# Properties

 Label 3150.2.g.a.2899.2 Level 3150 Weight 2 Character 3150.2899 Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1050) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.2 Root $$-1.00000i$$ Character $$\chi$$ = 3150.2899 Dual form 3150.2.g.a.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} -6.00000 q^{11} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +4.00000 q^{19} -6.00000i q^{22} -3.00000i q^{23} -1.00000 q^{26} -1.00000i q^{28} +3.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} -10.0000i q^{37} +4.00000i q^{38} -9.00000 q^{41} +1.00000i q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000 q^{49} -1.00000i q^{52} +9.00000i q^{53} +1.00000 q^{56} +3.00000i q^{58} +9.00000 q^{59} +11.0000 q^{61} +5.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.00000i q^{68} +12.0000 q^{71} +10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} -6.00000i q^{77} +10.0000 q^{79} -9.00000i q^{82} +9.00000i q^{83} -1.00000 q^{86} +6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} +3.00000i q^{92} +14.0000i q^{97} -1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 12q^{11} - 2q^{14} + 2q^{16} + 8q^{19} - 2q^{26} + 6q^{29} + 10q^{31} + 6q^{34} - 18q^{41} + 12q^{44} + 6q^{46} - 2q^{49} + 2q^{56} + 18q^{59} + 22q^{61} - 2q^{64} + 24q^{71} + 20q^{74} - 8q^{76} + 20q^{79} - 2q^{86} - 12q^{89} - 2q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$7$$ 1.00000i 0.377964i
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 6.00000i − 1.27920i
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 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
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 1.00000i − 0.138675i
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 3.00000i 0.393919i
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 0 0
$$61$$ 11.0000 1.40841 0.704203 0.709999i $$-0.251305\pi$$
0.704203 + 0.709999i $$0.251305\pi$$
$$62$$ 5.00000i 0.635001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 3.00000i 0.363803i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 6.00000i − 0.683763i
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 9.00000i − 0.993884i
$$83$$ 9.00000i 0.987878i 0.869496 + 0.493939i $$0.164443\pi$$
−0.869496 + 0.493939i $$0.835557\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 0 0
$$88$$ 6.00000i 0.639602i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 3.00000i 0.312772i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ − 17.0000i − 1.67506i −0.546392 0.837530i $$-0.683999\pi$$
0.546392 0.837530i $$-0.316001\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ − 18.0000i − 1.74013i −0.492941 0.870063i $$-0.664078\pi$$
0.492941 0.870063i $$-0.335922\pi$$
$$108$$ 0 0
$$109$$ −8.00000 −0.766261 −0.383131 0.923694i $$-0.625154\pi$$
−0.383131 + 0.923694i $$0.625154\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.00000 −0.278543
$$117$$ 0 0
$$118$$ 9.00000i 0.828517i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 11.0000i 0.995893i
$$123$$ 0 0
$$124$$ −5.00000 −0.449013
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 10.0000i − 0.887357i −0.896186 0.443678i $$-0.853673\pi$$
0.896186 0.443678i $$-0.146327\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.0000i 1.00702i
$$143$$ − 6.00000i − 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ 10.0000i 0.821995i
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 0 0
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ − 5.00000i − 0.391630i −0.980641 0.195815i $$-0.937265\pi$$
0.980641 0.195815i $$-0.0627352\pi$$
$$164$$ 9.00000 0.702782
$$165$$ 0 0
$$166$$ −9.00000 −0.698535
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 1.00000i − 0.0762493i
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −6.00000 −0.452267
$$177$$ 0 0
$$178$$ − 6.00000i − 0.449719i
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ − 1.00000i − 0.0741249i
$$183$$ 0 0
$$184$$ −3.00000 −0.221163
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 18.0000i 1.31629i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ 0 0
$$193$$ 22.0000i 1.58359i 0.610784 + 0.791797i $$0.290854\pi$$
−0.610784 + 0.791797i $$0.709146\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 15.0000i 1.06871i 0.845262 + 0.534353i $$0.179445\pi$$
−0.845262 + 0.534353i $$0.820555\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 12.0000i 0.844317i
$$203$$ 3.00000i 0.210559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 17.0000 1.18445
$$207$$ 0 0
$$208$$ 1.00000i 0.0693375i
$$209$$ −24.0000 −1.66011
$$210$$ 0 0
$$211$$ −25.0000 −1.72107 −0.860535 0.509390i $$-0.829871\pi$$
−0.860535 + 0.509390i $$0.829871\pi$$
$$212$$ − 9.00000i − 0.618123i
$$213$$ 0 0
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5.00000i 0.339422i
$$218$$ − 8.00000i − 0.541828i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 19.0000i 1.27233i 0.771551 + 0.636167i $$0.219481\pi$$
−0.771551 + 0.636167i $$0.780519\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 3.00000i − 0.199117i −0.995032 0.0995585i $$-0.968257\pi$$
0.995032 0.0995585i $$-0.0317430\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 3.00000i − 0.196960i
$$233$$ 12.0000i 0.786146i 0.919507 + 0.393073i $$0.128588\pi$$
−0.919507 + 0.393073i $$0.871412\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −9.00000 −0.585850
$$237$$ 0 0
$$238$$ 3.00000i 0.194461i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 25.0000i 1.60706i
$$243$$ 0 0
$$244$$ −11.0000 −0.704203
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ − 5.00000i − 0.317500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 27.0000 1.70422 0.852112 0.523359i $$-0.175321\pi$$
0.852112 + 0.523359i $$0.175321\pi$$
$$252$$ 0 0
$$253$$ 18.0000i 1.13165i
$$254$$ 10.0000 0.627456
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 15.0000i − 0.935674i −0.883815 0.467837i $$-0.845033\pi$$
0.883815 0.467837i $$-0.154967\pi$$
$$258$$ 0 0
$$259$$ 10.0000 0.621370
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 9.00000i 0.554964i 0.960731 + 0.277482i $$0.0894999\pi$$
−0.960731 + 0.277482i $$0.910500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ 0 0
$$268$$ 4.00000i 0.244339i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ − 3.00000i − 0.181902i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 16.0000i 0.959616i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ − 9.00000i − 0.531253i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 10.0000i − 0.585206i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ − 9.00000i − 0.521356i
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ 2.00000i 0.115087i
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 27.0000i 1.51647i 0.651981 + 0.758236i $$0.273938\pi$$
−0.651981 + 0.758236i $$0.726062\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3.00000i 0.167183i
$$323$$ − 12.0000i − 0.667698i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 5.00000 0.276924
$$327$$ 0 0
$$328$$ 9.00000i 0.496942i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ − 9.00000i − 0.493939i
$$333$$ 0 0
$$334$$ 18.0000 0.984916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 13.0000i − 0.708155i −0.935216 0.354078i $$-0.884795\pi$$
0.935216 0.354078i $$-0.115205\pi$$
$$338$$ 12.0000i 0.652714i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −30.0000 −1.62459
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ − 18.0000i − 0.966291i −0.875540 0.483145i $$-0.839494\pi$$
0.875540 0.483145i $$-0.160506\pi$$
$$348$$ 0 0
$$349$$ −17.0000 −0.909989 −0.454995 0.890494i $$-0.650359\pi$$
−0.454995 + 0.890494i $$0.650359\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 6.00000i − 0.319801i
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 18.0000i 0.951330i
$$359$$ −3.00000 −0.158334 −0.0791670 0.996861i $$-0.525226\pi$$
−0.0791670 + 0.996861i $$0.525226\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 2.00000i 0.105118i
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 37.0000i − 1.93138i −0.259690 0.965692i $$-0.583620\pi$$
0.259690 0.965692i $$-0.416380\pi$$
$$368$$ − 3.00000i − 0.156386i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ 0 0
$$373$$ − 8.00000i − 0.414224i −0.978317 0.207112i $$-0.933593\pi$$
0.978317 0.207112i $$-0.0664065\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.00000i 0.154508i
$$378$$ 0 0
$$379$$ −35.0000 −1.79783 −0.898915 0.438124i $$-0.855643\pi$$
−0.898915 + 0.438124i $$0.855643\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 15.0000i 0.767467i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22.0000 −1.11977
$$387$$ 0 0
$$388$$ − 14.0000i − 0.710742i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −9.00000 −0.455150
$$392$$ 1.00000i 0.0505076i
$$393$$ 0 0
$$394$$ −15.0000 −0.755689
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 29.0000i 1.45547i 0.685859 + 0.727734i $$0.259427\pi$$
−0.685859 + 0.727734i $$0.740573\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 5.00000i 0.249068i
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ −3.00000 −0.148888
$$407$$ 60.0000i 2.97409i
$$408$$ 0 0
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 17.0000i 0.837530i
$$413$$ 9.00000i 0.442861i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ − 24.0000i − 1.17388i
$$419$$ −21.0000 −1.02592 −0.512959 0.858413i $$-0.671451\pi$$
−0.512959 + 0.858413i $$0.671451\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ − 25.0000i − 1.21698i
$$423$$ 0 0
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 11.0000i 0.532327i
$$428$$ 18.0000i 0.870063i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.0000 0.722525 0.361262 0.932464i $$-0.382346\pi$$
0.361262 + 0.932464i $$0.382346\pi$$
$$432$$ 0 0
$$433$$ − 38.0000i − 1.82616i −0.407777 0.913082i $$-0.633696\pi$$
0.407777 0.913082i $$-0.366304\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ − 12.0000i − 0.574038i
$$438$$ 0 0
$$439$$ −41.0000 −1.95682 −0.978412 0.206666i $$-0.933739\pi$$
−0.978412 + 0.206666i $$0.933739\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3.00000i 0.142695i
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −19.0000 −0.899676
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ 24.0000 1.13263 0.566315 0.824189i $$-0.308369\pi$$
0.566315 + 0.824189i $$0.308369\pi$$
$$450$$ 0 0
$$451$$ 54.0000 2.54276
$$452$$ 6.00000i 0.282216i
$$453$$ 0 0
$$454$$ 3.00000 0.140797
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 25.0000i − 1.16945i −0.811231 0.584725i $$-0.801202\pi$$
0.811231 0.584725i $$-0.198798\pi$$
$$458$$ 22.0000i 1.02799i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ −12.0000 −0.555889
$$467$$ 3.00000i 0.138823i 0.997588 + 0.0694117i $$0.0221122\pi$$
−0.997588 + 0.0694117i $$0.977888\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 9.00000i − 0.414259i
$$473$$ − 6.00000i − 0.275880i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ − 11.0000i − 0.497947i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −42.0000 −1.89543 −0.947717 0.319113i $$-0.896615\pi$$
−0.947717 + 0.319113i $$0.896615\pi$$
$$492$$ 0 0
$$493$$ − 9.00000i − 0.405340i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ 19.0000 0.850557 0.425278 0.905063i $$-0.360176\pi$$
0.425278 + 0.905063i $$0.360176\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 27.0000i 1.20507i
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −18.0000 −0.800198
$$507$$ 0 0
$$508$$ 10.0000i 0.443678i
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 15.0000 0.661622
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 10.0000i 0.439375i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 21.0000 0.920027 0.460013 0.887912i $$-0.347845\pi$$
0.460013 + 0.887912i $$0.347845\pi$$
$$522$$ 0 0
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −9.00000 −0.392419
$$527$$ − 15.0000i − 0.653410i
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 9.00000i − 0.389833i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ − 18.0000i − 0.776035i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 17.0000i 0.726868i 0.931620 + 0.363434i $$0.118396\pi$$
−0.931620 + 0.363434i $$0.881604\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 10.0000i 0.425243i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 24.0000i 1.01238i
$$563$$ 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
$$568$$ − 12.0000i − 0.503509i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −13.0000 −0.544033 −0.272017 0.962293i $$-0.587691\pi$$
−0.272017 + 0.962293i $$0.587691\pi$$
$$572$$ 6.00000i 0.250873i
$$573$$ 0 0
$$574$$ 9.00000 0.375653
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ 8.00000i 0.332756i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −9.00000 −0.373383
$$582$$ 0 0
$$583$$ − 54.0000i − 2.23645i
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 33.0000i − 1.36206i −0.732257 0.681028i $$-0.761533\pi$$
0.732257 0.681028i $$-0.238467\pi$$
$$588$$ 0 0
$$589$$ 20.0000 0.824086
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 10.0000i − 0.410997i
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 9.00000 0.368654
$$597$$ 0 0
$$598$$ 3.00000i 0.122679i
$$599$$ 9.00000 0.367730 0.183865 0.982952i $$-0.441139\pi$$
0.183865 + 0.982952i $$0.441139\pi$$
$$600$$ 0 0
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ − 1.00000i − 0.0407570i
$$603$$ 0 0
$$604$$ −2.00000 −0.0813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ −26.0000 −1.04503 −0.522514 0.852631i $$-0.675006\pi$$
−0.522514 + 0.852631i $$0.675006\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000i 0.721734i
$$623$$ − 6.00000i − 0.240385i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ 10.0000i 0.399043i
$$629$$ −30.0000 −1.19618
$$630$$ 0 0
$$631$$ −22.0000 −0.875806 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ 0 0
$$634$$ −27.0000 −1.07231
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 1.00000i − 0.0396214i
$$638$$ − 18.0000i − 0.712627i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ − 6.00000i − 0.235884i −0.993020 0.117942i $$-0.962370\pi$$
0.993020 0.117942i $$-0.0376297\pi$$
$$648$$ 0 0
$$649$$ −54.0000 −2.11969
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5.00000i 0.195815i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 42.0000 1.63609 0.818044 0.575156i $$-0.195059\pi$$
0.818044 + 0.575156i $$0.195059\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ − 19.0000i − 0.738456i
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 9.00000i − 0.348481i
$$668$$ 18.0000i 0.696441i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −66.0000 −2.54790
$$672$$ 0 0
$$673$$ 37.0000i 1.42625i 0.701039 + 0.713123i $$0.252720\pi$$
−0.701039 + 0.713123i $$0.747280\pi$$
$$674$$ 13.0000 0.500741
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 12.0000i 0.461197i 0.973049 + 0.230599i $$0.0740685\pi$$
−0.973049 + 0.230599i $$0.925932\pi$$
$$678$$ 0 0
$$679$$ −14.0000 −0.537271
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 30.0000i − 1.14876i
$$683$$ 18.0000i 0.688751i 0.938832 + 0.344375i $$0.111909\pi$$
−0.938832 + 0.344375i $$0.888091\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 1.00000i 0.0381246i
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ −34.0000 −1.29342 −0.646710 0.762736i $$-0.723856\pi$$
−0.646710 + 0.762736i $$0.723856\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ 0 0
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 27.0000i 1.02270i
$$698$$ − 17.0000i − 0.643459i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3.00000 −0.113308 −0.0566542 0.998394i $$-0.518043\pi$$
−0.0566542 + 0.998394i $$0.518043\pi$$
$$702$$ 0 0
$$703$$ − 40.0000i − 1.50863i
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ 40.0000 1.50223 0.751116 0.660171i $$-0.229516\pi$$
0.751116 + 0.660171i $$0.229516\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000i 0.224860i
$$713$$ − 15.0000i − 0.561754i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −18.0000 −0.672692
$$717$$ 0 0
$$718$$ − 3.00000i − 0.111959i
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ 17.0000 0.633113
$$722$$ − 3.00000i − 0.111648i
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 7.00000i − 0.259616i −0.991539 0.129808i $$-0.958564\pi$$
0.991539 0.129808i $$-0.0414360\pi$$
$$728$$ 1.00000i 0.0370625i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 0 0
$$733$$ − 29.0000i − 1.07114i −0.844491 0.535570i $$-0.820097\pi$$
0.844491 0.535570i $$-0.179903\pi$$
$$734$$ 37.0000 1.36569
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ 24.0000i 0.884051i
$$738$$ 0 0
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 9.00000i − 0.330400i
$$743$$ − 9.00000i − 0.330178i −0.986279 0.165089i $$-0.947209\pi$$
0.986279 0.165089i $$-0.0527911\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 8.00000 0.292901
$$747$$ 0 0
$$748$$ − 18.0000i − 0.658145i
$$749$$ 18.0000 0.657706
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −3.00000 −0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.0000i 0.508839i 0.967094 + 0.254419i $$0.0818843\pi$$
−0.967094 + 0.254419i $$0.918116\pi$$
$$758$$ − 35.0000i − 1.27126i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ − 8.00000i − 0.289619i
$$764$$ −15.0000 −0.542681
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 9.00000i 0.324971i
$$768$$ 0 0
$$769$$ 4.00000 0.144244 0.0721218 0.997396i $$-0.477023\pi$$
0.0721218 + 0.997396i $$0.477023\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 22.0000i − 0.791797i
$$773$$ − 48.0000i − 1.72644i −0.504828 0.863220i $$-0.668444\pi$$
0.504828 0.863220i $$-0.331556\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ − 6.00000i − 0.215110i
$$779$$ −36.0000 −1.28983
$$780$$ 0 0
$$781$$ −72.0000 −2.57636
$$782$$ − 9.00000i − 0.321839i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 26.0000i 0.926800i 0.886149 + 0.463400i $$0.153371\pi$$
−0.886149 + 0.463400i $$0.846629\pi$$
$$788$$ − 15.0000i − 0.534353i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 11.0000i 0.390621i
$$794$$ −29.0000 −1.02917
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12.0000i 0.423735i
$$803$$ − 60.0000i − 2.11735i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −5.00000 −0.176117
$$807$$ 0 0
$$808$$ − 12.0000i − 0.422159i
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −34.0000 −1.19390 −0.596951 0.802278i $$-0.703621\pi$$
−0.596951 + 0.802278i $$0.703621\pi$$
$$812$$ − 3.00000i − 0.105279i
$$813$$ 0 0
$$814$$ −60.0000 −2.10300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.00000i 0.139942i
$$818$$ − 38.0000i − 1.32864i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 0 0
$$823$$ 34.0000i 1.18517i 0.805510 + 0.592583i $$0.201892\pi$$
−0.805510 + 0.592583i $$0.798108\pi$$
$$824$$ −17.0000 −0.592223
$$825$$ 0 0
$$826$$ −9.00000 −0.313150
$$827$$ − 6.00000i − 0.208640i −0.994544 0.104320i $$-0.966733\pi$$
0.994544 0.104320i $$-0.0332667\pi$$
$$828$$ 0 0
$$829$$ −29.0000 −1.00721 −0.503606 0.863934i $$-0.667994\pi$$
−0.503606 + 0.863934i $$0.667994\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 1.00000i − 0.0346688i
$$833$$ 3.00000i 0.103944i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 24.0000 0.830057
$$837$$ 0 0
$$838$$ − 21.0000i − 0.725433i
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ − 34.0000i − 1.17172i
$$843$$ 0 0
$$844$$ 25.0000 0.860535
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.0000i 0.859010i
$$848$$ 9.00000i 0.309061i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −30.0000 −1.02839
$$852$$ 0 0
$$853$$ 19.0000i 0.650548i 0.945620 + 0.325274i $$0.105456\pi$$
−0.945620 + 0.325274i $$0.894544\pi$$
$$854$$ −11.0000 −0.376412
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 15.0000i 0.510902i
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 38.0000 1.29129
$$867$$ 0 0
$$868$$ − 5.00000i − 0.169711i
$$869$$ −60.0000 −2.03536
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 8.00000i 0.270914i
$$873$$ 0 0
$$874$$ 12.0000 0.405906
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000i 1.08056i 0.841484 + 0.540282i $$0.181682\pi$$
−0.841484 + 0.540282i $$0.818318\pi$$
$$878$$ − 41.0000i − 1.38368i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −33.0000 −1.11180 −0.555899 0.831250i $$-0.687626\pi$$
−0.555899 + 0.831250i $$0.687626\pi$$
$$882$$ 0 0
$$883$$ − 5.00000i − 0.168263i −0.996455 0.0841317i $$-0.973188\pi$$
0.996455 0.0841317i $$-0.0268116\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ − 54.0000i − 1.81314i −0.422053 0.906571i $$-0.638690\pi$$
0.422053 0.906571i $$-0.361310\pi$$
$$888$$ 0 0
$$889$$ 10.0000 0.335389
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 19.0000i − 0.636167i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 24.0000i 0.800890i
$$899$$ 15.0000 0.500278
$$900$$ 0 0
$$901$$ 27.0000 0.899500
$$902$$ 54.0000i 1.79800i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 53.0000i 1.75984i 0.475125 + 0.879918i $$0.342403\pi$$
−0.475125 + 0.879918i $$0.657597\pi$$
$$908$$ 3.00000i 0.0995585i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ 0 0
$$913$$ − 54.0000i − 1.78714i
$$914$$ 25.0000 0.826927
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 12.0000i − 0.395199i
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 32.0000 1.05159
$$927$$ 0 0
$$928$$ 3.00000i 0.0984798i
$$929$$ −27.0000 −0.885841 −0.442921 0.896561i $$-0.646058\pi$$
−0.442921 + 0.896561i $$0.646058\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ − 12.0000i − 0.393073i
$$933$$ 0 0
$$934$$ −3.00000 −0.0981630
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ 4.00000i 0.130605i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ 27.0000i 0.879241i
$$944$$ 9.00000 0.292925
$$945$$ 0 0
$$946$$ 6.00000 0.195077
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 3.00000i − 0.0972306i
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 24.0000i 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 10.0000i 0.322413i
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 28.0000i − 0.900419i −0.892923 0.450210i $$-0.851349\pi$$
0.892923 0.450210i $$-0.148651\pi$$
$$968$$ − 25.0000i − 0.803530i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ 16.0000i 0.512936i
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 11.0000 0.352101
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 42.0000i − 1.34027i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 9.00000 0.286618
$$987$$ 0 0
$$988$$ − 4.00000i − 0.127257i
$$989$$ 3.00000 0.0953945
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 5.00000i 0.158750i
$$993$$ 0 0
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 19.0000i 0.601434i
$$999$$ 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 3150.2.g.a.2899.2 2
3.2 odd 2 1050.2.g.e.799.1 2
5.2 odd 4 3150.2.a.a.1.1 1
5.3 odd 4 3150.2.a.bg.1.1 1
5.4 even 2 inner 3150.2.g.a.2899.1 2
15.2 even 4 1050.2.a.l.1.1 yes 1
15.8 even 4 1050.2.a.j.1.1 1
15.14 odd 2 1050.2.g.e.799.2 2
60.23 odd 4 8400.2.a.a.1.1 1
60.47 odd 4 8400.2.a.ci.1.1 1
105.62 odd 4 7350.2.a.cz.1.1 1
105.83 odd 4 7350.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.j.1.1 1 15.8 even 4
1050.2.a.l.1.1 yes 1 15.2 even 4
1050.2.g.e.799.1 2 3.2 odd 2
1050.2.g.e.799.2 2 15.14 odd 2
3150.2.a.a.1.1 1 5.2 odd 4
3150.2.a.bg.1.1 1 5.3 odd 4
3150.2.g.a.2899.1 2 5.4 even 2 inner
3150.2.g.a.2899.2 2 1.1 even 1 trivial
7350.2.a.r.1.1 1 105.83 odd 4
7350.2.a.cz.1.1 1 105.62 odd 4
8400.2.a.a.1.1 1 60.23 odd 4
8400.2.a.ci.1.1 1 60.47 odd 4