LMFDB - Embedded newform 165.6.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [165,6,Mod(1,165)] 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("165.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(165, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	165.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,2,-27,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.4633302691$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.18257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.78415$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.78415 q^{2} -9.00000 q^{3} +1.45634 q^{4} +25.0000 q^{5} -52.0573 q^{6} +17.6498 q^{7} -176.669 q^{8} +81.0000 q^{9} +144.604 q^{10} +121.000 q^{11} -13.1070 q^{12} +674.396 q^{13} +102.089 q^{14} -225.000 q^{15} -1068.48 q^{16} -2117.62 q^{17} +468.516 q^{18} -2307.79 q^{19} +36.4084 q^{20} -158.848 q^{21} +699.882 q^{22} -3072.47 q^{23} +1590.02 q^{24} +625.000 q^{25} +3900.80 q^{26} -729.000 q^{27} +25.7040 q^{28} -1437.44 q^{29} -1301.43 q^{30} -5157.66 q^{31} -526.846 q^{32} -1089.00 q^{33} -12248.6 q^{34} +441.244 q^{35} +117.963 q^{36} +6928.88 q^{37} -13348.6 q^{38} -6069.56 q^{39} -4416.72 q^{40} +2844.78 q^{41} -918.799 q^{42} -11665.3 q^{43} +176.217 q^{44} +2025.00 q^{45} -17771.6 q^{46} +3451.40 q^{47} +9616.34 q^{48} -16495.5 q^{49} +3615.09 q^{50} +19058.6 q^{51} +982.146 q^{52} -18167.6 q^{53} -4216.64 q^{54} +3025.00 q^{55} -3118.17 q^{56} +20770.1 q^{57} -8314.36 q^{58} -8976.88 q^{59} -327.675 q^{60} -378.820 q^{61} -29832.7 q^{62} +1429.63 q^{63} +31144.1 q^{64} +16859.9 q^{65} -6298.93 q^{66} -12233.7 q^{67} -3083.97 q^{68} +27652.2 q^{69} +2552.22 q^{70} +55867.0 q^{71} -14310.2 q^{72} +19739.6 q^{73} +40077.6 q^{74} -5625.00 q^{75} -3360.92 q^{76} +2135.62 q^{77} -35107.2 q^{78} -13495.8 q^{79} -26712.0 q^{80} +6561.00 q^{81} +16454.6 q^{82} +42078.3 q^{83} -231.336 q^{84} -52940.5 q^{85} -67474.0 q^{86} +12937.0 q^{87} -21376.9 q^{88} -81223.2 q^{89} +11712.9 q^{90} +11902.9 q^{91} -4474.55 q^{92} +46419.0 q^{93} +19963.4 q^{94} -57694.8 q^{95} +4741.62 q^{96} +152101. q^{97} -95412.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 378 q^{12} + 290 q^{13} + 916 q^{14} - 675 q^{15} - 590 q^{16} + 434 q^{17} + 162 q^{18}+ \cdots + 29403 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 - 27 * q^3 - 42 * q^4 + 75 * q^5 - 18 * q^6 - 68 * q^7 - 24 * q^8 + 243 * q^9 + 50 * q^10 + 363 * q^11 + 378 * q^12 + 290 * q^13 + 916 * q^14 - 675 * q^15 - 590 * q^16 + 434 * q^17 + 162 * q^18 - 2856 * q^19 - 1050 * q^20 + 612 * q^21 + 242 * q^22 - 640 * q^23 + 216 * q^24 + 1875 * q^25 + 2132 * q^26 - 2187 * q^27 - 580 * q^28 - 4538 * q^29 - 450 * q^30 - 14968 * q^31 - 2496 * q^32 - 3267 * q^33 - 13704 * q^34 - 1700 * q^35 - 3402 * q^36 - 6190 * q^37 - 11668 * q^38 - 2610 * q^39 - 600 * q^40 - 8926 * q^41 - 8244 * q^42 - 33592 * q^43 - 5082 * q^44 + 6075 * q^45 - 35680 * q^46 - 24640 * q^47 + 5310 * q^48 - 14693 * q^49 + 1250 * q^50 - 3906 * q^51 + 18780 * q^52 - 22934 * q^53 - 1458 * q^54 + 9075 * q^55 - 40012 * q^56 + 25704 * q^57 - 32304 * q^58 - 13756 * q^59 + 9450 * q^60 + 24602 * q^61 - 7704 * q^62 - 5508 * q^63 + 35474 * q^64 + 7250 * q^65 - 2178 * q^66 + 16868 * q^67 - 71288 * q^68 + 5760 * q^69 + 22900 * q^70 + 4856 * q^71 - 1944 * q^72 + 1910 * q^73 + 29404 * q^74 - 16875 * q^75 + 6116 * q^76 - 8228 * q^77 - 19188 * q^78 - 36844 * q^79 - 14750 * q^80 + 19683 * q^81 + 84000 * q^82 - 48796 * q^83 + 5220 * q^84 + 10850 * q^85 - 83492 * q^86 + 40842 * q^87 - 2904 * q^88 - 188978 * q^89 + 4050 * q^90 - 93208 * q^91 - 6976 * q^92 + 134712 * q^93 + 70472 * q^94 - 71400 * q^95 + 22464 * q^96 + 247526 * q^97 - 154654 * q^98 + 29403 * 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 $

$2$	5.78415	1.02250	0.511251	−	0.859431i	$-0.329182\pi$
$2$	5.78415	1.02250	0.511251	+	0.859431i	$0.329182\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	1.45634	0.0455105
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−52.0573	−0.590342
$7$	17.6498	0.136143	0.0680713	−	0.997680i	$-0.478315\pi$
$7$	17.6498	0.136143	0.0680713	+	0.997680i	$0.478315\pi$
$8$	−176.669	−0.975968
$9$	81.0000	0.333333
$10$	144.604	0.457277
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−13.1070	−0.0262755
$13$	674.396	1.10677	0.553384	−	0.832926i	$-0.313336\pi$
$13$	674.396	1.10677	0.553384	+	0.832926i	$0.313336\pi$
$14$	102.089	0.139206
$15$	−225.000	−0.258199
$16$	−1068.48	−1.04344
$17$	−2117.62	−1.77716	−0.888579	−	0.458724i	$-0.848307\pi$
$17$	−2117.62	−1.77716	−0.888579	+	0.458724i	$0.848307\pi$
$18$	468.516	0.340834
$19$	−2307.79	−1.46660	−0.733302	−	0.679903i	$-0.762022\pi$
$19$	−2307.79	−1.46660	−0.733302	+	0.679903i	$0.762022\pi$
$20$	36.4084	0.0203529
$21$	−158.848	−0.0786019
$22$	699.882	0.308296
$23$	−3072.47	−1.21107	−0.605534	−	0.795820i	$-0.707040\pi$
$23$	−3072.47	−1.21107	−0.605534	+	0.795820i	$0.707040\pi$
$24$	1590.02	0.563475
$25$	625.000	0.200000
$26$	3900.80	1.13167
$27$	−729.000	−0.192450
$28$	25.7040	0.00619591
$29$	−1437.44	−0.317391	−0.158696	−	0.987328i	$-0.550729\pi$
$29$	−1437.44	−0.317391	−0.158696	+	0.987328i	$0.550729\pi$
$30$	−1301.43	−0.264009
$31$	−5157.66	−0.963937	−0.481969	−	0.876188i	$-0.660078\pi$
$31$	−5157.66	−0.963937	−0.481969	+	0.876188i	$0.660078\pi$
$32$	−526.846	−0.0909513
$33$	−1089.00	−0.174078
$34$	−12248.6	−1.81715
$35$	441.244	0.0608848
$36$	117.963	0.0151702
$37$	6928.88	0.832067	0.416034	−	0.909349i	$-0.363420\pi$
$37$	6928.88	0.832067	0.416034	+	0.909349i	$0.363420\pi$
$38$	−13348.6	−1.49961
$39$	−6069.56	−0.638993
$40$	−4416.72	−0.436466
$41$	2844.78	0.264295	0.132147	−	0.991230i	$-0.457813\pi$
$41$	2844.78	0.264295	0.132147	+	0.991230i	$0.457813\pi$
$42$	−918.799	−0.0803706
$43$	−11665.3	−0.962113	−0.481057	−	0.876689i	$-0.659747\pi$
$43$	−11665.3	−0.962113	−0.481057	+	0.876689i	$0.659747\pi$
$44$	176.217	0.0137219
$45$	2025.00	0.149071
$46$	−17771.6	−1.23832
$47$	3451.40	0.227903	0.113952	−	0.993486i	$-0.463649\pi$
$47$	3451.40	0.227903	0.113952	+	0.993486i	$0.463649\pi$
$48$	9616.34	0.602430
$49$	−16495.5	−0.981465
$50$	3615.09	0.204500
$51$	19058.6	1.02604
$52$	982.146	0.0503695
$53$	−18167.6	−0.888401	−0.444200	−	0.895927i	$-0.646512\pi$
$53$	−18167.6	−0.888401	−0.444200	+	0.895927i	$0.646512\pi$
$54$	−4216.64	−0.196781
$55$	3025.00	0.134840
$56$	−3118.17	−0.132871
$57$	20770.1	0.846744
$58$	−8314.36	−0.324533
$59$	−8976.88	−0.335734	−0.167867	−	0.985810i	$-0.553688\pi$
$59$	−8976.88	−0.335734	−0.167867	+	0.985810i	$0.553688\pi$
$60$	−327.675	−0.0117508
$61$	−378.820	−0.0130349	−0.00651746	−	0.999979i	$-0.502075\pi$
$61$	−378.820	−0.0130349	−0.00651746	+	0.999979i	$0.502075\pi$
$62$	−29832.7	−0.985628
$63$	1429.63	0.0453808
$64$	31144.1	0.950441
$65$	16859.9	0.494962
$66$	−6298.93	−0.177995
$67$	−12233.7	−0.332945	−0.166473	−	0.986046i	$-0.553238\pi$
$67$	−12233.7	−0.332945	−0.166473	+	0.986046i	$0.553238\pi$
$68$	−3083.97	−0.0808793
$69$	27652.2	0.699210
$70$	2552.22	0.0622548
$71$	55867.0	1.31525	0.657627	−	0.753344i	$-0.271560\pi$
$71$	55867.0	1.31525	0.657627	+	0.753344i	$0.271560\pi$
$72$	−14310.2	−0.325323
$73$	19739.6	0.433541	0.216771	−	0.976223i	$-0.430448\pi$
$73$	19739.6	0.433541	0.216771	+	0.976223i	$0.430448\pi$
$74$	40077.6	0.850791
$75$	−5625.00	−0.115470
$76$	−3360.92	−0.0667459
$77$	2135.62	0.0410485
$78$	−35107.2	−0.653371
$79$	−13495.8	−0.243293	−0.121647	−	0.992573i	$-0.538817\pi$
$79$	−13495.8	−0.243293	−0.121647	+	0.992573i	$0.538817\pi$
$80$	−26712.0	−0.466640
$81$	6561.00	0.111111
$82$	16454.6	0.270242
$83$	42078.3	0.670444	0.335222	−	0.942139i	$-0.391189\pi$
$83$	42078.3	0.670444	0.335222	+	0.942139i	$0.391189\pi$
$84$	−231.336	−0.00357721
$85$	−52940.5	−0.794769
$86$	−67474.0	−0.983763
$87$	12937.0	0.183246
$88$	−21376.9	−0.294265
$89$	−81223.2	−1.08694	−0.543469	−	0.839429i	$-0.682890\pi$
$89$	−81223.2	−1.08694	−0.543469	+	0.839429i	$0.682890\pi$
$90$	11712.9	0.152426
$91$	11902.9	0.150678
$92$	−4474.55	−0.0551163
$93$	46419.0	0.556530
$94$	19963.4	0.233031
$95$	−57694.8	−0.655885
$96$	4741.62	0.0525108
$97$	152101.	1.64136	0.820680	−	0.571388i	$-0.193595\pi$
$97$	152101.	1.64136	0.820680	+	0.571388i	$0.193595\pi$
$98$	−95412.3	−1.00355
$99$	9801.00	0.100504
$100$	910.210	0.00910210
$101$	95207.6	0.928684	0.464342	−	0.885656i	$-0.346291\pi$
$101$	95207.6	0.928684	0.464342	+	0.885656i	$0.346291\pi$
$102$	110238.	1.04913
$103$	−150824.	−1.40081	−0.700403	−	0.713748i	$-0.746996\pi$
$103$	−150824.	−1.40081	−0.700403	+	0.713748i	$0.746996\pi$
$104$	−119145.	−1.08017
$105$	−3971.20	−0.0351518
$106$	−105084.	−0.908392
$107$	192242.	1.62327	0.811633	−	0.584168i	$-0.198579\pi$
$107$	192242.	1.62327	0.811633	+	0.584168i	$0.198579\pi$
$108$	−1061.67	−0.00875850
$109$	96302.8	0.776377	0.388188	−	0.921580i	$-0.373101\pi$
$109$	96302.8	0.776377	0.388188	+	0.921580i	$0.373101\pi$
$110$	17497.0	0.137874
$111$	−62359.9	−0.480394
$112$	−18858.5	−0.142056
$113$	−210371.	−1.54985	−0.774926	−	0.632052i	$-0.782213\pi$
$113$	−210371.	−1.54985	−0.774926	+	0.632052i	$0.782213\pi$
$114$	120138.	0.865798
$115$	−76811.8	−0.541606
$116$	−2093.39	−0.0144446
$117$	54626.1	0.368923
$118$	−51923.6	−0.343289
$119$	−37375.5	−0.241947
$120$	39750.5	0.251994
$121$	14641.0	0.0909091
$122$	−2191.15	−0.0133282
$123$	−25603.0	−0.152591
$124$	−7511.29	−0.0438692
$125$	15625.0	0.0894427
$126$	8269.19	0.0464020
$127$	−63529.8	−0.349517	−0.174758	−	0.984611i	$-0.555914\pi$
$127$	−63529.8	−0.349517	−0.174758	+	0.984611i	$0.555914\pi$
$128$	197001.	1.06278
$129$	104988.	0.555476
$130$	97520.1	0.506099
$131$	−88396.0	−0.450043	−0.225022	−	0.974354i	$-0.572245\pi$
$131$	−88396.0	−0.450043	−0.225022	+	0.974354i	$0.572245\pi$
$132$	−1585.95	−0.00792236
$133$	−40732.0	−0.199667
$134$	−70761.8	−0.340437
$135$	−18225.0	−0.0860663
$136$	374118.	1.73445
$137$	−42642.4	−0.194106	−0.0970532	−	0.995279i	$-0.530942\pi$
$137$	−42642.4	−0.194106	−0.0970532	+	0.995279i	$0.530942\pi$
$138$	159945.	0.714944
$139$	169503.	0.744114	0.372057	−	0.928210i	$-0.378653\pi$
$139$	169503.	0.744114	0.372057	+	0.928210i	$0.378653\pi$
$140$	642.599	0.00277090
$141$	−31062.6	−0.131580
$142$	323143.	1.34485
$143$	81601.9	0.333703
$144$	−86547.0	−0.347813
$145$	−35936.0	−0.141942
$146$	114176.	0.443297
$147$	148459.	0.566649
$148$	10090.8	0.0378678
$149$	−72462.8	−0.267393	−0.133696	−	0.991022i	$-0.542685\pi$
$149$	−72462.8	−0.267393	−0.133696	+	0.991022i	$0.542685\pi$
$150$	−32535.8	−0.118068
$151$	−61316.2	−0.218843	−0.109422	−	0.993995i	$-0.534900\pi$
$151$	−61316.2	−0.218843	−0.109422	+	0.993995i	$0.534900\pi$
$152$	407716.	1.43136
$153$	−171527.	−0.592386
$154$	12352.7	0.0419722
$155$	−128942.	−0.431086
$156$	−8839.32	−0.0290809
$157$	505789.	1.63765	0.818824	−	0.574045i	$-0.194627\pi$
$157$	505789.	1.63765	0.818824	+	0.574045i	$0.194627\pi$
$158$	−78061.5	−0.248768
$159$	163509.	0.512919
$160$	−13171.2	−0.0406747
$161$	−54228.4	−0.164878
$162$	37949.8	0.113611
$163$	387702.	1.14295	0.571477	−	0.820618i	$-0.306371\pi$
$163$	387702.	1.14295	0.571477	+	0.820618i	$0.306371\pi$
$164$	4142.95	0.0120282
$165$	−27225.0	−0.0778499
$166$	243387.	0.685530
$167$	−635422.	−1.76308	−0.881538	−	0.472113i	$-0.843491\pi$
$167$	−635422.	−1.76308	−0.881538	+	0.472113i	$0.843491\pi$
$168$	28063.5	0.0767129
$169$	83516.8	0.224935
$170$	−306216.	−0.812653
$171$	−186931.	−0.488868
$172$	−16988.6	−0.0437862
$173$	−679820.	−1.72695	−0.863473	−	0.504395i	$-0.831716\pi$
$173$	−679820.	−1.72695	−0.863473	+	0.504395i	$0.831716\pi$
$174$	74829.3	0.187369
$175$	11031.1	0.0272285
$176$	−129286.	−0.314609
$177$	80792.0	0.193836
$178$	−469807.	−1.11140
$179$	−63083.3	−0.147157	−0.0735787	−	0.997289i	$-0.523442\pi$
$179$	−63083.3	−0.147157	−0.0735787	+	0.997289i	$0.523442\pi$
$180$	2949.08	0.00678430
$181$	523614.	1.18800	0.593998	−	0.804466i	$-0.297549\pi$
$181$	523614.	1.18800	0.593998	+	0.804466i	$0.297549\pi$
$182$	68848.3	0.154069
$183$	3409.38	0.00752571
$184$	542810.	1.18196
$185$	173222.	0.372112
$186$	268494.	0.569053
$187$	−256232.	−0.535833
$188$	5026.39	0.0103720
$189$	−12866.7	−0.0262006
$190$	−333715.	−0.670644
$191$	−684311.	−1.35728	−0.678641	−	0.734470i	$-0.737431\pi$
$191$	−684311.	−1.35728	−0.678641	+	0.734470i	$0.737431\pi$
$192$	−280297.	−0.548738
$193$	527417.	1.01920	0.509601	−	0.860411i	$-0.329793\pi$
$193$	527417.	1.01920	0.509601	+	0.860411i	$0.329793\pi$
$194$	879777.	1.67829
$195$	−151739.	−0.285766
$196$	−24023.0	−0.0446669
$197$	−467139.	−0.857591	−0.428796	−	0.903402i	$-0.641062\pi$
$197$	−467139.	−0.857591	−0.428796	+	0.903402i	$0.641062\pi$
$198$	56690.4	0.102765
$199$	−436613.	−0.781563	−0.390782	−	0.920483i	$-0.627795\pi$
$199$	−436613.	−0.781563	−0.390782	+	0.920483i	$0.627795\pi$
$200$	−110418.	−0.195194
$201$	110104.	0.192226
$202$	550694.	0.949581
$203$	−25370.5	−0.0432104
$204$	27755.7	0.0466957
$205$	71119.4	0.118196
$206$	−872389.	−1.43233
$207$	−248870.	−0.403689
$208$	−720580.	−1.15484
$209$	−279243.	−0.442198
$210$	−22970.0	−0.0359428
$211$	−747312.	−1.15557	−0.577785	−	0.816189i	$-0.696083\pi$
$211$	−747312.	−1.15557	−0.577785	+	0.816189i	$0.696083\pi$
$212$	−26458.2	−0.0404315
$213$	−502803.	−0.759362
$214$	1.11196e6	1.65979
$215$	−291634.	−0.430270
$216$	128792.	0.187825
$217$	−91031.6	−0.131233
$218$	557029.	0.793847
$219$	−177656.	−0.250305
$220$	4405.41	0.00613663
$221$	−1.42812e6	−1.96690
$222$	−360699.	−0.491204
$223$	870654.	1.17242	0.586211	−	0.810159i	$-0.300619\pi$
$223$	870654.	1.17242	0.586211	+	0.810159i	$0.300619\pi$
$224$	−9298.71	−0.0123823
$225$	50625.0	0.0666667
$226$	−1.21682e6	−1.58473
$227$	82323.8	0.106038	0.0530189	−	0.998594i	$-0.483116\pi$
$227$	82323.8	0.106038	0.0530189	+	0.998594i	$0.483116\pi$
$228$	30248.3	0.0385357
$229$	−340134.	−0.428609	−0.214305	−	0.976767i	$-0.568748\pi$
$229$	−340134.	−0.428609	−0.214305	+	0.976767i	$0.568748\pi$
$230$	−444291.	−0.553793
$231$	−19220.6	−0.0236994
$232$	253951.	0.309763
$233$	−248384.	−0.299732	−0.149866	−	0.988706i	$-0.547884\pi$
$233$	−248384.	−0.299732	−0.149866	+	0.988706i	$0.547884\pi$
$234$	315965.	0.377224
$235$	86284.9	0.101921
$236$	−13073.4	−0.0152794
$237$	121462.	0.140465
$238$	−216185.	−0.247391
$239$	−1.37978e6	−1.56248	−0.781242	−	0.624229i	$-0.785413\pi$
$239$	−1.37978e6	−1.56248	−0.781242	+	0.624229i	$0.785413\pi$
$240$	240408.	0.269415
$241$	−1.53382e6	−1.70110	−0.850552	−	0.525891i	$-0.823732\pi$
$241$	−1.53382e6	−1.70110	−0.850552	+	0.525891i	$0.823732\pi$
$242$	84685.7	0.0929547
$243$	−59049.0	−0.0641500
$244$	−551.689	−0.000593225 0
$245$	−412387.	−0.438925
$246$	−148091.	−0.156024
$247$	−1.55637e6	−1.62319
$248$	911199.	0.940772
$249$	−378704.	−0.387081
$250$	90377.3	0.0914554
$251$	338591.	0.339227	0.169614	−	0.985511i	$-0.445748\pi$
$251$	338591.	0.339227	0.169614	+	0.985511i	$0.445748\pi$
$252$	2082.02	0.00206530
$253$	−371769.	−0.365151
$254$	−367466.	−0.357382
$255$	476465.	0.458860
$256$	142872.	0.136253
$257$	1.60985e6	1.52038	0.760189	−	0.649702i	$-0.225106\pi$
$257$	1.60985e6	1.52038	0.760189	+	0.649702i	$0.225106\pi$
$258$	607266.	0.567976
$259$	122293.	0.113280
$260$	24553.7	0.0225259
$261$	−116433.	−0.105797
$262$	−511295.	−0.460170
$263$	458118.	0.408402	0.204201	−	0.978929i	$-0.434540\pi$
$263$	458118.	0.408402	0.204201	+	0.978929i	$0.434540\pi$
$264$	192393.	0.169894
$265$	−454191.	−0.397305
$266$	−235600.	−0.204160
$267$	731009.	0.627544
$268$	−17816.4	−0.0151525
$269$	−692380.	−0.583397	−0.291698	−	0.956510i	$-0.594220\pi$
$269$	−692380.	−0.583397	−0.291698	+	0.956510i	$0.594220\pi$
$270$	−105416.	−0.0880030
$271$	−486690.	−0.402558	−0.201279	−	0.979534i	$-0.564510\pi$
$271$	−486690.	−0.402558	−0.201279	+	0.979534i	$0.564510\pi$
$272$	2.26264e6	1.85436
$273$	−107126.	−0.0869941
$274$	−246650.	−0.198474
$275$	75625.0	0.0603023
$276$	40270.9	0.0318214
$277$	2.38271e6	1.86583	0.932915	−	0.360098i	$-0.117257\pi$
$277$	2.38271e6	1.86583	0.932915	+	0.360098i	$0.117257\pi$
$278$	980428.	0.760858
$279$	−417771.	−0.321312
$280$	−77954.1	−0.0594216
$281$	1.12154e6	0.847320	0.423660	−	0.905821i	$-0.360745\pi$
$281$	1.12154e6	0.847320	0.423660	+	0.905821i	$0.360745\pi$
$282$	−179670.	−0.134541
$283$	965956.	0.716954	0.358477	−	0.933539i	$-0.383296\pi$
$283$	965956.	0.716954	0.358477	+	0.933539i	$0.383296\pi$
$284$	81361.1	0.0598578
$285$	519254.	0.378676
$286$	471997.	0.341212
$287$	50209.6	0.0359817
$288$	−42674.5	−0.0303171
$289$	3.06446e6	2.15829
$290$	−207859.	−0.145136
$291$	−1.36891e6	−0.947640
$292$	28747.4	0.0197307
$293$	2.20819e6	1.50269	0.751343	−	0.659912i	$-0.229406\pi$
$293$	2.20819e6	1.50269	0.751343	+	0.659912i	$0.229406\pi$
$294$	858711.	0.579400
$295$	−224422.	−0.150145
$296$	−1.22412e6	−0.812071
$297$	−88209.0	−0.0580259
$298$	−419135.	−0.273410
$299$	−2.07206e6	−1.34037
$300$	−8191.89	−0.00525510
$301$	−205891.	−0.130985
$302$	−354662.	−0.223768
$303$	−856868.	−0.536176
$304$	2.46584e6	1.53031
$305$	−9470.49	−0.00582939
$306$	−992139.	−0.605716
$307$	805480.	0.487763	0.243881	−	0.969805i	$-0.421579\pi$
$307$	805480.	0.487763	0.243881	+	0.969805i	$0.421579\pi$
$308$	3110.18	0.00186814
$309$	1.35742e6	0.808756
$310$	−745817.	−0.440786
$311$	−870766.	−0.510505	−0.255253	−	0.966874i	$-0.582159\pi$
$311$	−870766.	−0.510505	−0.255253	+	0.966874i	$0.582159\pi$
$312$	1.07230e6	0.623636
$313$	−3.16630e6	−1.82680	−0.913400	−	0.407063i	$-0.866553\pi$
$313$	−3.16630e6	−1.82680	−0.913400	+	0.407063i	$0.866553\pi$
$314$	2.92556e6	1.67450
$315$	35740.8	0.0202949
$316$	−19654.4	−0.0110724
$317$	1.72288e6	0.962955	0.481478	−	0.876458i	$-0.340100\pi$
$317$	1.72288e6	0.962955	0.481478	+	0.876458i	$0.340100\pi$
$318$	945759.	0.524460
$319$	−173930.	−0.0956970
$320$	778602.	0.425050
$321$	−1.73018e6	−0.937193
$322$	−313665.	−0.168588
$323$	4.88703e6	2.60639
$324$	9555.02	0.00505672
$325$	421497.	0.221354
$326$	2.24252e6	1.16867
$327$	−866725.	−0.448241
$328$	−502584.	−0.257943
$329$	60916.3	0.0310273
$330$	−157473.	−0.0796017
$331$	−3.20493e6	−1.60786	−0.803932	−	0.594722i	$-0.797262\pi$
$331$	−3.20493e6	−1.60786	−0.803932	+	0.594722i	$0.797262\pi$
$332$	61280.1	0.0305122
$333$	561239.	0.277356
$334$	−3.67537e6	−1.80275
$335$	−305844.	−0.148898
$336$	169726.	0.0820163
$337$	2.44638e6	1.17341	0.586703	−	0.809802i	$-0.300425\pi$
$337$	2.44638e6	1.17341	0.586703	+	0.809802i	$0.300425\pi$
$338$	483073.	0.229996
$339$	1.89334e6	0.894807
$340$	−77099.2	−0.0361703
$341$	−624077.	−0.290638
$342$	−1.08124e6	−0.499869
$343$	−587781.	−0.269762
$344$	2.06090e6	0.938992
$345$	691306.	0.312696
$346$	−3.93218e6	−1.76581
$347$	−3.05055e6	−1.36005	−0.680025	−	0.733189i	$-0.738031\pi$
$347$	−3.05055e6	−1.36005	−0.680025	+	0.733189i	$0.738031\pi$
$348$	18840.5	0.00833960
$349$	−3.08755e6	−1.35691	−0.678454	−	0.734643i	$-0.737350\pi$
$349$	−3.08755e6	−1.35691	−0.678454	+	0.734643i	$0.737350\pi$
$350$	63805.5	0.0278412
$351$	−491635.	−0.212998
$352$	−63748.4	−0.0274228
$353$	2.98932e6	1.27684	0.638418	−	0.769690i	$-0.279589\pi$
$353$	2.98932e6	1.27684	0.638418	+	0.769690i	$0.279589\pi$
$354$	467312.	0.198198
$355$	1.39668e6	0.588200
$356$	−118288.	−0.0494671
$357$	336380.	0.139688
$358$	−364883.	−0.150469
$359$	−2.29991e6	−0.941836	−0.470918	−	0.882177i	$-0.656077\pi$
$359$	−2.29991e6	−0.941836	−0.470918	+	0.882177i	$0.656077\pi$
$360$	−357755.	−0.145489
$361$	2.84981e6	1.15093
$362$	3.02866e6	1.21473
$363$	−131769.	−0.0524864
$364$	17334.7	0.00685743
$365$	493489.	0.193885
$366$	19720.3	0.00769505
$367$	3.58207e6	1.38825	0.694127	−	0.719852i	$-0.255790\pi$
$367$	3.58207e6	1.38825	0.694127	+	0.719852i	$0.255790\pi$
$368$	3.28288e6	1.26368
$369$	230427.	0.0880982
$370$	1.00194e6	0.380485
$371$	−320655.	−0.120949
$372$	67601.6	0.0253279
$373$	511254.	0.190268	0.0951338	−	0.995464i	$-0.469672\pi$
$373$	511254.	0.190268	0.0951338	+	0.995464i	$0.469672\pi$
$374$	−1.48208e6	−0.547891
$375$	−140625.	−0.0516398
$376$	−609755.	−0.222426
$377$	−969404.	−0.351278
$378$	−74422.7	−0.0267902
$379$	2.45209e6	0.876877	0.438438	−	0.898761i	$-0.355532\pi$
$379$	2.45209e6	0.876877	0.438438	+	0.898761i	$0.355532\pi$
$380$	−84023.0	−0.0298497
$381$	571768.	0.201794
$382$	−3.95815e6	−1.38782
$383$	−1.46315e6	−0.509674	−0.254837	−	0.966984i	$-0.582022\pi$
$383$	−1.46315e6	−0.509674	−0.254837	+	0.966984i	$0.582022\pi$
$384$	−1.77301e6	−0.613596
$385$	53390.5	0.0183575
$386$	3.05065e6	1.04214
$387$	−944893.	−0.320704
$388$	221511.	0.0746991
$389$	−803978.	−0.269383	−0.134691	−	0.990888i	$-0.543004\pi$
$389$	−803978.	−0.269383	−0.134691	+	0.990888i	$0.543004\pi$
$390$	−877681.	−0.292197
$391$	6.50633e6	2.15226
$392$	2.91424e6	0.957878
$393$	795564.	0.259833
$394$	−2.70200e6	−0.876889
$395$	−337394.	−0.108804
$396$	14273.5	0.00457397
$397$	−2.87304e6	−0.914883	−0.457441	−	0.889240i	$-0.651234\pi$
$397$	−2.87304e6	−0.914883	−0.457441	+	0.889240i	$0.651234\pi$
$398$	−2.52543e6	−0.799150
$399$	366588.	0.115278
$400$	−667801.	−0.208688
$401$	−3.45072e6	−1.07164	−0.535820	−	0.844332i	$-0.679998\pi$
$401$	−3.45072e6	−1.07164	−0.535820	+	0.844332i	$0.679998\pi$
$402$	636856.	0.196551
$403$	−3.47831e6	−1.06685
$404$	138654.	0.0422649
$405$	164025.	0.0496904
$406$	−146747.	−0.0441827
$407$	838394.	0.250878
$408$	−3.36706e6	−1.00138
$409$	−1.17776e6	−0.348136	−0.174068	−	0.984734i	$-0.555691\pi$
$409$	−1.17776e6	−0.348136	−0.174068	+	0.984734i	$0.555691\pi$
$410$	411365.	0.120856
$411$	383781.	0.112067
$412$	−219651.	−0.0637513
$413$	−158440.	−0.0457077
$414$	−1.43950e6	−0.412773
$415$	1.05196e6	0.299832
$416$	−355303.	−0.100662
$417$	−1.52552e6	−0.429614
$418$	−1.61518e6	−0.452148
$419$	−443011.	−0.123276	−0.0616381	−	0.998099i	$-0.519632\pi$
$419$	−443011.	−0.123276	−0.0616381	+	0.998099i	$0.519632\pi$
$420$	−5783.39	−0.00159978
$421$	−3.41894e6	−0.940127	−0.470063	−	0.882633i	$-0.655769\pi$
$421$	−3.41894e6	−0.940127	−0.470063	+	0.882633i	$0.655769\pi$
$422$	−4.32256e6	−1.18157
$423$	279563.	0.0759677
$424$	3.20966e6	0.867050
$425$	−1.32351e6	−0.355432
$426$	−2.90829e6	−0.776450
$427$	−6686.08	−0.00177461
$428$	279969.	0.0738756
$429$	−734417.	−0.192664
$430$	−1.68685e6	−0.439952
$431$	−4.24640e6	−1.10110	−0.550551	−	0.834801i	$-0.685583\pi$
$431$	−4.24640e6	−1.10110	−0.550551	+	0.834801i	$0.685583\pi$
$432$	778923.	0.200810
$433$	−1.03407e6	−0.265050	−0.132525	−	0.991180i	$-0.542309\pi$
$433$	−1.03407e6	−0.265050	−0.132525	+	0.991180i	$0.542309\pi$
$434$	−526540.	−0.134186
$435$	323424.	0.0819500
$436$	140249.	0.0353333
$437$	7.09063e6	1.77616
$438$	−1.02759e6	−0.255937
$439$	3.55155e6	0.879542	0.439771	−	0.898110i	$-0.355060\pi$
$439$	3.55155e6	0.879542	0.439771	+	0.898110i	$0.355060\pi$
$440$	−534424.	−0.131599
$441$	−1.33613e6	−0.327155
$442$	−8.26043e6	−2.01116
$443$	2.35135e6	0.569257	0.284629	−	0.958638i	$-0.408130\pi$
$443$	2.35135e6	0.569257	0.284629	+	0.958638i	$0.408130\pi$
$444$	−90816.9	−0.0218630
$445$	−2.03058e6	−0.486094
$446$	5.03599e6	1.19880
$447$	652165.	0.154379
$448$	549685.	0.129395
$449$	3.01947e6	0.706831	0.353415	−	0.935467i	$-0.385020\pi$
$449$	3.01947e6	0.706831	0.353415	+	0.935467i	$0.385020\pi$
$450$	292822.	0.0681668
$451$	344218.	0.0796878
$452$	−306371.	−0.0705345
$453$	551846.	0.126349
$454$	476173.	0.108424
$455$	297573.	0.0673853
$456$	−3.66944e6	−0.826395
$457$	3.80671e6	0.852626	0.426313	−	0.904576i	$-0.359812\pi$
$457$	3.80671e6	0.852626	0.426313	+	0.904576i	$0.359812\pi$
$458$	−1.96738e6	−0.438254
$459$	1.54375e6	0.342014
$460$	−111864.	−0.0246487
$461$	7.43481e6	1.62936	0.814680	−	0.579910i	$-0.196912\pi$
$461$	7.43481e6	1.62936	0.814680	+	0.579910i	$0.196912\pi$
$462$	−111175.	−0.0242327
$463$	−108747.	−0.0235757	−0.0117879	−	0.999931i	$-0.503752\pi$
$463$	−108747.	−0.0235757	−0.0117879	+	0.999931i	$0.503752\pi$
$464$	1.53588e6	0.331178
$465$	1.16047e6	0.248888
$466$	−1.43669e6	−0.306477
$467$	−6.85473e6	−1.45445	−0.727224	−	0.686400i	$-0.759190\pi$
$467$	−6.85473e6	−1.45445	−0.727224	+	0.686400i	$0.759190\pi$
$468$	79553.9	0.0167898
$469$	−215923.	−0.0453280
$470$	499085.	0.104215
$471$	−4.55210e6	−0.945496
$472$	1.58594e6	0.327666
$473$	−1.41151e6	−0.290088
$474$	702554.	0.143626
$475$	−1.44237e6	−0.293321
$476$	−54431.3	−0.0110111
$477$	−1.47158e6	−0.296134
$478$	−7.98085e6	−1.59764
$479$	−9.48659e6	−1.88917	−0.944585	−	0.328266i	$-0.893536\pi$
$479$	−9.48659e6	−1.88917	−0.944585	+	0.328266i	$0.893536\pi$
$480$	118540.	0.0234835
$481$	4.67281e6	0.920905
$482$	−8.87182e6	−1.73938
$483$	488056.	0.0951922
$484$	21322.2	0.00413732
$485$	3.80253e6	0.734038
$486$	−341548.	−0.0655935
$487$	9.79771e6	1.87198	0.935992	−	0.352021i	$-0.114506\pi$
$487$	9.79771e6	1.87198	0.935992	+	0.352021i	$0.114506\pi$
$488$	66925.7	0.0127216
$489$	−3.48932e6	−0.659885
$490$	−2.38531e6	−0.448801
$491$	9.17094e6	1.71676	0.858381	−	0.513012i	$-0.171470\pi$
$491$	9.17094e6	1.71676	0.858381	+	0.513012i	$0.171470\pi$
$492$	−37286.5	−0.00694447
$493$	3.04395e6	0.564054
$494$	−9.00225e6	−1.65972
$495$	245025.	0.0449467
$496$	5.51087e6	1.00581
$497$	986040.	0.179062
$498$	−2.19048e6	−0.395791
$499$	−1.91031e6	−0.343441	−0.171720	−	0.985146i	$-0.554933\pi$
$499$	−1.91031e6	−0.343441	−0.171720	+	0.985146i	$0.554933\pi$
$500$	22755.2	0.00407058
$501$	5.71880e6	1.01791
$502$	1.95846e6	0.346861
$503$	7.08413e6	1.24844	0.624219	−	0.781250i	$-0.285417\pi$
$503$	7.08413e6	1.24844	0.624219	+	0.781250i	$0.285417\pi$
$504$	−252571.	−0.0442902
$505$	2.38019e6	0.415320
$506$	−2.15037e6	−0.373367
$507$	−751651.	−0.129866
$508$	−92520.7	−0.0159067
$509$	−7.51321e6	−1.28538	−0.642689	−	0.766127i	$-0.722181\pi$
$509$	−7.51321e6	−1.28538	−0.642689	+	0.766127i	$0.722181\pi$
$510$	2.75594e6	0.469186
$511$	348398.	0.0590234
$512$	−5.47764e6	−0.923461
$513$	1.68238e6	0.282248
$514$	9.31158e6	1.55459
$515$	−3.77060e6	−0.626459
$516$	152898.	0.0252800
$517$	417619.	0.0687154
$518$	707361.	0.115829
$519$	6.11838e6	0.997053
$520$	−2.97862e6	−0.483066
$521$	−1.03994e7	−1.67847	−0.839235	−	0.543770i	$-0.816997\pi$
$521$	−1.03994e7	−1.67847	−0.839235	+	0.543770i	$0.816997\pi$
$522$	−673463.	−0.108178
$523$	−9.14037e6	−1.46120	−0.730600	−	0.682806i	$-0.760759\pi$
$523$	−9.14037e6	−1.46120	−0.730600	+	0.682806i	$0.760759\pi$
$524$	−128734.	−0.0204817
$525$	−99279.9	−0.0157204
$526$	2.64982e6	0.417592
$527$	1.09220e7	1.71307
$528$	1.16358e6	0.181639
$529$	3.00374e6	0.466684
$530$	−2.62711e6	−0.406245
$531$	−727128.	−0.111911
$532$	−59319.5	−0.00908695
$533$	1.91850e6	0.292513
$534$	4.22826e6	0.641665
$535$	4.80606e6	0.725947
$536$	2.16132e6	0.324944
$537$	567750.	0.0849613
$538$	−4.00483e6	−0.596524
$539$	−1.99595e6	−0.295923
$540$	−26541.7	−0.00391692
$541$	6.47380e6	0.950969	0.475484	−	0.879724i	$-0.342273\pi$
$541$	6.47380e6	0.950969	0.475484	+	0.879724i	$0.342273\pi$
$542$	−2.81508e6	−0.411617
$543$	−4.71253e6	−0.685890
$544$	1.11566e6	0.161635
$545$	2.40757e6	0.347206
$546$	−619634.	−0.0889516
$547$	−226463.	−0.0323616	−0.0161808	−	0.999869i	$-0.505151\pi$
$547$	−226463.	−0.0323616	−0.0161808	+	0.999869i	$0.505151\pi$
$548$	−62101.6	−0.00883388
$549$	−30684.4	−0.00434497
$550$	437426.	0.0616592
$551$	3.31731e6	0.465487
$552$	−4.88529e6	−0.682406
$553$	−238197.	−0.0331226
$554$	1.37819e7	1.90781
$555$	−1.55900e6	−0.214839
$556$	246853.	0.0338650
$557$	−5.45555e6	−0.745076	−0.372538	−	0.928017i	$-0.621512\pi$
$557$	−5.45555e6	−0.745076	−0.372538	+	0.928017i	$0.621512\pi$
$558$	−2.41645e6	−0.328543
$559$	−7.86706e6	−1.06484
$560$	−471461.	−0.0635296
$561$	2.30609e6	0.309364
$562$	6.48712e6	0.866386
$563$	−1.30266e7	−1.73205	−0.866024	−	0.500002i	$-0.833333\pi$
$563$	−1.30266e7	−1.73205	−0.866024	+	0.500002i	$0.833333\pi$
$564$	−45237.5	−0.00598827
$565$	−5.25928e6	−0.693115
$566$	5.58723e6	0.733087
$567$	115800.	0.0151269
$568$	−9.86997e6	−1.28365
$569$	3.02736e6	0.391998	0.195999	−	0.980604i	$-0.437205\pi$
$569$	3.02736e6	0.391998	0.195999	+	0.980604i	$0.437205\pi$
$570$	3.00344e6	0.387197
$571$	9.79560e6	1.25731	0.628653	−	0.777686i	$-0.283607\pi$
$571$	9.79560e6	1.25731	0.628653	+	0.777686i	$0.283607\pi$
$572$	118840.	0.0151870
$573$	6.15880e6	0.783627
$574$	290420.	0.0367914
$575$	−1.92029e6	−0.242213
$576$	2.52267e6	0.316814
$577$	56282.7	0.00703778	0.00351889	−	0.999994i	$-0.498880\pi$
$577$	56282.7	0.00703778	0.00351889	+	0.999994i	$0.498880\pi$
$578$	1.77253e7	2.20686
$579$	−4.74675e6	−0.588437
$580$	−52334.9	−0.00645983
$581$	742671.	0.0912759
$582$	−7.91799e6	−0.968963
$583$	−2.19829e6	−0.267863
$584$	−3.48737e6	−0.423122
$585$	1.36565e6	0.164987
$586$	1.27725e7	1.53650
$587$	−7.92842e6	−0.949711	−0.474855	−	0.880064i	$-0.657500\pi$
$587$	−7.92842e6	−0.949711	−0.474855	+	0.880064i	$0.657500\pi$
$588$	216207.	0.0257885
$589$	1.19028e7	1.41371
$590$	−1.29809e6	−0.153523
$591$	4.20425e6	0.495130
$592$	−7.40338e6	−0.868212
$593$	−5.11910e6	−0.597801	−0.298900	−	0.954284i	$-0.596620\pi$
$593$	−5.11910e6	−0.597801	−0.298900	+	0.954284i	$0.596620\pi$
$594$	−510214.	−0.0593316
$595$	−934388.	−0.108202
$596$	−105530.	−0.0121692
$597$	3.92952e6	0.451236
$598$	−1.19851e7	−1.37053
$599$	−1.00831e7	−1.14822	−0.574110	−	0.818778i	$-0.694652\pi$
$599$	−1.00831e7	−1.14822	−0.574110	+	0.818778i	$0.694652\pi$
$600$	993763.	0.112695
$601$	1.32233e7	1.49332	0.746659	−	0.665207i	$-0.231657\pi$
$601$	1.32233e7	1.49332	0.746659	+	0.665207i	$0.231657\pi$
$602$	−1.19090e6	−0.133932
$603$	−990934.	−0.110982
$604$	−89296.9	−0.00995965
$605$	366025.	0.0406558
$606$	−4.95625e6	−0.548241
$607$	937160.	0.103239	0.0516193	−	0.998667i	$-0.483562\pi$
$607$	937160.	0.103239	0.0516193	+	0.998667i	$0.483562\pi$
$608$	1.21585e6	0.133390
$609$	228334.	0.0249475
$610$	−54778.7	−0.00596056
$611$	2.32761e6	0.252236
$612$	−249801.	−0.0269598
$613$	3.05919e6	0.328818	0.164409	−	0.986392i	$-0.447428\pi$
$613$	3.05919e6	0.328818	0.164409	+	0.986392i	$0.447428\pi$
$614$	4.65901e6	0.498738
$615$	−640074.	−0.0682406
$616$	−377298.	−0.0400620
$617$	8.74474e6	0.924771	0.462385	−	0.886679i	$-0.346994\pi$
$617$	8.74474e6	0.924771	0.462385	+	0.886679i	$0.346994\pi$
$618$	7.85150e6	0.826954
$619$	−4.14486e6	−0.434793	−0.217397	−	0.976083i	$-0.569757\pi$
$619$	−4.14486e6	−0.434793	−0.217397	+	0.976083i	$0.569757\pi$
$620$	−187782.	−0.0196189
$621$	2.23983e6	0.233070
$622$	−5.03664e6	−0.521993
$623$	−1.43357e6	−0.147979
$624$	6.48522e6	0.666750
$625$	390625.	0.0400000
$626$	−1.83143e7	−1.86791
$627$	2.51319e6	0.255303
$628$	736598.	0.0745301
$629$	−1.46727e7	−1.47872
$630$	206730.	0.0207516
$631$	−4.44508e6	−0.444433	−0.222216	−	0.974997i	$-0.571329\pi$
$631$	−4.44508e6	−0.444433	−0.222216	+	0.974997i	$0.571329\pi$
$632$	2.38429e6	0.237446
$633$	6.72581e6	0.667168
$634$	9.96537e6	0.984624
$635$	−1.58825e6	−0.156309
$636$	238124.	0.0233432
$637$	−1.11245e7	−1.08625
$638$	−1.00604e6	−0.0978504
$639$	4.52523e6	0.438418
$640$	4.92502e6	0.475289
$641$	1.08278e7	1.04086	0.520432	−	0.853903i	$-0.325771\pi$
$641$	1.08278e7	1.04086	0.520432	+	0.853903i	$0.325771\pi$
$642$	−1.00076e7	−0.958282
$643$	−1.23064e7	−1.17383	−0.586915	−	0.809649i	$-0.699658\pi$
$643$	−1.23064e7	−1.17383	−0.586915	+	0.809649i	$0.699658\pi$
$644$	−78974.7	−0.00750367
$645$	2.62470e6	0.248417
$646$	2.82673e7	2.66504
$647$	−1.26477e7	−1.18782	−0.593912	−	0.804530i	$-0.702417\pi$
$647$	−1.26477e7	−1.18782	−0.593912	+	0.804530i	$0.702417\pi$
$648$	−1.15913e6	−0.108441
$649$	−1.08620e6	−0.101228
$650$	2.43800e6	0.226334
$651$	819284.	0.0757673
$652$	564624.	0.0520164
$653$	−578855.	−0.0531235	−0.0265618	−	0.999647i	$-0.508456\pi$
$653$	−578855.	−0.0531235	−0.0265618	+	0.999647i	$0.508456\pi$
$654$	−5.01326e6	−0.458328
$655$	−2.20990e6	−0.201265
$656$	−3.03959e6	−0.275775
$657$	1.59890e6	0.144514
$658$	352349.	0.0317255
$659$	1.71481e7	1.53816	0.769081	−	0.639151i	$-0.220714\pi$
$659$	1.71481e7	1.53816	0.769081	+	0.639151i	$0.220714\pi$
$660$	−39648.7	−0.00354299
$661$	6.99329e6	0.622555	0.311278	−	0.950319i	$-0.399243\pi$
$661$	6.99329e6	0.622555	0.311278	+	0.950319i	$0.399243\pi$
$662$	−1.85378e7	−1.64404
$663$	1.28530e7	1.13559
$664$	−7.43392e6	−0.654332
$665$	−1.01830e6	−0.0892939
$666$	3.24629e6	0.283597
$667$	4.41649e6	0.384382
$668$	−925387.	−0.0802384
$669$	−7.83589e6	−0.676898
$670$	−1.76904e6	−0.152248
$671$	−45837.2	−0.00393017
$672$	83688.4	0.00714895
$673$	1.98109e7	1.68604	0.843018	−	0.537885i	$-0.180777\pi$
$673$	1.98109e7	1.68604	0.843018	+	0.537885i	$0.180777\pi$
$674$	1.41502e7	1.19981
$675$	−455625.	−0.0384900
$676$	121628.	0.0102369
$677$	−2.85012e6	−0.238996	−0.119498	−	0.992834i	$-0.538129\pi$
$677$	−2.85012e6	−0.238996	−0.119498	+	0.992834i	$0.538129\pi$
$678$	1.09514e7	0.914942
$679$	2.68455e6	0.223459
$680$	9.35295e6	0.775669
$681$	−740914.	−0.0612210
$682$	−3.60975e6	−0.297178
$683$	4.58865e6	0.376386	0.188193	−	0.982132i	$-0.439737\pi$
$683$	4.58865e6	0.376386	0.188193	+	0.982132i	$0.439737\pi$
$684$	−272235.	−0.0222486
$685$	−1.06606e6	−0.0868070
$686$	−3.39981e6	−0.275832
$687$	3.06121e6	0.247458
$688$	1.24642e7	1.00391
$689$	−1.22522e7	−0.983254
$690$	3.99862e6	0.319733
$691$	−2.65609e6	−0.211616	−0.105808	−	0.994387i	$-0.533743\pi$
$691$	−2.65609e6	−0.211616	−0.105808	+	0.994387i	$0.533743\pi$
$692$	−990046.	−0.0785942
$693$	172985.	0.0136828
$694$	−1.76448e7	−1.39065
$695$	4.23757e6	0.332778
$696$	−2.28556e6	−0.178842
$697$	−6.02416e6	−0.469693
$698$	−1.78588e7	−1.38744
$699$	2.23546e6	0.173051
$700$	16065.0	0.00123918
$701$	−7.90907e6	−0.607898	−0.303949	−	0.952688i	$-0.598305\pi$
$701$	−7.90907e6	−0.607898	−0.303949	+	0.952688i	$0.598305\pi$
$702$	−2.84369e6	−0.217790
$703$	−1.59904e7	−1.22031
$704$	3.76843e6	0.286569
$705$	−776564.	−0.0588443
$706$	1.72906e7	1.30557
$707$	1.68039e6	0.126433
$708$	117660.	0.00882158
$709$	−2.38057e7	−1.77855	−0.889273	−	0.457378i	$-0.848789\pi$
$709$	−2.38057e7	−1.77855	−0.889273	+	0.457378i	$0.848789\pi$
$710$	8.07857e6	0.601435
$711$	−1.09316e6	−0.0810978
$712$	1.43496e7	1.06082
$713$	1.58468e7	1.16739
$714$	1.94567e6	0.142831
$715$	2.04005e6	0.149237
$716$	−91870.5	−0.00669720
$717$	1.24180e7	0.902100
$718$	−1.33030e7	−0.963029
$719$	−2.00866e7	−1.44905	−0.724527	−	0.689246i	$-0.757942\pi$
$719$	−2.00866e7	−1.44905	−0.724527	+	0.689246i	$0.757942\pi$
$720$	−2.16368e6	−0.155547
$721$	−2.66201e6	−0.190709
$722$	1.64837e7	1.17683
$723$	1.38043e7	0.982133
$724$	762558.	0.0540663
$725$	−898400.	−0.0634782
$726$	−762171.	−0.0536674
$727$	−1.00897e7	−0.708012	−0.354006	−	0.935243i	$-0.615181\pi$
$727$	−1.00897e7	−0.708012	−0.354006	+	0.935243i	$0.615181\pi$
$728$	−2.10288e6	−0.147057
$729$	531441.	0.0370370
$730$	2.85441e6	0.198248
$731$	2.47028e7	1.70983
$732$	4965.20	0.000342499 0
$733$	−1.48230e7	−1.01901	−0.509504	−	0.860468i	$-0.670171\pi$
$733$	−1.48230e7	−1.01901	−0.509504	+	0.860468i	$0.670171\pi$
$734$	2.07192e7	1.41949
$735$	3.71148e6	0.253413
$736$	1.61872e6	0.110148
$737$	−1.48028e6	−0.100387
$738$	1.33282e6	0.0900806
$739$	−1.85393e7	−1.24877	−0.624383	−	0.781118i	$-0.714650\pi$
$739$	−1.85393e7	−1.24877	−0.624383	+	0.781118i	$0.714650\pi$
$740$	252269.	0.0169350
$741$	1.40073e7	0.937149
$742$	−1.85471e6	−0.123671
$743$	−8.68829e6	−0.577381	−0.288690	−	0.957423i	$-0.593220\pi$
$743$	−8.68829e6	−0.577381	−0.288690	+	0.957423i	$0.593220\pi$
$744$	−8.20079e6	−0.543155
$745$	−1.81157e6	−0.119582
$746$	2.95717e6	0.194549
$747$	3.40834e6	0.223481
$748$	−373160.	−0.0243860
$749$	3.39303e6	0.220995
$750$	−813395.	−0.0528018
$751$	−1.37377e7	−0.888821	−0.444411	−	0.895823i	$-0.646587\pi$
$751$	−1.37377e7	−0.888821	−0.444411	+	0.895823i	$0.646587\pi$
$752$	−3.68776e6	−0.237803
$753$	−3.04732e6	−0.195853
$754$	−5.60717e6	−0.359183
$755$	−1.53291e6	−0.0978696
$756$	−18738.2	−0.00119240
$757$	5.59097e6	0.354607	0.177304	−	0.984156i	$-0.443263\pi$
$757$	5.59097e6	0.354607	0.177304	+	0.984156i	$0.443263\pi$
$758$	1.41832e7	0.896608
$759$	3.34592e6	0.210820
$760$	1.01929e7	0.640123
$761$	−2.44147e7	−1.52823	−0.764117	−	0.645078i	$-0.776825\pi$
$761$	−2.44147e7	−1.52823	−0.764117	+	0.645078i	$0.776825\pi$
$762$	3.30719e6	0.206334
$763$	1.69972e6	0.105698
$764$	−996586.	−0.0617706
$765$	−4.28818e6	−0.264923
$766$	−8.46309e6	−0.521143
$767$	−6.05397e6	−0.371580
$768$	−1.28584e6	−0.0786657
$769$	1.99742e7	1.21802	0.609008	−	0.793164i	$-0.291568\pi$
$769$	1.99742e7	1.21802	0.609008	+	0.793164i	$0.291568\pi$
$770$	308819.	0.0187705
$771$	−1.44886e7	−0.877790
$772$	768096.	0.0463844
$773$	−2.64430e7	−1.59170	−0.795852	−	0.605492i	$-0.792976\pi$
$773$	−2.64430e7	−1.59170	−0.795852	+	0.605492i	$0.792976\pi$
$774$	−5.46540e6	−0.327921
$775$	−3.22354e6	−0.192787
$776$	−2.68716e7	−1.60191
$777$	−1.10064e6	−0.0654021
$778$	−4.65032e6	−0.275445
$779$	−6.56515e6	−0.387616
$780$	−220983.	−0.0130054
$781$	6.75991e6	0.396564
$782$	3.76336e7	2.20069
$783$	1.04789e6	0.0610819
$784$	1.76251e7	1.02410
$785$	1.26447e7	0.732378
$786$	4.60166e6	0.265679
$787$	−2.87892e6	−0.165689	−0.0828443	−	0.996562i	$-0.526400\pi$
$787$	−2.87892e6	−0.165689	−0.0828443	+	0.996562i	$0.526400\pi$
$788$	−680310.	−0.0390294
$789$	−4.12306e6	−0.235791
$790$	−1.95154e6	−0.111252
$791$	−3.71300e6	−0.211001
$792$	−1.73153e6	−0.0980884
$793$	−255474.	−0.0144266
$794$	−1.66181e7	−0.935469
$795$	4.08772e6	0.229384
$796$	−635855.	−0.0355693
$797$	−2.55634e7	−1.42552	−0.712759	−	0.701409i	$-0.752554\pi$
$797$	−2.55634e7	−1.42552	−0.712759	+	0.701409i	$0.752554\pi$
$798$	2.12040e6	0.117872
$799$	−7.30875e6	−0.405020
$800$	−329279.	−0.0181903
$801$	−6.57908e6	−0.362313
$802$	−1.99595e7	−1.09575
$803$	2.38849e6	0.130718
$804$	160348.	0.00874829
$805$	−1.35571e6	−0.0737356
$806$	−2.01190e7	−1.09086
$807$	6.23142e6	0.336824
$808$	−1.68202e7	−0.906365
$809$	−2.32458e6	−0.124874	−0.0624371	−	0.998049i	$-0.519887\pi$
$809$	−2.32458e6	−0.124874	−0.0624371	+	0.998049i	$0.519887\pi$
$810$	948744.	0.0508085
$811$	1.44240e7	0.770076	0.385038	−	0.922901i	$-0.374188\pi$
$811$	1.44240e7	0.770076	0.385038	+	0.922901i	$0.374188\pi$
$812$	−36947.9	−0.00196653
$813$	4.38021e6	0.232417
$814$	4.84939e6	0.256523
$815$	9.69255e6	0.511145
$816$	−2.03638e7	−1.07061
$817$	2.69212e7	1.41104
$818$	−6.81233e6	−0.355969
$819$	964137.	0.0502260
$820$	103574.	0.00537916
$821$	9.95289e6	0.515337	0.257668	−	0.966233i	$-0.417046\pi$
$821$	9.95289e6	0.515337	0.257668	+	0.966233i	$0.417046\pi$
$822$	2.21985e6	0.114589
$823$	2.46452e7	1.26833	0.634165	−	0.773198i	$-0.281344\pi$
$823$	2.46452e7	1.26833	0.634165	+	0.773198i	$0.281344\pi$
$824$	2.66460e7	1.36714
$825$	−680625.	−0.0348155
$826$	−916439.	−0.0467362
$827$	−2.09014e7	−1.06270	−0.531351	−	0.847152i	$-0.678315\pi$
$827$	−2.09014e7	−1.06270	−0.531351	+	0.847152i	$0.678315\pi$
$828$	−362438.	−0.0183721
$829$	−8.31387e6	−0.420162	−0.210081	−	0.977684i	$-0.567373\pi$
$829$	−8.31387e6	−0.420162	−0.210081	+	0.977684i	$0.567373\pi$
$830$	6.08467e6	0.306579
$831$	−2.14444e7	−1.07724
$832$	2.10034e7	1.05192
$833$	3.49312e7	1.74422
$834$	−8.82385e6	−0.439282
$835$	−1.58855e7	−0.788472
$836$	−406671.	−0.0201246
$837$	3.75994e6	0.185510
$838$	−2.56244e6	−0.126050
$839$	−4.03176e7	−1.97738	−0.988690	−	0.149977i	$-0.952080\pi$
$839$	−4.03176e7	−1.97738	−0.988690	+	0.149977i	$0.952080\pi$
$840$	701587.	0.0343071
$841$	−1.84449e7	−0.899263
$842$	−1.97757e7	−0.961282
$843$	−1.00938e7	−0.489200
$844$	−1.08834e6	−0.0525905
$845$	2.08792e6	0.100594
$846$	1.61703e6	0.0776771
$847$	258410.	0.0123766
$848$	1.94118e7	0.926992
$849$	−8.69361e6	−0.413934
$850$	−7.65539e6	−0.363430
$851$	−2.12888e7	−1.00769
$852$	−732250.	−0.0345589
$853$	−2.11646e7	−0.995951	−0.497976	−	0.867191i	$-0.665923\pi$
$853$	−2.11646e7	−0.995951	−0.497976	+	0.867191i	$0.665923\pi$
$854$	−38673.3	−0.00181454
$855$	−4.67328e6	−0.218628
$856$	−3.39633e7	−1.58425
$857$	−1.03622e7	−0.481946	−0.240973	−	0.970532i	$-0.577467\pi$
$857$	−1.03622e7	−0.481946	−0.240973	+	0.970532i	$0.577467\pi$
$858$	−4.24797e6	−0.196999
$859$	−1.77409e7	−0.820336	−0.410168	−	0.912010i	$-0.634530\pi$
$859$	−1.77409e7	−0.820336	−0.410168	+	0.912010i	$0.634530\pi$
$860$	−424716.	−0.0195818
$861$	−451886.	−0.0207741
$862$	−2.45618e7	−1.12588
$863$	−2.62577e6	−0.120014	−0.0600068	−	0.998198i	$-0.519112\pi$
$863$	−2.62577e6	−0.120014	−0.0600068	+	0.998198i	$0.519112\pi$
$864$	384071.	0.0175036
$865$	−1.69955e7	−0.772314
$866$	−5.98118e6	−0.271014
$867$	−2.75802e7	−1.24609
$868$	−132572.	−0.00597247
$869$	−1.63299e6	−0.0733557
$870$	1.87073e6	0.0837941
$871$	−8.25039e6	−0.368493
$872$	−1.70137e7	−0.757718
$873$	1.23202e7	0.547120
$874$	4.10132e7	1.81612
$875$	275778.	0.0121770
$876$	−258727.	−0.0113915
$877$	7.30171e6	0.320572	0.160286	−	0.987071i	$-0.448758\pi$
$877$	7.30171e6	0.320572	0.160286	+	0.987071i	$0.448758\pi$
$878$	2.05427e7	0.899334
$879$	−1.98737e7	−0.867576
$880$	−3.23216e6	−0.140697
$881$	6.08524e6	0.264142	0.132071	−	0.991240i	$-0.457837\pi$
$881$	6.08524e6	0.264142	0.132071	+	0.991240i	$0.457837\pi$
$882$	−7.72840e6	−0.334517
$883$	1.66311e7	0.717828	0.358914	−	0.933371i	$-0.383147\pi$
$883$	1.66311e7	0.717828	0.358914	+	0.933371i	$0.383147\pi$
$884$	−2.07981e6	−0.0895146
$885$	2.01980e6	0.0866862
$886$	1.36006e7	0.582067
$887$	−1.47741e7	−0.630511	−0.315256	−	0.949007i	$-0.602090\pi$
$887$	−1.47741e7	−0.630511	−0.315256	+	0.949007i	$0.602090\pi$
$888$	1.10171e7	0.468849
$889$	−1.12129e6	−0.0475841
$890$	−1.17452e7	−0.497032
$891$	793881.	0.0335013
$892$	1.26796e6	0.0533574
$893$	−7.96511e6	−0.334244
$894$	3.77222e6	0.157853
$895$	−1.57708e6	−0.0658108
$896$	3.47702e6	0.144689
$897$	1.86486e7	0.773863
$898$	1.74651e7	0.722736
$899$	7.41383e6	0.305945
$900$	73727.0	0.00303403
$901$	3.84722e7	1.57883
$902$	1.99101e6	0.0814810
$903$	1.85301e6	0.0756240
$904$	3.71661e7	1.51261
$905$	1.30904e7	0.531288
$906$	3.19196e6	0.129192
$907$	−9.23123e6	−0.372599	−0.186299	−	0.982493i	$-0.559649\pi$
$907$	−9.23123e6	−0.372599	−0.186299	+	0.982493i	$0.559649\pi$
$908$	119891.	0.00482583
$909$	7.71181e6	0.309561
$910$	1.72121e6	0.0689016
$911$	−1.55931e7	−0.622495	−0.311247	−	0.950329i	$-0.600747\pi$
$911$	−1.55931e7	−0.622495	−0.311247	+	0.950329i	$0.600747\pi$
$912$	−2.21925e7	−0.883526
$913$	5.09147e6	0.202146
$914$	2.20185e7	0.871812
$915$	85234.5	0.00336560
$916$	−495349.	−0.0195062
$917$	−1.56017e6	−0.0612700
$918$	8.92925e6	0.349710
$919$	−4.21091e6	−0.164470	−0.0822351	−	0.996613i	$-0.526206\pi$
$919$	−4.21091e6	−0.164470	−0.0822351	+	0.996613i	$0.526206\pi$
$920$	1.35703e7	0.528590
$921$	−7.24932e6	−0.281610
$922$	4.30040e7	1.66602
$923$	3.76765e7	1.45568
$924$	−27991.6	−0.00107857
$925$	4.33055e6	0.166413
$926$	−629009.	−0.0241062
$927$	−1.22168e7	−0.466935
$928$	757310.	0.0288671
$929$	4.10363e7	1.56002	0.780008	−	0.625770i	$-0.215215\pi$
$929$	4.10363e7	1.56002	0.780008	+	0.625770i	$0.215215\pi$
$930$	6.71235e6	0.254488
$931$	3.80682e7	1.43942
$932$	−361730.	−0.0136410
$933$	7.83689e6	0.294740
$934$	−3.96488e7	−1.48718
$935$	−6.40581e6	−0.239632
$936$	−9.65073e6	−0.360056
$937$	2.41565e7	0.898844	0.449422	−	0.893320i	$-0.351630\pi$
$937$	2.41565e7	0.898844	0.449422	+	0.893320i	$0.351630\pi$
$938$	−1.24893e6	−0.0463479
$939$	2.84967e7	1.05470
$940$	125660.	0.00463849
$941$	3.13080e7	1.15261	0.576304	−	0.817236i	$-0.304495\pi$
$941$	3.13080e7	1.15261	0.576304	+	0.817236i	$0.304495\pi$
$942$	−2.63300e7	−0.966772
$943$	−8.74049e6	−0.320079
$944$	9.59164e6	0.350318
$945$	−321667.	−0.0117173
$946$	−8.16436e6	−0.296616
$947$	5699.21	0.000206509 0	0.000103255	−	1.00000i	$-0.499967\pi$
$947$	5699.21	0.000206509 0	0.000103255		1.00000i	$0.499967\pi$
$948$	176889.	0.00639265
$949$	1.33123e7	0.479829
$950$	−8.34288e6	−0.299921
$951$	−1.55059e7	−0.555963
$952$	6.60310e6	0.236132
$953$	−4.24159e7	−1.51285	−0.756427	−	0.654078i	$-0.773057\pi$
$953$	−4.24159e7	−1.51285	−0.756427	+	0.654078i	$0.773057\pi$
$954$	−8.51183e6	−0.302797
$955$	−1.71078e7	−0.606995
$956$	−2.00942e6	−0.0711094
$957$	1.56537e6	0.0552507
$958$	−5.48718e7	−1.93168
$959$	−752628.	−0.0264261
$960$	−7.00741e6	−0.245403
$961$	−2.02765e6	−0.0708247
$962$	2.70282e7	0.941628
$963$	1.55716e7	0.541089
$964$	−2.23375e6	−0.0774180
$965$	1.31854e7	0.455801
$966$	2.82298e6	0.0973342
$967$	1.43147e7	0.492284	0.246142	−	0.969234i	$-0.420837\pi$
$967$	1.43147e7	0.492284	0.246142	+	0.969234i	$0.420837\pi$
$968$	−2.58661e6	−0.0887243
$969$	−4.39833e7	−1.50480
$970$	2.19944e7	0.750556
$971$	−1.06931e7	−0.363960	−0.181980	−	0.983302i	$-0.558251\pi$
$971$	−1.06931e7	−0.363960	−0.181980	+	0.983302i	$0.558251\pi$
$972$	−85995.1	−0.00291950
$973$	2.99168e6	0.101306
$974$	5.66714e7	1.91411
$975$	−3.79348e6	−0.127799
$976$	404762.	0.0136011
$977$	5.44429e7	1.82476	0.912378	−	0.409349i	$-0.134244\pi$
$977$	5.44429e7	1.82476	0.912378	+	0.409349i	$0.134244\pi$
$978$	−2.01827e7	−0.674734
$979$	−9.82800e6	−0.327724
$980$	−600574.	−0.0199757
$981$	7.80052e6	0.258792
$982$	5.30461e7	1.75539
$983$	4.16457e7	1.37463	0.687316	−	0.726359i	$-0.258789\pi$
$983$	4.16457e7	1.37463	0.687316	+	0.726359i	$0.258789\pi$
$984$	4.52325e6	0.148923
$985$	−1.16785e7	−0.383526
$986$	1.76067e7	0.576746
$987$	−548247.	−0.0179136
$988$	−2.26659e6	−0.0738722
$989$	3.58414e7	1.16518
$990$	1.41726e6	0.0459581
$991$	−9.62055e6	−0.311183	−0.155591	−	0.987821i	$-0.549728\pi$
$991$	−9.62055e6	−0.311183	−0.155591	+	0.987821i	$0.549728\pi$
$992$	2.71730e6	0.0876714
$993$	2.88444e7	0.928300
$994$	5.70340e6	0.183091
$995$	−1.09153e7	−0.349526
$996$	−551520.	−0.0176162
$997$	3.05865e7	0.974523	0.487262	−	0.873256i	$-0.337996\pi$
$997$	3.05865e7	0.974523	0.487262	+	0.873256i	$0.337996\pi$
$998$	−1.10495e7	−0.351169
$999$	−5.05115e6	−0.160131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.6.a.c.1.3	✓	3
3.2	odd	2		495.6.a.b.1.1		3
5.4	even	2		825.6.a.g.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.c.1.3	✓	3	1.1	even	1	trivial
495.6.a.b.1.1		3	3.2	odd	2
825.6.a.g.1.1		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q+5.78415 q^{2} -9.00000 q^{3} +1.45634 q^{4} +25.0000 q^{5} -52.0573 q^{6} +17.6498 q^{7} -176.669 q^{8} +81.0000 q^{9} +144.604 q^{10} +121.000 q^{11} -13.1070 q^{12} +674.396 q^{13} +102.089 q^{14} -225.000 q^{15} -1068.48 q^{16} -2117.62 q^{17} +468.516 q^{18} -2307.79 q^{19} +36.4084 q^{20} -158.848 q^{21} +699.882 q^{22} -3072.47 q^{23} +1590.02 q^{24} +625.000 q^{25} +3900.80 q^{26} -729.000 q^{27} +25.7040 q^{28} -1437.44 q^{29} -1301.43 q^{30} -5157.66 q^{31} -526.846 q^{32} -1089.00 q^{33} -12248.6 q^{34} +441.244 q^{35} +117.963 q^{36} +6928.88 q^{37} -13348.6 q^{38} -6069.56 q^{39} -4416.72 q^{40} +2844.78 q^{41} -918.799 q^{42} -11665.3 q^{43} +176.217 q^{44} +2025.00 q^{45} -17771.6 q^{46} +3451.40 q^{47} +9616.34 q^{48} -16495.5 q^{49} +3615.09 q^{50} +19058.6 q^{51} +982.146 q^{52} -18167.6 q^{53} -4216.64 q^{54} +3025.00 q^{55} -3118.17 q^{56} +20770.1 q^{57} -8314.36 q^{58} -8976.88 q^{59} -327.675 q^{60} -378.820 q^{61} -29832.7 q^{62} +1429.63 q^{63} +31144.1 q^{64} +16859.9 q^{65} -6298.93 q^{66} -12233.7 q^{67} -3083.97 q^{68} +27652.2 q^{69} +2552.22 q^{70} +55867.0 q^{71} -14310.2 q^{72} +19739.6 q^{73} +40077.6 q^{74} -5625.00 q^{75} -3360.92 q^{76} +2135.62 q^{77} -35107.2 q^{78} -13495.8 q^{79} -26712.0 q^{80} +6561.00 q^{81} +16454.6 q^{82} +42078.3 q^{83} -231.336 q^{84} -52940.5 q^{85} -67474.0 q^{86} +12937.0 q^{87} -21376.9 q^{88} -81223.2 q^{89} +11712.9 q^{90} +11902.9 q^{91} -4474.55 q^{92} +46419.0 q^{93} +19963.4 q^{94} -57694.8 q^{95} +4741.62 q^{96} +152101. q^{97} -95412.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 378 q^{12} + 290 q^{13} + 916 q^{14} - 675 q^{15} - 590 q^{16} + 434 q^{17} + 162 q^{18}+ \cdots + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 - 42 * q^4 + 75 * q^5 - 18 * q^6 - 68 * q^7 - 24 * q^8 + 243 * q^9 + 50 * q^10 + 363 * q^11 + 378 * q^12 + 290 * q^13 + 916 * q^14 - 675 * q^15 - 590 * q^16 + 434 * q^17 + 162 * q^18 - 2856 * q^19 - 1050 * q^20 + 612 * q^21 + 242 * q^22 - 640 * q^23 + 216 * q^24 + 1875 * q^25 + 2132 * q^26 - 2187 * q^27 - 580 * q^28 - 4538 * q^29 - 450 * q^30 - 14968 * q^31 - 2496 * q^32 - 3267 * q^33 - 13704 * q^34 - 1700 * q^35 - 3402 * q^36 - 6190 * q^37 - 11668 * q^38 - 2610 * q^39 - 600 * q^40 - 8926 * q^41 - 8244 * q^42 - 33592 * q^43 - 5082 * q^44 + 6075 * q^45 - 35680 * q^46 - 24640 * q^47 + 5310 * q^48 - 14693 * q^49 + 1250 * q^50 - 3906 * q^51 + 18780 * q^52 - 22934 * q^53 - 1458 * q^54 + 9075 * q^55 - 40012 * q^56 + 25704 * q^57 - 32304 * q^58 - 13756 * q^59 + 9450 * q^60 + 24602 * q^61 - 7704 * q^62 - 5508 * q^63 + 35474 * q^64 + 7250 * q^65 - 2178 * q^66 + 16868 * q^67 - 71288 * q^68 + 5760 * q^69 + 22900 * q^70 + 4856 * q^71 - 1944 * q^72 + 1910 * q^73 + 29404 * q^74 - 16875 * q^75 + 6116 * q^76 - 8228 * q^77 - 19188 * q^78 - 36844 * q^79 - 14750 * q^80 + 19683 * q^81 + 84000 * q^82 - 48796 * q^83 + 5220 * q^84 + 10850 * q^85 - 83492 * q^86 + 40842 * q^87 - 2904 * q^88 - 188978 * q^89 + 4050 * q^90 - 93208 * q^91 - 6976 * q^92 + 134712 * q^93 + 70472 * q^94 - 71400 * q^95 + 22464 * q^96 + 247526 * q^97 - 154654 * q^98 + 29403 * q^99

Introduction

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