LMFDB - Embedded newform 177.6.a.b.1.3

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$28.3879361069$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$8.80731$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

$f(q)$	$=$	$q-8.80731 q^{2} -9.00000 q^{3} +45.5687 q^{4} -88.4760 q^{5} +79.2658 q^{6} -134.279 q^{7} -119.504 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-8.80731 q^{2} -9.00000 q^{3} +45.5687 q^{4} -88.4760 q^{5} +79.2658 q^{6} -134.279 q^{7} -119.504 q^{8} +81.0000 q^{9} +779.236 q^{10} +423.269 q^{11} -410.118 q^{12} -713.936 q^{13} +1182.64 q^{14} +796.284 q^{15} -405.693 q^{16} +924.109 q^{17} -713.392 q^{18} -64.3204 q^{19} -4031.74 q^{20} +1208.51 q^{21} -3727.86 q^{22} +3703.37 q^{23} +1075.53 q^{24} +4703.01 q^{25} +6287.86 q^{26} -729.000 q^{27} -6118.92 q^{28} +4990.17 q^{29} -7013.12 q^{30} +3097.37 q^{31} +7397.18 q^{32} -3809.42 q^{33} -8138.92 q^{34} +11880.5 q^{35} +3691.06 q^{36} -4978.34 q^{37} +566.489 q^{38} +6425.43 q^{39} +10573.2 q^{40} +996.199 q^{41} -10643.7 q^{42} +7715.17 q^{43} +19287.8 q^{44} -7166.56 q^{45} -32616.7 q^{46} -17274.4 q^{47} +3651.24 q^{48} +1223.84 q^{49} -41420.8 q^{50} -8316.98 q^{51} -32533.1 q^{52} -34756.3 q^{53} +6420.53 q^{54} -37449.1 q^{55} +16046.8 q^{56} +578.883 q^{57} -43950.0 q^{58} -3481.00 q^{59} +36285.6 q^{60} +33316.0 q^{61} -27279.5 q^{62} -10876.6 q^{63} -52167.1 q^{64} +63166.3 q^{65} +33550.7 q^{66} +50498.8 q^{67} +42110.4 q^{68} -33330.3 q^{69} -104635. q^{70} -16903.5 q^{71} -9679.79 q^{72} -84572.2 q^{73} +43845.8 q^{74} -42327.1 q^{75} -2930.99 q^{76} -56836.1 q^{77} -56590.7 q^{78} +9984.38 q^{79} +35894.1 q^{80} +6561.00 q^{81} -8773.83 q^{82} +11880.4 q^{83} +55070.2 q^{84} -81761.5 q^{85} -67949.9 q^{86} -44911.6 q^{87} -50582.1 q^{88} -8140.62 q^{89} +63118.1 q^{90} +95866.6 q^{91} +168758. q^{92} -27876.3 q^{93} +152141. q^{94} +5690.81 q^{95} -66574.6 q^{96} -137348. q^{97} -10778.7 q^{98} +34284.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) $	\( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−8.80731	−1.55693	−0.778463	−	0.627690i	$-0.784001\pi$
$2$	−8.80731	−1.55693	−0.778463	+	0.627690i	$0.784001\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	45.5687	1.42402
$5$	−88.4760	−1.58271	−0.791354	−	0.611359i	$-0.790623\pi$
$5$	−88.4760	−1.58271	−0.791354	+	0.611359i	$0.790623\pi$
$6$	79.2658	0.898892
$7$	−134.279	−1.03577	−0.517884	−	0.855451i	$-0.673280\pi$
$7$	−134.279	−1.03577	−0.517884	+	0.855451i	$0.673280\pi$
$8$	−119.504	−0.660170
$9$	81.0000	0.333333
$10$	779.236	2.46416
$11$	423.269	1.05471	0.527357	−	0.849644i	$-0.323183\pi$
$11$	423.269	1.05471	0.527357	+	0.849644i	$0.323183\pi$
$12$	−410.118	−0.822159
$13$	−713.936	−1.17166	−0.585829	−	0.810434i	$-0.699231\pi$
$13$	−713.936	−1.17166	−0.585829	+	0.810434i	$0.699231\pi$
$14$	1182.64	1.61262
$15$	796.284	0.913776
$16$	−405.693	−0.396184
$17$	924.109	0.775534	0.387767	−	0.921757i	$-0.373246\pi$
$17$	924.109	0.775534	0.387767	+	0.921757i	$0.373246\pi$
$18$	−713.392	−0.518976
$19$	−64.3204	−0.0408756	−0.0204378	−	0.999791i	$-0.506506\pi$
$19$	−64.3204	−0.0408756	−0.0204378	+	0.999791i	$0.506506\pi$
$20$	−4031.74	−2.25381
$21$	1208.51	0.598001
$22$	−3727.86	−1.64211
$23$	3703.37	1.45975	0.729873	−	0.683583i	$-0.239579\pi$
$23$	3703.37	1.45975	0.729873	+	0.683583i	$0.239579\pi$
$24$	1075.53	0.381150
$25$	4703.01	1.50496
$26$	6287.86	1.82419
$27$	−729.000	−0.192450
$28$	−6118.92	−1.47496
$29$	4990.17	1.10185	0.550923	−	0.834556i	$-0.314276\pi$
$29$	4990.17	1.10185	0.550923	+	0.834556i	$0.314276\pi$
$30$	−7013.12	−1.42268
$31$	3097.37	0.578880	0.289440	−	0.957196i	$-0.406531\pi$
$31$	3097.37	0.578880	0.289440	+	0.957196i	$0.406531\pi$
$32$	7397.18	1.27700
$33$	−3809.42	−0.608939
$34$	−8138.92	−1.20745
$35$	11880.5	1.63932
$36$	3691.06	0.474674
$37$	−4978.34	−0.597834	−0.298917	−	0.954279i	$-0.596625\pi$
$37$	−4978.34	−0.597834	−0.298917	+	0.954279i	$0.596625\pi$
$38$	566.489	0.0636404
$39$	6425.43	0.676458
$40$	10573.2	1.04486
$41$	996.199	0.0925522	0.0462761	−	0.998929i	$-0.485265\pi$
$41$	996.199	0.0925522	0.0462761	+	0.998929i	$0.485265\pi$
$42$	−10643.7	−0.931044
$43$	7715.17	0.636318	0.318159	−	0.948037i	$-0.396935\pi$
$43$	7715.17	0.636318	0.318159	+	0.948037i	$0.396935\pi$
$44$	19287.8	1.50193
$45$	−7166.56	−0.527569
$46$	−32616.7	−2.27272
$47$	−17274.4	−1.14067	−0.570333	−	0.821414i	$-0.693186\pi$
$47$	−17274.4	−1.14067	−0.570333	+	0.821414i	$0.693186\pi$
$48$	3651.24	0.228737
$49$	1223.84	0.0728171
$50$	−41420.8	−2.34312
$51$	−8316.98	−0.447755
$52$	−32533.1	−1.66847
$53$	−34756.3	−1.69959	−0.849796	−	0.527112i	$-0.823275\pi$
$53$	−34756.3	−1.69959	−0.849796	+	0.527112i	$0.823275\pi$
$54$	6420.53	0.299631
$55$	−37449.1	−1.66930
$56$	16046.8	0.683784
$57$	578.883	0.0235996
$58$	−43950.0	−1.71549
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	36285.6	1.30124
$61$	33316.0	1.14638	0.573190	−	0.819422i	$-0.305706\pi$
$61$	33316.0	1.14638	0.573190	+	0.819422i	$0.305706\pi$
$62$	−27279.5	−0.901274
$63$	−10876.6	−0.345256
$64$	−52167.1	−1.59201
$65$	63166.3	1.85439
$66$	33550.7	0.948073
$67$	50498.8	1.37434	0.687169	−	0.726497i	$-0.258853\pi$
$67$	50498.8	1.37434	0.687169	+	0.726497i	$0.258853\pi$
$68$	42110.4	1.10438
$69$	−33330.3	−0.842784
$70$	−104635.	−2.55230
$71$	−16903.5	−0.397951	−0.198975	−	0.980004i	$-0.563761\pi$
$71$	−16903.5	−0.397951	−0.198975	+	0.980004i	$0.563761\pi$
$72$	−9679.79	−0.220057
$73$	−84572.2	−1.85746	−0.928732	−	0.370752i	$-0.879100\pi$
$73$	−84572.2	−1.85746	−0.928732	+	0.370752i	$0.879100\pi$
$74$	43845.8	0.930783
$75$	−42327.1	−0.868890
$76$	−2930.99	−0.0582078
$77$	−56836.1	−1.09244
$78$	−56590.7	−1.05320
$79$	9984.38	0.179992	0.0899960	−	0.995942i	$-0.471315\pi$
$79$	9984.38	0.179992	0.0899960	+	0.995942i	$0.471315\pi$
$80$	35894.1	0.627044
$81$	6561.00	0.111111
$82$	−8773.83	−0.144097
$83$	11880.4	0.189293	0.0946467	−	0.995511i	$-0.469828\pi$
$83$	11880.4	0.189293	0.0946467	+	0.995511i	$0.469828\pi$
$84$	55070.2	0.851567
$85$	−81761.5	−1.22744
$86$	−67949.9	−0.990701
$87$	−44911.6	−0.636151
$88$	−50582.1	−0.696290
$89$	−8140.62	−0.108939	−0.0544694	−	0.998515i	$-0.517347\pi$
$89$	−8140.62	−0.108939	−0.0544694	+	0.998515i	$0.517347\pi$
$90$	63118.1	0.821387
$91$	95866.6	1.21357
$92$	168758.	2.07871
$93$	−27876.3	−0.334216
$94$	152141.	1.77593
$95$	5690.81	0.0646942
$96$	−66574.6	−0.737277
$97$	−137348.	−1.48215	−0.741077	−	0.671420i	$-0.765685\pi$
$97$	−137348.	−1.48215	−0.741077	+	0.671420i	$0.765685\pi$
$98$	−10778.7	−0.113371
$99$	34284.8	0.351571
$100$	214310.	2.14310
$101$	−32965.3	−0.321553	−0.160777	−	0.986991i	$-0.551400\pi$
$101$	−32965.3	−0.321553	−0.160777	+	0.986991i	$0.551400\pi$
$102$	73250.2	0.697122
$103$	168560.	1.56553	0.782764	−	0.622319i	$-0.213809\pi$
$103$	168560.	1.56553	0.782764	+	0.622319i	$0.213809\pi$
$104$	85318.0	0.773494
$105$	−106924.	−0.946461
$106$	306110.	2.64614
$107$	−36729.3	−0.310137	−0.155069	−	0.987904i	$-0.549560\pi$
$107$	−36729.3	−0.310137	−0.155069	+	0.987904i	$0.549560\pi$
$108$	−33219.6	−0.274053
$109$	171480.	1.38244	0.691222	−	0.722642i	$-0.257073\pi$
$109$	171480.	1.38244	0.691222	+	0.722642i	$0.257073\pi$
$110$	329826.	2.59898
$111$	44805.1	0.345159
$112$	54476.0	0.410356
$113$	10840.2	0.0798624	0.0399312	−	0.999202i	$-0.487286\pi$
$113$	10840.2	0.0798624	0.0399312	+	0.999202i	$0.487286\pi$
$114$	−5098.40	−0.0367428
$115$	−327659.	−2.31035
$116$	227396.	1.56905
$117$	−57828.9	−0.390553
$118$	30658.2	0.202695
$119$	−124088.	−0.803274
$120$	−95158.8	−0.603248
$121$	18105.3	0.112420
$122$	−293425.	−1.78483
$123$	−8965.79	−0.0534350
$124$	141143.	0.824337
$125$	−139616.	−0.799208
$126$	95793.5	0.537539
$127$	−266071.	−1.46382	−0.731911	−	0.681400i	$-0.761371\pi$
$127$	−266071.	−1.46382	−0.731911	+	0.681400i	$0.761371\pi$
$128$	222742.	1.20165
$129$	−69436.5	−0.367379
$130$	−556325.	−2.88715
$131$	344872.	1.75582	0.877910	−	0.478826i	$-0.158938\pi$
$131$	344872.	1.75582	0.877910	+	0.478826i	$0.158938\pi$
$132$	−173590.	−0.867142
$133$	8636.87	0.0423377
$134$	−444758.	−2.13975
$135$	64499.0	0.304592
$136$	−110434.	−0.511985
$137$	299475.	1.36320	0.681600	−	0.731725i	$-0.261284\pi$
$137$	299475.	1.36320	0.681600	+	0.731725i	$0.261284\pi$
$138$	293550.	1.31215
$139$	−279442.	−1.22675	−0.613373	−	0.789793i	$-0.710188\pi$
$139$	−279442.	−1.22675	−0.613373	+	0.789793i	$0.710188\pi$
$140$	541377.	2.33443
$141$	155470.	0.658563
$142$	148874.	0.619581
$143$	−302187.	−1.23576
$144$	−32861.1	−0.132061
$145$	−441511.	−1.74390
$146$	744853.	2.89193
$147$	−11014.5	−0.0420410
$148$	−226856.	−0.851328
$149$	522957.	1.92974	0.964872	−	0.262719i	$-0.0846191\pi$
$149$	522957.	1.92974	0.964872	+	0.262719i	$0.0846191\pi$
$150$	372788.	1.35280
$151$	−225544.	−0.804988	−0.402494	−	0.915423i	$-0.631857\pi$
$151$	−225544.	−0.804988	−0.402494	+	0.915423i	$0.631857\pi$
$152$	7686.52	0.0269849
$153$	74852.9	0.258511
$154$	500573.	1.70085
$155$	−274043.	−0.916197
$156$	292798.	0.963290
$157$	169574.	0.549047	0.274524	−	0.961580i	$-0.411480\pi$
$157$	169574.	0.549047	0.274524	+	0.961580i	$0.411480\pi$
$158$	−87935.5	−0.280234
$159$	312807.	0.981259
$160$	−654473.	−2.02112
$161$	−497284.	−1.51196
$162$	−57784.8	−0.172992
$163$	247010.	0.728192	0.364096	−	0.931361i	$-0.381378\pi$
$163$	247010.	0.728192	0.364096	+	0.931361i	$0.381378\pi$
$164$	45395.5	0.131796
$165$	337042.	0.963772
$166$	−104634.	−0.294716
$167$	−37137.6	−0.103044	−0.0515220	−	0.998672i	$-0.516407\pi$
$167$	−37137.6	−0.103044	−0.0515220	+	0.998672i	$0.516407\pi$
$168$	−144421.	−0.394783
$169$	138412.	0.372785
$170$	720099.	1.91104
$171$	−5209.95	−0.0136252
$172$	351570.	0.906131
$173$	−49966.1	−0.126929	−0.0634644	−	0.997984i	$-0.520215\pi$
$173$	−49966.1	−0.126929	−0.0634644	+	0.997984i	$0.520215\pi$
$174$	395550.	0.990440
$175$	−631515.	−1.55879
$176$	−171717.	−0.417861
$177$	31329.0	0.0751646
$178$	71697.0	0.169610
$179$	−53182.0	−0.124060	−0.0620300	−	0.998074i	$-0.519757\pi$
$179$	−53182.0	−0.124060	−0.0620300	+	0.998074i	$0.519757\pi$
$180$	−326571.	−0.751270
$181$	−228956.	−0.519463	−0.259732	−	0.965681i	$-0.583634\pi$
$181$	−228956.	−0.519463	−0.259732	+	0.965681i	$0.583634\pi$
$182$	−844327.	−1.88944
$183$	−299844.	−0.661863
$184$	−442566.	−0.963681
$185$	440464.	0.946195
$186$	245515.	0.520351
$187$	391146.	0.817966
$188$	−787171.	−1.62433
$189$	97889.4	0.199334
$190$	−50120.7	−0.100724
$191$	383018.	0.759689	0.379845	−	0.925050i	$-0.375977\pi$
$191$	383018.	0.759689	0.379845	+	0.925050i	$0.375977\pi$
$192$	469503.	0.919149
$193$	−671956.	−1.29852	−0.649258	−	0.760568i	$-0.724921\pi$
$193$	−671956.	−1.29852	−0.649258	+	0.760568i	$0.724921\pi$
$194$	1.20967e6	2.30761
$195$	−568496.	−1.07063
$196$	55768.6	0.103693
$197$	−517980.	−0.950927	−0.475464	−	0.879735i	$-0.657720\pi$
$197$	−517980.	−0.950927	−0.475464	+	0.879735i	$0.657720\pi$
$198$	−301956.	−0.547370
$199$	437764.	0.783624	0.391812	−	0.920045i	$-0.371848\pi$
$199$	437764.	0.783624	0.391812	+	0.920045i	$0.371848\pi$
$200$	−562026.	−0.993531
$201$	−454489.	−0.793475
$202$	290335.	0.500635
$203$	−670075.	−1.14126
$204$	−378994.	−0.637613
$205$	−88139.8	−0.146483
$206$	−1.48456e6	−2.43741
$207$	299973.	0.486582
$208$	289639.	0.464193
$209$	−27224.8	−0.0431121
$210$	941715.	1.47357
$211$	795724.	1.23043	0.615214	−	0.788360i	$-0.289070\pi$
$211$	795724.	1.23043	0.615214	+	0.788360i	$0.289070\pi$
$212$	−1.58380e6	−2.42025
$213$	152131.	0.229757
$214$	323487.	0.482861
$215$	−682608.	−1.00711
$216$	87118.1	0.127050
$217$	−415911.	−0.599586
$218$	−1.51028e6	−2.15237
$219$	761149.	1.07241
$220$	−1.70651e6	−2.37712
$221$	−659755.	−0.908662
$222$	−394612.	−0.537388
$223$	−1.44232e6	−1.94222	−0.971110	−	0.238632i	$-0.923301\pi$
$223$	−1.44232e6	−1.94222	−0.971110	+	0.238632i	$0.923301\pi$
$224$	−993285.	−1.32268
$225$	380944.	0.501654
$226$	−95473.3	−0.124340
$227$	−421642.	−0.543099	−0.271549	−	0.962425i	$-0.587536\pi$
$227$	−421642.	−0.543099	−0.271549	+	0.962425i	$0.587536\pi$
$228$	26379.0	0.0336063
$229$	−1.41705e6	−1.78565	−0.892823	−	0.450407i	$-0.851279\pi$
$229$	−1.41705e6	−1.78565	−0.892823	+	0.450407i	$0.851279\pi$
$230$	2.88580e6	3.59705
$231$	511525.	0.630720
$232$	−596344.	−0.727406
$233$	673154.	0.812315	0.406158	−	0.913803i	$-0.366868\pi$
$233$	673154.	0.812315	0.406158	+	0.913803i	$0.366868\pi$
$234$	509317.	0.608062
$235$	1.52837e6	1.80534
$236$	−158625.	−0.185392
$237$	−89859.4	−0.103918
$238$	1.09289e6	1.25064
$239$	−760337.	−0.861016	−0.430508	−	0.902587i	$-0.641666\pi$
$239$	−760337.	−0.861016	−0.430508	+	0.902587i	$0.641666\pi$
$240$	−323047.	−0.362024
$241$	470539.	0.521859	0.260929	−	0.965358i	$-0.415971\pi$
$241$	470539.	0.521859	0.260929	+	0.965358i	$0.415971\pi$
$242$	−159459.	−0.175029
$243$	−59049.0	−0.0641500
$244$	1.51817e6	1.63247
$245$	−108280.	−0.115248
$246$	78964.5	0.0831944
$247$	45920.7	0.0478923
$248$	−370147.	−0.382159
$249$	−106923.	−0.109289
$250$	1.22964e6	1.24431
$251$	962815.	0.964625	0.482313	−	0.875999i	$-0.339797\pi$
$251$	962815.	0.964625	0.482313	+	0.875999i	$0.339797\pi$
$252$	−495632.	−0.491652
$253$	1.56752e6	1.53961
$254$	2.34337e6	2.27906
$255$	735854.	0.708665
$256$	−292409.	−0.278863
$257$	−931917.	−0.880125	−0.440062	−	0.897967i	$-0.645044\pi$
$257$	−931917.	−0.880125	−0.440062	+	0.897967i	$0.645044\pi$
$258$	611549.	0.571982
$259$	668486.	0.619217
$260$	2.87840e6	2.64070
$261$	404204.	0.367282
$262$	−3.03740e6	−2.73368
$263$	−1.26124e6	−1.12437	−0.562184	−	0.827012i	$-0.690039\pi$
$263$	−1.26124e6	−1.12437	−0.562184	+	0.827012i	$0.690039\pi$
$264$	455239.	0.402003
$265$	3.07510e6	2.68996
$266$	−76067.6	−0.0659167
$267$	73265.6	0.0628958
$268$	2.30116e6	1.95709
$269$	−688569.	−0.580185	−0.290093	−	0.956999i	$-0.593686\pi$
$269$	−688569.	−0.580185	−0.290093	+	0.956999i	$0.593686\pi$
$270$	−568063.	−0.474228
$271$	−733019.	−0.606306	−0.303153	−	0.952942i	$-0.598039\pi$
$271$	−733019.	−0.606306	−0.303153	+	0.952942i	$0.598039\pi$
$272$	−374905.	−0.307255
$273$	−862800.	−0.700654
$274$	−2.63757e6	−2.12240
$275$	1.99064e6	1.58730
$276$	−1.51882e6	−1.20014
$277$	−176065.	−0.137871	−0.0689356	−	0.997621i	$-0.521960\pi$
$277$	−176065.	−0.137871	−0.0689356	+	0.997621i	$0.521960\pi$
$278$	2.46113e6	1.90995
$279$	250887.	0.192960
$280$	−1.41976e6	−1.08223
$281$	−1.04068e6	−0.786232	−0.393116	−	0.919489i	$-0.628603\pi$
$281$	−1.04068e6	−0.786232	−0.393116	+	0.919489i	$0.628603\pi$
$282$	−1.36927e6	−1.02534
$283$	715219.	0.530851	0.265426	−	0.964131i	$-0.414488\pi$
$283$	715219.	0.530851	0.265426	+	0.964131i	$0.414488\pi$
$284$	−770268.	−0.566691
$285$	−51217.3	−0.0373512
$286$	2.66145e6	1.92399
$287$	−133769.	−0.0958626
$288$	599171.	0.425667
$289$	−565879.	−0.398546
$290$	3.88852e6	2.71512
$291$	1.23613e6	0.855722
$292$	−3.85384e6	−2.64507
$293$	2.03867e6	1.38732	0.693662	−	0.720301i	$-0.255996\pi$
$293$	2.03867e6	1.38732	0.693662	+	0.720301i	$0.255996\pi$
$294$	97008.4	0.0654547
$295$	307985.	0.206051
$296$	594930.	0.394672
$297$	−308563.	−0.202980
$298$	−4.60584e6	−3.00447
$299$	−2.64397e6	−1.71032
$300$	−1.92879e6	−1.23732
$301$	−1.03599e6	−0.659079
$302$	1.98644e6	1.25331
$303$	296687.	0.185649
$304$	26094.3	0.0161943
$305$	−2.94767e6	−1.81439
$306$	−659252.	−0.402483
$307$	−1.35132e6	−0.818301	−0.409151	−	0.912467i	$-0.634175\pi$
$307$	−1.35132e6	−0.818301	−0.409151	+	0.912467i	$0.634175\pi$
$308$	−2.58994e6	−1.55566
$309$	−1.51704e6	−0.903858
$310$	2.41358e6	1.42645
$311$	−1.87621e6	−1.09997	−0.549985	−	0.835175i	$-0.685367\pi$
$311$	−1.87621e6	−1.09997	−0.549985	+	0.835175i	$0.685367\pi$
$312$	−767862.	−0.446577
$313$	2.45653e6	1.41730	0.708649	−	0.705561i	$-0.249305\pi$
$313$	2.45653e6	1.41730	0.708649	+	0.705561i	$0.249305\pi$
$314$	−1.49349e6	−0.854826
$315$	962318.	0.546440
$316$	454975.	0.256312
$317$	−1.75984e6	−0.983615	−0.491808	−	0.870704i	$-0.663664\pi$
$317$	−1.75984e6	−0.983615	−0.491808	+	0.870704i	$0.663664\pi$
$318$	−2.75499e6	−1.52775
$319$	2.11218e6	1.16213
$320$	4.61553e6	2.51969
$321$	330564.	0.179058
$322$	4.37974e6	2.35401
$323$	−59439.1	−0.0317005
$324$	298976.	0.158225
$325$	−3.35765e6	−1.76330
$326$	−2.17549e6	−1.13374
$327$	−1.54332e6	−0.798155
$328$	−119049.	−0.0611002
$329$	2.31959e6	1.18147
$330$	−2.96843e6	−1.50052
$331$	−2.62035e6	−1.31459	−0.657293	−	0.753635i	$-0.728299\pi$
$331$	−2.62035e6	−1.31459	−0.657293	+	0.753635i	$0.728299\pi$
$332$	541374.	0.269558
$333$	−403246.	−0.199278
$334$	327082.	0.160432
$335$	−4.46793e6	−2.17518
$336$	−490284.	−0.236919
$337$	2.56955e6	1.23249	0.616244	−	0.787555i	$-0.288653\pi$
$337$	2.56955e6	1.23249	0.616244	+	0.787555i	$0.288653\pi$
$338$	−1.21904e6	−0.580398
$339$	−97562.1	−0.0461086
$340$	−3.72577e6	−1.74791
$341$	1.31102e6	0.610552
$342$	45885.6	0.0212135
$343$	2.09249e6	0.960347
$344$	−921991.	−0.420079
$345$	2.94893e6	1.33388
$346$	440067.	0.197619
$347$	1.69640e6	0.756318	0.378159	−	0.925741i	$-0.376557\pi$
$347$	1.69640e6	0.756318	0.378159	+	0.925741i	$0.376557\pi$
$348$	−2.04656e6	−0.905892
$349$	−2.36923e6	−1.04122	−0.520612	−	0.853794i	$-0.674296\pi$
$349$	−2.36923e6	−1.04122	−0.520612	+	0.853794i	$0.674296\pi$
$350$	5.56195e6	2.42693
$351$	520460.	0.225486
$352$	3.13099e6	1.34687
$353$	4.33080e6	1.84983	0.924914	−	0.380175i	$-0.124136\pi$
$353$	4.33080e6	1.84983	0.924914	+	0.380175i	$0.124136\pi$
$354$	−275924.	−0.117026
$355$	1.49555e6	0.629840
$356$	−370957.	−0.155131
$357$	1.11680e6	0.463771
$358$	468390.	0.193152
$359$	−1.01517e6	−0.415723	−0.207862	−	0.978158i	$-0.566650\pi$
$359$	−1.01517e6	−0.415723	−0.207862	+	0.978158i	$0.566650\pi$
$360$	856429.	0.348285
$361$	−2.47196e6	−0.998329
$362$	2.01648e6	0.808766
$363$	−162948.	−0.0649055
$364$	4.36852e6	1.72815
$365$	7.48261e6	2.93982
$366$	2.64082e6	1.03047
$367$	−2.21426e6	−0.858151	−0.429076	−	0.903269i	$-0.641161\pi$
$367$	−2.21426e6	−0.858151	−0.429076	+	0.903269i	$0.641161\pi$
$368$	−1.50243e6	−0.578328
$369$	80692.1	0.0308507
$370$	−3.87930e6	−1.47316
$371$	4.66705e6	1.76038
$372$	−1.27029e6	−0.475931
$373$	−2.01840e6	−0.751166	−0.375583	−	0.926789i	$-0.622557\pi$
$373$	−2.01840e6	−0.751166	−0.375583	+	0.926789i	$0.622557\pi$
$374$	−3.44495e6	−1.27351
$375$	1.25654e6	0.461423
$376$	2.06435e6	0.753033
$377$	−3.56267e6	−1.29099
$378$	−862142.	−0.310348
$379$	4.76169e6	1.70280	0.851398	−	0.524520i	$-0.175755\pi$
$379$	4.76169e6	1.70280	0.851398	+	0.524520i	$0.175755\pi$
$380$	259323.	0.0921259
$381$	2.39464e6	0.845138
$382$	−3.37336e6	−1.18278
$383$	1.00203e6	0.349046	0.174523	−	0.984653i	$-0.444162\pi$
$383$	1.00203e6	0.349046	0.174523	+	0.984653i	$0.444162\pi$
$384$	−2.00467e6	−0.693771
$385$	5.02863e6	1.72901
$386$	5.91812e6	2.02170
$387$	624929.	0.212106
$388$	−6.25877e6	−2.11062
$389$	−5.08858e6	−1.70499	−0.852496	−	0.522734i	$-0.824912\pi$
$389$	−5.08858e6	−1.70499	−0.852496	+	0.522734i	$0.824912\pi$
$390$	5.00692e6	1.66690
$391$	3.42232e6	1.13208
$392$	−146253.	−0.0480717
$393$	−3.10385e6	−1.01372
$394$	4.56201e6	1.48052
$395$	−883378.	−0.284875
$396$	1.56231e6	0.500645
$397$	−4.16172e6	−1.32525	−0.662624	−	0.748953i	$-0.730557\pi$
$397$	−4.16172e6	−1.32525	−0.662624	+	0.748953i	$0.730557\pi$
$398$	−3.85552e6	−1.22004
$399$	−77731.9	−0.0244437
$400$	−1.90798e6	−0.596243
$401$	6.12721e6	1.90284	0.951420	−	0.307897i	$-0.0996252\pi$
$401$	6.12721e6	1.90284	0.951420	+	0.307897i	$0.0996252\pi$
$402$	4.00282e6	1.23538
$403$	−2.21132e6	−0.678250
$404$	−1.50218e6	−0.457899
$405$	−580491.	−0.175856
$406$	5.90156e6	1.77685
$407$	−2.10718e6	−0.630543
$408$	993910.	0.295595
$409$	−1.39272e6	−0.411675	−0.205837	−	0.978586i	$-0.565992\pi$
$409$	−1.39272e6	−0.411675	−0.205837	+	0.978586i	$0.565992\pi$
$410$	776274.	0.228063
$411$	−2.69528e6	−0.787044
$412$	7.68104e6	2.22934
$413$	467425.	0.134846
$414$	−2.64195e6	−0.757572
$415$	−1.05113e6	−0.299596
$416$	−5.28111e6	−1.49621
$417$	2.51498e6	0.708262
$418$	239777.	0.0671223
$419$	−4.89970e6	−1.36343	−0.681717	−	0.731616i	$-0.738767\pi$
$419$	−4.89970e6	−1.36343	−0.681717	+	0.731616i	$0.738767\pi$
$420$	−4.87240e6	−1.34778
$421$	−1.47517e6	−0.405637	−0.202818	−	0.979216i	$-0.565010\pi$
$421$	−1.47517e6	−0.405637	−0.202818	+	0.979216i	$0.565010\pi$
$422$	−7.00819e6	−1.91569
$423$	−1.39923e6	−0.380222
$424$	4.15351e6	1.12202
$425$	4.34609e6	1.16715
$426$	−1.33987e6	−0.357715
$427$	−4.47364e6	−1.18739
$428$	−1.67371e6	−0.441642
$429$	2.71968e6	0.713469
$430$	6.01194e6	1.56799
$431$	−6.48538e6	−1.68168	−0.840838	−	0.541286i	$-0.817937\pi$
$431$	−6.48538e6	−1.68168	−0.840838	+	0.541286i	$0.817937\pi$
$432$	295750.	0.0762457
$433$	5.86229e6	1.50262	0.751308	−	0.659952i	$-0.229423\pi$
$433$	5.86229e6	1.50262	0.751308	+	0.659952i	$0.229423\pi$
$434$	3.66306e6	0.933511
$435$	3.97360e6	1.00684
$436$	7.81413e6	1.96863
$437$	−238202.	−0.0596680
$438$	−6.70368e6	−1.66966
$439$	−4.42815e6	−1.09663	−0.548316	−	0.836271i	$-0.684731\pi$
$439$	−4.42815e6	−1.09663	−0.548316	+	0.836271i	$0.684731\pi$
$440$	4.47530e6	1.10202
$441$	99130.8	0.0242724
$442$	5.81067e6	1.41472
$443$	−7.50913e6	−1.81794	−0.908971	−	0.416859i	$-0.863131\pi$
$443$	−7.50913e6	−1.81794	−0.908971	+	0.416859i	$0.863131\pi$
$444$	2.04171e6	0.491514
$445$	720250.	0.172418
$446$	1.27029e7	3.02389
$447$	−4.70661e6	−1.11414
$448$	7.00494e6	1.64896
$449$	1.95240e6	0.457040	0.228520	−	0.973539i	$-0.426611\pi$
$449$	1.95240e6	0.457040	0.228520	+	0.973539i	$0.426611\pi$
$450$	−3.35509e6	−0.781039
$451$	421660.	0.0976160
$452$	493975.	0.113726
$453$	2.02990e6	0.464760
$454$	3.71353e6	0.845565
$455$	−8.48190e6	−1.92072
$456$	−69178.6	−0.0155797
$457$	−5.94530e6	−1.33163	−0.665814	−	0.746118i	$-0.731916\pi$
$457$	−5.94530e6	−1.33163	−0.665814	+	0.746118i	$0.731916\pi$
$458$	1.24804e7	2.78012
$459$	−673676.	−0.149252
$460$	−1.49310e7	−3.28999
$461$	6.52618e6	1.43023	0.715116	−	0.699006i	$-0.246374\pi$
$461$	6.52618e6	1.43023	0.715116	+	0.699006i	$0.246374\pi$
$462$	−4.50515e6	−0.981985
$463$	−1.21024e6	−0.262373	−0.131187	−	0.991358i	$-0.541879\pi$
$463$	−1.21024e6	−0.262373	−0.131187	+	0.991358i	$0.541879\pi$
$464$	−2.02448e6	−0.436534
$465$	2.46638e6	0.528967
$466$	−5.92868e6	−1.26472
$467$	−3.93565e6	−0.835073	−0.417536	−	0.908660i	$-0.637106\pi$
$467$	−3.93565e6	−0.835073	−0.417536	+	0.908660i	$0.637106\pi$
$468$	−2.63518e6	−0.556156
$469$	−6.78092e6	−1.42350
$470$	−1.34608e7	−2.81078
$471$	−1.52616e6	−0.316993
$472$	415992.	0.0859469
$473$	3.26559e6	0.671133
$474$	791419.	0.161793
$475$	−302499.	−0.0615163
$476$	−5.65455e6	−1.14388
$477$	−2.81526e6	−0.566530
$478$	6.69652e6	1.34054
$479$	−242633.	−0.0483183	−0.0241592	−	0.999708i	$-0.507691\pi$
$479$	−242633.	−0.0483183	−0.0241592	+	0.999708i	$0.507691\pi$
$480$	5.89026e6	1.16689
$481$	3.55422e6	0.700457
$482$	−4.14418e6	−0.812496
$483$	4.47556e6	0.872930
$484$	825035.	0.160088
$485$	1.21520e7	2.34582
$486$	520063.	0.0998769
$487$	−2.57797e6	−0.492556	−0.246278	−	0.969199i	$-0.579208\pi$
$487$	−2.57797e6	−0.492556	−0.246278	+	0.969199i	$0.579208\pi$
$488$	−3.98139e6	−0.756807
$489$	−2.22309e6	−0.420422
$490$	953657.	0.179433
$491$	−287405.	−0.0538010	−0.0269005	−	0.999638i	$-0.508564\pi$
$491$	−287405.	−0.0538010	−0.0269005	+	0.999638i	$0.508564\pi$
$492$	−408559.	−0.0760926
$493$	4.61147e6	0.854519
$494$	−404437.	−0.0745648
$495$	−3.03338e6	−0.556434
$496$	−1.25658e6	−0.229343
$497$	2.26978e6	0.412185
$498$	941708.	0.170154
$499$	4.52048e6	0.812706	0.406353	−	0.913716i	$-0.366800\pi$
$499$	4.52048e6	0.812706	0.406353	+	0.913716i	$0.366800\pi$
$500$	−6.36211e6	−1.13809
$501$	334238.	0.0594925
$502$	−8.47981e6	−1.50185
$503$	−2.75026e6	−0.484679	−0.242339	−	0.970192i	$-0.577915\pi$
$503$	−2.75026e6	−0.484679	−0.242339	+	0.970192i	$0.577915\pi$
$504$	1.29979e6	0.227928
$505$	2.91664e6	0.508925
$506$	−1.38056e7	−2.39706
$507$	−1.24571e6	−0.215227
$508$	−1.21245e7	−2.08451
$509$	2.32843e6	0.398354	0.199177	−	0.979964i	$-0.436173\pi$
$509$	2.32843e6	0.398354	0.199177	+	0.979964i	$0.436173\pi$
$510$	−6.48089e6	−1.10334
$511$	1.13563e7	1.92390
$512$	−4.55240e6	−0.767477
$513$	46889.6	0.00786652
$514$	8.20768e6	1.37029
$515$	−1.49135e7	−2.47777
$516$	−3.16413e6	−0.523155
$517$	−7.31171e6	−1.20307
$518$	−5.88757e6	−0.964076
$519$	449695.	0.0732823
$520$	−7.54860e6	−1.22422
$521$	1.70375e6	0.274986	0.137493	−	0.990503i	$-0.456096\pi$
$521$	1.70375e6	0.274986	0.137493	+	0.990503i	$0.456096\pi$
$522$	−3.55995e6	−0.571831
$523$	−4.39690e6	−0.702898	−0.351449	−	0.936207i	$-0.614311\pi$
$523$	−4.39690e6	−0.702898	−0.351449	+	0.936207i	$0.614311\pi$
$524$	1.57154e7	2.50032
$525$	5.68363e6	0.899970
$526$	1.11081e7	1.75056
$527$	2.86231e6	0.448941
$528$	1.54545e6	0.241252
$529$	7.27858e6	1.13086
$530$	−2.70834e7	−4.18806
$531$	−281961.	−0.0433963
$532$	393571.	0.0602898
$533$	−711223.	−0.108440
$534$	−645273.	−0.0979242
$535$	3.24967e6	0.490856
$536$	−6.03478e6	−0.907298
$537$	478638.	0.0716261
$538$	6.06444e6	0.903306
$539$	518012.	0.0768011
$540$	2.93914e6	0.433746
$541$	−2.96004e6	−0.434815	−0.217407	−	0.976081i	$-0.569760\pi$
$541$	−2.96004e6	−0.434815	−0.217407	+	0.976081i	$0.569760\pi$
$542$	6.45593e6	0.943975
$543$	2.06060e6	0.299912
$544$	6.83580e6	0.990358
$545$	−1.51719e7	−2.18801
$546$	7.59894e6	1.09087
$547$	−3.64363e6	−0.520673	−0.260337	−	0.965518i	$-0.583834\pi$
$547$	−3.64363e6	−0.520673	−0.260337	+	0.965518i	$0.583834\pi$
$548$	1.36467e7	1.94123
$549$	2.69860e6	0.382127
$550$	−1.75321e7	−2.47132
$551$	−320970.	−0.0450386
$552$	3.98309e6	0.556381
$553$	−1.34069e6	−0.186430
$554$	1.55066e6	0.214655
$555$	−3.96417e6	−0.546286
$556$	−1.27338e7	−1.74691
$557$	−5.39423e6	−0.736701	−0.368350	−	0.929687i	$-0.620077\pi$
$557$	−5.39423e6	−0.736701	−0.368350	+	0.929687i	$0.620077\pi$
$558$	−2.20964e6	−0.300425
$559$	−5.50814e6	−0.745548
$560$	−4.81982e6	−0.649473
$561$	−3.52032e6	−0.472253
$562$	9.16558e6	1.22411
$563$	−8.18538e6	−1.08835	−0.544174	−	0.838973i	$-0.683157\pi$
$563$	−8.18538e6	−1.08835	−0.544174	+	0.838973i	$0.683157\pi$
$564$	7.08454e6	0.937808
$565$	−959101.	−0.126399
$566$	−6.29915e6	−0.826496
$567$	−881004.	−0.115085
$568$	2.02002e6	0.262715
$569$	1.23194e7	1.59518	0.797590	−	0.603201i	$-0.206108\pi$
$569$	1.23194e7	1.59518	0.797590	+	0.603201i	$0.206108\pi$
$570$	451087.	0.0581531
$571$	1.09533e7	1.40590	0.702948	−	0.711241i	$-0.251866\pi$
$571$	1.09533e7	1.40590	0.702948	+	0.711241i	$0.251866\pi$
$572$	−1.37703e7	−1.75975
$573$	−3.44716e6	−0.438607
$574$	1.17814e6	0.149251
$575$	1.74170e7	2.19686
$576$	−4.22553e6	−0.530671
$577$	−803138.	−0.100427	−0.0502135	−	0.998739i	$-0.515990\pi$
$577$	−803138.	−0.100427	−0.0502135	+	0.998739i	$0.515990\pi$
$578$	4.98387e6	0.620508
$579$	6.04760e6	0.749699
$580$	−2.01191e7	−2.48335
$581$	−1.59529e6	−0.196064
$582$	−1.08870e7	−1.33230
$583$	−1.47113e7	−1.79258
$584$	1.01067e7	1.22624
$585$	5.11647e6	0.618131
$586$	−1.79552e7	−2.15996
$587$	6.51982e6	0.780980	0.390490	−	0.920607i	$-0.372306\pi$
$587$	6.51982e6	0.780980	0.390490	+	0.920607i	$0.372306\pi$
$588$	−501918.	−0.0598672
$589$	−199224.	−0.0236621
$590$	−2.71252e6	−0.320806
$591$	4.66182e6	0.549018
$592$	2.01968e6	0.236852
$593$	−2.02927e6	−0.236975	−0.118488	−	0.992956i	$-0.537805\pi$
$593$	−2.02927e6	−0.236975	−0.118488	+	0.992956i	$0.537805\pi$
$594$	2.71761e6	0.316024
$595$	1.09789e7	1.27135
$596$	2.38304e7	2.74800
$597$	−3.93988e6	−0.452425
$598$	2.32862e7	2.66285
$599$	667254.	0.0759843	0.0379922	−	0.999278i	$-0.487904\pi$
$599$	667254.	0.0759843	0.0379922	+	0.999278i	$0.487904\pi$
$600$	5.05824e6	0.573616
$601$	−7.42028e6	−0.837981	−0.418991	−	0.907991i	$-0.637616\pi$
$601$	−7.42028e6	−0.837981	−0.418991	+	0.907991i	$0.637616\pi$
$602$	9.12424e6	1.02614
$603$	4.09040e6	0.458113
$604$	−1.02778e7	−1.14632
$605$	−1.60189e6	−0.177927
$606$	−2.61302e6	−0.289042
$607$	−2.14891e6	−0.236727	−0.118363	−	0.992970i	$-0.537765\pi$
$607$	−2.14891e6	−0.236727	−0.118363	+	0.992970i	$0.537765\pi$
$608$	−475789.	−0.0521982
$609$	6.03068e6	0.658905
$610$	2.59611e7	2.82487
$611$	1.23328e7	1.33647
$612$	3.41095e6	0.368126
$613$	1.69964e7	1.82686	0.913429	−	0.406999i	$-0.133425\pi$
$613$	1.69964e7	1.82686	0.913429	+	0.406999i	$0.133425\pi$
$614$	1.19015e7	1.27404
$615$	793258.	0.0845720
$616$	6.79211e6	0.721196
$617$	−3.94765e6	−0.417470	−0.208735	−	0.977972i	$-0.566935\pi$
$617$	−3.94765e6	−0.417470	−0.208735	+	0.977972i	$0.566935\pi$
$618$	1.33610e7	1.40724
$619$	−4.73731e6	−0.496942	−0.248471	−	0.968639i	$-0.579928\pi$
$619$	−4.73731e6	−0.496942	−0.248471	+	0.968639i	$0.579928\pi$
$620$	−1.24878e7	−1.30468
$621$	−2.69975e6	−0.280928
$622$	1.65244e7	1.71257
$623$	1.09311e6	0.112835
$624$	−2.60675e6	−0.268002
$625$	−2.34425e6	−0.240051
$626$	−2.16354e7	−2.20663
$627$	245023.	0.0248908
$628$	7.72726e6	0.781855
$629$	−4.60053e6	−0.463640
$630$	−8.47543e6	−0.850767
$631$	−1.34194e7	−1.34172	−0.670859	−	0.741585i	$-0.734074\pi$
$631$	−1.34194e7	−1.34172	−0.670859	+	0.741585i	$0.734074\pi$
$632$	−1.19317e6	−0.118825
$633$	−7.16152e6	−0.710388
$634$	1.54995e7	1.53142
$635$	2.35409e7	2.31680
$636$	1.42542e7	1.39733
$637$	−873742.	−0.0853168
$638$	−1.86027e7	−1.80935
$639$	−1.36918e6	−0.132650
$640$	−1.97073e7	−1.90185
$641$	1.36524e7	1.31239	0.656196	−	0.754591i	$-0.272165\pi$
$641$	1.36524e7	1.31239	0.656196	+	0.754591i	$0.272165\pi$
$642$	−2.91138e6	−0.278780
$643$	−918910.	−0.0876487	−0.0438244	−	0.999039i	$-0.513954\pi$
$643$	−918910.	−0.0876487	−0.0438244	+	0.999039i	$0.513954\pi$
$644$	−2.26606e7	−2.15306
$645$	6.14347e6	0.581453
$646$	523498.	0.0493553
$647$	−1.68376e7	−1.58132	−0.790658	−	0.612258i	$-0.790262\pi$
$647$	−1.68376e7	−1.58132	−0.790658	+	0.612258i	$0.790262\pi$
$648$	−784063.	−0.0733523
$649$	−1.47340e6	−0.137312
$650$	2.95718e7	2.74533
$651$	3.74320e6	0.346171
$652$	1.12559e7	1.03696
$653$	7.85260e6	0.720661	0.360330	−	0.932825i	$-0.382664\pi$
$653$	7.85260e6	0.720661	0.360330	+	0.932825i	$0.382664\pi$
$654$	1.35925e7	1.24267
$655$	−3.05129e7	−2.77895
$656$	−404151.	−0.0366677
$657$	−6.85034e6	−0.619154
$658$	−2.04293e7	−1.83946
$659$	1.37264e7	1.23124	0.615622	−	0.788042i	$-0.288905\pi$
$659$	1.37264e7	1.23124	0.615622	+	0.788042i	$0.288905\pi$
$660$	1.53586e7	1.37243
$661$	−1.55788e7	−1.38686	−0.693428	−	0.720526i	$-0.743901\pi$
$661$	−1.55788e7	−1.38686	−0.693428	+	0.720526i	$0.743901\pi$
$662$	2.30782e7	2.04672
$663$	5.93780e6	0.524616
$664$	−1.41975e6	−0.124966
$665$	−764156.	−0.0670082
$666$	3.55151e6	0.310261
$667$	1.84804e7	1.60841
$668$	−1.69231e6	−0.146737
$669$	1.29809e7	1.12134
$670$	3.93504e7	3.38659
$671$	1.41016e7	1.20910
$672$	8.93957e6	0.763648
$673$	9.65010e6	0.821285	0.410643	−	0.911796i	$-0.365304\pi$
$673$	9.65010e6	0.821285	0.410643	+	0.911796i	$0.365304\pi$
$674$	−2.26308e7	−1.91889
$675$	−3.42849e6	−0.289630
$676$	6.30727e6	0.530853
$677$	2.85548e6	0.239446	0.119723	−	0.992807i	$-0.461799\pi$
$677$	2.85548e6	0.239446	0.119723	+	0.992807i	$0.461799\pi$
$678$	859260.	0.0717877
$679$	1.84430e7	1.53517
$680$	9.77080e6	0.810322
$681$	3.79477e6	0.313558
$682$	−1.15465e7	−0.950585
$683$	−4.39043e6	−0.360127	−0.180063	−	0.983655i	$-0.557630\pi$
$683$	−4.39043e6	−0.360127	−0.180063	+	0.983655i	$0.557630\pi$
$684$	−237411.	−0.0194026
$685$	−2.64964e7	−2.15755
$686$	−1.84292e7	−1.49519
$687$	1.27534e7	1.03094
$688$	−3.12999e6	−0.252099
$689$	2.48138e7	1.99134
$690$	−2.59722e7	−2.07676
$691$	−8.79380e6	−0.700618	−0.350309	−	0.936634i	$-0.613923\pi$
$691$	−8.79380e6	−0.700618	−0.350309	+	0.936634i	$0.613923\pi$
$692$	−2.27689e6	−0.180749
$693$	−4.60372e6	−0.364146
$694$	−1.49407e7	−1.17753
$695$	2.47239e7	1.94158
$696$	5.36709e6	0.419968
$697$	920597.	0.0717774
$698$	2.08666e7	1.62111
$699$	−6.05839e6	−0.468991
$700$	−2.87773e7	−2.21975
$701$	1.57360e7	1.20948	0.604742	−	0.796421i	$-0.293276\pi$
$701$	1.57360e7	1.20948	0.604742	+	0.796421i	$0.293276\pi$
$702$	−4.58385e6	−0.351065
$703$	320209.	0.0244368
$704$	−2.20807e7	−1.67912
$705$	−1.37553e7	−1.04231
$706$	−3.81427e7	−2.88005
$707$	4.42654e6	0.333055
$708$	1.42762e6	0.107036
$709$	2.11378e7	1.57922	0.789611	−	0.613607i	$-0.210282\pi$
$709$	2.11378e7	1.57922	0.789611	+	0.613607i	$0.210282\pi$
$710$	−1.31718e7	−0.980615
$711$	808734.	0.0599973
$712$	972834.	0.0719182
$713$	1.14707e7	0.845017
$714$	−9.83597e6	−0.722057
$715$	2.67363e7	1.95585
$716$	−2.42343e6	−0.176664
$717$	6.84303e6	0.497108
$718$	8.94095e6	0.647250
$719$	−2.29229e7	−1.65366	−0.826831	−	0.562451i	$-0.809859\pi$
$719$	−2.29229e7	−1.65366	−0.826831	+	0.562451i	$0.809859\pi$
$720$	2.90742e6	0.209015
$721$	−2.26340e7	−1.62152
$722$	2.17713e7	1.55433
$723$	−4.23485e6	−0.301295
$724$	−1.04332e7	−0.739727
$725$	2.34688e7	1.65824
$726$	1.43513e6	0.101053
$727$	6.81555e6	0.478261	0.239130	−	0.970987i	$-0.423138\pi$
$727$	6.81555e6	0.478261	0.239130	+	0.970987i	$0.423138\pi$
$728$	−1.14564e7	−0.801161
$729$	531441.	0.0370370
$730$	−6.59016e7	−4.57709
$731$	7.12966e6	0.493487
$732$	−1.36635e7	−0.942507
$733$	1.68755e7	1.16011	0.580053	−	0.814578i	$-0.303032\pi$
$733$	1.68755e7	1.16011	0.580053	+	0.814578i	$0.303032\pi$
$734$	1.95017e7	1.33608
$735$	974522.	0.0665385
$736$	2.73945e7	1.86410
$737$	2.13745e7	1.44953
$738$	−710681.	−0.0480323
$739$	6.09722e6	0.410696	0.205348	−	0.978689i	$-0.434167\pi$
$739$	6.09722e6	0.410696	0.205348	+	0.978689i	$0.434167\pi$
$740$	2.00714e7	1.34740
$741$	−413286.	−0.0276506
$742$	−4.11041e7	−2.74079
$743$	5.82391e6	0.387028	0.193514	−	0.981097i	$-0.438011\pi$
$743$	5.82391e6	0.387028	0.193514	+	0.981097i	$0.438011\pi$
$744$	3.33132e6	0.220640
$745$	−4.62691e7	−3.05422
$746$	1.77767e7	1.16951
$747$	962311.	0.0630978
$748$	1.78240e7	1.16480
$749$	4.93198e6	0.321230
$750$	−1.10668e7	−0.718401
$751$	−1.45028e7	−0.938321	−0.469160	−	0.883113i	$-0.655443\pi$
$751$	−1.45028e7	−0.938321	−0.469160	+	0.883113i	$0.655443\pi$
$752$	7.00810e6	0.451914
$753$	−8.66534e6	−0.556927
$754$	3.13775e7	2.00997
$755$	1.99553e7	1.27406
$756$	4.46069e6	0.283856
$757$	2.63506e7	1.67128	0.835642	−	0.549274i	$-0.185096\pi$
$757$	2.63506e7	1.67128	0.835642	+	0.549274i	$0.185096\pi$
$758$	−4.19376e7	−2.65113
$759$	−1.41077e7	−0.888896
$760$	−680072.	−0.0427092
$761$	−8.31384e6	−0.520403	−0.260202	−	0.965554i	$-0.583789\pi$
$761$	−8.31384e6	−0.520403	−0.260202	+	0.965554i	$0.583789\pi$
$762$	−2.10903e7	−1.31582
$763$	−2.30262e7	−1.43189
$764$	1.74536e7	1.08181
$765$	−6.62268e6	−0.409148
$766$	−8.82517e6	−0.543439
$767$	2.48521e6	0.152537
$768$	2.63168e6	0.161001
$769$	1.06046e7	0.646666	0.323333	−	0.946285i	$-0.395197\pi$
$769$	1.06046e7	0.646666	0.323333	+	0.946285i	$0.395197\pi$
$770$	−4.42887e7	−2.69194
$771$	8.38725e6	0.508140
$772$	−3.06202e7	−1.84912
$773$	−2.65938e7	−1.60078	−0.800390	−	0.599479i	$-0.795374\pi$
$773$	−2.65938e7	−1.60078	−0.800390	+	0.599479i	$0.795374\pi$
$774$	−5.50394e6	−0.330234
$775$	1.45669e7	0.871192
$776$	1.64136e7	0.978474
$777$	−6.01638e6	−0.357505
$778$	4.48167e7	2.65455
$779$	−64075.9	−0.00378313
$780$	−2.59056e7	−1.52461
$781$	−7.15470e6	−0.419724
$782$	−3.01414e7	−1.76257
$783$	−3.63784e6	−0.212050
$784$	−496502.	−0.0288490
$785$	−1.50032e7	−0.868981
$786$	2.73366e7	1.57829
$787$	−3.05442e7	−1.75789	−0.878945	−	0.476923i	$-0.841752\pi$
$787$	−3.05442e7	−1.75789	−0.878945	+	0.476923i	$0.841752\pi$
$788$	−2.36037e7	−1.35414
$789$	1.13512e7	0.649154
$790$	7.78018e6	0.443529
$791$	−1.45562e6	−0.0827190
$792$	−4.09715e6	−0.232097
$793$	−2.37855e7	−1.34317
$794$	3.66536e7	2.06331
$795$	−2.76759e7	−1.55305
$796$	1.99483e7	1.11590
$797$	−2.58869e7	−1.44356	−0.721779	−	0.692123i	$-0.756675\pi$
$797$	−2.58869e7	−1.44356	−0.721779	+	0.692123i	$0.756675\pi$
$798$	684608.	0.0380570
$799$	−1.59634e7	−0.884625
$800$	3.47890e7	1.92184
$801$	−659390.	−0.0363129
$802$	−5.39643e7	−2.96258
$803$	−3.57967e7	−1.95909
$804$	−2.07105e7	−1.12993
$805$	4.39977e7	2.39299
$806$	1.94758e7	1.05599
$807$	6.19712e6	0.334970
$808$	3.93947e6	0.212280
$809$	6.74823e6	0.362509	0.181254	−	0.983436i	$-0.441984\pi$
$809$	6.74823e6	0.362509	0.181254	+	0.983436i	$0.441984\pi$
$810$	5.11257e6	0.273796
$811$	−2.22287e7	−1.18676	−0.593378	−	0.804924i	$-0.702206\pi$
$811$	−2.22287e7	−1.18676	−0.593378	+	0.804924i	$0.702206\pi$
$812$	−3.05345e7	−1.62517
$813$	6.59717e6	0.350051
$814$	1.85585e7	0.981709
$815$	−2.18545e7	−1.15251
$816$	3.37414e6	0.177394
$817$	−496243.	−0.0260099
$818$	1.22661e7	0.640947
$819$	7.76520e6	0.404523
$820$	−4.01641e6	−0.208595
$821$	−1.57133e7	−0.813599	−0.406799	−	0.913517i	$-0.633355\pi$
$821$	−1.57133e7	−0.813599	−0.406799	+	0.913517i	$0.633355\pi$
$822$	2.37382e7	1.22537
$823$	−2.22414e7	−1.14462	−0.572311	−	0.820036i	$-0.693953\pi$
$823$	−2.22414e7	−1.14462	−0.572311	+	0.820036i	$0.693953\pi$
$824$	−2.01435e7	−1.03351
$825$	−1.79157e7	−0.916430
$826$	−4.11676e6	−0.209945
$827$	1.20770e7	0.614038	0.307019	−	0.951703i	$-0.400668\pi$
$827$	1.20770e7	0.614038	0.307019	+	0.951703i	$0.400668\pi$
$828$	1.36694e7	0.692903
$829$	−1.91099e6	−0.0965765	−0.0482882	−	0.998833i	$-0.515377\pi$
$829$	−1.91099e6	−0.0965765	−0.0482882	+	0.998833i	$0.515377\pi$
$830$	9.25762e6	0.466449
$831$	1.58458e6	0.0795999
$832$	3.72440e7	1.86530
$833$	1.13096e6	0.0564722
$834$	−2.21502e7	−1.10271
$835$	3.28579e6	0.163088
$836$	−1.24060e6	−0.0613925
$837$	−2.25798e6	−0.111405
$838$	4.31532e7	2.12277
$839$	3.50301e7	1.71805	0.859025	−	0.511933i	$-0.171070\pi$
$839$	3.50301e7	1.71805	0.859025	+	0.511933i	$0.171070\pi$
$840$	1.27778e7	0.624826
$841$	4.39070e6	0.214064
$842$	1.29923e7	0.631547
$843$	9.36611e6	0.453931
$844$	3.62601e7	1.75216
$845$	−1.22462e7	−0.590009
$846$	1.23234e7	0.591977
$847$	−2.43116e6	−0.116441
$848$	1.41004e7	0.673352
$849$	−6.43697e6	−0.306487
$850$	−3.82774e7	−1.81717
$851$	−1.84366e7	−0.872685
$852$	6.93241e6	0.327179
$853$	−2.10044e7	−0.988412	−0.494206	−	0.869345i	$-0.664541\pi$
$853$	−2.10044e7	−0.988412	−0.494206	+	0.869345i	$0.664541\pi$
$854$	3.94008e7	1.84867
$855$	460956.	0.0215647
$856$	4.38929e6	0.204743
$857$	2.13586e7	0.993390	0.496695	−	0.867925i	$-0.334547\pi$
$857$	2.13586e7	0.993390	0.496695	+	0.867925i	$0.334547\pi$
$858$	−2.39531e7	−1.11082
$859$	2.37853e7	1.09983	0.549915	−	0.835221i	$-0.314660\pi$
$859$	2.37853e7	1.09983	0.549915	+	0.835221i	$0.314660\pi$
$860$	−3.11055e7	−1.43414
$861$	1.20392e6	0.0553463
$862$	5.71188e7	2.61825
$863$	−345138.	−0.0157749	−0.00788744	−	0.999969i	$-0.502511\pi$
$863$	−345138.	−0.0157749	−0.00788744	+	0.999969i	$0.502511\pi$
$864$	−5.39254e6	−0.245759
$865$	4.42080e6	0.200891
$866$	−5.16310e7	−2.33946
$867$	5.09291e6	0.230101
$868$	−1.89525e7	−0.853823
$869$	4.22607e6	0.189840
$870$	−3.49967e7	−1.56758
$871$	−3.60529e7	−1.61026
$872$	−2.04925e7	−0.912649
$873$	−1.11252e7	−0.494051
$874$	2.09792e6	0.0928988
$875$	1.87475e7	0.827794
$876$	3.46846e7	1.52713
$877$	−2.10330e7	−0.923425	−0.461712	−	0.887030i	$-0.652765\pi$
$877$	−2.10330e7	−0.923425	−0.461712	+	0.887030i	$0.652765\pi$
$878$	3.90001e7	1.70738
$879$	−1.83480e7	−0.800971
$880$	1.51928e7	0.661352
$881$	−3.40176e7	−1.47660	−0.738302	−	0.674470i	$-0.764372\pi$
$881$	−3.40176e7	−1.47660	−0.738302	+	0.674470i	$0.764372\pi$
$882$	−873075.	−0.0377903
$883$	9.95646e6	0.429737	0.214869	−	0.976643i	$-0.431068\pi$
$883$	9.95646e6	0.429737	0.214869	+	0.976643i	$0.431068\pi$
$884$	−3.00642e7	−1.29395
$885$	−2.77187e6	−0.118964
$886$	6.61352e7	2.83040
$887$	−1.73362e7	−0.739850	−0.369925	−	0.929062i	$-0.620617\pi$
$887$	−1.73362e7	−0.739850	−0.369925	+	0.929062i	$0.620617\pi$
$888$	−5.35437e6	−0.227864
$889$	3.57277e7	1.51618
$890$	−6.34346e6	−0.268443
$891$	2.77707e6	0.117190
$892$	−6.57245e7	−2.76576
$893$	1.11110e6	0.0466254
$894$	4.14526e7	1.73463
$895$	4.70533e6	0.196351
$896$	−2.99095e7	−1.24463
$897$	2.37957e7	0.987456
$898$	−1.71954e7	−0.711577
$899$	1.54564e7	0.637836
$900$	1.73591e7	0.714366
$901$	−3.21187e7	−1.31809
$902$	−3.71369e6	−0.151981
$903$	9.32387e6	0.380519
$904$	−1.29545e6	−0.0527228
$905$	2.02571e7	0.822158
$906$	−1.78779e7	−0.723598
$907$	7.69488e6	0.310587	0.155294	−	0.987868i	$-0.450368\pi$
$907$	7.69488e6	0.310587	0.155294	+	0.987868i	$0.450368\pi$
$908$	−1.92137e7	−0.773384
$909$	−2.67019e6	−0.107184
$910$	7.47027e7	2.99042
$911$	−1.61662e7	−0.645376	−0.322688	−	0.946505i	$-0.604587\pi$
$911$	−1.61662e7	−0.645376	−0.322688	+	0.946505i	$0.604587\pi$
$912$	−234849.	−0.00934978
$913$	5.02859e6	0.199650
$914$	5.23621e7	2.07325
$915$	2.65290e7	1.04754
$916$	−6.45730e7	−2.54280
$917$	−4.63091e7	−1.81862
$918$	5.93327e6	0.232374
$919$	−3.62199e7	−1.41468	−0.707341	−	0.706873i	$-0.750105\pi$
$919$	−3.62199e7	−1.41468	−0.707341	+	0.706873i	$0.750105\pi$
$920$	3.91564e7	1.52522
$921$	1.21619e7	0.472447
$922$	−5.74780e7	−2.22677
$923$	1.20680e7	0.466263
$924$	2.33095e7	0.898159
$925$	−2.34132e7	−0.899717
$926$	1.06590e7	0.408496
$927$	1.36533e7	0.521842
$928$	3.69132e7	1.40706
$929$	−2.11322e6	−0.0803350	−0.0401675	−	0.999193i	$-0.512789\pi$
$929$	−2.11322e6	−0.0803350	−0.0401675	+	0.999193i	$0.512789\pi$
$930$	−2.17222e7	−0.823563
$931$	−78717.6	−0.00297644
$932$	3.06747e7	1.15675
$933$	1.68859e7	0.635068
$934$	3.46625e7	1.30015
$935$	−3.46071e7	−1.29460
$936$	6.91076e6	0.257831
$937$	−1.46913e6	−0.0546652	−0.0273326	−	0.999626i	$-0.508701\pi$
$937$	−1.46913e6	−0.0546652	−0.0273326	+	0.999626i	$0.508701\pi$
$938$	5.97217e7	2.21628
$939$	−2.21088e7	−0.818278
$940$	6.96458e7	2.57084
$941$	−3.58757e7	−1.32077	−0.660384	−	0.750928i	$-0.729607\pi$
$941$	−3.58757e7	−1.32077	−0.660384	+	0.750928i	$0.729607\pi$
$942$	1.34414e7	0.493534
$943$	3.68929e6	0.135103
$944$	1.41222e6	0.0515788
$945$	−8.66086e6	−0.315487
$946$	−2.87611e7	−1.04491
$947$	−2.42642e7	−0.879208	−0.439604	−	0.898192i	$-0.644881\pi$
$947$	−2.42642e7	−0.879208	−0.439604	+	0.898192i	$0.644881\pi$
$948$	−4.09477e6	−0.147982
$949$	6.03792e7	2.17631
$950$	2.66420e6	0.0957764
$951$	1.58386e7	0.567890
$952$	1.48290e7	0.530298
$953$	−3.58333e7	−1.27807	−0.639036	−	0.769177i	$-0.720666\pi$
$953$	−3.58333e7	−1.27807	−0.639036	+	0.769177i	$0.720666\pi$
$954$	2.47949e7	0.882046
$955$	−3.38879e7	−1.20237
$956$	−3.46475e7	−1.22611
$957$	−1.90097e7	−0.670957
$958$	2.13695e6	0.0752281
$959$	−4.02132e7	−1.41196
$960$	−4.15398e7	−1.45474
$961$	−1.90355e7	−0.664898
$962$	−3.13031e7	−1.09056
$963$	−2.97508e6	−0.103379
$964$	2.14418e7	0.743138
$965$	5.94520e7	2.05517
$966$	−3.94176e7	−1.35909
$967$	−3.27974e7	−1.12791	−0.563953	−	0.825807i	$-0.690720\pi$
$967$	−3.27974e7	−1.12791	−0.563953	+	0.825807i	$0.690720\pi$
$968$	−2.16365e6	−0.0742161
$969$	534952.	0.0183023
$970$	−1.07027e8	−3.65226
$971$	5.64198e7	1.92036	0.960182	−	0.279376i	$-0.0901275\pi$
$971$	5.64198e7	1.92036	0.960182	+	0.279376i	$0.0901275\pi$
$972$	−2.69079e6	−0.0913510
$973$	3.75232e7	1.27063
$974$	2.27050e7	0.766874
$975$	3.02188e7	1.01804
$976$	−1.35161e7	−0.454178
$977$	−3.59696e7	−1.20559	−0.602794	−	0.797897i	$-0.705946\pi$
$977$	−3.59696e7	−1.20559	−0.602794	+	0.797897i	$0.705946\pi$
$978$	1.95794e7	0.654566
$979$	−3.44567e6	−0.114899
$980$	−4.93419e6	−0.164116
$981$	1.38899e7	0.460815
$982$	2.53126e6	0.0837642
$983$	−119682.	−0.00395044	−0.00197522	−	0.999998i	$-0.500629\pi$
$983$	−119682.	−0.00395044	−0.00197522	+	0.999998i	$0.500629\pi$
$984$	1.07144e6	0.0352762
$985$	4.58288e7	1.50504
$986$	−4.06146e7	−1.33042
$987$	−2.08763e7	−0.682119
$988$	2.09254e6	0.0681997
$989$	2.85721e7	0.928863
$990$	2.67159e7	0.866327
$991$	−7.51751e6	−0.243159	−0.121579	−	0.992582i	$-0.538796\pi$
$991$	−7.51751e6	−0.243159	−0.121579	+	0.992582i	$0.538796\pi$
$992$	2.29118e7	0.739230
$993$	2.35831e7	0.758977
$994$	−1.99906e7	−0.641742
$995$	−3.87316e7	−1.24025
$996$	−4.87236e6	−0.155629
$997$	3.03530e7	0.967084	0.483542	−	0.875321i	$-0.339350\pi$
$997$	3.03530e7	0.967084	0.483542	+	0.875321i	$0.339350\pi$
$998$	−3.98133e7	−1.26532
$999$	3.62921e6	0.115053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.6.a.b.1.3	✓	12
3.2	odd	2		531.6.a.d.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.3	✓	12	1.1	even	1	trivial
531.6.a.d.1.10		12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-8.80731 q^{2} -9.00000 q^{3} +45.5687 q^{4} -88.4760 q^{5} +79.2658 q^{6} -134.279 q^{7} -119.504 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-8.80731 q^{2} -9.00000 q^{3} +45.5687 q^{4} -88.4760 q^{5} +79.2658 q^{6} -134.279 q^{7} -119.504 q^{8} +81.0000 q^{9} +779.236 q^{10} +423.269 q^{11} -410.118 q^{12} -713.936 q^{13} +1182.64 q^{14} +796.284 q^{15} -405.693 q^{16} +924.109 q^{17} -713.392 q^{18} -64.3204 q^{19} -4031.74 q^{20} +1208.51 q^{21} -3727.86 q^{22} +3703.37 q^{23} +1075.53 q^{24} +4703.01 q^{25} +6287.86 q^{26} -729.000 q^{27} -6118.92 q^{28} +4990.17 q^{29} -7013.12 q^{30} +3097.37 q^{31} +7397.18 q^{32} -3809.42 q^{33} -8138.92 q^{34} +11880.5 q^{35} +3691.06 q^{36} -4978.34 q^{37} +566.489 q^{38} +6425.43 q^{39} +10573.2 q^{40} +996.199 q^{41} -10643.7 q^{42} +7715.17 q^{43} +19287.8 q^{44} -7166.56 q^{45} -32616.7 q^{46} -17274.4 q^{47} +3651.24 q^{48} +1223.84 q^{49} -41420.8 q^{50} -8316.98 q^{51} -32533.1 q^{52} -34756.3 q^{53} +6420.53 q^{54} -37449.1 q^{55} +16046.8 q^{56} +578.883 q^{57} -43950.0 q^{58} -3481.00 q^{59} +36285.6 q^{60} +33316.0 q^{61} -27279.5 q^{62} -10876.6 q^{63} -52167.1 q^{64} +63166.3 q^{65} +33550.7 q^{66} +50498.8 q^{67} +42110.4 q^{68} -33330.3 q^{69} -104635. q^{70} -16903.5 q^{71} -9679.79 q^{72} -84572.2 q^{73} +43845.8 q^{74} -42327.1 q^{75} -2930.99 q^{76} -56836.1 q^{77} -56590.7 q^{78} +9984.38 q^{79} +35894.1 q^{80} +6561.00 q^{81} -8773.83 q^{82} +11880.4 q^{83} +55070.2 q^{84} -81761.5 q^{85} -67949.9 q^{86} -44911.6 q^{87} -50582.1 q^{88} -8140.62 q^{89} +63118.1 q^{90} +95866.6 q^{91} +168758. q^{92} -27876.3 q^{93} +152141. q^{94} +5690.81 q^{95} -66574.6 q^{96} -137348. q^{97} -10778.7 q^{98} +34284.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \)	\( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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