576150232064653v^14 - 14048872855624473822839249077v^13 - 137830199378147105383155445835v^12 - 74928387499425691294020292334v^11 - 535405452337688503498263213275v^10 - 194316406449902361061359672873v^9 - 1263318221681322418919740924622v^8 - 413874631360924329377040494481v^7 - 2135494708725142942610441600454v^6 - 228522385091799319907148667435v^5 - 2067125596055756936774004264952v^4 + 54517431287208531193487858704v^3 - 1428880717892210696007911572128v^2 + 444849362815506583194211259840*v - 411721182109718592463546725376) / 189979886178118155596469165184
$\beta_{6}$	$=$	$( - 25\!\cdots\!87 \nu^{19} + \cdots + 61\!\cdots\!04 ) / 21\!\cdots\!68$ (-259452189909998150352587v^19 + 881522394478719334159974v^18 - 5334410785805341180432587v^17 + 12586524471128021679162605v^16 - 54477940078228623650786524v^15 + 115958717733081333240255702v^14 - 353695709848460721734672066v^13 + 606095286825193751669158304v^12 - 1445814845079825611672307321v^11 + 2193611869479082435484537023v^10 - 4092076683568351842770491132v^9 + 4904356215336203412056463929v^8 - 6820076890148476862047349447v^7 + 6823558583478235076247787473v^6 - 7496671111618008142832638473v^5 + 5725672519976575434315705045v^4 - 2114508143680176388906166032v^3 + 1641560766361188888095050960v^2 - 525605783319186629225038240*v + 6140098906893643533642188304) / 2158862342933160859050785968
$\beta_{7}$	$=$	$( 32\!\cdots\!00 \nu^{19} + \cdots - 10\!\cdots\!40 ) / 21\!\cdots\!68$ (328601681548670940927900v^19 - 1245257234556010973136287v^18 + 7453556025452138152717974v^17 - 19464283092398191640332287v^16 + 80607072551702906451237905v^15 - 186247214379245670962874424v^14 + 559899589505335774433848602v^13 - 1055260299954873180615738566v^12 + 2483725295194299508131178904v^11 - 4069042068882865733099733021v^10 + 7688817790017506580621808723v^9 - 10242842958796374515058923332v^8 + 15047304319699029345677953229v^7 - 16259488794315598311142204847v^6 + 19263432441866270886955297773v^5 - 15661765694739383683009097673v^4 + 12744604437856186732535649045v^3 - 4575077535116624394574281232v^2 + 1417505124386602529227740192*v - 1030337966177945194490292640) / 2158862342933160859050785968
$\beta_{8}$	$=$	$( - 32\!\cdots\!45 \nu^{19} + \cdots + 40\!\cdots\!76 ) / 69\!\cdots\!76$ (-32198061443060787327821645v^19 + 86078930519624891873772135v^18 - 604112997355423395416071716v^17 + 1146002849237145434209355567v^16 - 6042141659756840844368447331v^15 + 10331996317012882712016866685v^14 - 37539670149439262209075060827v^13 + 50826074316764036163016056811v^12 - 150211393487093397011469245418v^11 + 177557915053736680977802467107v^10 - 408238835308053843023969912663v^9 + 356641144810651606622267671974v^8 - 666086587881473337965985169591v^7 + 443407774782995237717310070942v^6 - 698618370631853079912793459661v^5 + 183627592597453775139142706224v^4 - 186574090192118139465979211664v^3 - 166728257925758799930412291680v^2 - 43179704653009935159836846336*v + 4096058396170088400246360576) / 69083594973861147489625150976
$\beta_{9}$	$=$	$( 23\!\cdots\!81 \nu^{19} + \cdots + 15\!\cdots\!04 ) / 37\!\cdots\!68$ (234769172967493576671619381v^19 + 210253291989595740553589925v^18 + 2953457343801953655469216056v^17 + 5652966090588689921887749545v^16 + 26845993627721032759574568271v^15 + 61930434933073089864142959831v^14 + 122692658728239110574377908655v^13 + 429854434174775399783438876185v^12 + 414211217820645078895302855198v^11 + 1836483806148112913887845527565v^10 + 633258696327916857739128756307v^9 + 5345782064872470366659718921334v^8 + 551490512114601642377663549431v^7 + 9566905670082525321577948536126v^6 - 515939151842746662448294109283v^5 + 11090311267101273049403780571348v^4 - 628866803023512229467790767728v^3 + 5524727177131556521065427445856v^2 - 96080251120544047326892671488*v + 1555261381801076466068643707904) / 379959772356236311192938330368
$\beta_{10}$	$=$	$( 71\!\cdots\!03 \nu^{19} + \cdots + 29\!\cdots\!64 ) / 75\!\cdots\!36$ (715631038526513445387924503v^19 - 528315964557739947408370541v^18 + 10369180242743313926440780452v^17 - 1351453668609591809698377149v^16 + 96674674902211096841424617905v^15 + 6969962982995785637443526913v^14 + 501604043050366478234516898953v^13 + 279494025951638880440366041431v^12 + 1812381938038778552885857503286v^11 + 1574321522576306070940662420983v^10 + 3850825341414607193658496370669v^9 + 6464236634870519145709290073206v^8 + 5349683219155119402321760876901v^7 + 13236366598868973889988129319438v^6 + 3268726617886835116054512453767v^5 + 19116939610952413924767840044360v^4 + 46109776251322894537016536752v^3 + 10184760153423567065997138716448v^2 - 1422874930823844345724255291648*v + 2973006765454172626777205310464) / 759919544712472622385876660736
$\beta_{11}$	$=$	$( - 20\!\cdots\!79 \nu^{19} + \cdots - 25\!\cdots\!56 ) / 18\!\cdots\!84$ (-201325991944996513292065379v^19 + 184801675169102356977131501v^18 - 2942962391898185889412467300v^17 + 1205542970541777300678630501v^16 - 27271904984624393892893196233v^15 + 7132907079305022408675918211v^14 - 139368151260477771215559266813v^13 - 13328627995913257989767808963v^12 - 480131458208260623794914115254v^11 - 148018541141328848337330903591v^10 - 890485076291805802373407004333v^9 - 879334460208747012943876859958v^8 - 773317547880539449887934597701v^7 - 1978580763053721502888932727722v^6 + 592505259663027413746789839761v^5 - 3190527308699625367779012298372v^4 + 1722878854483998981319471040340v^3 - 1788388430317731357591197613408v^2 + 531625880540024336486136499008*v - 257939200028572612628518153856) / 189979886178118155596469165184
$\beta_{12}$	$=$	$( 88\!\cdots\!51 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 69\!\cdots\!76$ (88291714252062421172182151v^19 - 230028074935555656928197237v^18 + 1641637846920499268474372364v^17 - 3035239764635339608894863341v^16 + 16383130896767206576280173225v^15 - 27285309983580045472362417591v^14 + 101300747733167043531715676369v^13 - 133083173771885908435635104705v^12 + 404368105418725771460893901982v^11 - 462478165337879404997902216585v^10 + 1093770052049974270103254021765v^9 - 912984035541196310680996170194v^8 + 1780017303159668754312440326469v^7 - 1111869449677682190712001013690v^6 + 1855961636323782979167735947847v^5 - 367661257153110911519325557232v^4 + 492058009978588773952940321968v^3 + 483631123326973296720653354784v^2 + 112719728892815833344309315328*v + 184194989832086327875810944000) / 69083594973861147489625150976
$\beta_{13}$	$=$	$( 97\!\cdots\!23 \nu^{19} + \cdots + 40\!\cdots\!84 ) / 75\!\cdots\!36$ (972131669173918168740350423v^19 - 923404102414821011715666693v^18 + 15180720498519144510395171628v^17 - 6435292049544714710981079773v^16 + 146149659602546838451050297657v^15 - 36759924143845520170124652903v^14 + 815428930571471698865070494369v^13 + 72580253937629206797633830319v^12 + 3108389974464716241868105870494v^11 + 903852242253260025516974284679v^10 + 7436590784786746987213686285525v^9 + 5216599439673766842841852015854v^8 + 11244972620324791283924012866293v^7 + 12328717789713078757938584707974v^6 + 9463743121949414028585812369751v^5 + 20294796119517766395472351525488v^4 + 1551160174221480857840363114928v^3 + 13472463505429280842370206516512v^2 - 172284129417300451599602778368*v + 4063531674376514182001850000384) / 759919544712472622385876660736
$\beta_{14}$	$=$	$( - 27\!\cdots\!12 \nu^{19} + \cdots + 59\!\cdots\!36 ) / 11\!\cdots\!24$ (-27036215861078456016004612v^19 + 77338021377920904290288913v^18 - 503100777132732750449510754v^17 + 1043480038449384178603263025v^16 - 4998288289532839720285168215v^15 + 9552923206219033203915408808v^14 - 30760747795087389993662229502v^13 + 48211649104930086323049669786v^12 - 121845823913586201455527963712v^11 + 174840390301265210902265742443v^10 - 326778680289430427271396675669v^9 + 377484384476499389254970612732v^8 - 526081860623075131726586698155v^7 + 546929880542951145284901057441v^6 - 541869832078421293089386423955v^5 + 396495880125409892719057799223v^4 - 133611568190610158967198559371v^3 + 112571393106971914555170354672v^2 - 37140122368672536857694112736*v + 59664285275728036470400719336) / 11873742886132384724779322824
$\beta_{15}$	$=$	$( - 21\!\cdots\!55 \nu^{19} + \cdots + 27\!\cdots\!76 ) / 94\!\cdots\!92$ (-219935463497777823180041355v^19 + 646221841327745451106725292v^18 - 4166319959625212937314733407v^17 + 8861587673461861455722568027v^16 - 41770555272095999190501921126v^15 + 81615571885319850837361744918v^14 - 260704971477805192309089274566v^13 + 418383597311234174381272340876v^12 - 1048407134755050109639806140273v^11 + 1533607192006187329886087578201v^10 - 2871616242218838625427966346426v^9 + 3398864091369850024709771849049v^8 - 4785391321597199373456928591521v^7 + 5040221719521444492028296522015v^6 - 5154941297716676616972679053251v^5 + 3865106802456840518202394420935v^4 - 1794894531327991726030174757794v^3 + 1105076983317016132511832061680v^2 - 359163359424184663440669644256*v + 277945241731414135094416758976) / 94989943089059077798234582592
$\beta_{16}$	$=$	$( - 22\!\cdots\!43 \nu^{19} + \cdots + 38\!\cdots\!96 ) / 94\!\cdots\!92$ (-222058135805364883538352743v^19 + 626075495389712770413601196v^18 - 4109859655771951810845957803v^17 + 8456278801508802029904087047v^16 - 40889626589850080904446851534v^15 + 77681540628537417316350228430v^14 - 251599235680036874041916979614v^13 + 394110537134340200683516633308v^12 - 1001802018687500856729696158981v^11 + 1441543286993077368600226839293v^10 - 2701154295174784334888192753490v^9 + 3154142304492431632265997465677v^8 - 4435884034504835012518684456421v^7 + 4658333937042803335797911224411v^6 - 4670571604909170325886628996911v^5 + 3452233455222747980881791424707v^4 - 1423196821497452236023470631290v^3 + 983634812115501386299486195888v^2 - 322815385649360360744753031520*v + 380014379036096608901946709696) / 94989943089059077798234582592
$\beta_{17}$	$=$	$( 19\!\cdots\!87 \nu^{19} + \cdots + 63\!\cdots\!68 ) / 75\!\cdots\!36$ (1964192472774229884179887487v^19 - 4203151466731529589921232877v^18 + 33786627698279502222719652268v^17 - 50482710375244132128183900677v^16 + 328497411480157120859455460897v^15 - 440352794409758424846326229455v^14 + 1929388884490083824929384923161v^13 - 1948949726825232699817175937033v^12 + 7418378204056165461443133169262v^11 - 6428005969095911698532323042977v^10 + 18798061884017667361759107094269v^9 - 10422008667844950404475507569154v^8 + 28515551364166900885145820168125v^7 - 10992769486724590046822784952090v^6 + 26177649841202895895965649102671v^5 + 2682577939962694663201226867744v^4 + 2213649965514021048333532064448v^3 + 5551510144687691004752355989664v^2 + 122117536990477609830104634368*v + 632233359225331084342060986368) / 759919544712472622385876660736
$\beta_{18}$	$=$	$( 10\!\cdots\!19 \nu^{19} + \cdots + 20\!\cdots\!28 ) / 37\!\cdots\!68$ (1043026613884485736909089019v^19 - 2702414509276241831712914425v^18 + 18917489087641323696976988052v^17 - 34943204472093862335068798089v^16 + 186996975547645223831553965357v^15 - 313366171399414978906075538883v^14 + 1137633549017052114415154397349v^13 - 1500201889025732649782971012165v^12 + 4493874511313494828908909225566v^11 - 5182772951230592535545959688357v^10 + 11967910659583250225219715991929v^9 - 9802180965353095045108815695202v^8 + 19365350002236758076431311399569v^7 - 11613297524948169414763047440938v^6 + 19811210920599121765148287934379v^5 - 1927435004372131805992844453720v^4 + 5229950130531744735044102133744v^3 + 5349375287385271341036549612192v^2 + 399515089800266450913236497920*v + 2018708792979121920656513430528) / 379959772356236311192938330368
$\beta_{19}$	$=$	$( 40\!\cdots\!27 \nu^{19} + \cdots - 64\!\cdots\!56 ) / 94\!\cdots\!92$ (408773043459706119631186927v^19 - 1303563254541649547410349822v^18 + 8172527852531490792761760191v^17 - 18644175559718549039243335705v^16 + 83844677584651181130490464956v^15 - 173878753095494852617480629854v^14 + 542640877296150943913975928298v^13 - 923042017652427542233239692304v^12 + 2256770786401006566998750793973v^11 - 3445243286702170062358508259603v^10 + 6478787942296119999058238802364v^9 - 8019659277028548486500309055365v^8 + 11475141028396881980194997324763v^7 - 12232166965627091063312747665101v^6 + 13324145692213341274274219059397v^5 - 10379560214511656429288868301281v^4 + 5942054753538102975071787598976v^3 - 2998248197189802089262651682704v^2 + 952049096360007876996469712672*v - 641957727768116460257011691456) / 94989943089059077798234582592

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{12} - 3\beta_{8} + \beta_{6}$ -b12 - 3*b8 + b6
$\nu^{3}$	$=$	$-\beta_{19} - \beta_{16} - \beta_{14} + \beta_{11} + 5\beta_{7} + \beta_{6} - \beta_{5} + 1$ -b19 - b16 - b14 + b11 + 5*b7 + b6 - b5 + 1
$\nu^{4}$	$=$	$-\beta_{18} + 7\beta_{12} + \beta_{10} - \beta_{9} + 14\beta_{8} - 14$ -b18 + 7b12 + b10 - b9 + 14b8 - 14
$\nu^{5}$	$=$	$8 \beta_{19} + 2 \beta_{17} + 9 \beta_{16} + \beta_{15} + 9 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} + \cdots - 29 \beta_1$ 8b19 + 2b17 + 9b16 + b15 + 9b14 - 9b13 + 8b12 - 10b11 - 9b9 - 7b8 - 29b7 - 8b6 - 8b4 + b2 - 29*b1
$\nu^{6}$	$=$	$\beta_{19} + 11 \beta_{17} + 13 \beta_{16} + \beta_{15} - 2 \beta_{14} - \beta_{11} - 11 \beta_{10} + \cdots + 78$ b19 + 11b17 + 13b16 + b15 - 2b14 - b11 - 11b10 - 46b6 + b5 + 10b3 + 78
$\nu^{7}$	$=$	$\beta_{18} + 66 \beta_{13} - 59 \beta_{12} - 24 \beta_{10} + 69 \beta_{9} + 40 \beta_{8} + 79 \beta_{5} + \cdots - 40$ b18 + 66b13 - 59b12 - 24b10 + 69b9 + 40b8 + 79b5 + 59b4 - 16b2 + 178*b1 - 40
$\nu^{8}$	$=$	$- 17 \beta_{19} + 79 \beta_{18} - 93 \beta_{17} - 123 \beta_{16} - 19 \beta_{15} + 25 \beta_{14} + \cdots + 4 \beta_1$ -17b19 + 79b18 - 93b17 - 123b16 - 19b15 + 25b14 - 25b13 - 306b12 + 19b11 + 123b9 - 473b8 + 4b7 + 306b6 + 17b4 - 79b3 - 19b2 + 4*b1
$\nu^{9}$	$=$	$- 424 \beta_{19} - 216 \beta_{17} - 511 \beta_{16} - 167 \beta_{15} - 453 \beta_{14} + 582 \beta_{11} + \cdots + 208$ -424b19 - 216b17 - 511b16 - 167b15 - 453b14 + 582b11 + 216b10 + 1128b7 + 433b6 - 582b5 - 19*b3 + 208
$\nu^{10}$	$=$	$- 582 \beta_{18} + 210 \beta_{13} + 2072 \beta_{12} + 727 \beta_{10} - 1032 \beta_{9} + 2999 \beta_{8} + \cdots - 2999$ -582b18 + 210b13 + 2072b12 + 727b10 - 1032b9 + 2999b8 - 234b5 - 196b4 + 225b2 - 86b1 - 2999
$\nu^{11}$	$=$	$3009 \beta_{19} - 234 \beta_{18} + 1759 \beta_{17} + 3751 \beta_{16} + 1467 \beta_{15} + \cdots - 7304 \beta_1$ 3009b19 - 234b18 + 1759b17 + 3751b16 + 1467b15 + 3031b14 - 3031b13 + 3190b12 - 4173b11 - 3751b9 - 967b8 - 7304b7 - 3190b6 - 3009b4 + 234b3 + 1467b2 - 7304*b1
$\nu^{12}$	$=$	$1918 \beta_{19} + 5510 \beta_{17} + 8175 \beta_{16} + 2187 \beta_{15} - 1474 \beta_{14} - 2386 \beta_{11} + \cdots + 19511$ 1918b19 + 5510b17 + 8175b16 + 2187b15 - 1474b14 - 2386b11 - 5510b10 - 1171b7 - 14245b6 + 2386b5 + 4173*b3 + 19511
$\nu^{13}$	$=$	$2386 \beta_{18} + 20102 \beta_{13} - 23591 \beta_{12} - 13685 \beta_{10} + 27452 \beta_{9} + 3537 \beta_{8} + \cdots - 3537$ 2386b18 + 20102b13 - 23591b12 - 13685b10 + 27452b9 + 3537b8 + 29575b5 + 21229b4 - 11836b2 + 48069b1 - 3537
$\nu^{14}$	$=$	$- 17174 \beta_{19} + 29575 \beta_{18} - 41137 \beta_{17} - 62797 \beta_{16} - 19186 \beta_{15} + \cdots + 12966 \beta_1$ -17174b19 + 29575b18 - 41137b17 - 62797b16 - 19186b15 + 9211b14 - 9211b13 - 99112b12 + 21946b11 + 62797b9 - 129070b8 + 12966b7 + 99112b6 + 17174b4 - 29575b3 - 19186b2 + 12966*b1
$\nu^{15}$	$=$	$- 149460 \beta_{19} - 103934 \beta_{17} - 200572 \beta_{16} - 91194 \beta_{15} - 133178 \beta_{14} + \cdots + 3367$ -149460b19 - 103934b17 - 200572b16 - 91194b15 - 133178b14 + 208610b11 + 103934b10 + 320509b7 + 174875b6 - 208610b5 - 21946*b3 + 3367
$\nu^{16}$	$=$	$- 208610 \beta_{18} + 51871 \beta_{13} + 695956 \beta_{12} + 304506 \beta_{10} - 473937 \beta_{9} + \cdots - 864144$ -208610b18 + 51871b13 + 695956b12 + 304506b10 - 473937b9 + 864144b8 - 189523b5 - 145631b4 + 158588b2 - 127675b1 - 864144
$\nu^{17}$	$=$	$1052333 \beta_{19} - 189523 \beta_{18} + 778443 \beta_{17} + 1463556 \beta_{16} + 684396 \beta_{15} + \cdots - 2160447 \beta_1$ 1052333b19 - 189523b18 + 778443b17 + 1463556b16 + 684396b15 + 884761b14 - 884761b13 + 1297568b12 - 1469553b11 - 1463556b9 + 118963b8 - 2160447b7 - 1297568b6 - 1052333b4 + 189523b3 + 684396b2 - 2160447*b1
$\nu^{18}$	$=$	$1191250 \beta_{19} + 2241999 \beta_{17} + 3538850 \beta_{16} + 1263191 \beta_{15} - 255635 \beta_{14} + \cdots + 5840221$ 1191250b19 + 2241999b17 + 3538850b16 + 1263191b15 - 255635b14 - 1570296b11 - 2241999b10 - 1166893b7 - 4921571b6 + 1570296b5 + 1469553*b3 + 5840221
$\nu^{19}$	$=$	$1570296 \beta_{18} + 5905237 \beta_{13} - 9627314 \beta_{12} - 5780849 \beta_{10} + 10668760 \beta_{9} + \cdots + 1788185$ 1570296b18 + 5905237b13 - 9627314b12 - 5780849b10 + 10668760b9 - 1788185b8 + 10358311b5 + 7419205b4 - 5057676b2 + 14698475b1 + 1788185

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/665\mathbb{Z}\right)^\times$ .

$n$	$211$	$267$	$381$
$\chi(n)$	$-\beta_{8}$	$1$	$1$

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}

106.1

−1.26008	+	2.18252i
−0.994744	+	1.72295i
−0.731711	+	1.26736i
−0.278420	+	0.482238i
0.302407	−	0.523784i
0.449475	−	0.778513i
0.745668	−	1.29153i
0.801658	−	1.38851i
1.11529	−	1.93174i
1.35046	−	2.33906i
−1.26008	−	2.18252i
−0.994744	−	1.72295i
−0.731711	−	1.26736i
−0.278420	−	0.482238i
0.302407	+	0.523784i
0.449475	+	0.778513i
0.745668	+	1.29153i
0.801658	+	1.38851i
1.11529	+	1.93174i
1.35046	+	2.33906i

−1.26008

2.18252i

−0.424874

0.735903i

−2.17561

−

3.76826i

0.500000

−

0.866025i

−1.07075

−

1.85460i

−1.00000

5.92545

1.13896

1.97274i

1.26008

2.18252i

106.2

−0.994744

1.72295i

0.457864

−

0.793043i

−0.979030

−

1.69573i

0.500000

−

0.866025i

0.910914

1.57775i

−1.00000

−0.0834388

1.08072

1.87186i

0.994744

1.72295i

106.3

−0.731711

1.26736i

−1.07446

1.86101i

−0.0708008

−

0.122631i

0.500000

−

0.866025i

−1.57238

−

2.72345i

−1.00000

−2.71962

−0.808916

−

1.40108i

0.731711

1.26736i

106.4

−0.278420

0.482238i

0.0391618

−

0.0678302i

0.844964

1.46352i

0.500000

−

0.866025i

0.0218069

0.0377706i

−1.00000

−2.05470

1.49693

2.59276i

0.278420

0.482238i

106.5

0.302407

−

0.523784i

0.354617

−

0.614215i

0.817100

1.41526i

0.500000

−

0.866025i

−0.214477

−

0.371485i

−1.00000

2.19801

1.24849

2.16245i

−0.302407

−

0.523784i

106.6

0.449475

−

0.778513i

−1.28912

2.23283i

0.595945

1.03221i

0.500000

−

0.866025i

1.15886

2.00720i

−1.00000

2.86935

−1.82367

−

3.15869i

−0.449475

−

0.778513i

106.7

0.745668

−

1.29153i

1.62892

−

2.82136i

−0.112041

−

0.194061i

0.500000

−

0.866025i

−2.42926

−

4.20760i

−1.00000

2.64849

−3.80673

−

6.59345i

−0.745668

−

1.29153i

106.8

0.801658

−

1.38851i

0.392253

−

0.679403i

−0.285310

−

0.494171i

0.500000

−

0.866025i

−0.628906

−

1.08930i

−1.00000

2.29175

1.19227

2.06508i

−0.801658

−

1.38851i

106.9

1.11529

−

1.93174i

−1.60334

2.77707i

−1.48775

−

2.57686i

0.500000

−

0.866025i

3.57639

6.19449i

−1.00000

−2.17594

−3.64141

−

6.30711i

−1.11529

−

1.93174i

106.10

1.35046

−

2.33906i

1.01898

−

1.76493i

−2.64747

−

4.58555i

0.500000

−

0.866025i

−2.75219

−

4.76693i

−1.00000

−8.89934

−0.576655

−

0.998797i

−1.35046

−

2.33906i

596.1

−1.26008

−

2.18252i

−0.424874

−

0.735903i

−2.17561

3.76826i

0.500000

0.866025i

−1.07075

1.85460i

−1.00000

5.92545

1.13896

−

1.97274i

1.26008

−

2.18252i

596.2

−0.994744

−

1.72295i

0.457864

0.793043i

−0.979030

1.69573i

0.500000

0.866025i

0.910914

−

1.57775i

−1.00000

−0.0834388

1.08072

−

1.87186i

0.994744

−

1.72295i

596.3

−0.731711

−

1.26736i

−1.07446

−

1.86101i

−0.0708008

0.122631i

0.500000

0.866025i

−1.57238

2.72345i

−1.00000

−2.71962

−0.808916

1.40108i

0.731711

−

1.26736i

596.4

−0.278420

−

0.482238i

0.0391618

0.0678302i

0.844964

−

1.46352i

0.500000

0.866025i

0.0218069

−

0.0377706i

−1.00000

−2.05470

1.49693

−

2.59276i

0.278420

−

0.482238i

596.5

0.302407

0.523784i

0.354617

0.614215i

0.817100

−

1.41526i

0.500000

0.866025i

−0.214477

0.371485i

−1.00000

2.19801

1.24849

−

2.16245i

−0.302407

0.523784i

596.6

0.449475

0.778513i

−1.28912

−

2.23283i

0.595945

−

1.03221i

0.500000

0.866025i

1.15886

−

2.00720i

−1.00000

2.86935

−1.82367

3.15869i

−0.449475

0.778513i

596.7

0.745668

1.29153i

1.62892

2.82136i

−0.112041

0.194061i

0.500000

0.866025i

−2.42926

4.20760i

−1.00000

2.64849

−3.80673

6.59345i

−0.745668

1.29153i

596.8

0.801658

1.38851i

0.392253

0.679403i

−0.285310

0.494171i

0.500000

0.866025i

−0.628906

1.08930i

−1.00000

2.29175

1.19227

−

2.06508i

−0.801658

1.38851i

596.9

1.11529

1.93174i

−1.60334

−

2.77707i

−1.48775

2.57686i

0.500000

0.866025i

3.57639

−

6.19449i

−1.00000

−2.17594

−3.64141

6.30711i

−1.11529

1.93174i

596.10

1.35046

2.33906i

1.01898

1.76493i

−2.64747

4.58555i

0.500000

0.866025i

−2.75219

4.76693i

−1.00000

−8.89934

−0.576655

0.998797i

−1.35046

2.33906i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	665.2.i.h	✓	20
19.c	even	3	1	inner	665.2.i.h	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
665.2.i.h	✓	20	1.a	even	1	1	trivial
665.2.i.h	✓	20	19.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(665, [\chi])$ :

T_{2}^{20} - 3 T_{2}^{19} + 20 T_{2}^{18} - 43 T_{2}^{17} + 207 T_{2}^{16} - 401 T_{2}^{15} + \cdots + 1024

T_{3}^{20} + T_{3}^{19} + 20 T_{3}^{18} + 9 T_{3}^{17} + 265 T_{3}^{16} + 75 T_{3}^{15} + 1797 T_{3}^{14} + \cdots + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20} - 3 T^{19} + \cdots + 1024$ T^20 - 3T^19 + 20T^18 - 43T^17 + 207T^16 - 401T^15 + 1351T^14 - 2135T^13 + 5714T^12 - 7983T^11 + 16723T^10 - 18718T^9 + 30867T^8 - 28726T^7 + 37857T^6 - 24848T^5 + 21360T^4 - 7488T^3 + 5888T^2 - 1536*T + 1024
$3$	$T^{20} + T^{19} + \cdots + 16$ T^20 + T^19 + 20T^18 + 9T^17 + 265T^16 + 75T^15 + 1797T^14 - 599T^13 + 8164T^12 - 4331T^11 + 22983T^10 - 22258T^9 + 40555T^8 - 28906T^7 + 32681T^6 - 19894T^5 + 17160T^4 - 7152T^3 + 3160T^2 - 240T + 16
$5$	$(T^{2} - T + 1)^{10}$ (T^2 - T + 1)^10
$7$	$(T + 1)^{20}$ (T + 1)^20
$11$	$(T^{10} - T^{9} - 69 T^{8} + \cdots - 479)^{2}$ (T^10 - T^9 - 69T^8 + 66T^7 + 1552T^6 - 1283T^5 - 11639T^4 + 7467T^3 + 3566T^2 - 1493T - 479)^2
$13$	$T^{20} - 6 T^{19} + \cdots + 12559936$ T^20 - 6T^19 + 90T^18 - 368T^17 + 4376T^16 - 16242T^15 + 121810T^14 - 297883T^13 + 1822364T^12 - 3534202T^11 + 18194502T^10 - 18874026T^9 + 80712947T^8 - 27730805T^7 + 259465583T^6 - 12303970T^5 + 399145136T^4 + 134836248T^3 + 439331136T^2 + 74268064*T + 12559936
$17$	$T^{20} - 8 T^{19} + \cdots + 21307456$ T^20 - 8T^19 + 118T^18 - 470T^17 + 5668T^16 - 17996T^15 + 180045T^14 - 327015T^13 + 3161772T^12 - 3358730T^11 + 40797785T^10 - 14747010T^9 + 259491718T^8 - 41449278T^7 + 1169796557T^6 + 106183998T^5 + 1114473256T^4 - 645344296T^3 + 641460768T^2 - 121807008*T + 21307456
$19$	$T^{20} + \cdots + 6131066257801$ T^20 - 17T^19 + 115T^18 - 370T^17 + 1037T^16 - 9073T^15 + 50819T^14 - 45559T^13 - 481180T^12 - 205279T^11 + 11842097T^10 - 3900301T^9 - 173705980T^8 - 312489181T^7 + 6622782899T^6 - 22465646227T^5 + 48786578597T^4 - 330732543430T^3 + 1953109749715T^2 - 5485690862243*T + 6131066257801
$23$	$T^{20} - 3 T^{19} + \cdots + 1628176$ T^20 - 3T^19 + 107T^18 - 152T^17 + 7455T^16 - 7786T^15 + 268323T^14 + 128602T^13 + 6260456T^12 + 4522040T^11 + 78782139T^10 + 165466544T^9 + 527698161T^8 + 503378947T^7 + 587932165T^6 + 330184620T^5 + 329716596T^4 + 159197704T^3 + 97621104T^2 + 13484768*T + 1628176
$29$	$T^{20} + \cdots + 128247448480321$ T^20 - T^19 + 209T^18 - 370T^17 + 28850T^16 - 59040T^15 + 2306755T^14 - 6068299T^13 + 135043990T^12 - 381914983T^11 + 5028138387T^10 - 16179655977T^9 + 139052204649T^8 - 435482030985T^7 + 2422917371870T^6 - 7538506649444T^5 + 29632880435633T^4 - 68036387292688T^3 + 139720509254365T^2 - 150695023335092T + 128247448480321
$31$	$(T^{10} - 5 T^{9} + \cdots - 1901)^{2}$ (T^10 - 5T^9 - 121T^8 + 487T^7 + 3885T^6 - 9983T^5 - 31081T^4 - 720T^3 + 19685T^2 + 3605*T - 1901)^2
$37$	$(T^{10} + 33 T^{9} + \cdots + 3060676)^{2}$ (T^10 + 33T^9 + 306T^8 - 1362T^7 - 42648T^6 - 244640T^5 + 169206T^4 + 7195907T^3 + 28455908T^2 + 38020220*T + 3060676)^2
$41$	$T^{20} + \cdots + 250275184327744$ T^20 + 29T^19 + 738T^18 + 11637T^17 + 180468T^16 + 2186974T^15 + 26531048T^14 + 259831418T^13 + 2474788018T^12 + 19324432253T^11 + 148520631828T^10 + 930763015199T^9 + 5554094393868T^8 + 24204881006453T^7 + 94393903485107T^6 + 201319645803498T^5 + 482286248319216T^4 + 110123246245736T^3 + 3036445878250272T^2 - 855422596009312*T + 250275184327744
$43$	$T^{20} + \cdots + 357737964544$ T^20 - T^19 + 163T^18 - 874T^17 + 19338T^16 - 107340T^15 + 1358896T^14 - 8103257T^13 + 68376522T^12 - 310967370T^11 + 1602402210T^10 - 4720165040T^9 + 18216755108T^8 - 40436421234T^7 + 137280042617T^6 - 190702683064T^5 + 452566158480T^4 - 276040239680T^3 + 944736553472T^2 - 534195359232T + 357737964544
$47$	$T^{20} + \cdots + 245805829814416$ T^20 + 7T^19 + 229T^18 + 970T^17 + 29631T^16 + 118258T^15 + 2219881T^14 + 8534630T^13 + 118709378T^12 + 447045980T^11 + 4443251067T^10 + 15570178736T^9 + 121696990727T^8 + 393311654421T^7 + 2320127658255T^6 + 6341661130746T^5 + 29458802941712T^4 + 67116808108408T^3 + 208230974705448T^2 + 234773459565920*T + 245805829814416
$53$	$T^{20} + \cdots + 12\!\cdots\!96$ T^20 - 40T^19 + 1196T^18 - 24774T^17 + 453522T^16 - 6992552T^15 + 98746553T^14 - 1205832573T^13 + 13204401266T^12 - 124460471292T^11 + 1050557589753T^10 - 7726691744226T^9 + 51720107468480T^8 - 300324742075416T^7 + 1553549346499353T^6 - 6420236052231890T^5 + 21677837508134924T^4 - 45469000164137992T^3 + 66551950505358136T^2 + 35539239318057696*T + 128345667707220496
$59$	$T^{20} + \cdots + 129037263334849$ T^20 + 9T^19 + 439T^18 + 2598T^17 + 109288T^16 + 548864T^15 + 17275656T^14 + 63933585T^13 + 1899567880T^12 + 5973987120T^11 + 142502095909T^10 + 312680261014T^9 + 7075343350330T^8 + 14167497406294T^7 + 195893077155885T^6 + 95723145191174T^5 + 2256697130407560T^4 + 3434508380405150T^3 + 10816158814722378T^2 + 1199725052526136*T + 129037263334849
$61$	$T^{20} + \cdots + 21\!\cdots\!81$ T^20 - 10T^19 + 403T^18 - 1822T^17 + 82250T^16 - 269114T^15 + 10868827T^14 - 21539955T^13 + 988039087T^12 - 1915514099T^11 + 61617001374T^10 - 118646137997T^9 + 2778994819665T^8 - 6435921142548T^7 + 82542925732650T^6 - 198049876358889T^5 + 1697867792729849T^4 - 3834116298831979T^3 + 14119169912145904T^2 - 5760899080197585*T + 2172113736495481
$67$	$T^{20} + \cdots + 12031457344$ T^20 - 22T^19 + 487T^18 - 8350T^17 + 143626T^16 - 1972206T^15 + 22122027T^14 - 192656887T^13 + 1338123358T^12 - 7315833795T^11 + 31968245085T^10 - 110011794064T^9 + 300149253645T^8 - 632544753285T^7 + 1023261066547T^6 - 1186719217190T^5 + 957201824832T^4 - 400342460600T^3 + 118712861248T^2 + 10551546848*T + 12031457344
$71$	$T^{20} + \cdots + 150675172561$ T^20 + 8T^19 + 303T^18 + 1014T^17 + 55874T^16 + 196046T^15 + 4350076T^14 + 4245569T^13 + 180738520T^12 + 175012241T^11 + 4366244913T^10 + 1110255330T^9 + 62319026601T^8 + 41968147979T^7 + 498787188229T^6 + 329226971979T^5 + 2706229363150T^4 + 2274796670374T^3 + 5863443858347T^2 - 902478174578*T + 150675172561
$73$	$T^{20} + \cdots + 30\!\cdots\!84$ T^20 - 14T^19 + 433T^18 - 3580T^17 + 82608T^16 - 513633T^15 + 10541420T^14 - 42997514T^13 + 903048649T^12 - 2064879475T^11 + 58809685133T^10 - 36930290934T^9 + 2819852271632T^8 + 2517797542604T^7 + 105930846773577T^6 + 198189914284388T^5 + 2784385500248880T^4 + 6947638706161160T^3 + 51088998911539344T^2 + 97701077682278400*T + 300418025509824784
$79$	$T^{20} + \cdots + 14\!\cdots\!41$ T^20 - 9T^19 + 537T^18 - 4162T^17 + 180660T^16 - 1259448T^15 + 34038215T^14 - 162578915T^13 + 4080971502T^12 - 14543607523T^11 + 334302544857T^10 - 600419578533T^9 + 17759970227999T^8 - 12850572586745T^7 + 662096482413994T^6 + 503969072996402T^5 + 14628586067188221T^4 + 24265458517616686T^3 + 249216483242519071T^2 + 429342112505034046*T + 1478842975638380641
$83$	$(T^{10} + 27 T^{9} + \cdots + 36896)^{2}$ (T^10 + 27T^9 + 137T^8 - 1593T^7 - 13490T^6 + 18936T^5 + 302186T^4 + 125445T^3 - 1781832T^2 - 1361760*T + 36896)^2
$89$	$T^{20} + \cdots + 16\!\cdots\!61$ T^20 + 25T^19 + 799T^18 + 9622T^17 + 201734T^16 + 1696488T^15 + 33968804T^14 + 188323217T^13 + 3525243092T^12 + 9988030216T^11 + 263259983839T^10 + 235060935158T^9 + 11760713296350T^8 - 20170962824120T^7 + 346208818211005T^6 - 243221134048834T^5 + 1765561960537952T^4 + 2532389933908318T^3 + 6409471208090718T^2 + 3370709402359282*T + 1658247159139561
$97$	$T^{20} + \cdots + 41\!\cdots\!76$ T^20 - 29T^19 + 855T^18 - 13580T^17 + 264409T^16 - 3791916T^15 + 53444657T^14 - 595116040T^13 + 6310443770T^12 - 58148334784T^11 + 491916131749T^10 - 3494655652338T^9 + 20936882377371T^8 - 101008242345971T^7 + 395849288568041T^6 - 1226369680561784T^5 + 3016269321065104T^4 - 5650535403406912T^3 + 7956171381774848T^2 - 7439477349147136*T + 4193886612530176
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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 106.10
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	665.2.i.h

Level	$665$
Weight	$2$
Character orbit	665.i
Analytic conductor	$5.310$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$