Properties

 Label 1950.2.e.g.1249.2 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +1.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +8.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} +1.00000i q^{52} +10.0000i q^{53} +1.00000 q^{54} -4.00000i q^{57} -6.00000i q^{58} -4.00000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +6.00000i q^{68} +8.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} +4.00000 q^{76} +1.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} +8.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -6.00000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} + 2q^{16} - 8q^{19} + 2q^{24} + 2q^{26} - 12q^{29} - 16q^{31} + 12q^{34} + 2q^{36} + 2q^{39} - 12q^{41} - 8q^{44} + 16q^{46} + 14q^{49} + 12q^{51} + 2q^{54} - 8q^{59} - 4q^{61} - 2q^{64} - 8q^{66} + 16q^{69} + 32q^{71} + 20q^{74} + 8q^{76} + 32q^{79} + 2q^{81} + 8q^{86} - 20q^{89} - 2q^{96} - 8q^{99} + O(q^{100})$$

Character values

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

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

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 1.00000i − 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ − 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
$$26$$ 1.00000 0.196116
$$27$$ − 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
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 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
$$51$$ 6.00000 0.840168
$$52$$ 1.00000i 0.138675i
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ − 6.00000i − 0.787839i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 1.00000i 0.113228i
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 6.00000i − 0.643268i
$$88$$ − 4.00000i − 0.426401i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 8.00000i 0.834058i
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 1.00000i 0.0924500i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 6.00000i − 0.541002i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000i 1.34269i
$$143$$ − 4.00000i − 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 7.00000i 0.577350i
$$148$$ 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
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 16.0000i 1.27289i
$$159$$ −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 $$-0.0500701\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 4.00000i − 0.300658i
$$178$$ − 10.0000i − 0.749532i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ − 2.00000i − 0.147844i
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 24.0000i − 1.75505i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 10.0000i − 0.719816i −0.932988 0.359908i $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ − 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
$$201$$ 12.0000 0.846415
$$202$$ − 2.00000i − 0.140720i
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.00000i 0.556038i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 16.0000i 1.09630i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 10.0000i 0.677285i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 10.0000i 0.671156i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 14.0000i 0.917170i 0.888650 + 0.458585i $$0.151644\pi$$
−0.888650 + 0.458585i $$0.848356\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 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
$$246$$ 6.00000 0.382546
$$247$$ 4.00000i 0.254514i
$$248$$ 8.00000i 0.508001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ − 32.0000i − 2.01182i
$$254$$ −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 $$-0.0599201\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ − 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
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ 12.0000i 0.733017i
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ 8.00000 0.478947
$$280$$ 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 $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 2.00000i 0.117041i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 6.00000i − 0.347571i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 2.00000i − 0.114897i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 10.0000i 0.553001i
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 10.0000i 0.547997i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 4.00000i 0.216295i
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 20.0000i 1.07366i 0.843692 + 0.536828i $$0.180378\pi$$
−0.843692 + 0.536828i $$0.819622\pi$$
$$348$$ 6.00000i 0.321634i
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 4.00000i 0.213201i
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 6.00000i 0.315353i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 24.0000i − 1.25279i −0.779506 0.626395i $$-0.784530\pi$$
0.779506 0.626395i $$-0.215470\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000i 0.414781i
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000i 0.309016i
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ − 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
$$386$$ 10.0000 0.508987
$$387$$ 4.00000i 0.203331i
$$388$$ 6.00000i 0.304604i
$$389$$ 34.0000 1.72387 0.861934 0.507020i $$-0.169253\pi$$
0.861934 + 0.507020i $$0.169253\pi$$
$$390$$ 0 0
$$391$$ −48.0000 −2.42746
$$392$$ − 7.00000i − 0.353553i
$$393$$ 12.0000i 0.605320i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 30.0000i 1.50566i 0.658217 + 0.752828i $$0.271311\pi$$
−0.658217 + 0.752828i $$0.728689\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 34.0000 1.69788 0.848939 0.528490i $$-0.177242\pi$$
0.848939 + 0.528490i $$0.177242\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ 8.00000i 0.398508i
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 40.0000i − 1.98273i
$$408$$ − 6.00000i − 0.297044i
$$409$$ 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
$$412$$ 0 0
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ − 12.0000i − 0.587643i
$$418$$ − 16.0000i − 0.782586i
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 32.0000i 1.53077i
$$438$$ 2.00000i 0.0955637i
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 6.00000i − 0.285391i
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ − 6.00000i − 0.283790i
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 10.0000i 0.470360i
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 34.0000i 1.59045i 0.606313 + 0.795226i $$0.292648\pi$$
−0.606313 + 0.795226i $$0.707352\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 4.00000i 0.184115i
$$473$$ − 16.0000i − 0.735681i
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 10.0000i − 0.457869i
$$478$$ 8.00000i 0.365911i
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ −10.0000 −0.455961
$$482$$ 18.0000i 0.819878i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 36.0000i 1.62136i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ − 28.0000i − 1.24970i
$$503$$ − 8.00000i − 0.356702i −0.983967 0.178351i $$-0.942924\pi$$
0.983967 0.178351i $$-0.0570763\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ − 1.00000i − 0.0444116i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ − 36.0000i − 1.57417i −0.616844 0.787085i $$-0.711589\pi$$
0.616844 0.787085i $$-0.288411\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 48.0000i 2.09091i
$$528$$ 4.00000i 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 6.00000i 0.259889i
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ − 12.0000i − 0.517838i
$$538$$ 26.0000i 1.12094i
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 24.0000i 1.03089i
$$543$$ 6.00000i 0.257485i
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ − 10.0000i − 0.427179i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ − 8.00000i − 0.340503i
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ 10.0000i 0.421825i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ − 16.0000i − 0.671345i
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 42.0000i 1.74848i 0.485491 + 0.874241i $$0.338641\pi$$
−0.485491 + 0.874241i $$0.661359\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 6.00000i 0.248708i
$$583$$ 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 $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 10.0000i − 0.410997i
$$593$$ − 34.0000i − 1.39621i −0.715994 0.698106i $$-0.754026\pi$$
0.715994 0.698106i $$-0.245974\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 8.00000i 0.327418i
$$598$$ − 8.00000i − 0.327144i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ − 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
$$612$$ − 6.00000i − 0.242536i
$$613$$ 10.0000i 0.403896i 0.979396 + 0.201948i $$0.0647272\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ − 16.0000i − 0.638978i
$$628$$ 2.00000i 0.0798087i
$$629$$ −60.0000 −2.39236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ − 16.0000i − 0.636446i
$$633$$ − 12.0000i − 0.476957i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ − 7.00000i − 0.277350i
$$638$$ − 24.0000i − 0.950169i
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ 34.0000i 1.33052i 0.746611 + 0.665261i $$0.231680\pi$$
−0.746611 + 0.665261i $$0.768320\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ − 6.00000i − 0.233021i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 48.0000i 1.85857i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 10.0000i 0.384048i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ − 32.0000i − 1.22534i
$$683$$ − 28.0000i − 1.07139i −0.844411 0.535695i $$-0.820050\pi$$
0.844411 0.535695i $$-0.179950\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 14.0000i − 0.534133i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 36.0000i 1.36360i
$$698$$ − 6.00000i − 0.227103i
$$699$$ −14.0000 −0.529529
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ − 1.00000i − 0.0377426i
$$703$$ 40.0000i 1.50863i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ 4.00000i 0.150329i
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 10.0000i 0.374766i
$$713$$ 64.0000i 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 8.00000i 0.298765i
$$718$$ − 16.0000i − 0.597115i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 18.0000i 0.669427i
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 2.00000i 0.0739221i
$$733$$ 50.0000i 1.84679i 0.383849 + 0.923396i $$0.374598\pi$$
−0.383849 + 0.923396i $$0.625402\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ − 48.0000i − 1.76810i
$$738$$ 6.00000i 0.220863i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ − 12.0000i − 0.439057i
$$748$$ 24.0000i 0.877527i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ − 28.0000i − 1.02038i
$$754$$ −6.00000 −0.218507
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 42.0000i − 1.52652i −0.646094 0.763258i $$-0.723599\pi$$
0.646094 0.763258i $$-0.276401\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 32.0000 1.15621
$$767$$ 4.00000i 0.144432i
$$768$$ 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
$$771$$ 6.00000 0.216085
$$772$$ 10.0000i 0.359908i
$$773$$ − 38.0000i − 1.36677i −0.730061 0.683383i $$-0.760508\pi$$
0.730061 0.683383i $$-0.239492\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 34.0000i 1.21896i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 64.0000 2.29010
$$782$$ − 48.0000i − 1.71648i
$$783$$ 6.00000i 0.214423i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 4.00000i 0.142134i
$$793$$ 2.00000i 0.0710221i
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ − 2.00000i − 0.0708436i −0.999372 0.0354218i $$-0.988723\pi$$
0.999372 0.0354218i $$-0.0112775\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 34.0000i 1.20058i
$$803$$ − 8.00000i − 0.282314i
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 26.0000i 0.915243i
$$808$$ 2.00000i 0.0703598i
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 16.0000i 0.559769i
$$818$$ 22.0000i 0.769212i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ − 10.0000i − 0.348790i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ − 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
$$831$$ 10.0000 0.346896
$$832$$ 1.00000i 0.0346688i
$$833$$ − 42.0000i − 1.45521i
$$834$$ 12.0000 0.415526
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 8.00000i 0.276520i
$$838$$ 4.00000i 0.138178i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 34.0000i − 1.17172i
$$843$$ 10.0000i 0.344418i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 10.0000i 0.343401i
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ −80.0000 −2.74236
$$852$$ − 16.0000i − 0.548151i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 24.0000i − 0.817443i
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 64.0000 2.17105
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ − 10.0000i − 0.338643i
$$873$$ 6.00000i 0.203069i
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ −26.0000 −0.876958
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 8.00000i 0.268614i 0.990940 + 0.134307i $$0.0428808\pi$$
−0.990940 + 0.134307i $$0.957119\pi$$
$$888$$ − 10.0000i − 0.335578i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 8.00000i 0.267860i
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 8.00000i − 0.267112i
$$898$$ − 18.0000i − 0.600668i
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 60.0000 1.99889
$$902$$ − 24.0000i − 0.799113i
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 12.0000i − 0.398453i −0.979953 0.199227i $$-0.936157\pi$$
0.979953 0.199227i $$-0.0638430\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 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
$$913$$ 48.0000i 1.58857i
$$914$$ −34.0000 −1.12462
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ − 6.00000i − 0.198030i
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 18.0000i − 0.592798i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ 0 0
$$928$$ − 6.00000i − 0.196960i
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 14.0000i − 0.458585i
$$933$$ − 24.0000i − 0.785725i
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ 0 0
$$939$$ −22.0000 −0.717943
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 2.00000i 0.0651635i
$$943$$ 48.0000i 1.56310i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 52.0000i 1.68977i 0.534946 + 0.844886i $$0.320332\pi$$
−0.534946 + 0.844886i $$0.679668\pi$$
$$948$$ − 16.0000i − 0.519656i
$$949$$ −2.00000 −0.0649227
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 0 0
$$953$$ − 34.0000i − 1.10137i −0.834714 0.550684i $$-0.814367\pi$$
0.834714 0.550684i $$-0.185633\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ − 24.0000i − 0.775810i
$$958$$ 8.00000i 0.258468i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 10.0000i − 0.322413i
$$963$$ 12.0000i 0.386695i
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 4.00000i 0.127645i
$$983$$ − 8.00000i − 0.255160i −0.991828 0.127580i $$-0.959279\pi$$
0.991828 0.127580i $$-0.0407210\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ − 4.00000i − 0.127257i
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ −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 1950.2.e.g.1249.2 2
3.2 odd 2 5850.2.e.e.5149.1 2
5.2 odd 4 1950.2.a.k.1.1 1
5.3 odd 4 390.2.a.f.1.1 1
5.4 even 2 inner 1950.2.e.g.1249.1 2
15.2 even 4 5850.2.a.bo.1.1 1
15.8 even 4 1170.2.a.a.1.1 1
15.14 odd 2 5850.2.e.e.5149.2 2
20.3 even 4 3120.2.a.w.1.1 1
60.23 odd 4 9360.2.a.p.1.1 1
65.8 even 4 5070.2.b.d.1351.2 2
65.18 even 4 5070.2.b.d.1351.1 2
65.38 odd 4 5070.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.f.1.1 1 5.3 odd 4
1170.2.a.a.1.1 1 15.8 even 4
1950.2.a.k.1.1 1 5.2 odd 4
1950.2.e.g.1249.1 2 5.4 even 2 inner
1950.2.e.g.1249.2 2 1.1 even 1 trivial
3120.2.a.w.1.1 1 20.3 even 4
5070.2.a.a.1.1 1 65.38 odd 4
5070.2.b.d.1351.1 2 65.18 even 4
5070.2.b.d.1351.2 2 65.8 even 4
5850.2.a.bo.1.1 1 15.2 even 4
5850.2.e.e.5149.1 2 3.2 odd 2
5850.2.e.e.5149.2 2 15.14 odd 2
9360.2.a.p.1.1 1 60.23 odd 4