# Properties

 Label 4650.2.d.n.3349.1 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ 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 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.n.3349.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -4.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} +1.00000 q^{81} -10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +14.0000 q^{89} +8.00000i q^{92} -1.00000i q^{93} +1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 8q^{19} - 2q^{24} + 12q^{26} - 12q^{29} - 2q^{31} - 4q^{34} + 2q^{36} - 12q^{39} + 20q^{41} + 8q^{44} - 16q^{46} + 14q^{49} + 4q^{51} - 2q^{54} + 24q^{59} - 4q^{61} - 2q^{64} - 8q^{66} + 16q^{69} + 4q^{74} + 8q^{76} + 2q^{81} - 8q^{86} + 28q^{89} + 2q^{96} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$-0.706037\pi$$
−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 $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 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 $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ − 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
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ − 6.00000i − 0.832050i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 6.00000i 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 1.00000i 0.127000i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 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
$$82$$ − 10.0000i − 1.10432i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 6.00000i − 0.643268i
$$88$$ − 4.00000i − 0.426401i
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 8.00000i 0.834058i
$$93$$ − 1.00000i − 0.103695i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 18.0000i − 1.82762i −0.406138 0.913812i $$-0.633125\pi$$
0.406138 0.913812i $$-0.366875\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ − 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
$$116$$ 6.00000 0.557086
$$117$$ − 6.00000i − 0.554700i
$$118$$ − 12.0000i − 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 2.00000i 0.181071i
$$123$$ 10.0000i 0.901670i
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 22.0000i 1.87959i 0.341743 + 0.939793i $$0.388983\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ − 8.00000i − 0.681005i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 24.0000i − 2.00698i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 7.00000i 0.577350i
$$148$$ − 2.00000i − 0.164399i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000i 0.901975i
$$178$$ − 14.0000i − 1.04934i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ − 2.00000i − 0.147844i
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 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 $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ − 6.00000i − 0.422159i
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ 8.00000i 0.556038i
$$208$$ 6.00000i 0.416025i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ − 18.0000i − 1.21911i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 2.00000i 0.134231i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ − 24.0000i − 1.52708i
$$248$$ − 1.00000i − 0.0635001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 4.00000i 0.247121i
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 14.0000i 0.856786i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ 22.0000 1.32907
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 26.0000i − 1.51894i −0.650545 0.759468i $$-0.725459\pi$$
0.650545 0.759468i $$-0.274541\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 4.00000i 0.232104i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 8.00000i − 0.460348i
$$303$$ 6.00000i 0.344691i
$$304$$ −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 $$-0.294649\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ 18.0000i 1.01742i 0.860938 + 0.508710i $$0.169877\pi$$
−0.860938 + 0.508710i $$0.830123\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 18.0000i 0.995402i
$$328$$ 10.0000i 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 2.00000i − 0.109599i
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ − 4.00000i − 0.216295i
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 28.0000i − 1.50312i −0.659665 0.751559i $$-0.729302\pi$$
0.659665 0.751559i $$-0.270698\pi$$
$$348$$ 6.00000i 0.321634i
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 4.00000i 0.213201i
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 4.00000i 0.211407i
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000i 0.525588i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 32.0000i − 1.67039i −0.549957 0.835193i $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 1.00000i 0.0518476i
$$373$$ 38.0000i 1.96757i 0.179364 + 0.983783i $$0.442596\pi$$
−0.179364 + 0.983783i $$0.557404\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 24.0000i 1.22795i
$$383$$ − 32.0000i − 1.63512i −0.575841 0.817562i $$-0.695325\pi$$
0.575841 0.817562i $$-0.304675\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 4.00000i 0.203331i
$$388$$ 18.0000i 0.913812i
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 7.00000i 0.353553i
$$393$$ − 4.00000i − 0.201773i
$$394$$ 10.0000 0.503793
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 18.0000i 0.903394i 0.892171 + 0.451697i $$0.149181\pi$$
−0.892171 + 0.451697i $$0.850819\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ − 6.00000i − 0.298881i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 2.00000i 0.0990148i
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ − 16.0000i − 0.788263i
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 4.00000i 0.195881i
$$418$$ − 16.0000i − 0.782586i
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 4.00000i − 0.193347i
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 22.0000i − 1.05725i −0.848855 0.528626i $$-0.822707\pi$$
0.848855 0.528626i $$-0.177293\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 32.0000i 1.53077i
$$438$$ 2.00000i 0.0955637i
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 12.0000i − 0.570782i
$$443$$ 28.0000i 1.33032i 0.746701 + 0.665160i $$0.231637\pi$$
−0.746701 + 0.665160i $$0.768363\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 10.0000i 0.472984i
$$448$$ 0 0
$$449$$ −42.0000 −1.98210 −0.991051 0.133482i $$-0.957384\pi$$
−0.991051 + 0.133482i $$0.957384\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 14.0000i 0.658505i
$$453$$ 8.00000i 0.375873i
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 2.00000i − 0.0935561i −0.998905 0.0467780i $$-0.985105\pi$$
0.998905 0.0467780i $$-0.0148953\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 12.0000i 0.552345i
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10.0000i 0.457869i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ − 18.0000i − 0.819878i
$$483$$ 0 0
$$484$$ −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 $$-0.0580167\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ 12.0000i 0.540453i
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ 0 0
$$498$$ 4.00000i 0.179244i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ − 12.0000i − 0.535586i
$$503$$ − 24.0000i − 1.07011i −0.844818 0.535054i $$-0.820291\pi$$
0.844818 0.535054i $$-0.179709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ − 23.0000i − 1.02147i
$$508$$ 0 0
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 2.00000i 0.0871214i
$$528$$ − 4.00000i − 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 60.0000i 2.59889i
$$534$$ 14.0000 0.605839
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ − 4.00000i − 0.172613i
$$538$$ − 10.0000i − 0.431131i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ − 10.0000i − 0.429141i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ − 22.0000i − 0.939793i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 8.00000i 0.340503i
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ − 1.00000i − 0.0423334i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ − 10.0000i − 0.421825i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 24.0000i 1.00349i
$$573$$ − 24.0000i − 1.00261i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 18.0000i − 0.746124i
$$583$$ 40.0000i 1.65663i
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ 2.00000i 0.0821995i
$$593$$ 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
$$596$$ −10.0000 −0.409616
$$597$$ 8.00000i 0.327418i
$$598$$ − 48.0000i − 1.96287i
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 14.0000i 0.565455i 0.959200 + 0.282727i $$0.0912392\pi$$
−0.959200 + 0.282727i $$0.908761\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ −12.0000 −0.482321 −0.241160 0.970485i $$-0.577528\pi$$
−0.241160 + 0.970485i $$0.577528\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ 18.0000 0.719425
$$627$$ 16.0000i 0.638978i
$$628$$ 14.0000i 0.558661i
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ 4.00000i 0.158986i
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ 42.0000i 1.66410i
$$638$$ − 24.0000i − 0.950169i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 4.00000i 0.157867i
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 0 0
$$645$$ 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
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ 18.0000 0.703856
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 12.0000i 0.466041i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 48.0000i 1.85857i
$$668$$ 24.0000i 0.928588i
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 26.0000i 0.999261i 0.866239 + 0.499631i $$0.166531\pi$$
−0.866239 + 0.499631i $$0.833469\pi$$
$$678$$ − 14.0000i − 0.537667i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ − 4.00000i − 0.153168i
$$683$$ 44.0000i 1.68361i 0.539779 + 0.841807i $$0.318508\pi$$
−0.539779 + 0.841807i $$0.681492\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ − 14.0000i − 0.532200i
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ − 20.0000i − 0.757554i
$$698$$ 30.0000i 1.13552i
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ − 6.00000i − 0.226455i
$$703$$ − 8.00000i − 0.301726i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 0 0
$$708$$ − 12.0000i − 0.450988i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 14.0000i 0.524672i
$$713$$ 8.00000i 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ − 16.0000i − 0.597115i
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ 18.0000i 0.669427i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ 48.0000i 1.78022i 0.455744 + 0.890111i $$0.349373\pi$$
−0.455744 + 0.890111i $$0.650627\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 2.00000i 0.0739221i
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ − 16.0000i − 0.589368i
$$738$$ 10.0000i 0.368105i
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ 38.0000 1.39128
$$747$$ − 4.00000i − 0.146352i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 0 0
$$753$$ 12.0000i 0.437304i
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 14.0000i − 0.508839i −0.967094 0.254419i $$-0.918116\pi$$
0.967094 0.254419i $$-0.0818843\pi$$
$$758$$ 12.0000i 0.435860i
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −32.0000 −1.15621
$$767$$ 72.0000i 2.59977i
$$768$$ 1.00000i 0.0360844i
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ 14.0000i 0.503871i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ 0 0
$$778$$ − 18.0000i − 0.645331i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 16.0000i 0.572159i
$$783$$ 6.00000i 0.214423i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ 12.0000i 0.427754i 0.976861 + 0.213877i $$0.0686091\pi$$
−0.976861 + 0.213877i $$0.931391\pi$$
$$788$$ − 10.0000i − 0.356235i
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 4.00000i 0.142134i
$$793$$ − 12.0000i − 0.426132i
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ − 26.0000i − 0.918092i
$$803$$ − 8.00000i − 0.282314i
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −6.00000 −0.211341
$$807$$ 10.0000i 0.352017i
$$808$$ 6.00000i 0.211079i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 16.0000i 0.559769i
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 22.0000i 0.767338i
$$823$$ − 40.0000i − 1.39431i −0.716919 0.697156i $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ − 8.00000i − 0.278019i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 14.0000i − 0.485071i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 1.00000i 0.0345651i
$$838$$ − 20.0000i − 0.690889i
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 38.0000i − 1.30957i
$$843$$ 10.0000i 0.344418i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 10.0000i − 0.343401i
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ 0 0
$$853$$ 22.0000i 0.753266i 0.926363 + 0.376633i $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 6.00000i 0.204956i 0.994735 + 0.102478i $$0.0326771\pi$$
−0.994735 + 0.102478i $$0.967323\pi$$
$$858$$ − 24.0000i − 0.819346i
$$859$$ 52.0000 1.77422 0.887109 0.461561i $$-0.152710\pi$$
0.887109 + 0.461561i $$0.152710\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 24.0000i − 0.817443i
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −22.0000 −0.747590
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 18.0000i 0.609557i
$$873$$ 18.0000i 0.609208i
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ −22.0000 −0.741199 −0.370599 0.928793i $$-0.620848\pi$$
−0.370599 + 0.928793i $$0.620848\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 48.0000i 1.60267i
$$898$$ 42.0000i 1.40156i
$$899$$ 6.00000 0.200111
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 40.0000i 1.33185i
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 16.0000i − 0.529523i
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ − 6.00000i − 0.197599i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ − 16.0000i − 0.525509i
$$928$$ 6.00000i 0.196960i
$$929$$ 38.0000 1.24674 0.623370 0.781927i $$-0.285763\pi$$
0.623370 + 0.781927i $$0.285763\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 10.0000i − 0.327561i
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ 0 0
$$939$$ −18.0000 −0.587408
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ − 14.0000i − 0.456145i
$$943$$ − 80.0000i − 2.60516i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 24.0000i 0.775810i
$$958$$ 24.0000i 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 12.0000i 0.386896i
$$963$$ − 4.00000i − 0.128898i
$$964$$ −18.0000 −0.579741
$$965$$ 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
$$969$$ −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
$$973$$ 0 0
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ −56.0000 −1.78977
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ 20.0000i 0.638226i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 24.0000i 0.763542i
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 1.00000i 0.0317500i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 42.0000i 1.33015i 0.746775 + 0.665077i $$0.231601\pi$$
−0.746775 + 0.665077i $$0.768399\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 2.00000 0.0632772
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 4650.2.d.n.3349.1 2
5.2 odd 4 930.2.a.o.1.1 1
5.3 odd 4 4650.2.a.h.1.1 1
5.4 even 2 inner 4650.2.d.n.3349.2 2
15.2 even 4 2790.2.a.c.1.1 1
20.7 even 4 7440.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.o.1.1 1 5.2 odd 4
2790.2.a.c.1.1 1 15.2 even 4
4650.2.a.h.1.1 1 5.3 odd 4
4650.2.d.n.3349.1 2 1.1 even 1 trivial
4650.2.d.n.3349.2 2 5.4 even 2 inner
7440.2.a.j.1.1 1 20.7 even 4