Properties

 Label 5850.2.e.f.5149.1 Level $5850$ Weight $2$ Character 5850.5149 Analytic conductor $46.712$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$46.7124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1950) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 5149.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5850.5149 Dual form 5850.2.e.f.5149.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} -1.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{26} -3.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +5.00000i q^{37} +1.00000i q^{38} -9.00000 q^{41} +2.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -3.00000i q^{47} +7.00000 q^{49} +1.00000i q^{52} -1.00000i q^{53} +3.00000i q^{58} +10.0000 q^{59} +4.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +9.00000i q^{67} -7.00000 q^{71} +4.00000i q^{73} +5.00000 q^{74} +1.00000 q^{76} -11.0000 q^{79} +9.00000i q^{82} -6.00000i q^{83} +2.00000 q^{86} -4.00000i q^{88} +10.0000 q^{89} +4.00000i q^{92} -3.00000 q^{94} +12.0000i q^{97} -7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 8q^{11} + 2q^{16} - 2q^{19} - 2q^{26} - 6q^{29} + 8q^{31} - 18q^{41} + 8q^{44} - 8q^{46} + 14q^{49} + 20q^{59} + 8q^{61} - 2q^{64} - 14q^{71} + 10q^{74} + 2q^{76} - 22q^{79} + 4q^{86} + 20q^{89} - 6q^{94} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$$.

 $$n$$ $$2251$$ $$3251$$ $$3277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.00000i 0.821995i 0.911636 + 0.410997i $$0.134819\pi$$
−0.911636 + 0.410997i $$0.865181\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ − 3.00000i − 0.437595i −0.975770 0.218797i $$-0.929787\pi$$
0.975770 0.218797i $$-0.0702134\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ − 1.00000i − 0.137361i −0.997639 0.0686803i $$-0.978121\pi$$
0.997639 0.0686803i $$-0.0218788\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 3.00000i 0.393919i
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 9.00000i 1.09952i 0.835321 + 0.549762i $$0.185282\pi$$
−0.835321 + 0.549762i $$0.814718\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −7.00000 −0.830747 −0.415374 0.909651i $$-0.636349\pi$$
−0.415374 + 0.909651i $$0.636349\pi$$
$$72$$ 0 0
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 5.00000 0.581238
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 9.00000i 0.993884i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ − 4.00000i − 0.426401i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000i 0.417029i
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 12.0000i 1.21842i 0.793011 + 0.609208i $$0.208512\pi$$
−0.793011 + 0.609208i $$0.791488\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ 9.00000i 0.870063i 0.900415 + 0.435031i $$0.143263\pi$$
−0.900415 + 0.435031i $$0.856737\pi$$
$$108$$ 0 0
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ − 10.0000i − 0.920575i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 4.00000i − 0.362143i
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 17.0000i 1.50851i 0.656584 + 0.754253i $$0.272001\pi$$
−0.656584 + 0.754253i $$0.727999\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 9.00000 0.777482
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 23.0000i 1.96502i 0.186203 + 0.982511i $$0.440382\pi$$
−0.186203 + 0.982511i $$0.559618\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 7.00000i 0.587427i
$$143$$ 4.00000i 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ − 5.00000i − 0.410997i
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 11.0000i 0.875113i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ 9.00000 0.702782
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 7.00000i 0.541676i 0.962625 + 0.270838i $$0.0873008\pi$$
−0.962625 + 0.270838i $$0.912699\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 2.00000i − 0.152499i
$$173$$ 3.00000i 0.228086i 0.993476 + 0.114043i $$0.0363801\pi$$
−0.993476 + 0.114043i $$0.963620\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 3.00000i 0.218797i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 20.0000i 1.43963i 0.694165 + 0.719816i $$0.255774\pi$$
−0.694165 + 0.719816i $$0.744226\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ −1.00000 −0.0708881 −0.0354441 0.999372i $$-0.511285\pi$$
−0.0354441 + 0.999372i $$0.511285\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 2.00000i − 0.140720i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ 1.00000i 0.0686803i
$$213$$ 0 0
$$214$$ 9.00000 0.615227
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 1.00000i − 0.0677285i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ − 10.0000i − 0.663723i −0.943328 0.331862i $$-0.892323\pi$$
0.943328 0.331862i $$-0.107677\pi$$
$$228$$ 0 0
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 3.00000i − 0.196960i
$$233$$ − 8.00000i − 0.524097i −0.965055 0.262049i $$-0.915602\pi$$
0.965055 0.262049i $$-0.0843981\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.00000i 0.0636285i
$$248$$ 4.00000i 0.254000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −23.0000 −1.45175 −0.725874 0.687828i $$-0.758564\pi$$
−0.725874 + 0.687828i $$0.758564\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 17.0000 1.06667
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 24.0000i 1.49708i 0.663090 + 0.748539i $$0.269245\pi$$
−0.663090 + 0.748539i $$0.730755\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 3.00000i − 0.185341i
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 9.00000i − 0.549762i
$$269$$ −11.0000 −0.670682 −0.335341 0.942097i $$-0.608852\pi$$
−0.335341 + 0.942097i $$0.608852\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 23.0000 1.38948
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.00000 0.298275 0.149137 0.988816i $$-0.452350\pi$$
0.149137 + 0.988816i $$0.452350\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 7.00000 0.415374
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 4.00000i − 0.234082i
$$293$$ 16.0000i 0.934730i 0.884064 + 0.467365i $$0.154797\pi$$
−0.884064 + 0.467365i $$0.845203\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −5.00000 −0.290619
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 12.0000i − 0.690522i
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 3.00000i − 0.171219i −0.996329 0.0856095i $$-0.972716\pi$$
0.996329 0.0856095i $$-0.0272838\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 1.00000i 0.0565233i 0.999601 + 0.0282617i $$0.00899717\pi$$
−0.999601 + 0.0282617i $$0.991003\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 11.0000 0.618798
$$317$$ 32.0000i 1.79730i 0.438667 + 0.898650i $$0.355451\pi$$
−0.438667 + 0.898650i $$0.644549\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 0 0
$$328$$ − 9.00000i − 0.496942i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 0 0
$$334$$ 7.00000 0.383023
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 3.00000 0.161281
$$347$$ − 17.0000i − 0.912608i −0.889824 0.456304i $$-0.849173\pi$$
0.889824 0.456304i $$-0.150827\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000i 0.213201i
$$353$$ 15.0000i 0.798369i 0.916871 + 0.399185i $$0.130707\pi$$
−0.916871 + 0.399185i $$0.869293\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ −17.0000 −0.897226 −0.448613 0.893726i $$-0.648082\pi$$
−0.448613 + 0.893726i $$0.648082\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 12.0000i 0.630706i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 21.0000i 1.09619i 0.836416 + 0.548096i $$0.184647\pi$$
−0.836416 + 0.548096i $$0.815353\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ 3.00000i 0.154508i
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 18.0000i − 0.920960i
$$383$$ 5.00000i 0.255488i 0.991807 + 0.127744i $$0.0407736\pi$$
−0.991807 + 0.127744i $$0.959226\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 20.0000 1.01797
$$387$$ 0 0
$$388$$ − 12.0000i − 0.609208i
$$389$$ −19.0000 −0.963338 −0.481669 0.876353i $$-0.659969\pi$$
−0.481669 + 0.876353i $$0.659969\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 7.00000i 0.353553i
$$393$$ 0 0
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.00000i − 0.150566i −0.997162 0.0752828i $$-0.976014\pi$$
0.997162 0.0752828i $$-0.0239860\pi$$
$$398$$ 1.00000i 0.0501255i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 0 0
$$403$$ − 4.00000i − 0.199254i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.0000i − 0.991363i
$$408$$ 0 0
$$409$$ 40.0000 1.97787 0.988936 0.148340i $$-0.0473931\pi$$
0.988936 + 0.148340i $$0.0473931\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ − 4.00000i − 0.195646i
$$419$$ −37.0000 −1.80757 −0.903784 0.427989i $$-0.859222\pi$$
−0.903784 + 0.427989i $$0.859222\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ − 24.0000i − 1.16830i
$$423$$ 0 0
$$424$$ 1.00000 0.0485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 9.00000i − 0.435031i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ − 19.0000i − 0.913082i −0.889702 0.456541i $$-0.849088\pi$$
0.889702 0.456541i $$-0.150912\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1.00000 −0.0478913
$$437$$ 4.00000i 0.191346i
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3.00000i 0.142534i 0.997457 + 0.0712672i $$0.0227043\pi$$
−0.997457 + 0.0712672i $$0.977296\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 2.00000i 0.0940721i
$$453$$ 0 0
$$454$$ −10.0000 −0.469323
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 32.0000i − 1.49690i −0.663193 0.748448i $$-0.730799\pi$$
0.663193 0.748448i $$-0.269201\pi$$
$$458$$ 5.00000i 0.233635i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 0 0
$$463$$ − 14.0000i − 0.650635i −0.945605 0.325318i $$-0.894529\pi$$
0.945605 0.325318i $$-0.105471\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ −8.00000 −0.370593
$$467$$ 15.0000i 0.694117i 0.937843 + 0.347059i $$0.112820\pi$$
−0.937843 + 0.347059i $$0.887180\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 10.0000i 0.460287i
$$473$$ − 8.00000i − 0.367840i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 8.00000i 0.365911i
$$479$$ 25.0000 1.14228 0.571140 0.820853i $$-0.306501\pi$$
0.571140 + 0.820853i $$0.306501\pi$$
$$480$$ 0 0
$$481$$ 5.00000 0.227980
$$482$$ 12.0000i 0.546585i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.0000i 1.72194i 0.508652 + 0.860972i $$0.330144\pi$$
−0.508652 + 0.860972i $$0.669856\pi$$
$$488$$ 4.00000i 0.181071i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 32.0000 1.44414 0.722070 0.691820i $$-0.243191\pi$$
0.722070 + 0.691820i $$0.243191\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 1.00000 0.0449921
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.00000 0.0447661 0.0223831 0.999749i $$-0.492875\pi$$
0.0223831 + 0.999749i $$0.492875\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 23.0000i 1.02654i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ 0 0
$$508$$ − 17.0000i − 0.754253i
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 12.0000i 0.527759i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ − 30.0000i − 1.31181i −0.754844 0.655904i $$-0.772288\pi$$
0.754844 0.655904i $$-0.227712\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 9.00000i 0.389833i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −9.00000 −0.388741
$$537$$ 0 0
$$538$$ 11.0000i 0.474244i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −26.0000 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$542$$ − 24.0000i − 1.03089i
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 22.0000i 0.940652i 0.882493 + 0.470326i $$0.155864\pi$$
−0.882493 + 0.470326i $$0.844136\pi$$
$$548$$ − 23.0000i − 0.982511i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.00000 0.127804
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 42.0000i 1.77960i 0.456354 + 0.889799i $$0.349155\pi$$
−0.456354 + 0.889799i $$0.650845\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 5.00000i − 0.210912i
$$563$$ 21.0000i 0.885044i 0.896758 + 0.442522i $$0.145916\pi$$
−0.896758 + 0.442522i $$0.854084\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ − 7.00000i − 0.293713i
$$569$$ −8.00000 −0.335377 −0.167689 0.985840i $$-0.553630\pi$$
−0.167689 + 0.985840i $$0.553630\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 12.0000i − 0.499567i −0.968302 0.249783i $$-0.919641\pi$$
0.968302 0.249783i $$-0.0803594\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000i 0.165663i
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 16.0000 0.660954
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 5.00000i 0.205499i
$$593$$ 1.00000i 0.0410651i 0.999789 + 0.0205325i $$0.00653617\pi$$
−0.999789 + 0.0205325i $$0.993464\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 4.00000i 0.163572i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 45.0000 1.83559 0.917794 0.397057i $$-0.129968\pi$$
0.917794 + 0.397057i $$0.129968\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.00000i 0.284121i 0.989858 + 0.142061i $$0.0453728\pi$$
−0.989858 + 0.142061i $$0.954627\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ −3.00000 −0.121070
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 27.0000i − 1.08698i −0.839416 0.543490i $$-0.817103\pi$$
0.839416 0.543490i $$-0.182897\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 30.0000i − 1.20289i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ 2.00000i 0.0798087i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ − 11.0000i − 0.437557i
$$633$$ 0 0
$$634$$ 32.0000 1.27088
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 7.00000i − 0.277350i
$$638$$ − 12.0000i − 0.475085i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ − 19.0000i − 0.749287i −0.927169 0.374643i $$-0.877765\pi$$
0.927169 0.374643i $$-0.122235\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 0 0
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 8.00000i − 0.313304i
$$653$$ 26.0000i 1.01746i 0.860927 + 0.508729i $$0.169885\pi$$
−0.860927 + 0.508729i $$0.830115\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 13.0000 0.506408 0.253204 0.967413i $$-0.418516\pi$$
0.253204 + 0.967413i $$0.418516\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.0000i 0.464642i
$$668$$ − 7.00000i − 0.270838i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −16.0000 −0.617673
$$672$$ 0 0
$$673$$ 11.0000i 0.424019i 0.977268 + 0.212009i $$0.0680008\pi$$
−0.977268 + 0.212009i $$0.931999\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 2.00000i 0.0768662i 0.999261 + 0.0384331i $$0.0122367\pi$$
−0.999261 + 0.0384331i $$0.987763\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 16.0000i 0.612672i
$$683$$ 28.0000i 1.07139i 0.844411 + 0.535695i $$0.179950\pi$$
−0.844411 + 0.535695i $$0.820050\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.00000i 0.0762493i
$$689$$ −1.00000 −0.0380970
$$690$$ 0 0
$$691$$ −3.00000 −0.114125 −0.0570627 0.998371i $$-0.518173\pi$$
−0.0570627 + 0.998371i $$0.518173\pi$$
$$692$$ − 3.00000i − 0.114043i
$$693$$ 0 0
$$694$$ −17.0000 −0.645311
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 18.0000i 0.681310i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 38.0000 1.43524 0.717620 0.696435i $$-0.245231\pi$$
0.717620 + 0.696435i $$0.245231\pi$$
$$702$$ 0 0
$$703$$ − 5.00000i − 0.188579i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000i 0.374766i
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 17.0000i 0.634434i
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000i 0.669891i
$$723$$ 0 0
$$724$$ 12.0000 0.445976
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 52.0000i − 1.92857i −0.264861 0.964287i $$-0.585326\pi$$
0.264861 0.964287i $$-0.414674\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ − 49.0000i − 1.80986i −0.425564 0.904928i $$-0.639924\pi$$
0.425564 0.904928i $$-0.360076\pi$$
$$734$$ 21.0000 0.775124
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ − 36.0000i − 1.32608i
$$738$$ 0 0
$$739$$ −53.0000 −1.94964 −0.974818 0.223001i $$-0.928415\pi$$
−0.974818 + 0.223001i $$0.928415\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 27.0000i 0.990534i 0.868741 + 0.495267i $$0.164930\pi$$
−0.868741 + 0.495267i $$0.835070\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −43.0000 −1.56909 −0.784546 0.620070i $$-0.787104\pi$$
−0.784546 + 0.620070i $$0.787104\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ 0 0
$$754$$ 3.00000 0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ − 28.0000i − 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 15.0000 0.543750 0.271875 0.962333i $$-0.412356\pi$$
0.271875 + 0.962333i $$0.412356\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 5.00000 0.180657
$$767$$ − 10.0000i − 0.361079i
$$768$$ 0 0
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 20.0000i − 0.719816i
$$773$$ 32.0000i 1.15096i 0.817816 + 0.575480i $$0.195185\pi$$
−0.817816 + 0.575480i $$0.804815\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ 0 0
$$778$$ 19.0000i 0.681183i
$$779$$ 9.00000 0.322458
$$780$$ 0 0
$$781$$ 28.0000 1.00192
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 4.00000i − 0.142044i
$$794$$ −3.00000 −0.106466
$$795$$ 0 0
$$796$$ 1.00000 0.0354441
$$797$$ − 22.0000i − 0.779280i −0.920967 0.389640i $$-0.872599\pi$$
0.920967 0.389640i $$-0.127401\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 26.0000i − 0.918092i
$$803$$ − 16.0000i − 0.564628i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 2.00000i 0.0703598i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −20.0000 −0.701000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 2.00000i − 0.0699711i
$$818$$ − 40.0000i − 1.39857i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ − 31.0000i − 1.08059i −0.841475 0.540296i $$-0.818312\pi$$
0.841475 0.540296i $$-0.181688\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000i 0.0346688i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ 37.0000i 1.27814i
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ − 2.00000i − 0.0689246i
$$843$$ 0 0
$$844$$ −24.0000 −0.826114
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 1.00000i − 0.0343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ 0 0
$$853$$ 41.0000i 1.40381i 0.712269 + 0.701907i $$0.247668\pi$$
−0.712269 + 0.701907i $$0.752332\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ 42.0000i 1.43469i 0.696717 + 0.717346i $$0.254643\pi$$
−0.696717 + 0.717346i $$0.745357\pi$$
$$858$$ 0 0
$$859$$ −22.0000 −0.750630 −0.375315 0.926897i $$-0.622466\pi$$
−0.375315 + 0.926897i $$0.622466\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 3.00000i − 0.102180i
$$863$$ − 57.0000i − 1.94030i −0.242500 0.970151i $$-0.577968\pi$$
0.242500 0.970151i $$-0.422032\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −19.0000 −0.645646
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 44.0000 1.49260
$$870$$ 0 0
$$871$$ 9.00000 0.304953
$$872$$ 1.00000i 0.0338643i
$$873$$ 0 0
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.0000i 0.776655i 0.921521 + 0.388327i $$0.126947\pi$$
−0.921521 + 0.388327i $$0.873053\pi$$
$$878$$ − 19.0000i − 0.641219i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −56.0000 −1.88669 −0.943344 0.331816i $$-0.892339\pi$$
−0.943344 + 0.331816i $$0.892339\pi$$
$$882$$ 0 0
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 3.00000 0.100787
$$887$$ − 44.0000i − 1.47738i −0.674048 0.738688i $$-0.735446\pi$$
0.674048 0.738688i $$-0.264554\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.0000i 0.468755i
$$893$$ 3.00000i 0.100391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 9.00000i 0.300334i
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 0 0
$$902$$ − 36.0000i − 1.19867i
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 30.0000i 0.996134i 0.867139 + 0.498067i $$0.165957\pi$$
−0.867139 + 0.498067i $$0.834043\pi$$
$$908$$ 10.0000i 0.331862i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ 24.0000i 0.794284i
$$914$$ −32.0000 −1.05847
$$915$$ 0 0
$$916$$ 5.00000 0.165205
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1.00000 −0.0329870 −0.0164935 0.999864i $$-0.505250\pi$$
−0.0164935 + 0.999864i $$0.505250\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 24.0000i − 0.790398i
$$923$$ 7.00000i 0.230408i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −14.0000 −0.460069
$$927$$ 0 0
$$928$$ 3.00000i 0.0984798i
$$929$$ 1.00000 0.0328089 0.0164045 0.999865i $$-0.494778\pi$$
0.0164045 + 0.999865i $$0.494778\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ 8.00000i 0.262049i
$$933$$ 0 0
$$934$$ 15.0000 0.490815
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 34.0000i − 1.11073i −0.831606 0.555366i $$-0.812578\pi$$
0.831606 0.555366i $$-0.187422\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ 36.0000i 1.17232i
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 56.0000i − 1.81402i −0.421111 0.907009i $$-0.638360\pi$$
0.421111 0.907009i $$-0.361640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ − 25.0000i − 0.807713i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 5.00000i − 0.161206i
$$963$$ 0 0
$$964$$ 12.0000 0.386494
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −15.0000 −0.481373 −0.240686 0.970603i $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 38.0000 1.21760
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ 0 0
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 32.0000i − 1.02116i
$$983$$ 32.0000i 1.02064i 0.859984 + 0.510321i $$0.170473\pi$$
−0.859984 + 0.510321i $$0.829527\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 1.00000i − 0.0318142i
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ −7.00000 −0.222362 −0.111181 0.993800i $$-0.535463\pi$$
−0.111181 + 0.993800i $$0.535463\pi$$
$$992$$ − 4.00000i − 0.127000i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 14.0000i 0.443384i 0.975117 + 0.221692i $$0.0711580\pi$$
−0.975117 + 0.221692i $$0.928842\pi$$
$$998$$ − 1.00000i − 0.0316544i
$$999$$ 0 0
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 5850.2.e.f.5149.1 2
3.2 odd 2 1950.2.e.h.1249.2 2
5.2 odd 4 5850.2.a.bp.1.1 1
5.3 odd 4 5850.2.a.l.1.1 1
5.4 even 2 inner 5850.2.e.f.5149.2 2
15.2 even 4 1950.2.a.j.1.1 1
15.8 even 4 1950.2.a.s.1.1 yes 1
15.14 odd 2 1950.2.e.h.1249.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.j.1.1 1 15.2 even 4
1950.2.a.s.1.1 yes 1 15.8 even 4
1950.2.e.h.1249.1 2 15.14 odd 2
1950.2.e.h.1249.2 2 3.2 odd 2
5850.2.a.l.1.1 1 5.3 odd 4
5850.2.a.bp.1.1 1 5.2 odd 4
5850.2.e.f.5149.1 2 1.1 even 1 trivial
5850.2.e.f.5149.2 2 5.4 even 2 inner