Properties

 Label 4998.2.a.bp.1.1 Level $4998$ Weight $2$ Character 4998.1 Self dual yes Analytic conductor $39.909$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4998.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$39.9092309302$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 714) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4998.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -8.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +12.0000 q^{43} +2.00000 q^{45} -8.00000 q^{46} +1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -6.00000 q^{58} +8.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +12.0000 q^{67} -1.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} -1.00000 q^{75} +6.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -16.0000 q^{83} -2.00000 q^{85} +12.0000 q^{86} -6.00000 q^{87} -2.00000 q^{89} +2.00000 q^{90} -8.00000 q^{92} +8.00000 q^{93} +1.00000 q^{96} -2.00000 q^{97} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 2.00000 0.316228
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ −8.00000 −1.17954
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ −1.00000 −0.140028
$$52$$ 6.00000 0.832050
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 12.0000 1.48842
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 10.0000 1.16248
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 6.00000 0.679366
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 12.0000 1.29399
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 0 0
$$92$$ −8.00000 −0.834058
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −16.0000 −1.49201
$$116$$ −6.00000 −0.557086
$$117$$ 6.00000 0.554700
$$118$$ 8.00000 0.736460
$$119$$ 0 0
$$120$$ 2.00000 0.182574
$$121$$ −11.0000 −1.00000
$$122$$ −6.00000 −0.543214
$$123$$ 6.00000 0.541002
$$124$$ 8.00000 0.718421
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 12.0000 1.05247
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 2.00000 0.172133
$$136$$ −1.00000 −0.0857493
$$137$$ −22.0000 −1.87959 −0.939793 0.341743i $$-0.888983\pi$$
−0.939793 + 0.341743i $$0.888983\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −12.0000 −0.996546
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 6.00000 0.480384
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −10.0000 −0.793052
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −16.0000 −1.24184
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −2.00000 −0.153393
$$171$$ 0 0
$$172$$ 12.0000 0.914991
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 8.00000 0.601317
$$178$$ −2.00000 −0.149906
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ −6.00000 −0.443533
$$184$$ −8.00000 −0.589768
$$185$$ 20.0000 1.47043
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 12.0000 0.859338
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 12.0000 0.846415
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 12.0000 0.838116
$$206$$ −16.0000 −1.11477
$$207$$ −8.00000 −0.556038
$$208$$ 6.00000 0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 0 0
$$214$$ 16.0000 1.09374
$$215$$ 24.0000 1.63679
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 10.0000 0.671156
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ −6.00000 −0.399114
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −16.0000 −1.05501
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −30.0000 −1.96537 −0.982683 0.185296i $$-0.940675\pi$$
−0.982683 + 0.185296i $$0.940675\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ −16.0000 −1.01396
$$250$$ −12.0000 −0.758947
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ −2.00000 −0.125245
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 0 0
$$260$$ 12.0000 0.744208
$$261$$ −6.00000 −0.371391
$$262$$ 4.00000 0.247121
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −20.0000 −1.22859
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ 12.0000 0.733017
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 2.00000 0.121716
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −22.0000 −1.32907
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ −12.0000 −0.704664
$$291$$ −2.00000 −0.117242
$$292$$ 6.00000 0.351123
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ −48.0000 −2.77591
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ −16.0000 −0.920697
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ −1.00000 −0.0571662
$$307$$ 32.0000 1.82634 0.913168 0.407583i $$-0.133628\pi$$
0.913168 + 0.407583i $$0.133628\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 16.0000 0.908739
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 6.00000 0.339683
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ 0 0
$$320$$ 2.00000 0.111803
$$321$$ 16.0000 0.893033
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −6.00000 −0.332820
$$326$$ 8.00000 0.443079
$$327$$ −6.00000 −0.331801
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −36.0000 −1.97874 −0.989369 0.145424i $$-0.953545\pi$$
−0.989369 + 0.145424i $$0.953545\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 10.0000 0.547997
$$334$$ 8.00000 0.437741
$$335$$ 24.0000 1.31126
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −6.00000 −0.325875
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ −16.0000 −0.861411
$$346$$ 2.00000 0.107521
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −19.0000 −1.00000
$$362$$ 10.0000 0.525588
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ 12.0000 0.628109
$$366$$ −6.00000 −0.313625
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ 6.00000 0.312348
$$370$$ 20.0000 1.03975
$$371$$ 0 0
$$372$$ 8.00000 0.414781
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ −36.0000 −1.85409
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 16.0000 0.818631
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 12.0000 0.609994
$$388$$ −2.00000 −0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 12.0000 0.607644
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ 18.0000 0.906827
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 24.0000 1.20301
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 48.0000 2.39105
$$404$$ 6.00000 0.298511
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −1.00000 −0.0495074
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 12.0000 0.592638
$$411$$ −22.0000 −1.08518
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ −32.0000 −1.57082
$$416$$ 6.00000 0.294174
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 0 0
$$424$$ −10.0000 −0.485643
$$425$$ 1.00000 0.0485071
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 16.0000 0.773389
$$429$$ 0 0
$$430$$ 24.0000 1.15738
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 22.0000 1.05725 0.528626 0.848855i $$-0.322707\pi$$
0.528626 + 0.848855i $$0.322707\pi$$
$$434$$ 0 0
$$435$$ −12.0000 −0.575356
$$436$$ −6.00000 −0.287348
$$437$$ 0 0
$$438$$ 6.00000 0.286691
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6.00000 −0.285391
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 10.0000 0.474579
$$445$$ −4.00000 −0.189618
$$446$$ 8.00000 0.378811
$$447$$ −18.0000 −0.851371
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ −16.0000 −0.751746
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −1.00000 −0.0466760
$$460$$ −16.0000 −0.746004
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 16.0000 0.741982
$$466$$ −30.0000 −1.38972
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 8.00000 0.368230
$$473$$ 0 0
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 24.0000 1.09773
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 60.0000 2.73576
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ −4.00000 −0.181631
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 6.00000 0.270226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ −16.0000 −0.716977
$$499$$ 8.00000 0.358129 0.179065 0.983837i $$-0.442693\pi$$
0.179065 + 0.983837i $$0.442693\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 8.00000 0.357414
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 8.00000 0.354943
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ −32.0000 −1.41009
$$516$$ 12.0000 0.528271
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 12.0000 0.526235
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ −20.0000 −0.868744
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ 36.0000 1.55933
$$534$$ −2.00000 −0.0865485
$$535$$ 32.0000 1.38348
$$536$$ 12.0000 0.518321
$$537$$ 4.00000 0.172613
$$538$$ 18.0000 0.776035
$$539$$ 0 0
$$540$$ 2.00000 0.0860663
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 10.0000 0.429141
$$544$$ −1.00000 −0.0428746
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −22.0000 −0.939793
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −8.00000 −0.340503
$$553$$ 0 0
$$554$$ −6.00000 −0.254916
$$555$$ 20.0000 0.848953
$$556$$ 4.00000 0.169638
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 72.0000 3.04528
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −22.0000 −0.928014
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ −12.0000 −0.504844
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 16.0000 0.668410
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 1.00000 0.0416667
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −6.00000 −0.249351
$$580$$ −12.0000 −0.498273
$$581$$ 0 0
$$582$$ −2.00000 −0.0829027
$$583$$ 0 0
$$584$$ 6.00000 0.248282
$$585$$ 12.0000 0.496139
$$586$$ 14.0000 0.578335
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 16.0000 0.658710
$$591$$ 18.0000 0.740421
$$592$$ 10.0000 0.410997
$$593$$ −26.0000 −1.06769 −0.533846 0.845582i $$-0.679254\pi$$
−0.533846 + 0.845582i $$0.679254\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 24.0000 0.982255
$$598$$ −48.0000 −1.96287
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ −16.0000 −0.651031
$$605$$ −22.0000 −0.894427
$$606$$ 6.00000 0.243733
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −12.0000 −0.485866
$$611$$ 0 0
$$612$$ −1.00000 −0.0404226
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ −14.0000 −0.563619 −0.281809 0.959470i $$-0.590935\pi$$
−0.281809 + 0.959470i $$0.590935\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 16.0000 0.642575
$$621$$ −8.00000 −0.321029
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ −19.0000 −0.760000
$$626$$ −26.0000 −1.03917
$$627$$ 0 0
$$628$$ −18.0000 −0.718278
$$629$$ −10.0000 −0.398726
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 8.00000 0.317971
$$634$$ −6.00000 −0.238290
$$635$$ 16.0000 0.634941
$$636$$ −10.0000 −0.396526
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 16.0000 0.631470
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ 0 0
$$645$$ 24.0000 0.944999
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 8.00000 0.312586
$$656$$ 6.00000 0.234261
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −36.0000 −1.39918
$$663$$ −6.00000 −0.233021
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 48.0000 1.85857
$$668$$ 8.00000 0.309529
$$669$$ 8.00000 0.309298
$$670$$ 24.0000 0.927201
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ −1.00000 −0.0384900
$$676$$ 23.0000 0.884615
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 0 0
$$680$$ −2.00000 −0.0766965
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ −44.0000 −1.68115
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ 12.0000 0.457496
$$689$$ −60.0000 −2.28582
$$690$$ −16.0000 −0.609110
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ −24.0000 −0.911028
$$695$$ 8.00000 0.303457
$$696$$ −6.00000 −0.227429
$$697$$ −6.00000 −0.227266
$$698$$ 22.0000 0.832712
$$699$$ −30.0000 −1.13470
$$700$$ 0 0
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ 6.00000 0.226455
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 0 0
$$708$$ 8.00000 0.300658
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −2.00000 −0.0749532
$$713$$ −64.0000 −2.39682
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 24.0000 0.896296
$$718$$ 16.0000 0.597115
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 0 0
$$722$$ −19.0000 −0.707107
$$723$$ −2.00000 −0.0743808
$$724$$ 10.0000 0.371647
$$725$$ 6.00000 0.222834
$$726$$ −11.0000 −0.408248
$$727$$ −40.0000 −1.48352 −0.741759 0.670667i $$-0.766008\pi$$
−0.741759 + 0.670667i $$0.766008\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ −12.0000 −0.443836
$$732$$ −6.00000 −0.221766
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 20.0000 0.735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 8.00000 0.293294
$$745$$ −36.0000 −1.31894
$$746$$ −2.00000 −0.0732252
$$747$$ −16.0000 −0.585409
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −12.0000 −0.438178
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ −32.0000 −1.16460
$$756$$ 0 0
$$757$$ 30.0000 1.09037 0.545184 0.838316i $$-0.316460\pi$$
0.545184 + 0.838316i $$0.316460\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 0 0
$$764$$ 16.0000 0.578860
$$765$$ −2.00000 −0.0723102
$$766$$ −8.00000 −0.289052
$$767$$ 48.0000 1.73318
$$768$$ 1.00000 0.0360844
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ −6.00000 −0.215945
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −8.00000 −0.287368
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ 0 0
$$780$$ 12.0000 0.429669
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ −6.00000 −0.214423
$$784$$ 0 0
$$785$$ −36.0000 −1.28490
$$786$$ 4.00000 0.142675
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 18.0000 0.641223
$$789$$ −16.0000 −0.569615
$$790$$ −16.0000 −0.569254
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −36.0000 −1.27840
$$794$$ 18.0000 0.638796
$$795$$ −20.0000 −0.709327
$$796$$ 24.0000 0.850657
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ −2.00000 −0.0706665
$$802$$ 2.00000 0.0706225
$$803$$ 0 0
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ 18.0000 0.633630
$$808$$ 6.00000 0.211079
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ −24.0000 −0.841717
$$814$$ 0 0
$$815$$ 16.0000 0.560456
$$816$$ −1.00000 −0.0350070
$$817$$ 0 0
$$818$$ −34.0000 −1.18878
$$819$$ 0 0
$$820$$ 12.0000 0.419058
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ −22.0000 −0.767338
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ −32.0000 −1.11074
$$831$$ −6.00000 −0.208138
$$832$$ 6.00000 0.208013
$$833$$ 0 0
$$834$$ 4.00000 0.138509
$$835$$ 16.0000 0.553703
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 12.0000 0.414533
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −34.0000 −1.17172
$$843$$ −22.0000 −0.757720
$$844$$ 8.00000 0.275371
$$845$$ 46.0000 1.58245
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 4.00000 0.137280
$$850$$ 1.00000 0.0342997
$$851$$ −80.0000 −2.74236
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 16.0000 0.546869
$$857$$ 54.0000 1.84460 0.922302 0.386469i $$-0.126305\pi$$
0.922302 + 0.386469i $$0.126305\pi$$
$$858$$ 0 0
$$859$$ 56.0000 1.91070 0.955348 0.295484i $$-0.0954809\pi$$
0.955348 + 0.295484i $$0.0954809\pi$$
$$860$$ 24.0000 0.818393
$$861$$ 0 0
$$862$$ −16.0000 −0.544962
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 4.00000 0.136004
$$866$$ 22.0000 0.747590
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −12.0000 −0.406838
$$871$$ 72.0000 2.43963
$$872$$ −6.00000 −0.203186
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 16.0000 0.537834
$$886$$ 4.00000 0.134383
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 10.0000 0.335578
$$889$$ 0 0
$$890$$ −4.00000 −0.134080
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ 0 0
$$894$$ −18.0000 −0.602010
$$895$$ 8.00000 0.267411
$$896$$ 0 0
$$897$$ −48.0000 −1.60267
$$898$$ 34.0000 1.13459
$$899$$ −48.0000 −1.60089
$$900$$ −1.00000 −0.0333333
$$901$$ 10.0000 0.333148
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 20.0000 0.664822
$$906$$ −16.0000 −0.531564
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 4.00000 0.132745
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −38.0000 −1.25693
$$915$$ −12.0000 −0.396708
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ −1.00000 −0.0330049
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −16.0000 −0.527504
$$921$$ 32.0000 1.05444
$$922$$ 6.00000 0.197599
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −16.0000 −0.525793
$$927$$ −16.0000 −0.525509
$$928$$ −6.00000 −0.196960
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 16.0000 0.524661
$$931$$ 0 0
$$932$$ −30.0000 −0.982683
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ −18.0000 −0.586472
$$943$$ −48.0000 −1.56310
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −32.0000 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 32.0000 1.03550
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ −40.0000 −1.29234
$$959$$ 0 0
$$960$$ 2.00000 0.0645497
$$961$$ 33.0000 1.06452
$$962$$ 60.0000 1.93448
$$963$$ 16.0000 0.515593
$$964$$ −2.00000 −0.0644157
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 0 0
$$970$$ −4.00000 −0.128432
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ −6.00000 −0.192154
$$976$$ −6.00000 −0.192055
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 8.00000 0.255812
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ −20.0000 −0.638226
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 36.0000 1.14706
$$986$$ 6.00000 0.191079
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −96.0000 −3.05262
$$990$$ 0 0
$$991$$ −56.0000 −1.77890 −0.889449 0.457034i $$-0.848912\pi$$
−0.889449 + 0.457034i $$0.848912\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −36.0000 −1.14243
$$994$$ 0 0
$$995$$ 48.0000 1.52170
$$996$$ −16.0000 −0.506979
$$997$$ −30.0000 −0.950110 −0.475055 0.879956i $$-0.657572\pi$$
−0.475055 + 0.879956i $$0.657572\pi$$
$$998$$ 8.00000 0.253236
$$999$$ 10.0000 0.316386
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 4998.2.a.bp.1.1 1
7.6 odd 2 714.2.a.e.1.1 1
21.20 even 2 2142.2.a.g.1.1 1
28.27 even 2 5712.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.a.e.1.1 1 7.6 odd 2
2142.2.a.g.1.1 1 21.20 even 2
4998.2.a.bp.1.1 1 1.1 even 1 trivial
5712.2.a.q.1.1 1 28.27 even 2