Properties

 Label 3850.2.c.j.1849.1 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ 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 154) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 1849.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.j.1849.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -3.00000i q^{18} +6.00000 q^{19} +1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +1.00000i q^{28} +2.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} -4.00000 q^{34} -3.00000 q^{36} +10.0000i q^{37} -6.00000i q^{38} +4.00000 q^{41} +8.00000i q^{43} +1.00000 q^{44} -4.00000 q^{46} +2.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{56} -2.00000i q^{58} +12.0000 q^{59} -14.0000 q^{61} +2.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} +4.00000i q^{68} -8.00000 q^{71} +3.00000i q^{72} -4.00000i q^{73} +10.0000 q^{74} -6.00000 q^{76} +1.00000i q^{77} +9.00000 q^{81} -4.00000i q^{82} +6.00000i q^{83} +8.00000 q^{86} -1.00000i q^{88} +6.00000 q^{89} -2.00000 q^{91} +4.00000i q^{92} +2.00000 q^{94} -14.0000i q^{97} +1.00000i q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 6 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{26} + 4 q^{29} - 4 q^{31} - 8 q^{34} - 6 q^{36} + 8 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49} + 2 q^{56} + 24 q^{59} - 28 q^{61} - 2 q^{64} - 16 q^{71} + 20 q^{74} - 12 q^{76} + 18 q^{81} + 16 q^{86} + 12 q^{89} - 4 q^{91} + 4 q^{94} - 6 q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ − 3.00000i − 0.707107i
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.00000i 0.213201i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 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
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 6.00000i − 0.973329i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ − 2.00000i − 0.262613i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ − 3.00000i − 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000i 0.353553i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 1.00000i 0.113961i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ − 4.00000i − 0.441726i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ − 1.00000i − 0.106600i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 4.00000i 0.417029i
$$93$$ 0 0
$$94$$ 2.00000 0.206284
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ − 18.0000i − 1.77359i −0.462160 0.886796i $$-0.652926\pi$$
0.462160 0.886796i $$-0.347074\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ − 6.00000i − 0.554700i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 14.0000i 1.26750i
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −3.00000 −0.267261
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ − 6.00000i − 0.520266i
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ 2.00000i 0.167248i
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ − 10.0000i − 0.821995i
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 6.00000i 0.486664i
$$153$$ − 12.0000i − 0.970143i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 8.00000i − 0.638470i −0.947676 0.319235i $$-0.896574\pi$$
0.947676 0.319235i $$-0.103426\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ − 9.00000i − 0.707107i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 4.00000i 0.309529i 0.987951 + 0.154765i $$0.0494619\pi$$
−0.987951 + 0.154765i $$0.950538\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 18.0000 1.37649
$$172$$ − 8.00000i − 0.609994i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ − 6.00000i − 0.449719i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ − 2.00000i − 0.145865i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 3.00000i 0.213201i
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 6.00000i − 0.422159i
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −18.0000 −1.25412
$$207$$ − 12.0000i − 0.834058i
$$208$$ − 2.00000i − 0.138675i
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.00000i 0.135769i
$$218$$ − 14.0000i − 0.948200i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ 2.00000i 0.133930i 0.997755 + 0.0669650i $$0.0213316\pi$$
−0.997755 + 0.0669650i $$0.978668\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 2.00000i − 0.132745i −0.997795 0.0663723i $$-0.978857\pi$$
0.997795 0.0663723i $$-0.0211425\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ − 30.0000i − 1.96537i −0.185296 0.982683i $$-0.559325\pi$$
0.185296 0.982683i $$-0.440675\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 4.00000i 0.259281i
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 12.0000i − 0.763542i
$$248$$ − 2.00000i − 0.127000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 3.00000i 0.188982i
$$253$$ 4.00000i 0.251478i
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ 0 0
$$259$$ 10.0000 0.621370
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ − 6.00000i − 0.370681i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ 12.0000i 0.733017i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ − 4.00000i − 0.242536i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 30.0000i − 1.80253i −0.433273 0.901263i $$-0.642641\pi$$
0.433273 0.901263i $$-0.357359\pi$$
$$278$$ 14.0000i 0.839664i
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ − 6.00000i − 0.356663i −0.983970 0.178331i $$-0.942930\pi$$
0.983970 0.178331i $$-0.0570699\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ − 4.00000i − 0.236113i
$$288$$ − 3.00000i − 0.176777i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4.00000i 0.234082i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ 2.00000i 0.115857i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 24.0000i 1.38104i
$$303$$ 0 0
$$304$$ 6.00000 0.344124
$$305$$ 0 0
$$306$$ −12.0000 −0.685994
$$307$$ − 10.0000i − 0.570730i −0.958419 0.285365i $$-0.907885\pi$$
0.958419 0.285365i $$-0.0921148\pi$$
$$308$$ − 1.00000i − 0.0569803i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ 2.00000i 0.113047i 0.998401 + 0.0565233i $$0.0180015\pi$$
−0.998401 + 0.0565233i $$0.981998\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 4.00000i 0.222911i
$$323$$ − 24.0000i − 1.33540i
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 4.00000i 0.220863i
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ 30.0000i 1.64399i
$$334$$ 4.00000 0.218870
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 18.0000i − 0.980522i −0.871576 0.490261i $$-0.836901\pi$$
0.871576 0.490261i $$-0.163099\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.00000 0.108306
$$342$$ − 18.0000i − 0.973329i
$$343$$ 1.00000i 0.0539949i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 8.00000i 0.429463i 0.976673 + 0.214731i $$0.0688876\pi$$
−0.976673 + 0.214731i $$0.931112\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.00000i 0.0533002i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 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
$$361$$ 17.0000 0.894737
$$362$$ − 20.0000i − 1.05118i
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 22.0000i 1.14839i 0.818718 + 0.574195i $$0.194685\pi$$
−0.818718 + 0.574195i $$0.805315\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ −2.00000 −0.103142
$$377$$ − 4.00000i − 0.206010i
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 4.00000i 0.204658i
$$383$$ 10.0000i 0.510976i 0.966812 + 0.255488i $$0.0822362\pi$$
−0.966812 + 0.255488i $$0.917764\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 24.0000i 1.21999i
$$388$$ 14.0000i 0.710742i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ 24.0000i 1.20453i 0.798298 + 0.602263i $$0.205734\pi$$
−0.798298 + 0.602263i $$0.794266\pi$$
$$398$$ − 14.0000i − 0.701757i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 4.00000i 0.199254i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ − 10.0000i − 0.495682i
$$408$$ 0 0
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 18.0000i 0.886796i
$$413$$ − 12.0000i − 0.590481i
$$414$$ −12.0000 −0.589768
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 6.00000i 0.293470i
$$419$$ 32.0000 1.56330 0.781651 0.623716i $$-0.214378\pi$$
0.781651 + 0.623716i $$0.214378\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 6.00000i 0.291730i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 14.0000i 0.677507i
$$428$$ 16.0000i 0.773389i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ − 24.0000i − 1.14808i
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 8.00000i 0.380521i
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ 14.0000i 0.658505i
$$453$$ 0 0
$$454$$ −2.00000 −0.0938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ 20.0000i 0.934539i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 32.0000i 1.48717i 0.668644 + 0.743583i $$0.266875\pi$$
−0.668644 + 0.743583i $$0.733125\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −30.0000 −1.38972
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 12.0000i 0.552345i
$$473$$ − 8.00000i − 0.367840i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ − 18.0000i − 0.824163i
$$478$$ 16.0000i 0.731823i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ − 12.0000i − 0.546585i
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 28.0000i − 1.26880i −0.773004 0.634401i $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ − 14.0000i − 0.633750i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ − 8.00000i − 0.360302i
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 8.00000i 0.358849i
$$498$$ 0 0
$$499$$ −44.0000 −1.96971 −0.984855 0.173379i $$-0.944532\pi$$
−0.984855 + 0.173379i $$0.944532\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ 36.0000i 1.60516i 0.596544 + 0.802580i $$0.296540\pi$$
−0.596544 + 0.802580i $$0.703460\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ 0 0
$$508$$ − 8.00000i − 0.354943i
$$509$$ 28.0000 1.24108 0.620539 0.784176i $$-0.286914\pi$$
0.620539 + 0.784176i $$0.286914\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 2.00000i − 0.0879599i
$$518$$ − 10.0000i − 0.439375i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 34.0000i 1.48672i 0.668894 + 0.743358i $$0.266768\pi$$
−0.668894 + 0.743358i $$0.733232\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000i 0.348485i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 36.0000 1.56227
$$532$$ 6.00000i 0.260133i
$$533$$ − 8.00000i − 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 12.0000i 0.517357i
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 12.0000i − 0.513083i −0.966533 0.256541i $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −42.0000 −1.79252
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ 14.0000i 0.593199i 0.955002 + 0.296600i $$0.0958526\pi$$
−0.955002 + 0.296600i $$0.904147\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 10.0000i 0.421825i
$$563$$ 34.0000i 1.43293i 0.697623 + 0.716465i $$0.254241\pi$$
−0.697623 + 0.716465i $$0.745759\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −6.00000 −0.252199
$$567$$ − 9.00000i − 0.377964i
$$568$$ − 8.00000i − 0.335673i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ − 2.00000i − 0.0836242i
$$573$$ 0 0
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ 14.0000i 0.582828i 0.956597 + 0.291414i $$0.0941257\pi$$
−0.956597 + 0.291414i $$0.905874\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ 6.00000i 0.248495i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000i 0.410997i
$$593$$ 12.0000i 0.492781i 0.969171 + 0.246390i $$0.0792446\pi$$
−0.969171 + 0.246390i $$0.920755\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 0 0
$$598$$ 8.00000i 0.327144i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ − 36.0000i − 1.46603i
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000i 0.324710i 0.986732 + 0.162355i $$0.0519090\pi$$
−0.986732 + 0.162355i $$0.948091\pi$$
$$608$$ − 6.00000i − 0.243332i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4.00000 0.161823
$$612$$ 12.0000i 0.485071i
$$613$$ − 46.0000i − 1.85792i −0.370177 0.928961i $$-0.620703\pi$$
0.370177 0.928961i $$-0.379297\pi$$
$$614$$ −10.0000 −0.403567
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 14.0000i − 0.561349i
$$623$$ − 6.00000i − 0.240385i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 2.00000 0.0799361
$$627$$ 0 0
$$628$$ 8.00000i 0.319235i
$$629$$ 40.0000 1.59490
$$630$$ 0 0
$$631$$ −12.0000 −0.477712 −0.238856 0.971055i $$-0.576772\pi$$
−0.238856 + 0.971055i $$0.576772\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ 2.00000i 0.0791808i
$$639$$ −24.0000 −0.949425
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 0 0
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 6.00000i 0.235884i 0.993020 + 0.117942i $$0.0376297\pi$$
−0.993020 + 0.117942i $$0.962370\pi$$
$$648$$ 9.00000i 0.353553i
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 10.0000i − 0.391330i −0.980671 0.195665i $$-0.937313\pi$$
0.980671 0.195665i $$-0.0626866\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 4.00000 0.156174
$$657$$ − 12.0000i − 0.468165i
$$658$$ − 2.00000i − 0.0779681i
$$659$$ 8.00000 0.311636 0.155818 0.987786i $$-0.450199\pi$$
0.155818 + 0.987786i $$0.450199\pi$$
$$660$$ 0 0
$$661$$ 20.0000 0.777910 0.388955 0.921257i $$-0.372836\pi$$
0.388955 + 0.921257i $$0.372836\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 30.0000 1.16248
$$667$$ − 8.00000i − 0.309761i
$$668$$ − 4.00000i − 0.154765i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ 0 0
$$673$$ − 22.0000i − 0.848038i −0.905653 0.424019i $$-0.860619\pi$$
0.905653 0.424019i $$-0.139381\pi$$
$$674$$ −18.0000 −0.693334
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 26.0000i 0.999261i 0.866239 + 0.499631i $$0.166531\pi$$
−0.866239 + 0.499631i $$0.833469\pi$$
$$678$$ 0 0
$$679$$ −14.0000 −0.537271
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 2.00000i − 0.0765840i
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ −18.0000 −0.688247
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ − 14.0000i − 0.532200i
$$693$$ 3.00000i 0.113961i
$$694$$ 8.00000 0.303676
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 16.0000i − 0.606043i
$$698$$ − 10.0000i − 0.378506i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 60.0000i 2.26294i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ −18.0000 −0.676004 −0.338002 0.941145i $$-0.609751\pi$$
−0.338002 + 0.941145i $$0.609751\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000i 0.224860i
$$713$$ 8.00000i 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ − 16.0000i − 0.597115i
$$719$$ 26.0000 0.969636 0.484818 0.874615i $$-0.338886\pi$$
0.484818 + 0.874615i $$0.338886\pi$$
$$720$$ 0 0
$$721$$ −18.0000 −0.670355
$$722$$ − 17.0000i − 0.632674i
$$723$$ 0 0
$$724$$ −20.0000 −0.743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 10.0000i − 0.370879i −0.982656 0.185440i $$-0.940629\pi$$
0.982656 0.185440i $$-0.0593710\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ 22.0000 0.812035
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 12.0000i 0.442026i
$$738$$ − 12.0000i − 0.441726i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 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
$$746$$ 10.0000 0.366126
$$747$$ 18.0000i 0.658586i
$$748$$ − 4.00000i − 0.146254i
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ 2.00000i 0.0729325i
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000i 0.944986i 0.881334 + 0.472493i $$0.156646\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ 0 0
$$763$$ − 14.0000i − 0.506834i
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ 10.0000 0.361315
$$767$$ − 24.0000i − 0.866590i
$$768$$ 0 0
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000i 0.0719816i
$$773$$ 48.0000i 1.72644i 0.504828 + 0.863220i $$0.331556\pi$$
−0.504828 + 0.863220i $$0.668444\pi$$
$$774$$ 24.0000 0.862662
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ − 30.0000i − 1.07555i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 16.0000i 0.572159i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 22.0000i − 0.784215i −0.919919 0.392108i $$-0.871746\pi$$
0.919919 0.392108i $$-0.128254\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ − 3.00000i − 0.106600i
$$793$$ 28.0000i 0.994309i
$$794$$ 24.0000 0.851728
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ − 16.0000i − 0.566749i −0.959009 0.283375i $$-0.908546\pi$$
0.959009 0.283375i $$-0.0914540\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 18.0000i 0.635602i
$$803$$ 4.00000i 0.141157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 6.00000i 0.211079i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ 2.00000i 0.0701862i
$$813$$ 0 0
$$814$$ −10.0000 −0.350500
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 48.0000i 1.67931i
$$818$$ 16.0000i 0.559427i
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 18.0000 0.627060
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 20.0000i 0.695468i 0.937593 + 0.347734i $$0.113049\pi$$
−0.937593 + 0.347734i $$0.886951\pi$$
$$828$$ 12.0000i 0.417029i
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.00000i 0.0693375i
$$833$$ 4.00000i 0.138592i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ 0 0
$$838$$ − 32.0000i − 1.10542i
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 2.00000i 0.0689246i
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 40.0000 1.37118
$$852$$ 0 0
$$853$$ − 2.00000i − 0.0684787i −0.999414 0.0342393i $$-0.989099\pi$$
0.999414 0.0342393i $$-0.0109009\pi$$
$$854$$ 14.0000 0.479070
$$855$$ 0 0
$$856$$ 16.0000 0.546869
$$857$$ 32.0000i 1.09310i 0.837427 + 0.546550i $$0.184059\pi$$
−0.837427 + 0.546550i $$0.815941\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 16.0000i − 0.544962i
$$863$$ 44.0000i 1.49778i 0.662696 + 0.748889i $$0.269412\pi$$
−0.662696 + 0.748889i $$0.730588\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ − 2.00000i − 0.0678844i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 14.0000i 0.474100i
$$873$$ − 42.0000i − 1.42148i
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000i 1.14810i 0.818821 + 0.574049i $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 28.0000i 0.944954i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 16.0000i 0.537227i 0.963248 + 0.268614i $$0.0865655\pi$$
−0.963248 + 0.268614i $$0.913434\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ − 2.00000i − 0.0669650i
$$893$$ 12.0000i 0.401565i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 18.0000i − 0.600668i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 4.00000i 0.133185i
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 52.0000i − 1.72663i −0.504664 0.863316i $$-0.668384\pi$$
0.504664 0.863316i $$-0.331616\pi$$
$$908$$ 2.00000i 0.0663723i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ − 6.00000i − 0.198571i
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ − 6.00000i − 0.198137i
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 30.0000i − 0.987997i
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 32.0000 1.05159
$$927$$ − 54.0000i − 1.77359i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 30.0000i 0.982683i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 12.0000i 0.392023i 0.980602 + 0.196011i $$0.0627990\pi$$
−0.980602 + 0.196011i $$0.937201\pi$$
$$938$$ 12.0000i 0.391814i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 0 0
$$943$$ − 16.0000i − 0.521032i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 4.00000i − 0.129641i
$$953$$ 22.0000i 0.712650i 0.934362 + 0.356325i $$0.115970\pi$$
−0.934362 + 0.356325i $$0.884030\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ − 16.0000i − 0.516937i
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 20.0000i − 0.644826i
$$963$$ − 48.0000i − 1.54678i
$$964$$ −12.0000 −0.386494
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 56.0000 1.79713 0.898563 0.438845i $$-0.144612\pi$$
0.898563 + 0.438845i $$0.144612\pi$$
$$972$$ 0 0
$$973$$ 14.0000i 0.448819i
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ 2.00000i 0.0639857i 0.999488 + 0.0319928i $$0.0101854\pi$$
−0.999488 + 0.0319928i $$0.989815\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 42.0000 1.34096
$$982$$ 36.0000i 1.14881i
$$983$$ − 18.0000i − 0.574111i −0.957914 0.287055i $$-0.907324\pi$$
0.957914 0.287055i $$-0.0926764\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 12.0000i 0.381771i
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ 0 0
$$994$$ 8.00000 0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 42.0000i − 1.33015i −0.746775 0.665077i $$-0.768399\pi$$
0.746775 0.665077i $$-0.231601\pi$$
$$998$$ 44.0000i 1.39280i
$$999$$ 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 3850.2.c.j.1849.1 2
5.2 odd 4 3850.2.a.u.1.1 1
5.3 odd 4 154.2.a.a.1.1 1
5.4 even 2 inner 3850.2.c.j.1849.2 2
15.8 even 4 1386.2.a.l.1.1 1
20.3 even 4 1232.2.a.e.1.1 1
35.3 even 12 1078.2.e.i.177.1 2
35.13 even 4 1078.2.a.d.1.1 1
35.18 odd 12 1078.2.e.j.177.1 2
35.23 odd 12 1078.2.e.j.67.1 2
35.33 even 12 1078.2.e.i.67.1 2
40.3 even 4 4928.2.a.w.1.1 1
40.13 odd 4 4928.2.a.v.1.1 1
55.43 even 4 1694.2.a.g.1.1 1
105.83 odd 4 9702.2.a.ba.1.1 1
140.83 odd 4 8624.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.a.1.1 1 5.3 odd 4
1078.2.a.d.1.1 1 35.13 even 4
1078.2.e.i.67.1 2 35.33 even 12
1078.2.e.i.177.1 2 35.3 even 12
1078.2.e.j.67.1 2 35.23 odd 12
1078.2.e.j.177.1 2 35.18 odd 12
1232.2.a.e.1.1 1 20.3 even 4
1386.2.a.l.1.1 1 15.8 even 4
1694.2.a.g.1.1 1 55.43 even 4
3850.2.a.u.1.1 1 5.2 odd 4
3850.2.c.j.1849.1 2 1.1 even 1 trivial
3850.2.c.j.1849.2 2 5.4 even 2 inner
4928.2.a.v.1.1 1 40.13 odd 4
4928.2.a.w.1.1 1 40.3 even 4
8624.2.a.r.1.1 1 140.83 odd 4
9702.2.a.ba.1.1 1 105.83 odd 4