# Properties

 Label 3920.2.a.be.1.1 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -3.00000 q^{11} +5.00000 q^{13} -2.00000 q^{15} +6.00000 q^{17} +1.00000 q^{19} -3.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} -6.00000 q^{33} +11.0000 q^{37} +10.0000 q^{39} +3.00000 q^{41} +10.0000 q^{43} -1.00000 q^{45} -3.00000 q^{47} +12.0000 q^{51} +3.00000 q^{53} +3.00000 q^{55} +2.00000 q^{57} -4.00000 q^{61} -5.00000 q^{65} +4.00000 q^{67} -6.00000 q^{69} -12.0000 q^{71} -4.00000 q^{73} +2.00000 q^{75} +10.0000 q^{79} -11.0000 q^{81} +12.0000 q^{83} -6.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} +8.00000 q^{93} -1.00000 q^{95} +14.0000 q^{97} -3.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.0000 1.80839 0.904194 0.427121i $$-0.140472\pi$$
0.904194 + 0.427121i $$0.140472\pi$$
$$38$$ 0 0
$$39$$ 10.0000 1.60128
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 12.0000 1.68034
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −5.00000 −0.620174
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ 2.00000 0.230940
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −6.00000 −0.650791
$$86$$ 0 0
$$87$$ −12.0000 −1.28654
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 22.0000 2.08815
$$112$$ 0 0
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ 3.00000 0.279751
$$116$$ 0 0
$$117$$ 5.00000 0.462250
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 19.0000 1.68598 0.842989 0.537931i $$-0.180794\pi$$
0.842989 + 0.537931i $$0.180794\pi$$
$$128$$ 0 0
$$129$$ 20.0000 1.76090
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −15.0000 −1.25436
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 6.00000 0.467099
$$166$$ 0 0
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 3.00000 0.228086 0.114043 0.993476i $$-0.463620\pi$$
0.114043 + 0.993476i $$0.463620\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ −8.00000 −0.591377
$$184$$ 0 0
$$185$$ −11.0000 −0.808736
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ −10.0000 −0.716115
$$196$$ 0 0
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −3.00000 −0.209529
$$206$$ 0 0
$$207$$ −3.00000 −0.208514
$$208$$ 0 0
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 1.00000 0.0688428 0.0344214 0.999407i $$-0.489041\pi$$
0.0344214 + 0.999407i $$0.489041\pi$$
$$212$$ 0 0
$$213$$ −24.0000 −1.64445
$$214$$ 0 0
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ −28.0000 −1.85029 −0.925146 0.379611i $$-0.876058\pi$$
−0.925146 + 0.379611i $$0.876058\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 3.00000 0.195698
$$236$$ 0 0
$$237$$ 20.0000 1.29914
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 0 0
$$253$$ 9.00000 0.565825
$$254$$ 0 0
$$255$$ −12.0000 −0.751469
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −3.00000 −0.184289
$$266$$ 0 0
$$267$$ 12.0000 0.734388
$$268$$ 0 0
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −3.00000 −0.178965 −0.0894825 0.995988i $$-0.528521\pi$$
−0.0894825 + 0.995988i $$0.528521\pi$$
$$282$$ 0 0
$$283$$ −26.0000 −1.54554 −0.772770 0.634686i $$-0.781129\pi$$
−0.772770 + 0.634686i $$0.781129\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 28.0000 1.64139
$$292$$ 0 0
$$293$$ −27.0000 −1.57736 −0.788678 0.614806i $$-0.789234\pi$$
−0.788678 + 0.614806i $$0.789234\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 12.0000 0.696311
$$298$$ 0 0
$$299$$ −15.0000 −0.867472
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −24.0000 −1.37876
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 24.0000 1.33955
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ 5.00000 0.277350
$$326$$ 0 0
$$327$$ −8.00000 −0.442401
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 0 0
$$333$$ 11.0000 0.602796
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 24.0000 1.30350
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 6.00000 0.323029
$$346$$ 0 0
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −20.0000 −1.06752
$$352$$ 0 0
$$353$$ 12.0000 0.638696 0.319348 0.947638i $$-0.396536\pi$$
0.319348 + 0.947638i $$0.396536\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −4.00000 −0.209946
$$364$$ 0 0
$$365$$ 4.00000 0.209370
$$366$$ 0 0
$$367$$ 1.00000 0.0521996 0.0260998 0.999659i $$-0.491691\pi$$
0.0260998 + 0.999659i $$0.491691\pi$$
$$368$$ 0 0
$$369$$ 3.00000 0.156174
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ −2.00000 −0.103280
$$376$$ 0 0
$$377$$ −30.0000 −1.54508
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 38.0000 1.94680
$$382$$ 0 0
$$383$$ −15.0000 −0.766464 −0.383232 0.923652i $$-0.625189\pi$$
−0.383232 + 0.923652i $$0.625189\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.0000 0.508329
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ −10.0000 −0.503155
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 21.0000 1.04869 0.524345 0.851506i $$-0.324310\pi$$
0.524345 + 0.851506i $$0.324310\pi$$
$$402$$ 0 0
$$403$$ 20.0000 0.996271
$$404$$ 0 0
$$405$$ 11.0000 0.546594
$$406$$ 0 0
$$407$$ −33.0000 −1.63575
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ 24.0000 1.18383
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −15.0000 −0.732798 −0.366399 0.930458i $$-0.619409\pi$$
−0.366399 + 0.930458i $$0.619409\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ −3.00000 −0.145865
$$424$$ 0 0
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0 0
$$435$$ 12.0000 0.575356
$$436$$ 0 0
$$437$$ −3.00000 −0.143509
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ −6.00000 −0.284427
$$446$$ 0 0
$$447$$ 36.0000 1.70274
$$448$$ 0 0
$$449$$ −3.00000 −0.141579 −0.0707894 0.997491i $$-0.522552\pi$$
−0.0707894 + 0.997491i $$0.522552\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 0 0
$$453$$ −28.0000 −1.31555
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ −24.0000 −1.12022
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 19.0000 0.883005 0.441502 0.897260i $$-0.354446\pi$$
0.441502 + 0.897260i $$0.354446\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ −30.0000 −1.37940
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 3.00000 0.137361
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 55.0000 2.50778
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −14.0000 −0.635707
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 0 0
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −36.0000 −1.62136
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ −18.0000 −0.804181
$$502$$ 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
$$507$$ 24.0000 1.06588
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 9.00000 0.395820
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 33.0000 1.44576 0.722878 0.690976i $$-0.242819\pi$$
0.722878 + 0.690976i $$0.242819\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 15.0000 0.649722
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ 6.00000 0.258919
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 0 0
$$543$$ 4.00000 0.171656
$$544$$ 0 0
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −22.0000 −0.933848
$$556$$ 0 0
$$557$$ −27.0000 −1.14403 −0.572013 0.820244i $$-0.693837\pi$$
−0.572013 + 0.820244i $$0.693837\pi$$
$$558$$ 0 0
$$559$$ 50.0000 2.11477
$$560$$ 0 0
$$561$$ −36.0000 −1.51992
$$562$$ 0 0
$$563$$ 18.0000 0.758610 0.379305 0.925272i $$-0.376163\pi$$
0.379305 + 0.925272i $$0.376163\pi$$
$$564$$ 0 0
$$565$$ −12.0000 −0.504844
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −3.00000 −0.125767 −0.0628833 0.998021i $$-0.520030\pi$$
−0.0628833 + 0.998021i $$0.520030\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ 0 0
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ −5.00000 −0.206725
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −42.0000 −1.71607 −0.858037 0.513588i $$-0.828316\pi$$
−0.858037 + 0.513588i $$0.828316\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 19.0000 0.771186 0.385593 0.922669i $$-0.373997\pi$$
0.385593 + 0.922669i $$0.373997\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −15.0000 −0.606835
$$612$$ 0 0
$$613$$ 47.0000 1.89831 0.949156 0.314806i $$-0.101939\pi$$
0.949156 + 0.314806i $$0.101939\pi$$
$$614$$ 0 0
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 1.00000 0.0401934 0.0200967 0.999798i $$-0.493603\pi$$
0.0200967 + 0.999798i $$0.493603\pi$$
$$620$$ 0 0
$$621$$ 12.0000 0.481543
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −6.00000 −0.239617
$$628$$ 0 0
$$629$$ 66.0000 2.63159
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 2.00000 0.0794929
$$634$$ 0 0
$$635$$ −19.0000 −0.753992
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −45.0000 −1.77739 −0.888697 0.458496i $$-0.848388\pi$$
−0.888697 + 0.458496i $$0.848388\pi$$
$$642$$ 0 0
$$643$$ −38.0000 −1.49857 −0.749287 0.662246i $$-0.769604\pi$$
−0.749287 + 0.662246i $$0.769604\pi$$
$$644$$ 0 0
$$645$$ −20.0000 −0.787499
$$646$$ 0 0
$$647$$ 21.0000 0.825595 0.412798 0.910823i $$-0.364552\pi$$
0.412798 + 0.910823i $$0.364552\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 21.0000 0.821794 0.410897 0.911682i $$-0.365216\pi$$
0.410897 + 0.911682i $$0.365216\pi$$
$$654$$ 0 0
$$655$$ 3.00000 0.117220
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 44.0000 1.71140 0.855701 0.517471i $$-0.173126\pi$$
0.855701 + 0.517471i $$0.173126\pi$$
$$662$$ 0 0
$$663$$ 60.0000 2.33021
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 18.0000 0.696963
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ −3.00000 −0.115299 −0.0576497 0.998337i $$-0.518361\pi$$
−0.0576497 + 0.998337i $$0.518361\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 48.0000 1.83936
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ −56.0000 −2.13653
$$688$$ 0 0
$$689$$ 15.0000 0.571454
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 0 0
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 11.0000 0.414873
$$704$$ 0 0
$$705$$ 6.00000 0.225973
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 0 0
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ 15.0000 0.560968
$$716$$ 0 0
$$717$$ −12.0000 −0.448148
$$718$$ 0 0
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −50.0000 −1.85952
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −29.0000 −1.07555 −0.537775 0.843088i $$-0.680735\pi$$
−0.537775 + 0.843088i $$0.680735\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 60.0000 2.21918
$$732$$ 0 0
$$733$$ 47.0000 1.73598 0.867992 0.496578i $$-0.165410\pi$$
0.867992 + 0.496578i $$0.165410\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 10.0000 0.367359
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 0 0
$$753$$ 30.0000 1.09326
$$754$$ 0 0
$$755$$ 14.0000 0.509512
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 0 0
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ −51.0000 −1.84875 −0.924374 0.381487i $$-0.875412\pi$$
−0.924374 + 0.381487i $$0.875412\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −49.0000 −1.76699 −0.883493 0.468445i $$-0.844814\pi$$
−0.883493 + 0.468445i $$0.844814\pi$$
$$770$$ 0 0
$$771$$ −24.0000 −0.864339
$$772$$ 0 0
$$773$$ 39.0000 1.40273 0.701366 0.712801i $$-0.252574\pi$$
0.701366 + 0.712801i $$0.252574\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 3.00000 0.107486
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ 0 0
$$785$$ −5.00000 −0.178458
$$786$$ 0 0
$$787$$ 34.0000 1.21197 0.605985 0.795476i $$-0.292779\pi$$
0.605985 + 0.795476i $$0.292779\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 12.0000 0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −24.0000 −0.844840
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ −47.0000 −1.65039 −0.825197 0.564846i $$-0.808936\pi$$
−0.825197 + 0.564846i $$0.808936\pi$$
$$812$$ 0 0
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 10.0000 0.349856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ 0 0
$$825$$ −6.00000 −0.208893
$$826$$ 0 0
$$827$$ 54.0000 1.87776 0.938882 0.344239i $$-0.111863\pi$$
0.938882 + 0.344239i $$0.111863\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 4.00000 0.138758
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 9.00000 0.311458
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ 0 0
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −6.00000 −0.206651
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −52.0000 −1.78464
$$850$$ 0 0
$$851$$ −33.0000 −1.13123
$$852$$ 0 0
$$853$$ −1.00000 −0.0342393 −0.0171197 0.999853i $$-0.505450\pi$$
−0.0171197 + 0.999853i $$0.505450\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ 0 0
$$865$$ −3.00000 −0.102003
$$866$$ 0 0
$$867$$ 38.0000 1.29055
$$868$$ 0 0
$$869$$ −30.0000 −1.01768
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.00000 −0.236373 −0.118187 0.992991i $$-0.537708\pi$$
−0.118187 + 0.992991i $$0.537708\pi$$
$$878$$ 0 0
$$879$$ −54.0000 −1.82137
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ 0 0
$$883$$ −8.00000 −0.269221 −0.134611 0.990899i $$-0.542978\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 33.0000 1.10554
$$892$$ 0 0
$$893$$ −3.00000 −0.100391
$$894$$ 0 0
$$895$$ −3.00000 −0.100279
$$896$$ 0 0
$$897$$ −30.0000 −1.00167
$$898$$ 0 0
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 18.0000 0.599667
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −2.00000 −0.0664822
$$906$$ 0 0
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ 0 0
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ 8.00000 0.264472
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −38.0000 −1.25350 −0.626752 0.779219i $$-0.715616\pi$$
−0.626752 + 0.779219i $$0.715616\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ −60.0000 −1.97492
$$924$$ 0 0
$$925$$ 11.0000 0.361678
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 33.0000 1.08269 0.541347 0.840799i $$-0.317914\pi$$
0.541347 + 0.840799i $$0.317914\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 18.0000 0.588663
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ 16.0000 0.522140
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ −9.00000 −0.293080
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ 36.0000 1.16738
$$952$$ 0 0
$$953$$ −12.0000 −0.388718 −0.194359 0.980930i $$-0.562263\pi$$
−0.194359 + 0.980930i $$0.562263\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 36.0000 1.16371
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 10.0000 0.320256
$$976$$ 0 0
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ −57.0000 −1.81802 −0.909009 0.416777i $$-0.863160\pi$$
−0.909009 + 0.416777i $$0.863160\pi$$
$$984$$ 0 0
$$985$$ 3.00000 0.0955879
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −30.0000 −0.953945
$$990$$ 0 0
$$991$$ −20.0000 −0.635321 −0.317660 0.948205i $$-0.602897\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ 0 0
$$993$$ 14.0000 0.444277
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 0 0
$$999$$ −44.0000 −1.39210
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 3920.2.a.be.1.1 1
4.3 odd 2 490.2.a.g.1.1 1
7.2 even 3 560.2.q.d.81.1 2
7.4 even 3 560.2.q.d.401.1 2
7.6 odd 2 3920.2.a.g.1.1 1
12.11 even 2 4410.2.a.m.1.1 1
20.3 even 4 2450.2.c.f.99.1 2
20.7 even 4 2450.2.c.f.99.2 2
20.19 odd 2 2450.2.a.p.1.1 1
28.3 even 6 490.2.e.a.471.1 2
28.11 odd 6 70.2.e.b.51.1 yes 2
28.19 even 6 490.2.e.a.361.1 2
28.23 odd 6 70.2.e.b.11.1 2
28.27 even 2 490.2.a.j.1.1 1
84.11 even 6 630.2.k.e.541.1 2
84.23 even 6 630.2.k.e.361.1 2
84.83 odd 2 4410.2.a.c.1.1 1
140.23 even 12 350.2.j.a.249.1 4
140.27 odd 4 2450.2.c.p.99.2 2
140.39 odd 6 350.2.e.h.51.1 2
140.67 even 12 350.2.j.a.149.1 4
140.79 odd 6 350.2.e.h.151.1 2
140.83 odd 4 2450.2.c.p.99.1 2
140.107 even 12 350.2.j.a.249.2 4
140.123 even 12 350.2.j.a.149.2 4
140.139 even 2 2450.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.b.11.1 2 28.23 odd 6
70.2.e.b.51.1 yes 2 28.11 odd 6
350.2.e.h.51.1 2 140.39 odd 6
350.2.e.h.151.1 2 140.79 odd 6
350.2.j.a.149.1 4 140.67 even 12
350.2.j.a.149.2 4 140.123 even 12
350.2.j.a.249.1 4 140.23 even 12
350.2.j.a.249.2 4 140.107 even 12
490.2.a.g.1.1 1 4.3 odd 2
490.2.a.j.1.1 1 28.27 even 2
490.2.e.a.361.1 2 28.19 even 6
490.2.e.a.471.1 2 28.3 even 6
560.2.q.d.81.1 2 7.2 even 3
560.2.q.d.401.1 2 7.4 even 3
630.2.k.e.361.1 2 84.23 even 6
630.2.k.e.541.1 2 84.11 even 6
2450.2.a.f.1.1 1 140.139 even 2
2450.2.a.p.1.1 1 20.19 odd 2
2450.2.c.f.99.1 2 20.3 even 4
2450.2.c.f.99.2 2 20.7 even 4
2450.2.c.p.99.1 2 140.83 odd 4
2450.2.c.p.99.2 2 140.27 odd 4
3920.2.a.g.1.1 1 7.6 odd 2
3920.2.a.be.1.1 1 1.1 even 1 trivial
4410.2.a.c.1.1 1 84.83 odd 2
4410.2.a.m.1.1 1 12.11 even 2