LMFDB - Embedded newform 6027.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} -4.24914 q^{5} +0.470683 q^{6} -1.77846 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} -4.24914 q^{5} +0.470683 q^{6} -1.77846 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.47068 q^{11} -1.77846 q^{12} +0.249141 q^{13} -4.24914 q^{15} +2.71982 q^{16} +5.02760 q^{17} +0.470683 q^{18} +2.24914 q^{19} +7.55691 q^{20} -0.692226 q^{22} -6.24914 q^{23} -1.77846 q^{24} +13.0552 q^{25} +0.117266 q^{26} +1.00000 q^{27} -2.41205 q^{29} -2.00000 q^{30} +2.89572 q^{31} +4.83709 q^{32} -1.47068 q^{33} +2.36641 q^{34} -1.77846 q^{36} +9.71982 q^{37} +1.05863 q^{38} +0.249141 q^{39} +7.55691 q^{40} -1.00000 q^{41} -10.8337 q^{43} +2.61555 q^{44} -4.24914 q^{45} -2.94137 q^{46} +4.08623 q^{47} +2.71982 q^{48} +6.14486 q^{50} +5.02760 q^{51} -0.443086 q^{52} -1.43965 q^{53} +0.470683 q^{54} +6.24914 q^{55} +2.24914 q^{57} -1.13531 q^{58} -2.61555 q^{59} +7.55691 q^{60} +5.71982 q^{61} +1.36297 q^{62} -3.16291 q^{64} -1.05863 q^{65} -0.692226 q^{66} +15.9379 q^{67} -8.94137 q^{68} -6.24914 q^{69} -10.5293 q^{71} -1.77846 q^{72} -9.39400 q^{73} +4.57496 q^{74} +13.0552 q^{75} -4.00000 q^{76} +0.117266 q^{78} -0.560352 q^{79} -11.5569 q^{80} +1.00000 q^{81} -0.470683 q^{82} -3.80605 q^{83} -21.3630 q^{85} -5.09922 q^{86} -2.41205 q^{87} +2.61555 q^{88} -4.11727 q^{89} -2.00000 q^{90} +11.1138 q^{92} +2.89572 q^{93} +1.92332 q^{94} -9.55691 q^{95} +4.83709 q^{96} +7.19051 q^{97} -1.47068 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + 3 q^{4} - 4 q^{5} + q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 3 * q^3 + 3 * q^4 - 4 * q^5 + q^6 + 3 * q^8 + 3 * q^9	$ 3 q + q^{2} + 3 q^{3} + 3 q^{4} - 4 q^{5} + q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 4 q^{11} + 3 q^{12} - 8 q^{13} - 4 q^{15} - q^{16} - 2 q^{17} + q^{18} - 2 q^{19} + 6 q^{20} - 10 q^{22} - 10 q^{23} + 3 q^{24} + 5 q^{25} + 2 q^{26} + 3 q^{27} - 6 q^{29} - 6 q^{30} + 2 q^{31} + 7 q^{32} - 4 q^{33} + 3 q^{36} + 20 q^{37} + 4 q^{38} - 8 q^{39} + 6 q^{40} - 3 q^{41} + 10 q^{43} - 8 q^{44} - 4 q^{45} - 8 q^{46} - 4 q^{47} - q^{48} + 3 q^{50} - 2 q^{51} - 18 q^{52} + 14 q^{53} + q^{54} + 10 q^{55} - 2 q^{57} - 28 q^{58} + 8 q^{59} + 6 q^{60} + 8 q^{61} - 38 q^{62} - 17 q^{64} - 4 q^{65} - 10 q^{66} + 12 q^{67} - 26 q^{68} - 10 q^{69} - 32 q^{71} + 3 q^{72} - 4 q^{73} + 20 q^{74} + 5 q^{75} - 12 q^{76} + 2 q^{78} - 20 q^{79} - 18 q^{80} + 3 q^{81} - q^{82} + 14 q^{83} - 22 q^{85} + 6 q^{86} - 6 q^{87} - 8 q^{88} - 14 q^{89} - 6 q^{90} + 2 q^{93} - 18 q^{94} - 12 q^{95} + 7 q^{96} + 12 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + 3 * q^4 - 4 * q^5 + q^6 + 3 * q^8 + 3 * q^9 - 6 * q^10 - 4 * q^11 + 3 * q^12 - 8 * q^13 - 4 * q^15 - q^16 - 2 * q^17 + q^18 - 2 * q^19 + 6 * q^20 - 10 * q^22 - 10 * q^23 + 3 * q^24 + 5 * q^25 + 2 * q^26 + 3 * q^27 - 6 * q^29 - 6 * q^30 + 2 * q^31 + 7 * q^32 - 4 * q^33 + 3 * q^36 + 20 * q^37 + 4 * q^38 - 8 * q^39 + 6 * q^40 - 3 * q^41 + 10 * q^43 - 8 * q^44 - 4 * q^45 - 8 * q^46 - 4 * q^47 - q^48 + 3 * q^50 - 2 * q^51 - 18 * q^52 + 14 * q^53 + q^54 + 10 * q^55 - 2 * q^57 - 28 * q^58 + 8 * q^59 + 6 * q^60 + 8 * q^61 - 38 * q^62 - 17 * q^64 - 4 * q^65 - 10 * q^66 + 12 * q^67 - 26 * q^68 - 10 * q^69 - 32 * q^71 + 3 * q^72 - 4 * q^73 + 20 * q^74 + 5 * q^75 - 12 * q^76 + 2 * q^78 - 20 * q^79 - 18 * q^80 + 3 * q^81 - q^82 + 14 * q^83 - 22 * q^85 + 6 * q^86 - 6 * q^87 - 8 * q^88 - 14 * q^89 - 6 * q^90 + 2 * q^93 - 18 * q^94 - 12 * q^95 + 7 * q^96 + 12 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.470683	0.332823	0.166412	−	0.986056i	$-0.446782\pi$
$2$	0.470683	0.332823	0.166412	+	0.986056i	$0.446782\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.77846	−0.889229
$5$	−4.24914	−1.90027	−0.950137	−	0.311834i	$-0.899057\pi$
$5$	−4.24914	−1.90027	−0.950137	+	0.311834i	$0.899057\pi$
$6$	0.470683	0.192156
$7$	0	0
$7$	0	0
$8$	−1.77846	−0.628780
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−1.47068	−0.443428	−0.221714	−	0.975112i	$-0.571165\pi$
$11$	−1.47068	−0.443428	−0.221714	+	0.975112i	$0.571165\pi$
$12$	−1.77846	−0.513396
$13$	0.249141	0.0690992	0.0345496	−	0.999403i	$-0.489000\pi$
$13$	0.249141	0.0690992	0.0345496	+	0.999403i	$0.489000\pi$
$14$	0	0
$15$	−4.24914	−1.09712
$16$	2.71982	0.679956
$17$	5.02760	1.21937	0.609686	−	0.792643i	$-0.291296\pi$
$17$	5.02760	1.21937	0.609686	+	0.792643i	$0.291296\pi$
$18$	0.470683	0.110941
$19$	2.24914	0.515988	0.257994	−	0.966146i	$-0.416938\pi$
$19$	2.24914	0.515988	0.257994	+	0.966146i	$0.416938\pi$
$20$	7.55691	1.68978
$21$	0	0
$22$	−0.692226	−0.147583
$23$	−6.24914	−1.30304	−0.651518	−	0.758633i	$-0.725867\pi$
$23$	−6.24914	−1.30304	−0.651518	+	0.758633i	$0.725867\pi$
$24$	−1.77846	−0.363026
$25$	13.0552	2.61104
$26$	0.117266	0.0229978
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.41205	−0.447906	−0.223953	−	0.974600i	$-0.571896\pi$
$29$	−2.41205	−0.447906	−0.223953	+	0.974600i	$0.571896\pi$
$30$	−2.00000	−0.365148
$31$	2.89572	0.520087	0.260044	−	0.965597i	$-0.416263\pi$
$31$	2.89572	0.520087	0.260044	+	0.965597i	$0.416263\pi$
$32$	4.83709	0.855085
$33$	−1.47068	−0.256013
$34$	2.36641	0.405835
$35$	0	0
$36$	−1.77846	−0.296410
$37$	9.71982	1.59793	0.798965	−	0.601378i	$-0.205381\pi$
$37$	9.71982	1.59793	0.798965	+	0.601378i	$0.205381\pi$
$38$	1.05863	0.171733
$39$	0.249141	0.0398944
$40$	7.55691	1.19485
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−10.8337	−1.65212	−0.826058	−	0.563585i	$-0.809422\pi$
$43$	−10.8337	−1.65212	−0.826058	+	0.563585i	$0.809422\pi$
$44$	2.61555	0.394309
$45$	−4.24914	−0.633424
$46$	−2.94137	−0.433681
$47$	4.08623	0.596038	0.298019	−	0.954560i	$-0.403674\pi$
$47$	4.08623	0.596038	0.298019	+	0.954560i	$0.403674\pi$
$48$	2.71982	0.392573
$49$	0	0
$50$	6.14486	0.869015
$51$	5.02760	0.704004
$52$	−0.443086	−0.0614449
$53$	−1.43965	−0.197751	−0.0988754	−	0.995100i	$-0.531525\pi$
$53$	−1.43965	−0.197751	−0.0988754	+	0.995100i	$0.531525\pi$
$54$	0.470683	0.0640519
$55$	6.24914	0.842634
$56$	0	0
$57$	2.24914	0.297906
$58$	−1.13531	−0.149074
$59$	−2.61555	−0.340515	−0.170258	−	0.985400i	$-0.554460\pi$
$59$	−2.61555	−0.340515	−0.170258	+	0.985400i	$0.554460\pi$
$60$	7.55691	0.975593
$61$	5.71982	0.732348	0.366174	−	0.930546i	$-0.380667\pi$
$61$	5.71982	0.732348	0.366174	+	0.930546i	$0.380667\pi$
$62$	1.36297	0.173097
$63$	0	0
$64$	−3.16291	−0.395364
$65$	−1.05863	−0.131307
$66$	−0.692226	−0.0852072
$67$	15.9379	1.94713	0.973564	−	0.228415i	$-0.0733542\pi$
$67$	15.9379	1.94713	0.973564	+	0.228415i	$0.0733542\pi$
$68$	−8.94137	−1.08430
$69$	−6.24914	−0.752308
$70$	0	0
$71$	−10.5293	−1.24960	−0.624800	−	0.780785i	$-0.714819\pi$
$71$	−10.5293	−1.24960	−0.624800	+	0.780785i	$0.714819\pi$
$72$	−1.77846	−0.209593
$73$	−9.39400	−1.09949	−0.549743	−	0.835334i	$-0.685274\pi$
$73$	−9.39400	−1.09949	−0.549743	+	0.835334i	$0.685274\pi$
$74$	4.57496	0.531828
$75$	13.0552	1.50748
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0.117266	0.0132778
$79$	−0.560352	−0.0630445	−0.0315223	−	0.999503i	$-0.510036\pi$
$79$	−0.560352	−0.0630445	−0.0315223	+	0.999503i	$0.510036\pi$
$80$	−11.5569	−1.29210
$81$	1.00000	0.111111
$82$	−0.470683	−0.0519783
$83$	−3.80605	−0.417769	−0.208884	−	0.977940i	$-0.566983\pi$
$83$	−3.80605	−0.417769	−0.208884	+	0.977940i	$0.566983\pi$
$84$	0	0
$85$	−21.3630	−2.31714
$86$	−5.09922	−0.549863
$87$	−2.41205	−0.258599
$88$	2.61555	0.278818
$89$	−4.11727	−0.436429	−0.218215	−	0.975901i	$-0.570023\pi$
$89$	−4.11727	−0.436429	−0.218215	+	0.975901i	$0.570023\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	11.1138	1.15870
$93$	2.89572	0.300273
$94$	1.92332	0.198375
$95$	−9.55691	−0.980519
$96$	4.83709	0.493683
$97$	7.19051	0.730085	0.365043	−	0.930991i	$-0.381054\pi$
$97$	7.19051	0.730085	0.365043	+	0.930991i	$0.381054\pi$
$98$	0	0
$99$	−1.47068	−0.147809
$100$	−23.2181	−2.32181
$101$	11.1449	1.10896	0.554478	−	0.832199i	$-0.312918\pi$
$101$	11.1449	1.10896	0.554478	+	0.832199i	$0.312918\pi$
$102$	2.36641	0.234309
$103$	−13.2767	−1.30820	−0.654098	−	0.756410i	$-0.726952\pi$
$103$	−13.2767	−1.30820	−0.654098	+	0.756410i	$0.726952\pi$
$104$	−0.443086	−0.0434481
$105$	0	0
$106$	−0.677618	−0.0658161
$107$	9.32238	0.901229	0.450614	−	0.892719i	$-0.351205\pi$
$107$	9.32238	0.901229	0.450614	+	0.892719i	$0.351205\pi$
$108$	−1.77846	−0.171132
$109$	−0.249141	−0.0238633	−0.0119317	−	0.999929i	$-0.503798\pi$
$109$	−0.249141	−0.0238633	−0.0119317	+	0.999929i	$0.503798\pi$
$110$	2.94137	0.280448
$111$	9.71982	0.922565
$112$	0	0
$113$	9.24570	0.869763	0.434881	−	0.900488i	$-0.356790\pi$
$113$	9.24570	0.869763	0.434881	+	0.900488i	$0.356790\pi$
$114$	1.05863	0.0991501
$115$	26.5535	2.47612
$116$	4.28973	0.398291
$117$	0.249141	0.0230331
$118$	−1.23109	−0.113331
$119$	0	0
$120$	7.55691	0.689849
$121$	−8.83709	−0.803372
$122$	2.69223	0.243743
$123$	−1.00000	−0.0901670
$124$	−5.14992	−0.462476
$125$	−34.2277	−3.06141
$126$	0	0
$127$	−0.824101	−0.0731271	−0.0365635	−	0.999331i	$-0.511641\pi$
$127$	−0.824101	−0.0731271	−0.0365635	+	0.999331i	$0.511641\pi$
$128$	−11.1629	−0.986671
$129$	−10.8337	−0.953850
$130$	−0.498281	−0.0437021
$131$	−8.13187	−0.710485	−0.355243	−	0.934774i	$-0.615602\pi$
$131$	−8.13187	−0.710485	−0.355243	+	0.934774i	$0.615602\pi$
$132$	2.61555	0.227654
$133$	0	0
$134$	7.50172	0.648050
$135$	−4.24914	−0.365708
$136$	−8.94137	−0.766716
$137$	−12.2035	−1.04262	−0.521308	−	0.853369i	$-0.674556\pi$
$137$	−12.2035	−1.04262	−0.521308	+	0.853369i	$0.674556\pi$
$138$	−2.94137	−0.250386
$139$	−14.2897	−1.21204	−0.606019	−	0.795450i	$-0.707235\pi$
$139$	−14.2897	−1.21204	−0.606019	+	0.795450i	$0.707235\pi$
$140$	0	0
$141$	4.08623	0.344123
$142$	−4.95597	−0.415896
$143$	−0.366407	−0.0306405
$144$	2.71982	0.226652
$145$	10.2491	0.851145
$146$	−4.42160	−0.365934
$147$	0	0
$148$	−17.2863	−1.42092
$149$	−22.1104	−1.81135	−0.905677	−	0.423969i	$-0.860637\pi$
$149$	−22.1104	−1.81135	−0.905677	+	0.423969i	$0.860637\pi$
$150$	6.14486	0.501726
$151$	10.1173	0.823331	0.411666	−	0.911335i	$-0.364947\pi$
$151$	10.1173	0.823331	0.411666	+	0.911335i	$0.364947\pi$
$152$	−4.00000	−0.324443
$153$	5.02760	0.406457
$154$	0	0
$155$	−12.3043	−0.988308
$156$	−0.443086	−0.0354753
$157$	−18.7620	−1.49737	−0.748686	−	0.662924i	$-0.769315\pi$
$157$	−18.7620	−1.49737	−0.748686	+	0.662924i	$0.769315\pi$
$158$	−0.263748	−0.0209827
$159$	−1.43965	−0.114172
$160$	−20.5535	−1.62489
$161$	0	0
$162$	0.470683	0.0369804
$163$	8.54392	0.669212	0.334606	−	0.942358i	$-0.391397\pi$
$163$	8.54392	0.669212	0.334606	+	0.942358i	$0.391397\pi$
$164$	1.77846	0.138874
$165$	6.24914	0.486495
$166$	−1.79145	−0.139043
$167$	−5.05863	−0.391449	−0.195724	−	0.980659i	$-0.562706\pi$
$167$	−5.05863	−0.391449	−0.195724	+	0.980659i	$0.562706\pi$
$168$	0	0
$169$	−12.9379	−0.995225
$170$	−10.0552	−0.771198
$171$	2.24914	0.171996
$172$	19.2672	1.46911
$173$	−18.8647	−1.43426	−0.717128	−	0.696942i	$-0.754544\pi$
$173$	−18.8647	−1.43426	−0.717128	+	0.696942i	$0.754544\pi$
$174$	−1.13531	−0.0860678
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−2.61555	−0.196597
$178$	−1.93793	−0.145254
$179$	−2.85514	−0.213403	−0.106701	−	0.994291i	$-0.534029\pi$
$179$	−2.85514	−0.213403	−0.106701	+	0.994291i	$0.534029\pi$
$180$	7.55691	0.563259
$181$	−13.4396	−0.998961	−0.499481	−	0.866325i	$-0.666476\pi$
$181$	−13.4396	−0.998961	−0.499481	+	0.866325i	$0.666476\pi$
$182$	0	0
$183$	5.71982	0.422822
$184$	11.1138	0.819322
$185$	−41.3009	−3.03650
$186$	1.36297	0.0999377
$187$	−7.39400	−0.540703
$188$	−7.26719	−0.530014
$189$	0	0
$190$	−4.49828	−0.326340
$191$	13.4948	0.976453	0.488226	−	0.872717i	$-0.337644\pi$
$191$	13.4948	0.976453	0.488226	+	0.872717i	$0.337644\pi$
$192$	−3.16291	−0.228263
$193$	−8.01461	−0.576904	−0.288452	−	0.957494i	$-0.593141\pi$
$193$	−8.01461	−0.576904	−0.288452	+	0.957494i	$0.593141\pi$
$194$	3.38445	0.242990
$195$	−1.05863	−0.0758103
$196$	0	0
$197$	13.9785	0.995928	0.497964	−	0.867198i	$-0.334081\pi$
$197$	13.9785	0.995928	0.497964	+	0.867198i	$0.334081\pi$
$198$	−0.692226	−0.0491944
$199$	4.86469	0.344849	0.172424	−	0.985023i	$-0.444840\pi$
$199$	4.86469	0.344849	0.172424	+	0.985023i	$0.444840\pi$
$200$	−23.2181	−1.64177
$201$	15.9379	1.12417
$202$	5.24570	0.369086
$203$	0	0
$204$	−8.94137	−0.626021
$205$	4.24914	0.296773
$206$	−6.24914	−0.435398
$207$	−6.24914	−0.434345
$208$	0.677618	0.0469844
$209$	−3.30777	−0.228803
$210$	0	0
$211$	−10.6155	−0.730804	−0.365402	−	0.930850i	$-0.619069\pi$
$211$	−10.6155	−0.730804	−0.365402	+	0.930850i	$0.619069\pi$
$212$	2.56035	0.175846
$213$	−10.5293	−0.721457
$214$	4.38789	0.299950
$215$	46.0337	3.13947
$216$	−1.77846	−0.121009
$217$	0	0
$218$	−0.117266	−0.00794228
$219$	−9.39400	−0.634788
$220$	−11.1138	−0.749294
$221$	1.25258	0.0842575
$222$	4.57496	0.307051
$223$	17.9931	1.20491	0.602454	−	0.798153i	$-0.294190\pi$
$223$	17.9931	1.20491	0.602454	+	0.798153i	$0.294190\pi$
$224$	0	0
$225$	13.0552	0.870346
$226$	4.35180	0.289477
$227$	0.910331	0.0604208	0.0302104	−	0.999544i	$-0.490382\pi$
$227$	0.910331	0.0604208	0.0302104	+	0.999544i	$0.490382\pi$
$228$	−4.00000	−0.264906
$229$	7.42504	0.490660	0.245330	−	0.969440i	$-0.421104\pi$
$229$	7.42504	0.490660	0.245330	+	0.969440i	$0.421104\pi$
$230$	12.4983	0.824112
$231$	0	0
$232$	4.28973	0.281634
$233$	14.7620	0.967093	0.483546	−	0.875319i	$-0.339348\pi$
$233$	14.7620	0.967093	0.483546	+	0.875319i	$0.339348\pi$
$234$	0.117266	0.00766594
$235$	−17.3630	−1.13264
$236$	4.65164	0.302796
$237$	−0.560352	−0.0363988
$238$	0	0
$239$	−1.38445	−0.0895528	−0.0447764	−	0.998997i	$-0.514258\pi$
$239$	−1.38445	−0.0895528	−0.0447764	+	0.998997i	$0.514258\pi$
$240$	−11.5569	−0.745996
$241$	5.95436	0.383554	0.191777	−	0.981439i	$-0.438575\pi$
$241$	5.95436	0.383554	0.191777	+	0.981439i	$0.438575\pi$
$242$	−4.15947	−0.267381
$243$	1.00000	0.0641500
$244$	−10.1725	−0.651225
$245$	0	0
$246$	−0.470683	−0.0300097
$247$	0.560352	0.0356543
$248$	−5.14992	−0.327020
$249$	−3.80605	−0.241199
$250$	−16.1104	−1.01891
$251$	−2.70683	−0.170854	−0.0854269	−	0.996344i	$-0.527225\pi$
$251$	−2.70683	−0.170854	−0.0854269	+	0.996344i	$0.527225\pi$
$252$	0	0
$253$	9.19051	0.577802
$254$	−0.387890	−0.0243384
$255$	−21.3630	−1.33780
$256$	1.07162	0.0669764
$257$	0.793065	0.0494700	0.0247350	−	0.999694i	$-0.492126\pi$
$257$	0.793065	0.0494700	0.0247350	+	0.999694i	$0.492126\pi$
$258$	−5.09922	−0.317464
$259$	0	0
$260$	1.88273	0.116762
$261$	−2.41205	−0.149302
$262$	−3.82754	−0.236466
$263$	−15.5879	−0.961194	−0.480597	−	0.876942i	$-0.659580\pi$
$263$	−15.5879	−0.961194	−0.480597	+	0.876942i	$0.659580\pi$
$264$	2.61555	0.160976
$265$	6.11727	0.375781
$266$	0	0
$267$	−4.11727	−0.251973
$268$	−28.3449	−1.73144
$269$	−3.05863	−0.186488	−0.0932441	−	0.995643i	$-0.529724\pi$
$269$	−3.05863	−0.186488	−0.0932441	+	0.995643i	$0.529724\pi$
$270$	−2.00000	−0.121716
$271$	−12.2802	−0.745968	−0.372984	−	0.927838i	$-0.621665\pi$
$271$	−12.2802	−0.745968	−0.372984	+	0.927838i	$0.621665\pi$
$272$	13.6742	0.829119
$273$	0	0
$274$	−5.74398	−0.347007
$275$	−19.2001	−1.15781
$276$	11.1138	0.668974
$277$	−8.56990	−0.514916	−0.257458	−	0.966290i	$-0.582885\pi$
$277$	−8.56990	−0.514916	−0.257458	+	0.966290i	$0.582885\pi$
$278$	−6.72594	−0.403395
$279$	2.89572	0.173362
$280$	0	0
$281$	−14.5845	−0.870039	−0.435020	−	0.900421i	$-0.643259\pi$
$281$	−14.5845	−0.870039	−0.435020	+	0.900421i	$0.643259\pi$
$282$	1.92332	0.114532
$283$	20.9509	1.24540	0.622701	−	0.782460i	$-0.286035\pi$
$283$	20.9509	1.24540	0.622701	+	0.782460i	$0.286035\pi$
$284$	18.7259	1.11118
$285$	−9.55691	−0.566103
$286$	−0.172462	−0.0101979
$287$	0	0
$288$	4.83709	0.285028
$289$	8.27674	0.486867
$290$	4.82410	0.283281
$291$	7.19051	0.421515
$292$	16.7068	0.977694
$293$	−31.4086	−1.83491	−0.917455	−	0.397839i	$-0.869760\pi$
$293$	−31.4086	−1.83491	−0.917455	+	0.397839i	$0.869760\pi$
$294$	0	0
$295$	11.1138	0.647072
$296$	−17.2863	−1.00475
$297$	−1.47068	−0.0853377
$298$	−10.4070	−0.602861
$299$	−1.55691	−0.0900387
$300$	−23.2181	−1.34050
$301$	0	0
$302$	4.76203	0.274024
$303$	11.1449	0.640256
$304$	6.11727	0.350849
$305$	−24.3043	−1.39166
$306$	2.36641	0.135278
$307$	−8.54392	−0.487628	−0.243814	−	0.969822i	$-0.578399\pi$
$307$	−8.54392	−0.487628	−0.243814	+	0.969822i	$0.578399\pi$
$308$	0	0
$309$	−13.2767	−0.755287
$310$	−5.79145	−0.328932
$311$	5.49484	0.311584	0.155792	−	0.987790i	$-0.450207\pi$
$311$	5.49484	0.311584	0.155792	+	0.987790i	$0.450207\pi$
$312$	−0.443086	−0.0250848
$313$	−5.48024	−0.309761	−0.154881	−	0.987933i	$-0.549499\pi$
$313$	−5.48024	−0.309761	−0.154881	+	0.987933i	$0.549499\pi$
$314$	−8.83098	−0.498361
$315$	0	0
$316$	0.996562	0.0560610
$317$	−11.3173	−0.635644	−0.317822	−	0.948150i	$-0.602951\pi$
$317$	−11.3173	−0.635644	−0.317822	+	0.948150i	$0.602951\pi$
$318$	−0.677618	−0.0379990
$319$	3.54736	0.198614
$320$	13.4396	0.751299
$321$	9.32238	0.520325
$322$	0	0
$323$	11.3078	0.629181
$324$	−1.77846	−0.0988032
$325$	3.25258	0.180421
$326$	4.02148	0.222729
$327$	−0.249141	−0.0137775
$328$	1.77846	0.0981989
$329$	0	0
$330$	2.94137	0.161917
$331$	33.8613	1.86118	0.930591	−	0.366060i	$-0.119293\pi$
$331$	33.8613	1.86118	0.930591	+	0.366060i	$0.119293\pi$
$332$	6.76891	0.371492
$333$	9.71982	0.532643
$334$	−2.38101	−0.130283
$335$	−67.7225	−3.70008
$336$	0	0
$337$	−6.77846	−0.369246	−0.184623	−	0.982809i	$-0.559106\pi$
$337$	−6.77846	−0.369246	−0.184623	+	0.982809i	$0.559106\pi$
$338$	−6.08967	−0.331234
$339$	9.24570	0.502158
$340$	37.9931	2.06047
$341$	−4.25869	−0.230621
$342$	1.05863	0.0572443
$343$	0	0
$344$	19.2672	1.03882
$345$	26.5535	1.42959
$346$	−8.87930	−0.477354
$347$	19.2001	1.03071	0.515357	−	0.856976i	$-0.327659\pi$
$347$	19.2001	1.03071	0.515357	+	0.856976i	$0.327659\pi$
$348$	4.28973	0.229954
$349$	6.54392	0.350288	0.175144	−	0.984543i	$-0.443961\pi$
$349$	6.54392	0.350288	0.175144	+	0.984543i	$0.443961\pi$
$350$	0	0
$351$	0.249141	0.0132981
$352$	−7.11383	−0.379168
$353$	15.9931	0.851228	0.425614	−	0.904905i	$-0.360058\pi$
$353$	15.9931	0.851228	0.425614	+	0.904905i	$0.360058\pi$
$354$	−1.23109	−0.0654320
$355$	44.7405	2.37458
$356$	7.32238	0.388085
$357$	0	0
$358$	−1.34387	−0.0710255
$359$	−12.1319	−0.640296	−0.320148	−	0.947368i	$-0.603733\pi$
$359$	−12.1319	−0.640296	−0.320148	+	0.947368i	$0.603733\pi$
$360$	7.55691	0.398284
$361$	−13.9414	−0.733756
$362$	−6.32582	−0.332478
$363$	−8.83709	−0.463827
$364$	0	0
$365$	39.9164	2.08932
$366$	2.69223	0.140725
$367$	−6.39744	−0.333944	−0.166972	−	0.985962i	$-0.553399\pi$
$367$	−6.39744	−0.333944	−0.166972	+	0.985962i	$0.553399\pi$
$368$	−16.9966	−0.886007
$369$	−1.00000	−0.0520579
$370$	−19.4396	−1.01062
$371$	0	0
$372$	−5.14992	−0.267011
$373$	−7.54049	−0.390432	−0.195216	−	0.980760i	$-0.562541\pi$
$373$	−7.54049	−0.390432	−0.195216	+	0.980760i	$0.562541\pi$
$374$	−3.48024	−0.179959
$375$	−34.2277	−1.76751
$376$	−7.26719	−0.374777
$377$	−0.600939	−0.0309500
$378$	0	0
$379$	−0.0620710	−0.00318837	−0.00159419	−	0.999999i	$-0.500507\pi$
$379$	−0.0620710	−0.00318837	−0.00159419	+	0.999999i	$0.500507\pi$
$380$	16.9966	0.871905
$381$	−0.824101	−0.0422199
$382$	6.35180	0.324986
$383$	−23.9329	−1.22291	−0.611456	−	0.791278i	$-0.709416\pi$
$383$	−23.9329	−1.22291	−0.611456	+	0.791278i	$0.709416\pi$
$384$	−11.1629	−0.569655
$385$	0	0
$386$	−3.77234	−0.192007
$387$	−10.8337	−0.550706
$388$	−12.7880	−0.649213
$389$	−23.8613	−1.20981	−0.604907	−	0.796296i	$-0.706790\pi$
$389$	−23.8613	−1.20981	−0.604907	+	0.796296i	$0.706790\pi$
$390$	−0.498281	−0.0252314
$391$	−31.4182	−1.58888
$392$	0	0
$393$	−8.13187	−0.410199
$394$	6.57946	0.331468
$395$	2.38101	0.119802
$396$	2.61555	0.131436
$397$	−14.6302	−0.734266	−0.367133	−	0.930168i	$-0.619661\pi$
$397$	−14.6302	−0.734266	−0.367133	+	0.930168i	$0.619661\pi$
$398$	2.28973	0.114774
$399$	0	0
$400$	35.5078	1.77539
$401$	−11.3224	−0.565413	−0.282706	−	0.959206i	$-0.591232\pi$
$401$	−11.3224	−0.565413	−0.282706	+	0.959206i	$0.591232\pi$
$402$	7.50172	0.374152
$403$	0.721442	0.0359376
$404$	−19.8207	−0.986115
$405$	−4.24914	−0.211141
$406$	0	0
$407$	−14.2948	−0.708566
$408$	−8.94137	−0.442664
$409$	33.8923	1.67587	0.837933	−	0.545773i	$-0.183764\pi$
$409$	33.8923	1.67587	0.837933	+	0.545773i	$0.183764\pi$
$410$	2.00000	0.0987730
$411$	−12.2035	−0.601954
$412$	23.6121	1.16329
$413$	0	0
$414$	−2.94137	−0.144560
$415$	16.1725	0.793875
$416$	1.20512	0.0590856
$417$	−14.2897	−0.699771
$418$	−1.55691	−0.0761512
$419$	1.98539	0.0969928	0.0484964	−	0.998823i	$-0.484557\pi$
$419$	1.98539	0.0969928	0.0484964	+	0.998823i	$0.484557\pi$
$420$	0	0
$421$	−5.67418	−0.276543	−0.138271	−	0.990394i	$-0.544155\pi$
$421$	−5.67418	−0.276543	−0.138271	+	0.990394i	$0.544155\pi$
$422$	−4.99656	−0.243229
$423$	4.08623	0.198679
$424$	2.56035	0.124342
$425$	65.6363	3.18383
$426$	−4.95597	−0.240118
$427$	0	0
$428$	−16.5795	−0.801398
$429$	−0.366407	−0.0176903
$430$	21.6673	1.04489
$431$	1.75086	0.0843359	0.0421680	−	0.999111i	$-0.486574\pi$
$431$	1.75086	0.0843359	0.0421680	+	0.999111i	$0.486574\pi$
$432$	2.71982	0.130858
$433$	34.0647	1.63705	0.818524	−	0.574473i	$-0.194793\pi$
$433$	34.0647	1.63705	0.818524	+	0.574473i	$0.194793\pi$
$434$	0	0
$435$	10.2491	0.491409
$436$	0.443086	0.0212200
$437$	−14.0552	−0.672351
$438$	−4.42160	−0.211272
$439$	4.32582	0.206460	0.103230	−	0.994658i	$-0.467082\pi$
$439$	4.32582	0.206460	0.103230	+	0.994658i	$0.467082\pi$
$440$	−11.1138	−0.529831
$441$	0	0
$442$	0.589568	0.0280429
$443$	36.9966	1.75776	0.878880	−	0.477043i	$-0.158291\pi$
$443$	36.9966	1.75776	0.878880	+	0.477043i	$0.158291\pi$
$444$	−17.2863	−0.820371
$445$	17.4948	0.829335
$446$	8.46907	0.401022
$447$	−22.1104	−1.04579
$448$	0	0
$449$	−15.9448	−0.752482	−0.376241	−	0.926522i	$-0.622784\pi$
$449$	−15.9448	−0.752482	−0.376241	+	0.926522i	$0.622784\pi$
$450$	6.14486	0.289672
$451$	1.47068	0.0692518
$452$	−16.4431	−0.773418
$453$	10.1173	0.475351
$454$	0.428478	0.0201095
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−29.4396	−1.37713	−0.688564	−	0.725175i	$-0.741759\pi$
$457$	−29.4396	−1.37713	−0.688564	+	0.725175i	$0.741759\pi$
$458$	3.49484	0.163303
$459$	5.02760	0.234668
$460$	−47.2242	−2.20184
$461$	−21.8759	−1.01886	−0.509430	−	0.860512i	$-0.670144\pi$
$461$	−21.8759	−1.01886	−0.509430	+	0.860512i	$0.670144\pi$
$462$	0	0
$463$	−18.6561	−0.867024	−0.433512	−	0.901148i	$-0.642726\pi$
$463$	−18.6561	−0.867024	−0.433512	+	0.901148i	$0.642726\pi$
$464$	−6.56035	−0.304557
$465$	−12.3043	−0.570600
$466$	6.94824	0.321871
$467$	−8.67074	−0.401234	−0.200617	−	0.979670i	$-0.564295\pi$
$467$	−8.67074	−0.401234	−0.200617	+	0.979670i	$0.564295\pi$
$468$	−0.443086	−0.0204816
$469$	0	0
$470$	−8.17246	−0.376968
$471$	−18.7620	−0.864509
$472$	4.65164	0.214109
$473$	15.9329	0.732594
$474$	−0.263748	−0.0121144
$475$	29.3630	1.34727
$476$	0	0
$477$	−1.43965	−0.0659169
$478$	−0.651639	−0.0298053
$479$	20.4312	0.933523	0.466762	−	0.884383i	$-0.345421\pi$
$479$	20.4312	0.933523	0.466762	+	0.884383i	$0.345421\pi$
$480$	−20.5535	−0.938134
$481$	2.42160	0.110416
$482$	2.80262	0.127656
$483$	0	0
$484$	15.7164	0.714381
$485$	−30.5535	−1.38736
$486$	0.470683	0.0213506
$487$	−7.00611	−0.317477	−0.158739	−	0.987321i	$-0.550743\pi$
$487$	−7.00611	−0.317477	−0.158739	+	0.987321i	$0.550743\pi$
$488$	−10.1725	−0.460486
$489$	8.54392	0.386370
$490$	0	0
$491$	−15.5715	−0.702733	−0.351366	−	0.936238i	$-0.614283\pi$
$491$	−15.5715	−0.702733	−0.351366	+	0.936238i	$0.614283\pi$
$492$	1.77846	0.0801790
$493$	−12.1268	−0.546164
$494$	0.263748	0.0118666
$495$	6.24914	0.280878
$496$	7.87586	0.353636
$497$	0	0
$498$	−1.79145	−0.0802767
$499$	−1.61899	−0.0724757	−0.0362379	−	0.999343i	$-0.511537\pi$
$499$	−1.61899	−0.0724757	−0.0362379	+	0.999343i	$0.511537\pi$
$500$	60.8724	2.72230
$501$	−5.05863	−0.226003
$502$	−1.27406	−0.0568642
$503$	39.3725	1.75553	0.877767	−	0.479088i	$-0.159032\pi$
$503$	39.3725	1.75553	0.877767	+	0.479088i	$0.159032\pi$
$504$	0	0
$505$	−47.3561	−2.10732
$506$	4.32582	0.192306
$507$	−12.9379	−0.574594
$508$	1.46563	0.0650267
$509$	8.63971	0.382948	0.191474	−	0.981498i	$-0.438673\pi$
$509$	8.63971	0.382948	0.191474	+	0.981498i	$0.438673\pi$
$510$	−10.0552	−0.445252
$511$	0	0
$512$	22.8302	1.00896
$513$	2.24914	0.0993020
$514$	0.373283	0.0164648
$515$	56.4147	2.48593
$516$	19.2672	0.848191
$517$	−6.00955	−0.264300
$518$	0	0
$519$	−18.8647	−0.828068
$520$	1.88273	0.0825633
$521$	−30.7862	−1.34877	−0.674384	−	0.738381i	$-0.735591\pi$
$521$	−30.7862	−1.34877	−0.674384	+	0.738381i	$0.735591\pi$
$522$	−1.13531	−0.0496913
$523$	−17.4036	−0.761004	−0.380502	−	0.924780i	$-0.624249\pi$
$523$	−17.4036	−0.761004	−0.380502	+	0.924780i	$0.624249\pi$
$524$	14.4622	0.631784
$525$	0	0
$526$	−7.33699	−0.319908
$527$	14.5585	0.634180
$528$	−4.00000	−0.174078
$529$	16.0518	0.697902
$530$	2.87930	0.125069
$531$	−2.61555	−0.113505
$532$	0	0
$533$	−0.249141	−0.0107915
$534$	−1.93793	−0.0838624
$535$	−39.6121	−1.71258
$536$	−28.3449	−1.22431
$537$	−2.85514	−0.123208
$538$	−1.43965	−0.0620676
$539$	0	0
$540$	7.55691	0.325198
$541$	−9.87586	−0.424596	−0.212298	−	0.977205i	$-0.568095\pi$
$541$	−9.87586	−0.424596	−0.212298	+	0.977205i	$0.568095\pi$
$542$	−5.78008	−0.248275
$543$	−13.4396	−0.576750
$544$	24.3189	1.04267
$545$	1.05863	0.0453469
$546$	0	0
$547$	−18.1465	−0.775888	−0.387944	−	0.921683i	$-0.626815\pi$
$547$	−18.1465	−0.775888	−0.387944	+	0.921683i	$0.626815\pi$
$548$	21.7034	0.927123
$549$	5.71982	0.244116
$550$	−9.03715	−0.385345
$551$	−5.42504	−0.231114
$552$	11.1138	0.473036
$553$	0	0
$554$	−4.03371	−0.171376
$555$	−41.3009	−1.75313
$556$	25.4137	1.07778
$557$	37.3725	1.58352	0.791762	−	0.610829i	$-0.209164\pi$
$557$	37.3725	1.58352	0.791762	+	0.610829i	$0.209164\pi$
$558$	1.36297	0.0576991
$559$	−2.69910	−0.114160
$560$	0	0
$561$	−7.39400	−0.312175
$562$	−6.86469	−0.289569
$563$	−22.8742	−0.964034	−0.482017	−	0.876162i	$-0.660096\pi$
$563$	−22.8742	−0.964034	−0.482017	+	0.876162i	$0.660096\pi$
$564$	−7.26719	−0.306004
$565$	−39.2863	−1.65279
$566$	9.86125	0.414499
$567$	0	0
$568$	18.7259	0.785723
$569$	34.4147	1.44274	0.721370	−	0.692550i	$-0.243513\pi$
$569$	34.4147	1.44274	0.721370	+	0.692550i	$0.243513\pi$
$570$	−4.49828	−0.188412
$571$	36.3449	1.52099	0.760494	−	0.649345i	$-0.224957\pi$
$571$	36.3449	1.52099	0.760494	+	0.649345i	$0.224957\pi$
$572$	0.651639	0.0272464
$573$	13.4948	0.563755
$574$	0	0
$575$	−81.5838	−3.40228
$576$	−3.16291	−0.131788
$577$	−26.1725	−1.08957	−0.544787	−	0.838575i	$-0.683389\pi$
$577$	−26.1725	−1.08957	−0.544787	+	0.838575i	$0.683389\pi$
$578$	3.89572	0.162041
$579$	−8.01461	−0.333076
$580$	−18.2277	−0.756862
$581$	0	0
$582$	3.38445	0.140290
$583$	2.11727	0.0876882
$584$	16.7068	0.691334
$585$	−1.05863	−0.0437691
$586$	−14.7835	−0.610701
$587$	17.8156	0.735329	0.367664	−	0.929959i	$-0.380158\pi$
$587$	17.8156	0.735329	0.367664	+	0.929959i	$0.380158\pi$
$588$	0	0
$589$	6.51289	0.268359
$590$	5.23109	0.215361
$591$	13.9785	0.574999
$592$	26.4362	1.08652
$593$	32.7018	1.34290	0.671451	−	0.741049i	$-0.265672\pi$
$593$	32.7018	1.34290	0.671451	+	0.741049i	$0.265672\pi$
$594$	−0.692226	−0.0284024
$595$	0	0
$596$	39.3224	1.61071
$597$	4.86469	0.199098
$598$	−0.732814	−0.0299670
$599$	10.2784	0.419962	0.209981	−	0.977705i	$-0.432660\pi$
$599$	10.2784	0.419962	0.209981	+	0.977705i	$0.432660\pi$
$600$	−23.2181	−0.947875
$601$	−31.8398	−1.29877	−0.649386	−	0.760459i	$-0.724974\pi$
$601$	−31.8398	−1.29877	−0.649386	+	0.760459i	$0.724974\pi$
$602$	0	0
$603$	15.9379	0.649043
$604$	−17.9931	−0.732130
$605$	37.5500	1.52663
$606$	5.24570	0.213092
$607$	−10.2086	−0.414352	−0.207176	−	0.978304i	$-0.566427\pi$
$607$	−10.2086	−0.414352	−0.207176	+	0.978304i	$0.566427\pi$
$608$	10.8793	0.441214
$609$	0	0
$610$	−11.4396	−0.463178
$611$	1.01805	0.0411857
$612$	−8.94137	−0.361433
$613$	−4.94137	−0.199580	−0.0997900	−	0.995009i	$-0.531817\pi$
$613$	−4.94137	−0.199580	−0.0997900	+	0.995009i	$0.531817\pi$
$614$	−4.02148	−0.162294
$615$	4.24914	0.171342
$616$	0	0
$617$	15.2526	0.614046	0.307023	−	0.951702i	$-0.400667\pi$
$617$	15.2526	0.614046	0.307023	+	0.951702i	$0.400667\pi$
$618$	−6.24914	−0.251377
$619$	20.6543	0.830167	0.415084	−	0.909783i	$-0.363752\pi$
$619$	20.6543	0.830167	0.415084	+	0.909783i	$0.363752\pi$
$620$	21.8827	0.878832
$621$	−6.24914	−0.250769
$622$	2.58633	0.103702
$623$	0	0
$624$	0.677618	0.0271264
$625$	80.1621	3.20649
$626$	−2.57946	−0.103096
$627$	−3.30777	−0.132100
$628$	33.3675	1.33151
$629$	48.8674	1.94847
$630$	0	0
$631$	5.83709	0.232371	0.116185	−	0.993228i	$-0.462933\pi$
$631$	5.83709	0.232371	0.116185	+	0.993228i	$0.462933\pi$
$632$	0.996562	0.0396411
$633$	−10.6155	−0.421930
$634$	−5.32688	−0.211557
$635$	3.50172	0.138961
$636$	2.56035	0.101525
$637$	0	0
$638$	1.66968	0.0661034
$639$	−10.5293	−0.416533
$640$	47.4328	1.87494
$641$	42.6328	1.68390	0.841948	−	0.539559i	$-0.181409\pi$
$641$	42.6328	1.68390	0.841948	+	0.539559i	$0.181409\pi$
$642$	4.38789	0.173176
$643$	−30.7259	−1.21171	−0.605856	−	0.795574i	$-0.707169\pi$
$643$	−30.7259	−1.21171	−0.605856	+	0.795574i	$0.707169\pi$
$644$	0	0
$645$	46.0337	1.81258
$646$	5.32238	0.209406
$647$	−18.3595	−0.721788	−0.360894	−	0.932607i	$-0.617528\pi$
$647$	−18.3595	−0.721788	−0.360894	+	0.932607i	$0.617528\pi$
$648$	−1.77846	−0.0698644
$649$	3.84664	0.150994
$650$	1.53093	0.0600482
$651$	0	0
$652$	−15.1950	−0.595082
$653$	−35.5811	−1.39240	−0.696198	−	0.717850i	$-0.745126\pi$
$653$	−35.5811	−1.39240	−0.696198	+	0.717850i	$0.745126\pi$
$654$	−0.117266	−0.00458548
$655$	34.5535	1.35012
$656$	−2.71982	−0.106191
$657$	−9.39400	−0.366495
$658$	0	0
$659$	−6.38101	−0.248569	−0.124285	−	0.992247i	$-0.539664\pi$
$659$	−6.38101	−0.248569	−0.124285	+	0.992247i	$0.539664\pi$
$660$	−11.1138	−0.432605
$661$	−30.9053	−1.20208	−0.601038	−	0.799220i	$-0.705246\pi$
$661$	−30.9053	−1.20208	−0.601038	+	0.799220i	$0.705246\pi$
$662$	15.9379	0.619445
$663$	1.25258	0.0486461
$664$	6.76891	0.262684
$665$	0	0
$666$	4.57496	0.177276
$667$	15.0732	0.583638
$668$	8.99656	0.348087
$669$	17.9931	0.695654
$670$	−31.8759	−1.23147
$671$	−8.41205	−0.324744
$672$	0	0
$673$	36.7259	1.41568	0.707840	−	0.706372i	$-0.249670\pi$
$673$	36.7259	1.41568	0.707840	+	0.706372i	$0.249670\pi$
$674$	−3.19051	−0.122894
$675$	13.0552	0.502495
$676$	23.0096	0.884983
$677$	5.66281	0.217639	0.108820	−	0.994062i	$-0.465293\pi$
$677$	5.66281	0.217639	0.108820	+	0.994062i	$0.465293\pi$
$678$	4.35180	0.167130
$679$	0	0
$680$	37.9931	1.45697
$681$	0.910331	0.0348840
$682$	−2.00450	−0.0767561
$683$	−29.1690	−1.11612	−0.558061	−	0.829800i	$-0.688454\pi$
$683$	−29.1690	−1.11612	−0.558061	+	0.829800i	$0.688454\pi$
$684$	−4.00000	−0.152944
$685$	51.8544	1.98125
$686$	0	0
$687$	7.42504	0.283283
$688$	−29.4656	−1.12337
$689$	−0.358675	−0.0136644
$690$	12.4983	0.475801
$691$	−1.68879	−0.0642445	−0.0321223	−	0.999484i	$-0.510227\pi$
$691$	−1.68879	−0.0642445	−0.0321223	+	0.999484i	$0.510227\pi$
$692$	33.5500	1.27538
$693$	0	0
$694$	9.03715	0.343046
$695$	60.7191	2.30321
$696$	4.28973	0.162602
$697$	−5.02760	−0.190434
$698$	3.08012	0.116584
$699$	14.7620	0.558351
$700$	0	0
$701$	−20.3810	−0.769780	−0.384890	−	0.922962i	$-0.625761\pi$
$701$	−20.3810	−0.769780	−0.384890	+	0.922962i	$0.625761\pi$
$702$	0.117266	0.00442593
$703$	21.8613	0.824513
$704$	4.65164	0.175315
$705$	−17.3630	−0.653927
$706$	7.52770	0.283309
$707$	0	0
$708$	4.65164	0.174819
$709$	−40.8363	−1.53364	−0.766820	−	0.641862i	$-0.778162\pi$
$709$	−40.8363	−1.53364	−0.766820	+	0.641862i	$0.778162\pi$
$710$	21.0586	0.790316
$711$	−0.560352	−0.0210148
$712$	7.32238	0.274418
$713$	−18.0958	−0.677692
$714$	0	0
$715$	1.55691	0.0582253
$716$	5.07774	0.189764
$717$	−1.38445	−0.0517033
$718$	−5.71027	−0.213105
$719$	−9.14486	−0.341046	−0.170523	−	0.985354i	$-0.554546\pi$
$719$	−9.14486	−0.341046	−0.170523	+	0.985354i	$0.554546\pi$
$720$	−11.5569	−0.430701
$721$	0	0
$722$	−6.56197	−0.244211
$723$	5.95436	0.221445
$724$	23.9018	0.888305
$725$	−31.4898	−1.16950
$726$	−4.15947	−0.154372
$727$	2.13875	0.0793218	0.0396609	−	0.999213i	$-0.487372\pi$
$727$	2.13875	0.0793218	0.0396609	+	0.999213i	$0.487372\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	18.7880	0.695375
$731$	−54.4672	−2.01454
$732$	−10.1725	−0.375985
$733$	−15.3388	−0.566552	−0.283276	−	0.959038i	$-0.591421\pi$
$733$	−15.3388	−0.566552	−0.283276	+	0.959038i	$0.591421\pi$
$734$	−3.01117	−0.111144
$735$	0	0
$736$	−30.2277	−1.11421
$737$	−23.4396	−0.863411
$738$	−0.470683	−0.0173261
$739$	−15.8111	−0.581621	−0.290811	−	0.956781i	$-0.593925\pi$
$739$	−15.8111	−0.581621	−0.290811	+	0.956781i	$0.593925\pi$
$740$	73.4519	2.70014
$741$	0.560352	0.0205850
$742$	0	0
$743$	−20.2017	−0.741128	−0.370564	−	0.928807i	$-0.620836\pi$
$743$	−20.2017	−0.741128	−0.370564	+	0.928807i	$0.620836\pi$
$744$	−5.14992	−0.188805
$745$	93.9502	3.44207
$746$	−3.54918	−0.129945
$747$	−3.80605	−0.139256
$748$	13.1499	0.480809
$749$	0	0
$750$	−16.1104	−0.588268
$751$	32.9053	1.20073	0.600365	−	0.799726i	$-0.295022\pi$
$751$	32.9053	1.20073	0.600365	+	0.799726i	$0.295022\pi$
$752$	11.1138	0.405280
$753$	−2.70683	−0.0986425
$754$	−0.282852	−0.0103009
$755$	−42.9897	−1.56455
$756$	0	0
$757$	11.3009	0.410738	0.205369	−	0.978685i	$-0.434161\pi$
$757$	11.3009	0.410738	0.205369	+	0.978685i	$0.434161\pi$
$758$	−0.0292158	−0.00106117
$759$	9.19051	0.333594
$760$	16.9966	0.616530
$761$	2.93449	0.106375	0.0531876	−	0.998585i	$-0.483062\pi$
$761$	2.93449	0.106375	0.0531876	+	0.998585i	$0.483062\pi$
$762$	−0.387890	−0.0140518
$763$	0	0
$764$	−24.0000	−0.868290
$765$	−21.3630	−0.772380
$766$	−11.2648	−0.407014
$767$	−0.651639	−0.0235293
$768$	1.07162	0.0386689
$769$	22.7785	0.821412	0.410706	−	0.911768i	$-0.365282\pi$
$769$	22.7785	0.821412	0.410706	+	0.911768i	$0.365282\pi$
$770$	0	0
$771$	0.793065	0.0285615
$772$	14.2536	0.513000
$773$	−10.5553	−0.379648	−0.189824	−	0.981818i	$-0.560792\pi$
$773$	−10.5553	−0.379648	−0.189824	+	0.981818i	$0.560792\pi$
$774$	−5.09922	−0.183288
$775$	37.8042	1.35797
$776$	−12.7880	−0.459063
$777$	0	0
$778$	−11.2311	−0.402654
$779$	−2.24914	−0.0805838
$780$	1.88273	0.0674127
$781$	15.4853	0.554107
$782$	−14.7880	−0.528818
$783$	−2.41205	−0.0861996
$784$	0	0
$785$	79.7225	2.84542
$786$	−3.82754	−0.136524
$787$	23.0682	0.822292	0.411146	−	0.911570i	$-0.365129\pi$
$787$	23.0682	0.822292	0.411146	+	0.911570i	$0.365129\pi$
$788$	−24.8602	−0.885608
$789$	−15.5879	−0.554946
$790$	1.12070	0.0398729
$791$	0	0
$792$	2.61555	0.0929394
$793$	1.42504	0.0506047
$794$	−6.88617	−0.244381
$795$	6.11727	0.216957
$796$	−8.65164	−0.306649
$797$	−0.0360915	−0.00127843	−0.000639213	−	1.00000i	$-0.500203\pi$
$797$	−0.0360915	−0.00127843	−0.000639213		1.00000i	$0.500203\pi$
$798$	0	0
$799$	20.5439	0.726792
$800$	63.1492	2.23266
$801$	−4.11727	−0.145476
$802$	−5.32926	−0.188183
$803$	13.8156	0.487542
$804$	−28.3449	−0.999648
$805$	0	0
$806$	0.339571	0.0119609
$807$	−3.05863	−0.107669
$808$	−19.8207	−0.697288
$809$	33.1430	1.16525	0.582624	−	0.812742i	$-0.302026\pi$
$809$	33.1430	1.16525	0.582624	+	0.812742i	$0.302026\pi$
$810$	−2.00000	−0.0702728
$811$	1.10428	0.0387764	0.0193882	−	0.999812i	$-0.493828\pi$
$811$	1.10428	0.0387764	0.0193882	+	0.999812i	$0.493828\pi$
$812$	0	0
$813$	−12.2802	−0.430685
$814$	−6.72832	−0.235827
$815$	−36.3043	−1.27169
$816$	13.6742	0.478692
$817$	−24.3664	−0.852473
$818$	15.9525	0.557767
$819$	0	0
$820$	−7.55691	−0.263899
$821$	−7.42504	−0.259136	−0.129568	−	0.991571i	$-0.541359\pi$
$821$	−7.42504	−0.259136	−0.129568	+	0.991571i	$0.541359\pi$
$822$	−5.74398	−0.200344
$823$	42.1656	1.46980	0.734900	−	0.678176i	$-0.237229\pi$
$823$	42.1656	1.46980	0.734900	+	0.678176i	$0.237229\pi$
$824$	23.6121	0.822567
$825$	−19.2001	−0.668460
$826$	0	0
$827$	−54.4863	−1.89468	−0.947338	−	0.320235i	$-0.896238\pi$
$827$	−54.4863	−1.89468	−0.947338	+	0.320235i	$0.896238\pi$
$828$	11.1138	0.386232
$829$	−24.0388	−0.834901	−0.417450	−	0.908700i	$-0.637076\pi$
$829$	−24.0388	−0.834901	−0.417450	+	0.908700i	$0.637076\pi$
$830$	7.61211	0.264220
$831$	−8.56990	−0.297287
$832$	−0.788009	−0.0273193
$833$	0	0
$834$	−6.72594	−0.232900
$835$	21.4948	0.743860
$836$	5.88273	0.203459
$837$	2.89572	0.100091
$838$	0.934491	0.0322815
$839$	13.6190	0.470180	0.235090	−	0.971974i	$-0.424462\pi$
$839$	13.6190	0.470180	0.235090	+	0.971974i	$0.424462\pi$
$840$	0	0
$841$	−23.1820	−0.799380
$842$	−2.67074	−0.0920399
$843$	−14.5845	−0.502317
$844$	18.8793	0.649852
$845$	54.9751	1.89120
$846$	1.92332	0.0661251
$847$	0	0
$848$	−3.91559	−0.134462
$849$	20.9509	0.719034
$850$	30.8939	1.05965
$851$	−60.7405	−2.08216
$852$	18.7259	0.641540
$853$	6.93449	0.237432	0.118716	−	0.992928i	$-0.462122\pi$
$853$	6.93449	0.237432	0.118716	+	0.992928i	$0.462122\pi$
$854$	0	0
$855$	−9.55691	−0.326840
$856$	−16.5795	−0.566674
$857$	34.9345	1.19334	0.596670	−	0.802487i	$-0.296490\pi$
$857$	34.9345	1.19334	0.596670	+	0.802487i	$0.296490\pi$
$858$	−0.172462	−0.00588774
$859$	−4.39057	−0.149804	−0.0749021	−	0.997191i	$-0.523864\pi$
$859$	−4.39057	−0.149804	−0.0749021	+	0.997191i	$0.523864\pi$
$860$	−81.8690	−2.79171
$861$	0	0
$862$	0.824101	0.0280690
$863$	15.4611	0.526303	0.263152	−	0.964755i	$-0.415238\pi$
$863$	15.4611	0.526303	0.263152	+	0.964755i	$0.415238\pi$
$864$	4.83709	0.164561
$865$	80.1587	2.72548
$866$	16.0337	0.544848
$867$	8.27674	0.281093
$868$	0	0
$869$	0.824101	0.0279557
$870$	4.82410	0.163552
$871$	3.97078	0.134545
$872$	0.443086	0.0150048
$873$	7.19051	0.243362
$874$	−6.61555	−0.223774
$875$	0	0
$876$	16.7068	0.564472
$877$	−55.2338	−1.86511	−0.932556	−	0.361025i	$-0.882427\pi$
$877$	−55.2338	−1.86511	−0.932556	+	0.361025i	$0.882427\pi$
$878$	2.03609	0.0687148
$879$	−31.4086	−1.05939
$880$	16.9966	0.572954
$881$	28.4431	0.958272	0.479136	−	0.877741i	$-0.340950\pi$
$881$	28.4431	0.958272	0.479136	+	0.877741i	$0.340950\pi$
$882$	0	0
$883$	−24.1319	−0.812102	−0.406051	−	0.913850i	$-0.633095\pi$
$883$	−24.1319	−0.812102	−0.406051	+	0.913850i	$0.633095\pi$
$884$	−2.22766	−0.0749242
$885$	11.1138	0.373587
$886$	17.4137	0.585024
$887$	2.06025	0.0691765	0.0345882	−	0.999402i	$-0.488988\pi$
$887$	2.06025	0.0691765	0.0345882	+	0.999402i	$0.488988\pi$
$888$	−17.2863	−0.580090
$889$	0	0
$890$	8.23453	0.276022
$891$	−1.47068	−0.0492697
$892$	−32.0000	−1.07144
$893$	9.19051	0.307549
$894$	−10.4070	−0.348062
$895$	12.1319	0.405524
$896$	0	0
$897$	−1.55691	−0.0519839
$898$	−7.50496	−0.250444
$899$	−6.98463	−0.232950
$900$	−23.2181	−0.773937
$901$	−7.23797	−0.241132
$902$	0.692226	0.0230486
$903$	0	0
$904$	−16.4431	−0.546889
$905$	57.1070	1.89830
$906$	4.76203	0.158208
$907$	−19.5208	−0.648178	−0.324089	−	0.946027i	$-0.605058\pi$
$907$	−19.5208	−0.648178	−0.324089	+	0.946027i	$0.605058\pi$
$908$	−1.61899	−0.0537279
$909$	11.1449	0.369652
$910$	0	0
$911$	−39.9785	−1.32455	−0.662274	−	0.749262i	$-0.730408\pi$
$911$	−39.9785	−1.32455	−0.662274	+	0.749262i	$0.730408\pi$
$912$	6.11727	0.202563
$913$	5.59750	0.185250
$914$	−13.8568	−0.458341
$915$	−24.3043	−0.803477
$916$	−13.2051	−0.436309
$917$	0	0
$918$	2.36641	0.0781031
$919$	−26.4508	−0.872532	−0.436266	−	0.899818i	$-0.643699\pi$
$919$	−26.4508	−0.872532	−0.436266	+	0.899818i	$0.643699\pi$
$920$	−47.2242	−1.55694
$921$	−8.54392	−0.281532
$922$	−10.2966	−0.339101
$923$	−2.62328	−0.0863463
$924$	0	0
$925$	126.894	4.17226
$926$	−8.78113	−0.288566
$927$	−13.2767	−0.436065
$928$	−11.6673	−0.382998
$929$	−33.2913	−1.09225	−0.546127	−	0.837703i	$-0.683898\pi$
$929$	−33.2913	−1.09225	−0.546127	+	0.837703i	$0.683898\pi$
$930$	−5.79145	−0.189909
$931$	0	0
$932$	−26.2536	−0.859966
$933$	5.49484	0.179893
$934$	−4.08117	−0.133540
$935$	31.4182	1.02748
$936$	−0.443086	−0.0144827
$937$	38.7811	1.26692	0.633462	−	0.773774i	$-0.281633\pi$
$937$	38.7811	1.26692	0.633462	+	0.773774i	$0.281633\pi$
$938$	0	0
$939$	−5.48024	−0.178841
$940$	30.8793	1.00717
$941$	−15.6482	−0.510117	−0.255058	−	0.966926i	$-0.582095\pi$
$941$	−15.6482	−0.510117	−0.255058	+	0.966926i	$0.582095\pi$
$942$	−8.83098	−0.287729
$943$	6.24914	0.203500
$944$	−7.11383	−0.231535
$945$	0	0
$946$	7.49934	0.243825
$947$	−47.5500	−1.54517	−0.772584	−	0.634912i	$-0.781036\pi$
$947$	−47.5500	−1.54517	−0.772584	+	0.634912i	$0.781036\pi$
$948$	0.996562	0.0323668
$949$	−2.34043	−0.0759735
$950$	13.8207	0.448402
$951$	−11.3173	−0.366989
$952$	0	0
$953$	−29.6819	−0.961491	−0.480746	−	0.876860i	$-0.659634\pi$
$953$	−29.6819	−0.961491	−0.480746	+	0.876860i	$0.659634\pi$
$954$	−0.677618	−0.0219387
$955$	−57.3415	−1.85553
$956$	2.46219	0.0796329
$957$	3.54736	0.114670
$958$	9.61661	0.310698
$959$	0	0
$960$	13.4396	0.433763
$961$	−22.6148	−0.729509
$962$	1.13981	0.0367489
$963$	9.32238	0.300410
$964$	−10.5896	−0.341067
$965$	34.0552	1.09628
$966$	0	0
$967$	−1.21199	−0.0389750	−0.0194875	−	0.999810i	$-0.506203\pi$
$967$	−1.21199	−0.0389750	−0.0194875	+	0.999810i	$0.506203\pi$
$968$	15.7164	0.505144
$969$	11.3078	0.363258
$970$	−14.3810	−0.461747
$971$	−12.5362	−0.402306	−0.201153	−	0.979560i	$-0.564469\pi$
$971$	−12.5362	−0.402306	−0.201153	+	0.979560i	$0.564469\pi$
$972$	−1.77846	−0.0570440
$973$	0	0
$974$	−3.29766	−0.105664
$975$	3.25258	0.104166
$976$	15.5569	0.497965
$977$	15.2069	0.486513	0.243256	−	0.969962i	$-0.421784\pi$
$977$	15.2069	0.486513	0.243256	+	0.969962i	$0.421784\pi$
$978$	4.02148	0.128593
$979$	6.05520	0.193525
$980$	0	0
$981$	−0.249141	−0.00795445
$982$	−7.32926	−0.233886
$983$	−39.0303	−1.24487	−0.622436	−	0.782671i	$-0.713857\pi$
$983$	−39.0303	−1.24487	−0.622436	+	0.782671i	$0.713857\pi$
$984$	1.77846	0.0566951
$985$	−59.3967	−1.89254
$986$	−5.70789	−0.181776
$987$	0	0
$988$	−0.996562	−0.0317049
$989$	67.7010	2.15277
$990$	2.94137	0.0934828
$991$	−8.91539	−0.283207	−0.141603	−	0.989923i	$-0.545226\pi$
$991$	−8.91539	−0.283207	−0.141603	+	0.989923i	$0.545226\pi$
$992$	14.0069	0.444719
$993$	33.8613	1.07455
$994$	0	0
$995$	−20.6707	−0.655307
$996$	6.76891	0.214481
$997$	33.6336	1.06519	0.532593	−	0.846371i	$-0.321218\pi$
$997$	33.6336	1.06519	0.532593	+	0.846371i	$0.321218\pi$
$998$	−0.762030	−0.0241216
$999$	9.71982	0.307522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.s.1.2		3
7.6	odd	2		123.2.a.d.1.2	✓	3
21.20	even	2		369.2.a.e.1.2		3
28.27	even	2		1968.2.a.w.1.3		3
35.34	odd	2		3075.2.a.t.1.2		3
56.13	odd	2		7872.2.a.bx.1.1		3
56.27	even	2		7872.2.a.bs.1.1		3
84.83	odd	2		5904.2.a.bd.1.1		3
105.104	even	2		9225.2.a.bx.1.2		3
287.286	odd	2		5043.2.a.n.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.2	✓	3	7.6	odd	2
369.2.a.e.1.2		3	21.20	even	2
1968.2.a.w.1.3		3	28.27	even	2
3075.2.a.t.1.2		3	35.34	odd	2
5043.2.a.n.1.2		3	287.286	odd	2
5904.2.a.bd.1.1		3	84.83	odd	2
6027.2.a.s.1.2		3	1.1	even	1	trivial
7872.2.a.bs.1.1		3	56.27	even	2
7872.2.a.bx.1.1		3	56.13	odd	2
9225.2.a.bx.1.2		3	105.104	even	2

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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} -4.24914 q^{5} +0.470683 q^{6} -1.77846 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} -4.24914 q^{5} +0.470683 q^{6} -1.77846 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.47068 q^{11} -1.77846 q^{12} +0.249141 q^{13} -4.24914 q^{15} +2.71982 q^{16} +5.02760 q^{17} +0.470683 q^{18} +2.24914 q^{19} +7.55691 q^{20} -0.692226 q^{22} -6.24914 q^{23} -1.77846 q^{24} +13.0552 q^{25} +0.117266 q^{26} +1.00000 q^{27} -2.41205 q^{29} -2.00000 q^{30} +2.89572 q^{31} +4.83709 q^{32} -1.47068 q^{33} +2.36641 q^{34} -1.77846 q^{36} +9.71982 q^{37} +1.05863 q^{38} +0.249141 q^{39} +7.55691 q^{40} -1.00000 q^{41} -10.8337 q^{43} +2.61555 q^{44} -4.24914 q^{45} -2.94137 q^{46} +4.08623 q^{47} +2.71982 q^{48} +6.14486 q^{50} +5.02760 q^{51} -0.443086 q^{52} -1.43965 q^{53} +0.470683 q^{54} +6.24914 q^{55} +2.24914 q^{57} -1.13531 q^{58} -2.61555 q^{59} +7.55691 q^{60} +5.71982 q^{61} +1.36297 q^{62} -3.16291 q^{64} -1.05863 q^{65} -0.692226 q^{66} +15.9379 q^{67} -8.94137 q^{68} -6.24914 q^{69} -10.5293 q^{71} -1.77846 q^{72} -9.39400 q^{73} +4.57496 q^{74} +13.0552 q^{75} -4.00000 q^{76} +0.117266 q^{78} -0.560352 q^{79} -11.5569 q^{80} +1.00000 q^{81} -0.470683 q^{82} -3.80605 q^{83} -21.3630 q^{85} -5.09922 q^{86} -2.41205 q^{87} +2.61555 q^{88} -4.11727 q^{89} -2.00000 q^{90} +11.1138 q^{92} +2.89572 q^{93} +1.92332 q^{94} -9.55691 q^{95} +4.83709 q^{96} +7.19051 q^{97} -1.47068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} - 4 q^{5} + q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 - 4 * q^5 + q^6 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} - 4 q^{5} + q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 4 q^{11} + 3 q^{12} - 8 q^{13} - 4 q^{15} - q^{16} - 2 q^{17} + q^{18} - 2 q^{19} + 6 q^{20} - 10 q^{22} - 10 q^{23} + 3 q^{24} + 5 q^{25} + 2 q^{26} + 3 q^{27} - 6 q^{29} - 6 q^{30} + 2 q^{31} + 7 q^{32} - 4 q^{33} + 3 q^{36} + 20 q^{37} + 4 q^{38} - 8 q^{39} + 6 q^{40} - 3 q^{41} + 10 q^{43} - 8 q^{44} - 4 q^{45} - 8 q^{46} - 4 q^{47} - q^{48} + 3 q^{50} - 2 q^{51} - 18 q^{52} + 14 q^{53} + q^{54} + 10 q^{55} - 2 q^{57} - 28 q^{58} + 8 q^{59} + 6 q^{60} + 8 q^{61} - 38 q^{62} - 17 q^{64} - 4 q^{65} - 10 q^{66} + 12 q^{67} - 26 q^{68} - 10 q^{69} - 32 q^{71} + 3 q^{72} - 4 q^{73} + 20 q^{74} + 5 q^{75} - 12 q^{76} + 2 q^{78} - 20 q^{79} - 18 q^{80} + 3 q^{81} - q^{82} + 14 q^{83} - 22 q^{85} + 6 q^{86} - 6 q^{87} - 8 q^{88} - 14 q^{89} - 6 q^{90} + 2 q^{93} - 18 q^{94} - 12 q^{95} + 7 q^{96} + 12 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 - 4 * q^5 + q^6 + 3 * q^8 + 3 * q^9 - 6 * q^10 - 4 * q^11 + 3 * q^12 - 8 * q^13 - 4 * q^15 - q^16 - 2 * q^17 + q^18 - 2 * q^19 + 6 * q^20 - 10 * q^22 - 10 * q^23 + 3 * q^24 + 5 * q^25 + 2 * q^26 + 3 * q^27 - 6 * q^29 - 6 * q^30 + 2 * q^31 + 7 * q^32 - 4 * q^33 + 3 * q^36 + 20 * q^37 + 4 * q^38 - 8 * q^39 + 6 * q^40 - 3 * q^41 + 10 * q^43 - 8 * q^44 - 4 * q^45 - 8 * q^46 - 4 * q^47 - q^48 + 3 * q^50 - 2 * q^51 - 18 * q^52 + 14 * q^53 + q^54 + 10 * q^55 - 2 * q^57 - 28 * q^58 + 8 * q^59 + 6 * q^60 + 8 * q^61 - 38 * q^62 - 17 * q^64 - 4 * q^65 - 10 * q^66 + 12 * q^67 - 26 * q^68 - 10 * q^69 - 32 * q^71 + 3 * q^72 - 4 * q^73 + 20 * q^74 + 5 * q^75 - 12 * q^76 + 2 * q^78 - 20 * q^79 - 18 * q^80 + 3 * q^81 - q^82 + 14 * q^83 - 22 * q^85 + 6 * q^86 - 6 * q^87 - 8 * q^88 - 14 * q^89 - 6 * q^90 + 2 * q^93 - 18 * q^94 - 12 * q^95 + 7 * q^96 + 12 * q^97 - 4 * q^99

Introduction

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