# Properties

 Label 3626.2.a.a.1.1 Level $3626$ Weight $2$ Character 3626.1 Self dual yes Analytic conductor $28.954$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3626 = 2 \cdot 7^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3626.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.9537557729$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 3626.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} +2.30278 q^{5} +3.30278 q^{6} -1.00000 q^{8} +7.90833 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} +2.30278 q^{5} +3.30278 q^{6} -1.00000 q^{8} +7.90833 q^{9} -2.30278 q^{10} -2.30278 q^{11} -3.30278 q^{12} -1.30278 q^{13} -7.60555 q^{15} +1.00000 q^{16} +6.00000 q^{17} -7.90833 q^{18} -2.00000 q^{19} +2.30278 q^{20} +2.30278 q^{22} +3.90833 q^{23} +3.30278 q^{24} +0.302776 q^{25} +1.30278 q^{26} -16.2111 q^{27} -3.90833 q^{29} +7.60555 q^{30} +0.302776 q^{31} -1.00000 q^{32} +7.60555 q^{33} -6.00000 q^{34} +7.90833 q^{36} +1.00000 q^{37} +2.00000 q^{38} +4.30278 q^{39} -2.30278 q^{40} -9.90833 q^{41} +0.605551 q^{43} -2.30278 q^{44} +18.2111 q^{45} -3.90833 q^{46} -4.60555 q^{47} -3.30278 q^{48} -0.302776 q^{50} -19.8167 q^{51} -1.30278 q^{52} -6.00000 q^{53} +16.2111 q^{54} -5.30278 q^{55} +6.60555 q^{57} +3.90833 q^{58} -10.6056 q^{59} -7.60555 q^{60} -7.51388 q^{61} -0.302776 q^{62} +1.00000 q^{64} -3.00000 q^{65} -7.60555 q^{66} -3.51388 q^{67} +6.00000 q^{68} -12.9083 q^{69} +6.00000 q^{71} -7.90833 q^{72} +12.3028 q^{73} -1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -4.30278 q^{78} +9.11943 q^{79} +2.30278 q^{80} +29.8167 q^{81} +9.90833 q^{82} -2.78890 q^{83} +13.8167 q^{85} -0.605551 q^{86} +12.9083 q^{87} +2.30278 q^{88} +9.21110 q^{89} -18.2111 q^{90} +3.90833 q^{92} -1.00000 q^{93} +4.60555 q^{94} -4.60555 q^{95} +3.30278 q^{96} +16.4222 q^{97} -18.2111 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 + q^5 + 3 * q^6 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9} - q^{10} - q^{11} - 3 q^{12} + q^{13} - 8 q^{15} + 2 q^{16} + 12 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + q^{22} - 3 q^{23} + 3 q^{24} - 3 q^{25} - q^{26} - 18 q^{27} + 3 q^{29} + 8 q^{30} - 3 q^{31} - 2 q^{32} + 8 q^{33} - 12 q^{34} + 5 q^{36} + 2 q^{37} + 4 q^{38} + 5 q^{39} - q^{40} - 9 q^{41} - 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 3 q^{50} - 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 6 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} + 3 q^{61} + 3 q^{62} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} + 12 q^{71} - 5 q^{72} + 21 q^{73} - 2 q^{74} - 2 q^{75} - 4 q^{76} - 5 q^{78} - 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} - 20 q^{83} + 6 q^{85} + 6 q^{86} + 15 q^{87} + q^{88} + 4 q^{89} - 22 q^{90} - 3 q^{92} - 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} - 22 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 + q^5 + 3 * q^6 - 2 * q^8 + 5 * q^9 - q^10 - q^11 - 3 * q^12 + q^13 - 8 * q^15 + 2 * q^16 + 12 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + q^22 - 3 * q^23 + 3 * q^24 - 3 * q^25 - q^26 - 18 * q^27 + 3 * q^29 + 8 * q^30 - 3 * q^31 - 2 * q^32 + 8 * q^33 - 12 * q^34 + 5 * q^36 + 2 * q^37 + 4 * q^38 + 5 * q^39 - q^40 - 9 * q^41 - 6 * q^43 - q^44 + 22 * q^45 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 3 * q^50 - 18 * q^51 + q^52 - 12 * q^53 + 18 * q^54 - 7 * q^55 + 6 * q^57 - 3 * q^58 - 14 * q^59 - 8 * q^60 + 3 * q^61 + 3 * q^62 + 2 * q^64 - 6 * q^65 - 8 * q^66 + 11 * q^67 + 12 * q^68 - 15 * q^69 + 12 * q^71 - 5 * q^72 + 21 * q^73 - 2 * q^74 - 2 * q^75 - 4 * q^76 - 5 * q^78 - 7 * q^79 + q^80 + 38 * q^81 + 9 * q^82 - 20 * q^83 + 6 * q^85 + 6 * q^86 + 15 * q^87 + q^88 + 4 * q^89 - 22 * q^90 - 3 * q^92 - 2 * q^93 + 2 * q^94 - 2 * q^95 + 3 * q^96 + 4 * q^97 - 22 * q^99

## 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 $$-0.902474\pi$$
−0.953429 + 0.301617i $$0.902474\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 2.30278 1.02983 0.514916 0.857240i $$-0.327823\pi$$
0.514916 + 0.857240i $$0.327823\pi$$
$$6$$ 3.30278 1.34835
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 7.90833 2.63611
$$10$$ −2.30278 −0.728202
$$11$$ −2.30278 −0.694313 −0.347156 0.937807i $$-0.612853\pi$$
−0.347156 + 0.937807i $$0.612853\pi$$
$$12$$ −3.30278 −0.953429
$$13$$ −1.30278 −0.361325 −0.180662 0.983545i $$-0.557824\pi$$
−0.180662 + 0.983545i $$0.557824\pi$$
$$14$$ 0 0
$$15$$ −7.60555 −1.96374
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −7.90833 −1.86401
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 2.30278 0.514916
$$21$$ 0 0
$$22$$ 2.30278 0.490953
$$23$$ 3.90833 0.814942 0.407471 0.913218i $$-0.366411\pi$$
0.407471 + 0.913218i $$0.366411\pi$$
$$24$$ 3.30278 0.674176
$$25$$ 0.302776 0.0605551
$$26$$ 1.30278 0.255495
$$27$$ −16.2111 −3.11983
$$28$$ 0 0
$$29$$ −3.90833 −0.725758 −0.362879 0.931836i $$-0.618206\pi$$
−0.362879 + 0.931836i $$0.618206\pi$$
$$30$$ 7.60555 1.38858
$$31$$ 0.302776 0.0543801 0.0271901 0.999630i $$-0.491344\pi$$
0.0271901 + 0.999630i $$0.491344\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 7.60555 1.32396
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 7.90833 1.31805
$$37$$ 1.00000 0.164399
$$38$$ 2.00000 0.324443
$$39$$ 4.30278 0.688996
$$40$$ −2.30278 −0.364101
$$41$$ −9.90833 −1.54742 −0.773710 0.633540i $$-0.781601\pi$$
−0.773710 + 0.633540i $$0.781601\pi$$
$$42$$ 0 0
$$43$$ 0.605551 0.0923457 0.0461729 0.998933i $$-0.485297\pi$$
0.0461729 + 0.998933i $$0.485297\pi$$
$$44$$ −2.30278 −0.347156
$$45$$ 18.2111 2.71475
$$46$$ −3.90833 −0.576251
$$47$$ −4.60555 −0.671789 −0.335894 0.941900i $$-0.609039\pi$$
−0.335894 + 0.941900i $$0.609039\pi$$
$$48$$ −3.30278 −0.476715
$$49$$ 0 0
$$50$$ −0.302776 −0.0428189
$$51$$ −19.8167 −2.77489
$$52$$ −1.30278 −0.180662
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 16.2111 2.20605
$$55$$ −5.30278 −0.715026
$$56$$ 0 0
$$57$$ 6.60555 0.874927
$$58$$ 3.90833 0.513188
$$59$$ −10.6056 −1.38073 −0.690363 0.723464i $$-0.742549\pi$$
−0.690363 + 0.723464i $$0.742549\pi$$
$$60$$ −7.60555 −0.981872
$$61$$ −7.51388 −0.962054 −0.481027 0.876706i $$-0.659736\pi$$
−0.481027 + 0.876706i $$0.659736\pi$$
$$62$$ −0.302776 −0.0384525
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.00000 −0.372104
$$66$$ −7.60555 −0.936179
$$67$$ −3.51388 −0.429289 −0.214644 0.976692i $$-0.568859\pi$$
−0.214644 + 0.976692i $$0.568859\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −12.9083 −1.55398
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −7.90833 −0.932005
$$73$$ 12.3028 1.43993 0.719965 0.694010i $$-0.244158\pi$$
0.719965 + 0.694010i $$0.244158\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ −1.00000 −0.115470
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ −4.30278 −0.487193
$$79$$ 9.11943 1.02602 0.513008 0.858384i $$-0.328531\pi$$
0.513008 + 0.858384i $$0.328531\pi$$
$$80$$ 2.30278 0.257458
$$81$$ 29.8167 3.31296
$$82$$ 9.90833 1.09419
$$83$$ −2.78890 −0.306121 −0.153061 0.988217i $$-0.548913\pi$$
−0.153061 + 0.988217i $$0.548913\pi$$
$$84$$ 0 0
$$85$$ 13.8167 1.49863
$$86$$ −0.605551 −0.0652983
$$87$$ 12.9083 1.38392
$$88$$ 2.30278 0.245477
$$89$$ 9.21110 0.976375 0.488187 0.872739i $$-0.337658\pi$$
0.488187 + 0.872739i $$0.337658\pi$$
$$90$$ −18.2111 −1.91962
$$91$$ 0 0
$$92$$ 3.90833 0.407471
$$93$$ −1.00000 −0.103695
$$94$$ 4.60555 0.475026
$$95$$ −4.60555 −0.472520
$$96$$ 3.30278 0.337088
$$97$$ 16.4222 1.66742 0.833711 0.552201i $$-0.186212\pi$$
0.833711 + 0.552201i $$0.186212\pi$$
$$98$$ 0 0
$$99$$ −18.2111 −1.83028
$$100$$ 0.302776 0.0302776
$$101$$ 12.4222 1.23606 0.618028 0.786156i $$-0.287932\pi$$
0.618028 + 0.786156i $$0.287932\pi$$
$$102$$ 19.8167 1.96214
$$103$$ 0.302776 0.0298334 0.0149167 0.999889i $$-0.495252\pi$$
0.0149167 + 0.999889i $$0.495252\pi$$
$$104$$ 1.30278 0.127748
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 0.697224 0.0674032 0.0337016 0.999432i $$-0.489270\pi$$
0.0337016 + 0.999432i $$0.489270\pi$$
$$108$$ −16.2111 −1.55991
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 5.30278 0.505600
$$111$$ −3.30278 −0.313486
$$112$$ 0 0
$$113$$ −3.21110 −0.302075 −0.151038 0.988528i $$-0.548261\pi$$
−0.151038 + 0.988528i $$0.548261\pi$$
$$114$$ −6.60555 −0.618667
$$115$$ 9.00000 0.839254
$$116$$ −3.90833 −0.362879
$$117$$ −10.3028 −0.952492
$$118$$ 10.6056 0.976320
$$119$$ 0 0
$$120$$ 7.60555 0.694289
$$121$$ −5.69722 −0.517929
$$122$$ 7.51388 0.680275
$$123$$ 32.7250 2.95071
$$124$$ 0.302776 0.0271901
$$125$$ −10.8167 −0.967471
$$126$$ 0 0
$$127$$ −19.2111 −1.70471 −0.852355 0.522964i $$-0.824826\pi$$
−0.852355 + 0.522964i $$0.824826\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −2.00000 −0.176090
$$130$$ 3.00000 0.263117
$$131$$ −10.6056 −0.926611 −0.463306 0.886199i $$-0.653337\pi$$
−0.463306 + 0.886199i $$0.653337\pi$$
$$132$$ 7.60555 0.661978
$$133$$ 0 0
$$134$$ 3.51388 0.303553
$$135$$ −37.3305 −3.21290
$$136$$ −6.00000 −0.514496
$$137$$ 0.908327 0.0776036 0.0388018 0.999247i $$-0.487646\pi$$
0.0388018 + 0.999247i $$0.487646\pi$$
$$138$$ 12.9083 1.09883
$$139$$ 1.90833 0.161862 0.0809311 0.996720i $$-0.474211\pi$$
0.0809311 + 0.996720i $$0.474211\pi$$
$$140$$ 0 0
$$141$$ 15.2111 1.28101
$$142$$ −6.00000 −0.503509
$$143$$ 3.00000 0.250873
$$144$$ 7.90833 0.659027
$$145$$ −9.00000 −0.747409
$$146$$ −12.3028 −1.01818
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ 19.8167 1.62344 0.811722 0.584044i $$-0.198531\pi$$
0.811722 + 0.584044i $$0.198531\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −20.6056 −1.67686 −0.838428 0.545012i $$-0.816525\pi$$
−0.838428 + 0.545012i $$0.816525\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 47.4500 3.83610
$$154$$ 0 0
$$155$$ 0.697224 0.0560024
$$156$$ 4.30278 0.344498
$$157$$ 7.21110 0.575509 0.287754 0.957704i $$-0.407091\pi$$
0.287754 + 0.957704i $$0.407091\pi$$
$$158$$ −9.11943 −0.725503
$$159$$ 19.8167 1.57156
$$160$$ −2.30278 −0.182050
$$161$$ 0 0
$$162$$ −29.8167 −2.34262
$$163$$ 8.42221 0.659678 0.329839 0.944037i $$-0.393006\pi$$
0.329839 + 0.944037i $$0.393006\pi$$
$$164$$ −9.90833 −0.773710
$$165$$ 17.5139 1.36345
$$166$$ 2.78890 0.216460
$$167$$ 5.51388 0.426677 0.213338 0.976978i $$-0.431566\pi$$
0.213338 + 0.976978i $$0.431566\pi$$
$$168$$ 0 0
$$169$$ −11.3028 −0.869444
$$170$$ −13.8167 −1.05969
$$171$$ −15.8167 −1.20953
$$172$$ 0.605551 0.0461729
$$173$$ 8.78890 0.668207 0.334104 0.942536i $$-0.391566\pi$$
0.334104 + 0.942536i $$0.391566\pi$$
$$174$$ −12.9083 −0.978578
$$175$$ 0 0
$$176$$ −2.30278 −0.173578
$$177$$ 35.0278 2.63285
$$178$$ −9.21110 −0.690401
$$179$$ −13.8167 −1.03271 −0.516353 0.856376i $$-0.672711\pi$$
−0.516353 + 0.856376i $$0.672711\pi$$
$$180$$ 18.2111 1.35738
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 24.8167 1.83450
$$184$$ −3.90833 −0.288126
$$185$$ 2.30278 0.169303
$$186$$ 1.00000 0.0733236
$$187$$ −13.8167 −1.01037
$$188$$ −4.60555 −0.335894
$$189$$ 0 0
$$190$$ 4.60555 0.334122
$$191$$ −5.51388 −0.398970 −0.199485 0.979901i $$-0.563927\pi$$
−0.199485 + 0.979901i $$0.563927\pi$$
$$192$$ −3.30278 −0.238357
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −16.4222 −1.17905
$$195$$ 9.90833 0.709550
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 18.2111 1.29421
$$199$$ −26.4222 −1.87302 −0.936510 0.350640i $$-0.885964\pi$$
−0.936510 + 0.350640i $$0.885964\pi$$
$$200$$ −0.302776 −0.0214095
$$201$$ 11.6056 0.818592
$$202$$ −12.4222 −0.874023
$$203$$ 0 0
$$204$$ −19.8167 −1.38744
$$205$$ −22.8167 −1.59358
$$206$$ −0.302776 −0.0210954
$$207$$ 30.9083 2.14828
$$208$$ −1.30278 −0.0903312
$$209$$ 4.60555 0.318573
$$210$$ 0 0
$$211$$ 10.3028 0.709272 0.354636 0.935004i $$-0.384605\pi$$
0.354636 + 0.935004i $$0.384605\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −19.8167 −1.35781
$$214$$ −0.697224 −0.0476613
$$215$$ 1.39445 0.0951006
$$216$$ 16.2111 1.10303
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ −40.6333 −2.74574
$$220$$ −5.30278 −0.357513
$$221$$ −7.81665 −0.525805
$$222$$ 3.30278 0.221668
$$223$$ 5.81665 0.389512 0.194756 0.980852i $$-0.437609\pi$$
0.194756 + 0.980852i $$0.437609\pi$$
$$224$$ 0 0
$$225$$ 2.39445 0.159630
$$226$$ 3.21110 0.213599
$$227$$ −13.8167 −0.917044 −0.458522 0.888683i $$-0.651621\pi$$
−0.458522 + 0.888683i $$0.651621\pi$$
$$228$$ 6.60555 0.437463
$$229$$ −24.6056 −1.62598 −0.812990 0.582277i $$-0.802162\pi$$
−0.812990 + 0.582277i $$0.802162\pi$$
$$230$$ −9.00000 −0.593442
$$231$$ 0 0
$$232$$ 3.90833 0.256594
$$233$$ 8.51388 0.557763 0.278881 0.960326i $$-0.410036\pi$$
0.278881 + 0.960326i $$0.410036\pi$$
$$234$$ 10.3028 0.673514
$$235$$ −10.6056 −0.691830
$$236$$ −10.6056 −0.690363
$$237$$ −30.1194 −1.95647
$$238$$ 0 0
$$239$$ −17.5139 −1.13288 −0.566439 0.824103i $$-0.691679\pi$$
−0.566439 + 0.824103i $$0.691679\pi$$
$$240$$ −7.60555 −0.490936
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 5.69722 0.366231
$$243$$ −49.8444 −3.19752
$$244$$ −7.51388 −0.481027
$$245$$ 0 0
$$246$$ −32.7250 −2.08647
$$247$$ 2.60555 0.165787
$$248$$ −0.302776 −0.0192263
$$249$$ 9.21110 0.583730
$$250$$ 10.8167 0.684105
$$251$$ 21.2111 1.33883 0.669416 0.742887i $$-0.266544\pi$$
0.669416 + 0.742887i $$0.266544\pi$$
$$252$$ 0 0
$$253$$ −9.00000 −0.565825
$$254$$ 19.2111 1.20541
$$255$$ −45.6333 −2.85767
$$256$$ 1.00000 0.0625000
$$257$$ −3.21110 −0.200303 −0.100152 0.994972i $$-0.531933\pi$$
−0.100152 + 0.994972i $$0.531933\pi$$
$$258$$ 2.00000 0.124515
$$259$$ 0 0
$$260$$ −3.00000 −0.186052
$$261$$ −30.9083 −1.91318
$$262$$ 10.6056 0.655213
$$263$$ 13.8167 0.851971 0.425986 0.904730i $$-0.359927\pi$$
0.425986 + 0.904730i $$0.359927\pi$$
$$264$$ −7.60555 −0.468089
$$265$$ −13.8167 −0.848750
$$266$$ 0 0
$$267$$ −30.4222 −1.86181
$$268$$ −3.51388 −0.214644
$$269$$ 21.2111 1.29326 0.646632 0.762802i $$-0.276177\pi$$
0.646632 + 0.762802i $$0.276177\pi$$
$$270$$ 37.3305 2.27186
$$271$$ 22.4222 1.36205 0.681026 0.732259i $$-0.261534\pi$$
0.681026 + 0.732259i $$0.261534\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −0.908327 −0.0548740
$$275$$ −0.697224 −0.0420442
$$276$$ −12.9083 −0.776990
$$277$$ 0.119429 0.00717582 0.00358791 0.999994i $$-0.498858\pi$$
0.00358791 + 0.999994i $$0.498858\pi$$
$$278$$ −1.90833 −0.114454
$$279$$ 2.39445 0.143352
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ −15.2111 −0.905808
$$283$$ −24.6056 −1.46265 −0.731324 0.682030i $$-0.761097\pi$$
−0.731324 + 0.682030i $$0.761097\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 15.2111 0.901028
$$286$$ −3.00000 −0.177394
$$287$$ 0 0
$$288$$ −7.90833 −0.466003
$$289$$ 19.0000 1.11765
$$290$$ 9.00000 0.528498
$$291$$ −54.2389 −3.17954
$$292$$ 12.3028 0.719965
$$293$$ −11.0278 −0.644248 −0.322124 0.946697i $$-0.604397\pi$$
−0.322124 + 0.946697i $$0.604397\pi$$
$$294$$ 0 0
$$295$$ −24.4222 −1.42192
$$296$$ −1.00000 −0.0581238
$$297$$ 37.3305 2.16614
$$298$$ −19.8167 −1.14795
$$299$$ −5.09167 −0.294459
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 20.6056 1.18572
$$303$$ −41.0278 −2.35698
$$304$$ −2.00000 −0.114708
$$305$$ −17.3028 −0.990754
$$306$$ −47.4500 −2.71253
$$307$$ −17.9083 −1.02208 −0.511041 0.859556i $$-0.670740\pi$$
−0.511041 + 0.859556i $$0.670740\pi$$
$$308$$ 0 0
$$309$$ −1.00000 −0.0568880
$$310$$ −0.697224 −0.0395997
$$311$$ −15.9083 −0.902078 −0.451039 0.892504i $$-0.648947\pi$$
−0.451039 + 0.892504i $$0.648947\pi$$
$$312$$ −4.30278 −0.243597
$$313$$ 9.02776 0.510279 0.255139 0.966904i $$-0.417879\pi$$
0.255139 + 0.966904i $$0.417879\pi$$
$$314$$ −7.21110 −0.406946
$$315$$ 0 0
$$316$$ 9.11943 0.513008
$$317$$ 9.21110 0.517347 0.258674 0.965965i $$-0.416715\pi$$
0.258674 + 0.965965i $$0.416715\pi$$
$$318$$ −19.8167 −1.11126
$$319$$ 9.00000 0.503903
$$320$$ 2.30278 0.128729
$$321$$ −2.30278 −0.128528
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 29.8167 1.65648
$$325$$ −0.394449 −0.0218801
$$326$$ −8.42221 −0.466463
$$327$$ −6.60555 −0.365288
$$328$$ 9.90833 0.547096
$$329$$ 0 0
$$330$$ −17.5139 −0.964107
$$331$$ −13.2111 −0.726148 −0.363074 0.931760i $$-0.618273\pi$$
−0.363074 + 0.931760i $$0.618273\pi$$
$$332$$ −2.78890 −0.153061
$$333$$ 7.90833 0.433374
$$334$$ −5.51388 −0.301706
$$335$$ −8.09167 −0.442095
$$336$$ 0 0
$$337$$ 6.11943 0.333347 0.166673 0.986012i $$-0.446697\pi$$
0.166673 + 0.986012i $$0.446697\pi$$
$$338$$ 11.3028 0.614790
$$339$$ 10.6056 0.576014
$$340$$ 13.8167 0.749313
$$341$$ −0.697224 −0.0377568
$$342$$ 15.8167 0.855267
$$343$$ 0 0
$$344$$ −0.605551 −0.0326491
$$345$$ −29.7250 −1.60034
$$346$$ −8.78890 −0.472494
$$347$$ 10.1833 0.546671 0.273335 0.961919i $$-0.411873\pi$$
0.273335 + 0.961919i $$0.411873\pi$$
$$348$$ 12.9083 0.691959
$$349$$ −28.2389 −1.51159 −0.755796 0.654807i $$-0.772750\pi$$
−0.755796 + 0.654807i $$0.772750\pi$$
$$350$$ 0 0
$$351$$ 21.1194 1.12727
$$352$$ 2.30278 0.122738
$$353$$ −10.1833 −0.542005 −0.271002 0.962579i $$-0.587355\pi$$
−0.271002 + 0.962579i $$0.587355\pi$$
$$354$$ −35.0278 −1.86170
$$355$$ 13.8167 0.733312
$$356$$ 9.21110 0.488187
$$357$$ 0 0
$$358$$ 13.8167 0.730233
$$359$$ 3.21110 0.169476 0.0847378 0.996403i $$-0.472995\pi$$
0.0847378 + 0.996403i $$0.472995\pi$$
$$360$$ −18.2111 −0.959809
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000 1.05118
$$363$$ 18.8167 0.987618
$$364$$ 0 0
$$365$$ 28.3305 1.48289
$$366$$ −24.8167 −1.29719
$$367$$ −3.81665 −0.199228 −0.0996139 0.995026i $$-0.531761\pi$$
−0.0996139 + 0.995026i $$0.531761\pi$$
$$368$$ 3.90833 0.203736
$$369$$ −78.3583 −4.07917
$$370$$ −2.30278 −0.119716
$$371$$ 0 0
$$372$$ −1.00000 −0.0518476
$$373$$ −17.8167 −0.922511 −0.461256 0.887267i $$-0.652601\pi$$
−0.461256 + 0.887267i $$0.652601\pi$$
$$374$$ 13.8167 0.714442
$$375$$ 35.7250 1.84483
$$376$$ 4.60555 0.237513
$$377$$ 5.09167 0.262235
$$378$$ 0 0
$$379$$ 24.3305 1.24978 0.624888 0.780715i $$-0.285145\pi$$
0.624888 + 0.780715i $$0.285145\pi$$
$$380$$ −4.60555 −0.236260
$$381$$ 63.4500 3.25064
$$382$$ 5.51388 0.282115
$$383$$ 36.8444 1.88266 0.941331 0.337486i $$-0.109576\pi$$
0.941331 + 0.337486i $$0.109576\pi$$
$$384$$ 3.30278 0.168544
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 4.78890 0.243433
$$388$$ 16.4222 0.833711
$$389$$ −37.1194 −1.88203 −0.941015 0.338365i $$-0.890126\pi$$
−0.941015 + 0.338365i $$0.890126\pi$$
$$390$$ −9.90833 −0.501728
$$391$$ 23.4500 1.18592
$$392$$ 0 0
$$393$$ 35.0278 1.76692
$$394$$ 6.00000 0.302276
$$395$$ 21.0000 1.05662
$$396$$ −18.2111 −0.915142
$$397$$ −6.18335 −0.310333 −0.155167 0.987888i $$-0.549591\pi$$
−0.155167 + 0.987888i $$0.549591\pi$$
$$398$$ 26.4222 1.32443
$$399$$ 0 0
$$400$$ 0.302776 0.0151388
$$401$$ −7.81665 −0.390345 −0.195173 0.980769i $$-0.562527\pi$$
−0.195173 + 0.980769i $$0.562527\pi$$
$$402$$ −11.6056 −0.578832
$$403$$ −0.394449 −0.0196489
$$404$$ 12.4222 0.618028
$$405$$ 68.6611 3.41180
$$406$$ 0 0
$$407$$ −2.30278 −0.114144
$$408$$ 19.8167 0.981071
$$409$$ −31.0278 −1.53422 −0.767112 0.641513i $$-0.778307\pi$$
−0.767112 + 0.641513i $$0.778307\pi$$
$$410$$ 22.8167 1.12683
$$411$$ −3.00000 −0.147979
$$412$$ 0.302776 0.0149167
$$413$$ 0 0
$$414$$ −30.9083 −1.51906
$$415$$ −6.42221 −0.315254
$$416$$ 1.30278 0.0638738
$$417$$ −6.30278 −0.308648
$$418$$ −4.60555 −0.225265
$$419$$ −36.1472 −1.76591 −0.882953 0.469462i $$-0.844448\pi$$
−0.882953 + 0.469462i $$0.844448\pi$$
$$420$$ 0 0
$$421$$ −3.72498 −0.181544 −0.0907722 0.995872i $$-0.528934\pi$$
−0.0907722 + 0.995872i $$0.528934\pi$$
$$422$$ −10.3028 −0.501531
$$423$$ −36.4222 −1.77091
$$424$$ 6.00000 0.291386
$$425$$ 1.81665 0.0881207
$$426$$ 19.8167 0.960120
$$427$$ 0 0
$$428$$ 0.697224 0.0337016
$$429$$ −9.90833 −0.478379
$$430$$ −1.39445 −0.0672463
$$431$$ 9.21110 0.443683 0.221842 0.975083i $$-0.428793\pi$$
0.221842 + 0.975083i $$0.428793\pi$$
$$432$$ −16.2111 −0.779957
$$433$$ −34.9361 −1.67892 −0.839461 0.543421i $$-0.817129\pi$$
−0.839461 + 0.543421i $$0.817129\pi$$
$$434$$ 0 0
$$435$$ 29.7250 1.42520
$$436$$ 2.00000 0.0957826
$$437$$ −7.81665 −0.373921
$$438$$ 40.6333 1.94153
$$439$$ −30.3305 −1.44760 −0.723799 0.690011i $$-0.757606\pi$$
−0.723799 + 0.690011i $$0.757606\pi$$
$$440$$ 5.30278 0.252800
$$441$$ 0 0
$$442$$ 7.81665 0.371800
$$443$$ 32.7250 1.55481 0.777405 0.629000i $$-0.216535\pi$$
0.777405 + 0.629000i $$0.216535\pi$$
$$444$$ −3.30278 −0.156743
$$445$$ 21.2111 1.00550
$$446$$ −5.81665 −0.275427
$$447$$ −65.4500 −3.09568
$$448$$ 0 0
$$449$$ −15.2111 −0.717856 −0.358928 0.933365i $$-0.616858\pi$$
−0.358928 + 0.933365i $$0.616858\pi$$
$$450$$ −2.39445 −0.112875
$$451$$ 22.8167 1.07439
$$452$$ −3.21110 −0.151038
$$453$$ 68.0555 3.19753
$$454$$ 13.8167 0.648448
$$455$$ 0 0
$$456$$ −6.60555 −0.309333
$$457$$ −2.60555 −0.121883 −0.0609413 0.998141i $$-0.519410\pi$$
−0.0609413 + 0.998141i $$0.519410\pi$$
$$458$$ 24.6056 1.14974
$$459$$ −97.2666 −4.54002
$$460$$ 9.00000 0.419627
$$461$$ −12.4222 −0.578560 −0.289280 0.957245i $$-0.593416\pi$$
−0.289280 + 0.957245i $$0.593416\pi$$
$$462$$ 0 0
$$463$$ 26.6972 1.24073 0.620363 0.784315i $$-0.286985\pi$$
0.620363 + 0.784315i $$0.286985\pi$$
$$464$$ −3.90833 −0.181440
$$465$$ −2.30278 −0.106789
$$466$$ −8.51388 −0.394398
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ −10.3028 −0.476246
$$469$$ 0 0
$$470$$ 10.6056 0.489198
$$471$$ −23.8167 −1.09741
$$472$$ 10.6056 0.488160
$$473$$ −1.39445 −0.0641168
$$474$$ 30.1194 1.38343
$$475$$ −0.605551 −0.0277846
$$476$$ 0 0
$$477$$ −47.4500 −2.17258
$$478$$ 17.5139 0.801066
$$479$$ 13.1194 0.599442 0.299721 0.954027i $$-0.403106\pi$$
0.299721 + 0.954027i $$0.403106\pi$$
$$480$$ 7.60555 0.347144
$$481$$ −1.30278 −0.0594015
$$482$$ 8.00000 0.364390
$$483$$ 0 0
$$484$$ −5.69722 −0.258965
$$485$$ 37.8167 1.71717
$$486$$ 49.8444 2.26099
$$487$$ −37.2111 −1.68620 −0.843098 0.537760i $$-0.819271\pi$$
−0.843098 + 0.537760i $$0.819271\pi$$
$$488$$ 7.51388 0.340137
$$489$$ −27.8167 −1.25791
$$490$$ 0 0
$$491$$ 17.7250 0.799917 0.399959 0.916533i $$-0.369025\pi$$
0.399959 + 0.916533i $$0.369025\pi$$
$$492$$ 32.7250 1.47536
$$493$$ −23.4500 −1.05613
$$494$$ −2.60555 −0.117229
$$495$$ −41.9361 −1.88489
$$496$$ 0.302776 0.0135950
$$497$$ 0 0
$$498$$ −9.21110 −0.412759
$$499$$ −42.2389 −1.89087 −0.945436 0.325809i $$-0.894363\pi$$
−0.945436 + 0.325809i $$0.894363\pi$$
$$500$$ −10.8167 −0.483735
$$501$$ −18.2111 −0.813612
$$502$$ −21.2111 −0.946698
$$503$$ 6.48612 0.289202 0.144601 0.989490i $$-0.453810\pi$$
0.144601 + 0.989490i $$0.453810\pi$$
$$504$$ 0 0
$$505$$ 28.6056 1.27293
$$506$$ 9.00000 0.400099
$$507$$ 37.3305 1.65791
$$508$$ −19.2111 −0.852355
$$509$$ 4.18335 0.185424 0.0927118 0.995693i $$-0.470446\pi$$
0.0927118 + 0.995693i $$0.470446\pi$$
$$510$$ 45.6333 2.02068
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 32.4222 1.43148
$$514$$ 3.21110 0.141636
$$515$$ 0.697224 0.0307234
$$516$$ −2.00000 −0.0880451
$$517$$ 10.6056 0.466432
$$518$$ 0 0
$$519$$ −29.0278 −1.27418
$$520$$ 3.00000 0.131559
$$521$$ −33.6333 −1.47350 −0.736751 0.676164i $$-0.763641\pi$$
−0.736751 + 0.676164i $$0.763641\pi$$
$$522$$ 30.9083 1.35282
$$523$$ 18.2389 0.797530 0.398765 0.917053i $$-0.369439\pi$$
0.398765 + 0.917053i $$0.369439\pi$$
$$524$$ −10.6056 −0.463306
$$525$$ 0 0
$$526$$ −13.8167 −0.602435
$$527$$ 1.81665 0.0791347
$$528$$ 7.60555 0.330989
$$529$$ −7.72498 −0.335869
$$530$$ 13.8167 0.600157
$$531$$ −83.8722 −3.63974
$$532$$ 0 0
$$533$$ 12.9083 0.559122
$$534$$ 30.4222 1.31650
$$535$$ 1.60555 0.0694140
$$536$$ 3.51388 0.151776
$$537$$ 45.6333 1.96922
$$538$$ −21.2111 −0.914476
$$539$$ 0 0
$$540$$ −37.3305 −1.60645
$$541$$ 25.9361 1.11508 0.557540 0.830150i $$-0.311745\pi$$
0.557540 + 0.830150i $$0.311745\pi$$
$$542$$ −22.4222 −0.963116
$$543$$ 66.0555 2.83471
$$544$$ −6.00000 −0.257248
$$545$$ 4.60555 0.197280
$$546$$ 0 0
$$547$$ −20.6056 −0.881030 −0.440515 0.897745i $$-0.645204\pi$$
−0.440515 + 0.897745i $$0.645204\pi$$
$$548$$ 0.908327 0.0388018
$$549$$ −59.4222 −2.53608
$$550$$ 0.697224 0.0297297
$$551$$ 7.81665 0.333001
$$552$$ 12.9083 0.549415
$$553$$ 0 0
$$554$$ −0.119429 −0.00507407
$$555$$ −7.60555 −0.322838
$$556$$ 1.90833 0.0809311
$$557$$ 11.5139 0.487859 0.243929 0.969793i $$-0.421564\pi$$
0.243929 + 0.969793i $$0.421564\pi$$
$$558$$ −2.39445 −0.101365
$$559$$ −0.788897 −0.0333668
$$560$$ 0 0
$$561$$ 45.6333 1.92664
$$562$$ 12.0000 0.506189
$$563$$ 28.0555 1.18240 0.591199 0.806525i $$-0.298655\pi$$
0.591199 + 0.806525i $$0.298655\pi$$
$$564$$ 15.2111 0.640503
$$565$$ −7.39445 −0.311087
$$566$$ 24.6056 1.03425
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ 18.4222 0.772299 0.386150 0.922436i $$-0.373805\pi$$
0.386150 + 0.922436i $$0.373805\pi$$
$$570$$ −15.2111 −0.637123
$$571$$ −16.6972 −0.698757 −0.349379 0.936982i $$-0.613607\pi$$
−0.349379 + 0.936982i $$0.613607\pi$$
$$572$$ 3.00000 0.125436
$$573$$ 18.2111 0.760780
$$574$$ 0 0
$$575$$ 1.18335 0.0493489
$$576$$ 7.90833 0.329514
$$577$$ −22.2389 −0.925816 −0.462908 0.886406i $$-0.653194\pi$$
−0.462908 + 0.886406i $$0.653194\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 13.2111 0.549035
$$580$$ −9.00000 −0.373705
$$581$$ 0 0
$$582$$ 54.2389 2.24827
$$583$$ 13.8167 0.572227
$$584$$ −12.3028 −0.509092
$$585$$ −23.7250 −0.980907
$$586$$ 11.0278 0.455552
$$587$$ 45.6333 1.88349 0.941744 0.336330i $$-0.109186\pi$$
0.941744 + 0.336330i $$0.109186\pi$$
$$588$$ 0 0
$$589$$ −0.605551 −0.0249513
$$590$$ 24.4222 1.00545
$$591$$ 19.8167 0.815148
$$592$$ 1.00000 0.0410997
$$593$$ −18.4861 −0.759134 −0.379567 0.925164i $$-0.623927\pi$$
−0.379567 + 0.925164i $$0.623927\pi$$
$$594$$ −37.3305 −1.53169
$$595$$ 0 0
$$596$$ 19.8167 0.811722
$$597$$ 87.2666 3.57158
$$598$$ 5.09167 0.208214
$$599$$ −20.7889 −0.849411 −0.424706 0.905331i $$-0.639622\pi$$
−0.424706 + 0.905331i $$0.639622\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 24.3028 0.991331 0.495665 0.868514i $$-0.334924\pi$$
0.495665 + 0.868514i $$0.334924\pi$$
$$602$$ 0 0
$$603$$ −27.7889 −1.13165
$$604$$ −20.6056 −0.838428
$$605$$ −13.1194 −0.533381
$$606$$ 41.0278 1.66664
$$607$$ 13.4861 0.547385 0.273692 0.961817i $$-0.411755\pi$$
0.273692 + 0.961817i $$0.411755\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 17.3028 0.700569
$$611$$ 6.00000 0.242734
$$612$$ 47.4500 1.91805
$$613$$ −29.8167 −1.20428 −0.602142 0.798389i $$-0.705686\pi$$
−0.602142 + 0.798389i $$0.705686\pi$$
$$614$$ 17.9083 0.722721
$$615$$ 75.3583 3.03874
$$616$$ 0 0
$$617$$ −42.5694 −1.71378 −0.856890 0.515500i $$-0.827606\pi$$
−0.856890 + 0.515500i $$0.827606\pi$$
$$618$$ 1.00000 0.0402259
$$619$$ 6.30278 0.253330 0.126665 0.991946i $$-0.459573\pi$$
0.126665 + 0.991946i $$0.459573\pi$$
$$620$$ 0.697224 0.0280012
$$621$$ −63.3583 −2.54248
$$622$$ 15.9083 0.637866
$$623$$ 0 0
$$624$$ 4.30278 0.172249
$$625$$ −26.4222 −1.05689
$$626$$ −9.02776 −0.360822
$$627$$ −15.2111 −0.607473
$$628$$ 7.21110 0.287754
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 14.6972 0.585087 0.292544 0.956252i $$-0.405498\pi$$
0.292544 + 0.956252i $$0.405498\pi$$
$$632$$ −9.11943 −0.362751
$$633$$ −34.0278 −1.35248
$$634$$ −9.21110 −0.365820
$$635$$ −44.2389 −1.75557
$$636$$ 19.8167 0.785781
$$637$$ 0 0
$$638$$ −9.00000 −0.356313
$$639$$ 47.4500 1.87709
$$640$$ −2.30278 −0.0910252
$$641$$ −20.5139 −0.810249 −0.405125 0.914261i $$-0.632772\pi$$
−0.405125 + 0.914261i $$0.632772\pi$$
$$642$$ 2.30278 0.0908833
$$643$$ 8.18335 0.322720 0.161360 0.986896i $$-0.448412\pi$$
0.161360 + 0.986896i $$0.448412\pi$$
$$644$$ 0 0
$$645$$ −4.60555 −0.181343
$$646$$ 12.0000 0.472134
$$647$$ −20.9361 −0.823082 −0.411541 0.911391i $$-0.635009\pi$$
−0.411541 + 0.911391i $$0.635009\pi$$
$$648$$ −29.8167 −1.17131
$$649$$ 24.4222 0.958655
$$650$$ 0.394449 0.0154716
$$651$$ 0 0
$$652$$ 8.42221 0.329839
$$653$$ −3.90833 −0.152945 −0.0764723 0.997072i $$-0.524366\pi$$
−0.0764723 + 0.997072i $$0.524366\pi$$
$$654$$ 6.60555 0.258297
$$655$$ −24.4222 −0.954255
$$656$$ −9.90833 −0.386855
$$657$$ 97.2944 3.79581
$$658$$ 0 0
$$659$$ −16.8806 −0.657574 −0.328787 0.944404i $$-0.606640\pi$$
−0.328787 + 0.944404i $$0.606640\pi$$
$$660$$ 17.5139 0.681727
$$661$$ 30.5139 1.18685 0.593426 0.804888i $$-0.297775\pi$$
0.593426 + 0.804888i $$0.297775\pi$$
$$662$$ 13.2111 0.513464
$$663$$ 25.8167 1.00264
$$664$$ 2.78890 0.108230
$$665$$ 0 0
$$666$$ −7.90833 −0.306441
$$667$$ −15.2750 −0.591451
$$668$$ 5.51388 0.213338
$$669$$ −19.2111 −0.742744
$$670$$ 8.09167 0.312609
$$671$$ 17.3028 0.667966
$$672$$ 0 0
$$673$$ 20.6972 0.797819 0.398910 0.916990i $$-0.369389\pi$$
0.398910 + 0.916990i $$0.369389\pi$$
$$674$$ −6.11943 −0.235712
$$675$$ −4.90833 −0.188922
$$676$$ −11.3028 −0.434722
$$677$$ −14.2389 −0.547244 −0.273622 0.961837i $$-0.588222\pi$$
−0.273622 + 0.961837i $$0.588222\pi$$
$$678$$ −10.6056 −0.407304
$$679$$ 0 0
$$680$$ −13.8167 −0.529844
$$681$$ 45.6333 1.74867
$$682$$ 0.697224 0.0266981
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ −15.8167 −0.604765
$$685$$ 2.09167 0.0799187
$$686$$ 0 0
$$687$$ 81.2666 3.10051
$$688$$ 0.605551 0.0230864
$$689$$ 7.81665 0.297791
$$690$$ 29.7250 1.13161
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 8.78890 0.334104
$$693$$ 0 0
$$694$$ −10.1833 −0.386555
$$695$$ 4.39445 0.166691
$$696$$ −12.9083 −0.489289
$$697$$ −59.4500 −2.25183
$$698$$ 28.2389 1.06886
$$699$$ −28.1194 −1.06357
$$700$$ 0 0
$$701$$ 40.1194 1.51529 0.757645 0.652667i $$-0.226350\pi$$
0.757645 + 0.652667i $$0.226350\pi$$
$$702$$ −21.1194 −0.797101
$$703$$ −2.00000 −0.0754314
$$704$$ −2.30278 −0.0867891
$$705$$ 35.0278 1.31922
$$706$$ 10.1833 0.383255
$$707$$ 0 0
$$708$$ 35.0278 1.31642
$$709$$ −41.3305 −1.55220 −0.776100 0.630609i $$-0.782805\pi$$
−0.776100 + 0.630609i $$0.782805\pi$$
$$710$$ −13.8167 −0.518530
$$711$$ 72.1194 2.70469
$$712$$ −9.21110 −0.345201
$$713$$ 1.18335 0.0443167
$$714$$ 0 0
$$715$$ 6.90833 0.258357
$$716$$ −13.8167 −0.516353
$$717$$ 57.8444 2.16024
$$718$$ −3.21110 −0.119837
$$719$$ 51.6333 1.92560 0.962799 0.270220i $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 18.2111 0.678688
$$721$$ 0 0
$$722$$ 15.0000 0.558242
$$723$$ 26.4222 0.982652
$$724$$ −20.0000 −0.743294
$$725$$ −1.18335 −0.0439484
$$726$$ −18.8167 −0.698352
$$727$$ −19.0917 −0.708071 −0.354035 0.935232i $$-0.615191\pi$$
−0.354035 + 0.935232i $$0.615191\pi$$
$$728$$ 0 0
$$729$$ 75.1749 2.78426
$$730$$ −28.3305 −1.04856
$$731$$ 3.63331 0.134383
$$732$$ 24.8167 0.917250
$$733$$ 13.6333 0.503558 0.251779 0.967785i $$-0.418984\pi$$
0.251779 + 0.967785i $$0.418984\pi$$
$$734$$ 3.81665 0.140875
$$735$$ 0 0
$$736$$ −3.90833 −0.144063
$$737$$ 8.09167 0.298061
$$738$$ 78.3583 2.88441
$$739$$ −2.66947 −0.0981980 −0.0490990 0.998794i $$-0.515635\pi$$
−0.0490990 + 0.998794i $$0.515635\pi$$
$$740$$ 2.30278 0.0846517
$$741$$ −8.60555 −0.316133
$$742$$ 0 0
$$743$$ −29.4500 −1.08041 −0.540207 0.841532i $$-0.681654\pi$$
−0.540207 + 0.841532i $$0.681654\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 45.6333 1.67188
$$746$$ 17.8167 0.652314
$$747$$ −22.0555 −0.806969
$$748$$ −13.8167 −0.505187
$$749$$ 0 0
$$750$$ −35.7250 −1.30449
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ −4.60555 −0.167947
$$753$$ −70.0555 −2.55296
$$754$$ −5.09167 −0.185428
$$755$$ −47.4500 −1.72688
$$756$$ 0 0
$$757$$ 5.69722 0.207069 0.103535 0.994626i $$-0.466985\pi$$
0.103535 + 0.994626i $$0.466985\pi$$
$$758$$ −24.3305 −0.883725
$$759$$ 29.7250 1.07895
$$760$$ 4.60555 0.167061
$$761$$ −16.8806 −0.611920 −0.305960 0.952044i $$-0.598977\pi$$
−0.305960 + 0.952044i $$0.598977\pi$$
$$762$$ −63.4500 −2.29855
$$763$$ 0 0
$$764$$ −5.51388 −0.199485
$$765$$ 109.267 3.95054
$$766$$ −36.8444 −1.33124
$$767$$ 13.8167 0.498890
$$768$$ −3.30278 −0.119179
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 10.6056 0.381950
$$772$$ −4.00000 −0.143963
$$773$$ −22.0555 −0.793282 −0.396641 0.917974i $$-0.629824\pi$$
−0.396641 + 0.917974i $$0.629824\pi$$
$$774$$ −4.78890 −0.172133
$$775$$ 0.0916731 0.00329299
$$776$$ −16.4222 −0.589523
$$777$$ 0 0
$$778$$ 37.1194 1.33080
$$779$$ 19.8167 0.710005
$$780$$ 9.90833 0.354775
$$781$$ −13.8167 −0.494399
$$782$$ −23.4500 −0.838569
$$783$$ 63.3583 2.26424
$$784$$ 0 0
$$785$$ 16.6056 0.592678
$$786$$ −35.0278 −1.24940
$$787$$ −10.7889 −0.384583 −0.192291 0.981338i $$-0.561592\pi$$
−0.192291 + 0.981338i $$0.561592\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −45.6333 −1.62459
$$790$$ −21.0000 −0.747146
$$791$$ 0 0
$$792$$ 18.2111 0.647103
$$793$$ 9.78890 0.347614
$$794$$ 6.18335 0.219439
$$795$$ 45.6333 1.61845
$$796$$ −26.4222 −0.936510
$$797$$ −22.3305 −0.790988 −0.395494 0.918469i $$-0.629427\pi$$
−0.395494 + 0.918469i $$0.629427\pi$$
$$798$$ 0 0
$$799$$ −27.6333 −0.977596
$$800$$ −0.302776 −0.0107047
$$801$$ 72.8444 2.57383
$$802$$ 7.81665 0.276016
$$803$$ −28.3305 −0.999763
$$804$$ 11.6056 0.409296
$$805$$ 0 0
$$806$$ 0.394449 0.0138939
$$807$$ −70.0555 −2.46607
$$808$$ −12.4222 −0.437012
$$809$$ −35.4500 −1.24635 −0.623177 0.782081i $$-0.714158\pi$$
−0.623177 + 0.782081i $$0.714158\pi$$
$$810$$ −68.6611 −2.41250
$$811$$ 7.14719 0.250972 0.125486 0.992095i $$-0.459951\pi$$
0.125486 + 0.992095i $$0.459951\pi$$
$$812$$ 0 0
$$813$$ −74.0555 −2.59724
$$814$$ 2.30278 0.0807122
$$815$$ 19.3944 0.679358
$$816$$ −19.8167 −0.693722
$$817$$ −1.21110 −0.0423711
$$818$$ 31.0278 1.08486
$$819$$ 0 0
$$820$$ −22.8167 −0.796792
$$821$$ −3.21110 −0.112068 −0.0560341 0.998429i $$-0.517846\pi$$
−0.0560341 + 0.998429i $$0.517846\pi$$
$$822$$ 3.00000 0.104637
$$823$$ 44.8444 1.56318 0.781589 0.623794i $$-0.214410\pi$$
0.781589 + 0.623794i $$0.214410\pi$$
$$824$$ −0.302776 −0.0105477
$$825$$ 2.30278 0.0801724
$$826$$ 0 0
$$827$$ 34.6056 1.20335 0.601676 0.798740i $$-0.294500\pi$$
0.601676 + 0.798740i $$0.294500\pi$$
$$828$$ 30.9083 1.07414
$$829$$ 27.7250 0.962928 0.481464 0.876466i $$-0.340105\pi$$
0.481464 + 0.876466i $$0.340105\pi$$
$$830$$ 6.42221 0.222918
$$831$$ −0.394449 −0.0136833
$$832$$ −1.30278 −0.0451656
$$833$$ 0 0
$$834$$ 6.30278 0.218247
$$835$$ 12.6972 0.439406
$$836$$ 4.60555 0.159286
$$837$$ −4.90833 −0.169657
$$838$$ 36.1472 1.24868
$$839$$ 12.9722 0.447852 0.223926 0.974606i $$-0.428113\pi$$
0.223926 + 0.974606i $$0.428113\pi$$
$$840$$ 0 0
$$841$$ −13.7250 −0.473275
$$842$$ 3.72498 0.128371
$$843$$ 39.6333 1.36504
$$844$$ 10.3028 0.354636
$$845$$ −26.0278 −0.895382
$$846$$ 36.4222 1.25222
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 81.2666 2.78906
$$850$$ −1.81665 −0.0623107
$$851$$ 3.90833 0.133976
$$852$$ −19.8167 −0.678907
$$853$$ −42.5416 −1.45660 −0.728299 0.685260i $$-0.759689\pi$$
−0.728299 + 0.685260i $$0.759689\pi$$
$$854$$ 0 0
$$855$$ −36.4222 −1.24561
$$856$$ −0.697224 −0.0238306
$$857$$ −42.8444 −1.46354 −0.731769 0.681553i $$-0.761305\pi$$
−0.731769 + 0.681553i $$0.761305\pi$$
$$858$$ 9.90833 0.338265
$$859$$ −48.0555 −1.63963 −0.819816 0.572626i $$-0.805925\pi$$
−0.819816 + 0.572626i $$0.805925\pi$$
$$860$$ 1.39445 0.0475503
$$861$$ 0 0
$$862$$ −9.21110 −0.313731
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 16.2111 0.551513
$$865$$ 20.2389 0.688142
$$866$$ 34.9361 1.18718
$$867$$ −62.7527 −2.13119
$$868$$ 0 0
$$869$$ −21.0000 −0.712376
$$870$$ −29.7250 −1.00777
$$871$$ 4.57779 0.155113
$$872$$ −2.00000 −0.0677285
$$873$$ 129.872 4.39551
$$874$$ 7.81665 0.264402
$$875$$ 0 0
$$876$$ −40.6333 −1.37287
$$877$$ −7.21110 −0.243502 −0.121751 0.992561i $$-0.538851\pi$$
−0.121751 + 0.992561i $$0.538851\pi$$
$$878$$ 30.3305 1.02361
$$879$$ 36.4222 1.22849
$$880$$ −5.30278 −0.178757
$$881$$ 28.5416 0.961592 0.480796 0.876832i $$-0.340348\pi$$
0.480796 + 0.876832i $$0.340348\pi$$
$$882$$ 0 0
$$883$$ 26.4222 0.889178 0.444589 0.895735i $$-0.353350\pi$$
0.444589 + 0.895735i $$0.353350\pi$$
$$884$$ −7.81665 −0.262903
$$885$$ 80.6611 2.71139
$$886$$ −32.7250 −1.09942
$$887$$ 0.422205 0.0141763 0.00708813 0.999975i $$-0.497744\pi$$
0.00708813 + 0.999975i $$0.497744\pi$$
$$888$$ 3.30278 0.110834
$$889$$ 0 0
$$890$$ −21.2111 −0.710998
$$891$$ −68.6611 −2.30023
$$892$$ 5.81665 0.194756
$$893$$ 9.21110 0.308238
$$894$$ 65.4500 2.18897
$$895$$ −31.8167 −1.06351
$$896$$ 0 0
$$897$$ 16.8167 0.561492
$$898$$ 15.2111 0.507601
$$899$$ −1.18335 −0.0394668
$$900$$ 2.39445 0.0798150
$$901$$ −36.0000 −1.19933
$$902$$ −22.8167 −0.759711
$$903$$ 0 0
$$904$$ 3.21110 0.106800
$$905$$ −46.0555 −1.53094
$$906$$ −68.0555 −2.26099
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ −13.8167 −0.458522
$$909$$ 98.2389 3.25838
$$910$$ 0 0
$$911$$ −17.5778 −0.582378 −0.291189 0.956665i $$-0.594051\pi$$
−0.291189 + 0.956665i $$0.594051\pi$$
$$912$$ 6.60555 0.218732
$$913$$ 6.42221 0.212544
$$914$$ 2.60555 0.0861840
$$915$$ 57.1472 1.88923
$$916$$ −24.6056 −0.812990
$$917$$ 0 0
$$918$$ 97.2666 3.21028
$$919$$ −9.57779 −0.315942 −0.157971 0.987444i $$-0.550495\pi$$
−0.157971 + 0.987444i $$0.550495\pi$$
$$920$$ −9.00000 −0.296721
$$921$$ 59.1472 1.94897
$$922$$ 12.4222 0.409104
$$923$$ −7.81665 −0.257288
$$924$$ 0 0
$$925$$ 0.302776 0.00995520
$$926$$ −26.6972 −0.877325
$$927$$ 2.39445 0.0786440
$$928$$ 3.90833 0.128297
$$929$$ 18.4861 0.606510 0.303255 0.952909i $$-0.401927\pi$$
0.303255 + 0.952909i $$0.401927\pi$$
$$930$$ 2.30278 0.0755110
$$931$$ 0 0
$$932$$ 8.51388 0.278881
$$933$$ 52.5416 1.72014
$$934$$ 0 0
$$935$$ −31.8167 −1.04052
$$936$$ 10.3028 0.336757
$$937$$ 18.0917 0.591029 0.295515 0.955338i $$-0.404509\pi$$
0.295515 + 0.955338i $$0.404509\pi$$
$$938$$ 0 0
$$939$$ −29.8167 −0.973030
$$940$$ −10.6056 −0.345915
$$941$$ −13.8167 −0.450410 −0.225205 0.974311i $$-0.572305\pi$$
−0.225205 + 0.974311i $$0.572305\pi$$
$$942$$ 23.8167 0.775989
$$943$$ −38.7250 −1.26106
$$944$$ −10.6056 −0.345181
$$945$$ 0 0
$$946$$ 1.39445 0.0453374
$$947$$ −3.63331 −0.118067 −0.0590333 0.998256i $$-0.518802\pi$$
−0.0590333 + 0.998256i $$0.518802\pi$$
$$948$$ −30.1194 −0.978234
$$949$$ −16.0278 −0.520283
$$950$$ 0.605551 0.0196467
$$951$$ −30.4222 −0.986508
$$952$$ 0 0
$$953$$ 49.7527 1.61165 0.805825 0.592154i $$-0.201722\pi$$
0.805825 + 0.592154i $$0.201722\pi$$
$$954$$ 47.4500 1.53625
$$955$$ −12.6972 −0.410873
$$956$$ −17.5139 −0.566439
$$957$$ −29.7250 −0.960872
$$958$$ −13.1194 −0.423870
$$959$$ 0 0
$$960$$ −7.60555 −0.245468
$$961$$ −30.9083 −0.997043
$$962$$ 1.30278 0.0420032
$$963$$ 5.51388 0.177682
$$964$$ −8.00000 −0.257663
$$965$$ −9.21110 −0.296516
$$966$$ 0 0
$$967$$ −6.72498 −0.216261 −0.108130 0.994137i $$-0.534486\pi$$
−0.108130 + 0.994137i $$0.534486\pi$$
$$968$$ 5.69722 0.183116
$$969$$ 39.6333 1.27321
$$970$$ −37.8167 −1.21422
$$971$$ 22.5416 0.723395 0.361698 0.932295i $$-0.382197\pi$$
0.361698 + 0.932295i $$0.382197\pi$$
$$972$$ −49.8444 −1.59876
$$973$$ 0 0
$$974$$ 37.2111 1.19232
$$975$$ 1.30278 0.0417222
$$976$$ −7.51388 −0.240513
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 27.8167 0.889479
$$979$$ −21.2111 −0.677910
$$980$$ 0 0
$$981$$ 15.8167 0.504987
$$982$$ −17.7250 −0.565627
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ −32.7250 −1.04323
$$985$$ −13.8167 −0.440235
$$986$$ 23.4500 0.746799
$$987$$ 0 0
$$988$$ 2.60555 0.0828936
$$989$$ 2.36669 0.0752564
$$990$$ 41.9361 1.33282
$$991$$ 50.6972 1.61045 0.805225 0.592969i $$-0.202044\pi$$
0.805225 + 0.592969i $$0.202044\pi$$
$$992$$ −0.302776 −0.00961314
$$993$$ 43.6333 1.38466
$$994$$ 0 0
$$995$$ −60.8444 −1.92890
$$996$$ 9.21110 0.291865
$$997$$ 52.4222 1.66023 0.830114 0.557594i $$-0.188275\pi$$
0.830114 + 0.557594i $$0.188275\pi$$
$$998$$ 42.2389 1.33705
$$999$$ −16.2111 −0.512897
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 3626.2.a.a.1.1 2
7.6 odd 2 74.2.a.a.1.2 2
21.20 even 2 666.2.a.j.1.2 2
28.27 even 2 592.2.a.f.1.1 2
35.13 even 4 1850.2.b.i.149.4 4
35.27 even 4 1850.2.b.i.149.1 4
35.34 odd 2 1850.2.a.u.1.1 2
56.13 odd 2 2368.2.a.s.1.1 2
56.27 even 2 2368.2.a.ba.1.2 2
77.76 even 2 8954.2.a.p.1.2 2
84.83 odd 2 5328.2.a.bf.1.2 2
259.258 odd 2 2738.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 7.6 odd 2
592.2.a.f.1.1 2 28.27 even 2
666.2.a.j.1.2 2 21.20 even 2
1850.2.a.u.1.1 2 35.34 odd 2
1850.2.b.i.149.1 4 35.27 even 4
1850.2.b.i.149.4 4 35.13 even 4
2368.2.a.s.1.1 2 56.13 odd 2
2368.2.a.ba.1.2 2 56.27 even 2
2738.2.a.l.1.2 2 259.258 odd 2
3626.2.a.a.1.1 2 1.1 even 1 trivial
5328.2.a.bf.1.2 2 84.83 odd 2
8954.2.a.p.1.2 2 77.76 even 2