LMFDB - Embedded newform 85.6.a.a.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [85,6,Mod(1,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	85.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.6326246841$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.95319$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.95319 q^{2} -9.94571 q^{3} -7.46587 q^{4} +25.0000 q^{5} +49.2631 q^{6} +13.5050 q^{7} +195.482 q^{8} -144.083 q^{9} -123.830 q^{10} +742.882 q^{11} +74.2534 q^{12} -179.248 q^{13} -66.8928 q^{14} -248.643 q^{15} -729.353 q^{16} +289.000 q^{17} +713.670 q^{18} -1855.97 q^{19} -186.647 q^{20} -134.317 q^{21} -3679.64 q^{22} -2557.44 q^{23} -1944.21 q^{24} +625.000 q^{25} +887.850 q^{26} +3849.81 q^{27} -100.826 q^{28} -1108.67 q^{29} +1231.58 q^{30} -8774.81 q^{31} -2642.80 q^{32} -7388.50 q^{33} -1431.47 q^{34} +337.625 q^{35} +1075.70 q^{36} -5237.19 q^{37} +9192.99 q^{38} +1782.75 q^{39} +4887.05 q^{40} -16598.4 q^{41} +665.297 q^{42} +14484.7 q^{43} -5546.26 q^{44} -3602.07 q^{45} +12667.5 q^{46} +16088.0 q^{47} +7253.94 q^{48} -16624.6 q^{49} -3095.75 q^{50} -2874.31 q^{51} +1338.24 q^{52} +11804.4 q^{53} -19068.9 q^{54} +18572.1 q^{55} +2639.98 q^{56} +18459.0 q^{57} +5491.46 q^{58} -29390.1 q^{59} +1856.33 q^{60} +29333.5 q^{61} +43463.3 q^{62} -1945.84 q^{63} +36429.6 q^{64} -4481.20 q^{65} +36596.7 q^{66} +4122.11 q^{67} -2157.64 q^{68} +25435.6 q^{69} -1672.32 q^{70} -13649.5 q^{71} -28165.6 q^{72} -33582.1 q^{73} +25940.8 q^{74} -6216.07 q^{75} +13856.4 q^{76} +10032.6 q^{77} -8830.30 q^{78} -86281.4 q^{79} -18233.8 q^{80} -3277.05 q^{81} +82215.3 q^{82} -73620.8 q^{83} +1002.79 q^{84} +7225.00 q^{85} -71745.7 q^{86} +11026.5 q^{87} +145220. q^{88} -100199. q^{89} +17841.7 q^{90} -2420.74 q^{91} +19093.5 q^{92} +87271.8 q^{93} -79687.2 q^{94} -46399.3 q^{95} +26284.5 q^{96} +4407.67 q^{97} +82344.9 q^{98} -107037. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 7 q^{2} - 36 q^{3} + 43 q^{4} + 125 q^{5} - 105 q^{6} - 204 q^{7} - 63 q^{8} + 531 q^{9} - 175 q^{10} - 792 q^{11} + 785 q^{12} + 88 q^{13} + 860 q^{14} - 900 q^{15} - 2365 q^{16} + 1445 q^{17} - 2052 q^{18}+ \cdots - 535112 q^{99}+O(q^{100}) $ 5 * q - 7 * q^2 - 36 * q^3 + 43 * q^4 + 125 * q^5 - 105 * q^6 - 204 * q^7 - 63 * q^8 + 531 * q^9 - 175 * q^10 - 792 * q^11 + 785 * q^12 + 88 * q^13 + 860 * q^14 - 900 * q^15 - 2365 * q^16 + 1445 * q^17 - 2052 * q^18 - 5160 * q^19 + 1075 * q^20 - 6428 * q^21 - 3058 * q^22 - 6140 * q^23 - 11751 * q^24 + 3125 * q^25 - 9643 * q^26 - 13380 * q^27 - 4568 * q^28 - 7328 * q^29 - 2625 * q^30 - 8968 * q^31 - 4703 * q^32 + 13072 * q^33 - 2023 * q^34 - 5100 * q^35 + 2048 * q^36 + 9834 * q^37 + 10625 * q^38 + 10108 * q^39 - 1575 * q^40 - 5302 * q^41 + 16222 * q^42 - 4916 * q^43 - 31514 * q^44 + 13275 * q^45 - 16768 * q^46 - 20364 * q^47 + 45557 * q^48 - 26483 * q^49 - 4375 * q^50 - 10404 * q^51 - 11525 * q^52 + 56492 * q^53 + 64635 * q^54 - 19800 * q^55 + 73228 * q^56 + 67554 * q^57 + 54213 * q^58 - 96200 * q^59 + 19625 * q^60 - 7284 * q^61 + 122335 * q^62 + 64108 * q^63 + 28483 * q^64 + 2200 * q^65 + 115232 * q^66 - 60308 * q^67 + 12427 * q^68 - 89076 * q^69 + 21500 * q^70 - 25304 * q^71 + 118722 * q^72 + 163328 * q^73 - 82768 * q^74 - 22500 * q^75 + 92027 * q^76 + 19912 * q^77 + 134895 * q^78 - 50940 * q^79 - 59125 * q^80 + 92445 * q^81 + 191390 * q^82 - 179060 * q^83 - 128054 * q^84 + 36125 * q^85 + 87086 * q^86 + 55764 * q^87 + 167634 * q^88 - 387356 * q^89 - 51300 * q^90 - 115200 * q^91 - 293228 * q^92 - 202558 * q^93 + 100515 * q^94 - 129000 * q^95 + 428113 * q^96 - 106076 * q^97 + 49329 * q^98 - 535112 * q^99

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 $

$2$	−4.95319	−0.875609	−0.437805	−	0.899070i	$-0.644244\pi$
$2$	−4.95319	−0.875609	−0.437805	+	0.899070i	$0.644244\pi$
$3$	−9.94571	−0.638018	−0.319009	−	0.947752i	$-0.603350\pi$
$3$	−9.94571	−0.638018	−0.319009	+	0.947752i	$0.603350\pi$
$4$	−7.46587	−0.233308
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	49.2631	0.558654
$7$	13.5050	0.104172	0.0520858	−	0.998643i	$-0.483413\pi$
$7$	13.5050	0.104172	0.0520858	+	0.998643i	$0.483413\pi$
$8$	195.482	1.07990
$9$	−144.083	−0.592933
$10$	−123.830	−0.391584
$11$	742.882	1.85114	0.925568	−	0.378581i	$-0.123588\pi$
$11$	742.882	1.85114	0.925568	+	0.378581i	$0.123588\pi$
$12$	74.2534	0.148855
$13$	−179.248	−0.294168	−0.147084	−	0.989124i	$-0.546989\pi$
$13$	−179.248	−0.294168	−0.147084	+	0.989124i	$0.546989\pi$
$14$	−66.8928	−0.0912135
$15$	−248.643	−0.285330
$16$	−729.353	−0.712259
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	713.670	0.519178
$19$	−1855.97	−1.17947	−0.589735	−	0.807596i	$-0.700768\pi$
$19$	−1855.97	−1.17947	−0.589735	+	0.807596i	$0.700768\pi$
$20$	−186.647	−0.104339
$21$	−134.317	−0.0664633
$22$	−3679.64	−1.62087
$23$	−2557.44	−1.00806	−0.504029	−	0.863687i	$-0.668150\pi$
$23$	−2557.44	−1.00806	−0.504029	+	0.863687i	$0.668150\pi$
$24$	−1944.21	−0.688993
$25$	625.000	0.200000
$26$	887.850	0.257577
$27$	3849.81	1.01632
$28$	−100.826	−0.0243041
$29$	−1108.67	−0.244798	−0.122399	−	0.992481i	$-0.539059\pi$
$29$	−1108.67	−0.244798	−0.122399	+	0.992481i	$0.539059\pi$
$30$	1231.58	0.249838
$31$	−8774.81	−1.63996	−0.819980	−	0.572391i	$-0.806016\pi$
$31$	−8774.81	−1.63996	−0.819980	+	0.572391i	$0.806016\pi$
$32$	−2642.80	−0.456236
$33$	−7388.50	−1.18106
$34$	−1431.47	−0.212366
$35$	337.625	0.0465869
$36$	1075.70	0.138336
$37$	−5237.19	−0.628919	−0.314459	−	0.949271i	$-0.601823\pi$
$37$	−5237.19	−0.628919	−0.314459	+	0.949271i	$0.601823\pi$
$38$	9192.99	1.03276
$39$	1782.75	0.187685
$40$	4887.05	0.482944
$41$	−16598.4	−1.54208	−0.771042	−	0.636785i	$-0.780264\pi$
$41$	−16598.4	−1.54208	−0.771042	+	0.636785i	$0.780264\pi$
$42$	665.297	0.0581959
$43$	14484.7	1.19465	0.597324	−	0.802000i	$-0.296231\pi$
$43$	14484.7	1.19465	0.597324	+	0.802000i	$0.296231\pi$
$44$	−5546.26	−0.431886
$45$	−3602.07	−0.265168
$46$	12667.5	0.882665
$47$	16088.0	1.06233	0.531164	−	0.847269i	$-0.321755\pi$
$47$	16088.0	1.06233	0.531164	+	0.847269i	$0.321755\pi$
$48$	7253.94	0.454434
$49$	−16624.6	−0.989148
$50$	−3095.75	−0.175122
$51$	−2874.31	−0.154742
$52$	1338.24	0.0686319
$53$	11804.4	0.577237	0.288618	−	0.957444i	$-0.406804\pi$
$53$	11804.4	0.577237	0.288618	+	0.957444i	$0.406804\pi$
$54$	−19068.9	−0.889899
$55$	18572.1	0.827853
$56$	2639.98	0.112494
$57$	18459.0	0.752524
$58$	5491.46	0.214347
$59$	−29390.1	−1.09919	−0.549593	−	0.835433i	$-0.685217\pi$
$59$	−29390.1	−1.09919	−0.549593	+	0.835433i	$0.685217\pi$
$60$	1856.33	0.0665699
$61$	29333.5	1.00934	0.504671	−	0.863311i	$-0.331614\pi$
$61$	29333.5	1.00934	0.504671	+	0.863311i	$0.331614\pi$
$62$	43463.3	1.43597
$63$	−1945.84	−0.0617667
$64$	36429.6	1.11174
$65$	−4481.20	−0.131556
$66$	36596.7	1.03415
$67$	4122.11	0.112184	0.0560922	−	0.998426i	$-0.482136\pi$
$67$	4122.11	0.112184	0.0560922	+	0.998426i	$0.482136\pi$
$68$	−2157.64	−0.0565856
$69$	25435.6	0.643159
$70$	−1672.32	−0.0407919
$71$	−13649.5	−0.321343	−0.160672	−	0.987008i	$-0.551366\pi$
$71$	−13649.5	−0.321343	−0.160672	+	0.987008i	$0.551366\pi$
$72$	−28165.6	−0.640306
$73$	−33582.1	−0.737566	−0.368783	−	0.929516i	$-0.620225\pi$
$73$	−33582.1	−0.737566	−0.368783	+	0.929516i	$0.620225\pi$
$74$	25940.8	0.550687
$75$	−6216.07	−0.127604
$76$	13856.4	0.275180
$77$	10032.6	0.192836
$78$	−8830.30	−0.164338
$79$	−86281.4	−1.55543	−0.777713	−	0.628619i	$-0.783621\pi$
$79$	−86281.4	−1.55543	−0.777713	+	0.628619i	$0.783621\pi$
$80$	−18233.8	−0.318532
$81$	−3277.05	−0.0554971
$82$	82215.3	1.35026
$83$	−73620.8	−1.17302	−0.586510	−	0.809942i	$-0.699499\pi$
$83$	−73620.8	−1.17302	−0.586510	+	0.809942i	$0.699499\pi$
$84$	1002.79	0.0155064
$85$	7225.00	0.108465
$86$	−71745.7	−1.04604
$87$	11026.5	0.156185
$88$	145220.	1.99903
$89$	−100199.	−1.34087	−0.670435	−	0.741968i	$-0.733893\pi$
$89$	−100199.	−1.34087	−0.670435	+	0.741968i	$0.733893\pi$
$90$	17841.7	0.232183
$91$	−2420.74	−0.0306440
$92$	19093.5	0.235188
$93$	87271.8	1.04632
$94$	−79687.2	−0.930184
$95$	−46399.3	−0.527475
$96$	26284.5	0.291087
$97$	4407.67	0.0475642	0.0237821	−	0.999717i	$-0.492429\pi$
$97$	4407.67	0.0475642	0.0237821	+	0.999717i	$0.492429\pi$
$98$	82344.9	0.866107
$99$	−107037.	−1.09760
$100$	−4666.17	−0.0466617
$101$	42746.3	0.416960	0.208480	−	0.978027i	$-0.433148\pi$
$101$	42746.3	0.416960	0.208480	+	0.978027i	$0.433148\pi$
$102$	14237.0	0.135494
$103$	−192307.	−1.78609	−0.893043	−	0.449972i	$-0.851434\pi$
$103$	−192307.	−1.78609	−0.893043	+	0.449972i	$0.851434\pi$
$104$	−35039.8	−0.317671
$105$	−3357.92	−0.0297233
$106$	−58469.4	−0.505434
$107$	−114157.	−0.963927	−0.481963	−	0.876191i	$-0.660076\pi$
$107$	−114157.	−0.963927	−0.481963	+	0.876191i	$0.660076\pi$
$108$	−28742.2	−0.237116
$109$	−98113.3	−0.790973	−0.395487	−	0.918472i	$-0.629424\pi$
$109$	−98113.3	−0.790973	−0.395487	+	0.918472i	$0.629424\pi$
$110$	−91991.0	−0.724876
$111$	52087.6	0.401261
$112$	−9849.90	−0.0741971
$113$	133620.	0.984412	0.492206	−	0.870479i	$-0.336191\pi$
$113$	133620.	0.984412	0.492206	+	0.870479i	$0.336191\pi$
$114$	−91430.8	−0.658917
$115$	−63936.0	−0.450817
$116$	8277.19	0.0571134
$117$	25826.5	0.174422
$118$	145575.	0.962457
$119$	3902.94	0.0252653
$120$	−48605.2	−0.308127
$121$	390823.	2.42670
$122$	−145294.	−0.883790
$123$	165083.	0.983877
$124$	65511.6	0.382617
$125$	15625.0	0.0894427
$126$	9638.10	0.0540835
$127$	60761.4	0.334286	0.167143	−	0.985933i	$-0.446546\pi$
$127$	60761.4	0.334286	0.167143	+	0.985933i	$0.446546\pi$
$128$	−95873.3	−0.517217
$129$	−144061.	−0.762206
$130$	22196.3	0.115192
$131$	−283562.	−1.44368	−0.721838	−	0.692062i	$-0.756703\pi$
$131$	−283562.	−1.44368	−0.721838	+	0.692062i	$0.756703\pi$
$132$	55161.5	0.275551
$133$	−25064.9	−0.122867
$134$	−20417.6	−0.0982296
$135$	96245.4	0.454512
$136$	56494.3	0.261913
$137$	−116549.	−0.530528	−0.265264	−	0.964176i	$-0.585459\pi$
$137$	−116549.	−0.530528	−0.265264	+	0.964176i	$0.585459\pi$
$138$	−125987.	−0.563156
$139$	212209.	0.931595	0.465798	−	0.884891i	$-0.345767\pi$
$139$	212209.	0.931595	0.465798	+	0.884891i	$0.345767\pi$
$140$	−2520.66	−0.0108691
$141$	−160007.	−0.677784
$142$	67608.4	0.281371
$143$	−133160.	−0.544546
$144$	105087.	0.422322
$145$	−27716.8	−0.109477
$146$	166339.	0.645820
$147$	165344.	0.631094
$148$	39100.2	0.146732
$149$	8825.81	0.0325678	0.0162839	−	0.999867i	$-0.494816\pi$
$149$	8825.81	0.0325678	0.0162839	+	0.999867i	$0.494816\pi$
$150$	30789.4	0.111731
$151$	353306.	1.26098	0.630491	−	0.776196i	$-0.282853\pi$
$151$	353306.	1.26098	0.630491	+	0.776196i	$0.282853\pi$
$152$	−362809.	−1.27371
$153$	−41639.9	−0.143807
$154$	−49693.5	−0.168849
$155$	−219370.	−0.733413
$156$	−13309.8	−0.0437884
$157$	55282.7	0.178995	0.0894974	−	0.995987i	$-0.471474\pi$
$157$	55282.7	0.178995	0.0894974	+	0.995987i	$0.471474\pi$
$158$	427369.	1.36195
$159$	−117403.	−0.368287
$160$	−66070.0	−0.204035
$161$	−34538.2	−0.105011
$162$	16231.9	0.0485938
$163$	306844.	0.904585	0.452292	−	0.891870i	$-0.350606\pi$
$163$	306844.	0.904585	0.452292	+	0.891870i	$0.350606\pi$
$164$	123922.	0.359781
$165$	−184712.	−0.528185
$166$	364658.	1.02711
$167$	−589063.	−1.63444	−0.817222	−	0.576322i	$-0.804487\pi$
$167$	−589063.	−1.63444	−0.817222	+	0.576322i	$0.804487\pi$
$168$	−26256.5	−0.0717735
$169$	−339163.	−0.913465
$170$	−35786.8	−0.0949732
$171$	267413.	0.699347
$172$	−108141.	−0.278721
$173$	647356.	1.64448	0.822238	−	0.569143i	$-0.192725\pi$
$173$	647356.	1.64448	0.822238	+	0.569143i	$0.192725\pi$
$174$	−54616.5	−0.136757
$175$	8440.62	0.0208343
$176$	−541824.	−1.31849
$177$	292306.	0.701300
$178$	496303.	1.17408
$179$	385367.	0.898963	0.449482	−	0.893290i	$-0.351609\pi$
$179$	385367.	0.898963	0.449482	+	0.893290i	$0.351609\pi$
$180$	26892.6	0.0618659
$181$	94431.6	0.214250	0.107125	−	0.994246i	$-0.465836\pi$
$181$	94431.6	0.214250	0.107125	+	0.994246i	$0.465836\pi$
$182$	11990.4	0.0268321
$183$	−291742.	−0.643979
$184$	−499933.	−1.08860
$185$	−130930.	−0.281261
$186$	−432274.	−0.916171
$187$	214693.	0.448966
$188$	−120111.	−0.247850
$189$	51991.7	0.105872
$190$	229825.	0.461862
$191$	−284007.	−0.563308	−0.281654	−	0.959516i	$-0.590883\pi$
$191$	−284007.	−0.563308	−0.281654	+	0.959516i	$0.590883\pi$
$192$	−362318.	−0.709312
$193$	847831.	1.63838	0.819192	−	0.573519i	$-0.194422\pi$
$193$	847831.	1.63838	0.819192	+	0.573519i	$0.194422\pi$
$194$	−21832.1	−0.0416476
$195$	44568.7	0.0839351
$196$	124117.	0.230777
$197$	812264.	1.49119	0.745593	−	0.666402i	$-0.232167\pi$
$197$	812264.	1.49119	0.745593	+	0.666402i	$0.232167\pi$
$198$	530173.	0.961069
$199$	413483.	0.740160	0.370080	−	0.929000i	$-0.379330\pi$
$199$	413483.	0.740160	0.370080	+	0.929000i	$0.379330\pi$
$200$	122176.	0.215979
$201$	−40997.3	−0.0715756
$202$	−211731.	−0.365094
$203$	−14972.6	−0.0255010
$204$	21459.2	0.0361026
$205$	−414961.	−0.689641
$206$	952534.	1.56391
$207$	368483.	0.597711
$208$	130735.	0.209524
$209$	−1.37877e6	−2.18336
$210$	16632.4	0.0260260
$211$	−1.08516e6	−1.67798	−0.838990	−	0.544146i	$-0.816854\pi$
$211$	−1.08516e6	−1.67798	−0.838990	+	0.544146i	$0.816854\pi$
$212$	−88130.0	−0.134674
$213$	135754.	0.205023
$214$	565443.	0.844023
$215$	362119.	0.534262
$216$	752570.	1.09752
$217$	−118504.	−0.170837
$218$	485974.	0.692584
$219$	333998.	0.470580
$220$	−138657.	−0.193145
$221$	−51802.7	−0.0713463
$222$	−258000.	−0.351348
$223$	−1.31798e6	−1.77479	−0.887395	−	0.461010i	$-0.847487\pi$
$223$	−1.31798e6	−1.77479	−0.887395	+	0.461010i	$0.847487\pi$
$224$	−35691.0	−0.0475268
$225$	−90051.7	−0.118587
$226$	−661848.	−0.861960
$227$	−813956.	−1.04842	−0.524211	−	0.851588i	$-0.675640\pi$
$227$	−813956.	−1.04842	−0.524211	+	0.851588i	$0.675640\pi$
$228$	−137812.	−0.175570
$229$	1.19764e6	1.50917	0.754586	−	0.656201i	$-0.227838\pi$
$229$	1.19764e6	1.50917	0.754586	+	0.656201i	$0.227838\pi$
$230$	316687.	0.394740
$231$	−99781.5	−0.123033
$232$	−216725.	−0.264356
$233$	−1.07508e6	−1.29733	−0.648663	−	0.761076i	$-0.724672\pi$
$233$	−1.07508e6	−1.29733	−0.648663	+	0.761076i	$0.724672\pi$
$234$	−127924.	−0.152726
$235$	402201.	0.475087
$236$	219423.	0.256449
$237$	858131.	0.992390
$238$	−19332.0	−0.0221225
$239$	−1.50327e6	−1.70232	−0.851162	−	0.524903i	$-0.824101\pi$
$239$	−1.50327e6	−1.70232	−0.851162	+	0.524903i	$0.824101\pi$
$240$	181348.	0.203229
$241$	−259714.	−0.288040	−0.144020	−	0.989575i	$-0.546003\pi$
$241$	−259714.	−0.288040	−0.144020	+	0.989575i	$0.546003\pi$
$242$	−1.93582e6	−2.12485
$243$	−902912.	−0.980912
$244$	−219000.	−0.235488
$245$	−415615.	−0.442361
$246$	−817690.	−0.861491
$247$	332679.	0.346963
$248$	−1.71532e6	−1.77099
$249$	732212.	0.748408
$250$	−77393.7	−0.0783169
$251$	417745.	0.418530	0.209265	−	0.977859i	$-0.432893\pi$
$251$	417745.	0.418530	0.209265	+	0.977859i	$0.432893\pi$
$252$	14527.4	0.0144107
$253$	−1.89988e6	−1.86605
$254$	−300963.	−0.292704
$255$	−71857.8	−0.0692028
$256$	−690868.	−0.658863
$257$	1.23266e6	1.16415	0.582076	−	0.813134i	$-0.302240\pi$
$257$	1.23266e6	1.16415	0.582076	+	0.813134i	$0.302240\pi$
$258$	713563.	0.667395
$259$	−70728.2	−0.0655154
$260$	33456.0	0.0306931
$261$	159740.	0.145149
$262$	1.40454e6	1.26410
$263$	210447.	0.187609	0.0938044	−	0.995591i	$-0.470097\pi$
$263$	210447.	0.187609	0.0938044	+	0.995591i	$0.470097\pi$
$264$	−1.44432e6	−1.27542
$265$	295110.	0.258148
$266$	124151.	0.107584
$267$	996547.	0.855499
$268$	−30775.1	−0.0261735
$269$	1.19459e6	1.00656	0.503280	−	0.864123i	$-0.332126\pi$
$269$	1.19459e6	1.00656	0.503280	+	0.864123i	$0.332126\pi$
$270$	−476722.	−0.397975
$271$	−486392.	−0.402312	−0.201156	−	0.979559i	$-0.564470\pi$
$271$	−486392.	−0.402312	−0.201156	+	0.979559i	$0.564470\pi$
$272$	−210783.	−0.172748
$273$	24076.0	0.0195514
$274$	577292.	0.464535
$275$	464301.	0.370227
$276$	−189898.	−0.150054
$277$	−681163.	−0.533399	−0.266699	−	0.963780i	$-0.585933\pi$
$277$	−681163.	−0.533399	−0.266699	+	0.963780i	$0.585933\pi$
$278$	−1.05111e6	−0.815713
$279$	1.26430e6	0.972387
$280$	65999.6	0.0503090
$281$	1.22238e6	0.923509	0.461754	−	0.887008i	$-0.347220\pi$
$281$	1.22238e6	0.923509	0.461754	+	0.887008i	$0.347220\pi$
$282$	792546.	0.593474
$283$	44528.8	0.0330503	0.0165251	−	0.999863i	$-0.494740\pi$
$283$	44528.8	0.0330503	0.0165251	+	0.999863i	$0.494740\pi$
$284$	101905.	0.0749721
$285$	461474.	0.336539
$286$	659568.	0.476809
$287$	−224162.	−0.160641
$288$	380782.	0.270517
$289$	83521.0	0.0588235
$290$	137287.	0.0958590
$291$	−43837.4	−0.0303468
$292$	250720.	0.172080
$293$	151289.	0.102953	0.0514763	−	0.998674i	$-0.483607\pi$
$293$	151289.	0.102953	0.0514763	+	0.998674i	$0.483607\pi$
$294$	−818979.	−0.552592
$295$	−734753.	−0.491571
$296$	−1.02378e6	−0.679167
$297$	2.85996e6	1.88135
$298$	−43715.9	−0.0285167
$299$	458416.	0.296539
$300$	46408.4	0.0297710
$301$	195616.	0.124448
$302$	−1.74999e6	−1.10413
$303$	−425142.	−0.266028
$304$	1.35366e6	0.840089
$305$	733337.	0.451392
$306$	206251.	0.125919
$307$	20159.6	0.0122077	0.00610387	−	0.999981i	$-0.498057\pi$
$307$	20159.6	0.0122077	0.00610387	+	0.999981i	$0.498057\pi$
$308$	−74902.2	−0.0449902
$309$	1.91263e6	1.13955
$310$	1.08658e6	0.642183
$311$	1.75609e6	1.02954	0.514772	−	0.857327i	$-0.327877\pi$
$311$	1.75609e6	1.02954	0.514772	+	0.857327i	$0.327877\pi$
$312$	348496.	0.202680
$313$	683041.	0.394081	0.197041	−	0.980395i	$-0.436867\pi$
$313$	683041.	0.394081	0.197041	+	0.980395i	$0.436867\pi$
$314$	−273826.	−0.156729
$315$	−48645.9	−0.0276229
$316$	644166.	0.362894
$317$	87223.8	0.0487514	0.0243757	−	0.999703i	$-0.492240\pi$
$317$	87223.8	0.0487514	0.0243757	+	0.999703i	$0.492240\pi$
$318$	581520.	0.322476
$319$	−823612.	−0.453154
$320$	910740.	0.497187
$321$	1.13538e6	0.615002
$322$	171074.	0.0919486
$323$	−536376.	−0.286064
$324$	24466.0	0.0129479
$325$	−112030.	−0.0588337
$326$	−1.51986e6	−0.792063
$327$	975807.	0.504655
$328$	−3.24470e6	−1.66529
$329$	217269.	0.110664
$330$	914916.	0.462484
$331$	2.57320e6	1.29093	0.645467	−	0.763788i	$-0.276663\pi$
$331$	2.57320e6	1.29093	0.645467	+	0.763788i	$0.276663\pi$
$332$	549643.	0.273675
$333$	754589.	0.372907
$334$	2.91774e6	1.43114
$335$	103053.	0.0501704
$336$	97964.3	0.0473391
$337$	−1.70829e6	−0.819384	−0.409692	−	0.912224i	$-0.634364\pi$
$337$	−1.70829e6	−0.819384	−0.409692	+	0.912224i	$0.634364\pi$
$338$	1.67994e6	0.799838
$339$	−1.32895e6	−0.628072
$340$	−53940.9	−0.0253058
$341$	−6.51865e6	−3.03579
$342$	−1.32455e6	−0.612355
$343$	−451493.	−0.207213
$344$	2.83151e6	1.29009
$345$	635889.	0.287630
$346$	−3.20648e6	−1.43992
$347$	1.00945e6	0.450051	0.225025	−	0.974353i	$-0.427753\pi$
$347$	1.00945e6	0.450051	0.225025	+	0.974353i	$0.427753\pi$
$348$	−82322.6	−0.0364394
$349$	1.26376e6	0.555394	0.277697	−	0.960669i	$-0.410429\pi$
$349$	1.26376e6	0.555394	0.277697	+	0.960669i	$0.410429\pi$
$350$	−41808.0	−0.0182427
$351$	−690072.	−0.298969
$352$	−1.96329e6	−0.844555
$353$	1.36753e6	0.584117	0.292059	−	0.956400i	$-0.405660\pi$
$353$	1.36753e6	0.584117	0.292059	+	0.956400i	$0.405660\pi$
$354$	−1.44785e6	−0.614065
$355$	−341236.	−0.143709
$356$	748070.	0.312836
$357$	−38817.5	−0.0161197
$358$	−1.90880e6	−0.787141
$359$	−509814.	−0.208774	−0.104387	−	0.994537i	$-0.533288\pi$
$359$	−509814.	−0.208774	−0.104387	+	0.994537i	$0.533288\pi$
$360$	−704140.	−0.286354
$361$	968531.	0.391152
$362$	−467738.	−0.187599
$363$	−3.88702e6	−1.54828
$364$	18072.9	0.00714949
$365$	−839553.	−0.329850
$366$	1.44506e6	0.563874
$367$	1.26468e6	0.490136	0.245068	−	0.969506i	$-0.421190\pi$
$367$	1.26468e6	0.490136	0.245068	+	0.969506i	$0.421190\pi$
$368$	1.86528e6	0.717998
$369$	2.39155e6	0.914352
$370$	648521.	0.246275
$371$	159418.	0.0601316
$372$	−651559.	−0.244116
$373$	−2.53071e6	−0.941827	−0.470913	−	0.882179i	$-0.656076\pi$
$373$	−2.53071e6	−0.941827	−0.470913	+	0.882179i	$0.656076\pi$
$374$	−1.06342e6	−0.393119
$375$	−155402.	−0.0570661
$376$	3.14492e6	1.14720
$377$	198727.	0.0720118
$378$	−257525.	−0.0927021
$379$	1.23704e6	0.442370	0.221185	−	0.975232i	$-0.429007\pi$
$379$	1.23704e6	0.442370	0.221185	+	0.975232i	$0.429007\pi$
$380$	346411.	0.123064
$381$	−604315.	−0.213280
$382$	1.40674e6	0.493237
$383$	−118138.	−0.0411522	−0.0205761	−	0.999788i	$-0.506550\pi$
$383$	−118138.	−0.0411522	−0.0205761	+	0.999788i	$0.506550\pi$
$384$	953528.	0.329994
$385$	250815.	0.0862387
$386$	−4.19947e6	−1.43458
$387$	−2.08700e6	−0.708346
$388$	−32907.1	−0.0110971
$389$	752781.	0.252229	0.126114	−	0.992016i	$-0.459749\pi$
$389$	752781.	0.252229	0.126114	+	0.992016i	$0.459749\pi$
$390$	−220758.	−0.0734944
$391$	−739100.	−0.244490
$392$	−3.24981e6	−1.06818
$393$	2.82023e6	0.921091
$394$	−4.02330e6	−1.30570
$395$	−2.15704e6	−0.695608
$396$	799121.	0.256079
$397$	−3.41832e6	−1.08852	−0.544261	−	0.838916i	$-0.683190\pi$
$397$	−3.41832e6	−1.08852	−0.544261	+	0.838916i	$0.683190\pi$
$398$	−2.04806e6	−0.648091
$399$	249288.	0.0783915
$400$	−455846.	−0.142452
$401$	−349725.	−0.108609	−0.0543044	−	0.998524i	$-0.517294\pi$
$401$	−349725.	−0.108609	−0.0543044	+	0.998524i	$0.517294\pi$
$402$	203068.	0.0626723
$403$	1.57287e6	0.482425
$404$	−319138.	−0.0972803
$405$	−81926.3	−0.0248191
$406$	74162.1	0.0223289
$407$	−3.89062e6	−1.16421
$408$	−561876.	−0.167105
$409$	1.04602e6	0.309193	0.154597	−	0.987978i	$-0.450592\pi$
$409$	1.04602e6	0.309193	0.154597	+	0.987978i	$0.450592\pi$
$410$	2.05538e6	0.603856
$411$	1.15917e6	0.338487
$412$	1.43574e6	0.416709
$413$	−396913.	−0.114504
$414$	−1.82517e6	−0.523361
$415$	−1.84052e6	−0.524590
$416$	473717.	0.134210
$417$	−2.11057e6	−0.594374
$418$	6.82931e6	1.91177
$419$	−1.09315e6	−0.304189	−0.152095	−	0.988366i	$-0.548602\pi$
$419$	−1.09315e6	−0.304189	−0.152095	+	0.988366i	$0.548602\pi$
$420$	25069.8	0.00693469
$421$	−3.13270e6	−0.861416	−0.430708	−	0.902491i	$-0.641736\pi$
$421$	−3.13270e6	−0.861416	−0.430708	+	0.902491i	$0.641736\pi$
$422$	5.37500e6	1.46926
$423$	−2.31801e6	−0.629889
$424$	2.30755e6	0.623356
$425$	180625.	0.0485071
$426$	−672414.	−0.179520
$427$	396148.	0.105145
$428$	852283.	0.224892
$429$	1.32437e6	0.347430
$430$	−1.79364e6	−0.467805
$431$	−5.19507e6	−1.34710	−0.673548	−	0.739144i	$-0.735230\pi$
$431$	−5.19507e6	−1.34710	−0.673548	+	0.739144i	$0.735230\pi$
$432$	−2.80787e6	−0.723883
$433$	3.87822e6	0.994061	0.497030	−	0.867733i	$-0.334424\pi$
$433$	3.87822e6	0.994061	0.497030	+	0.867733i	$0.334424\pi$
$434$	586972.	0.149587
$435$	275663.	0.0698482
$436$	732501.	0.184541
$437$	4.74653e6	1.18898
$438$	−1.65436e6	−0.412045
$439$	1.95323e6	0.483717	0.241858	−	0.970312i	$-0.422243\pi$
$439$	1.95323e6	0.483717	0.241858	+	0.970312i	$0.422243\pi$
$440$	3.63051e6	0.893996
$441$	2.39532e6	0.586499
$442$	256589.	0.0624715
$443$	386945.	0.0936786	0.0468393	−	0.998902i	$-0.485085\pi$
$443$	386945.	0.0936786	0.0468393	+	0.998902i	$0.485085\pi$
$444$	−388880.	−0.0936176
$445$	−2.50497e6	−0.599656
$446$	6.52822e6	1.55402
$447$	−87779.0	−0.0207789
$448$	491981.	0.115812
$449$	3.34050e6	0.781980	0.390990	−	0.920395i	$-0.372133\pi$
$449$	3.34050e6	0.781980	0.390990	+	0.920395i	$0.372133\pi$
$450$	446044.	0.103836
$451$	−1.23307e7	−2.85461
$452$	−997593.	−0.229672
$453$	−3.51388e6	−0.804529
$454$	4.03168e6	0.918009
$455$	−60518.5	−0.0137044
$456$	3.60840e6	0.812647
$457$	−4.35918e6	−0.976370	−0.488185	−	0.872740i	$-0.662341\pi$
$457$	−4.35918e6	−0.976370	−0.488185	+	0.872740i	$0.662341\pi$
$458$	−5.93216e6	−1.32145
$459$	1.11260e6	0.246494
$460$	477337.	0.105179
$461$	−3.93227e6	−0.861769	−0.430884	−	0.902407i	$-0.641798\pi$
$461$	−3.93227e6	−0.861769	−0.430884	+	0.902407i	$0.641798\pi$
$462$	494237.	0.107728
$463$	−4.47600e6	−0.970370	−0.485185	−	0.874412i	$-0.661248\pi$
$463$	−4.47600e6	−0.970370	−0.485185	+	0.874412i	$0.661248\pi$
$464$	808612.	0.174359
$465$	2.18179e6	0.467931
$466$	5.32506e6	1.13595
$467$	−5.93018e6	−1.25828	−0.629138	−	0.777294i	$-0.716592\pi$
$467$	−5.93018e6	−1.25828	−0.629138	+	0.777294i	$0.716592\pi$
$468$	−192818.	−0.0406941
$469$	55669.0	0.0116864
$470$	−1.99218e6	−0.415991
$471$	−549826.	−0.114202
$472$	−5.74524e6	−1.18701
$473$	1.07605e7	2.21145
$474$	−4.25049e6	−0.868946
$475$	−1.15998e6	−0.235894
$476$	−29138.8	−0.00589461
$477$	−1.70081e6	−0.342263
$478$	7.44599e6	1.49057
$479$	2.00176e6	0.398632	0.199316	−	0.979935i	$-0.436128\pi$
$479$	2.00176e6	0.398632	0.199316	+	0.979935i	$0.436128\pi$
$480$	657113.	0.130178
$481$	938757.	0.185008
$482$	1.28641e6	0.252211
$483$	343507.	0.0669989
$484$	−2.91783e6	−0.566171
$485$	110192.	0.0212713
$486$	4.47230e6	0.858895
$487$	7.29169e6	1.39317	0.696587	−	0.717472i	$-0.254701\pi$
$487$	7.29169e6	1.39317	0.696587	+	0.717472i	$0.254701\pi$
$488$	5.73417e6	1.08999
$489$	−3.05179e6	−0.577141
$490$	2.05862e6	0.387335
$491$	−814260.	−0.152426	−0.0762130	−	0.997092i	$-0.524283\pi$
$491$	−814260.	−0.152426	−0.0762130	+	0.997092i	$0.524283\pi$
$492$	−1.23249e6	−0.229547
$493$	−320406.	−0.0593722
$494$	−1.64782e6	−0.303804
$495$	−2.67591e6	−0.490862
$496$	6.39993e6	1.16808
$497$	−184336.	−0.0334748
$498$	−3.62679e6	−0.655313
$499$	−1.65977e6	−0.298399	−0.149199	−	0.988807i	$-0.547670\pi$
$499$	−1.65977e6	−0.298399	−0.149199	+	0.988807i	$0.547670\pi$
$500$	−116654.	−0.0208677
$501$	5.85865e6	1.04281
$502$	−2.06917e6	−0.366469
$503$	1.07132e7	1.88799	0.943997	−	0.329955i	$-0.107034\pi$
$503$	1.07132e7	1.88799	0.943997	+	0.329955i	$0.107034\pi$
$504$	−380376.	−0.0667017
$505$	1.06866e6	0.186470
$506$	9.41045e6	1.63393
$507$	3.37322e6	0.582807
$508$	−453636.	−0.0779917
$509$	−2.37549e6	−0.406405	−0.203202	−	0.979137i	$-0.565135\pi$
$509$	−2.37549e6	−0.406405	−0.203202	+	0.979137i	$0.565135\pi$
$510$	355926.	0.0605946
$511$	−453526.	−0.0768334
$512$	6.48995e6	1.09412
$513$	−7.14515e6	−1.19872
$514$	−6.10560e6	−1.01934
$515$	−4.80768e6	−0.798762
$516$	1.07554e6	0.177829
$517$	1.19515e7	1.96651
$518$	350331.	0.0573659
$519$	−6.43841e6	−1.04921
$520$	−875994.	−0.142067
$521$	−1.78521e6	−0.288135	−0.144067	−	0.989568i	$-0.546018\pi$
$521$	−1.78521e6	−0.288135	−0.144067	+	0.989568i	$0.546018\pi$
$522$	−791225.	−0.127094
$523$	−3.77324e6	−0.603199	−0.301599	−	0.953435i	$-0.597521\pi$
$523$	−3.77324e6	−0.603199	−0.301599	+	0.953435i	$0.597521\pi$
$524$	2.11704e6	0.336822
$525$	−83948.0	−0.0132927
$526$	−1.04238e6	−0.164272
$527$	−2.53592e6	−0.397749
$528$	5.38882e6	0.841219
$529$	104148.	0.0161812
$530$	−1.46174e6	−0.226037
$531$	4.23461e6	0.651744
$532$	187131.	0.0286660
$533$	2.97524e6	0.453632
$534$	−4.93609e6	−0.749083
$535$	−2.85393e6	−0.431081
$536$	805798.	0.121147
$537$	−3.83275e6	−0.573555
$538$	−5.91706e6	−0.881354
$539$	−1.23501e7	−1.83105
$540$	−718555.	−0.106041
$541$	1.60332e6	0.235519	0.117760	−	0.993042i	$-0.462429\pi$
$541$	1.60332e6	0.235519	0.117760	+	0.993042i	$0.462429\pi$
$542$	2.40919e6	0.352268
$543$	−939189.	−0.136695
$544$	−763769.	−0.110653
$545$	−2.45283e6	−0.353734
$546$	−119253.	−0.0171194
$547$	1.05509e7	1.50772	0.753861	−	0.657034i	$-0.228189\pi$
$547$	1.05509e7	1.50772	0.753861	+	0.657034i	$0.228189\pi$
$548$	870142.	0.123777
$549$	−4.22645e6	−0.598473
$550$	−2.29978e6	−0.324174
$551$	2.05766e6	0.288732
$552$	4.97220e6	0.694545
$553$	−1.16523e6	−0.162031
$554$	3.37393e6	0.467049
$555$	1.30219e6	0.179450
$556$	−1.58433e6	−0.217349
$557$	5.14612e6	0.702817	0.351409	−	0.936222i	$-0.385703\pi$
$557$	5.14612e6	0.702817	0.351409	+	0.936222i	$0.385703\pi$
$558$	−6.26232e6	−0.851431
$559$	−2.59636e6	−0.351427
$560$	−246248.	−0.0331819
$561$	−2.13528e6	−0.286449
$562$	−6.05469e6	−0.808633
$563$	1.22483e7	1.62856	0.814282	−	0.580470i	$-0.197131\pi$
$563$	1.22483e7	1.62856	0.814282	+	0.580470i	$0.197131\pi$
$564$	1.19459e6	0.158133
$565$	3.34051e6	0.440242
$566$	−220560.	−0.0289391
$567$	−44256.5	−0.00578122
$568$	−2.66822e6	−0.347018
$569$	1.16148e6	0.150395	0.0751973	−	0.997169i	$-0.476041\pi$
$569$	1.16148e6	0.150395	0.0751973	+	0.997169i	$0.476041\pi$
$570$	−2.28577e6	−0.294676
$571$	7.48367e6	0.960560	0.480280	−	0.877115i	$-0.340535\pi$
$571$	7.48367e6	0.960560	0.480280	+	0.877115i	$0.340535\pi$
$572$	994156.	0.127047
$573$	2.82465e6	0.359400
$574$	1.11032e6	0.140659
$575$	−1.59840e6	−0.201612
$576$	−5.24888e6	−0.659189
$577$	5.14705e6	0.643604	0.321802	−	0.946807i	$-0.395712\pi$
$577$	5.14705e6	0.643604	0.321802	+	0.946807i	$0.395712\pi$
$578$	−413696.	−0.0515064
$579$	−8.43228e6	−1.04532
$580$	206930.	0.0255419
$581$	−994248.	−0.122195
$582$	217135.	0.0265719
$583$	8.76927e6	1.06854
$584$	−6.56470e6	−0.796495
$585$	645664.	0.0780040
$586$	−749362.	−0.0901463
$587$	−1.18473e7	−1.41913	−0.709566	−	0.704639i	$-0.751109\pi$
$587$	−1.18473e7	−1.41913	−0.709566	+	0.704639i	$0.751109\pi$
$588$	−1.23443e6	−0.147240
$589$	1.62858e7	1.93429
$590$	3.63937e6	0.430424
$591$	−8.07855e6	−0.951403
$592$	3.81976e6	0.447953
$593$	−1.17061e7	−1.36703	−0.683513	−	0.729938i	$-0.739549\pi$
$593$	−1.17061e7	−1.36703	−0.683513	+	0.729938i	$0.739549\pi$
$594$	−1.41659e7	−1.64732
$595$	97573.5	0.0112990
$596$	−65892.3	−0.00759835
$597$	−4.11239e6	−0.472235
$598$	−2.27062e6	−0.259652
$599$	1.16813e7	1.33022	0.665110	−	0.746746i	$-0.268385\pi$
$599$	1.16813e7	1.33022	0.665110	+	0.746746i	$0.268385\pi$
$600$	−1.21513e6	−0.137799
$601$	5.65882e6	0.639057	0.319529	−	0.947577i	$-0.396475\pi$
$601$	5.65882e6	0.639057	0.319529	+	0.947577i	$0.396475\pi$
$602$	−968925.	−0.108968
$603$	−593924.	−0.0665178
$604$	−2.63774e6	−0.294198
$605$	9.77058e6	1.08526
$606$	2.10581e6	0.232937
$607$	3.58036e6	0.394416	0.197208	−	0.980362i	$-0.436813\pi$
$607$	3.58036e6	0.394416	0.197208	+	0.980362i	$0.436813\pi$
$608$	4.90496e6	0.538117
$609$	148913.	0.0162701
$610$	−3.63236e6	−0.395243
$611$	−2.88375e6	−0.312503
$612$	310878.	0.0335515
$613$	4.02237e6	0.432345	0.216173	−	0.976355i	$-0.430643\pi$
$613$	4.02237e6	0.432345	0.216173	+	0.976355i	$0.430643\pi$
$614$	−99854.2	−0.0106892
$615$	4.12709e6	0.440003
$616$	1.96120e6	0.208243
$617$	1.12013e7	1.18456	0.592280	−	0.805732i	$-0.298228\pi$
$617$	1.12013e7	1.18456	0.592280	+	0.805732i	$0.298228\pi$
$618$	−9.47363e6	−0.997804
$619$	−2.08785e6	−0.219015	−0.109507	−	0.993986i	$-0.534927\pi$
$619$	−2.08785e6	−0.219015	−0.109507	+	0.993986i	$0.534927\pi$
$620$	1.63779e6	0.171111
$621$	−9.84566e6	−1.02451
$622$	−8.69824e6	−0.901478
$623$	−1.35318e6	−0.139681
$624$	−1.30025e6	−0.133680
$625$	390625.	0.0400000
$626$	−3.38323e6	−0.345061
$627$	1.37128e7	1.39302
$628$	−412733.	−0.0417610
$629$	−1.51355e6	−0.152535
$630$	240953.	0.0241869
$631$	−1.62010e7	−1.61982	−0.809911	−	0.586553i	$-0.800484\pi$
$631$	−1.62010e7	−1.61982	−0.809911	+	0.586553i	$0.800484\pi$
$632$	−1.68665e7	−1.67970
$633$	1.07927e7	1.07058
$634$	−432036.	−0.0426872
$635$	1.51903e6	0.149497
$636$	876516.	0.0859245
$637$	2.97993e6	0.290976
$638$	4.07951e6	0.396786
$639$	1.96665e6	0.190535
$640$	−2.39683e6	−0.231306
$641$	−8.40847e6	−0.808299	−0.404149	−	0.914693i	$-0.632432\pi$
$641$	−8.40847e6	−0.808299	−0.404149	+	0.914693i	$0.632432\pi$
$642$	−5.62373e6	−0.538502
$643$	−1.66230e7	−1.58556	−0.792780	−	0.609508i	$-0.791367\pi$
$643$	−1.66230e7	−1.58556	−0.792780	+	0.609508i	$0.791367\pi$
$644$	257857.	0.0244999
$645$	−3.60153e6	−0.340869
$646$	2.65677e6	0.250480
$647$	5.98066e6	0.561680	0.280840	−	0.959755i	$-0.409387\pi$
$647$	5.98066e6	0.561680	0.280840	+	0.959755i	$0.409387\pi$
$648$	−640605.	−0.0599312
$649$	−2.18334e7	−2.03474
$650$	554906.	0.0515153
$651$	1.17860e6	0.108997
$652$	−2.29086e6	−0.211047
$653$	1.16116e7	1.06564	0.532819	−	0.846229i	$-0.321133\pi$
$653$	1.16116e7	1.06564	0.532819	+	0.846229i	$0.321133\pi$
$654$	−4.83336e6	−0.441881
$655$	−7.08905e6	−0.645632
$656$	1.21061e7	1.09836
$657$	4.83860e6	0.437327
$658$	−1.07617e6	−0.0968986
$659$	−2.21305e7	−1.98508	−0.992541	−	0.121910i	$-0.961098\pi$
$659$	−2.21305e7	−1.98508	−0.992541	+	0.121910i	$0.961098\pi$
$660$	1.37904e6	0.123230
$661$	−9.38416e6	−0.835395	−0.417697	−	0.908586i	$-0.637163\pi$
$661$	−9.38416e6	−0.835395	−0.417697	+	0.908586i	$0.637163\pi$
$662$	−1.27456e7	−1.13035
$663$	515215.	0.0455202
$664$	−1.43916e7	−1.26674
$665$	−626622.	−0.0549479
$666$	−3.73763e6	−0.326521
$667$	2.83536e6	0.246770
$668$	4.39786e6	0.381330
$669$	1.31083e7	1.13235
$670$	−510440.	−0.0439296
$671$	2.17913e7	1.86843
$672$	354972.	0.0303229
$673$	1.69192e6	0.143993	0.0719965	−	0.997405i	$-0.477063\pi$
$673$	1.69192e6	0.143993	0.0719965	+	0.997405i	$0.477063\pi$
$674$	8.46150e6	0.717460
$675$	2.40613e6	0.203264
$676$	2.53215e6	0.213119
$677$	−1.60015e7	−1.34180	−0.670902	−	0.741546i	$-0.734093\pi$
$677$	−1.60015e7	−1.34180	−0.670902	+	0.741546i	$0.734093\pi$
$678$	6.58255e6	0.549946
$679$	59525.5	0.00495483
$680$	1.41236e6	0.117131
$681$	8.09538e6	0.668912
$682$	3.22881e7	2.65817
$683$	1.85149e7	1.51870	0.759348	−	0.650685i	$-0.225518\pi$
$683$	1.85149e7	1.51870	0.759348	+	0.650685i	$0.225518\pi$
$684$	−1.99647e6	−0.163164
$685$	−2.91373e6	−0.237259
$686$	2.23633e6	0.181437
$687$	−1.19114e7	−0.962879
$688$	−1.05645e7	−0.850898
$689$	−2.11591e6	−0.169805
$690$	−3.14968e6	−0.251851
$691$	−2.22261e7	−1.77079	−0.885397	−	0.464835i	$-0.846114\pi$
$691$	−2.22261e7	−1.77079	−0.885397	+	0.464835i	$0.846114\pi$
$692$	−4.83307e6	−0.383670
$693$	−1.44553e6	−0.114339
$694$	−5.00001e6	−0.394069
$695$	5.30523e6	0.416622
$696$	2.15549e6	0.168664
$697$	−4.79695e6	−0.374010
$698$	−6.25965e6	−0.486308
$699$	1.06924e7	0.827717
$700$	−63016.5	−0.00486082
$701$	−1.66904e7	−1.28284	−0.641419	−	0.767190i	$-0.721654\pi$
$701$	−1.66904e7	−1.28284	−0.641419	+	0.767190i	$0.721654\pi$
$702$	3.41806e6	0.261780
$703$	9.72008e6	0.741791
$704$	2.70629e7	2.05799
$705$	−4.00018e6	−0.303114
$706$	−6.77364e6	−0.511458
$707$	577288.	0.0434354
$708$	−2.18232e6	−0.163619
$709$	9.38491e6	0.701156	0.350578	−	0.936534i	$-0.385985\pi$
$709$	9.38491e6	0.701156	0.350578	+	0.936534i	$0.385985\pi$
$710$	1.69021e6	0.125833
$711$	1.24317e7	0.922264
$712$	−1.95870e7	−1.44800
$713$	2.24410e7	1.65318
$714$	192271.	0.0141146
$715$	−3.32900e6	−0.243528
$716$	−2.87710e6	−0.209736
$717$	1.49511e7	1.08611
$718$	2.52521e6	0.182804
$719$	−2.29753e6	−0.165745	−0.0828724	−	0.996560i	$-0.526409\pi$
$719$	−2.29753e6	−0.165745	−0.0828724	+	0.996560i	$0.526409\pi$
$720$	2.62718e6	0.188868
$721$	−2.59710e6	−0.186059
$722$	−4.79732e6	−0.342496
$723$	2.58304e6	0.183775
$724$	−705014.	−0.0499863
$725$	−692919.	−0.0489596
$726$	1.92531e7	1.35569
$727$	2.21473e6	0.155412	0.0777059	−	0.996976i	$-0.475240\pi$
$727$	2.21473e6	0.155412	0.0777059	+	0.996976i	$0.475240\pi$
$728$	−473212.	−0.0330923
$729$	9.77643e6	0.681336
$730$	4.15847e6	0.288819
$731$	4.18609e6	0.289744
$732$	2.17811e6	0.150246
$733$	−6.33827e6	−0.435724	−0.217862	−	0.975980i	$-0.569908\pi$
$733$	−6.33827e6	−0.435724	−0.217862	+	0.975980i	$0.569908\pi$
$734$	−6.26422e6	−0.429168
$735$	4.13359e6	0.282234
$736$	6.75880e6	0.459912
$737$	3.06224e6	0.207668
$738$	−1.18458e7	−0.800615
$739$	−7.68253e6	−0.517480	−0.258740	−	0.965947i	$-0.583307\pi$
$739$	−7.68253e6	−0.517480	−0.258740	+	0.965947i	$0.583307\pi$
$740$	977505.	0.0656205
$741$	−3.30873e6	−0.221369
$742$	−789629.	−0.0526518
$743$	−2.47626e7	−1.64560	−0.822801	−	0.568330i	$-0.807590\pi$
$743$	−2.47626e7	−1.64560	−0.822801	+	0.568330i	$0.807590\pi$
$744$	1.70601e7	1.12992
$745$	220645.	0.0145648
$746$	1.25351e7	0.824672
$747$	1.06075e7	0.695522
$748$	−1.60287e6	−0.104748
$749$	−1.54169e6	−0.100414
$750$	769735.	0.0499676
$751$	−4.97740e6	−0.322035	−0.161017	−	0.986952i	$-0.551478\pi$
$751$	−4.97740e6	−0.322035	−0.161017	+	0.986952i	$0.551478\pi$
$752$	−1.17339e7	−0.756652
$753$	−4.15477e6	−0.267030
$754$	−984333.	−0.0630542
$755$	8.83266e6	0.563929
$756$	−388163.	−0.0247007
$757$	2.57073e7	1.63049	0.815244	−	0.579118i	$-0.196603\pi$
$757$	2.57073e7	1.63049	0.815244	+	0.579118i	$0.196603\pi$
$758$	−6.12731e6	−0.387344
$759$	1.88956e7	1.19058
$760$	−9.07023e6	−0.569619
$761$	2.72075e7	1.70305	0.851524	−	0.524316i	$-0.175679\pi$
$761$	2.72075e7	1.70305	0.851524	+	0.524316i	$0.175679\pi$
$762$	2.99329e6	0.186750
$763$	−1.32502e6	−0.0823969
$764$	2.12036e6	0.131424
$765$	−1.04100e6	−0.0643126
$766$	585161.	0.0360333
$767$	5.26812e6	0.323346
$768$	6.87118e6	0.420367
$769$	1.70602e7	1.04032	0.520161	−	0.854068i	$-0.325872\pi$
$769$	1.70602e7	1.04032	0.520161	+	0.854068i	$0.325872\pi$
$770$	−1.24234e6	−0.0755114
$771$	−1.22597e7	−0.742750
$772$	−6.32979e6	−0.382249
$773$	2.42134e7	1.45749	0.728746	−	0.684784i	$-0.240103\pi$
$773$	2.42134e7	1.45749	0.728746	+	0.684784i	$0.240103\pi$
$774$	1.03373e7	0.620234
$775$	−5.48426e6	−0.327992
$776$	861621.	0.0513644
$777$	703443.	0.0418000
$778$	−3.72867e6	−0.220854
$779$	3.08062e7	1.81884
$780$	−332744.	−0.0195828
$781$	−1.01399e7	−0.594850
$782$	3.66090e6	0.214078
$783$	−4.26818e6	−0.248793
$784$	1.21252e7	0.704530
$785$	1.38207e6	0.0800489
$786$	−1.39691e7	−0.806516
$787$	−801866.	−0.0461493	−0.0230746	−	0.999734i	$-0.507346\pi$
$787$	−801866.	−0.0461493	−0.0230746	+	0.999734i	$0.507346\pi$
$788$	−6.06426e6	−0.347906
$789$	−2.09304e6	−0.119698
$790$	1.06842e7	0.609081
$791$	1.80454e6	0.102548
$792$	−2.09237e7	−1.18529
$793$	−5.25796e6	−0.296917
$794$	1.69316e7	0.953119
$795$	−2.93508e6	−0.164703
$796$	−3.08701e6	−0.172685
$797$	−1.85153e7	−1.03249	−0.516245	−	0.856441i	$-0.672670\pi$
$797$	−1.85153e7	−1.03249	−0.516245	+	0.856441i	$0.672670\pi$
$798$	−1.23477e6	−0.0686403
$799$	4.64944e6	0.257652
$800$	−1.65175e6	−0.0912472
$801$	1.44369e7	0.795047
$802$	1.73225e6	0.0950990
$803$	−2.49476e7	−1.36534
$804$	306080.	0.0166992
$805$	−863454.	−0.0469623
$806$	−7.79072e6	−0.422415
$807$	−1.18811e7	−0.642204
$808$	8.35613e6	0.450274
$809$	−1.25719e7	−0.675349	−0.337674	−	0.941263i	$-0.609640\pi$
$809$	−1.25719e7	−0.675349	−0.337674	+	0.941263i	$0.609640\pi$
$810$	405797.	0.0217318
$811$	6.53924e6	0.349120	0.174560	−	0.984647i	$-0.444150\pi$
$811$	6.53924e6	0.349120	0.174560	+	0.984647i	$0.444150\pi$
$812$	111783.	0.00594959
$813$	4.83751e6	0.256682
$814$	1.92710e7	1.01940
$815$	7.67111e6	0.404543
$816$	2.09639e6	0.110216
$817$	−2.68833e7	−1.40905
$818$	−5.18112e6	−0.270732
$819$	348787.	0.0181698
$820$	3.09805e6	0.160899
$821$	−3.60810e7	−1.86819	−0.934095	−	0.357024i	$-0.883792\pi$
$821$	−3.60810e7	−1.86819	−0.934095	+	0.357024i	$0.883792\pi$
$822$	−5.74158e6	−0.296382
$823$	−1.53622e7	−0.790595	−0.395297	−	0.918553i	$-0.629358\pi$
$823$	−1.53622e7	−0.790595	−0.395297	+	0.918553i	$0.629358\pi$
$824$	−3.75926e7	−1.92879
$825$	−4.61781e6	−0.236212
$826$	1.96599e6	0.100261
$827$	−1.40297e7	−0.713319	−0.356659	−	0.934234i	$-0.616084\pi$
$827$	−1.40297e7	−0.713319	−0.356659	+	0.934234i	$0.616084\pi$
$828$	−2.75104e6	−0.139451
$829$	−1.95476e6	−0.0987886	−0.0493943	−	0.998779i	$-0.515729\pi$
$829$	−1.95476e6	−0.0987886	−0.0493943	+	0.998779i	$0.515729\pi$
$830$	9.11645e6	0.459336
$831$	6.77466e6	0.340318
$832$	−6.52993e6	−0.327040
$833$	−4.80451e6	−0.239904
$834$	1.04541e7	0.520440
$835$	−1.47266e7	−0.730946
$836$	1.02937e7	0.509397
$837$	−3.37814e7	−1.66672
$838$	5.41457e6	0.266351
$839$	3.06449e7	1.50298	0.751490	−	0.659744i	$-0.229335\pi$
$839$	3.06449e7	1.50298	0.751490	+	0.659744i	$0.229335\pi$
$840$	−656413.	−0.0320981
$841$	−1.92820e7	−0.940074
$842$	1.55169e7	0.754264
$843$	−1.21575e7	−0.589215
$844$	8.10165e6	0.391487
$845$	−8.47908e6	−0.408514
$846$	1.14815e7	0.551537
$847$	5.27806e6	0.252794
$848$	−8.60957e6	−0.411142
$849$	−442871.	−0.0210867
$850$	−894671.	−0.0424733
$851$	1.33938e7	0.633986
$852$	−1.01352e6	−0.0478335
$853$	1.39171e7	0.654902	0.327451	−	0.944868i	$-0.393810\pi$
$853$	1.39171e7	0.654902	0.327451	+	0.944868i	$0.393810\pi$
$854$	−1.96220e6	−0.0920657
$855$	6.68534e6	0.312758
$856$	−2.23157e7	−1.04094
$857$	−4.54122e6	−0.211213	−0.105607	−	0.994408i	$-0.533678\pi$
$857$	−4.54122e6	−0.211213	−0.105607	+	0.994408i	$0.533678\pi$
$858$	−6.55988e6	−0.304213
$859$	1.93388e7	0.894226	0.447113	−	0.894478i	$-0.352452\pi$
$859$	1.93388e7	0.894226	0.447113	+	0.894478i	$0.352452\pi$
$860$	−2.70353e6	−0.124648
$861$	2.22945e6	0.102492
$862$	2.57322e7	1.17953
$863$	−2.61691e7	−1.19608	−0.598042	−	0.801465i	$-0.704054\pi$
$863$	−2.61691e7	−1.19608	−0.598042	+	0.801465i	$0.704054\pi$
$864$	−1.01743e7	−0.463682
$865$	1.61839e7	0.735432
$866$	−1.92096e7	−0.870409
$867$	−830676.	−0.0375305
$868$	884733.	0.0398578
$869$	−6.40970e7	−2.87931
$870$	−1.36541e6	−0.0611598
$871$	−738879.	−0.0330011
$872$	−1.91794e7	−0.854169
$873$	−635069.	−0.0282024
$874$	−2.35105e7	−1.04108
$875$	211015.	0.00931738
$876$	−2.49359e6	−0.109790
$877$	8.15287e6	0.357941	0.178970	−	0.983854i	$-0.442723\pi$
$877$	8.15287e6	0.357941	0.178970	+	0.983854i	$0.442723\pi$
$878$	−9.67471e6	−0.423547
$879$	−1.50467e6	−0.0656856
$880$	−1.35456e7	−0.589646
$881$	−2.60347e7	−1.13009	−0.565046	−	0.825060i	$-0.691141\pi$
$881$	−2.60347e7	−1.13009	−0.565046	+	0.825060i	$0.691141\pi$
$882$	−1.18645e7	−0.513544
$883$	9.01584e6	0.389139	0.194569	−	0.980889i	$-0.437669\pi$
$883$	9.01584e6	0.389139	0.194569	+	0.980889i	$0.437669\pi$
$884$	386752.	0.0166457
$885$	7.30764e6	0.313631
$886$	−1.91662e6	−0.0820258
$887$	−1.25216e7	−0.534380	−0.267190	−	0.963644i	$-0.586095\pi$
$887$	−1.25216e7	−0.534380	−0.267190	+	0.963644i	$0.586095\pi$
$888$	1.01822e7	0.433321
$889$	820581.	0.0348231
$890$	1.24076e7	0.525064
$891$	−2.43446e6	−0.102733
$892$	9.83987e6	0.414073
$893$	−2.98589e7	−1.25298
$894$	434786.	0.0181942
$895$	9.63418e6	0.402029
$896$	−1.29477e6	−0.0538793
$897$	−4.55927e6	−0.189197
$898$	−1.65462e7	−0.684709
$899$	9.72838e6	0.401459
$900$	672314.	0.0276673
$901$	3.41147e6	0.140000
$902$	6.10763e7	2.49952
$903$	−1.94554e6	−0.0794002
$904$	2.61204e7	1.06306
$905$	2.36079e6	0.0958155
$906$	1.74050e7	0.704453
$907$	−5.20915e6	−0.210256	−0.105128	−	0.994459i	$-0.533525\pi$
$907$	−5.20915e6	−0.210256	−0.105128	+	0.994459i	$0.533525\pi$
$908$	6.07689e6	0.244606
$909$	−6.15900e6	−0.247230
$910$	299760.	0.0119997
$911$	1.33715e7	0.533808	0.266904	−	0.963723i	$-0.413999\pi$
$911$	1.33715e7	0.533808	0.266904	+	0.963723i	$0.413999\pi$
$912$	−1.34631e7	−0.535992
$913$	−5.46916e7	−2.17142
$914$	2.15919e7	0.854919
$915$	−7.29356e6	−0.287996
$916$	−8.94145e6	−0.352103
$917$	−3.82950e6	−0.150390
$918$	−5.51091e6	−0.215832
$919$	3.60916e7	1.40967	0.704835	−	0.709371i	$-0.251021\pi$
$919$	3.60916e7	1.40967	0.704835	+	0.709371i	$0.251021\pi$
$920$	−1.24983e7	−0.486836
$921$	−200501.	−0.00778875
$922$	1.94773e7	0.754573
$923$	2.44664e6	0.0945290
$924$	744956.	0.0287045
$925$	−3.27325e6	−0.125784
$926$	2.21705e7	0.849665
$927$	2.77081e7	1.05903
$928$	2.93000e6	0.111686
$929$	2.41737e7	0.918977	0.459489	−	0.888184i	$-0.348033\pi$
$929$	2.41737e7	0.918977	0.459489	+	0.888184i	$0.348033\pi$
$930$	−1.08068e7	−0.409724
$931$	3.08548e7	1.16667
$932$	8.02637e6	0.302677
$933$	−1.74655e7	−0.656868
$934$	2.93733e7	1.10176
$935$	5.36733e6	0.200784
$936$	5.04863e6	0.188358
$937$	−3.26851e7	−1.21619	−0.608094	−	0.793865i	$-0.708066\pi$
$937$	−3.26851e7	−1.21619	−0.608094	+	0.793865i	$0.708066\pi$
$938$	−275739.	−0.0102327
$939$	−6.79333e6	−0.251431
$940$	−3.00278e6	−0.110842
$941$	−3.02020e7	−1.11189	−0.555944	−	0.831220i	$-0.687643\pi$
$941$	−3.02020e7	−1.11189	−0.555944	+	0.831220i	$0.687643\pi$
$942$	2.72340e6	0.0999962
$943$	4.24495e7	1.55451
$944$	2.14358e7	0.782905
$945$	1.29979e6	0.0473472
$946$	−5.32986e7	−1.93637
$947$	8.76265e6	0.317512	0.158756	−	0.987318i	$-0.449252\pi$
$947$	8.76265e6	0.317512	0.158756	+	0.987318i	$0.449252\pi$
$948$	−6.40669e6	−0.231533
$949$	6.01953e6	0.216969
$950$	5.74562e6	0.206551
$951$	−867503.	−0.0311043
$952$	762955.	0.0272839
$953$	267287.	0.00953334	0.00476667	−	0.999989i	$-0.498483\pi$
$953$	267287.	0.00953334	0.00476667	+	0.999989i	$0.498483\pi$
$954$	8.42444e6	0.299688
$955$	−7.10017e6	−0.251919
$956$	1.12232e7	0.397166
$957$	8.19141e6	0.289120
$958$	−9.91509e6	−0.349046
$959$	−1.57400e6	−0.0552659
$960$	−9.05796e6	−0.317214
$961$	4.83681e7	1.68947
$962$	−4.64984e6	−0.161995
$963$	1.64481e7	0.571544
$964$	1.93899e6	0.0672022
$965$	2.11958e7	0.732708
$966$	−1.70146e6	−0.0586648
$967$	4.05337e7	1.39396	0.696980	−	0.717090i	$-0.254526\pi$
$967$	4.05337e7	1.39396	0.696980	+	0.717090i	$0.254526\pi$
$968$	7.63990e7	2.62059
$969$	5.33464e6	0.182514
$970$	−545801.	−0.0186254
$971$	−4.41436e7	−1.50252	−0.751259	−	0.660007i	$-0.770553\pi$
$971$	−4.41436e7	−1.50252	−0.751259	+	0.660007i	$0.770553\pi$
$972$	6.74102e6	0.228855
$973$	2.86588e6	0.0970457
$974$	−3.61171e7	−1.21988
$975$	1.11422e6	0.0375369
$976$	−2.13944e7	−0.718913
$977$	−7.20419e6	−0.241462	−0.120731	−	0.992685i	$-0.538524\pi$
$977$	−7.20419e6	−0.241462	−0.120731	+	0.992685i	$0.538524\pi$
$978$	1.51161e7	0.505350
$979$	−7.44358e7	−2.48213
$980$	3.10293e6	0.103206
$981$	1.41364e7	0.468994
$982$	4.03319e6	0.133466
$983$	−4.44557e7	−1.46738	−0.733692	−	0.679482i	$-0.762204\pi$
$983$	−4.44557e7	−1.46738	−0.733692	+	0.679482i	$0.762204\pi$
$984$	3.22709e7	1.06248
$985$	2.03066e7	0.666878
$986$	1.58703e6	0.0519868
$987$	−2.16089e6	−0.0706058
$988$	−2.48374e6	−0.0809494
$989$	−3.70438e7	−1.20427
$990$	1.32543e7	0.429803
$991$	4.66195e7	1.50794	0.753970	−	0.656909i	$-0.228137\pi$
$991$	4.66195e7	1.50794	0.753970	+	0.656909i	$0.228137\pi$
$992$	2.31901e7	0.748209
$993$	−2.55923e7	−0.823639
$994$	913050.	0.0293109
$995$	1.03371e7	0.331009
$996$	−5.46660e6	−0.174610
$997$	6.01277e7	1.91574	0.957870	−	0.287202i	$-0.0927251\pi$
$997$	6.01277e7	1.91574	0.957870	+	0.287202i	$0.0927251\pi$
$998$	8.22117e6	0.261281
$999$	−2.01622e7	−0.639182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.6.a.a.1.2	✓	5
3.2	odd	2		765.6.a.g.1.4		5
5.4	even	2		425.6.a.d.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.2	✓	5	1.1	even	1	trivial
425.6.a.d.1.4		5	5.4	even	2
765.6.a.g.1.4		5	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

\(f(q)\)	\(=\)	\(q-4.95319 q^{2} -9.94571 q^{3} -7.46587 q^{4} +25.0000 q^{5} +49.2631 q^{6} +13.5050 q^{7} +195.482 q^{8} -144.083 q^{9} -123.830 q^{10} +742.882 q^{11} +74.2534 q^{12} -179.248 q^{13} -66.8928 q^{14} -248.643 q^{15} -729.353 q^{16} +289.000 q^{17} +713.670 q^{18} -1855.97 q^{19} -186.647 q^{20} -134.317 q^{21} -3679.64 q^{22} -2557.44 q^{23} -1944.21 q^{24} +625.000 q^{25} +887.850 q^{26} +3849.81 q^{27} -100.826 q^{28} -1108.67 q^{29} +1231.58 q^{30} -8774.81 q^{31} -2642.80 q^{32} -7388.50 q^{33} -1431.47 q^{34} +337.625 q^{35} +1075.70 q^{36} -5237.19 q^{37} +9192.99 q^{38} +1782.75 q^{39} +4887.05 q^{40} -16598.4 q^{41} +665.297 q^{42} +14484.7 q^{43} -5546.26 q^{44} -3602.07 q^{45} +12667.5 q^{46} +16088.0 q^{47} +7253.94 q^{48} -16624.6 q^{49} -3095.75 q^{50} -2874.31 q^{51} +1338.24 q^{52} +11804.4 q^{53} -19068.9 q^{54} +18572.1 q^{55} +2639.98 q^{56} +18459.0 q^{57} +5491.46 q^{58} -29390.1 q^{59} +1856.33 q^{60} +29333.5 q^{61} +43463.3 q^{62} -1945.84 q^{63} +36429.6 q^{64} -4481.20 q^{65} +36596.7 q^{66} +4122.11 q^{67} -2157.64 q^{68} +25435.6 q^{69} -1672.32 q^{70} -13649.5 q^{71} -28165.6 q^{72} -33582.1 q^{73} +25940.8 q^{74} -6216.07 q^{75} +13856.4 q^{76} +10032.6 q^{77} -8830.30 q^{78} -86281.4 q^{79} -18233.8 q^{80} -3277.05 q^{81} +82215.3 q^{82} -73620.8 q^{83} +1002.79 q^{84} +7225.00 q^{85} -71745.7 q^{86} +11026.5 q^{87} +145220. q^{88} -100199. q^{89} +17841.7 q^{90} -2420.74 q^{91} +19093.5 q^{92} +87271.8 q^{93} -79687.2 q^{94} -46399.3 q^{95} +26284.5 q^{96} +4407.67 q^{97} +82344.9 q^{98} -107037. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 7 q^{2} - 36 q^{3} + 43 q^{4} + 125 q^{5} - 105 q^{6} - 204 q^{7} - 63 q^{8} + 531 q^{9} - 175 q^{10} - 792 q^{11} + 785 q^{12} + 88 q^{13} + 860 q^{14} - 900 q^{15} - 2365 q^{16} + 1445 q^{17} - 2052 q^{18}+ \cdots - 535112 q^{99}+O(q^{100}) \) 5 * q - 7 * q^2 - 36 * q^3 + 43 * q^4 + 125 * q^5 - 105 * q^6 - 204 * q^7 - 63 * q^8 + 531 * q^9 - 175 * q^10 - 792 * q^11 + 785 * q^12 + 88 * q^13 + 860 * q^14 - 900 * q^15 - 2365 * q^16 + 1445 * q^17 - 2052 * q^18 - 5160 * q^19 + 1075 * q^20 - 6428 * q^21 - 3058 * q^22 - 6140 * q^23 - 11751 * q^24 + 3125 * q^25 - 9643 * q^26 - 13380 * q^27 - 4568 * q^28 - 7328 * q^29 - 2625 * q^30 - 8968 * q^31 - 4703 * q^32 + 13072 * q^33 - 2023 * q^34 - 5100 * q^35 + 2048 * q^36 + 9834 * q^37 + 10625 * q^38 + 10108 * q^39 - 1575 * q^40 - 5302 * q^41 + 16222 * q^42 - 4916 * q^43 - 31514 * q^44 + 13275 * q^45 - 16768 * q^46 - 20364 * q^47 + 45557 * q^48 - 26483 * q^49 - 4375 * q^50 - 10404 * q^51 - 11525 * q^52 + 56492 * q^53 + 64635 * q^54 - 19800 * q^55 + 73228 * q^56 + 67554 * q^57 + 54213 * q^58 - 96200 * q^59 + 19625 * q^60 - 7284 * q^61 + 122335 * q^62 + 64108 * q^63 + 28483 * q^64 + 2200 * q^65 + 115232 * q^66 - 60308 * q^67 + 12427 * q^68 - 89076 * q^69 + 21500 * q^70 - 25304 * q^71 + 118722 * q^72 + 163328 * q^73 - 82768 * q^74 - 22500 * q^75 + 92027 * q^76 + 19912 * q^77 + 134895 * q^78 - 50940 * q^79 - 59125 * q^80 + 92445 * q^81 + 191390 * q^82 - 179060 * q^83 - 128054 * q^84 + 36125 * q^85 + 87086 * q^86 + 55764 * q^87 + 167634 * q^88 - 387356 * q^89 - 51300 * q^90 - 115200 * q^91 - 293228 * q^92 - 202558 * q^93 + 100515 * q^94 - 129000 * q^95 + 428113 * q^96 - 106076 * q^97 + 49329 * q^98 - 535112 * q^99

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