# Properties

 Label 3381.2.a.l.1.1 Level $3381$ Weight $2$ Character 3381.1 Self dual yes Analytic conductor $26.997$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3381 = 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9974209234$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3381.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} -8.00000 q^{10} -5.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +4.00000 q^{15} -4.00000 q^{16} +2.00000 q^{18} +5.00000 q^{19} -8.00000 q^{20} -10.0000 q^{22} -1.00000 q^{23} +11.0000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +8.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +5.00000 q^{33} +2.00000 q^{36} +6.00000 q^{37} +10.0000 q^{38} -2.00000 q^{39} -5.00000 q^{41} +8.00000 q^{43} -10.0000 q^{44} -4.00000 q^{45} -2.00000 q^{46} +9.00000 q^{47} +4.00000 q^{48} +22.0000 q^{50} +4.00000 q^{52} +9.00000 q^{53} -2.00000 q^{54} +20.0000 q^{55} -5.00000 q^{57} -4.00000 q^{58} -9.00000 q^{59} +8.00000 q^{60} +5.00000 q^{61} -12.0000 q^{62} -8.00000 q^{64} -8.00000 q^{65} +10.0000 q^{66} +4.00000 q^{67} +1.00000 q^{69} +12.0000 q^{71} +12.0000 q^{74} -11.0000 q^{75} +10.0000 q^{76} -4.00000 q^{78} -10.0000 q^{79} +16.0000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +18.0000 q^{83} +16.0000 q^{86} +2.00000 q^{87} -10.0000 q^{89} -8.00000 q^{90} -2.00000 q^{92} +6.00000 q^{93} +18.0000 q^{94} -20.0000 q^{95} +8.00000 q^{96} +18.0000 q^{97} -5.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.00000 1.00000
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ −2.00000 −0.816497
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ −8.00000 −2.52982
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ −8.00000 −1.78885
$$21$$ 0 0
$$22$$ −10.0000 −2.13201
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 4.00000 0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 8.00000 1.46059
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 5.00000 0.870388
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 10.0000 1.62221
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −10.0000 −1.50756
$$45$$ −4.00000 −0.596285
$$46$$ −2.00000 −0.294884
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 0 0
$$50$$ 22.0000 3.11127
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 20.0000 2.69680
$$56$$ 0 0
$$57$$ −5.00000 −0.662266
$$58$$ −4.00000 −0.525226
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 8.00000 1.03280
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ −12.0000 −1.52400
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ −8.00000 −0.992278
$$66$$ 10.0000 1.23091
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 12.0000 1.39497
$$75$$ −11.0000 −1.27017
$$76$$ 10.0000 1.14708
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 16.0000 1.78885
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ 18.0000 1.97576 0.987878 0.155230i $$-0.0496119\pi$$
0.987878 + 0.155230i $$0.0496119\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 16.0000 1.72532
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ −8.00000 −0.843274
$$91$$ 0 0
$$92$$ −2.00000 −0.208514
$$93$$ 6.00000 0.622171
$$94$$ 18.0000 1.85656
$$95$$ −20.0000 −2.05196
$$96$$ 8.00000 0.816497
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.502519
$$100$$ 22.0000 2.20000
$$101$$ −5.00000 −0.497519 −0.248759 0.968565i $$-0.580023\pi$$
−0.248759 + 0.968565i $$0.580023\pi$$
$$102$$ 0 0
$$103$$ 19.0000 1.87213 0.936063 0.351833i $$-0.114441\pi$$
0.936063 + 0.351833i $$0.114441\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 18.0000 1.74831
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 40.0000 3.81385
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ −10.0000 −0.936586
$$115$$ 4.00000 0.373002
$$116$$ −4.00000 −0.371391
$$117$$ 2.00000 0.184900
$$118$$ −18.0000 −1.65703
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 10.0000 0.905357
$$123$$ 5.00000 0.450835
$$124$$ −12.0000 −1.07763
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ 9.00000 0.798621 0.399310 0.916816i $$-0.369250\pi$$
0.399310 + 0.916816i $$0.369250\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ −16.0000 −1.40329
$$131$$ 5.00000 0.436852 0.218426 0.975854i $$-0.429908\pi$$
0.218426 + 0.975854i $$0.429908\pi$$
$$132$$ 10.0000 0.870388
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ 2.00000 0.170251
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 24.0000 2.01404
$$143$$ −10.0000 −0.836242
$$144$$ −4.00000 −0.333333
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 12.0000 0.986394
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ −22.0000 −1.79629
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 24.0000 1.92773
$$156$$ −4.00000 −0.320256
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ −20.0000 −1.59111
$$159$$ −9.00000 −0.713746
$$160$$ 32.0000 2.52982
$$161$$ 0 0
$$162$$ 2.00000 0.157135
$$163$$ 13.0000 1.01824 0.509119 0.860696i $$-0.329971\pi$$
0.509119 + 0.860696i $$0.329971\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ −20.0000 −1.55700
$$166$$ 36.0000 2.79414
$$167$$ −19.0000 −1.47026 −0.735132 0.677924i $$-0.762880\pi$$
−0.735132 + 0.677924i $$0.762880\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 5.00000 0.382360
$$172$$ 16.0000 1.21999
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ 20.0000 1.50756
$$177$$ 9.00000 0.676481
$$178$$ −20.0000 −1.49906
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ −8.00000 −0.596285
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ −5.00000 −0.369611
$$184$$ 0 0
$$185$$ −24.0000 −1.76452
$$186$$ 12.0000 0.879883
$$187$$ 0 0
$$188$$ 18.0000 1.31278
$$189$$ 0 0
$$190$$ −40.0000 −2.90191
$$191$$ −23.0000 −1.66422 −0.832111 0.554609i $$-0.812868\pi$$
−0.832111 + 0.554609i $$0.812868\pi$$
$$192$$ 8.00000 0.577350
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 36.0000 2.58465
$$195$$ 8.00000 0.572892
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −10.0000 −0.710669
$$199$$ −3.00000 −0.212664 −0.106332 0.994331i $$-0.533911\pi$$
−0.106332 + 0.994331i $$0.533911\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 38.0000 2.64759
$$207$$ −1.00000 −0.0695048
$$208$$ −8.00000 −0.554700
$$209$$ −25.0000 −1.72929
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 18.0000 1.23625
$$213$$ −12.0000 −0.822226
$$214$$ 8.00000 0.546869
$$215$$ −32.0000 −2.18238
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −24.0000 −1.62549
$$219$$ 0 0
$$220$$ 40.0000 2.69680
$$221$$ 0 0
$$222$$ −12.0000 −0.805387
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ 11.0000 0.733333
$$226$$ −4.00000 −0.266076
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ −10.0000 −0.662266
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −12.0000 −0.786146 −0.393073 0.919507i $$-0.628588\pi$$
−0.393073 + 0.919507i $$0.628588\pi$$
$$234$$ 4.00000 0.261488
$$235$$ −36.0000 −2.34838
$$236$$ −18.0000 −1.17170
$$237$$ 10.0000 0.649570
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ −16.0000 −1.03280
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ 28.0000 1.79991
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 10.0000 0.636285
$$248$$ 0 0
$$249$$ −18.0000 −1.14070
$$250$$ −48.0000 −3.03579
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 0 0
$$253$$ 5.00000 0.314347
$$254$$ 18.0000 1.12942
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −17.0000 −1.06043 −0.530215 0.847863i $$-0.677889\pi$$
−0.530215 + 0.847863i $$0.677889\pi$$
$$258$$ −16.0000 −0.996116
$$259$$ 0 0
$$260$$ −16.0000 −0.992278
$$261$$ −2.00000 −0.123797
$$262$$ 10.0000 0.617802
$$263$$ 7.00000 0.431638 0.215819 0.976433i $$-0.430758\pi$$
0.215819 + 0.976433i $$0.430758\pi$$
$$264$$ 0 0
$$265$$ −36.0000 −2.21146
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ 8.00000 0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 8.00000 0.486864
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ −55.0000 −3.31662
$$276$$ 2.00000 0.120386
$$277$$ 7.00000 0.420589 0.210295 0.977638i $$-0.432558\pi$$
0.210295 + 0.977638i $$0.432558\pi$$
$$278$$ −24.0000 −1.43942
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ −18.0000 −1.07188
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 24.0000 1.42414
$$285$$ 20.0000 1.18470
$$286$$ −20.0000 −1.18262
$$287$$ 0 0
$$288$$ −8.00000 −0.471405
$$289$$ −17.0000 −1.00000
$$290$$ 16.0000 0.939552
$$291$$ −18.0000 −1.05518
$$292$$ 0 0
$$293$$ 4.00000 0.233682 0.116841 0.993151i $$-0.462723\pi$$
0.116841 + 0.993151i $$0.462723\pi$$
$$294$$ 0 0
$$295$$ 36.0000 2.09600
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 6.00000 0.347571
$$299$$ −2.00000 −0.115663
$$300$$ −22.0000 −1.27017
$$301$$ 0 0
$$302$$ −38.0000 −2.18665
$$303$$ 5.00000 0.287242
$$304$$ −20.0000 −1.14708
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ −19.0000 −1.08087
$$310$$ 48.0000 2.72622
$$311$$ 15.0000 0.850572 0.425286 0.905059i $$-0.360174\pi$$
0.425286 + 0.905059i $$0.360174\pi$$
$$312$$ 0 0
$$313$$ 25.0000 1.41308 0.706542 0.707671i $$-0.250254\pi$$
0.706542 + 0.707671i $$0.250254\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ −20.0000 −1.12509
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ −18.0000 −1.00939
$$319$$ 10.0000 0.559893
$$320$$ 32.0000 1.78885
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 2.00000 0.111111
$$325$$ 22.0000 1.22034
$$326$$ 26.0000 1.44001
$$327$$ 12.0000 0.663602
$$328$$ 0 0
$$329$$ 0 0
$$330$$ −40.0000 −2.20193
$$331$$ 29.0000 1.59398 0.796992 0.603990i $$-0.206423\pi$$
0.796992 + 0.603990i $$0.206423\pi$$
$$332$$ 36.0000 1.97576
$$333$$ 6.00000 0.328798
$$334$$ −38.0000 −2.07927
$$335$$ −16.0000 −0.874173
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ −18.0000 −0.979071
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 30.0000 1.62459
$$342$$ 10.0000 0.540738
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ −4.00000 −0.215041
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 4.00000 0.214423
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 40.0000 2.13201
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 18.0000 0.956689
$$355$$ −48.0000 −2.54758
$$356$$ −20.0000 −1.06000
$$357$$ 0 0
$$358$$ 16.0000 0.845626
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 28.0000 1.47165
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −5.00000 −0.260290
$$370$$ −48.0000 −2.49540
$$371$$ 0 0
$$372$$ 12.0000 0.622171
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ 24.0000 1.23935
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ −40.0000 −2.05196
$$381$$ −9.00000 −0.461084
$$382$$ −46.0000 −2.35356
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −38.0000 −1.93415
$$387$$ 8.00000 0.406663
$$388$$ 36.0000 1.82762
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 16.0000 0.810191
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −5.00000 −0.252217
$$394$$ −4.00000 −0.201517
$$395$$ 40.0000 2.01262
$$396$$ −10.0000 −0.502519
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −6.00000 −0.300753
$$399$$ 0 0
$$400$$ −44.0000 −2.20000
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ −12.0000 −0.597763
$$404$$ −10.0000 −0.497519
$$405$$ −4.00000 −0.198762
$$406$$ 0 0
$$407$$ −30.0000 −1.48704
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 40.0000 1.97546
$$411$$ −9.00000 −0.443937
$$412$$ 38.0000 1.87213
$$413$$ 0 0
$$414$$ −2.00000 −0.0982946
$$415$$ −72.0000 −3.53434
$$416$$ −16.0000 −0.784465
$$417$$ 12.0000 0.587643
$$418$$ −50.0000 −2.44558
$$419$$ 18.0000 0.879358 0.439679 0.898155i $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 26.0000 1.26566
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ 0 0
$$428$$ 8.00000 0.386695
$$429$$ 10.0000 0.482805
$$430$$ −64.0000 −3.08635
$$431$$ 27.0000 1.30054 0.650272 0.759701i $$-0.274655\pi$$
0.650272 + 0.759701i $$0.274655\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ −24.0000 −1.14939
$$437$$ −5.00000 −0.239182
$$438$$ 0 0
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −12.0000 −0.569495
$$445$$ 40.0000 1.89618
$$446$$ −20.0000 −0.947027
$$447$$ −3.00000 −0.141895
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 22.0000 1.03709
$$451$$ 25.0000 1.17720
$$452$$ −4.00000 −0.188144
$$453$$ 19.0000 0.892698
$$454$$ 36.0000 1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 26.0000 1.21490
$$459$$ 0 0
$$460$$ 8.00000 0.373002
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ −35.0000 −1.62659 −0.813294 0.581853i $$-0.802328\pi$$
−0.813294 + 0.581853i $$0.802328\pi$$
$$464$$ 8.00000 0.371391
$$465$$ −24.0000 −1.11297
$$466$$ −24.0000 −1.11178
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 0 0
$$470$$ −72.0000 −3.32111
$$471$$ 7.00000 0.322543
$$472$$ 0 0
$$473$$ −40.0000 −1.83920
$$474$$ 20.0000 0.918630
$$475$$ 55.0000 2.52357
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ 24.0000 1.09773
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ −32.0000 −1.46059
$$481$$ 12.0000 0.547153
$$482$$ −34.0000 −1.54866
$$483$$ 0 0
$$484$$ 28.0000 1.27273
$$485$$ −72.0000 −3.26935
$$486$$ −2.00000 −0.0907218
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ −13.0000 −0.587880
$$490$$ 0 0
$$491$$ 14.0000 0.631811 0.315906 0.948791i $$-0.397692\pi$$
0.315906 + 0.948791i $$0.397692\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 0 0
$$494$$ 20.0000 0.899843
$$495$$ 20.0000 0.898933
$$496$$ 24.0000 1.07763
$$497$$ 0 0
$$498$$ −36.0000 −1.61320
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ −48.0000 −2.14663
$$501$$ 19.0000 0.848857
$$502$$ −20.0000 −0.892644
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 10.0000 0.444554
$$507$$ 9.00000 0.399704
$$508$$ 18.0000 0.798621
$$509$$ 19.0000 0.842160 0.421080 0.907023i $$-0.361651\pi$$
0.421080 + 0.907023i $$0.361651\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 32.0000 1.41421
$$513$$ −5.00000 −0.220755
$$514$$ −34.0000 −1.49968
$$515$$ −76.0000 −3.34896
$$516$$ −16.0000 −0.704361
$$517$$ −45.0000 −1.97910
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ −19.0000 −0.830812 −0.415406 0.909636i $$-0.636360\pi$$
−0.415406 + 0.909636i $$0.636360\pi$$
$$524$$ 10.0000 0.436852
$$525$$ 0 0
$$526$$ 14.0000 0.610429
$$527$$ 0 0
$$528$$ −20.0000 −0.870388
$$529$$ 1.00000 0.0434783
$$530$$ −72.0000 −3.12748
$$531$$ −9.00000 −0.390567
$$532$$ 0 0
$$533$$ −10.0000 −0.433148
$$534$$ 20.0000 0.865485
$$535$$ −16.0000 −0.691740
$$536$$ 0 0
$$537$$ −8.00000 −0.345225
$$538$$ 36.0000 1.55207
$$539$$ 0 0
$$540$$ 8.00000 0.344265
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −14.0000 −0.600798
$$544$$ 0 0
$$545$$ 48.0000 2.05609
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 5.00000 0.213395
$$550$$ −110.000 −4.69042
$$551$$ −10.0000 −0.426014
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 14.0000 0.594803
$$555$$ 24.0000 1.01874
$$556$$ −24.0000 −1.01783
$$557$$ 46.0000 1.94908 0.974541 0.224208i $$-0.0719796\pi$$
0.974541 + 0.224208i $$0.0719796\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 4.00000 0.168730
$$563$$ −40.0000 −1.68580 −0.842900 0.538071i $$-0.819153\pi$$
−0.842900 + 0.538071i $$0.819153\pi$$
$$564$$ −18.0000 −0.757937
$$565$$ 8.00000 0.336563
$$566$$ 40.0000 1.68133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −17.0000 −0.712677 −0.356339 0.934357i $$-0.615975\pi$$
−0.356339 + 0.934357i $$0.615975\pi$$
$$570$$ 40.0000 1.67542
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ −20.0000 −0.836242
$$573$$ 23.0000 0.960839
$$574$$ 0 0
$$575$$ −11.0000 −0.458732
$$576$$ −8.00000 −0.333333
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −34.0000 −1.41421
$$579$$ 19.0000 0.789613
$$580$$ 16.0000 0.664364
$$581$$ 0 0
$$582$$ −36.0000 −1.49225
$$583$$ −45.0000 −1.86371
$$584$$ 0 0
$$585$$ −8.00000 −0.330759
$$586$$ 8.00000 0.330477
$$587$$ 17.0000 0.701665 0.350833 0.936438i $$-0.385899\pi$$
0.350833 + 0.936438i $$0.385899\pi$$
$$588$$ 0 0
$$589$$ −30.0000 −1.23613
$$590$$ 72.0000 2.96419
$$591$$ 2.00000 0.0822690
$$592$$ −24.0000 −0.986394
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 10.0000 0.410305
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 3.00000 0.122782
$$598$$ −4.00000 −0.163572
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ −38.0000 −1.54620
$$605$$ −56.0000 −2.27672
$$606$$ 10.0000 0.406222
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ 0 0
$$610$$ −40.0000 −1.61955
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 16.0000 0.645707
$$615$$ −20.0000 −0.806478
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ −38.0000 −1.52858
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 48.0000 1.92773
$$621$$ 1.00000 0.0401286
$$622$$ 30.0000 1.20289
$$623$$ 0 0
$$624$$ 8.00000 0.320256
$$625$$ 41.0000 1.64000
$$626$$ 50.0000 1.99840
$$627$$ 25.0000 0.998404
$$628$$ −14.0000 −0.558661
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −10.0000 −0.398094 −0.199047 0.979990i $$-0.563785\pi$$
−0.199047 + 0.979990i $$0.563785\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 60.0000 2.38290
$$635$$ −36.0000 −1.42862
$$636$$ −18.0000 −0.713746
$$637$$ 0 0
$$638$$ 20.0000 0.791808
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ −8.00000 −0.315735
$$643$$ 31.0000 1.22252 0.611260 0.791430i $$-0.290663\pi$$
0.611260 + 0.791430i $$0.290663\pi$$
$$644$$ 0 0
$$645$$ 32.0000 1.26000
$$646$$ 0 0
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ 45.0000 1.76640
$$650$$ 44.0000 1.72582
$$651$$ 0 0
$$652$$ 26.0000 1.01824
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ 24.0000 0.938474
$$655$$ −20.0000 −0.781465
$$656$$ 20.0000 0.780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ −40.0000 −1.55700
$$661$$ 5.00000 0.194477 0.0972387 0.995261i $$-0.468999\pi$$
0.0972387 + 0.995261i $$0.468999\pi$$
$$662$$ 58.0000 2.25423
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 12.0000 0.464991
$$667$$ 2.00000 0.0774403
$$668$$ −38.0000 −1.47026
$$669$$ 10.0000 0.386622
$$670$$ −32.0000 −1.23627
$$671$$ −25.0000 −0.965114
$$672$$ 0 0
$$673$$ 49.0000 1.88881 0.944406 0.328783i $$-0.106638\pi$$
0.944406 + 0.328783i $$0.106638\pi$$
$$674$$ 44.0000 1.69482
$$675$$ −11.0000 −0.423390
$$676$$ −18.0000 −0.692308
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ 4.00000 0.153619
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 60.0000 2.29752
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 10.0000 0.382360
$$685$$ −36.0000 −1.37549
$$686$$ 0 0
$$687$$ −13.0000 −0.495981
$$688$$ −32.0000 −1.21999
$$689$$ 18.0000 0.685745
$$690$$ −8.00000 −0.304555
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ −4.00000 −0.152057
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 48.0000 1.82074
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 4.00000 0.151402
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ −15.0000 −0.566542 −0.283271 0.959040i $$-0.591420\pi$$
−0.283271 + 0.959040i $$0.591420\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 30.0000 1.13147
$$704$$ 40.0000 1.50756
$$705$$ 36.0000 1.35584
$$706$$ 28.0000 1.05379
$$707$$ 0 0
$$708$$ 18.0000 0.676481
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ −96.0000 −3.60282
$$711$$ −10.0000 −0.375029
$$712$$ 0 0
$$713$$ 6.00000 0.224702
$$714$$ 0 0
$$715$$ 40.0000 1.49592
$$716$$ 16.0000 0.597948
$$717$$ −12.0000 −0.448148
$$718$$ 64.0000 2.38846
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 16.0000 0.596285
$$721$$ 0 0
$$722$$ 12.0000 0.446594
$$723$$ 17.0000 0.632237
$$724$$ 28.0000 1.04061
$$725$$ −22.0000 −0.817059
$$726$$ −28.0000 −1.03918
$$727$$ −41.0000 −1.52061 −0.760303 0.649569i $$-0.774949\pi$$
−0.760303 + 0.649569i $$0.774949\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −10.0000 −0.369611
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ −54.0000 −1.99318
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ −20.0000 −0.736709
$$738$$ −10.0000 −0.368105
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −48.0000 −1.76452
$$741$$ −10.0000 −0.367359
$$742$$ 0 0
$$743$$ 15.0000 0.550297 0.275148 0.961402i $$-0.411273\pi$$
0.275148 + 0.961402i $$0.411273\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ −68.0000 −2.48966
$$747$$ 18.0000 0.658586
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 48.0000 1.75271
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ −36.0000 −1.31278
$$753$$ 10.0000 0.364420
$$754$$ −8.00000 −0.291343
$$755$$ 76.0000 2.76592
$$756$$ 0 0
$$757$$ 8.00000 0.290765 0.145382 0.989376i $$-0.453559\pi$$
0.145382 + 0.989376i $$0.453559\pi$$
$$758$$ −60.0000 −2.17930
$$759$$ −5.00000 −0.181489
$$760$$ 0 0
$$761$$ 21.0000 0.761249 0.380625 0.924730i $$-0.375709\pi$$
0.380625 + 0.924730i $$0.375709\pi$$
$$762$$ −18.0000 −0.652071
$$763$$ 0 0
$$764$$ −46.0000 −1.66422
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ −18.0000 −0.649942
$$768$$ −16.0000 −0.577350
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 17.0000 0.612240
$$772$$ −38.0000 −1.36765
$$773$$ −20.0000 −0.719350 −0.359675 0.933078i $$-0.617112\pi$$
−0.359675 + 0.933078i $$0.617112\pi$$
$$774$$ 16.0000 0.575108
$$775$$ −66.0000 −2.37079
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 12.0000 0.430221
$$779$$ −25.0000 −0.895718
$$780$$ 16.0000 0.572892
$$781$$ −60.0000 −2.14697
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 28.0000 0.999363
$$786$$ −10.0000 −0.356688
$$787$$ 23.0000 0.819861 0.409931 0.912117i $$-0.365553\pi$$
0.409931 + 0.912117i $$0.365553\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ −7.00000 −0.249207
$$790$$ 80.0000 2.84627
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ −60.0000 −2.12932
$$795$$ 36.0000 1.27679
$$796$$ −6.00000 −0.212664
$$797$$ −46.0000 −1.62940 −0.814702 0.579880i $$-0.803099\pi$$
−0.814702 + 0.579880i $$0.803099\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −88.0000 −3.11127
$$801$$ −10.0000 −0.353333
$$802$$ 6.00000 0.211867
$$803$$ 0 0
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ −24.0000 −0.845364
$$807$$ −18.0000 −0.633630
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ −8.00000 −0.281091
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ −60.0000 −2.10300
$$815$$ −52.0000 −1.82148
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ −28.0000 −0.978997
$$819$$ 0 0
$$820$$ 40.0000 1.39686
$$821$$ −4.00000 −0.139601 −0.0698005 0.997561i $$-0.522236\pi$$
−0.0698005 + 0.997561i $$0.522236\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 27.0000 0.941161 0.470580 0.882357i $$-0.344045\pi$$
0.470580 + 0.882357i $$0.344045\pi$$
$$824$$ 0 0
$$825$$ 55.0000 1.91485
$$826$$ 0 0
$$827$$ −21.0000 −0.730242 −0.365121 0.930960i $$-0.618972\pi$$
−0.365121 + 0.930960i $$0.618972\pi$$
$$828$$ −2.00000 −0.0695048
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ −144.000 −4.99831
$$831$$ −7.00000 −0.242827
$$832$$ −16.0000 −0.554700
$$833$$ 0 0
$$834$$ 24.0000 0.831052
$$835$$ 76.0000 2.63009
$$836$$ −50.0000 −1.72929
$$837$$ 6.00000 0.207390
$$838$$ 36.0000 1.24360
$$839$$ −46.0000 −1.58810 −0.794048 0.607855i $$-0.792030\pi$$
−0.794048 + 0.607855i $$0.792030\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 16.0000 0.551396
$$843$$ −2.00000 −0.0688837
$$844$$ 26.0000 0.894957
$$845$$ 36.0000 1.23844
$$846$$ 18.0000 0.618853
$$847$$ 0 0
$$848$$ −36.0000 −1.23625
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ −24.0000 −0.822226
$$853$$ −8.00000 −0.273915 −0.136957 0.990577i $$-0.543732\pi$$
−0.136957 + 0.990577i $$0.543732\pi$$
$$854$$ 0 0
$$855$$ −20.0000 −0.683986
$$856$$ 0 0
$$857$$ −1.00000 −0.0341593 −0.0170797 0.999854i $$-0.505437\pi$$
−0.0170797 + 0.999854i $$0.505437\pi$$
$$858$$ 20.0000 0.682789
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ −64.0000 −2.18238
$$861$$ 0 0
$$862$$ 54.0000 1.83925
$$863$$ −30.0000 −1.02121 −0.510606 0.859815i $$-0.670579\pi$$
−0.510606 + 0.859815i $$0.670579\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 8.00000 0.272008
$$866$$ 22.0000 0.747590
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ 50.0000 1.69613
$$870$$ −16.0000 −0.542451
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ 18.0000 0.609208
$$874$$ −10.0000 −0.338255
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17.0000 −0.574049 −0.287025 0.957923i $$-0.592666\pi$$
−0.287025 + 0.957923i $$0.592666\pi$$
$$878$$ 72.0000 2.42988
$$879$$ −4.00000 −0.134917
$$880$$ −80.0000 −2.69680
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ 0 0
$$885$$ −36.0000 −1.21013
$$886$$ −72.0000 −2.41889
$$887$$ 16.0000 0.537227 0.268614 0.963248i $$-0.413434\pi$$
0.268614 + 0.963248i $$0.413434\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 80.0000 2.68161
$$891$$ −5.00000 −0.167506
$$892$$ −20.0000 −0.669650
$$893$$ 45.0000 1.50587
$$894$$ −6.00000 −0.200670
$$895$$ −32.0000 −1.06964
$$896$$ 0 0
$$897$$ 2.00000 0.0667781
$$898$$ 12.0000 0.400445
$$899$$ 12.0000 0.400222
$$900$$ 22.0000 0.733333
$$901$$ 0 0
$$902$$ 50.0000 1.66482
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −56.0000 −1.86150
$$906$$ 38.0000 1.26247
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 36.0000 1.19470
$$909$$ −5.00000 −0.165840
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 20.0000 0.662266
$$913$$ −90.0000 −2.97857
$$914$$ 20.0000 0.661541
$$915$$ 20.0000 0.661180
$$916$$ 26.0000 0.859064
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 2.00000 0.0659739 0.0329870 0.999456i $$-0.489498\pi$$
0.0329870 + 0.999456i $$0.489498\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ −4.00000 −0.131733
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 66.0000 2.17007
$$926$$ −70.0000 −2.30034
$$927$$ 19.0000 0.624042
$$928$$ 16.0000 0.525226
$$929$$ −26.0000 −0.853032 −0.426516 0.904480i $$-0.640259\pi$$
−0.426516 + 0.904480i $$0.640259\pi$$
$$930$$ −48.0000 −1.57398
$$931$$ 0 0
$$932$$ −24.0000 −0.786146
$$933$$ −15.0000 −0.491078
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 27.0000 0.882052 0.441026 0.897494i $$-0.354615\pi$$
0.441026 + 0.897494i $$0.354615\pi$$
$$938$$ 0 0
$$939$$ −25.0000 −0.815844
$$940$$ −72.0000 −2.34838
$$941$$ −20.0000 −0.651981 −0.325991 0.945373i $$-0.605698\pi$$
−0.325991 + 0.945373i $$0.605698\pi$$
$$942$$ 14.0000 0.456145
$$943$$ 5.00000 0.162822
$$944$$ 36.0000 1.17170
$$945$$ 0 0
$$946$$ −80.0000 −2.60102
$$947$$ 32.0000 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$948$$ 20.0000 0.649570
$$949$$ 0 0
$$950$$ 110.000 3.56887
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ −55.0000 −1.78162 −0.890812 0.454371i $$-0.849864\pi$$
−0.890812 + 0.454371i $$0.849864\pi$$
$$954$$ 18.0000 0.582772
$$955$$ 92.0000 2.97705
$$956$$ 24.0000 0.776215
$$957$$ −10.0000 −0.323254
$$958$$ −64.0000 −2.06775
$$959$$ 0 0
$$960$$ −32.0000 −1.03280
$$961$$ 5.00000 0.161290
$$962$$ 24.0000 0.773791
$$963$$ 4.00000 0.128898
$$964$$ −34.0000 −1.09507
$$965$$ 76.0000 2.44653
$$966$$ 0 0
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −144.000 −4.62356
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ 0 0
$$974$$ −40.0000 −1.28168
$$975$$ −22.0000 −0.704564
$$976$$ −20.0000 −0.640184
$$977$$ −3.00000 −0.0959785 −0.0479893 0.998848i $$-0.515281\pi$$
−0.0479893 + 0.998848i $$0.515281\pi$$
$$978$$ −26.0000 −0.831388
$$979$$ 50.0000 1.59801
$$980$$ 0 0
$$981$$ −12.0000 −0.383131
$$982$$ 28.0000 0.893516
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ 0 0
$$985$$ 8.00000 0.254901
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 20.0000 0.636285
$$989$$ −8.00000 −0.254385
$$990$$ 40.0000 1.27128
$$991$$ 5.00000 0.158830 0.0794151 0.996842i $$-0.474695\pi$$
0.0794151 + 0.996842i $$0.474695\pi$$
$$992$$ 48.0000 1.52400
$$993$$ −29.0000 −0.920287
$$994$$ 0 0
$$995$$ 12.0000 0.380426
$$996$$ −36.0000 −1.14070
$$997$$ 56.0000 1.77354 0.886769 0.462213i $$-0.152944\pi$$
0.886769 + 0.462213i $$0.152944\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ −6.00000 −0.189832
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 3381.2.a.l.1.1 1
7.6 odd 2 483.2.a.b.1.1 1
21.20 even 2 1449.2.a.a.1.1 1
28.27 even 2 7728.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.b.1.1 1 7.6 odd 2
1449.2.a.a.1.1 1 21.20 even 2
3381.2.a.l.1.1 1 1.1 even 1 trivial
7728.2.a.l.1.1 1 28.27 even 2