# Properties

 Label 2001.2.a.i.1.1 Level $2001$ Weight $2$ Character 2001.1 Self dual yes Analytic conductor $15.978$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9780654445$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.37349$$ of defining polynomial Character $$\chi$$ $$=$$ 2001.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.37349 q^{2} +1.00000 q^{3} +3.63344 q^{4} -1.31800 q^{5} -2.37349 q^{6} -0.859291 q^{7} -3.87694 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.37349 q^{2} +1.00000 q^{3} +3.63344 q^{4} -1.31800 q^{5} -2.37349 q^{6} -0.859291 q^{7} -3.87694 q^{8} +1.00000 q^{9} +3.12824 q^{10} +4.30847 q^{11} +3.63344 q^{12} -4.15542 q^{13} +2.03951 q^{14} -1.31800 q^{15} +1.93498 q^{16} +2.08920 q^{17} -2.37349 q^{18} -2.45615 q^{19} -4.78885 q^{20} -0.859291 q^{21} -10.2261 q^{22} +1.00000 q^{23} -3.87694 q^{24} -3.26289 q^{25} +9.86283 q^{26} +1.00000 q^{27} -3.12218 q^{28} -1.00000 q^{29} +3.12824 q^{30} -4.70746 q^{31} +3.16122 q^{32} +4.30847 q^{33} -4.95870 q^{34} +1.13254 q^{35} +3.63344 q^{36} -3.94232 q^{37} +5.82963 q^{38} -4.15542 q^{39} +5.10979 q^{40} -3.17442 q^{41} +2.03951 q^{42} +11.0153 q^{43} +15.6545 q^{44} -1.31800 q^{45} -2.37349 q^{46} +11.1103 q^{47} +1.93498 q^{48} -6.26162 q^{49} +7.74442 q^{50} +2.08920 q^{51} -15.0984 q^{52} -14.0165 q^{53} -2.37349 q^{54} -5.67854 q^{55} +3.33142 q^{56} -2.45615 q^{57} +2.37349 q^{58} +8.47222 q^{59} -4.78885 q^{60} -6.39249 q^{61} +11.1731 q^{62} -0.859291 q^{63} -11.3731 q^{64} +5.47682 q^{65} -10.2261 q^{66} +4.74715 q^{67} +7.59099 q^{68} +1.00000 q^{69} -2.68807 q^{70} -12.4909 q^{71} -3.87694 q^{72} +8.54106 q^{73} +9.35703 q^{74} -3.26289 q^{75} -8.92426 q^{76} -3.70223 q^{77} +9.86283 q^{78} -13.0092 q^{79} -2.55030 q^{80} +1.00000 q^{81} +7.53445 q^{82} -14.2415 q^{83} -3.12218 q^{84} -2.75356 q^{85} -26.1447 q^{86} -1.00000 q^{87} -16.7037 q^{88} +0.795808 q^{89} +3.12824 q^{90} +3.57071 q^{91} +3.63344 q^{92} -4.70746 q^{93} -26.3702 q^{94} +3.23719 q^{95} +3.16122 q^{96} +3.92369 q^{97} +14.8619 q^{98} +4.30847 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + O(q^{10})$$ $$7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + 3q^{10} - 4q^{11} + 5q^{12} - 18q^{13} - 2q^{14} - 3q^{15} - 7q^{16} - 3q^{17} - 3q^{18} - 4q^{19} - 2q^{20} - 5q^{21} - 26q^{22} + 7q^{23} - 6q^{24} - 8q^{25} - 7q^{26} + 7q^{27} - 6q^{28} - 7q^{29} + 3q^{30} - 22q^{31} + 5q^{32} - 4q^{33} + 9q^{34} + 3q^{35} + 5q^{36} - 25q^{37} + 14q^{38} - 18q^{39} - 10q^{40} - 13q^{41} - 2q^{42} - 2q^{43} + 4q^{44} - 3q^{45} - 3q^{46} - 25q^{47} - 7q^{48} - 8q^{49} + 19q^{50} - 3q^{51} - 12q^{52} - 5q^{53} - 3q^{54} - 15q^{55} + 18q^{56} - 4q^{57} + 3q^{58} + 11q^{59} - 2q^{60} - 33q^{61} + 28q^{62} - 5q^{63} - 14q^{64} - 2q^{65} - 26q^{66} + 8q^{67} + 12q^{68} + 7q^{69} - 22q^{70} - 6q^{71} - 6q^{72} + 15q^{73} + 34q^{74} - 8q^{75} - 28q^{76} - q^{77} - 7q^{78} - 15q^{79} - 12q^{80} + 7q^{81} - 14q^{82} + 21q^{83} - 6q^{84} - 28q^{85} - 12q^{86} - 7q^{87} - 13q^{88} + 8q^{89} + 3q^{90} + 6q^{91} + 5q^{92} - 22q^{93} - 35q^{94} - 25q^{95} + 5q^{96} + 13q^{97} + q^{98} - 4q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.37349 −1.67831 −0.839154 0.543894i $$-0.816949\pi$$
−0.839154 + 0.543894i $$0.816949\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 3.63344 1.81672
$$5$$ −1.31800 −0.589426 −0.294713 0.955586i $$-0.595224\pi$$
−0.294713 + 0.955586i $$0.595224\pi$$
$$6$$ −2.37349 −0.968972
$$7$$ −0.859291 −0.324781 −0.162391 0.986727i $$-0.551920\pi$$
−0.162391 + 0.986727i $$0.551920\pi$$
$$8$$ −3.87694 −1.37070
$$9$$ 1.00000 0.333333
$$10$$ 3.12824 0.989238
$$11$$ 4.30847 1.29905 0.649526 0.760339i $$-0.274967\pi$$
0.649526 + 0.760339i $$0.274967\pi$$
$$12$$ 3.63344 1.04888
$$13$$ −4.15542 −1.15251 −0.576253 0.817272i $$-0.695486\pi$$
−0.576253 + 0.817272i $$0.695486\pi$$
$$14$$ 2.03951 0.545083
$$15$$ −1.31800 −0.340305
$$16$$ 1.93498 0.483746
$$17$$ 2.08920 0.506706 0.253353 0.967374i $$-0.418467\pi$$
0.253353 + 0.967374i $$0.418467\pi$$
$$18$$ −2.37349 −0.559436
$$19$$ −2.45615 −0.563479 −0.281740 0.959491i $$-0.590911\pi$$
−0.281740 + 0.959491i $$0.590911\pi$$
$$20$$ −4.78885 −1.07082
$$21$$ −0.859291 −0.187513
$$22$$ −10.2261 −2.18021
$$23$$ 1.00000 0.208514
$$24$$ −3.87694 −0.791376
$$25$$ −3.26289 −0.652577
$$26$$ 9.86283 1.93426
$$27$$ 1.00000 0.192450
$$28$$ −3.12218 −0.590036
$$29$$ −1.00000 −0.185695
$$30$$ 3.12824 0.571137
$$31$$ −4.70746 −0.845484 −0.422742 0.906250i $$-0.638932\pi$$
−0.422742 + 0.906250i $$0.638932\pi$$
$$32$$ 3.16122 0.558830
$$33$$ 4.30847 0.750008
$$34$$ −4.95870 −0.850409
$$35$$ 1.13254 0.191434
$$36$$ 3.63344 0.605573
$$37$$ −3.94232 −0.648113 −0.324056 0.946038i $$-0.605047\pi$$
−0.324056 + 0.946038i $$0.605047\pi$$
$$38$$ 5.82963 0.945692
$$39$$ −4.15542 −0.665399
$$40$$ 5.10979 0.807928
$$41$$ −3.17442 −0.495762 −0.247881 0.968791i $$-0.579734\pi$$
−0.247881 + 0.968791i $$0.579734\pi$$
$$42$$ 2.03951 0.314704
$$43$$ 11.0153 1.67982 0.839909 0.542727i $$-0.182608\pi$$
0.839909 + 0.542727i $$0.182608\pi$$
$$44$$ 15.6545 2.36001
$$45$$ −1.31800 −0.196475
$$46$$ −2.37349 −0.349951
$$47$$ 11.1103 1.62061 0.810303 0.586012i $$-0.199303\pi$$
0.810303 + 0.586012i $$0.199303\pi$$
$$48$$ 1.93498 0.279291
$$49$$ −6.26162 −0.894517
$$50$$ 7.74442 1.09523
$$51$$ 2.08920 0.292547
$$52$$ −15.0984 −2.09378
$$53$$ −14.0165 −1.92531 −0.962657 0.270723i $$-0.912737\pi$$
−0.962657 + 0.270723i $$0.912737\pi$$
$$54$$ −2.37349 −0.322991
$$55$$ −5.67854 −0.765695
$$56$$ 3.33142 0.445179
$$57$$ −2.45615 −0.325325
$$58$$ 2.37349 0.311654
$$59$$ 8.47222 1.10299 0.551495 0.834178i $$-0.314058\pi$$
0.551495 + 0.834178i $$0.314058\pi$$
$$60$$ −4.78885 −0.618238
$$61$$ −6.39249 −0.818475 −0.409237 0.912428i $$-0.634205\pi$$
−0.409237 + 0.912428i $$0.634205\pi$$
$$62$$ 11.1731 1.41898
$$63$$ −0.859291 −0.108260
$$64$$ −11.3731 −1.42163
$$65$$ 5.47682 0.679316
$$66$$ −10.2261 −1.25874
$$67$$ 4.74715 0.579956 0.289978 0.957033i $$-0.406352\pi$$
0.289978 + 0.957033i $$0.406352\pi$$
$$68$$ 7.59099 0.920542
$$69$$ 1.00000 0.120386
$$70$$ −2.68807 −0.321286
$$71$$ −12.4909 −1.48240 −0.741201 0.671283i $$-0.765743\pi$$
−0.741201 + 0.671283i $$0.765743\pi$$
$$72$$ −3.87694 −0.456901
$$73$$ 8.54106 0.999655 0.499827 0.866125i $$-0.333397\pi$$
0.499827 + 0.866125i $$0.333397\pi$$
$$74$$ 9.35703 1.08773
$$75$$ −3.26289 −0.376766
$$76$$ −8.92426 −1.02368
$$77$$ −3.70223 −0.421908
$$78$$ 9.86283 1.11675
$$79$$ −13.0092 −1.46365 −0.731827 0.681491i $$-0.761332\pi$$
−0.731827 + 0.681491i $$0.761332\pi$$
$$80$$ −2.55030 −0.285132
$$81$$ 1.00000 0.111111
$$82$$ 7.53445 0.832041
$$83$$ −14.2415 −1.56321 −0.781606 0.623772i $$-0.785599\pi$$
−0.781606 + 0.623772i $$0.785599\pi$$
$$84$$ −3.12218 −0.340657
$$85$$ −2.75356 −0.298666
$$86$$ −26.1447 −2.81925
$$87$$ −1.00000 −0.107211
$$88$$ −16.7037 −1.78062
$$89$$ 0.795808 0.0843554 0.0421777 0.999110i $$-0.486570\pi$$
0.0421777 + 0.999110i $$0.486570\pi$$
$$90$$ 3.12824 0.329746
$$91$$ 3.57071 0.374312
$$92$$ 3.63344 0.378812
$$93$$ −4.70746 −0.488141
$$94$$ −26.3702 −2.71988
$$95$$ 3.23719 0.332129
$$96$$ 3.16122 0.322640
$$97$$ 3.92369 0.398390 0.199195 0.979960i $$-0.436167\pi$$
0.199195 + 0.979960i $$0.436167\pi$$
$$98$$ 14.8619 1.50128
$$99$$ 4.30847 0.433017
$$100$$ −11.8555 −1.18555
$$101$$ 3.90652 0.388713 0.194357 0.980931i $$-0.437738\pi$$
0.194357 + 0.980931i $$0.437738\pi$$
$$102$$ −4.95870 −0.490984
$$103$$ 1.68054 0.165589 0.0827944 0.996567i $$-0.473616\pi$$
0.0827944 + 0.996567i $$0.473616\pi$$
$$104$$ 16.1103 1.57974
$$105$$ 1.13254 0.110525
$$106$$ 33.2680 3.23127
$$107$$ 11.2977 1.09219 0.546096 0.837722i $$-0.316113\pi$$
0.546096 + 0.837722i $$0.316113\pi$$
$$108$$ 3.63344 0.349627
$$109$$ −17.5897 −1.68479 −0.842394 0.538863i $$-0.818854\pi$$
−0.842394 + 0.538863i $$0.818854\pi$$
$$110$$ 13.4779 1.28507
$$111$$ −3.94232 −0.374188
$$112$$ −1.66271 −0.157112
$$113$$ −3.98253 −0.374645 −0.187323 0.982298i $$-0.559981\pi$$
−0.187323 + 0.982298i $$0.559981\pi$$
$$114$$ 5.82963 0.545995
$$115$$ −1.31800 −0.122904
$$116$$ −3.63344 −0.337356
$$117$$ −4.15542 −0.384169
$$118$$ −20.1087 −1.85116
$$119$$ −1.79523 −0.164569
$$120$$ 5.10979 0.466458
$$121$$ 7.56290 0.687537
$$122$$ 15.1725 1.37365
$$123$$ −3.17442 −0.286228
$$124$$ −17.1042 −1.53601
$$125$$ 10.8905 0.974072
$$126$$ 2.03951 0.181694
$$127$$ 16.7386 1.48531 0.742654 0.669675i $$-0.233567\pi$$
0.742654 + 0.669675i $$0.233567\pi$$
$$128$$ 20.6714 1.82711
$$129$$ 11.0153 0.969844
$$130$$ −12.9992 −1.14010
$$131$$ −22.3630 −1.95386 −0.976930 0.213559i $$-0.931494\pi$$
−0.976930 + 0.213559i $$0.931494\pi$$
$$132$$ 15.6545 1.36255
$$133$$ 2.11055 0.183008
$$134$$ −11.2673 −0.973345
$$135$$ −1.31800 −0.113435
$$136$$ −8.09971 −0.694544
$$137$$ −12.8745 −1.09995 −0.549973 0.835182i $$-0.685362\pi$$
−0.549973 + 0.835182i $$0.685362\pi$$
$$138$$ −2.37349 −0.202045
$$139$$ 12.8067 1.08625 0.543123 0.839653i $$-0.317242\pi$$
0.543123 + 0.839653i $$0.317242\pi$$
$$140$$ 4.11502 0.347782
$$141$$ 11.1103 0.935657
$$142$$ 29.6471 2.48793
$$143$$ −17.9035 −1.49716
$$144$$ 1.93498 0.161249
$$145$$ 1.31800 0.109454
$$146$$ −20.2721 −1.67773
$$147$$ −6.26162 −0.516450
$$148$$ −14.3242 −1.17744
$$149$$ 11.7517 0.962740 0.481370 0.876518i $$-0.340139\pi$$
0.481370 + 0.876518i $$0.340139\pi$$
$$150$$ 7.74442 0.632329
$$151$$ −11.1096 −0.904086 −0.452043 0.891996i $$-0.649305\pi$$
−0.452043 + 0.891996i $$0.649305\pi$$
$$152$$ 9.52233 0.772363
$$153$$ 2.08920 0.168902
$$154$$ 8.78718 0.708091
$$155$$ 6.20441 0.498350
$$156$$ −15.0984 −1.20884
$$157$$ −19.4984 −1.55614 −0.778072 0.628175i $$-0.783802\pi$$
−0.778072 + 0.628175i $$0.783802\pi$$
$$158$$ 30.8772 2.45646
$$159$$ −14.0165 −1.11158
$$160$$ −4.16647 −0.329389
$$161$$ −0.859291 −0.0677216
$$162$$ −2.37349 −0.186479
$$163$$ 21.4125 1.67716 0.838579 0.544779i $$-0.183387\pi$$
0.838579 + 0.544779i $$0.183387\pi$$
$$164$$ −11.5341 −0.900659
$$165$$ −5.67854 −0.442074
$$166$$ 33.8021 2.62355
$$167$$ −3.76831 −0.291600 −0.145800 0.989314i $$-0.546576\pi$$
−0.145800 + 0.989314i $$0.546576\pi$$
$$168$$ 3.33142 0.257024
$$169$$ 4.26750 0.328269
$$170$$ 6.53554 0.501253
$$171$$ −2.45615 −0.187826
$$172$$ 40.0234 3.05176
$$173$$ 18.5637 1.41137 0.705686 0.708525i $$-0.250639\pi$$
0.705686 + 0.708525i $$0.250639\pi$$
$$174$$ 2.37349 0.179934
$$175$$ 2.80377 0.211945
$$176$$ 8.33681 0.628411
$$177$$ 8.47222 0.636811
$$178$$ −1.88884 −0.141574
$$179$$ −21.5082 −1.60760 −0.803798 0.594902i $$-0.797191\pi$$
−0.803798 + 0.594902i $$0.797191\pi$$
$$180$$ −4.78885 −0.356940
$$181$$ −12.8583 −0.955754 −0.477877 0.878427i $$-0.658593\pi$$
−0.477877 + 0.878427i $$0.658593\pi$$
$$182$$ −8.47503 −0.628211
$$183$$ −6.39249 −0.472547
$$184$$ −3.87694 −0.285812
$$185$$ 5.19596 0.382014
$$186$$ 11.1731 0.819250
$$187$$ 9.00127 0.658238
$$188$$ 40.3686 2.94418
$$189$$ −0.859291 −0.0625042
$$190$$ −7.68344 −0.557415
$$191$$ 1.67695 0.121340 0.0606700 0.998158i $$-0.480676\pi$$
0.0606700 + 0.998158i $$0.480676\pi$$
$$192$$ −11.3731 −0.820781
$$193$$ −13.8369 −0.996002 −0.498001 0.867177i $$-0.665932\pi$$
−0.498001 + 0.867177i $$0.665932\pi$$
$$194$$ −9.31282 −0.668622
$$195$$ 5.47682 0.392204
$$196$$ −22.7512 −1.62509
$$197$$ −27.8816 −1.98648 −0.993241 0.116074i $$-0.962969\pi$$
−0.993241 + 0.116074i $$0.962969\pi$$
$$198$$ −10.2261 −0.726737
$$199$$ −17.7635 −1.25922 −0.629609 0.776912i $$-0.716785\pi$$
−0.629609 + 0.776912i $$0.716785\pi$$
$$200$$ 12.6500 0.894490
$$201$$ 4.74715 0.334838
$$202$$ −9.27207 −0.652380
$$203$$ 0.859291 0.0603104
$$204$$ 7.59099 0.531475
$$205$$ 4.18388 0.292215
$$206$$ −3.98875 −0.277909
$$207$$ 1.00000 0.0695048
$$208$$ −8.04066 −0.557520
$$209$$ −10.5822 −0.731989
$$210$$ −2.68807 −0.185495
$$211$$ −21.7285 −1.49585 −0.747926 0.663783i $$-0.768950\pi$$
−0.747926 + 0.663783i $$0.768950\pi$$
$$212$$ −50.9281 −3.49775
$$213$$ −12.4909 −0.855865
$$214$$ −26.8150 −1.83304
$$215$$ −14.5181 −0.990128
$$216$$ −3.87694 −0.263792
$$217$$ 4.04507 0.274598
$$218$$ 41.7489 2.82759
$$219$$ 8.54106 0.577151
$$220$$ −20.6326 −1.39105
$$221$$ −8.68152 −0.583982
$$222$$ 9.35703 0.628003
$$223$$ −0.162927 −0.0109104 −0.00545520 0.999985i $$-0.501736\pi$$
−0.00545520 + 0.999985i $$0.501736\pi$$
$$224$$ −2.71641 −0.181497
$$225$$ −3.26289 −0.217526
$$226$$ 9.45248 0.628770
$$227$$ −14.9422 −0.991752 −0.495876 0.868393i $$-0.665153\pi$$
−0.495876 + 0.868393i $$0.665153\pi$$
$$228$$ −8.92426 −0.591023
$$229$$ −9.10145 −0.601441 −0.300720 0.953712i $$-0.597227\pi$$
−0.300720 + 0.953712i $$0.597227\pi$$
$$230$$ 3.12824 0.206270
$$231$$ −3.70223 −0.243589
$$232$$ 3.87694 0.254533
$$233$$ −12.6237 −0.827007 −0.413504 0.910502i $$-0.635695\pi$$
−0.413504 + 0.910502i $$0.635695\pi$$
$$234$$ 9.86283 0.644753
$$235$$ −14.6433 −0.955227
$$236$$ 30.7833 2.00382
$$237$$ −13.0092 −0.845041
$$238$$ 4.26096 0.276197
$$239$$ 16.8514 1.09003 0.545013 0.838428i $$-0.316525\pi$$
0.545013 + 0.838428i $$0.316525\pi$$
$$240$$ −2.55030 −0.164621
$$241$$ −17.4282 −1.12265 −0.561323 0.827597i $$-0.689708\pi$$
−0.561323 + 0.827597i $$0.689708\pi$$
$$242$$ −17.9504 −1.15390
$$243$$ 1.00000 0.0641500
$$244$$ −23.2267 −1.48694
$$245$$ 8.25279 0.527251
$$246$$ 7.53445 0.480379
$$247$$ 10.2063 0.649413
$$248$$ 18.2505 1.15891
$$249$$ −14.2415 −0.902521
$$250$$ −25.8483 −1.63479
$$251$$ 10.2885 0.649407 0.324703 0.945816i $$-0.394736\pi$$
0.324703 + 0.945816i $$0.394736\pi$$
$$252$$ −3.12218 −0.196679
$$253$$ 4.30847 0.270871
$$254$$ −39.7288 −2.49280
$$255$$ −2.75356 −0.172435
$$256$$ −26.3171 −1.64482
$$257$$ −5.27151 −0.328827 −0.164414 0.986391i $$-0.552573\pi$$
−0.164414 + 0.986391i $$0.552573\pi$$
$$258$$ −26.1447 −1.62770
$$259$$ 3.38760 0.210495
$$260$$ 19.8997 1.23413
$$261$$ −1.00000 −0.0618984
$$262$$ 53.0782 3.27918
$$263$$ −4.72941 −0.291628 −0.145814 0.989312i $$-0.546580\pi$$
−0.145814 + 0.989312i $$0.546580\pi$$
$$264$$ −16.7037 −1.02804
$$265$$ 18.4737 1.13483
$$266$$ −5.00935 −0.307143
$$267$$ 0.795808 0.0487026
$$268$$ 17.2485 1.05362
$$269$$ −21.2129 −1.29338 −0.646688 0.762754i $$-0.723846\pi$$
−0.646688 + 0.762754i $$0.723846\pi$$
$$270$$ 3.12824 0.190379
$$271$$ −11.0667 −0.672253 −0.336127 0.941817i $$-0.609117\pi$$
−0.336127 + 0.941817i $$0.609117\pi$$
$$272$$ 4.04257 0.245117
$$273$$ 3.57071 0.216109
$$274$$ 30.5575 1.84605
$$275$$ −14.0580 −0.847732
$$276$$ 3.63344 0.218707
$$277$$ −7.98577 −0.479818 −0.239909 0.970795i $$-0.577118\pi$$
−0.239909 + 0.970795i $$0.577118\pi$$
$$278$$ −30.3964 −1.82306
$$279$$ −4.70746 −0.281828
$$280$$ −4.39079 −0.262400
$$281$$ 6.63114 0.395581 0.197790 0.980244i $$-0.436623\pi$$
0.197790 + 0.980244i $$0.436623\pi$$
$$282$$ −26.3702 −1.57032
$$283$$ 2.98274 0.177306 0.0886528 0.996063i $$-0.471744\pi$$
0.0886528 + 0.996063i $$0.471744\pi$$
$$284$$ −45.3850 −2.69311
$$285$$ 3.23719 0.191755
$$286$$ 42.4937 2.51270
$$287$$ 2.72775 0.161014
$$288$$ 3.16122 0.186277
$$289$$ −12.6352 −0.743249
$$290$$ −3.12824 −0.183697
$$291$$ 3.92369 0.230011
$$292$$ 31.0334 1.81609
$$293$$ 13.4117 0.783521 0.391760 0.920067i $$-0.371866\pi$$
0.391760 + 0.920067i $$0.371866\pi$$
$$294$$ 14.8619 0.866762
$$295$$ −11.1664 −0.650130
$$296$$ 15.2841 0.888371
$$297$$ 4.30847 0.250003
$$298$$ −27.8926 −1.61577
$$299$$ −4.15542 −0.240314
$$300$$ −11.8555 −0.684477
$$301$$ −9.46535 −0.545574
$$302$$ 26.3685 1.51734
$$303$$ 3.90652 0.224424
$$304$$ −4.75261 −0.272581
$$305$$ 8.42528 0.482430
$$306$$ −4.95870 −0.283470
$$307$$ −7.23125 −0.412709 −0.206355 0.978477i $$-0.566160\pi$$
−0.206355 + 0.978477i $$0.566160\pi$$
$$308$$ −13.4518 −0.766488
$$309$$ 1.68054 0.0956028
$$310$$ −14.7261 −0.836385
$$311$$ 0.648787 0.0367894 0.0183947 0.999831i $$-0.494144\pi$$
0.0183947 + 0.999831i $$0.494144\pi$$
$$312$$ 16.1103 0.912066
$$313$$ 18.3081 1.03483 0.517416 0.855734i $$-0.326894\pi$$
0.517416 + 0.855734i $$0.326894\pi$$
$$314$$ 46.2793 2.61169
$$315$$ 1.13254 0.0638115
$$316$$ −47.2682 −2.65905
$$317$$ −21.4173 −1.20292 −0.601458 0.798904i $$-0.705413\pi$$
−0.601458 + 0.798904i $$0.705413\pi$$
$$318$$ 33.2680 1.86558
$$319$$ −4.30847 −0.241228
$$320$$ 14.9897 0.837948
$$321$$ 11.2977 0.630578
$$322$$ 2.03951 0.113658
$$323$$ −5.13140 −0.285518
$$324$$ 3.63344 0.201858
$$325$$ 13.5587 0.752099
$$326$$ −50.8223 −2.81479
$$327$$ −17.5897 −0.972712
$$328$$ 12.3070 0.679542
$$329$$ −9.54699 −0.526342
$$330$$ 13.4779 0.741936
$$331$$ 2.79629 0.153698 0.0768490 0.997043i $$-0.475514\pi$$
0.0768490 + 0.997043i $$0.475514\pi$$
$$332$$ −51.7457 −2.83992
$$333$$ −3.94232 −0.216038
$$334$$ 8.94403 0.489395
$$335$$ −6.25672 −0.341841
$$336$$ −1.66271 −0.0907084
$$337$$ −31.0998 −1.69411 −0.847057 0.531501i $$-0.821628\pi$$
−0.847057 + 0.531501i $$0.821628\pi$$
$$338$$ −10.1288 −0.550937
$$339$$ −3.98253 −0.216301
$$340$$ −10.0049 −0.542591
$$341$$ −20.2819 −1.09833
$$342$$ 5.82963 0.315231
$$343$$ 11.3956 0.615304
$$344$$ −42.7056 −2.30253
$$345$$ −1.31800 −0.0709585
$$346$$ −44.0607 −2.36872
$$347$$ 34.9752 1.87757 0.938785 0.344504i $$-0.111953\pi$$
0.938785 + 0.344504i $$0.111953\pi$$
$$348$$ −3.63344 −0.194773
$$349$$ 28.8052 1.54191 0.770953 0.636892i $$-0.219780\pi$$
0.770953 + 0.636892i $$0.219780\pi$$
$$350$$ −6.65470 −0.355709
$$351$$ −4.15542 −0.221800
$$352$$ 13.6200 0.725949
$$353$$ −11.5890 −0.616819 −0.308409 0.951254i $$-0.599797\pi$$
−0.308409 + 0.951254i $$0.599797\pi$$
$$354$$ −20.1087 −1.06877
$$355$$ 16.4630 0.873766
$$356$$ 2.89152 0.153250
$$357$$ −1.79523 −0.0950138
$$358$$ 51.0493 2.69804
$$359$$ 35.7439 1.88649 0.943246 0.332095i $$-0.107755\pi$$
0.943246 + 0.332095i $$0.107755\pi$$
$$360$$ 5.10979 0.269309
$$361$$ −12.9673 −0.682491
$$362$$ 30.5191 1.60405
$$363$$ 7.56290 0.396949
$$364$$ 12.9740 0.680020
$$365$$ −11.2571 −0.589222
$$366$$ 15.1725 0.793079
$$367$$ −27.4350 −1.43209 −0.716046 0.698053i $$-0.754050\pi$$
−0.716046 + 0.698053i $$0.754050\pi$$
$$368$$ 1.93498 0.100868
$$369$$ −3.17442 −0.165254
$$370$$ −12.3325 −0.641138
$$371$$ 12.0443 0.625306
$$372$$ −17.1042 −0.886814
$$373$$ 13.3266 0.690027 0.345013 0.938598i $$-0.387874\pi$$
0.345013 + 0.938598i $$0.387874\pi$$
$$374$$ −21.3644 −1.10473
$$375$$ 10.8905 0.562380
$$376$$ −43.0740 −2.22137
$$377$$ 4.15542 0.214015
$$378$$ 2.03951 0.104901
$$379$$ −5.84883 −0.300434 −0.150217 0.988653i $$-0.547997\pi$$
−0.150217 + 0.988653i $$0.547997\pi$$
$$380$$ 11.7621 0.603385
$$381$$ 16.7386 0.857543
$$382$$ −3.98022 −0.203646
$$383$$ 13.9417 0.712388 0.356194 0.934412i $$-0.384074\pi$$
0.356194 + 0.934412i $$0.384074\pi$$
$$384$$ 20.6714 1.05488
$$385$$ 4.87952 0.248683
$$386$$ 32.8417 1.67160
$$387$$ 11.0153 0.559940
$$388$$ 14.2565 0.723763
$$389$$ 24.2834 1.23122 0.615608 0.788052i $$-0.288910\pi$$
0.615608 + 0.788052i $$0.288910\pi$$
$$390$$ −12.9992 −0.658238
$$391$$ 2.08920 0.105656
$$392$$ 24.2759 1.22612
$$393$$ −22.3630 −1.12806
$$394$$ 66.1766 3.33393
$$395$$ 17.1461 0.862715
$$396$$ 15.6545 0.786670
$$397$$ −33.5680 −1.68473 −0.842364 0.538909i $$-0.818837\pi$$
−0.842364 + 0.538909i $$0.818837\pi$$
$$398$$ 42.1613 2.11336
$$399$$ 2.11055 0.105659
$$400$$ −6.31363 −0.315681
$$401$$ 16.7895 0.838426 0.419213 0.907888i $$-0.362306\pi$$
0.419213 + 0.907888i $$0.362306\pi$$
$$402$$ −11.2673 −0.561961
$$403$$ 19.5615 0.974425
$$404$$ 14.1941 0.706182
$$405$$ −1.31800 −0.0654917
$$406$$ −2.03951 −0.101219
$$407$$ −16.9854 −0.841933
$$408$$ −8.09971 −0.400995
$$409$$ 33.9540 1.67892 0.839458 0.543424i $$-0.182872\pi$$
0.839458 + 0.543424i $$0.182872\pi$$
$$410$$ −9.93037 −0.490426
$$411$$ −12.8745 −0.635054
$$412$$ 6.10615 0.300828
$$413$$ −7.28010 −0.358230
$$414$$ −2.37349 −0.116650
$$415$$ 18.7703 0.921398
$$416$$ −13.1362 −0.644054
$$417$$ 12.8067 0.627145
$$418$$ 25.1168 1.22850
$$419$$ 1.25305 0.0612157 0.0306079 0.999531i $$-0.490256\pi$$
0.0306079 + 0.999531i $$0.490256\pi$$
$$420$$ 4.11502 0.200792
$$421$$ 19.7566 0.962880 0.481440 0.876479i $$-0.340114\pi$$
0.481440 + 0.876479i $$0.340114\pi$$
$$422$$ 51.5723 2.51050
$$423$$ 11.1103 0.540202
$$424$$ 54.3411 2.63904
$$425$$ −6.81684 −0.330665
$$426$$ 29.6471 1.43640
$$427$$ 5.49301 0.265825
$$428$$ 41.0496 1.98421
$$429$$ −17.9035 −0.864389
$$430$$ 34.4586 1.66174
$$431$$ −28.0855 −1.35283 −0.676416 0.736520i $$-0.736468\pi$$
−0.676416 + 0.736520i $$0.736468\pi$$
$$432$$ 1.93498 0.0930969
$$433$$ −37.9606 −1.82427 −0.912135 0.409890i $$-0.865567\pi$$
−0.912135 + 0.409890i $$0.865567\pi$$
$$434$$ −9.60093 −0.460859
$$435$$ 1.31800 0.0631931
$$436$$ −63.9110 −3.06078
$$437$$ −2.45615 −0.117494
$$438$$ −20.2721 −0.968637
$$439$$ 33.0666 1.57818 0.789091 0.614277i $$-0.210552\pi$$
0.789091 + 0.614277i $$0.210552\pi$$
$$440$$ 22.0154 1.04954
$$441$$ −6.26162 −0.298172
$$442$$ 20.6055 0.980101
$$443$$ −4.67818 −0.222267 −0.111134 0.993805i $$-0.535448\pi$$
−0.111134 + 0.993805i $$0.535448\pi$$
$$444$$ −14.3242 −0.679794
$$445$$ −1.04887 −0.0497213
$$446$$ 0.386705 0.0183110
$$447$$ 11.7517 0.555838
$$448$$ 9.77278 0.461720
$$449$$ 0.0578209 0.00272873 0.00136437 0.999999i $$-0.499566\pi$$
0.00136437 + 0.999999i $$0.499566\pi$$
$$450$$ 7.74442 0.365075
$$451$$ −13.6769 −0.644020
$$452$$ −14.4703 −0.680624
$$453$$ −11.1096 −0.521974
$$454$$ 35.4652 1.66446
$$455$$ −4.70618 −0.220629
$$456$$ 9.52233 0.445924
$$457$$ −9.68884 −0.453225 −0.226612 0.973985i $$-0.572765\pi$$
−0.226612 + 0.973985i $$0.572765\pi$$
$$458$$ 21.6022 1.00940
$$459$$ 2.08920 0.0975157
$$460$$ −4.78885 −0.223281
$$461$$ 13.9639 0.650365 0.325182 0.945651i $$-0.394574\pi$$
0.325182 + 0.945651i $$0.394574\pi$$
$$462$$ 8.78718 0.408817
$$463$$ 31.3995 1.45926 0.729629 0.683844i $$-0.239693\pi$$
0.729629 + 0.683844i $$0.239693\pi$$
$$464$$ −1.93498 −0.0898293
$$465$$ 6.20441 0.287723
$$466$$ 29.9622 1.38797
$$467$$ −20.9278 −0.968422 −0.484211 0.874951i $$-0.660893\pi$$
−0.484211 + 0.874951i $$0.660893\pi$$
$$468$$ −15.0984 −0.697926
$$469$$ −4.07918 −0.188359
$$470$$ 34.7558 1.60316
$$471$$ −19.4984 −0.898440
$$472$$ −32.8463 −1.51187
$$473$$ 47.4591 2.18217
$$474$$ 30.8772 1.41824
$$475$$ 8.01414 0.367714
$$476$$ −6.52286 −0.298975
$$477$$ −14.0165 −0.641772
$$478$$ −39.9965 −1.82940
$$479$$ 41.1688 1.88105 0.940525 0.339724i $$-0.110334\pi$$
0.940525 + 0.339724i $$0.110334\pi$$
$$480$$ −4.16647 −0.190173
$$481$$ 16.3820 0.746954
$$482$$ 41.3655 1.88415
$$483$$ −0.859291 −0.0390991
$$484$$ 27.4793 1.24906
$$485$$ −5.17141 −0.234821
$$486$$ −2.37349 −0.107664
$$487$$ −16.7097 −0.757190 −0.378595 0.925562i $$-0.623593\pi$$
−0.378595 + 0.925562i $$0.623593\pi$$
$$488$$ 24.7833 1.12189
$$489$$ 21.4125 0.968308
$$490$$ −19.5879 −0.884890
$$491$$ 18.7578 0.846526 0.423263 0.906007i $$-0.360885\pi$$
0.423263 + 0.906007i $$0.360885\pi$$
$$492$$ −11.5341 −0.519996
$$493$$ −2.08920 −0.0940930
$$494$$ −24.2246 −1.08991
$$495$$ −5.67854 −0.255232
$$496$$ −9.10885 −0.408999
$$497$$ 10.7333 0.481456
$$498$$ 33.8021 1.51471
$$499$$ 4.60344 0.206078 0.103039 0.994677i $$-0.467143\pi$$
0.103039 + 0.994677i $$0.467143\pi$$
$$500$$ 39.5698 1.76961
$$501$$ −3.76831 −0.168356
$$502$$ −24.4197 −1.08990
$$503$$ −10.6369 −0.474276 −0.237138 0.971476i $$-0.576209\pi$$
−0.237138 + 0.971476i $$0.576209\pi$$
$$504$$ 3.33142 0.148393
$$505$$ −5.14878 −0.229117
$$506$$ −10.2261 −0.454605
$$507$$ 4.26750 0.189526
$$508$$ 60.8185 2.69839
$$509$$ 12.0067 0.532186 0.266093 0.963947i $$-0.414267\pi$$
0.266093 + 0.963947i $$0.414267\pi$$
$$510$$ 6.53554 0.289399
$$511$$ −7.33925 −0.324669
$$512$$ 21.1205 0.933404
$$513$$ −2.45615 −0.108442
$$514$$ 12.5118 0.551874
$$515$$ −2.21495 −0.0976023
$$516$$ 40.0234 1.76193
$$517$$ 47.8684 2.10525
$$518$$ −8.04041 −0.353275
$$519$$ 18.5637 0.814856
$$520$$ −21.2333 −0.931142
$$521$$ 11.2329 0.492123 0.246061 0.969254i $$-0.420864\pi$$
0.246061 + 0.969254i $$0.420864\pi$$
$$522$$ 2.37349 0.103885
$$523$$ −30.6342 −1.33954 −0.669770 0.742569i $$-0.733607\pi$$
−0.669770 + 0.742569i $$0.733607\pi$$
$$524$$ −81.2544 −3.54961
$$525$$ 2.80377 0.122366
$$526$$ 11.2252 0.489442
$$527$$ −9.83484 −0.428412
$$528$$ 8.33681 0.362813
$$529$$ 1.00000 0.0434783
$$530$$ −43.8471 −1.90459
$$531$$ 8.47222 0.367663
$$532$$ 7.66853 0.332473
$$533$$ 13.1911 0.571368
$$534$$ −1.88884 −0.0817380
$$535$$ −14.8904 −0.643766
$$536$$ −18.4044 −0.794948
$$537$$ −21.5082 −0.928146
$$538$$ 50.3486 2.17068
$$539$$ −26.9780 −1.16202
$$540$$ −4.78885 −0.206079
$$541$$ −24.1023 −1.03624 −0.518119 0.855308i $$-0.673368\pi$$
−0.518119 + 0.855308i $$0.673368\pi$$
$$542$$ 26.2666 1.12825
$$543$$ −12.8583 −0.551805
$$544$$ 6.60443 0.283163
$$545$$ 23.1831 0.993057
$$546$$ −8.47503 −0.362698
$$547$$ 13.8709 0.593075 0.296538 0.955021i $$-0.404168\pi$$
0.296538 + 0.955021i $$0.404168\pi$$
$$548$$ −46.7788 −1.99829
$$549$$ −6.39249 −0.272825
$$550$$ 33.3666 1.42276
$$551$$ 2.45615 0.104635
$$552$$ −3.87694 −0.165013
$$553$$ 11.1787 0.475367
$$554$$ 18.9541 0.805283
$$555$$ 5.19596 0.220556
$$556$$ 46.5322 1.97340
$$557$$ 33.6906 1.42752 0.713758 0.700392i $$-0.246991\pi$$
0.713758 + 0.700392i $$0.246991\pi$$
$$558$$ 11.1731 0.472994
$$559$$ −45.7732 −1.93600
$$560$$ 2.19145 0.0926056
$$561$$ 9.00127 0.380034
$$562$$ −15.7389 −0.663906
$$563$$ 41.3582 1.74304 0.871520 0.490360i $$-0.163135\pi$$
0.871520 + 0.490360i $$0.163135\pi$$
$$564$$ 40.3686 1.69982
$$565$$ 5.24896 0.220825
$$566$$ −7.07949 −0.297573
$$567$$ −0.859291 −0.0360868
$$568$$ 48.4266 2.03193
$$569$$ 14.0644 0.589612 0.294806 0.955557i $$-0.404745\pi$$
0.294806 + 0.955557i $$0.404745\pi$$
$$570$$ −7.68344 −0.321824
$$571$$ −5.85298 −0.244940 −0.122470 0.992472i $$-0.539081\pi$$
−0.122470 + 0.992472i $$0.539081\pi$$
$$572$$ −65.0512 −2.71993
$$573$$ 1.67695 0.0700556
$$574$$ −6.47428 −0.270231
$$575$$ −3.26289 −0.136072
$$576$$ −11.3731 −0.473878
$$577$$ −27.3870 −1.14014 −0.570069 0.821597i $$-0.693083\pi$$
−0.570069 + 0.821597i $$0.693083\pi$$
$$578$$ 29.9895 1.24740
$$579$$ −13.8369 −0.575042
$$580$$ 4.78885 0.198846
$$581$$ 12.2376 0.507702
$$582$$ −9.31282 −0.386029
$$583$$ −60.3897 −2.50108
$$584$$ −33.1131 −1.37023
$$585$$ 5.47682 0.226439
$$586$$ −31.8325 −1.31499
$$587$$ −45.3687 −1.87257 −0.936284 0.351245i $$-0.885758\pi$$
−0.936284 + 0.351245i $$0.885758\pi$$
$$588$$ −22.7512 −0.938243
$$589$$ 11.5622 0.476413
$$590$$ 26.5032 1.09112
$$591$$ −27.8816 −1.14690
$$592$$ −7.62832 −0.313522
$$593$$ 17.9412 0.736756 0.368378 0.929676i $$-0.379913\pi$$
0.368378 + 0.929676i $$0.379913\pi$$
$$594$$ −10.2261 −0.419582
$$595$$ 2.36611 0.0970011
$$596$$ 42.6992 1.74903
$$597$$ −17.7635 −0.727010
$$598$$ 9.86283 0.403321
$$599$$ −38.0747 −1.55569 −0.777845 0.628456i $$-0.783687\pi$$
−0.777845 + 0.628456i $$0.783687\pi$$
$$600$$ 12.6500 0.516434
$$601$$ −25.1262 −1.02492 −0.512461 0.858711i $$-0.671266\pi$$
−0.512461 + 0.858711i $$0.671266\pi$$
$$602$$ 22.4659 0.915641
$$603$$ 4.74715 0.193319
$$604$$ −40.3660 −1.64247
$$605$$ −9.96787 −0.405252
$$606$$ −9.27207 −0.376652
$$607$$ 19.6470 0.797447 0.398723 0.917071i $$-0.369453\pi$$
0.398723 + 0.917071i $$0.369453\pi$$
$$608$$ −7.76442 −0.314889
$$609$$ 0.859291 0.0348202
$$610$$ −19.9973 −0.809666
$$611$$ −46.1680 −1.86776
$$612$$ 7.59099 0.306847
$$613$$ 17.5314 0.708086 0.354043 0.935229i $$-0.384807\pi$$
0.354043 + 0.935229i $$0.384807\pi$$
$$614$$ 17.1633 0.692653
$$615$$ 4.18388 0.168710
$$616$$ 14.3533 0.578311
$$617$$ 28.1894 1.13486 0.567430 0.823421i $$-0.307938\pi$$
0.567430 + 0.823421i $$0.307938\pi$$
$$618$$ −3.98875 −0.160451
$$619$$ 12.6807 0.509680 0.254840 0.966983i $$-0.417977\pi$$
0.254840 + 0.966983i $$0.417977\pi$$
$$620$$ 22.5433 0.905362
$$621$$ 1.00000 0.0401286
$$622$$ −1.53989 −0.0617439
$$623$$ −0.683830 −0.0273971
$$624$$ −8.04066 −0.321884
$$625$$ 1.96086 0.0784345
$$626$$ −43.4539 −1.73677
$$627$$ −10.5822 −0.422614
$$628$$ −70.8463 −2.82708
$$629$$ −8.23630 −0.328403
$$630$$ −2.68807 −0.107095
$$631$$ 24.4176 0.972050 0.486025 0.873945i $$-0.338446\pi$$
0.486025 + 0.873945i $$0.338446\pi$$
$$632$$ 50.4360 2.00624
$$633$$ −21.7285 −0.863630
$$634$$ 50.8337 2.01886
$$635$$ −22.0614 −0.875479
$$636$$ −50.9281 −2.01943
$$637$$ 26.0196 1.03094
$$638$$ 10.2261 0.404855
$$639$$ −12.4909 −0.494134
$$640$$ −27.2448 −1.07695
$$641$$ −39.9966 −1.57977 −0.789886 0.613254i $$-0.789860\pi$$
−0.789886 + 0.613254i $$0.789860\pi$$
$$642$$ −26.8150 −1.05830
$$643$$ −21.9596 −0.866002 −0.433001 0.901393i $$-0.642545\pi$$
−0.433001 + 0.901393i $$0.642545\pi$$
$$644$$ −3.12218 −0.123031
$$645$$ −14.5181 −0.571651
$$646$$ 12.1793 0.479188
$$647$$ 5.75236 0.226149 0.113074 0.993587i $$-0.463930\pi$$
0.113074 + 0.993587i $$0.463930\pi$$
$$648$$ −3.87694 −0.152300
$$649$$ 36.5023 1.43284
$$650$$ −32.1813 −1.26225
$$651$$ 4.04507 0.158539
$$652$$ 77.8010 3.04692
$$653$$ −16.3800 −0.640998 −0.320499 0.947249i $$-0.603851\pi$$
−0.320499 + 0.947249i $$0.603851\pi$$
$$654$$ 41.7489 1.63251
$$655$$ 29.4743 1.15166
$$656$$ −6.14245 −0.239823
$$657$$ 8.54106 0.333218
$$658$$ 22.6596 0.883365
$$659$$ −7.33501 −0.285731 −0.142866 0.989742i $$-0.545632\pi$$
−0.142866 + 0.989742i $$0.545632\pi$$
$$660$$ −20.6326 −0.803124
$$661$$ 23.2593 0.904681 0.452340 0.891845i $$-0.350589\pi$$
0.452340 + 0.891845i $$0.350589\pi$$
$$662$$ −6.63695 −0.257953
$$663$$ −8.68152 −0.337162
$$664$$ 55.2136 2.14270
$$665$$ −2.78169 −0.107869
$$666$$ 9.35703 0.362578
$$667$$ −1.00000 −0.0387202
$$668$$ −13.6919 −0.529756
$$669$$ −0.162927 −0.00629913
$$670$$ 14.8502 0.573715
$$671$$ −27.5418 −1.06324
$$672$$ −2.71641 −0.104788
$$673$$ 28.8875 1.11353 0.556766 0.830670i $$-0.312042\pi$$
0.556766 + 0.830670i $$0.312042\pi$$
$$674$$ 73.8150 2.84325
$$675$$ −3.26289 −0.125589
$$676$$ 15.5057 0.596372
$$677$$ 27.0586 1.03995 0.519974 0.854182i $$-0.325942\pi$$
0.519974 + 0.854182i $$0.325942\pi$$
$$678$$ 9.45248 0.363020
$$679$$ −3.37159 −0.129390
$$680$$ 10.6754 0.409382
$$681$$ −14.9422 −0.572588
$$682$$ 48.1389 1.84333
$$683$$ 33.2775 1.27333 0.636664 0.771141i $$-0.280314\pi$$
0.636664 + 0.771141i $$0.280314\pi$$
$$684$$ −8.92426 −0.341228
$$685$$ 16.9686 0.648337
$$686$$ −27.0473 −1.03267
$$687$$ −9.10145 −0.347242
$$688$$ 21.3144 0.812605
$$689$$ 58.2444 2.21894
$$690$$ 3.12824 0.119090
$$691$$ 13.0933 0.498093 0.249046 0.968492i $$-0.419883\pi$$
0.249046 + 0.968492i $$0.419883\pi$$
$$692$$ 67.4500 2.56406
$$693$$ −3.70223 −0.140636
$$694$$ −83.0133 −3.15114
$$695$$ −16.8791 −0.640262
$$696$$ 3.87694 0.146955
$$697$$ −6.63202 −0.251206
$$698$$ −68.3687 −2.58779
$$699$$ −12.6237 −0.477473
$$700$$ 10.1873 0.385044
$$701$$ −5.48506 −0.207168 −0.103584 0.994621i $$-0.533031\pi$$
−0.103584 + 0.994621i $$0.533031\pi$$
$$702$$ 9.86283 0.372248
$$703$$ 9.68292 0.365198
$$704$$ −49.0005 −1.84678
$$705$$ −14.6433 −0.551500
$$706$$ 27.5063 1.03521
$$707$$ −3.35683 −0.126247
$$708$$ 30.7833 1.15691
$$709$$ 9.81311 0.368539 0.184270 0.982876i $$-0.441008\pi$$
0.184270 + 0.982876i $$0.441008\pi$$
$$710$$ −39.0747 −1.46645
$$711$$ −13.0092 −0.487885
$$712$$ −3.08530 −0.115626
$$713$$ −4.70746 −0.176296
$$714$$ 4.26096 0.159462
$$715$$ 23.5967 0.882467
$$716$$ −78.1485 −2.92055
$$717$$ 16.8514 0.629327
$$718$$ −84.8377 −3.16611
$$719$$ −24.9155 −0.929193 −0.464596 0.885522i $$-0.653801\pi$$
−0.464596 + 0.885522i $$0.653801\pi$$
$$720$$ −2.55030 −0.0950440
$$721$$ −1.44408 −0.0537802
$$722$$ 30.7778 1.14543
$$723$$ −17.4282 −0.648160
$$724$$ −46.7200 −1.73633
$$725$$ 3.26289 0.121181
$$726$$ −17.9504 −0.666203
$$727$$ 36.6602 1.35965 0.679825 0.733374i $$-0.262056\pi$$
0.679825 + 0.733374i $$0.262056\pi$$
$$728$$ −13.8434 −0.513071
$$729$$ 1.00000 0.0370370
$$730$$ 26.7185 0.988897
$$731$$ 23.0132 0.851175
$$732$$ −23.2267 −0.858484
$$733$$ 18.9303 0.699206 0.349603 0.936898i $$-0.386317\pi$$
0.349603 + 0.936898i $$0.386317\pi$$
$$734$$ 65.1165 2.40349
$$735$$ 8.25279 0.304409
$$736$$ 3.16122 0.116524
$$737$$ 20.4529 0.753393
$$738$$ 7.53445 0.277347
$$739$$ 8.84818 0.325485 0.162743 0.986669i $$-0.447966\pi$$
0.162743 + 0.986669i $$0.447966\pi$$
$$740$$ 18.8792 0.694012
$$741$$ 10.2063 0.374939
$$742$$ −28.5869 −1.04946
$$743$$ 4.67217 0.171405 0.0857026 0.996321i $$-0.472687\pi$$
0.0857026 + 0.996321i $$0.472687\pi$$
$$744$$ 18.2505 0.669096
$$745$$ −15.4887 −0.567464
$$746$$ −31.6306 −1.15808
$$747$$ −14.2415 −0.521071
$$748$$ 32.7055 1.19583
$$749$$ −9.70803 −0.354724
$$750$$ −25.8483 −0.943848
$$751$$ −23.8022 −0.868555 −0.434277 0.900779i $$-0.642996\pi$$
−0.434277 + 0.900779i $$0.642996\pi$$
$$752$$ 21.4983 0.783961
$$753$$ 10.2885 0.374935
$$754$$ −9.86283 −0.359183
$$755$$ 14.6424 0.532892
$$756$$ −3.12218 −0.113552
$$757$$ −0.980413 −0.0356337 −0.0178169 0.999841i $$-0.505672\pi$$
−0.0178169 + 0.999841i $$0.505672\pi$$
$$758$$ 13.8821 0.504221
$$759$$ 4.30847 0.156388
$$760$$ −12.5504 −0.455251
$$761$$ 28.6974 1.04028 0.520140 0.854081i $$-0.325880\pi$$
0.520140 + 0.854081i $$0.325880\pi$$
$$762$$ −39.7288 −1.43922
$$763$$ 15.1147 0.547187
$$764$$ 6.09309 0.220440
$$765$$ −2.75356 −0.0995553
$$766$$ −33.0904 −1.19561
$$767$$ −35.2056 −1.27120
$$768$$ −26.3171 −0.949637
$$769$$ 34.6389 1.24911 0.624556 0.780980i $$-0.285280\pi$$
0.624556 + 0.780980i $$0.285280\pi$$
$$770$$ −11.5815 −0.417367
$$771$$ −5.27151 −0.189849
$$772$$ −50.2755 −1.80945
$$773$$ 3.18514 0.114561 0.0572807 0.998358i $$-0.481757\pi$$
0.0572807 + 0.998358i $$0.481757\pi$$
$$774$$ −26.1447 −0.939751
$$775$$ 15.3599 0.551744
$$776$$ −15.2119 −0.546075
$$777$$ 3.38760 0.121529
$$778$$ −57.6363 −2.06636
$$779$$ 7.79686 0.279351
$$780$$ 19.8997 0.712523
$$781$$ −53.8168 −1.92572
$$782$$ −4.95870 −0.177323
$$783$$ −1.00000 −0.0357371
$$784$$ −12.1161 −0.432719
$$785$$ 25.6989 0.917232
$$786$$ 53.0782 1.89324
$$787$$ 47.6246 1.69764 0.848818 0.528686i $$-0.177315\pi$$
0.848818 + 0.528686i $$0.177315\pi$$
$$788$$ −101.306 −3.60888
$$789$$ −4.72941 −0.168371
$$790$$ −40.6961 −1.44790
$$791$$ 3.42215 0.121678
$$792$$ −16.7037 −0.593539
$$793$$ 26.5635 0.943297
$$794$$ 79.6731 2.82749
$$795$$ 18.4737 0.655194
$$796$$ −64.5424 −2.28764
$$797$$ 44.5050 1.57645 0.788223 0.615389i $$-0.211001\pi$$
0.788223 + 0.615389i $$0.211001\pi$$
$$798$$ −5.00935 −0.177329
$$799$$ 23.2117 0.821171
$$800$$ −10.3147 −0.364680
$$801$$ 0.795808 0.0281185
$$802$$ −39.8496 −1.40714
$$803$$ 36.7989 1.29860
$$804$$ 17.2485 0.608306
$$805$$ 1.13254 0.0399168
$$806$$ −46.4288 −1.63539
$$807$$ −21.2129 −0.746731
$$808$$ −15.1453 −0.532811
$$809$$ −34.0983 −1.19883 −0.599416 0.800438i $$-0.704600\pi$$
−0.599416 + 0.800438i $$0.704600\pi$$
$$810$$ 3.12824 0.109915
$$811$$ −10.5525 −0.370550 −0.185275 0.982687i $$-0.559318\pi$$
−0.185275 + 0.982687i $$0.559318\pi$$
$$812$$ 3.12218 0.109567
$$813$$ −11.0667 −0.388126
$$814$$ 40.3145 1.41302
$$815$$ −28.2216 −0.988561
$$816$$ 4.04257 0.141518
$$817$$ −27.0552 −0.946543
$$818$$ −80.5893 −2.81774
$$819$$ 3.57071 0.124771
$$820$$ 15.2018 0.530872
$$821$$ 30.2775 1.05669 0.528345 0.849030i $$-0.322813\pi$$
0.528345 + 0.849030i $$0.322813\pi$$
$$822$$ 30.5575 1.06582
$$823$$ −40.7346 −1.41992 −0.709960 0.704242i $$-0.751287\pi$$
−0.709960 + 0.704242i $$0.751287\pi$$
$$824$$ −6.51536 −0.226973
$$825$$ −14.0580 −0.489438
$$826$$ 17.2792 0.601221
$$827$$ 7.88557 0.274208 0.137104 0.990557i $$-0.456221\pi$$
0.137104 + 0.990557i $$0.456221\pi$$
$$828$$ 3.63344 0.126271
$$829$$ 8.87941 0.308395 0.154197 0.988040i $$-0.450721\pi$$
0.154197 + 0.988040i $$0.450721\pi$$
$$830$$ −44.5510 −1.54639
$$831$$ −7.98577 −0.277023
$$832$$ 47.2599 1.63844
$$833$$ −13.0818 −0.453257
$$834$$ −30.3964 −1.05254
$$835$$ 4.96662 0.171877
$$836$$ −38.4499 −1.32982
$$837$$ −4.70746 −0.162714
$$838$$ −2.97411 −0.102739
$$839$$ −0.405359 −0.0139946 −0.00699728 0.999976i $$-0.502227\pi$$
−0.00699728 + 0.999976i $$0.502227\pi$$
$$840$$ −4.39079 −0.151497
$$841$$ 1.00000 0.0344828
$$842$$ −46.8921 −1.61601
$$843$$ 6.63114 0.228389
$$844$$ −78.9491 −2.71754
$$845$$ −5.62455 −0.193490
$$846$$ −26.3702 −0.906625
$$847$$ −6.49873 −0.223299
$$848$$ −27.1217 −0.931363
$$849$$ 2.98274 0.102367
$$850$$ 16.1797 0.554958
$$851$$ −3.94232 −0.135141
$$852$$ −45.3850 −1.55486
$$853$$ 38.4556 1.31669 0.658347 0.752715i $$-0.271256\pi$$
0.658347 + 0.752715i $$0.271256\pi$$
$$854$$ −13.0376 −0.446137
$$855$$ 3.23719 0.110710
$$856$$ −43.8006 −1.49707
$$857$$ −4.08648 −0.139592 −0.0697958 0.997561i $$-0.522235\pi$$
−0.0697958 + 0.997561i $$0.522235\pi$$
$$858$$ 42.4937 1.45071
$$859$$ −26.7695 −0.913362 −0.456681 0.889631i $$-0.650962\pi$$
−0.456681 + 0.889631i $$0.650962\pi$$
$$860$$ −52.7507 −1.79878
$$861$$ 2.72775 0.0929615
$$862$$ 66.6606 2.27047
$$863$$ 26.1278 0.889402 0.444701 0.895679i $$-0.353310\pi$$
0.444701 + 0.895679i $$0.353310\pi$$
$$864$$ 3.16122 0.107547
$$865$$ −24.4669 −0.831899
$$866$$ 90.0989 3.06169
$$867$$ −12.6352 −0.429115
$$868$$ 14.6975 0.498866
$$869$$ −56.0499 −1.90136
$$870$$ −3.12824 −0.106057
$$871$$ −19.7264 −0.668403
$$872$$ 68.1941 2.30934
$$873$$ 3.92369 0.132797
$$874$$ 5.82963 0.197190
$$875$$ −9.35806 −0.316360
$$876$$ 31.0334 1.04852
$$877$$ −4.82414 −0.162900 −0.0814498 0.996677i $$-0.525955\pi$$
−0.0814498 + 0.996677i $$0.525955\pi$$
$$878$$ −78.4831 −2.64867
$$879$$ 13.4117 0.452366
$$880$$ −10.9879 −0.370402
$$881$$ −24.2045 −0.815469 −0.407734 0.913101i $$-0.633681\pi$$
−0.407734 + 0.913101i $$0.633681\pi$$
$$882$$ 14.8619 0.500425
$$883$$ 18.1602 0.611139 0.305570 0.952170i $$-0.401153\pi$$
0.305570 + 0.952170i $$0.401153\pi$$
$$884$$ −31.5437 −1.06093
$$885$$ −11.1664 −0.375353
$$886$$ 11.1036 0.373033
$$887$$ −40.7965 −1.36981 −0.684907 0.728630i $$-0.740157\pi$$
−0.684907 + 0.728630i $$0.740157\pi$$
$$888$$ 15.2841 0.512901
$$889$$ −14.3833 −0.482400
$$890$$ 2.48948 0.0834476
$$891$$ 4.30847 0.144339
$$892$$ −0.591985 −0.0198211
$$893$$ −27.2886 −0.913177
$$894$$ −27.8926 −0.932867
$$895$$ 28.3477 0.947558
$$896$$ −17.7627 −0.593411
$$897$$ −4.15542 −0.138745
$$898$$ −0.137237 −0.00457966
$$899$$ 4.70746 0.157002
$$900$$ −11.8555 −0.395183
$$901$$ −29.2833 −0.975569
$$902$$ 32.4619 1.08086
$$903$$ −9.46535 −0.314987
$$904$$ 15.4400 0.513528
$$905$$ 16.9473 0.563346
$$906$$ 26.3685 0.876034
$$907$$ 20.5919 0.683744 0.341872 0.939747i $$-0.388939\pi$$
0.341872 + 0.939747i $$0.388939\pi$$
$$908$$ −54.2917 −1.80173
$$909$$ 3.90652 0.129571
$$910$$ 11.1701 0.370284
$$911$$ 0.555205 0.0183948 0.00919738 0.999958i $$-0.497072\pi$$
0.00919738 + 0.999958i $$0.497072\pi$$
$$912$$ −4.75261 −0.157374
$$913$$ −61.3593 −2.03069
$$914$$ 22.9963 0.760651
$$915$$ 8.42528 0.278531
$$916$$ −33.0695 −1.09265
$$917$$ 19.2163 0.634577
$$918$$ −4.95870 −0.163661
$$919$$ 51.8269 1.70961 0.854805 0.518949i $$-0.173676\pi$$
0.854805 + 0.518949i $$0.173676\pi$$
$$920$$ 5.10979 0.168465
$$921$$ −7.23125 −0.238278
$$922$$ −33.1432 −1.09151
$$923$$ 51.9051 1.70848
$$924$$ −13.4518 −0.442532
$$925$$ 12.8633 0.422944
$$926$$ −74.5262 −2.44908
$$927$$ 1.68054 0.0551963
$$928$$ −3.16122 −0.103772
$$929$$ 24.3448 0.798727 0.399364 0.916793i $$-0.369231\pi$$
0.399364 + 0.916793i $$0.369231\pi$$
$$930$$ −14.7261 −0.482887
$$931$$ 15.3795 0.504042
$$932$$ −45.8675 −1.50244
$$933$$ 0.648787 0.0212403
$$934$$ 49.6718 1.62531
$$935$$ −11.8636 −0.387982
$$936$$ 16.1103 0.526581
$$937$$ 2.62171 0.0856475 0.0428237 0.999083i $$-0.486365\pi$$
0.0428237 + 0.999083i $$0.486365\pi$$
$$938$$ 9.68187 0.316124
$$939$$ 18.3081 0.597461
$$940$$ −53.2057 −1.73538
$$941$$ −27.1040 −0.883566 −0.441783 0.897122i $$-0.645654\pi$$
−0.441783 + 0.897122i $$0.645654\pi$$
$$942$$ 46.2793 1.50786
$$943$$ −3.17442 −0.103373
$$944$$ 16.3936 0.533566
$$945$$ 1.13254 0.0368416
$$946$$ −112.644 −3.66236
$$947$$ 1.67384 0.0543924 0.0271962 0.999630i $$-0.491342\pi$$
0.0271962 + 0.999630i $$0.491342\pi$$
$$948$$ −47.2682 −1.53520
$$949$$ −35.4917 −1.15211
$$950$$ −19.0214 −0.617137
$$951$$ −21.4173 −0.694504
$$952$$ 6.96001 0.225575
$$953$$ −39.1266 −1.26743 −0.633717 0.773565i $$-0.718472\pi$$
−0.633717 + 0.773565i $$0.718472\pi$$
$$954$$ 33.2680 1.07709
$$955$$ −2.21021 −0.0715209
$$956$$ 61.2284 1.98027
$$957$$ −4.30847 −0.139273
$$958$$ −97.7136 −3.15698
$$959$$ 11.0630 0.357242
$$960$$ 14.9897 0.483789
$$961$$ −8.83984 −0.285156
$$962$$ −38.8824 −1.25362
$$963$$ 11.2977 0.364064
$$964$$ −63.3241 −2.03953
$$965$$ 18.2370 0.587069
$$966$$ 2.03951 0.0656203
$$967$$ −46.2331 −1.48676 −0.743378 0.668872i $$-0.766777\pi$$
−0.743378 + 0.668872i $$0.766777\pi$$
$$968$$ −29.3209 −0.942409
$$969$$ −5.13140 −0.164844
$$970$$ 12.2743 0.394103
$$971$$ 39.1168 1.25532 0.627660 0.778488i $$-0.284013\pi$$
0.627660 + 0.778488i $$0.284013\pi$$
$$972$$ 3.63344 0.116542
$$973$$ −11.0046 −0.352793
$$974$$ 39.6603 1.27080
$$975$$ 13.5587 0.434225
$$976$$ −12.3694 −0.395934
$$977$$ −11.4002 −0.364725 −0.182363 0.983231i $$-0.558374\pi$$
−0.182363 + 0.983231i $$0.558374\pi$$
$$978$$ −50.8223 −1.62512
$$979$$ 3.42871 0.109582
$$980$$ 29.9860 0.957867
$$981$$ −17.5897 −0.561596
$$982$$ −44.5213 −1.42073
$$983$$ 47.4586 1.51369 0.756846 0.653593i $$-0.226739\pi$$
0.756846 + 0.653593i $$0.226739\pi$$
$$984$$ 12.3070 0.392334
$$985$$ 36.7478 1.17088
$$986$$ 4.95870 0.157917
$$987$$ −9.54699 −0.303884
$$988$$ 37.0840 1.17980
$$989$$ 11.0153 0.350266
$$990$$ 13.4779 0.428357
$$991$$ 52.3146 1.66183 0.830914 0.556401i $$-0.187818\pi$$
0.830914 + 0.556401i $$0.187818\pi$$
$$992$$ −14.8813 −0.472482
$$993$$ 2.79629 0.0887376
$$994$$ −25.4754 −0.808032
$$995$$ 23.4122 0.742215
$$996$$ −51.7457 −1.63963
$$997$$ −46.9541 −1.48705 −0.743525 0.668708i $$-0.766848\pi$$
−0.743525 + 0.668708i $$0.766848\pi$$
$$998$$ −10.9262 −0.345863
$$999$$ −3.94232 −0.124729
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 2001.2.a.i.1.1 7
3.2 odd 2 6003.2.a.j.1.7 7

By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.i.1.1 7 1.1 even 1 trivial
6003.2.a.j.1.7 7 3.2 odd 2