LMFDB - Embedded newform 8022.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8022,2,Mod(1,8022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8022, 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("8022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8022 = 2 \cdot 3 \cdot 7 \cdot 191 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8022.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:	$64.0559925015$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + \cdots + 4704 $ x^15 - 53x^13 - x^12 + 1068x^11 + 45x^10 - 10139x^9 - 615x^8 + 45390x^7 + 2130x^6 - 84842x^5 + 7822x^4 + 62828x^3 - 16144x^2 - 13616x + 4704
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.31212$ of defining polynomial
Character	$\chi$	$=$	8022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.31212 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.31212 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.31212 q^{10} -6.38474 q^{11} +1.00000 q^{12} -3.56515 q^{13} -1.00000 q^{14} -3.31212 q^{15} +1.00000 q^{16} -3.41128 q^{17} -1.00000 q^{18} +1.81383 q^{19} -3.31212 q^{20} +1.00000 q^{21} +6.38474 q^{22} -4.94502 q^{23} -1.00000 q^{24} +5.97012 q^{25} +3.56515 q^{26} +1.00000 q^{27} +1.00000 q^{28} +2.06774 q^{29} +3.31212 q^{30} -6.29759 q^{31} -1.00000 q^{32} -6.38474 q^{33} +3.41128 q^{34} -3.31212 q^{35} +1.00000 q^{36} -4.60103 q^{37} -1.81383 q^{38} -3.56515 q^{39} +3.31212 q^{40} +3.36340 q^{41} -1.00000 q^{42} -10.5332 q^{43} -6.38474 q^{44} -3.31212 q^{45} +4.94502 q^{46} -0.528430 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.97012 q^{50} -3.41128 q^{51} -3.56515 q^{52} +6.04890 q^{53} -1.00000 q^{54} +21.1470 q^{55} -1.00000 q^{56} +1.81383 q^{57} -2.06774 q^{58} -9.14905 q^{59} -3.31212 q^{60} +8.30484 q^{61} +6.29759 q^{62} +1.00000 q^{63} +1.00000 q^{64} +11.8082 q^{65} +6.38474 q^{66} +3.79329 q^{67} -3.41128 q^{68} -4.94502 q^{69} +3.31212 q^{70} -7.57141 q^{71} -1.00000 q^{72} +2.22589 q^{73} +4.60103 q^{74} +5.97012 q^{75} +1.81383 q^{76} -6.38474 q^{77} +3.56515 q^{78} -7.84545 q^{79} -3.31212 q^{80} +1.00000 q^{81} -3.36340 q^{82} -13.5794 q^{83} +1.00000 q^{84} +11.2986 q^{85} +10.5332 q^{86} +2.06774 q^{87} +6.38474 q^{88} -12.9315 q^{89} +3.31212 q^{90} -3.56515 q^{91} -4.94502 q^{92} -6.29759 q^{93} +0.528430 q^{94} -6.00760 q^{95} -1.00000 q^{96} +3.68748 q^{97} -1.00000 q^{98} -6.38474 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9	$ 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9 - 7 * q^11 + 15 * q^12 + 10 * q^13 - 15 * q^14 + 15 * q^16 - 3 * q^17 - 15 * q^18 + 12 * q^19 + 15 * q^21 + 7 * q^22 - q^23 - 15 * q^24 + 31 * q^25 - 10 * q^26 + 15 * q^27 + 15 * q^28 - 3 * q^29 + 19 * q^31 - 15 * q^32 - 7 * q^33 + 3 * q^34 + 15 * q^36 + 25 * q^37 - 12 * q^38 + 10 * q^39 + 8 * q^41 - 15 * q^42 + 25 * q^43 - 7 * q^44 + q^46 + 11 * q^47 + 15 * q^48 + 15 * q^49 - 31 * q^50 - 3 * q^51 + 10 * q^52 - 4 * q^53 - 15 * q^54 + 9 * q^55 - 15 * q^56 + 12 * q^57 + 3 * q^58 + 27 * q^61 - 19 * q^62 + 15 * q^63 + 15 * q^64 - 2 * q^65 + 7 * q^66 + 31 * q^67 - 3 * q^68 - q^69 - 16 * q^71 - 15 * q^72 + 26 * q^73 - 25 * q^74 + 31 * q^75 + 12 * q^76 - 7 * q^77 - 10 * q^78 + 32 * q^79 + 15 * q^81 - 8 * q^82 + 7 * q^83 + 15 * q^84 + 26 * q^85 - 25 * q^86 - 3 * q^87 + 7 * q^88 - 11 * q^89 + 10 * q^91 - q^92 + 19 * q^93 - 11 * q^94 - 8 * q^95 - 15 * q^96 + 30 * q^97 - 15 * q^98 - 7 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.31212	−1.48122	−0.740612	−	0.671933i	$-0.765464\pi$
$5$	−3.31212	−1.48122	−0.740612	+	0.671933i	$0.765464\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.31212	1.04738
$11$	−6.38474	−1.92507	−0.962536	−	0.271155i	$-0.912594\pi$
$11$	−6.38474	−1.92507	−0.962536	+	0.271155i	$0.912594\pi$
$12$	1.00000	0.288675
$13$	−3.56515	−0.988796	−0.494398	−	0.869236i	$-0.664611\pi$
$13$	−3.56515	−0.988796	−0.494398	+	0.869236i	$0.664611\pi$
$14$	−1.00000	−0.267261
$15$	−3.31212	−0.855185
$16$	1.00000	0.250000
$17$	−3.41128	−0.827358	−0.413679	−	0.910423i	$-0.635756\pi$
$17$	−3.41128	−0.827358	−0.413679	+	0.910423i	$0.635756\pi$
$18$	−1.00000	−0.235702
$19$	1.81383	0.416120	0.208060	−	0.978116i	$-0.433285\pi$
$19$	1.81383	0.416120	0.208060	+	0.978116i	$0.433285\pi$
$20$	−3.31212	−0.740612
$21$	1.00000	0.218218
$22$	6.38474	1.36123
$23$	−4.94502	−1.03111	−0.515554	−	0.856857i	$-0.672414\pi$
$23$	−4.94502	−1.03111	−0.515554	+	0.856857i	$0.672414\pi$
$24$	−1.00000	−0.204124
$25$	5.97012	1.19402
$26$	3.56515	0.699184
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	2.06774	0.383970	0.191985	−	0.981398i	$-0.438507\pi$
$29$	2.06774	0.383970	0.191985	+	0.981398i	$0.438507\pi$
$30$	3.31212	0.604707
$31$	−6.29759	−1.13108	−0.565541	−	0.824720i	$-0.691332\pi$
$31$	−6.29759	−1.13108	−0.565541	+	0.824720i	$0.691332\pi$
$32$	−1.00000	−0.176777
$33$	−6.38474	−1.11144
$34$	3.41128	0.585030
$35$	−3.31212	−0.559850
$36$	1.00000	0.166667
$37$	−4.60103	−0.756405	−0.378203	−	0.925723i	$-0.623458\pi$
$37$	−4.60103	−0.756405	−0.378203	+	0.925723i	$0.623458\pi$
$38$	−1.81383	−0.294242
$39$	−3.56515	−0.570881
$40$	3.31212	0.523692
$41$	3.36340	0.525275	0.262638	−	0.964895i	$-0.415408\pi$
$41$	3.36340	0.525275	0.262638	+	0.964895i	$0.415408\pi$
$42$	−1.00000	−0.154303
$43$	−10.5332	−1.60630	−0.803151	−	0.595775i	$-0.796845\pi$
$43$	−10.5332	−1.60630	−0.803151	+	0.595775i	$0.796845\pi$
$44$	−6.38474	−0.962536
$45$	−3.31212	−0.493741
$46$	4.94502	0.729103
$47$	−0.528430	−0.0770795	−0.0385397	−	0.999257i	$-0.512271\pi$
$47$	−0.528430	−0.0770795	−0.0385397	+	0.999257i	$0.512271\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−5.97012	−0.844302
$51$	−3.41128	−0.477675
$52$	−3.56515	−0.494398
$53$	6.04890	0.830881	0.415440	−	0.909620i	$-0.363627\pi$
$53$	6.04890	0.830881	0.415440	+	0.909620i	$0.363627\pi$
$54$	−1.00000	−0.136083
$55$	21.1470	2.85146
$56$	−1.00000	−0.133631
$57$	1.81383	0.240247
$58$	−2.06774	−0.271508
$59$	−9.14905	−1.19110	−0.595552	−	0.803317i	$-0.703067\pi$
$59$	−9.14905	−1.19110	−0.595552	+	0.803317i	$0.703067\pi$
$60$	−3.31212	−0.427592
$61$	8.30484	1.06333	0.531663	−	0.846956i	$-0.321567\pi$
$61$	8.30484	1.06333	0.531663	+	0.846956i	$0.321567\pi$
$62$	6.29759	0.799795
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	11.8082	1.46463
$66$	6.38474	0.785907
$67$	3.79329	0.463424	0.231712	−	0.972784i	$-0.425567\pi$
$67$	3.79329	0.463424	0.231712	+	0.972784i	$0.425567\pi$
$68$	−3.41128	−0.413679
$69$	−4.94502	−0.595310
$70$	3.31212	0.395874
$71$	−7.57141	−0.898561	−0.449281	−	0.893391i	$-0.648320\pi$
$71$	−7.57141	−0.898561	−0.449281	+	0.893391i	$0.648320\pi$
$72$	−1.00000	−0.117851
$73$	2.22589	0.260521	0.130261	−	0.991480i	$-0.458419\pi$
$73$	2.22589	0.260521	0.130261	+	0.991480i	$0.458419\pi$
$74$	4.60103	0.534859
$75$	5.97012	0.689370
$76$	1.81383	0.208060
$77$	−6.38474	−0.727609
$78$	3.56515	0.403674
$79$	−7.84545	−0.882682	−0.441341	−	0.897339i	$-0.645497\pi$
$79$	−7.84545	−0.882682	−0.441341	+	0.897339i	$0.645497\pi$
$80$	−3.31212	−0.370306
$81$	1.00000	0.111111
$82$	−3.36340	−0.371426
$83$	−13.5794	−1.49054	−0.745269	−	0.666764i	$-0.767679\pi$
$83$	−13.5794	−1.49054	−0.745269	+	0.666764i	$0.767679\pi$
$84$	1.00000	0.109109
$85$	11.2986	1.22550
$86$	10.5332	1.13583
$87$	2.06774	0.221685
$88$	6.38474	0.680616
$89$	−12.9315	−1.37073	−0.685367	−	0.728198i	$-0.740358\pi$
$89$	−12.9315	−1.37073	−0.685367	+	0.728198i	$0.740358\pi$
$90$	3.31212	0.349128
$91$	−3.56515	−0.373730
$92$	−4.94502	−0.515554
$93$	−6.29759	−0.653030
$94$	0.528430	0.0545034
$95$	−6.00760	−0.616367
$96$	−1.00000	−0.102062
$97$	3.68748	0.374407	0.187204	−	0.982321i	$-0.440058\pi$
$97$	3.68748	0.374407	0.187204	+	0.982321i	$0.440058\pi$
$98$	−1.00000	−0.101015
$99$	−6.38474	−0.641690
$100$	5.97012	0.597012
$101$	1.23297	0.122685	0.0613426	−	0.998117i	$-0.480462\pi$
$101$	1.23297	0.122685	0.0613426	+	0.998117i	$0.480462\pi$
$102$	3.41128	0.337767
$103$	0.310898	0.0306337	0.0153168	−	0.999883i	$-0.495124\pi$
$103$	0.310898	0.0306337	0.0153168	+	0.999883i	$0.495124\pi$
$104$	3.56515	0.349592
$105$	−3.31212	−0.323229
$106$	−6.04890	−0.587521
$107$	−0.341629	−0.0330265	−0.0165133	−	0.999864i	$-0.505257\pi$
$107$	−0.341629	−0.0330265	−0.0165133	+	0.999864i	$0.505257\pi$
$108$	1.00000	0.0962250
$109$	−2.65067	−0.253888	−0.126944	−	0.991910i	$-0.540517\pi$
$109$	−2.65067	−0.253888	−0.126944	+	0.991910i	$0.540517\pi$
$110$	−21.1470	−2.01629
$111$	−4.60103	−0.436711
$112$	1.00000	0.0944911
$113$	16.1304	1.51742	0.758708	−	0.651430i	$-0.225831\pi$
$113$	16.1304	1.51742	0.758708	+	0.651430i	$0.225831\pi$
$114$	−1.81383	−0.169880
$115$	16.3785	1.52730
$116$	2.06774	0.191985
$117$	−3.56515	−0.329599
$118$	9.14905	0.842238
$119$	−3.41128	−0.312712
$120$	3.31212	0.302353
$121$	29.7649	2.70590
$122$	−8.30484	−0.751885
$123$	3.36340	0.303268
$124$	−6.29759	−0.565541
$125$	−3.21314	−0.287392
$126$	−1.00000	−0.0890871
$127$	9.54153	0.846674	0.423337	−	0.905972i	$-0.360859\pi$
$127$	9.54153	0.846674	0.423337	+	0.905972i	$0.360859\pi$
$128$	−1.00000	−0.0883883
$129$	−10.5332	−0.927399
$130$	−11.8082	−1.03565
$131$	2.32765	0.203367	0.101684	−	0.994817i	$-0.467577\pi$
$131$	2.32765	0.203367	0.101684	+	0.994817i	$0.467577\pi$
$132$	−6.38474	−0.555720
$133$	1.81383	0.157279
$134$	−3.79329	−0.327690
$135$	−3.31212	−0.285062
$136$	3.41128	0.292515
$137$	−21.7668	−1.85966	−0.929830	−	0.367990i	$-0.880046\pi$
$137$	−21.7668	−1.85966	−0.929830	+	0.367990i	$0.880046\pi$
$138$	4.94502	0.420948
$139$	7.63589	0.647667	0.323834	−	0.946114i	$-0.395028\pi$
$139$	7.63589	0.647667	0.323834	+	0.946114i	$0.395028\pi$
$140$	−3.31212	−0.279925
$141$	−0.528430	−0.0445018
$142$	7.57141	0.635379
$143$	22.7626	1.90350
$144$	1.00000	0.0833333
$145$	−6.84860	−0.568746
$146$	−2.22589	−0.184216
$147$	1.00000	0.0824786
$148$	−4.60103	−0.378203
$149$	20.0880	1.64567	0.822835	−	0.568281i	$-0.192391\pi$
$149$	20.0880	1.64567	0.822835	+	0.568281i	$0.192391\pi$
$150$	−5.97012	−0.487458
$151$	−12.7004	−1.03355	−0.516773	−	0.856122i	$-0.672867\pi$
$151$	−12.7004	−1.03355	−0.516773	+	0.856122i	$0.672867\pi$
$152$	−1.81383	−0.147121
$153$	−3.41128	−0.275786
$154$	6.38474	0.514497
$155$	20.8584	1.67538
$156$	−3.56515	−0.285441
$157$	20.0621	1.60113	0.800564	−	0.599247i	$-0.204533\pi$
$157$	20.0621	1.60113	0.800564	+	0.599247i	$0.204533\pi$
$158$	7.84545	0.624150
$159$	6.04890	0.479709
$160$	3.31212	0.261846
$161$	−4.94502	−0.389722
$162$	−1.00000	−0.0785674
$163$	17.4523	1.36697	0.683483	−	0.729967i	$-0.260464\pi$
$163$	17.4523	1.36697	0.683483	+	0.729967i	$0.260464\pi$
$164$	3.36340	0.262638
$165$	21.1470	1.64629
$166$	13.5794	1.05397
$167$	−22.0422	−1.70567	−0.852837	−	0.522177i	$-0.825120\pi$
$167$	−22.0422	−1.70567	−0.852837	+	0.522177i	$0.825120\pi$
$168$	−1.00000	−0.0771517
$169$	−0.289681	−0.0222832
$170$	−11.2986	−0.866560
$171$	1.81383	0.138707
$172$	−10.5332	−0.803151
$173$	18.5059	1.40698	0.703489	−	0.710706i	$-0.251625\pi$
$173$	18.5059	1.40698	0.703489	+	0.710706i	$0.251625\pi$
$174$	−2.06774	−0.156755
$175$	5.97012	0.451298
$176$	−6.38474	−0.481268
$177$	−9.14905	−0.687684
$178$	12.9315	0.969255
$179$	−16.5151	−1.23440	−0.617200	−	0.786806i	$-0.711733\pi$
$179$	−16.5151	−1.23440	−0.617200	+	0.786806i	$0.711733\pi$
$180$	−3.31212	−0.246871
$181$	−13.5795	−1.00935	−0.504677	−	0.863308i	$-0.668388\pi$
$181$	−13.5795	−1.00935	−0.504677	+	0.863308i	$0.668388\pi$
$182$	3.56515	0.264267
$183$	8.30484	0.613911
$184$	4.94502	0.364551
$185$	15.2392	1.12041
$186$	6.29759	0.461762
$187$	21.7802	1.59272
$188$	−0.528430	−0.0385397
$189$	1.00000	0.0727393
$190$	6.00760	0.435837
$191$	1.00000	0.0723575
$191$	1.00000	0.0723575
$192$	1.00000	0.0721688
$193$	16.1694	1.16390	0.581950	−	0.813225i	$-0.302290\pi$
$193$	16.1694	1.16390	0.581950	+	0.813225i	$0.302290\pi$
$194$	−3.68748	−0.264746
$195$	11.8082	0.845603
$196$	1.00000	0.0714286
$197$	−9.57180	−0.681963	−0.340981	−	0.940070i	$-0.610759\pi$
$197$	−9.57180	−0.681963	−0.340981	+	0.940070i	$0.610759\pi$
$198$	6.38474	0.453744
$199$	13.6172	0.965301	0.482650	−	0.875813i	$-0.339674\pi$
$199$	13.6172	0.965301	0.482650	+	0.875813i	$0.339674\pi$
$200$	−5.97012	−0.422151
$201$	3.79329	0.267558
$202$	−1.23297	−0.0867516
$203$	2.06774	0.145127
$204$	−3.41128	−0.238838
$205$	−11.1400	−0.778050
$206$	−0.310898	−0.0216613
$207$	−4.94502	−0.343702
$208$	−3.56515	−0.247199
$209$	−11.5808	−0.801061
$210$	3.31212	0.228558
$211$	2.26871	0.156184	0.0780922	−	0.996946i	$-0.475117\pi$
$211$	2.26871	0.156184	0.0780922	+	0.996946i	$0.475117\pi$
$212$	6.04890	0.415440
$213$	−7.57141	−0.518785
$214$	0.341629	0.0233533
$215$	34.8873	2.37929
$216$	−1.00000	−0.0680414
$217$	−6.29759	−0.427509
$218$	2.65067	0.179526
$219$	2.22589	0.150412
$220$	21.1470	1.42573
$221$	12.1617	0.818088
$222$	4.60103	0.308801
$223$	−11.7462	−0.786583	−0.393292	−	0.919414i	$-0.628664\pi$
$223$	−11.7462	−0.786583	−0.393292	+	0.919414i	$0.628664\pi$
$224$	−1.00000	−0.0668153
$225$	5.97012	0.398008
$226$	−16.1304	−1.07298
$227$	18.9515	1.25785	0.628926	−	0.777465i	$-0.283495\pi$
$227$	18.9515	1.25785	0.628926	+	0.777465i	$0.283495\pi$
$228$	1.81383	0.120124
$229$	6.67424	0.441046	0.220523	−	0.975382i	$-0.429224\pi$
$229$	6.67424	0.441046	0.220523	+	0.975382i	$0.429224\pi$
$230$	−16.3785	−1.07996
$231$	−6.38474	−0.420085
$232$	−2.06774	−0.135754
$233$	−0.547062	−0.0358392	−0.0179196	−	0.999839i	$-0.505704\pi$
$233$	−0.547062	−0.0358392	−0.0179196	+	0.999839i	$0.505704\pi$
$234$	3.56515	0.233061
$235$	1.75022	0.114172
$236$	−9.14905	−0.595552
$237$	−7.84545	−0.509617
$238$	3.41128	0.221121
$239$	5.56677	0.360084	0.180042	−	0.983659i	$-0.442377\pi$
$239$	5.56677	0.360084	0.180042	+	0.983659i	$0.442377\pi$
$240$	−3.31212	−0.213796
$241$	−27.6640	−1.78200	−0.890998	−	0.454008i	$-0.849994\pi$
$241$	−27.6640	−1.78200	−0.890998	+	0.454008i	$0.849994\pi$
$242$	−29.7649	−1.91336
$243$	1.00000	0.0641500
$244$	8.30484	0.531663
$245$	−3.31212	−0.211603
$246$	−3.36340	−0.214443
$247$	−6.46657	−0.411458
$248$	6.29759	0.399898
$249$	−13.5794	−0.860562
$250$	3.21314	0.203217
$251$	3.78162	0.238694	0.119347	−	0.992853i	$-0.461920\pi$
$251$	3.78162	0.238694	0.119347	+	0.992853i	$0.461920\pi$
$252$	1.00000	0.0629941
$253$	31.5726	1.98496
$254$	−9.54153	−0.598689
$255$	11.2986	0.707544
$256$	1.00000	0.0625000
$257$	−6.70756	−0.418406	−0.209203	−	0.977872i	$-0.567087\pi$
$257$	−6.70756	−0.418406	−0.209203	+	0.977872i	$0.567087\pi$
$258$	10.5332	0.655770
$259$	−4.60103	−0.285894
$260$	11.8082	0.732314
$261$	2.06774	0.127990
$262$	−2.32765	−0.143803
$263$	−24.9422	−1.53800	−0.769001	−	0.639248i	$-0.779246\pi$
$263$	−24.9422	−1.53800	−0.769001	+	0.639248i	$0.779246\pi$
$264$	6.38474	0.392954
$265$	−20.0347	−1.23072
$266$	−1.81383	−0.111213
$267$	−12.9315	−0.791393
$268$	3.79329	0.231712
$269$	23.4152	1.42765	0.713826	−	0.700323i	$-0.246961\pi$
$269$	23.4152	1.42765	0.713826	+	0.700323i	$0.246961\pi$
$270$	3.31212	0.201569
$271$	−29.4519	−1.78907	−0.894537	−	0.446993i	$-0.852495\pi$
$271$	−29.4519	−1.78907	−0.894537	+	0.446993i	$0.852495\pi$
$272$	−3.41128	−0.206839
$273$	−3.56515	−0.215773
$274$	21.7668	1.31498
$275$	−38.1176	−2.29858
$276$	−4.94502	−0.297655
$277$	1.29421	0.0777615	0.0388808	−	0.999244i	$-0.487621\pi$
$277$	1.29421	0.0777615	0.0388808	+	0.999244i	$0.487621\pi$
$278$	−7.63589	−0.457970
$279$	−6.29759	−0.377027
$280$	3.31212	0.197937
$281$	−8.60945	−0.513597	−0.256798	−	0.966465i	$-0.582668\pi$
$281$	−8.60945	−0.513597	−0.256798	+	0.966465i	$0.582668\pi$
$282$	0.528430	0.0314676
$283$	−13.4866	−0.801697	−0.400849	−	0.916144i	$-0.631285\pi$
$283$	−13.4866	−0.801697	−0.400849	+	0.916144i	$0.631285\pi$
$284$	−7.57141	−0.449281
$285$	−6.00760	−0.355860
$286$	−22.7626	−1.34598
$287$	3.36340	0.198535
$288$	−1.00000	−0.0589256
$289$	−5.36315	−0.315479
$290$	6.84860	0.402164
$291$	3.68748	0.216164
$292$	2.22589	0.130261
$293$	23.9085	1.39675	0.698374	−	0.715733i	$-0.253907\pi$
$293$	23.9085	1.39675	0.698374	+	0.715733i	$0.253907\pi$
$294$	−1.00000	−0.0583212
$295$	30.3027	1.76429
$296$	4.60103	0.267430
$297$	−6.38474	−0.370480
$298$	−20.0880	−1.16366
$299$	17.6297	1.01955
$300$	5.97012	0.344685
$301$	−10.5332	−0.607125
$302$	12.7004	0.730828
$303$	1.23297	0.0708324
$304$	1.81383	0.104030
$305$	−27.5066	−1.57502
$306$	3.41128	0.195010
$307$	12.1166	0.691529	0.345764	−	0.938321i	$-0.387620\pi$
$307$	12.1166	0.691529	0.345764	+	0.938321i	$0.387620\pi$
$308$	−6.38474	−0.363804
$309$	0.310898	0.0176863
$310$	−20.8584	−1.18468
$311$	5.92659	0.336066	0.168033	−	0.985781i	$-0.446258\pi$
$311$	5.92659	0.336066	0.168033	+	0.985781i	$0.446258\pi$
$312$	3.56515	0.201837
$313$	−5.21064	−0.294523	−0.147261	−	0.989098i	$-0.547046\pi$
$313$	−5.21064	−0.294523	−0.147261	+	0.989098i	$0.547046\pi$
$314$	−20.0621	−1.13217
$315$	−3.31212	−0.186617
$316$	−7.84545	−0.441341
$317$	13.2714	0.745395	0.372697	−	0.927953i	$-0.378433\pi$
$317$	13.2714	0.745395	0.372697	+	0.927953i	$0.378433\pi$
$318$	−6.04890	−0.339206
$319$	−13.2020	−0.739170
$320$	−3.31212	−0.185153
$321$	−0.341629	−0.0190679
$322$	4.94502	0.275575
$323$	−6.18747	−0.344280
$324$	1.00000	0.0555556
$325$	−21.2844	−1.18065
$326$	−17.4523	−0.966591
$327$	−2.65067	−0.146583
$328$	−3.36340	−0.185713
$329$	−0.528430	−0.0291333
$330$	−21.1470	−1.16410
$331$	24.0280	1.32070	0.660349	−	0.750959i	$-0.270408\pi$
$331$	24.0280	1.32070	0.660349	+	0.750959i	$0.270408\pi$
$332$	−13.5794	−0.745269
$333$	−4.60103	−0.252135
$334$	22.0422	1.20609
$335$	−12.5638	−0.686434
$336$	1.00000	0.0545545
$337$	6.87409	0.374455	0.187228	−	0.982317i	$-0.440050\pi$
$337$	6.87409	0.374455	0.187228	+	0.982317i	$0.440050\pi$
$338$	0.289681	0.0157566
$339$	16.1304	0.876081
$340$	11.2986	0.612751
$341$	40.2085	2.17741
$342$	−1.81383	−0.0980805
$343$	1.00000	0.0539949
$344$	10.5332	0.567914
$345$	16.3785	0.881787
$346$	−18.5059	−0.994883
$347$	−6.46553	−0.347088	−0.173544	−	0.984826i	$-0.555522\pi$
$347$	−6.46553	−0.347088	−0.173544	+	0.984826i	$0.555522\pi$
$348$	2.06774	0.110843
$349$	12.1779	0.651870	0.325935	−	0.945392i	$-0.394321\pi$
$349$	12.1779	0.651870	0.325935	+	0.945392i	$0.394321\pi$
$350$	−5.97012	−0.319116
$351$	−3.56515	−0.190294
$352$	6.38474	0.340308
$353$	20.6409	1.09861	0.549303	−	0.835623i	$-0.314893\pi$
$353$	20.6409	1.09861	0.549303	+	0.835623i	$0.314893\pi$
$354$	9.14905	0.486266
$355$	25.0774	1.33097
$356$	−12.9315	−0.685367
$357$	−3.41128	−0.180544
$358$	16.5151	0.872853
$359$	1.64957	0.0870609	0.0435305	−	0.999052i	$-0.486139\pi$
$359$	1.64957	0.0870609	0.0435305	+	0.999052i	$0.486139\pi$
$360$	3.31212	0.174564
$361$	−15.7100	−0.826844
$362$	13.5795	0.713721
$363$	29.7649	1.56225
$364$	−3.56515	−0.186865
$365$	−7.37242	−0.385890
$366$	−8.30484	−0.434101
$367$	−5.83621	−0.304648	−0.152324	−	0.988331i	$-0.548676\pi$
$367$	−5.83621	−0.304648	−0.152324	+	0.988331i	$0.548676\pi$
$368$	−4.94502	−0.257777
$369$	3.36340	0.175092
$370$	−15.2392	−0.792246
$371$	6.04890	0.314043
$372$	−6.29759	−0.326515
$373$	9.02608	0.467353	0.233676	−	0.972314i	$-0.424924\pi$
$373$	9.02608	0.467353	0.233676	+	0.972314i	$0.424924\pi$
$374$	−21.7802	−1.12622
$375$	−3.21314	−0.165926
$376$	0.528430	0.0272517
$377$	−7.37182	−0.379668
$378$	−1.00000	−0.0514344
$379$	−18.2490	−0.937387	−0.468694	−	0.883361i	$-0.655275\pi$
$379$	−18.2490	−0.937387	−0.468694	+	0.883361i	$0.655275\pi$
$380$	−6.00760	−0.308184
$381$	9.54153	0.488827
$382$	−1.00000	−0.0511645
$383$	33.1742	1.69512	0.847562	−	0.530696i	$-0.178069\pi$
$383$	33.1742	1.69512	0.847562	+	0.530696i	$0.178069\pi$
$384$	−1.00000	−0.0510310
$385$	21.1470	1.07775
$386$	−16.1694	−0.823002
$387$	−10.5332	−0.535434
$388$	3.68748	0.187204
$389$	10.7704	0.546083	0.273041	−	0.962002i	$-0.411970\pi$
$389$	10.7704	0.546083	0.273041	+	0.962002i	$0.411970\pi$
$390$	−11.8082	−0.597932
$391$	16.8688	0.853094
$392$	−1.00000	−0.0505076
$393$	2.32765	0.117414
$394$	9.57180	0.482220
$395$	25.9850	1.30745
$396$	−6.38474	−0.320845
$397$	−3.63234	−0.182302	−0.0911509	−	0.995837i	$-0.529055\pi$
$397$	−3.63234	−0.182302	−0.0911509	+	0.995837i	$0.529055\pi$
$398$	−13.6172	−0.682571
$399$	1.81383	0.0908049
$400$	5.97012	0.298506
$401$	27.0386	1.35024	0.675120	−	0.737708i	$-0.264092\pi$
$401$	27.0386	1.35024	0.675120	+	0.737708i	$0.264092\pi$
$402$	−3.79329	−0.189192
$403$	22.4519	1.11841
$404$	1.23297	0.0613426
$405$	−3.31212	−0.164580
$406$	−2.06774	−0.102620
$407$	29.3764	1.45613
$408$	3.41128	0.168884
$409$	23.6115	1.16751	0.583757	−	0.811928i	$-0.301582\pi$
$409$	23.6115	1.16751	0.583757	+	0.811928i	$0.301582\pi$
$410$	11.1400	0.550165
$411$	−21.7668	−1.07367
$412$	0.310898	0.0153168
$413$	−9.14905	−0.450195
$414$	4.94502	0.243034
$415$	44.9767	2.20782
$416$	3.56515	0.174796
$417$	7.63589	0.373931
$418$	11.5808	0.566436
$419$	25.3058	1.23627	0.618134	−	0.786073i	$-0.287889\pi$
$419$	25.3058	1.23627	0.618134	+	0.786073i	$0.287889\pi$
$420$	−3.31212	−0.161615
$421$	4.39287	0.214096	0.107048	−	0.994254i	$-0.465860\pi$
$421$	4.39287	0.214096	0.107048	+	0.994254i	$0.465860\pi$
$422$	−2.26871	−0.110439
$423$	−0.528430	−0.0256932
$424$	−6.04890	−0.293761
$425$	−20.3658	−0.987884
$426$	7.57141	0.366836
$427$	8.30484	0.401899
$428$	−0.341629	−0.0165133
$429$	22.7626	1.09899
$430$	−34.8873	−1.68241
$431$	−19.3472	−0.931922	−0.465961	−	0.884805i	$-0.654291\pi$
$431$	−19.3472	−0.931922	−0.465961	+	0.884805i	$0.654291\pi$
$432$	1.00000	0.0481125
$433$	−25.5837	−1.22947	−0.614737	−	0.788732i	$-0.710738\pi$
$433$	−25.5837	−1.22947	−0.614737	+	0.788732i	$0.710738\pi$
$434$	6.29759	0.302294
$435$	−6.84860	−0.328365
$436$	−2.65067	−0.126944
$437$	−8.96940	−0.429065
$438$	−2.22589	−0.106357
$439$	−31.5534	−1.50596	−0.752980	−	0.658043i	$-0.771385\pi$
$439$	−31.5534	−1.50596	−0.752980	+	0.658043i	$0.771385\pi$
$440$	−21.1470	−1.00814
$441$	1.00000	0.0476190
$442$	−12.1617	−0.578475
$443$	6.91771	0.328670	0.164335	−	0.986405i	$-0.447452\pi$
$443$	6.91771	0.328670	0.164335	+	0.986405i	$0.447452\pi$
$444$	−4.60103	−0.218355
$445$	42.8305	2.03036
$446$	11.7462	0.556198
$447$	20.0880	0.950128
$448$	1.00000	0.0472456
$449$	0.0802325	0.00378641	0.00189320	−	0.999998i	$-0.499397\pi$
$449$	0.0802325	0.00378641	0.00189320	+	0.999998i	$0.499397\pi$
$450$	−5.97012	−0.281434
$451$	−21.4745	−1.01119
$452$	16.1304	0.758708
$453$	−12.7004	−0.596718
$454$	−18.9515	−0.889436
$455$	11.8082	0.553577
$456$	−1.81383	−0.0849402
$457$	−5.21124	−0.243772	−0.121886	−	0.992544i	$-0.538894\pi$
$457$	−5.21124	−0.243772	−0.121886	+	0.992544i	$0.538894\pi$
$458$	−6.67424	−0.311867
$459$	−3.41128	−0.159225
$460$	16.3785	0.763650
$461$	−30.0442	−1.39930	−0.699650	−	0.714486i	$-0.746661\pi$
$461$	−30.0442	−1.39930	−0.699650	+	0.714486i	$0.746661\pi$
$462$	6.38474	0.297045
$463$	−0.643402	−0.0299014	−0.0149507	−	0.999888i	$-0.504759\pi$
$463$	−0.643402	−0.0299014	−0.0149507	+	0.999888i	$0.504759\pi$
$464$	2.06774	0.0959925
$465$	20.8584	0.967283
$466$	0.547062	0.0253422
$467$	14.3241	0.662842	0.331421	−	0.943483i	$-0.392472\pi$
$467$	14.3241	0.662842	0.331421	+	0.943483i	$0.392472\pi$
$468$	−3.56515	−0.164799
$469$	3.79329	0.175158
$470$	−1.75022	−0.0807317
$471$	20.0621	0.924412
$472$	9.14905	0.421119
$473$	67.2519	3.09225
$474$	7.84545	0.360353
$475$	10.8288	0.496857
$476$	−3.41128	−0.156356
$477$	6.04890	0.276960
$478$	−5.56677	−0.254618
$479$	−10.7700	−0.492096	−0.246048	−	0.969258i	$-0.579132\pi$
$479$	−10.7700	−0.492096	−0.246048	+	0.969258i	$0.579132\pi$
$480$	3.31212	0.151177
$481$	16.4034	0.747930
$482$	27.6640	1.26006
$483$	−4.94502	−0.225006
$484$	29.7649	1.35295
$485$	−12.2134	−0.554581
$486$	−1.00000	−0.0453609
$487$	−38.0672	−1.72499	−0.862494	−	0.506067i	$-0.831099\pi$
$487$	−38.0672	−1.72499	−0.862494	+	0.506067i	$0.831099\pi$
$488$	−8.30484	−0.375942
$489$	17.4523	0.789218
$490$	3.31212	0.149626
$491$	−30.0943	−1.35814	−0.679068	−	0.734075i	$-0.737616\pi$
$491$	−30.0943	−1.35814	−0.679068	+	0.734075i	$0.737616\pi$
$492$	3.36340	0.151634
$493$	−7.05365	−0.317681
$494$	6.46657	0.290945
$495$	21.1470	0.950487
$496$	−6.29759	−0.282770
$497$	−7.57141	−0.339624
$498$	13.5794	0.608509
$499$	37.0948	1.66059	0.830295	−	0.557325i	$-0.188172\pi$
$499$	37.0948	1.66059	0.830295	+	0.557325i	$0.188172\pi$
$500$	−3.21314	−0.143696
$501$	−22.0422	−0.984772
$502$	−3.78162	−0.168782
$503$	−10.0144	−0.446518	−0.223259	−	0.974759i	$-0.571670\pi$
$503$	−10.0144	−0.446518	−0.223259	+	0.974759i	$0.571670\pi$
$504$	−1.00000	−0.0445435
$505$	−4.08375	−0.181724
$506$	−31.5726	−1.40358
$507$	−0.289681	−0.0128652
$508$	9.54153	0.423337
$509$	−12.5186	−0.554878	−0.277439	−	0.960743i	$-0.589486\pi$
$509$	−12.5186	−0.554878	−0.277439	+	0.960743i	$0.589486\pi$
$510$	−11.2986	−0.500309
$511$	2.22589	0.0984677
$512$	−1.00000	−0.0441942
$513$	1.81383	0.0800824
$514$	6.70756	0.295858
$515$	−1.02973	−0.0453753
$516$	−10.5332	−0.463700
$517$	3.37389	0.148383
$518$	4.60103	0.202158
$519$	18.5059	0.812319
$520$	−11.8082	−0.517824
$521$	−17.5869	−0.770495	−0.385247	−	0.922813i	$-0.625884\pi$
$521$	−17.5869	−0.770495	−0.385247	+	0.922813i	$0.625884\pi$
$522$	−2.06774	−0.0905026
$523$	34.3332	1.50129	0.750644	−	0.660707i	$-0.229744\pi$
$523$	34.3332	1.50129	0.750644	+	0.660707i	$0.229744\pi$
$524$	2.32765	0.101684
$525$	5.97012	0.260557
$526$	24.9422	1.08753
$527$	21.4829	0.935809
$528$	−6.38474	−0.277860
$529$	1.45319	0.0631821
$530$	20.0347	0.870251
$531$	−9.14905	−0.397035
$532$	1.81383	0.0786393
$533$	−11.9910	−0.519390
$534$	12.9315	0.559599
$535$	1.13152	0.0489197
$536$	−3.79329	−0.163845
$537$	−16.5151	−0.712681
$538$	−23.4152	−1.00950
$539$	−6.38474	−0.275010
$540$	−3.31212	−0.142531
$541$	7.13840	0.306904	0.153452	−	0.988156i	$-0.450961\pi$
$541$	7.13840	0.306904	0.153452	+	0.988156i	$0.450961\pi$
$542$	29.4519	1.26507
$543$	−13.5795	−0.582751
$544$	3.41128	0.146258
$545$	8.77934	0.376065
$546$	3.56515	0.152574
$547$	19.3452	0.827139	0.413569	−	0.910473i	$-0.364282\pi$
$547$	19.3452	0.827139	0.413569	+	0.910473i	$0.364282\pi$
$548$	−21.7668	−0.929830
$549$	8.30484	0.354442
$550$	38.1176	1.62534
$551$	3.75053	0.159778
$552$	4.94502	0.210474
$553$	−7.84545	−0.333622
$554$	−1.29421	−0.0549857
$555$	15.2392	0.646866
$556$	7.63589	0.323834
$557$	27.9400	1.18386	0.591929	−	0.805990i	$-0.298367\pi$
$557$	27.9400	1.18386	0.591929	+	0.805990i	$0.298367\pi$
$558$	6.29759	0.266598
$559$	37.5526	1.58831
$560$	−3.31212	−0.139962
$561$	21.7802	0.919559
$562$	8.60945	0.363168
$563$	14.8490	0.625811	0.312905	−	0.949784i	$-0.398698\pi$
$563$	14.8490	0.625811	0.312905	+	0.949784i	$0.398698\pi$
$564$	−0.528430	−0.0222509
$565$	−53.4256	−2.24763
$566$	13.4866	0.566886
$567$	1.00000	0.0419961
$568$	7.57141	0.317689
$569$	−3.07979	−0.129112	−0.0645558	−	0.997914i	$-0.520563\pi$
$569$	−3.07979	−0.129112	−0.0645558	+	0.997914i	$0.520563\pi$
$570$	6.00760	0.251631
$571$	−44.1288	−1.84673	−0.923367	−	0.383918i	$-0.874574\pi$
$571$	−44.1288	−1.84673	−0.923367	+	0.383918i	$0.874574\pi$
$572$	22.7626	0.951751
$573$	1.00000	0.0417756
$574$	−3.36340	−0.140386
$575$	−29.5223	−1.23117
$576$	1.00000	0.0416667
$577$	−44.4442	−1.85024	−0.925118	−	0.379681i	$-0.876034\pi$
$577$	−44.4442	−1.85024	−0.925118	+	0.379681i	$0.876034\pi$
$578$	5.36315	0.223078
$579$	16.1694	0.671978
$580$	−6.84860	−0.284373
$581$	−13.5794	−0.563370
$582$	−3.68748	−0.152851
$583$	−38.6207	−1.59950
$584$	−2.22589	−0.0921081
$585$	11.8082	0.488209
$586$	−23.9085	−0.987650
$587$	1.41322	0.0583299	0.0291650	−	0.999575i	$-0.490715\pi$
$587$	1.41322	0.0583299	0.0291650	+	0.999575i	$0.490715\pi$
$588$	1.00000	0.0412393
$589$	−11.4227	−0.470666
$590$	−30.3027	−1.24754
$591$	−9.57180	−0.393731
$592$	−4.60103	−0.189101
$593$	−23.5738	−0.968061	−0.484030	−	0.875051i	$-0.660828\pi$
$593$	−23.5738	−0.968061	−0.484030	+	0.875051i	$0.660828\pi$
$594$	6.38474	0.261969
$595$	11.2986	0.463196
$596$	20.0880	0.822835
$597$	13.6172	0.557317
$598$	−17.6297	−0.720934
$599$	1.75090	0.0715400	0.0357700	−	0.999360i	$-0.488612\pi$
$599$	1.75090	0.0715400	0.0357700	+	0.999360i	$0.488612\pi$
$600$	−5.97012	−0.243729
$601$	27.6436	1.12761	0.563804	−	0.825909i	$-0.309337\pi$
$601$	27.6436	1.12761	0.563804	+	0.825909i	$0.309337\pi$
$602$	10.5332	0.429303
$603$	3.79329	0.154475
$604$	−12.7004	−0.516773
$605$	−98.5848	−4.00804
$606$	−1.23297	−0.0500861
$607$	21.6496	0.878732	0.439366	−	0.898308i	$-0.355203\pi$
$607$	21.6496	0.878732	0.439366	+	0.898308i	$0.355203\pi$
$608$	−1.81383	−0.0735604
$609$	2.06774	0.0837892
$610$	27.5066	1.11371
$611$	1.88393	0.0762158
$612$	−3.41128	−0.137893
$613$	−10.7277	−0.433286	−0.216643	−	0.976251i	$-0.569511\pi$
$613$	−10.7277	−0.433286	−0.216643	+	0.976251i	$0.569511\pi$
$614$	−12.1166	−0.488985
$615$	−11.1400	−0.449207
$616$	6.38474	0.257248
$617$	−9.64658	−0.388357	−0.194178	−	0.980966i	$-0.562204\pi$
$617$	−9.64658	−0.388357	−0.194178	+	0.980966i	$0.562204\pi$
$618$	−0.310898	−0.0125061
$619$	5.47495	0.220057	0.110028	−	0.993928i	$-0.464906\pi$
$619$	5.47495	0.220057	0.110028	+	0.993928i	$0.464906\pi$
$620$	20.8584	0.837692
$621$	−4.94502	−0.198437
$622$	−5.92659	−0.237635
$623$	−12.9315	−0.518088
$624$	−3.56515	−0.142720
$625$	−19.2083	−0.768332
$626$	5.21064	0.208259
$627$	−11.5808	−0.462493
$628$	20.0621	0.800564
$629$	15.6954	0.625818
$630$	3.31212	0.131958
$631$	47.2389	1.88055	0.940275	−	0.340415i	$-0.110568\pi$
$631$	47.2389	1.88055	0.940275	+	0.340415i	$0.110568\pi$
$632$	7.84545	0.312075
$633$	2.26871	0.0901732
$634$	−13.2714	−0.527074
$635$	−31.6027	−1.25411
$636$	6.04890	0.239855
$637$	−3.56515	−0.141257
$638$	13.2020	0.522672
$639$	−7.57141	−0.299520
$640$	3.31212	0.130923
$641$	−39.4888	−1.55971	−0.779856	−	0.625959i	$-0.784708\pi$
$641$	−39.4888	−1.55971	−0.779856	+	0.625959i	$0.784708\pi$
$642$	0.341629	0.0134830
$643$	−10.0115	−0.394814	−0.197407	−	0.980322i	$-0.563252\pi$
$643$	−10.0115	−0.394814	−0.197407	+	0.980322i	$0.563252\pi$
$644$	−4.94502	−0.194861
$645$	34.8873	1.37369
$646$	6.18747	0.243443
$647$	16.7125	0.657038	0.328519	−	0.944497i	$-0.393450\pi$
$647$	16.7125	0.657038	0.328519	+	0.944497i	$0.393450\pi$
$648$	−1.00000	−0.0392837
$649$	58.4143	2.29296
$650$	21.2844	0.834842
$651$	−6.29759	−0.246822
$652$	17.4523	0.683483
$653$	−32.0881	−1.25571	−0.627853	−	0.778332i	$-0.716066\pi$
$653$	−32.0881	−1.25571	−0.627853	+	0.778332i	$0.716066\pi$
$654$	2.65067	0.103649
$655$	−7.70944	−0.301233
$656$	3.36340	0.131319
$657$	2.22589	0.0868403
$658$	0.528430	0.0206004
$659$	−12.2457	−0.477026	−0.238513	−	0.971139i	$-0.576660\pi$
$659$	−12.2457	−0.477026	−0.238513	+	0.971139i	$0.576660\pi$
$660$	21.1470	0.823146
$661$	23.7859	0.925166	0.462583	−	0.886576i	$-0.346923\pi$
$661$	23.7859	0.925166	0.462583	+	0.886576i	$0.346923\pi$
$662$	−24.0280	−0.933875
$663$	12.1617	0.472323
$664$	13.5794	0.526985
$665$	−6.00760	−0.232965
$666$	4.60103	0.178286
$667$	−10.2250	−0.395914
$668$	−22.0422	−0.852837
$669$	−11.7462	−0.454134
$670$	12.5638	0.485382
$671$	−53.0242	−2.04698
$672$	−1.00000	−0.0385758
$673$	5.27582	0.203368	0.101684	−	0.994817i	$-0.467577\pi$
$673$	5.27582	0.203368	0.101684	+	0.994817i	$0.467577\pi$
$674$	−6.87409	−0.264780
$675$	5.97012	0.229790
$676$	−0.289681	−0.0111416
$677$	1.56851	0.0602827	0.0301413	−	0.999546i	$-0.490404\pi$
$677$	1.56851	0.0602827	0.0301413	+	0.999546i	$0.490404\pi$
$678$	−16.1304	−0.619483
$679$	3.68748	0.141513
$680$	−11.2986	−0.433280
$681$	18.9515	0.726221
$682$	−40.2085	−1.53966
$683$	−7.59130	−0.290473	−0.145237	−	0.989397i	$-0.546394\pi$
$683$	−7.59130	−0.290473	−0.145237	+	0.989397i	$0.546394\pi$
$684$	1.81383	0.0693534
$685$	72.0940	2.75457
$686$	−1.00000	−0.0381802
$687$	6.67424	0.254638
$688$	−10.5332	−0.401576
$689$	−21.5653	−0.821571
$690$	−16.3785	−0.623518
$691$	18.6002	0.707585	0.353792	−	0.935324i	$-0.384892\pi$
$691$	18.6002	0.707585	0.353792	+	0.935324i	$0.384892\pi$
$692$	18.5059	0.703489
$693$	−6.38474	−0.242536
$694$	6.46553	0.245428
$695$	−25.2909	−0.959340
$696$	−2.06774	−0.0783776
$697$	−11.4735	−0.434590
$698$	−12.1779	−0.460941
$699$	−0.547062	−0.0206918
$700$	5.97012	0.225649
$701$	29.9365	1.13069	0.565344	−	0.824855i	$-0.308743\pi$
$701$	29.9365	1.13069	0.565344	+	0.824855i	$0.308743\pi$
$702$	3.56515	0.134558
$703$	−8.34548	−0.314756
$704$	−6.38474	−0.240634
$705$	1.75022	0.0659172
$706$	−20.6409	−0.776832
$707$	1.23297	0.0463707
$708$	−9.14905	−0.343842
$709$	−2.02396	−0.0760113	−0.0380057	−	0.999278i	$-0.512100\pi$
$709$	−2.02396	−0.0760113	−0.0380057	+	0.999278i	$0.512100\pi$
$710$	−25.0774	−0.941138
$711$	−7.84545	−0.294227
$712$	12.9315	0.484627
$713$	31.1417	1.16627
$714$	3.41128	0.127664
$715$	−75.3923	−2.81951
$716$	−16.5151	−0.617200
$717$	5.56677	0.207895
$718$	−1.64957	−0.0615614
$719$	38.2801	1.42761	0.713803	−	0.700346i	$-0.246971\pi$
$719$	38.2801	1.42761	0.713803	+	0.700346i	$0.246971\pi$
$720$	−3.31212	−0.123435
$721$	0.310898	0.0115784
$722$	15.7100	0.584667
$723$	−27.6640	−1.02884
$724$	−13.5795	−0.504677
$725$	12.3447	0.458469
$726$	−29.7649	−1.10468
$727$	30.1681	1.11887	0.559436	−	0.828874i	$-0.311018\pi$
$727$	30.1681	1.11887	0.559436	+	0.828874i	$0.311018\pi$
$728$	3.56515	0.132133
$729$	1.00000	0.0370370
$730$	7.37242	0.272865
$731$	35.9318	1.32899
$732$	8.30484	0.306956
$733$	6.79920	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$733$	6.79920	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$734$	5.83621	0.215418
$735$	−3.31212	−0.122169
$736$	4.94502	0.182276
$737$	−24.2191	−0.892124
$738$	−3.36340	−0.123809
$739$	−15.9636	−0.587231	−0.293615	−	0.955924i	$-0.594858\pi$
$739$	−15.9636	−0.587231	−0.293615	+	0.955924i	$0.594858\pi$
$740$	15.2392	0.560203
$741$	−6.46657	−0.237555
$742$	−6.04890	−0.222062
$743$	10.8683	0.398717	0.199359	−	0.979927i	$-0.436114\pi$
$743$	10.8683	0.398717	0.199359	+	0.979927i	$0.436114\pi$
$744$	6.29759	0.230881
$745$	−66.5336	−2.43760
$746$	−9.02608	−0.330468
$747$	−13.5794	−0.496846
$748$	21.7802	0.796361
$749$	−0.341629	−0.0124829
$750$	3.21314	0.117327
$751$	36.7505	1.34105	0.670523	−	0.741889i	$-0.266070\pi$
$751$	36.7505	1.34105	0.670523	+	0.741889i	$0.266070\pi$
$752$	−0.528430	−0.0192699
$753$	3.78162	0.137810
$754$	7.37182	0.268466
$755$	42.0653	1.53091
$756$	1.00000	0.0363696
$757$	29.9297	1.08781	0.543907	−	0.839146i	$-0.316944\pi$
$757$	29.9297	1.08781	0.543907	+	0.839146i	$0.316944\pi$
$758$	18.2490	0.662833
$759$	31.5726	1.14601
$760$	6.00760	0.217919
$761$	26.0340	0.943732	0.471866	−	0.881670i	$-0.343581\pi$
$761$	26.0340	0.943732	0.471866	+	0.881670i	$0.343581\pi$
$762$	−9.54153	−0.345653
$763$	−2.65067	−0.0959608
$764$	1.00000	0.0361787
$765$	11.2986	0.408501
$766$	−33.1742	−1.19863
$767$	32.6178	1.17776
$768$	1.00000	0.0360844
$769$	−16.3565	−0.589830	−0.294915	−	0.955523i	$-0.595291\pi$
$769$	−16.3565	−0.589830	−0.294915	+	0.955523i	$0.595291\pi$
$770$	−21.1470	−0.762085
$771$	−6.70756	−0.241567
$772$	16.1694	0.581950
$773$	35.7804	1.28693	0.643465	−	0.765475i	$-0.277496\pi$
$773$	35.7804	1.28693	0.643465	+	0.765475i	$0.277496\pi$
$774$	10.5332	0.378609
$775$	−37.5974	−1.35054
$776$	−3.68748	−0.132373
$777$	−4.60103	−0.165061
$778$	−10.7704	−0.386139
$779$	6.10063	0.218578
$780$	11.8082	0.422802
$781$	48.3415	1.72979
$782$	−16.8688	−0.603229
$783$	2.06774	0.0738951
$784$	1.00000	0.0357143
$785$	−66.4479	−2.37163
$786$	−2.32765	−0.0830244
$787$	−34.6232	−1.23418	−0.617092	−	0.786891i	$-0.711689\pi$
$787$	−34.6232	−1.23418	−0.617092	+	0.786891i	$0.711689\pi$
$788$	−9.57180	−0.340981
$789$	−24.9422	−0.887965
$790$	−25.9850	−0.924506
$791$	16.1304	0.573530
$792$	6.38474	0.226872
$793$	−29.6080	−1.05141
$794$	3.63234	0.128907
$795$	−20.0347	−0.710557
$796$	13.6172	0.482650
$797$	−19.4499	−0.688950	−0.344475	−	0.938796i	$-0.611943\pi$
$797$	−19.4499	−0.688950	−0.344475	+	0.938796i	$0.611943\pi$
$798$	−1.81383	−0.0642088
$799$	1.80262	0.0637723
$800$	−5.97012	−0.211075
$801$	−12.9315	−0.456911
$802$	−27.0386	−0.954764
$803$	−14.2117	−0.501522
$804$	3.79329	0.133779
$805$	16.3785	0.577265
$806$	−22.4519	−0.790834
$807$	23.4152	0.824256
$808$	−1.23297	−0.0433758
$809$	−22.6621	−0.796756	−0.398378	−	0.917221i	$-0.630427\pi$
$809$	−22.6621	−0.796756	−0.398378	+	0.917221i	$0.630427\pi$
$810$	3.31212	0.116376
$811$	−3.93941	−0.138331	−0.0691657	−	0.997605i	$-0.522034\pi$
$811$	−3.93941	−0.138331	−0.0691657	+	0.997605i	$0.522034\pi$
$812$	2.06774	0.0725635
$813$	−29.4519	−1.03292
$814$	−29.3764	−1.02964
$815$	−57.8039	−2.02478
$816$	−3.41128	−0.119419
$817$	−19.1055	−0.668415
$818$	−23.6115	−0.825558
$819$	−3.56515	−0.124577
$820$	−11.1400	−0.389025
$821$	−18.3491	−0.640390	−0.320195	−	0.947352i	$-0.603748\pi$
$821$	−18.3491	−0.640390	−0.320195	+	0.947352i	$0.603748\pi$
$822$	21.7668	0.759203
$823$	−19.6216	−0.683964	−0.341982	−	0.939706i	$-0.611098\pi$
$823$	−19.6216	−0.683964	−0.341982	+	0.939706i	$0.611098\pi$
$824$	−0.310898	−0.0108306
$825$	−38.1176	−1.32709
$826$	9.14905	0.318336
$827$	38.4592	1.33736	0.668679	−	0.743551i	$-0.266860\pi$
$827$	38.4592	1.33736	0.668679	+	0.743551i	$0.266860\pi$
$828$	−4.94502	−0.171851
$829$	−29.8562	−1.03695	−0.518475	−	0.855093i	$-0.673500\pi$
$829$	−29.8562	−1.03695	−0.518475	+	0.855093i	$0.673500\pi$
$830$	−44.9767	−1.56116
$831$	1.29421	0.0448956
$832$	−3.56515	−0.123599
$833$	−3.41128	−0.118194
$834$	−7.63589	−0.264409
$835$	73.0063	2.52649
$836$	−11.5808	−0.400531
$837$	−6.29759	−0.217677
$838$	−25.3058	−0.874173
$839$	16.2400	0.560666	0.280333	−	0.959903i	$-0.409555\pi$
$839$	16.2400	0.560666	0.280333	+	0.959903i	$0.409555\pi$
$840$	3.31212	0.114279
$841$	−24.7244	−0.852567
$842$	−4.39287	−0.151388
$843$	−8.60945	−0.296525
$844$	2.26871	0.0780922
$845$	0.959459	0.0330064
$846$	0.528430	0.0181678
$847$	29.7649	1.02273
$848$	6.04890	0.207720
$849$	−13.4866	−0.462860
$850$	20.3658	0.698540
$851$	22.7522	0.779935
$852$	−7.57141	−0.259392
$853$	24.4506	0.837172	0.418586	−	0.908177i	$-0.362526\pi$
$853$	24.4506	0.837172	0.418586	+	0.908177i	$0.362526\pi$
$854$	−8.30484	−0.284186
$855$	−6.00760	−0.205456
$856$	0.341629	0.0116766
$857$	−43.4943	−1.48574	−0.742869	−	0.669437i	$-0.766535\pi$
$857$	−43.4943	−1.48574	−0.742869	+	0.669437i	$0.766535\pi$
$858$	−22.7626	−0.777102
$859$	−35.7634	−1.22023	−0.610116	−	0.792312i	$-0.708877\pi$
$859$	−35.7634	−1.22023	−0.610116	+	0.792312i	$0.708877\pi$
$860$	34.8873	1.18965
$861$	3.36340	0.114624
$862$	19.3472	0.658968
$863$	33.7176	1.14776	0.573880	−	0.818940i	$-0.305438\pi$
$863$	33.7176	1.14776	0.573880	+	0.818940i	$0.305438\pi$
$864$	−1.00000	−0.0340207
$865$	−61.2937	−2.08405
$866$	25.5837	0.869370
$867$	−5.36315	−0.182142
$868$	−6.29759	−0.213754
$869$	50.0912	1.69923
$870$	6.84860	0.232189
$871$	−13.5236	−0.458231
$872$	2.65067	0.0897631
$873$	3.68748	0.124802
$874$	8.96940	0.303395
$875$	−3.21314	−0.108624
$876$	2.22589	0.0752059
$877$	−10.3082	−0.348084	−0.174042	−	0.984738i	$-0.555683\pi$
$877$	−10.3082	−0.348084	−0.174042	+	0.984738i	$0.555683\pi$
$878$	31.5534	1.06488
$879$	23.9085	0.806413
$880$	21.1470	0.712865
$881$	4.71401	0.158819	0.0794096	−	0.996842i	$-0.474697\pi$
$881$	4.71401	0.158819	0.0794096	+	0.996842i	$0.474697\pi$
$882$	−1.00000	−0.0336718
$883$	−11.1090	−0.373846	−0.186923	−	0.982375i	$-0.559852\pi$
$883$	−11.1090	−0.373846	−0.186923	+	0.982375i	$0.559852\pi$
$884$	12.1617	0.409044
$885$	30.3027	1.01861
$886$	−6.91771	−0.232405
$887$	59.1289	1.98535	0.992677	−	0.120798i	$-0.0385454\pi$
$887$	59.1289	1.98535	0.992677	+	0.120798i	$0.0385454\pi$
$888$	4.60103	0.154401
$889$	9.54153	0.320013
$890$	−42.8305	−1.43568
$891$	−6.38474	−0.213897
$892$	−11.7462	−0.393292
$893$	−0.958481	−0.0320743
$894$	−20.0880	−0.671842
$895$	54.7001	1.82842
$896$	−1.00000	−0.0334077
$897$	17.6297	0.588640
$898$	−0.0802325	−0.00267739
$899$	−13.0218	−0.434301
$900$	5.97012	0.199004
$901$	−20.6345	−0.687436
$902$	21.4745	0.715021
$903$	−10.5332	−0.350524
$904$	−16.1304	−0.536488
$905$	44.9768	1.49508
$906$	12.7004	0.421944
$907$	2.29025	0.0760464	0.0380232	−	0.999277i	$-0.487894\pi$
$907$	2.29025	0.0760464	0.0380232	+	0.999277i	$0.487894\pi$
$908$	18.9515	0.628926
$909$	1.23297	0.0408951
$910$	−11.8082	−0.391438
$911$	0.235291	0.00779553	0.00389776	−	0.999992i	$-0.498759\pi$
$911$	0.235291	0.00779553	0.00389776	+	0.999992i	$0.498759\pi$
$912$	1.81383	0.0600618
$913$	86.7012	2.86939
$914$	5.21124	0.172372
$915$	−27.5066	−0.909340
$916$	6.67424	0.220523
$917$	2.32765	0.0768657
$918$	3.41128	0.112589
$919$	7.34643	0.242336	0.121168	−	0.992632i	$-0.461336\pi$
$919$	7.34643	0.242336	0.121168	+	0.992632i	$0.461336\pi$
$920$	−16.3785	−0.539982
$921$	12.1166	0.399254
$922$	30.0442	0.989454
$923$	26.9932	0.888493
$924$	−6.38474	−0.210043
$925$	−27.4687	−0.903165
$926$	0.643402	0.0211435
$927$	0.310898	0.0102112
$928$	−2.06774	−0.0678770
$929$	27.1790	0.891714	0.445857	−	0.895104i	$-0.352899\pi$
$929$	27.1790	0.891714	0.445857	+	0.895104i	$0.352899\pi$
$930$	−20.8584	−0.683973
$931$	1.81383	0.0594458
$932$	−0.547062	−0.0179196
$933$	5.92659	0.194028
$934$	−14.3241	−0.468700
$935$	−72.1384	−2.35918
$936$	3.56515	0.116531
$937$	35.6079	1.16326	0.581630	−	0.813453i	$-0.302415\pi$
$937$	35.6079	1.16326	0.581630	+	0.813453i	$0.302415\pi$
$938$	−3.79329	−0.123855
$939$	−5.21064	−0.170043
$940$	1.75022	0.0570860
$941$	42.3559	1.38076	0.690381	−	0.723446i	$-0.257443\pi$
$941$	42.3559	1.38076	0.690381	+	0.723446i	$0.257443\pi$
$942$	−20.0621	−0.653658
$943$	−16.6321	−0.541615
$944$	−9.14905	−0.297776
$945$	−3.31212	−0.107743
$946$	−67.2519	−2.18655
$947$	−60.4995	−1.96597	−0.982984	−	0.183690i	$-0.941196\pi$
$947$	−60.4995	−1.96597	−0.982984	+	0.183690i	$0.941196\pi$
$948$	−7.84545	−0.254808
$949$	−7.93565	−0.257602
$950$	−10.8288	−0.351331
$951$	13.2714	0.430354
$952$	3.41128	0.110560
$953$	30.1044	0.975178	0.487589	−	0.873073i	$-0.337876\pi$
$953$	30.1044	0.975178	0.487589	+	0.873073i	$0.337876\pi$
$954$	−6.04890	−0.195840
$955$	−3.31212	−0.107178
$956$	5.56677	0.180042
$957$	−13.2020	−0.426760
$958$	10.7700	0.347964
$959$	−21.7668	−0.702885
$960$	−3.31212	−0.106898
$961$	8.65968	0.279345
$962$	−16.4034	−0.528867
$963$	−0.341629	−0.0110088
$964$	−27.6640	−0.890998
$965$	−53.5550	−1.72400
$966$	4.94502	0.159103
$967$	17.5263	0.563608	0.281804	−	0.959472i	$-0.409067\pi$
$967$	17.5263	0.563608	0.281804	+	0.959472i	$0.409067\pi$
$968$	−29.7649	−0.956680
$969$	−6.18747	−0.198770
$970$	12.2134	0.392148
$971$	54.7581	1.75727	0.878636	−	0.477493i	$-0.158454\pi$
$971$	54.7581	1.75727	0.878636	+	0.477493i	$0.158454\pi$
$972$	1.00000	0.0320750
$973$	7.63589	0.244795
$974$	38.0672	1.21975
$975$	−21.2844	−0.681646
$976$	8.30484	0.265831
$977$	−38.2648	−1.22420	−0.612100	−	0.790780i	$-0.709675\pi$
$977$	−38.2648	−1.22420	−0.612100	+	0.790780i	$0.709675\pi$
$978$	−17.4523	−0.558061
$979$	82.5641	2.63876
$980$	−3.31212	−0.105802
$981$	−2.65067	−0.0846294
$982$	30.0943	0.960347
$983$	32.6646	1.04184	0.520920	−	0.853605i	$-0.325589\pi$
$983$	32.6646	1.04184	0.520920	+	0.853605i	$0.325589\pi$
$984$	−3.36340	−0.107221
$985$	31.7029	1.01014
$986$	7.05365	0.224634
$987$	−0.528430	−0.0168201
$988$	−6.46657	−0.205729
$989$	52.0870	1.65627
$990$	−21.1470	−0.672096
$991$	−50.7631	−1.61254	−0.806272	−	0.591545i	$-0.798518\pi$
$991$	−50.7631	−1.61254	−0.806272	+	0.591545i	$0.798518\pi$
$992$	6.29759	0.199949
$993$	24.0280	0.762506
$994$	7.57141	0.240151
$995$	−45.1019	−1.42983
$996$	−13.5794	−0.430281
$997$	−22.9990	−0.728386	−0.364193	−	0.931324i	$-0.618655\pi$
$997$	−22.9990	−0.728386	−0.364193	+	0.931324i	$0.618655\pi$
$998$	−37.0948	−1.17421
$999$	−4.60103	−0.145570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8022.2.a.z.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8022.2.a.z.1.3	✓	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.31212 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.31212 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.31212 q^{10} -6.38474 q^{11} +1.00000 q^{12} -3.56515 q^{13} -1.00000 q^{14} -3.31212 q^{15} +1.00000 q^{16} -3.41128 q^{17} -1.00000 q^{18} +1.81383 q^{19} -3.31212 q^{20} +1.00000 q^{21} +6.38474 q^{22} -4.94502 q^{23} -1.00000 q^{24} +5.97012 q^{25} +3.56515 q^{26} +1.00000 q^{27} +1.00000 q^{28} +2.06774 q^{29} +3.31212 q^{30} -6.29759 q^{31} -1.00000 q^{32} -6.38474 q^{33} +3.41128 q^{34} -3.31212 q^{35} +1.00000 q^{36} -4.60103 q^{37} -1.81383 q^{38} -3.56515 q^{39} +3.31212 q^{40} +3.36340 q^{41} -1.00000 q^{42} -10.5332 q^{43} -6.38474 q^{44} -3.31212 q^{45} +4.94502 q^{46} -0.528430 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.97012 q^{50} -3.41128 q^{51} -3.56515 q^{52} +6.04890 q^{53} -1.00000 q^{54} +21.1470 q^{55} -1.00000 q^{56} +1.81383 q^{57} -2.06774 q^{58} -9.14905 q^{59} -3.31212 q^{60} +8.30484 q^{61} +6.29759 q^{62} +1.00000 q^{63} +1.00000 q^{64} +11.8082 q^{65} +6.38474 q^{66} +3.79329 q^{67} -3.41128 q^{68} -4.94502 q^{69} +3.31212 q^{70} -7.57141 q^{71} -1.00000 q^{72} +2.22589 q^{73} +4.60103 q^{74} +5.97012 q^{75} +1.81383 q^{76} -6.38474 q^{77} +3.56515 q^{78} -7.84545 q^{79} -3.31212 q^{80} +1.00000 q^{81} -3.36340 q^{82} -13.5794 q^{83} +1.00000 q^{84} +11.2986 q^{85} +10.5332 q^{86} +2.06774 q^{87} +6.38474 q^{88} -12.9315 q^{89} +3.31212 q^{90} -3.56515 q^{91} -4.94502 q^{92} -6.29759 q^{93} +0.528430 q^{94} -6.00760 q^{95} -1.00000 q^{96} +3.68748 q^{97} -1.00000 q^{98} -6.38474 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9	\( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9 - 7 * q^11 + 15 * q^12 + 10 * q^13 - 15 * q^14 + 15 * q^16 - 3 * q^17 - 15 * q^18 + 12 * q^19 + 15 * q^21 + 7 * q^22 - q^23 - 15 * q^24 + 31 * q^25 - 10 * q^26 + 15 * q^27 + 15 * q^28 - 3 * q^29 + 19 * q^31 - 15 * q^32 - 7 * q^33 + 3 * q^34 + 15 * q^36 + 25 * q^37 - 12 * q^38 + 10 * q^39 + 8 * q^41 - 15 * q^42 + 25 * q^43 - 7 * q^44 + q^46 + 11 * q^47 + 15 * q^48 + 15 * q^49 - 31 * q^50 - 3 * q^51 + 10 * q^52 - 4 * q^53 - 15 * q^54 + 9 * q^55 - 15 * q^56 + 12 * q^57 + 3 * q^58 + 27 * q^61 - 19 * q^62 + 15 * q^63 + 15 * q^64 - 2 * q^65 + 7 * q^66 + 31 * q^67 - 3 * q^68 - q^69 - 16 * q^71 - 15 * q^72 + 26 * q^73 - 25 * q^74 + 31 * q^75 + 12 * q^76 - 7 * q^77 - 10 * q^78 + 32 * q^79 + 15 * q^81 - 8 * q^82 + 7 * q^83 + 15 * q^84 + 26 * q^85 - 25 * q^86 - 3 * q^87 + 7 * q^88 - 11 * q^89 + 10 * q^91 - q^92 + 19 * q^93 - 11 * q^94 - 8 * q^95 - 15 * q^96 + 30 * q^97 - 15 * q^98 - 7 * q^99

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