# Properties

 Label 4030.2.a.a.1.1 Level 4030 Weight 2 Character 4030.1 Self dual yes Analytic conductor 32.180 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4030 = 2 \cdot 5 \cdot 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4030.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1797120146$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0$$ of $$x$$ Character $$\chi$$ $$=$$ 4030.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} +3.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -3.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} -3.00000 q^{45} +2.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} +10.0000 q^{53} -2.00000 q^{55} +12.0000 q^{59} -4.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +4.00000 q^{67} +4.00000 q^{68} +3.00000 q^{72} -2.00000 q^{74} +4.00000 q^{76} -4.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{85} +8.00000 q^{86} +2.00000 q^{88} +6.00000 q^{89} +3.00000 q^{90} -2.00000 q^{92} +12.0000 q^{94} +4.00000 q^{95} +2.00000 q^{97} +7.00000 q^{98} +6.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −3.00000 −1.00000
$$10$$ −1.00000 −0.316228
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 3.00000 0.707107
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ −3.00000 −0.447214
$$46$$ 2.00000 0.294884
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ −1.00000 −0.127000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 9.00000 1.00000
$$82$$ −6.00000 −0.662589
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 2.00000 0.213201
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 3.00000 0.316228
$$91$$ 0 0
$$92$$ −2.00000 −0.208514
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 7.00000 0.707107
$$99$$ 6.00000 0.603023
$$100$$ 1.00000 0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ −2.00000 −0.186501
$$116$$ 0 0
$$117$$ −3.00000 −0.277350
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 4.00000 0.362143
$$123$$ 0 0
$$124$$ 1.00000 0.0898027
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 18.0000 1.59724 0.798621 0.601834i $$-0.205563\pi$$
0.798621 + 0.601834i $$0.205563\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −1.00000 −0.0877058
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 22.0000 1.86602 0.933008 0.359856i $$-0.117174\pi$$
0.933008 + 0.359856i $$0.117174\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −12.0000 −0.970143
$$154$$ 0 0
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ −9.00000 −0.707107
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −4.00000 −0.306786
$$171$$ −12.0000 −0.917663
$$172$$ −8.00000 −0.609994
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ −3.00000 −0.223607
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2.00000 0.147442
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ −6.00000 −0.426401
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ 4.00000 0.278693
$$207$$ 6.00000 0.417029
$$208$$ 1.00000 0.0693375
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 6.00000 0.406371
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ −14.0000 −0.931266
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ −12.0000 −0.792982 −0.396491 0.918039i $$-0.629772\pi$$
−0.396491 + 0.918039i $$0.629772\pi$$
$$230$$ 2.00000 0.131876
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ 3.00000 0.196116
$$235$$ −12.0000 −0.782794
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ −7.00000 −0.447214
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ −1.00000 −0.0635001
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ −18.0000 −1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 1.00000 0.0620174
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −22.0000 −1.31947
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 3.00000 0.176777
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 22.0000 1.28525 0.642627 0.766179i $$-0.277845\pi$$
0.642627 + 0.766179i $$0.277845\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 12.0000 0.690522
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ −4.00000 −0.229039
$$306$$ 12.0000 0.685994
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1.00000 −0.0567962
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 28.0000 1.58265 0.791327 0.611393i $$-0.209391\pi$$
0.791327 + 0.611393i $$0.209391\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 34.0000 1.90963 0.954815 0.297200i $$-0.0960529\pi$$
0.954815 + 0.297200i $$0.0960529\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 16.0000 0.890264
$$324$$ 9.00000 0.500000
$$325$$ 1.00000 0.0554700
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −30.0000 −1.64895 −0.824475 0.565899i $$-0.808529\pi$$
−0.824475 + 0.565899i $$0.808529\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −6.00000 −0.328798
$$334$$ −6.00000 −0.328305
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 24.0000 1.30736 0.653682 0.756770i $$-0.273224\pi$$
0.653682 + 0.756770i $$0.273224\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ −2.00000 −0.108306
$$342$$ 12.0000 0.648886
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000 0.106600
$$353$$ 36.0000 1.91609 0.958043 0.286623i $$-0.0925328\pi$$
0.958043 + 0.286623i $$0.0925328\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 10.0000 0.528516
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 3.00000 0.158114
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −14.0000 −0.730794 −0.365397 0.930852i $$-0.619067\pi$$
−0.365397 + 0.930852i $$0.619067\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ −18.0000 −0.937043
$$370$$ −2.00000 −0.103975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 8.00000 0.409316
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ 24.0000 1.21999
$$388$$ 2.00000 0.101535
$$389$$ −20.0000 −1.01404 −0.507020 0.861934i $$-0.669253\pi$$
−0.507020 + 0.861934i $$0.669253\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 7.00000 0.353553
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ −4.00000 −0.201262
$$396$$ 6.00000 0.301511
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ 1.00000 0.0498135
$$404$$ 10.0000 0.497519
$$405$$ 9.00000 0.447214
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ −6.00000 −0.294884
$$415$$ 4.00000 0.196352
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 8.00000 0.391293
$$419$$ 32.0000 1.56330 0.781651 0.623716i $$-0.214378\pi$$
0.781651 + 0.623716i $$0.214378\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 36.0000 1.75038
$$424$$ −10.0000 −0.485643
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 0 0
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ −8.00000 −0.382692
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 21.0000 1.00000
$$442$$ −4.00000 −0.190261
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ −2.00000 −0.0947027
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 3.00000 0.141421
$$451$$ −12.0000 −0.565058
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4.00000 −0.187112 −0.0935561 0.995614i $$-0.529823\pi$$
−0.0935561 + 0.995614i $$0.529823\pi$$
$$458$$ 12.0000 0.560723
$$459$$ 0 0
$$460$$ −2.00000 −0.0932505
$$461$$ −36.0000 −1.67669 −0.838344 0.545142i $$-0.816476\pi$$
−0.838344 + 0.545142i $$0.816476\pi$$
$$462$$ 0 0
$$463$$ −30.0000 −1.39422 −0.697109 0.716965i $$-0.745531\pi$$
−0.697109 + 0.716965i $$0.745531\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 2.00000 0.0926482
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ −3.00000 −0.138675
$$469$$ 0 0
$$470$$ 12.0000 0.553519
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −30.0000 −1.37361
$$478$$ 16.0000 0.731823
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 0 0
$$490$$ 7.00000 0.316228
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −4.00000 −0.179969
$$495$$ 6.00000 0.269680
$$496$$ 1.00000 0.0449013
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.00000 −0.0895323 −0.0447661 0.998997i $$-0.514254\pi$$
−0.0447661 + 0.998997i $$0.514254\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 6.00000 0.267793
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ −4.00000 −0.177822
$$507$$ 0 0
$$508$$ 18.0000 0.798621
$$509$$ 40.0000 1.77297 0.886484 0.462758i $$-0.153140\pi$$
0.886484 + 0.462758i $$0.153140\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −1.00000 −0.0438529
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ 32.0000 1.39926 0.699631 0.714504i $$-0.253348\pi$$
0.699631 + 0.714504i $$0.253348\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 18.0000 0.784837
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ −10.0000 −0.434372
$$531$$ −36.0000 −1.56227
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ −24.0000 −1.03471
$$539$$ 14.0000 0.603023
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 12.0000 0.512148
$$550$$ 2.00000 0.0852803
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 22.0000 0.933008
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 3.00000 0.127000
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 26.0000 1.09674
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 14.0000 0.588984
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −38.0000 −1.59025 −0.795125 0.606445i $$-0.792595\pi$$
−0.795125 + 0.606445i $$0.792595\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.00000 −0.0834058
$$576$$ −3.00000 −0.125000
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −20.0000 −0.828315
$$584$$ 0 0
$$585$$ −3.00000 −0.124035
$$586$$ −22.0000 −0.908812
$$587$$ 20.0000 0.825488 0.412744 0.910847i $$-0.364570\pi$$
0.412744 + 0.910847i $$0.364570\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ −12.0000 −0.494032
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ 2.00000 0.0817861
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ −12.0000 −0.488273
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ −12.0000 −0.485468
$$612$$ −12.0000 −0.485071
$$613$$ −22.0000 −0.888572 −0.444286 0.895885i $$-0.646543\pi$$
−0.444286 + 0.895885i $$0.646543\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ 1.00000 0.0401610
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −28.0000 −1.11911
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ −48.0000 −1.91085 −0.955425 0.295234i $$-0.904602\pi$$
−0.955425 + 0.295234i $$0.904602\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 0 0
$$634$$ −34.0000 −1.35031
$$635$$ 18.0000 0.714308
$$636$$ 0 0
$$637$$ −7.00000 −0.277350
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 0 0
$$643$$ −8.00000 −0.315489 −0.157745 0.987480i $$-0.550422\pi$$
−0.157745 + 0.987480i $$0.550422\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −16.0000 −0.629512
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −24.0000 −0.942082
$$650$$ −1.00000 −0.0392232
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 30.0000 1.16598
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ 6.00000 0.232147
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ −16.0000 −0.616755 −0.308377 0.951264i $$-0.599786\pi$$
−0.308377 + 0.951264i $$0.599786\pi$$
$$674$$ −24.0000 −0.924445
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 46.0000 1.76792 0.883962 0.467559i $$-0.154866\pi$$
0.883962 + 0.467559i $$0.154866\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ 2.00000 0.0765840
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 22.0000 0.834508
$$696$$ 0 0
$$697$$ 24.0000 0.909065
$$698$$ −14.0000 −0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −36.0000 −1.35488
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −16.0000 −0.600893 −0.300446 0.953799i $$-0.597136\pi$$
−0.300446 + 0.953799i $$0.597136\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ −6.00000 −0.224860
$$713$$ −2.00000 −0.0749006
$$714$$ 0 0
$$715$$ −2.00000 −0.0747958
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ −32.0000 −1.19423
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −32.0000 −1.18356
$$732$$ 0 0
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ 14.0000 0.516749
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ −8.00000 −0.294684
$$738$$ 18.0000 0.662589
$$739$$ 10.0000 0.367856 0.183928 0.982940i $$-0.441119\pi$$
0.183928 + 0.982940i $$0.441119\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −22.0000 −0.807102 −0.403551 0.914957i $$-0.632224\pi$$
−0.403551 + 0.914957i $$0.632224\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 10.0000 0.366126
$$747$$ −12.0000 −0.439057
$$748$$ −8.00000 −0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ 34.0000 1.23575 0.617876 0.786276i $$-0.287994\pi$$
0.617876 + 0.786276i $$0.287994\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ −4.00000 −0.145095
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −8.00000 −0.289430
$$765$$ −12.0000 −0.433861
$$766$$ −6.00000 −0.216789
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10.0000 0.359908
$$773$$ 2.00000 0.0719350 0.0359675 0.999353i $$-0.488549\pi$$
0.0359675 + 0.999353i $$0.488549\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ 1.00000 0.0359211
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 20.0000 0.717035
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 22.0000 0.783718
$$789$$ 0 0
$$790$$ 4.00000 0.142314
$$791$$ 0 0
$$792$$ −6.00000 −0.213201
$$793$$ −4.00000 −0.142044
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 10.0000 0.354218 0.177109 0.984191i $$-0.443325\pi$$
0.177109 + 0.984191i $$0.443325\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ −1.00000 −0.0353553
$$801$$ −18.0000 −0.635999
$$802$$ 2.00000 0.0706225
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.00000 −0.0352235
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ −14.0000 −0.492214 −0.246107 0.969243i $$-0.579151\pi$$
−0.246107 + 0.969243i $$0.579151\pi$$
$$810$$ −9.00000 −0.316228
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4.00000 0.140200
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ −32.0000 −1.11954
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 36.0000 1.25641 0.628204 0.778048i $$-0.283790\pi$$
0.628204 + 0.778048i $$0.283790\pi$$
$$822$$ 0 0
$$823$$ 2.00000 0.0697156 0.0348578 0.999392i $$-0.488902\pi$$
0.0348578 + 0.999392i $$0.488902\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 32.0000 1.11141 0.555703 0.831381i $$-0.312449\pi$$
0.555703 + 0.831381i $$0.312449\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ −28.0000 −0.970143
$$834$$ 0 0
$$835$$ 6.00000 0.207639
$$836$$ −8.00000 −0.276686
$$837$$ 0 0
$$838$$ −32.0000 −1.10542
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −38.0000 −1.30957
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 1.00000 0.0344010
$$846$$ −36.0000 −1.23771
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ −12.0000 −0.410391
$$856$$ −12.0000 −0.410152
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ 8.00000 0.272481
$$863$$ −18.0000 −0.612727 −0.306364 0.951915i $$-0.599112\pi$$
−0.306364 + 0.951915i $$0.599112\pi$$
$$864$$ 0 0
$$865$$ 2.00000 0.0680020
$$866$$ 4.00000 0.135926
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 6.00000 0.203186
$$873$$ −6.00000 −0.203069
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ −21.0000 −0.707107
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −6.00000 −0.201120
$$891$$ −18.0000 −0.603023
$$892$$ 2.00000 0.0669650
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ −10.0000 −0.334263
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 0 0
$$900$$ −3.00000 −0.100000
$$901$$ 40.0000 1.33259
$$902$$ 12.0000 0.399556
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −36.0000 −1.19536 −0.597680 0.801735i $$-0.703911\pi$$
−0.597680 + 0.801735i $$0.703911\pi$$
$$908$$ 4.00000 0.132745
$$909$$ −30.0000 −0.995037
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ 4.00000 0.132308
$$915$$ 0 0
$$916$$ −12.0000 −0.396491
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −48.0000 −1.58337 −0.791687 0.610927i $$-0.790797\pi$$
−0.791687 + 0.610927i $$0.790797\pi$$
$$920$$ 2.00000 0.0659380
$$921$$ 0 0
$$922$$ 36.0000 1.18560
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 30.0000 0.985861
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ −2.00000 −0.0655122
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ −8.00000 −0.261628
$$936$$ 3.00000 0.0980581
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −12.0000 −0.391397
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ −12.0000 −0.390774
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −4.00000 −0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ 30.0000 0.971286
$$955$$ −8.00000 −0.258874
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 40.0000 1.29234
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ −2.00000 −0.0644826
$$963$$ −36.0000 −1.16008
$$964$$ −22.0000 −0.708572
$$965$$ 10.0000 0.321911
$$966$$ 0 0
$$967$$ −50.0000 −1.60789 −0.803946 0.594703i $$-0.797270\pi$$
−0.803946 + 0.594703i $$0.797270\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 0 0
$$979$$ −12.0000 −0.383522
$$980$$ −7.00000 −0.223607
$$981$$ 18.0000 0.574696
$$982$$ 18.0000 0.574403
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ 22.0000 0.700978
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 4.00000 0.127257
$$989$$ 16.0000 0.508770
$$990$$ −6.00000 −0.190693
$$991$$ −20.0000 −0.635321 −0.317660 0.948205i $$-0.602897\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ −1.00000 −0.0317500
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 20.0000 0.634043
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 2.00000 0.0633089
$$999$$ 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 4030.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4030.2.a.a.1.1 1 1.1 even 1 trivial