LMFDB - Embedded newform 177.4.a.b.1.3

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$10.4433380710$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
Defining polynomial:	$x^{7} - 41 x^{5} - 7 x^{4} + 484 x^{3} + 63 x^{2} - 1736 x - 44$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.61892$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

$f(q)$	$=$	$q-2.61892 q^{2} -3.00000 q^{3} -1.14126 q^{4} -7.96684 q^{5} +7.85676 q^{6} +16.8751 q^{7} +23.9402 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.61892 q^{2} -3.00000 q^{3} -1.14126 q^{4} -7.96684 q^{5} +7.85676 q^{6} +16.8751 q^{7} +23.9402 q^{8} +9.00000 q^{9} +20.8645 q^{10} +62.7682 q^{11} +3.42378 q^{12} -75.2191 q^{13} -44.1946 q^{14} +23.9005 q^{15} -53.5675 q^{16} -51.9804 q^{17} -23.5703 q^{18} +24.2087 q^{19} +9.09222 q^{20} -50.6254 q^{21} -164.385 q^{22} -9.85642 q^{23} -71.8207 q^{24} -61.5295 q^{25} +196.993 q^{26} -27.0000 q^{27} -19.2589 q^{28} +160.344 q^{29} -62.5935 q^{30} -125.861 q^{31} -51.2329 q^{32} -188.305 q^{33} +136.132 q^{34} -134.441 q^{35} -10.2713 q^{36} -224.138 q^{37} -63.4006 q^{38} +225.657 q^{39} -190.728 q^{40} -467.472 q^{41} +132.584 q^{42} +90.9608 q^{43} -71.6348 q^{44} -71.7015 q^{45} +25.8132 q^{46} +437.881 q^{47} +160.702 q^{48} -58.2301 q^{49} +161.141 q^{50} +155.941 q^{51} +85.8445 q^{52} -353.688 q^{53} +70.7108 q^{54} -500.064 q^{55} +403.994 q^{56} -72.6261 q^{57} -419.929 q^{58} +59.0000 q^{59} -27.2767 q^{60} -697.325 q^{61} +329.620 q^{62} +151.876 q^{63} +562.715 q^{64} +599.258 q^{65} +493.154 q^{66} -292.026 q^{67} +59.3231 q^{68} +29.5693 q^{69} +352.091 q^{70} +223.496 q^{71} +215.462 q^{72} +122.800 q^{73} +586.998 q^{74} +184.589 q^{75} -27.6284 q^{76} +1059.22 q^{77} -590.978 q^{78} -440.541 q^{79} +426.763 q^{80} +81.0000 q^{81} +1224.27 q^{82} -662.365 q^{83} +57.7767 q^{84} +414.119 q^{85} -238.219 q^{86} -481.033 q^{87} +1502.68 q^{88} +1012.63 q^{89} +187.781 q^{90} -1269.33 q^{91} +11.2487 q^{92} +377.583 q^{93} -1146.77 q^{94} -192.867 q^{95} +153.699 q^{96} -397.912 q^{97} +152.500 q^{98} +564.914 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} + O(q^{10}) $	\( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} - 71q^{10} - 5q^{11} - 78q^{12} - 67q^{13} - 65q^{14} + 6q^{15} - 94q^{16} - 23q^{17} - 176q^{19} - 207q^{20} + 177q^{21} - 704q^{22} - 218q^{23} + 63q^{24} - 183q^{25} + 58q^{26} - 189q^{27} - 938q^{28} + 168q^{29} + 213q^{30} - 604q^{31} - 448q^{32} + 15q^{33} - 610q^{34} - 336q^{35} + 234q^{36} - 505q^{37} - 453q^{38} + 201q^{39} - 1080q^{40} - 265q^{41} + 195q^{42} - 493q^{43} + 504q^{44} - 18q^{45} + 381q^{46} - 244q^{47} + 282q^{48} + 770q^{49} + 1639q^{50} + 69q^{51} + 160q^{52} + 686q^{53} - 116q^{55} + 2190q^{56} + 528q^{57} + 1584q^{58} + 413q^{59} + 621q^{60} - 838q^{61} + 286q^{62} - 531q^{63} + 205q^{64} + 490q^{65} + 2112q^{66} - 1504q^{67} + 3047q^{68} + 654q^{69} + 1530q^{70} - 1267q^{71} - 189q^{72} - 666q^{73} + 528q^{74} + 549q^{75} - 64q^{76} + 1109q^{77} - 174q^{78} - 2741q^{79} + 1213q^{80} + 567q^{81} + 953q^{82} - 2025q^{83} + 2814q^{84} - 1274q^{85} + 4394q^{86} - 504q^{87} - 1639q^{88} + 616q^{89} - 639q^{90} - 2415q^{91} + 218q^{92} + 1812q^{93} + 900q^{94} + 2554q^{95} + 1344q^{96} - 1298q^{97} - 172q^{98} - 45q^{99} + O(q^{100}) \)

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 $

$2$	−2.61892	−0.925928	−0.462964	−	0.886377i	$-0.653214\pi$
$2$	−2.61892	−0.925928	−0.462964	+	0.886377i	$0.653214\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−1.14126	−0.142657
$5$	−7.96684	−0.712575	−0.356288	−	0.934376i	$-0.615958\pi$
$5$	−7.96684	−0.712575	−0.356288	+	0.934376i	$0.615958\pi$
$6$	7.85676	0.534585
$7$	16.8751	0.911171	0.455586	−	0.890192i	$-0.349430\pi$
$7$	16.8751	0.911171	0.455586	+	0.890192i	$0.349430\pi$
$8$	23.9402	1.05802
$9$	9.00000	0.333333
$10$	20.8645	0.659793
$11$	62.7682	1.72048	0.860242	−	0.509886i	$-0.170313\pi$
$11$	62.7682	1.72048	0.860242	+	0.509886i	$0.170313\pi$
$12$	3.42378	0.0823633
$13$	−75.2191	−1.60477	−0.802385	−	0.596807i	$-0.796436\pi$
$13$	−75.2191	−1.60477	−0.802385	+	0.596807i	$0.796436\pi$
$14$	−44.1946	−0.843679
$15$	23.9005	0.411406
$16$	−53.5675	−0.836991
$17$	−51.9804	−0.741593	−0.370797	−	0.928714i	$-0.620915\pi$
$17$	−51.9804	−0.741593	−0.370797	+	0.928714i	$0.620915\pi$
$18$	−23.5703	−0.308643
$19$	24.2087	0.292308	0.146154	−	0.989262i	$-0.453310\pi$
$19$	24.2087	0.292308	0.146154	+	0.989262i	$0.453310\pi$
$20$	9.09222	0.101654
$21$	−50.6254	−0.526065
$22$	−164.385	−1.59304
$23$	−9.85642	−0.0893568	−0.0446784	−	0.999001i	$-0.514226\pi$
$23$	−9.85642	−0.0893568	−0.0446784	+	0.999001i	$0.514226\pi$
$24$	−71.8207	−0.610847
$25$	−61.5295	−0.492236
$26$	196.993	1.48590
$27$	−27.0000	−0.192450
$28$	−19.2589	−0.129985
$29$	160.344	1.02673	0.513365	−	0.858170i	$-0.328399\pi$
$29$	160.344	1.02673	0.513365	+	0.858170i	$0.328399\pi$
$30$	−62.5935	−0.380932
$31$	−125.861	−0.729203	−0.364602	−	0.931164i	$-0.618795\pi$
$31$	−125.861	−0.729203	−0.364602	+	0.931164i	$0.618795\pi$
$32$	−51.2329	−0.283025
$33$	−188.305	−0.993322
$34$	136.132	0.686662
$35$	−134.441	−0.649278
$36$	−10.2713	−0.0475525
$37$	−224.138	−0.995892	−0.497946	−	0.867208i	$-0.665912\pi$
$37$	−224.138	−0.995892	−0.497946	+	0.867208i	$0.665912\pi$
$38$	−63.4006	−0.270656
$39$	225.657	0.926515
$40$	−190.728	−0.753918
$41$	−467.472	−1.78066	−0.890328	−	0.455319i	$-0.849525\pi$
$41$	−467.472	−1.78066	−0.890328	+	0.455319i	$0.849525\pi$
$42$	132.584	0.487098
$43$	90.9608	0.322591	0.161295	−	0.986906i	$-0.448433\pi$
$43$	90.9608	0.322591	0.161295	+	0.986906i	$0.448433\pi$
$44$	−71.6348	−0.245440
$45$	−71.7015	−0.237525
$46$	25.8132	0.0827379
$47$	437.881	1.35897	0.679484	−	0.733691i	$-0.262204\pi$
$47$	437.881	1.35897	0.679484	+	0.733691i	$0.262204\pi$
$48$	160.702	0.483237
$49$	−58.2301	−0.169767
$50$	161.141	0.455775
$51$	155.941	0.428159
$52$	85.8445	0.228932
$53$	−353.688	−0.916655	−0.458328	−	0.888783i	$-0.651551\pi$
$53$	−353.688	−0.916655	−0.458328	+	0.888783i	$0.651551\pi$
$54$	70.7108	0.178195
$55$	−500.064	−1.22597
$56$	403.994	0.964036
$57$	−72.6261	−0.168764
$58$	−419.929	−0.950678
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−27.2767	−0.0586901
$61$	−697.325	−1.46366	−0.731831	−	0.681487i	$-0.761334\pi$
$61$	−697.325	−1.46366	−0.731831	+	0.681487i	$0.761334\pi$
$62$	329.620	0.675190
$63$	151.876	0.303724
$64$	562.715	1.09905
$65$	599.258	1.14352
$66$	493.154	0.919744
$67$	−292.026	−0.532487	−0.266244	−	0.963906i	$-0.585783\pi$
$67$	−292.026	−0.532487	−0.266244	+	0.963906i	$0.585783\pi$
$68$	59.3231	0.105794
$69$	29.5693	0.0515902
$70$	352.091	0.601185
$71$	223.496	0.373579	0.186789	−	0.982400i	$-0.440192\pi$
$71$	223.496	0.373579	0.186789	+	0.982400i	$0.440192\pi$
$72$	215.462	0.352673
$73$	122.800	0.196885	0.0984426	−	0.995143i	$-0.468614\pi$
$73$	122.800	0.196885	0.0984426	+	0.995143i	$0.468614\pi$
$74$	586.998	0.922124
$75$	184.589	0.284193
$76$	−27.6284	−0.0416999
$77$	1059.22	1.56766
$78$	−590.978	−0.857886
$79$	−440.541	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$79$	−440.541	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$80$	426.763	0.596420
$81$	81.0000	0.111111
$82$	1224.27	1.64876
$83$	−662.365	−0.875951	−0.437976	−	0.898987i	$-0.644304\pi$
$83$	−662.365	−0.875951	−0.437976	+	0.898987i	$0.644304\pi$
$84$	57.7767	0.0750471
$85$	414.119	0.528441
$86$	−238.219	−0.298696
$87$	−481.033	−0.592783
$88$	1502.68	1.82030
$89$	1012.63	1.20606	0.603029	−	0.797720i	$-0.293960\pi$
$89$	1012.63	1.20606	0.603029	+	0.797720i	$0.293960\pi$
$90$	187.781	0.219931
$91$	−1269.33	−1.46222
$92$	11.2487	0.0127474
$93$	377.583	0.421006
$94$	−1146.77	−1.25831
$95$	−192.867	−0.208292
$96$	153.699	0.163404
$97$	−397.912	−0.416513	−0.208257	−	0.978074i	$-0.566779\pi$
$97$	−397.912	−0.416513	−0.208257	+	0.978074i	$0.566779\pi$
$98$	152.500	0.157192
$99$	564.914	0.573495
$100$	70.2212	0.0702212
$101$	−961.752	−0.947504	−0.473752	−	0.880658i	$-0.657101\pi$
$101$	−961.752	−0.947504	−0.473752	+	0.880658i	$0.657101\pi$
$102$	−408.397	−0.396445
$103$	−1085.47	−1.03839	−0.519197	−	0.854654i	$-0.673769\pi$
$103$	−1085.47	−1.03839	−0.519197	+	0.854654i	$0.673769\pi$
$104$	−1800.76	−1.69788
$105$	403.324	0.374861
$106$	926.279	0.848757
$107$	39.2036	0.0354201	0.0177101	−	0.999843i	$-0.494362\pi$
$107$	39.2036	0.0354201	0.0177101	+	0.999843i	$0.494362\pi$
$108$	30.8140	0.0274544
$109$	−210.156	−0.184672	−0.0923361	−	0.995728i	$-0.529433\pi$
$109$	−210.156	−0.184672	−0.0923361	+	0.995728i	$0.529433\pi$
$110$	1309.63	1.13516
$111$	672.413	0.574978
$112$	−903.958	−0.762643
$113$	−889.542	−0.740540	−0.370270	−	0.928924i	$-0.620735\pi$
$113$	−889.542	−0.740540	−0.370270	+	0.928924i	$0.620735\pi$
$114$	190.202	0.156264
$115$	78.5245	0.0636734
$116$	−182.994	−0.146471
$117$	−676.972	−0.534923
$118$	−154.516	−0.120546
$119$	−877.175	−0.675719
$120$	572.183	0.435275
$121$	2608.84	1.96006
$122$	1826.24	1.35524
$123$	1402.42	1.02806
$124$	143.640	0.104026
$125$	1486.05	1.06333
$126$	−397.751	−0.281226
$127$	−2040.11	−1.42543	−0.712717	−	0.701451i	$-0.752536\pi$
$127$	−2040.11	−1.42543	−0.712717	+	0.701451i	$0.752536\pi$
$128$	−1063.84	−0.734618
$129$	−272.883	−0.186248
$130$	−1569.41	−1.05882
$131$	−2233.61	−1.48971	−0.744853	−	0.667228i	$-0.767481\pi$
$131$	−2233.61	−1.48971	−0.744853	+	0.667228i	$0.767481\pi$
$132$	214.904	0.141705
$133$	408.525	0.266343
$134$	764.792	0.493045
$135$	215.105	0.137135
$136$	−1244.42	−0.784620
$137$	−2125.70	−1.32563	−0.662815	−	0.748784i	$-0.730638\pi$
$137$	−2125.70	−1.32563	−0.662815	+	0.748784i	$0.730638\pi$
$138$	−77.4395	−0.0477688
$139$	2710.99	1.65427	0.827133	−	0.562006i	$-0.189970\pi$
$139$	2710.99	1.65427	0.827133	+	0.562006i	$0.189970\pi$
$140$	153.432	0.0926244
$141$	−1313.64	−0.784600
$142$	−585.318	−0.345907
$143$	−4721.36	−2.76098
$144$	−482.107	−0.278997
$145$	−1277.44	−0.731622
$146$	−321.603	−0.182301
$147$	174.690	0.0980150
$148$	255.799	0.142071
$149$	1041.19	0.572470	0.286235	−	0.958160i	$-0.407596\pi$
$149$	1041.19	0.572470	0.286235	+	0.958160i	$0.407596\pi$
$150$	−483.423	−0.263142
$151$	1448.13	0.780444	0.390222	−	0.920721i	$-0.372398\pi$
$151$	1448.13	0.780444	0.390222	+	0.920721i	$0.372398\pi$
$152$	579.562	0.309267
$153$	−467.823	−0.247198
$154$	−2774.02	−1.45154
$155$	1002.71	0.519612
$156$	−257.533	−0.132174
$157$	−2438.90	−1.23978	−0.619891	−	0.784688i	$-0.712823\pi$
$157$	−2438.90	−1.23978	−0.619891	+	0.784688i	$0.712823\pi$
$158$	1153.74	0.580928
$159$	1061.06	0.529231
$160$	408.164	0.201676
$161$	−166.328	−0.0814193
$162$	−212.133	−0.102881
$163$	3332.07	1.60115	0.800575	−	0.599232i	$-0.204527\pi$
$163$	3332.07	1.60115	0.800575	+	0.599232i	$0.204527\pi$
$164$	533.507	0.254024
$165$	1500.19	0.707817
$166$	1734.68	0.811068
$167$	−2643.67	−1.22499	−0.612494	−	0.790475i	$-0.709834\pi$
$167$	−2643.67	−1.22499	−0.612494	+	0.790475i	$0.709834\pi$
$168$	−1211.98	−0.556586
$169$	3460.91	1.57529
$170$	−1084.54	−0.489299
$171$	217.878	0.0974361
$172$	−103.810	−0.0460200
$173$	−1207.93	−0.530849	−0.265424	−	0.964132i	$-0.585512\pi$
$173$	−1207.93	−0.530849	−0.265424	+	0.964132i	$0.585512\pi$
$174$	1259.79	0.548874
$175$	−1038.32	−0.448512
$176$	−3362.33	−1.44003
$177$	−177.000	−0.0751646
$178$	−2652.01	−1.11672
$179$	−1379.61	−0.576074	−0.288037	−	0.957619i	$-0.593003\pi$
$179$	−1379.61	−0.576074	−0.288037	+	0.957619i	$0.593003\pi$
$180$	81.8300	0.0338847
$181$	−2223.79	−0.913221	−0.456611	−	0.889667i	$-0.650937\pi$
$181$	−2223.79	−0.913221	−0.456611	+	0.889667i	$0.650937\pi$
$182$	3324.28	1.35391
$183$	2091.98	0.845045
$184$	−235.965	−0.0945411
$185$	1785.67	0.709648
$186$	−988.860	−0.389821
$187$	−3262.71	−1.27590
$188$	−499.735	−0.193867
$189$	−455.628	−0.175355
$190$	505.102	0.192863
$191$	−1419.51	−0.537762	−0.268881	−	0.963173i	$-0.586654\pi$
$191$	−1419.51	−0.537762	−0.268881	+	0.963173i	$0.586654\pi$
$192$	−1688.14	−0.634538
$193$	2797.83	1.04348	0.521741	−	0.853104i	$-0.325283\pi$
$193$	2797.83	1.04348	0.521741	+	0.853104i	$0.325283\pi$
$194$	1042.10	0.385661
$195$	−1797.77	−0.660212
$196$	66.4556	0.0242185
$197$	4134.50	1.49528	0.747641	−	0.664103i	$-0.231186\pi$
$197$	4134.50	1.49528	0.747641	+	0.664103i	$0.231186\pi$
$198$	−1479.46	−0.531015
$199$	−5049.42	−1.79871	−0.899357	−	0.437214i	$-0.855965\pi$
$199$	−5049.42	−1.79871	−0.899357	+	0.437214i	$0.855965\pi$
$200$	−1473.03	−0.520795
$201$	876.078	0.307432
$202$	2518.75	0.877320
$203$	2705.83	0.935527
$204$	−177.969	−0.0610801
$205$	3724.28	1.26885
$206$	2842.76	0.961479
$207$	−88.7078	−0.0297856
$208$	4029.29	1.34318
$209$	1519.54	0.502911
$210$	−1056.27	−0.347094
$211$	4910.79	1.60224	0.801121	−	0.598503i	$-0.204237\pi$
$211$	4910.79	1.60224	0.801121	+	0.598503i	$0.204237\pi$
$212$	403.649	0.130768
$213$	−670.488	−0.215686
$214$	−102.671	−0.0327965
$215$	−724.670	−0.229870
$216$	−646.386	−0.203616
$217$	−2123.92	−0.664429
$218$	550.381	0.170993
$219$	−368.399	−0.113672
$220$	570.702	0.174894
$221$	3909.91	1.19009
$222$	−1760.99	−0.532388
$223$	−47.4526	−0.0142496	−0.00712480	−	0.999975i	$-0.502268\pi$
$223$	−47.4526	−0.0142496	−0.00712480	+	0.999975i	$0.502268\pi$
$224$	−864.562	−0.257884
$225$	−553.766	−0.164079
$226$	2329.64	0.685687
$227$	3097.71	0.905737	0.452869	−	0.891577i	$-0.350401\pi$
$227$	3097.71	0.905737	0.452869	+	0.891577i	$0.350401\pi$
$228$	82.8852	0.0240755
$229$	2273.82	0.656149	0.328074	−	0.944652i	$-0.393600\pi$
$229$	2273.82	0.656149	0.328074	+	0.944652i	$0.393600\pi$
$230$	−205.649	−0.0589570
$231$	−3177.66	−0.905086
$232$	3838.68	1.08630
$233$	−3433.68	−0.965442	−0.482721	−	0.875774i	$-0.660351\pi$
$233$	−3433.68	−0.965442	−0.482721	+	0.875774i	$0.660351\pi$
$234$	1772.93	0.495301
$235$	−3488.52	−0.968367
$236$	−67.3343	−0.0185724
$237$	1321.62	0.362230
$238$	2297.25	0.625667
$239$	5489.29	1.48566	0.742830	−	0.669480i	$-0.233483\pi$
$239$	5489.29	1.48566	0.742830	+	0.669480i	$0.233483\pi$
$240$	−1280.29	−0.344343
$241$	71.4197	0.0190894	0.00954470	−	0.999954i	$-0.496962\pi$
$241$	71.4197	0.0190894	0.00954470	+	0.999954i	$0.496962\pi$
$242$	−6832.35	−1.81488
$243$	−243.000	−0.0641500
$244$	795.829	0.208802
$245$	463.909	0.120972
$246$	−3672.82	−0.951912
$247$	−1820.96	−0.469088
$248$	−3013.14	−0.771511
$249$	1987.09	0.505731
$250$	−3891.85	−0.984568
$251$	7078.52	1.78005	0.890024	−	0.455913i	$-0.150687\pi$
$251$	7078.52	1.78005	0.890024	+	0.455913i	$0.150687\pi$
$252$	−173.330	−0.0433284
$253$	−618.670	−0.153737
$254$	5342.88	1.31985
$255$	−1242.36	−0.305096
$256$	−1715.60	−0.418848
$257$	−5131.54	−1.24551	−0.622756	−	0.782416i	$-0.713987\pi$
$257$	−5131.54	−1.24551	−0.622756	+	0.782416i	$0.713987\pi$
$258$	714.657	0.172452
$259$	−3782.35	−0.907428
$260$	−683.909	−0.163132
$261$	1443.10	0.342243
$262$	5849.65	1.37936
$263$	−2899.64	−0.679846	−0.339923	−	0.940453i	$-0.610401\pi$
$263$	−2899.64	−0.679846	−0.339923	+	0.940453i	$0.610401\pi$
$264$	−4508.05	−1.05095
$265$	2817.77	0.653186
$266$	−1069.89	−0.246614
$267$	−3037.90	−0.696317
$268$	333.277	0.0759632
$269$	2624.44	0.594852	0.297426	−	0.954745i	$-0.403872\pi$
$269$	2624.44	0.594852	0.297426	+	0.954745i	$0.403872\pi$
$270$	−563.342	−0.126977
$271$	8503.68	1.90613	0.953066	−	0.302762i	$-0.0979088\pi$
$271$	8503.68	1.90613	0.953066	+	0.302762i	$0.0979088\pi$
$272$	2784.46	0.620707
$273$	3807.99	0.844213
$274$	5567.05	1.22744
$275$	−3862.10	−0.846884
$276$	−33.7462	−0.00735972
$277$	−3533.33	−0.766415	−0.383208	−	0.923662i	$-0.625181\pi$
$277$	−3533.33	−0.766415	−0.383208	+	0.923662i	$0.625181\pi$
$278$	−7099.86	−1.53173
$279$	−1132.75	−0.243068
$280$	−3218.56	−0.686948
$281$	−2767.76	−0.587584	−0.293792	−	0.955869i	$-0.594917\pi$
$281$	−2767.76	−0.587584	−0.293792	+	0.955869i	$0.594917\pi$
$282$	3440.32	0.726483
$283$	−3853.33	−0.809388	−0.404694	−	0.914452i	$-0.632622\pi$
$283$	−3853.33	−0.809388	−0.404694	+	0.914452i	$0.632622\pi$
$284$	−255.067	−0.0532938
$285$	578.600	0.120257
$286$	12364.9	2.55647
$287$	−7888.66	−1.62248
$288$	−461.096	−0.0943416
$289$	−2211.04	−0.450039
$290$	3345.50	0.677430
$291$	1193.73	0.240474
$292$	−140.146	−0.0280871
$293$	6720.24	1.33993	0.669967	−	0.742391i	$-0.266308\pi$
$293$	6720.24	1.33993	0.669967	+	0.742391i	$0.266308\pi$
$294$	−457.500	−0.0907548
$295$	−470.043	−0.0927694
$296$	−5365.90	−1.05367
$297$	−1694.74	−0.331107
$298$	−2726.80	−0.530066
$299$	741.391	0.143397
$300$	−210.663	−0.0405422
$301$	1534.98	0.293935
$302$	−3792.53	−0.722635
$303$	2885.26	0.547042
$304$	−1296.80	−0.244659
$305$	5555.47	1.04297
$306$	1225.19	0.228887
$307$	−3154.37	−0.586416	−0.293208	−	0.956049i	$-0.594723\pi$
$307$	−3154.37	−0.586416	−0.293208	+	0.956049i	$0.594723\pi$
$308$	−1208.85	−0.223638
$309$	3256.41	0.599518
$310$	−2626.03	−0.481124
$311$	4204.57	0.766622	0.383311	−	0.923619i	$-0.374784\pi$
$311$	4204.57	0.766622	0.383311	+	0.923619i	$0.374784\pi$
$312$	5402.28	0.980270
$313$	−6497.54	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$313$	−6497.54	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$314$	6387.29	1.14795
$315$	−1209.97	−0.216426
$316$	502.771	0.0895034
$317$	5621.97	0.996092	0.498046	−	0.867151i	$-0.334051\pi$
$317$	5621.97	0.996092	0.498046	+	0.867151i	$0.334051\pi$
$318$	−2778.84	−0.490030
$319$	10064.5	1.76647
$320$	−4483.05	−0.783157
$321$	−117.611	−0.0204498
$322$	435.601	0.0753884
$323$	−1258.38	−0.216774
$324$	−92.4420	−0.0158508
$325$	4628.19	0.789926
$326$	−8726.41	−1.48255
$327$	630.467	0.106620
$328$	−11191.4	−1.88397
$329$	7389.29	1.23825
$330$	−3928.88	−0.655387
$331$	−7641.62	−1.26895	−0.634473	−	0.772945i	$-0.718783\pi$
$331$	−7641.62	−1.26895	−0.634473	+	0.772945i	$0.718783\pi$
$332$	755.930	0.124961
$333$	−2017.24	−0.331964
$334$	6923.55	1.13425
$335$	2326.52	0.379437
$336$	2711.87	0.440312
$337$	7138.77	1.15393	0.576964	−	0.816770i	$-0.304237\pi$
$337$	7138.77	1.15393	0.576964	+	0.816770i	$0.304237\pi$
$338$	−9063.84	−1.45860
$339$	2668.63	0.427551
$340$	−472.617	−0.0753861
$341$	−7900.07	−1.25458
$342$	−570.606	−0.0902188
$343$	−6770.81	−1.06586
$344$	2177.62	0.341307
$345$	−235.573	−0.0367619
$346$	3163.46	0.491528
$347$	3416.69	0.528580	0.264290	−	0.964443i	$-0.414862\pi$
$347$	3416.69	0.528580	0.264290	+	0.964443i	$0.414862\pi$
$348$	548.983	0.0845649
$349$	8930.15	1.36968	0.684842	−	0.728691i	$-0.259871\pi$
$349$	8930.15	1.36968	0.684842	+	0.728691i	$0.259871\pi$
$350$	2719.27	0.415289
$351$	2030.91	0.308838
$352$	−3215.80	−0.486939
$353$	9550.53	1.44001	0.720005	−	0.693969i	$-0.244139\pi$
$353$	9550.53	1.44001	0.720005	+	0.693969i	$0.244139\pi$
$354$	463.549	0.0695970
$355$	−1780.56	−0.266203
$356$	−1155.68	−0.172053
$357$	2631.53	0.390126
$358$	3613.10	0.533403
$359$	−7990.13	−1.17466	−0.587330	−	0.809347i	$-0.699821\pi$
$359$	−7990.13	−1.17466	−0.587330	+	0.809347i	$0.699821\pi$
$360$	−1716.55	−0.251306
$361$	−6272.94	−0.914556
$362$	5823.93	0.845577
$363$	−7826.53	−1.13164
$364$	1448.64	0.208597
$365$	−978.325	−0.140296
$366$	−5478.72	−0.782451
$367$	−9255.46	−1.31643	−0.658216	−	0.752829i	$-0.728689\pi$
$367$	−9255.46	−1.31643	−0.658216	+	0.752829i	$0.728689\pi$
$368$	527.983	0.0747908
$369$	−4207.25	−0.593552
$370$	−4676.52	−0.657083
$371$	−5968.52	−0.835230
$372$	−430.920	−0.0600596
$373$	−5532.87	−0.768047	−0.384023	−	0.923323i	$-0.625462\pi$
$373$	−5532.87	−0.768047	−0.384023	+	0.923323i	$0.625462\pi$
$374$	8544.78	1.18139
$375$	−4458.15	−0.613914
$376$	10483.0	1.43781
$377$	−12060.9	−1.64767
$378$	1193.25	0.162366
$379$	12355.4	1.67456	0.837278	−	0.546778i	$-0.184146\pi$
$379$	12355.4	1.67456	0.837278	+	0.546778i	$0.184146\pi$
$380$	220.111	0.0297143
$381$	6120.32	0.822975
$382$	3717.59	0.497929
$383$	−12814.3	−1.70961	−0.854807	−	0.518946i	$-0.826325\pi$
$383$	−12814.3	−1.70961	−0.854807	+	0.518946i	$0.826325\pi$
$384$	3191.52	0.424132
$385$	−8438.64	−1.11707
$386$	−7327.28	−0.966188
$387$	818.648	0.107530
$388$	454.120	0.0594187
$389$	−1715.48	−0.223595	−0.111797	−	0.993731i	$-0.535661\pi$
$389$	−1715.48	−0.223595	−0.111797	+	0.993731i	$0.535661\pi$
$390$	4708.22	0.611308
$391$	512.340	0.0662664
$392$	−1394.04	−0.179617
$393$	6700.84	0.860083
$394$	−10827.9	−1.38452
$395$	3509.71	0.447071
$396$	−644.713	−0.0818132
$397$	1097.81	0.138785	0.0693924	−	0.997589i	$-0.477894\pi$
$397$	1097.81	0.138785	0.0693924	+	0.997589i	$0.477894\pi$
$398$	13224.0	1.66548
$399$	−1225.57	−0.153773
$400$	3295.98	0.411998
$401$	−3949.81	−0.491880	−0.245940	−	0.969285i	$-0.579097\pi$
$401$	−3949.81	−0.491880	−0.245940	+	0.969285i	$0.579097\pi$
$402$	−2294.38	−0.284660
$403$	9467.15	1.17020
$404$	1097.61	0.135168
$405$	−645.314	−0.0791750
$406$	−7086.35	−0.866230
$407$	−14068.7	−1.71342
$408$	3733.26	0.453000
$409$	−1709.43	−0.206665	−0.103332	−	0.994647i	$-0.532951\pi$
$409$	−1709.43	−0.206665	−0.103332	+	0.994647i	$0.532951\pi$
$410$	−9753.58	−1.17487
$411$	6377.11	0.765352
$412$	1238.80	0.148135
$413$	995.633	0.118624
$414$	232.319	0.0275793
$415$	5276.95	0.624181
$416$	3853.69	0.454190
$417$	−8132.97	−0.955091
$418$	−3979.54	−0.465660
$419$	−280.723	−0.0327308	−0.0163654	−	0.999866i	$-0.505209\pi$
$419$	−280.723	−0.0327308	−0.0163654	+	0.999866i	$0.505209\pi$
$420$	−460.297	−0.0534767
$421$	−11642.2	−1.34776	−0.673878	−	0.738842i	$-0.735373\pi$
$421$	−11642.2	−1.34776	−0.673878	+	0.738842i	$0.735373\pi$
$422$	−12861.0	−1.48356
$423$	3940.93	0.452989
$424$	−8467.36	−0.969838
$425$	3198.33	0.365039
$426$	1755.95	0.199709
$427$	−11767.5	−1.33365
$428$	−44.7414	−0.00505294
$429$	14164.1	1.59405
$430$	1897.85	0.212843
$431$	−1887.48	−0.210943	−0.105472	−	0.994422i	$-0.533635\pi$
$431$	−1887.48	−0.210943	−0.105472	+	0.994422i	$0.533635\pi$
$432$	1446.32	0.161079
$433$	−2668.89	−0.296210	−0.148105	−	0.988972i	$-0.547317\pi$
$433$	−2668.89	−0.296210	−0.148105	+	0.988972i	$0.547317\pi$
$434$	5562.38	0.615213
$435$	3832.31	0.422402
$436$	239.842	0.0263448
$437$	−238.611	−0.0261197
$438$	964.808	0.105252
$439$	11389.6	1.23826	0.619131	−	0.785287i	$-0.287485\pi$
$439$	11389.6	1.23826	0.619131	+	0.785287i	$0.287485\pi$
$440$	−11971.6	−1.29710
$441$	−524.071	−0.0565890
$442$	−10239.8	−1.10193
$443$	14630.8	1.56914	0.784569	−	0.620041i	$-0.212884\pi$
$443$	14630.8	1.56914	0.784569	+	0.620041i	$0.212884\pi$
$444$	−767.397	−0.0820249
$445$	−8067.50	−0.859407
$446$	124.274	0.0131941
$447$	−3123.58	−0.330515
$448$	9495.88	1.00142
$449$	10838.9	1.13924	0.569620	−	0.821908i	$-0.307090\pi$
$449$	10838.9	1.13924	0.569620	+	0.821908i	$0.307090\pi$
$450$	1450.27	0.151925
$451$	−29342.4	−3.06359
$452$	1015.20	0.105644
$453$	−4344.39	−0.450589
$454$	−8112.66	−0.838648
$455$	10112.6	1.04194
$456$	−1738.68	−0.178556
$457$	−7768.54	−0.795179	−0.397590	−	0.917563i	$-0.630153\pi$
$457$	−7768.54	−0.795179	−0.397590	+	0.917563i	$0.630153\pi$
$458$	−5954.95	−0.607547
$459$	1403.47	0.142720
$460$	−89.6168	−0.00908349
$461$	7262.50	0.733727	0.366864	−	0.930275i	$-0.380432\pi$
$461$	7262.50	0.733727	0.366864	+	0.930275i	$0.380432\pi$
$462$	8322.05	0.838045
$463$	8935.85	0.896942	0.448471	−	0.893797i	$-0.351969\pi$
$463$	8935.85	0.896942	0.448471	+	0.893797i	$0.351969\pi$
$464$	−8589.23	−0.859364
$465$	−3008.14	−0.299998
$466$	8992.53	0.893929
$467$	17921.0	1.77577	0.887885	−	0.460065i	$-0.152174\pi$
$467$	17921.0	1.77577	0.887885	+	0.460065i	$0.152174\pi$
$468$	772.600	0.0763108
$469$	−4927.97	−0.485187
$470$	9136.16	0.896638
$471$	7316.71	0.715788
$472$	1412.47	0.137742
$473$	5709.45	0.555012
$474$	−3461.22	−0.335399
$475$	−1489.55	−0.143885
$476$	1001.08	0.0963963
$477$	−3183.19	−0.305552
$478$	−14376.0	−1.37561
$479$	1733.15	0.165323	0.0826616	−	0.996578i	$-0.473658\pi$
$479$	1733.15	0.165323	0.0826616	+	0.996578i	$0.473658\pi$
$480$	−1224.49	−0.116438
$481$	16859.4	1.59818
$482$	−187.042	−0.0176754
$483$	498.985	0.0470075
$484$	−2977.37	−0.279618
$485$	3170.10	0.296797
$486$	636.398	0.0593983
$487$	−2597.17	−0.241661	−0.120831	−	0.992673i	$-0.538556\pi$
$487$	−2597.17	−0.241661	−0.120831	+	0.992673i	$0.538556\pi$
$488$	−16694.1	−1.54858
$489$	−9996.20	−0.924425
$490$	−1214.94	−0.112011
$491$	7523.32	0.691492	0.345746	−	0.938328i	$-0.387626\pi$
$491$	7523.32	0.691492	0.345746	+	0.938328i	$0.387626\pi$
$492$	−1600.52	−0.146661
$493$	−8334.75	−0.761416
$494$	4768.94	0.434341
$495$	−4500.57	−0.408658
$496$	6742.05	0.610337
$497$	3771.52	0.340394
$498$	−5204.04	−0.468270
$499$	9392.79	0.842643	0.421322	−	0.906911i	$-0.361566\pi$
$499$	9392.79	0.842643	0.421322	+	0.906911i	$0.361566\pi$
$500$	−1695.97	−0.151692
$501$	7931.00	0.707248
$502$	−18538.1	−1.64820
$503$	2687.97	0.238272	0.119136	−	0.992878i	$-0.461988\pi$
$503$	2687.97	0.238272	0.119136	+	0.992878i	$0.461988\pi$
$504$	3635.95	0.321345
$505$	7662.12	0.675168
$506$	1620.25	0.142349
$507$	−10382.7	−0.909493
$508$	2328.29	0.203349
$509$	10267.4	0.894098	0.447049	−	0.894510i	$-0.352475\pi$
$509$	10267.4	0.894098	0.447049	+	0.894510i	$0.352475\pi$
$510$	3253.63	0.282497
$511$	2072.26	0.179396
$512$	13003.8	1.12244
$513$	−653.635	−0.0562547
$514$	13439.1	1.15325
$515$	8647.77	0.739935
$516$	311.430	0.0265696
$517$	27485.0	2.33808
$518$	9905.67	0.840213
$519$	3623.78	0.306486
$520$	14346.4	1.20987
$521$	−14725.4	−1.23826	−0.619128	−	0.785290i	$-0.712514\pi$
$521$	−14725.4	−1.23826	−0.619128	+	0.785290i	$0.712514\pi$
$522$	−3779.36	−0.316893
$523$	5672.69	0.474282	0.237141	−	0.971475i	$-0.423790\pi$
$523$	5672.69	0.474282	0.237141	+	0.971475i	$0.423790\pi$
$524$	2549.13	0.212518
$525$	3114.96	0.258948
$526$	7593.93	0.629488
$527$	6542.30	0.540772
$528$	10087.0	0.831402
$529$	−12069.9	−0.992015
$530$	−7379.52	−0.604803
$531$	531.000	0.0433963
$532$	−466.233	−0.0379958
$533$	35162.8	2.85755
$534$	7956.03	0.644740
$535$	−312.328	−0.0252395
$536$	−6991.17	−0.563381
$537$	4138.84	0.332596
$538$	−6873.20	−0.550790
$539$	−3655.00	−0.292081
$540$	−245.490	−0.0195634
$541$	−13483.4	−1.07153	−0.535766	−	0.844367i	$-0.679977\pi$
$541$	−13483.4	−1.07153	−0.535766	+	0.844367i	$0.679977\pi$
$542$	−22270.5	−1.76494
$543$	6671.37	0.527248
$544$	2663.11	0.209889
$545$	1674.28	0.131593
$546$	−9972.83	−0.781681
$547$	3503.56	0.273860	0.136930	−	0.990581i	$-0.456277\pi$
$547$	3503.56	0.273860	0.136930	+	0.990581i	$0.456277\pi$
$548$	2425.98	0.189111
$549$	−6275.93	−0.487887
$550$	10114.5	0.784154
$551$	3881.72	0.300122
$552$	707.895	0.0545833
$553$	−7434.18	−0.571670
$554$	9253.50	0.709645
$555$	−5357.00	−0.409715
$556$	−3093.94	−0.235993
$557$	785.734	0.0597713	0.0298856	−	0.999553i	$-0.490486\pi$
$557$	785.734	0.0597713	0.0298856	+	0.999553i	$0.490486\pi$
$558$	2966.58	0.225063
$559$	−6841.99	−0.517684
$560$	7201.68	0.543440
$561$	9788.14	0.736641
$562$	7248.55	0.544060
$563$	−711.755	−0.0532805	−0.0266402	−	0.999645i	$-0.508481\pi$
$563$	−711.755	−0.0532805	−0.0266402	+	0.999645i	$0.508481\pi$
$564$	1499.21	0.111929
$565$	7086.83	0.527691
$566$	10091.6	0.749435
$567$	1366.89	0.101241
$568$	5350.54	0.395253
$569$	−12752.6	−0.939574	−0.469787	−	0.882780i	$-0.655669\pi$
$569$	−12752.6	−0.939574	−0.469787	+	0.882780i	$0.655669\pi$
$570$	−1515.31	−0.111350
$571$	568.844	0.0416907	0.0208454	−	0.999783i	$-0.493364\pi$
$571$	568.844	0.0416907	0.0208454	+	0.999783i	$0.493364\pi$
$572$	5388.30	0.393874
$573$	4258.54	0.310477
$574$	20659.8	1.50230
$575$	606.461	0.0439846
$576$	5064.43	0.366351
$577$	−4814.50	−0.347366	−0.173683	−	0.984802i	$-0.555567\pi$
$577$	−4814.50	−0.347366	−0.173683	+	0.984802i	$0.555567\pi$
$578$	5790.54	0.416704
$579$	−8393.48	−0.602454
$580$	1457.89	0.104371
$581$	−11177.5	−0.798142
$582$	−3126.29	−0.222662
$583$	−22200.3	−1.57709
$584$	2939.85	0.208308
$585$	5393.32	0.381173
$586$	−17599.8	−1.24068
$587$	−3790.32	−0.266513	−0.133256	−	0.991082i	$-0.542543\pi$
$587$	−3790.32	−0.266513	−0.133256	+	0.991082i	$0.542543\pi$
$588$	−199.367	−0.0139826
$589$	−3046.93	−0.213152
$590$	1231.01	0.0858978
$591$	−12403.5	−0.863302
$592$	12006.5	0.833553
$593$	15605.2	1.08065	0.540326	−	0.841456i	$-0.318301\pi$
$593$	15605.2	1.08065	0.540326	+	0.841456i	$0.318301\pi$
$594$	4438.39	0.306581
$595$	6988.31	0.481500
$596$	−1188.27	−0.0816670
$597$	15148.3	1.03849
$598$	−1941.64	−0.132775
$599$	−26016.3	−1.77462	−0.887310	−	0.461174i	$-0.847428\pi$
$599$	−26016.3	−1.77462	−0.887310	+	0.461174i	$0.847428\pi$
$600$	4419.09	0.300681
$601$	−15990.4	−1.08530	−0.542648	−	0.839960i	$-0.682578\pi$
$601$	−15990.4	−1.08530	−0.542648	+	0.839960i	$0.682578\pi$
$602$	−4019.98	−0.272163
$603$	−2628.23	−0.177496
$604$	−1652.69	−0.111336
$605$	−20784.2	−1.39669
$606$	−7556.25	−0.506521
$607$	8461.94	0.565831	0.282915	−	0.959145i	$-0.408698\pi$
$607$	8461.94	0.565831	0.282915	+	0.959145i	$0.408698\pi$
$608$	−1240.28	−0.0827304
$609$	−8117.49	−0.540127
$610$	−14549.3	−0.965714
$611$	−32937.0	−2.18083
$612$	533.908	0.0352646
$613$	−3306.42	−0.217855	−0.108927	−	0.994050i	$-0.534742\pi$
$613$	−3306.42	−0.217855	−0.108927	+	0.994050i	$0.534742\pi$
$614$	8261.05	0.542979
$615$	−11172.8	−0.732572
$616$	25358.0	1.65861
$617$	−2038.03	−0.132979	−0.0664894	−	0.997787i	$-0.521180\pi$
$617$	−2038.03	−0.132979	−0.0664894	+	0.997787i	$0.521180\pi$
$618$	−8528.29	−0.555110
$619$	−1376.65	−0.0893896	−0.0446948	−	0.999001i	$-0.514232\pi$
$619$	−1376.65	−0.0893896	−0.0446948	+	0.999001i	$0.514232\pi$
$620$	−1144.36	−0.0741266
$621$	266.123	0.0171967
$622$	−11011.4	−0.709837
$623$	17088.3	1.09892
$624$	−12087.9	−0.775485
$625$	−4147.92	−0.265467
$626$	17016.5	1.08645
$627$	−4558.61	−0.290356
$628$	2783.42	0.176864
$629$	11650.8	0.738547
$630$	3168.82	0.200395
$631$	−12856.2	−0.811091	−0.405546	−	0.914075i	$-0.632918\pi$
$631$	−12856.2	−0.811091	−0.405546	+	0.914075i	$0.632918\pi$
$632$	−10546.6	−0.663802
$633$	−14732.4	−0.925054
$634$	−14723.5	−0.922309
$635$	16253.2	1.01573
$636$	−1210.95	−0.0754987
$637$	4380.01	0.272437
$638$	−26358.2	−1.63563
$639$	2011.46	0.124526
$640$	8475.44	0.523471
$641$	6936.45	0.427416	0.213708	−	0.976898i	$-0.431446\pi$
$641$	6936.45	0.427416	0.213708	+	0.976898i	$0.431446\pi$
$642$	308.013	0.0189351
$643$	−30055.9	−1.84337	−0.921685	−	0.387938i	$-0.873187\pi$
$643$	−30055.9	−1.84337	−0.921685	+	0.387938i	$0.873187\pi$
$644$	189.824	0.0116151
$645$	2174.01	0.132716
$646$	3295.59	0.200717
$647$	−22193.5	−1.34856	−0.674280	−	0.738476i	$-0.735546\pi$
$647$	−22193.5	−1.34856	−0.674280	+	0.738476i	$0.735546\pi$
$648$	1939.16	0.117558
$649$	3703.32	0.223988
$650$	−12120.9	−0.731415
$651$	6371.76	0.383608
$652$	−3802.75	−0.228416
$653$	−4334.02	−0.259729	−0.129865	−	0.991532i	$-0.541454\pi$
$653$	−4334.02	−0.259729	−0.129865	+	0.991532i	$0.541454\pi$
$654$	−1651.14	−0.0987229
$655$	17794.8	1.06153
$656$	25041.3	1.49039
$657$	1105.20	0.0656284
$658$	−19352.0	−1.14653
$659$	10735.4	0.634584	0.317292	−	0.948328i	$-0.397226\pi$
$659$	10735.4	0.634584	0.317292	+	0.948328i	$0.397226\pi$
$660$	−1712.11	−0.100975
$661$	−31199.2	−1.83587	−0.917934	−	0.396734i	$-0.870144\pi$
$661$	−31199.2	−1.83587	−0.917934	+	0.396734i	$0.870144\pi$
$662$	20012.8	1.17495
$663$	−11729.7	−0.687097
$664$	−15857.2	−0.926773
$665$	−3254.65	−0.189789
$666$	5282.98	0.307375
$667$	−1580.42	−0.0917453
$668$	3017.11	0.174754
$669$	142.358	0.00822701
$670$	−6092.97	−0.351332
$671$	−43769.8	−2.51821
$672$	2593.69	0.148889
$673$	4517.81	0.258765	0.129383	−	0.991595i	$-0.458700\pi$
$673$	4517.81	0.258765	0.129383	+	0.991595i	$0.458700\pi$
$674$	−18695.9	−1.06845
$675$	1661.30	0.0947309
$676$	−3949.79	−0.224726
$677$	4980.54	0.282744	0.141372	−	0.989957i	$-0.454849\pi$
$677$	4980.54	0.282744	0.141372	+	0.989957i	$0.454849\pi$
$678$	−6988.92	−0.395882
$679$	−6714.81	−0.379515
$680$	9914.10	0.559101
$681$	−9293.14	−0.522928
$682$	20689.6	1.16165
$683$	−11878.0	−0.665446	−0.332723	−	0.943025i	$-0.607967\pi$
$683$	−11878.0	−0.665446	−0.332723	+	0.943025i	$0.607967\pi$
$684$	−248.656	−0.0139000
$685$	16935.1	0.944611
$686$	17732.2	0.986908
$687$	−6821.45	−0.378828
$688$	−4872.54	−0.270006
$689$	26604.1	1.47102
$690$	616.948	0.0340388
$691$	3069.47	0.168984	0.0844921	−	0.996424i	$-0.473073\pi$
$691$	3069.47	0.168984	0.0844921	+	0.996424i	$0.473073\pi$
$692$	1378.56	0.0757295
$693$	9532.99	0.522552
$694$	−8948.03	−0.489427
$695$	−21598.0	−1.17879
$696$	−11516.0	−0.627175
$697$	24299.4	1.32052
$698$	−23387.3	−1.26823
$699$	10301.0	0.557398
$700$	1184.99	0.0639835
$701$	−14816.8	−0.798321	−0.399161	−	0.916881i	$-0.630698\pi$
$701$	−14816.8	−0.798321	−0.399161	+	0.916881i	$0.630698\pi$
$702$	−5318.80	−0.285962
$703$	−5426.08	−0.291107
$704$	35320.6	1.89090
$705$	10465.6	0.559087
$706$	−25012.1	−1.33335
$707$	−16229.7	−0.863338
$708$	202.003	0.0107228
$709$	29464.2	1.56072	0.780361	−	0.625329i	$-0.215035\pi$
$709$	29464.2	1.56072	0.780361	+	0.625329i	$0.215035\pi$
$710$	4663.13	0.246485
$711$	−3964.87	−0.209134
$712$	24242.7	1.27603
$713$	1240.54	0.0651592
$714$	−6891.75	−0.361229
$715$	37614.3	1.96741
$716$	1574.50	0.0821812
$717$	−16467.9	−0.857746
$718$	20925.5	1.08765
$719$	16747.1	0.868654	0.434327	−	0.900755i	$-0.356986\pi$
$719$	16747.1	0.868654	0.434327	+	0.900755i	$0.356986\pi$
$720$	3840.87	0.198807
$721$	−18317.5	−0.946156
$722$	16428.3	0.846813
$723$	−214.259	−0.0110213
$724$	2537.92	0.130278
$725$	−9865.90	−0.505394
$726$	20497.1	1.04782
$727$	13485.0	0.687936	0.343968	−	0.938981i	$-0.388229\pi$
$727$	13485.0	0.687936	0.343968	+	0.938981i	$0.388229\pi$
$728$	−30388.1	−1.54706
$729$	729.000	0.0370370
$730$	2562.15	0.129904
$731$	−4728.18	−0.239231
$732$	−2387.49	−0.120552
$733$	−25581.8	−1.28906	−0.644532	−	0.764577i	$-0.722948\pi$
$733$	−25581.8	−1.28906	−0.644532	+	0.764577i	$0.722948\pi$
$734$	24239.3	1.21892
$735$	−1391.73	−0.0698431
$736$	504.973	0.0252902
$737$	−18329.9	−0.916135
$738$	11018.5	0.549587
$739$	−8290.39	−0.412675	−0.206338	−	0.978481i	$-0.566155\pi$
$739$	−8290.39	−0.412675	−0.206338	+	0.978481i	$0.566155\pi$
$740$	−2037.91	−0.101237
$741$	5462.87	0.270828
$742$	15631.1	0.773363
$743$	−29226.2	−1.44308	−0.721539	−	0.692374i	$-0.756565\pi$
$743$	−29226.2	−1.44308	−0.721539	+	0.692374i	$0.756565\pi$
$744$	9039.42	0.445432
$745$	−8295.02	−0.407928
$746$	14490.2	0.711156
$747$	−5961.28	−0.291984
$748$	3723.60	0.182016
$749$	661.565	0.0322738
$750$	11675.5	0.568440
$751$	4511.16	0.219194	0.109597	−	0.993976i	$-0.465044\pi$
$751$	4511.16	0.219194	0.109597	+	0.993976i	$0.465044\pi$
$752$	−23456.2	−1.13744
$753$	−21235.6	−1.02771
$754$	31586.6	1.52562
$755$	−11537.0	−0.556125
$756$	519.990	0.0250157
$757$	−14064.1	−0.675256	−0.337628	−	0.941280i	$-0.609625\pi$
$757$	−14064.1	−0.675256	−0.337628	+	0.941280i	$0.609625\pi$
$758$	−32357.9	−1.55052
$759$	1856.01	0.0887600
$760$	−4617.27	−0.220376
$761$	−31754.7	−1.51262	−0.756312	−	0.654211i	$-0.773001\pi$
$761$	−31754.7	−1.51262	−0.756312	+	0.654211i	$0.773001\pi$
$762$	−16028.6	−0.762016
$763$	−3546.40	−0.168268
$764$	1620.03	0.0767157
$765$	3727.07	0.176147
$766$	33559.7	1.58298
$767$	−4437.92	−0.208923
$768$	5146.81	0.241822
$769$	23548.0	1.10424	0.552121	−	0.833764i	$-0.313819\pi$
$769$	23548.0	1.10424	0.552121	+	0.833764i	$0.313819\pi$
$770$	22100.1	1.03433
$771$	15394.6	0.719096
$772$	−3193.04	−0.148860
$773$	−14079.0	−0.655092	−0.327546	−	0.944835i	$-0.606222\pi$
$773$	−14079.0	−0.655092	−0.327546	+	0.944835i	$0.606222\pi$
$774$	−2143.97	−0.0995652
$775$	7744.17	0.358940
$776$	−9526.09	−0.440679
$777$	11347.1	0.523904
$778$	4492.71	0.207033
$779$	−11316.9	−0.520501
$780$	2051.73	0.0941841
$781$	14028.4	0.642736
$782$	−1341.78	−0.0613579
$783$	−4329.29	−0.197594
$784$	3119.24	0.142093
$785$	19430.3	0.883438
$786$	−17549.0	−0.796375
$787$	−11840.6	−0.536305	−0.268152	−	0.963377i	$-0.586413\pi$
$787$	−11840.6	−0.536305	−0.268152	+	0.963377i	$0.586413\pi$
$788$	−4718.53	−0.213313
$789$	8698.92	0.392509
$790$	−9191.66	−0.413955
$791$	−15011.1	−0.674759
$792$	13524.2	0.606768
$793$	52452.1	2.34884
$794$	−2875.08	−0.128505
$795$	−8453.31	−0.377117
$796$	5762.70	0.256600
$797$	4057.07	0.180312	0.0901561	−	0.995928i	$-0.471263\pi$
$797$	4057.07	0.180312	0.0901561	+	0.995928i	$0.471263\pi$
$798$	3209.68	0.142383
$799$	−22761.2	−1.00780
$800$	3152.34	0.139315
$801$	9113.71	0.402019
$802$	10344.2	0.455446
$803$	7707.91	0.338738
$804$	−999.832	−0.0438574
$805$	1325.11	0.0580174
$806$	−24793.7	−1.08352
$807$	−7873.33	−0.343438
$808$	−23024.6	−1.00248
$809$	23041.8	1.00137	0.500684	−	0.865630i	$-0.333082\pi$
$809$	23041.8	1.00137	0.500684	+	0.865630i	$0.333082\pi$
$810$	1690.02	0.0733104
$811$	34179.5	1.47991	0.739953	−	0.672659i	$-0.234848\pi$
$811$	34179.5	1.47991	0.739953	+	0.672659i	$0.234848\pi$
$812$	−3088.05	−0.133460
$813$	−25511.0	−1.10051
$814$	36844.8	1.58650
$815$	−26546.0	−1.14094
$816$	−8353.37	−0.358366
$817$	2202.04	0.0942959
$818$	4476.86	0.191357
$819$	−11424.0	−0.487407
$820$	−4250.36	−0.181011
$821$	5743.98	0.244173	0.122087	−	0.992519i	$-0.461041\pi$
$821$	5743.98	0.244173	0.122087	+	0.992519i	$0.461041\pi$
$822$	−16701.1	−0.708661
$823$	28540.8	1.20883	0.604416	−	0.796669i	$-0.293406\pi$
$823$	28540.8	1.20883	0.604416	+	0.796669i	$0.293406\pi$
$824$	−25986.4	−1.09864
$825$	11586.3	0.488949
$826$	−2607.48	−0.109838
$827$	33133.5	1.39319	0.696594	−	0.717466i	$-0.254698\pi$
$827$	33133.5	1.39319	0.696594	+	0.717466i	$0.254698\pi$
$828$	101.239	0.00424913
$829$	24606.9	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$829$	24606.9	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$830$	−13819.9	−0.577947
$831$	10600.0	0.442490
$832$	−42326.9	−1.76373
$833$	3026.82	0.125898
$834$	21299.6	0.884346
$835$	21061.7	0.872897
$836$	−1734.18	−0.0717440
$837$	3398.25	0.140335
$838$	735.190	0.0303063
$839$	−9141.86	−0.376177	−0.188088	−	0.982152i	$-0.560229\pi$
$839$	−9141.86	−0.376177	−0.188088	+	0.982152i	$0.560229\pi$
$840$	9655.67	0.396610
$841$	1321.26	0.0541743
$842$	30490.0	1.24793
$843$	8303.29	0.339242
$844$	−5604.49	−0.228572
$845$	−27572.5	−1.12251
$846$	−10321.0	−0.419435
$847$	44024.6	1.78595
$848$	18946.1	0.767233
$849$	11560.0	0.467300
$850$	−8376.16	−0.338000
$851$	2209.19	0.0889897
$852$	765.200	0.0307692
$853$	−1471.84	−0.0590794	−0.0295397	−	0.999564i	$-0.509404\pi$
$853$	−1471.84	−0.0590794	−0.0295397	+	0.999564i	$0.509404\pi$
$854$	30818.0	1.23486
$855$	−1735.80	−0.0694305
$856$	938.542	0.0374751
$857$	35038.0	1.39659	0.698294	−	0.715811i	$-0.253943\pi$
$857$	35038.0	1.39659	0.698294	+	0.715811i	$0.253943\pi$
$858$	−37094.6	−1.47598
$859$	6739.91	0.267710	0.133855	−	0.991001i	$-0.457264\pi$
$859$	6739.91	0.267710	0.133855	+	0.991001i	$0.457264\pi$
$860$	827.036	0.0327927
$861$	23666.0	0.936741
$862$	4943.15	0.195318
$863$	265.457	0.0104708	0.00523538	−	0.999986i	$-0.498334\pi$
$863$	265.457	0.0104708	0.00523538	+	0.999986i	$0.498334\pi$
$864$	1383.29	0.0544681
$865$	9623.34	0.378270
$866$	6989.62	0.274269
$867$	6633.13	0.259830
$868$	2423.94	0.0947857
$869$	−27651.9	−1.07943
$870$	−10036.5	−0.391114
$871$	21965.9	0.854520
$872$	−5031.17	−0.195386
$873$	−3581.20	−0.138838
$874$	624.903	0.0241850
$875$	25077.3	0.968877
$876$	420.439	0.0162161
$877$	−46175.2	−1.77791	−0.888955	−	0.457995i	$-0.848568\pi$
$877$	−46175.2	−1.77791	−0.888955	+	0.457995i	$0.848568\pi$
$878$	−29828.5	−1.14654
$879$	−20160.7	−0.773611
$880$	26787.1	1.02613
$881$	−31385.6	−1.20024	−0.600118	−	0.799912i	$-0.704880\pi$
$881$	−31385.6	−1.20024	−0.600118	+	0.799912i	$0.704880\pi$
$882$	1372.50	0.0523973
$883$	16931.3	0.645281	0.322641	−	0.946522i	$-0.395429\pi$
$883$	16931.3	0.645281	0.322641	+	0.946522i	$0.395429\pi$
$884$	−4462.23	−0.169775
$885$	1410.13	0.0535604
$886$	−38316.8	−1.45291
$887$	34774.3	1.31635	0.658177	−	0.752863i	$-0.271328\pi$
$887$	34774.3	1.31635	0.658177	+	0.752863i	$0.271328\pi$
$888$	16097.7	0.608338
$889$	−34427.1	−1.29882
$890$	21128.1	0.795749
$891$	5084.22	0.191165
$892$	54.1557	0.00203281
$893$	10600.5	0.397237
$894$	8180.41	0.306034
$895$	10991.2	0.410496
$896$	−17952.4	−0.669363
$897$	−2224.17	−0.0827903
$898$	−28386.2	−1.05485
$899$	−20181.1	−0.748695
$900$	631.990	0.0234071
$901$	18384.8	0.679786
$902$	76845.4	2.83666
$903$	−4604.93	−0.169704
$904$	−21295.8	−0.783505
$905$	17716.6	0.650739
$906$	11377.6	0.417213
$907$	−35925.5	−1.31520	−0.657601	−	0.753367i	$-0.728429\pi$
$907$	−35925.5	−1.31520	−0.657601	+	0.753367i	$0.728429\pi$
$908$	−3535.29	−0.129210
$909$	−8655.77	−0.315835
$910$	−26484.0	−0.964764
$911$	48786.7	1.77429	0.887143	−	0.461495i	$-0.152687\pi$
$911$	48786.7	1.77429	0.887143	+	0.461495i	$0.152687\pi$
$912$	3890.39	0.141254
$913$	−41575.4	−1.50706
$914$	20345.2	0.736279
$915$	−16666.4	−0.602158
$916$	−2595.02	−0.0936045
$917$	−37692.5	−1.35738
$918$	−3675.57	−0.132148
$919$	−18322.3	−0.657669	−0.328834	−	0.944388i	$-0.606656\pi$
$919$	−18322.3	−0.657669	−0.328834	+	0.944388i	$0.606656\pi$
$920$	1879.89	0.0673677
$921$	9463.12	0.338567
$922$	−19019.9	−0.679378
$923$	−16811.2	−0.599508
$924$	3626.54	0.129117
$925$	13791.1	0.490214
$926$	−23402.3	−0.830504
$927$	−9769.24	−0.346132
$928$	−8214.90	−0.290590
$929$	30350.8	1.07188	0.535940	−	0.844256i	$-0.319957\pi$
$929$	30350.8	1.07188	0.535940	+	0.844256i	$0.319957\pi$
$930$	7878.08	0.277777
$931$	−1409.67	−0.0496243
$932$	3918.72	0.137727
$933$	−12613.7	−0.442609
$934$	−46933.7	−1.64424
$935$	25993.5	0.909174
$936$	−16206.9	−0.565959
$937$	52314.3	1.82394	0.911971	−	0.410256i	$-0.134560\pi$
$937$	52314.3	1.82394	0.911971	+	0.410256i	$0.134560\pi$
$938$	12906.0	0.449248
$939$	19492.6	0.677442
$940$	3981.31	0.138145
$941$	38821.1	1.34488	0.672440	−	0.740152i	$-0.265247\pi$
$941$	38821.1	1.34488	0.672440	+	0.740152i	$0.265247\pi$
$942$	−19161.9	−0.662768
$943$	4607.60	0.159114
$944$	−3160.48	−0.108967
$945$	3629.92	0.124954
$946$	−14952.6	−0.513901
$947$	4992.59	0.171317	0.0856586	−	0.996325i	$-0.472701\pi$
$947$	4992.59	0.171317	0.0856586	+	0.996325i	$0.472701\pi$
$948$	−1508.31	−0.0516748
$949$	−9236.88	−0.315955
$950$	3901.01	0.133227
$951$	−16865.9	−0.575094
$952$	−20999.8	−0.714923
$953$	−18808.2	−0.639304	−0.319652	−	0.947535i	$-0.603566\pi$
$953$	−18808.2	−0.639304	−0.319652	+	0.947535i	$0.603566\pi$
$954$	8336.52	0.282919
$955$	11309.0	0.383196
$956$	−6264.71	−0.211940
$957$	−30193.5	−1.01987
$958$	−4538.99	−0.153077
$959$	−35871.5	−1.20788
$960$	13449.2	0.452156
$961$	−13950.0	−0.468262
$962$	−44153.5	−1.47980
$963$	352.832	0.0118067
$964$	−81.5084	−0.00272324
$965$	−22289.8	−0.743559
$966$	−1306.80	−0.0435255
$967$	113.323	0.00376857	0.00188428	−	0.999998i	$-0.499400\pi$
$967$	113.323	0.00376857	0.00188428	+	0.999998i	$0.499400\pi$
$968$	62456.3	2.07378
$969$	3775.13	0.125154
$970$	−8302.23	−0.274813
$971$	18935.8	0.625827	0.312913	−	0.949782i	$-0.398695\pi$
$971$	18935.8	0.625827	0.312913	+	0.949782i	$0.398695\pi$
$972$	277.326	0.00915148
$973$	45748.3	1.50732
$974$	6801.78	0.223761
$975$	−13884.6	−0.456064
$976$	37353.9	1.22507
$977$	9255.46	0.303079	0.151540	−	0.988451i	$-0.451577\pi$
$977$	9255.46	0.303079	0.151540	+	0.988451i	$0.451577\pi$
$978$	26179.2	0.855951
$979$	63561.3	2.07500
$980$	−529.441	−0.0172575
$981$	−1891.40	−0.0615574
$982$	−19703.0	−0.640272
$983$	21559.8	0.699544	0.349772	−	0.936835i	$-0.386259\pi$
$983$	21559.8	0.699544	0.349772	+	0.936835i	$0.386259\pi$
$984$	33574.2	1.08771
$985$	−32938.9	−1.06550
$986$	21828.0	0.705016
$987$	−22167.9	−0.714905
$988$	2078.18	0.0669188
$989$	−896.548	−0.0288257
$990$	11786.6	0.378388
$991$	35161.9	1.12710	0.563550	−	0.826082i	$-0.309435\pi$
$991$	35161.9	1.12710	0.563550	+	0.826082i	$0.309435\pi$
$992$	6448.23	0.206383
$993$	22924.9	0.732627
$994$	−9877.31	−0.315180
$995$	40227.9	1.28172
$996$	−2267.79	−0.0721462
$997$	−48076.0	−1.52716	−0.763582	−	0.645711i	$-0.776561\pi$
$997$	−48076.0	−1.52716	−0.763582	+	0.645711i	$0.776561\pi$
$998$	−24599.0	−0.780227
$999$	6051.71	0.191659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.b.1.3	✓	7
3.2	odd	2		531.4.a.c.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.3	✓	7	1.1	even	1	trivial
531.4.a.c.1.5		7	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61892 q^{2} -3.00000 q^{3} -1.14126 q^{4} -7.96684 q^{5} +7.85676 q^{6} +16.8751 q^{7} +23.9402 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.61892 q^{2} -3.00000 q^{3} -1.14126 q^{4} -7.96684 q^{5} +7.85676 q^{6} +16.8751 q^{7} +23.9402 q^{8} +9.00000 q^{9} +20.8645 q^{10} +62.7682 q^{11} +3.42378 q^{12} -75.2191 q^{13} -44.1946 q^{14} +23.9005 q^{15} -53.5675 q^{16} -51.9804 q^{17} -23.5703 q^{18} +24.2087 q^{19} +9.09222 q^{20} -50.6254 q^{21} -164.385 q^{22} -9.85642 q^{23} -71.8207 q^{24} -61.5295 q^{25} +196.993 q^{26} -27.0000 q^{27} -19.2589 q^{28} +160.344 q^{29} -62.5935 q^{30} -125.861 q^{31} -51.2329 q^{32} -188.305 q^{33} +136.132 q^{34} -134.441 q^{35} -10.2713 q^{36} -224.138 q^{37} -63.4006 q^{38} +225.657 q^{39} -190.728 q^{40} -467.472 q^{41} +132.584 q^{42} +90.9608 q^{43} -71.6348 q^{44} -71.7015 q^{45} +25.8132 q^{46} +437.881 q^{47} +160.702 q^{48} -58.2301 q^{49} +161.141 q^{50} +155.941 q^{51} +85.8445 q^{52} -353.688 q^{53} +70.7108 q^{54} -500.064 q^{55} +403.994 q^{56} -72.6261 q^{57} -419.929 q^{58} +59.0000 q^{59} -27.2767 q^{60} -697.325 q^{61} +329.620 q^{62} +151.876 q^{63} +562.715 q^{64} +599.258 q^{65} +493.154 q^{66} -292.026 q^{67} +59.3231 q^{68} +29.5693 q^{69} +352.091 q^{70} +223.496 q^{71} +215.462 q^{72} +122.800 q^{73} +586.998 q^{74} +184.589 q^{75} -27.6284 q^{76} +1059.22 q^{77} -590.978 q^{78} -440.541 q^{79} +426.763 q^{80} +81.0000 q^{81} +1224.27 q^{82} -662.365 q^{83} +57.7767 q^{84} +414.119 q^{85} -238.219 q^{86} -481.033 q^{87} +1502.68 q^{88} +1012.63 q^{89} +187.781 q^{90} -1269.33 q^{91} +11.2487 q^{92} +377.583 q^{93} -1146.77 q^{94} -192.867 q^{95} +153.699 q^{96} -397.912 q^{97} +152.500 q^{98} +564.914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} + O(q^{10}) \)	\( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} - 71q^{10} - 5q^{11} - 78q^{12} - 67q^{13} - 65q^{14} + 6q^{15} - 94q^{16} - 23q^{17} - 176q^{19} - 207q^{20} + 177q^{21} - 704q^{22} - 218q^{23} + 63q^{24} - 183q^{25} + 58q^{26} - 189q^{27} - 938q^{28} + 168q^{29} + 213q^{30} - 604q^{31} - 448q^{32} + 15q^{33} - 610q^{34} - 336q^{35} + 234q^{36} - 505q^{37} - 453q^{38} + 201q^{39} - 1080q^{40} - 265q^{41} + 195q^{42} - 493q^{43} + 504q^{44} - 18q^{45} + 381q^{46} - 244q^{47} + 282q^{48} + 770q^{49} + 1639q^{50} + 69q^{51} + 160q^{52} + 686q^{53} - 116q^{55} + 2190q^{56} + 528q^{57} + 1584q^{58} + 413q^{59} + 621q^{60} - 838q^{61} + 286q^{62} - 531q^{63} + 205q^{64} + 490q^{65} + 2112q^{66} - 1504q^{67} + 3047q^{68} + 654q^{69} + 1530q^{70} - 1267q^{71} - 189q^{72} - 666q^{73} + 528q^{74} + 549q^{75} - 64q^{76} + 1109q^{77} - 174q^{78} - 2741q^{79} + 1213q^{80} + 567q^{81} + 953q^{82} - 2025q^{83} + 2814q^{84} - 1274q^{85} + 4394q^{86} - 504q^{87} - 1639q^{88} + 616q^{89} - 639q^{90} - 2415q^{91} + 218q^{92} + 1812q^{93} + 900q^{94} + 2554q^{95} + 1344q^{96} - 1298q^{97} - 172q^{98} - 45q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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