# Properties

 Label 1950.2.e.h.1249.1 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1249 Dual form 1950.2.e.h.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} +1.00000 q^{16} +1.00000i q^{18} -1.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} +1.00000 q^{36} -5.00000i q^{37} +1.00000i q^{38} +1.00000 q^{39} +9.00000 q^{41} -2.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -3.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -1.00000i q^{52} -1.00000i q^{53} +1.00000 q^{54} +1.00000i q^{57} -3.00000i q^{58} -10.0000 q^{59} +4.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -9.00000i q^{67} -4.00000 q^{69} +7.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -5.00000 q^{74} +1.00000 q^{76} -1.00000i q^{78} -11.0000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -6.00000i q^{83} -2.00000 q^{86} -3.00000i q^{87} +4.00000i q^{88} -10.0000 q^{89} +4.00000i q^{92} -4.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} -12.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} + 2q^{16} - 2q^{19} + 2q^{24} + 2q^{26} + 6q^{29} + 8q^{31} + 2q^{36} + 2q^{39} + 18q^{41} - 8q^{44} - 8q^{46} + 14q^{49} + 2q^{54} - 20q^{59} + 8q^{61} - 2q^{64} - 8q^{66} - 8q^{69} + 14q^{71} - 10q^{74} + 2q^{76} - 22q^{79} + 2q^{81} - 4q^{86} - 20q^{89} - 6q^{94} - 2q^{96} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 1.00000i 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −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 $$-0.136929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 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
$$33$$ − 4.00000i − 0.696311i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 5.00000i − 0.821995i −0.911636 0.410997i $$-0.865181\pi$$
0.911636 0.410997i $$-0.134819\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ −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 $$-0.0702134\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 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 $$-0.0218788\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000i 0.132453i
$$58$$ − 3.00000i − 0.393919i
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 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
$$66$$ −4.00000 −0.492366
$$67$$ − 9.00000i − 1.09952i −0.835321 0.549762i $$-0.814718\pi$$
0.835321 0.549762i $$-0.185282\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ −5.00000 −0.581238
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ − 1.00000i − 0.113228i
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 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
$$87$$ − 3.00000i − 0.321634i
$$88$$ 4.00000i 0.426401i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000i 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 12.0000i − 1.21842i −0.793011 0.609208i $$-0.791488\pi$$
0.793011 0.609208i $$-0.208512\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 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 $$0.856737\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ −5.00000 −0.474579
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −3.00000 −0.278543
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 10.0000i 0.920575i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 4.00000i − 0.362143i
$$123$$ − 9.00000i − 0.811503i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 17.0000i − 1.50851i −0.656584 0.754253i $$-0.727999\pi$$
0.656584 0.754253i $$-0.272001\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 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 $$0.559618\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ − 7.00000i − 0.587427i
$$143$$ 4.00000i 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ − 7.00000i − 0.577350i
$$148$$ 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
$$156$$ −1.00000 −0.0800641
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 11.0000i 0.875113i
$$159$$ −1.00000 −0.0793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ −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
$$171$$ 1.00000 0.0764719
$$172$$ 2.00000i 0.152499i
$$173$$ 3.00000i 0.228086i 0.993476 + 0.114043i $$0.0363801\pi$$
−0.993476 + 0.114043i $$0.963620\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 10.0000i 0.751646i
$$178$$ 10.0000i 0.749532i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 3.00000i 0.218797i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 20.0000i − 1.43963i −0.694165 0.719816i $$-0.744226\pi$$
0.694165 0.719816i $$-0.255774\pi$$
$$194$$ −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 $$0.859401\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ −1.00000 −0.0708881 −0.0354441 0.999372i $$-0.511285\pi$$
−0.0354441 + 0.999372i $$0.511285\pi$$
$$200$$ 0 0
$$201$$ −9.00000 −0.634811
$$202$$ 2.00000i 0.140720i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 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
$$213$$ − 7.00000i − 0.479632i
$$214$$ 9.00000 0.615227
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 1.00000i − 0.0677285i
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 5.00000i 0.335578i
$$223$$ 14.0000i 0.937509i 0.883328 + 0.468755i $$0.155297\pi$$
−0.883328 + 0.468755i $$0.844703\pi$$
$$224$$ 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 $$-0.107677\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −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 $$-0.0843981\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ 11.0000i 0.714527i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 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
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ −9.00000 −0.573819
$$247$$ − 1.00000i − 0.0636285i
$$248$$ 4.00000i 0.254000i
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 23.0000 1.45175 0.725874 0.687828i $$-0.241436\pi$$
0.725874 + 0.687828i $$0.241436\pi$$
$$252$$ 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 $$0.730755\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −3.00000 −0.185695
$$262$$ 3.00000i 0.185341i
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000i 0.611990i
$$268$$ 9.00000i 0.549762i
$$269$$ 11.0000 0.670682 0.335341 0.942097i $$-0.391148\pi$$
0.335341 + 0.942097i $$0.391148\pi$$
$$270$$ 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
$$276$$ 4.00000 0.240772
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −5.00000 −0.298275 −0.149137 0.988816i $$-0.547650\pi$$
−0.149137 + 0.988816i $$0.547650\pi$$
$$282$$ 3.00000i 0.178647i
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ −7.00000 −0.415374
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 4.00000i 0.234082i
$$293$$ 16.0000i 0.934730i 0.884064 + 0.467365i $$0.154797\pi$$
−0.884064 + 0.467365i $$0.845203\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ 5.00000 0.290619
$$297$$ 4.00000i 0.232104i
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 12.0000i − 0.690522i
$$303$$ 2.00000i 0.114897i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3.00000i 0.171219i 0.996329 + 0.0856095i $$0.0272838\pi$$
−0.996329 + 0.0856095i $$0.972716\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ 1.00000i 0.0566139i
$$313$$ − 1.00000i − 0.0565233i −0.999601 0.0282617i $$-0.991003\pi$$
0.999601 0.0282617i $$-0.00899717\pi$$
$$314$$ 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 $$0.644549\pi$$
$$318$$ 1.00000i 0.0560772i
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ − 1.00000i − 0.0553001i
$$328$$ 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
$$333$$ 5.00000i 0.273998i
$$334$$ 7.00000 0.383023
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ − 1.00000i − 0.0540738i
$$343$$ 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 $$-0.150827\pi$$
$$348$$ 3.00000i 0.160817i
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ − 4.00000i − 0.213201i
$$353$$ 15.0000i 0.798369i 0.916871 + 0.399185i $$0.130707\pi$$
−0.916871 + 0.399185i $$0.869293\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ 17.0000 0.897226 0.448613 0.893726i $$-0.351918\pi$$
0.448613 + 0.893726i $$0.351918\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 12.0000i 0.630706i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −4.00000 −0.209083
$$367$$ − 21.0000i − 1.09619i −0.836416 0.548096i $$-0.815353\pi$$
0.836416 0.548096i $$-0.184647\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −9.00000 −0.468521
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 4.00000i 0.207390i
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 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
$$381$$ −17.0000 −0.870936
$$382$$ 18.0000i 0.920960i
$$383$$ 5.00000i 0.255488i 0.991807 + 0.127744i $$0.0407736\pi$$
−0.991807 + 0.127744i $$0.959226\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −20.0000 −1.01797
$$387$$ 2.00000i 0.101666i
$$388$$ 12.0000i 0.609208i
$$389$$ 19.0000 0.963338 0.481669 0.876353i $$-0.340031\pi$$
0.481669 + 0.876353i $$0.340031\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 7.00000i 0.353553i
$$393$$ 3.00000i 0.151330i
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 3.00000i 0.150566i 0.997162 + 0.0752828i $$0.0239860\pi$$
−0.997162 + 0.0752828i $$0.976014\pi$$
$$398$$ 1.00000i 0.0501255i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −26.0000 −1.29838 −0.649189 0.760627i $$-0.724892\pi$$
−0.649189 + 0.760627i $$0.724892\pi$$
$$402$$ 9.00000i 0.448879i
$$403$$ 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
$$411$$ 23.0000 1.13451
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ − 12.0000i − 0.587643i
$$418$$ 4.00000i 0.195646i
$$419$$ 37.0000 1.80757 0.903784 0.427989i $$-0.140778\pi$$
0.903784 + 0.427989i $$0.140778\pi$$
$$420$$ 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
$$423$$ 3.00000i 0.145865i
$$424$$ 1.00000 0.0485643
$$425$$ 0 0
$$426$$ −7.00000 −0.339151
$$427$$ 0 0
$$428$$ − 9.00000i − 0.435031i
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −3.00000 −0.144505 −0.0722525 0.997386i $$-0.523019\pi$$
−0.0722525 + 0.997386i $$0.523019\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 19.0000i 0.913082i 0.889702 + 0.456541i $$0.150912\pi$$
−0.889702 + 0.456541i $$0.849088\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1.00000 −0.0478913
$$437$$ 4.00000i 0.191346i
$$438$$ 4.00000i 0.191127i
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ 3.00000i 0.142534i 0.997457 + 0.0712672i $$0.0227043\pi$$
−0.997457 + 0.0712672i $$0.977296\pi$$
$$444$$ 5.00000 0.237289
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9.00000 0.424736 0.212368 0.977190i $$-0.431882\pi$$
0.212368 + 0.977190i $$0.431882\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 2.00000i 0.0940721i
$$453$$ − 12.0000i − 0.563809i
$$454$$ −10.0000 −0.469323
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 32.0000i 1.49690i 0.663193 + 0.748448i $$0.269201\pi$$
−0.663193 + 0.748448i $$0.730799\pi$$
$$458$$ 5.00000i 0.233635i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −24.0000 −1.11779 −0.558896 0.829238i $$-0.688775\pi$$
−0.558896 + 0.829238i $$0.688775\pi$$
$$462$$ 0 0
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ 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 $$0.887180\pi$$
$$468$$ 1.00000i 0.0462250i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ − 10.0000i − 0.460287i
$$473$$ − 8.00000i − 0.367840i
$$474$$ 11.0000 0.505247
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.00000i 0.0457869i
$$478$$ − 8.00000i − 0.365911i
$$479$$ −25.0000 −1.14228 −0.571140 0.820853i $$-0.693499\pi$$
−0.571140 + 0.820853i $$0.693499\pi$$
$$480$$ 0 0
$$481$$ 5.00000 0.227980
$$482$$ 12.0000i 0.546585i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 38.0000i − 1.72194i −0.508652 0.860972i $$-0.669856\pi$$
0.508652 0.860972i $$-0.330144\pi$$
$$488$$ 4.00000i 0.181071i
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ 9.00000i 0.405751i
$$493$$ 0 0
$$494$$ −1.00000 −0.0449921
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 6.00000i 0.268866i
$$499$$ 1.00000 0.0447661 0.0223831 0.999749i $$-0.492875\pi$$
0.0223831 + 0.999749i $$0.492875\pi$$
$$500$$ 0 0
$$501$$ 7.00000 0.312737
$$502$$ − 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
$$507$$ 1.00000i 0.0444116i
$$508$$ 17.0000i 0.754253i
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ − 12.0000i − 0.527759i
$$518$$ 0 0
$$519$$ 3.00000 0.131685
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 3.00000i 0.131306i
$$523$$ 30.0000i 1.31181i 0.754844 + 0.655904i $$0.227712\pi$$
−0.754844 + 0.655904i $$0.772288\pi$$
$$524$$ 3.00000 0.131056
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ 0 0
$$528$$ − 4.00000i − 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ 9.00000i 0.389833i
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 9.00000 0.388741
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 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
$$543$$ 12.0000i 0.514969i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 22.0000i − 0.940652i −0.882493 0.470326i $$-0.844136\pi$$
0.882493 0.470326i $$-0.155864\pi$$
$$548$$ − 23.0000i − 0.982511i
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ −3.00000 −0.127804
$$552$$ − 4.00000i − 0.170251i
$$553$$ 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 $$0.650845\pi$$
$$558$$ 4.00000i 0.169334i
$$559$$ 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 $$0.854084\pi$$
$$564$$ 3.00000 0.126323
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 7.00000i 0.293713i
$$569$$ 8.00000 0.335377 0.167689 0.985840i $$-0.446370\pi$$
0.167689 + 0.985840i $$0.446370\pi$$
$$570$$ 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
$$573$$ 18.0000i 0.751961i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 12.0000i 0.499567i 0.968302 + 0.249783i $$0.0803594\pi$$
−0.968302 + 0.249783i $$0.919641\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ −20.0000 −0.831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 12.0000i 0.497416i
$$583$$ − 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 $$0.920343\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ − 5.00000i − 0.205499i
$$593$$ 1.00000i 0.0410651i 0.999789 + 0.0205325i $$0.00653617\pi$$
−0.999789 + 0.0205325i $$0.993464\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.00000i 0.0409273i
$$598$$ − 4.00000i − 0.163572i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 45.0000 1.83559 0.917794 0.397057i $$-0.129968\pi$$
0.917794 + 0.397057i $$0.129968\pi$$
$$602$$ 0 0
$$603$$ 9.00000i 0.366508i
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ − 7.00000i − 0.284121i −0.989858 0.142061i $$-0.954627\pi$$
0.989858 0.142061i $$-0.0453728\pi$$
$$608$$ 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 $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 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
$$621$$ 4.00000 0.160514
$$622$$ 30.0000i 1.20289i
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −1.00000 −0.0399680
$$627$$ 4.00000i 0.159745i
$$628$$ − 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
$$633$$ − 24.0000i − 0.953914i
$$634$$ 32.0000 1.27088
$$635$$ 0 0
$$636$$ 1.00000 0.0396526
$$637$$ 7.00000i 0.277350i
$$638$$ − 12.0000i − 0.475085i
$$639$$ −7.00000 −0.276916
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ − 9.00000i − 0.355202i
$$643$$ 19.0000i 0.749287i 0.927169 + 0.374643i $$0.122235\pi$$
−0.927169 + 0.374643i $$0.877765\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −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 $$0.830115\pi$$
$$654$$ −1.00000 −0.0391031
$$655$$ 0 0
$$656$$ 9.00000 0.351391
$$657$$ 4.00000i 0.156055i
$$658$$ 0 0
$$659$$ −13.0000 −0.506408 −0.253204 0.967413i $$-0.581484\pi$$
−0.253204 + 0.967413i $$0.581484\pi$$
$$660$$ 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
$$666$$ 5.00000 0.193746
$$667$$ − 12.0000i − 0.464642i
$$668$$ − 7.00000i − 0.270838i
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 16.0000 0.617673
$$672$$ 0 0
$$673$$ − 11.0000i − 0.424019i −0.977268 0.212009i $$-0.931999\pi$$
0.977268 0.212009i $$-0.0680008\pi$$
$$674$$ 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 $$0.987763\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −10.0000 −0.383201
$$682$$ − 16.0000i − 0.612672i
$$683$$ 28.0000i 1.07139i 0.844411 + 0.535695i $$0.179950\pi$$
−0.844411 + 0.535695i $$0.820050\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 5.00000i 0.190762i
$$688$$ − 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
$$696$$ 3.00000 0.113715
$$697$$ 0 0
$$698$$ 18.0000i 0.681310i
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ 1.00000i 0.0377426i
$$703$$ 5.00000i 0.188579i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ 0 0
$$708$$ − 10.0000i − 0.375823i
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 11.0000 0.412532
$$712$$ − 10.0000i − 0.374766i
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 8.00000i − 0.298765i
$$718$$ − 17.0000i − 0.634434i
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000i 0.669891i
$$723$$ 12.0000i 0.446285i
$$724$$ 12.0000 0.445976
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 52.0000i 1.92857i 0.264861 + 0.964287i $$0.414674\pi$$
−0.264861 + 0.964287i $$0.585326\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 4.00000i 0.147844i
$$733$$ 49.0000i 1.80986i 0.425564 + 0.904928i $$0.360076\pi$$
−0.425564 + 0.904928i $$0.639924\pi$$
$$734$$ −21.0000 −0.775124
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ − 36.0000i − 1.32608i
$$738$$ 9.00000i 0.331295i
$$739$$ −53.0000 −1.94964 −0.974818 0.223001i $$-0.928415\pi$$
−0.974818 + 0.223001i $$0.928415\pi$$
$$740$$ 0 0
$$741$$ −1.00000 −0.0367359
$$742$$ 0 0
$$743$$ 27.0000i 0.990534i 0.868741 + 0.495267i $$0.164930\pi$$
−0.868741 + 0.495267i $$0.835070\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 6.00000i 0.219529i
$$748$$ 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
$$753$$ − 23.0000i − 0.838167i
$$754$$ 3.00000 0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ − 28.0000i − 1.01701i
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ −15.0000 −0.543750 −0.271875 0.962333i $$-0.587644\pi$$
−0.271875 + 0.962333i $$0.587644\pi$$
$$762$$ 17.0000i 0.615845i
$$763$$ 0 0
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 5.00000 0.180657
$$767$$ − 10.0000i − 0.361079i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ 24.0000 0.864339
$$772$$ 20.0000i 0.719816i
$$773$$ 32.0000i 1.15096i 0.817816 + 0.575480i $$0.195185\pi$$
−0.817816 + 0.575480i $$0.804815\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 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
$$783$$ 3.00000i 0.107211i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 3.00000 0.107006
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ 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
$$801$$ 10.0000 0.353333
$$802$$ 26.0000i 0.918092i
$$803$$ − 16.0000i − 0.564628i
$$804$$ 9.00000 0.317406
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ − 11.0000i − 0.387218i
$$808$$ − 2.00000i − 0.0703598i
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ − 24.0000i − 0.841717i
$$814$$ −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 $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ − 23.0000i − 0.802217i
$$823$$ 31.0000i 1.08059i 0.841475 + 0.540296i $$0.181688\pi$$
−0.841475 + 0.540296i $$0.818312\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ − 1.00000i − 0.0346688i
$$833$$ 0 0
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 4.00000i 0.138260i
$$838$$ − 37.0000i − 1.27814i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ − 2.00000i − 0.0689246i
$$843$$ 5.00000i 0.172209i
$$844$$ −24.0000 −0.826114
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ 0 0
$$848$$ − 1.00000i − 0.0343401i
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −20.0000 −0.685591
$$852$$ 7.00000i 0.239816i
$$853$$ − 41.0000i − 1.40381i −0.712269 0.701907i $$-0.752332\pi$$
0.712269 0.701907i $$-0.247668\pi$$
$$854$$ 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 $$0.745357\pi$$
$$858$$ − 4.00000i − 0.136558i
$$859$$ −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 $$-0.422032\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 19.0000 0.645646
$$867$$ − 17.0000i − 0.577350i
$$868$$ 0 0
$$869$$ −44.0000 −1.49260
$$870$$ 0 0
$$871$$ 9.00000 0.304953
$$872$$ 1.00000i 0.0338643i
$$873$$ 12.0000i 0.406138i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ − 23.0000i − 0.776655i −0.921521 0.388327i $$-0.873053\pi$$
0.921521 0.388327i $$-0.126947\pi$$
$$878$$ − 19.0000i − 0.641219i
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ 56.0000 1.88669 0.943344 0.331816i $$-0.107661\pi$$
0.943344 + 0.331816i $$0.107661\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ − 36.0000i − 1.21150i −0.795656 0.605748i $$-0.792874\pi$$
0.795656 0.605748i $$-0.207126\pi$$
$$884$$ 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 $$-0.264554\pi$$
$$888$$ − 5.00000i − 0.167789i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ − 14.0000i − 0.468755i
$$893$$ 3.00000i 0.100391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 4.00000i − 0.133556i
$$898$$ − 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
$$906$$ −12.0000 −0.398673
$$907$$ − 30.0000i − 0.996134i −0.867139 0.498067i $$-0.834043\pi$$
0.867139 0.498067i $$-0.165957\pi$$
$$908$$ 10.0000i 0.331862i
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ − 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
$$921$$ 3.00000 0.0988534
$$922$$ 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 $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ 8.00000i 0.262049i
$$933$$ 30.0000i 0.982156i
$$934$$ 15.0000 0.490815
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 34.0000i 1.11073i 0.831606 + 0.555366i $$0.187422\pi$$
−0.831606 + 0.555366i $$0.812578\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ − 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 $$-0.0207020\pi$$
$$948$$ − 11.0000i − 0.357263i
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 32.0000 1.03767
$$952$$ 0 0
$$953$$ − 56.0000i − 1.81402i −0.421111 0.907009i $$-0.638360\pi$$
0.421111 0.907009i $$-0.361640\pi$$
$$954$$ 1.00000 0.0323762
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ − 12.0000i − 0.387905i
$$958$$ 25.0000i 0.807713i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 5.00000i − 0.161206i
$$963$$ − 9.00000i − 0.290021i
$$964$$ 12.0000 0.386494
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 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 $$0.840676\pi$$
$$978$$ 8.00000i 0.255812i
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ −1.00000 −0.0319275
$$982$$ 32.0000i 1.02116i
$$983$$ 32.0000i 1.02064i 0.859984 + 0.510321i $$0.170473\pi$$
−0.859984 + 0.510321i $$0.829527\pi$$
$$984$$ 9.00000 0.286910
$$985$$ 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
$$993$$ − 4.00000i − 0.126936i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ − 14.0000i − 0.443384i −0.975117 0.221692i $$-0.928842\pi$$
0.975117 0.221692i $$-0.0711580\pi$$
$$998$$ − 1.00000i − 0.0316544i
$$999$$ 5.00000 0.158193
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 1950.2.e.h.1249.1 2
3.2 odd 2 5850.2.e.f.5149.2 2
5.2 odd 4 1950.2.a.s.1.1 yes 1
5.3 odd 4 1950.2.a.j.1.1 1
5.4 even 2 inner 1950.2.e.h.1249.2 2
15.2 even 4 5850.2.a.l.1.1 1
15.8 even 4 5850.2.a.bp.1.1 1
15.14 odd 2 5850.2.e.f.5149.1 2

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