Properties

 Label 3450.2.a.be.1.2 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.23607 q^{11} -1.00000 q^{12} -4.47214 q^{13} -4.47214 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +5.70820 q^{19} -4.47214 q^{21} +5.23607 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.47214 q^{26} -1.00000 q^{27} +4.47214 q^{28} -4.47214 q^{29} -2.47214 q^{31} -1.00000 q^{32} +5.23607 q^{33} -4.00000 q^{34} +1.00000 q^{36} -11.2361 q^{37} -5.70820 q^{38} +4.47214 q^{39} -2.00000 q^{41} +4.47214 q^{42} +4.76393 q^{43} -5.23607 q^{44} +1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +13.0000 q^{49} -4.00000 q^{51} -4.47214 q^{52} -5.23607 q^{53} +1.00000 q^{54} -4.47214 q^{56} -5.70820 q^{57} +4.47214 q^{58} -8.94427 q^{59} +0.763932 q^{61} +2.47214 q^{62} +4.47214 q^{63} +1.00000 q^{64} -5.23607 q^{66} -9.70820 q^{67} +4.00000 q^{68} +1.00000 q^{69} +8.94427 q^{71} -1.00000 q^{72} +4.47214 q^{73} +11.2361 q^{74} +5.70820 q^{76} -23.4164 q^{77} -4.47214 q^{78} +4.47214 q^{79} +1.00000 q^{81} +2.00000 q^{82} -13.2361 q^{83} -4.47214 q^{84} -4.76393 q^{86} +4.47214 q^{87} +5.23607 q^{88} -10.4721 q^{89} -20.0000 q^{91} -1.00000 q^{92} +2.47214 q^{93} +4.00000 q^{94} +1.00000 q^{96} -0.472136 q^{97} -13.0000 q^{98} -5.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} - 6q^{11} - 2q^{12} + 2q^{16} + 8q^{17} - 2q^{18} - 2q^{19} + 6q^{22} - 2q^{23} + 2q^{24} - 2q^{27} + 4q^{31} - 2q^{32} + 6q^{33} - 8q^{34} + 2q^{36} - 18q^{37} + 2q^{38} - 4q^{41} + 14q^{43} - 6q^{44} + 2q^{46} - 8q^{47} - 2q^{48} + 26q^{49} - 8q^{51} - 6q^{53} + 2q^{54} + 2q^{57} + 6q^{61} - 4q^{62} + 2q^{64} - 6q^{66} - 6q^{67} + 8q^{68} + 2q^{69} - 2q^{72} + 18q^{74} - 2q^{76} - 20q^{77} + 2q^{81} + 4q^{82} - 22q^{83} - 14q^{86} + 6q^{88} - 12q^{89} - 40q^{91} - 2q^{92} - 4q^{93} + 8q^{94} + 2q^{96} + 8q^{97} - 26q^{98} - 6q^{99} + O(q^{100})$$

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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 4.47214 1.69031 0.845154 0.534522i $$-0.179509\pi$$
0.845154 + 0.534522i $$0.179509\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.23607 −1.57873 −0.789367 0.613922i $$-0.789591\pi$$
−0.789367 + 0.613922i $$0.789591\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −4.47214 −1.24035 −0.620174 0.784465i $$-0.712938\pi$$
−0.620174 + 0.784465i $$0.712938\pi$$
$$14$$ −4.47214 −1.19523
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 5.70820 1.30955 0.654776 0.755823i $$-0.272763\pi$$
0.654776 + 0.755823i $$0.272763\pi$$
$$20$$ 0 0
$$21$$ −4.47214 −0.975900
$$22$$ 5.23607 1.11633
$$23$$ −1.00000 −0.208514
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 4.47214 0.877058
$$27$$ −1.00000 −0.192450
$$28$$ 4.47214 0.845154
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 0 0
$$31$$ −2.47214 −0.444009 −0.222004 0.975046i $$-0.571260\pi$$
−0.222004 + 0.975046i $$0.571260\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.23607 0.911482
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −11.2361 −1.84720 −0.923599 0.383360i $$-0.874767\pi$$
−0.923599 + 0.383360i $$0.874767\pi$$
$$38$$ −5.70820 −0.925993
$$39$$ 4.47214 0.716115
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 4.47214 0.690066
$$43$$ 4.76393 0.726493 0.363246 0.931693i $$-0.381668\pi$$
0.363246 + 0.931693i $$0.381668\pi$$
$$44$$ −5.23607 −0.789367
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 13.0000 1.85714
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ −4.47214 −0.620174
$$53$$ −5.23607 −0.719229 −0.359615 0.933101i $$-0.617092\pi$$
−0.359615 + 0.933101i $$0.617092\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.47214 −0.597614
$$57$$ −5.70820 −0.756070
$$58$$ 4.47214 0.587220
$$59$$ −8.94427 −1.16445 −0.582223 0.813029i $$-0.697817\pi$$
−0.582223 + 0.813029i $$0.697817\pi$$
$$60$$ 0 0
$$61$$ 0.763932 0.0978115 0.0489057 0.998803i $$-0.484427\pi$$
0.0489057 + 0.998803i $$0.484427\pi$$
$$62$$ 2.47214 0.313962
$$63$$ 4.47214 0.563436
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.23607 −0.644515
$$67$$ −9.70820 −1.18605 −0.593023 0.805186i $$-0.702066\pi$$
−0.593023 + 0.805186i $$0.702066\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 8.94427 1.06149 0.530745 0.847532i $$-0.321912\pi$$
0.530745 + 0.847532i $$0.321912\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 4.47214 0.523424 0.261712 0.965146i $$-0.415713\pi$$
0.261712 + 0.965146i $$0.415713\pi$$
$$74$$ 11.2361 1.30617
$$75$$ 0 0
$$76$$ 5.70820 0.654776
$$77$$ −23.4164 −2.66855
$$78$$ −4.47214 −0.506370
$$79$$ 4.47214 0.503155 0.251577 0.967837i $$-0.419051\pi$$
0.251577 + 0.967837i $$0.419051\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ −13.2361 −1.45285 −0.726424 0.687247i $$-0.758819\pi$$
−0.726424 + 0.687247i $$0.758819\pi$$
$$84$$ −4.47214 −0.487950
$$85$$ 0 0
$$86$$ −4.76393 −0.513708
$$87$$ 4.47214 0.479463
$$88$$ 5.23607 0.558167
$$89$$ −10.4721 −1.11004 −0.555022 0.831836i $$-0.687290\pi$$
−0.555022 + 0.831836i $$0.687290\pi$$
$$90$$ 0 0
$$91$$ −20.0000 −2.09657
$$92$$ −1.00000 −0.104257
$$93$$ 2.47214 0.256349
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −0.472136 −0.0479381 −0.0239691 0.999713i $$-0.507630\pi$$
−0.0239691 + 0.999713i $$0.507630\pi$$
$$98$$ −13.0000 −1.31320
$$99$$ −5.23607 −0.526245
$$100$$ 0 0
$$101$$ 4.47214 0.444994 0.222497 0.974933i $$-0.428579\pi$$
0.222497 + 0.974933i $$0.428579\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 4.47214 0.438529
$$105$$ 0 0
$$106$$ 5.23607 0.508572
$$107$$ 12.6525 1.22316 0.611581 0.791182i $$-0.290534\pi$$
0.611581 + 0.791182i $$0.290534\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.76393 0.456302 0.228151 0.973626i $$-0.426732\pi$$
0.228151 + 0.973626i $$0.426732\pi$$
$$110$$ 0 0
$$111$$ 11.2361 1.06648
$$112$$ 4.47214 0.422577
$$113$$ 5.52786 0.520018 0.260009 0.965606i $$-0.416275\pi$$
0.260009 + 0.965606i $$0.416275\pi$$
$$114$$ 5.70820 0.534622
$$115$$ 0 0
$$116$$ −4.47214 −0.415227
$$117$$ −4.47214 −0.413449
$$118$$ 8.94427 0.823387
$$119$$ 17.8885 1.63984
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ −0.763932 −0.0691632
$$123$$ 2.00000 0.180334
$$124$$ −2.47214 −0.222004
$$125$$ 0 0
$$126$$ −4.47214 −0.398410
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.76393 −0.419441
$$130$$ 0 0
$$131$$ −9.52786 −0.832453 −0.416227 0.909261i $$-0.636648\pi$$
−0.416227 + 0.909261i $$0.636648\pi$$
$$132$$ 5.23607 0.455741
$$133$$ 25.5279 2.21355
$$134$$ 9.70820 0.838661
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ −3.05573 −0.261068 −0.130534 0.991444i $$-0.541669\pi$$
−0.130534 + 0.991444i $$0.541669\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ −16.9443 −1.43719 −0.718597 0.695427i $$-0.755215\pi$$
−0.718597 + 0.695427i $$0.755215\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ −8.94427 −0.750587
$$143$$ 23.4164 1.95818
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −4.47214 −0.370117
$$147$$ −13.0000 −1.07222
$$148$$ −11.2361 −0.923599
$$149$$ −11.7082 −0.959173 −0.479587 0.877494i $$-0.659213\pi$$
−0.479587 + 0.877494i $$0.659213\pi$$
$$150$$ 0 0
$$151$$ −14.4721 −1.17773 −0.588863 0.808233i $$-0.700424\pi$$
−0.588863 + 0.808233i $$0.700424\pi$$
$$152$$ −5.70820 −0.462996
$$153$$ 4.00000 0.323381
$$154$$ 23.4164 1.88695
$$155$$ 0 0
$$156$$ 4.47214 0.358057
$$157$$ 6.65248 0.530925 0.265463 0.964121i $$-0.414475\pi$$
0.265463 + 0.964121i $$0.414475\pi$$
$$158$$ −4.47214 −0.355784
$$159$$ 5.23607 0.415247
$$160$$ 0 0
$$161$$ −4.47214 −0.352454
$$162$$ −1.00000 −0.0785674
$$163$$ 2.47214 0.193633 0.0968163 0.995302i $$-0.469134\pi$$
0.0968163 + 0.995302i $$0.469134\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 13.2361 1.02732
$$167$$ −16.9443 −1.31119 −0.655594 0.755114i $$-0.727582\pi$$
−0.655594 + 0.755114i $$0.727582\pi$$
$$168$$ 4.47214 0.345033
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 5.70820 0.436517
$$172$$ 4.76393 0.363246
$$173$$ 17.4164 1.32414 0.662072 0.749440i $$-0.269677\pi$$
0.662072 + 0.749440i $$0.269677\pi$$
$$174$$ −4.47214 −0.339032
$$175$$ 0 0
$$176$$ −5.23607 −0.394683
$$177$$ 8.94427 0.672293
$$178$$ 10.4721 0.784920
$$179$$ 19.4164 1.45125 0.725625 0.688090i $$-0.241551\pi$$
0.725625 + 0.688090i $$0.241551\pi$$
$$180$$ 0 0
$$181$$ −11.2361 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$182$$ 20.0000 1.48250
$$183$$ −0.763932 −0.0564715
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ −2.47214 −0.181266
$$187$$ −20.9443 −1.53160
$$188$$ −4.00000 −0.291730
$$189$$ −4.47214 −0.325300
$$190$$ 0 0
$$191$$ 6.47214 0.468307 0.234154 0.972200i $$-0.424768\pi$$
0.234154 + 0.972200i $$0.424768\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −23.8885 −1.71954 −0.859768 0.510686i $$-0.829392\pi$$
−0.859768 + 0.510686i $$0.829392\pi$$
$$194$$ 0.472136 0.0338974
$$195$$ 0 0
$$196$$ 13.0000 0.928571
$$197$$ −2.94427 −0.209771 −0.104885 0.994484i $$-0.533448\pi$$
−0.104885 + 0.994484i $$0.533448\pi$$
$$198$$ 5.23607 0.372111
$$199$$ 17.4164 1.23462 0.617308 0.786721i $$-0.288223\pi$$
0.617308 + 0.786721i $$0.288223\pi$$
$$200$$ 0 0
$$201$$ 9.70820 0.684764
$$202$$ −4.47214 −0.314658
$$203$$ −20.0000 −1.40372
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ −6.00000 −0.418040
$$207$$ −1.00000 −0.0695048
$$208$$ −4.47214 −0.310087
$$209$$ −29.8885 −2.06743
$$210$$ 0 0
$$211$$ −23.4164 −1.61205 −0.806026 0.591880i $$-0.798386\pi$$
−0.806026 + 0.591880i $$0.798386\pi$$
$$212$$ −5.23607 −0.359615
$$213$$ −8.94427 −0.612851
$$214$$ −12.6525 −0.864905
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −11.0557 −0.750512
$$218$$ −4.76393 −0.322654
$$219$$ −4.47214 −0.302199
$$220$$ 0 0
$$221$$ −17.8885 −1.20331
$$222$$ −11.2361 −0.754116
$$223$$ −19.4164 −1.30022 −0.650109 0.759841i $$-0.725277\pi$$
−0.650109 + 0.759841i $$0.725277\pi$$
$$224$$ −4.47214 −0.298807
$$225$$ 0 0
$$226$$ −5.52786 −0.367708
$$227$$ 9.23607 0.613019 0.306510 0.951868i $$-0.400839\pi$$
0.306510 + 0.951868i $$0.400839\pi$$
$$228$$ −5.70820 −0.378035
$$229$$ 17.7082 1.17019 0.585096 0.810964i $$-0.301057\pi$$
0.585096 + 0.810964i $$0.301057\pi$$
$$230$$ 0 0
$$231$$ 23.4164 1.54069
$$232$$ 4.47214 0.293610
$$233$$ −19.8885 −1.30294 −0.651471 0.758674i $$-0.725848\pi$$
−0.651471 + 0.758674i $$0.725848\pi$$
$$234$$ 4.47214 0.292353
$$235$$ 0 0
$$236$$ −8.94427 −0.582223
$$237$$ −4.47214 −0.290496
$$238$$ −17.8885 −1.15954
$$239$$ −4.94427 −0.319818 −0.159909 0.987132i $$-0.551120\pi$$
−0.159909 + 0.987132i $$0.551120\pi$$
$$240$$ 0 0
$$241$$ −12.4721 −0.803401 −0.401700 0.915771i $$-0.631581\pi$$
−0.401700 + 0.915771i $$0.631581\pi$$
$$242$$ −16.4164 −1.05529
$$243$$ −1.00000 −0.0641500
$$244$$ 0.763932 0.0489057
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ −25.5279 −1.62430
$$248$$ 2.47214 0.156981
$$249$$ 13.2361 0.838802
$$250$$ 0 0
$$251$$ −19.7082 −1.24397 −0.621985 0.783029i $$-0.713674\pi$$
−0.621985 + 0.783029i $$0.713674\pi$$
$$252$$ 4.47214 0.281718
$$253$$ 5.23607 0.329189
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 11.8885 0.741587 0.370793 0.928715i $$-0.379086\pi$$
0.370793 + 0.928715i $$0.379086\pi$$
$$258$$ 4.76393 0.296589
$$259$$ −50.2492 −3.12233
$$260$$ 0 0
$$261$$ −4.47214 −0.276818
$$262$$ 9.52786 0.588633
$$263$$ −24.9443 −1.53813 −0.769065 0.639171i $$-0.779278\pi$$
−0.769065 + 0.639171i $$0.779278\pi$$
$$264$$ −5.23607 −0.322258
$$265$$ 0 0
$$266$$ −25.5279 −1.56521
$$267$$ 10.4721 0.640884
$$268$$ −9.70820 −0.593023
$$269$$ −13.0557 −0.796022 −0.398011 0.917381i $$-0.630299\pi$$
−0.398011 + 0.917381i $$0.630299\pi$$
$$270$$ 0 0
$$271$$ −16.9443 −1.02929 −0.514646 0.857403i $$-0.672076\pi$$
−0.514646 + 0.857403i $$0.672076\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 20.0000 1.21046
$$274$$ 3.05573 0.184603
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ 20.4721 1.23005 0.615026 0.788507i $$-0.289146\pi$$
0.615026 + 0.788507i $$0.289146\pi$$
$$278$$ 16.9443 1.01625
$$279$$ −2.47214 −0.148003
$$280$$ 0 0
$$281$$ −13.5279 −0.807005 −0.403502 0.914979i $$-0.632207\pi$$
−0.403502 + 0.914979i $$0.632207\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ 3.81966 0.227055 0.113528 0.993535i $$-0.463785\pi$$
0.113528 + 0.993535i $$0.463785\pi$$
$$284$$ 8.94427 0.530745
$$285$$ 0 0
$$286$$ −23.4164 −1.38464
$$287$$ −8.94427 −0.527964
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0.472136 0.0276771
$$292$$ 4.47214 0.261712
$$293$$ −0.291796 −0.0170469 −0.00852345 0.999964i $$-0.502713\pi$$
−0.00852345 + 0.999964i $$0.502713\pi$$
$$294$$ 13.0000 0.758175
$$295$$ 0 0
$$296$$ 11.2361 0.653083
$$297$$ 5.23607 0.303827
$$298$$ 11.7082 0.678238
$$299$$ 4.47214 0.258630
$$300$$ 0 0
$$301$$ 21.3050 1.22800
$$302$$ 14.4721 0.832778
$$303$$ −4.47214 −0.256917
$$304$$ 5.70820 0.327388
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ −15.4164 −0.879861 −0.439930 0.898032i $$-0.644997\pi$$
−0.439930 + 0.898032i $$0.644997\pi$$
$$308$$ −23.4164 −1.33427
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ −20.9443 −1.18764 −0.593820 0.804598i $$-0.702381\pi$$
−0.593820 + 0.804598i $$0.702381\pi$$
$$312$$ −4.47214 −0.253185
$$313$$ −15.5279 −0.877687 −0.438843 0.898564i $$-0.644612\pi$$
−0.438843 + 0.898564i $$0.644612\pi$$
$$314$$ −6.65248 −0.375421
$$315$$ 0 0
$$316$$ 4.47214 0.251577
$$317$$ −19.5279 −1.09679 −0.548397 0.836218i $$-0.684762\pi$$
−0.548397 + 0.836218i $$0.684762\pi$$
$$318$$ −5.23607 −0.293624
$$319$$ 23.4164 1.31107
$$320$$ 0 0
$$321$$ −12.6525 −0.706192
$$322$$ 4.47214 0.249222
$$323$$ 22.8328 1.27045
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −2.47214 −0.136919
$$327$$ −4.76393 −0.263446
$$328$$ 2.00000 0.110432
$$329$$ −17.8885 −0.986227
$$330$$ 0 0
$$331$$ −10.4721 −0.575601 −0.287800 0.957690i $$-0.592924\pi$$
−0.287800 + 0.957690i $$0.592924\pi$$
$$332$$ −13.2361 −0.726424
$$333$$ −11.2361 −0.615733
$$334$$ 16.9443 0.927149
$$335$$ 0 0
$$336$$ −4.47214 −0.243975
$$337$$ 19.8885 1.08340 0.541699 0.840573i $$-0.317781\pi$$
0.541699 + 0.840573i $$0.317781\pi$$
$$338$$ −7.00000 −0.380750
$$339$$ −5.52786 −0.300232
$$340$$ 0 0
$$341$$ 12.9443 0.700972
$$342$$ −5.70820 −0.308664
$$343$$ 26.8328 1.44884
$$344$$ −4.76393 −0.256854
$$345$$ 0 0
$$346$$ −17.4164 −0.936312
$$347$$ −30.4721 −1.63583 −0.817915 0.575339i $$-0.804870\pi$$
−0.817915 + 0.575339i $$0.804870\pi$$
$$348$$ 4.47214 0.239732
$$349$$ 3.88854 0.208149 0.104074 0.994570i $$-0.466812\pi$$
0.104074 + 0.994570i $$0.466812\pi$$
$$350$$ 0 0
$$351$$ 4.47214 0.238705
$$352$$ 5.23607 0.279083
$$353$$ 3.88854 0.206966 0.103483 0.994631i $$-0.467001\pi$$
0.103483 + 0.994631i $$0.467001\pi$$
$$354$$ −8.94427 −0.475383
$$355$$ 0 0
$$356$$ −10.4721 −0.555022
$$357$$ −17.8885 −0.946762
$$358$$ −19.4164 −1.02619
$$359$$ 29.3050 1.54666 0.773328 0.634006i $$-0.218591\pi$$
0.773328 + 0.634006i $$0.218591\pi$$
$$360$$ 0 0
$$361$$ 13.5836 0.714926
$$362$$ 11.2361 0.590555
$$363$$ −16.4164 −0.861638
$$364$$ −20.0000 −1.04828
$$365$$ 0 0
$$366$$ 0.763932 0.0399314
$$367$$ −9.41641 −0.491532 −0.245766 0.969329i $$-0.579040\pi$$
−0.245766 + 0.969329i $$0.579040\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ −23.4164 −1.21572
$$372$$ 2.47214 0.128174
$$373$$ 35.5967 1.84313 0.921565 0.388224i $$-0.126911\pi$$
0.921565 + 0.388224i $$0.126911\pi$$
$$374$$ 20.9443 1.08300
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ 20.0000 1.03005
$$378$$ 4.47214 0.230022
$$379$$ 13.7082 0.704143 0.352072 0.935973i $$-0.385477\pi$$
0.352072 + 0.935973i $$0.385477\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ −6.47214 −0.331143
$$383$$ −7.05573 −0.360531 −0.180265 0.983618i $$-0.557696\pi$$
−0.180265 + 0.983618i $$0.557696\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 23.8885 1.21589
$$387$$ 4.76393 0.242164
$$388$$ −0.472136 −0.0239691
$$389$$ 23.7082 1.20205 0.601027 0.799229i $$-0.294758\pi$$
0.601027 + 0.799229i $$0.294758\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ −13.0000 −0.656599
$$393$$ 9.52786 0.480617
$$394$$ 2.94427 0.148330
$$395$$ 0 0
$$396$$ −5.23607 −0.263122
$$397$$ −26.9443 −1.35229 −0.676147 0.736767i $$-0.736352\pi$$
−0.676147 + 0.736767i $$0.736352\pi$$
$$398$$ −17.4164 −0.873006
$$399$$ −25.5279 −1.27799
$$400$$ 0 0
$$401$$ 8.94427 0.446656 0.223328 0.974743i $$-0.428308\pi$$
0.223328 + 0.974743i $$0.428308\pi$$
$$402$$ −9.70820 −0.484201
$$403$$ 11.0557 0.550725
$$404$$ 4.47214 0.222497
$$405$$ 0 0
$$406$$ 20.0000 0.992583
$$407$$ 58.8328 2.91623
$$408$$ 4.00000 0.198030
$$409$$ −35.8885 −1.77457 −0.887287 0.461217i $$-0.847413\pi$$
−0.887287 + 0.461217i $$0.847413\pi$$
$$410$$ 0 0
$$411$$ 3.05573 0.150728
$$412$$ 6.00000 0.295599
$$413$$ −40.0000 −1.96827
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 4.47214 0.219265
$$417$$ 16.9443 0.829765
$$418$$ 29.8885 1.46190
$$419$$ 28.0689 1.37125 0.685627 0.727953i $$-0.259528\pi$$
0.685627 + 0.727953i $$0.259528\pi$$
$$420$$ 0 0
$$421$$ −0.763932 −0.0372318 −0.0186159 0.999827i $$-0.505926\pi$$
−0.0186159 + 0.999827i $$0.505926\pi$$
$$422$$ 23.4164 1.13989
$$423$$ −4.00000 −0.194487
$$424$$ 5.23607 0.254286
$$425$$ 0 0
$$426$$ 8.94427 0.433351
$$427$$ 3.41641 0.165332
$$428$$ 12.6525 0.611581
$$429$$ −23.4164 −1.13055
$$430$$ 0 0
$$431$$ −23.4164 −1.12793 −0.563964 0.825799i $$-0.690724\pi$$
−0.563964 + 0.825799i $$0.690724\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 21.4164 1.02921 0.514603 0.857428i $$-0.327939\pi$$
0.514603 + 0.857428i $$0.327939\pi$$
$$434$$ 11.0557 0.530692
$$435$$ 0 0
$$436$$ 4.76393 0.228151
$$437$$ −5.70820 −0.273060
$$438$$ 4.47214 0.213687
$$439$$ 19.0557 0.909480 0.454740 0.890624i $$-0.349732\pi$$
0.454740 + 0.890624i $$0.349732\pi$$
$$440$$ 0 0
$$441$$ 13.0000 0.619048
$$442$$ 17.8885 0.850871
$$443$$ 15.0557 0.715319 0.357660 0.933852i $$-0.383575\pi$$
0.357660 + 0.933852i $$0.383575\pi$$
$$444$$ 11.2361 0.533240
$$445$$ 0 0
$$446$$ 19.4164 0.919394
$$447$$ 11.7082 0.553779
$$448$$ 4.47214 0.211289
$$449$$ 15.8885 0.749827 0.374913 0.927060i $$-0.377672\pi$$
0.374913 + 0.927060i $$0.377672\pi$$
$$450$$ 0 0
$$451$$ 10.4721 0.493114
$$452$$ 5.52786 0.260009
$$453$$ 14.4721 0.679960
$$454$$ −9.23607 −0.433470
$$455$$ 0 0
$$456$$ 5.70820 0.267311
$$457$$ −27.5279 −1.28770 −0.643850 0.765152i $$-0.722664\pi$$
−0.643850 + 0.765152i $$0.722664\pi$$
$$458$$ −17.7082 −0.827450
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ −23.4164 −1.08943
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −4.47214 −0.207614
$$465$$ 0 0
$$466$$ 19.8885 0.921319
$$467$$ 17.8197 0.824596 0.412298 0.911049i $$-0.364726\pi$$
0.412298 + 0.911049i $$0.364726\pi$$
$$468$$ −4.47214 −0.206725
$$469$$ −43.4164 −2.00478
$$470$$ 0 0
$$471$$ −6.65248 −0.306530
$$472$$ 8.94427 0.411693
$$473$$ −24.9443 −1.14694
$$474$$ 4.47214 0.205412
$$475$$ 0 0
$$476$$ 17.8885 0.819920
$$477$$ −5.23607 −0.239743
$$478$$ 4.94427 0.226146
$$479$$ −28.9443 −1.32250 −0.661249 0.750167i $$-0.729973\pi$$
−0.661249 + 0.750167i $$0.729973\pi$$
$$480$$ 0 0
$$481$$ 50.2492 2.29117
$$482$$ 12.4721 0.568090
$$483$$ 4.47214 0.203489
$$484$$ 16.4164 0.746200
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −9.88854 −0.448093 −0.224046 0.974578i $$-0.571927\pi$$
−0.224046 + 0.974578i $$0.571927\pi$$
$$488$$ −0.763932 −0.0345816
$$489$$ −2.47214 −0.111794
$$490$$ 0 0
$$491$$ 29.3050 1.32251 0.661257 0.750159i $$-0.270023\pi$$
0.661257 + 0.750159i $$0.270023\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ −17.8885 −0.805659
$$494$$ 25.5279 1.14855
$$495$$ 0 0
$$496$$ −2.47214 −0.111002
$$497$$ 40.0000 1.79425
$$498$$ −13.2361 −0.593122
$$499$$ 0.583592 0.0261252 0.0130626 0.999915i $$-0.495842\pi$$
0.0130626 + 0.999915i $$0.495842\pi$$
$$500$$ 0 0
$$501$$ 16.9443 0.757014
$$502$$ 19.7082 0.879620
$$503$$ −10.4721 −0.466929 −0.233465 0.972365i $$-0.575006\pi$$
−0.233465 + 0.972365i $$0.575006\pi$$
$$504$$ −4.47214 −0.199205
$$505$$ 0 0
$$506$$ −5.23607 −0.232772
$$507$$ −7.00000 −0.310881
$$508$$ 4.00000 0.177471
$$509$$ −33.4164 −1.48116 −0.740578 0.671970i $$-0.765448\pi$$
−0.740578 + 0.671970i $$0.765448\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ −1.00000 −0.0441942
$$513$$ −5.70820 −0.252023
$$514$$ −11.8885 −0.524381
$$515$$ 0 0
$$516$$ −4.76393 −0.209720
$$517$$ 20.9443 0.921128
$$518$$ 50.2492 2.20782
$$519$$ −17.4164 −0.764495
$$520$$ 0 0
$$521$$ 16.0000 0.700973 0.350486 0.936568i $$-0.386016\pi$$
0.350486 + 0.936568i $$0.386016\pi$$
$$522$$ 4.47214 0.195740
$$523$$ −25.7082 −1.12414 −0.562071 0.827089i $$-0.689995\pi$$
−0.562071 + 0.827089i $$0.689995\pi$$
$$524$$ −9.52786 −0.416227
$$525$$ 0 0
$$526$$ 24.9443 1.08762
$$527$$ −9.88854 −0.430752
$$528$$ 5.23607 0.227871
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −8.94427 −0.388148
$$532$$ 25.5279 1.10677
$$533$$ 8.94427 0.387419
$$534$$ −10.4721 −0.453174
$$535$$ 0 0
$$536$$ 9.70820 0.419331
$$537$$ −19.4164 −0.837880
$$538$$ 13.0557 0.562872
$$539$$ −68.0689 −2.93193
$$540$$ 0 0
$$541$$ 8.11146 0.348739 0.174369 0.984680i $$-0.444211\pi$$
0.174369 + 0.984680i $$0.444211\pi$$
$$542$$ 16.9443 0.727819
$$543$$ 11.2361 0.482186
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ −20.0000 −0.855921
$$547$$ −41.3050 −1.76607 −0.883036 0.469305i $$-0.844504\pi$$
−0.883036 + 0.469305i $$0.844504\pi$$
$$548$$ −3.05573 −0.130534
$$549$$ 0.763932 0.0326038
$$550$$ 0 0
$$551$$ −25.5279 −1.08752
$$552$$ −1.00000 −0.0425628
$$553$$ 20.0000 0.850487
$$554$$ −20.4721 −0.869778
$$555$$ 0 0
$$556$$ −16.9443 −0.718597
$$557$$ 15.1246 0.640850 0.320425 0.947274i $$-0.396174\pi$$
0.320425 + 0.947274i $$0.396174\pi$$
$$558$$ 2.47214 0.104654
$$559$$ −21.3050 −0.901103
$$560$$ 0 0
$$561$$ 20.9443 0.884268
$$562$$ 13.5279 0.570639
$$563$$ −24.6525 −1.03898 −0.519489 0.854477i $$-0.673878\pi$$
−0.519489 + 0.854477i $$0.673878\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ −3.81966 −0.160552
$$567$$ 4.47214 0.187812
$$568$$ −8.94427 −0.375293
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ −14.2918 −0.598093 −0.299047 0.954239i $$-0.596669\pi$$
−0.299047 + 0.954239i $$0.596669\pi$$
$$572$$ 23.4164 0.979089
$$573$$ −6.47214 −0.270377
$$574$$ 8.94427 0.373327
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 34.3607 1.43045 0.715227 0.698892i $$-0.246323\pi$$
0.715227 + 0.698892i $$0.246323\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 23.8885 0.992774
$$580$$ 0 0
$$581$$ −59.1935 −2.45576
$$582$$ −0.472136 −0.0195707
$$583$$ 27.4164 1.13547
$$584$$ −4.47214 −0.185058
$$585$$ 0 0
$$586$$ 0.291796 0.0120540
$$587$$ −6.47214 −0.267134 −0.133567 0.991040i $$-0.542643\pi$$
−0.133567 + 0.991040i $$0.542643\pi$$
$$588$$ −13.0000 −0.536111
$$589$$ −14.1115 −0.581452
$$590$$ 0 0
$$591$$ 2.94427 0.121111
$$592$$ −11.2361 −0.461800
$$593$$ −33.7771 −1.38706 −0.693529 0.720428i $$-0.743945\pi$$
−0.693529 + 0.720428i $$0.743945\pi$$
$$594$$ −5.23607 −0.214838
$$595$$ 0 0
$$596$$ −11.7082 −0.479587
$$597$$ −17.4164 −0.712806
$$598$$ −4.47214 −0.182879
$$599$$ 33.8885 1.38465 0.692324 0.721587i $$-0.256587\pi$$
0.692324 + 0.721587i $$0.256587\pi$$
$$600$$ 0 0
$$601$$ −10.3607 −0.422621 −0.211310 0.977419i $$-0.567773\pi$$
−0.211310 + 0.977419i $$0.567773\pi$$
$$602$$ −21.3050 −0.868325
$$603$$ −9.70820 −0.395349
$$604$$ −14.4721 −0.588863
$$605$$ 0 0
$$606$$ 4.47214 0.181668
$$607$$ 17.5279 0.711434 0.355717 0.934594i $$-0.384237\pi$$
0.355717 + 0.934594i $$0.384237\pi$$
$$608$$ −5.70820 −0.231498
$$609$$ 20.0000 0.810441
$$610$$ 0 0
$$611$$ 17.8885 0.723693
$$612$$ 4.00000 0.161690
$$613$$ 25.1246 1.01477 0.507387 0.861718i $$-0.330612\pi$$
0.507387 + 0.861718i $$0.330612\pi$$
$$614$$ 15.4164 0.622156
$$615$$ 0 0
$$616$$ 23.4164 0.943474
$$617$$ −20.3607 −0.819690 −0.409845 0.912155i $$-0.634417\pi$$
−0.409845 + 0.912155i $$0.634417\pi$$
$$618$$ 6.00000 0.241355
$$619$$ 18.2918 0.735209 0.367605 0.929982i $$-0.380178\pi$$
0.367605 + 0.929982i $$0.380178\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 20.9443 0.839789
$$623$$ −46.8328 −1.87632
$$624$$ 4.47214 0.179029
$$625$$ 0 0
$$626$$ 15.5279 0.620618
$$627$$ 29.8885 1.19363
$$628$$ 6.65248 0.265463
$$629$$ −44.9443 −1.79205
$$630$$ 0 0
$$631$$ −6.00000 −0.238856 −0.119428 0.992843i $$-0.538106\pi$$
−0.119428 + 0.992843i $$0.538106\pi$$
$$632$$ −4.47214 −0.177892
$$633$$ 23.4164 0.930719
$$634$$ 19.5279 0.775551
$$635$$ 0 0
$$636$$ 5.23607 0.207624
$$637$$ −58.1378 −2.30350
$$638$$ −23.4164 −0.927064
$$639$$ 8.94427 0.353830
$$640$$ 0 0
$$641$$ 4.00000 0.157991 0.0789953 0.996875i $$-0.474829\pi$$
0.0789953 + 0.996875i $$0.474829\pi$$
$$642$$ 12.6525 0.499353
$$643$$ 32.5410 1.28329 0.641646 0.767001i $$-0.278252\pi$$
0.641646 + 0.767001i $$0.278252\pi$$
$$644$$ −4.47214 −0.176227
$$645$$ 0 0
$$646$$ −22.8328 −0.898345
$$647$$ −12.9443 −0.508892 −0.254446 0.967087i $$-0.581893\pi$$
−0.254446 + 0.967087i $$0.581893\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 46.8328 1.83835
$$650$$ 0 0
$$651$$ 11.0557 0.433308
$$652$$ 2.47214 0.0968163
$$653$$ −9.41641 −0.368493 −0.184246 0.982880i $$-0.558984\pi$$
−0.184246 + 0.982880i $$0.558984\pi$$
$$654$$ 4.76393 0.186284
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 4.47214 0.174475
$$658$$ 17.8885 0.697368
$$659$$ −32.6525 −1.27196 −0.635980 0.771706i $$-0.719404\pi$$
−0.635980 + 0.771706i $$0.719404\pi$$
$$660$$ 0 0
$$661$$ 3.81966 0.148568 0.0742838 0.997237i $$-0.476333\pi$$
0.0742838 + 0.997237i $$0.476333\pi$$
$$662$$ 10.4721 0.407011
$$663$$ 17.8885 0.694733
$$664$$ 13.2361 0.513659
$$665$$ 0 0
$$666$$ 11.2361 0.435389
$$667$$ 4.47214 0.173162
$$668$$ −16.9443 −0.655594
$$669$$ 19.4164 0.750682
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 4.47214 0.172516
$$673$$ −4.47214 −0.172388 −0.0861941 0.996278i $$-0.527471\pi$$
−0.0861941 + 0.996278i $$0.527471\pi$$
$$674$$ −19.8885 −0.766078
$$675$$ 0 0
$$676$$ 7.00000 0.269231
$$677$$ −19.1246 −0.735019 −0.367509 0.930020i $$-0.619789\pi$$
−0.367509 + 0.930020i $$0.619789\pi$$
$$678$$ 5.52786 0.212296
$$679$$ −2.11146 −0.0810303
$$680$$ 0 0
$$681$$ −9.23607 −0.353927
$$682$$ −12.9443 −0.495662
$$683$$ −14.4721 −0.553761 −0.276880 0.960904i $$-0.589301\pi$$
−0.276880 + 0.960904i $$0.589301\pi$$
$$684$$ 5.70820 0.218259
$$685$$ 0 0
$$686$$ −26.8328 −1.02448
$$687$$ −17.7082 −0.675610
$$688$$ 4.76393 0.181623
$$689$$ 23.4164 0.892094
$$690$$ 0 0
$$691$$ 16.5836 0.630870 0.315435 0.948947i $$-0.397850\pi$$
0.315435 + 0.948947i $$0.397850\pi$$
$$692$$ 17.4164 0.662072
$$693$$ −23.4164 −0.889516
$$694$$ 30.4721 1.15671
$$695$$ 0 0
$$696$$ −4.47214 −0.169516
$$697$$ −8.00000 −0.303022
$$698$$ −3.88854 −0.147184
$$699$$ 19.8885 0.752254
$$700$$ 0 0
$$701$$ −3.12461 −0.118015 −0.0590075 0.998258i $$-0.518794\pi$$
−0.0590075 + 0.998258i $$0.518794\pi$$
$$702$$ −4.47214 −0.168790
$$703$$ −64.1378 −2.41900
$$704$$ −5.23607 −0.197342
$$705$$ 0 0
$$706$$ −3.88854 −0.146347
$$707$$ 20.0000 0.752177
$$708$$ 8.94427 0.336146
$$709$$ 35.0132 1.31495 0.657473 0.753478i $$-0.271625\pi$$
0.657473 + 0.753478i $$0.271625\pi$$
$$710$$ 0 0
$$711$$ 4.47214 0.167718
$$712$$ 10.4721 0.392460
$$713$$ 2.47214 0.0925822
$$714$$ 17.8885 0.669462
$$715$$ 0 0
$$716$$ 19.4164 0.725625
$$717$$ 4.94427 0.184647
$$718$$ −29.3050 −1.09365
$$719$$ −3.05573 −0.113959 −0.0569797 0.998375i $$-0.518147\pi$$
−0.0569797 + 0.998375i $$0.518147\pi$$
$$720$$ 0 0
$$721$$ 26.8328 0.999306
$$722$$ −13.5836 −0.505529
$$723$$ 12.4721 0.463844
$$724$$ −11.2361 −0.417585
$$725$$ 0 0
$$726$$ 16.4164 0.609270
$$727$$ 19.3050 0.715981 0.357991 0.933725i $$-0.383462\pi$$
0.357991 + 0.933725i $$0.383462\pi$$
$$728$$ 20.0000 0.741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 19.0557 0.704802
$$732$$ −0.763932 −0.0282357
$$733$$ 21.7082 0.801811 0.400905 0.916119i $$-0.368696\pi$$
0.400905 + 0.916119i $$0.368696\pi$$
$$734$$ 9.41641 0.347566
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 50.8328 1.87245
$$738$$ 2.00000 0.0736210
$$739$$ −32.9443 −1.21187 −0.605937 0.795512i $$-0.707202\pi$$
−0.605937 + 0.795512i $$0.707202\pi$$
$$740$$ 0 0
$$741$$ 25.5279 0.937790
$$742$$ 23.4164 0.859643
$$743$$ 4.00000 0.146746 0.0733729 0.997305i $$-0.476624\pi$$
0.0733729 + 0.997305i $$0.476624\pi$$
$$744$$ −2.47214 −0.0906329
$$745$$ 0 0
$$746$$ −35.5967 −1.30329
$$747$$ −13.2361 −0.484282
$$748$$ −20.9443 −0.765798
$$749$$ 56.5836 2.06752
$$750$$ 0 0
$$751$$ −18.0000 −0.656829 −0.328415 0.944534i $$-0.606514\pi$$
−0.328415 + 0.944534i $$0.606514\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 19.7082 0.718207
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ −4.47214 −0.162650
$$757$$ 1.34752 0.0489766 0.0244883 0.999700i $$-0.492204\pi$$
0.0244883 + 0.999700i $$0.492204\pi$$
$$758$$ −13.7082 −0.497904
$$759$$ −5.23607 −0.190057
$$760$$ 0 0
$$761$$ −5.05573 −0.183270 −0.0916350 0.995793i $$-0.529209\pi$$
−0.0916350 + 0.995793i $$0.529209\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 21.3050 0.771291
$$764$$ 6.47214 0.234154
$$765$$ 0 0
$$766$$ 7.05573 0.254934
$$767$$ 40.0000 1.44432
$$768$$ −1.00000 −0.0360844
$$769$$ 12.4721 0.449757 0.224878 0.974387i $$-0.427802\pi$$
0.224878 + 0.974387i $$0.427802\pi$$
$$770$$ 0 0
$$771$$ −11.8885 −0.428155
$$772$$ −23.8885 −0.859768
$$773$$ 38.1803 1.37325 0.686626 0.727011i $$-0.259091\pi$$
0.686626 + 0.727011i $$0.259091\pi$$
$$774$$ −4.76393 −0.171236
$$775$$ 0 0
$$776$$ 0.472136 0.0169487
$$777$$ 50.2492 1.80268
$$778$$ −23.7082 −0.849980
$$779$$ −11.4164 −0.409035
$$780$$ 0 0
$$781$$ −46.8328 −1.67581
$$782$$ 4.00000 0.143040
$$783$$ 4.47214 0.159821
$$784$$ 13.0000 0.464286
$$785$$ 0 0
$$786$$ −9.52786 −0.339848
$$787$$ −35.2361 −1.25603 −0.628015 0.778201i $$-0.716132\pi$$
−0.628015 + 0.778201i $$0.716132\pi$$
$$788$$ −2.94427 −0.104885
$$789$$ 24.9443 0.888040
$$790$$ 0 0
$$791$$ 24.7214 0.878990
$$792$$ 5.23607 0.186056
$$793$$ −3.41641 −0.121320
$$794$$ 26.9443 0.956216
$$795$$ 0 0
$$796$$ 17.4164 0.617308
$$797$$ −41.5967 −1.47343 −0.736716 0.676202i $$-0.763625\pi$$
−0.736716 + 0.676202i $$0.763625\pi$$
$$798$$ 25.5279 0.903677
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −10.4721 −0.370015
$$802$$ −8.94427 −0.315833
$$803$$ −23.4164 −0.826347
$$804$$ 9.70820 0.342382
$$805$$ 0 0
$$806$$ −11.0557 −0.389421
$$807$$ 13.0557 0.459583
$$808$$ −4.47214 −0.157329
$$809$$ 42.9443 1.50984 0.754920 0.655817i $$-0.227676\pi$$
0.754920 + 0.655817i $$0.227676\pi$$
$$810$$ 0 0
$$811$$ 41.3050 1.45041 0.725207 0.688531i $$-0.241744\pi$$
0.725207 + 0.688531i $$0.241744\pi$$
$$812$$ −20.0000 −0.701862
$$813$$ 16.9443 0.594262
$$814$$ −58.8328 −2.06209
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 27.1935 0.951380
$$818$$ 35.8885 1.25481
$$819$$ −20.0000 −0.698857
$$820$$ 0 0
$$821$$ 39.5279 1.37953 0.689766 0.724032i $$-0.257713\pi$$
0.689766 + 0.724032i $$0.257713\pi$$
$$822$$ −3.05573 −0.106581
$$823$$ 52.3607 1.82518 0.912589 0.408877i $$-0.134080\pi$$
0.912589 + 0.408877i $$0.134080\pi$$
$$824$$ −6.00000 −0.209020
$$825$$ 0 0
$$826$$ 40.0000 1.39178
$$827$$ 21.5967 0.750993 0.375496 0.926824i $$-0.377472\pi$$
0.375496 + 0.926824i $$0.377472\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 0 0
$$831$$ −20.4721 −0.710171
$$832$$ −4.47214 −0.155043
$$833$$ 52.0000 1.80169
$$834$$ −16.9443 −0.586732
$$835$$ 0 0
$$836$$ −29.8885 −1.03372
$$837$$ 2.47214 0.0854495
$$838$$ −28.0689 −0.969623
$$839$$ −45.8885 −1.58425 −0.792124 0.610360i $$-0.791025\pi$$
−0.792124 + 0.610360i $$0.791025\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0.763932 0.0263268
$$843$$ 13.5279 0.465924
$$844$$ −23.4164 −0.806026
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 73.4164 2.52262
$$848$$ −5.23607 −0.179807
$$849$$ −3.81966 −0.131090
$$850$$ 0 0
$$851$$ 11.2361 0.385167
$$852$$ −8.94427 −0.306426
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −3.41641 −0.116907
$$855$$ 0 0
$$856$$ −12.6525 −0.432453
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 23.4164 0.799423
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 8.94427 0.304820
$$862$$ 23.4164 0.797566
$$863$$ −14.8328 −0.504915 −0.252457 0.967608i $$-0.581239\pi$$
−0.252457 + 0.967608i $$0.581239\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −21.4164 −0.727759
$$867$$ 1.00000 0.0339618
$$868$$ −11.0557 −0.375256
$$869$$ −23.4164 −0.794347
$$870$$ 0 0
$$871$$ 43.4164 1.47111
$$872$$ −4.76393 −0.161327
$$873$$ −0.472136 −0.0159794
$$874$$ 5.70820 0.193083
$$875$$ 0 0
$$876$$ −4.47214 −0.151099
$$877$$ 48.2492 1.62926 0.814630 0.579981i $$-0.196940\pi$$
0.814630 + 0.579981i $$0.196940\pi$$
$$878$$ −19.0557 −0.643100
$$879$$ 0.291796 0.00984204
$$880$$ 0 0
$$881$$ 39.4164 1.32797 0.663986 0.747745i $$-0.268863\pi$$
0.663986 + 0.747745i $$0.268863\pi$$
$$882$$ −13.0000 −0.437733
$$883$$ −13.5279 −0.455249 −0.227624 0.973749i $$-0.573096\pi$$
−0.227624 + 0.973749i $$0.573096\pi$$
$$884$$ −17.8885 −0.601657
$$885$$ 0 0
$$886$$ −15.0557 −0.505807
$$887$$ 16.9443 0.568933 0.284466 0.958686i $$-0.408184\pi$$
0.284466 + 0.958686i $$0.408184\pi$$
$$888$$ −11.2361 −0.377058
$$889$$ 17.8885 0.599963
$$890$$ 0 0
$$891$$ −5.23607 −0.175415
$$892$$ −19.4164 −0.650109
$$893$$ −22.8328 −0.764071
$$894$$ −11.7082 −0.391581
$$895$$ 0 0
$$896$$ −4.47214 −0.149404
$$897$$ −4.47214 −0.149320
$$898$$ −15.8885 −0.530208
$$899$$ 11.0557 0.368729
$$900$$ 0 0
$$901$$ −20.9443 −0.697755
$$902$$ −10.4721 −0.348684
$$903$$ −21.3050 −0.708984
$$904$$ −5.52786 −0.183854
$$905$$ 0 0
$$906$$ −14.4721 −0.480805
$$907$$ −4.18034 −0.138806 −0.0694030 0.997589i $$-0.522109\pi$$
−0.0694030 + 0.997589i $$0.522109\pi$$
$$908$$ 9.23607 0.306510
$$909$$ 4.47214 0.148331
$$910$$ 0 0
$$911$$ 16.5836 0.549439 0.274719 0.961524i $$-0.411415\pi$$
0.274719 + 0.961524i $$0.411415\pi$$
$$912$$ −5.70820 −0.189018
$$913$$ 69.3050 2.29366
$$914$$ 27.5279 0.910541
$$915$$ 0 0
$$916$$ 17.7082 0.585096
$$917$$ −42.6099 −1.40710
$$918$$ 4.00000 0.132020
$$919$$ −16.4721 −0.543366 −0.271683 0.962387i $$-0.587580\pi$$
−0.271683 + 0.962387i $$0.587580\pi$$
$$920$$ 0 0
$$921$$ 15.4164 0.507988
$$922$$ −22.0000 −0.724531
$$923$$ −40.0000 −1.31662
$$924$$ 23.4164 0.770343
$$925$$ 0 0
$$926$$ 24.0000 0.788689
$$927$$ 6.00000 0.197066
$$928$$ 4.47214 0.146805
$$929$$ −24.8328 −0.814738 −0.407369 0.913264i $$-0.633554\pi$$
−0.407369 + 0.913264i $$0.633554\pi$$
$$930$$ 0 0
$$931$$ 74.2067 2.43202
$$932$$ −19.8885 −0.651471
$$933$$ 20.9443 0.685685
$$934$$ −17.8197 −0.583077
$$935$$ 0 0
$$936$$ 4.47214 0.146176
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 43.4164 1.41760
$$939$$ 15.5279 0.506733
$$940$$ 0 0
$$941$$ 38.1803 1.24464 0.622322 0.782762i $$-0.286190\pi$$
0.622322 + 0.782762i $$0.286190\pi$$
$$942$$ 6.65248 0.216749
$$943$$ 2.00000 0.0651290
$$944$$ −8.94427 −0.291111
$$945$$ 0 0
$$946$$ 24.9443 0.811008
$$947$$ 47.1935 1.53358 0.766791 0.641897i $$-0.221852\pi$$
0.766791 + 0.641897i $$0.221852\pi$$
$$948$$ −4.47214 −0.145248
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ 19.5279 0.633234
$$952$$ −17.8885 −0.579771
$$953$$ 47.7771 1.54765 0.773826 0.633398i $$-0.218341\pi$$
0.773826 + 0.633398i $$0.218341\pi$$
$$954$$ 5.23607 0.169524
$$955$$ 0 0
$$956$$ −4.94427 −0.159909
$$957$$ −23.4164 −0.756945
$$958$$ 28.9443 0.935147
$$959$$ −13.6656 −0.441286
$$960$$ 0 0
$$961$$ −24.8885 −0.802856
$$962$$ −50.2492 −1.62010
$$963$$ 12.6525 0.407720
$$964$$ −12.4721 −0.401700
$$965$$ 0 0
$$966$$ −4.47214 −0.143889
$$967$$ −15.4164 −0.495758 −0.247879 0.968791i $$-0.579734\pi$$
−0.247879 + 0.968791i $$0.579734\pi$$
$$968$$ −16.4164 −0.527643
$$969$$ −22.8328 −0.733496
$$970$$ 0 0
$$971$$ −19.1246 −0.613738 −0.306869 0.951752i $$-0.599281\pi$$
−0.306869 + 0.951752i $$0.599281\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −75.7771 −2.42930
$$974$$ 9.88854 0.316849
$$975$$ 0 0
$$976$$ 0.763932 0.0244529
$$977$$ −1.16718 −0.0373415 −0.0186708 0.999826i $$-0.505943\pi$$
−0.0186708 + 0.999826i $$0.505943\pi$$
$$978$$ 2.47214 0.0790502
$$979$$ 54.8328 1.75246
$$980$$ 0 0
$$981$$ 4.76393 0.152101
$$982$$ −29.3050 −0.935159
$$983$$ −0.583592 −0.0186137 −0.00930685 0.999957i $$-0.502963\pi$$
−0.00930685 + 0.999957i $$0.502963\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ 17.8885 0.569687
$$987$$ 17.8885 0.569399
$$988$$ −25.5279 −0.812150
$$989$$ −4.76393 −0.151484
$$990$$ 0 0
$$991$$ −10.4721 −0.332658 −0.166329 0.986070i $$-0.553191\pi$$
−0.166329 + 0.986070i $$0.553191\pi$$
$$992$$ 2.47214 0.0784904
$$993$$ 10.4721 0.332323
$$994$$ −40.0000 −1.26872
$$995$$ 0 0
$$996$$ 13.2361 0.419401
$$997$$ −5.05573 −0.160117 −0.0800583 0.996790i $$-0.525511\pi$$
−0.0800583 + 0.996790i $$0.525511\pi$$
$$998$$ −0.583592 −0.0184733
$$999$$ 11.2361 0.355493
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 3450.2.a.be.1.2 2
5.2 odd 4 3450.2.d.x.2899.2 4
5.3 odd 4 3450.2.d.x.2899.3 4
5.4 even 2 138.2.a.d.1.2 2
15.14 odd 2 414.2.a.f.1.1 2
20.19 odd 2 1104.2.a.j.1.2 2
35.34 odd 2 6762.2.a.cb.1.1 2
40.19 odd 2 4416.2.a.bl.1.1 2
40.29 even 2 4416.2.a.bh.1.1 2
60.59 even 2 3312.2.a.bc.1.1 2
115.114 odd 2 3174.2.a.s.1.1 2
345.344 even 2 9522.2.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.2 2 5.4 even 2
414.2.a.f.1.1 2 15.14 odd 2
1104.2.a.j.1.2 2 20.19 odd 2
3174.2.a.s.1.1 2 115.114 odd 2
3312.2.a.bc.1.1 2 60.59 even 2
3450.2.a.be.1.2 2 1.1 even 1 trivial
3450.2.d.x.2899.2 4 5.2 odd 4
3450.2.d.x.2899.3 4 5.3 odd 4
4416.2.a.bh.1.1 2 40.29 even 2
4416.2.a.bl.1.1 2 40.19 odd 2
6762.2.a.cb.1.1 2 35.34 odd 2
9522.2.a.q.1.2 2 345.344 even 2