# Properties

 Label 4650.2.a.by.1.1 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.1304369399$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.627719 q^{11} -1.00000 q^{12} +2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} +4.74456 q^{17} -1.00000 q^{18} -0.627719 q^{19} +3.37228 q^{21} -0.627719 q^{22} -3.37228 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -3.37228 q^{28} -8.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} -0.627719 q^{33} -4.74456 q^{34} +1.00000 q^{36} +0.744563 q^{37} +0.627719 q^{38} -2.00000 q^{39} +0.744563 q^{41} -3.37228 q^{42} -0.627719 q^{43} +0.627719 q^{44} +3.37228 q^{46} +6.74456 q^{47} -1.00000 q^{48} +4.37228 q^{49} -4.74456 q^{51} +2.00000 q^{52} -1.37228 q^{53} +1.00000 q^{54} +3.37228 q^{56} +0.627719 q^{57} +8.74456 q^{58} +2.74456 q^{59} +11.4891 q^{61} +1.00000 q^{62} -3.37228 q^{63} +1.00000 q^{64} +0.627719 q^{66} -10.7446 q^{67} +4.74456 q^{68} +3.37228 q^{69} +3.37228 q^{71} -1.00000 q^{72} +8.11684 q^{73} -0.744563 q^{74} -0.627719 q^{76} -2.11684 q^{77} +2.00000 q^{78} -4.62772 q^{79} +1.00000 q^{81} -0.744563 q^{82} +12.0000 q^{83} +3.37228 q^{84} +0.627719 q^{86} +8.74456 q^{87} -0.627719 q^{88} -1.37228 q^{89} -6.74456 q^{91} -3.37228 q^{92} +1.00000 q^{93} -6.74456 q^{94} +1.00000 q^{96} -2.00000 q^{97} -4.37228 q^{98} +0.627719 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{12} + 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 7 q^{19} + q^{21} - 7 q^{22} - q^{23} + 2 q^{24} - 4 q^{26} - 2 q^{27} - q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} - 7 q^{33} + 2 q^{34} + 2 q^{36} - 10 q^{37} + 7 q^{38} - 4 q^{39} - 10 q^{41} - q^{42} - 7 q^{43} + 7 q^{44} + q^{46} + 2 q^{47} - 2 q^{48} + 3 q^{49} + 2 q^{51} + 4 q^{52} + 3 q^{53} + 2 q^{54} + q^{56} + 7 q^{57} + 6 q^{58} - 6 q^{59} + 2 q^{62} - q^{63} + 2 q^{64} + 7 q^{66} - 10 q^{67} - 2 q^{68} + q^{69} + q^{71} - 2 q^{72} - q^{73} + 10 q^{74} - 7 q^{76} + 13 q^{77} + 4 q^{78} - 15 q^{79} + 2 q^{81} + 10 q^{82} + 24 q^{83} + q^{84} + 7 q^{86} + 6 q^{87} - 7 q^{88} + 3 q^{89} - 2 q^{91} - q^{92} + 2 q^{93} - 2 q^{94} + 2 q^{96} - 4 q^{97} - 3 q^{98} + 7 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 7 * q^11 - 2 * q^12 + 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 7 * q^19 + q^21 - 7 * q^22 - q^23 + 2 * q^24 - 4 * q^26 - 2 * q^27 - q^28 - 6 * q^29 - 2 * q^31 - 2 * q^32 - 7 * q^33 + 2 * q^34 + 2 * q^36 - 10 * q^37 + 7 * q^38 - 4 * q^39 - 10 * q^41 - q^42 - 7 * q^43 + 7 * q^44 + q^46 + 2 * q^47 - 2 * q^48 + 3 * q^49 + 2 * q^51 + 4 * q^52 + 3 * q^53 + 2 * q^54 + q^56 + 7 * q^57 + 6 * q^58 - 6 * q^59 + 2 * q^62 - q^63 + 2 * q^64 + 7 * q^66 - 10 * q^67 - 2 * q^68 + q^69 + q^71 - 2 * q^72 - q^73 + 10 * q^74 - 7 * q^76 + 13 * q^77 + 4 * q^78 - 15 * q^79 + 2 * q^81 + 10 * q^82 + 24 * q^83 + q^84 + 7 * q^86 + 6 * q^87 - 7 * q^88 + 3 * q^89 - 2 * q^91 - q^92 + 2 * q^93 - 2 * q^94 + 2 * q^96 - 4 * q^97 - 3 * q^98 + 7 * 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −3.37228 −1.27460 −0.637301 0.770615i $$-0.719949\pi$$
−0.637301 + 0.770615i $$0.719949\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.627719 0.189264 0.0946322 0.995512i $$-0.469833\pi$$
0.0946322 + 0.995512i $$0.469833\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 3.37228 0.901280
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.74456 1.15073 0.575363 0.817898i $$-0.304861\pi$$
0.575363 + 0.817898i $$0.304861\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −0.627719 −0.144009 −0.0720043 0.997404i $$-0.522940\pi$$
−0.0720043 + 0.997404i $$0.522940\pi$$
$$20$$ 0 0
$$21$$ 3.37228 0.735892
$$22$$ −0.627719 −0.133830
$$23$$ −3.37228 −0.703169 −0.351585 0.936156i $$-0.614357\pi$$
−0.351585 + 0.936156i $$0.614357\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −3.37228 −0.637301
$$29$$ −8.74456 −1.62382 −0.811912 0.583779i $$-0.801573\pi$$
−0.811912 + 0.583779i $$0.801573\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ −1.00000 −0.176777
$$33$$ −0.627719 −0.109272
$$34$$ −4.74456 −0.813686
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 0.744563 0.122405 0.0612027 0.998125i $$-0.480506\pi$$
0.0612027 + 0.998125i $$0.480506\pi$$
$$38$$ 0.627719 0.101829
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 0.744563 0.116281 0.0581406 0.998308i $$-0.481483\pi$$
0.0581406 + 0.998308i $$0.481483\pi$$
$$42$$ −3.37228 −0.520354
$$43$$ −0.627719 −0.0957262 −0.0478631 0.998854i $$-0.515241\pi$$
−0.0478631 + 0.998854i $$0.515241\pi$$
$$44$$ 0.627719 0.0946322
$$45$$ 0 0
$$46$$ 3.37228 0.497216
$$47$$ 6.74456 0.983796 0.491898 0.870653i $$-0.336303\pi$$
0.491898 + 0.870653i $$0.336303\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 4.37228 0.624612
$$50$$ 0 0
$$51$$ −4.74456 −0.664372
$$52$$ 2.00000 0.277350
$$53$$ −1.37228 −0.188497 −0.0942487 0.995549i $$-0.530045\pi$$
−0.0942487 + 0.995549i $$0.530045\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 3.37228 0.450640
$$57$$ 0.627719 0.0831434
$$58$$ 8.74456 1.14822
$$59$$ 2.74456 0.357312 0.178656 0.983912i $$-0.442825\pi$$
0.178656 + 0.983912i $$0.442825\pi$$
$$60$$ 0 0
$$61$$ 11.4891 1.47103 0.735516 0.677507i $$-0.236940\pi$$
0.735516 + 0.677507i $$0.236940\pi$$
$$62$$ 1.00000 0.127000
$$63$$ −3.37228 −0.424868
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.627719 0.0772668
$$67$$ −10.7446 −1.31266 −0.656329 0.754475i $$-0.727892\pi$$
−0.656329 + 0.754475i $$0.727892\pi$$
$$68$$ 4.74456 0.575363
$$69$$ 3.37228 0.405975
$$70$$ 0 0
$$71$$ 3.37228 0.400216 0.200108 0.979774i $$-0.435871\pi$$
0.200108 + 0.979774i $$0.435871\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 8.11684 0.950005 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$74$$ −0.744563 −0.0865536
$$75$$ 0 0
$$76$$ −0.627719 −0.0720043
$$77$$ −2.11684 −0.241237
$$78$$ 2.00000 0.226455
$$79$$ −4.62772 −0.520659 −0.260330 0.965520i $$-0.583831\pi$$
−0.260330 + 0.965520i $$0.583831\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −0.744563 −0.0822232
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 3.37228 0.367946
$$85$$ 0 0
$$86$$ 0.627719 0.0676886
$$87$$ 8.74456 0.937516
$$88$$ −0.627719 −0.0669150
$$89$$ −1.37228 −0.145462 −0.0727308 0.997352i $$-0.523171\pi$$
−0.0727308 + 0.997352i $$0.523171\pi$$
$$90$$ 0 0
$$91$$ −6.74456 −0.707022
$$92$$ −3.37228 −0.351585
$$93$$ 1.00000 0.103695
$$94$$ −6.74456 −0.695649
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −4.37228 −0.441667
$$99$$ 0.627719 0.0630881
$$100$$ 0 0
$$101$$ 8.11684 0.807656 0.403828 0.914835i $$-0.367679\pi$$
0.403828 + 0.914835i $$0.367679\pi$$
$$102$$ 4.74456 0.469782
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 1.37228 0.133288
$$107$$ −0.627719 −0.0606839 −0.0303419 0.999540i $$-0.509660\pi$$
−0.0303419 + 0.999540i $$0.509660\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.74456 0.454447 0.227223 0.973843i $$-0.427035\pi$$
0.227223 + 0.973843i $$0.427035\pi$$
$$110$$ 0 0
$$111$$ −0.744563 −0.0706708
$$112$$ −3.37228 −0.318651
$$113$$ 2.62772 0.247195 0.123597 0.992332i $$-0.460557\pi$$
0.123597 + 0.992332i $$0.460557\pi$$
$$114$$ −0.627719 −0.0587912
$$115$$ 0 0
$$116$$ −8.74456 −0.811912
$$117$$ 2.00000 0.184900
$$118$$ −2.74456 −0.252657
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ −10.6060 −0.964179
$$122$$ −11.4891 −1.04018
$$123$$ −0.744563 −0.0671350
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 3.37228 0.300427
$$127$$ −13.4891 −1.19697 −0.598483 0.801135i $$-0.704230\pi$$
−0.598483 + 0.801135i $$0.704230\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0.627719 0.0552675
$$130$$ 0 0
$$131$$ −16.2337 −1.41834 −0.709172 0.705036i $$-0.750931\pi$$
−0.709172 + 0.705036i $$0.750931\pi$$
$$132$$ −0.627719 −0.0546359
$$133$$ 2.11684 0.183554
$$134$$ 10.7446 0.928189
$$135$$ 0 0
$$136$$ −4.74456 −0.406843
$$137$$ 3.48913 0.298096 0.149048 0.988830i $$-0.452379\pi$$
0.149048 + 0.988830i $$0.452379\pi$$
$$138$$ −3.37228 −0.287068
$$139$$ −10.7446 −0.911342 −0.455671 0.890148i $$-0.650601\pi$$
−0.455671 + 0.890148i $$0.650601\pi$$
$$140$$ 0 0
$$141$$ −6.74456 −0.567995
$$142$$ −3.37228 −0.282996
$$143$$ 1.25544 0.104985
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −8.11684 −0.671755
$$147$$ −4.37228 −0.360620
$$148$$ 0.744563 0.0612027
$$149$$ −12.1168 −0.992651 −0.496325 0.868137i $$-0.665318\pi$$
−0.496325 + 0.868137i $$0.665318\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0.627719 0.0509147
$$153$$ 4.74456 0.383575
$$154$$ 2.11684 0.170580
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −9.37228 −0.747989 −0.373995 0.927431i $$-0.622012\pi$$
−0.373995 + 0.927431i $$0.622012\pi$$
$$158$$ 4.62772 0.368162
$$159$$ 1.37228 0.108829
$$160$$ 0 0
$$161$$ 11.3723 0.896261
$$162$$ −1.00000 −0.0785674
$$163$$ −24.2337 −1.89813 −0.949064 0.315082i $$-0.897968\pi$$
−0.949064 + 0.315082i $$0.897968\pi$$
$$164$$ 0.744563 0.0581406
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 12.6277 0.977162 0.488581 0.872518i $$-0.337515\pi$$
0.488581 + 0.872518i $$0.337515\pi$$
$$168$$ −3.37228 −0.260177
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −0.627719 −0.0480028
$$172$$ −0.627719 −0.0478631
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −8.74456 −0.662924
$$175$$ 0 0
$$176$$ 0.627719 0.0473161
$$177$$ −2.74456 −0.206294
$$178$$ 1.37228 0.102857
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 14.8614 1.10464 0.552320 0.833632i $$-0.313743\pi$$
0.552320 + 0.833632i $$0.313743\pi$$
$$182$$ 6.74456 0.499940
$$183$$ −11.4891 −0.849301
$$184$$ 3.37228 0.248608
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ 2.97825 0.217791
$$188$$ 6.74456 0.491898
$$189$$ 3.37228 0.245297
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −15.4891 −1.11493 −0.557466 0.830200i $$-0.688226\pi$$
−0.557466 + 0.830200i $$0.688226\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 4.37228 0.312306
$$197$$ −19.4891 −1.38854 −0.694271 0.719713i $$-0.744273\pi$$
−0.694271 + 0.719713i $$0.744273\pi$$
$$198$$ −0.627719 −0.0446100
$$199$$ −12.6277 −0.895155 −0.447578 0.894245i $$-0.647713\pi$$
−0.447578 + 0.894245i $$0.647713\pi$$
$$200$$ 0 0
$$201$$ 10.7446 0.757863
$$202$$ −8.11684 −0.571099
$$203$$ 29.4891 2.06973
$$204$$ −4.74456 −0.332186
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ −3.37228 −0.234390
$$208$$ 2.00000 0.138675
$$209$$ −0.394031 −0.0272557
$$210$$ 0 0
$$211$$ −0.627719 −0.0432139 −0.0216070 0.999767i $$-0.506878\pi$$
−0.0216070 + 0.999767i $$0.506878\pi$$
$$212$$ −1.37228 −0.0942487
$$213$$ −3.37228 −0.231065
$$214$$ 0.627719 0.0429100
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 3.37228 0.228925
$$218$$ −4.74456 −0.321342
$$219$$ −8.11684 −0.548485
$$220$$ 0 0
$$221$$ 9.48913 0.638308
$$222$$ 0.744563 0.0499718
$$223$$ −9.25544 −0.619790 −0.309895 0.950771i $$-0.600294\pi$$
−0.309895 + 0.950771i $$0.600294\pi$$
$$224$$ 3.37228 0.225320
$$225$$ 0 0
$$226$$ −2.62772 −0.174793
$$227$$ −6.11684 −0.405989 −0.202995 0.979180i $$-0.565067\pi$$
−0.202995 + 0.979180i $$0.565067\pi$$
$$228$$ 0.627719 0.0415717
$$229$$ 3.88316 0.256606 0.128303 0.991735i $$-0.459047\pi$$
0.128303 + 0.991735i $$0.459047\pi$$
$$230$$ 0 0
$$231$$ 2.11684 0.139278
$$232$$ 8.74456 0.574109
$$233$$ 14.8614 0.973603 0.486802 0.873513i $$-0.338163\pi$$
0.486802 + 0.873513i $$0.338163\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 2.74456 0.178656
$$237$$ 4.62772 0.300603
$$238$$ 16.0000 1.03713
$$239$$ −29.4891 −1.90749 −0.953746 0.300612i $$-0.902809\pi$$
−0.953746 + 0.300612i $$0.902809\pi$$
$$240$$ 0 0
$$241$$ 4.51087 0.290571 0.145285 0.989390i $$-0.453590\pi$$
0.145285 + 0.989390i $$0.453590\pi$$
$$242$$ 10.6060 0.681778
$$243$$ −1.00000 −0.0641500
$$244$$ 11.4891 0.735516
$$245$$ 0 0
$$246$$ 0.744563 0.0474716
$$247$$ −1.25544 −0.0798816
$$248$$ 1.00000 0.0635001
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ −3.37228 −0.212434
$$253$$ −2.11684 −0.133085
$$254$$ 13.4891 0.846383
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −13.3723 −0.834140 −0.417070 0.908874i $$-0.636943\pi$$
−0.417070 + 0.908874i $$0.636943\pi$$
$$258$$ −0.627719 −0.0390801
$$259$$ −2.51087 −0.156018
$$260$$ 0 0
$$261$$ −8.74456 −0.541275
$$262$$ 16.2337 1.00292
$$263$$ 18.9783 1.17025 0.585125 0.810943i $$-0.301046\pi$$
0.585125 + 0.810943i $$0.301046\pi$$
$$264$$ 0.627719 0.0386334
$$265$$ 0 0
$$266$$ −2.11684 −0.129792
$$267$$ 1.37228 0.0839823
$$268$$ −10.7446 −0.656329
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ −13.8832 −0.843342 −0.421671 0.906749i $$-0.638556\pi$$
−0.421671 + 0.906749i $$0.638556\pi$$
$$272$$ 4.74456 0.287681
$$273$$ 6.74456 0.408200
$$274$$ −3.48913 −0.210786
$$275$$ 0 0
$$276$$ 3.37228 0.202987
$$277$$ −15.2554 −0.916610 −0.458305 0.888795i $$-0.651543\pi$$
−0.458305 + 0.888795i $$0.651543\pi$$
$$278$$ 10.7446 0.644416
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 16.7446 0.998897 0.499448 0.866344i $$-0.333536\pi$$
0.499448 + 0.866344i $$0.333536\pi$$
$$282$$ 6.74456 0.401633
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 3.37228 0.200108
$$285$$ 0 0
$$286$$ −1.25544 −0.0742356
$$287$$ −2.51087 −0.148212
$$288$$ −1.00000 −0.0589256
$$289$$ 5.51087 0.324169
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 8.11684 0.475002
$$293$$ 7.48913 0.437519 0.218760 0.975779i $$-0.429799\pi$$
0.218760 + 0.975779i $$0.429799\pi$$
$$294$$ 4.37228 0.254997
$$295$$ 0 0
$$296$$ −0.744563 −0.0432768
$$297$$ −0.627719 −0.0364239
$$298$$ 12.1168 0.701910
$$299$$ −6.74456 −0.390048
$$300$$ 0 0
$$301$$ 2.11684 0.122013
$$302$$ 8.00000 0.460348
$$303$$ −8.11684 −0.466301
$$304$$ −0.627719 −0.0360021
$$305$$ 0 0
$$306$$ −4.74456 −0.271229
$$307$$ −30.9783 −1.76802 −0.884011 0.467466i $$-0.845167\pi$$
−0.884011 + 0.467466i $$0.845167\pi$$
$$308$$ −2.11684 −0.120618
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 2.00000 0.113228
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 9.37228 0.528908
$$315$$ 0 0
$$316$$ −4.62772 −0.260330
$$317$$ 24.7446 1.38979 0.694897 0.719110i $$-0.255450\pi$$
0.694897 + 0.719110i $$0.255450\pi$$
$$318$$ −1.37228 −0.0769537
$$319$$ −5.48913 −0.307332
$$320$$ 0 0
$$321$$ 0.627719 0.0350358
$$322$$ −11.3723 −0.633752
$$323$$ −2.97825 −0.165714
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 24.2337 1.34218
$$327$$ −4.74456 −0.262375
$$328$$ −0.744563 −0.0411116
$$329$$ −22.7446 −1.25395
$$330$$ 0 0
$$331$$ −1.48913 −0.0818497 −0.0409249 0.999162i $$-0.513030\pi$$
−0.0409249 + 0.999162i $$0.513030\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0.744563 0.0408018
$$334$$ −12.6277 −0.690958
$$335$$ 0 0
$$336$$ 3.37228 0.183973
$$337$$ −28.9783 −1.57855 −0.789273 0.614043i $$-0.789542\pi$$
−0.789273 + 0.614043i $$0.789542\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −2.62772 −0.142718
$$340$$ 0 0
$$341$$ −0.627719 −0.0339929
$$342$$ 0.627719 0.0339431
$$343$$ 8.86141 0.478471
$$344$$ 0.627719 0.0338443
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −36.4674 −1.95767 −0.978836 0.204648i $$-0.934395\pi$$
−0.978836 + 0.204648i $$0.934395\pi$$
$$348$$ 8.74456 0.468758
$$349$$ 7.25544 0.388375 0.194187 0.980964i $$-0.437793\pi$$
0.194187 + 0.980964i $$0.437793\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ −0.627719 −0.0334575
$$353$$ −15.4891 −0.824403 −0.412201 0.911093i $$-0.635240\pi$$
−0.412201 + 0.911093i $$0.635240\pi$$
$$354$$ 2.74456 0.145872
$$355$$ 0 0
$$356$$ −1.37228 −0.0727308
$$357$$ 16.0000 0.846810
$$358$$ 12.0000 0.634220
$$359$$ 3.37228 0.177982 0.0889911 0.996032i $$-0.471636\pi$$
0.0889911 + 0.996032i $$0.471636\pi$$
$$360$$ 0 0
$$361$$ −18.6060 −0.979262
$$362$$ −14.8614 −0.781098
$$363$$ 10.6060 0.556669
$$364$$ −6.74456 −0.353511
$$365$$ 0 0
$$366$$ 11.4891 0.600546
$$367$$ 20.2337 1.05619 0.528095 0.849185i $$-0.322907\pi$$
0.528095 + 0.849185i $$0.322907\pi$$
$$368$$ −3.37228 −0.175792
$$369$$ 0.744563 0.0387604
$$370$$ 0 0
$$371$$ 4.62772 0.240259
$$372$$ 1.00000 0.0518476
$$373$$ −14.8614 −0.769494 −0.384747 0.923022i $$-0.625711\pi$$
−0.384747 + 0.923022i $$0.625711\pi$$
$$374$$ −2.97825 −0.154002
$$375$$ 0 0
$$376$$ −6.74456 −0.347824
$$377$$ −17.4891 −0.900736
$$378$$ −3.37228 −0.173451
$$379$$ −28.8614 −1.48251 −0.741255 0.671223i $$-0.765769\pi$$
−0.741255 + 0.671223i $$0.765769\pi$$
$$380$$ 0 0
$$381$$ 13.4891 0.691069
$$382$$ −16.0000 −0.818631
$$383$$ −13.4891 −0.689262 −0.344631 0.938738i $$-0.611996\pi$$
−0.344631 + 0.938738i $$0.611996\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 15.4891 0.788376
$$387$$ −0.627719 −0.0319087
$$388$$ −2.00000 −0.101535
$$389$$ 16.9783 0.860831 0.430416 0.902631i $$-0.358367\pi$$
0.430416 + 0.902631i $$0.358367\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ −4.37228 −0.220834
$$393$$ 16.2337 0.818881
$$394$$ 19.4891 0.981848
$$395$$ 0 0
$$396$$ 0.627719 0.0315441
$$397$$ 1.60597 0.0806013 0.0403006 0.999188i $$-0.487168\pi$$
0.0403006 + 0.999188i $$0.487168\pi$$
$$398$$ 12.6277 0.632970
$$399$$ −2.11684 −0.105975
$$400$$ 0 0
$$401$$ 22.6277 1.12997 0.564987 0.825100i $$-0.308881\pi$$
0.564987 + 0.825100i $$0.308881\pi$$
$$402$$ −10.7446 −0.535890
$$403$$ −2.00000 −0.0996271
$$404$$ 8.11684 0.403828
$$405$$ 0 0
$$406$$ −29.4891 −1.46352
$$407$$ 0.467376 0.0231670
$$408$$ 4.74456 0.234891
$$409$$ −10.2337 −0.506023 −0.253012 0.967463i $$-0.581421\pi$$
−0.253012 + 0.967463i $$0.581421\pi$$
$$410$$ 0 0
$$411$$ −3.48913 −0.172106
$$412$$ 8.00000 0.394132
$$413$$ −9.25544 −0.455430
$$414$$ 3.37228 0.165739
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 10.7446 0.526163
$$418$$ 0.394031 0.0192727
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 19.4891 0.949842 0.474921 0.880028i $$-0.342477\pi$$
0.474921 + 0.880028i $$0.342477\pi$$
$$422$$ 0.627719 0.0305569
$$423$$ 6.74456 0.327932
$$424$$ 1.37228 0.0666439
$$425$$ 0 0
$$426$$ 3.37228 0.163388
$$427$$ −38.7446 −1.87498
$$428$$ −0.627719 −0.0303419
$$429$$ −1.25544 −0.0606131
$$430$$ 0 0
$$431$$ −26.9783 −1.29950 −0.649748 0.760149i $$-0.725126\pi$$
−0.649748 + 0.760149i $$0.725126\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −36.1168 −1.73566 −0.867832 0.496857i $$-0.834487\pi$$
−0.867832 + 0.496857i $$0.834487\pi$$
$$434$$ −3.37228 −0.161875
$$435$$ 0 0
$$436$$ 4.74456 0.227223
$$437$$ 2.11684 0.101262
$$438$$ 8.11684 0.387838
$$439$$ 14.7446 0.703720 0.351860 0.936053i $$-0.385549\pi$$
0.351860 + 0.936053i $$0.385549\pi$$
$$440$$ 0 0
$$441$$ 4.37228 0.208204
$$442$$ −9.48913 −0.451352
$$443$$ −34.3505 −1.63204 −0.816022 0.578022i $$-0.803825\pi$$
−0.816022 + 0.578022i $$0.803825\pi$$
$$444$$ −0.744563 −0.0353354
$$445$$ 0 0
$$446$$ 9.25544 0.438258
$$447$$ 12.1168 0.573107
$$448$$ −3.37228 −0.159325
$$449$$ −8.97825 −0.423710 −0.211855 0.977301i $$-0.567950\pi$$
−0.211855 + 0.977301i $$0.567950\pi$$
$$450$$ 0 0
$$451$$ 0.467376 0.0220079
$$452$$ 2.62772 0.123597
$$453$$ 8.00000 0.375873
$$454$$ 6.11684 0.287078
$$455$$ 0 0
$$456$$ −0.627719 −0.0293956
$$457$$ −34.4674 −1.61232 −0.806158 0.591700i $$-0.798457\pi$$
−0.806158 + 0.591700i $$0.798457\pi$$
$$458$$ −3.88316 −0.181448
$$459$$ −4.74456 −0.221457
$$460$$ 0 0
$$461$$ −19.7228 −0.918583 −0.459291 0.888286i $$-0.651897\pi$$
−0.459291 + 0.888286i $$0.651897\pi$$
$$462$$ −2.11684 −0.0984845
$$463$$ −2.51087 −0.116690 −0.0583451 0.998296i $$-0.518582\pi$$
−0.0583451 + 0.998296i $$0.518582\pi$$
$$464$$ −8.74456 −0.405956
$$465$$ 0 0
$$466$$ −14.8614 −0.688441
$$467$$ −6.51087 −0.301287 −0.150644 0.988588i $$-0.548135\pi$$
−0.150644 + 0.988588i $$0.548135\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 36.2337 1.67312
$$470$$ 0 0
$$471$$ 9.37228 0.431852
$$472$$ −2.74456 −0.126329
$$473$$ −0.394031 −0.0181176
$$474$$ −4.62772 −0.212558
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ −1.37228 −0.0628324
$$478$$ 29.4891 1.34880
$$479$$ 8.86141 0.404888 0.202444 0.979294i $$-0.435112\pi$$
0.202444 + 0.979294i $$0.435112\pi$$
$$480$$ 0 0
$$481$$ 1.48913 0.0678983
$$482$$ −4.51087 −0.205465
$$483$$ −11.3723 −0.517457
$$484$$ −10.6060 −0.482090
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 37.4891 1.69879 0.849397 0.527754i $$-0.176966\pi$$
0.849397 + 0.527754i $$0.176966\pi$$
$$488$$ −11.4891 −0.520088
$$489$$ 24.2337 1.09589
$$490$$ 0 0
$$491$$ 27.6060 1.24584 0.622920 0.782286i $$-0.285946\pi$$
0.622920 + 0.782286i $$0.285946\pi$$
$$492$$ −0.744563 −0.0335675
$$493$$ −41.4891 −1.86858
$$494$$ 1.25544 0.0564848
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ −11.3723 −0.510117
$$498$$ 12.0000 0.537733
$$499$$ 33.4891 1.49918 0.749590 0.661903i $$-0.230251\pi$$
0.749590 + 0.661903i $$0.230251\pi$$
$$500$$ 0 0
$$501$$ −12.6277 −0.564165
$$502$$ 4.00000 0.178529
$$503$$ −5.48913 −0.244748 −0.122374 0.992484i $$-0.539051\pi$$
−0.122374 + 0.992484i $$0.539051\pi$$
$$504$$ 3.37228 0.150213
$$505$$ 0 0
$$506$$ 2.11684 0.0941052
$$507$$ 9.00000 0.399704
$$508$$ −13.4891 −0.598483
$$509$$ −27.2554 −1.20808 −0.604038 0.796956i $$-0.706442\pi$$
−0.604038 + 0.796956i $$0.706442\pi$$
$$510$$ 0 0
$$511$$ −27.3723 −1.21088
$$512$$ −1.00000 −0.0441942
$$513$$ 0.627719 0.0277145
$$514$$ 13.3723 0.589826
$$515$$ 0 0
$$516$$ 0.627719 0.0276338
$$517$$ 4.23369 0.186197
$$518$$ 2.51087 0.110322
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ 36.9783 1.62005 0.810023 0.586398i $$-0.199454\pi$$
0.810023 + 0.586398i $$0.199454\pi$$
$$522$$ 8.74456 0.382739
$$523$$ 12.8614 0.562390 0.281195 0.959651i $$-0.409269\pi$$
0.281195 + 0.959651i $$0.409269\pi$$
$$524$$ −16.2337 −0.709172
$$525$$ 0 0
$$526$$ −18.9783 −0.827491
$$527$$ −4.74456 −0.206676
$$528$$ −0.627719 −0.0273179
$$529$$ −11.6277 −0.505553
$$530$$ 0 0
$$531$$ 2.74456 0.119104
$$532$$ 2.11684 0.0917768
$$533$$ 1.48913 0.0645012
$$534$$ −1.37228 −0.0593844
$$535$$ 0 0
$$536$$ 10.7446 0.464094
$$537$$ 12.0000 0.517838
$$538$$ 2.00000 0.0862261
$$539$$ 2.74456 0.118217
$$540$$ 0 0
$$541$$ −15.4891 −0.665930 −0.332965 0.942939i $$-0.608049\pi$$
−0.332965 + 0.942939i $$0.608049\pi$$
$$542$$ 13.8832 0.596333
$$543$$ −14.8614 −0.637764
$$544$$ −4.74456 −0.203421
$$545$$ 0 0
$$546$$ −6.74456 −0.288641
$$547$$ 2.74456 0.117349 0.0586745 0.998277i $$-0.481313\pi$$
0.0586745 + 0.998277i $$0.481313\pi$$
$$548$$ 3.48913 0.149048
$$549$$ 11.4891 0.490344
$$550$$ 0 0
$$551$$ 5.48913 0.233845
$$552$$ −3.37228 −0.143534
$$553$$ 15.6060 0.663633
$$554$$ 15.2554 0.648141
$$555$$ 0 0
$$556$$ −10.7446 −0.455671
$$557$$ −11.8832 −0.503505 −0.251753 0.967792i $$-0.581007\pi$$
−0.251753 + 0.967792i $$0.581007\pi$$
$$558$$ 1.00000 0.0423334
$$559$$ −1.25544 −0.0530993
$$560$$ 0 0
$$561$$ −2.97825 −0.125742
$$562$$ −16.7446 −0.706327
$$563$$ 6.97825 0.294098 0.147049 0.989129i $$-0.453022\pi$$
0.147049 + 0.989129i $$0.453022\pi$$
$$564$$ −6.74456 −0.283997
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ −3.37228 −0.141623
$$568$$ −3.37228 −0.141498
$$569$$ −1.37228 −0.0575290 −0.0287645 0.999586i $$-0.509157\pi$$
−0.0287645 + 0.999586i $$0.509157\pi$$
$$570$$ 0 0
$$571$$ −26.7446 −1.11923 −0.559613 0.828754i $$-0.689050\pi$$
−0.559613 + 0.828754i $$0.689050\pi$$
$$572$$ 1.25544 0.0524925
$$573$$ −16.0000 −0.668410
$$574$$ 2.51087 0.104802
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.23369 0.0929896 0.0464948 0.998919i $$-0.485195\pi$$
0.0464948 + 0.998919i $$0.485195\pi$$
$$578$$ −5.51087 −0.229222
$$579$$ 15.4891 0.643706
$$580$$ 0 0
$$581$$ −40.4674 −1.67887
$$582$$ −2.00000 −0.0829027
$$583$$ −0.861407 −0.0356758
$$584$$ −8.11684 −0.335877
$$585$$ 0 0
$$586$$ −7.48913 −0.309373
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −4.37228 −0.180310
$$589$$ 0.627719 0.0258647
$$590$$ 0 0
$$591$$ 19.4891 0.801675
$$592$$ 0.744563 0.0306013
$$593$$ −44.9783 −1.84704 −0.923518 0.383556i $$-0.874699\pi$$
−0.923518 + 0.383556i $$0.874699\pi$$
$$594$$ 0.627719 0.0257556
$$595$$ 0 0
$$596$$ −12.1168 −0.496325
$$597$$ 12.6277 0.516818
$$598$$ 6.74456 0.275806
$$599$$ 10.1168 0.413363 0.206682 0.978408i $$-0.433734\pi$$
0.206682 + 0.978408i $$0.433734\pi$$
$$600$$ 0 0
$$601$$ 12.5109 0.510329 0.255165 0.966898i $$-0.417870\pi$$
0.255165 + 0.966898i $$0.417870\pi$$
$$602$$ −2.11684 −0.0862761
$$603$$ −10.7446 −0.437552
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 8.11684 0.329724
$$607$$ 4.62772 0.187833 0.0939167 0.995580i $$-0.470061\pi$$
0.0939167 + 0.995580i $$0.470061\pi$$
$$608$$ 0.627719 0.0254574
$$609$$ −29.4891 −1.19496
$$610$$ 0 0
$$611$$ 13.4891 0.545712
$$612$$ 4.74456 0.191788
$$613$$ −35.4891 −1.43339 −0.716696 0.697386i $$-0.754347\pi$$
−0.716696 + 0.697386i $$0.754347\pi$$
$$614$$ 30.9783 1.25018
$$615$$ 0 0
$$616$$ 2.11684 0.0852901
$$617$$ −7.88316 −0.317364 −0.158682 0.987330i $$-0.550724\pi$$
−0.158682 + 0.987330i $$0.550724\pi$$
$$618$$ 8.00000 0.321807
$$619$$ 16.2337 0.652487 0.326244 0.945286i $$-0.394217\pi$$
0.326244 + 0.945286i $$0.394217\pi$$
$$620$$ 0 0
$$621$$ 3.37228 0.135325
$$622$$ −8.00000 −0.320771
$$623$$ 4.62772 0.185406
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0.394031 0.0157361
$$628$$ −9.37228 −0.373995
$$629$$ 3.53262 0.140855
$$630$$ 0 0
$$631$$ −46.3505 −1.84519 −0.922593 0.385775i $$-0.873934\pi$$
−0.922593 + 0.385775i $$0.873934\pi$$
$$632$$ 4.62772 0.184081
$$633$$ 0.627719 0.0249496
$$634$$ −24.7446 −0.982732
$$635$$ 0 0
$$636$$ 1.37228 0.0544145
$$637$$ 8.74456 0.346472
$$638$$ 5.48913 0.217317
$$639$$ 3.37228 0.133405
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −0.627719 −0.0247741
$$643$$ 25.0951 0.989654 0.494827 0.868992i $$-0.335231\pi$$
0.494827 + 0.868992i $$0.335231\pi$$
$$644$$ 11.3723 0.448131
$$645$$ 0 0
$$646$$ 2.97825 0.117178
$$647$$ −50.5842 −1.98867 −0.994335 0.106287i $$-0.966104\pi$$
−0.994335 + 0.106287i $$0.966104\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 1.72281 0.0676263
$$650$$ 0 0
$$651$$ −3.37228 −0.132170
$$652$$ −24.2337 −0.949064
$$653$$ −40.9783 −1.60360 −0.801801 0.597591i $$-0.796125\pi$$
−0.801801 + 0.597591i $$0.796125\pi$$
$$654$$ 4.74456 0.185527
$$655$$ 0 0
$$656$$ 0.744563 0.0290703
$$657$$ 8.11684 0.316668
$$658$$ 22.7446 0.886675
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −4.97825 −0.193632 −0.0968158 0.995302i $$-0.530866\pi$$
−0.0968158 + 0.995302i $$0.530866\pi$$
$$662$$ 1.48913 0.0578765
$$663$$ −9.48913 −0.368527
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −0.744563 −0.0288512
$$667$$ 29.4891 1.14182
$$668$$ 12.6277 0.488581
$$669$$ 9.25544 0.357836
$$670$$ 0 0
$$671$$ 7.21194 0.278414
$$672$$ −3.37228 −0.130089
$$673$$ 0.510875 0.0196928 0.00984639 0.999952i $$-0.496866\pi$$
0.00984639 + 0.999952i $$0.496866\pi$$
$$674$$ 28.9783 1.11620
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 41.6060 1.59905 0.799524 0.600635i $$-0.205085\pi$$
0.799524 + 0.600635i $$0.205085\pi$$
$$678$$ 2.62772 0.100917
$$679$$ 6.74456 0.258833
$$680$$ 0 0
$$681$$ 6.11684 0.234398
$$682$$ 0.627719 0.0240366
$$683$$ 12.8614 0.492128 0.246064 0.969254i $$-0.420863\pi$$
0.246064 + 0.969254i $$0.420863\pi$$
$$684$$ −0.627719 −0.0240014
$$685$$ 0 0
$$686$$ −8.86141 −0.338330
$$687$$ −3.88316 −0.148152
$$688$$ −0.627719 −0.0239316
$$689$$ −2.74456 −0.104560
$$690$$ 0 0
$$691$$ −19.1386 −0.728066 −0.364033 0.931386i $$-0.618601\pi$$
−0.364033 + 0.931386i $$0.618601\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −2.11684 −0.0804123
$$694$$ 36.4674 1.38428
$$695$$ 0 0
$$696$$ −8.74456 −0.331462
$$697$$ 3.53262 0.133808
$$698$$ −7.25544 −0.274622
$$699$$ −14.8614 −0.562110
$$700$$ 0 0
$$701$$ 45.6060 1.72251 0.861257 0.508170i $$-0.169678\pi$$
0.861257 + 0.508170i $$0.169678\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −0.467376 −0.0176274
$$704$$ 0.627719 0.0236580
$$705$$ 0 0
$$706$$ 15.4891 0.582941
$$707$$ −27.3723 −1.02944
$$708$$ −2.74456 −0.103147
$$709$$ 40.1168 1.50662 0.753310 0.657666i $$-0.228456\pi$$
0.753310 + 0.657666i $$0.228456\pi$$
$$710$$ 0 0
$$711$$ −4.62772 −0.173553
$$712$$ 1.37228 0.0514284
$$713$$ 3.37228 0.126293
$$714$$ −16.0000 −0.598785
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 29.4891 1.10129
$$718$$ −3.37228 −0.125852
$$719$$ 38.7446 1.44493 0.722464 0.691408i $$-0.243009\pi$$
0.722464 + 0.691408i $$0.243009\pi$$
$$720$$ 0 0
$$721$$ −26.9783 −1.00472
$$722$$ 18.6060 0.692442
$$723$$ −4.51087 −0.167761
$$724$$ 14.8614 0.552320
$$725$$ 0 0
$$726$$ −10.6060 −0.393624
$$727$$ −19.3723 −0.718478 −0.359239 0.933246i $$-0.616964\pi$$
−0.359239 + 0.933246i $$0.616964\pi$$
$$728$$ 6.74456 0.249970
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −2.97825 −0.110155
$$732$$ −11.4891 −0.424650
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −20.2337 −0.746839
$$735$$ 0 0
$$736$$ 3.37228 0.124304
$$737$$ −6.74456 −0.248439
$$738$$ −0.744563 −0.0274077
$$739$$ −22.9783 −0.845269 −0.422634 0.906300i $$-0.638895\pi$$
−0.422634 + 0.906300i $$0.638895\pi$$
$$740$$ 0 0
$$741$$ 1.25544 0.0461196
$$742$$ −4.62772 −0.169889
$$743$$ 35.3723 1.29768 0.648842 0.760924i $$-0.275254\pi$$
0.648842 + 0.760924i $$0.275254\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 14.8614 0.544115
$$747$$ 12.0000 0.439057
$$748$$ 2.97825 0.108896
$$749$$ 2.11684 0.0773478
$$750$$ 0 0
$$751$$ −38.7446 −1.41381 −0.706905 0.707309i $$-0.749909\pi$$
−0.706905 + 0.707309i $$0.749909\pi$$
$$752$$ 6.74456 0.245949
$$753$$ 4.00000 0.145768
$$754$$ 17.4891 0.636916
$$755$$ 0 0
$$756$$ 3.37228 0.122649
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 28.8614 1.04829
$$759$$ 2.11684 0.0768366
$$760$$ 0 0
$$761$$ −46.8614 −1.69872 −0.849362 0.527810i $$-0.823013\pi$$
−0.849362 + 0.527810i $$0.823013\pi$$
$$762$$ −13.4891 −0.488659
$$763$$ −16.0000 −0.579239
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 13.4891 0.487382
$$767$$ 5.48913 0.198201
$$768$$ −1.00000 −0.0360844
$$769$$ −9.37228 −0.337973 −0.168987 0.985618i $$-0.554049\pi$$
−0.168987 + 0.985618i $$0.554049\pi$$
$$770$$ 0 0
$$771$$ 13.3723 0.481591
$$772$$ −15.4891 −0.557466
$$773$$ −21.6060 −0.777113 −0.388556 0.921425i $$-0.627026\pi$$
−0.388556 + 0.921425i $$0.627026\pi$$
$$774$$ 0.627719 0.0225629
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 2.51087 0.0900771
$$778$$ −16.9783 −0.608700
$$779$$ −0.467376 −0.0167455
$$780$$ 0 0
$$781$$ 2.11684 0.0757466
$$782$$ 16.0000 0.572159
$$783$$ 8.74456 0.312505
$$784$$ 4.37228 0.156153
$$785$$ 0 0
$$786$$ −16.2337 −0.579036
$$787$$ 9.88316 0.352296 0.176148 0.984364i $$-0.443636\pi$$
0.176148 + 0.984364i $$0.443636\pi$$
$$788$$ −19.4891 −0.694271
$$789$$ −18.9783 −0.675644
$$790$$ 0 0
$$791$$ −8.86141 −0.315075
$$792$$ −0.627719 −0.0223050
$$793$$ 22.9783 0.815982
$$794$$ −1.60597 −0.0569937
$$795$$ 0 0
$$796$$ −12.6277 −0.447578
$$797$$ 20.5109 0.726532 0.363266 0.931685i $$-0.381662\pi$$
0.363266 + 0.931685i $$0.381662\pi$$
$$798$$ 2.11684 0.0749355
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ −1.37228 −0.0484872
$$802$$ −22.6277 −0.799013
$$803$$ 5.09509 0.179802
$$804$$ 10.7446 0.378932
$$805$$ 0 0
$$806$$ 2.00000 0.0704470
$$807$$ 2.00000 0.0704033
$$808$$ −8.11684 −0.285550
$$809$$ 9.60597 0.337728 0.168864 0.985639i $$-0.445990\pi$$
0.168864 + 0.985639i $$0.445990\pi$$
$$810$$ 0 0
$$811$$ −40.6277 −1.42663 −0.713316 0.700842i $$-0.752808\pi$$
−0.713316 + 0.700842i $$0.752808\pi$$
$$812$$ 29.4891 1.03487
$$813$$ 13.8832 0.486904
$$814$$ −0.467376 −0.0163815
$$815$$ 0 0
$$816$$ −4.74456 −0.166093
$$817$$ 0.394031 0.0137854
$$818$$ 10.2337 0.357813
$$819$$ −6.74456 −0.235674
$$820$$ 0 0
$$821$$ 15.2554 0.532418 0.266209 0.963915i $$-0.414229\pi$$
0.266209 + 0.963915i $$0.414229\pi$$
$$822$$ 3.48913 0.121697
$$823$$ 17.2554 0.601487 0.300743 0.953705i $$-0.402765\pi$$
0.300743 + 0.953705i $$0.402765\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 9.25544 0.322038
$$827$$ 17.4891 0.608156 0.304078 0.952647i $$-0.401652\pi$$
0.304078 + 0.952647i $$0.401652\pi$$
$$828$$ −3.37228 −0.117195
$$829$$ −31.0951 −1.07998 −0.539989 0.841672i $$-0.681571\pi$$
−0.539989 + 0.841672i $$0.681571\pi$$
$$830$$ 0 0
$$831$$ 15.2554 0.529205
$$832$$ 2.00000 0.0693375
$$833$$ 20.7446 0.718756
$$834$$ −10.7446 −0.372054
$$835$$ 0 0
$$836$$ −0.394031 −0.0136278
$$837$$ 1.00000 0.0345651
$$838$$ 12.0000 0.414533
$$839$$ 16.8614 0.582120 0.291060 0.956705i $$-0.405992\pi$$
0.291060 + 0.956705i $$0.405992\pi$$
$$840$$ 0 0
$$841$$ 47.4674 1.63681
$$842$$ −19.4891 −0.671640
$$843$$ −16.7446 −0.576713
$$844$$ −0.627719 −0.0216070
$$845$$ 0 0
$$846$$ −6.74456 −0.231883
$$847$$ 35.7663 1.22895
$$848$$ −1.37228 −0.0471243
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ −2.51087 −0.0860717
$$852$$ −3.37228 −0.115532
$$853$$ 39.0951 1.33859 0.669295 0.742997i $$-0.266596\pi$$
0.669295 + 0.742997i $$0.266596\pi$$
$$854$$ 38.7446 1.32581
$$855$$ 0 0
$$856$$ 0.627719 0.0214550
$$857$$ −7.48913 −0.255824 −0.127912 0.991786i $$-0.540827\pi$$
−0.127912 + 0.991786i $$0.540827\pi$$
$$858$$ 1.25544 0.0428599
$$859$$ −17.4891 −0.596721 −0.298361 0.954453i $$-0.596440\pi$$
−0.298361 + 0.954453i $$0.596440\pi$$
$$860$$ 0 0
$$861$$ 2.51087 0.0855704
$$862$$ 26.9783 0.918883
$$863$$ 6.35053 0.216175 0.108087 0.994141i $$-0.465527\pi$$
0.108087 + 0.994141i $$0.465527\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 36.1168 1.22730
$$867$$ −5.51087 −0.187159
$$868$$ 3.37228 0.114463
$$869$$ −2.90491 −0.0985422
$$870$$ 0 0
$$871$$ −21.4891 −0.728131
$$872$$ −4.74456 −0.160671
$$873$$ −2.00000 −0.0676897
$$874$$ −2.11684 −0.0716033
$$875$$ 0 0
$$876$$ −8.11684 −0.274243
$$877$$ 44.9783 1.51881 0.759404 0.650620i $$-0.225491\pi$$
0.759404 + 0.650620i $$0.225491\pi$$
$$878$$ −14.7446 −0.497605
$$879$$ −7.48913 −0.252602
$$880$$ 0 0
$$881$$ −3.02175 −0.101805 −0.0509027 0.998704i $$-0.516210\pi$$
−0.0509027 + 0.998704i $$0.516210\pi$$
$$882$$ −4.37228 −0.147222
$$883$$ −44.8614 −1.50971 −0.754853 0.655894i $$-0.772292\pi$$
−0.754853 + 0.655894i $$0.772292\pi$$
$$884$$ 9.48913 0.319154
$$885$$ 0 0
$$886$$ 34.3505 1.15403
$$887$$ 30.7446 1.03230 0.516151 0.856498i $$-0.327364\pi$$
0.516151 + 0.856498i $$0.327364\pi$$
$$888$$ 0.744563 0.0249859
$$889$$ 45.4891 1.52566
$$890$$ 0 0
$$891$$ 0.627719 0.0210294
$$892$$ −9.25544 −0.309895
$$893$$ −4.23369 −0.141675
$$894$$ −12.1168 −0.405248
$$895$$ 0 0
$$896$$ 3.37228 0.112660
$$897$$ 6.74456 0.225194
$$898$$ 8.97825 0.299608
$$899$$ 8.74456 0.291647
$$900$$ 0 0
$$901$$ −6.51087 −0.216909
$$902$$ −0.467376 −0.0155619
$$903$$ −2.11684 −0.0704442
$$904$$ −2.62772 −0.0873966
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 17.4891 0.580717 0.290358 0.956918i $$-0.406225\pi$$
0.290358 + 0.956918i $$0.406225\pi$$
$$908$$ −6.11684 −0.202995
$$909$$ 8.11684 0.269219
$$910$$ 0 0
$$911$$ −2.51087 −0.0831890 −0.0415945 0.999135i $$-0.513244\pi$$
−0.0415945 + 0.999135i $$0.513244\pi$$
$$912$$ 0.627719 0.0207858
$$913$$ 7.53262 0.249293
$$914$$ 34.4674 1.14008
$$915$$ 0 0
$$916$$ 3.88316 0.128303
$$917$$ 54.7446 1.80782
$$918$$ 4.74456 0.156594
$$919$$ −12.2337 −0.403552 −0.201776 0.979432i $$-0.564671\pi$$
−0.201776 + 0.979432i $$0.564671\pi$$
$$920$$ 0 0
$$921$$ 30.9783 1.02077
$$922$$ 19.7228 0.649536
$$923$$ 6.74456 0.222000
$$924$$ 2.11684 0.0696391
$$925$$ 0 0
$$926$$ 2.51087 0.0825125
$$927$$ 8.00000 0.262754
$$928$$ 8.74456 0.287054
$$929$$ −16.1168 −0.528776 −0.264388 0.964416i $$-0.585170\pi$$
−0.264388 + 0.964416i $$0.585170\pi$$
$$930$$ 0 0
$$931$$ −2.74456 −0.0899494
$$932$$ 14.8614 0.486802
$$933$$ −8.00000 −0.261908
$$934$$ 6.51087 0.213042
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 31.2554 1.02107 0.510535 0.859857i $$-0.329447\pi$$
0.510535 + 0.859857i $$0.329447\pi$$
$$938$$ −36.2337 −1.18307
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −19.7228 −0.642945 −0.321473 0.946919i $$-0.604178\pi$$
−0.321473 + 0.946919i $$0.604178\pi$$
$$942$$ −9.37228 −0.305365
$$943$$ −2.51087 −0.0817653
$$944$$ 2.74456 0.0893279
$$945$$ 0 0
$$946$$ 0.394031 0.0128110
$$947$$ 2.74456 0.0891863 0.0445932 0.999005i $$-0.485801\pi$$
0.0445932 + 0.999005i $$0.485801\pi$$
$$948$$ 4.62772 0.150301
$$949$$ 16.2337 0.526968
$$950$$ 0 0
$$951$$ −24.7446 −0.802397
$$952$$ 16.0000 0.518563
$$953$$ −11.7228 −0.379739 −0.189870 0.981809i $$-0.560807\pi$$
−0.189870 + 0.981809i $$0.560807\pi$$
$$954$$ 1.37228 0.0444292
$$955$$ 0 0
$$956$$ −29.4891 −0.953746
$$957$$ 5.48913 0.177438
$$958$$ −8.86141 −0.286299
$$959$$ −11.7663 −0.379954
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ −1.48913 −0.0480113
$$963$$ −0.627719 −0.0202280
$$964$$ 4.51087 0.145285
$$965$$ 0 0
$$966$$ 11.3723 0.365897
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 10.6060 0.340889
$$969$$ 2.97825 0.0956752
$$970$$ 0 0
$$971$$ −48.7011 −1.56289 −0.781446 0.623973i $$-0.785517\pi$$
−0.781446 + 0.623973i $$0.785517\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 36.2337 1.16160
$$974$$ −37.4891 −1.20123
$$975$$ 0 0
$$976$$ 11.4891 0.367758
$$977$$ 46.0000 1.47167 0.735835 0.677161i $$-0.236790\pi$$
0.735835 + 0.677161i $$0.236790\pi$$
$$978$$ −24.2337 −0.774908
$$979$$ −0.861407 −0.0275307
$$980$$ 0 0
$$981$$ 4.74456 0.151482
$$982$$ −27.6060 −0.880942
$$983$$ 2.97825 0.0949914 0.0474957 0.998871i $$-0.484876\pi$$
0.0474957 + 0.998871i $$0.484876\pi$$
$$984$$ 0.744563 0.0237358
$$985$$ 0 0
$$986$$ 41.4891 1.32128
$$987$$ 22.7446 0.723967
$$988$$ −1.25544 −0.0399408
$$989$$ 2.11684 0.0673117
$$990$$ 0 0
$$991$$ −47.6060 −1.51225 −0.756127 0.654425i $$-0.772911\pi$$
−0.756127 + 0.654425i $$0.772911\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ 1.48913 0.0472560
$$994$$ 11.3723 0.360707
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ −6.00000 −0.190022 −0.0950110 0.995476i $$-0.530289\pi$$
−0.0950110 + 0.995476i $$0.530289\pi$$
$$998$$ −33.4891 −1.06008
$$999$$ −0.744563 −0.0235569
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 4650.2.a.by.1.1 2
5.2 odd 4 4650.2.d.bh.3349.1 4
5.3 odd 4 4650.2.d.bh.3349.4 4
5.4 even 2 930.2.a.r.1.2 2
15.14 odd 2 2790.2.a.bd.1.2 2
20.19 odd 2 7440.2.a.bg.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.r.1.2 2 5.4 even 2
2790.2.a.bd.1.2 2 15.14 odd 2
4650.2.a.by.1.1 2 1.1 even 1 trivial
4650.2.d.bh.3349.1 4 5.2 odd 4
4650.2.d.bh.3349.4 4 5.3 odd 4
7440.2.a.bg.1.1 2 20.19 odd 2