Properties

 Label 4410.2.a.m.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 4410.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} +5.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} +3.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} -5.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +11.0000 q^{37} +1.00000 q^{38} -1.00000 q^{40} -3.00000 q^{41} -10.0000 q^{43} -3.00000 q^{44} +3.00000 q^{46} -3.00000 q^{47} -1.00000 q^{50} +5.00000 q^{52} -3.00000 q^{53} -3.00000 q^{55} -6.00000 q^{58} -4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +5.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -12.0000 q^{71} -4.00000 q^{73} -11.0000 q^{74} -1.00000 q^{76} -10.0000 q^{79} +1.00000 q^{80} +3.00000 q^{82} +12.0000 q^{83} -6.00000 q^{85} +10.0000 q^{86} +3.00000 q^{88} -6.00000 q^{89} -3.00000 q^{92} +3.00000 q^{94} -1.00000 q^{95} +14.0000 q^{97} +O(q^{100})$$

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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ −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
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 3.00000 0.639602
$$23$$ −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
$$26$$ −5.00000 −0.980581
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.0000 1.80839 0.904194 0.427121i $$-0.140472\pi$$
0.904194 + 0.427121i $$0.140472\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 5.00000 0.693375
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 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 $$0.582429\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 5.00000 0.620174
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 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 $$0.575209\pi$$
$$74$$ −11.0000 −1.27872
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 3.00000 0.331295
$$83$$ 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
$$86$$ 10.0000 1.07833
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −3.00000 −0.312772
$$93$$ 0 0
$$94$$ 3.00000 0.309426
$$95$$ −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
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −5.00000 −0.490290
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ 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 $$0.561356\pi$$
$$110$$ 3.00000 0.286039
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 4.00000 0.362143
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −19.0000 −1.68598 −0.842989 0.537931i $$-0.819206\pi$$
−0.842989 + 0.537931i $$0.819206\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −5.00000 −0.438529
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.0000 1.00702
$$143$$ −15.0000 −1.25436
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ 11.0000 0.904194
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 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 $$0.436061\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −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
$$170$$ 6.00000 0.460179
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 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
$$183$$ 0 0
$$184$$ 3.00000 0.221163
$$185$$ 11.0000 0.808736
$$186$$ 0 0
$$187$$ 18.0000 1.31629
$$188$$ −3.00000 −0.218797
$$189$$ 0 0
$$190$$ 1.00000 0.0725476
$$191$$ −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 $$0.545985\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −12.0000 −0.844317
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −3.00000 −0.209529
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 5.00000 0.346688
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ −1.00000 −0.0688428 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ −3.00000 −0.202260
$$221$$ −30.0000 −2.01802
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ 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 $$0.876058\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −3.00000 −0.195698
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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 $$0.797940\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 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
$$254$$ 19.0000 1.19217
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 5.00000 0.310087
$$261$$ 0 0
$$262$$ 3.00000 0.185341
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −3.00000 −0.184289
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3.00000 0.178965 0.0894825 0.995988i $$-0.471479\pi$$
0.0894825 + 0.995988i $$0.471479\pi$$
$$282$$ 0 0
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 15.0000 0.886969
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ 27.0000 1.57736 0.788678 0.614806i $$-0.210766\pi$$
0.788678 + 0.614806i $$0.210766\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −11.0000 −0.639362
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ −15.0000 −0.867472
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −14.0000 −0.805609
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ −4.00000 −0.229039
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ −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 $$0.427405\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ 5.00000 0.277350
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 3.00000 0.165647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 9.00000 0.492458
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 0 0
$$340$$ −6.00000 −0.325396
$$341$$ 12.0000 0.649836
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 10.0000 0.539164
$$345$$ 0 0
$$346$$ 3.00000 0.161281
$$347$$ −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
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −3.00000 −0.158555
$$359$$ −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
$$362$$ −2.00000 −0.105118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4.00000 −0.209370
$$366$$ 0 0
$$367$$ −1.00000 −0.0521996 −0.0260998 0.999659i $$-0.508309\pi$$
−0.0260998 + 0.999659i $$0.508309\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 0 0
$$370$$ −11.0000 −0.571863
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ 30.0000 1.54508
$$378$$ 0 0
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ −1.00000 −0.0512989
$$381$$ 0 0
$$382$$ 12.0000 0.613973
$$383$$ −15.0000 −0.766464 −0.383232 0.923652i $$-0.625189\pi$$
−0.383232 + 0.923652i $$0.625189\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −3.00000 −0.151138
$$395$$ −10.0000 −0.503155
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −21.0000 −1.04869 −0.524345 0.851506i $$-0.675690\pi$$
−0.524345 + 0.851506i $$0.675690\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 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 $$0.683059\pi$$
$$410$$ 3.00000 0.148159
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ −5.00000 −0.245145
$$417$$ 0 0
$$418$$ −3.00000 −0.146735
$$419$$ −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 $$0.810822\pi$$
$$422$$ 1.00000 0.0486792
$$423$$ 0 0
$$424$$ 3.00000 0.145693
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 10.0000 0.482243
$$431$$ 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
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 3.00000 0.143509
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 0 0
$$442$$ 30.0000 1.42695
$$443$$ 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
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 3.00000 0.141579 0.0707894 0.997491i $$-0.477448\pi$$
0.0707894 + 0.997491i $$0.477448\pi$$
$$450$$ 0 0
$$451$$ 9.00000 0.423793
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 28.0000 1.30835
$$459$$ 0 0
$$460$$ −3.00000 −0.139876
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ −19.0000 −0.883005 −0.441502 0.897260i $$-0.645554\pi$$
−0.441502 + 0.897260i $$0.645554\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 3.00000 0.138380
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 30.0000 1.37940
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ −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
$$482$$ 25.0000 1.13872
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 0 0
$$490$$ 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
$$494$$ 5.00000 0.224961
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −15.0000 −0.669483
$$503$$ −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
$$506$$ −9.00000 −0.400099
$$507$$ 0 0
$$508$$ −19.0000 −0.842989
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 9.00000 0.395820
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −5.00000 −0.219265
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 3.00000 0.130312
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −15.0000 −0.649722
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 6.00000 0.257248
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 0 0
$$550$$ 3.00000 0.127920
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 27.0000 1.14403 0.572013 0.820244i $$-0.306163\pi$$
0.572013 + 0.820244i $$0.306163\pi$$
$$558$$ 0 0
$$559$$ −50.0000 −2.11477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −3.00000 −0.126547
$$563$$ 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
$$566$$ −26.0000 −1.09286
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ 3.00000 0.125767 0.0628833 0.998021i $$-0.479970\pi$$
0.0628833 + 0.998021i $$0.479970\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ −15.0000 −0.627182
$$573$$ 0 0
$$574$$ 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 $$0.363325\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 9.00000 0.372742
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −27.0000 −1.11536
$$587$$ −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
$$591$$ 0 0
$$592$$ 11.0000 0.452097
$$593$$ 36.0000 1.47834 0.739171 0.673517i $$-0.235217\pi$$
0.739171 + 0.673517i $$0.235217\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ 15.0000 0.613396
$$599$$ −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
$$603$$ 0 0
$$604$$ 14.0000 0.569652
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ −19.0000 −0.771186 −0.385593 0.922669i $$-0.626003\pi$$
−0.385593 + 0.922669i $$0.626003\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ −15.0000 −0.606835
$$612$$ 0 0
$$613$$ 47.0000 1.89831 0.949156 0.314806i $$-0.101939\pi$$
0.949156 + 0.314806i $$0.101939\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −1.00000 −0.0401934 −0.0200967 0.999798i $$-0.506397\pi$$
−0.0200967 + 0.999798i $$0.506397\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 5.00000 0.199522
$$629$$ −66.0000 −2.63159
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ −19.0000 −0.753992
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 18.0000 0.712627
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 45.0000 1.77739 0.888697 0.458496i $$-0.151612\pi$$
0.888697 + 0.458496i $$0.151612\pi$$
$$642$$ 0 0
$$643$$ 38.0000 1.49857 0.749287 0.662246i $$-0.230396\pi$$
0.749287 + 0.662246i $$0.230396\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ 21.0000 0.825595 0.412798 0.910823i $$-0.364552\pi$$
0.412798 + 0.910823i $$0.364552\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −5.00000 −0.196116
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −21.0000 −0.821794 −0.410897 0.911682i $$-0.634784\pi$$
−0.410897 + 0.911682i $$0.634784\pi$$
$$654$$ 0 0
$$655$$ −3.00000 −0.117220
$$656$$ −3.00000 −0.117130
$$657$$ 0 0
$$658$$ 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 $$0.173126\pi$$
$$662$$ 7.00000 0.272063
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −18.0000 −0.696963
$$668$$ −9.00000 −0.348220
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ 3.00000 0.115299 0.0576497 0.998337i $$-0.481639\pi$$
0.0576497 + 0.998337i $$0.481639\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ 0 0
$$682$$ −12.0000 −0.459504
$$683$$ 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
$$687$$ 0 0
$$688$$ −10.0000 −0.381246
$$689$$ −15.0000 −0.571454
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ −3.00000 −0.114043
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ −11.0000 −0.414873
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ −15.0000 −0.560968
$$716$$ 3.00000 0.112115
$$717$$ 0 0
$$718$$ 6.00000 0.223918
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000 0.669891
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ 29.0000 1.07555 0.537775 0.843088i $$-0.319265\pi$$
0.537775 + 0.843088i $$0.319265\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 4.00000 0.148047
$$731$$ 60.0000 2.21918
$$732$$ 0 0
$$733$$ 47.0000 1.73598 0.867992 0.496578i $$-0.165410\pi$$
0.867992 + 0.496578i $$0.165410\pi$$
$$734$$ 1.00000 0.0369107
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ 12.0000 0.442026
$$738$$ 0 0
$$739$$ −37.0000 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 11.0000 0.404368
$$741$$ 0 0
$$742$$ 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
$$746$$ 34.0000 1.24483
$$747$$ 0 0
$$748$$ 18.0000 0.658145
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 0 0
$$754$$ −30.0000 −1.09254
$$755$$ 14.0000 0.509512
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 25.0000 0.908041
$$759$$ 0 0
$$760$$ 1.00000 0.0362738
$$761$$ 51.0000 1.84875 0.924374 0.381487i $$-0.124588\pi$$
0.924374 + 0.381487i $$0.124588\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 15.0000 0.541972
$$767$$ 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
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −39.0000 −1.40273 −0.701366 0.712801i $$-0.747426\pi$$
−0.701366 + 0.712801i $$0.747426\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ −24.0000 −0.860442
$$779$$ 3.00000 0.107486
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −18.0000 −0.643679
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5.00000 0.178458
$$786$$ 0 0
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ 3.00000 0.106871
$$789$$ 0 0
$$790$$ 10.0000 0.355784
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 21.0000 0.741536
$$803$$ 12.0000 0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 20.0000 0.704470
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ 47.0000 1.65039 0.825197 0.564846i $$-0.191064\pi$$
0.825197 + 0.564846i $$0.191064\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 33.0000 1.15665
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 10.0000 0.349856
$$818$$ 22.0000 0.769212
$$819$$ 0 0
$$820$$ −3.00000 −0.104765
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 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 $$0.421829\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 0 0
$$832$$ 5.00000 0.173344
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −9.00000 −0.311458
$$836$$ 3.00000 0.103757
$$837$$ 0 0
$$838$$ 15.0000 0.518166
$$839$$ 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
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ −1.00000 −0.0344214
$$845$$ 12.0000 0.412813
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −3.00000 −0.103020
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ −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
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ −10.0000 −0.340997
$$861$$ 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
$$866$$ 16.0000 0.543702
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 30.0000 1.01768
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 4.00000 0.135457
$$873$$ 0 0
$$874$$ −3.00000 −0.101477
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.00000 −0.236373 −0.118187 0.992991i $$-0.537708\pi$$
−0.118187 + 0.992991i $$0.537708\pi$$
$$878$$ 10.0000 0.337484
$$879$$ 0 0
$$880$$ −3.00000 −0.101130
$$881$$ −33.0000 −1.11180 −0.555899 0.831250i $$-0.687626\pi$$
−0.555899 + 0.831250i $$0.687626\pi$$
$$882$$ 0 0
$$883$$ 8.00000 0.269221 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$884$$ −30.0000 −1.00901
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ 3.00000 0.100391
$$894$$ 0 0
$$895$$ 3.00000 0.100279
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −3.00000 −0.100111
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 18.0000 0.599667
$$902$$ −9.00000 −0.299667
$$903$$ 0 0
$$904$$ 12.0000 0.399114
$$905$$ 2.00000 0.0664822
$$906$$ 0 0
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 0 0
$$910$$ 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
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ −28.0000 −0.925146
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 3.00000 0.0989071
$$921$$ 0 0
$$922$$ 6.00000 0.197599
$$923$$ −60.0000 −1.97492
$$924$$ 0 0
$$925$$ 11.0000 0.361678
$$926$$ 19.0000 0.624379
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −33.0000 −1.08269 −0.541347 0.840799i $$-0.682086\pi$$
−0.541347 + 0.840799i $$0.682086\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −18.0000 −0.588978
$$935$$ 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
$$939$$ 0 0
$$940$$ −3.00000 −0.0978492
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ 9.00000 0.293080
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −30.0000 −0.975384
$$947$$ −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
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 12.0000 0.388718 0.194359 0.980930i $$-0.437737\pi$$
0.194359 + 0.980930i $$0.437737\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ −6.00000 −0.194054
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −55.0000 −1.77327
$$963$$ 0 0
$$964$$ −25.0000 −0.805196
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ −14.0000 −0.449513
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ −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
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ −5.00000 −0.159071
$$989$$ 30.0000 0.953945
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ 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 $$0.428842\pi$$
$$998$$ 28.0000 0.886325
$$999$$ 0 0
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 4410.2.a.m.1.1 1
3.2 odd 2 490.2.a.g.1.1 1
7.2 even 3 630.2.k.e.361.1 2
7.4 even 3 630.2.k.e.541.1 2
7.6 odd 2 4410.2.a.c.1.1 1
12.11 even 2 3920.2.a.be.1.1 1
15.2 even 4 2450.2.c.f.99.2 2
15.8 even 4 2450.2.c.f.99.1 2
15.14 odd 2 2450.2.a.p.1.1 1
21.2 odd 6 70.2.e.b.11.1 2
21.5 even 6 490.2.e.a.361.1 2
21.11 odd 6 70.2.e.b.51.1 yes 2
21.17 even 6 490.2.e.a.471.1 2
21.20 even 2 490.2.a.j.1.1 1
84.11 even 6 560.2.q.d.401.1 2
84.23 even 6 560.2.q.d.81.1 2
84.83 odd 2 3920.2.a.g.1.1 1
105.2 even 12 350.2.j.a.249.2 4
105.23 even 12 350.2.j.a.249.1 4
105.32 even 12 350.2.j.a.149.1 4
105.44 odd 6 350.2.e.h.151.1 2
105.53 even 12 350.2.j.a.149.2 4
105.62 odd 4 2450.2.c.p.99.2 2
105.74 odd 6 350.2.e.h.51.1 2
105.83 odd 4 2450.2.c.p.99.1 2
105.104 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 21.2 odd 6
70.2.e.b.51.1 yes 2 21.11 odd 6
350.2.e.h.51.1 2 105.74 odd 6
350.2.e.h.151.1 2 105.44 odd 6
350.2.j.a.149.1 4 105.32 even 12
350.2.j.a.149.2 4 105.53 even 12
350.2.j.a.249.1 4 105.23 even 12
350.2.j.a.249.2 4 105.2 even 12
490.2.a.g.1.1 1 3.2 odd 2
490.2.a.j.1.1 1 21.20 even 2
490.2.e.a.361.1 2 21.5 even 6
490.2.e.a.471.1 2 21.17 even 6
560.2.q.d.81.1 2 84.23 even 6
560.2.q.d.401.1 2 84.11 even 6
630.2.k.e.361.1 2 7.2 even 3
630.2.k.e.541.1 2 7.4 even 3
2450.2.a.f.1.1 1 105.104 even 2
2450.2.a.p.1.1 1 15.14 odd 2
2450.2.c.f.99.1 2 15.8 even 4
2450.2.c.f.99.2 2 15.2 even 4
2450.2.c.p.99.1 2 105.83 odd 4
2450.2.c.p.99.2 2 105.62 odd 4
3920.2.a.g.1.1 1 84.83 odd 2
3920.2.a.be.1.1 1 12.11 even 2
4410.2.a.c.1.1 1 7.6 odd 2
4410.2.a.m.1.1 1 1.1 even 1 trivial