# Properties

 Label 1050.2.g.a.799.1 Level $1050$ Weight $2$ Character 1050.799 Analytic conductor $8.384$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ 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 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1050.799 Dual form 1050.2.g.a.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +1.00000 q^{21} +4.00000i q^{22} +8.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} -1.00000i q^{48} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} -1.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} -4.00000i q^{77} -6.00000i q^{78} +1.00000 q^{81} +6.00000i q^{82} -4.00000i q^{83} -1.00000 q^{84} -4.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} -6.00000 q^{91} -8.00000i q^{92} -1.00000 q^{96} +14.0000i q^{97} +1.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} + 2q^{14} + 2q^{16} + 8q^{19} + 2q^{21} + 2q^{24} + 12q^{26} + 4q^{29} - 4q^{34} + 2q^{36} + 12q^{39} - 12q^{41} + 8q^{44} + 16q^{46} - 2q^{49} - 4q^{51} + 2q^{54} - 2q^{56} - 8q^{59} + 12q^{61} - 2q^{64} + 8q^{66} + 16q^{69} + 16q^{71} + 20q^{74} - 8q^{76} + 2q^{81} - 2q^{84} - 8q^{86} + 12q^{89} - 12q^{91} - 2q^{96} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$7$$ 1.00000i 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 4.00000i 0.852803i
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 1.00000i 0.192450i
$$28$$ − 1.00000i − 0.188982i
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ − 1.00000i − 0.154303i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ − 6.00000i − 0.832050i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ − 4.00000i − 0.529813i
$$58$$ − 2.00000i − 0.262613i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ − 1.00000i − 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 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
$$77$$ − 4.00000i − 0.455842i
$$78$$ − 6.00000i − 0.679366i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 2.00000i − 0.214423i
$$88$$ − 4.00000i − 0.426401i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ − 8.00000i − 0.834058i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 1.00000i 0.0944911i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ − 6.00000i − 0.554700i
$$118$$ 4.00000i 0.368230i
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 6.00000i − 0.543214i
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 4.00000i 0.346844i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ − 8.00000i − 0.681005i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ − 24.0000i − 2.00698i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ 1.00000i 0.0824786i
$$148$$ − 10.0000i − 0.821995i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 2.00000i 0.161690i
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 1.00000i 0.0771517i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 4.00000i 0.300658i
$$178$$ − 6.00000i − 0.449719i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 6.00000i 0.444750i
$$183$$ − 6.00000i − 0.443533i
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 10.0000i 0.712470i 0.934396 + 0.356235i $$0.115940\pi$$
−0.934396 + 0.356235i $$0.884060\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 2.00000i 0.140720i
$$203$$ 2.00000i 0.140372i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ − 8.00000i − 0.556038i
$$208$$ 6.00000i 0.416025i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 8.00000i − 0.548151i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 2.00000i − 0.135457i
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ − 10.0000i − 0.671156i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 2.00000i 0.131306i
$$233$$ − 22.0000i − 1.44127i −0.693316 0.720634i $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ − 2.00000i − 0.129641i
$$239$$ 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 $$0.479482\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 24.0000i 1.52708i
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 1.00000i 0.0629941i
$$253$$ − 32.0000i − 2.01182i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000i 1.87135i 0.352865 + 0.935674i $$0.385208\pi$$
−0.352865 + 0.935674i $$0.614792\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 20.0000i 1.23560i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ − 6.00000i − 0.367194i
$$268$$ 4.00000i 0.244339i
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 6.00000i 0.363137i
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ − 6.00000i − 0.354169i
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ − 10.0000i − 0.585206i
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ − 4.00000i − 0.232104i
$$298$$ 6.00000i 0.347571i
$$299$$ −48.0000 −2.77591
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 8.00000i 0.460348i
$$303$$ 2.00000i 0.114897i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ 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
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 8.00000i 0.445823i
$$323$$ − 8.00000i − 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 10.0000i − 0.547997i
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ − 18.0000i − 0.980522i −0.871576 0.490261i $$-0.836901\pi$$
0.871576 0.490261i $$-0.163099\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 4.00000i 0.213201i
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ − 2.00000i − 0.105851i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 18.0000i 0.946059i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ − 32.0000i − 1.67039i −0.549957 0.835193i $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 1.00000i 0.0514344i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 16.0000i − 0.817562i −0.912633 0.408781i $$-0.865954\pi$$
0.912633 0.408781i $$-0.134046\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 4.00000i 0.203331i
$$388$$ − 14.0000i − 0.710742i
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 20.0000i 1.00887i
$$394$$ 10.0000 0.503793
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 0 0
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ − 40.0000i − 1.98273i
$$408$$ − 2.00000i − 0.0990148i
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ − 8.00000i − 0.394132i
$$413$$ − 4.00000i − 0.196827i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 4.00000i 0.195881i
$$418$$ 16.0000i 0.782586i
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 6.00000i 0.290360i
$$428$$ 12.0000i 0.580042i
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 32.0000i 1.53077i
$$438$$ − 10.0000i − 0.477818i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ − 12.0000i − 0.570782i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 6.00000i 0.283790i
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 14.0000i 0.658505i
$$453$$ 8.00000i 0.375873i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ − 2.00000i − 0.0934539i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ 4.00000i 0.186097i
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ − 4.00000i − 0.184115i
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ − 2.00000i − 0.0910975i
$$483$$ 8.00000i 0.364013i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ − 4.00000i − 0.180151i
$$494$$ 24.0000 1.07981
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000i 0.358849i
$$498$$ 4.00000i 0.179244i
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 12.0000i 0.535586i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ 23.0000i 1.02147i
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 10.0000i 0.439375i
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ 4.00000i 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 36.0000i − 1.55933i
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ − 12.0000i − 0.517838i
$$538$$ 22.0000i 0.948487i
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 0 0
$$543$$ 18.0000i 0.772454i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 8.00000i 0.340503i
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ − 26.0000i − 1.09674i
$$563$$ 44.0000i 1.85438i 0.374593 + 0.927189i $$0.377783\pi$$
−0.374593 + 0.927189i $$0.622217\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 1.00000i 0.0419961i
$$568$$ 8.00000i 0.335673i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 24.0000i 1.00349i
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 2.00000 0.0831172
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ − 14.0000i − 0.580319i
$$583$$ − 24.0000i − 0.993978i
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ − 1.00000i − 0.0412393i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 10.0000i 0.410997i
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 8.00000i 0.327418i
$$598$$ 48.0000i 1.96287i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ − 4.00000i − 0.163028i
$$603$$ 4.00000i 0.162893i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ − 48.0000i − 1.94826i −0.225989 0.974130i $$-0.572561\pi$$
0.225989 0.974130i $$-0.427439\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 42.0000i − 1.69636i −0.529705 0.848182i $$-0.677697\pi$$
0.529705 0.848182i $$-0.322303\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 8.00000i 0.320771i
$$623$$ 6.00000i 0.240385i
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 16.0000i 0.638978i
$$628$$ − 10.0000i − 0.399043i
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ − 20.0000i − 0.794929i
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ − 6.00000i − 0.237729i
$$638$$ 8.00000i 0.316723i
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 20.0000i − 0.783260i
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ − 10.0000i − 0.390137i
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 4.00000i 0.155464i
$$663$$ − 12.0000i − 0.466041i
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 16.0000i 0.619522i
$$668$$ − 8.00000i − 0.309529i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ − 1.00000i − 0.0385758i
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ −18.0000 −0.693334
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ −14.0000 −0.537271
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ − 2.00000i − 0.0763048i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ − 22.0000i − 0.836315i
$$693$$ 4.00000i 0.151947i
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 12.0000i 0.454532i
$$698$$ 22.0000i 0.832712i
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ 40.0000i 1.50863i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ − 2.00000i − 0.0752177i
$$708$$ − 4.00000i − 0.150329i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000i 0.224860i
$$713$$ 0 0
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ − 8.00000i − 0.298557i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 3.00000i 0.111648i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ − 6.00000i − 0.222375i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 6.00000i 0.221766i
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 16.0000i 0.589368i
$$738$$ − 6.00000i − 0.220863i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 6.00000i 0.220267i
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ 4.00000i 0.146352i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ 12.0000i 0.437304i
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ − 6.00000i − 0.218074i −0.994038 0.109037i $$-0.965223\pi$$
0.994038 0.109037i $$-0.0347767\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ − 24.0000i − 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 2.00000i − 0.0719350i −0.999353 0.0359675i $$-0.988549\pi$$
0.999353 0.0359675i $$-0.0114513\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 10.0000i 0.358748i
$$778$$ − 26.0000i − 0.932145i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ − 16.0000i − 0.572159i
$$783$$ 2.00000i 0.0714742i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ 36.0000i 1.28326i 0.767014 + 0.641631i $$0.221742\pi$$
−0.767014 + 0.641631i $$0.778258\pi$$
$$788$$ − 10.0000i − 0.356235i
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 4.00000i 0.142134i
$$793$$ 36.0000i 1.27840i
$$794$$ −6.00000 −0.212932
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 18.0000i − 0.635602i
$$803$$ − 40.0000i − 1.41157i
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 22.0000i 0.774437i
$$808$$ − 2.00000i − 0.0703598i
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ − 2.00000i − 0.0701862i
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 22.0000i − 0.769212i
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 10.0000i 0.348790i
$$823$$ − 56.0000i − 1.95204i −0.217687 0.976019i $$-0.569851\pi$$
0.217687 0.976019i $$-0.430149\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −4.00000 −0.139178
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ − 6.00000i − 0.208013i
$$833$$ 2.00000i 0.0692959i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ − 36.0000i − 1.24360i
$$839$$ −56.0000 −1.93333 −0.966667 0.256036i $$-0.917584\pi$$
−0.966667 + 0.256036i $$0.917584\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 6.00000i − 0.206774i
$$843$$ − 26.0000i − 0.895488i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.00000i 0.171802i
$$848$$ 6.00000i 0.206041i
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ −80.0000 −2.74236
$$852$$ 8.00000i 0.274075i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 24.0000i 0.819346i
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 2.00000i 0.0677285i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ 2.00000i 0.0675352i 0.999430 + 0.0337676i $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ − 1.00000i − 0.0336718i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ 10.0000i 0.335578i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 48.0000i 1.60267i
$$898$$ 34.0000i 1.13459i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 24.0000i − 0.799113i
$$903$$ − 4.00000i − 0.133112i
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ − 12.0000i − 0.398453i −0.979953 0.199227i $$-0.936157\pi$$
0.979953 0.199227i $$-0.0638430\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 16.0000i 0.529523i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ − 20.0000i − 0.660458i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ − 22.0000i − 0.724531i
$$923$$ 48.0000i 1.57994i
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ − 8.00000i − 0.262754i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 22.0000i 0.720634i
$$933$$ 8.00000i 0.261908i
$$934$$ −28.0000 −0.916188
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −26.0000 −0.847576 −0.423788 0.905761i $$-0.639300\pi$$
−0.423788 + 0.905761i $$0.639300\pi$$
$$942$$ − 10.0000i − 0.325818i
$$943$$ − 48.0000i − 1.56310i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 2.00000i 0.0648204i
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8.00000i 0.258603i
$$958$$ − 16.0000i − 0.516937i
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 60.0000i 1.93448i
$$963$$ 12.0000i 0.386695i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 4.00000i − 0.128234i
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 20.0000i − 0.639529i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ − 12.0000i − 0.382935i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ − 24.0000i − 0.763542i
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 4.00000i 0.126936i
$$994$$ 8.00000 0.253745
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ − 14.0000i − 0.443384i −0.975117 0.221692i $$-0.928842\pi$$
0.975117 0.221692i $$-0.0711580\pi$$
$$998$$ − 44.0000i − 1.39280i
$$999$$ −10.0000 −0.316386
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 1050.2.g.a.799.1 2
3.2 odd 2 3150.2.g.r.2899.2 2
5.2 odd 4 42.2.a.a.1.1 1
5.3 odd 4 1050.2.a.i.1.1 1
5.4 even 2 inner 1050.2.g.a.799.2 2
15.2 even 4 126.2.a.a.1.1 1
15.8 even 4 3150.2.a.bo.1.1 1
15.14 odd 2 3150.2.g.r.2899.1 2
20.3 even 4 8400.2.a.k.1.1 1
20.7 even 4 336.2.a.d.1.1 1
35.2 odd 12 294.2.e.c.67.1 2
35.12 even 12 294.2.e.a.67.1 2
35.13 even 4 7350.2.a.f.1.1 1
35.17 even 12 294.2.e.a.79.1 2
35.27 even 4 294.2.a.g.1.1 1
35.32 odd 12 294.2.e.c.79.1 2
40.27 even 4 1344.2.a.i.1.1 1
40.37 odd 4 1344.2.a.q.1.1 1
45.2 even 12 1134.2.f.j.757.1 2
45.7 odd 12 1134.2.f.g.757.1 2
45.22 odd 12 1134.2.f.g.379.1 2
45.32 even 12 1134.2.f.j.379.1 2
55.32 even 4 5082.2.a.d.1.1 1
60.47 odd 4 1008.2.a.j.1.1 1
65.12 odd 4 7098.2.a.f.1.1 1
80.27 even 4 5376.2.c.e.2689.1 2
80.37 odd 4 5376.2.c.bc.2689.2 2
80.67 even 4 5376.2.c.e.2689.2 2
80.77 odd 4 5376.2.c.bc.2689.1 2
105.2 even 12 882.2.g.h.361.1 2
105.17 odd 12 882.2.g.j.667.1 2
105.32 even 12 882.2.g.h.667.1 2
105.47 odd 12 882.2.g.j.361.1 2
105.62 odd 4 882.2.a.b.1.1 1
120.77 even 4 4032.2.a.e.1.1 1
120.107 odd 4 4032.2.a.m.1.1 1
140.27 odd 4 2352.2.a.l.1.1 1
140.47 odd 12 2352.2.q.n.1537.1 2
140.67 even 12 2352.2.q.i.961.1 2
140.87 odd 12 2352.2.q.n.961.1 2
140.107 even 12 2352.2.q.i.1537.1 2
280.27 odd 4 9408.2.a.bw.1.1 1
280.237 even 4 9408.2.a.n.1.1 1
420.167 even 4 7056.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 5.2 odd 4
126.2.a.a.1.1 1 15.2 even 4
294.2.a.g.1.1 1 35.27 even 4
294.2.e.a.67.1 2 35.12 even 12
294.2.e.a.79.1 2 35.17 even 12
294.2.e.c.67.1 2 35.2 odd 12
294.2.e.c.79.1 2 35.32 odd 12
336.2.a.d.1.1 1 20.7 even 4
882.2.a.b.1.1 1 105.62 odd 4
882.2.g.h.361.1 2 105.2 even 12
882.2.g.h.667.1 2 105.32 even 12
882.2.g.j.361.1 2 105.47 odd 12
882.2.g.j.667.1 2 105.17 odd 12
1008.2.a.j.1.1 1 60.47 odd 4
1050.2.a.i.1.1 1 5.3 odd 4
1050.2.g.a.799.1 2 1.1 even 1 trivial
1050.2.g.a.799.2 2 5.4 even 2 inner
1134.2.f.g.379.1 2 45.22 odd 12
1134.2.f.g.757.1 2 45.7 odd 12
1134.2.f.j.379.1 2 45.32 even 12
1134.2.f.j.757.1 2 45.2 even 12
1344.2.a.i.1.1 1 40.27 even 4
1344.2.a.q.1.1 1 40.37 odd 4
2352.2.a.l.1.1 1 140.27 odd 4
2352.2.q.i.961.1 2 140.67 even 12
2352.2.q.i.1537.1 2 140.107 even 12
2352.2.q.n.961.1 2 140.87 odd 12
2352.2.q.n.1537.1 2 140.47 odd 12
3150.2.a.bo.1.1 1 15.8 even 4
3150.2.g.r.2899.1 2 15.14 odd 2
3150.2.g.r.2899.2 2 3.2 odd 2
4032.2.a.e.1.1 1 120.77 even 4
4032.2.a.m.1.1 1 120.107 odd 4
5082.2.a.d.1.1 1 55.32 even 4
5376.2.c.e.2689.1 2 80.27 even 4
5376.2.c.e.2689.2 2 80.67 even 4
5376.2.c.bc.2689.1 2 80.77 odd 4
5376.2.c.bc.2689.2 2 80.37 odd 4
7056.2.a.k.1.1 1 420.167 even 4
7098.2.a.f.1.1 1 65.12 odd 4
7350.2.a.f.1.1 1 35.13 even 4
8400.2.a.k.1.1 1 20.3 even 4
9408.2.a.n.1.1 1 280.237 even 4
9408.2.a.bw.1.1 1 280.27 odd 4