LMFDB - Newform orbit 6195.2.a.bd

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6195,2,Mod(1,6195)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6195.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6195, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6195 = 3 \cdot 5 \cdot 7 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6195.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,2,18,26,18,2,18,6,18,2,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.4673240522$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} - 29 x^{16} + 56 x^{15} + 347 x^{14} - 639 x^{13} - 2227 x^{12} + 3844 x^{11} + \cdots - 1024 $ x^18 - 2x^17 - 29x^16 + 56x^15 + 347x^14 - 639x^13 - 2227x^12 + 3844x^11 + 8342x^10 - 13156x^9 - 18599x^8 + 25708x^7 + 24171x^6 - 27079x^5 - 17413x^4 + 13300x^3 + 6567x^2 - 2296*x - 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 2 x^{17} - 29 x^{16} + 56 x^{15} + 347 x^{14} - 639 x^{13} - 2227 x^{12} + 3844 x^{11} + \cdots - 1024 $ x^18 - 2*x^17 - 29*x^16 + 56*x^15 + 347*x^14 - 639*x^13 - 2227*x^12 + 3844*x^11 + 8342*x^10 - 13156*x^9 - 18599*x^8 + 25708*x^7 + 24171*x^6 - 27079*x^5 - 17413*x^4 + 13300*x^3 + 6567*x^2 - 2296*x - 1024 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - 5\nu $ v^3 - 5*v
$\beta_{4}$	$=$	$ \nu^{4} - 8\nu^{2} + 9 $ v^4 - 8*v^2 + 9
$\beta_{5}$	$=$	$ ( - 75683 \nu^{17} + 349024 \nu^{16} + 1389493 \nu^{15} - 8509816 \nu^{14} - 5688025 \nu^{13} + \cdots - 74750328 ) / 2577160 $ (-75683v^17 + 349024v^16 + 1389493v^15 - 8509816v^14 - 5688025v^13 + 79539787v^12 - 38546231v^11 - 352792606v^10 + 413848120v^9 + 717579698v^8 - 1370663801v^7 - 398970768v^6 + 1901731615v^5 - 508750153v^4 - 922056713v^3 + 457581578v^2 + 105331441*v - 74750328) / 2577160
$\beta_{6}$	$=$	$ ( 124729 \nu^{17} - 7777 \nu^{16} - 4548444 \nu^{15} + 2002048 \nu^{14} + 66338695 \nu^{13} + \cdots - 613335576 ) / 2577160 $ (124729v^17 - 7777v^16 - 4548444v^15 + 2002048v^14 + 66338695v^13 - 47828556v^12 - 498632637v^11 + 467863773v^10 + 2064630915v^9 - 2308251679v^8 - 4627194242v^7 + 5961157784v^6 + 5035570145v^5 - 7585347596v^4 - 1931439921v^3 + 3916234171v^2 + 102211432*v - 613335576) / 2577160
$\beta_{7}$	$=$	$ ( - 75423 \nu^{17} + 141044 \nu^{16} + 2296633 \nu^{15} - 4275496 \nu^{14} - 28598885 \nu^{13} + \cdots + 230699072 ) / 1288580 $ (-75423v^17 + 141044v^16 + 2296633v^15 - 4275496v^14 - 28598885v^13 + 53225007v^12 + 187027469v^11 - 351020626v^10 - 683869660v^9 + 1316654458v^8 + 1365633339v^7 - 2792758148v^6 - 1316682325v^5 + 3108774667v^4 + 409961587v^3 - 1508603422v^2 + 13167401*v + 230699072) / 1288580
$\beta_{8}$	$=$	$ ( - 41592 \nu^{17} + 114246 \nu^{16} + 1139342 \nu^{15} - 3160989 \nu^{14} - 12668835 \nu^{13} + \cdots + 62230498 ) / 644290 $ (-41592v^17 + 114246v^16 + 1139342v^15 - 3160989v^14 - 12668835v^13 + 35492613v^12 + 73977241v^11 - 208754374v^10 - 245085085v^9 + 691387417v^8 + 463424766v^7 - 1284197262v^6 - 472547535v^5 + 1240368878v^4 + 221764733v^3 - 512407463v^2 - 34436211*v + 62230498) / 644290
$\beta_{9}$	$=$	$ ( 102724 \nu^{17} - 272127 \nu^{16} - 2720509 \nu^{15} + 7308928 \nu^{14} + 28758360 \nu^{13} + \cdots - 103901836 ) / 1288580 $ (102724v^17 - 272127v^16 - 2720509v^15 + 7308928v^14 + 28758360v^13 - 78874161v^12 - 155265492v^11 + 440035443v^10 + 453424265v^9 - 1359966709v^8 - 693927587v^7 + 2318055894v^6 + 486701320v^5 - 2045023091v^4 - 112129156v^3 + 812779291v^2 + 4599227*v - 103901836) / 1288580
$\beta_{10}$	$=$	$ ( - 49156 \nu^{17} + 163055 \nu^{16} + 1228865 \nu^{15} - 4363208 \nu^{14} - 11923552 \nu^{13} + \cdots + 74886488 ) / 515432 $ (-49156v^17 + 163055v^16 + 1228865v^15 - 4363208v^14 - 11923552v^13 + 46933625v^12 + 56131320v^11 - 261440303v^10 - 127241841v^9 + 810069109v^8 + 98130259v^7 - 1395581706v^6 + 81785924v^5 + 1261357495v^4 - 146635404v^3 - 524758179v^2 + 47128161*v + 74886488) / 515432
$\beta_{11}$	$=$	$ ( - 279573 \nu^{17} + 1009919 \nu^{16} + 6789908 \nu^{15} - 26896056 \nu^{14} - 62833675 \nu^{13} + \cdots + 486865872 ) / 2577160 $ (-279573v^17 + 1009919v^16 + 6789908v^15 - 26896056v^14 - 62833675v^13 + 287553882v^12 + 270444879v^11 - 1589644181v^10 - 485312345v^9 + 4882279063v^8 - 52674496v^7 - 8344965358v^6 + 1255801005v^5 + 7542768992v^4 - 1283921383v^3 - 3222278797v^2 + 354557896*v + 486865872) / 2577160
$\beta_{12}$	$=$	$ ( 174252 \nu^{17} - 543701 \nu^{16} - 4432417 \nu^{15} + 14562484 \nu^{14} + 44188060 \nu^{13} + \cdots - 269448768 ) / 1288580 $ (174252v^17 - 543701v^16 - 4432417v^15 + 14562484v^14 + 44188060v^13 - 156771143v^12 - 217961046v^11 + 873618479v^10 + 544816735v^9 - 2705800417v^8 - 592289441v^7 + 4658290852v^6 + 37597270v^5 - 4228737063v^4 + 322121152v^3 + 1809774273v^2 - 138715229*v - 269448768) / 1288580
$\beta_{13}$	$=$	$ ( - 193198 \nu^{17} + 583009 \nu^{16} + 5335983 \nu^{15} - 16481646 \nu^{14} - 59668260 \nu^{13} + \cdots + 578031532 ) / 1288580 $ (-193198v^17 + 583009v^16 + 5335983v^15 - 16481646v^14 - 59668260v^13 + 190117507v^12 + 347386234v^11 - 1156369771v^10 - 1121131385v^9 + 3993260163v^8 + 1943264519v^7 - 7820400788v^6 - 1513602270v^5 + 8120354637v^4 + 164587392v^3 - 3779355627v^2 + 152515281*v + 578031532) / 1288580
$\beta_{14}$	$=$	$ ( 46027 \nu^{17} - 128222 \nu^{16} - 1257445 \nu^{15} + 3570206 \nu^{14} + 13873251 \nu^{13} + \cdots - 104914548 ) / 257716 $ (46027v^17 - 128222v^16 - 1257445v^15 + 3570206v^14 + 13873251v^13 - 40421443v^12 - 79535501v^11 + 240422482v^10 + 253086460v^9 - 809052600v^8 - 438168945v^7 + 1540277526v^6 + 362792385v^5 - 1554645279v^4 - 86167593v^3 + 706340382v^2 - 13656147*v - 104914548) / 257716
$\beta_{15}$	$=$	$ ( 245167 \nu^{17} - 572726 \nu^{16} - 6672357 \nu^{15} + 15549874 \nu^{14} + 73212755 \nu^{13} + \cdots - 336338108 ) / 1288580 $ (245167v^17 - 572726v^16 - 6672357v^15 + 15549874v^14 + 73212755v^13 - 170589023v^12 - 416360661v^11 + 976150634v^10 + 1309271760v^9 - 3137254392v^8 - 2229626261v^7 + 5671374862v^6 + 1816010385v^5 - 5426780043v^4 - 457522233v^3 + 2354921378v^2 - 37040679*v - 336338108) / 1288580
$\beta_{16}$	$=$	$ ( 615506 \nu^{17} - 1383693 \nu^{16} - 17636111 \nu^{15} + 39424332 \nu^{14} + 206566590 \nu^{13} + \cdots - 1432631224 ) / 2577160 $ (615506v^17 - 1383693v^16 - 17636111v^15 + 39424332v^14 + 206566590v^13 - 459247789v^12 - 1275737748v^11 + 2827842947v^10 + 4451226995v^9 - 9911863901v^8 - 8651067083v^7 + 19735460656v^6 + 8425475480v^5 - 20785113109v^4 - 2986594554v^3 + 9661553349v^2 + 99953493*v - 1432631224) / 2577160
$\beta_{17}$	$=$	$ ( 63912 \nu^{17} - 181169 \nu^{16} - 1674955 \nu^{15} + 4882328 \nu^{14} + 17471276 \nu^{13} + \cdots - 97106616 ) / 257716 $ (63912v^17 - 181169v^16 - 1674955v^15 + 4882328v^14 + 17471276v^13 - 53010531v^12 - 92732996v^11 + 298975597v^10 + 264678607v^9 - 941772987v^8 - 389807845v^7 + 1658446154v^6 + 242178112v^5 - 1543143385v^4 - 7717492v^3 + 664201233v^2 - 28563035*v - 97106616) / 257716

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{3} + 5\beta_1 $ b3 + 5*b1
$\nu^{4}$	$=$	$ \beta_{4} + 8\beta_{2} + 15 $ b4 + 8*b2 + 15
$\nu^{5}$	$=$	$ \beta_{14} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + 9\beta_{3} - \beta_{2} + 30\beta_1 $ b14 - b12 + b9 + b8 + b7 + 9b3 - b2 + 30b1
$\nu^{6}$	$=$	$ - \beta_{17} + \beta_{16} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} - \beta_{6} + 11 \beta_{4} + \cdots + 88 $ -b17 + b16 + b12 + b11 - b10 + b8 - b6 + 11b4 - b3 + 59b2 + 88
$\nu^{7}$	$=$	$ 3 \beta_{17} - \beta_{16} - \beta_{15} + 13 \beta_{14} - 15 \beta_{12} + \beta_{11} - \beta_{10} + \cdots + 195 \beta_1 $ 3b17 - b16 - b15 + 13b14 - 15b12 + b11 - b10 + 12b9 + 13b8 + 13b7 + b6 + 69b3 - 16b2 + 195*b1
$\nu^{8}$	$=$	$ - 16 \beta_{17} + 16 \beta_{16} - 2 \beta_{14} + 16 \beta_{12} + 16 \beta_{11} - 18 \beta_{10} + \cdots + 557 $ -16b17 + 16b16 - 2b14 + 16b12 + 16b11 - 18b10 - b9 + 14b8 - 2b7 - 16b6 + 94b4 - 16b3 + 432b2 - 6*b1 + 557
$\nu^{9}$	$=$	$ 50 \beta_{17} - 17 \beta_{16} - 18 \beta_{15} + 125 \beta_{14} - 159 \beta_{12} + 17 \beta_{11} + \cdots - 8 $ 50b17 - 17b16 - 18b15 + 125b14 - 159b12 + 17b11 - 17b10 + 110b9 + 124b8 + 125b7 + 19b6 - 4b5 - 3b4 + 510b3 - 181b2 + 1324b1 - 8
$\nu^{10}$	$=$	$ - 177 \beta_{17} + 178 \beta_{16} - \beta_{15} - 39 \beta_{14} + 2 \beta_{13} + 178 \beta_{12} + \cdots + 3686 $ -177b17 + 178b16 - b15 - 39b14 + 2b13 + 178b12 + 172b11 - 210b10 - 20b9 + 138b8 - 43b7 - 178b6 + 744b4 - 183b3 + 3176b2 - 122*b1 + 3686
$\nu^{11}$	$=$	$ 570 \beta_{17} - 196 \beta_{16} - 217 \beta_{15} + 1069 \beta_{14} - 2 \beta_{13} - 1468 \beta_{12} + \cdots - 175 $ 570b17 - 196b16 - 217b15 + 1069b14 - 2b13 - 1468b12 + 193b11 - 195b10 + 922b9 + 1054b8 + 1075b7 + 238b6 - 79b5 - 65b4 + 3743b3 - 1784b2 + 9231*b1 - 175
$\nu^{12}$	$=$	$ - 1686 \beta_{17} + 1709 \beta_{16} - 23 \beta_{15} - 506 \beta_{14} + 46 \beta_{13} + 1718 \beta_{12} + \cdots + 25110 $ -1686b17 + 1709b16 - 23b15 - 506b14 + 46b13 + 1718b12 + 1570b11 - 2046b10 - 261b9 + 1182b8 - 593b7 - 1711b6 + 11b5 + 5719b4 - 1834b3 + 23494b2 - 1635*b1 + 25110
$\nu^{13}$	$=$	$ 5560 \beta_{17} - 1924 \beta_{16} - 2208 \beta_{15} + 8620 \beta_{14} - 43 \beta_{13} - 12629 \beta_{12} + \cdots - 2501 $ 5560b17 - 1924b16 - 2208b15 + 8620b14 - 43b13 - 12629b12 + 1840b11 - 1884b10 + 7428b9 + 8471b8 + 8764b7 + 2492b6 - 1008b5 - 913b4 + 27525b3 - 16380b2 + 65525*b1 - 2501
$\nu^{14}$	$=$	$ - 14884 \beta_{17} + 15196 \beta_{16} - 328 \beta_{15} - 5505 \beta_{14} + 669 \beta_{13} + \cdots + 174632 $ -14884b17 + 15196b16 - 328b15 - 5505b14 + 669b13 + 15467b12 + 13170b11 - 18154b10 - 2837b9 + 9398b8 - 6700b7 - 15238b6 + 282b5 + 43424b4 - 17144b3 + 174849b2 - 18273*b1 + 174632
$\nu^{15}$	$=$	$ 49998 \beta_{17} - 17393 \beta_{16} - 20508 \beta_{15} + 67287 \beta_{14} - 568 \beta_{13} + \cdots - 29545 $ 49998b17 - 17393b16 - 20508b15 + 67287b14 - 568b13 - 104296b12 + 15926b11 - 16540b10 + 58620b9 + 65992b8 + 69433b7 + 23715b6 - 10611b5 - 10583b4 + 203370b3 - 144256b2 + 471380*b1 - 29545
$\nu^{16}$	$=$	$ - 125876 \beta_{17} + 129126 \beta_{16} - 3729 \beta_{15} - 54350 \beta_{14} + 7900 \beta_{13} + \cdots + 1233964 $ -125876b17 + 129126b16 - 3729b15 - 54350b14 + 7900b13 + 134008b12 + 105220b11 - 152558b10 - 27976b9 + 71335b8 - 67675b7 - 129648b6 + 4424b5 + 327982b4 - 153544b3 + 1308274b2 - 184677*b1 + 1233964
$\nu^{17}$	$=$	$ 428627 \beta_{17} - 150123 \beta_{16} - 180193 \beta_{15} + 515650 \beta_{14} - 5947 \beta_{13} + \cdots - 313088 $ 428627b17 - 150123b16 - 180193b15 + 515650b14 - 5947b13 - 839779b12 + 129724b11 - 136660b10 + 457108b9 + 504537b8 + 541299b7 + 213261b6 - 100471b5 - 110185b4 + 1510571b3 - 1236957b2 + 3427248*b1 - 313088

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

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−2.79257
−2.42913
−2.25913
−1.75540
−1.63309
−1.60767
−0.802107
−0.478560
−0.463960
0.641632
0.854847
1.32502
1.61510
1.82351
2.22262
2.41885
2.62503
2.69500

−2.79257

1.00000

5.79844

1.00000

−2.79257

1.00000

−10.6074

1.00000

−2.79257

1.2

−2.42913

1.00000

3.90066

1.00000

−2.42913

1.00000

−4.61694

1.00000

−2.42913

1.3

−2.25913

1.00000

3.10367

1.00000

−2.25913

1.00000

−2.49334

1.00000

−2.25913

1.4

−1.75540

1.00000

1.08145

1.00000

−1.75540

1.00000

1.61244

1.00000

−1.75540

1.5

−1.63309

1.00000

0.666972

1.00000

−1.63309

1.00000

2.17695

1.00000

−1.63309

1.6

−1.60767

1.00000

0.584588

1.00000

−1.60767

1.00000

2.27551

1.00000

−1.60767

1.7

−0.802107

1.00000

−1.35662

1.00000

−0.802107

1.00000

2.69237

1.00000

−0.802107

1.8

−0.478560

1.00000

−1.77098

1.00000

−0.478560

1.00000

1.80464

1.00000

−0.478560

1.9

−0.463960

1.00000

−1.78474

1.00000

−0.463960

1.00000

1.75597

1.00000

−0.463960

1.10

0.641632

1.00000

−1.58831

1.00000

0.641632

1.00000

−2.30237

1.00000

0.641632

1.11

0.854847

1.00000

−1.26924

1.00000

0.854847

1.00000

−2.79470

1.00000

0.854847

1.12

1.32502

1.00000

−0.244324

1.00000

1.32502

1.00000

−2.97377

1.00000

1.32502

1.13

1.61510

1.00000

0.608562

1.00000

1.61510

1.00000

−2.24732

1.00000

1.61510

1.14

1.82351

1.00000

1.32519

1.00000

1.82351

1.00000

−1.23053

1.00000

1.82351

1.15

2.22262

1.00000

2.94002

1.00000

2.22262

1.00000

2.08931

1.00000

2.22262

1.16

2.41885

1.00000

3.85082

1.00000

2.41885

1.00000

4.47686

1.00000

2.41885

1.17

2.62503

1.00000

4.89081

1.00000

2.62503

1.00000

7.58846

1.00000

2.62503

1.18

2.69500

1.00000

5.26303

1.00000

2.69500

1.00000

8.79388

1.00000

2.69500

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$5$	$ -1 $
$7$	$ -1 $
$59$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6195.2.a.bd	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6195.2.a.bd	✓	18	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6195))$:

$ T_{2}^{18} - 2 T_{2}^{17} - 29 T_{2}^{16} + 56 T_{2}^{15} + 347 T_{2}^{14} - 639 T_{2}^{13} + \cdots - 1024 $

T2^18 - 2*T2^17 - 29*T2^16 + 56*T2^15 + 347*T2^14 - 639*T2^13 - 2227*T2^12 + 3844*T2^11 + 8342*T2^10 - 13156*T2^9 - 18599*T2^8 + 25708*T2^7 + 24171*T2^6 - 27079*T2^5 - 17413*T2^4 + 13300*T2^3 + 6567*T2^2 - 2296*T2 - 1024

$ T_{11}^{18} - 9 T_{11}^{17} - 75 T_{11}^{16} + 888 T_{11}^{15} + 1311 T_{11}^{14} - 33400 T_{11}^{13} + \cdots + 655360 $

T11^18 - 9*T11^17 - 75*T11^16 + 888*T11^15 + 1311*T11^14 - 33400*T11^13 + 28473*T11^12 + 596013*T11^11 - 1299998*T11^10 - 4925573*T11^9 + 17295984*T11^8 + 11649785*T11^7 - 96512724*T11^6 + 61704580*T11^5 + 173426272*T11^4 - 277096656*T11^3 + 125581760*T11^2 - 16479040*T11 + 655360

$ T_{13}^{18} - 21 T_{13}^{17} + 99 T_{13}^{16} + 711 T_{13}^{15} - 6558 T_{13}^{14} - 5707 T_{13}^{13} + \cdots + 123410176 $

T13^18 - 21*T13^17 + 99*T13^16 + 711*T13^15 - 6558*T13^14 - 5707*T13^13 + 157356*T13^12 - 71688*T13^11 - 2087967*T13^10 + 1616436*T13^9 + 17145836*T13^8 - 10806360*T13^7 - 87309936*T13^6 + 23260896*T13^5 + 249422704*T13^4 + 24765248*T13^3 - 309919616*T13^2 - 65944320*T13 + 123410176

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} - 2 T^{17} + \cdots - 1024 $ T^18 - 2T^17 - 29T^16 + 56T^15 + 347T^14 - 639T^13 - 2227T^12 + 3844T^11 + 8342T^10 - 13156T^9 - 18599T^8 + 25708T^7 + 24171T^6 - 27079T^5 - 17413T^4 + 13300T^3 + 6567T^2 - 2296*T - 1024
$3$	$ (T - 1)^{18} $ (T - 1)^18
$5$	$ (T - 1)^{18} $ (T - 1)^18
$7$	$ (T - 1)^{18} $ (T - 1)^18
$11$	$ T^{18} - 9 T^{17} + \cdots + 655360 $ T^18 - 9T^17 - 75T^16 + 888T^15 + 1311T^14 - 33400T^13 + 28473T^12 + 596013T^11 - 1299998T^10 - 4925573T^9 + 17295984T^8 + 11649785T^7 - 96512724T^6 + 61704580T^5 + 173426272T^4 - 277096656T^3 + 125581760T^2 - 16479040*T + 655360
$13$	$ T^{18} + \cdots + 123410176 $ T^18 - 21T^17 + 99T^16 + 711T^15 - 6558T^14 - 5707T^13 + 157356T^12 - 71688T^11 - 2087967T^10 + 1616436T^9 + 17145836T^8 - 10806360T^7 - 87309936T^6 + 23260896T^5 + 249422704T^4 + 24765248T^3 - 309919616T^2 - 65944320*T + 123410176
$17$	$ T^{18} + \cdots - 232396544 $ T^18 - 20T^17 + 39T^16 + 1624T^15 - 9881T^14 - 37401T^13 + 427797T^12 - 64391T^11 - 7846449T^10 + 13665736T^9 + 65573617T^8 - 190600832T^7 - 189618064T^6 + 1000008704T^5 - 289038880T^4 - 1672928512T^3 + 1435146240T^2 + 93321216*T - 232396544
$19$	$ T^{18} + \cdots - 33651384320 $ T^18 - 23T^17 + 72T^16 + 2327T^15 - 22195T^14 - 22053T^13 + 1180976T^12 - 4406568T^11 - 16681507T^10 + 162057861T^9 - 247490247T^8 - 1535326700T^7 + 7342458448T^6 - 7209547040T^5 - 29664548352T^4 + 110707959552T^3 - 163081384704T^2 + 117184007168*T - 33651384320
$23$	$ T^{18} - 3 T^{17} + \cdots - 82247680 $ T^18 - 3T^17 - 227T^16 + 553T^15 + 20599T^14 - 38449T^13 - 962848T^12 + 1253529T^11 + 24567054T^10 - 18815436T^9 - 328977088T^8 + 101544000T^7 + 1985900704T^6 - 111700624T^5 - 3926811168T^4 + 1037750592T^3 + 1009068160T^2 - 162513920*T - 82247680
$29$	$ T^{18} + \cdots + 6270845440 $ T^18 - 4T^17 - 251T^16 + 776T^15 + 25674T^14 - 57010T^13 - 1383367T^12 + 1975901T^11 + 42123939T^10 - 32497657T^9 - 719635830T^8 + 230980656T^7 + 6454951328T^6 - 1100460704T^5 - 27085516864T^4 + 10031062784T^3 + 41948681472T^2 - 35693037824*T + 6270845440
$31$	$ T^{18} + \cdots + 43748786176 $ T^18 - 12T^17 - 255T^16 + 3495T^15 + 20947T^14 - 391832T^13 - 271023T^12 + 20669338T^11 - 46252019T^10 - 477318916T^9 + 2284548976T^8 + 1990115008T^7 - 33123099296T^6 + 62732757888T^5 + 60903026176T^4 - 378206555648T^3 + 530300138752T^2 - 293818782720*T + 43748786176
$37$	$ T^{18} + \cdots - 10415391040 $ T^18 - 34T^17 + 256T^16 + 3052T^15 - 46787T^14 - 2904T^13 + 2433024T^12 - 6552533T^11 - 52040563T^10 + 225832209T^9 + 404542591T^8 - 2974229760T^7 + 579386868T^6 + 15841712792T^5 - 18385973856T^4 - 24225056304T^3 + 48541347280T^2 - 10420882880*T - 10415391040
$41$	$ T^{18} + \cdots - 13216307200 $ T^18 - 13T^17 - 239T^16 + 3836T^15 + 15402T^14 - 422753T^13 + 376714T^12 + 21022269T^11 - 79497157T^10 - 404585768T^9 + 2988058256T^8 - 1309571024T^7 - 34003341248T^6 + 98678030528T^5 - 50308934272T^4 - 153738184704T^3 + 171120088320T^2 + 19492326400*T - 13216307200
$43$	$ T^{18} + \cdots + 198551151872 $ T^18 - 8T^17 - 375T^16 + 3228T^15 + 52542T^14 - 510522T^13 - 3232495T^12 + 39413543T^11 + 64123397T^10 - 1492153135T^9 + 1418245272T^8 + 23626129360T^7 - 58393912248T^6 - 84635556160T^5 + 329500067824T^4 - 11032453136T^3 - 468596538048T^2 + 88938032832*T + 198551151872
$47$	$ T^{18} + \cdots + 414580736 $ T^18 - 15T^17 - 283T^16 + 4657T^15 + 28748T^14 - 548983T^13 - 1251329T^12 + 30943961T^11 + 21466460T^10 - 871850280T^9 - 114905256T^8 + 11894088432T^7 + 1650695184T^6 - 68199936912T^5 - 28065588608T^4 + 98961745344T^3 + 10954449664T^2 - 30351132672*T + 414580736
$53$	$ T^{18} + 15 T^{17} + \cdots + 66993152 $ T^18 + 15T^17 - 233T^16 - 4257T^15 + 8799T^14 + 319510T^13 - 27102T^12 - 11179995T^11 - 1646843T^10 + 202965267T^9 - 26440174T^8 - 1779847044T^7 + 653254744T^6 + 5887562128T^5 + 84402336T^4 - 7014497216T^3 - 4461049984T^2 - 654528512*T + 66993152
$59$	$ (T + 1)^{18} $ (T + 1)^18
$61$	$ T^{18} + \cdots - 1926297012224 $ T^18 - 8T^17 - 469T^16 + 3732T^15 + 79535T^14 - 573184T^13 - 6951583T^12 + 41773021T^11 + 348549114T^10 - 1615563483T^9 - 10118081850T^8 + 34490837220T^7 + 162000001624T^6 - 417771040160T^5 - 1320191502656T^4 + 2747447707904T^3 + 4432711836160T^2 - 7622677806080*T - 1926297012224
$67$	$ T^{18} - 39 T^{17} + \cdots - 43379968 $ T^18 - 39T^17 + 428T^16 + 2004T^15 - 77390T^14 + 539634T^13 - 63653T^12 - 16116424T^11 + 70701296T^10 - 41975425T^9 - 393081668T^8 + 765446232T^7 + 373364776T^6 - 1649789072T^5 + 130130048T^4 + 1189524560T^3 - 93224384T^2 - 285452992*T - 43379968
$71$	$ T^{18} + \cdots + 10079142412288 $ T^18 - 15T^17 - 398T^16 + 7616T^15 + 34577T^14 - 1234240T^13 + 1886427T^12 + 81996539T^11 - 351018294T^10 - 2380860712T^9 + 15759643808T^8 + 25254790848T^7 - 303402730752T^6 + 50800924416T^5 + 2680103797248T^4 - 2271549839360T^3 - 9980355649536T^2 + 9272198135808*T + 10079142412288
$73$	$ T^{18} + \cdots - 1281892867840 $ T^18 - 14T^17 - 397T^16 + 6539T^15 + 51883T^14 - 1160737T^13 - 1763533T^12 + 99653317T^11 - 149538857T^10 - 4333352380T^9 + 14376520060T^8 + 88375259080T^7 - 432928643456T^6 - 567192854768T^5 + 4909232942800T^4 - 2785809438976T^3 - 11325740297600T^2 + 7796426594560*T - 1281892867840
$79$	$ T^{18} + \cdots - 496251699200 $ T^18 - 25T^17 - 250T^16 + 10256T^15 - 6410T^14 - 1516792T^13 + 6377985T^12 + 98192586T^11 - 622741440T^10 - 2478380135T^9 + 21996847304T^8 + 8682028612T^7 - 232757608800T^6 - 39326227456T^5 + 1048095851520T^4 + 708286697472T^3 - 1337184944128T^2 - 1661538336768*T - 496251699200
$83$	$ T^{18} + \cdots - 14938828981120 $ T^18 - 36T^17 - 205T^16 + 21416T^15 - 142511T^14 - 3964540T^13 + 50956025T^12 + 201136355T^11 - 5553311627T^10 + 9629244441T^9 + 218839374375T^8 - 806471243698T^7 - 3707833423036T^6 + 16823457110304T^5 + 30421552306864T^4 - 120866601216016T^3 - 134634855198320T^2 + 123243197456480*T - 14938828981120
$89$	$ T^{18} + \cdots + 15\!\cdots\!20 $ T^18 + 9T^17 - 814T^16 - 7736T^15 + 251547T^14 + 2621970T^13 - 36419064T^12 - 439621197T^11 + 2377356021T^10 + 37951970770T^9 - 40634191037T^8 - 1630550887702T^7 - 2020651357604T^6 + 31272902004632T^5 + 74724624326960T^4 - 233304681038880T^3 - 684150825550528T^2 + 523928311917696*T + 1533914840235520
$97$	$ T^{18} + \cdots + 16511214803968 $ T^18 - 50T^17 + 338T^16 + 21976T^15 - 406173T^14 - 1529328T^13 + 87288600T^12 - 371705665T^11 - 6676210350T^10 + 55992832272T^9 + 177337135736T^8 - 2776897373824T^7 + 698894042272T^6 + 57027228275408T^5 - 81741652869024T^4 - 408589923728960T^3 + 684270150273408T^2 - 288122302433280*T + 16511214803968
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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

Level:	\( N \)	\(=\)	\( 6195 = 3 \cdot 5 \cdot 7 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6195.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	6195.2.a.bd

Level	$6195$
Weight	$2$
Character orbit	6195.a
Self dual	yes
Analytic conductor	$49.467$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(49.4673240522\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - 2 x^{17} - 29 x^{16} + 56 x^{15} + 347 x^{14} - 639 x^{13} - 2227 x^{12} + 3844 x^{11} + \cdots - 1024 \) x^18 - 2x^17 - 29x^16 + 56x^15 + 347x^14 - 639x^13 - 2227x^12 + 3844x^11 + 8342x^10 - 13156x^9 - 18599x^8 + 25708x^7 + 24171x^6 - 27079x^5 - 17413x^4 + 13300x^3 + 6567x^2 - 2296*x - 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \) v^2 - 3
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 5\nu \) v^3 - 5*v
\(\beta_{4}\)	\(=\)	\( \nu^{4} - 8\nu^{2} + 9 \) v^4 - 8*v^2 + 9
\(\beta_{5}\)	\(=\)	\( ( - 75683 \nu^{17} + 349024 \nu^{16} + 1389493 \nu^{15} - 8509816 \nu^{14} - 5688025 \nu^{13} + \cdots - 74750328 ) / 2577160 \) (-75683v^17 + 349024v^16 + 1389493v^15 - 8509816v^14 - 5688025v^13 + 79539787v^12 - 38546231v^11 - 352792606v^10 + 413848120v^9 + 717579698v^8 - 1370663801v^7 - 398970768v^6 + 1901731615v^5 - 508750153v^4 - 922056713v^3 + 457581578v^2 + 105331441*v - 74750328) / 2577160
\(\beta_{6}\)	\(=\)	\( ( 124729 \nu^{17} - 7777 \nu^{16} - 4548444 \nu^{15} + 2002048 \nu^{14} + 66338695 \nu^{13} + \cdots - 613335576 ) / 2577160 \) (124729v^17 - 7777v^16 - 4548444v^15 + 2002048v^14 + 66338695v^13 - 47828556v^12 - 498632637v^11 + 467863773v^10 + 2064630915v^9 - 2308251679v^8 - 4627194242v^7 + 5961157784v^6 + 5035570145v^5 - 7585347596v^4 - 1931439921v^3 + 3916234171v^2 + 102211432*v - 613335576) / 2577160
\(\beta_{7}\)	\(=\)	\( ( - 75423 \nu^{17} + 141044 \nu^{16} + 2296633 \nu^{15} - 4275496 \nu^{14} - 28598885 \nu^{13} + \cdots + 230699072 ) / 1288580 \) (-75423v^17 + 141044v^16 + 2296633v^15 - 4275496v^14 - 28598885v^13 + 53225007v^12 + 187027469v^11 - 351020626v^10 - 683869660v^9 + 1316654458v^8 + 1365633339v^7 - 2792758148v^6 - 1316682325v^5 + 3108774667v^4 + 409961587v^3 - 1508603422v^2 + 13167401*v + 230699072) / 1288580
\(\beta_{8}\)	\(=\)	\( ( - 41592 \nu^{17} + 114246 \nu^{16} + 1139342 \nu^{15} - 3160989 \nu^{14} - 12668835 \nu^{13} + \cdots + 62230498 ) / 644290 \) (-41592v^17 + 114246v^16 + 1139342v^15 - 3160989v^14 - 12668835v^13 + 35492613v^12 + 73977241v^11 - 208754374v^10 - 245085085v^9 + 691387417v^8 + 463424766v^7 - 1284197262v^6 - 472547535v^5 + 1240368878v^4 + 221764733v^3 - 512407463v^2 - 34436211*v + 62230498) / 644290
\(\beta_{9}\)	\(=\)	\( ( 102724 \nu^{17} - 272127 \nu^{16} - 2720509 \nu^{15} + 7308928 \nu^{14} + 28758360 \nu^{13} + \cdots - 103901836 ) / 1288580 \) (102724v^17 - 272127v^16 - 2720509v^15 + 7308928v^14 + 28758360v^13 - 78874161v^12 - 155265492v^11 + 440035443v^10 + 453424265v^9 - 1359966709v^8 - 693927587v^7 + 2318055894v^6 + 486701320v^5 - 2045023091v^4 - 112129156v^3 + 812779291v^2 + 4599227*v - 103901836) / 1288580
\(\beta_{10}\)	\(=\)	\( ( - 49156 \nu^{17} + 163055 \nu^{16} + 1228865 \nu^{15} - 4363208 \nu^{14} - 11923552 \nu^{13} + \cdots + 74886488 ) / 515432 \) (-49156v^17 + 163055v^16 + 1228865v^15 - 4363208v^14 - 11923552v^13 + 46933625v^12 + 56131320v^11 - 261440303v^10 - 127241841v^9 + 810069109v^8 + 98130259v^7 - 1395581706v^6 + 81785924v^5 + 1261357495v^4 - 146635404v^3 - 524758179v^2 + 47128161*v + 74886488) / 515432
\(\beta_{11}\)	\(=\)	\( ( - 279573 \nu^{17} + 1009919 \nu^{16} + 6789908 \nu^{15} - 26896056 \nu^{14} - 62833675 \nu^{13} + \cdots + 486865872 ) / 2577160 \) (-279573v^17 + 1009919v^16 + 6789908v^15 - 26896056v^14 - 62833675v^13 + 287553882v^12 + 270444879v^11 - 1589644181v^10 - 485312345v^9 + 4882279063v^8 - 52674496v^7 - 8344965358v^6 + 1255801005v^5 + 7542768992v^4 - 1283921383v^3 - 3222278797v^2 + 354557896*v + 486865872) / 2577160
\(\beta_{12}\)	\(=\)	\( ( 174252 \nu^{17} - 543701 \nu^{16} - 4432417 \nu^{15} + 14562484 \nu^{14} + 44188060 \nu^{13} + \cdots - 269448768 ) / 1288580 \) (174252v^17 - 543701v^16 - 4432417v^15 + 14562484v^14 + 44188060v^13 - 156771143v^12 - 217961046v^11 + 873618479v^10 + 544816735v^9 - 2705800417v^8 - 592289441v^7 + 4658290852v^6 + 37597270v^5 - 4228737063v^4 + 322121152v^3 + 1809774273v^2 - 138715229*v - 269448768) / 1288580
\(\beta_{13}\)	\(=\)	\( ( - 193198 \nu^{17} + 583009 \nu^{16} + 5335983 \nu^{15} - 16481646 \nu^{14} - 59668260 \nu^{13} + \cdots + 578031532 ) / 1288580 \) (-193198v^17 + 583009v^16 + 5335983v^15 - 16481646v^14 - 59668260v^13 + 190117507v^12 + 347386234v^11 - 1156369771v^10 - 1121131385v^9 + 3993260163v^8 + 1943264519v^7 - 7820400788v^6 - 1513602270v^5 + 8120354637v^4 + 164587392v^3 - 3779355627v^2 + 152515281*v + 578031532) / 1288580
\(\beta_{14}\)	\(=\)	\( ( 46027 \nu^{17} - 128222 \nu^{16} - 1257445 \nu^{15} + 3570206 \nu^{14} + 13873251 \nu^{13} + \cdots - 104914548 ) / 257716 \) (46027v^17 - 128222v^16 - 1257445v^15 + 3570206v^14 + 13873251v^13 - 40421443v^12 - 79535501v^11 + 240422482v^10 + 253086460v^9 - 809052600v^8 - 438168945v^7 + 1540277526v^6 + 362792385v^5 - 1554645279v^4 - 86167593v^3 + 706340382v^2 - 13656147*v - 104914548) / 257716
\(\beta_{15}\)	\(=\)	\( ( 245167 \nu^{17} - 572726 \nu^{16} - 6672357 \nu^{15} + 15549874 \nu^{14} + 73212755 \nu^{13} + \cdots - 336338108 ) / 1288580 \) (245167v^17 - 572726v^16 - 6672357v^15 + 15549874v^14 + 73212755v^13 - 170589023v^12 - 416360661v^11 + 976150634v^10 + 1309271760v^9 - 3137254392v^8 - 2229626261v^7 + 5671374862v^6 + 1816010385v^5 - 5426780043v^4 - 457522233v^3 + 2354921378v^2 - 37040679*v - 336338108) / 1288580
\(\beta_{16}\)	\(=\)	\( ( 615506 \nu^{17} - 1383693 \nu^{16} - 17636111 \nu^{15} + 39424332 \nu^{14} + 206566590 \nu^{13} + \cdots - 1432631224 ) / 2577160 \) (615506v^17 - 1383693v^16 - 17636111v^15 + 39424332v^14 + 206566590v^13 - 459247789v^12 - 1275737748v^11 + 2827842947v^10 + 4451226995v^9 - 9911863901v^8 - 8651067083v^7 + 19735460656v^6 + 8425475480v^5 - 20785113109v^4 - 2986594554v^3 + 9661553349v^2 + 99953493*v - 1432631224) / 2577160
\(\beta_{17}\)	\(=\)	\( ( 63912 \nu^{17} - 181169 \nu^{16} - 1674955 \nu^{15} + 4882328 \nu^{14} + 17471276 \nu^{13} + \cdots - 97106616 ) / 257716 \) (63912v^17 - 181169v^16 - 1674955v^15 + 4882328v^14 + 17471276v^13 - 53010531v^12 - 92732996v^11 + 298975597v^10 + 264678607v^9 - 941772987v^8 - 389807845v^7 + 1658446154v^6 + 242178112v^5 - 1543143385v^4 - 7717492v^3 + 664201233v^2 - 28563035*v - 97106616) / 257716

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 5\beta_1 \) b3 + 5*b1
\(\nu^{4}\)	\(=\)	\( \beta_{4} + 8\beta_{2} + 15 \) b4 + 8*b2 + 15
\(\nu^{5}\)	\(=\)	\( \beta_{14} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + 9\beta_{3} - \beta_{2} + 30\beta_1 \) b14 - b12 + b9 + b8 + b7 + 9b3 - b2 + 30b1
\(\nu^{6}\)	\(=\)	\( - \beta_{17} + \beta_{16} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} - \beta_{6} + 11 \beta_{4} + \cdots + 88 \) -b17 + b16 + b12 + b11 - b10 + b8 - b6 + 11b4 - b3 + 59b2 + 88
\(\nu^{7}\)	\(=\)	\( 3 \beta_{17} - \beta_{16} - \beta_{15} + 13 \beta_{14} - 15 \beta_{12} + \beta_{11} - \beta_{10} + \cdots + 195 \beta_1 \) 3b17 - b16 - b15 + 13b14 - 15b12 + b11 - b10 + 12b9 + 13b8 + 13b7 + b6 + 69b3 - 16b2 + 195*b1
\(\nu^{8}\)	\(=\)	\( - 16 \beta_{17} + 16 \beta_{16} - 2 \beta_{14} + 16 \beta_{12} + 16 \beta_{11} - 18 \beta_{10} + \cdots + 557 \) -16b17 + 16b16 - 2b14 + 16b12 + 16b11 - 18b10 - b9 + 14b8 - 2b7 - 16b6 + 94b4 - 16b3 + 432b2 - 6*b1 + 557
\(\nu^{9}\)	\(=\)	\( 50 \beta_{17} - 17 \beta_{16} - 18 \beta_{15} + 125 \beta_{14} - 159 \beta_{12} + 17 \beta_{11} + \cdots - 8 \) 50b17 - 17b16 - 18b15 + 125b14 - 159b12 + 17b11 - 17b10 + 110b9 + 124b8 + 125b7 + 19b6 - 4b5 - 3b4 + 510b3 - 181b2 + 1324b1 - 8
\(\nu^{10}\)	\(=\)	\( - 177 \beta_{17} + 178 \beta_{16} - \beta_{15} - 39 \beta_{14} + 2 \beta_{13} + 178 \beta_{12} + \cdots + 3686 \) -177b17 + 178b16 - b15 - 39b14 + 2b13 + 178b12 + 172b11 - 210b10 - 20b9 + 138b8 - 43b7 - 178b6 + 744b4 - 183b3 + 3176b2 - 122*b1 + 3686
\(\nu^{11}\)	\(=\)	\( 570 \beta_{17} - 196 \beta_{16} - 217 \beta_{15} + 1069 \beta_{14} - 2 \beta_{13} - 1468 \beta_{12} + \cdots - 175 \) 570b17 - 196b16 - 217b15 + 1069b14 - 2b13 - 1468b12 + 193b11 - 195b10 + 922b9 + 1054b8 + 1075b7 + 238b6 - 79b5 - 65b4 + 3743b3 - 1784b2 + 9231*b1 - 175
\(\nu^{12}\)	\(=\)	\( - 1686 \beta_{17} + 1709 \beta_{16} - 23 \beta_{15} - 506 \beta_{14} + 46 \beta_{13} + 1718 \beta_{12} + \cdots + 25110 \) -1686b17 + 1709b16 - 23b15 - 506b14 + 46b13 + 1718b12 + 1570b11 - 2046b10 - 261b9 + 1182b8 - 593b7 - 1711b6 + 11b5 + 5719b4 - 1834b3 + 23494b2 - 1635*b1 + 25110
\(\nu^{13}\)	\(=\)	\( 5560 \beta_{17} - 1924 \beta_{16} - 2208 \beta_{15} + 8620 \beta_{14} - 43 \beta_{13} - 12629 \beta_{12} + \cdots - 2501 \) 5560b17 - 1924b16 - 2208b15 + 8620b14 - 43b13 - 12629b12 + 1840b11 - 1884b10 + 7428b9 + 8471b8 + 8764b7 + 2492b6 - 1008b5 - 913b4 + 27525b3 - 16380b2 + 65525*b1 - 2501
\(\nu^{14}\)	\(=\)	\( - 14884 \beta_{17} + 15196 \beta_{16} - 328 \beta_{15} - 5505 \beta_{14} + 669 \beta_{13} + \cdots + 174632 \) -14884b17 + 15196b16 - 328b15 - 5505b14 + 669b13 + 15467b12 + 13170b11 - 18154b10 - 2837b9 + 9398b8 - 6700b7 - 15238b6 + 282b5 + 43424b4 - 17144b3 + 174849b2 - 18273*b1 + 174632
\(\nu^{15}\)	\(=\)	\( 49998 \beta_{17} - 17393 \beta_{16} - 20508 \beta_{15} + 67287 \beta_{14} - 568 \beta_{13} + \cdots - 29545 \) 49998b17 - 17393b16 - 20508b15 + 67287b14 - 568b13 - 104296b12 + 15926b11 - 16540b10 + 58620b9 + 65992b8 + 69433b7 + 23715b6 - 10611b5 - 10583b4 + 203370b3 - 144256b2 + 471380*b1 - 29545
\(\nu^{16}\)	\(=\)	\( - 125876 \beta_{17} + 129126 \beta_{16} - 3729 \beta_{15} - 54350 \beta_{14} + 7900 \beta_{13} + \cdots + 1233964 \) -125876b17 + 129126b16 - 3729b15 - 54350b14 + 7900b13 + 134008b12 + 105220b11 - 152558b10 - 27976b9 + 71335b8 - 67675b7 - 129648b6 + 4424b5 + 327982b4 - 153544b3 + 1308274b2 - 184677*b1 + 1233964
\(\nu^{17}\)	\(=\)	\( 428627 \beta_{17} - 150123 \beta_{16} - 180193 \beta_{15} + 515650 \beta_{14} - 5947 \beta_{13} + \cdots - 313088 \) 428627b17 - 150123b16 - 180193b15 + 515650b14 - 5947b13 - 839779b12 + 129724b11 - 136660b10 + 457108b9 + 504537b8 + 541299b7 + 213261b6 - 100471b5 - 110185b4 + 1510571b3 - 1236957b2 + 3427248*b1 - 313088

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.18
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} - 2 T^{17} + \cdots - 1024 \) T^18 - 2T^17 - 29T^16 + 56T^15 + 347T^14 - 639T^13 - 2227T^12 + 3844T^11 + 8342T^10 - 13156T^9 - 18599T^8 + 25708T^7 + 24171T^6 - 27079T^5 - 17413T^4 + 13300T^3 + 6567T^2 - 2296*T - 1024
$3$	\( (T - 1)^{18} \) (T - 1)^18
$5$	\( (T - 1)^{18} \) (T - 1)^18
$7$	\( (T - 1)^{18} \) (T - 1)^18
$11$	\( T^{18} - 9 T^{17} + \cdots + 655360 \) T^18 - 9T^17 - 75T^16 + 888T^15 + 1311T^14 - 33400T^13 + 28473T^12 + 596013T^11 - 1299998T^10 - 4925573T^9 + 17295984T^8 + 11649785T^7 - 96512724T^6 + 61704580T^5 + 173426272T^4 - 277096656T^3 + 125581760T^2 - 16479040*T + 655360
$13$	\( T^{18} + \cdots + 123410176 \) T^18 - 21T^17 + 99T^16 + 711T^15 - 6558T^14 - 5707T^13 + 157356T^12 - 71688T^11 - 2087967T^10 + 1616436T^9 + 17145836T^8 - 10806360T^7 - 87309936T^6 + 23260896T^5 + 249422704T^4 + 24765248T^3 - 309919616T^2 - 65944320*T + 123410176
$17$	\( T^{18} + \cdots - 232396544 \) T^18 - 20T^17 + 39T^16 + 1624T^15 - 9881T^14 - 37401T^13 + 427797T^12 - 64391T^11 - 7846449T^10 + 13665736T^9 + 65573617T^8 - 190600832T^7 - 189618064T^6 + 1000008704T^5 - 289038880T^4 - 1672928512T^3 + 1435146240T^2 + 93321216*T - 232396544
$19$	\( T^{18} + \cdots - 33651384320 \) T^18 - 23T^17 + 72T^16 + 2327T^15 - 22195T^14 - 22053T^13 + 1180976T^12 - 4406568T^11 - 16681507T^10 + 162057861T^9 - 247490247T^8 - 1535326700T^7 + 7342458448T^6 - 7209547040T^5 - 29664548352T^4 + 110707959552T^3 - 163081384704T^2 + 117184007168*T - 33651384320
$23$	\( T^{18} - 3 T^{17} + \cdots - 82247680 \) T^18 - 3T^17 - 227T^16 + 553T^15 + 20599T^14 - 38449T^13 - 962848T^12 + 1253529T^11 + 24567054T^10 - 18815436T^9 - 328977088T^8 + 101544000T^7 + 1985900704T^6 - 111700624T^5 - 3926811168T^4 + 1037750592T^3 + 1009068160T^2 - 162513920*T - 82247680
$29$	\( T^{18} + \cdots + 6270845440 \) T^18 - 4T^17 - 251T^16 + 776T^15 + 25674T^14 - 57010T^13 - 1383367T^12 + 1975901T^11 + 42123939T^10 - 32497657T^9 - 719635830T^8 + 230980656T^7 + 6454951328T^6 - 1100460704T^5 - 27085516864T^4 + 10031062784T^3 + 41948681472T^2 - 35693037824*T + 6270845440
$31$	\( T^{18} + \cdots + 43748786176 \) T^18 - 12T^17 - 255T^16 + 3495T^15 + 20947T^14 - 391832T^13 - 271023T^12 + 20669338T^11 - 46252019T^10 - 477318916T^9 + 2284548976T^8 + 1990115008T^7 - 33123099296T^6 + 62732757888T^5 + 60903026176T^4 - 378206555648T^3 + 530300138752T^2 - 293818782720*T + 43748786176
$37$	\( T^{18} + \cdots - 10415391040 \) T^18 - 34T^17 + 256T^16 + 3052T^15 - 46787T^14 - 2904T^13 + 2433024T^12 - 6552533T^11 - 52040563T^10 + 225832209T^9 + 404542591T^8 - 2974229760T^7 + 579386868T^6 + 15841712792T^5 - 18385973856T^4 - 24225056304T^3 + 48541347280T^2 - 10420882880*T - 10415391040
$41$	\( T^{18} + \cdots - 13216307200 \) T^18 - 13T^17 - 239T^16 + 3836T^15 + 15402T^14 - 422753T^13 + 376714T^12 + 21022269T^11 - 79497157T^10 - 404585768T^9 + 2988058256T^8 - 1309571024T^7 - 34003341248T^6 + 98678030528T^5 - 50308934272T^4 - 153738184704T^3 + 171120088320T^2 + 19492326400*T - 13216307200
$43$	\( T^{18} + \cdots + 198551151872 \) T^18 - 8T^17 - 375T^16 + 3228T^15 + 52542T^14 - 510522T^13 - 3232495T^12 + 39413543T^11 + 64123397T^10 - 1492153135T^9 + 1418245272T^8 + 23626129360T^7 - 58393912248T^6 - 84635556160T^5 + 329500067824T^4 - 11032453136T^3 - 468596538048T^2 + 88938032832*T + 198551151872
$47$	\( T^{18} + \cdots + 414580736 \) T^18 - 15T^17 - 283T^16 + 4657T^15 + 28748T^14 - 548983T^13 - 1251329T^12 + 30943961T^11 + 21466460T^10 - 871850280T^9 - 114905256T^8 + 11894088432T^7 + 1650695184T^6 - 68199936912T^5 - 28065588608T^4 + 98961745344T^3 + 10954449664T^2 - 30351132672*T + 414580736
$53$	\( T^{18} + 15 T^{17} + \cdots + 66993152 \) T^18 + 15T^17 - 233T^16 - 4257T^15 + 8799T^14 + 319510T^13 - 27102T^12 - 11179995T^11 - 1646843T^10 + 202965267T^9 - 26440174T^8 - 1779847044T^7 + 653254744T^6 + 5887562128T^5 + 84402336T^4 - 7014497216T^3 - 4461049984T^2 - 654528512*T + 66993152
$59$	\( (T + 1)^{18} \) (T + 1)^18
$61$	\( T^{18} + \cdots - 1926297012224 \) T^18 - 8T^17 - 469T^16 + 3732T^15 + 79535T^14 - 573184T^13 - 6951583T^12 + 41773021T^11 + 348549114T^10 - 1615563483T^9 - 10118081850T^8 + 34490837220T^7 + 162000001624T^6 - 417771040160T^5 - 1320191502656T^4 + 2747447707904T^3 + 4432711836160T^2 - 7622677806080*T - 1926297012224
$67$	\( T^{18} - 39 T^{17} + \cdots - 43379968 \) T^18 - 39T^17 + 428T^16 + 2004T^15 - 77390T^14 + 539634T^13 - 63653T^12 - 16116424T^11 + 70701296T^10 - 41975425T^9 - 393081668T^8 + 765446232T^7 + 373364776T^6 - 1649789072T^5 + 130130048T^4 + 1189524560T^3 - 93224384T^2 - 285452992*T - 43379968
$71$	\( T^{18} + \cdots + 10079142412288 \) T^18 - 15T^17 - 398T^16 + 7616T^15 + 34577T^14 - 1234240T^13 + 1886427T^12 + 81996539T^11 - 351018294T^10 - 2380860712T^9 + 15759643808T^8 + 25254790848T^7 - 303402730752T^6 + 50800924416T^5 + 2680103797248T^4 - 2271549839360T^3 - 9980355649536T^2 + 9272198135808*T + 10079142412288
$73$	\( T^{18} + \cdots - 1281892867840 \) T^18 - 14T^17 - 397T^16 + 6539T^15 + 51883T^14 - 1160737T^13 - 1763533T^12 + 99653317T^11 - 149538857T^10 - 4333352380T^9 + 14376520060T^8 + 88375259080T^7 - 432928643456T^6 - 567192854768T^5 + 4909232942800T^4 - 2785809438976T^3 - 11325740297600T^2 + 7796426594560*T - 1281892867840
$79$	\( T^{18} + \cdots - 496251699200 \) T^18 - 25T^17 - 250T^16 + 10256T^15 - 6410T^14 - 1516792T^13 + 6377985T^12 + 98192586T^11 - 622741440T^10 - 2478380135T^9 + 21996847304T^8 + 8682028612T^7 - 232757608800T^6 - 39326227456T^5 + 1048095851520T^4 + 708286697472T^3 - 1337184944128T^2 - 1661538336768*T - 496251699200
$83$	\( T^{18} + \cdots - 14938828981120 \) T^18 - 36T^17 - 205T^16 + 21416T^15 - 142511T^14 - 3964540T^13 + 50956025T^12 + 201136355T^11 - 5553311627T^10 + 9629244441T^9 + 218839374375T^8 - 806471243698T^7 - 3707833423036T^6 + 16823457110304T^5 + 30421552306864T^4 - 120866601216016T^3 - 134634855198320T^2 + 123243197456480*T - 14938828981120
$89$	\( T^{18} + \cdots + 15\!\cdots\!20 \) T^18 + 9T^17 - 814T^16 - 7736T^15 + 251547T^14 + 2621970T^13 - 36419064T^12 - 439621197T^11 + 2377356021T^10 + 37951970770T^9 - 40634191037T^8 - 1630550887702T^7 - 2020651357604T^6 + 31272902004632T^5 + 74724624326960T^4 - 233304681038880T^3 - 684150825550528T^2 + 523928311917696*T + 1533914840235520
$97$	\( T^{18} + \cdots + 16511214803968 \) T^18 - 50T^17 + 338T^16 + 21976T^15 - 406173T^14 - 1529328T^13 + 87288600T^12 - 371705665T^11 - 6676210350T^10 + 55992832272T^9 + 177337135736T^8 - 2776897373824T^7 + 698894042272T^6 + 57027228275408T^5 - 81741652869024T^4 - 408589923728960T^3 + 684270150273408T^2 - 288122302433280*T + 16511214803968
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\( p \)	Sign
\(3\)	\( -1 \)
\(5\)	\( -1 \)
\(7\)	\( -1 \)
\(59\)	\( +1 \)