# Properties

 Label 825.2.c.e.199.1 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.e.199.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} +0.828427i q^{7} +4.41421i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} +0.828427i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.00000 q^{11} -3.82843i q^{12} +5.65685i q^{13} +2.00000 q^{14} +3.00000 q^{16} -1.17157i q^{17} +2.41421i q^{18} +6.82843 q^{19} -0.828427 q^{21} +2.41421i q^{22} +4.00000i q^{23} -4.41421 q^{24} +13.6569 q^{26} -1.00000i q^{27} -3.17157i q^{28} +4.82843 q^{29} +1.58579i q^{32} -1.00000i q^{33} -2.82843 q^{34} +3.82843 q^{36} +11.6569i q^{37} -16.4853i q^{38} -5.65685 q^{39} +4.82843 q^{41} +2.00000i q^{42} +8.82843i q^{43} +3.82843 q^{44} +9.65685 q^{46} -4.00000i q^{47} +3.00000i q^{48} +6.31371 q^{49} +1.17157 q^{51} -21.6569i q^{52} -9.31371i q^{53} -2.41421 q^{54} -3.65685 q^{56} +6.82843i q^{57} -11.6569i q^{58} +4.00000 q^{59} -11.6569 q^{61} -0.828427i q^{63} +9.82843 q^{64} -2.41421 q^{66} -5.65685i q^{67} +4.48528i q^{68} -4.00000 q^{69} +2.34315 q^{71} -4.41421i q^{72} -11.3137i q^{73} +28.1421 q^{74} -26.1421 q^{76} -0.828427i q^{77} +13.6569i q^{78} -8.48528 q^{79} +1.00000 q^{81} -11.6569i q^{82} +10.0000i q^{83} +3.17157 q^{84} +21.3137 q^{86} +4.82843i q^{87} -4.41421i q^{88} -3.65685 q^{89} -4.68629 q^{91} -15.3137i q^{92} -9.65685 q^{94} -1.58579 q^{96} +11.6569i q^{97} -15.2426i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} - 4q^{11} + 8q^{14} + 12q^{16} + 16q^{19} + 8q^{21} - 12q^{24} + 32q^{26} + 8q^{29} + 4q^{36} + 8q^{41} + 4q^{44} + 16q^{46} - 20q^{49} + 16q^{51} - 4q^{54} + 8q^{56} + 16q^{59} - 24q^{61} + 28q^{64} - 4q^{66} - 16q^{69} + 32q^{71} + 56q^{74} - 48q^{76} + 4q^{81} + 24q^{84} + 40q^{86} + 8q^{89} - 64q^{91} - 16q^{94} - 12q^{96} + 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.41421i − 1.70711i −0.521005 0.853553i $$-0.674443\pi$$
0.521005 0.853553i $$-0.325557\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −3.82843 −1.91421
$$5$$ 0 0
$$6$$ 2.41421 0.985599
$$7$$ 0.828427i 0.313116i 0.987669 + 0.156558i $$0.0500398\pi$$
−0.987669 + 0.156558i $$0.949960\pi$$
$$8$$ 4.41421i 1.56066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ − 3.82843i − 1.10517i
$$13$$ 5.65685i 1.56893i 0.620174 + 0.784465i $$0.287062\pi$$
−0.620174 + 0.784465i $$0.712938\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ − 1.17157i − 0.284148i −0.989856 0.142074i $$-0.954623\pi$$
0.989856 0.142074i $$-0.0453771\pi$$
$$18$$ 2.41421i 0.569036i
$$19$$ 6.82843 1.56655 0.783274 0.621676i $$-0.213548\pi$$
0.783274 + 0.621676i $$0.213548\pi$$
$$20$$ 0 0
$$21$$ −0.828427 −0.180778
$$22$$ 2.41421i 0.514712i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −4.41421 −0.901048
$$25$$ 0 0
$$26$$ 13.6569 2.67833
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 3.17157i − 0.599371i
$$29$$ 4.82843 0.896616 0.448308 0.893879i $$-0.352027\pi$$
0.448308 + 0.893879i $$0.352027\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.58579i 0.280330i
$$33$$ − 1.00000i − 0.174078i
$$34$$ −2.82843 −0.485071
$$35$$ 0 0
$$36$$ 3.82843 0.638071
$$37$$ 11.6569i 1.91638i 0.286141 + 0.958188i $$0.407627\pi$$
−0.286141 + 0.958188i $$0.592373\pi$$
$$38$$ − 16.4853i − 2.67427i
$$39$$ −5.65685 −0.905822
$$40$$ 0 0
$$41$$ 4.82843 0.754074 0.377037 0.926198i $$-0.376943\pi$$
0.377037 + 0.926198i $$0.376943\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ 8.82843i 1.34632i 0.739496 + 0.673161i $$0.235064\pi$$
−0.739496 + 0.673161i $$0.764936\pi$$
$$44$$ 3.82843 0.577157
$$45$$ 0 0
$$46$$ 9.65685 1.42383
$$47$$ − 4.00000i − 0.583460i −0.956501 0.291730i $$-0.905769\pi$$
0.956501 0.291730i $$-0.0942309\pi$$
$$48$$ 3.00000i 0.433013i
$$49$$ 6.31371 0.901958
$$50$$ 0 0
$$51$$ 1.17157 0.164053
$$52$$ − 21.6569i − 3.00327i
$$53$$ − 9.31371i − 1.27934i −0.768651 0.639668i $$-0.779072\pi$$
0.768651 0.639668i $$-0.220928\pi$$
$$54$$ −2.41421 −0.328533
$$55$$ 0 0
$$56$$ −3.65685 −0.488668
$$57$$ 6.82843i 0.904447i
$$58$$ − 11.6569i − 1.53062i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −11.6569 −1.49251 −0.746254 0.665662i $$-0.768149\pi$$
−0.746254 + 0.665662i $$0.768149\pi$$
$$62$$ 0 0
$$63$$ − 0.828427i − 0.104372i
$$64$$ 9.82843 1.22855
$$65$$ 0 0
$$66$$ −2.41421 −0.297169
$$67$$ − 5.65685i − 0.691095i −0.938401 0.345547i $$-0.887693\pi$$
0.938401 0.345547i $$-0.112307\pi$$
$$68$$ 4.48528i 0.543920i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 2.34315 0.278080 0.139040 0.990287i $$-0.455598\pi$$
0.139040 + 0.990287i $$0.455598\pi$$
$$72$$ − 4.41421i − 0.520220i
$$73$$ − 11.3137i − 1.32417i −0.749429 0.662085i $$-0.769672\pi$$
0.749429 0.662085i $$-0.230328\pi$$
$$74$$ 28.1421 3.27146
$$75$$ 0 0
$$76$$ −26.1421 −2.99871
$$77$$ − 0.828427i − 0.0944080i
$$78$$ 13.6569i 1.54633i
$$79$$ −8.48528 −0.954669 −0.477334 0.878722i $$-0.658397\pi$$
−0.477334 + 0.878722i $$0.658397\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 11.6569i − 1.28728i
$$83$$ 10.0000i 1.09764i 0.835940 + 0.548821i $$0.184923\pi$$
−0.835940 + 0.548821i $$0.815077\pi$$
$$84$$ 3.17157 0.346047
$$85$$ 0 0
$$86$$ 21.3137 2.29832
$$87$$ 4.82843i 0.517662i
$$88$$ − 4.41421i − 0.470557i
$$89$$ −3.65685 −0.387626 −0.193813 0.981039i $$-0.562085\pi$$
−0.193813 + 0.981039i $$0.562085\pi$$
$$90$$ 0 0
$$91$$ −4.68629 −0.491257
$$92$$ − 15.3137i − 1.59656i
$$93$$ 0 0
$$94$$ −9.65685 −0.996028
$$95$$ 0 0
$$96$$ −1.58579 −0.161849
$$97$$ 11.6569i 1.18357i 0.806094 + 0.591787i $$0.201577\pi$$
−0.806094 + 0.591787i $$0.798423\pi$$
$$98$$ − 15.2426i − 1.53974i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −0.828427 −0.0824316 −0.0412158 0.999150i $$-0.513123\pi$$
−0.0412158 + 0.999150i $$0.513123\pi$$
$$102$$ − 2.82843i − 0.280056i
$$103$$ 3.31371i 0.326509i 0.986584 + 0.163255i $$0.0521992\pi$$
−0.986584 + 0.163255i $$0.947801\pi$$
$$104$$ −24.9706 −2.44857
$$105$$ 0 0
$$106$$ −22.4853 −2.18396
$$107$$ 17.3137i 1.67378i 0.547372 + 0.836890i $$0.315628\pi$$
−0.547372 + 0.836890i $$0.684372\pi$$
$$108$$ 3.82843i 0.368391i
$$109$$ −17.3137 −1.65835 −0.829176 0.558987i $$-0.811190\pi$$
−0.829176 + 0.558987i $$0.811190\pi$$
$$110$$ 0 0
$$111$$ −11.6569 −1.10642
$$112$$ 2.48528i 0.234837i
$$113$$ 18.9706i 1.78460i 0.451442 + 0.892300i $$0.350910\pi$$
−0.451442 + 0.892300i $$0.649090\pi$$
$$114$$ 16.4853 1.54399
$$115$$ 0 0
$$116$$ −18.4853 −1.71632
$$117$$ − 5.65685i − 0.522976i
$$118$$ − 9.65685i − 0.888985i
$$119$$ 0.970563 0.0889713
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 28.1421i 2.54787i
$$123$$ 4.82843i 0.435365i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 14.4853i − 1.28536i −0.766134 0.642680i $$-0.777822\pi$$
0.766134 0.642680i $$-0.222178\pi$$
$$128$$ − 20.5563i − 1.81694i
$$129$$ −8.82843 −0.777300
$$130$$ 0 0
$$131$$ 3.31371 0.289520 0.144760 0.989467i $$-0.453759\pi$$
0.144760 + 0.989467i $$0.453759\pi$$
$$132$$ 3.82843i 0.333222i
$$133$$ 5.65685i 0.490511i
$$134$$ −13.6569 −1.17977
$$135$$ 0 0
$$136$$ 5.17157 0.443459
$$137$$ − 13.3137i − 1.13747i −0.822522 0.568733i $$-0.807434\pi$$
0.822522 0.568733i $$-0.192566\pi$$
$$138$$ 9.65685i 0.822046i
$$139$$ −0.485281 −0.0411610 −0.0205805 0.999788i $$-0.506551\pi$$
−0.0205805 + 0.999788i $$0.506551\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ − 5.65685i − 0.474713i
$$143$$ − 5.65685i − 0.473050i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ −27.3137 −2.26050
$$147$$ 6.31371i 0.520746i
$$148$$ − 44.6274i − 3.66835i
$$149$$ −1.51472 −0.124091 −0.0620453 0.998073i $$-0.519762\pi$$
−0.0620453 + 0.998073i $$0.519762\pi$$
$$150$$ 0 0
$$151$$ 16.4853 1.34155 0.670777 0.741659i $$-0.265961\pi$$
0.670777 + 0.741659i $$0.265961\pi$$
$$152$$ 30.1421i 2.44485i
$$153$$ 1.17157i 0.0947161i
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 21.6569 1.73394
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 20.4853i 1.62972i
$$159$$ 9.31371 0.738625
$$160$$ 0 0
$$161$$ −3.31371 −0.261157
$$162$$ − 2.41421i − 0.189679i
$$163$$ 7.31371i 0.572854i 0.958102 + 0.286427i $$0.0924676\pi$$
−0.958102 + 0.286427i $$0.907532\pi$$
$$164$$ −18.4853 −1.44346
$$165$$ 0 0
$$166$$ 24.1421 1.87379
$$167$$ − 13.3137i − 1.03025i −0.857116 0.515123i $$-0.827746\pi$$
0.857116 0.515123i $$-0.172254\pi$$
$$168$$ − 3.65685i − 0.282132i
$$169$$ −19.0000 −1.46154
$$170$$ 0 0
$$171$$ −6.82843 −0.522183
$$172$$ − 33.7990i − 2.57715i
$$173$$ − 2.82843i − 0.215041i −0.994203 0.107521i $$-0.965709\pi$$
0.994203 0.107521i $$-0.0342912\pi$$
$$174$$ 11.6569 0.883704
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 4.00000i 0.300658i
$$178$$ 8.82843i 0.661719i
$$179$$ 17.6569 1.31974 0.659868 0.751382i $$-0.270612\pi$$
0.659868 + 0.751382i $$0.270612\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 11.3137i 0.838628i
$$183$$ − 11.6569i − 0.861699i
$$184$$ −17.6569 −1.30168
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.17157i 0.0856739i
$$188$$ 15.3137i 1.11687i
$$189$$ 0.828427 0.0602592
$$190$$ 0 0
$$191$$ 5.65685 0.409316 0.204658 0.978834i $$-0.434392\pi$$
0.204658 + 0.978834i $$0.434392\pi$$
$$192$$ 9.82843i 0.709306i
$$193$$ − 13.6569i − 0.983042i −0.870866 0.491521i $$-0.836441\pi$$
0.870866 0.491521i $$-0.163559\pi$$
$$194$$ 28.1421 2.02049
$$195$$ 0 0
$$196$$ −24.1716 −1.72654
$$197$$ − 8.48528i − 0.604551i −0.953221 0.302276i $$-0.902254\pi$$
0.953221 0.302276i $$-0.0977463\pi$$
$$198$$ − 2.41421i − 0.171571i
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ 0 0
$$201$$ 5.65685 0.399004
$$202$$ 2.00000i 0.140720i
$$203$$ 4.00000i 0.280745i
$$204$$ −4.48528 −0.314033
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ − 4.00000i − 0.278019i
$$208$$ 16.9706i 1.17670i
$$209$$ −6.82843 −0.472332
$$210$$ 0 0
$$211$$ 1.17157 0.0806544 0.0403272 0.999187i $$-0.487160\pi$$
0.0403272 + 0.999187i $$0.487160\pi$$
$$212$$ 35.6569i 2.44892i
$$213$$ 2.34315i 0.160550i
$$214$$ 41.7990 2.85732
$$215$$ 0 0
$$216$$ 4.41421 0.300349
$$217$$ 0 0
$$218$$ 41.7990i 2.83098i
$$219$$ 11.3137 0.764510
$$220$$ 0 0
$$221$$ 6.62742 0.445808
$$222$$ 28.1421i 1.88878i
$$223$$ 6.34315i 0.424768i 0.977186 + 0.212384i $$0.0681228\pi$$
−0.977186 + 0.212384i $$0.931877\pi$$
$$224$$ −1.31371 −0.0877758
$$225$$ 0 0
$$226$$ 45.7990 3.04650
$$227$$ − 14.0000i − 0.929213i −0.885517 0.464606i $$-0.846196\pi$$
0.885517 0.464606i $$-0.153804\pi$$
$$228$$ − 26.1421i − 1.73131i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0.828427 0.0545065
$$232$$ 21.3137i 1.39931i
$$233$$ 18.8284i 1.23349i 0.787163 + 0.616746i $$0.211549\pi$$
−0.787163 + 0.616746i $$0.788451\pi$$
$$234$$ −13.6569 −0.892776
$$235$$ 0 0
$$236$$ −15.3137 −0.996838
$$237$$ − 8.48528i − 0.551178i
$$238$$ − 2.34315i − 0.151884i
$$239$$ −17.6569 −1.14213 −0.571063 0.820906i $$-0.693469\pi$$
−0.571063 + 0.820906i $$0.693469\pi$$
$$240$$ 0 0
$$241$$ −12.3431 −0.795092 −0.397546 0.917582i $$-0.630138\pi$$
−0.397546 + 0.917582i $$0.630138\pi$$
$$242$$ − 2.41421i − 0.155192i
$$243$$ 1.00000i 0.0641500i
$$244$$ 44.6274 2.85698
$$245$$ 0 0
$$246$$ 11.6569 0.743214
$$247$$ 38.6274i 2.45780i
$$248$$ 0 0
$$249$$ −10.0000 −0.633724
$$250$$ 0 0
$$251$$ 20.9706 1.32365 0.661825 0.749658i $$-0.269782\pi$$
0.661825 + 0.749658i $$0.269782\pi$$
$$252$$ 3.17157i 0.199790i
$$253$$ − 4.00000i − 0.251478i
$$254$$ −34.9706 −2.19425
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ − 16.3431i − 1.01946i −0.860335 0.509729i $$-0.829746\pi$$
0.860335 0.509729i $$-0.170254\pi$$
$$258$$ 21.3137i 1.32693i
$$259$$ −9.65685 −0.600048
$$260$$ 0 0
$$261$$ −4.82843 −0.298872
$$262$$ − 8.00000i − 0.494242i
$$263$$ − 18.0000i − 1.10993i −0.831875 0.554964i $$-0.812732\pi$$
0.831875 0.554964i $$-0.187268\pi$$
$$264$$ 4.41421 0.271676
$$265$$ 0 0
$$266$$ 13.6569 0.837355
$$267$$ − 3.65685i − 0.223796i
$$268$$ 21.6569i 1.32290i
$$269$$ −20.6274 −1.25768 −0.628838 0.777536i $$-0.716469\pi$$
−0.628838 + 0.777536i $$0.716469\pi$$
$$270$$ 0 0
$$271$$ −11.7990 −0.716738 −0.358369 0.933580i $$-0.616667\pi$$
−0.358369 + 0.933580i $$0.616667\pi$$
$$272$$ − 3.51472i − 0.213111i
$$273$$ − 4.68629i − 0.283627i
$$274$$ −32.1421 −1.94178
$$275$$ 0 0
$$276$$ 15.3137 0.921777
$$277$$ − 2.34315i − 0.140786i −0.997519 0.0703930i $$-0.977575\pi$$
0.997519 0.0703930i $$-0.0224253\pi$$
$$278$$ 1.17157i 0.0702663i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −11.1716 −0.666440 −0.333220 0.942849i $$-0.608135\pi$$
−0.333220 + 0.942849i $$0.608135\pi$$
$$282$$ − 9.65685i − 0.575057i
$$283$$ 8.82843i 0.524796i 0.964960 + 0.262398i $$0.0845132\pi$$
−0.964960 + 0.262398i $$0.915487\pi$$
$$284$$ −8.97056 −0.532305
$$285$$ 0 0
$$286$$ −13.6569 −0.807547
$$287$$ 4.00000i 0.236113i
$$288$$ − 1.58579i − 0.0934434i
$$289$$ 15.6274 0.919260
$$290$$ 0 0
$$291$$ −11.6569 −0.683337
$$292$$ 43.3137i 2.53474i
$$293$$ 6.82843i 0.398921i 0.979906 + 0.199460i $$0.0639190\pi$$
−0.979906 + 0.199460i $$0.936081\pi$$
$$294$$ 15.2426 0.888969
$$295$$ 0 0
$$296$$ −51.4558 −2.99081
$$297$$ 1.00000i 0.0580259i
$$298$$ 3.65685i 0.211836i
$$299$$ −22.6274 −1.30858
$$300$$ 0 0
$$301$$ −7.31371 −0.421555
$$302$$ − 39.7990i − 2.29017i
$$303$$ − 0.828427i − 0.0475919i
$$304$$ 20.4853 1.17491
$$305$$ 0 0
$$306$$ 2.82843 0.161690
$$307$$ − 3.17157i − 0.181011i −0.995896 0.0905056i $$-0.971152\pi$$
0.995896 0.0905056i $$-0.0288483\pi$$
$$308$$ 3.17157i 0.180717i
$$309$$ −3.31371 −0.188510
$$310$$ 0 0
$$311$$ −3.31371 −0.187903 −0.0939516 0.995577i $$-0.529950\pi$$
−0.0939516 + 0.995577i $$0.529950\pi$$
$$312$$ − 24.9706i − 1.41368i
$$313$$ − 15.6569i − 0.884978i −0.896774 0.442489i $$-0.854096\pi$$
0.896774 0.442489i $$-0.145904\pi$$
$$314$$ 43.4558 2.45236
$$315$$ 0 0
$$316$$ 32.4853 1.82744
$$317$$ − 26.2843i − 1.47627i −0.674652 0.738136i $$-0.735706\pi$$
0.674652 0.738136i $$-0.264294\pi$$
$$318$$ − 22.4853i − 1.26091i
$$319$$ −4.82843 −0.270340
$$320$$ 0 0
$$321$$ −17.3137 −0.966357
$$322$$ 8.00000i 0.445823i
$$323$$ − 8.00000i − 0.445132i
$$324$$ −3.82843 −0.212690
$$325$$ 0 0
$$326$$ 17.6569 0.977923
$$327$$ − 17.3137i − 0.957450i
$$328$$ 21.3137i 1.17685i
$$329$$ 3.31371 0.182691
$$330$$ 0 0
$$331$$ 6.34315 0.348651 0.174325 0.984688i $$-0.444226\pi$$
0.174325 + 0.984688i $$0.444226\pi$$
$$332$$ − 38.2843i − 2.10112i
$$333$$ − 11.6569i − 0.638792i
$$334$$ −32.1421 −1.75874
$$335$$ 0 0
$$336$$ −2.48528 −0.135583
$$337$$ 3.31371i 0.180509i 0.995919 + 0.0902546i $$0.0287681\pi$$
−0.995919 + 0.0902546i $$0.971232\pi$$
$$338$$ 45.8701i 2.49500i
$$339$$ −18.9706 −1.03034
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 16.4853i 0.891422i
$$343$$ 11.0294i 0.595534i
$$344$$ −38.9706 −2.10115
$$345$$ 0 0
$$346$$ −6.82843 −0.367099
$$347$$ − 29.3137i − 1.57364i −0.617181 0.786821i $$-0.711725\pi$$
0.617181 0.786821i $$-0.288275\pi$$
$$348$$ − 18.4853i − 0.990915i
$$349$$ −10.9706 −0.587241 −0.293620 0.955922i $$-0.594860\pi$$
−0.293620 + 0.955922i $$0.594860\pi$$
$$350$$ 0 0
$$351$$ 5.65685 0.301941
$$352$$ − 1.58579i − 0.0845227i
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 9.65685 0.513256
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ 0.970563i 0.0513676i
$$358$$ − 42.6274i − 2.25293i
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 27.6274 1.45407
$$362$$ 33.7990i 1.77644i
$$363$$ 1.00000i 0.0524864i
$$364$$ 17.9411 0.940370
$$365$$ 0 0
$$366$$ −28.1421 −1.47101
$$367$$ 9.65685i 0.504084i 0.967716 + 0.252042i $$0.0811021\pi$$
−0.967716 + 0.252042i $$0.918898\pi$$
$$368$$ 12.0000i 0.625543i
$$369$$ −4.82843 −0.251358
$$370$$ 0 0
$$371$$ 7.71573 0.400581
$$372$$ 0 0
$$373$$ − 10.6274i − 0.550267i −0.961406 0.275133i $$-0.911278\pi$$
0.961406 0.275133i $$-0.0887220\pi$$
$$374$$ 2.82843 0.146254
$$375$$ 0 0
$$376$$ 17.6569 0.910583
$$377$$ 27.3137i 1.40673i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 23.3137 1.19754 0.598772 0.800919i $$-0.295655\pi$$
0.598772 + 0.800919i $$0.295655\pi$$
$$380$$ 0 0
$$381$$ 14.4853 0.742103
$$382$$ − 13.6569i − 0.698745i
$$383$$ 8.00000i 0.408781i 0.978889 + 0.204390i $$0.0655212\pi$$
−0.978889 + 0.204390i $$0.934479\pi$$
$$384$$ 20.5563 1.04901
$$385$$ 0 0
$$386$$ −32.9706 −1.67816
$$387$$ − 8.82843i − 0.448774i
$$388$$ − 44.6274i − 2.26561i
$$389$$ 23.6569 1.19945 0.599725 0.800206i $$-0.295277\pi$$
0.599725 + 0.800206i $$0.295277\pi$$
$$390$$ 0 0
$$391$$ 4.68629 0.236996
$$392$$ 27.8701i 1.40765i
$$393$$ 3.31371i 0.167154i
$$394$$ −20.4853 −1.03203
$$395$$ 0 0
$$396$$ −3.82843 −0.192386
$$397$$ − 14.9706i − 0.751351i −0.926751 0.375676i $$-0.877411\pi$$
0.926751 0.375676i $$-0.122589\pi$$
$$398$$ − 52.2843i − 2.62077i
$$399$$ −5.65685 −0.283197
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ − 13.6569i − 0.681142i
$$403$$ 0 0
$$404$$ 3.17157 0.157792
$$405$$ 0 0
$$406$$ 9.65685 0.479262
$$407$$ − 11.6569i − 0.577809i
$$408$$ 5.17157i 0.256031i
$$409$$ 19.6569 0.971969 0.485984 0.873967i $$-0.338461\pi$$
0.485984 + 0.873967i $$0.338461\pi$$
$$410$$ 0 0
$$411$$ 13.3137 0.656717
$$412$$ − 12.6863i − 0.625009i
$$413$$ 3.31371i 0.163057i
$$414$$ −9.65685 −0.474608
$$415$$ 0 0
$$416$$ −8.97056 −0.439818
$$417$$ − 0.485281i − 0.0237643i
$$418$$ 16.4853i 0.806321i
$$419$$ 36.9706 1.80613 0.903065 0.429504i $$-0.141311\pi$$
0.903065 + 0.429504i $$0.141311\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ − 2.82843i − 0.137686i
$$423$$ 4.00000i 0.194487i
$$424$$ 41.1127 1.99661
$$425$$ 0 0
$$426$$ 5.65685 0.274075
$$427$$ − 9.65685i − 0.467328i
$$428$$ − 66.2843i − 3.20397i
$$429$$ 5.65685 0.273115
$$430$$ 0 0
$$431$$ 21.6569 1.04317 0.521587 0.853198i $$-0.325340\pi$$
0.521587 + 0.853198i $$0.325340\pi$$
$$432$$ − 3.00000i − 0.144338i
$$433$$ 15.6569i 0.752420i 0.926534 + 0.376210i $$0.122773\pi$$
−0.926534 + 0.376210i $$0.877227\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 66.2843 3.17444
$$437$$ 27.3137i 1.30659i
$$438$$ − 27.3137i − 1.30510i
$$439$$ 20.4853 0.977709 0.488855 0.872365i $$-0.337415\pi$$
0.488855 + 0.872365i $$0.337415\pi$$
$$440$$ 0 0
$$441$$ −6.31371 −0.300653
$$442$$ − 16.0000i − 0.761042i
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 44.6274 2.11792
$$445$$ 0 0
$$446$$ 15.3137 0.725125
$$447$$ − 1.51472i − 0.0716437i
$$448$$ 8.14214i 0.384680i
$$449$$ −30.9706 −1.46159 −0.730796 0.682596i $$-0.760851\pi$$
−0.730796 + 0.682596i $$0.760851\pi$$
$$450$$ 0 0
$$451$$ −4.82843 −0.227362
$$452$$ − 72.6274i − 3.41611i
$$453$$ 16.4853i 0.774546i
$$454$$ −33.7990 −1.58627
$$455$$ 0 0
$$456$$ −30.1421 −1.41153
$$457$$ − 23.3137i − 1.09057i −0.838251 0.545285i $$-0.816422\pi$$
0.838251 0.545285i $$-0.183578\pi$$
$$458$$ − 4.82843i − 0.225618i
$$459$$ −1.17157 −0.0546843
$$460$$ 0 0
$$461$$ 0.142136 0.00661992 0.00330996 0.999995i $$-0.498946\pi$$
0.00330996 + 0.999995i $$0.498946\pi$$
$$462$$ − 2.00000i − 0.0930484i
$$463$$ − 4.97056i − 0.231002i −0.993307 0.115501i $$-0.963153\pi$$
0.993307 0.115501i $$-0.0368473\pi$$
$$464$$ 14.4853 0.672462
$$465$$ 0 0
$$466$$ 45.4558 2.10570
$$467$$ − 22.6274i − 1.04707i −0.852004 0.523536i $$-0.824613\pi$$
0.852004 0.523536i $$-0.175387\pi$$
$$468$$ 21.6569i 1.00109i
$$469$$ 4.68629 0.216393
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 17.6569i 0.812723i
$$473$$ − 8.82843i − 0.405932i
$$474$$ −20.4853 −0.940920
$$475$$ 0 0
$$476$$ −3.71573 −0.170310
$$477$$ 9.31371i 0.426445i
$$478$$ 42.6274i 1.94973i
$$479$$ −36.9706 −1.68923 −0.844614 0.535376i $$-0.820170\pi$$
−0.844614 + 0.535376i $$0.820170\pi$$
$$480$$ 0 0
$$481$$ −65.9411 −3.00666
$$482$$ 29.7990i 1.35731i
$$483$$ − 3.31371i − 0.150779i
$$484$$ −3.82843 −0.174019
$$485$$ 0 0
$$486$$ 2.41421 0.109511
$$487$$ − 12.9706i − 0.587752i −0.955844 0.293876i $$-0.905055\pi$$
0.955844 0.293876i $$-0.0949453\pi$$
$$488$$ − 51.4558i − 2.32930i
$$489$$ −7.31371 −0.330737
$$490$$ 0 0
$$491$$ 14.3431 0.647297 0.323649 0.946177i $$-0.395090\pi$$
0.323649 + 0.946177i $$0.395090\pi$$
$$492$$ − 18.4853i − 0.833381i
$$493$$ − 5.65685i − 0.254772i
$$494$$ 93.2548 4.19573
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.94113i 0.0870714i
$$498$$ 24.1421i 1.08183i
$$499$$ 22.3431 1.00022 0.500108 0.865963i $$-0.333294\pi$$
0.500108 + 0.865963i $$0.333294\pi$$
$$500$$ 0 0
$$501$$ 13.3137 0.594813
$$502$$ − 50.6274i − 2.25961i
$$503$$ − 17.3137i − 0.771980i −0.922503 0.385990i $$-0.873860\pi$$
0.922503 0.385990i $$-0.126140\pi$$
$$504$$ 3.65685 0.162889
$$505$$ 0 0
$$506$$ −9.65685 −0.429300
$$507$$ − 19.0000i − 0.843820i
$$508$$ 55.4558i 2.46046i
$$509$$ −18.6863 −0.828255 −0.414128 0.910219i $$-0.635913\pi$$
−0.414128 + 0.910219i $$0.635913\pi$$
$$510$$ 0 0
$$511$$ 9.37258 0.414619
$$512$$ 31.2426i 1.38074i
$$513$$ − 6.82843i − 0.301482i
$$514$$ −39.4558 −1.74032
$$515$$ 0 0
$$516$$ 33.7990 1.48792
$$517$$ 4.00000i 0.175920i
$$518$$ 23.3137i 1.02435i
$$519$$ 2.82843 0.124154
$$520$$ 0 0
$$521$$ −32.6274 −1.42943 −0.714717 0.699414i $$-0.753444\pi$$
−0.714717 + 0.699414i $$0.753444\pi$$
$$522$$ 11.6569i 0.510207i
$$523$$ 9.51472i 0.416050i 0.978124 + 0.208025i $$0.0667035\pi$$
−0.978124 + 0.208025i $$0.933297\pi$$
$$524$$ −12.6863 −0.554203
$$525$$ 0 0
$$526$$ −43.4558 −1.89476
$$527$$ 0 0
$$528$$ − 3.00000i − 0.130558i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ − 21.6569i − 0.938944i
$$533$$ 27.3137i 1.18309i
$$534$$ −8.82843 −0.382043
$$535$$ 0 0
$$536$$ 24.9706 1.07856
$$537$$ 17.6569i 0.761950i
$$538$$ 49.7990i 2.14699i
$$539$$ −6.31371 −0.271951
$$540$$ 0 0
$$541$$ 17.3137 0.744374 0.372187 0.928158i $$-0.378608\pi$$
0.372187 + 0.928158i $$0.378608\pi$$
$$542$$ 28.4853i 1.22355i
$$543$$ − 14.0000i − 0.600798i
$$544$$ 1.85786 0.0796553
$$545$$ 0 0
$$546$$ −11.3137 −0.484182
$$547$$ − 8.14214i − 0.348133i −0.984734 0.174066i $$-0.944309\pi$$
0.984734 0.174066i $$-0.0556907\pi$$
$$548$$ 50.9706i 2.17735i
$$549$$ 11.6569 0.497502
$$550$$ 0 0
$$551$$ 32.9706 1.40459
$$552$$ − 17.6569i − 0.751526i
$$553$$ − 7.02944i − 0.298922i
$$554$$ −5.65685 −0.240337
$$555$$ 0 0
$$556$$ 1.85786 0.0787910
$$557$$ 5.17157i 0.219127i 0.993980 + 0.109563i $$0.0349452\pi$$
−0.993980 + 0.109563i $$0.965055\pi$$
$$558$$ 0 0
$$559$$ −49.9411 −2.11228
$$560$$ 0 0
$$561$$ −1.17157 −0.0494638
$$562$$ 26.9706i 1.13768i
$$563$$ − 31.6569i − 1.33418i −0.744978 0.667089i $$-0.767540\pi$$
0.744978 0.667089i $$-0.232460\pi$$
$$564$$ −15.3137 −0.644823
$$565$$ 0 0
$$566$$ 21.3137 0.895882
$$567$$ 0.828427i 0.0347907i
$$568$$ 10.3431i 0.433989i
$$569$$ 35.4558 1.48639 0.743193 0.669077i $$-0.233310\pi$$
0.743193 + 0.669077i $$0.233310\pi$$
$$570$$ 0 0
$$571$$ −16.4853 −0.689888 −0.344944 0.938623i $$-0.612102\pi$$
−0.344944 + 0.938623i $$0.612102\pi$$
$$572$$ 21.6569i 0.905519i
$$573$$ 5.65685i 0.236318i
$$574$$ 9.65685 0.403069
$$575$$ 0 0
$$576$$ −9.82843 −0.409518
$$577$$ 14.0000i 0.582828i 0.956597 + 0.291414i $$0.0941257\pi$$
−0.956597 + 0.291414i $$0.905874\pi$$
$$578$$ − 37.7279i − 1.56927i
$$579$$ 13.6569 0.567559
$$580$$ 0 0
$$581$$ −8.28427 −0.343689
$$582$$ 28.1421i 1.16653i
$$583$$ 9.31371i 0.385734i
$$584$$ 49.9411 2.06658
$$585$$ 0 0
$$586$$ 16.4853 0.681001
$$587$$ 14.6274i 0.603738i 0.953349 + 0.301869i $$0.0976105\pi$$
−0.953349 + 0.301869i $$0.902389\pi$$
$$588$$ − 24.1716i − 0.996819i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 8.48528 0.349038
$$592$$ 34.9706i 1.43728i
$$593$$ 22.8284i 0.937451i 0.883344 + 0.468726i $$0.155287\pi$$
−0.883344 + 0.468726i $$0.844713\pi$$
$$594$$ 2.41421 0.0990564
$$595$$ 0 0
$$596$$ 5.79899 0.237536
$$597$$ 21.6569i 0.886356i
$$598$$ 54.6274i 2.23388i
$$599$$ −27.3137 −1.11601 −0.558004 0.829838i $$-0.688433\pi$$
−0.558004 + 0.829838i $$0.688433\pi$$
$$600$$ 0 0
$$601$$ −5.31371 −0.216751 −0.108375 0.994110i $$-0.534565\pi$$
−0.108375 + 0.994110i $$0.534565\pi$$
$$602$$ 17.6569i 0.719640i
$$603$$ 5.65685i 0.230365i
$$604$$ −63.1127 −2.56802
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 1.51472i 0.0614805i 0.999527 + 0.0307403i $$0.00978647\pi$$
−0.999527 + 0.0307403i $$0.990214\pi$$
$$608$$ 10.8284i 0.439151i
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 22.6274 0.915407
$$612$$ − 4.48528i − 0.181307i
$$613$$ 45.9411i 1.85554i 0.373147 + 0.927772i $$0.378279\pi$$
−0.373147 + 0.927772i $$0.621721\pi$$
$$614$$ −7.65685 −0.309005
$$615$$ 0 0
$$616$$ 3.65685 0.147339
$$617$$ 0.343146i 0.0138145i 0.999976 + 0.00690726i $$0.00219867\pi$$
−0.999976 + 0.00690726i $$0.997801\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ 14.3431 0.576500 0.288250 0.957555i $$-0.406927\pi$$
0.288250 + 0.957555i $$0.406927\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 8.00000i 0.320771i
$$623$$ − 3.02944i − 0.121372i
$$624$$ −16.9706 −0.679366
$$625$$ 0 0
$$626$$ −37.7990 −1.51075
$$627$$ − 6.82843i − 0.272701i
$$628$$ − 68.9117i − 2.74988i
$$629$$ 13.6569 0.544534
$$630$$ 0 0
$$631$$ 45.6569 1.81757 0.908785 0.417264i $$-0.137011\pi$$
0.908785 + 0.417264i $$0.137011\pi$$
$$632$$ − 37.4558i − 1.48991i
$$633$$ 1.17157i 0.0465658i
$$634$$ −63.4558 −2.52015
$$635$$ 0 0
$$636$$ −35.6569 −1.41389
$$637$$ 35.7157i 1.41511i
$$638$$ 11.6569i 0.461499i
$$639$$ −2.34315 −0.0926934
$$640$$ 0 0
$$641$$ 6.97056 0.275321 0.137660 0.990479i $$-0.456042\pi$$
0.137660 + 0.990479i $$0.456042\pi$$
$$642$$ 41.7990i 1.64967i
$$643$$ − 37.9411i − 1.49625i −0.663557 0.748126i $$-0.730954\pi$$
0.663557 0.748126i $$-0.269046\pi$$
$$644$$ 12.6863 0.499910
$$645$$ 0 0
$$646$$ −19.3137 −0.759888
$$647$$ 4.68629i 0.184237i 0.995748 + 0.0921186i $$0.0293639\pi$$
−0.995748 + 0.0921186i $$0.970636\pi$$
$$648$$ 4.41421i 0.173407i
$$649$$ −4.00000 −0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 28.0000i − 1.09656i
$$653$$ − 6.97056i − 0.272779i −0.990655 0.136390i $$-0.956450\pi$$
0.990655 0.136390i $$-0.0435499\pi$$
$$654$$ −41.7990 −1.63447
$$655$$ 0 0
$$656$$ 14.4853 0.565555
$$657$$ 11.3137i 0.441390i
$$658$$ − 8.00000i − 0.311872i
$$659$$ 15.3137 0.596537 0.298269 0.954482i $$-0.403591\pi$$
0.298269 + 0.954482i $$0.403591\pi$$
$$660$$ 0 0
$$661$$ 9.31371 0.362261 0.181131 0.983459i $$-0.442024\pi$$
0.181131 + 0.983459i $$0.442024\pi$$
$$662$$ − 15.3137i − 0.595184i
$$663$$ 6.62742i 0.257388i
$$664$$ −44.1421 −1.71305
$$665$$ 0 0
$$666$$ −28.1421 −1.09049
$$667$$ 19.3137i 0.747830i
$$668$$ 50.9706i 1.97211i
$$669$$ −6.34315 −0.245240
$$670$$ 0 0
$$671$$ 11.6569 0.450008
$$672$$ − 1.31371i − 0.0506774i
$$673$$ − 18.3431i − 0.707076i −0.935420 0.353538i $$-0.884978\pi$$
0.935420 0.353538i $$-0.115022\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ 72.7401 2.79770
$$677$$ 29.4558i 1.13208i 0.824378 + 0.566040i $$0.191525\pi$$
−0.824378 + 0.566040i $$0.808475\pi$$
$$678$$ 45.7990i 1.75890i
$$679$$ −9.65685 −0.370596
$$680$$ 0 0
$$681$$ 14.0000 0.536481
$$682$$ 0 0
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 26.1421 0.999570
$$685$$ 0 0
$$686$$ 26.6274 1.01664
$$687$$ 2.00000i 0.0763048i
$$688$$ 26.4853i 1.00974i
$$689$$ 52.6863 2.00719
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 10.8284i 0.411635i
$$693$$ 0.828427i 0.0314693i
$$694$$ −70.7696 −2.68638
$$695$$ 0 0
$$696$$ −21.3137 −0.807894
$$697$$ − 5.65685i − 0.214269i
$$698$$ 26.4853i 1.00248i
$$699$$ −18.8284 −0.712157
$$700$$ 0 0
$$701$$ 36.1421 1.36507 0.682535 0.730853i $$-0.260878\pi$$
0.682535 + 0.730853i $$0.260878\pi$$
$$702$$ − 13.6569i − 0.515445i
$$703$$ 79.5980i 3.00209i
$$704$$ −9.82843 −0.370423
$$705$$ 0 0
$$706$$ −62.7696 −2.36236
$$707$$ − 0.686292i − 0.0258106i
$$708$$ − 15.3137i − 0.575524i
$$709$$ −6.68629 −0.251109 −0.125554 0.992087i $$-0.540071\pi$$
−0.125554 + 0.992087i $$0.540071\pi$$
$$710$$ 0 0
$$711$$ 8.48528 0.318223
$$712$$ − 16.1421i − 0.604952i
$$713$$ 0 0
$$714$$ 2.34315 0.0876900
$$715$$ 0 0
$$716$$ −67.5980 −2.52626
$$717$$ − 17.6569i − 0.659407i
$$718$$ 28.9706i 1.08117i
$$719$$ 47.5980 1.77511 0.887553 0.460706i $$-0.152404\pi$$
0.887553 + 0.460706i $$0.152404\pi$$
$$720$$ 0 0
$$721$$ −2.74517 −0.102235
$$722$$ − 66.6985i − 2.48226i
$$723$$ − 12.3431i − 0.459047i
$$724$$ 53.5980 1.99195
$$725$$ 0 0
$$726$$ 2.41421 0.0895999
$$727$$ 33.9411i 1.25881i 0.777079 + 0.629403i $$0.216701\pi$$
−0.777079 + 0.629403i $$0.783299\pi$$
$$728$$ − 20.6863i − 0.766685i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 10.3431 0.382555
$$732$$ 44.6274i 1.64948i
$$733$$ 6.34315i 0.234289i 0.993115 + 0.117145i $$0.0373741\pi$$
−0.993115 + 0.117145i $$0.962626\pi$$
$$734$$ 23.3137 0.860525
$$735$$ 0 0
$$736$$ −6.34315 −0.233811
$$737$$ 5.65685i 0.208373i
$$738$$ 11.6569i 0.429095i
$$739$$ 15.1127 0.555930 0.277965 0.960591i $$-0.410340\pi$$
0.277965 + 0.960591i $$0.410340\pi$$
$$740$$ 0 0
$$741$$ −38.6274 −1.41901
$$742$$ − 18.6274i − 0.683834i
$$743$$ − 36.3431i − 1.33330i −0.745371 0.666650i $$-0.767727\pi$$
0.745371 0.666650i $$-0.232273\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −25.6569 −0.939364
$$747$$ − 10.0000i − 0.365881i
$$748$$ − 4.48528i − 0.163998i
$$749$$ −14.3431 −0.524087
$$750$$ 0 0
$$751$$ 20.2843 0.740184 0.370092 0.928995i $$-0.379326\pi$$
0.370092 + 0.928995i $$0.379326\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 20.9706i 0.764210i
$$754$$ 65.9411 2.40143
$$755$$ 0 0
$$756$$ −3.17157 −0.115349
$$757$$ − 36.6274i − 1.33125i −0.746288 0.665623i $$-0.768166\pi$$
0.746288 0.665623i $$-0.231834\pi$$
$$758$$ − 56.2843i − 2.04434i
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ −28.8284 −1.04503 −0.522515 0.852630i $$-0.675006\pi$$
−0.522515 + 0.852630i $$0.675006\pi$$
$$762$$ − 34.9706i − 1.26685i
$$763$$ − 14.3431i − 0.519257i
$$764$$ −21.6569 −0.783517
$$765$$ 0 0
$$766$$ 19.3137 0.697833
$$767$$ 22.6274i 0.817029i
$$768$$ − 29.9706i − 1.08147i
$$769$$ −10.6863 −0.385358 −0.192679 0.981262i $$-0.561718\pi$$
−0.192679 + 0.981262i $$0.561718\pi$$
$$770$$ 0 0
$$771$$ 16.3431 0.588584
$$772$$ 52.2843i 1.88175i
$$773$$ − 3.65685i − 0.131528i −0.997835 0.0657640i $$-0.979052\pi$$
0.997835 0.0657640i $$-0.0209484\pi$$
$$774$$ −21.3137 −0.766105
$$775$$ 0 0
$$776$$ −51.4558 −1.84716
$$777$$ − 9.65685i − 0.346438i
$$778$$ − 57.1127i − 2.04759i
$$779$$ 32.9706 1.18129
$$780$$ 0 0
$$781$$ −2.34315 −0.0838443
$$782$$ − 11.3137i − 0.404577i
$$783$$ − 4.82843i − 0.172554i
$$784$$ 18.9411 0.676469
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ − 20.1421i − 0.717990i −0.933340 0.358995i $$-0.883120\pi$$
0.933340 0.358995i $$-0.116880\pi$$
$$788$$ 32.4853i 1.15724i
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ −15.7157 −0.558787
$$792$$ 4.41421i 0.156852i
$$793$$ − 65.9411i − 2.34164i
$$794$$ −36.1421 −1.28264
$$795$$ 0 0
$$796$$ −82.9117 −2.93873
$$797$$ 34.9706i 1.23872i 0.785107 + 0.619360i $$0.212608\pi$$
−0.785107 + 0.619360i $$0.787392\pi$$
$$798$$ 13.6569i 0.483447i
$$799$$ −4.68629 −0.165789
$$800$$ 0 0
$$801$$ 3.65685 0.129209
$$802$$ 16.1421i 0.569999i
$$803$$ 11.3137i 0.399252i
$$804$$ −21.6569 −0.763778
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 20.6274i − 0.726119i
$$808$$ − 3.65685i − 0.128648i
$$809$$ −28.4264 −0.999419 −0.499710 0.866193i $$-0.666560\pi$$
−0.499710 + 0.866193i $$0.666560\pi$$
$$810$$ 0 0
$$811$$ 0.485281 0.0170405 0.00852027 0.999964i $$-0.497288\pi$$
0.00852027 + 0.999964i $$0.497288\pi$$
$$812$$ − 15.3137i − 0.537406i
$$813$$ − 11.7990i − 0.413809i
$$814$$ −28.1421 −0.986381
$$815$$ 0 0
$$816$$ 3.51472 0.123040
$$817$$ 60.2843i 2.10908i
$$818$$ − 47.4558i − 1.65925i
$$819$$ 4.68629 0.163752
$$820$$ 0 0
$$821$$ −12.8284 −0.447715 −0.223858 0.974622i $$-0.571865\pi$$
−0.223858 + 0.974622i $$0.571865\pi$$
$$822$$ − 32.1421i − 1.12109i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ −14.6274 −0.509570
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 41.3137i 1.43662i 0.695724 + 0.718309i $$0.255084\pi$$
−0.695724 + 0.718309i $$0.744916\pi$$
$$828$$ 15.3137i 0.532188i
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ 2.34315 0.0812828
$$832$$ 55.5980i 1.92751i
$$833$$ − 7.39697i − 0.256290i
$$834$$ −1.17157 −0.0405683
$$835$$ 0 0
$$836$$ 26.1421 0.904145
$$837$$ 0 0
$$838$$ − 89.2548i − 3.08326i
$$839$$ 22.6274 0.781185 0.390593 0.920564i $$-0.372270\pi$$
0.390593 + 0.920564i $$0.372270\pi$$
$$840$$ 0 0
$$841$$ −5.68629 −0.196079
$$842$$ 14.4853i 0.499196i
$$843$$ − 11.1716i − 0.384769i
$$844$$ −4.48528 −0.154390
$$845$$ 0 0
$$846$$ 9.65685 0.332009
$$847$$ 0.828427i 0.0284651i
$$848$$ − 27.9411i − 0.959502i
$$849$$ −8.82843 −0.302991
$$850$$ 0 0
$$851$$ −46.6274 −1.59837
$$852$$ − 8.97056i − 0.307326i
$$853$$ 8.68629i 0.297413i 0.988881 + 0.148706i $$0.0475110\pi$$
−0.988881 + 0.148706i $$0.952489\pi$$
$$854$$ −23.3137 −0.797779
$$855$$ 0 0
$$856$$ −76.4264 −2.61220
$$857$$ − 28.4853i − 0.973039i −0.873670 0.486519i $$-0.838266\pi$$
0.873670 0.486519i $$-0.161734\pi$$
$$858$$ − 13.6569i − 0.466237i
$$859$$ 52.9706 1.80733 0.903666 0.428238i $$-0.140865\pi$$
0.903666 + 0.428238i $$0.140865\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ − 52.2843i − 1.78081i
$$863$$ 20.6863i 0.704170i 0.935968 + 0.352085i $$0.114527\pi$$
−0.935968 + 0.352085i $$0.885473\pi$$
$$864$$ 1.58579 0.0539496
$$865$$ 0 0
$$866$$ 37.7990 1.28446
$$867$$ 15.6274i 0.530735i
$$868$$ 0 0
$$869$$ 8.48528 0.287843
$$870$$ 0 0
$$871$$ 32.0000 1.08428
$$872$$ − 76.4264i − 2.58812i
$$873$$ − 11.6569i − 0.394525i
$$874$$ 65.9411 2.23049
$$875$$ 0 0
$$876$$ −43.3137 −1.46343
$$877$$ 2.62742i 0.0887216i 0.999016 + 0.0443608i $$0.0141251\pi$$
−0.999016 + 0.0443608i $$0.985875\pi$$
$$878$$ − 49.4558i − 1.66905i
$$879$$ −6.82843 −0.230317
$$880$$ 0 0
$$881$$ −46.9706 −1.58248 −0.791239 0.611507i $$-0.790564\pi$$
−0.791239 + 0.611507i $$0.790564\pi$$
$$882$$ 15.2426i 0.513246i
$$883$$ 5.37258i 0.180802i 0.995905 + 0.0904009i $$0.0288148\pi$$
−0.995905 + 0.0904009i $$0.971185\pi$$
$$884$$ −25.3726 −0.853372
$$885$$ 0 0
$$886$$ −28.9706 −0.973285
$$887$$ − 15.6569i − 0.525706i −0.964836 0.262853i $$-0.915337\pi$$
0.964836 0.262853i $$-0.0846634\pi$$
$$888$$ − 51.4558i − 1.72675i
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ − 24.2843i − 0.813098i
$$893$$ − 27.3137i − 0.914018i
$$894$$ −3.65685 −0.122304
$$895$$ 0 0
$$896$$ 17.0294 0.568914
$$897$$ − 22.6274i − 0.755507i
$$898$$ 74.7696i 2.49509i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −10.9117 −0.363521
$$902$$ 11.6569i 0.388131i
$$903$$ − 7.31371i − 0.243385i
$$904$$ −83.7401 −2.78515
$$905$$ 0 0
$$906$$ 39.7990 1.32223
$$907$$ 40.9706i 1.36041i 0.733024 + 0.680203i $$0.238108\pi$$
−0.733024 + 0.680203i $$0.761892\pi$$
$$908$$ 53.5980i 1.77871i
$$909$$ 0.828427 0.0274772
$$910$$ 0 0
$$911$$ −48.9706 −1.62247 −0.811234 0.584722i $$-0.801203\pi$$
−0.811234 + 0.584722i $$0.801203\pi$$
$$912$$ 20.4853i 0.678335i
$$913$$ − 10.0000i − 0.330952i
$$914$$ −56.2843 −1.86172
$$915$$ 0 0
$$916$$ −7.65685 −0.252990
$$917$$ 2.74517i 0.0906534i
$$918$$ 2.82843i 0.0933520i
$$919$$ −11.5147 −0.379836 −0.189918 0.981800i $$-0.560822\pi$$
−0.189918 + 0.981800i $$0.560822\pi$$
$$920$$ 0 0
$$921$$ 3.17157 0.104507
$$922$$ − 0.343146i − 0.0113009i
$$923$$ 13.2548i 0.436288i
$$924$$ −3.17157 −0.104337
$$925$$ 0 0
$$926$$ −12.0000 −0.394344
$$927$$ − 3.31371i − 0.108836i
$$928$$ 7.65685i 0.251349i
$$929$$ 45.5980 1.49602 0.748011 0.663687i $$-0.231009\pi$$
0.748011 + 0.663687i $$0.231009\pi$$
$$930$$ 0 0
$$931$$ 43.1127 1.41296
$$932$$ − 72.0833i − 2.36117i
$$933$$ − 3.31371i − 0.108486i
$$934$$ −54.6274 −1.78746
$$935$$ 0 0
$$936$$ 24.9706 0.816188
$$937$$ 11.0294i 0.360316i 0.983638 + 0.180158i $$0.0576609\pi$$
−0.983638 + 0.180158i $$0.942339\pi$$
$$938$$ − 11.3137i − 0.369406i
$$939$$ 15.6569 0.510942
$$940$$ 0 0
$$941$$ −34.7696 −1.13346 −0.566728 0.823905i $$-0.691791\pi$$
−0.566728 + 0.823905i $$0.691791\pi$$
$$942$$ 43.4558i 1.41587i
$$943$$ 19.3137i 0.628941i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −21.3137 −0.692968
$$947$$ 6.62742i 0.215362i 0.994185 + 0.107681i $$0.0343425\pi$$
−0.994185 + 0.107681i $$0.965657\pi$$
$$948$$ 32.4853i 1.05507i
$$949$$ 64.0000 2.07753
$$950$$ 0 0
$$951$$ 26.2843 0.852326
$$952$$ 4.28427i 0.138854i
$$953$$ 11.7990i 0.382207i 0.981570 + 0.191103i $$0.0612066\pi$$
−0.981570 + 0.191103i $$0.938793\pi$$
$$954$$ 22.4853 0.727988
$$955$$ 0 0
$$956$$ 67.5980 2.18627
$$957$$ − 4.82843i − 0.156081i
$$958$$ 89.2548i 2.88369i
$$959$$ 11.0294 0.356159
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 159.196i 5.13268i
$$963$$ − 17.3137i − 0.557926i
$$964$$ 47.2548 1.52198
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 11.4558i 0.368395i 0.982889 + 0.184198i $$0.0589686\pi$$
−0.982889 + 0.184198i $$0.941031\pi$$
$$968$$ 4.41421i 0.141878i
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −34.6274 −1.11125 −0.555623 0.831434i $$-0.687520\pi$$
−0.555623 + 0.831434i $$0.687520\pi$$
$$972$$ − 3.82843i − 0.122797i
$$973$$ − 0.402020i − 0.0128882i
$$974$$ −31.3137 −1.00336
$$975$$ 0 0
$$976$$ −34.9706 −1.11938
$$977$$ 2.68629i 0.0859421i 0.999076 + 0.0429710i $$0.0136823\pi$$
−0.999076 + 0.0429710i $$0.986318\pi$$
$$978$$ 17.6569i 0.564604i
$$979$$ 3.65685 0.116874
$$980$$ 0 0
$$981$$ 17.3137 0.552784
$$982$$ − 34.6274i − 1.10501i
$$983$$ 30.6274i 0.976863i 0.872602 + 0.488431i $$0.162431\pi$$
−0.872602 + 0.488431i $$0.837569\pi$$
$$984$$ −21.3137 −0.679456
$$985$$ 0 0
$$986$$ −13.6569 −0.434923
$$987$$ 3.31371i 0.105477i
$$988$$ − 147.882i − 4.70476i
$$989$$ −35.3137 −1.12291
$$990$$ 0 0
$$991$$ −30.6274 −0.972912 −0.486456 0.873705i $$-0.661711\pi$$
−0.486456 + 0.873705i $$0.661711\pi$$
$$992$$ 0 0
$$993$$ 6.34315i 0.201294i
$$994$$ 4.68629 0.148640
$$995$$ 0 0
$$996$$ 38.2843 1.21308
$$997$$ − 39.3137i − 1.24508i −0.782589 0.622539i $$-0.786101\pi$$
0.782589 0.622539i $$-0.213899\pi$$
$$998$$ − 53.9411i − 1.70748i
$$999$$ 11.6569 0.368807
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 825.2.c.e.199.1 4
3.2 odd 2 2475.2.c.m.199.4 4
5.2 odd 4 825.2.a.g.1.2 2
5.3 odd 4 165.2.a.a.1.1 2
5.4 even 2 inner 825.2.c.e.199.4 4
15.2 even 4 2475.2.a.m.1.1 2
15.8 even 4 495.2.a.d.1.2 2
15.14 odd 2 2475.2.c.m.199.1 4
20.3 even 4 2640.2.a.bb.1.1 2
35.13 even 4 8085.2.a.ba.1.1 2
55.32 even 4 9075.2.a.v.1.1 2
55.43 even 4 1815.2.a.k.1.2 2
60.23 odd 4 7920.2.a.cg.1.1 2
165.98 odd 4 5445.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 5.3 odd 4
495.2.a.d.1.2 2 15.8 even 4
825.2.a.g.1.2 2 5.2 odd 4
825.2.c.e.199.1 4 1.1 even 1 trivial
825.2.c.e.199.4 4 5.4 even 2 inner
1815.2.a.k.1.2 2 55.43 even 4
2475.2.a.m.1.1 2 15.2 even 4
2475.2.c.m.199.1 4 15.14 odd 2
2475.2.c.m.199.4 4 3.2 odd 2
2640.2.a.bb.1.1 2 20.3 even 4
5445.2.a.m.1.1 2 165.98 odd 4
7920.2.a.cg.1.1 2 60.23 odd 4
8085.2.a.ba.1.1 2 35.13 even 4
9075.2.a.v.1.1 2 55.32 even 4