LMFDB - Newform orbit 99.2.g.b

Newspace parameters

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	99.g (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.790518980011$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ 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_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{3} + ( -1 - \beta_{2} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{4} + ( -\beta_{6} + \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{6} + ( -\beta_{4} + \beta_{10} ) q^{7} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{10} ) q^{8} + ( 1 + 2 \beta_{2} - \beta_{9} - 2 \beta_{13} ) q^{9} +O(q^{10})$	\( q + \beta_{1} q^{2} + ( \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{3} + ( -1 - \beta_{2} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{4} + ( -\beta_{6} + \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{6} + ( -\beta_{4} + \beta_{10} ) q^{7} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{10} ) q^{8} + ( 1 + 2 \beta_{2} - \beta_{9} - 2 \beta_{13} ) q^{9} + ( -\beta_{1} + \beta_{10} - \beta_{11} ) q^{10} + ( \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{11} + ( -1 - 2 \beta_{2} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{13} + 3 \beta_{14} ) q^{12} + ( -\beta_{4} - 2 \beta_{5} - \beta_{8} - \beta_{10} ) q^{13} + ( -2 + 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - 3 \beta_{9} - 3 \beta_{12} - 4 \beta_{13} ) q^{14} + ( -3 - \beta_{6} - \beta_{9} - \beta_{12} ) q^{15} + ( -2 \beta_{2} + \beta_{6} - \beta_{7} + 2 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{15} ) q^{17} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{10} ) q^{18} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{11} ) q^{19} + ( 4 + 2 \beta_{2} + 2 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{12} - 2 \beta_{14} ) q^{20} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{21} + ( 3 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{22} + ( 1 - \beta_{2} + 3 \beta_{6} - \beta_{7} + 2 \beta_{13} ) q^{23} + ( -\beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{15} ) q^{24} + ( \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{25} + ( -1 - 2 \beta_{2} - 2 \beta_{7} + 2 \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{26} + ( 3 + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{28} + ( -\beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} ) q^{29} + ( -2 \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{30} + ( -2 - 2 \beta_{2} - 2 \beta_{7} - \beta_{9} + \beta_{12} + \beta_{13} ) q^{31} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{11} - \beta_{15} ) q^{32} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{33} + ( 3 \beta_{2} - 4 \beta_{6} + 3 \beta_{7} + \beta_{9} - 4 \beta_{13} - 4 \beta_{14} ) q^{34} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{11} + 2 \beta_{15} ) q^{35} + ( -2 - 4 \beta_{2} + 2 \beta_{6} + \beta_{7} + 3 \beta_{9} - 2 \beta_{12} - \beta_{14} ) q^{36} + ( -4 + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{37} + ( 4 + 2 \beta_{2} - 3 \beta_{6} - 3 \beta_{9} - 3 \beta_{13} + 3 \beta_{14} ) q^{38} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{8} + \beta_{11} + \beta_{15} ) q^{39} + ( \beta_{1} - \beta_{3} - \beta_{8} - \beta_{10} ) q^{40} + ( -\beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{15} ) q^{41} + ( 1 + 2 \beta_{2} - \beta_{6} + 4 \beta_{7} + 3 \beta_{9} + \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{42} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{15} ) q^{43} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{44} + ( -2 + 2 \beta_{2} - 3 \beta_{6} + 3 \beta_{7} - \beta_{9} - 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{45} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - 2 \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{46} + ( -4 - 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{47} + ( -\beta_{9} - 4 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{48} + ( 2 + 2 \beta_{2} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 2 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{11} - \beta_{15} ) q^{50} + ( 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} + \beta_{10} ) q^{51} + ( \beta_{1} + 3 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{15} ) q^{52} + ( -2 - 4 \beta_{2} + 2 \beta_{7} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{53} + ( -\beta_{3} - \beta_{5} - 5 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{15} ) q^{54} + ( 2 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{55} + ( -6 - 3 \beta_{2} + 7 \beta_{6} - \beta_{7} + 6 \beta_{9} + 3 \beta_{12} + 4 \beta_{13} - 7 \beta_{14} ) q^{56} + ( \beta_{1} + \beta_{3} + \beta_{8} - 2 \beta_{11} - \beta_{15} ) q^{57} + ( -3 - 3 \beta_{2} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{58} + ( -\beta_{6} - \beta_{7} + 4 \beta_{9} + 4 \beta_{12} + 2 \beta_{13} ) q^{59} + ( 5 + 4 \beta_{2} + 3 \beta_{6} - \beta_{7} + 2 \beta_{9} + 4 \beta_{12} - 5 \beta_{14} ) q^{60} + ( -2 \beta_{1} - 6 \beta_{3} - \beta_{4} + \beta_{10} + 2 \beta_{15} ) q^{61} + ( -\beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{10} ) q^{62} + ( -\beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{8} - \beta_{11} - 2 \beta_{15} ) q^{63} + ( 1 - 6 \beta_{6} + 5 \beta_{7} - 2 \beta_{9} - 4 \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{64} + ( -3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{15} ) q^{65} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{66} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{15} ) q^{68} + ( 7 + 2 \beta_{2} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{14} ) q^{69} + ( 3 \beta_{6} - 2 \beta_{7} - \beta_{9} - 3 \beta_{12} + 3 \beta_{14} ) q^{70} + ( \beta_{7} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{71} + ( 2 \beta_{1} + 5 \beta_{3} + 3 \beta_{4} - \beta_{8} - 3 \beta_{10} + 2 \beta_{11} + \beta_{15} ) q^{72} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{11} ) q^{73} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{8} + 2 \beta_{11} + \beta_{15} ) q^{74} + ( -3 \beta_{2} - 3 \beta_{13} ) q^{75} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{8} + 3 \beta_{10} - \beta_{11} - \beta_{15} ) q^{76} + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} + \beta_{15} ) q^{77} + ( -4 - 5 \beta_{2} + \beta_{6} - 4 \beta_{7} + 2 \beta_{12} + 4 \beta_{14} ) q^{78} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{8} - 3 \beta_{15} ) q^{79} + ( -1 - 2 \beta_{2} + 2 \beta_{12} - 2 \beta_{14} ) q^{80} + ( 3 \beta_{2} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{14} ) q^{81} + ( 3 + 8 \beta_{6} - 6 \beta_{7} + 3 \beta_{9} + 5 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{82} + ( -2 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} - 3 \beta_{10} ) q^{83} + ( -7 \beta_{1} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{84} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{8} - 4 \beta_{10} + \beta_{11} + \beta_{15} ) q^{85} + ( -1 + \beta_{2} + 4 \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{12} + 7 \beta_{13} ) q^{86} + ( 2 \beta_{1} - \beta_{5} - \beta_{8} + \beta_{11} + \beta_{15} ) q^{87} + ( -3 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{88} + ( 4 + 8 \beta_{2} + 4 \beta_{7} - 2 \beta_{9} - 4 \beta_{13} - 2 \beta_{14} ) q^{89} + ( -5 \beta_{1} + 2 \beta_{3} + \beta_{5} + 3 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} - 3 \beta_{15} ) q^{90} + ( -3 - 2 \beta_{6} + \beta_{7} - 3 \beta_{9} - 3 \beta_{12} - \beta_{13} + \beta_{14} ) q^{91} + ( -6 - 3 \beta_{2} - 8 \beta_{6} - \beta_{7} - 9 \beta_{9} - 5 \beta_{12} - 3 \beta_{13} + 8 \beta_{14} ) q^{92} + ( 3 \beta_{2} + \beta_{6} + 2 \beta_{7} - 3 \beta_{12} - 2 \beta_{13} ) q^{93} + ( -3 \beta_{1} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} - 2 \beta_{15} ) q^{94} + ( 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{95} + ( 5 \beta_{1} + 4 \beta_{3} - \beta_{5} - 2 \beta_{10} ) q^{96} + ( -\beta_{2} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} + 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{97} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} - 2 \beta_{11} - 4 \beta_{15} ) q^{98} + ( -1 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q - 6q^{3} - 14q^{4} + 6q^{9} + O(q^{10}) $	$ 16q - 6q^{3} - 14q^{4} + 6q^{9} - 12q^{11} + 12q^{12} - 6q^{14} - 30q^{15} - 2q^{16} + 36q^{20} + 6q^{22} + 12q^{23} - 12q^{25} + 18q^{27} - 4q^{31} + 18q^{33} - 18q^{36} - 28q^{37} + 66q^{38} - 54q^{42} - 42q^{45} - 30q^{47} + 42q^{48} + 10q^{49} + 20q^{55} - 120q^{56} - 6q^{58} - 36q^{59} + 30q^{60} + 40q^{64} + 54q^{66} + 8q^{67} + 96q^{69} + 24q^{75} + 72q^{77} - 42q^{78} + 30q^{81} + 12q^{82} - 72q^{86} - 6q^{88} - 12q^{91} + 18q^{92} - 24q^{93} - 4q^{97} - 36q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 15 x^{14} + 150 x^{12} + 837 x^{10} + 3372 x^{8} + 8010 x^{6} + 13761 x^{4} + 13392 x^{2} + 8649$:

$\beta_{0}$	$=$	$ 1 $
$\beta_{1}$	$=$	$ \nu $
$\beta_{2}$	$=$	$($$-2126395 \nu^{14} - 29523650 \nu^{12} - 288594750 \nu^{10} - 1489331264 \nu^{8} - 5823700650 \nu^{6} - 11995771800 \nu^{4} - 23155939764 \nu^{2} - 22396351233$$)/ 13131109983 $
$\beta_{3}$	$=$	$($$ 2126395 \nu^{15} + 29523650 \nu^{13} + 288594750 \nu^{11} + 1489331264 \nu^{9} + 5823700650 \nu^{7} + 11995771800 \nu^{5} + 23155939764 \nu^{3} + 9265241250 \nu $$)/ 13131109983 $
$\beta_{4}$	$=$	$($$-4873266 \nu^{15} - 160857665 \nu^{13} - 1890026425 \nu^{11} - 14824095809 \nu^{9} - 66244476882 \nu^{7} - 206255555451 \nu^{5} - 292920249753 \nu^{3} - 290195282991 \nu$$)/ 13131109983 $
$\beta_{5}$	$=$	$($$47500907 \nu^{15} + 812018893 \nu^{13} + 8261503794 \nu^{11} + 49665627558 \nu^{9} + 195642483900 \nu^{7} + 477061297716 \nu^{5} + 607209996627 \nu^{3} + 551339033751 \nu$$)/ 39393329949 $
$\beta_{6}$	$=$	$($$-49032332 \nu^{14} - 864786042 \nu^{12} - 8872595664 \nu^{10} - 54770706318 \nu^{8} - 220344752319 \nu^{6} - 565719472953 \nu^{4} - 758542460580 \nu^{2} - 671327583345$$)/ 39393329949 $
$\beta_{7}$	$=$	$($$-54617732 \nu^{14} - 903112393 \nu^{12} - 9132887847 \nu^{10} - 53705137428 \nu^{8} - 210752440350 \nu^{6} - 495377443209 \nu^{4} - 625450988448 \nu^{2} - 448939285020$$)/ 39393329949 $
$\beta_{8}$	$=$	$($$54617732 \nu^{15} + 903112393 \nu^{13} + 9132887847 \nu^{11} + 53705137428 \nu^{9} + 210752440350 \nu^{7} + 495377443209 \nu^{5} + 625450988448 \nu^{3} + 409545955071 \nu$$)/ 39393329949 $
$\beta_{9}$	$=$	$($$-58397849 \nu^{14} - 683729698 \nu^{12} - 6054561519 \nu^{10} - 23065315299 \nu^{8} - 66793461054 \nu^{6} - 2243892963 \nu^{4} + 33072805413 \nu^{2} + 208063920222$$)/ 39393329949 $
$\beta_{10}$	$=$	$($$64777034 \nu^{15} + 772300648 \nu^{13} + 6920345769 \nu^{11} + 27533309091 \nu^{9} + 84264563004 \nu^{7} + 38231208363 \nu^{5} + 36395013879 \nu^{3} - 180268196472 \nu$$)/ 39393329949 $
$\beta_{11}$	$=$	$($$70362434 \nu^{15} + 810626999 \nu^{13} + 7180637952 \nu^{11} + 26467740201 \nu^{9} + 74672251035 \nu^{7} - 32110821381 \nu^{5} - 96696458253 \nu^{3} - 442049824746 \nu$$)/ 39393329949 $
$\beta_{12}$	$=$	$($$-93878026 \nu^{14} - 1321129241 \nu^{12} - 12590096616 \nu^{10} - 63366471351 \nu^{8} - 225132595554 \nu^{6} - 389659389972 \nu^{4} - 423368055108 \nu^{2} - 157488193548$$)/ 39393329949 $
$\beta_{13}$	$=$	$($$105898756 \nu^{14} + 1495748591 \nu^{12} + 14316065313 \nu^{10} + 72730942857 \nu^{8} + 262435944954 \nu^{6} + 479305190679 \nu^{4} + 574137191214 \nu^{2} + 303881783580$$)/ 39393329949 $
$\beta_{14}$	$=$	$($$-131099776 \nu^{14} - 1868178342 \nu^{12} - 17994696072 \nu^{10} - 92965220511 \nu^{8} - 341617204785 \nu^{6} - 654888451842 \nu^{4} - 813584976426 \nu^{2} - 446373561123$$)/ 39393329949 $
$\beta_{15}$	$=$	$($$145719574 \nu^{15} + 2350751337 \nu^{13} + 23664775347 \nu^{11} + 137437507938 \nu^{9} + 540350635431 \nu^{7} + 1273655118195 \nu^{5} + 1692345725685 \nu^{3} + 1277566080147 \nu$$)/ 39393329949 $

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

$1$	$=$	$\beta_0$
$\nu$	$=$	$\beta_{1}$
$\nu^{2}$	$=$	$-\beta_{12} + \beta_{9} + \beta_{7} - 3 \beta_{2} - 3$
$\nu^{3}$	$=$	$-\beta_{10} + \beta_{5} + \beta_{4} + 5 \beta_{3}$
$\nu^{4}$	$=$	$\beta_{14} + 9 \beta_{13} + 8 \beta_{12} - \beta_{7} + \beta_{6} + 12 \beta_{2}$
$\nu^{5}$	$=$	$-\beta_{15} + \beta_{11} + 8 \beta_{10} - 8 \beta_{8} + \beta_{5} - 9 \beta_{4} - 29 \beta_{3} - 29 \beta_{1}$
$\nu^{6}$	$=$	$13 \beta_{14} - 67 \beta_{13} - 14 \beta_{12} - 68 \beta_{9} - 41 \beta_{7} - 26 \beta_{6} + 57$
$\nu^{7}$	$=$	$-13 \beta_{15} - 26 \beta_{11} + 27 \beta_{10} + 81 \beta_{8} - 53 \beta_{5} + \beta_{4} + 13 \beta_{3} + 166 \beta_{1}$
$\nu^{8}$	$=$	$-240 \beta_{14} - 15 \beta_{13} - 234 \beta_{12} + 474 \beta_{9} + 369 \beta_{7} + 120 \beta_{6} - 294 \beta_{2} - 294$
$\nu^{9}$	$=$	$240 \beta_{15} + 120 \beta_{11} - 609 \beta_{10} - 255 \beta_{8} + 219 \beta_{5} + 474 \beta_{4} + 1017 \beta_{3} + 120 \beta_{1}$
$\nu^{10}$	$=$	$969 \beta_{14} + 3438 \beta_{13} + 2469 \beta_{12} + 150 \beta_{9} - 1119 \beta_{7} + 969 \beta_{6} + 1584 \beta_{2}$
$\nu^{11}$	$=$	$-969 \beta_{15} + 969 \beta_{11} + 2319 \beta_{10} - 2319 \beta_{8} + 969 \beta_{5} - 3438 \beta_{4} - 7341 \beta_{3} - 7341 \beta_{1}$
$\nu^{12}$	$=$	$7314 \beta_{14} - 22581 \beta_{13} - 8583 \beta_{12} - 23850 \beta_{9} - 7953 \beta_{7} - 14628 \beta_{6} + 8802$
$\nu^{13}$	$=$	$-7314 \beta_{15} - 14628 \beta_{11} + 15897 \beta_{10} + 31164 \beta_{8} - 13998 \beta_{5} + 1269 \beta_{4} + 7314 \beta_{3} + 40605 \beta_{1}$
$\nu^{14}$	$=$	$-106212 \beta_{14} - 9852 \beta_{13} - 47928 \beta_{12} + 154140 \beta_{9} + 110886 \beta_{7} + 53106 \beta_{6} - 50265 \beta_{2} - 50265$
$\nu^{15}$	$=$	$106212 \beta_{15} + 53106 \beta_{11} - 217098 \beta_{10} - 116064 \beta_{8} + 38076 \beta_{5} + 154140 \beta_{4} + 262185 \beta_{3} + 53106 \beta_{1}$

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$1 + \beta_{2}$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

32.1

−1.29716	−	2.24675i
−1.10617	−	1.91594i
−0.679041	−	1.17613i
−0.618600	−	1.07145i
0.618600	+	1.07145i
0.679041	+	1.17613i
1.10617	+	1.91594i
1.29716	+	2.24675i
−1.29716	+	2.24675i
−1.10617	+	1.91594i
−0.679041	+	1.17613i
−0.618600	+	1.07145i
0.618600	−	1.07145i
0.679041	−	1.17613i
1.10617	−	1.91594i
1.29716	−	2.24675i

−1.29716

−

2.24675i

−1.72785

0.120512i

−2.36526

4.09675i

−0.137404

−

0.0793301i

2.51207

3.72573i

−2.91814

1.68479i

7.08384

2.97095

−

0.416454i

0.411616i

32.2

−1.10617

−

1.91594i

1.59999

−

0.663339i

−1.44722

2.50665i

−2.54721

−

1.47063i

−3.04078

−

2.33173i

1.72189

−

0.994132i

1.97879

2.11996

−

2.12268i

6.50707i

32.3

−0.679041

−

1.17613i

−0.323333

−

1.70160i

0.0778064

−

0.134765i

0.901139

0.520273i

−1.78176

1.53574i

0.600962

−

0.346965i

−2.92750

−2.79091

1.10037i

−

1.41315i

32.4

−0.618600

−

1.07145i

−1.04881

1.37841i

0.234668

−

0.406456i

1.78348

1.02969i

2.12568

0.271060i

3.59283

−

2.07432i

−3.05506

−0.800005

−

2.89137i

−

2.54787i

32.5

0.618600

1.07145i

−1.04881

1.37841i

0.234668

−

0.406456i

1.78348

1.02969i

−2.12568

−

0.271060i

−3.59283

2.07432i

3.05506

−0.800005

−

2.89137i

2.54787i

32.6

0.679041

1.17613i

−0.323333

−

1.70160i

0.0778064

−

0.134765i

0.901139

0.520273i

1.78176

−

1.53574i

−0.600962

0.346965i

2.92750

−2.79091

1.10037i

1.41315i

32.7

1.10617

1.91594i

1.59999

−

0.663339i

−1.44722

2.50665i

−2.54721

−

1.47063i

3.04078

2.33173i

−1.72189

0.994132i

−1.97879

2.11996

−

2.12268i

−

6.50707i

32.8

1.29716

2.24675i

−1.72785

0.120512i

−2.36526

4.09675i

−0.137404

−

0.0793301i

−2.51207

−

3.72573i

2.91814

−

1.68479i

−7.08384

2.97095

−

0.416454i

−

0.411616i

65.1

−1.29716

2.24675i

−1.72785

−

0.120512i

−2.36526

−

4.09675i

−0.137404

0.0793301i

2.51207

−

3.72573i

−2.91814

−

1.68479i

7.08384

2.97095

0.416454i

−

0.411616i

65.2

−1.10617

1.91594i

1.59999

0.663339i

−1.44722

−

2.50665i

−2.54721

1.47063i

−3.04078

2.33173i

1.72189

0.994132i

1.97879

2.11996

2.12268i

−

6.50707i

65.3

−0.679041

1.17613i

−0.323333

1.70160i

0.0778064

0.134765i

0.901139

−

0.520273i

−1.78176

−

1.53574i

0.600962

0.346965i

−2.92750

−2.79091

−

1.10037i

1.41315i

65.4

−0.618600

1.07145i

−1.04881

−

1.37841i

0.234668

0.406456i

1.78348

−

1.02969i

2.12568

−

0.271060i

3.59283

2.07432i

−3.05506

−0.800005

2.89137i

2.54787i

65.5

0.618600

−

1.07145i

−1.04881

−

1.37841i

0.234668

0.406456i

1.78348

−

1.02969i

−2.12568

0.271060i

−3.59283

−

2.07432i

3.05506

−0.800005

2.89137i

−

2.54787i

65.6

0.679041

−

1.17613i

−0.323333

1.70160i

0.0778064

0.134765i

0.901139

−

0.520273i

1.78176

1.53574i

−0.600962

−

0.346965i

2.92750

−2.79091

−

1.10037i

−

1.41315i

65.7

1.10617

−

1.91594i

1.59999

0.663339i

−1.44722

−

2.50665i

−2.54721

1.47063i

3.04078

−

2.33173i

−1.72189

−

0.994132i

−1.97879

2.11996

2.12268i

6.50707i

65.8

1.29716

−

2.24675i

−1.72785

−

0.120512i

−2.36526

−

4.09675i

−0.137404

0.0793301i

−2.51207

3.72573i

2.91814

1.68479i

−7.08384

2.97095

0.416454i

0.411616i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
11.b	odd	2	1	inner
99.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.2.g.b	✓	16
3.b	odd	2	1		297.2.g.b		16
9.c	even	3	1		297.2.g.b		16
9.c	even	3	1		891.2.d.b		16
9.d	odd	6	1	inner	99.2.g.b	✓	16
9.d	odd	6	1		891.2.d.b		16
11.b	odd	2	1	inner	99.2.g.b	✓	16
33.d	even	2	1		297.2.g.b		16
99.g	even	6	1	inner	99.2.g.b	✓	16
99.g	even	6	1		891.2.d.b		16
99.h	odd	6	1		297.2.g.b		16
99.h	odd	6	1		891.2.d.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.2.g.b	✓	16	1.a	even	1	1	trivial
99.2.g.b	✓	16	9.d	odd	6	1	inner
99.2.g.b	✓	16	11.b	odd	2	1	inner
99.2.g.b	✓	16	99.g	even	6	1	inner
297.2.g.b		16	3.b	odd	2	1
297.2.g.b		16	9.c	even	3	1
297.2.g.b		16	33.d	even	2	1
297.2.g.b		16	99.h	odd	6	1
891.2.d.b		16	9.c	even	3	1
891.2.d.b		16	9.d	odd	6	1
891.2.d.b		16	99.g	even	6	1
891.2.d.b		16	99.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{16} + \cdots$ acting on $S_{2}^{\mathrm{new}}(99, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ 1 - T^{2} - 6 T^{4} + T^{6} + 8 T^{8} + 30 T^{10} - 35 T^{12} - 100 T^{14} + 393 T^{16} - 400 T^{18} - 560 T^{20} + 1920 T^{22} + 2048 T^{24} + 1024 T^{26} - 24576 T^{28} - 16384 T^{30} + 65536 T^{32} $
$3$	$ ( 1 + 3 T + 3 T^{2} - 3 T^{3} - 15 T^{4} - 9 T^{5} + 27 T^{6} + 81 T^{7} + 81 T^{8} )^{2} $
$5$	$ ( 1 + 13 T^{2} + 88 T^{4} + 72 T^{5} + 430 T^{6} + 729 T^{7} + 1846 T^{8} + 3645 T^{9} + 10750 T^{10} + 9000 T^{11} + 55000 T^{12} + 203125 T^{14} + 390625 T^{16} )^{2} $
$7$	$ 1 + 23 T^{2} + 219 T^{4} + 922 T^{6} + 284 T^{8} - 9168 T^{10} + 53788 T^{12} + 1337633 T^{14} + 12397533 T^{16} + 65544017 T^{18} + 129144988 T^{20} - 1078606032 T^{22} + 1637203484 T^{24} + 260442179578 T^{26} + 3031241897019 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32} $
$11$	$ 1 + 12 T + 88 T^{2} + 480 T^{3} + 2146 T^{4} + 8004 T^{5} + 25888 T^{6} + 79956 T^{7} + 254659 T^{8} + 879516 T^{9} + 3132448 T^{10} + 10653324 T^{11} + 31419586 T^{12} + 77304480 T^{13} + 155897368 T^{14} + 233846052 T^{15} + 214358881 T^{16} $
$13$	$ 1 + 53 T^{2} + 1500 T^{4} + 25933 T^{6} + 263753 T^{8} + 468144 T^{10} - 38632730 T^{12} - 890236186 T^{14} - 13299736152 T^{16} - 150449915434 T^{18} - 1103389401530 T^{20} + 2259641672496 T^{22} + 215151424855913 T^{24} + 3575084269120117 T^{26} + 34947127683721500 T^{28} + 208680948442062317 T^{30} + 665416609183179841 T^{32} $
$17$	$ ( 1 + 43 T^{2} + 1684 T^{4} + 38266 T^{6} + 797440 T^{8} + 11058874 T^{10} + 140649364 T^{12} + 1037915467 T^{14} + 6975757441 T^{16} )^{2} $
$19$	$ ( 1 - 89 T^{2} + 4078 T^{4} - 125462 T^{6} + 2786848 T^{8} - 45291782 T^{10} + 531449038 T^{12} - 4187083409 T^{14} + 16983563041 T^{16} )^{2} $
$23$	$ ( 1 - 6 T + 67 T^{2} - 330 T^{3} + 1765 T^{4} - 8928 T^{5} + 54913 T^{6} - 259650 T^{7} + 1729831 T^{8} - 5971950 T^{9} + 29048977 T^{10} - 108626976 T^{11} + 493919365 T^{12} - 2123993190 T^{13} + 9918404563 T^{14} - 20428952682 T^{15} + 78310985281 T^{16} )^{2} $
$29$	$ 1 - 202 T^{2} + 22161 T^{4} - 1720736 T^{6} + 104609393 T^{8} - 5221521276 T^{10} + 219859920283 T^{12} - 7931725055308 T^{14} + 247189558914249 T^{16} - 6670580771514028 T^{18} + 155502744277680523 T^{20} - 3105882626062477596 T^{22} + 52330473610277542673 T^{24} - $$72\!\cdots\!36$$ T^{26} + $$78\!\cdots\!01$$ T^{28} - $$60\!\cdots\!62$$ T^{30} + $$25\!\cdots\!21$$ T^{32} $
$31$	$ ( 1 + 2 T - 105 T^{2} - 140 T^{3} + 6686 T^{4} + 5712 T^{5} - 296792 T^{6} - 74779 T^{7} + 10300464 T^{8} - 2318149 T^{9} - 285217112 T^{10} + 170166192 T^{11} + 6174661406 T^{12} - 4008081140 T^{13} - 93187886505 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2} $
$37$	$ ( 1 + 7 T + 97 T^{2} + 829 T^{3} + 4471 T^{4} + 30673 T^{5} + 132793 T^{6} + 354571 T^{7} + 1874161 T^{8} )^{4} $
$41$	$ 1 - 196 T^{2} + 19095 T^{4} - 1280834 T^{6} + 68413043 T^{8} - 3073917978 T^{10} + 120877869925 T^{12} - 4533161979472 T^{14} + 178294470855873 T^{16} - 7620245287492432 T^{18} + 341571970597137925 T^{20} - 14601430823783944698 T^{22} + $$54\!\cdots\!03$$ T^{24} - $$17\!\cdots\!34$$ T^{26} + $$43\!\cdots\!95$$ T^{28} - $$74\!\cdots\!56$$ T^{30} + $$63\!\cdots\!41$$ T^{32} $
$43$	$ 1 + 161 T^{2} + 12045 T^{4} + 451720 T^{6} + 4744646 T^{8} - 258318474 T^{10} + 8737088662 T^{12} + 2425239363077 T^{14} + 153815372427309 T^{16} + 4484267582329373 T^{18} + 29870367454734262 T^{20} - 1632924856417667226 T^{22} + 55456372694318474246 T^{24} + $$97\!\cdots\!80$$ T^{26} + $$48\!\cdots\!45$$ T^{28} + $$11\!\cdots\!89$$ T^{30} + $$13\!\cdots\!01$$ T^{32} $
$47$	$ ( 1 + 15 T + 238 T^{2} + 2445 T^{3} + 25363 T^{4} + 223068 T^{5} + 1824427 T^{6} + 13850433 T^{7} + 95884210 T^{8} + 650970351 T^{9} + 4030159243 T^{10} + 23159588964 T^{11} + 123763349203 T^{12} + 560748542115 T^{13} + 2565453248302 T^{14} + 7599346806945 T^{15} + 23811286661761 T^{16} )^{2} $
$53$	$ ( 1 - 278 T^{2} + 37843 T^{4} - 3330905 T^{6} + 207574231 T^{8} - 9356512145 T^{10} + 298599472483 T^{12} - 6161692393862 T^{14} + 62259690411361 T^{16} )^{2} $
$59$	$ ( 1 + 18 T + 307 T^{2} + 3582 T^{3} + 38122 T^{4} + 338940 T^{5} + 2896270 T^{6} + 22582341 T^{7} + 177100792 T^{8} + 1332358119 T^{9} + 10081915870 T^{10} + 69611158260 T^{11} + 461938036042 T^{12} + 2560858839018 T^{13} + 12949423827787 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} )^{2} $
$61$	$ 1 + 11 T^{2} - 10329 T^{4} - 236306 T^{6} + 57256364 T^{8} + 1388511372 T^{10} - 209614040984 T^{12} - 2716193993095 T^{14} + 695408199928701 T^{16} - 10106957848306495 T^{18} - 2902282682831947544 T^{20} + 71536625689945733292 T^{22} + $$10\!\cdots\!84$$ T^{24} - $$16\!\cdots\!06$$ T^{26} - $$27\!\cdots\!09$$ T^{28} + $$10\!\cdots\!51$$ T^{30} + $$36\!\cdots\!61$$ T^{32} $
$67$	$ ( 1 - 4 T - 165 T^{2} + 1330 T^{3} + 13514 T^{4} - 133584 T^{5} - 335648 T^{6} + 4908017 T^{7} + 931644 T^{8} + 328837139 T^{9} - 1506723872 T^{10} - 40177124592 T^{11} + 272322249194 T^{12} + 1795666392310 T^{13} - 14925633057885 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} )^{2} $
$71$	$ ( 1 - 515 T^{2} + 119305 T^{4} - 16242779 T^{6} + 1421466937 T^{8} - 81879848939 T^{10} + 3031740601705 T^{12} - 65971646219315 T^{14} + 645753531245761 T^{16} )^{2} $
$73$	$ ( 1 - 176 T^{2} + 28675 T^{4} - 2789687 T^{6} + 246500362 T^{8} - 14866242023 T^{10} + 814319560675 T^{12} - 26634823826864 T^{14} + 806460091894081 T^{16} )^{2} $
$79$	$ 1 + 353 T^{2} + 60567 T^{4} + 6638158 T^{6} + 550474730 T^{8} + 44065155330 T^{10} + 4241918300434 T^{12} + 431883347780411 T^{14} + 37912769819643561 T^{16} + 2695383973497545051 T^{18} + $$16\!\cdots\!54$$ T^{20} + $$10\!\cdots\!30$$ T^{22} + $$83\!\cdots\!30$$ T^{24} + $$62\!\cdots\!58$$ T^{26} + $$35\!\cdots\!47$$ T^{28} + $$13\!\cdots\!93$$ T^{30} + $$23\!\cdots\!21$$ T^{32} $
$83$	$ 1 - 370 T^{2} + 62193 T^{4} - 7697168 T^{6} + 924406361 T^{8} - 98104079436 T^{10} + 8874940013659 T^{12} - 811506409257412 T^{14} + 72427176409933185 T^{16} - 5590467653374311268 T^{18} + $$42\!\cdots\!39$$ T^{20} - $$32\!\cdots\!84$$ T^{22} + $$20\!\cdots\!01$$ T^{24} - $$11\!\cdots\!32$$ T^{26} + $$66\!\cdots\!73$$ T^{28} - $$27\!\cdots\!30$$ T^{30} + $$50\!\cdots\!81$$ T^{32} $
$89$	$ ( 1 - 404 T^{2} + 80980 T^{4} - 10737836 T^{6} + 1074473494 T^{8} - 85054398956 T^{10} + 5080866676180 T^{12} - 200780441548244 T^{14} + 3936588805702081 T^{16} )^{2} $
$97$	$ ( 1 + 2 T - 264 T^{2} - 224 T^{3} + 37139 T^{4} + 3000 T^{5} - 3977996 T^{6} + 3110 T^{7} + 379878912 T^{8} + 301670 T^{9} - 37428964364 T^{10} + 2738019000 T^{11} + 3287888967059 T^{12} - 1923564217568 T^{13} - 219904609301256 T^{14} + 161596568956226 T^{15} + 7837433594376961 T^{16} )^{2} $
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Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{1} q^{2} + ( \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{3} + ( -1 - \beta_{2} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{4} + ( -\beta_{6} + \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{6} + ( -\beta_{4} + \beta_{10} ) q^{7} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{10} ) q^{8} + ( 1 + 2 \beta_{2} - \beta_{9} - 2 \beta_{13} ) q^{9} +O(q^{10})\)	\( q + \beta_{1} q^{2} + ( \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{3} + ( -1 - \beta_{2} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{4} + ( -\beta_{6} + \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{6} + ( -\beta_{4} + \beta_{10} ) q^{7} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{10} ) q^{8} + ( 1 + 2 \beta_{2} - \beta_{9} - 2 \beta_{13} ) q^{9} + ( -\beta_{1} + \beta_{10} - \beta_{11} ) q^{10} + ( \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{11} + ( -1 - 2 \beta_{2} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{13} + 3 \beta_{14} ) q^{12} + ( -\beta_{4} - 2 \beta_{5} - \beta_{8} - \beta_{10} ) q^{13} + ( -2 + 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - 3 \beta_{9} - 3 \beta_{12} - 4 \beta_{13} ) q^{14} + ( -3 - \beta_{6} - \beta_{9} - \beta_{12} ) q^{15} + ( -2 \beta_{2} + \beta_{6} - \beta_{7} + 2 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{15} ) q^{17} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{10} ) q^{18} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{11} ) q^{19} + ( 4 + 2 \beta_{2} + 2 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{12} - 2 \beta_{14} ) q^{20} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{21} + ( 3 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{22} + ( 1 - \beta_{2} + 3 \beta_{6} - \beta_{7} + 2 \beta_{13} ) q^{23} + ( -\beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{15} ) q^{24} + ( \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{25} + ( -1 - 2 \beta_{2} - 2 \beta_{7} + 2 \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{26} + ( 3 + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{28} + ( -\beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} ) q^{29} + ( -2 \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{30} + ( -2 - 2 \beta_{2} - 2 \beta_{7} - \beta_{9} + \beta_{12} + \beta_{13} ) q^{31} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{11} - \beta_{15} ) q^{32} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{33} + ( 3 \beta_{2} - 4 \beta_{6} + 3 \beta_{7} + \beta_{9} - 4 \beta_{13} - 4 \beta_{14} ) q^{34} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{11} + 2 \beta_{15} ) q^{35} + ( -2 - 4 \beta_{2} + 2 \beta_{6} + \beta_{7} + 3 \beta_{9} - 2 \beta_{12} - \beta_{14} ) q^{36} + ( -4 + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{37} + ( 4 + 2 \beta_{2} - 3 \beta_{6} - 3 \beta_{9} - 3 \beta_{13} + 3 \beta_{14} ) q^{38} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{8} + \beta_{11} + \beta_{15} ) q^{39} + ( \beta_{1} - \beta_{3} - \beta_{8} - \beta_{10} ) q^{40} + ( -\beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{15} ) q^{41} + ( 1 + 2 \beta_{2} - \beta_{6} + 4 \beta_{7} + 3 \beta_{9} + \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{42} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{15} ) q^{43} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{44} + ( -2 + 2 \beta_{2} - 3 \beta_{6} + 3 \beta_{7} - \beta_{9} - 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{45} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - 2 \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{46} + ( -4 - 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{47} + ( -\beta_{9} - 4 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{48} + ( 2 + 2 \beta_{2} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 2 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{11} - \beta_{15} ) q^{50} + ( 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} + \beta_{10} ) q^{51} + ( \beta_{1} + 3 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{15} ) q^{52} + ( -2 - 4 \beta_{2} + 2 \beta_{7} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{53} + ( -\beta_{3} - \beta_{5} - 5 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{15} ) q^{54} + ( 2 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{55} + ( -6 - 3 \beta_{2} + 7 \beta_{6} - \beta_{7} + 6 \beta_{9} + 3 \beta_{12} + 4 \beta_{13} - 7 \beta_{14} ) q^{56} + ( \beta_{1} + \beta_{3} + \beta_{8} - 2 \beta_{11} - \beta_{15} ) q^{57} + ( -3 - 3 \beta_{2} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{58} + ( -\beta_{6} - \beta_{7} + 4 \beta_{9} + 4 \beta_{12} + 2 \beta_{13} ) q^{59} + ( 5 + 4 \beta_{2} + 3 \beta_{6} - \beta_{7} + 2 \beta_{9} + 4 \beta_{12} - 5 \beta_{14} ) q^{60} + ( -2 \beta_{1} - 6 \beta_{3} - \beta_{4} + \beta_{10} + 2 \beta_{15} ) q^{61} + ( -\beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{10} ) q^{62} + ( -\beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{8} - \beta_{11} - 2 \beta_{15} ) q^{63} + ( 1 - 6 \beta_{6} + 5 \beta_{7} - 2 \beta_{9} - 4 \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{64} + ( -3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{15} ) q^{65} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{66} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{15} ) q^{68} + ( 7 + 2 \beta_{2} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{14} ) q^{69} + ( 3 \beta_{6} - 2 \beta_{7} - \beta_{9} - 3 \beta_{12} + 3 \beta_{14} ) q^{70} + ( \beta_{7} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{71} + ( 2 \beta_{1} + 5 \beta_{3} + 3 \beta_{4} - \beta_{8} - 3 \beta_{10} + 2 \beta_{11} + \beta_{15} ) q^{72} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{11} ) q^{73} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{8} + 2 \beta_{11} + \beta_{15} ) q^{74} + ( -3 \beta_{2} - 3 \beta_{13} ) q^{75} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{8} + 3 \beta_{10} - \beta_{11} - \beta_{15} ) q^{76} + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} + \beta_{15} ) q^{77} + ( -4 - 5 \beta_{2} + \beta_{6} - 4 \beta_{7} + 2 \beta_{12} + 4 \beta_{14} ) q^{78} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{8} - 3 \beta_{15} ) q^{79} + ( -1 - 2 \beta_{2} + 2 \beta_{12} - 2 \beta_{14} ) q^{80} + ( 3 \beta_{2} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{14} ) q^{81} + ( 3 + 8 \beta_{6} - 6 \beta_{7} + 3 \beta_{9} + 5 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{82} + ( -2 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} - 3 \beta_{10} ) q^{83} + ( -7 \beta_{1} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{84} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{8} - 4 \beta_{10} + \beta_{11} + \beta_{15} ) q^{85} + ( -1 + \beta_{2} + 4 \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{12} + 7 \beta_{13} ) q^{86} + ( 2 \beta_{1} - \beta_{5} - \beta_{8} + \beta_{11} + \beta_{15} ) q^{87} + ( -3 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{88} + ( 4 + 8 \beta_{2} + 4 \beta_{7} - 2 \beta_{9} - 4 \beta_{13} - 2 \beta_{14} ) q^{89} + ( -5 \beta_{1} + 2 \beta_{3} + \beta_{5} + 3 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} - 3 \beta_{15} ) q^{90} + ( -3 - 2 \beta_{6} + \beta_{7} - 3 \beta_{9} - 3 \beta_{12} - \beta_{13} + \beta_{14} ) q^{91} + ( -6 - 3 \beta_{2} - 8 \beta_{6} - \beta_{7} - 9 \beta_{9} - 5 \beta_{12} - 3 \beta_{13} + 8 \beta_{14} ) q^{92} + ( 3 \beta_{2} + \beta_{6} + 2 \beta_{7} - 3 \beta_{12} - 2 \beta_{13} ) q^{93} + ( -3 \beta_{1} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} - 2 \beta_{15} ) q^{94} + ( 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{95} + ( 5 \beta_{1} + 4 \beta_{3} - \beta_{5} - 2 \beta_{10} ) q^{96} + ( -\beta_{2} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} + 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{97} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} - 2 \beta_{11} - 4 \beta_{15} ) q^{98} + ( -1 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 6q^{3} - 14q^{4} + 6q^{9} + O(q^{10}) \)	\( 16q - 6q^{3} - 14q^{4} + 6q^{9} - 12q^{11} + 12q^{12} - 6q^{14} - 30q^{15} - 2q^{16} + 36q^{20} + 6q^{22} + 12q^{23} - 12q^{25} + 18q^{27} - 4q^{31} + 18q^{33} - 18q^{36} - 28q^{37} + 66q^{38} - 54q^{42} - 42q^{45} - 30q^{47} + 42q^{48} + 10q^{49} + 20q^{55} - 120q^{56} - 6q^{58} - 36q^{59} + 30q^{60} + 40q^{64} + 54q^{66} + 8q^{67} + 96q^{69} + 24q^{75} + 72q^{77} - 42q^{78} + 30q^{81} + 12q^{82} - 72q^{86} - 6q^{88} - 12q^{91} + 18q^{92} - 24q^{93} - 4q^{97} - 36q^{99} + O(q^{100}) \)

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\((\)\(-2126395 \nu^{14} - 29523650 \nu^{12} - 288594750 \nu^{10} - 1489331264 \nu^{8} - 5823700650 \nu^{6} - 11995771800 \nu^{4} - 23155939764 \nu^{2} - 22396351233\)\()/ 13131109983 \)
\(\beta_{3}\)	\(=\)	\((\)\( 2126395 \nu^{15} + 29523650 \nu^{13} + 288594750 \nu^{11} + 1489331264 \nu^{9} + 5823700650 \nu^{7} + 11995771800 \nu^{5} + 23155939764 \nu^{3} + 9265241250 \nu \)\()/ 13131109983 \)
\(\beta_{4}\)	\(=\)	\((\)\(-4873266 \nu^{15} - 160857665 \nu^{13} - 1890026425 \nu^{11} - 14824095809 \nu^{9} - 66244476882 \nu^{7} - 206255555451 \nu^{5} - 292920249753 \nu^{3} - 290195282991 \nu\)\()/ 13131109983 \)
\(\beta_{5}\)	\(=\)	\((\)\(47500907 \nu^{15} + 812018893 \nu^{13} + 8261503794 \nu^{11} + 49665627558 \nu^{9} + 195642483900 \nu^{7} + 477061297716 \nu^{5} + 607209996627 \nu^{3} + 551339033751 \nu\)\()/ 39393329949 \)
\(\beta_{6}\)	\(=\)	\((\)\(-49032332 \nu^{14} - 864786042 \nu^{12} - 8872595664 \nu^{10} - 54770706318 \nu^{8} - 220344752319 \nu^{6} - 565719472953 \nu^{4} - 758542460580 \nu^{2} - 671327583345\)\()/ 39393329949 \)
\(\beta_{7}\)	\(=\)	\((\)\(-54617732 \nu^{14} - 903112393 \nu^{12} - 9132887847 \nu^{10} - 53705137428 \nu^{8} - 210752440350 \nu^{6} - 495377443209 \nu^{4} - 625450988448 \nu^{2} - 448939285020\)\()/ 39393329949 \)
\(\beta_{8}\)	\(=\)	\((\)\(54617732 \nu^{15} + 903112393 \nu^{13} + 9132887847 \nu^{11} + 53705137428 \nu^{9} + 210752440350 \nu^{7} + 495377443209 \nu^{5} + 625450988448 \nu^{3} + 409545955071 \nu\)\()/ 39393329949 \)
\(\beta_{9}\)	\(=\)	\((\)\(-58397849 \nu^{14} - 683729698 \nu^{12} - 6054561519 \nu^{10} - 23065315299 \nu^{8} - 66793461054 \nu^{6} - 2243892963 \nu^{4} + 33072805413 \nu^{2} + 208063920222\)\()/ 39393329949 \)
\(\beta_{10}\)	\(=\)	\((\)\(64777034 \nu^{15} + 772300648 \nu^{13} + 6920345769 \nu^{11} + 27533309091 \nu^{9} + 84264563004 \nu^{7} + 38231208363 \nu^{5} + 36395013879 \nu^{3} - 180268196472 \nu\)\()/ 39393329949 \)
\(\beta_{11}\)	\(=\)	\((\)\(70362434 \nu^{15} + 810626999 \nu^{13} + 7180637952 \nu^{11} + 26467740201 \nu^{9} + 74672251035 \nu^{7} - 32110821381 \nu^{5} - 96696458253 \nu^{3} - 442049824746 \nu\)\()/ 39393329949 \)
\(\beta_{12}\)	\(=\)	\((\)\(-93878026 \nu^{14} - 1321129241 \nu^{12} - 12590096616 \nu^{10} - 63366471351 \nu^{8} - 225132595554 \nu^{6} - 389659389972 \nu^{4} - 423368055108 \nu^{2} - 157488193548\)\()/ 39393329949 \)
\(\beta_{13}\)	\(=\)	\((\)\(105898756 \nu^{14} + 1495748591 \nu^{12} + 14316065313 \nu^{10} + 72730942857 \nu^{8} + 262435944954 \nu^{6} + 479305190679 \nu^{4} + 574137191214 \nu^{2} + 303881783580\)\()/ 39393329949 \)
\(\beta_{14}\)	\(=\)	\((\)\(-131099776 \nu^{14} - 1868178342 \nu^{12} - 17994696072 \nu^{10} - 92965220511 \nu^{8} - 341617204785 \nu^{6} - 654888451842 \nu^{4} - 813584976426 \nu^{2} - 446373561123\)\()/ 39393329949 \)
\(\beta_{15}\)	\(=\)	\((\)\(145719574 \nu^{15} + 2350751337 \nu^{13} + 23664775347 \nu^{11} + 137437507938 \nu^{9} + 540350635431 \nu^{7} + 1273655118195 \nu^{5} + 1692345725685 \nu^{3} + 1277566080147 \nu\)\()/ 39393329949 \)

\(1\)	\(=\)	\(\beta_0\)
\(\nu\)	\(=\)	\(\beta_{1}\)
\(\nu^{2}\)	\(=\)	\(-\beta_{12} + \beta_{9} + \beta_{7} - 3 \beta_{2} - 3\)
\(\nu^{3}\)	\(=\)	\(-\beta_{10} + \beta_{5} + \beta_{4} + 5 \beta_{3}\)
\(\nu^{4}\)	\(=\)	\(\beta_{14} + 9 \beta_{13} + 8 \beta_{12} - \beta_{7} + \beta_{6} + 12 \beta_{2}\)
\(\nu^{5}\)	\(=\)	\(-\beta_{15} + \beta_{11} + 8 \beta_{10} - 8 \beta_{8} + \beta_{5} - 9 \beta_{4} - 29 \beta_{3} - 29 \beta_{1}\)
\(\nu^{6}\)	\(=\)	\(13 \beta_{14} - 67 \beta_{13} - 14 \beta_{12} - 68 \beta_{9} - 41 \beta_{7} - 26 \beta_{6} + 57\)
\(\nu^{7}\)	\(=\)	\(-13 \beta_{15} - 26 \beta_{11} + 27 \beta_{10} + 81 \beta_{8} - 53 \beta_{5} + \beta_{4} + 13 \beta_{3} + 166 \beta_{1}\)
\(\nu^{8}\)	\(=\)	\(-240 \beta_{14} - 15 \beta_{13} - 234 \beta_{12} + 474 \beta_{9} + 369 \beta_{7} + 120 \beta_{6} - 294 \beta_{2} - 294\)
\(\nu^{9}\)	\(=\)	\(240 \beta_{15} + 120 \beta_{11} - 609 \beta_{10} - 255 \beta_{8} + 219 \beta_{5} + 474 \beta_{4} + 1017 \beta_{3} + 120 \beta_{1}\)
\(\nu^{10}\)	\(=\)	\(969 \beta_{14} + 3438 \beta_{13} + 2469 \beta_{12} + 150 \beta_{9} - 1119 \beta_{7} + 969 \beta_{6} + 1584 \beta_{2}\)
\(\nu^{11}\)	\(=\)	\(-969 \beta_{15} + 969 \beta_{11} + 2319 \beta_{10} - 2319 \beta_{8} + 969 \beta_{5} - 3438 \beta_{4} - 7341 \beta_{3} - 7341 \beta_{1}\)
\(\nu^{12}\)	\(=\)	\(7314 \beta_{14} - 22581 \beta_{13} - 8583 \beta_{12} - 23850 \beta_{9} - 7953 \beta_{7} - 14628 \beta_{6} + 8802\)
\(\nu^{13}\)	\(=\)	\(-7314 \beta_{15} - 14628 \beta_{11} + 15897 \beta_{10} + 31164 \beta_{8} - 13998 \beta_{5} + 1269 \beta_{4} + 7314 \beta_{3} + 40605 \beta_{1}\)
\(\nu^{14}\)	\(=\)	\(-106212 \beta_{14} - 9852 \beta_{13} - 47928 \beta_{12} + 154140 \beta_{9} + 110886 \beta_{7} + 53106 \beta_{6} - 50265 \beta_{2} - 50265\)
\(\nu^{15}\)	\(=\)	\(106212 \beta_{15} + 53106 \beta_{11} - 217098 \beta_{10} - 116064 \beta_{8} + 38076 \beta_{5} + 154140 \beta_{4} + 262185 \beta_{3} + 53106 \beta_{1}\)

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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	99.2.g.b

Level	99
Weight	2
Character orbit	99.g
Analytic conductor	0.791
Analytic rank	0
Dimension	16
CM	no
Inner twists	4

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 65.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( 1 - T^{2} - 6 T^{4} + T^{6} + 8 T^{8} + 30 T^{10} - 35 T^{12} - 100 T^{14} + 393 T^{16} - 400 T^{18} - 560 T^{20} + 1920 T^{22} + 2048 T^{24} + 1024 T^{26} - 24576 T^{28} - 16384 T^{30} + 65536 T^{32} \)
$3$	\( ( 1 + 3 T + 3 T^{2} - 3 T^{3} - 15 T^{4} - 9 T^{5} + 27 T^{6} + 81 T^{7} + 81 T^{8} )^{2} \)
$5$	\( ( 1 + 13 T^{2} + 88 T^{4} + 72 T^{5} + 430 T^{6} + 729 T^{7} + 1846 T^{8} + 3645 T^{9} + 10750 T^{10} + 9000 T^{11} + 55000 T^{12} + 203125 T^{14} + 390625 T^{16} )^{2} \)
$7$	\( 1 + 23 T^{2} + 219 T^{4} + 922 T^{6} + 284 T^{8} - 9168 T^{10} + 53788 T^{12} + 1337633 T^{14} + 12397533 T^{16} + 65544017 T^{18} + 129144988 T^{20} - 1078606032 T^{22} + 1637203484 T^{24} + 260442179578 T^{26} + 3031241897019 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32} \)
$11$	\( 1 + 12 T + 88 T^{2} + 480 T^{3} + 2146 T^{4} + 8004 T^{5} + 25888 T^{6} + 79956 T^{7} + 254659 T^{8} + 879516 T^{9} + 3132448 T^{10} + 10653324 T^{11} + 31419586 T^{12} + 77304480 T^{13} + 155897368 T^{14} + 233846052 T^{15} + 214358881 T^{16} \)
$13$	\( 1 + 53 T^{2} + 1500 T^{4} + 25933 T^{6} + 263753 T^{8} + 468144 T^{10} - 38632730 T^{12} - 890236186 T^{14} - 13299736152 T^{16} - 150449915434 T^{18} - 1103389401530 T^{20} + 2259641672496 T^{22} + 215151424855913 T^{24} + 3575084269120117 T^{26} + 34947127683721500 T^{28} + 208680948442062317 T^{30} + 665416609183179841 T^{32} \)
$17$	\( ( 1 + 43 T^{2} + 1684 T^{4} + 38266 T^{6} + 797440 T^{8} + 11058874 T^{10} + 140649364 T^{12} + 1037915467 T^{14} + 6975757441 T^{16} )^{2} \)
$19$	\( ( 1 - 89 T^{2} + 4078 T^{4} - 125462 T^{6} + 2786848 T^{8} - 45291782 T^{10} + 531449038 T^{12} - 4187083409 T^{14} + 16983563041 T^{16} )^{2} \)
$23$	\( ( 1 - 6 T + 67 T^{2} - 330 T^{3} + 1765 T^{4} - 8928 T^{5} + 54913 T^{6} - 259650 T^{7} + 1729831 T^{8} - 5971950 T^{9} + 29048977 T^{10} - 108626976 T^{11} + 493919365 T^{12} - 2123993190 T^{13} + 9918404563 T^{14} - 20428952682 T^{15} + 78310985281 T^{16} )^{2} \)
$29$	\( 1 - 202 T^{2} + 22161 T^{4} - 1720736 T^{6} + 104609393 T^{8} - 5221521276 T^{10} + 219859920283 T^{12} - 7931725055308 T^{14} + 247189558914249 T^{16} - 6670580771514028 T^{18} + 155502744277680523 T^{20} - 3105882626062477596 T^{22} + 52330473610277542673 T^{24} - \)\(72\!\cdots\!36\)\( T^{26} + \)\(78\!\cdots\!01\)\( T^{28} - \)\(60\!\cdots\!62\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$	\( ( 1 + 2 T - 105 T^{2} - 140 T^{3} + 6686 T^{4} + 5712 T^{5} - 296792 T^{6} - 74779 T^{7} + 10300464 T^{8} - 2318149 T^{9} - 285217112 T^{10} + 170166192 T^{11} + 6174661406 T^{12} - 4008081140 T^{13} - 93187886505 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2} \)
$37$	\( ( 1 + 7 T + 97 T^{2} + 829 T^{3} + 4471 T^{4} + 30673 T^{5} + 132793 T^{6} + 354571 T^{7} + 1874161 T^{8} )^{4} \)
$41$	\( 1 - 196 T^{2} + 19095 T^{4} - 1280834 T^{6} + 68413043 T^{8} - 3073917978 T^{10} + 120877869925 T^{12} - 4533161979472 T^{14} + 178294470855873 T^{16} - 7620245287492432 T^{18} + 341571970597137925 T^{20} - 14601430823783944698 T^{22} + \)\(54\!\cdots\!03\)\( T^{24} - \)\(17\!\cdots\!34\)\( T^{26} + \)\(43\!\cdots\!95\)\( T^{28} - \)\(74\!\cdots\!56\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$	\( 1 + 161 T^{2} + 12045 T^{4} + 451720 T^{6} + 4744646 T^{8} - 258318474 T^{10} + 8737088662 T^{12} + 2425239363077 T^{14} + 153815372427309 T^{16} + 4484267582329373 T^{18} + 29870367454734262 T^{20} - 1632924856417667226 T^{22} + 55456372694318474246 T^{24} + \)\(97\!\cdots\!80\)\( T^{26} + \)\(48\!\cdots\!45\)\( T^{28} + \)\(11\!\cdots\!89\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$	\( ( 1 + 15 T + 238 T^{2} + 2445 T^{3} + 25363 T^{4} + 223068 T^{5} + 1824427 T^{6} + 13850433 T^{7} + 95884210 T^{8} + 650970351 T^{9} + 4030159243 T^{10} + 23159588964 T^{11} + 123763349203 T^{12} + 560748542115 T^{13} + 2565453248302 T^{14} + 7599346806945 T^{15} + 23811286661761 T^{16} )^{2} \)
$53$	\( ( 1 - 278 T^{2} + 37843 T^{4} - 3330905 T^{6} + 207574231 T^{8} - 9356512145 T^{10} + 298599472483 T^{12} - 6161692393862 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$	\( ( 1 + 18 T + 307 T^{2} + 3582 T^{3} + 38122 T^{4} + 338940 T^{5} + 2896270 T^{6} + 22582341 T^{7} + 177100792 T^{8} + 1332358119 T^{9} + 10081915870 T^{10} + 69611158260 T^{11} + 461938036042 T^{12} + 2560858839018 T^{13} + 12949423827787 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} )^{2} \)
$61$	\( 1 + 11 T^{2} - 10329 T^{4} - 236306 T^{6} + 57256364 T^{8} + 1388511372 T^{10} - 209614040984 T^{12} - 2716193993095 T^{14} + 695408199928701 T^{16} - 10106957848306495 T^{18} - 2902282682831947544 T^{20} + 71536625689945733292 T^{22} + \)\(10\!\cdots\!84\)\( T^{24} - \)\(16\!\cdots\!06\)\( T^{26} - \)\(27\!\cdots\!09\)\( T^{28} + \)\(10\!\cdots\!51\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$	\( ( 1 - 4 T - 165 T^{2} + 1330 T^{3} + 13514 T^{4} - 133584 T^{5} - 335648 T^{6} + 4908017 T^{7} + 931644 T^{8} + 328837139 T^{9} - 1506723872 T^{10} - 40177124592 T^{11} + 272322249194 T^{12} + 1795666392310 T^{13} - 14925633057885 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$	\( ( 1 - 515 T^{2} + 119305 T^{4} - 16242779 T^{6} + 1421466937 T^{8} - 81879848939 T^{10} + 3031740601705 T^{12} - 65971646219315 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$	\( ( 1 - 176 T^{2} + 28675 T^{4} - 2789687 T^{6} + 246500362 T^{8} - 14866242023 T^{10} + 814319560675 T^{12} - 26634823826864 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$	\( 1 + 353 T^{2} + 60567 T^{4} + 6638158 T^{6} + 550474730 T^{8} + 44065155330 T^{10} + 4241918300434 T^{12} + 431883347780411 T^{14} + 37912769819643561 T^{16} + 2695383973497545051 T^{18} + \)\(16\!\cdots\!54\)\( T^{20} + \)\(10\!\cdots\!30\)\( T^{22} + \)\(83\!\cdots\!30\)\( T^{24} + \)\(62\!\cdots\!58\)\( T^{26} + \)\(35\!\cdots\!47\)\( T^{28} + \)\(13\!\cdots\!93\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$	\( 1 - 370 T^{2} + 62193 T^{4} - 7697168 T^{6} + 924406361 T^{8} - 98104079436 T^{10} + 8874940013659 T^{12} - 811506409257412 T^{14} + 72427176409933185 T^{16} - 5590467653374311268 T^{18} + \)\(42\!\cdots\!39\)\( T^{20} - \)\(32\!\cdots\!84\)\( T^{22} + \)\(20\!\cdots\!01\)\( T^{24} - \)\(11\!\cdots\!32\)\( T^{26} + \)\(66\!\cdots\!73\)\( T^{28} - \)\(27\!\cdots\!30\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$	\( ( 1 - 404 T^{2} + 80980 T^{4} - 10737836 T^{6} + 1074473494 T^{8} - 85054398956 T^{10} + 5080866676180 T^{12} - 200780441548244 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$	\( ( 1 + 2 T - 264 T^{2} - 224 T^{3} + 37139 T^{4} + 3000 T^{5} - 3977996 T^{6} + 3110 T^{7} + 379878912 T^{8} + 301670 T^{9} - 37428964364 T^{10} + 2738019000 T^{11} + 3287888967059 T^{12} - 1923564217568 T^{13} - 219904609301256 T^{14} + 161596568956226 T^{15} + 7837433594376961 T^{16} )^{2} \)
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\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(-1\)	\(1 + \beta_{2}\)