# Properties

 Label 2541.2.a.t.1.2 Level $2541$ Weight $2$ Character 2541.1 Self dual yes Analytic conductor $20.290$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2541 = 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2541.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2899871536$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 2541.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{10} -1.61803 q^{12} -3.47214 q^{13} -0.618034 q^{14} +1.00000 q^{15} +1.85410 q^{16} -5.23607 q^{17} +0.618034 q^{18} +6.70820 q^{19} -1.61803 q^{20} -1.00000 q^{21} +5.70820 q^{23} -2.23607 q^{24} -4.00000 q^{25} -2.14590 q^{26} +1.00000 q^{27} +1.61803 q^{28} -5.00000 q^{29} +0.618034 q^{30} -5.23607 q^{31} +5.61803 q^{32} -3.23607 q^{34} -1.00000 q^{35} -1.61803 q^{36} -7.00000 q^{37} +4.14590 q^{38} -3.47214 q^{39} -2.23607 q^{40} +2.47214 q^{41} -0.618034 q^{42} -5.70820 q^{43} +1.00000 q^{45} +3.52786 q^{46} +0.236068 q^{47} +1.85410 q^{48} +1.00000 q^{49} -2.47214 q^{50} -5.23607 q^{51} +5.61803 q^{52} -12.1803 q^{53} +0.618034 q^{54} +2.23607 q^{56} +6.70820 q^{57} -3.09017 q^{58} -11.1803 q^{59} -1.61803 q^{60} -2.00000 q^{61} -3.23607 q^{62} -1.00000 q^{63} -0.236068 q^{64} -3.47214 q^{65} -9.76393 q^{67} +8.47214 q^{68} +5.70820 q^{69} -0.618034 q^{70} -2.47214 q^{71} -2.23607 q^{72} -4.52786 q^{73} -4.32624 q^{74} -4.00000 q^{75} -10.8541 q^{76} -2.14590 q^{78} +14.4721 q^{79} +1.85410 q^{80} +1.00000 q^{81} +1.52786 q^{82} -6.76393 q^{83} +1.61803 q^{84} -5.23607 q^{85} -3.52786 q^{86} -5.00000 q^{87} -4.47214 q^{89} +0.618034 q^{90} +3.47214 q^{91} -9.23607 q^{92} -5.23607 q^{93} +0.145898 q^{94} +6.70820 q^{95} +5.61803 q^{96} +9.70820 q^{97} +0.618034 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} - q^{10} - q^{12} + 2 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} - 6 q^{17} - q^{18} - q^{20} - 2 q^{21} - 2 q^{23} - 8 q^{25} - 11 q^{26} + 2 q^{27} + q^{28} - 10 q^{29} - q^{30} - 6 q^{31} + 9 q^{32} - 2 q^{34} - 2 q^{35} - q^{36} - 14 q^{37} + 15 q^{38} + 2 q^{39} - 4 q^{41} + q^{42} + 2 q^{43} + 2 q^{45} + 16 q^{46} - 4 q^{47} - 3 q^{48} + 2 q^{49} + 4 q^{50} - 6 q^{51} + 9 q^{52} - 2 q^{53} - q^{54} + 5 q^{58} - q^{60} - 4 q^{61} - 2 q^{62} - 2 q^{63} + 4 q^{64} + 2 q^{65} - 24 q^{67} + 8 q^{68} - 2 q^{69} + q^{70} + 4 q^{71} - 18 q^{73} + 7 q^{74} - 8 q^{75} - 15 q^{76} - 11 q^{78} + 20 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} - 18 q^{83} + q^{84} - 6 q^{85} - 16 q^{86} - 10 q^{87} - q^{90} - 2 q^{91} - 14 q^{92} - 6 q^{93} + 7 q^{94} + 9 q^{96} + 6 q^{97} - q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.437016 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.61803 −0.809017
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0.618034 0.252311
$$7$$ −1.00000 −0.377964
$$8$$ −2.23607 −0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0.618034 0.195440
$$11$$ 0 0
$$12$$ −1.61803 −0.467086
$$13$$ −3.47214 −0.962997 −0.481499 0.876447i $$-0.659907\pi$$
−0.481499 + 0.876447i $$0.659907\pi$$
$$14$$ −0.618034 −0.165177
$$15$$ 1.00000 0.258199
$$16$$ 1.85410 0.463525
$$17$$ −5.23607 −1.26993 −0.634967 0.772540i $$-0.718986\pi$$
−0.634967 + 0.772540i $$0.718986\pi$$
$$18$$ 0.618034 0.145672
$$19$$ 6.70820 1.53897 0.769484 0.638666i $$-0.220514\pi$$
0.769484 + 0.638666i $$0.220514\pi$$
$$20$$ −1.61803 −0.361803
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 5.70820 1.19024 0.595121 0.803636i $$-0.297104\pi$$
0.595121 + 0.803636i $$0.297104\pi$$
$$24$$ −2.23607 −0.456435
$$25$$ −4.00000 −0.800000
$$26$$ −2.14590 −0.420845
$$27$$ 1.00000 0.192450
$$28$$ 1.61803 0.305780
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0.618034 0.112837
$$31$$ −5.23607 −0.940426 −0.470213 0.882553i $$-0.655823\pi$$
−0.470213 + 0.882553i $$0.655823\pi$$
$$32$$ 5.61803 0.993137
$$33$$ 0 0
$$34$$ −3.23607 −0.554981
$$35$$ −1.00000 −0.169031
$$36$$ −1.61803 −0.269672
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 4.14590 0.672553
$$39$$ −3.47214 −0.555987
$$40$$ −2.23607 −0.353553
$$41$$ 2.47214 0.386083 0.193041 0.981191i $$-0.438165\pi$$
0.193041 + 0.981191i $$0.438165\pi$$
$$42$$ −0.618034 −0.0953647
$$43$$ −5.70820 −0.870493 −0.435246 0.900311i $$-0.643339\pi$$
−0.435246 + 0.900311i $$0.643339\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 3.52786 0.520155
$$47$$ 0.236068 0.0344341 0.0172170 0.999852i $$-0.494519\pi$$
0.0172170 + 0.999852i $$0.494519\pi$$
$$48$$ 1.85410 0.267617
$$49$$ 1.00000 0.142857
$$50$$ −2.47214 −0.349613
$$51$$ −5.23607 −0.733196
$$52$$ 5.61803 0.779081
$$53$$ −12.1803 −1.67310 −0.836549 0.547892i $$-0.815431\pi$$
−0.836549 + 0.547892i $$0.815431\pi$$
$$54$$ 0.618034 0.0841038
$$55$$ 0 0
$$56$$ 2.23607 0.298807
$$57$$ 6.70820 0.888523
$$58$$ −3.09017 −0.405759
$$59$$ −11.1803 −1.45556 −0.727778 0.685813i $$-0.759447\pi$$
−0.727778 + 0.685813i $$0.759447\pi$$
$$60$$ −1.61803 −0.208887
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −3.23607 −0.410981
$$63$$ −1.00000 −0.125988
$$64$$ −0.236068 −0.0295085
$$65$$ −3.47214 −0.430665
$$66$$ 0 0
$$67$$ −9.76393 −1.19285 −0.596427 0.802667i $$-0.703414\pi$$
−0.596427 + 0.802667i $$0.703414\pi$$
$$68$$ 8.47214 1.02740
$$69$$ 5.70820 0.687187
$$70$$ −0.618034 −0.0738692
$$71$$ −2.47214 −0.293389 −0.146694 0.989182i $$-0.546863\pi$$
−0.146694 + 0.989182i $$0.546863\pi$$
$$72$$ −2.23607 −0.263523
$$73$$ −4.52786 −0.529946 −0.264973 0.964256i $$-0.585363\pi$$
−0.264973 + 0.964256i $$0.585363\pi$$
$$74$$ −4.32624 −0.502915
$$75$$ −4.00000 −0.461880
$$76$$ −10.8541 −1.24505
$$77$$ 0 0
$$78$$ −2.14590 −0.242975
$$79$$ 14.4721 1.62824 0.814121 0.580695i $$-0.197219\pi$$
0.814121 + 0.580695i $$0.197219\pi$$
$$80$$ 1.85410 0.207295
$$81$$ 1.00000 0.111111
$$82$$ 1.52786 0.168724
$$83$$ −6.76393 −0.742438 −0.371219 0.928545i $$-0.621060\pi$$
−0.371219 + 0.928545i $$0.621060\pi$$
$$84$$ 1.61803 0.176542
$$85$$ −5.23607 −0.567931
$$86$$ −3.52786 −0.380419
$$87$$ −5.00000 −0.536056
$$88$$ 0 0
$$89$$ −4.47214 −0.474045 −0.237023 0.971504i $$-0.576172\pi$$
−0.237023 + 0.971504i $$0.576172\pi$$
$$90$$ 0.618034 0.0651465
$$91$$ 3.47214 0.363979
$$92$$ −9.23607 −0.962927
$$93$$ −5.23607 −0.542955
$$94$$ 0.145898 0.0150482
$$95$$ 6.70820 0.688247
$$96$$ 5.61803 0.573388
$$97$$ 9.70820 0.985719 0.492859 0.870109i $$-0.335952\pi$$
0.492859 + 0.870109i $$0.335952\pi$$
$$98$$ 0.618034 0.0624309
$$99$$ 0 0
$$100$$ 6.47214 0.647214
$$101$$ −18.1803 −1.80901 −0.904506 0.426461i $$-0.859760\pi$$
−0.904506 + 0.426461i $$0.859760\pi$$
$$102$$ −3.23607 −0.320418
$$103$$ 17.4164 1.71609 0.858045 0.513575i $$-0.171679\pi$$
0.858045 + 0.513575i $$0.171679\pi$$
$$104$$ 7.76393 0.761316
$$105$$ −1.00000 −0.0975900
$$106$$ −7.52786 −0.731171
$$107$$ 4.23607 0.409516 0.204758 0.978813i $$-0.434359\pi$$
0.204758 + 0.978813i $$0.434359\pi$$
$$108$$ −1.61803 −0.155695
$$109$$ −2.76393 −0.264737 −0.132368 0.991201i $$-0.542258\pi$$
−0.132368 + 0.991201i $$0.542258\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ −1.85410 −0.175196
$$113$$ −0.472136 −0.0444148 −0.0222074 0.999753i $$-0.507069\pi$$
−0.0222074 + 0.999753i $$0.507069\pi$$
$$114$$ 4.14590 0.388299
$$115$$ 5.70820 0.532293
$$116$$ 8.09017 0.751153
$$117$$ −3.47214 −0.320999
$$118$$ −6.90983 −0.636101
$$119$$ 5.23607 0.479990
$$120$$ −2.23607 −0.204124
$$121$$ 0 0
$$122$$ −1.23607 −0.111908
$$123$$ 2.47214 0.222905
$$124$$ 8.47214 0.760820
$$125$$ −9.00000 −0.804984
$$126$$ −0.618034 −0.0550588
$$127$$ 12.6525 1.12273 0.561363 0.827570i $$-0.310277\pi$$
0.561363 + 0.827570i $$0.310277\pi$$
$$128$$ −11.3820 −1.00603
$$129$$ −5.70820 −0.502579
$$130$$ −2.14590 −0.188208
$$131$$ −0.944272 −0.0825014 −0.0412507 0.999149i $$-0.513134\pi$$
−0.0412507 + 0.999149i $$0.513134\pi$$
$$132$$ 0 0
$$133$$ −6.70820 −0.581675
$$134$$ −6.03444 −0.521296
$$135$$ 1.00000 0.0860663
$$136$$ 11.7082 1.00397
$$137$$ 19.7082 1.68379 0.841893 0.539645i $$-0.181441\pi$$
0.841893 + 0.539645i $$0.181441\pi$$
$$138$$ 3.52786 0.300312
$$139$$ −14.4721 −1.22751 −0.613755 0.789496i $$-0.710342\pi$$
−0.613755 + 0.789496i $$0.710342\pi$$
$$140$$ 1.61803 0.136749
$$141$$ 0.236068 0.0198805
$$142$$ −1.52786 −0.128216
$$143$$ 0 0
$$144$$ 1.85410 0.154508
$$145$$ −5.00000 −0.415227
$$146$$ −2.79837 −0.231595
$$147$$ 1.00000 0.0824786
$$148$$ 11.3262 0.931011
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ −2.47214 −0.201849
$$151$$ 14.1803 1.15398 0.576990 0.816751i $$-0.304227\pi$$
0.576990 + 0.816751i $$0.304227\pi$$
$$152$$ −15.0000 −1.21666
$$153$$ −5.23607 −0.423311
$$154$$ 0 0
$$155$$ −5.23607 −0.420571
$$156$$ 5.61803 0.449803
$$157$$ −15.4164 −1.23036 −0.615182 0.788385i $$-0.710917\pi$$
−0.615182 + 0.788385i $$0.710917\pi$$
$$158$$ 8.94427 0.711568
$$159$$ −12.1803 −0.965964
$$160$$ 5.61803 0.444145
$$161$$ −5.70820 −0.449869
$$162$$ 0.618034 0.0485573
$$163$$ −22.7082 −1.77864 −0.889322 0.457282i $$-0.848823\pi$$
−0.889322 + 0.457282i $$0.848823\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ −4.18034 −0.324457
$$167$$ 22.6525 1.75290 0.876451 0.481492i $$-0.159905\pi$$
0.876451 + 0.481492i $$0.159905\pi$$
$$168$$ 2.23607 0.172516
$$169$$ −0.944272 −0.0726363
$$170$$ −3.23607 −0.248195
$$171$$ 6.70820 0.512989
$$172$$ 9.23607 0.704244
$$173$$ 1.52786 0.116161 0.0580807 0.998312i $$-0.481502\pi$$
0.0580807 + 0.998312i $$0.481502\pi$$
$$174$$ −3.09017 −0.234265
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ −11.1803 −0.840366
$$178$$ −2.76393 −0.207165
$$179$$ 8.94427 0.668526 0.334263 0.942480i $$-0.391513\pi$$
0.334263 + 0.942480i $$0.391513\pi$$
$$180$$ −1.61803 −0.120601
$$181$$ −0.763932 −0.0567826 −0.0283913 0.999597i $$-0.509038\pi$$
−0.0283913 + 0.999597i $$0.509038\pi$$
$$182$$ 2.14590 0.159065
$$183$$ −2.00000 −0.147844
$$184$$ −12.7639 −0.940970
$$185$$ −7.00000 −0.514650
$$186$$ −3.23607 −0.237280
$$187$$ 0 0
$$188$$ −0.381966 −0.0278577
$$189$$ −1.00000 −0.0727393
$$190$$ 4.14590 0.300775
$$191$$ −10.7639 −0.778851 −0.389425 0.921058i $$-0.627326\pi$$
−0.389425 + 0.921058i $$0.627326\pi$$
$$192$$ −0.236068 −0.0170367
$$193$$ −14.6525 −1.05471 −0.527354 0.849646i $$-0.676816\pi$$
−0.527354 + 0.849646i $$0.676816\pi$$
$$194$$ 6.00000 0.430775
$$195$$ −3.47214 −0.248645
$$196$$ −1.61803 −0.115574
$$197$$ 16.4721 1.17359 0.586796 0.809735i $$-0.300389\pi$$
0.586796 + 0.809735i $$0.300389\pi$$
$$198$$ 0 0
$$199$$ −3.81966 −0.270769 −0.135384 0.990793i $$-0.543227\pi$$
−0.135384 + 0.990793i $$0.543227\pi$$
$$200$$ 8.94427 0.632456
$$201$$ −9.76393 −0.688695
$$202$$ −11.2361 −0.790567
$$203$$ 5.00000 0.350931
$$204$$ 8.47214 0.593168
$$205$$ 2.47214 0.172661
$$206$$ 10.7639 0.749959
$$207$$ 5.70820 0.396748
$$208$$ −6.43769 −0.446374
$$209$$ 0 0
$$210$$ −0.618034 −0.0426484
$$211$$ −5.41641 −0.372881 −0.186440 0.982466i $$-0.559695\pi$$
−0.186440 + 0.982466i $$0.559695\pi$$
$$212$$ 19.7082 1.35357
$$213$$ −2.47214 −0.169388
$$214$$ 2.61803 0.178965
$$215$$ −5.70820 −0.389296
$$216$$ −2.23607 −0.152145
$$217$$ 5.23607 0.355447
$$218$$ −1.70820 −0.115694
$$219$$ −4.52786 −0.305965
$$220$$ 0 0
$$221$$ 18.1803 1.22294
$$222$$ −4.32624 −0.290358
$$223$$ −6.00000 −0.401790 −0.200895 0.979613i $$-0.564385\pi$$
−0.200895 + 0.979613i $$0.564385\pi$$
$$224$$ −5.61803 −0.375371
$$225$$ −4.00000 −0.266667
$$226$$ −0.291796 −0.0194100
$$227$$ 2.00000 0.132745 0.0663723 0.997795i $$-0.478857\pi$$
0.0663723 + 0.997795i $$0.478857\pi$$
$$228$$ −10.8541 −0.718830
$$229$$ 7.23607 0.478173 0.239086 0.970998i $$-0.423152\pi$$
0.239086 + 0.970998i $$0.423152\pi$$
$$230$$ 3.52786 0.232620
$$231$$ 0 0
$$232$$ 11.1803 0.734025
$$233$$ 14.9443 0.979032 0.489516 0.871994i $$-0.337174\pi$$
0.489516 + 0.871994i $$0.337174\pi$$
$$234$$ −2.14590 −0.140282
$$235$$ 0.236068 0.0153994
$$236$$ 18.0902 1.17757
$$237$$ 14.4721 0.940066
$$238$$ 3.23607 0.209763
$$239$$ −10.1246 −0.654907 −0.327453 0.944867i $$-0.606190\pi$$
−0.327453 + 0.944867i $$0.606190\pi$$
$$240$$ 1.85410 0.119682
$$241$$ −25.9443 −1.67122 −0.835609 0.549325i $$-0.814885\pi$$
−0.835609 + 0.549325i $$0.814885\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 3.23607 0.207168
$$245$$ 1.00000 0.0638877
$$246$$ 1.52786 0.0974131
$$247$$ −23.2918 −1.48202
$$248$$ 11.7082 0.743472
$$249$$ −6.76393 −0.428647
$$250$$ −5.56231 −0.351791
$$251$$ 12.1246 0.765299 0.382649 0.923894i $$-0.375012\pi$$
0.382649 + 0.923894i $$0.375012\pi$$
$$252$$ 1.61803 0.101927
$$253$$ 0 0
$$254$$ 7.81966 0.490649
$$255$$ −5.23607 −0.327895
$$256$$ −6.56231 −0.410144
$$257$$ −7.00000 −0.436648 −0.218324 0.975876i $$-0.570059\pi$$
−0.218324 + 0.975876i $$0.570059\pi$$
$$258$$ −3.52786 −0.219635
$$259$$ 7.00000 0.434959
$$260$$ 5.61803 0.348416
$$261$$ −5.00000 −0.309492
$$262$$ −0.583592 −0.0360544
$$263$$ 26.1246 1.61091 0.805456 0.592655i $$-0.201920\pi$$
0.805456 + 0.592655i $$0.201920\pi$$
$$264$$ 0 0
$$265$$ −12.1803 −0.748232
$$266$$ −4.14590 −0.254201
$$267$$ −4.47214 −0.273690
$$268$$ 15.7984 0.965039
$$269$$ 1.05573 0.0643689 0.0321844 0.999482i $$-0.489754\pi$$
0.0321844 + 0.999482i $$0.489754\pi$$
$$270$$ 0.618034 0.0376124
$$271$$ −5.29180 −0.321454 −0.160727 0.986999i $$-0.551384\pi$$
−0.160727 + 0.986999i $$0.551384\pi$$
$$272$$ −9.70820 −0.588646
$$273$$ 3.47214 0.210143
$$274$$ 12.1803 0.735841
$$275$$ 0 0
$$276$$ −9.23607 −0.555946
$$277$$ 6.47214 0.388873 0.194436 0.980915i $$-0.437712\pi$$
0.194436 + 0.980915i $$0.437712\pi$$
$$278$$ −8.94427 −0.536442
$$279$$ −5.23607 −0.313475
$$280$$ 2.23607 0.133631
$$281$$ −11.4721 −0.684370 −0.342185 0.939633i $$-0.611167\pi$$
−0.342185 + 0.939633i $$0.611167\pi$$
$$282$$ 0.145898 0.00868810
$$283$$ 13.7639 0.818181 0.409090 0.912494i $$-0.365846\pi$$
0.409090 + 0.912494i $$0.365846\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 6.70820 0.397360
$$286$$ 0 0
$$287$$ −2.47214 −0.145926
$$288$$ 5.61803 0.331046
$$289$$ 10.4164 0.612730
$$290$$ −3.09017 −0.181461
$$291$$ 9.70820 0.569105
$$292$$ 7.32624 0.428736
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 0.618034 0.0360445
$$295$$ −11.1803 −0.650945
$$296$$ 15.6525 0.909782
$$297$$ 0 0
$$298$$ −3.09017 −0.179009
$$299$$ −19.8197 −1.14620
$$300$$ 6.47214 0.373669
$$301$$ 5.70820 0.329015
$$302$$ 8.76393 0.504308
$$303$$ −18.1803 −1.04443
$$304$$ 12.4377 0.713351
$$305$$ −2.00000 −0.114520
$$306$$ −3.23607 −0.184994
$$307$$ 20.9443 1.19535 0.597676 0.801737i $$-0.296091\pi$$
0.597676 + 0.801737i $$0.296091\pi$$
$$308$$ 0 0
$$309$$ 17.4164 0.990785
$$310$$ −3.23607 −0.183796
$$311$$ 9.88854 0.560728 0.280364 0.959894i $$-0.409545\pi$$
0.280364 + 0.959894i $$0.409545\pi$$
$$312$$ 7.76393 0.439546
$$313$$ 24.6525 1.39344 0.696720 0.717343i $$-0.254642\pi$$
0.696720 + 0.717343i $$0.254642\pi$$
$$314$$ −9.52786 −0.537688
$$315$$ −1.00000 −0.0563436
$$316$$ −23.4164 −1.31728
$$317$$ 24.1803 1.35810 0.679052 0.734091i $$-0.262391\pi$$
0.679052 + 0.734091i $$0.262391\pi$$
$$318$$ −7.52786 −0.422142
$$319$$ 0 0
$$320$$ −0.236068 −0.0131966
$$321$$ 4.23607 0.236434
$$322$$ −3.52786 −0.196600
$$323$$ −35.1246 −1.95439
$$324$$ −1.61803 −0.0898908
$$325$$ 13.8885 0.770398
$$326$$ −14.0344 −0.777296
$$327$$ −2.76393 −0.152846
$$328$$ −5.52786 −0.305225
$$329$$ −0.236068 −0.0130148
$$330$$ 0 0
$$331$$ −11.4164 −0.627503 −0.313751 0.949505i $$-0.601586\pi$$
−0.313751 + 0.949505i $$0.601586\pi$$
$$332$$ 10.9443 0.600645
$$333$$ −7.00000 −0.383598
$$334$$ 14.0000 0.766046
$$335$$ −9.76393 −0.533461
$$336$$ −1.85410 −0.101150
$$337$$ 8.18034 0.445612 0.222806 0.974863i $$-0.428478\pi$$
0.222806 + 0.974863i $$0.428478\pi$$
$$338$$ −0.583592 −0.0317432
$$339$$ −0.472136 −0.0256429
$$340$$ 8.47214 0.459466
$$341$$ 0 0
$$342$$ 4.14590 0.224184
$$343$$ −1.00000 −0.0539949
$$344$$ 12.7639 0.688185
$$345$$ 5.70820 0.307319
$$346$$ 0.944272 0.0507644
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 8.09017 0.433679
$$349$$ −1.58359 −0.0847677 −0.0423839 0.999101i $$-0.513495\pi$$
−0.0423839 + 0.999101i $$0.513495\pi$$
$$350$$ 2.47214 0.132141
$$351$$ −3.47214 −0.185329
$$352$$ 0 0
$$353$$ 24.5279 1.30549 0.652743 0.757579i $$-0.273618\pi$$
0.652743 + 0.757579i $$0.273618\pi$$
$$354$$ −6.90983 −0.367253
$$355$$ −2.47214 −0.131207
$$356$$ 7.23607 0.383511
$$357$$ 5.23607 0.277122
$$358$$ 5.52786 0.292157
$$359$$ −23.4164 −1.23587 −0.617935 0.786229i $$-0.712031\pi$$
−0.617935 + 0.786229i $$0.712031\pi$$
$$360$$ −2.23607 −0.117851
$$361$$ 26.0000 1.36842
$$362$$ −0.472136 −0.0248149
$$363$$ 0 0
$$364$$ −5.61803 −0.294465
$$365$$ −4.52786 −0.236999
$$366$$ −1.23607 −0.0646103
$$367$$ −19.8885 −1.03817 −0.519087 0.854722i $$-0.673728\pi$$
−0.519087 + 0.854722i $$0.673728\pi$$
$$368$$ 10.5836 0.551708
$$369$$ 2.47214 0.128694
$$370$$ −4.32624 −0.224910
$$371$$ 12.1803 0.632372
$$372$$ 8.47214 0.439260
$$373$$ −4.65248 −0.240896 −0.120448 0.992720i $$-0.538433\pi$$
−0.120448 + 0.992720i $$0.538433\pi$$
$$374$$ 0 0
$$375$$ −9.00000 −0.464758
$$376$$ −0.527864 −0.0272225
$$377$$ 17.3607 0.894120
$$378$$ −0.618034 −0.0317882
$$379$$ −31.1803 −1.60163 −0.800813 0.598914i $$-0.795599\pi$$
−0.800813 + 0.598914i $$0.795599\pi$$
$$380$$ −10.8541 −0.556804
$$381$$ 12.6525 0.648206
$$382$$ −6.65248 −0.340370
$$383$$ 32.9443 1.68337 0.841687 0.539966i $$-0.181563\pi$$
0.841687 + 0.539966i $$0.181563\pi$$
$$384$$ −11.3820 −0.580834
$$385$$ 0 0
$$386$$ −9.05573 −0.460924
$$387$$ −5.70820 −0.290164
$$388$$ −15.7082 −0.797463
$$389$$ 11.0557 0.560548 0.280274 0.959920i $$-0.409575\pi$$
0.280274 + 0.959920i $$0.409575\pi$$
$$390$$ −2.14590 −0.108662
$$391$$ −29.8885 −1.51153
$$392$$ −2.23607 −0.112938
$$393$$ −0.944272 −0.0476322
$$394$$ 10.1803 0.512878
$$395$$ 14.4721 0.728172
$$396$$ 0 0
$$397$$ 23.1246 1.16059 0.580295 0.814406i $$-0.302937\pi$$
0.580295 + 0.814406i $$0.302937\pi$$
$$398$$ −2.36068 −0.118330
$$399$$ −6.70820 −0.335830
$$400$$ −7.41641 −0.370820
$$401$$ −29.7082 −1.48356 −0.741778 0.670645i $$-0.766017\pi$$
−0.741778 + 0.670645i $$0.766017\pi$$
$$402$$ −6.03444 −0.300971
$$403$$ 18.1803 0.905627
$$404$$ 29.4164 1.46352
$$405$$ 1.00000 0.0496904
$$406$$ 3.09017 0.153363
$$407$$ 0 0
$$408$$ 11.7082 0.579642
$$409$$ −21.0557 −1.04114 −0.520569 0.853819i $$-0.674280\pi$$
−0.520569 + 0.853819i $$0.674280\pi$$
$$410$$ 1.52786 0.0754558
$$411$$ 19.7082 0.972134
$$412$$ −28.1803 −1.38835
$$413$$ 11.1803 0.550149
$$414$$ 3.52786 0.173385
$$415$$ −6.76393 −0.332028
$$416$$ −19.5066 −0.956389
$$417$$ −14.4721 −0.708704
$$418$$ 0 0
$$419$$ −1.18034 −0.0576634 −0.0288317 0.999584i $$-0.509179\pi$$
−0.0288317 + 0.999584i $$0.509179\pi$$
$$420$$ 1.61803 0.0789520
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ −3.34752 −0.162955
$$423$$ 0.236068 0.0114780
$$424$$ 27.2361 1.32270
$$425$$ 20.9443 1.01595
$$426$$ −1.52786 −0.0740253
$$427$$ 2.00000 0.0967868
$$428$$ −6.85410 −0.331306
$$429$$ 0 0
$$430$$ −3.52786 −0.170129
$$431$$ −8.70820 −0.419459 −0.209730 0.977759i $$-0.567258\pi$$
−0.209730 + 0.977759i $$0.567258\pi$$
$$432$$ 1.85410 0.0892055
$$433$$ −10.4721 −0.503259 −0.251629 0.967824i $$-0.580966\pi$$
−0.251629 + 0.967824i $$0.580966\pi$$
$$434$$ 3.23607 0.155336
$$435$$ −5.00000 −0.239732
$$436$$ 4.47214 0.214176
$$437$$ 38.2918 1.83175
$$438$$ −2.79837 −0.133711
$$439$$ −11.1803 −0.533609 −0.266804 0.963751i $$-0.585968\pi$$
−0.266804 + 0.963751i $$0.585968\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 11.2361 0.534445
$$443$$ 18.4721 0.877638 0.438819 0.898576i $$-0.355397\pi$$
0.438819 + 0.898576i $$0.355397\pi$$
$$444$$ 11.3262 0.537519
$$445$$ −4.47214 −0.212000
$$446$$ −3.70820 −0.175589
$$447$$ −5.00000 −0.236492
$$448$$ 0.236068 0.0111532
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ −2.47214 −0.116538
$$451$$ 0 0
$$452$$ 0.763932 0.0359323
$$453$$ 14.1803 0.666250
$$454$$ 1.23607 0.0580115
$$455$$ 3.47214 0.162776
$$456$$ −15.0000 −0.702439
$$457$$ −15.2361 −0.712713 −0.356357 0.934350i $$-0.615981\pi$$
−0.356357 + 0.934350i $$0.615981\pi$$
$$458$$ 4.47214 0.208969
$$459$$ −5.23607 −0.244399
$$460$$ −9.23607 −0.430634
$$461$$ 28.0000 1.30409 0.652045 0.758180i $$-0.273911\pi$$
0.652045 + 0.758180i $$0.273911\pi$$
$$462$$ 0 0
$$463$$ −4.81966 −0.223989 −0.111994 0.993709i $$-0.535724\pi$$
−0.111994 + 0.993709i $$0.535724\pi$$
$$464$$ −9.27051 −0.430373
$$465$$ −5.23607 −0.242817
$$466$$ 9.23607 0.427853
$$467$$ 29.1803 1.35031 0.675153 0.737678i $$-0.264078\pi$$
0.675153 + 0.737678i $$0.264078\pi$$
$$468$$ 5.61803 0.259694
$$469$$ 9.76393 0.450856
$$470$$ 0.145898 0.00672977
$$471$$ −15.4164 −0.710351
$$472$$ 25.0000 1.15072
$$473$$ 0 0
$$474$$ 8.94427 0.410824
$$475$$ −26.8328 −1.23117
$$476$$ −8.47214 −0.388320
$$477$$ −12.1803 −0.557699
$$478$$ −6.25735 −0.286205
$$479$$ −11.7082 −0.534961 −0.267481 0.963563i $$-0.586191\pi$$
−0.267481 + 0.963563i $$0.586191\pi$$
$$480$$ 5.61803 0.256427
$$481$$ 24.3050 1.10821
$$482$$ −16.0344 −0.730349
$$483$$ −5.70820 −0.259732
$$484$$ 0 0
$$485$$ 9.70820 0.440827
$$486$$ 0.618034 0.0280346
$$487$$ 16.9443 0.767818 0.383909 0.923371i $$-0.374578\pi$$
0.383909 + 0.923371i $$0.374578\pi$$
$$488$$ 4.47214 0.202444
$$489$$ −22.7082 −1.02690
$$490$$ 0.618034 0.0279199
$$491$$ 18.1246 0.817952 0.408976 0.912545i $$-0.365886\pi$$
0.408976 + 0.912545i $$0.365886\pi$$
$$492$$ −4.00000 −0.180334
$$493$$ 26.1803 1.17910
$$494$$ −14.3951 −0.647667
$$495$$ 0 0
$$496$$ −9.70820 −0.435911
$$497$$ 2.47214 0.110890
$$498$$ −4.18034 −0.187326
$$499$$ −2.23607 −0.100100 −0.0500501 0.998747i $$-0.515938\pi$$
−0.0500501 + 0.998747i $$0.515938\pi$$
$$500$$ 14.5623 0.651246
$$501$$ 22.6525 1.01204
$$502$$ 7.49342 0.334448
$$503$$ −2.29180 −0.102186 −0.0510931 0.998694i $$-0.516271\pi$$
−0.0510931 + 0.998694i $$0.516271\pi$$
$$504$$ 2.23607 0.0996024
$$505$$ −18.1803 −0.809015
$$506$$ 0 0
$$507$$ −0.944272 −0.0419366
$$508$$ −20.4721 −0.908304
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ −3.23607 −0.143295
$$511$$ 4.52786 0.200301
$$512$$ 18.7082 0.826794
$$513$$ 6.70820 0.296174
$$514$$ −4.32624 −0.190822
$$515$$ 17.4164 0.767459
$$516$$ 9.23607 0.406595
$$517$$ 0 0
$$518$$ 4.32624 0.190084
$$519$$ 1.52786 0.0670658
$$520$$ 7.76393 0.340471
$$521$$ 38.3050 1.67817 0.839085 0.544000i $$-0.183091\pi$$
0.839085 + 0.544000i $$0.183091\pi$$
$$522$$ −3.09017 −0.135253
$$523$$ 21.6525 0.946797 0.473398 0.880848i $$-0.343027\pi$$
0.473398 + 0.880848i $$0.343027\pi$$
$$524$$ 1.52786 0.0667451
$$525$$ 4.00000 0.174574
$$526$$ 16.1459 0.703995
$$527$$ 27.4164 1.19428
$$528$$ 0 0
$$529$$ 9.58359 0.416678
$$530$$ −7.52786 −0.326990
$$531$$ −11.1803 −0.485185
$$532$$ 10.8541 0.470585
$$533$$ −8.58359 −0.371797
$$534$$ −2.76393 −0.119607
$$535$$ 4.23607 0.183141
$$536$$ 21.8328 0.943034
$$537$$ 8.94427 0.385974
$$538$$ 0.652476 0.0281302
$$539$$ 0 0
$$540$$ −1.61803 −0.0696291
$$541$$ 36.9443 1.58836 0.794179 0.607684i $$-0.207901\pi$$
0.794179 + 0.607684i $$0.207901\pi$$
$$542$$ −3.27051 −0.140480
$$543$$ −0.763932 −0.0327835
$$544$$ −29.4164 −1.26122
$$545$$ −2.76393 −0.118394
$$546$$ 2.14590 0.0918360
$$547$$ −14.8328 −0.634205 −0.317103 0.948391i $$-0.602710\pi$$
−0.317103 + 0.948391i $$0.602710\pi$$
$$548$$ −31.8885 −1.36221
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −33.5410 −1.42890
$$552$$ −12.7639 −0.543269
$$553$$ −14.4721 −0.615418
$$554$$ 4.00000 0.169944
$$555$$ −7.00000 −0.297133
$$556$$ 23.4164 0.993077
$$557$$ 12.5279 0.530823 0.265411 0.964135i $$-0.414492\pi$$
0.265411 + 0.964135i $$0.414492\pi$$
$$558$$ −3.23607 −0.136994
$$559$$ 19.8197 0.838282
$$560$$ −1.85410 −0.0783501
$$561$$ 0 0
$$562$$ −7.09017 −0.299081
$$563$$ −34.6525 −1.46043 −0.730214 0.683219i $$-0.760580\pi$$
−0.730214 + 0.683219i $$0.760580\pi$$
$$564$$ −0.381966 −0.0160837
$$565$$ −0.472136 −0.0198629
$$566$$ 8.50658 0.357558
$$567$$ −1.00000 −0.0419961
$$568$$ 5.52786 0.231944
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 4.14590 0.173653
$$571$$ −17.5279 −0.733518 −0.366759 0.930316i $$-0.619533\pi$$
−0.366759 + 0.930316i $$0.619533\pi$$
$$572$$ 0 0
$$573$$ −10.7639 −0.449670
$$574$$ −1.52786 −0.0637718
$$575$$ −22.8328 −0.952194
$$576$$ −0.236068 −0.00983617
$$577$$ 7.34752 0.305881 0.152941 0.988235i $$-0.451126\pi$$
0.152941 + 0.988235i $$0.451126\pi$$
$$578$$ 6.43769 0.267773
$$579$$ −14.6525 −0.608936
$$580$$ 8.09017 0.335926
$$581$$ 6.76393 0.280615
$$582$$ 6.00000 0.248708
$$583$$ 0 0
$$584$$ 10.1246 0.418959
$$585$$ −3.47214 −0.143555
$$586$$ 9.88854 0.408492
$$587$$ 46.0132 1.89917 0.949583 0.313515i $$-0.101507\pi$$
0.949583 + 0.313515i $$0.101507\pi$$
$$588$$ −1.61803 −0.0667266
$$589$$ −35.1246 −1.44728
$$590$$ −6.90983 −0.284473
$$591$$ 16.4721 0.677573
$$592$$ −12.9787 −0.533422
$$593$$ 22.8328 0.937631 0.468816 0.883296i $$-0.344681\pi$$
0.468816 + 0.883296i $$0.344681\pi$$
$$594$$ 0 0
$$595$$ 5.23607 0.214658
$$596$$ 8.09017 0.331386
$$597$$ −3.81966 −0.156328
$$598$$ −12.2492 −0.500908
$$599$$ −25.5279 −1.04304 −0.521520 0.853239i $$-0.674635\pi$$
−0.521520 + 0.853239i $$0.674635\pi$$
$$600$$ 8.94427 0.365148
$$601$$ −27.0000 −1.10135 −0.550676 0.834719i $$-0.685630\pi$$
−0.550676 + 0.834719i $$0.685630\pi$$
$$602$$ 3.52786 0.143785
$$603$$ −9.76393 −0.397618
$$604$$ −22.9443 −0.933589
$$605$$ 0 0
$$606$$ −11.2361 −0.456434
$$607$$ −6.81966 −0.276801 −0.138401 0.990376i $$-0.544196\pi$$
−0.138401 + 0.990376i $$0.544196\pi$$
$$608$$ 37.6869 1.52841
$$609$$ 5.00000 0.202610
$$610$$ −1.23607 −0.0500469
$$611$$ −0.819660 −0.0331599
$$612$$ 8.47214 0.342466
$$613$$ −13.5967 −0.549167 −0.274584 0.961563i $$-0.588540\pi$$
−0.274584 + 0.961563i $$0.588540\pi$$
$$614$$ 12.9443 0.522388
$$615$$ 2.47214 0.0996861
$$616$$ 0 0
$$617$$ 32.4721 1.30728 0.653639 0.756806i $$-0.273241\pi$$
0.653639 + 0.756806i $$0.273241\pi$$
$$618$$ 10.7639 0.432989
$$619$$ −44.0689 −1.77128 −0.885639 0.464374i $$-0.846279\pi$$
−0.885639 + 0.464374i $$0.846279\pi$$
$$620$$ 8.47214 0.340249
$$621$$ 5.70820 0.229062
$$622$$ 6.11146 0.245047
$$623$$ 4.47214 0.179172
$$624$$ −6.43769 −0.257714
$$625$$ 11.0000 0.440000
$$626$$ 15.2361 0.608956
$$627$$ 0 0
$$628$$ 24.9443 0.995385
$$629$$ 36.6525 1.46143
$$630$$ −0.618034 −0.0246231
$$631$$ 44.3607 1.76597 0.882985 0.469400i $$-0.155530\pi$$
0.882985 + 0.469400i $$0.155530\pi$$
$$632$$ −32.3607 −1.28724
$$633$$ −5.41641 −0.215283
$$634$$ 14.9443 0.593513
$$635$$ 12.6525 0.502098
$$636$$ 19.7082 0.781481
$$637$$ −3.47214 −0.137571
$$638$$ 0 0
$$639$$ −2.47214 −0.0977962
$$640$$ −11.3820 −0.449912
$$641$$ −46.5410 −1.83826 −0.919130 0.393955i $$-0.871107\pi$$
−0.919130 + 0.393955i $$0.871107\pi$$
$$642$$ 2.61803 0.103326
$$643$$ −47.9574 −1.89126 −0.945628 0.325250i $$-0.894552\pi$$
−0.945628 + 0.325250i $$0.894552\pi$$
$$644$$ 9.23607 0.363952
$$645$$ −5.70820 −0.224760
$$646$$ −21.7082 −0.854098
$$647$$ 12.3475 0.485431 0.242716 0.970097i $$-0.421962\pi$$
0.242716 + 0.970097i $$0.421962\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ 0 0
$$650$$ 8.58359 0.336676
$$651$$ 5.23607 0.205218
$$652$$ 36.7426 1.43895
$$653$$ −44.9443 −1.75881 −0.879403 0.476079i $$-0.842058\pi$$
−0.879403 + 0.476079i $$0.842058\pi$$
$$654$$ −1.70820 −0.0667961
$$655$$ −0.944272 −0.0368958
$$656$$ 4.58359 0.178959
$$657$$ −4.52786 −0.176649
$$658$$ −0.145898 −0.00568770
$$659$$ −23.5410 −0.917028 −0.458514 0.888687i $$-0.651618\pi$$
−0.458514 + 0.888687i $$0.651618\pi$$
$$660$$ 0 0
$$661$$ 40.5410 1.57686 0.788431 0.615123i $$-0.210894\pi$$
0.788431 + 0.615123i $$0.210894\pi$$
$$662$$ −7.05573 −0.274229
$$663$$ 18.1803 0.706066
$$664$$ 15.1246 0.586949
$$665$$ −6.70820 −0.260133
$$666$$ −4.32624 −0.167638
$$667$$ −28.5410 −1.10511
$$668$$ −36.6525 −1.41813
$$669$$ −6.00000 −0.231973
$$670$$ −6.03444 −0.233131
$$671$$ 0 0
$$672$$ −5.61803 −0.216720
$$673$$ −3.59675 −0.138644 −0.0693222 0.997594i $$-0.522084\pi$$
−0.0693222 + 0.997594i $$0.522084\pi$$
$$674$$ 5.05573 0.194739
$$675$$ −4.00000 −0.153960
$$676$$ 1.52786 0.0587640
$$677$$ −19.3050 −0.741950 −0.370975 0.928643i $$-0.620976\pi$$
−0.370975 + 0.928643i $$0.620976\pi$$
$$678$$ −0.291796 −0.0112064
$$679$$ −9.70820 −0.372567
$$680$$ 11.7082 0.448989
$$681$$ 2.00000 0.0766402
$$682$$ 0 0
$$683$$ 37.0132 1.41627 0.708135 0.706078i $$-0.249537\pi$$
0.708135 + 0.706078i $$0.249537\pi$$
$$684$$ −10.8541 −0.415017
$$685$$ 19.7082 0.753012
$$686$$ −0.618034 −0.0235966
$$687$$ 7.23607 0.276073
$$688$$ −10.5836 −0.403496
$$689$$ 42.2918 1.61119
$$690$$ 3.52786 0.134303
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ −2.47214 −0.0939765
$$693$$ 0 0
$$694$$ −17.3050 −0.656887
$$695$$ −14.4721 −0.548959
$$696$$ 11.1803 0.423790
$$697$$ −12.9443 −0.490299
$$698$$ −0.978714 −0.0370449
$$699$$ 14.9443 0.565244
$$700$$ −6.47214 −0.244624
$$701$$ −4.11146 −0.155288 −0.0776438 0.996981i $$-0.524740\pi$$
−0.0776438 + 0.996981i $$0.524740\pi$$
$$702$$ −2.14590 −0.0809917
$$703$$ −46.9574 −1.77103
$$704$$ 0 0
$$705$$ 0.236068 0.00889083
$$706$$ 15.1591 0.570519
$$707$$ 18.1803 0.683742
$$708$$ 18.0902 0.679870
$$709$$ −49.7214 −1.86732 −0.933662 0.358154i $$-0.883406\pi$$
−0.933662 + 0.358154i $$0.883406\pi$$
$$710$$ −1.52786 −0.0573397
$$711$$ 14.4721 0.542748
$$712$$ 10.0000 0.374766
$$713$$ −29.8885 −1.11933
$$714$$ 3.23607 0.121107
$$715$$ 0 0
$$716$$ −14.4721 −0.540849
$$717$$ −10.1246 −0.378111
$$718$$ −14.4721 −0.540095
$$719$$ 7.76393 0.289546 0.144773 0.989465i $$-0.453755\pi$$
0.144773 + 0.989465i $$0.453755\pi$$
$$720$$ 1.85410 0.0690983
$$721$$ −17.4164 −0.648621
$$722$$ 16.0689 0.598022
$$723$$ −25.9443 −0.964878
$$724$$ 1.23607 0.0459381
$$725$$ 20.0000 0.742781
$$726$$ 0 0
$$727$$ 24.1803 0.896799 0.448400 0.893833i $$-0.351994\pi$$
0.448400 + 0.893833i $$0.351994\pi$$
$$728$$ −7.76393 −0.287750
$$729$$ 1.00000 0.0370370
$$730$$ −2.79837 −0.103572
$$731$$ 29.8885 1.10547
$$732$$ 3.23607 0.119609
$$733$$ 8.11146 0.299603 0.149802 0.988716i $$-0.452136\pi$$
0.149802 + 0.988716i $$0.452136\pi$$
$$734$$ −12.2918 −0.453698
$$735$$ 1.00000 0.0368856
$$736$$ 32.0689 1.18207
$$737$$ 0 0
$$738$$ 1.52786 0.0562415
$$739$$ −34.0689 −1.25324 −0.626622 0.779323i $$-0.715563\pi$$
−0.626622 + 0.779323i $$0.715563\pi$$
$$740$$ 11.3262 0.416361
$$741$$ −23.2918 −0.855646
$$742$$ 7.52786 0.276357
$$743$$ −2.81966 −0.103443 −0.0517216 0.998662i $$-0.516471\pi$$
−0.0517216 + 0.998662i $$0.516471\pi$$
$$744$$ 11.7082 0.429244
$$745$$ −5.00000 −0.183186
$$746$$ −2.87539 −0.105275
$$747$$ −6.76393 −0.247479
$$748$$ 0 0
$$749$$ −4.23607 −0.154783
$$750$$ −5.56231 −0.203107
$$751$$ 39.7639 1.45101 0.725503 0.688219i $$-0.241607\pi$$
0.725503 + 0.688219i $$0.241607\pi$$
$$752$$ 0.437694 0.0159611
$$753$$ 12.1246 0.441845
$$754$$ 10.7295 0.390745
$$755$$ 14.1803 0.516075
$$756$$ 1.61803 0.0588473
$$757$$ −51.7214 −1.87984 −0.939922 0.341388i $$-0.889103\pi$$
−0.939922 + 0.341388i $$0.889103\pi$$
$$758$$ −19.2705 −0.699936
$$759$$ 0 0
$$760$$ −15.0000 −0.544107
$$761$$ −27.7771 −1.00692 −0.503459 0.864019i $$-0.667940\pi$$
−0.503459 + 0.864019i $$0.667940\pi$$
$$762$$ 7.81966 0.283276
$$763$$ 2.76393 0.100061
$$764$$ 17.4164 0.630104
$$765$$ −5.23607 −0.189310
$$766$$ 20.3607 0.735661
$$767$$ 38.8197 1.40170
$$768$$ −6.56231 −0.236797
$$769$$ −13.9443 −0.502843 −0.251422 0.967878i $$-0.580898\pi$$
−0.251422 + 0.967878i $$0.580898\pi$$
$$770$$ 0 0
$$771$$ −7.00000 −0.252099
$$772$$ 23.7082 0.853277
$$773$$ −5.47214 −0.196819 −0.0984095 0.995146i $$-0.531376\pi$$
−0.0984095 + 0.995146i $$0.531376\pi$$
$$774$$ −3.52786 −0.126806
$$775$$ 20.9443 0.752340
$$776$$ −21.7082 −0.779279
$$777$$ 7.00000 0.251124
$$778$$ 6.83282 0.244968
$$779$$ 16.5836 0.594169
$$780$$ 5.61803 0.201158
$$781$$ 0 0
$$782$$ −18.4721 −0.660562
$$783$$ −5.00000 −0.178685
$$784$$ 1.85410 0.0662179
$$785$$ −15.4164 −0.550235
$$786$$ −0.583592 −0.0208160
$$787$$ −3.65248 −0.130197 −0.0650984 0.997879i $$-0.520736\pi$$
−0.0650984 + 0.997879i $$0.520736\pi$$
$$788$$ −26.6525 −0.949455
$$789$$ 26.1246 0.930061
$$790$$ 8.94427 0.318223
$$791$$ 0.472136 0.0167872
$$792$$ 0 0
$$793$$ 6.94427 0.246598
$$794$$ 14.2918 0.507197
$$795$$ −12.1803 −0.431992
$$796$$ 6.18034 0.219056
$$797$$ −2.52786 −0.0895415 −0.0447708 0.998997i $$-0.514256\pi$$
−0.0447708 + 0.998997i $$0.514256\pi$$
$$798$$ −4.14590 −0.146763
$$799$$ −1.23607 −0.0437289
$$800$$ −22.4721 −0.794510
$$801$$ −4.47214 −0.158015
$$802$$ −18.3607 −0.648338
$$803$$ 0 0
$$804$$ 15.7984 0.557166
$$805$$ −5.70820 −0.201188
$$806$$ 11.2361 0.395774
$$807$$ 1.05573 0.0371634
$$808$$ 40.6525 1.43015
$$809$$ 56.3050 1.97958 0.989788 0.142545i $$-0.0455285\pi$$
0.989788 + 0.142545i $$0.0455285\pi$$
$$810$$ 0.618034 0.0217155
$$811$$ −15.2918 −0.536968 −0.268484 0.963284i $$-0.586523\pi$$
−0.268484 + 0.963284i $$0.586523\pi$$
$$812$$ −8.09017 −0.283909
$$813$$ −5.29180 −0.185591
$$814$$ 0 0
$$815$$ −22.7082 −0.795434
$$816$$ −9.70820 −0.339855
$$817$$ −38.2918 −1.33966
$$818$$ −13.0132 −0.454994
$$819$$ 3.47214 0.121326
$$820$$ −4.00000 −0.139686
$$821$$ 7.47214 0.260779 0.130390 0.991463i $$-0.458377\pi$$
0.130390 + 0.991463i $$0.458377\pi$$
$$822$$ 12.1803 0.424838
$$823$$ −17.1803 −0.598869 −0.299435 0.954117i $$-0.596798\pi$$
−0.299435 + 0.954117i $$0.596798\pi$$
$$824$$ −38.9443 −1.35669
$$825$$ 0 0
$$826$$ 6.90983 0.240424
$$827$$ −12.3475 −0.429365 −0.214683 0.976684i $$-0.568872\pi$$
−0.214683 + 0.976684i $$0.568872\pi$$
$$828$$ −9.23607 −0.320976
$$829$$ 35.7771 1.24259 0.621295 0.783577i $$-0.286607\pi$$
0.621295 + 0.783577i $$0.286607\pi$$
$$830$$ −4.18034 −0.145102
$$831$$ 6.47214 0.224516
$$832$$ 0.819660 0.0284166
$$833$$ −5.23607 −0.181419
$$834$$ −8.94427 −0.309715
$$835$$ 22.6525 0.783921
$$836$$ 0 0
$$837$$ −5.23607 −0.180985
$$838$$ −0.729490 −0.0251998
$$839$$ −23.5410 −0.812726 −0.406363 0.913712i $$-0.633203\pi$$
−0.406363 + 0.913712i $$0.633203\pi$$
$$840$$ 2.23607 0.0771517
$$841$$ −4.00000 −0.137931
$$842$$ −8.03444 −0.276885
$$843$$ −11.4721 −0.395121
$$844$$ 8.76393 0.301667
$$845$$ −0.944272 −0.0324839
$$846$$ 0.145898 0.00501608
$$847$$ 0 0
$$848$$ −22.5836 −0.775524
$$849$$ 13.7639 0.472377
$$850$$ 12.9443 0.443985
$$851$$ −39.9574 −1.36972
$$852$$ 4.00000 0.137038
$$853$$ 29.4164 1.00720 0.503599 0.863937i $$-0.332009\pi$$
0.503599 + 0.863937i $$0.332009\pi$$
$$854$$ 1.23607 0.0422974
$$855$$ 6.70820 0.229416
$$856$$ −9.47214 −0.323751
$$857$$ −0.111456 −0.00380727 −0.00190364 0.999998i $$-0.500606\pi$$
−0.00190364 + 0.999998i $$0.500606\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 9.23607 0.314947
$$861$$ −2.47214 −0.0842502
$$862$$ −5.38197 −0.183310
$$863$$ −43.2361 −1.47177 −0.735886 0.677105i $$-0.763234\pi$$
−0.735886 + 0.677105i $$0.763234\pi$$
$$864$$ 5.61803 0.191129
$$865$$ 1.52786 0.0519489
$$866$$ −6.47214 −0.219932
$$867$$ 10.4164 0.353760
$$868$$ −8.47214 −0.287563
$$869$$ 0 0
$$870$$ −3.09017 −0.104767
$$871$$ 33.9017 1.14872
$$872$$ 6.18034 0.209293
$$873$$ 9.70820 0.328573
$$874$$ 23.6656 0.800502
$$875$$ 9.00000 0.304256
$$876$$ 7.32624 0.247531
$$877$$ −4.58359 −0.154777 −0.0773885 0.997001i $$-0.524658\pi$$
−0.0773885 + 0.997001i $$0.524658\pi$$
$$878$$ −6.90983 −0.233195
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ 54.8885 1.84924 0.924621 0.380888i $$-0.124382\pi$$
0.924621 + 0.380888i $$0.124382\pi$$
$$882$$ 0.618034 0.0208103
$$883$$ −14.8197 −0.498721 −0.249361 0.968411i $$-0.580220\pi$$
−0.249361 + 0.968411i $$0.580220\pi$$
$$884$$ −29.4164 −0.989381
$$885$$ −11.1803 −0.375823
$$886$$ 11.4164 0.383542
$$887$$ −10.7639 −0.361417 −0.180709 0.983537i $$-0.557839\pi$$
−0.180709 + 0.983537i $$0.557839\pi$$
$$888$$ 15.6525 0.525263
$$889$$ −12.6525 −0.424350
$$890$$ −2.76393 −0.0926472
$$891$$ 0 0
$$892$$ 9.70820 0.325055
$$893$$ 1.58359 0.0529929
$$894$$ −3.09017 −0.103351
$$895$$ 8.94427 0.298974
$$896$$ 11.3820 0.380245
$$897$$ −19.8197 −0.661759
$$898$$ 12.3607 0.412481
$$899$$ 26.1803 0.873163
$$900$$ 6.47214 0.215738
$$901$$ 63.7771 2.12472
$$902$$ 0 0
$$903$$ 5.70820 0.189957
$$904$$ 1.05573 0.0351130
$$905$$ −0.763932 −0.0253940
$$906$$ 8.76393 0.291162
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ −3.23607 −0.107393
$$909$$ −18.1803 −0.603004
$$910$$ 2.14590 0.0711358
$$911$$ −14.5836 −0.483176 −0.241588 0.970379i $$-0.577668\pi$$
−0.241588 + 0.970379i $$0.577668\pi$$
$$912$$ 12.4377 0.411853
$$913$$ 0 0
$$914$$ −9.41641 −0.311467
$$915$$ −2.00000 −0.0661180
$$916$$ −11.7082 −0.386850
$$917$$ 0.944272 0.0311826
$$918$$ −3.23607 −0.106806
$$919$$ −39.5967 −1.30618 −0.653088 0.757282i $$-0.726527\pi$$
−0.653088 + 0.757282i $$0.726527\pi$$
$$920$$ −12.7639 −0.420814
$$921$$ 20.9443 0.690137
$$922$$ 17.3050 0.569908
$$923$$ 8.58359 0.282532
$$924$$ 0 0
$$925$$ 28.0000 0.920634
$$926$$ −2.97871 −0.0978866
$$927$$ 17.4164 0.572030
$$928$$ −28.0902 −0.922105
$$929$$ 12.8885 0.422859 0.211430 0.977393i $$-0.432188\pi$$
0.211430 + 0.977393i $$0.432188\pi$$
$$930$$ −3.23607 −0.106115
$$931$$ 6.70820 0.219853
$$932$$ −24.1803 −0.792053
$$933$$ 9.88854 0.323736
$$934$$ 18.0344 0.590105
$$935$$ 0 0
$$936$$ 7.76393 0.253772
$$937$$ 39.8885 1.30310 0.651551 0.758605i $$-0.274119\pi$$
0.651551 + 0.758605i $$0.274119\pi$$
$$938$$ 6.03444 0.197032
$$939$$ 24.6525 0.804503
$$940$$ −0.381966 −0.0124584
$$941$$ 60.3607 1.96770 0.983851 0.178990i $$-0.0572829\pi$$
0.983851 + 0.178990i $$0.0572829\pi$$
$$942$$ −9.52786 −0.310435
$$943$$ 14.1115 0.459532
$$944$$ −20.7295 −0.674687
$$945$$ −1.00000 −0.0325300
$$946$$ 0 0
$$947$$ −5.41641 −0.176010 −0.0880048 0.996120i $$-0.528049\pi$$
−0.0880048 + 0.996120i $$0.528049\pi$$
$$948$$ −23.4164 −0.760530
$$949$$ 15.7214 0.510337
$$950$$ −16.5836 −0.538043
$$951$$ 24.1803 0.784101
$$952$$ −11.7082 −0.379465
$$953$$ −44.7771 −1.45047 −0.725236 0.688500i $$-0.758269\pi$$
−0.725236 + 0.688500i $$0.758269\pi$$
$$954$$ −7.52786 −0.243724
$$955$$ −10.7639 −0.348313
$$956$$ 16.3820 0.529831
$$957$$ 0 0
$$958$$ −7.23607 −0.233787
$$959$$ −19.7082 −0.636411
$$960$$ −0.236068 −0.00761906
$$961$$ −3.58359 −0.115600
$$962$$ 15.0213 0.484306
$$963$$ 4.23607 0.136505
$$964$$ 41.9787 1.35204
$$965$$ −14.6525 −0.471680
$$966$$ −3.52786 −0.113507
$$967$$ 61.1935 1.96785 0.983925 0.178582i $$-0.0571509\pi$$
0.983925 + 0.178582i $$0.0571509\pi$$
$$968$$ 0 0
$$969$$ −35.1246 −1.12837
$$970$$ 6.00000 0.192648
$$971$$ 32.1246 1.03093 0.515464 0.856911i $$-0.327620\pi$$
0.515464 + 0.856911i $$0.327620\pi$$
$$972$$ −1.61803 −0.0518985
$$973$$ 14.4721 0.463955
$$974$$ 10.4721 0.335549
$$975$$ 13.8885 0.444789
$$976$$ −3.70820 −0.118697
$$977$$ 4.18034 0.133741 0.0668705 0.997762i $$-0.478699\pi$$
0.0668705 + 0.997762i $$0.478699\pi$$
$$978$$ −14.0344 −0.448772
$$979$$ 0 0
$$980$$ −1.61803 −0.0516862
$$981$$ −2.76393 −0.0882456
$$982$$ 11.2016 0.357458
$$983$$ 27.4164 0.874448 0.437224 0.899353i $$-0.355962\pi$$
0.437224 + 0.899353i $$0.355962\pi$$
$$984$$ −5.52786 −0.176222
$$985$$ 16.4721 0.524846
$$986$$ 16.1803 0.515287
$$987$$ −0.236068 −0.00751413
$$988$$ 37.6869 1.19898
$$989$$ −32.5836 −1.03610
$$990$$ 0 0
$$991$$ −29.1803 −0.926944 −0.463472 0.886112i $$-0.653397\pi$$
−0.463472 + 0.886112i $$0.653397\pi$$
$$992$$ −29.4164 −0.933972
$$993$$ −11.4164 −0.362289
$$994$$ 1.52786 0.0484609
$$995$$ −3.81966 −0.121091
$$996$$ 10.9443 0.346783
$$997$$ −26.9443 −0.853334 −0.426667 0.904409i $$-0.640312\pi$$
−0.426667 + 0.904409i $$0.640312\pi$$
$$998$$ −1.38197 −0.0437454
$$999$$ −7.00000 −0.221470
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 2541.2.a.t.1.2 2
3.2 odd 2 7623.2.a.bm.1.1 2
11.10 odd 2 231.2.a.c.1.1 2
33.32 even 2 693.2.a.f.1.2 2
44.43 even 2 3696.2.a.be.1.2 2
55.54 odd 2 5775.2.a.be.1.2 2
77.76 even 2 1617.2.a.p.1.1 2
231.230 odd 2 4851.2.a.w.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.1 2 11.10 odd 2
693.2.a.f.1.2 2 33.32 even 2
1617.2.a.p.1.1 2 77.76 even 2
2541.2.a.t.1.2 2 1.1 even 1 trivial
3696.2.a.be.1.2 2 44.43 even 2
4851.2.a.w.1.2 2 231.230 odd 2
5775.2.a.be.1.2 2 55.54 odd 2
7623.2.a.bm.1.1 2 3.2 odd 2