LMFDB - Newform orbit 9405.2.a.bt

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.a (trivial)

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:	yes
Analytic conductor:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 23 x^{13} + 21 x^{12} + 205 x^{11} - 168 x^{10} - 899 x^{9} + 647 x^{8} + 2048 x^{7} + \cdots + 90 $ x^15 - x^14 - 23x^13 + 21x^12 + 205x^11 - 168x^10 - 899x^9 + 647x^8 + 2048x^7 - 1264x^6 - 2379x^5 + 1258x^4 + 1276x^3 - 583x^2 - 237*x + 90
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
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 $1,\beta_1,\ldots,\beta_{14}$ 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} + 1) q^{4} + q^{5} + \beta_{9} q^{7} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b2 + 1) * q^4 + q^5 + b9 * q^7 + (b3 + b1) * q^8	$ q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_{9} q^{7} + (\beta_{3} + \beta_1) q^{8} + \beta_1 q^{10} - q^{11} + ( - \beta_{13} + \beta_{11} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{14} + \beta_{13} + \beta_{12} + \cdots - 2) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b2 + 1) * q^4 + q^5 + b9 * q^7 + (b3 + b1) * q^8 + b1 * q^10 - q^11 + (-b13 + b11 + b4 - b2 + 1) * q^13 + (-b14 + b4 + 1) * q^14 + (b11 - b9 - b7 + b5 + b2 + 2) * q^16 + (-b13 - b10 - b8 - b6 - b2) * q^17 - q^19 + (b2 + 1) * q^20 - b1 * q^22 + (b8 + b7 - b5 + b3 + b2 + 1) * q^23 + q^25 + (b13 - b12 + b10 + b9 + b8 + b6 + b4 + b2 + b1) * q^26 + (-b14 + b13 + b9 + b6 - b4 + b2 + b1) * q^28 + (-b14 - b11 - b10 + b9 - b8 - b4 - 1) * q^29 + (b14 - b13 - b12 + b11 + b10 + b4 + b1 + 2) * q^31 + (b13 - b12 + b11 + 2b10 + 2b8 + b6 - b5 + b3 + 2b2 + 3b1) * q^32 + (-b11 - b10 + b9 - b8 + b7 - b6 - b4 + b2 - b1 + 1) * q^34 + b9 * q^35 + (b14 - b12 + b10 + b9 + b4 - b3 + b2 + 1) * q^37 - b1 * q^38 + (b3 + b1) * q^40 + (-b13 - b9 - b7 - b2 + b1) * q^41 + (-b13 - b8 - b7 - b3 - b2 + 1) * q^43 + (-b2 - 1) * q^44 + (-b10 - b9 - b7 + b5 - b4 + b2 + 2b1) q^46 + (-b10 - b9 - b8 + b5 - b2 + b1 + 1) * q^47 + (b12 + b9 + b7 - 2b5 + b3 - b2 - b1 + 1) q^49 + b1 * q^50 + (-b14 + b13 + b12 - b9 + b8 - b7 + b6 + b5 + b2 + 1) * q^52 + (b12 - b11 - b10 - b9 - b8 - b6 - b1 + 1) * q^53 - q^55 + (b13 + b12 + b11 - b6 + b3 + 2) * q^56 + (-b14 + b12 - b10 + b9 - b8 + b7 - 2b6 + b4 - 2b1 + 2) * q^58 + (-b13 + b12 + b9 - b8 + b7 - b6 - b5 - b2 + b1) * q^59 + (b14 - b12 + b11 + b10 + b9 + 2b8 + b7 - b5 + b4 + b2 + 2b1) * q^61 + (b14 - b13 + b11 - 2b9 - b7 + 2b5 + b4 + 2b1 + 2) q^62 + (-b14 + b12 + b11 - 2b9 + 2b8 - b7 + b6 + b5 + b3 + b2 + 4) * q^64 + (-b13 + b11 + b4 - b2 + 1) * q^65 + (b12 - b10 - b2 - 2b1 + 2) q^67 + (-b14 + b13 + b12 - 2b11 - b10 + b9 - 2b8 + b7 - b6 - 2b4 + b1 - 2) q^68 + (-b14 + b4 + 1) * q^70 + (b14 - b13 - b11 - b10 - b9 - b8 - b7 + 2b5 - b3 - b2 + 1) q^71 + (-b13 - b11 - b7 + b5 - b3 + 1) * q^73 + (-2b13 + b12 - b10 - b9 - b8 + b5 + b4 - 4b2 + 2b1 - 1) q^74 + (-b2 - 1) * q^76 - b9 * q^77 + (-b13 + b7 + b4 + b3 - b2 + 2) * q^79 + (b11 - b9 - b7 + b5 + b2 + 2) * q^80 + (2b14 - b12 + b11 + 2b10 - b9 + 3b8 + b6 + b4 + 3b2 + 3) * q^82 + (b14 - b12 - b11 + b10 + b9 + b8 + b7 - b5 + b2 + 2b1 - 1) q^83 + (-b13 - b10 - b8 - b6 - b2) * q^85 + (b14 + b10 + b8 + b7 + 2b4 - b2 + b1) q^86 + (-b3 - b1) * q^88 + (b14 + b13 + b11 + b9 + b7 + b6 - b4 + b1 - 1) * q^89 + (b14 + b13 + b12 + b10 + b9 + b8 + b7 - 2b5 + b4 + 2b1 + 2) * q^91 + (b14 - b12 + b11 + 2b10 - b9 + b8 - b7 + b4 + b3 + 2b2 + 3b1 + 5) q^92 + (b14 + b9 - b8 + b7 - b6 - b5 + b2 + 3) * q^94 - q^95 + (-b14 + b13 + b12 - b8 - b7 + b6 - b2 + 1) * q^97 + (-b14 + b13 + b12 - b11 - 2b10 - b9 - 2b8 - b7 + 2b5 - 2b3 - b2 - 2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 17 q^{4} + 15 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) $ 15 * q + q^2 + 17 * q^4 + 15 * q^5 + 4 * q^7 + 3 * q^8	$ 15 q + q^{2} + 17 q^{4} + 15 q^{5} + 4 q^{7} + 3 q^{8} + q^{10} - 15 q^{11} + 3 q^{13} + 4 q^{14} + 25 q^{16} - 7 q^{17} - 15 q^{19} + 17 q^{20} - q^{22} + 24 q^{23} + 15 q^{25} + 6 q^{26} + 13 q^{28} - 10 q^{29} + 20 q^{31} + 12 q^{32} + 24 q^{34} + 4 q^{35} + 14 q^{37} - q^{38} + 3 q^{40} - 10 q^{41} + 4 q^{43} - 17 q^{44} + 7 q^{46} + 10 q^{47} + 27 q^{49} + q^{50} + 16 q^{52} + 13 q^{53} - 15 q^{55} + 37 q^{56} + 20 q^{58} - 3 q^{59} + 10 q^{61} + 14 q^{62} + 55 q^{64} + 3 q^{65} + 34 q^{67} - 18 q^{68} + 4 q^{70} + 5 q^{71} + 4 q^{73} - 39 q^{74} - 17 q^{76} - 4 q^{77} + 19 q^{79} + 25 q^{80} + 52 q^{82} - 3 q^{83} - 7 q^{85} - 7 q^{86} - 3 q^{88} + 9 q^{89} + 50 q^{91} + 72 q^{92} + 56 q^{94} - 15 q^{95} + 16 q^{97} - 38 q^{98}+O(q^{100}) $ 15 * q + q^2 + 17 * q^4 + 15 * q^5 + 4 * q^7 + 3 * q^8 + q^10 - 15 * q^11 + 3 * q^13 + 4 * q^14 + 25 * q^16 - 7 * q^17 - 15 * q^19 + 17 * q^20 - q^22 + 24 * q^23 + 15 * q^25 + 6 * q^26 + 13 * q^28 - 10 * q^29 + 20 * q^31 + 12 * q^32 + 24 * q^34 + 4 * q^35 + 14 * q^37 - q^38 + 3 * q^40 - 10 * q^41 + 4 * q^43 - 17 * q^44 + 7 * q^46 + 10 * q^47 + 27 * q^49 + q^50 + 16 * q^52 + 13 * q^53 - 15 * q^55 + 37 * q^56 + 20 * q^58 - 3 * q^59 + 10 * q^61 + 14 * q^62 + 55 * q^64 + 3 * q^65 + 34 * q^67 - 18 * q^68 + 4 * q^70 + 5 * q^71 + 4 * q^73 - 39 * q^74 - 17 * q^76 - 4 * q^77 + 19 * q^79 + 25 * q^80 + 52 * q^82 - 3 * q^83 - 7 * q^85 - 7 * q^86 - 3 * q^88 + 9 * q^89 + 50 * q^91 + 72 * q^92 + 56 * q^94 - 15 * q^95 + 16 * q^97 - 38 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{15} - x^{14} - 23 x^{13} + 21 x^{12} + 205 x^{11} - 168 x^{10} - 899 x^{9} + 647 x^{8} + 2048 x^{7} + \cdots + 90 $ x^15 - x^14 - 23*x^13 + 21*x^12 + 205*x^11 - 168*x^10 - 899*x^9 + 647*x^8 + 2048*x^7 - 1264*x^6 - 2379*x^5 + 1258*x^4 + 1276*x^3 - 583*x^2 - 237*x + 90 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - 5\nu $ v^3 - 5*v
$\beta_{4}$	$=$	$ ( 4439 \nu^{14} - 52877 \nu^{13} - 32602 \nu^{12} + 1099050 \nu^{11} - 463825 \nu^{10} + \cdots + 3044262 ) / 362094 $ (4439v^14 - 52877v^13 - 32602v^12 + 1099050v^11 - 463825v^10 - 8468514v^9 + 5975012v^8 + 29834563v^7 - 23358308v^6 - 48444227v^5 + 36126903v^4 + 34512977v^3 - 21366841v^2 - 9205694v + 3044262) / 362094
$\beta_{5}$	$=$	$ ( - 7322 \nu^{14} + 100352 \nu^{13} + 30202 \nu^{12} - 2130570 \nu^{11} + 1272397 \nu^{10} + \cdots - 8138904 ) / 181047 $ (-7322v^14 + 100352v^13 + 30202v^12 - 2130570v^11 + 1272397v^10 + 16920255v^9 - 13870331v^8 - 62313670v^7 + 52746803v^6 + 107829311v^5 - 81594021v^4 - 81419489v^3 + 48593164v^2 + 19952147v - 8138904) / 181047
$\beta_{6}$	$=$	$ ( 39506 \nu^{14} - 203243 \nu^{13} - 637357 \nu^{12} + 4236642 \nu^{11} + 2622242 \nu^{10} + \cdots + 10994718 ) / 362094 $ (39506v^14 - 203243v^13 - 637357v^12 + 4236642v^11 + 2622242v^10 - 32739489v^9 + 5155724v^8 + 115321114v^7 - 54881879v^6 - 184474520v^5 + 106231701v^4 + 122539373v^3 - 67914853v^2 - 25812839v + 10994718) / 362094
$\beta_{7}$	$=$	$ ( - 45445 \nu^{14} + 80746 \nu^{13} + 967127 \nu^{12} - 1666290 \nu^{11} - 7711333 \nu^{10} + \cdots - 1046916 ) / 362094 $ (-45445v^14 + 80746v^13 + 967127v^12 - 1666290v^11 - 7711333v^10 + 12823995v^9 + 28618856v^8 - 45689849v^7 - 50377295v^6 + 77026279v^5 + 39480186v^4 - 59802862v^3 - 9569344v^2 + 17378155v - 1046916) / 362094
$\beta_{8}$	$=$	$ ( - 82547 \nu^{14} + 230963 \nu^{13} + 1608118 \nu^{12} - 4751412 \nu^{11} - 11031041 \nu^{10} + \cdots - 8138838 ) / 362094 $ (-82547v^14 + 230963v^13 + 1608118v^12 - 4751412v^11 - 11031041v^10 + 36153378v^9 + 30166240v^8 - 124879183v^7 - 20238496v^6 + 194080595v^5 - 29121939v^4 - 122970143v^3 + 36711565v^2 + 23702924v - 8138838) / 362094
$\beta_{9}$	$=$	$ ( - 43972 \nu^{14} + 136369 \nu^{13} + 821594 \nu^{12} - 2805231 \nu^{11} - 5172502 \nu^{10} + \cdots - 5697114 ) / 181047 $ (-43972v^14 + 136369v^13 + 821594v^12 - 2805231v^11 - 5172502v^10 + 21352581v^9 + 10903403v^8 - 73885247v^7 + 6024274v^6 + 115580902v^5 - 38651493v^4 - 74785189v^3 + 30648548v^2 + 14773525v - 5697114) / 181047
$\beta_{10}$	$=$	$ ( 38883 \nu^{14} - 2580 \nu^{13} - 913685 \nu^{12} - 10870 \nu^{11} + 8323519 \nu^{10} + \cdots - 6056176 ) / 120698 $ (38883v^14 - 2580v^13 - 913685v^12 - 10870v^11 + 8323519v^10 + 717987v^9 - 37178600v^8 - 5676289v^7 + 85001427v^6 + 17341123v^5 - 94837332v^4 - 20976250v^3 + 44634360v^2 + 7801527v - 6056176) / 120698
$\beta_{11}$	$=$	$ ( - 118745 \nu^{14} + 152780 \nu^{13} + 2549911 \nu^{12} - 3015612 \nu^{11} - 20601131 \nu^{10} + \cdots + 5647134 ) / 362094 $ (-118745v^14 + 152780v^13 + 2549911v^12 - 3015612v^11 - 20601131v^10 + 21688647v^9 + 78166324v^8 - 68833003v^7 - 143822353v^6 + 92529461v^5 + 125727336v^4 - 46534262v^3 - 47993234v^2 + 7020911v + 5647134) / 362094
$\beta_{12}$	$=$	$ ( - 142711 \nu^{14} + 340702 \nu^{13} + 2795345 \nu^{12} - 6928746 \nu^{11} - 19299559 \nu^{10} + \cdots - 15689190 ) / 362094 $ (-142711v^14 + 340702v^13 + 2795345v^12 - 6928746v^11 - 19299559v^10 + 51924231v^9 + 53164340v^8 - 175656539v^7 - 35589383v^6 + 265552669v^5 - 54210114v^4 - 165532222v^3 + 67009100v^2 + 32795773v - 15689190) / 362094
$\beta_{13}$	$=$	$ ( - 146320 \nu^{14} + 145423 \nu^{13} + 3209069 \nu^{12} - 2842872 \nu^{11} - 26654908 \nu^{10} + \cdots + 6178446 ) / 362094 $ (-146320v^14 + 145423v^13 + 3209069v^12 - 2842872v^11 - 26654908v^10 + 20200905v^9 + 104840750v^8 - 62955890v^7 - 200923115v^6 + 81310516v^5 + 177910677v^4 - 35837371v^3 - 61849963v^2 + 2346301v + 6178446) / 362094
$\beta_{14}$	$=$	$ ( - 180355 \nu^{14} + 326647 \nu^{13} + 3731036 \nu^{12} - 6584466 \nu^{11} - 28394395 \nu^{10} + \cdots - 4508604 ) / 362094 $ (-180355v^14 + 326647v^13 + 3731036v^12 - 6584466v^11 - 28394395v^10 + 48786336v^9 + 96845738v^8 - 162323297v^7 - 143358896v^6 + 238077535v^5 + 75063729v^4 - 139000663v^3 + 357461v^2 + 23031262v - 4508604) / 362094

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_{11} - \beta_{9} - \beta_{7} + \beta_{5} + 7\beta_{2} + 16 $ b11 - b9 - b7 + b5 + 7*b2 + 16
$\nu^{5}$	$=$	$ \beta_{13} - \beta_{12} + \beta_{11} + 2\beta_{10} + 2\beta_{8} + \beta_{6} - \beta_{5} + 9\beta_{3} + 2\beta_{2} + 31\beta_1 $ b13 - b12 + b11 + 2b10 + 2b8 + b6 - b5 + 9b3 + 2b2 + 31*b1
$\nu^{6}$	$=$	$ - \beta_{14} + \beta_{12} + 11 \beta_{11} - 12 \beta_{9} + 2 \beta_{8} - 11 \beta_{7} + \beta_{6} + \cdots + 100 $ -b14 + b12 + 11b11 - 12b9 + 2b8 - 11b7 + b6 + 11b5 + b3 + 47b2 + 100
$\nu^{7}$	$=$	$ \beta_{14} + 13 \beta_{13} - 14 \beta_{12} + 12 \beta_{11} + 26 \beta_{10} + \beta_{9} + \cdots + 207 \beta_1 $ b14 + 13b13 - 14b12 + 12b11 + 26b10 + b9 + 27b8 - b7 + 14b6 - 13b5 + b4 + 69b3 + 27b2 + 207b1
$\nu^{8}$	$=$	$ - 11 \beta_{14} - 3 \beta_{13} + 13 \beta_{12} + 98 \beta_{11} + 2 \beta_{10} - 112 \beta_{9} + \cdots + 666 $ -11b14 - 3b13 + 13b12 + 98b11 + 2b10 - 112b9 + 31b8 - 95b7 + 15b6 + 95b5 + 6b4 + 14b3 + 318b2 + 4b1 + 666
$\nu^{9}$	$=$	$ 21 \beta_{14} + 122 \beta_{13} - 142 \beta_{12} + 111 \beta_{11} + 252 \beta_{10} + 12 \beta_{9} + \cdots + 7 $ 21b14 + 122b13 - 142b12 + 111b11 + 252b10 + 12b9 + 268b8 - 16b7 + 145b6 - 123b5 + 18b4 + 509b3 + 265b2 + 1434b1 + 7
$\nu^{10}$	$=$	$ - 79 \beta_{14} - 52 \beta_{13} + 119 \beta_{12} + 812 \beta_{11} + 42 \beta_{10} - 959 \beta_{9} + \cdots + 4587 $ -79b14 - 52b13 + 119b12 + 812b11 + 42b10 - 959b9 + 339b8 - 761b7 + 160b6 + 758b5 + 110b4 + 140b3 + 2183b2 + 79b1 + 4587
$\nu^{11}$	$=$	$ 280 \beta_{14} + 1010 \beta_{13} - 1268 \beta_{12} + 942 \beta_{11} + 2189 \beta_{10} + 84 \beta_{9} + \cdots + 137 $ 280b14 + 1010b13 - 1268b12 + 942b11 + 2189b10 + 84b9 + 2370b8 - 182b7 + 1329b6 - 1034b5 + 213b4 + 3714b3 + 2307b2 + 10153b1 + 137
$\nu^{12}$	$=$	$ - 427 \beta_{14} - 606 \beta_{13} + 942 \beta_{12} + 6493 \beta_{11} + 569 \beta_{10} - 7874 \beta_{9} + \cdots + 32250 $ -427b14 - 606b13 + 942b12 + 6493b11 + 569b10 - 7874b9 + 3234b8 - 5903b7 + 1497b6 + 5835b5 + 1334b4 + 1242b3 + 15212b2 + 1020b1 + 32250
$\nu^{13}$	$=$	$ 3057 \beta_{14} + 7862 \beta_{13} - 10617 \beta_{12} + 7700 \beta_{11} + 18017 \beta_{10} + 323 \beta_{9} + \cdots + 1762 $ 3057b14 + 7862b13 - 10617b12 + 7700b11 + 18017b10 + 323b9 + 19808b8 - 1811b7 + 11413b6 - 8200b5 + 2130b4 + 27039b3 + 18954b2 + 72902b1 + 1762
$\nu^{14}$	$=$	$ - 1415 \beta_{14} - 5974 \beta_{13} + 6871 \beta_{12} + 50880 \beta_{11} + 6353 \beta_{10} + \cdots + 229862 $ -1415b14 - 5974b13 + 6871b12 + 50880b11 + 6353b10 - 63064b9 + 28772b8 - 45056b7 + 13132b6 + 44065b5 + 13551b4 + 10448b3 + 107450b2 + 10968b1 + 229862

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

−2.69685
−2.38676
−2.14963
−1.32779
−1.30333
−0.935799
−0.585457
0.324697
0.724177
0.883896
1.23522
1.80019
2.11535
2.54855
2.75353

−2.69685

5.27299

1.00000

−0.774278

−8.82674

−2.69685

1.2

−2.38676

3.69660

1.00000

3.31861

−4.04938

−2.38676

1.3

−2.14963

2.62091

1.00000

−3.62382

−1.33472

−2.14963

1.4

−1.32779

−0.236976

1.00000

5.10516

2.97023

−1.32779

1.5

−1.30333

−0.301321

1.00000

−3.81926

2.99939

−1.30333

1.6

−0.935799

−1.12428

1.00000

0.773665

2.92370

−0.935799

1.7

−0.585457

−1.65724

1.00000

3.43570

2.14116

−0.585457

1.8

0.324697

−1.89457

1.00000

−1.43855

−1.26456

0.324697

1.9

0.724177

−1.47557

1.00000

−0.324231

−2.51693

0.724177

1.10

0.883896

−1.21873

1.00000

−4.25158

−2.84502

0.883896

1.11

1.23522

−0.474222

1.00000

1.27658

−3.05622

1.23522

1.12

1.80019

1.24067

1.00000

3.87783

−1.36693

1.80019

1.13

2.11535

2.47472

1.00000

−2.91413

1.00419

2.11535

1.14

2.54855

4.49510

1.00000

2.93500

6.35888

2.54855

1.15

2.75353

5.58193

1.00000

0.423309

9.86294

2.75353

Atkin-Lehner signs

$ p $	Sign
$3$	$1$
$5$	$-1$
$11$	$1$
$19$	$1$

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	9405.2.a.bt	yes	15
3.b	odd	2	1		9405.2.a.bs	✓	15

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9405.2.a.bs	✓	15	3.b	odd	2	1
9405.2.a.bt	yes	15	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 $S_{2}^{\mathrm{new}}(\Gamma_0(9405))$:

$ T_{2}^{15} - T_{2}^{14} - 23 T_{2}^{13} + 21 T_{2}^{12} + 205 T_{2}^{11} - 168 T_{2}^{10} - 899 T_{2}^{9} + \cdots + 90 $

$ T_{7}^{15} - 4 T_{7}^{14} - 58 T_{7}^{13} + 224 T_{7}^{12} + 1270 T_{7}^{11} - 4764 T_{7}^{10} + \cdots + 17152 $

$ T_{13}^{15} - 3 T_{13}^{14} - 102 T_{13}^{13} + 302 T_{13}^{12} + 3835 T_{13}^{11} - 10817 T_{13}^{10} + \cdots + 564992 $

$ T_{17}^{15} + 7 T_{17}^{14} - 90 T_{17}^{13} - 748 T_{17}^{12} + 2647 T_{17}^{11} + 31205 T_{17}^{10} + \cdots + 54526144 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{15} - T^{14} + \cdots + 90 $ T^15 - T^14 - 23T^13 + 21T^12 + 205T^11 - 168T^10 - 899T^9 + 647T^8 + 2048T^7 - 1264T^6 - 2379T^5 + 1258T^4 + 1276T^3 - 583T^2 - 237*T + 90
$3$	$ T^{15} $ T^15
$5$	$ (T - 1)^{15} $ (T - 1)^15
$7$	$ T^{15} - 4 T^{14} + \cdots + 17152 $ T^15 - 4T^14 - 58T^13 + 224T^12 + 1270T^11 - 4764T^10 - 12908T^9 + 47274T^8 + 59267T^7 - 213360T^6 - 96203T^5 + 351940T^4 + 28672T^3 - 172480T^2 + 6208T + 17152
$11$	$ (T + 1)^{15} $ (T + 1)^15
$13$	$ T^{15} - 3 T^{14} + \cdots + 564992 $ T^15 - 3T^14 - 102T^13 + 302T^12 + 3835T^11 - 10817T^10 - 68777T^9 + 177853T^8 + 634280T^7 - 1396100T^6 - 3001856T^5 + 4753696T^4 + 6655104T^3 - 4953152T^2 - 4105984T + 564992
$17$	$ T^{15} + 7 T^{14} + \cdots + 54526144 $ T^15 + 7T^14 - 90T^13 - 748T^12 + 2647T^11 + 31205T^10 - 12573T^9 - 627057T^8 - 792368T^7 + 5815900T^6 + 14989516T^5 - 14552100T^4 - 83861632T^3 - 69157168T^2 + 39999616T + 54526144
$19$	$ (T + 1)^{15} $ (T + 1)^15
$23$	$ T^{15} - 24 T^{14} + \cdots - 3754208 $ T^15 - 24T^14 + 109T^13 + 1640T^12 - 17564T^11 + 16716T^10 + 413060T^9 - 1422520T^8 - 2052768T^7 + 13779388T^6 - 1207059T^5 - 44275064T^4 + 6982020T^3 + 55174864T^2 + 12527108T - 3754208
$29$	$ T^{15} + 10 T^{14} + \cdots - 77038976 $ T^15 + 10T^14 - 163T^13 - 2004T^12 + 6416T^11 + 132004T^10 + 90011T^9 - 3299838T^8 - 7273204T^7 + 29899704T^6 + 83628220T^5 - 96020408T^4 - 251783072T^3 + 153188928T^2 + 134932672T - 77038976
$31$	$ T^{15} - 20 T^{14} + \cdots + 29299968 $ T^15 - 20T^14 - 26T^13 + 2870T^12 - 13873T^11 - 95352T^10 + 842063T^9 + 122084T^8 - 16099532T^7 + 33882384T^6 + 76703708T^5 - 353185840T^4 + 411231600T^3 - 116206272T^2 - 63249984T + 29299968
$37$	$ T^{15} - 14 T^{14} + \cdots + 6385408 $ T^15 - 14T^14 - 182T^13 + 2832T^12 + 8890T^11 - 180772T^10 - 28610T^9 + 3953636T^8 - 2081939T^7 - 35989616T^6 + 25511215T^5 + 128168508T^4 - 112657520T^3 - 127888704T^2 + 116047552T + 6385408
$41$	$ T^{15} + \cdots + 6429858560 $ T^15 + 10T^14 - 203T^13 - 2480T^12 + 10483T^11 + 201250T^10 + 55007T^9 - 6882492T^8 - 16101012T^7 + 98540872T^6 + 384834284T^5 - 390432528T^4 - 3126545616T^3 - 1937163456T^2 + 6037032384T + 6429858560
$43$	$ T^{15} + \cdots - 284923904 $ T^15 - 4T^14 - 213T^13 + 650T^12 + 17388T^11 - 33200T^10 - 690806T^9 + 439552T^8 + 13850966T^7 + 7141924T^6 - 123073727T^5 - 161890104T^4 + 349514208T^3 + 664326464T^2 + 16196160T - 284923904
$47$	$ T^{15} + \cdots - 1934701056 $ T^15 - 10T^14 - 187T^13 + 1634T^12 + 14525T^11 - 102070T^10 - 609108T^9 + 3062216T^8 + 14889268T^7 - 44584472T^6 - 206709552T^5 + 253223008T^4 + 1408661504T^3 + 103964416T^2 - 2968141056T - 1934701056
$53$	$ T^{15} + \cdots - 2616475648 $ T^15 - 13T^14 - 267T^13 + 3503T^12 + 23723T^11 - 322315T^10 - 786741T^9 + 12158897T^8 + 7987868T^7 - 208776368T^6 + 73399424T^5 + 1596926528T^4 - 1647311104T^3 - 4111915008T^2 + 6931668992T - 2616475648
$59$	$ T^{15} + \cdots + 33865917820 $ T^15 + 3T^14 - 473T^13 - 1671T^12 + 80682T^11 + 327722T^10 - 6009056T^9 - 29337318T^8 + 176322774T^7 + 1205600812T^6 - 475240657T^5 - 17271899853T^4 - 42270133908T^3 - 24492269908T^2 + 31619343196T + 33865917820
$61$	$ T^{15} + \cdots - 1775024640 $ T^15 - 10T^14 - 333T^13 + 3202T^12 + 33798T^11 - 283384T^10 - 1718691T^9 + 10108966T^8 + 48613300T^7 - 141960552T^6 - 671770336T^5 + 510381696T^4 + 3217595200T^3 + 427800448T^2 - 3456496896T - 1775024640
$67$	$ T^{15} + \cdots - 293959168 $ T^15 - 34T^14 + 238T^13 + 3294T^12 - 43728T^11 - 44424T^10 + 2144762T^9 - 2766918T^8 - 43619287T^7 + 83814386T^6 + 356467999T^5 - 697782760T^4 - 757852336T^3 + 1326432384T^2 + 343162048T - 293959168
$71$	$ T^{15} - 5 T^{14} + \cdots + 52858948 $ T^15 - 5T^14 - 592T^13 + 3294T^12 + 122226T^11 - 747128T^10 - 10208646T^9 + 68165422T^8 + 301027199T^7 - 2286660125T^6 - 1846179003T^5 + 23621722537T^4 - 13716029944T^3 - 33027785184T^2 - 3667593372T + 52858948
$73$	$ T^{15} + \cdots - 222789500928 $ T^15 - 4T^14 - 543T^13 + 3612T^12 + 102957T^11 - 962288T^10 - 7240176T^9 + 101348752T^8 + 37648420T^7 - 4070551936T^6 + 11216524736T^5 + 41774735872T^4 - 207882731520T^3 + 83848536064T^2 + 461198180352T - 222789500928
$79$	$ T^{15} + \cdots - 240480000 $ T^15 - 19T^14 - 171T^13 + 4323T^12 + 1884T^11 - 297874T^10 + 798686T^9 + 6303392T^8 - 27957468T^7 - 31231150T^6 + 292734727T^5 - 226253799T^4 - 802027600T^3 + 1393527200T^2 - 405312000T - 240480000
$83$	$ T^{15} + \cdots - 573908992 $ T^15 + 3T^14 - 365T^13 - 93T^12 + 39976T^11 - 12710T^10 - 1807935T^9 + 724129T^8 + 35943808T^7 - 28122012T^6 - 313917264T^5 + 484387120T^4 + 846973312T^3 - 2572124928T^2 + 2126991360T - 573908992
$89$	$ T^{15} + \cdots - 5244593067536 $ T^15 - 9T^14 - 848T^13 + 5252T^12 + 278532T^11 - 733528T^10 - 44914608T^9 - 37903996T^8 + 3303943511T^7 + 11623240457T^6 - 69957682281T^5 - 298097282321T^4 + 634779461200T^3 + 2443969438168T^2 - 2478816028464T - 5244593067536
$97$	$ T^{15} + \cdots - 447568064 $ T^15 - 16T^14 - 331T^13 + 6496T^12 + 11310T^11 - 593970T^10 + 776018T^9 + 21841058T^8 - 45450536T^7 - 366909164T^6 + 701789407T^5 + 2774620868T^4 - 3336993448T^3 - 6688814304T^2 + 3636023152T - 447568064
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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	9405.2.a.bt

Level	$9405$
Weight	$2$
Character orbit	9405.a
Self dual	yes
Analytic conductor	$75.099$
Analytic rank	$0$
Dimension	$15$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(75.0993031010\)
Analytic rank:	\(0\)
Dimension:	\(15\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{15} - x^{14} - 23 x^{13} + 21 x^{12} + 205 x^{11} - 168 x^{10} - 899 x^{9} + 647 x^{8} + 2048 x^{7} + \cdots + 90 \) x^15 - x^14 - 23x^13 + 21x^12 + 205x^11 - 168x^10 - 899x^9 + 647x^8 + 2048x^7 - 1264x^6 - 2379x^5 + 1258x^4 + 1276x^3 - 583x^2 - 237*x + 90
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2 \)
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
\(\beta_{4}\)	\(=\)	\( ( 4439 \nu^{14} - 52877 \nu^{13} - 32602 \nu^{12} + 1099050 \nu^{11} - 463825 \nu^{10} + \cdots + 3044262 ) / 362094 \) (4439v^14 - 52877v^13 - 32602v^12 + 1099050v^11 - 463825v^10 - 8468514v^9 + 5975012v^8 + 29834563v^7 - 23358308v^6 - 48444227v^5 + 36126903v^4 + 34512977v^3 - 21366841v^2 - 9205694v + 3044262) / 362094
\(\beta_{5}\)	\(=\)	\( ( - 7322 \nu^{14} + 100352 \nu^{13} + 30202 \nu^{12} - 2130570 \nu^{11} + 1272397 \nu^{10} + \cdots - 8138904 ) / 181047 \) (-7322v^14 + 100352v^13 + 30202v^12 - 2130570v^11 + 1272397v^10 + 16920255v^9 - 13870331v^8 - 62313670v^7 + 52746803v^6 + 107829311v^5 - 81594021v^4 - 81419489v^3 + 48593164v^2 + 19952147v - 8138904) / 181047
\(\beta_{6}\)	\(=\)	\( ( 39506 \nu^{14} - 203243 \nu^{13} - 637357 \nu^{12} + 4236642 \nu^{11} + 2622242 \nu^{10} + \cdots + 10994718 ) / 362094 \) (39506v^14 - 203243v^13 - 637357v^12 + 4236642v^11 + 2622242v^10 - 32739489v^9 + 5155724v^8 + 115321114v^7 - 54881879v^6 - 184474520v^5 + 106231701v^4 + 122539373v^3 - 67914853v^2 - 25812839v + 10994718) / 362094
\(\beta_{7}\)	\(=\)	\( ( - 45445 \nu^{14} + 80746 \nu^{13} + 967127 \nu^{12} - 1666290 \nu^{11} - 7711333 \nu^{10} + \cdots - 1046916 ) / 362094 \) (-45445v^14 + 80746v^13 + 967127v^12 - 1666290v^11 - 7711333v^10 + 12823995v^9 + 28618856v^8 - 45689849v^7 - 50377295v^6 + 77026279v^5 + 39480186v^4 - 59802862v^3 - 9569344v^2 + 17378155v - 1046916) / 362094
\(\beta_{8}\)	\(=\)	\( ( - 82547 \nu^{14} + 230963 \nu^{13} + 1608118 \nu^{12} - 4751412 \nu^{11} - 11031041 \nu^{10} + \cdots - 8138838 ) / 362094 \) (-82547v^14 + 230963v^13 + 1608118v^12 - 4751412v^11 - 11031041v^10 + 36153378v^9 + 30166240v^8 - 124879183v^7 - 20238496v^6 + 194080595v^5 - 29121939v^4 - 122970143v^3 + 36711565v^2 + 23702924v - 8138838) / 362094
\(\beta_{9}\)	\(=\)	\( ( - 43972 \nu^{14} + 136369 \nu^{13} + 821594 \nu^{12} - 2805231 \nu^{11} - 5172502 \nu^{10} + \cdots - 5697114 ) / 181047 \) (-43972v^14 + 136369v^13 + 821594v^12 - 2805231v^11 - 5172502v^10 + 21352581v^9 + 10903403v^8 - 73885247v^7 + 6024274v^6 + 115580902v^5 - 38651493v^4 - 74785189v^3 + 30648548v^2 + 14773525v - 5697114) / 181047
\(\beta_{10}\)	\(=\)	\( ( 38883 \nu^{14} - 2580 \nu^{13} - 913685 \nu^{12} - 10870 \nu^{11} + 8323519 \nu^{10} + \cdots - 6056176 ) / 120698 \) (38883v^14 - 2580v^13 - 913685v^12 - 10870v^11 + 8323519v^10 + 717987v^9 - 37178600v^8 - 5676289v^7 + 85001427v^6 + 17341123v^5 - 94837332v^4 - 20976250v^3 + 44634360v^2 + 7801527v - 6056176) / 120698
\(\beta_{11}\)	\(=\)	\( ( - 118745 \nu^{14} + 152780 \nu^{13} + 2549911 \nu^{12} - 3015612 \nu^{11} - 20601131 \nu^{10} + \cdots + 5647134 ) / 362094 \) (-118745v^14 + 152780v^13 + 2549911v^12 - 3015612v^11 - 20601131v^10 + 21688647v^9 + 78166324v^8 - 68833003v^7 - 143822353v^6 + 92529461v^5 + 125727336v^4 - 46534262v^3 - 47993234v^2 + 7020911v + 5647134) / 362094
\(\beta_{12}\)	\(=\)	\( ( - 142711 \nu^{14} + 340702 \nu^{13} + 2795345 \nu^{12} - 6928746 \nu^{11} - 19299559 \nu^{10} + \cdots - 15689190 ) / 362094 \) (-142711v^14 + 340702v^13 + 2795345v^12 - 6928746v^11 - 19299559v^10 + 51924231v^9 + 53164340v^8 - 175656539v^7 - 35589383v^6 + 265552669v^5 - 54210114v^4 - 165532222v^3 + 67009100v^2 + 32795773v - 15689190) / 362094
\(\beta_{13}\)	\(=\)	\( ( - 146320 \nu^{14} + 145423 \nu^{13} + 3209069 \nu^{12} - 2842872 \nu^{11} - 26654908 \nu^{10} + \cdots + 6178446 ) / 362094 \) (-146320v^14 + 145423v^13 + 3209069v^12 - 2842872v^11 - 26654908v^10 + 20200905v^9 + 104840750v^8 - 62955890v^7 - 200923115v^6 + 81310516v^5 + 177910677v^4 - 35837371v^3 - 61849963v^2 + 2346301v + 6178446) / 362094
\(\beta_{14}\)	\(=\)	\( ( - 180355 \nu^{14} + 326647 \nu^{13} + 3731036 \nu^{12} - 6584466 \nu^{11} - 28394395 \nu^{10} + \cdots - 4508604 ) / 362094 \) (-180355v^14 + 326647v^13 + 3731036v^12 - 6584466v^11 - 28394395v^10 + 48786336v^9 + 96845738v^8 - 162323297v^7 - 143358896v^6 + 238077535v^5 + 75063729v^4 - 139000663v^3 + 357461v^2 + 23031262v - 4508604) / 362094

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 5\beta_1 \) b3 + 5*b1
\(\nu^{4}\)	\(=\)	\( \beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} + 7\beta_{2} + 16 \) b11 - b9 - b7 + b5 + 7*b2 + 16
\(\nu^{5}\)	\(=\)	\( \beta_{13} - \beta_{12} + \beta_{11} + 2\beta_{10} + 2\beta_{8} + \beta_{6} - \beta_{5} + 9\beta_{3} + 2\beta_{2} + 31\beta_1 \) b13 - b12 + b11 + 2b10 + 2b8 + b6 - b5 + 9b3 + 2b2 + 31*b1
\(\nu^{6}\)	\(=\)	\( - \beta_{14} + \beta_{12} + 11 \beta_{11} - 12 \beta_{9} + 2 \beta_{8} - 11 \beta_{7} + \beta_{6} + \cdots + 100 \) -b14 + b12 + 11b11 - 12b9 + 2b8 - 11b7 + b6 + 11b5 + b3 + 47b2 + 100
\(\nu^{7}\)	\(=\)	\( \beta_{14} + 13 \beta_{13} - 14 \beta_{12} + 12 \beta_{11} + 26 \beta_{10} + \beta_{9} + \cdots + 207 \beta_1 \) b14 + 13b13 - 14b12 + 12b11 + 26b10 + b9 + 27b8 - b7 + 14b6 - 13b5 + b4 + 69b3 + 27b2 + 207b1
\(\nu^{8}\)	\(=\)	\( - 11 \beta_{14} - 3 \beta_{13} + 13 \beta_{12} + 98 \beta_{11} + 2 \beta_{10} - 112 \beta_{9} + \cdots + 666 \) -11b14 - 3b13 + 13b12 + 98b11 + 2b10 - 112b9 + 31b8 - 95b7 + 15b6 + 95b5 + 6b4 + 14b3 + 318b2 + 4b1 + 666
\(\nu^{9}\)	\(=\)	\( 21 \beta_{14} + 122 \beta_{13} - 142 \beta_{12} + 111 \beta_{11} + 252 \beta_{10} + 12 \beta_{9} + \cdots + 7 \) 21b14 + 122b13 - 142b12 + 111b11 + 252b10 + 12b9 + 268b8 - 16b7 + 145b6 - 123b5 + 18b4 + 509b3 + 265b2 + 1434b1 + 7
\(\nu^{10}\)	\(=\)	\( - 79 \beta_{14} - 52 \beta_{13} + 119 \beta_{12} + 812 \beta_{11} + 42 \beta_{10} - 959 \beta_{9} + \cdots + 4587 \) -79b14 - 52b13 + 119b12 + 812b11 + 42b10 - 959b9 + 339b8 - 761b7 + 160b6 + 758b5 + 110b4 + 140b3 + 2183b2 + 79b1 + 4587
\(\nu^{11}\)	\(=\)	\( 280 \beta_{14} + 1010 \beta_{13} - 1268 \beta_{12} + 942 \beta_{11} + 2189 \beta_{10} + 84 \beta_{9} + \cdots + 137 \) 280b14 + 1010b13 - 1268b12 + 942b11 + 2189b10 + 84b9 + 2370b8 - 182b7 + 1329b6 - 1034b5 + 213b4 + 3714b3 + 2307b2 + 10153b1 + 137
\(\nu^{12}\)	\(=\)	\( - 427 \beta_{14} - 606 \beta_{13} + 942 \beta_{12} + 6493 \beta_{11} + 569 \beta_{10} - 7874 \beta_{9} + \cdots + 32250 \) -427b14 - 606b13 + 942b12 + 6493b11 + 569b10 - 7874b9 + 3234b8 - 5903b7 + 1497b6 + 5835b5 + 1334b4 + 1242b3 + 15212b2 + 1020b1 + 32250
\(\nu^{13}\)	\(=\)	\( 3057 \beta_{14} + 7862 \beta_{13} - 10617 \beta_{12} + 7700 \beta_{11} + 18017 \beta_{10} + 323 \beta_{9} + \cdots + 1762 \) 3057b14 + 7862b13 - 10617b12 + 7700b11 + 18017b10 + 323b9 + 19808b8 - 1811b7 + 11413b6 - 8200b5 + 2130b4 + 27039b3 + 18954b2 + 72902b1 + 1762
\(\nu^{14}\)	\(=\)	\( - 1415 \beta_{14} - 5974 \beta_{13} + 6871 \beta_{12} + 50880 \beta_{11} + 6353 \beta_{10} + \cdots + 229862 \) -1415b14 - 5974b13 + 6871b12 + 50880b11 + 6353b10 - 63064b9 + 28772b8 - 45056b7 + 13132b6 + 44065b5 + 13551b4 + 10448b3 + 107450b2 + 10968b1 + 229862

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.15
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{15} - T^{14} + \cdots + 90 \) T^15 - T^14 - 23T^13 + 21T^12 + 205T^11 - 168T^10 - 899T^9 + 647T^8 + 2048T^7 - 1264T^6 - 2379T^5 + 1258T^4 + 1276T^3 - 583T^2 - 237*T + 90
$3$	\( T^{15} \) T^15
$5$	\( (T - 1)^{15} \) (T - 1)^15
$7$	\( T^{15} - 4 T^{14} + \cdots + 17152 \) T^15 - 4T^14 - 58T^13 + 224T^12 + 1270T^11 - 4764T^10 - 12908T^9 + 47274T^8 + 59267T^7 - 213360T^6 - 96203T^5 + 351940T^4 + 28672T^3 - 172480T^2 + 6208T + 17152
$11$	\( (T + 1)^{15} \) (T + 1)^15
$13$	\( T^{15} - 3 T^{14} + \cdots + 564992 \) T^15 - 3T^14 - 102T^13 + 302T^12 + 3835T^11 - 10817T^10 - 68777T^9 + 177853T^8 + 634280T^7 - 1396100T^6 - 3001856T^5 + 4753696T^4 + 6655104T^3 - 4953152T^2 - 4105984T + 564992
$17$	\( T^{15} + 7 T^{14} + \cdots + 54526144 \) T^15 + 7T^14 - 90T^13 - 748T^12 + 2647T^11 + 31205T^10 - 12573T^9 - 627057T^8 - 792368T^7 + 5815900T^6 + 14989516T^5 - 14552100T^4 - 83861632T^3 - 69157168T^2 + 39999616T + 54526144
$19$	\( (T + 1)^{15} \) (T + 1)^15
$23$	\( T^{15} - 24 T^{14} + \cdots - 3754208 \) T^15 - 24T^14 + 109T^13 + 1640T^12 - 17564T^11 + 16716T^10 + 413060T^9 - 1422520T^8 - 2052768T^7 + 13779388T^6 - 1207059T^5 - 44275064T^4 + 6982020T^3 + 55174864T^2 + 12527108T - 3754208
$29$	\( T^{15} + 10 T^{14} + \cdots - 77038976 \) T^15 + 10T^14 - 163T^13 - 2004T^12 + 6416T^11 + 132004T^10 + 90011T^9 - 3299838T^8 - 7273204T^7 + 29899704T^6 + 83628220T^5 - 96020408T^4 - 251783072T^3 + 153188928T^2 + 134932672T - 77038976
$31$	\( T^{15} - 20 T^{14} + \cdots + 29299968 \) T^15 - 20T^14 - 26T^13 + 2870T^12 - 13873T^11 - 95352T^10 + 842063T^9 + 122084T^8 - 16099532T^7 + 33882384T^6 + 76703708T^5 - 353185840T^4 + 411231600T^3 - 116206272T^2 - 63249984T + 29299968
$37$	\( T^{15} - 14 T^{14} + \cdots + 6385408 \) T^15 - 14T^14 - 182T^13 + 2832T^12 + 8890T^11 - 180772T^10 - 28610T^9 + 3953636T^8 - 2081939T^7 - 35989616T^6 + 25511215T^5 + 128168508T^4 - 112657520T^3 - 127888704T^2 + 116047552T + 6385408
$41$	\( T^{15} + \cdots + 6429858560 \) T^15 + 10T^14 - 203T^13 - 2480T^12 + 10483T^11 + 201250T^10 + 55007T^9 - 6882492T^8 - 16101012T^7 + 98540872T^6 + 384834284T^5 - 390432528T^4 - 3126545616T^3 - 1937163456T^2 + 6037032384T + 6429858560
$43$	\( T^{15} + \cdots - 284923904 \) T^15 - 4T^14 - 213T^13 + 650T^12 + 17388T^11 - 33200T^10 - 690806T^9 + 439552T^8 + 13850966T^7 + 7141924T^6 - 123073727T^5 - 161890104T^4 + 349514208T^3 + 664326464T^2 + 16196160T - 284923904
$47$	\( T^{15} + \cdots - 1934701056 \) T^15 - 10T^14 - 187T^13 + 1634T^12 + 14525T^11 - 102070T^10 - 609108T^9 + 3062216T^8 + 14889268T^7 - 44584472T^6 - 206709552T^5 + 253223008T^4 + 1408661504T^3 + 103964416T^2 - 2968141056T - 1934701056
$53$	\( T^{15} + \cdots - 2616475648 \) T^15 - 13T^14 - 267T^13 + 3503T^12 + 23723T^11 - 322315T^10 - 786741T^9 + 12158897T^8 + 7987868T^7 - 208776368T^6 + 73399424T^5 + 1596926528T^4 - 1647311104T^3 - 4111915008T^2 + 6931668992T - 2616475648
$59$	\( T^{15} + \cdots + 33865917820 \) T^15 + 3T^14 - 473T^13 - 1671T^12 + 80682T^11 + 327722T^10 - 6009056T^9 - 29337318T^8 + 176322774T^7 + 1205600812T^6 - 475240657T^5 - 17271899853T^4 - 42270133908T^3 - 24492269908T^2 + 31619343196T + 33865917820
$61$	\( T^{15} + \cdots - 1775024640 \) T^15 - 10T^14 - 333T^13 + 3202T^12 + 33798T^11 - 283384T^10 - 1718691T^9 + 10108966T^8 + 48613300T^7 - 141960552T^6 - 671770336T^5 + 510381696T^4 + 3217595200T^3 + 427800448T^2 - 3456496896T - 1775024640
$67$	\( T^{15} + \cdots - 293959168 \) T^15 - 34T^14 + 238T^13 + 3294T^12 - 43728T^11 - 44424T^10 + 2144762T^9 - 2766918T^8 - 43619287T^7 + 83814386T^6 + 356467999T^5 - 697782760T^4 - 757852336T^3 + 1326432384T^2 + 343162048T - 293959168
$71$	\( T^{15} - 5 T^{14} + \cdots + 52858948 \) T^15 - 5T^14 - 592T^13 + 3294T^12 + 122226T^11 - 747128T^10 - 10208646T^9 + 68165422T^8 + 301027199T^7 - 2286660125T^6 - 1846179003T^5 + 23621722537T^4 - 13716029944T^3 - 33027785184T^2 - 3667593372T + 52858948
$73$	\( T^{15} + \cdots - 222789500928 \) T^15 - 4T^14 - 543T^13 + 3612T^12 + 102957T^11 - 962288T^10 - 7240176T^9 + 101348752T^8 + 37648420T^7 - 4070551936T^6 + 11216524736T^5 + 41774735872T^4 - 207882731520T^3 + 83848536064T^2 + 461198180352T - 222789500928
$79$	\( T^{15} + \cdots - 240480000 \) T^15 - 19T^14 - 171T^13 + 4323T^12 + 1884T^11 - 297874T^10 + 798686T^9 + 6303392T^8 - 27957468T^7 - 31231150T^6 + 292734727T^5 - 226253799T^4 - 802027600T^3 + 1393527200T^2 - 405312000T - 240480000
$83$	\( T^{15} + \cdots - 573908992 \) T^15 + 3T^14 - 365T^13 - 93T^12 + 39976T^11 - 12710T^10 - 1807935T^9 + 724129T^8 + 35943808T^7 - 28122012T^6 - 313917264T^5 + 484387120T^4 + 846973312T^3 - 2572124928T^2 + 2126991360T - 573908992
$89$	\( T^{15} + \cdots - 5244593067536 \) T^15 - 9T^14 - 848T^13 + 5252T^12 + 278532T^11 - 733528T^10 - 44914608T^9 - 37903996T^8 + 3303943511T^7 + 11623240457T^6 - 69957682281T^5 - 298097282321T^4 + 634779461200T^3 + 2443969438168T^2 - 2478816028464T - 5244593067536
$97$	\( T^{15} + \cdots - 447568064 \) T^15 - 16T^14 - 331T^13 + 6496T^12 + 11310T^11 - 593970T^10 + 776018T^9 + 21841058T^8 - 45450536T^7 - 366909164T^6 + 701789407T^5 + 2774620868T^4 - 3336993448T^3 - 6688814304T^2 + 3636023152T - 447568064
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\( p \)	Sign
\(3\)	\(1\)
\(5\)	\(-1\)
\(11\)	\(1\)
\(19\)	\(1\)