# Properties

 Label 3381.2.a.bi.1.5 Level $3381$ Weight $2$ Character 3381.1 Self dual yes Analytic conductor $26.997$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3381 = 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9974209234$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-0.0262565$$ of defining polynomial Character $$\chi$$ $$=$$ 3381.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.0262565 q^{2} -1.00000 q^{3} -1.99931 q^{4} +2.77828 q^{5} +0.0262565 q^{6} +0.105008 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.0262565 q^{2} -1.00000 q^{3} -1.99931 q^{4} +2.77828 q^{5} +0.0262565 q^{6} +0.105008 q^{8} +1.00000 q^{9} -0.0729480 q^{10} +0.660156 q^{11} +1.99931 q^{12} +4.12488 q^{13} -2.77828 q^{15} +3.99586 q^{16} +4.12381 q^{17} -0.0262565 q^{18} +4.76829 q^{19} -5.55465 q^{20} -0.0173334 q^{22} +1.00000 q^{23} -0.105008 q^{24} +2.71885 q^{25} -0.108305 q^{26} -1.00000 q^{27} -1.78025 q^{29} +0.0729480 q^{30} -3.62429 q^{31} -0.314933 q^{32} -0.660156 q^{33} -0.108277 q^{34} -1.99931 q^{36} -8.06211 q^{37} -0.125199 q^{38} -4.12488 q^{39} +0.291742 q^{40} -7.99969 q^{41} +12.5842 q^{43} -1.31986 q^{44} +2.77828 q^{45} -0.0262565 q^{46} +11.7622 q^{47} -3.99586 q^{48} -0.0713876 q^{50} -4.12381 q^{51} -8.24691 q^{52} -0.0608257 q^{53} +0.0262565 q^{54} +1.83410 q^{55} -4.76829 q^{57} +0.0467430 q^{58} -8.98717 q^{59} +5.55465 q^{60} +2.71267 q^{61} +0.0951611 q^{62} -7.98346 q^{64} +11.4601 q^{65} +0.0173334 q^{66} -2.68583 q^{67} -8.24477 q^{68} -1.00000 q^{69} +12.4462 q^{71} +0.105008 q^{72} -14.2871 q^{73} +0.211683 q^{74} -2.71885 q^{75} -9.53329 q^{76} +0.108305 q^{78} -6.94797 q^{79} +11.1016 q^{80} +1.00000 q^{81} +0.210044 q^{82} +9.40856 q^{83} +11.4571 q^{85} -0.330417 q^{86} +1.78025 q^{87} +0.0693216 q^{88} -5.92430 q^{89} -0.0729480 q^{90} -1.99931 q^{92} +3.62429 q^{93} -0.308834 q^{94} +13.2477 q^{95} +0.314933 q^{96} +2.53740 q^{97} +0.660156 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10})$$ $$10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100})$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.0262565 −0.0185661 −0.00928307 0.999957i $$-0.502955\pi$$
−0.00928307 + 0.999957i $$0.502955\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.99931 −0.999655
$$5$$ 2.77828 1.24249 0.621243 0.783618i $$-0.286628\pi$$
0.621243 + 0.783618i $$0.286628\pi$$
$$6$$ 0.0262565 0.0107192
$$7$$ 0 0
$$8$$ 0.105008 0.0371259
$$9$$ 1.00000 0.333333
$$10$$ −0.0729480 −0.0230682
$$11$$ 0.660156 0.199045 0.0995223 0.995035i $$-0.468269\pi$$
0.0995223 + 0.995035i $$0.468269\pi$$
$$12$$ 1.99931 0.577151
$$13$$ 4.12488 1.14404 0.572018 0.820241i $$-0.306161\pi$$
0.572018 + 0.820241i $$0.306161\pi$$
$$14$$ 0 0
$$15$$ −2.77828 −0.717349
$$16$$ 3.99586 0.998966
$$17$$ 4.12381 1.00017 0.500085 0.865976i $$-0.333302\pi$$
0.500085 + 0.865976i $$0.333302\pi$$
$$18$$ −0.0262565 −0.00618871
$$19$$ 4.76829 1.09392 0.546960 0.837159i $$-0.315785\pi$$
0.546960 + 0.837159i $$0.315785\pi$$
$$20$$ −5.55465 −1.24206
$$21$$ 0 0
$$22$$ −0.0173334 −0.00369549
$$23$$ 1.00000 0.208514
$$24$$ −0.105008 −0.0214346
$$25$$ 2.71885 0.543771
$$26$$ −0.108305 −0.0212403
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −1.78025 −0.330583 −0.165292 0.986245i $$-0.552857\pi$$
−0.165292 + 0.986245i $$0.552857\pi$$
$$30$$ 0.0729480 0.0133184
$$31$$ −3.62429 −0.650941 −0.325471 0.945552i $$-0.605523\pi$$
−0.325471 + 0.945552i $$0.605523\pi$$
$$32$$ −0.314933 −0.0556728
$$33$$ −0.660156 −0.114918
$$34$$ −0.108277 −0.0185693
$$35$$ 0 0
$$36$$ −1.99931 −0.333218
$$37$$ −8.06211 −1.32540 −0.662701 0.748884i $$-0.730590\pi$$
−0.662701 + 0.748884i $$0.730590\pi$$
$$38$$ −0.125199 −0.0203099
$$39$$ −4.12488 −0.660509
$$40$$ 0.291742 0.0461284
$$41$$ −7.99969 −1.24934 −0.624671 0.780888i $$-0.714767\pi$$
−0.624671 + 0.780888i $$0.714767\pi$$
$$42$$ 0 0
$$43$$ 12.5842 1.91907 0.959536 0.281585i $$-0.0908601\pi$$
0.959536 + 0.281585i $$0.0908601\pi$$
$$44$$ −1.31986 −0.198976
$$45$$ 2.77828 0.414162
$$46$$ −0.0262565 −0.00387131
$$47$$ 11.7622 1.71569 0.857846 0.513908i $$-0.171803\pi$$
0.857846 + 0.513908i $$0.171803\pi$$
$$48$$ −3.99586 −0.576753
$$49$$ 0 0
$$50$$ −0.0713876 −0.0100957
$$51$$ −4.12381 −0.577448
$$52$$ −8.24691 −1.14364
$$53$$ −0.0608257 −0.00835505 −0.00417752 0.999991i $$-0.501330\pi$$
−0.00417752 + 0.999991i $$0.501330\pi$$
$$54$$ 0.0262565 0.00357306
$$55$$ 1.83410 0.247310
$$56$$ 0 0
$$57$$ −4.76829 −0.631575
$$58$$ 0.0467430 0.00613766
$$59$$ −8.98717 −1.17003 −0.585015 0.811022i $$-0.698911\pi$$
−0.585015 + 0.811022i $$0.698911\pi$$
$$60$$ 5.55465 0.717102
$$61$$ 2.71267 0.347322 0.173661 0.984806i $$-0.444440\pi$$
0.173661 + 0.984806i $$0.444440\pi$$
$$62$$ 0.0951611 0.0120855
$$63$$ 0 0
$$64$$ −7.98346 −0.997932
$$65$$ 11.4601 1.42145
$$66$$ 0.0173334 0.00213359
$$67$$ −2.68583 −0.328126 −0.164063 0.986450i $$-0.552460\pi$$
−0.164063 + 0.986450i $$0.552460\pi$$
$$68$$ −8.24477 −0.999825
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 12.4462 1.47709 0.738545 0.674205i $$-0.235513\pi$$
0.738545 + 0.674205i $$0.235513\pi$$
$$72$$ 0.105008 0.0123753
$$73$$ −14.2871 −1.67218 −0.836088 0.548596i $$-0.815163\pi$$
−0.836088 + 0.548596i $$0.815163\pi$$
$$74$$ 0.211683 0.0246076
$$75$$ −2.71885 −0.313946
$$76$$ −9.53329 −1.09354
$$77$$ 0 0
$$78$$ 0.108305 0.0122631
$$79$$ −6.94797 −0.781708 −0.390854 0.920453i $$-0.627820\pi$$
−0.390854 + 0.920453i $$0.627820\pi$$
$$80$$ 11.1016 1.24120
$$81$$ 1.00000 0.111111
$$82$$ 0.210044 0.0231955
$$83$$ 9.40856 1.03272 0.516362 0.856371i $$-0.327286\pi$$
0.516362 + 0.856371i $$0.327286\pi$$
$$84$$ 0 0
$$85$$ 11.4571 1.24270
$$86$$ −0.330417 −0.0356298
$$87$$ 1.78025 0.190862
$$88$$ 0.0693216 0.00738971
$$89$$ −5.92430 −0.627975 −0.313987 0.949427i $$-0.601665\pi$$
−0.313987 + 0.949427i $$0.601665\pi$$
$$90$$ −0.0729480 −0.00768939
$$91$$ 0 0
$$92$$ −1.99931 −0.208443
$$93$$ 3.62429 0.375821
$$94$$ −0.308834 −0.0318538
$$95$$ 13.2477 1.35918
$$96$$ 0.314933 0.0321427
$$97$$ 2.53740 0.257634 0.128817 0.991668i $$-0.458882\pi$$
0.128817 + 0.991668i $$0.458882\pi$$
$$98$$ 0 0
$$99$$ 0.660156 0.0663482
$$100$$ −5.43583 −0.543583
$$101$$ 6.71178 0.667847 0.333923 0.942600i $$-0.391627\pi$$
0.333923 + 0.942600i $$0.391627\pi$$
$$102$$ 0.108277 0.0107210
$$103$$ 10.7478 1.05901 0.529507 0.848306i $$-0.322377\pi$$
0.529507 + 0.848306i $$0.322377\pi$$
$$104$$ 0.433145 0.0424733
$$105$$ 0 0
$$106$$ 0.00159707 0.000155121 0
$$107$$ −0.484556 −0.0468438 −0.0234219 0.999726i $$-0.507456\pi$$
−0.0234219 + 0.999726i $$0.507456\pi$$
$$108$$ 1.99931 0.192384
$$109$$ 12.4652 1.19395 0.596975 0.802260i $$-0.296369\pi$$
0.596975 + 0.802260i $$0.296369\pi$$
$$110$$ −0.0481570 −0.00459159
$$111$$ 8.06211 0.765221
$$112$$ 0 0
$$113$$ 9.28875 0.873812 0.436906 0.899507i $$-0.356074\pi$$
0.436906 + 0.899507i $$0.356074\pi$$
$$114$$ 0.125199 0.0117259
$$115$$ 2.77828 0.259076
$$116$$ 3.55926 0.330469
$$117$$ 4.12488 0.381345
$$118$$ 0.235972 0.0217229
$$119$$ 0 0
$$120$$ −0.291742 −0.0266322
$$121$$ −10.5642 −0.960381
$$122$$ −0.0712252 −0.00644842
$$123$$ 7.99969 0.721308
$$124$$ 7.24608 0.650717
$$125$$ −6.33767 −0.566858
$$126$$ 0 0
$$127$$ 2.40074 0.213031 0.106515 0.994311i $$-0.466031\pi$$
0.106515 + 0.994311i $$0.466031\pi$$
$$128$$ 0.839484 0.0742006
$$129$$ −12.5842 −1.10798
$$130$$ −0.300901 −0.0263908
$$131$$ −20.1505 −1.76056 −0.880278 0.474458i $$-0.842644\pi$$
−0.880278 + 0.474458i $$0.842644\pi$$
$$132$$ 1.31986 0.114879
$$133$$ 0 0
$$134$$ 0.0705204 0.00609204
$$135$$ −2.77828 −0.239116
$$136$$ 0.433032 0.0371322
$$137$$ 7.00658 0.598613 0.299306 0.954157i $$-0.403245\pi$$
0.299306 + 0.954157i $$0.403245\pi$$
$$138$$ 0.0262565 0.00223510
$$139$$ −2.88510 −0.244711 −0.122356 0.992486i $$-0.539045\pi$$
−0.122356 + 0.992486i $$0.539045\pi$$
$$140$$ 0 0
$$141$$ −11.7622 −0.990555
$$142$$ −0.326793 −0.0274238
$$143$$ 2.72306 0.227714
$$144$$ 3.99586 0.332989
$$145$$ −4.94603 −0.410745
$$146$$ 0.375128 0.0310458
$$147$$ 0 0
$$148$$ 16.1187 1.32495
$$149$$ 19.0608 1.56152 0.780762 0.624829i $$-0.214831\pi$$
0.780762 + 0.624829i $$0.214831\pi$$
$$150$$ 0.0713876 0.00582877
$$151$$ −5.06290 −0.412013 −0.206006 0.978551i $$-0.566047\pi$$
−0.206006 + 0.978551i $$0.566047\pi$$
$$152$$ 0.500708 0.0406128
$$153$$ 4.12381 0.333390
$$154$$ 0 0
$$155$$ −10.0693 −0.808785
$$156$$ 8.24691 0.660281
$$157$$ −2.74388 −0.218985 −0.109493 0.993988i $$-0.534923\pi$$
−0.109493 + 0.993988i $$0.534923\pi$$
$$158$$ 0.182429 0.0145133
$$159$$ 0.0608257 0.00482379
$$160$$ −0.874973 −0.0691727
$$161$$ 0 0
$$162$$ −0.0262565 −0.00206290
$$163$$ 15.3088 1.19908 0.599539 0.800346i $$-0.295351\pi$$
0.599539 + 0.800346i $$0.295351\pi$$
$$164$$ 15.9939 1.24891
$$165$$ −1.83410 −0.142785
$$166$$ −0.247036 −0.0191737
$$167$$ 13.0016 1.00610 0.503049 0.864258i $$-0.332212\pi$$
0.503049 + 0.864258i $$0.332212\pi$$
$$168$$ 0 0
$$169$$ 4.01461 0.308816
$$170$$ −0.300823 −0.0230721
$$171$$ 4.76829 0.364640
$$172$$ −25.1597 −1.91841
$$173$$ −10.4220 −0.792372 −0.396186 0.918170i $$-0.629666\pi$$
−0.396186 + 0.918170i $$0.629666\pi$$
$$174$$ −0.0467430 −0.00354358
$$175$$ 0 0
$$176$$ 2.63789 0.198839
$$177$$ 8.98717 0.675517
$$178$$ 0.155551 0.0116591
$$179$$ 19.2231 1.43680 0.718402 0.695628i $$-0.244874\pi$$
0.718402 + 0.695628i $$0.244874\pi$$
$$180$$ −5.55465 −0.414019
$$181$$ −11.1552 −0.829162 −0.414581 0.910012i $$-0.636072\pi$$
−0.414581 + 0.910012i $$0.636072\pi$$
$$182$$ 0 0
$$183$$ −2.71267 −0.200526
$$184$$ 0.105008 0.00774128
$$185$$ −22.3988 −1.64679
$$186$$ −0.0951611 −0.00697755
$$187$$ 2.72236 0.199078
$$188$$ −23.5163 −1.71510
$$189$$ 0 0
$$190$$ −0.347837 −0.0252347
$$191$$ 19.2346 1.39177 0.695884 0.718154i $$-0.255013\pi$$
0.695884 + 0.718154i $$0.255013\pi$$
$$192$$ 7.98346 0.576157
$$193$$ 17.4527 1.25627 0.628136 0.778104i $$-0.283818\pi$$
0.628136 + 0.778104i $$0.283818\pi$$
$$194$$ −0.0666232 −0.00478326
$$195$$ −11.4601 −0.820673
$$196$$ 0 0
$$197$$ −1.16239 −0.0828168 −0.0414084 0.999142i $$-0.513184\pi$$
−0.0414084 + 0.999142i $$0.513184\pi$$
$$198$$ −0.0173334 −0.00123183
$$199$$ 16.9768 1.20346 0.601728 0.798701i $$-0.294479\pi$$
0.601728 + 0.798701i $$0.294479\pi$$
$$200$$ 0.285501 0.0201880
$$201$$ 2.68583 0.189444
$$202$$ −0.176228 −0.0123993
$$203$$ 0 0
$$204$$ 8.24477 0.577249
$$205$$ −22.2254 −1.55229
$$206$$ −0.282200 −0.0196618
$$207$$ 1.00000 0.0695048
$$208$$ 16.4824 1.14285
$$209$$ 3.14782 0.217739
$$210$$ 0 0
$$211$$ 11.6924 0.804938 0.402469 0.915434i $$-0.368152\pi$$
0.402469 + 0.915434i $$0.368152\pi$$
$$212$$ 0.121609 0.00835217
$$213$$ −12.4462 −0.852798
$$214$$ 0.0127227 0.000869708 0
$$215$$ 34.9625 2.38442
$$216$$ −0.105008 −0.00714488
$$217$$ 0 0
$$218$$ −0.327293 −0.0221671
$$219$$ 14.2871 0.965431
$$220$$ −3.66694 −0.247225
$$221$$ 17.0102 1.14423
$$222$$ −0.211683 −0.0142072
$$223$$ −15.5087 −1.03854 −0.519270 0.854610i $$-0.673796\pi$$
−0.519270 + 0.854610i $$0.673796\pi$$
$$224$$ 0 0
$$225$$ 2.71885 0.181257
$$226$$ −0.243890 −0.0162233
$$227$$ −15.3330 −1.01768 −0.508842 0.860860i $$-0.669926\pi$$
−0.508842 + 0.860860i $$0.669926\pi$$
$$228$$ 9.53329 0.631358
$$229$$ −6.40484 −0.423243 −0.211622 0.977352i $$-0.567874\pi$$
−0.211622 + 0.977352i $$0.567874\pi$$
$$230$$ −0.0729480 −0.00481005
$$231$$ 0 0
$$232$$ −0.186940 −0.0122732
$$233$$ −26.1430 −1.71269 −0.856344 0.516406i $$-0.827270\pi$$
−0.856344 + 0.516406i $$0.827270\pi$$
$$234$$ −0.108305 −0.00708011
$$235$$ 32.6787 2.13172
$$236$$ 17.9681 1.16963
$$237$$ 6.94797 0.451319
$$238$$ 0 0
$$239$$ 17.6773 1.14345 0.571724 0.820446i $$-0.306275\pi$$
0.571724 + 0.820446i $$0.306275\pi$$
$$240$$ −11.1016 −0.716608
$$241$$ 5.75969 0.371014 0.185507 0.982643i $$-0.440607\pi$$
0.185507 + 0.982643i $$0.440607\pi$$
$$242$$ 0.277379 0.0178306
$$243$$ −1.00000 −0.0641500
$$244$$ −5.42347 −0.347202
$$245$$ 0 0
$$246$$ −0.210044 −0.0133919
$$247$$ 19.6686 1.25148
$$248$$ −0.380579 −0.0241668
$$249$$ −9.40856 −0.596243
$$250$$ 0.166405 0.0105244
$$251$$ −10.3866 −0.655597 −0.327798 0.944748i $$-0.606307\pi$$
−0.327798 + 0.944748i $$0.606307\pi$$
$$252$$ 0 0
$$253$$ 0.660156 0.0415037
$$254$$ −0.0630349 −0.00395516
$$255$$ −11.4571 −0.717471
$$256$$ 15.9449 0.996555
$$257$$ −2.69848 −0.168327 −0.0841633 0.996452i $$-0.526822\pi$$
−0.0841633 + 0.996452i $$0.526822\pi$$
$$258$$ 0.330417 0.0205709
$$259$$ 0 0
$$260$$ −22.9122 −1.42096
$$261$$ −1.78025 −0.110194
$$262$$ 0.529081 0.0326867
$$263$$ −21.1490 −1.30411 −0.652053 0.758174i $$-0.726092\pi$$
−0.652053 + 0.758174i $$0.726092\pi$$
$$264$$ −0.0693216 −0.00426645
$$265$$ −0.168991 −0.0103810
$$266$$ 0 0
$$267$$ 5.92430 0.362561
$$268$$ 5.36981 0.328013
$$269$$ −28.5411 −1.74018 −0.870090 0.492893i $$-0.835939\pi$$
−0.870090 + 0.492893i $$0.835939\pi$$
$$270$$ 0.0729480 0.00443947
$$271$$ 15.1472 0.920130 0.460065 0.887885i $$-0.347826\pi$$
0.460065 + 0.887885i $$0.347826\pi$$
$$272$$ 16.4782 0.999136
$$273$$ 0 0
$$274$$ −0.183968 −0.0111139
$$275$$ 1.79487 0.108235
$$276$$ 1.99931 0.120344
$$277$$ 4.23717 0.254587 0.127294 0.991865i $$-0.459371\pi$$
0.127294 + 0.991865i $$0.459371\pi$$
$$278$$ 0.0757527 0.00454335
$$279$$ −3.62429 −0.216980
$$280$$ 0 0
$$281$$ 0.762123 0.0454645 0.0227322 0.999742i $$-0.492763\pi$$
0.0227322 + 0.999742i $$0.492763\pi$$
$$282$$ 0.308834 0.0183908
$$283$$ 17.1714 1.02073 0.510367 0.859957i $$-0.329510\pi$$
0.510367 + 0.859957i $$0.329510\pi$$
$$284$$ −24.8838 −1.47658
$$285$$ −13.2477 −0.784723
$$286$$ −0.0714981 −0.00422777
$$287$$ 0 0
$$288$$ −0.314933 −0.0185576
$$289$$ 0.00578278 0.000340164 0
$$290$$ 0.129865 0.00762595
$$291$$ −2.53740 −0.148745
$$292$$ 28.5643 1.67160
$$293$$ −1.15566 −0.0675144 −0.0337572 0.999430i $$-0.510747\pi$$
−0.0337572 + 0.999430i $$0.510747\pi$$
$$294$$ 0 0
$$295$$ −24.9689 −1.45375
$$296$$ −0.846585 −0.0492067
$$297$$ −0.660156 −0.0383061
$$298$$ −0.500470 −0.0289915
$$299$$ 4.12488 0.238548
$$300$$ 5.43583 0.313838
$$301$$ 0 0
$$302$$ 0.132934 0.00764949
$$303$$ −6.71178 −0.385581
$$304$$ 19.0534 1.09279
$$305$$ 7.53656 0.431542
$$306$$ −0.108277 −0.00618977
$$307$$ 31.0365 1.77135 0.885673 0.464310i $$-0.153698\pi$$
0.885673 + 0.464310i $$0.153698\pi$$
$$308$$ 0 0
$$309$$ −10.7478 −0.611422
$$310$$ 0.264384 0.0150160
$$311$$ 17.6141 0.998801 0.499401 0.866371i $$-0.333554\pi$$
0.499401 + 0.866371i $$0.333554\pi$$
$$312$$ −0.433145 −0.0245220
$$313$$ 15.7406 0.889713 0.444856 0.895602i $$-0.353255\pi$$
0.444856 + 0.895602i $$0.353255\pi$$
$$314$$ 0.0720446 0.00406571
$$315$$ 0 0
$$316$$ 13.8912 0.781438
$$317$$ −5.22139 −0.293263 −0.146631 0.989191i $$-0.546843\pi$$
−0.146631 + 0.989191i $$0.546843\pi$$
$$318$$ −0.00159707 −8.95592e−5 0
$$319$$ −1.17524 −0.0658008
$$320$$ −22.1803 −1.23992
$$321$$ 0.484556 0.0270453
$$322$$ 0 0
$$323$$ 19.6635 1.09411
$$324$$ −1.99931 −0.111073
$$325$$ 11.2149 0.622093
$$326$$ −0.401955 −0.0222623
$$327$$ −12.4652 −0.689328
$$328$$ −0.840030 −0.0463829
$$329$$ 0 0
$$330$$ 0.0481570 0.00265096
$$331$$ −13.9449 −0.766483 −0.383241 0.923648i $$-0.625192\pi$$
−0.383241 + 0.923648i $$0.625192\pi$$
$$332$$ −18.8106 −1.03237
$$333$$ −8.06211 −0.441801
$$334$$ −0.341378 −0.0186793
$$335$$ −7.46199 −0.407692
$$336$$ 0 0
$$337$$ 9.28645 0.505865 0.252933 0.967484i $$-0.418605\pi$$
0.252933 + 0.967484i $$0.418605\pi$$
$$338$$ −0.105410 −0.00573353
$$339$$ −9.28875 −0.504496
$$340$$ −22.9063 −1.24227
$$341$$ −2.39260 −0.129566
$$342$$ −0.125199 −0.00676996
$$343$$ 0 0
$$344$$ 1.32144 0.0712473
$$345$$ −2.77828 −0.149578
$$346$$ 0.273646 0.0147113
$$347$$ 7.67494 0.412013 0.206006 0.978551i $$-0.433953\pi$$
0.206006 + 0.978551i $$0.433953\pi$$
$$348$$ −3.55926 −0.190797
$$349$$ 5.71924 0.306144 0.153072 0.988215i $$-0.451083\pi$$
0.153072 + 0.988215i $$0.451083\pi$$
$$350$$ 0 0
$$351$$ −4.12488 −0.220170
$$352$$ −0.207905 −0.0110814
$$353$$ 13.4196 0.714253 0.357127 0.934056i $$-0.383757\pi$$
0.357127 + 0.934056i $$0.383757\pi$$
$$354$$ −0.235972 −0.0125417
$$355$$ 34.5790 1.83526
$$356$$ 11.8445 0.627758
$$357$$ 0 0
$$358$$ −0.504732 −0.0266759
$$359$$ 17.8658 0.942922 0.471461 0.881887i $$-0.343727\pi$$
0.471461 + 0.881887i $$0.343727\pi$$
$$360$$ 0.291742 0.0153761
$$361$$ 3.73658 0.196662
$$362$$ 0.292897 0.0153943
$$363$$ 10.5642 0.554476
$$364$$ 0 0
$$365$$ −39.6935 −2.07765
$$366$$ 0.0712252 0.00372300
$$367$$ 26.1522 1.36513 0.682566 0.730824i $$-0.260864\pi$$
0.682566 + 0.730824i $$0.260864\pi$$
$$368$$ 3.99586 0.208299
$$369$$ −7.99969 −0.416447
$$370$$ 0.588114 0.0305746
$$371$$ 0 0
$$372$$ −7.24608 −0.375692
$$373$$ 20.1300 1.04229 0.521147 0.853467i $$-0.325504\pi$$
0.521147 + 0.853467i $$0.325504\pi$$
$$374$$ −0.0714795 −0.00369612
$$375$$ 6.33767 0.327276
$$376$$ 1.23512 0.0636966
$$377$$ −7.34330 −0.378199
$$378$$ 0 0
$$379$$ 8.04896 0.413447 0.206724 0.978399i $$-0.433720\pi$$
0.206724 + 0.978399i $$0.433720\pi$$
$$380$$ −26.4862 −1.35871
$$381$$ −2.40074 −0.122993
$$382$$ −0.505034 −0.0258398
$$383$$ 20.0694 1.02550 0.512749 0.858539i $$-0.328627\pi$$
0.512749 + 0.858539i $$0.328627\pi$$
$$384$$ −0.839484 −0.0428397
$$385$$ 0 0
$$386$$ −0.458246 −0.0233241
$$387$$ 12.5842 0.639691
$$388$$ −5.07305 −0.257545
$$389$$ 12.4516 0.631322 0.315661 0.948872i $$-0.397774\pi$$
0.315661 + 0.948872i $$0.397774\pi$$
$$390$$ 0.300901 0.0152367
$$391$$ 4.12381 0.208550
$$392$$ 0 0
$$393$$ 20.1505 1.01646
$$394$$ 0.0305203 0.00153759
$$395$$ −19.3034 −0.971261
$$396$$ −1.31986 −0.0663253
$$397$$ −6.93528 −0.348072 −0.174036 0.984739i $$-0.555681\pi$$
−0.174036 + 0.984739i $$0.555681\pi$$
$$398$$ −0.445752 −0.0223435
$$399$$ 0 0
$$400$$ 10.8642 0.543208
$$401$$ −14.4161 −0.719908 −0.359954 0.932970i $$-0.617208\pi$$
−0.359954 + 0.932970i $$0.617208\pi$$
$$402$$ −0.0705204 −0.00351724
$$403$$ −14.9497 −0.744700
$$404$$ −13.4189 −0.667617
$$405$$ 2.77828 0.138054
$$406$$ 0 0
$$407$$ −5.32225 −0.263814
$$408$$ −0.433032 −0.0214383
$$409$$ 24.0265 1.18803 0.594016 0.804453i $$-0.297541\pi$$
0.594016 + 0.804453i $$0.297541\pi$$
$$410$$ 0.583561 0.0288200
$$411$$ −7.00658 −0.345609
$$412$$ −21.4882 −1.05865
$$413$$ 0 0
$$414$$ −0.0262565 −0.00129044
$$415$$ 26.1396 1.28314
$$416$$ −1.29906 −0.0636917
$$417$$ 2.88510 0.141284
$$418$$ −0.0826506 −0.00404257
$$419$$ 31.8946 1.55815 0.779076 0.626930i $$-0.215689\pi$$
0.779076 + 0.626930i $$0.215689\pi$$
$$420$$ 0 0
$$421$$ −18.8400 −0.918205 −0.459102 0.888383i $$-0.651829\pi$$
−0.459102 + 0.888383i $$0.651829\pi$$
$$422$$ −0.307002 −0.0149446
$$423$$ 11.7622 0.571897
$$424$$ −0.00638717 −0.000310189 0
$$425$$ 11.2120 0.543863
$$426$$ 0.326793 0.0158332
$$427$$ 0 0
$$428$$ 0.968777 0.0468276
$$429$$ −2.72306 −0.131471
$$430$$ −0.917992 −0.0442695
$$431$$ −36.3431 −1.75058 −0.875292 0.483595i $$-0.839331\pi$$
−0.875292 + 0.483595i $$0.839331\pi$$
$$432$$ −3.99586 −0.192251
$$433$$ 21.5486 1.03556 0.517780 0.855514i $$-0.326759\pi$$
0.517780 + 0.855514i $$0.326759\pi$$
$$434$$ 0 0
$$435$$ 4.94603 0.237144
$$436$$ −24.9218 −1.19354
$$437$$ 4.76829 0.228098
$$438$$ −0.375128 −0.0179243
$$439$$ −10.7375 −0.512473 −0.256237 0.966614i $$-0.582483\pi$$
−0.256237 + 0.966614i $$0.582483\pi$$
$$440$$ 0.192595 0.00918160
$$441$$ 0 0
$$442$$ −0.446628 −0.0212439
$$443$$ −33.9531 −1.61316 −0.806580 0.591125i $$-0.798684\pi$$
−0.806580 + 0.591125i $$0.798684\pi$$
$$444$$ −16.1187 −0.764957
$$445$$ −16.4594 −0.780250
$$446$$ 0.407204 0.0192817
$$447$$ −19.0608 −0.901546
$$448$$ 0 0
$$449$$ 6.94219 0.327622 0.163811 0.986492i $$-0.447621\pi$$
0.163811 + 0.986492i $$0.447621\pi$$
$$450$$ −0.0713876 −0.00336524
$$451$$ −5.28104 −0.248675
$$452$$ −18.5711 −0.873511
$$453$$ 5.06290 0.237876
$$454$$ 0.402590 0.0188945
$$455$$ 0 0
$$456$$ −0.500708 −0.0234478
$$457$$ −39.5509 −1.85011 −0.925057 0.379829i $$-0.875983\pi$$
−0.925057 + 0.379829i $$0.875983\pi$$
$$458$$ 0.168169 0.00785800
$$459$$ −4.12381 −0.192483
$$460$$ −5.55465 −0.258987
$$461$$ −4.90582 −0.228487 −0.114243 0.993453i $$-0.536444\pi$$
−0.114243 + 0.993453i $$0.536444\pi$$
$$462$$ 0 0
$$463$$ −0.772106 −0.0358828 −0.0179414 0.999839i $$-0.505711\pi$$
−0.0179414 + 0.999839i $$0.505711\pi$$
$$464$$ −7.11362 −0.330242
$$465$$ 10.0693 0.466952
$$466$$ 0.686425 0.0317980
$$467$$ −19.1977 −0.888365 −0.444182 0.895936i $$-0.646506\pi$$
−0.444182 + 0.895936i $$0.646506\pi$$
$$468$$ −8.24691 −0.381214
$$469$$ 0 0
$$470$$ −0.858027 −0.0395778
$$471$$ 2.74388 0.126431
$$472$$ −0.943724 −0.0434384
$$473$$ 8.30754 0.381981
$$474$$ −0.182429 −0.00837926
$$475$$ 12.9643 0.594842
$$476$$ 0 0
$$477$$ −0.0608257 −0.00278502
$$478$$ −0.464143 −0.0212294
$$479$$ −18.2104 −0.832056 −0.416028 0.909352i $$-0.636578\pi$$
−0.416028 + 0.909352i $$0.636578\pi$$
$$480$$ 0.874973 0.0399369
$$481$$ −33.2552 −1.51631
$$482$$ −0.151229 −0.00688830
$$483$$ 0 0
$$484$$ 21.1211 0.960050
$$485$$ 7.04961 0.320106
$$486$$ 0.0262565 0.00119102
$$487$$ −28.8813 −1.30873 −0.654367 0.756177i $$-0.727065\pi$$
−0.654367 + 0.756177i $$0.727065\pi$$
$$488$$ 0.284852 0.0128946
$$489$$ −15.3088 −0.692288
$$490$$ 0 0
$$491$$ −23.6817 −1.06874 −0.534369 0.845251i $$-0.679451\pi$$
−0.534369 + 0.845251i $$0.679451\pi$$
$$492$$ −15.9939 −0.721059
$$493$$ −7.34139 −0.330640
$$494$$ −0.516429 −0.0232352
$$495$$ 1.83410 0.0824367
$$496$$ −14.4822 −0.650268
$$497$$ 0 0
$$498$$ 0.247036 0.0110699
$$499$$ 25.3699 1.13571 0.567856 0.823128i $$-0.307773\pi$$
0.567856 + 0.823128i $$0.307773\pi$$
$$500$$ 12.6710 0.566663
$$501$$ −13.0016 −0.580871
$$502$$ 0.272716 0.0121719
$$503$$ −29.2636 −1.30480 −0.652400 0.757874i $$-0.726238\pi$$
−0.652400 + 0.757874i $$0.726238\pi$$
$$504$$ 0 0
$$505$$ 18.6472 0.829790
$$506$$ −0.0173334 −0.000770563 0
$$507$$ −4.01461 −0.178295
$$508$$ −4.79982 −0.212957
$$509$$ −40.6862 −1.80338 −0.901692 0.432379i $$-0.857674\pi$$
−0.901692 + 0.432379i $$0.857674\pi$$
$$510$$ 0.300823 0.0133207
$$511$$ 0 0
$$512$$ −2.09762 −0.0927028
$$513$$ −4.76829 −0.210525
$$514$$ 0.0708526 0.00312518
$$515$$ 29.8605 1.31581
$$516$$ 25.1597 1.10760
$$517$$ 7.76488 0.341499
$$518$$ 0 0
$$519$$ 10.4220 0.457476
$$520$$ 1.20340 0.0527725
$$521$$ −33.7965 −1.48065 −0.740326 0.672248i $$-0.765329\pi$$
−0.740326 + 0.672248i $$0.765329\pi$$
$$522$$ 0.0467430 0.00204589
$$523$$ 12.8567 0.562186 0.281093 0.959681i $$-0.409303\pi$$
0.281093 + 0.959681i $$0.409303\pi$$
$$524$$ 40.2871 1.75995
$$525$$ 0 0
$$526$$ 0.555300 0.0242122
$$527$$ −14.9459 −0.651052
$$528$$ −2.63789 −0.114800
$$529$$ 1.00000 0.0434783
$$530$$ 0.00443711 0.000192736 0
$$531$$ −8.98717 −0.390010
$$532$$ 0 0
$$533$$ −32.9977 −1.42929
$$534$$ −0.155551 −0.00673137
$$535$$ −1.34623 −0.0582027
$$536$$ −0.282033 −0.0121820
$$537$$ −19.2231 −0.829539
$$538$$ 0.749388 0.0323084
$$539$$ 0 0
$$540$$ 5.55465 0.239034
$$541$$ −11.5066 −0.494706 −0.247353 0.968925i $$-0.579561\pi$$
−0.247353 + 0.968925i $$0.579561\pi$$
$$542$$ −0.397714 −0.0170833
$$543$$ 11.1552 0.478717
$$544$$ −1.29872 −0.0556823
$$545$$ 34.6319 1.48347
$$546$$ 0 0
$$547$$ 8.26280 0.353292 0.176646 0.984274i $$-0.443475\pi$$
0.176646 + 0.984274i $$0.443475\pi$$
$$548$$ −14.0083 −0.598406
$$549$$ 2.71267 0.115774
$$550$$ −0.0471269 −0.00200950
$$551$$ −8.48873 −0.361632
$$552$$ −0.105008 −0.00446943
$$553$$ 0 0
$$554$$ −0.111253 −0.00472670
$$555$$ 22.3988 0.950776
$$556$$ 5.76822 0.244627
$$557$$ −22.6003 −0.957604 −0.478802 0.877923i $$-0.658929\pi$$
−0.478802 + 0.877923i $$0.658929\pi$$
$$558$$ 0.0951611 0.00402849
$$559$$ 51.9083 2.19549
$$560$$ 0 0
$$561$$ −2.72236 −0.114938
$$562$$ −0.0200107 −0.000844100 0
$$563$$ −10.1504 −0.427786 −0.213893 0.976857i $$-0.568614\pi$$
−0.213893 + 0.976857i $$0.568614\pi$$
$$564$$ 23.5163 0.990213
$$565$$ 25.8068 1.08570
$$566$$ −0.450860 −0.0189511
$$567$$ 0 0
$$568$$ 1.30695 0.0548382
$$569$$ 36.5753 1.53332 0.766659 0.642055i $$-0.221918\pi$$
0.766659 + 0.642055i $$0.221918\pi$$
$$570$$ 0.347837 0.0145693
$$571$$ 8.14486 0.340852 0.170426 0.985371i $$-0.445486\pi$$
0.170426 + 0.985371i $$0.445486\pi$$
$$572$$ −5.44425 −0.227635
$$573$$ −19.2346 −0.803538
$$574$$ 0 0
$$575$$ 2.71885 0.113384
$$576$$ −7.98346 −0.332644
$$577$$ −7.31887 −0.304689 −0.152344 0.988327i $$-0.548682\pi$$
−0.152344 + 0.988327i $$0.548682\pi$$
$$578$$ −0.000151836 0 −6.31553e−6 0
$$579$$ −17.4527 −0.725309
$$580$$ 9.88864 0.410604
$$581$$ 0 0
$$582$$ 0.0666232 0.00276162
$$583$$ −0.0401544 −0.00166303
$$584$$ −1.50025 −0.0620810
$$585$$ 11.4601 0.473816
$$586$$ 0.0303436 0.00125348
$$587$$ −46.8386 −1.93324 −0.966618 0.256223i $$-0.917522\pi$$
−0.966618 + 0.256223i $$0.917522\pi$$
$$588$$ 0 0
$$589$$ −17.2816 −0.712078
$$590$$ 0.655596 0.0269905
$$591$$ 1.16239 0.0478143
$$592$$ −32.2151 −1.32403
$$593$$ −16.0755 −0.660143 −0.330072 0.943956i $$-0.607073\pi$$
−0.330072 + 0.943956i $$0.607073\pi$$
$$594$$ 0.0173334 0.000711197 0
$$595$$ 0 0
$$596$$ −38.1085 −1.56099
$$597$$ −16.9768 −0.694816
$$598$$ −0.108305 −0.00442891
$$599$$ −27.5797 −1.12688 −0.563439 0.826158i $$-0.690522\pi$$
−0.563439 + 0.826158i $$0.690522\pi$$
$$600$$ −0.285501 −0.0116555
$$601$$ −32.9659 −1.34471 −0.672353 0.740231i $$-0.734716\pi$$
−0.672353 + 0.740231i $$0.734716\pi$$
$$602$$ 0 0
$$603$$ −2.68583 −0.109375
$$604$$ 10.1223 0.411871
$$605$$ −29.3503 −1.19326
$$606$$ 0.176228 0.00715876
$$607$$ −36.8404 −1.49531 −0.747653 0.664090i $$-0.768819\pi$$
−0.747653 + 0.664090i $$0.768819\pi$$
$$608$$ −1.50169 −0.0609017
$$609$$ 0 0
$$610$$ −0.197884 −0.00801208
$$611$$ 48.5176 1.96281
$$612$$ −8.24477 −0.333275
$$613$$ 22.1316 0.893887 0.446943 0.894562i $$-0.352513\pi$$
0.446943 + 0.894562i $$0.352513\pi$$
$$614$$ −0.814909 −0.0328871
$$615$$ 22.2254 0.896214
$$616$$ 0 0
$$617$$ 25.5349 1.02800 0.513998 0.857792i $$-0.328164\pi$$
0.513998 + 0.857792i $$0.328164\pi$$
$$618$$ 0.282200 0.0113517
$$619$$ 30.3835 1.22121 0.610607 0.791933i $$-0.290925\pi$$
0.610607 + 0.791933i $$0.290925\pi$$
$$620$$ 20.1316 0.808506
$$621$$ −1.00000 −0.0401286
$$622$$ −0.462483 −0.0185439
$$623$$ 0 0
$$624$$ −16.4824 −0.659826
$$625$$ −31.2021 −1.24808
$$626$$ −0.413294 −0.0165185
$$627$$ −3.14782 −0.125712
$$628$$ 5.48586 0.218910
$$629$$ −33.2466 −1.32563
$$630$$ 0 0
$$631$$ −23.5164 −0.936175 −0.468087 0.883682i $$-0.655057\pi$$
−0.468087 + 0.883682i $$0.655057\pi$$
$$632$$ −0.729592 −0.0290216
$$633$$ −11.6924 −0.464731
$$634$$ 0.137095 0.00544475
$$635$$ 6.66992 0.264688
$$636$$ −0.121609 −0.00482213
$$637$$ 0 0
$$638$$ 0.0308577 0.00122167
$$639$$ 12.4462 0.492363
$$640$$ 2.33232 0.0921932
$$641$$ 21.6111 0.853586 0.426793 0.904349i $$-0.359643\pi$$
0.426793 + 0.904349i $$0.359643\pi$$
$$642$$ −0.0127227 −0.000502126 0
$$643$$ −12.5132 −0.493472 −0.246736 0.969083i $$-0.579358\pi$$
−0.246736 + 0.969083i $$0.579358\pi$$
$$644$$ 0 0
$$645$$ −34.9625 −1.37665
$$646$$ −0.516295 −0.0203133
$$647$$ 18.8434 0.740811 0.370405 0.928870i $$-0.379219\pi$$
0.370405 + 0.928870i $$0.379219\pi$$
$$648$$ 0.105008 0.00412510
$$649$$ −5.93294 −0.232888
$$650$$ −0.294465 −0.0115499
$$651$$ 0 0
$$652$$ −30.6071 −1.19866
$$653$$ 32.6635 1.27822 0.639111 0.769115i $$-0.279303\pi$$
0.639111 + 0.769115i $$0.279303\pi$$
$$654$$ 0.327293 0.0127982
$$655$$ −55.9837 −2.18747
$$656$$ −31.9657 −1.24805
$$657$$ −14.2871 −0.557392
$$658$$ 0 0
$$659$$ −31.3480 −1.22114 −0.610571 0.791961i $$-0.709060\pi$$
−0.610571 + 0.791961i $$0.709060\pi$$
$$660$$ 3.66694 0.142735
$$661$$ 37.9924 1.47773 0.738866 0.673852i $$-0.235362\pi$$
0.738866 + 0.673852i $$0.235362\pi$$
$$662$$ 0.366145 0.0142306
$$663$$ −17.0102 −0.660621
$$664$$ 0.987973 0.0383408
$$665$$ 0 0
$$666$$ 0.211683 0.00820253
$$667$$ −1.78025 −0.0689314
$$668$$ −25.9943 −1.00575
$$669$$ 15.5087 0.599601
$$670$$ 0.195926 0.00756927
$$671$$ 1.79079 0.0691325
$$672$$ 0 0
$$673$$ 10.3170 0.397691 0.198845 0.980031i $$-0.436281\pi$$
0.198845 + 0.980031i $$0.436281\pi$$
$$674$$ −0.243830 −0.00939197
$$675$$ −2.71885 −0.104649
$$676$$ −8.02645 −0.308710
$$677$$ −27.3198 −1.04999 −0.524993 0.851106i $$-0.675932\pi$$
−0.524993 + 0.851106i $$0.675932\pi$$
$$678$$ 0.243890 0.00936654
$$679$$ 0 0
$$680$$ 1.20309 0.0461362
$$681$$ 15.3330 0.587560
$$682$$ 0.0628212 0.00240555
$$683$$ −21.3082 −0.815337 −0.407669 0.913130i $$-0.633658\pi$$
−0.407669 + 0.913130i $$0.633658\pi$$
$$684$$ −9.53329 −0.364514
$$685$$ 19.4663 0.743768
$$686$$ 0 0
$$687$$ 6.40484 0.244360
$$688$$ 50.2848 1.91709
$$689$$ −0.250898 −0.00955847
$$690$$ 0.0729480 0.00277708
$$691$$ −20.6702 −0.786332 −0.393166 0.919467i $$-0.628620\pi$$
−0.393166 + 0.919467i $$0.628620\pi$$
$$692$$ 20.8369 0.792099
$$693$$ 0 0
$$694$$ −0.201517 −0.00764949
$$695$$ −8.01563 −0.304050
$$696$$ 0.186940 0.00708594
$$697$$ −32.9892 −1.24955
$$698$$ −0.150167 −0.00568391
$$699$$ 26.1430 0.988821
$$700$$ 0 0
$$701$$ −1.93401 −0.0730465 −0.0365232 0.999333i $$-0.511628\pi$$
−0.0365232 + 0.999333i $$0.511628\pi$$
$$702$$ 0.108305 0.00408770
$$703$$ −38.4425 −1.44988
$$704$$ −5.27033 −0.198633
$$705$$ −32.6787 −1.23075
$$706$$ −0.352352 −0.0132609
$$707$$ 0 0
$$708$$ −17.9681 −0.675284
$$709$$ 25.3329 0.951396 0.475698 0.879609i $$-0.342196\pi$$
0.475698 + 0.879609i $$0.342196\pi$$
$$710$$ −0.907923 −0.0340737
$$711$$ −6.94797 −0.260569
$$712$$ −0.622098 −0.0233141
$$713$$ −3.62429 −0.135731
$$714$$ 0 0
$$715$$ 7.56544 0.282931
$$716$$ −38.4330 −1.43631
$$717$$ −17.6773 −0.660170
$$718$$ −0.469094 −0.0175064
$$719$$ −11.7275 −0.437361 −0.218681 0.975796i $$-0.570175\pi$$
−0.218681 + 0.975796i $$0.570175\pi$$
$$720$$ 11.1016 0.413734
$$721$$ 0 0
$$722$$ −0.0981095 −0.00365126
$$723$$ −5.75969 −0.214205
$$724$$ 22.3028 0.828876
$$725$$ −4.84023 −0.179762
$$726$$ −0.277379 −0.0102945
$$727$$ 13.4547 0.499007 0.249503 0.968374i $$-0.419733\pi$$
0.249503 + 0.968374i $$0.419733\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 1.04221 0.0385740
$$731$$ 51.8948 1.91940
$$732$$ 5.42347 0.200457
$$733$$ 20.7942 0.768051 0.384025 0.923323i $$-0.374538\pi$$
0.384025 + 0.923323i $$0.374538\pi$$
$$734$$ −0.686664 −0.0253452
$$735$$ 0 0
$$736$$ −0.314933 −0.0116086
$$737$$ −1.77307 −0.0653117
$$738$$ 0.210044 0.00773182
$$739$$ −23.3524 −0.859034 −0.429517 0.903059i $$-0.641316\pi$$
−0.429517 + 0.903059i $$0.641316\pi$$
$$740$$ 44.7822 1.64623
$$741$$ −19.6686 −0.722544
$$742$$ 0 0
$$743$$ 6.11578 0.224366 0.112183 0.993688i $$-0.464216\pi$$
0.112183 + 0.993688i $$0.464216\pi$$
$$744$$ 0.380579 0.0139527
$$745$$ 52.9564 1.94017
$$746$$ −0.528544 −0.0193514
$$747$$ 9.40856 0.344241
$$748$$ −5.44284 −0.199010
$$749$$ 0 0
$$750$$ −0.166405 −0.00607625
$$751$$ 24.5032 0.894136 0.447068 0.894500i $$-0.352468\pi$$
0.447068 + 0.894500i $$0.352468\pi$$
$$752$$ 47.0001 1.71392
$$753$$ 10.3866 0.378509
$$754$$ 0.192809 0.00702170
$$755$$ −14.0662 −0.511920
$$756$$ 0 0
$$757$$ 23.0058 0.836160 0.418080 0.908410i $$-0.362703\pi$$
0.418080 + 0.908410i $$0.362703\pi$$
$$758$$ −0.211337 −0.00767612
$$759$$ −0.660156 −0.0239621
$$760$$ 1.39111 0.0504608
$$761$$ −10.6470 −0.385953 −0.192976 0.981203i $$-0.561814\pi$$
−0.192976 + 0.981203i $$0.561814\pi$$
$$762$$ 0.0630349 0.00228351
$$763$$ 0 0
$$764$$ −38.4560 −1.39129
$$765$$ 11.4571 0.414232
$$766$$ −0.526952 −0.0190395
$$767$$ −37.0710 −1.33856
$$768$$ −15.9449 −0.575361
$$769$$ 17.0235 0.613883 0.306942 0.951728i $$-0.400694\pi$$
0.306942 + 0.951728i $$0.400694\pi$$
$$770$$ 0 0
$$771$$ 2.69848 0.0971834
$$772$$ −34.8933 −1.25584
$$773$$ −7.84327 −0.282103 −0.141051 0.990002i $$-0.545048\pi$$
−0.141051 + 0.990002i $$0.545048\pi$$
$$774$$ −0.330417 −0.0118766
$$775$$ −9.85391 −0.353963
$$776$$ 0.266447 0.00956488
$$777$$ 0 0
$$778$$ −0.326936 −0.0117212
$$779$$ −38.1448 −1.36668
$$780$$ 22.9122 0.820390
$$781$$ 8.21642 0.294007
$$782$$ −0.108277 −0.00387197
$$783$$ 1.78025 0.0636208
$$784$$ 0 0
$$785$$ −7.62327 −0.272086
$$786$$ −0.529081 −0.0188717
$$787$$ −40.9580 −1.45999 −0.729997 0.683450i $$-0.760479\pi$$
−0.729997 + 0.683450i $$0.760479\pi$$
$$788$$ 2.32398 0.0827883
$$789$$ 21.1490 0.752926
$$790$$ 0.506840 0.0180326
$$791$$ 0 0
$$792$$ 0.0693216 0.00246324
$$793$$ 11.1894 0.397348
$$794$$ 0.182096 0.00646235
$$795$$ 0.168991 0.00599349
$$796$$ −33.9420 −1.20304
$$797$$ −11.8092 −0.418303 −0.209151 0.977883i $$-0.567070\pi$$
−0.209151 + 0.977883i $$0.567070\pi$$
$$798$$ 0 0
$$799$$ 48.5050 1.71598
$$800$$ −0.856257 −0.0302733
$$801$$ −5.92430 −0.209325
$$802$$ 0.378517 0.0133659
$$803$$ −9.43170 −0.332837
$$804$$ −5.36981 −0.189378
$$805$$ 0 0
$$806$$ 0.392528 0.0138262
$$807$$ 28.5411 1.00469
$$808$$ 0.704789 0.0247944
$$809$$ 13.0478 0.458736 0.229368 0.973340i $$-0.426334\pi$$
0.229368 + 0.973340i $$0.426334\pi$$
$$810$$ −0.0729480 −0.00256313
$$811$$ −51.3449 −1.80296 −0.901481 0.432818i $$-0.857519\pi$$
−0.901481 + 0.432818i $$0.857519\pi$$
$$812$$ 0 0
$$813$$ −15.1472 −0.531237
$$814$$ 0.139744 0.00489801
$$815$$ 42.5322 1.48984
$$816$$ −16.4782 −0.576851
$$817$$ 60.0051 2.09931
$$818$$ −0.630851 −0.0220572
$$819$$ 0 0
$$820$$ 44.4355 1.55175
$$821$$ 56.3021 1.96496 0.982479 0.186374i $$-0.0596737\pi$$
0.982479 + 0.186374i $$0.0596737\pi$$
$$822$$ 0.183968 0.00641663
$$823$$ −13.5565 −0.472551 −0.236276 0.971686i $$-0.575927\pi$$
−0.236276 + 0.971686i $$0.575927\pi$$
$$824$$ 1.12860 0.0393168
$$825$$ −1.79487 −0.0624893
$$826$$ 0 0
$$827$$ 3.80574 0.132339 0.0661693 0.997808i $$-0.478922\pi$$
0.0661693 + 0.997808i $$0.478922\pi$$
$$828$$ −1.99931 −0.0694808
$$829$$ −38.8924 −1.35079 −0.675395 0.737457i $$-0.736027\pi$$
−0.675395 + 0.737457i $$0.736027\pi$$
$$830$$ −0.686335 −0.0238230
$$831$$ −4.23717 −0.146986
$$832$$ −32.9308 −1.14167
$$833$$ 0 0
$$834$$ −0.0757527 −0.00262310
$$835$$ 36.1222 1.25006
$$836$$ −6.29346 −0.217664
$$837$$ 3.62429 0.125274
$$838$$ −0.837440 −0.0289289
$$839$$ 5.77279 0.199299 0.0996495 0.995023i $$-0.468228\pi$$
0.0996495 + 0.995023i $$0.468228\pi$$
$$840$$ 0 0
$$841$$ −25.8307 −0.890715
$$842$$ 0.494672 0.0170475
$$843$$ −0.762123 −0.0262489
$$844$$ −23.3767 −0.804661
$$845$$ 11.1537 0.383700
$$846$$ −0.308834 −0.0106179
$$847$$ 0 0
$$848$$ −0.243051 −0.00834641
$$849$$ −17.1714 −0.589321
$$850$$ −0.294388 −0.0100974
$$851$$ −8.06211 −0.276365
$$852$$ 24.8838 0.852504
$$853$$ 6.63846 0.227296 0.113648 0.993521i $$-0.463746\pi$$
0.113648 + 0.993521i $$0.463746\pi$$
$$854$$ 0 0
$$855$$ 13.2477 0.453060
$$856$$ −0.0508822 −0.00173912
$$857$$ −25.7842 −0.880770 −0.440385 0.897809i $$-0.645158\pi$$
−0.440385 + 0.897809i $$0.645158\pi$$
$$858$$ 0.0714981 0.00244090
$$859$$ 21.1070 0.720162 0.360081 0.932921i $$-0.382749\pi$$
0.360081 + 0.932921i $$0.382749\pi$$
$$860$$ −69.9008 −2.38360
$$861$$ 0 0
$$862$$ 0.954242 0.0325016
$$863$$ 12.4548 0.423965 0.211982 0.977273i $$-0.432008\pi$$
0.211982 + 0.977273i $$0.432008\pi$$
$$864$$ 0.314933 0.0107142
$$865$$ −28.9553 −0.984511
$$866$$ −0.565791 −0.0192264
$$867$$ −0.00578278 −0.000196394 0
$$868$$ 0 0
$$869$$ −4.58675 −0.155595
$$870$$ −0.129865 −0.00440285
$$871$$ −11.0787 −0.375388
$$872$$ 1.30894 0.0443265
$$873$$ 2.53740 0.0858779
$$874$$ −0.125199 −0.00423490
$$875$$ 0 0
$$876$$ −28.5643 −0.965098
$$877$$ −45.9072 −1.55018 −0.775088 0.631854i $$-0.782294\pi$$
−0.775088 + 0.631854i $$0.782294\pi$$
$$878$$ 0.281929 0.00951465
$$879$$ 1.15566 0.0389794
$$880$$ 7.32881 0.247054
$$881$$ 17.7656 0.598539 0.299269 0.954169i $$-0.403257\pi$$
0.299269 + 0.954169i $$0.403257\pi$$
$$882$$ 0 0
$$883$$ 8.07698 0.271812 0.135906 0.990722i $$-0.456605\pi$$
0.135906 + 0.990722i $$0.456605\pi$$
$$884$$ −34.0087 −1.14384
$$885$$ 24.9689 0.839320
$$886$$ 0.891489 0.0299502
$$887$$ −51.6718 −1.73497 −0.867484 0.497464i $$-0.834265\pi$$
−0.867484 + 0.497464i $$0.834265\pi$$
$$888$$ 0.846585 0.0284095
$$889$$ 0 0
$$890$$ 0.432166 0.0144862
$$891$$ 0.660156 0.0221161
$$892$$ 31.0067 1.03818
$$893$$ 56.0855 1.87683
$$894$$ 0.500470 0.0167382
$$895$$ 53.4073 1.78521
$$896$$ 0 0
$$897$$ −4.12488 −0.137726
$$898$$ −0.182278 −0.00608268
$$899$$ 6.45212 0.215190
$$900$$ −5.43583 −0.181194
$$901$$ −0.250833 −0.00835647
$$902$$ 0.138662 0.00461693
$$903$$ 0 0
$$904$$ 0.975392 0.0324411
$$905$$ −30.9924 −1.03022
$$906$$ −0.132934 −0.00441643
$$907$$ 38.7795 1.28765 0.643826 0.765172i $$-0.277346\pi$$
0.643826 + 0.765172i $$0.277346\pi$$
$$908$$ 30.6553 1.01733
$$909$$ 6.71178 0.222616
$$910$$ 0 0
$$911$$ 9.13030 0.302500 0.151250 0.988496i $$-0.451670\pi$$
0.151250 + 0.988496i $$0.451670\pi$$
$$912$$ −19.0534 −0.630922
$$913$$ 6.21112 0.205558
$$914$$ 1.03847 0.0343495
$$915$$ −7.53656 −0.249151
$$916$$ 12.8053 0.423098
$$917$$ 0 0
$$918$$ 0.108277 0.00357366
$$919$$ 3.74922 0.123675 0.0618377 0.998086i $$-0.480304\pi$$
0.0618377 + 0.998086i $$0.480304\pi$$
$$920$$ 0.291742 0.00961843
$$921$$ −31.0365 −1.02269
$$922$$ 0.128810 0.00424212
$$923$$ 51.3389 1.68984
$$924$$ 0 0
$$925$$ −21.9197 −0.720715
$$926$$ 0.0202728 0.000666206 0
$$927$$ 10.7478 0.353004
$$928$$ 0.560658 0.0184045
$$929$$ 44.4320 1.45777 0.728883 0.684638i $$-0.240040\pi$$
0.728883 + 0.684638i $$0.240040\pi$$
$$930$$ −0.264384 −0.00866950
$$931$$ 0 0
$$932$$ 52.2681 1.71210
$$933$$ −17.6141 −0.576658
$$934$$ 0.504065 0.0164935
$$935$$ 7.56347 0.247352
$$936$$ 0.433145 0.0141578
$$937$$ 59.8482 1.95516 0.977579 0.210571i $$-0.0675324\pi$$
0.977579 + 0.210571i $$0.0675324\pi$$
$$938$$ 0 0
$$939$$ −15.7406 −0.513676
$$940$$ −65.3348 −2.13099
$$941$$ 21.7078 0.707654 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$942$$ −0.0720446 −0.00234734
$$943$$ −7.99969 −0.260506
$$944$$ −35.9115 −1.16882
$$945$$ 0 0
$$946$$ −0.218127 −0.00709191
$$947$$ 16.5648 0.538285 0.269143 0.963100i $$-0.413260\pi$$
0.269143 + 0.963100i $$0.413260\pi$$
$$948$$ −13.8912 −0.451164
$$949$$ −58.9324 −1.91303
$$950$$ −0.340396 −0.0110439
$$951$$ 5.22139 0.169315
$$952$$ 0 0
$$953$$ 51.8971 1.68111 0.840555 0.541726i $$-0.182229\pi$$
0.840555 + 0.541726i $$0.182229\pi$$
$$954$$ 0.00159707 5.17070e−5 0
$$955$$ 53.4392 1.72925
$$956$$ −35.3424 −1.14305
$$957$$ 1.17524 0.0379901
$$958$$ 0.478142 0.0154481
$$959$$ 0 0
$$960$$ 22.1803 0.715866
$$961$$ −17.8645 −0.576276
$$962$$ 0.873165 0.0281520
$$963$$ −0.484556 −0.0156146
$$964$$ −11.5154 −0.370886
$$965$$ 48.4885 1.56090
$$966$$ 0 0
$$967$$ −1.28492 −0.0413202 −0.0206601 0.999787i $$-0.506577\pi$$
−0.0206601 + 0.999787i $$0.506577\pi$$
$$968$$ −1.10932 −0.0356550
$$969$$ −19.6635 −0.631683
$$970$$ −0.185098 −0.00594314
$$971$$ −50.4972 −1.62053 −0.810266 0.586063i $$-0.800677\pi$$
−0.810266 + 0.586063i $$0.800677\pi$$
$$972$$ 1.99931 0.0641279
$$973$$ 0 0
$$974$$ 0.758321 0.0242982
$$975$$ −11.2149 −0.359165
$$976$$ 10.8395 0.346963
$$977$$ −5.77501 −0.184759 −0.0923794 0.995724i $$-0.529447\pi$$
−0.0923794 + 0.995724i $$0.529447\pi$$
$$978$$ 0.401955 0.0128531
$$979$$ −3.91096 −0.124995
$$980$$ 0 0
$$981$$ 12.4652 0.397983
$$982$$ 0.621797 0.0198423
$$983$$ 44.3068 1.41317 0.706584 0.707629i $$-0.250235\pi$$
0.706584 + 0.707629i $$0.250235\pi$$
$$984$$ 0.840030 0.0267792
$$985$$ −3.22945 −0.102899
$$986$$ 0.192759 0.00613870
$$987$$ 0 0
$$988$$ −39.3237 −1.25105
$$989$$ 12.5842 0.400154
$$990$$ −0.0481570 −0.00153053
$$991$$ 9.45792 0.300441 0.150220 0.988653i $$-0.452002\pi$$
0.150220 + 0.988653i $$0.452002\pi$$
$$992$$ 1.14141 0.0362397
$$993$$ 13.9449 0.442529
$$994$$ 0 0
$$995$$ 47.1665 1.49528
$$996$$ 18.8106 0.596038
$$997$$ −33.4467 −1.05927 −0.529635 0.848226i $$-0.677671\pi$$
−0.529635 + 0.848226i $$0.677671\pi$$
$$998$$ −0.666124 −0.0210858
$$999$$ 8.06211 0.255074
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3381.2.a.bi.1.5 10
7.2 even 3 483.2.i.h.277.6 20
7.4 even 3 483.2.i.h.415.6 yes 20
7.6 odd 2 3381.2.a.bj.1.5 10

By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.6 20 7.2 even 3
483.2.i.h.415.6 yes 20 7.4 even 3
3381.2.a.bi.1.5 10 1.1 even 1 trivial
3381.2.a.bj.1.5 10 7.6 odd 2