# Properties

 Label 5290.2.a.j.1.2 Level $5290$ Weight $2$ Character 5290.1 Self dual yes Analytic conductor $42.241$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5290 = 2 \cdot 5 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5290.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.2408626693$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 5290.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.30278 q^{6} +0.302776 q^{7} -1.00000 q^{8} +7.90833 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.30278 q^{6} +0.302776 q^{7} -1.00000 q^{8} +7.90833 q^{9} +1.00000 q^{10} +5.30278 q^{11} +3.30278 q^{12} -0.302776 q^{13} -0.302776 q^{14} -3.30278 q^{15} +1.00000 q^{16} +3.90833 q^{17} -7.90833 q^{18} +4.90833 q^{19} -1.00000 q^{20} +1.00000 q^{21} -5.30278 q^{22} -3.30278 q^{24} +1.00000 q^{25} +0.302776 q^{26} +16.2111 q^{27} +0.302776 q^{28} +4.60555 q^{29} +3.30278 q^{30} +2.90833 q^{31} -1.00000 q^{32} +17.5139 q^{33} -3.90833 q^{34} -0.302776 q^{35} +7.90833 q^{36} -8.00000 q^{37} -4.90833 q^{38} -1.00000 q^{39} +1.00000 q^{40} -9.90833 q^{41} -1.00000 q^{42} -5.21110 q^{43} +5.30278 q^{44} -7.90833 q^{45} +4.60555 q^{47} +3.30278 q^{48} -6.90833 q^{49} -1.00000 q^{50} +12.9083 q^{51} -0.302776 q^{52} -3.21110 q^{53} -16.2111 q^{54} -5.30278 q^{55} -0.302776 q^{56} +16.2111 q^{57} -4.60555 q^{58} -10.6056 q^{59} -3.30278 q^{60} +6.51388 q^{61} -2.90833 q^{62} +2.39445 q^{63} +1.00000 q^{64} +0.302776 q^{65} -17.5139 q^{66} +4.00000 q^{67} +3.90833 q^{68} +0.302776 q^{70} -12.6972 q^{71} -7.90833 q^{72} +15.8167 q^{73} +8.00000 q^{74} +3.30278 q^{75} +4.90833 q^{76} +1.60555 q^{77} +1.00000 q^{78} -14.4222 q^{79} -1.00000 q^{80} +29.8167 q^{81} +9.90833 q^{82} +3.21110 q^{83} +1.00000 q^{84} -3.90833 q^{85} +5.21110 q^{86} +15.2111 q^{87} -5.30278 q^{88} +7.90833 q^{90} -0.0916731 q^{91} +9.60555 q^{93} -4.60555 q^{94} -4.90833 q^{95} -3.30278 q^{96} -2.69722 q^{97} +6.90833 q^{98} +41.9361 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 3q^{7} - 2q^{8} + 5q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 3q^{7} - 2q^{8} + 5q^{9} + 2q^{10} + 7q^{11} + 3q^{12} + 3q^{13} + 3q^{14} - 3q^{15} + 2q^{16} - 3q^{17} - 5q^{18} - q^{19} - 2q^{20} + 2q^{21} - 7q^{22} - 3q^{24} + 2q^{25} - 3q^{26} + 18q^{27} - 3q^{28} + 2q^{29} + 3q^{30} - 5q^{31} - 2q^{32} + 17q^{33} + 3q^{34} + 3q^{35} + 5q^{36} - 16q^{37} + q^{38} - 2q^{39} + 2q^{40} - 9q^{41} - 2q^{42} + 4q^{43} + 7q^{44} - 5q^{45} + 2q^{47} + 3q^{48} - 3q^{49} - 2q^{50} + 15q^{51} + 3q^{52} + 8q^{53} - 18q^{54} - 7q^{55} + 3q^{56} + 18q^{57} - 2q^{58} - 14q^{59} - 3q^{60} - 5q^{61} + 5q^{62} + 12q^{63} + 2q^{64} - 3q^{65} - 17q^{66} + 8q^{67} - 3q^{68} - 3q^{70} - 29q^{71} - 5q^{72} + 10q^{73} + 16q^{74} + 3q^{75} - q^{76} - 4q^{77} + 2q^{78} - 2q^{80} + 38q^{81} + 9q^{82} - 8q^{83} + 2q^{84} + 3q^{85} - 4q^{86} + 16q^{87} - 7q^{88} + 5q^{90} - 11q^{91} + 12q^{93} - 2q^{94} + q^{95} - 3q^{96} - 9q^{97} + 3q^{98} + 37q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 3.30278 1.90686 0.953429 0.301617i $$-0.0975264\pi$$
0.953429 + 0.301617i $$0.0975264\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −3.30278 −1.34835
$$7$$ 0.302776 0.114438 0.0572192 0.998362i $$-0.481777\pi$$
0.0572192 + 0.998362i $$0.481777\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 7.90833 2.63611
$$10$$ 1.00000 0.316228
$$11$$ 5.30278 1.59885 0.799424 0.600768i $$-0.205138\pi$$
0.799424 + 0.600768i $$0.205138\pi$$
$$12$$ 3.30278 0.953429
$$13$$ −0.302776 −0.0839749 −0.0419874 0.999118i $$-0.513369\pi$$
−0.0419874 + 0.999118i $$0.513369\pi$$
$$14$$ −0.302776 −0.0809202
$$15$$ −3.30278 −0.852773
$$16$$ 1.00000 0.250000
$$17$$ 3.90833 0.947909 0.473954 0.880549i $$-0.342826\pi$$
0.473954 + 0.880549i $$0.342826\pi$$
$$18$$ −7.90833 −1.86401
$$19$$ 4.90833 1.12605 0.563024 0.826441i $$-0.309638\pi$$
0.563024 + 0.826441i $$0.309638\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 1.00000 0.218218
$$22$$ −5.30278 −1.13056
$$23$$ 0 0
$$24$$ −3.30278 −0.674176
$$25$$ 1.00000 0.200000
$$26$$ 0.302776 0.0593792
$$27$$ 16.2111 3.11983
$$28$$ 0.302776 0.0572192
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ 3.30278 0.603002
$$31$$ 2.90833 0.522351 0.261175 0.965291i $$-0.415890\pi$$
0.261175 + 0.965291i $$0.415890\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 17.5139 3.04877
$$34$$ −3.90833 −0.670273
$$35$$ −0.302776 −0.0511784
$$36$$ 7.90833 1.31805
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ −4.90833 −0.796236
$$39$$ −1.00000 −0.160128
$$40$$ 1.00000 0.158114
$$41$$ −9.90833 −1.54742 −0.773710 0.633540i $$-0.781601\pi$$
−0.773710 + 0.633540i $$0.781601\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −5.21110 −0.794686 −0.397343 0.917670i $$-0.630068\pi$$
−0.397343 + 0.917670i $$0.630068\pi$$
$$44$$ 5.30278 0.799424
$$45$$ −7.90833 −1.17890
$$46$$ 0 0
$$47$$ 4.60555 0.671789 0.335894 0.941900i $$-0.390961\pi$$
0.335894 + 0.941900i $$0.390961\pi$$
$$48$$ 3.30278 0.476715
$$49$$ −6.90833 −0.986904
$$50$$ −1.00000 −0.141421
$$51$$ 12.9083 1.80753
$$52$$ −0.302776 −0.0419874
$$53$$ −3.21110 −0.441079 −0.220539 0.975378i $$-0.570782\pi$$
−0.220539 + 0.975378i $$0.570782\pi$$
$$54$$ −16.2111 −2.20605
$$55$$ −5.30278 −0.715026
$$56$$ −0.302776 −0.0404601
$$57$$ 16.2111 2.14721
$$58$$ −4.60555 −0.604739
$$59$$ −10.6056 −1.38073 −0.690363 0.723464i $$-0.742549\pi$$
−0.690363 + 0.723464i $$0.742549\pi$$
$$60$$ −3.30278 −0.426387
$$61$$ 6.51388 0.834017 0.417008 0.908903i $$-0.363079\pi$$
0.417008 + 0.908903i $$0.363079\pi$$
$$62$$ −2.90833 −0.369358
$$63$$ 2.39445 0.301672
$$64$$ 1.00000 0.125000
$$65$$ 0.302776 0.0375547
$$66$$ −17.5139 −2.15581
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 3.90833 0.473954
$$69$$ 0 0
$$70$$ 0.302776 0.0361886
$$71$$ −12.6972 −1.50688 −0.753442 0.657515i $$-0.771608\pi$$
−0.753442 + 0.657515i $$0.771608\pi$$
$$72$$ −7.90833 −0.932005
$$73$$ 15.8167 1.85120 0.925600 0.378504i $$-0.123561\pi$$
0.925600 + 0.378504i $$0.123561\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 3.30278 0.381372
$$76$$ 4.90833 0.563024
$$77$$ 1.60555 0.182970
$$78$$ 1.00000 0.113228
$$79$$ −14.4222 −1.62262 −0.811312 0.584613i $$-0.801246\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 29.8167 3.31296
$$82$$ 9.90833 1.09419
$$83$$ 3.21110 0.352464 0.176232 0.984349i $$-0.443609\pi$$
0.176232 + 0.984349i $$0.443609\pi$$
$$84$$ 1.00000 0.109109
$$85$$ −3.90833 −0.423918
$$86$$ 5.21110 0.561928
$$87$$ 15.2111 1.63080
$$88$$ −5.30278 −0.565278
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 7.90833 0.833611
$$91$$ −0.0916731 −0.00960995
$$92$$ 0 0
$$93$$ 9.60555 0.996049
$$94$$ −4.60555 −0.475026
$$95$$ −4.90833 −0.503584
$$96$$ −3.30278 −0.337088
$$97$$ −2.69722 −0.273862 −0.136931 0.990581i $$-0.543724\pi$$
−0.136931 + 0.990581i $$0.543724\pi$$
$$98$$ 6.90833 0.697846
$$99$$ 41.9361 4.21473
$$100$$ 1.00000 0.100000
$$101$$ −4.60555 −0.458269 −0.229135 0.973395i $$-0.573590\pi$$
−0.229135 + 0.973395i $$0.573590\pi$$
$$102$$ −12.9083 −1.27811
$$103$$ 17.1194 1.68683 0.843414 0.537265i $$-0.180542\pi$$
0.843414 + 0.537265i $$0.180542\pi$$
$$104$$ 0.302776 0.0296896
$$105$$ −1.00000 −0.0975900
$$106$$ 3.21110 0.311890
$$107$$ −4.60555 −0.445235 −0.222618 0.974906i $$-0.571460\pi$$
−0.222618 + 0.974906i $$0.571460\pi$$
$$108$$ 16.2111 1.55991
$$109$$ −19.5139 −1.86909 −0.934545 0.355844i $$-0.884193\pi$$
−0.934545 + 0.355844i $$0.884193\pi$$
$$110$$ 5.30278 0.505600
$$111$$ −26.4222 −2.50788
$$112$$ 0.302776 0.0286096
$$113$$ −12.4222 −1.16858 −0.584291 0.811544i $$-0.698627\pi$$
−0.584291 + 0.811544i $$0.698627\pi$$
$$114$$ −16.2111 −1.51831
$$115$$ 0 0
$$116$$ 4.60555 0.427615
$$117$$ −2.39445 −0.221367
$$118$$ 10.6056 0.976320
$$119$$ 1.18335 0.108477
$$120$$ 3.30278 0.301501
$$121$$ 17.1194 1.55631
$$122$$ −6.51388 −0.589739
$$123$$ −32.7250 −2.95071
$$124$$ 2.90833 0.261175
$$125$$ −1.00000 −0.0894427
$$126$$ −2.39445 −0.213314
$$127$$ −11.8167 −1.04856 −0.524279 0.851546i $$-0.675665\pi$$
−0.524279 + 0.851546i $$0.675665\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −17.2111 −1.51535
$$130$$ −0.302776 −0.0265552
$$131$$ 3.21110 0.280555 0.140278 0.990112i $$-0.455200\pi$$
0.140278 + 0.990112i $$0.455200\pi$$
$$132$$ 17.5139 1.52439
$$133$$ 1.48612 0.128863
$$134$$ −4.00000 −0.345547
$$135$$ −16.2111 −1.39523
$$136$$ −3.90833 −0.335136
$$137$$ 6.90833 0.590218 0.295109 0.955464i $$-0.404644\pi$$
0.295109 + 0.955464i $$0.404644\pi$$
$$138$$ 0 0
$$139$$ −5.39445 −0.457551 −0.228776 0.973479i $$-0.573472\pi$$
−0.228776 + 0.973479i $$0.573472\pi$$
$$140$$ −0.302776 −0.0255892
$$141$$ 15.2111 1.28101
$$142$$ 12.6972 1.06553
$$143$$ −1.60555 −0.134263
$$144$$ 7.90833 0.659027
$$145$$ −4.60555 −0.382470
$$146$$ −15.8167 −1.30900
$$147$$ −22.8167 −1.88189
$$148$$ −8.00000 −0.657596
$$149$$ −9.69722 −0.794428 −0.397214 0.917726i $$-0.630023\pi$$
−0.397214 + 0.917726i $$0.630023\pi$$
$$150$$ −3.30278 −0.269671
$$151$$ −1.90833 −0.155297 −0.0776487 0.996981i $$-0.524741\pi$$
−0.0776487 + 0.996981i $$0.524741\pi$$
$$152$$ −4.90833 −0.398118
$$153$$ 30.9083 2.49879
$$154$$ −1.60555 −0.129379
$$155$$ −2.90833 −0.233602
$$156$$ −1.00000 −0.0800641
$$157$$ 11.3944 0.909376 0.454688 0.890651i $$-0.349751\pi$$
0.454688 + 0.890651i $$0.349751\pi$$
$$158$$ 14.4222 1.14737
$$159$$ −10.6056 −0.841075
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −29.8167 −2.34262
$$163$$ 5.69722 0.446241 0.223121 0.974791i $$-0.428376\pi$$
0.223121 + 0.974791i $$0.428376\pi$$
$$164$$ −9.90833 −0.773710
$$165$$ −17.5139 −1.36345
$$166$$ −3.21110 −0.249230
$$167$$ 21.2111 1.64136 0.820682 0.571385i $$-0.193594\pi$$
0.820682 + 0.571385i $$0.193594\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −12.9083 −0.992948
$$170$$ 3.90833 0.299755
$$171$$ 38.8167 2.96838
$$172$$ −5.21110 −0.397343
$$173$$ 23.3028 1.77168 0.885839 0.463993i $$-0.153584\pi$$
0.885839 + 0.463993i $$0.153584\pi$$
$$174$$ −15.2111 −1.15315
$$175$$ 0.302776 0.0228877
$$176$$ 5.30278 0.399712
$$177$$ −35.0278 −2.63285
$$178$$ 0 0
$$179$$ 16.6056 1.24116 0.620579 0.784144i $$-0.286898\pi$$
0.620579 + 0.784144i $$0.286898\pi$$
$$180$$ −7.90833 −0.589452
$$181$$ 8.11943 0.603512 0.301756 0.953385i $$-0.402427\pi$$
0.301756 + 0.953385i $$0.402427\pi$$
$$182$$ 0.0916731 0.00679526
$$183$$ 21.5139 1.59035
$$184$$ 0 0
$$185$$ 8.00000 0.588172
$$186$$ −9.60555 −0.704313
$$187$$ 20.7250 1.51556
$$188$$ 4.60555 0.335894
$$189$$ 4.90833 0.357028
$$190$$ 4.90833 0.356087
$$191$$ 1.39445 0.100899 0.0504494 0.998727i $$-0.483935\pi$$
0.0504494 + 0.998727i $$0.483935\pi$$
$$192$$ 3.30278 0.238357
$$193$$ 3.81665 0.274729 0.137364 0.990521i $$-0.456137\pi$$
0.137364 + 0.990521i $$0.456137\pi$$
$$194$$ 2.69722 0.193649
$$195$$ 1.00000 0.0716115
$$196$$ −6.90833 −0.493452
$$197$$ 0.697224 0.0496752 0.0248376 0.999691i $$-0.492093\pi$$
0.0248376 + 0.999691i $$0.492093\pi$$
$$198$$ −41.9361 −2.98027
$$199$$ −8.42221 −0.597034 −0.298517 0.954404i $$-0.596492\pi$$
−0.298517 + 0.954404i $$0.596492\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 13.2111 0.931839
$$202$$ 4.60555 0.324045
$$203$$ 1.39445 0.0978711
$$204$$ 12.9083 0.903764
$$205$$ 9.90833 0.692028
$$206$$ −17.1194 −1.19277
$$207$$ 0 0
$$208$$ −0.302776 −0.0209937
$$209$$ 26.0278 1.80038
$$210$$ 1.00000 0.0690066
$$211$$ −7.21110 −0.496433 −0.248216 0.968705i $$-0.579844\pi$$
−0.248216 + 0.968705i $$0.579844\pi$$
$$212$$ −3.21110 −0.220539
$$213$$ −41.9361 −2.87341
$$214$$ 4.60555 0.314829
$$215$$ 5.21110 0.355394
$$216$$ −16.2111 −1.10303
$$217$$ 0.880571 0.0597770
$$218$$ 19.5139 1.32165
$$219$$ 52.2389 3.52997
$$220$$ −5.30278 −0.357513
$$221$$ −1.18335 −0.0796005
$$222$$ 26.4222 1.77334
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −0.302776 −0.0202300
$$225$$ 7.90833 0.527222
$$226$$ 12.4222 0.826313
$$227$$ 7.39445 0.490787 0.245393 0.969424i $$-0.421083\pi$$
0.245393 + 0.969424i $$0.421083\pi$$
$$228$$ 16.2111 1.07361
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 5.30278 0.348897
$$232$$ −4.60555 −0.302369
$$233$$ 4.18335 0.274060 0.137030 0.990567i $$-0.456244\pi$$
0.137030 + 0.990567i $$0.456244\pi$$
$$234$$ 2.39445 0.156530
$$235$$ −4.60555 −0.300433
$$236$$ −10.6056 −0.690363
$$237$$ −47.6333 −3.09412
$$238$$ −1.18335 −0.0767049
$$239$$ −9.21110 −0.595817 −0.297908 0.954594i $$-0.596289\pi$$
−0.297908 + 0.954594i $$0.596289\pi$$
$$240$$ −3.30278 −0.213193
$$241$$ −14.4222 −0.929016 −0.464508 0.885569i $$-0.653769\pi$$
−0.464508 + 0.885569i $$0.653769\pi$$
$$242$$ −17.1194 −1.10048
$$243$$ 49.8444 3.19752
$$244$$ 6.51388 0.417008
$$245$$ 6.90833 0.441357
$$246$$ 32.7250 2.08647
$$247$$ −1.48612 −0.0945597
$$248$$ −2.90833 −0.184679
$$249$$ 10.6056 0.672100
$$250$$ 1.00000 0.0632456
$$251$$ 5.51388 0.348033 0.174016 0.984743i $$-0.444325\pi$$
0.174016 + 0.984743i $$0.444325\pi$$
$$252$$ 2.39445 0.150836
$$253$$ 0 0
$$254$$ 11.8167 0.741443
$$255$$ −12.9083 −0.808351
$$256$$ 1.00000 0.0625000
$$257$$ 19.8167 1.23613 0.618064 0.786127i $$-0.287917\pi$$
0.618064 + 0.786127i $$0.287917\pi$$
$$258$$ 17.2111 1.07152
$$259$$ −2.42221 −0.150509
$$260$$ 0.302776 0.0187773
$$261$$ 36.4222 2.25448
$$262$$ −3.21110 −0.198383
$$263$$ 14.5139 0.894964 0.447482 0.894293i $$-0.352321\pi$$
0.447482 + 0.894293i $$0.352321\pi$$
$$264$$ −17.5139 −1.07790
$$265$$ 3.21110 0.197256
$$266$$ −1.48612 −0.0911200
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ 25.8167 1.57407 0.787035 0.616909i $$-0.211615\pi$$
0.787035 + 0.616909i $$0.211615\pi$$
$$270$$ 16.2111 0.986576
$$271$$ −6.30278 −0.382866 −0.191433 0.981506i $$-0.561314\pi$$
−0.191433 + 0.981506i $$0.561314\pi$$
$$272$$ 3.90833 0.236977
$$273$$ −0.302776 −0.0183248
$$274$$ −6.90833 −0.417347
$$275$$ 5.30278 0.319769
$$276$$ 0 0
$$277$$ −12.7889 −0.768410 −0.384205 0.923248i $$-0.625524\pi$$
−0.384205 + 0.923248i $$0.625524\pi$$
$$278$$ 5.39445 0.323538
$$279$$ 23.0000 1.37697
$$280$$ 0.302776 0.0180943
$$281$$ 19.3944 1.15698 0.578488 0.815691i $$-0.303643\pi$$
0.578488 + 0.815691i $$0.303643\pi$$
$$282$$ −15.2111 −0.905808
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ −12.6972 −0.753442
$$285$$ −16.2111 −0.960263
$$286$$ 1.60555 0.0949382
$$287$$ −3.00000 −0.177084
$$288$$ −7.90833 −0.466003
$$289$$ −1.72498 −0.101469
$$290$$ 4.60555 0.270447
$$291$$ −8.90833 −0.522215
$$292$$ 15.8167 0.925600
$$293$$ −8.78890 −0.513453 −0.256726 0.966484i $$-0.582644\pi$$
−0.256726 + 0.966484i $$0.582644\pi$$
$$294$$ 22.8167 1.33069
$$295$$ 10.6056 0.617479
$$296$$ 8.00000 0.464991
$$297$$ 85.9638 4.98813
$$298$$ 9.69722 0.561745
$$299$$ 0 0
$$300$$ 3.30278 0.190686
$$301$$ −1.57779 −0.0909426
$$302$$ 1.90833 0.109812
$$303$$ −15.2111 −0.873855
$$304$$ 4.90833 0.281512
$$305$$ −6.51388 −0.372984
$$306$$ −30.9083 −1.76691
$$307$$ −15.3028 −0.873376 −0.436688 0.899613i $$-0.643849\pi$$
−0.436688 + 0.899613i $$0.643849\pi$$
$$308$$ 1.60555 0.0914848
$$309$$ 56.5416 3.21654
$$310$$ 2.90833 0.165182
$$311$$ −6.42221 −0.364170 −0.182085 0.983283i $$-0.558285\pi$$
−0.182085 + 0.983283i $$0.558285\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 12.7250 0.719258 0.359629 0.933095i $$-0.382903\pi$$
0.359629 + 0.933095i $$0.382903\pi$$
$$314$$ −11.3944 −0.643026
$$315$$ −2.39445 −0.134912
$$316$$ −14.4222 −0.811312
$$317$$ −14.7250 −0.827037 −0.413519 0.910496i $$-0.635700\pi$$
−0.413519 + 0.910496i $$0.635700\pi$$
$$318$$ 10.6056 0.594730
$$319$$ 24.4222 1.36738
$$320$$ −1.00000 −0.0559017
$$321$$ −15.2111 −0.849001
$$322$$ 0 0
$$323$$ 19.1833 1.06739
$$324$$ 29.8167 1.65648
$$325$$ −0.302776 −0.0167950
$$326$$ −5.69722 −0.315540
$$327$$ −64.4500 −3.56409
$$328$$ 9.90833 0.547096
$$329$$ 1.39445 0.0768784
$$330$$ 17.5139 0.964107
$$331$$ 9.39445 0.516366 0.258183 0.966096i $$-0.416876\pi$$
0.258183 + 0.966096i $$0.416876\pi$$
$$332$$ 3.21110 0.176232
$$333$$ −63.2666 −3.46699
$$334$$ −21.2111 −1.16062
$$335$$ −4.00000 −0.218543
$$336$$ 1.00000 0.0545545
$$337$$ 4.48612 0.244375 0.122187 0.992507i $$-0.461009\pi$$
0.122187 + 0.992507i $$0.461009\pi$$
$$338$$ 12.9083 0.702120
$$339$$ −41.0278 −2.22832
$$340$$ −3.90833 −0.211959
$$341$$ 15.4222 0.835159
$$342$$ −38.8167 −2.09896
$$343$$ −4.21110 −0.227378
$$344$$ 5.21110 0.280964
$$345$$ 0 0
$$346$$ −23.3028 −1.25276
$$347$$ −25.5416 −1.37115 −0.685573 0.728004i $$-0.740448\pi$$
−0.685573 + 0.728004i $$0.740448\pi$$
$$348$$ 15.2111 0.815401
$$349$$ −12.7889 −0.684574 −0.342287 0.939595i $$-0.611202\pi$$
−0.342287 + 0.939595i $$0.611202\pi$$
$$350$$ −0.302776 −0.0161840
$$351$$ −4.90833 −0.261987
$$352$$ −5.30278 −0.282639
$$353$$ 18.4222 0.980515 0.490258 0.871578i $$-0.336903\pi$$
0.490258 + 0.871578i $$0.336903\pi$$
$$354$$ 35.0278 1.86170
$$355$$ 12.6972 0.673899
$$356$$ 0 0
$$357$$ 3.90833 0.206851
$$358$$ −16.6056 −0.877631
$$359$$ 3.21110 0.169476 0.0847378 0.996403i $$-0.472995\pi$$
0.0847378 + 0.996403i $$0.472995\pi$$
$$360$$ 7.90833 0.416805
$$361$$ 5.09167 0.267983
$$362$$ −8.11943 −0.426748
$$363$$ 56.5416 2.96767
$$364$$ −0.0916731 −0.00480498
$$365$$ −15.8167 −0.827881
$$366$$ −21.5139 −1.12455
$$367$$ −29.2111 −1.52481 −0.762404 0.647102i $$-0.775981\pi$$
−0.762404 + 0.647102i $$0.775981\pi$$
$$368$$ 0 0
$$369$$ −78.3583 −4.07917
$$370$$ −8.00000 −0.415900
$$371$$ −0.972244 −0.0504764
$$372$$ 9.60555 0.498025
$$373$$ 2.60555 0.134910 0.0674552 0.997722i $$-0.478512\pi$$
0.0674552 + 0.997722i $$0.478512\pi$$
$$374$$ −20.7250 −1.07166
$$375$$ −3.30278 −0.170555
$$376$$ −4.60555 −0.237513
$$377$$ −1.39445 −0.0718178
$$378$$ −4.90833 −0.252457
$$379$$ −4.09167 −0.210175 −0.105088 0.994463i $$-0.533512\pi$$
−0.105088 + 0.994463i $$0.533512\pi$$
$$380$$ −4.90833 −0.251792
$$381$$ −39.0278 −1.99945
$$382$$ −1.39445 −0.0713462
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −3.30278 −0.168544
$$385$$ −1.60555 −0.0818265
$$386$$ −3.81665 −0.194263
$$387$$ −41.2111 −2.09488
$$388$$ −2.69722 −0.136931
$$389$$ −20.9361 −1.06150 −0.530751 0.847528i $$-0.678090\pi$$
−0.530751 + 0.847528i $$0.678090\pi$$
$$390$$ −1.00000 −0.0506370
$$391$$ 0 0
$$392$$ 6.90833 0.348923
$$393$$ 10.6056 0.534979
$$394$$ −0.697224 −0.0351257
$$395$$ 14.4222 0.725660
$$396$$ 41.9361 2.10737
$$397$$ −21.7250 −1.09035 −0.545173 0.838324i $$-0.683536\pi$$
−0.545173 + 0.838324i $$0.683536\pi$$
$$398$$ 8.42221 0.422167
$$399$$ 4.90833 0.245724
$$400$$ 1.00000 0.0500000
$$401$$ 1.39445 0.0696354 0.0348177 0.999394i $$-0.488915\pi$$
0.0348177 + 0.999394i $$0.488915\pi$$
$$402$$ −13.2111 −0.658910
$$403$$ −0.880571 −0.0438643
$$404$$ −4.60555 −0.229135
$$405$$ −29.8167 −1.48160
$$406$$ −1.39445 −0.0692053
$$407$$ −42.4222 −2.10279
$$408$$ −12.9083 −0.639057
$$409$$ −15.0917 −0.746235 −0.373118 0.927784i $$-0.621711\pi$$
−0.373118 + 0.927784i $$0.621711\pi$$
$$410$$ −9.90833 −0.489337
$$411$$ 22.8167 1.12546
$$412$$ 17.1194 0.843414
$$413$$ −3.21110 −0.158008
$$414$$ 0 0
$$415$$ −3.21110 −0.157627
$$416$$ 0.302776 0.0148448
$$417$$ −17.8167 −0.872485
$$418$$ −26.0278 −1.27306
$$419$$ 39.6333 1.93621 0.968107 0.250538i $$-0.0806073\pi$$
0.968107 + 0.250538i $$0.0806073\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ −34.3028 −1.67181 −0.835907 0.548870i $$-0.815058\pi$$
−0.835907 + 0.548870i $$0.815058\pi$$
$$422$$ 7.21110 0.351031
$$423$$ 36.4222 1.77091
$$424$$ 3.21110 0.155945
$$425$$ 3.90833 0.189582
$$426$$ 41.9361 2.03181
$$427$$ 1.97224 0.0954436
$$428$$ −4.60555 −0.222618
$$429$$ −5.30278 −0.256020
$$430$$ −5.21110 −0.251302
$$431$$ −20.2389 −0.974872 −0.487436 0.873159i $$-0.662068\pi$$
−0.487436 + 0.873159i $$0.662068\pi$$
$$432$$ 16.2111 0.779957
$$433$$ 34.9083 1.67759 0.838794 0.544450i $$-0.183261\pi$$
0.838794 + 0.544450i $$0.183261\pi$$
$$434$$ −0.880571 −0.0422687
$$435$$ −15.2111 −0.729317
$$436$$ −19.5139 −0.934545
$$437$$ 0 0
$$438$$ −52.2389 −2.49607
$$439$$ −18.3028 −0.873544 −0.436772 0.899572i $$-0.643878\pi$$
−0.436772 + 0.899572i $$0.643878\pi$$
$$440$$ 5.30278 0.252800
$$441$$ −54.6333 −2.60159
$$442$$ 1.18335 0.0562860
$$443$$ 35.5139 1.68732 0.843658 0.536882i $$-0.180398\pi$$
0.843658 + 0.536882i $$0.180398\pi$$
$$444$$ −26.4222 −1.25394
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ −32.0278 −1.51486
$$448$$ 0.302776 0.0143048
$$449$$ −12.9083 −0.609182 −0.304591 0.952483i $$-0.598520\pi$$
−0.304591 + 0.952483i $$0.598520\pi$$
$$450$$ −7.90833 −0.372802
$$451$$ −52.5416 −2.47409
$$452$$ −12.4222 −0.584291
$$453$$ −6.30278 −0.296130
$$454$$ −7.39445 −0.347039
$$455$$ 0.0916731 0.00429770
$$456$$ −16.2111 −0.759154
$$457$$ 3.57779 0.167362 0.0836811 0.996493i $$-0.473332\pi$$
0.0836811 + 0.996493i $$0.473332\pi$$
$$458$$ 2.00000 0.0934539
$$459$$ 63.3583 2.95731
$$460$$ 0 0
$$461$$ 31.8167 1.48185 0.740925 0.671588i $$-0.234388\pi$$
0.740925 + 0.671588i $$0.234388\pi$$
$$462$$ −5.30278 −0.246707
$$463$$ −25.6333 −1.19128 −0.595640 0.803251i $$-0.703102\pi$$
−0.595640 + 0.803251i $$0.703102\pi$$
$$464$$ 4.60555 0.213807
$$465$$ −9.60555 −0.445447
$$466$$ −4.18335 −0.193790
$$467$$ −19.8167 −0.917005 −0.458503 0.888693i $$-0.651614\pi$$
−0.458503 + 0.888693i $$0.651614\pi$$
$$468$$ −2.39445 −0.110683
$$469$$ 1.21110 0.0559235
$$470$$ 4.60555 0.212438
$$471$$ 37.6333 1.73405
$$472$$ 10.6056 0.488160
$$473$$ −27.6333 −1.27058
$$474$$ 47.6333 2.18787
$$475$$ 4.90833 0.225209
$$476$$ 1.18335 0.0542386
$$477$$ −25.3944 −1.16273
$$478$$ 9.21110 0.421306
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 3.30278 0.150750
$$481$$ 2.42221 0.110443
$$482$$ 14.4222 0.656913
$$483$$ 0 0
$$484$$ 17.1194 0.778156
$$485$$ 2.69722 0.122475
$$486$$ −49.8444 −2.26099
$$487$$ −11.8167 −0.535464 −0.267732 0.963493i $$-0.586274\pi$$
−0.267732 + 0.963493i $$0.586274\pi$$
$$488$$ −6.51388 −0.294869
$$489$$ 18.8167 0.850918
$$490$$ −6.90833 −0.312086
$$491$$ −25.8167 −1.16509 −0.582545 0.812799i $$-0.697943\pi$$
−0.582545 + 0.812799i $$0.697943\pi$$
$$492$$ −32.7250 −1.47536
$$493$$ 18.0000 0.810679
$$494$$ 1.48612 0.0668638
$$495$$ −41.9361 −1.88489
$$496$$ 2.90833 0.130588
$$497$$ −3.84441 −0.172445
$$498$$ −10.6056 −0.475246
$$499$$ 11.6333 0.520778 0.260389 0.965504i $$-0.416149\pi$$
0.260389 + 0.965504i $$0.416149\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 70.0555 3.12985
$$502$$ −5.51388 −0.246096
$$503$$ 2.72498 0.121501 0.0607504 0.998153i $$-0.480651\pi$$
0.0607504 + 0.998153i $$0.480651\pi$$
$$504$$ −2.39445 −0.106657
$$505$$ 4.60555 0.204944
$$506$$ 0 0
$$507$$ −42.6333 −1.89341
$$508$$ −11.8167 −0.524279
$$509$$ 29.4500 1.30535 0.652673 0.757639i $$-0.273647\pi$$
0.652673 + 0.757639i $$0.273647\pi$$
$$510$$ 12.9083 0.571590
$$511$$ 4.78890 0.211848
$$512$$ −1.00000 −0.0441942
$$513$$ 79.5694 3.51307
$$514$$ −19.8167 −0.874075
$$515$$ −17.1194 −0.754372
$$516$$ −17.2111 −0.757677
$$517$$ 24.4222 1.07409
$$518$$ 2.42221 0.106426
$$519$$ 76.9638 3.37834
$$520$$ −0.302776 −0.0132776
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ −36.4222 −1.59416
$$523$$ −8.42221 −0.368277 −0.184139 0.982900i $$-0.558950\pi$$
−0.184139 + 0.982900i $$0.558950\pi$$
$$524$$ 3.21110 0.140278
$$525$$ 1.00000 0.0436436
$$526$$ −14.5139 −0.632835
$$527$$ 11.3667 0.495141
$$528$$ 17.5139 0.762194
$$529$$ 0 0
$$530$$ −3.21110 −0.139481
$$531$$ −83.8722 −3.63974
$$532$$ 1.48612 0.0644316
$$533$$ 3.00000 0.129944
$$534$$ 0 0
$$535$$ 4.60555 0.199115
$$536$$ −4.00000 −0.172774
$$537$$ 54.8444 2.36671
$$538$$ −25.8167 −1.11303
$$539$$ −36.6333 −1.57791
$$540$$ −16.2111 −0.697615
$$541$$ −28.8444 −1.24012 −0.620059 0.784555i $$-0.712891\pi$$
−0.620059 + 0.784555i $$0.712891\pi$$
$$542$$ 6.30278 0.270727
$$543$$ 26.8167 1.15081
$$544$$ −3.90833 −0.167568
$$545$$ 19.5139 0.835883
$$546$$ 0.302776 0.0129576
$$547$$ 7.51388 0.321270 0.160635 0.987014i $$-0.448646\pi$$
0.160635 + 0.987014i $$0.448646\pi$$
$$548$$ 6.90833 0.295109
$$549$$ 51.5139 2.19856
$$550$$ −5.30278 −0.226111
$$551$$ 22.6056 0.963029
$$552$$ 0 0
$$553$$ −4.36669 −0.185691
$$554$$ 12.7889 0.543348
$$555$$ 26.4222 1.12156
$$556$$ −5.39445 −0.228776
$$557$$ 6.42221 0.272118 0.136059 0.990701i $$-0.456556\pi$$
0.136059 + 0.990701i $$0.456556\pi$$
$$558$$ −23.0000 −0.973668
$$559$$ 1.57779 0.0667336
$$560$$ −0.302776 −0.0127946
$$561$$ 68.4500 2.88996
$$562$$ −19.3944 −0.818105
$$563$$ −39.6333 −1.67034 −0.835172 0.549988i $$-0.814632\pi$$
−0.835172 + 0.549988i $$0.814632\pi$$
$$564$$ 15.2111 0.640503
$$565$$ 12.4222 0.522606
$$566$$ 2.00000 0.0840663
$$567$$ 9.02776 0.379130
$$568$$ 12.6972 0.532764
$$569$$ 0.422205 0.0176998 0.00884988 0.999961i $$-0.497183\pi$$
0.00884988 + 0.999961i $$0.497183\pi$$
$$570$$ 16.2111 0.679008
$$571$$ −9.11943 −0.381636 −0.190818 0.981625i $$-0.561114\pi$$
−0.190818 + 0.981625i $$0.561114\pi$$
$$572$$ −1.60555 −0.0671315
$$573$$ 4.60555 0.192400
$$574$$ 3.00000 0.125218
$$575$$ 0 0
$$576$$ 7.90833 0.329514
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 1.72498 0.0717497
$$579$$ 12.6056 0.523869
$$580$$ −4.60555 −0.191235
$$581$$ 0.972244 0.0403355
$$582$$ 8.90833 0.369262
$$583$$ −17.0278 −0.705218
$$584$$ −15.8167 −0.654498
$$585$$ 2.39445 0.0989983
$$586$$ 8.78890 0.363066
$$587$$ −37.5416 −1.54951 −0.774755 0.632262i $$-0.782127\pi$$
−0.774755 + 0.632262i $$0.782127\pi$$
$$588$$ −22.8167 −0.940943
$$589$$ 14.2750 0.588192
$$590$$ −10.6056 −0.436624
$$591$$ 2.30278 0.0947235
$$592$$ −8.00000 −0.328798
$$593$$ 19.8167 0.813772 0.406886 0.913479i $$-0.366615\pi$$
0.406886 + 0.913479i $$0.366615\pi$$
$$594$$ −85.9638 −3.52714
$$595$$ −1.18335 −0.0485125
$$596$$ −9.69722 −0.397214
$$597$$ −27.8167 −1.13846
$$598$$ 0 0
$$599$$ 4.33053 0.176941 0.0884704 0.996079i $$-0.471802\pi$$
0.0884704 + 0.996079i $$0.471802\pi$$
$$600$$ −3.30278 −0.134835
$$601$$ −3.93608 −0.160556 −0.0802781 0.996773i $$-0.525581\pi$$
−0.0802781 + 0.996773i $$0.525581\pi$$
$$602$$ 1.57779 0.0643061
$$603$$ 31.6333 1.28821
$$604$$ −1.90833 −0.0776487
$$605$$ −17.1194 −0.696004
$$606$$ 15.2111 0.617909
$$607$$ −26.0555 −1.05756 −0.528780 0.848759i $$-0.677350\pi$$
−0.528780 + 0.848759i $$0.677350\pi$$
$$608$$ −4.90833 −0.199059
$$609$$ 4.60555 0.186626
$$610$$ 6.51388 0.263739
$$611$$ −1.39445 −0.0564134
$$612$$ 30.9083 1.24940
$$613$$ −32.4222 −1.30952 −0.654760 0.755837i $$-0.727230\pi$$
−0.654760 + 0.755837i $$0.727230\pi$$
$$614$$ 15.3028 0.617570
$$615$$ 32.7250 1.31960
$$616$$ −1.60555 −0.0646895
$$617$$ −8.09167 −0.325758 −0.162879 0.986646i $$-0.552078\pi$$
−0.162879 + 0.986646i $$0.552078\pi$$
$$618$$ −56.5416 −2.27444
$$619$$ −27.3305 −1.09851 −0.549253 0.835656i $$-0.685088\pi$$
−0.549253 + 0.835656i $$0.685088\pi$$
$$620$$ −2.90833 −0.116801
$$621$$ 0 0
$$622$$ 6.42221 0.257507
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 1.00000 0.0400000
$$626$$ −12.7250 −0.508593
$$627$$ 85.9638 3.43307
$$628$$ 11.3944 0.454688
$$629$$ −31.2666 −1.24668
$$630$$ 2.39445 0.0953971
$$631$$ −30.6056 −1.21839 −0.609194 0.793021i $$-0.708507\pi$$
−0.609194 + 0.793021i $$0.708507\pi$$
$$632$$ 14.4222 0.573685
$$633$$ −23.8167 −0.946627
$$634$$ 14.7250 0.584804
$$635$$ 11.8167 0.468930
$$636$$ −10.6056 −0.420537
$$637$$ 2.09167 0.0828751
$$638$$ −24.4222 −0.966884
$$639$$ −100.414 −3.97231
$$640$$ 1.00000 0.0395285
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ 15.2111 0.600334
$$643$$ −16.2389 −0.640398 −0.320199 0.947350i $$-0.603750\pi$$
−0.320199 + 0.947350i $$0.603750\pi$$
$$644$$ 0 0
$$645$$ 17.2111 0.677687
$$646$$ −19.1833 −0.754759
$$647$$ −30.8444 −1.21262 −0.606309 0.795229i $$-0.707351\pi$$
−0.606309 + 0.795229i $$0.707351\pi$$
$$648$$ −29.8167 −1.17131
$$649$$ −56.2389 −2.20757
$$650$$ 0.302776 0.0118758
$$651$$ 2.90833 0.113986
$$652$$ 5.69722 0.223121
$$653$$ 9.27502 0.362960 0.181480 0.983395i $$-0.441911\pi$$
0.181480 + 0.983395i $$0.441911\pi$$
$$654$$ 64.4500 2.52019
$$655$$ −3.21110 −0.125468
$$656$$ −9.90833 −0.386855
$$657$$ 125.083 4.87996
$$658$$ −1.39445 −0.0543613
$$659$$ 27.6333 1.07644 0.538220 0.842804i $$-0.319097\pi$$
0.538220 + 0.842804i $$0.319097\pi$$
$$660$$ −17.5139 −0.681727
$$661$$ 24.0917 0.937057 0.468529 0.883448i $$-0.344784\pi$$
0.468529 + 0.883448i $$0.344784\pi$$
$$662$$ −9.39445 −0.365126
$$663$$ −3.90833 −0.151787
$$664$$ −3.21110 −0.124615
$$665$$ −1.48612 −0.0576293
$$666$$ 63.2666 2.45153
$$667$$ 0 0
$$668$$ 21.2111 0.820682
$$669$$ −13.2111 −0.510771
$$670$$ 4.00000 0.154533
$$671$$ 34.5416 1.33347
$$672$$ −1.00000 −0.0385758
$$673$$ 5.63331 0.217148 0.108574 0.994088i $$-0.465372\pi$$
0.108574 + 0.994088i $$0.465372\pi$$
$$674$$ −4.48612 −0.172799
$$675$$ 16.2111 0.623966
$$676$$ −12.9083 −0.496474
$$677$$ 12.4222 0.477424 0.238712 0.971090i $$-0.423275\pi$$
0.238712 + 0.971090i $$0.423275\pi$$
$$678$$ 41.0278 1.57566
$$679$$ −0.816654 −0.0313403
$$680$$ 3.90833 0.149877
$$681$$ 24.4222 0.935861
$$682$$ −15.4222 −0.590547
$$683$$ −32.7250 −1.25219 −0.626093 0.779748i $$-0.715347\pi$$
−0.626093 + 0.779748i $$0.715347\pi$$
$$684$$ 38.8167 1.48419
$$685$$ −6.90833 −0.263954
$$686$$ 4.21110 0.160781
$$687$$ −6.60555 −0.252018
$$688$$ −5.21110 −0.198671
$$689$$ 0.972244 0.0370395
$$690$$ 0 0
$$691$$ 30.1833 1.14823 0.574114 0.818775i $$-0.305347\pi$$
0.574114 + 0.818775i $$0.305347\pi$$
$$692$$ 23.3028 0.885839
$$693$$ 12.6972 0.482328
$$694$$ 25.5416 0.969547
$$695$$ 5.39445 0.204623
$$696$$ −15.2111 −0.576575
$$697$$ −38.7250 −1.46681
$$698$$ 12.7889 0.484067
$$699$$ 13.8167 0.522594
$$700$$ 0.302776 0.0114438
$$701$$ −42.9083 −1.62063 −0.810313 0.585998i $$-0.800703\pi$$
−0.810313 + 0.585998i $$0.800703\pi$$
$$702$$ 4.90833 0.185253
$$703$$ −39.2666 −1.48097
$$704$$ 5.30278 0.199856
$$705$$ −15.2111 −0.572883
$$706$$ −18.4222 −0.693329
$$707$$ −1.39445 −0.0524436
$$708$$ −35.0278 −1.31642
$$709$$ 41.1194 1.54427 0.772136 0.635457i $$-0.219188\pi$$
0.772136 + 0.635457i $$0.219188\pi$$
$$710$$ −12.6972 −0.476518
$$711$$ −114.056 −4.27742
$$712$$ 0 0
$$713$$ 0 0
$$714$$ −3.90833 −0.146265
$$715$$ 1.60555 0.0600442
$$716$$ 16.6056 0.620579
$$717$$ −30.4222 −1.13614
$$718$$ −3.21110 −0.119837
$$719$$ 14.3028 0.533404 0.266702 0.963779i $$-0.414066\pi$$
0.266702 + 0.963779i $$0.414066\pi$$
$$720$$ −7.90833 −0.294726
$$721$$ 5.18335 0.193038
$$722$$ −5.09167 −0.189492
$$723$$ −47.6333 −1.77150
$$724$$ 8.11943 0.301756
$$725$$ 4.60555 0.171046
$$726$$ −56.5416 −2.09846
$$727$$ 7.90833 0.293304 0.146652 0.989188i $$-0.453150\pi$$
0.146652 + 0.989188i $$0.453150\pi$$
$$728$$ 0.0916731 0.00339763
$$729$$ 75.1749 2.78426
$$730$$ 15.8167 0.585401
$$731$$ −20.3667 −0.753289
$$732$$ 21.5139 0.795176
$$733$$ 13.6333 0.503558 0.251779 0.967785i $$-0.418984\pi$$
0.251779 + 0.967785i $$0.418984\pi$$
$$734$$ 29.2111 1.07820
$$735$$ 22.8167 0.841605
$$736$$ 0 0
$$737$$ 21.2111 0.781321
$$738$$ 78.3583 2.88441
$$739$$ −7.63331 −0.280796 −0.140398 0.990095i $$-0.544838\pi$$
−0.140398 + 0.990095i $$0.544838\pi$$
$$740$$ 8.00000 0.294086
$$741$$ −4.90833 −0.180312
$$742$$ 0.972244 0.0356922
$$743$$ −7.33053 −0.268931 −0.134466 0.990918i $$-0.542932\pi$$
−0.134466 + 0.990918i $$0.542932\pi$$
$$744$$ −9.60555 −0.352157
$$745$$ 9.69722 0.355279
$$746$$ −2.60555 −0.0953960
$$747$$ 25.3944 0.929134
$$748$$ 20.7250 0.757780
$$749$$ −1.39445 −0.0509520
$$750$$ 3.30278 0.120600
$$751$$ −0.183346 −0.00669040 −0.00334520 0.999994i $$-0.501065\pi$$
−0.00334520 + 0.999994i $$0.501065\pi$$
$$752$$ 4.60555 0.167947
$$753$$ 18.2111 0.663649
$$754$$ 1.39445 0.0507828
$$755$$ 1.90833 0.0694511
$$756$$ 4.90833 0.178514
$$757$$ 1.21110 0.0440183 0.0220091 0.999758i $$-0.492994\pi$$
0.0220091 + 0.999758i $$0.492994\pi$$
$$758$$ 4.09167 0.148616
$$759$$ 0 0
$$760$$ 4.90833 0.178044
$$761$$ 4.54163 0.164634 0.0823171 0.996606i $$-0.473768\pi$$
0.0823171 + 0.996606i $$0.473768\pi$$
$$762$$ 39.0278 1.41383
$$763$$ −5.90833 −0.213896
$$764$$ 1.39445 0.0504494
$$765$$ −30.9083 −1.11749
$$766$$ 0 0
$$767$$ 3.21110 0.115946
$$768$$ 3.30278 0.119179
$$769$$ 41.2666 1.48811 0.744056 0.668117i $$-0.232900\pi$$
0.744056 + 0.668117i $$0.232900\pi$$
$$770$$ 1.60555 0.0578601
$$771$$ 65.4500 2.35712
$$772$$ 3.81665 0.137364
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 41.2111 1.48130
$$775$$ 2.90833 0.104470
$$776$$ 2.69722 0.0968247
$$777$$ −8.00000 −0.286998
$$778$$ 20.9361 0.750595
$$779$$ −48.6333 −1.74247
$$780$$ 1.00000 0.0358057
$$781$$ −67.3305 −2.40928
$$782$$ 0 0
$$783$$ 74.6611 2.66817
$$784$$ −6.90833 −0.246726
$$785$$ −11.3944 −0.406685
$$786$$ −10.6056 −0.378287
$$787$$ 27.4500 0.978485 0.489243 0.872148i $$-0.337273\pi$$
0.489243 + 0.872148i $$0.337273\pi$$
$$788$$ 0.697224 0.0248376
$$789$$ 47.9361 1.70657
$$790$$ −14.4222 −0.513119
$$791$$ −3.76114 −0.133731
$$792$$ −41.9361 −1.49013
$$793$$ −1.97224 −0.0700364
$$794$$ 21.7250 0.770991
$$795$$ 10.6056 0.376140
$$796$$ −8.42221 −0.298517
$$797$$ 19.8167 0.701942 0.350971 0.936386i $$-0.385852\pi$$
0.350971 + 0.936386i $$0.385852\pi$$
$$798$$ −4.90833 −0.173753
$$799$$ 18.0000 0.636794
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −1.39445 −0.0492397
$$803$$ 83.8722 2.95978
$$804$$ 13.2111 0.465920
$$805$$ 0 0
$$806$$ 0.880571 0.0310168
$$807$$ 85.2666 3.00153
$$808$$ 4.60555 0.162023
$$809$$ 18.2750 0.642515 0.321258 0.946992i $$-0.395894\pi$$
0.321258 + 0.946992i $$0.395894\pi$$
$$810$$ 29.8167 1.04765
$$811$$ −4.97224 −0.174599 −0.0872995 0.996182i $$-0.527824\pi$$
−0.0872995 + 0.996182i $$0.527824\pi$$
$$812$$ 1.39445 0.0489356
$$813$$ −20.8167 −0.730072
$$814$$ 42.4222 1.48690
$$815$$ −5.69722 −0.199565
$$816$$ 12.9083 0.451882
$$817$$ −25.5778 −0.894854
$$818$$ 15.0917 0.527668
$$819$$ −0.724981 −0.0253329
$$820$$ 9.90833 0.346014
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ −22.8167 −0.795822
$$823$$ −0.788897 −0.0274992 −0.0137496 0.999905i $$-0.504377\pi$$
−0.0137496 + 0.999905i $$0.504377\pi$$
$$824$$ −17.1194 −0.596384
$$825$$ 17.5139 0.609755
$$826$$ 3.21110 0.111729
$$827$$ −35.4500 −1.23272 −0.616358 0.787466i $$-0.711393\pi$$
−0.616358 + 0.787466i $$0.711393\pi$$
$$828$$ 0 0
$$829$$ 16.7889 0.583103 0.291551 0.956555i $$-0.405829\pi$$
0.291551 + 0.956555i $$0.405829\pi$$
$$830$$ 3.21110 0.111459
$$831$$ −42.2389 −1.46525
$$832$$ −0.302776 −0.0104969
$$833$$ −27.0000 −0.935495
$$834$$ 17.8167 0.616940
$$835$$ −21.2111 −0.734040
$$836$$ 26.0278 0.900189
$$837$$ 47.1472 1.62965
$$838$$ −39.6333 −1.36911
$$839$$ 22.1833 0.765854 0.382927 0.923779i $$-0.374916\pi$$
0.382927 + 0.923779i $$0.374916\pi$$
$$840$$ 1.00000 0.0345033
$$841$$ −7.78890 −0.268583
$$842$$ 34.3028 1.18215
$$843$$ 64.0555 2.20619
$$844$$ −7.21110 −0.248216
$$845$$ 12.9083 0.444060
$$846$$ −36.4222 −1.25222
$$847$$ 5.18335 0.178102
$$848$$ −3.21110 −0.110270
$$849$$ −6.60555 −0.226702
$$850$$ −3.90833 −0.134055
$$851$$ 0 0
$$852$$ −41.9361 −1.43671
$$853$$ 10.7250 0.367216 0.183608 0.983000i $$-0.441222\pi$$
0.183608 + 0.983000i $$0.441222\pi$$
$$854$$ −1.97224 −0.0674888
$$855$$ −38.8167 −1.32750
$$856$$ 4.60555 0.157415
$$857$$ −33.6333 −1.14889 −0.574446 0.818543i $$-0.694782\pi$$
−0.574446 + 0.818543i $$0.694782\pi$$
$$858$$ 5.30278 0.181034
$$859$$ −14.1833 −0.483930 −0.241965 0.970285i $$-0.577792\pi$$
−0.241965 + 0.970285i $$0.577792\pi$$
$$860$$ 5.21110 0.177697
$$861$$ −9.90833 −0.337675
$$862$$ 20.2389 0.689338
$$863$$ 23.4500 0.798246 0.399123 0.916897i $$-0.369315\pi$$
0.399123 + 0.916897i $$0.369315\pi$$
$$864$$ −16.2111 −0.551513
$$865$$ −23.3028 −0.792318
$$866$$ −34.9083 −1.18623
$$867$$ −5.69722 −0.193488
$$868$$ 0.880571 0.0298885
$$869$$ −76.4777 −2.59433
$$870$$ 15.2111 0.515705
$$871$$ −1.21110 −0.0410366
$$872$$ 19.5139 0.660823
$$873$$ −21.3305 −0.721929
$$874$$ 0 0
$$875$$ −0.302776 −0.0102357
$$876$$ 52.2389 1.76499
$$877$$ 49.1749 1.66052 0.830260 0.557376i $$-0.188192\pi$$
0.830260 + 0.557376i $$0.188192\pi$$
$$878$$ 18.3028 0.617689
$$879$$ −29.0278 −0.979082
$$880$$ −5.30278 −0.178757
$$881$$ 31.2666 1.05340 0.526700 0.850052i $$-0.323429\pi$$
0.526700 + 0.850052i $$0.323429\pi$$
$$882$$ 54.6333 1.83960
$$883$$ 40.7250 1.37050 0.685252 0.728306i $$-0.259692\pi$$
0.685252 + 0.728306i $$0.259692\pi$$
$$884$$ −1.18335 −0.0398002
$$885$$ 35.0278 1.17745
$$886$$ −35.5139 −1.19311
$$887$$ −15.6333 −0.524915 −0.262458 0.964944i $$-0.584533\pi$$
−0.262458 + 0.964944i $$0.584533\pi$$
$$888$$ 26.4222 0.886671
$$889$$ −3.57779 −0.119995
$$890$$ 0 0
$$891$$ 158.111 5.29692
$$892$$ −4.00000 −0.133930
$$893$$ 22.6056 0.756466
$$894$$ 32.0278 1.07117
$$895$$ −16.6056 −0.555062
$$896$$ −0.302776 −0.0101150
$$897$$ 0 0
$$898$$ 12.9083 0.430756
$$899$$ 13.3944 0.446730
$$900$$ 7.90833 0.263611
$$901$$ −12.5500 −0.418102
$$902$$ 52.5416 1.74945
$$903$$ −5.21110 −0.173415
$$904$$ 12.4222 0.413156
$$905$$ −8.11943 −0.269899
$$906$$ 6.30278 0.209396
$$907$$ 30.6611 1.01808 0.509042 0.860742i $$-0.330000\pi$$
0.509042 + 0.860742i $$0.330000\pi$$
$$908$$ 7.39445 0.245393
$$909$$ −36.4222 −1.20805
$$910$$ −0.0916731 −0.00303893
$$911$$ 25.8167 0.855344 0.427672 0.903934i $$-0.359334\pi$$
0.427672 + 0.903934i $$0.359334\pi$$
$$912$$ 16.2111 0.536803
$$913$$ 17.0278 0.563536
$$914$$ −3.57779 −0.118343
$$915$$ −21.5139 −0.711227
$$916$$ −2.00000 −0.0660819
$$917$$ 0.972244 0.0321063
$$918$$ −63.3583 −2.09114
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ 0 0
$$921$$ −50.5416 −1.66540
$$922$$ −31.8167 −1.04783
$$923$$ 3.84441 0.126540
$$924$$ 5.30278 0.174449
$$925$$ −8.00000 −0.263038
$$926$$ 25.6333 0.842363
$$927$$ 135.386 4.44666
$$928$$ −4.60555 −0.151185
$$929$$ −57.6333 −1.89089 −0.945444 0.325785i $$-0.894371\pi$$
−0.945444 + 0.325785i $$0.894371\pi$$
$$930$$ 9.60555 0.314978
$$931$$ −33.9083 −1.11130
$$932$$ 4.18335 0.137030
$$933$$ −21.2111 −0.694420
$$934$$ 19.8167 0.648421
$$935$$ −20.7250 −0.677779
$$936$$ 2.39445 0.0782650
$$937$$ 44.9638 1.46890 0.734452 0.678660i $$-0.237439\pi$$
0.734452 + 0.678660i $$0.237439\pi$$
$$938$$ −1.21110 −0.0395439
$$939$$ 42.0278 1.37152
$$940$$ −4.60555 −0.150217
$$941$$ 20.9361 0.682497 0.341248 0.939973i $$-0.389150\pi$$
0.341248 + 0.939973i $$0.389150\pi$$
$$942$$ −37.6333 −1.22616
$$943$$ 0 0
$$944$$ −10.6056 −0.345181
$$945$$ −4.90833 −0.159668
$$946$$ 27.6333 0.898436
$$947$$ 41.9361 1.36274 0.681370 0.731939i $$-0.261385\pi$$
0.681370 + 0.731939i $$0.261385\pi$$
$$948$$ −47.6333 −1.54706
$$949$$ −4.78890 −0.155454
$$950$$ −4.90833 −0.159247
$$951$$ −48.6333 −1.57704
$$952$$ −1.18335 −0.0383525
$$953$$ 1.66947 0.0540794 0.0270397 0.999634i $$-0.491392\pi$$
0.0270397 + 0.999634i $$0.491392\pi$$
$$954$$ 25.3944 0.822176
$$955$$ −1.39445 −0.0451233
$$956$$ −9.21110 −0.297908
$$957$$ 80.6611 2.60740
$$958$$ −30.0000 −0.969256
$$959$$ 2.09167 0.0675436
$$960$$ −3.30278 −0.106597
$$961$$ −22.5416 −0.727150
$$962$$ −2.42221 −0.0780950
$$963$$ −36.4222 −1.17369
$$964$$ −14.4222 −0.464508
$$965$$ −3.81665 −0.122862
$$966$$ 0 0
$$967$$ −5.39445 −0.173474 −0.0867369 0.996231i $$-0.527644\pi$$
−0.0867369 + 0.996231i $$0.527644\pi$$
$$968$$ −17.1194 −0.550239
$$969$$ 63.3583 2.03536
$$970$$ −2.69722 −0.0866027
$$971$$ 27.9083 0.895621 0.447810 0.894129i $$-0.352204\pi$$
0.447810 + 0.894129i $$0.352204\pi$$
$$972$$ 49.8444 1.59876
$$973$$ −1.63331 −0.0523614
$$974$$ 11.8167 0.378630
$$975$$ −1.00000 −0.0320256
$$976$$ 6.51388 0.208504
$$977$$ 11.5139 0.368362 0.184181 0.982892i $$-0.441037\pi$$
0.184181 + 0.982892i $$0.441037\pi$$
$$978$$ −18.8167 −0.601690
$$979$$ 0 0
$$980$$ 6.90833 0.220678
$$981$$ −154.322 −4.92713
$$982$$ 25.8167 0.823843
$$983$$ 19.5416 0.623281 0.311641 0.950200i $$-0.399121\pi$$
0.311641 + 0.950200i $$0.399121\pi$$
$$984$$ 32.7250 1.04323
$$985$$ −0.697224 −0.0222154
$$986$$ −18.0000 −0.573237
$$987$$ 4.60555 0.146596
$$988$$ −1.48612 −0.0472798
$$989$$ 0 0
$$990$$ 41.9361 1.33282
$$991$$ 24.3305 0.772885 0.386442 0.922314i $$-0.373704\pi$$
0.386442 + 0.922314i $$0.373704\pi$$
$$992$$ −2.90833 −0.0923395
$$993$$ 31.0278 0.984636
$$994$$ 3.84441 0.121937
$$995$$ 8.42221 0.267002
$$996$$ 10.6056 0.336050
$$997$$ −31.2111 −0.988466 −0.494233 0.869330i $$-0.664551\pi$$
−0.494233 + 0.869330i $$0.664551\pi$$
$$998$$ −11.6333 −0.368246
$$999$$ −129.689 −4.10317
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 5290.2.a.j.1.2 2
23.22 odd 2 230.2.a.b.1.2 2
69.68 even 2 2070.2.a.w.1.1 2
92.91 even 2 1840.2.a.j.1.1 2
115.22 even 4 1150.2.b.f.599.1 4
115.68 even 4 1150.2.b.f.599.4 4
115.114 odd 2 1150.2.a.m.1.1 2
184.45 odd 2 7360.2.a.bc.1.1 2
184.91 even 2 7360.2.a.bu.1.2 2
460.459 even 2 9200.2.a.ca.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 2 23.22 odd 2
1150.2.a.m.1.1 2 115.114 odd 2
1150.2.b.f.599.1 4 115.22 even 4
1150.2.b.f.599.4 4 115.68 even 4
1840.2.a.j.1.1 2 92.91 even 2
2070.2.a.w.1.1 2 69.68 even 2
5290.2.a.j.1.2 2 1.1 even 1 trivial
7360.2.a.bc.1.1 2 184.45 odd 2
7360.2.a.bu.1.2 2 184.91 even 2
9200.2.a.ca.1.2 2 460.459 even 2