# Properties

 Label 637.2.a.k.1.4 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.746052.1 Defining polynomial: $$x^{5} - x^{4} - 7x^{3} + 8x + 2$$ x^5 - x^4 - 7*x^3 + 8*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.21332$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.21332 q^{2} -2.47443 q^{3} +2.89879 q^{4} +2.12280 q^{5} -5.47671 q^{6} +1.98932 q^{8} +3.12280 q^{9} +O(q^{10})$$ $$q+2.21332 q^{2} -2.47443 q^{3} +2.89879 q^{4} +2.12280 q^{5} -5.47671 q^{6} +1.98932 q^{8} +3.12280 q^{9} +4.69843 q^{10} +4.78896 q^{11} -7.17286 q^{12} -1.00000 q^{13} -5.25271 q^{15} -1.39458 q^{16} +3.77828 q^{17} +6.91175 q^{18} +3.56723 q^{19} +6.15355 q^{20} +10.5995 q^{22} +4.47443 q^{23} -4.92243 q^{24} -0.493740 q^{25} -2.21332 q^{26} -0.303848 q^{27} -5.90107 q^{29} -11.6259 q^{30} +3.77116 q^{31} -7.06530 q^{32} -11.8499 q^{33} +8.36254 q^{34} +9.05234 q^{36} +5.62570 q^{37} +7.89544 q^{38} +2.47443 q^{39} +4.22292 q^{40} -10.3948 q^{41} +3.40733 q^{43} +13.8822 q^{44} +6.62906 q^{45} +9.90335 q^{46} +7.10876 q^{47} +3.45079 q^{48} -1.09280 q^{50} -9.34907 q^{51} -2.89879 q^{52} -12.3801 q^{53} -0.672514 q^{54} +10.1660 q^{55} -8.82686 q^{57} -13.0610 q^{58} -4.78896 q^{59} -15.2265 q^{60} -3.20697 q^{61} +8.34680 q^{62} -12.8486 q^{64} -2.12280 q^{65} -26.2277 q^{66} -2.89955 q^{67} +10.9524 q^{68} -11.0717 q^{69} -2.53876 q^{71} +6.21224 q^{72} -7.70071 q^{73} +12.4515 q^{74} +1.22172 q^{75} +10.3407 q^{76} +5.47671 q^{78} -5.17850 q^{79} -2.96041 q^{80} -8.61654 q^{81} -23.0071 q^{82} -3.46731 q^{83} +8.02051 q^{85} +7.54152 q^{86} +14.6018 q^{87} +9.52677 q^{88} -3.66432 q^{89} +14.6722 q^{90} +12.9704 q^{92} -9.33147 q^{93} +15.7340 q^{94} +7.57251 q^{95} +17.4826 q^{96} +5.40733 q^{97} +14.9549 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q + 4 * q^2 + 8 * q^4 - 2 * q^5 + 5 * q^6 + 9 * q^8 + 3 * q^9 $$5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} - 5 q^{13} + 10 q^{16} + 5 q^{17} + 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 4 q^{26} - 3 q^{29} - 13 q^{30} + 6 q^{31} + 22 q^{32} - 8 q^{33} + 22 q^{34} + 7 q^{36} + 4 q^{37} + 10 q^{38} - 28 q^{40} - 14 q^{41} + 2 q^{43} + 32 q^{45} + 3 q^{46} - q^{47} + 23 q^{48} + 9 q^{50} - 8 q^{51} - 8 q^{52} + 17 q^{53} - 23 q^{54} - 16 q^{57} - 27 q^{58} - 11 q^{59} - 29 q^{60} + 11 q^{61} + 23 q^{62} + 9 q^{64} + 2 q^{65} - 21 q^{66} + 13 q^{67} + 32 q^{68} - 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} + 20 q^{75} - 8 q^{76} - 5 q^{78} + 2 q^{79} - 55 q^{80} - 19 q^{81} - 34 q^{82} - 6 q^{83} - 22 q^{85} + 28 q^{86} + 8 q^{87} - 3 q^{88} + 4 q^{89} + 34 q^{90} + 21 q^{92} + 18 q^{93} - 20 q^{94} - 12 q^{95} + 37 q^{96} + 12 q^{97} + 11 q^{99}+O(q^{100})$$ 5 * q + 4 * q^2 + 8 * q^4 - 2 * q^5 + 5 * q^6 + 9 * q^8 + 3 * q^9 + 5 * q^10 + 11 * q^11 - 5 * q^12 - 5 * q^13 + 10 * q^16 + 5 * q^17 + 9 * q^18 - 9 * q^19 - q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 - 4 * q^26 - 3 * q^29 - 13 * q^30 + 6 * q^31 + 22 * q^32 - 8 * q^33 + 22 * q^34 + 7 * q^36 + 4 * q^37 + 10 * q^38 - 28 * q^40 - 14 * q^41 + 2 * q^43 + 32 * q^45 + 3 * q^46 - q^47 + 23 * q^48 + 9 * q^50 - 8 * q^51 - 8 * q^52 + 17 * q^53 - 23 * q^54 - 16 * q^57 - 27 * q^58 - 11 * q^59 - 29 * q^60 + 11 * q^61 + 23 * q^62 + 9 * q^64 + 2 * q^65 - 21 * q^66 + 13 * q^67 + 32 * q^68 - 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 + 20 * q^75 - 8 * q^76 - 5 * q^78 + 2 * q^79 - 55 * q^80 - 19 * q^81 - 34 * q^82 - 6 * q^83 - 22 * q^85 + 28 * q^86 + 8 * q^87 - 3 * q^88 + 4 * q^89 + 34 * q^90 + 21 * q^92 + 18 * q^93 - 20 * q^94 - 12 * q^95 + 37 * q^96 + 12 * q^97 + 11 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.21332 1.56505 0.782527 0.622616i $$-0.213930\pi$$
0.782527 + 0.622616i $$0.213930\pi$$
$$3$$ −2.47443 −1.42861 −0.714306 0.699834i $$-0.753257\pi$$
−0.714306 + 0.699834i $$0.753257\pi$$
$$4$$ 2.89879 1.44940
$$5$$ 2.12280 0.949343 0.474671 0.880163i $$-0.342567\pi$$
0.474671 + 0.880163i $$0.342567\pi$$
$$6$$ −5.47671 −2.23586
$$7$$ 0 0
$$8$$ 1.98932 0.703331
$$9$$ 3.12280 1.04093
$$10$$ 4.69843 1.48577
$$11$$ 4.78896 1.44392 0.721962 0.691932i $$-0.243240\pi$$
0.721962 + 0.691932i $$0.243240\pi$$
$$12$$ −7.17286 −2.07063
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −5.25271 −1.35624
$$16$$ −1.39458 −0.348645
$$17$$ 3.77828 0.916367 0.458183 0.888858i $$-0.348500\pi$$
0.458183 + 0.888858i $$0.348500\pi$$
$$18$$ 6.91175 1.62912
$$19$$ 3.56723 0.818379 0.409190 0.912449i $$-0.365811\pi$$
0.409190 + 0.912449i $$0.365811\pi$$
$$20$$ 6.15355 1.37597
$$21$$ 0 0
$$22$$ 10.5995 2.25982
$$23$$ 4.47443 0.932983 0.466491 0.884526i $$-0.345518\pi$$
0.466491 + 0.884526i $$0.345518\pi$$
$$24$$ −4.92243 −1.00479
$$25$$ −0.493740 −0.0987479
$$26$$ −2.21332 −0.434068
$$27$$ −0.303848 −0.0584757
$$28$$ 0 0
$$29$$ −5.90107 −1.09580 −0.547901 0.836543i $$-0.684573\pi$$
−0.547901 + 0.836543i $$0.684573\pi$$
$$30$$ −11.6259 −2.12259
$$31$$ 3.77116 0.677321 0.338660 0.940909i $$-0.390026\pi$$
0.338660 + 0.940909i $$0.390026\pi$$
$$32$$ −7.06530 −1.24898
$$33$$ −11.8499 −2.06281
$$34$$ 8.36254 1.43416
$$35$$ 0 0
$$36$$ 9.05234 1.50872
$$37$$ 5.62570 0.924859 0.462429 0.886656i $$-0.346978\pi$$
0.462429 + 0.886656i $$0.346978\pi$$
$$38$$ 7.89544 1.28081
$$39$$ 2.47443 0.396226
$$40$$ 4.22292 0.667702
$$41$$ −10.3948 −1.62340 −0.811698 0.584077i $$-0.801457\pi$$
−0.811698 + 0.584077i $$0.801457\pi$$
$$42$$ 0 0
$$43$$ 3.40733 0.519613 0.259807 0.965661i $$-0.416341\pi$$
0.259807 + 0.965661i $$0.416341\pi$$
$$44$$ 13.8822 2.09282
$$45$$ 6.62906 0.988201
$$46$$ 9.90335 1.46017
$$47$$ 7.10876 1.03692 0.518459 0.855102i $$-0.326506\pi$$
0.518459 + 0.855102i $$0.326506\pi$$
$$48$$ 3.45079 0.498079
$$49$$ 0 0
$$50$$ −1.09280 −0.154546
$$51$$ −9.34907 −1.30913
$$52$$ −2.89879 −0.401990
$$53$$ −12.3801 −1.70053 −0.850266 0.526354i $$-0.823559\pi$$
−0.850266 + 0.526354i $$0.823559\pi$$
$$54$$ −0.672514 −0.0915176
$$55$$ 10.1660 1.37078
$$56$$ 0 0
$$57$$ −8.82686 −1.16915
$$58$$ −13.0610 −1.71499
$$59$$ −4.78896 −0.623469 −0.311734 0.950169i $$-0.600910\pi$$
−0.311734 + 0.950169i $$0.600910\pi$$
$$60$$ −15.2265 −1.96573
$$61$$ −3.20697 −0.410610 −0.205305 0.978698i $$-0.565819\pi$$
−0.205305 + 0.978698i $$0.565819\pi$$
$$62$$ 8.34680 1.06004
$$63$$ 0 0
$$64$$ −12.8486 −1.60608
$$65$$ −2.12280 −0.263300
$$66$$ −26.2277 −3.22841
$$67$$ −2.89955 −0.354237 −0.177118 0.984190i $$-0.556678\pi$$
−0.177118 + 0.984190i $$0.556678\pi$$
$$68$$ 10.9524 1.32818
$$69$$ −11.0717 −1.33287
$$70$$ 0 0
$$71$$ −2.53876 −0.301295 −0.150648 0.988588i $$-0.548136\pi$$
−0.150648 + 0.988588i $$0.548136\pi$$
$$72$$ 6.21224 0.732120
$$73$$ −7.70071 −0.901300 −0.450650 0.892701i $$-0.648808\pi$$
−0.450650 + 0.892701i $$0.648808\pi$$
$$74$$ 12.4515 1.44745
$$75$$ 1.22172 0.141072
$$76$$ 10.3407 1.18616
$$77$$ 0 0
$$78$$ 5.47671 0.620115
$$79$$ −5.17850 −0.582626 −0.291313 0.956628i $$-0.594092\pi$$
−0.291313 + 0.956628i $$0.594092\pi$$
$$80$$ −2.96041 −0.330984
$$81$$ −8.61654 −0.957393
$$82$$ −23.0071 −2.54071
$$83$$ −3.46731 −0.380587 −0.190294 0.981727i $$-0.560944\pi$$
−0.190294 + 0.981727i $$0.560944\pi$$
$$84$$ 0 0
$$85$$ 8.02051 0.869946
$$86$$ 7.54152 0.813223
$$87$$ 14.6018 1.56548
$$88$$ 9.52677 1.01556
$$89$$ −3.66432 −0.388417 −0.194209 0.980960i $$-0.562214\pi$$
−0.194209 + 0.980960i $$0.562214\pi$$
$$90$$ 14.6722 1.54659
$$91$$ 0 0
$$92$$ 12.9704 1.35226
$$93$$ −9.33147 −0.967629
$$94$$ 15.7340 1.62283
$$95$$ 7.57251 0.776923
$$96$$ 17.4826 1.78431
$$97$$ 5.40733 0.549031 0.274516 0.961583i $$-0.411482\pi$$
0.274516 + 0.961583i $$0.411482\pi$$
$$98$$ 0 0
$$99$$ 14.9549 1.50303
$$100$$ −1.43125 −0.143125
$$101$$ −9.31724 −0.927100 −0.463550 0.886071i $$-0.653425\pi$$
−0.463550 + 0.886071i $$0.653425\pi$$
$$102$$ −20.6925 −2.04886
$$103$$ −7.30636 −0.719917 −0.359958 0.932968i $$-0.617209\pi$$
−0.359958 + 0.932968i $$0.617209\pi$$
$$104$$ −1.98932 −0.195069
$$105$$ 0 0
$$106$$ −27.4011 −2.66143
$$107$$ 6.74729 0.652286 0.326143 0.945321i $$-0.394251\pi$$
0.326143 + 0.945321i $$0.394251\pi$$
$$108$$ −0.880794 −0.0847545
$$109$$ 4.17645 0.400031 0.200016 0.979793i $$-0.435901\pi$$
0.200016 + 0.979793i $$0.435901\pi$$
$$110$$ 22.5006 2.14535
$$111$$ −13.9204 −1.32126
$$112$$ 0 0
$$113$$ 5.90107 0.555126 0.277563 0.960707i $$-0.410473\pi$$
0.277563 + 0.960707i $$0.410473\pi$$
$$114$$ −19.5367 −1.82978
$$115$$ 9.49830 0.885721
$$116$$ −17.1060 −1.58825
$$117$$ −3.12280 −0.288703
$$118$$ −10.5995 −0.975763
$$119$$ 0 0
$$120$$ −10.4493 −0.953888
$$121$$ 11.9341 1.08492
$$122$$ −7.09805 −0.642628
$$123$$ 25.7212 2.31920
$$124$$ 10.9318 0.981707
$$125$$ −11.6621 −1.04309
$$126$$ 0 0
$$127$$ −10.5268 −0.934100 −0.467050 0.884231i $$-0.654683\pi$$
−0.467050 + 0.884231i $$0.654683\pi$$
$$128$$ −14.3075 −1.26462
$$129$$ −8.43120 −0.742326
$$130$$ −4.69843 −0.412080
$$131$$ −5.42409 −0.473905 −0.236952 0.971521i $$-0.576149\pi$$
−0.236952 + 0.971521i $$0.576149\pi$$
$$132$$ −34.3505 −2.98983
$$133$$ 0 0
$$134$$ −6.41765 −0.554400
$$135$$ −0.645008 −0.0555135
$$136$$ 7.51620 0.644509
$$137$$ 22.2447 1.90050 0.950248 0.311494i $$-0.100829\pi$$
0.950248 + 0.311494i $$0.100829\pi$$
$$138$$ −24.5051 −2.08602
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −17.5901 −1.48135
$$142$$ −5.61909 −0.471544
$$143$$ −4.78896 −0.400473
$$144$$ −4.35499 −0.362916
$$145$$ −12.5268 −1.04029
$$146$$ −17.0441 −1.41058
$$147$$ 0 0
$$148$$ 16.3077 1.34049
$$149$$ 2.95472 0.242060 0.121030 0.992649i $$-0.461380\pi$$
0.121030 + 0.992649i $$0.461380\pi$$
$$150$$ 2.70407 0.220786
$$151$$ −18.5547 −1.50996 −0.754981 0.655747i $$-0.772354\pi$$
−0.754981 + 0.655747i $$0.772354\pi$$
$$152$$ 7.09637 0.575592
$$153$$ 11.7988 0.953875
$$154$$ 0 0
$$155$$ 8.00541 0.643010
$$156$$ 7.17286 0.574288
$$157$$ 9.79964 0.782096 0.391048 0.920370i $$-0.372113\pi$$
0.391048 + 0.920370i $$0.372113\pi$$
$$158$$ −11.4617 −0.911842
$$159$$ 30.6336 2.42940
$$160$$ −14.9982 −1.18571
$$161$$ 0 0
$$162$$ −19.0712 −1.49837
$$163$$ 13.8342 1.08358 0.541788 0.840515i $$-0.317747\pi$$
0.541788 + 0.840515i $$0.317747\pi$$
$$164$$ −30.1324 −2.35295
$$165$$ −25.1550 −1.95831
$$166$$ −7.67428 −0.595640
$$167$$ −17.3534 −1.34285 −0.671424 0.741073i $$-0.734317\pi$$
−0.671424 + 0.741073i $$0.734317\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 17.7520 1.36151
$$171$$ 11.1397 0.851877
$$172$$ 9.87716 0.753126
$$173$$ 2.96138 0.225149 0.112575 0.993643i $$-0.464090\pi$$
0.112575 + 0.993643i $$0.464090\pi$$
$$174$$ 32.3184 2.45005
$$175$$ 0 0
$$176$$ −6.67859 −0.503418
$$177$$ 11.8499 0.890695
$$178$$ −8.11032 −0.607894
$$179$$ −5.66888 −0.423712 −0.211856 0.977301i $$-0.567951\pi$$
−0.211856 + 0.977301i $$0.567951\pi$$
$$180$$ 19.2163 1.43230
$$181$$ −7.17645 −0.533421 −0.266711 0.963777i $$-0.585937\pi$$
−0.266711 + 0.963777i $$0.585937\pi$$
$$182$$ 0 0
$$183$$ 7.93541 0.586603
$$184$$ 8.90107 0.656196
$$185$$ 11.9422 0.878008
$$186$$ −20.6536 −1.51439
$$187$$ 18.0940 1.32316
$$188$$ 20.6068 1.50291
$$189$$ 0 0
$$190$$ 16.7604 1.21593
$$191$$ 11.8818 0.859734 0.429867 0.902892i $$-0.358560\pi$$
0.429867 + 0.902892i $$0.358560\pi$$
$$192$$ 31.7930 2.29446
$$193$$ 22.9702 1.65343 0.826714 0.562622i $$-0.190207\pi$$
0.826714 + 0.562622i $$0.190207\pi$$
$$194$$ 11.9682 0.859264
$$195$$ 5.25271 0.376154
$$196$$ 0 0
$$197$$ 16.9216 1.20561 0.602806 0.797888i $$-0.294049\pi$$
0.602806 + 0.797888i $$0.294049\pi$$
$$198$$ 33.1001 2.35232
$$199$$ −10.0591 −0.713068 −0.356534 0.934282i $$-0.616042\pi$$
−0.356534 + 0.934282i $$0.616042\pi$$
$$200$$ −0.982206 −0.0694525
$$201$$ 7.17474 0.506067
$$202$$ −20.6221 −1.45096
$$203$$ 0 0
$$204$$ −27.1010 −1.89745
$$205$$ −22.0661 −1.54116
$$206$$ −16.1713 −1.12671
$$207$$ 13.9727 0.971171
$$208$$ 1.39458 0.0966968
$$209$$ 17.0833 1.18168
$$210$$ 0 0
$$211$$ −24.4609 −1.68396 −0.841978 0.539512i $$-0.818609\pi$$
−0.841978 + 0.539512i $$0.818609\pi$$
$$212$$ −35.8872 −2.46475
$$213$$ 6.28198 0.430434
$$214$$ 14.9339 1.02086
$$215$$ 7.23307 0.493291
$$216$$ −0.604452 −0.0411277
$$217$$ 0 0
$$218$$ 9.24382 0.626071
$$219$$ 19.0548 1.28761
$$220$$ 29.4691 1.98680
$$221$$ −3.77828 −0.254154
$$222$$ −30.8103 −2.06785
$$223$$ 29.2625 1.95956 0.979780 0.200076i $$-0.0641188\pi$$
0.979780 + 0.200076i $$0.0641188\pi$$
$$224$$ 0 0
$$225$$ −1.54185 −0.102790
$$226$$ 13.0610 0.868803
$$227$$ 10.0737 0.668615 0.334307 0.942464i $$-0.391498\pi$$
0.334307 + 0.942464i $$0.391498\pi$$
$$228$$ −25.5873 −1.69456
$$229$$ 11.1399 0.736148 0.368074 0.929796i $$-0.380017\pi$$
0.368074 + 0.929796i $$0.380017\pi$$
$$230$$ 21.0228 1.38620
$$231$$ 0 0
$$232$$ −11.7391 −0.770711
$$233$$ −17.0833 −1.11917 −0.559583 0.828774i $$-0.689039\pi$$
−0.559583 + 0.828774i $$0.689039\pi$$
$$234$$ −6.91175 −0.451835
$$235$$ 15.0904 0.984392
$$236$$ −13.8822 −0.903654
$$237$$ 12.8138 0.832347
$$238$$ 0 0
$$239$$ 6.92142 0.447710 0.223855 0.974622i $$-0.428136\pi$$
0.223855 + 0.974622i $$0.428136\pi$$
$$240$$ 7.32532 0.472848
$$241$$ −6.49625 −0.418460 −0.209230 0.977866i $$-0.567096\pi$$
−0.209230 + 0.977866i $$0.567096\pi$$
$$242$$ 26.4140 1.69796
$$243$$ 22.2325 1.42622
$$244$$ −9.29634 −0.595137
$$245$$ 0 0
$$246$$ 56.9293 3.62968
$$247$$ −3.56723 −0.226978
$$248$$ 7.50205 0.476381
$$249$$ 8.57962 0.543711
$$250$$ −25.8119 −1.63249
$$251$$ 9.86804 0.622865 0.311433 0.950268i $$-0.399191\pi$$
0.311433 + 0.950268i $$0.399191\pi$$
$$252$$ 0 0
$$253$$ 21.4278 1.34716
$$254$$ −23.2991 −1.46192
$$255$$ −19.8462 −1.24282
$$256$$ −5.96994 −0.373121
$$257$$ −6.86468 −0.428207 −0.214104 0.976811i $$-0.568683\pi$$
−0.214104 + 0.976811i $$0.568683\pi$$
$$258$$ −18.6610 −1.16178
$$259$$ 0 0
$$260$$ −6.15355 −0.381627
$$261$$ −18.4278 −1.14065
$$262$$ −12.0052 −0.741687
$$263$$ −0.126551 −0.00780345 −0.00390172 0.999992i $$-0.501242\pi$$
−0.00390172 + 0.999992i $$0.501242\pi$$
$$264$$ −23.5733 −1.45084
$$265$$ −26.2803 −1.61439
$$266$$ 0 0
$$267$$ 9.06710 0.554897
$$268$$ −8.40521 −0.513430
$$269$$ 4.24308 0.258705 0.129353 0.991599i $$-0.458710\pi$$
0.129353 + 0.991599i $$0.458710\pi$$
$$270$$ −1.42761 −0.0868816
$$271$$ −1.56723 −0.0952026 −0.0476013 0.998866i $$-0.515158\pi$$
−0.0476013 + 0.998866i $$0.515158\pi$$
$$272$$ −5.26911 −0.319487
$$273$$ 0 0
$$274$$ 49.2348 2.97438
$$275$$ −2.36450 −0.142585
$$276$$ −32.0944 −1.93186
$$277$$ −12.7452 −0.765785 −0.382892 0.923793i $$-0.625072\pi$$
−0.382892 + 0.923793i $$0.625072\pi$$
$$278$$ 8.85329 0.530985
$$279$$ 11.7766 0.705045
$$280$$ 0 0
$$281$$ 4.62986 0.276194 0.138097 0.990419i $$-0.455901\pi$$
0.138097 + 0.990419i $$0.455901\pi$$
$$282$$ −38.9326 −2.31840
$$283$$ −3.64832 −0.216870 −0.108435 0.994104i $$-0.534584\pi$$
−0.108435 + 0.994104i $$0.534584\pi$$
$$284$$ −7.35934 −0.436697
$$285$$ −18.7376 −1.10992
$$286$$ −10.5995 −0.626762
$$287$$ 0 0
$$288$$ −22.0635 −1.30010
$$289$$ −2.72462 −0.160272
$$290$$ −27.7258 −1.62811
$$291$$ −13.3801 −0.784353
$$292$$ −22.3228 −1.30634
$$293$$ 21.0415 1.22926 0.614630 0.788816i $$-0.289305\pi$$
0.614630 + 0.788816i $$0.289305\pi$$
$$294$$ 0 0
$$295$$ −10.1660 −0.591886
$$296$$ 11.1913 0.650482
$$297$$ −1.45512 −0.0844345
$$298$$ 6.53976 0.378838
$$299$$ −4.47443 −0.258763
$$300$$ 3.54152 0.204470
$$301$$ 0 0
$$302$$ −41.0676 −2.36317
$$303$$ 23.0548 1.32447
$$304$$ −4.97480 −0.285324
$$305$$ −6.80774 −0.389810
$$306$$ 26.1145 1.49287
$$307$$ 4.95861 0.283003 0.141502 0.989938i $$-0.454807\pi$$
0.141502 + 0.989938i $$0.454807\pi$$
$$308$$ 0 0
$$309$$ 18.0791 1.02848
$$310$$ 17.7185 1.00635
$$311$$ 2.42158 0.137315 0.0686575 0.997640i $$-0.478128\pi$$
0.0686575 + 0.997640i $$0.478128\pi$$
$$312$$ 4.92243 0.278678
$$313$$ −13.9605 −0.789096 −0.394548 0.918875i $$-0.629099\pi$$
−0.394548 + 0.918875i $$0.629099\pi$$
$$314$$ 21.6897 1.22402
$$315$$ 0 0
$$316$$ −15.0114 −0.844457
$$317$$ 3.06862 0.172351 0.0861753 0.996280i $$-0.472535\pi$$
0.0861753 + 0.996280i $$0.472535\pi$$
$$318$$ 67.8019 3.80214
$$319$$ −28.2600 −1.58225
$$320$$ −27.2750 −1.52472
$$321$$ −16.6957 −0.931863
$$322$$ 0 0
$$323$$ 13.4780 0.749936
$$324$$ −24.9776 −1.38764
$$325$$ 0.493740 0.0273877
$$326$$ 30.6195 1.69586
$$327$$ −10.3343 −0.571489
$$328$$ −20.6786 −1.14179
$$329$$ 0 0
$$330$$ −55.6761 −3.06487
$$331$$ 13.6052 0.747810 0.373905 0.927467i $$-0.378019\pi$$
0.373905 + 0.927467i $$0.378019\pi$$
$$332$$ −10.0510 −0.551622
$$333$$ 17.5679 0.962715
$$334$$ −38.4087 −2.10163
$$335$$ −6.15516 −0.336292
$$336$$ 0 0
$$337$$ −35.1646 −1.91554 −0.957769 0.287538i $$-0.907163\pi$$
−0.957769 + 0.287538i $$0.907163\pi$$
$$338$$ 2.21332 0.120389
$$339$$ −14.6018 −0.793060
$$340$$ 23.2498 1.26090
$$341$$ 18.0599 0.978000
$$342$$ 24.6558 1.33323
$$343$$ 0 0
$$344$$ 6.77828 0.365460
$$345$$ −23.5029 −1.26535
$$346$$ 6.55448 0.352371
$$347$$ −5.47102 −0.293700 −0.146850 0.989159i $$-0.546913\pi$$
−0.146850 + 0.989159i $$0.546913\pi$$
$$348$$ 42.3276 2.26899
$$349$$ −4.34196 −0.232420 −0.116210 0.993225i $$-0.537075\pi$$
−0.116210 + 0.993225i $$0.537075\pi$$
$$350$$ 0 0
$$351$$ 0.303848 0.0162182
$$352$$ −33.8354 −1.80343
$$353$$ −27.5992 −1.46896 −0.734479 0.678631i $$-0.762573\pi$$
−0.734479 + 0.678631i $$0.762573\pi$$
$$354$$ 26.2277 1.39399
$$355$$ −5.38927 −0.286033
$$356$$ −10.6221 −0.562971
$$357$$ 0 0
$$358$$ −12.5470 −0.663132
$$359$$ 6.62855 0.349841 0.174921 0.984583i $$-0.444033\pi$$
0.174921 + 0.984583i $$0.444033\pi$$
$$360$$ 13.1873 0.695033
$$361$$ −6.27485 −0.330255
$$362$$ −15.8838 −0.834833
$$363$$ −29.5301 −1.54993
$$364$$ 0 0
$$365$$ −16.3470 −0.855643
$$366$$ 17.5636 0.918065
$$367$$ −31.2074 −1.62901 −0.814506 0.580156i $$-0.802992\pi$$
−0.814506 + 0.580156i $$0.802992\pi$$
$$368$$ −6.23995 −0.325280
$$369$$ −32.4609 −1.68985
$$370$$ 26.4319 1.37413
$$371$$ 0 0
$$372$$ −27.0500 −1.40248
$$373$$ −15.7746 −0.816778 −0.408389 0.912808i $$-0.633909\pi$$
−0.408389 + 0.912808i $$0.633909\pi$$
$$374$$ 40.0479 2.07083
$$375$$ 28.8570 1.49017
$$376$$ 14.1416 0.729297
$$377$$ 5.90107 0.303921
$$378$$ 0 0
$$379$$ 31.6512 1.62581 0.812907 0.582393i $$-0.197884\pi$$
0.812907 + 0.582393i $$0.197884\pi$$
$$380$$ 21.9511 1.12607
$$381$$ 26.0477 1.33447
$$382$$ 26.2982 1.34553
$$383$$ −12.3935 −0.633278 −0.316639 0.948546i $$-0.602554\pi$$
−0.316639 + 0.948546i $$0.602554\pi$$
$$384$$ 35.4030 1.80665
$$385$$ 0 0
$$386$$ 50.8404 2.58771
$$387$$ 10.6404 0.540882
$$388$$ 15.6747 0.795765
$$389$$ −14.0741 −0.713585 −0.356792 0.934184i $$-0.616130\pi$$
−0.356792 + 0.934184i $$0.616130\pi$$
$$390$$ 11.6259 0.588702
$$391$$ 16.9056 0.854954
$$392$$ 0 0
$$393$$ 13.4215 0.677026
$$394$$ 37.4529 1.88685
$$395$$ −10.9929 −0.553112
$$396$$ 43.3513 2.17848
$$397$$ 6.97305 0.349967 0.174984 0.984571i $$-0.444013\pi$$
0.174984 + 0.984571i $$0.444013\pi$$
$$398$$ −22.2639 −1.11599
$$399$$ 0 0
$$400$$ 0.688560 0.0344280
$$401$$ 2.73682 0.136670 0.0683352 0.997662i $$-0.478231\pi$$
0.0683352 + 0.997662i $$0.478231\pi$$
$$402$$ 15.8800 0.792023
$$403$$ −3.77116 −0.187855
$$404$$ −27.0088 −1.34374
$$405$$ −18.2911 −0.908894
$$406$$ 0 0
$$407$$ 26.9412 1.33543
$$408$$ −18.5983 −0.920753
$$409$$ 24.5154 1.21221 0.606104 0.795386i $$-0.292732\pi$$
0.606104 + 0.795386i $$0.292732\pi$$
$$410$$ −48.8393 −2.41200
$$411$$ −55.0430 −2.71507
$$412$$ −21.1796 −1.04345
$$413$$ 0 0
$$414$$ 30.9261 1.51994
$$415$$ −7.36040 −0.361308
$$416$$ 7.06530 0.346405
$$417$$ −9.89771 −0.484693
$$418$$ 37.8109 1.84939
$$419$$ 3.01252 0.147171 0.0735856 0.997289i $$-0.476556\pi$$
0.0735856 + 0.997289i $$0.476556\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −54.1398 −2.63548
$$423$$ 22.1992 1.07936
$$424$$ −24.6279 −1.19604
$$425$$ −1.86548 −0.0904893
$$426$$ 13.9040 0.673653
$$427$$ 0 0
$$428$$ 19.5590 0.945421
$$429$$ 11.8499 0.572120
$$430$$ 16.0091 0.772028
$$431$$ −18.7942 −0.905286 −0.452643 0.891692i $$-0.649519\pi$$
−0.452643 + 0.891692i $$0.649519\pi$$
$$432$$ 0.423741 0.0203873
$$433$$ 7.76911 0.373360 0.186680 0.982421i $$-0.440227\pi$$
0.186680 + 0.982421i $$0.440227\pi$$
$$434$$ 0 0
$$435$$ 30.9966 1.48617
$$436$$ 12.1067 0.579804
$$437$$ 15.9613 0.763534
$$438$$ 42.1745 2.01518
$$439$$ 37.9681 1.81212 0.906060 0.423150i $$-0.139076\pi$$
0.906060 + 0.423150i $$0.139076\pi$$
$$440$$ 20.2234 0.964112
$$441$$ 0 0
$$442$$ −8.36254 −0.397766
$$443$$ 35.6270 1.69269 0.846344 0.532637i $$-0.178799\pi$$
0.846344 + 0.532637i $$0.178799\pi$$
$$444$$ −40.3523 −1.91504
$$445$$ −7.77860 −0.368741
$$446$$ 64.7673 3.06682
$$447$$ −7.31125 −0.345810
$$448$$ 0 0
$$449$$ −8.05285 −0.380038 −0.190019 0.981780i $$-0.560855\pi$$
−0.190019 + 0.981780i $$0.560855\pi$$
$$450$$ −3.41261 −0.160872
$$451$$ −49.7803 −2.34406
$$452$$ 17.1060 0.804598
$$453$$ 45.9123 2.15715
$$454$$ 22.2963 1.04642
$$455$$ 0 0
$$456$$ −17.5595 −0.822297
$$457$$ 15.5976 0.729626 0.364813 0.931081i $$-0.381133\pi$$
0.364813 + 0.931081i $$0.381133\pi$$
$$458$$ 24.6563 1.15211
$$459$$ −1.14802 −0.0535852
$$460$$ 27.5336 1.28376
$$461$$ 25.6991 1.19692 0.598462 0.801151i $$-0.295779\pi$$
0.598462 + 0.801151i $$0.295779\pi$$
$$462$$ 0 0
$$463$$ −20.5209 −0.953685 −0.476842 0.878989i $$-0.658219\pi$$
−0.476842 + 0.878989i $$0.658219\pi$$
$$464$$ 8.22952 0.382046
$$465$$ −19.8088 −0.918611
$$466$$ −37.8109 −1.75156
$$467$$ −11.8248 −0.547187 −0.273594 0.961845i $$-0.588212\pi$$
−0.273594 + 0.961845i $$0.588212\pi$$
$$468$$ −9.05234 −0.418445
$$469$$ 0 0
$$470$$ 33.4000 1.54063
$$471$$ −24.2485 −1.11731
$$472$$ −9.52677 −0.438505
$$473$$ 16.3176 0.750283
$$474$$ 28.3611 1.30267
$$475$$ −1.76128 −0.0808133
$$476$$ 0 0
$$477$$ −38.6604 −1.77014
$$478$$ 15.3193 0.700690
$$479$$ −22.6552 −1.03514 −0.517571 0.855640i $$-0.673164\pi$$
−0.517571 + 0.855640i $$0.673164\pi$$
$$480$$ 37.1119 1.69392
$$481$$ −5.62570 −0.256510
$$482$$ −14.3783 −0.654913
$$483$$ 0 0
$$484$$ 34.5945 1.57248
$$485$$ 11.4787 0.521219
$$486$$ 49.2078 2.23211
$$487$$ −32.7167 −1.48254 −0.741268 0.671209i $$-0.765775\pi$$
−0.741268 + 0.671209i $$0.765775\pi$$
$$488$$ −6.37969 −0.288795
$$489$$ −34.2317 −1.54801
$$490$$ 0 0
$$491$$ 6.17281 0.278575 0.139288 0.990252i $$-0.455519\pi$$
0.139288 + 0.990252i $$0.455519\pi$$
$$492$$ 74.5605 3.36145
$$493$$ −22.2959 −1.00416
$$494$$ −7.89544 −0.355232
$$495$$ 31.7463 1.42689
$$496$$ −5.25919 −0.236145
$$497$$ 0 0
$$498$$ 18.9895 0.850938
$$499$$ −14.6387 −0.655317 −0.327659 0.944796i $$-0.606260\pi$$
−0.327659 + 0.944796i $$0.606260\pi$$
$$500$$ −33.8060 −1.51185
$$501$$ 42.9398 1.91841
$$502$$ 21.8412 0.974818
$$503$$ −12.7787 −0.569774 −0.284887 0.958561i $$-0.591956\pi$$
−0.284887 + 0.958561i $$0.591956\pi$$
$$504$$ 0 0
$$505$$ −19.7786 −0.880136
$$506$$ 47.4267 2.10837
$$507$$ −2.47443 −0.109893
$$508$$ −30.5149 −1.35388
$$509$$ −11.6853 −0.517940 −0.258970 0.965885i $$-0.583383\pi$$
−0.258970 + 0.965885i $$0.583383\pi$$
$$510$$ −43.9260 −1.94507
$$511$$ 0 0
$$512$$ 15.4017 0.680664
$$513$$ −1.08390 −0.0478553
$$514$$ −15.1938 −0.670168
$$515$$ −15.5099 −0.683448
$$516$$ −24.4403 −1.07592
$$517$$ 34.0435 1.49723
$$518$$ 0 0
$$519$$ −7.32772 −0.321651
$$520$$ −4.22292 −0.185187
$$521$$ −8.47675 −0.371373 −0.185687 0.982609i $$-0.559451\pi$$
−0.185687 + 0.982609i $$0.559451\pi$$
$$522$$ −40.7867 −1.78519
$$523$$ −32.7108 −1.43034 −0.715172 0.698949i $$-0.753652\pi$$
−0.715172 + 0.698949i $$0.753652\pi$$
$$524$$ −15.7233 −0.686876
$$525$$ 0 0
$$526$$ −0.280097 −0.0122128
$$527$$ 14.2485 0.620674
$$528$$ 16.5257 0.719188
$$529$$ −2.97949 −0.129543
$$530$$ −58.1668 −2.52661
$$531$$ −14.9549 −0.648989
$$532$$ 0 0
$$533$$ 10.3948 0.450249
$$534$$ 20.0684 0.868445
$$535$$ 14.3231 0.619243
$$536$$ −5.76814 −0.249146
$$537$$ 14.0272 0.605320
$$538$$ 9.39131 0.404888
$$539$$ 0 0
$$540$$ −1.86975 −0.0804610
$$541$$ −28.1705 −1.21115 −0.605573 0.795790i $$-0.707056\pi$$
−0.605573 + 0.795790i $$0.707056\pi$$
$$542$$ −3.46879 −0.148997
$$543$$ 17.7576 0.762052
$$544$$ −26.6947 −1.14452
$$545$$ 8.86574 0.379767
$$546$$ 0 0
$$547$$ −18.5377 −0.792615 −0.396307 0.918118i $$-0.629709\pi$$
−0.396307 + 0.918118i $$0.629709\pi$$
$$548$$ 64.4829 2.75457
$$549$$ −10.0147 −0.427417
$$550$$ −5.23339 −0.223153
$$551$$ −21.0505 −0.896781
$$552$$ −22.0251 −0.937449
$$553$$ 0 0
$$554$$ −28.2092 −1.19850
$$555$$ −29.5501 −1.25433
$$556$$ 11.5952 0.491745
$$557$$ 4.00283 0.169605 0.0848027 0.996398i $$-0.472974\pi$$
0.0848027 + 0.996398i $$0.472974\pi$$
$$558$$ 26.0653 1.10343
$$559$$ −3.40733 −0.144115
$$560$$ 0 0
$$561$$ −44.7723 −1.89029
$$562$$ 10.2474 0.432259
$$563$$ 17.8620 0.752794 0.376397 0.926459i $$-0.377163\pi$$
0.376397 + 0.926459i $$0.377163\pi$$
$$564$$ −50.9901 −2.14707
$$565$$ 12.5268 0.527005
$$566$$ −8.07490 −0.339413
$$567$$ 0 0
$$568$$ −5.05041 −0.211910
$$569$$ −37.4672 −1.57071 −0.785353 0.619048i $$-0.787519\pi$$
−0.785353 + 0.619048i $$0.787519\pi$$
$$570$$ −41.4724 −1.73709
$$571$$ 17.5703 0.735293 0.367646 0.929966i $$-0.380164\pi$$
0.367646 + 0.929966i $$0.380164\pi$$
$$572$$ −13.8822 −0.580444
$$573$$ −29.4006 −1.22823
$$574$$ 0 0
$$575$$ −2.20920 −0.0921301
$$576$$ −40.1236 −1.67182
$$577$$ 34.2494 1.42582 0.712910 0.701256i $$-0.247377\pi$$
0.712910 + 0.701256i $$0.247377\pi$$
$$578$$ −6.03047 −0.250835
$$579$$ −56.8380 −2.36211
$$580$$ −36.3125 −1.50780
$$581$$ 0 0
$$582$$ −29.6144 −1.22756
$$583$$ −59.2876 −2.45544
$$584$$ −15.3192 −0.633912
$$585$$ −6.62906 −0.274078
$$586$$ 46.5717 1.92386
$$587$$ 29.4494 1.21551 0.607754 0.794126i $$-0.292071\pi$$
0.607754 + 0.794126i $$0.292071\pi$$
$$588$$ 0 0
$$589$$ 13.4526 0.554305
$$590$$ −22.5006 −0.926334
$$591$$ −41.8712 −1.72235
$$592$$ −7.84549 −0.322448
$$593$$ 34.0001 1.39622 0.698109 0.715992i $$-0.254025\pi$$
0.698109 + 0.715992i $$0.254025\pi$$
$$594$$ −3.22064 −0.132145
$$595$$ 0 0
$$596$$ 8.56514 0.350842
$$597$$ 24.8904 1.01870
$$598$$ −9.90335 −0.404978
$$599$$ 21.4418 0.876087 0.438043 0.898954i $$-0.355672\pi$$
0.438043 + 0.898954i $$0.355672\pi$$
$$600$$ 2.43040 0.0992206
$$601$$ −40.4039 −1.64811 −0.824054 0.566511i $$-0.808293\pi$$
−0.824054 + 0.566511i $$0.808293\pi$$
$$602$$ 0 0
$$603$$ −9.05472 −0.368737
$$604$$ −53.7863 −2.18853
$$605$$ 25.3337 1.02996
$$606$$ 51.0278 2.07286
$$607$$ 43.8913 1.78149 0.890746 0.454501i $$-0.150182\pi$$
0.890746 + 0.454501i $$0.150182\pi$$
$$608$$ −25.2036 −1.02214
$$609$$ 0 0
$$610$$ −15.0677 −0.610074
$$611$$ −7.10876 −0.287590
$$612$$ 34.2022 1.38254
$$613$$ −14.3155 −0.578199 −0.289100 0.957299i $$-0.593356\pi$$
−0.289100 + 0.957299i $$0.593356\pi$$
$$614$$ 10.9750 0.442915
$$615$$ 54.6009 2.20172
$$616$$ 0 0
$$617$$ −36.9097 −1.48593 −0.742965 0.669330i $$-0.766581\pi$$
−0.742965 + 0.669330i $$0.766581\pi$$
$$618$$ 40.0148 1.60963
$$619$$ −14.2929 −0.574481 −0.287240 0.957859i $$-0.592738\pi$$
−0.287240 + 0.957859i $$0.592738\pi$$
$$620$$ 23.2060 0.931976
$$621$$ −1.35955 −0.0545568
$$622$$ 5.35973 0.214906
$$623$$ 0 0
$$624$$ −3.45079 −0.138142
$$625$$ −22.2875 −0.891501
$$626$$ −30.8991 −1.23498
$$627$$ −42.2715 −1.68816
$$628$$ 28.4071 1.13357
$$629$$ 21.2554 0.847510
$$630$$ 0 0
$$631$$ −0.0431064 −0.00171604 −0.000858019 1.00000i $$-0.500273\pi$$
−0.000858019 1.00000i $$0.500273\pi$$
$$632$$ −10.3017 −0.409779
$$633$$ 60.5267 2.40572
$$634$$ 6.79184 0.269738
$$635$$ −22.3462 −0.886781
$$636$$ 88.8004 3.52116
$$637$$ 0 0
$$638$$ −62.5484 −2.47632
$$639$$ −7.92803 −0.313628
$$640$$ −30.3720 −1.20056
$$641$$ 42.6655 1.68519 0.842594 0.538550i $$-0.181028\pi$$
0.842594 + 0.538550i $$0.181028\pi$$
$$642$$ −36.9530 −1.45842
$$643$$ 5.49737 0.216795 0.108398 0.994108i $$-0.465428\pi$$
0.108398 + 0.994108i $$0.465428\pi$$
$$644$$ 0 0
$$645$$ −17.8977 −0.704722
$$646$$ 29.8311 1.17369
$$647$$ 38.1867 1.50127 0.750637 0.660715i $$-0.229747\pi$$
0.750637 + 0.660715i $$0.229747\pi$$
$$648$$ −17.1411 −0.673364
$$649$$ −22.9341 −0.900242
$$650$$ 1.09280 0.0428633
$$651$$ 0 0
$$652$$ 40.1024 1.57053
$$653$$ 38.5019 1.50670 0.753349 0.657621i $$-0.228437\pi$$
0.753349 + 0.657621i $$0.228437\pi$$
$$654$$ −22.8732 −0.894412
$$655$$ −11.5142 −0.449898
$$656$$ 14.4964 0.565990
$$657$$ −24.0477 −0.938192
$$658$$ 0 0
$$659$$ 19.4843 0.759002 0.379501 0.925191i $$-0.376096\pi$$
0.379501 + 0.925191i $$0.376096\pi$$
$$660$$ −72.9191 −2.83837
$$661$$ 41.6667 1.62065 0.810324 0.585983i $$-0.199291\pi$$
0.810324 + 0.585983i $$0.199291\pi$$
$$662$$ 30.1127 1.17036
$$663$$ 9.34907 0.363088
$$664$$ −6.89760 −0.267679
$$665$$ 0 0
$$666$$ 38.8834 1.50670
$$667$$ −26.4039 −1.02236
$$668$$ −50.3040 −1.94632
$$669$$ −72.4079 −2.79945
$$670$$ −13.6234 −0.526316
$$671$$ −15.3580 −0.592890
$$672$$ 0 0
$$673$$ −14.3157 −0.551830 −0.275915 0.961182i $$-0.588981\pi$$
−0.275915 + 0.961182i $$0.588981\pi$$
$$674$$ −77.8306 −2.99792
$$675$$ 0.150022 0.00577435
$$676$$ 2.89879 0.111492
$$677$$ 29.5281 1.13486 0.567429 0.823423i $$-0.307938\pi$$
0.567429 + 0.823423i $$0.307938\pi$$
$$678$$ −32.3184 −1.24118
$$679$$ 0 0
$$680$$ 15.9554 0.611860
$$681$$ −24.9266 −0.955191
$$682$$ 39.9724 1.53062
$$683$$ 47.0699 1.80108 0.900539 0.434774i $$-0.143172\pi$$
0.900539 + 0.434774i $$0.143172\pi$$
$$684$$ 32.2918 1.23471
$$685$$ 47.2210 1.80422
$$686$$ 0 0
$$687$$ −27.5650 −1.05167
$$688$$ −4.75180 −0.181161
$$689$$ 12.3801 0.471643
$$690$$ −52.0194 −1.98034
$$691$$ 30.8668 1.17423 0.587113 0.809505i $$-0.300264\pi$$
0.587113 + 0.809505i $$0.300264\pi$$
$$692$$ 8.58442 0.326331
$$693$$ 0 0
$$694$$ −12.1091 −0.459656
$$695$$ 8.49118 0.322089
$$696$$ 29.0476 1.10105
$$697$$ −39.2745 −1.48763
$$698$$ −9.61016 −0.363750
$$699$$ 42.2715 1.59885
$$700$$ 0 0
$$701$$ 6.48958 0.245108 0.122554 0.992462i $$-0.460892\pi$$
0.122554 + 0.992462i $$0.460892\pi$$
$$702$$ 0.672514 0.0253824
$$703$$ 20.0682 0.756885
$$704$$ −61.5315 −2.31905
$$705$$ −37.3402 −1.40631
$$706$$ −61.0860 −2.29900
$$707$$ 0 0
$$708$$ 34.3505 1.29097
$$709$$ −13.3738 −0.502263 −0.251131 0.967953i $$-0.580803\pi$$
−0.251131 + 0.967953i $$0.580803\pi$$
$$710$$ −11.9282 −0.447657
$$711$$ −16.1714 −0.606474
$$712$$ −7.28951 −0.273186
$$713$$ 16.8738 0.631929
$$714$$ 0 0
$$715$$ −10.1660 −0.380186
$$716$$ −16.4329 −0.614126
$$717$$ −17.1266 −0.639603
$$718$$ 14.6711 0.547521
$$719$$ 16.7410 0.624333 0.312166 0.950027i $$-0.398945\pi$$
0.312166 + 0.950027i $$0.398945\pi$$
$$720$$ −9.24476 −0.344532
$$721$$ 0 0
$$722$$ −13.8883 −0.516868
$$723$$ 16.0745 0.597817
$$724$$ −20.8030 −0.773139
$$725$$ 2.91359 0.108208
$$726$$ −65.3596 −2.42572
$$727$$ −38.8138 −1.43952 −0.719761 0.694221i $$-0.755749\pi$$
−0.719761 + 0.694221i $$0.755749\pi$$
$$728$$ 0 0
$$729$$ −29.1632 −1.08012
$$730$$ −36.1812 −1.33913
$$731$$ 12.8738 0.476156
$$732$$ 23.0031 0.850220
$$733$$ −37.7279 −1.39351 −0.696756 0.717309i $$-0.745374\pi$$
−0.696756 + 0.717309i $$0.745374\pi$$
$$734$$ −69.0719 −2.54949
$$735$$ 0 0
$$736$$ −31.6132 −1.16528
$$737$$ −13.8858 −0.511491
$$738$$ −71.8464 −2.64470
$$739$$ −9.22952 −0.339514 −0.169757 0.985486i $$-0.554298\pi$$
−0.169757 + 0.985486i $$0.554298\pi$$
$$740$$ 34.6180 1.27258
$$741$$ 8.82686 0.324263
$$742$$ 0 0
$$743$$ −3.56327 −0.130724 −0.0653619 0.997862i $$-0.520820\pi$$
−0.0653619 + 0.997862i $$0.520820\pi$$
$$744$$ −18.5633 −0.680563
$$745$$ 6.27228 0.229798
$$746$$ −34.9143 −1.27830
$$747$$ −10.8277 −0.396165
$$748$$ 52.4508 1.91779
$$749$$ 0 0
$$750$$ 63.8698 2.33220
$$751$$ 51.2106 1.86870 0.934350 0.356357i $$-0.115981\pi$$
0.934350 + 0.356357i $$0.115981\pi$$
$$752$$ −9.91374 −0.361517
$$753$$ −24.4178 −0.889833
$$754$$ 13.0610 0.475653
$$755$$ −39.3879 −1.43347
$$756$$ 0 0
$$757$$ 25.2305 0.917019 0.458509 0.888690i $$-0.348384\pi$$
0.458509 + 0.888690i $$0.348384\pi$$
$$758$$ 70.0543 2.54449
$$759$$ −53.0217 −1.92456
$$760$$ 15.0641 0.546434
$$761$$ −3.64744 −0.132220 −0.0661099 0.997812i $$-0.521059\pi$$
−0.0661099 + 0.997812i $$0.521059\pi$$
$$762$$ 57.6520 2.08851
$$763$$ 0 0
$$764$$ 34.4428 1.24610
$$765$$ 25.0464 0.905555
$$766$$ −27.4308 −0.991115
$$767$$ 4.78896 0.172919
$$768$$ 14.7722 0.533045
$$769$$ −21.9882 −0.792914 −0.396457 0.918053i $$-0.629760\pi$$
−0.396457 + 0.918053i $$0.629760\pi$$
$$770$$ 0 0
$$771$$ 16.9862 0.611742
$$772$$ 66.5858 2.39647
$$773$$ −21.8590 −0.786215 −0.393108 0.919493i $$-0.628600\pi$$
−0.393108 + 0.919493i $$0.628600\pi$$
$$774$$ 23.5506 0.846510
$$775$$ −1.86197 −0.0668840
$$776$$ 10.7569 0.386151
$$777$$ 0 0
$$778$$ −31.1505 −1.11680
$$779$$ −37.0807 −1.32855
$$780$$ 15.2265 0.545197
$$781$$ −12.1580 −0.435048
$$782$$ 37.4176 1.33805
$$783$$ 1.79303 0.0640777
$$784$$ 0 0
$$785$$ 20.8026 0.742477
$$786$$ 29.7061 1.05958
$$787$$ −39.8673 −1.42111 −0.710557 0.703639i $$-0.751557\pi$$
−0.710557 + 0.703639i $$0.751557\pi$$
$$788$$ 49.0522 1.74741
$$789$$ 0.313141 0.0111481
$$790$$ −24.3308 −0.865651
$$791$$ 0 0
$$792$$ 29.7502 1.05713
$$793$$ 3.20697 0.113883
$$794$$ 15.4336 0.547718
$$795$$ 65.0288 2.30633
$$796$$ −29.1591 −1.03352
$$797$$ 40.1971 1.42385 0.711927 0.702253i $$-0.247822\pi$$
0.711927 + 0.702253i $$0.247822\pi$$
$$798$$ 0 0
$$799$$ 26.8589 0.950198
$$800$$ 3.48842 0.123334
$$801$$ −11.4429 −0.404316
$$802$$ 6.05747 0.213897
$$803$$ −36.8784 −1.30141
$$804$$ 20.7981 0.733492
$$805$$ 0 0
$$806$$ −8.34680 −0.294003
$$807$$ −10.4992 −0.369589
$$808$$ −18.5350 −0.652058
$$809$$ 2.53849 0.0892485 0.0446243 0.999004i $$-0.485791\pi$$
0.0446243 + 0.999004i $$0.485791\pi$$
$$810$$ −40.4842 −1.42247
$$811$$ 41.7062 1.46450 0.732251 0.681035i $$-0.238470\pi$$
0.732251 + 0.681035i $$0.238470\pi$$
$$812$$ 0 0
$$813$$ 3.87801 0.136008
$$814$$ 59.6296 2.09002
$$815$$ 29.3671 1.02869
$$816$$ 13.0380 0.456423
$$817$$ 12.1547 0.425241
$$818$$ 54.2604 1.89717
$$819$$ 0 0
$$820$$ −63.9650 −2.23375
$$821$$ 31.9304 1.11438 0.557189 0.830386i $$-0.311880\pi$$
0.557189 + 0.830386i $$0.311880\pi$$
$$822$$ −121.828 −4.24924
$$823$$ −34.2531 −1.19399 −0.596995 0.802245i $$-0.703639\pi$$
−0.596995 + 0.802245i $$0.703639\pi$$
$$824$$ −14.5347 −0.506340
$$825$$ 5.85078 0.203698
$$826$$ 0 0
$$827$$ 36.9755 1.28576 0.642882 0.765965i $$-0.277739\pi$$
0.642882 + 0.765965i $$0.277739\pi$$
$$828$$ 40.5040 1.40761
$$829$$ 19.9895 0.694263 0.347131 0.937817i $$-0.387156\pi$$
0.347131 + 0.937817i $$0.387156\pi$$
$$830$$ −16.2909 −0.565466
$$831$$ 31.5371 1.09401
$$832$$ 12.8486 0.445446
$$833$$ 0 0
$$834$$ −21.9068 −0.758571
$$835$$ −36.8378 −1.27482
$$836$$ 49.5210 1.71272
$$837$$ −1.14586 −0.0396068
$$838$$ 6.66768 0.230331
$$839$$ −12.8147 −0.442411 −0.221206 0.975227i $$-0.570999\pi$$
−0.221206 + 0.975227i $$0.570999\pi$$
$$840$$ 0 0
$$841$$ 5.82265 0.200781
$$842$$ −22.1332 −0.762761
$$843$$ −11.4562 −0.394574
$$844$$ −70.9070 −2.44072
$$845$$ 2.12280 0.0730264
$$846$$ 49.1340 1.68926
$$847$$ 0 0
$$848$$ 17.2650 0.592882
$$849$$ 9.02750 0.309823
$$850$$ −4.12892 −0.141621
$$851$$ 25.1718 0.862877
$$852$$ 18.2102 0.623870
$$853$$ 30.1839 1.03348 0.516739 0.856143i $$-0.327146\pi$$
0.516739 + 0.856143i $$0.327146\pi$$
$$854$$ 0 0
$$855$$ 23.6474 0.808724
$$856$$ 13.4225 0.458773
$$857$$ 53.2327 1.81839 0.909197 0.416366i $$-0.136696\pi$$
0.909197 + 0.416366i $$0.136696\pi$$
$$858$$ 26.2277 0.895399
$$859$$ 12.2719 0.418713 0.209357 0.977839i $$-0.432863\pi$$
0.209357 + 0.977839i $$0.432863\pi$$
$$860$$ 20.9672 0.714975
$$861$$ 0 0
$$862$$ −41.5977 −1.41682
$$863$$ 24.4453 0.832127 0.416064 0.909335i $$-0.363409\pi$$
0.416064 + 0.909335i $$0.363409\pi$$
$$864$$ 2.14678 0.0730349
$$865$$ 6.28640 0.213744
$$866$$ 17.1956 0.584329
$$867$$ 6.74189 0.228967
$$868$$ 0 0
$$869$$ −24.7996 −0.841268
$$870$$ 68.6054 2.32594
$$871$$ 2.89955 0.0982477
$$872$$ 8.30829 0.281354
$$873$$ 16.8860 0.571504
$$874$$ 35.3276 1.19497
$$875$$ 0 0
$$876$$ 55.2361 1.86625
$$877$$ −52.8753 −1.78547 −0.892736 0.450580i $$-0.851217\pi$$
−0.892736 + 0.450580i $$0.851217\pi$$
$$878$$ 84.0357 2.83607
$$879$$ −52.0658 −1.75614
$$880$$ −14.1773 −0.477916
$$881$$ −55.0118 −1.85339 −0.926697 0.375809i $$-0.877365\pi$$
−0.926697 + 0.375809i $$0.877365\pi$$
$$882$$ 0 0
$$883$$ 44.1730 1.48654 0.743269 0.668992i $$-0.233274\pi$$
0.743269 + 0.668992i $$0.233274\pi$$
$$884$$ −10.9524 −0.368371
$$885$$ 25.1550 0.845575
$$886$$ 78.8539 2.64915
$$887$$ 5.08659 0.170791 0.0853955 0.996347i $$-0.472785\pi$$
0.0853955 + 0.996347i $$0.472785\pi$$
$$888$$ −27.6921 −0.929286
$$889$$ 0 0
$$890$$ −17.2165 −0.577100
$$891$$ −41.2642 −1.38240
$$892$$ 84.8259 2.84018
$$893$$ 25.3586 0.848593
$$894$$ −16.1822 −0.541212
$$895$$ −12.0339 −0.402248
$$896$$ 0 0
$$897$$ 11.0717 0.369672
$$898$$ −17.8236 −0.594780
$$899$$ −22.2539 −0.742209
$$900$$ −4.46950 −0.148983
$$901$$ −46.7753 −1.55831
$$902$$ −110.180 −3.66859
$$903$$ 0 0
$$904$$ 11.7391 0.390437
$$905$$ −15.2341 −0.506400
$$906$$ 101.619 3.37606
$$907$$ −18.1253 −0.601840 −0.300920 0.953649i $$-0.597294\pi$$
−0.300920 + 0.953649i $$0.597294\pi$$
$$908$$ 29.2016 0.969088
$$909$$ −29.0958 −0.965048
$$910$$ 0 0
$$911$$ −9.65804 −0.319985 −0.159993 0.987118i $$-0.551147\pi$$
−0.159993 + 0.987118i $$0.551147\pi$$
$$912$$ 12.3098 0.407617
$$913$$ −16.6048 −0.549539
$$914$$ 34.5226 1.14190
$$915$$ 16.8453 0.556887
$$916$$ 32.2924 1.06697
$$917$$ 0 0
$$918$$ −2.54095 −0.0838637
$$919$$ 47.7603 1.57547 0.787733 0.616017i $$-0.211255\pi$$
0.787733 + 0.616017i $$0.211255\pi$$
$$920$$ 18.8952 0.622955
$$921$$ −12.2697 −0.404301
$$922$$ 56.8803 1.87325
$$923$$ 2.53876 0.0835643
$$924$$ 0 0
$$925$$ −2.77763 −0.0913279
$$926$$ −45.4193 −1.49257
$$927$$ −22.8163 −0.749384
$$928$$ 41.6928 1.36863
$$929$$ −33.9811 −1.11488 −0.557442 0.830216i $$-0.688217\pi$$
−0.557442 + 0.830216i $$0.688217\pi$$
$$930$$ −43.8433 −1.43768
$$931$$ 0 0
$$932$$ −49.5210 −1.62212
$$933$$ −5.99202 −0.196170
$$934$$ −26.1721 −0.856378
$$935$$ 38.4099 1.25614
$$936$$ −6.21224 −0.203053
$$937$$ 24.7948 0.810012 0.405006 0.914314i $$-0.367269\pi$$
0.405006 + 0.914314i $$0.367269\pi$$
$$938$$ 0 0
$$939$$ 34.5443 1.12731
$$940$$ 43.7441 1.42677
$$941$$ 8.24196 0.268680 0.134340 0.990935i $$-0.457109\pi$$
0.134340 + 0.990935i $$0.457109\pi$$
$$942$$ −53.6697 −1.74865
$$943$$ −46.5108 −1.51460
$$944$$ 6.67859 0.217370
$$945$$ 0 0
$$946$$ 36.1160 1.17423
$$947$$ 19.9729 0.649031 0.324515 0.945880i $$-0.394799\pi$$
0.324515 + 0.945880i $$0.394799\pi$$
$$948$$ 37.1446 1.20640
$$949$$ 7.70071 0.249976
$$950$$ −3.89829 −0.126477
$$951$$ −7.59307 −0.246222
$$952$$ 0 0
$$953$$ −21.5341 −0.697557 −0.348778 0.937205i $$-0.613403\pi$$
−0.348778 + 0.937205i $$0.613403\pi$$
$$954$$ −85.5679 −2.77036
$$955$$ 25.2225 0.816182
$$956$$ 20.0638 0.648909
$$957$$ 69.9273 2.26043
$$958$$ −50.1432 −1.62005
$$959$$ 0 0
$$960$$ 67.4900 2.17823
$$961$$ −16.7783 −0.541237
$$962$$ −12.4515 −0.401452
$$963$$ 21.0704 0.678985
$$964$$ −18.8313 −0.606515
$$965$$ 48.7610 1.56967
$$966$$ 0 0
$$967$$ 43.2887 1.39207 0.696036 0.718007i $$-0.254945\pi$$
0.696036 + 0.718007i $$0.254945\pi$$
$$968$$ 23.7408 0.763057
$$969$$ −33.3503 −1.07137
$$970$$ 25.4060 0.815737
$$971$$ 52.6713 1.69030 0.845151 0.534528i $$-0.179510\pi$$
0.845151 + 0.534528i $$0.179510\pi$$
$$972$$ 64.4476 2.06716
$$973$$ 0 0
$$974$$ −72.4127 −2.32025
$$975$$ −1.22172 −0.0391265
$$976$$ 4.47238 0.143157
$$977$$ 15.4061 0.492885 0.246442 0.969157i $$-0.420738\pi$$
0.246442 + 0.969157i $$0.420738\pi$$
$$978$$ −75.7658 −2.42272
$$979$$ −17.5483 −0.560845
$$980$$ 0 0
$$981$$ 13.0422 0.416405
$$982$$ 13.6624 0.435985
$$983$$ −7.58146 −0.241811 −0.120906 0.992664i $$-0.538580\pi$$
−0.120906 + 0.992664i $$0.538580\pi$$
$$984$$ 51.1677 1.63117
$$985$$ 35.9211 1.14454
$$986$$ −49.3480 −1.57156
$$987$$ 0 0
$$988$$ −10.3407 −0.328981
$$989$$ 15.2459 0.484790
$$990$$ 70.2647 2.23316
$$991$$ −19.0185 −0.604141 −0.302071 0.953286i $$-0.597678\pi$$
−0.302071 + 0.953286i $$0.597678\pi$$
$$992$$ −26.6444 −0.845960
$$993$$ −33.6651 −1.06833
$$994$$ 0 0
$$995$$ −21.3533 −0.676946
$$996$$ 24.8706 0.788054
$$997$$ −46.0998 −1.46000 −0.729998 0.683449i $$-0.760479\pi$$
−0.729998 + 0.683449i $$0.760479\pi$$
$$998$$ −32.4001 −1.02561
$$999$$ −1.70936 −0.0540817
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 637.2.a.k.1.4 5
3.2 odd 2 5733.2.a.bm.1.2 5
7.2 even 3 637.2.e.m.508.2 10
7.3 odd 6 91.2.e.c.79.2 yes 10
7.4 even 3 637.2.e.m.79.2 10
7.5 odd 6 91.2.e.c.53.2 10
7.6 odd 2 637.2.a.l.1.4 5
13.12 even 2 8281.2.a.bx.1.2 5
21.5 even 6 819.2.j.h.235.4 10
21.17 even 6 819.2.j.h.352.4 10
21.20 even 2 5733.2.a.bl.1.2 5
28.3 even 6 1456.2.r.p.625.5 10
28.19 even 6 1456.2.r.p.417.5 10
91.12 odd 6 1183.2.e.f.508.4 10
91.38 odd 6 1183.2.e.f.170.4 10
91.90 odd 2 8281.2.a.bw.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.2 10 7.5 odd 6
91.2.e.c.79.2 yes 10 7.3 odd 6
637.2.a.k.1.4 5 1.1 even 1 trivial
637.2.a.l.1.4 5 7.6 odd 2
637.2.e.m.79.2 10 7.4 even 3
637.2.e.m.508.2 10 7.2 even 3
819.2.j.h.235.4 10 21.5 even 6
819.2.j.h.352.4 10 21.17 even 6
1183.2.e.f.170.4 10 91.38 odd 6
1183.2.e.f.508.4 10 91.12 odd 6
1456.2.r.p.417.5 10 28.19 even 6
1456.2.r.p.625.5 10 28.3 even 6
5733.2.a.bl.1.2 5 21.20 even 2
5733.2.a.bm.1.2 5 3.2 odd 2
8281.2.a.bw.1.2 5 91.90 odd 2
8281.2.a.bx.1.2 5 13.12 even 2