# Properties

 Label 3150.2.g.v.2899.1 Level 3150 Weight 2 Character 3150.2899 Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.1 Root $$1.00000i$$ Character $$\chi$$ = 3150.2899 Dual form 3150.2.g.v.2899.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +5.00000 q^{11} +6.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +3.00000 q^{19} -5.00000i q^{22} +6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +1.00000 q^{34} +8.00000i q^{37} -3.00000i q^{38} -11.0000 q^{41} +8.00000i q^{43} -5.00000 q^{44} -2.00000i q^{47} -1.00000 q^{49} -6.00000i q^{52} +4.00000i q^{53} +1.00000 q^{56} +6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +9.00000i q^{67} -1.00000i q^{68} +10.0000 q^{71} +7.00000i q^{73} +8.00000 q^{74} -3.00000 q^{76} -5.00000i q^{77} +2.00000 q^{79} +11.0000i q^{82} +11.0000i q^{83} +8.00000 q^{86} +5.00000i q^{88} -11.0000 q^{89} +6.00000 q^{91} -2.00000 q^{94} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 10q^{11} - 2q^{14} + 2q^{16} + 6q^{19} + 12q^{26} - 12q^{29} - 8q^{31} + 2q^{34} - 22q^{41} - 10q^{44} - 2q^{49} + 2q^{56} + 8q^{59} - 4q^{61} - 2q^{64} + 20q^{71} + 16q^{74} - 6q^{76} + 4q^{79} + 16q^{86} - 22q^{89} + 12q^{91} - 4q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000i 0.242536i 0.992620 + 0.121268i $$0.0386960\pi$$
−0.992620 + 0.121268i $$0.961304\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 5.00000i − 1.06600i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ 1.00000i 0.188982i
$$29$$ −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 $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ − 3.00000i − 0.486664i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −11.0000 −1.71791 −0.858956 0.512050i $$-0.828886\pi$$
−0.858956 + 0.512050i $$0.828886\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 2.00000i − 0.291730i −0.989305 0.145865i $$-0.953403\pi$$
0.989305 0.145865i $$-0.0465965\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 6.00000i − 0.832050i
$$53$$ 4.00000i 0.549442i 0.961524 + 0.274721i $$0.0885855\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 9.00000i 1.09952i 0.835321 + 0.549762i $$0.185282\pi$$
−0.835321 + 0.549762i $$0.814718\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ 0 0
$$73$$ 7.00000i 0.819288i 0.912245 + 0.409644i $$0.134347\pi$$
−0.912245 + 0.409644i $$0.865653\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −3.00000 −0.344124
$$77$$ − 5.00000i − 0.569803i
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 11.0000i 1.21475i
$$83$$ 11.0000i 1.20741i 0.797209 + 0.603703i $$0.206309\pi$$
−0.797209 + 0.603703i $$0.793691\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 5.00000i 0.533002i
$$89$$ −11.0000 −1.16600 −0.582999 0.812473i $$-0.698121\pi$$
−0.582999 + 0.812473i $$0.698121\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −2.00000 −0.206284
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ − 3.00000i − 0.290021i −0.989430 0.145010i $$-0.953678\pi$$
0.989430 0.145010i $$-0.0463216\pi$$
$$108$$ 0 0
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 1.00000i − 0.0940721i −0.998893 0.0470360i $$-0.985022\pi$$
0.998893 0.0470360i $$-0.0149776\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ − 4.00000i − 0.368230i
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 14.0000i − 1.24230i −0.783692 0.621150i $$-0.786666\pi$$
0.783692 0.621150i $$-0.213334\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ − 3.00000i − 0.260133i
$$134$$ 9.00000 0.777482
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 3.00000i 0.256307i 0.991754 + 0.128154i $$0.0409051\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ 0 0
$$139$$ 11.0000 0.933008 0.466504 0.884519i $$-0.345513\pi$$
0.466504 + 0.884519i $$0.345513\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 10.0000i − 0.839181i
$$143$$ 30.0000i 2.50873i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ 0 0
$$148$$ − 8.00000i − 0.657596i
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 3.00000i 0.243332i
$$153$$ 0 0
$$154$$ −5.00000 −0.402911
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ − 2.00000i − 0.159111i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 19.0000i 1.48819i 0.668071 + 0.744097i $$0.267120\pi$$
−0.668071 + 0.744097i $$0.732880\pi$$
$$164$$ 11.0000 0.858956
$$165$$ 0 0
$$166$$ 11.0000 0.853766
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 8.00000i − 0.609994i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.00000 0.376889
$$177$$ 0 0
$$178$$ 11.0000i 0.824485i
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ − 6.00000i − 0.444750i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5.00000i 0.365636i
$$188$$ 2.00000i 0.145865i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 19.0000i 1.36765i 0.729646 + 0.683825i $$0.239685\pi$$
−0.729646 + 0.683825i $$0.760315\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ 6.00000i 0.416025i
$$209$$ 15.0000 1.03757
$$210$$ 0 0
$$211$$ 1.00000 0.0688428 0.0344214 0.999407i $$-0.489041\pi$$
0.0344214 + 0.999407i $$0.489041\pi$$
$$212$$ − 4.00000i − 0.274721i
$$213$$ 0 0
$$214$$ −3.00000 −0.205076
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000i 0.271538i
$$218$$ − 18.0000i − 1.21911i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ − 22.0000i − 1.47323i −0.676313 0.736614i $$-0.736423\pi$$
0.676313 0.736614i $$-0.263577\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −1.00000 −0.0665190
$$227$$ 28.0000i 1.85843i 0.369546 + 0.929213i $$0.379513\pi$$
−0.369546 + 0.929213i $$0.620487\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ − 1.00000i − 0.0648204i
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ −5.00000 −0.322078 −0.161039 0.986948i $$-0.551485\pi$$
−0.161039 + 0.986948i $$0.551485\pi$$
$$242$$ − 14.0000i − 0.899954i
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 18.0000i 1.14531i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −27.0000 −1.70422 −0.852112 0.523359i $$-0.824679\pi$$
−0.852112 + 0.523359i $$0.824679\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −14.0000 −0.878438
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 8.00000i 0.494242i
$$263$$ − 10.0000i − 0.616626i −0.951285 0.308313i $$-0.900236\pi$$
0.951285 0.308313i $$-0.0997645\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −3.00000 −0.183942
$$267$$ 0 0
$$268$$ − 9.00000i − 0.549762i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −6.00000 −0.364474 −0.182237 0.983255i $$-0.558334\pi$$
−0.182237 + 0.983255i $$0.558334\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ 0 0
$$274$$ 3.00000 0.181237
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 30.0000i − 1.80253i −0.433273 0.901263i $$-0.642641\pi$$
0.433273 0.901263i $$-0.357359\pi$$
$$278$$ − 11.0000i − 0.659736i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −14.0000 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$282$$ 0 0
$$283$$ 13.0000i 0.772770i 0.922338 + 0.386385i $$0.126276\pi$$
−0.922338 + 0.386385i $$0.873724\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 0 0
$$286$$ 30.0000 1.77394
$$287$$ 11.0000i 0.649309i
$$288$$ 0 0
$$289$$ 16.0000 0.941176
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 7.00000i − 0.409644i
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 0 0
$$298$$ − 12.0000i − 0.695141i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ − 8.00000i − 0.460348i
$$303$$ 0 0
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 13.0000i 0.741949i 0.928643 + 0.370975i $$0.120976\pi$$
−0.928643 + 0.370975i $$0.879024\pi$$
$$308$$ 5.00000i 0.284901i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ − 4.00000i − 0.224662i −0.993671 0.112331i $$-0.964168\pi$$
0.993671 0.112331i $$-0.0358318\pi$$
$$318$$ 0 0
$$319$$ −30.0000 −1.67968
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.00000i 0.166924i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.0000 1.05231
$$327$$ 0 0
$$328$$ − 11.0000i − 0.607373i
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ −17.0000 −0.934405 −0.467202 0.884150i $$-0.654738\pi$$
−0.467202 + 0.884150i $$0.654738\pi$$
$$332$$ − 11.0000i − 0.603703i
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 29.0000i − 1.57973i −0.613280 0.789865i $$-0.710150\pi$$
0.613280 0.789865i $$-0.289850\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −20.0000 −1.08306
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ − 19.0000i − 1.01997i −0.860182 0.509987i $$-0.829650\pi$$
0.860182 0.509987i $$-0.170350\pi$$
$$348$$ 0 0
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 5.00000i − 0.266501i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 11.0000 0.582999
$$357$$ 0 0
$$358$$ − 3.00000i − 0.158555i
$$359$$ 26.0000 1.37223 0.686114 0.727494i $$-0.259315\pi$$
0.686114 + 0.727494i $$0.259315\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ − 10.0000i − 0.525588i
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 5.00000 0.258544
$$375$$ 0 0
$$376$$ 2.00000 0.103142
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ −9.00000 −0.462299 −0.231149 0.972918i $$-0.574249\pi$$
−0.231149 + 0.972918i $$0.574249\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 6.00000i − 0.306987i
$$383$$ − 6.00000i − 0.306586i −0.988181 0.153293i $$-0.951012\pi$$
0.988181 0.153293i $$-0.0489878\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 19.0000 0.967075
$$387$$ 0 0
$$388$$ 10.0000i 0.507673i
$$389$$ 8.00000 0.405616 0.202808 0.979219i $$-0.434993\pi$$
0.202808 + 0.979219i $$0.434993\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 10.0000i 0.501886i 0.968002 + 0.250943i $$0.0807406\pi$$
−0.968002 + 0.250943i $$0.919259\pi$$
$$398$$ − 10.0000i − 0.501255i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −37.0000 −1.84769 −0.923846 0.382765i $$-0.874972\pi$$
−0.923846 + 0.382765i $$0.874972\pi$$
$$402$$ 0 0
$$403$$ − 24.0000i − 1.19553i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 40.0000i 1.98273i
$$408$$ 0 0
$$409$$ 21.0000 1.03838 0.519192 0.854658i $$-0.326233\pi$$
0.519192 + 0.854658i $$0.326233\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.00000i 0.197066i
$$413$$ − 4.00000i − 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 0 0
$$418$$ − 15.0000i − 0.733674i
$$419$$ −39.0000 −1.90527 −0.952637 0.304109i $$-0.901641\pi$$
−0.952637 + 0.304109i $$0.901641\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ − 1.00000i − 0.0486792i
$$423$$ 0 0
$$424$$ −4.00000 −0.194257
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.00000i 0.0967868i
$$428$$ 3.00000i 0.145010i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 1.00000i 0.0480569i 0.999711 + 0.0240285i $$0.00764923\pi$$
−0.999711 + 0.0240285i $$0.992351\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.00000i 0.285391i
$$443$$ 37.0000i 1.75792i 0.476893 + 0.878962i $$0.341763\pi$$
−0.476893 + 0.878962i $$0.658237\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −22.0000 −1.04173
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 33.0000 1.55737 0.778683 0.627417i $$-0.215888\pi$$
0.778683 + 0.627417i $$0.215888\pi$$
$$450$$ 0 0
$$451$$ −55.0000 −2.58985
$$452$$ 1.00000i 0.0470360i
$$453$$ 0 0
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 25.0000i 1.16945i 0.811231 + 0.584725i $$0.198798\pi$$
−0.811231 + 0.584725i $$0.801202\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 38.0000 1.76984 0.884918 0.465746i $$-0.154214\pi$$
0.884918 + 0.465746i $$0.154214\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 4.00000i 0.185098i 0.995708 + 0.0925490i $$0.0295015\pi$$
−0.995708 + 0.0925490i $$0.970499\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000i 0.184115i
$$473$$ 40.0000i 1.83920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.00000 −0.0458349
$$477$$ 0 0
$$478$$ − 4.00000i − 0.182956i
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ −48.0000 −2.18861
$$482$$ 5.00000i 0.227744i
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 34.0000i − 1.54069i −0.637629 0.770344i $$-0.720085\pi$$
0.637629 0.770344i $$-0.279915\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 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
$$493$$ − 6.00000i − 0.270226i
$$494$$ 18.0000 0.809858
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ − 10.0000i − 0.448561i
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 27.0000i 1.20507i
$$503$$ − 30.0000i − 1.33763i −0.743427 0.668817i $$-0.766801\pi$$
0.743427 0.668817i $$-0.233199\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 14.0000i 0.621150i
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 10.0000i − 0.439799i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11.0000 −0.481919 −0.240959 0.970535i $$-0.577462\pi$$
−0.240959 + 0.970535i $$0.577462\pi$$
$$522$$ 0 0
$$523$$ − 13.0000i − 0.568450i −0.958758 0.284225i $$-0.908264\pi$$
0.958758 0.284225i $$-0.0917363\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ −10.0000 −0.436021
$$527$$ − 4.00000i − 0.174243i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 3.00000i 0.130066i
$$533$$ − 66.0000i − 2.85878i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −9.00000 −0.388741
$$537$$ 0 0
$$538$$ − 18.0000i − 0.776035i
$$539$$ −5.00000 −0.215365
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ 6.00000i 0.257722i
$$543$$ 0 0
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 27.0000i 1.15444i 0.816590 + 0.577218i $$0.195862\pi$$
−0.816590 + 0.577218i $$0.804138\pi$$
$$548$$ − 3.00000i − 0.128154i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −18.0000 −0.766826
$$552$$ 0 0
$$553$$ − 2.00000i − 0.0850487i
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ −11.0000 −0.466504
$$557$$ − 4.00000i − 0.169485i −0.996403 0.0847427i $$-0.972993\pi$$
0.996403 0.0847427i $$-0.0270068\pi$$
$$558$$ 0 0
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14.0000i 0.590554i
$$563$$ 20.0000i 0.842900i 0.906852 + 0.421450i $$0.138479\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 13.0000 0.546431
$$567$$ 0 0
$$568$$ 10.0000i 0.419591i
$$569$$ 21.0000 0.880366 0.440183 0.897908i $$-0.354914\pi$$
0.440183 + 0.897908i $$0.354914\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ − 30.0000i − 1.25436i
$$573$$ 0 0
$$574$$ 11.0000 0.459131
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.0000i 0.541197i 0.962692 + 0.270599i $$0.0872216\pi$$
−0.962692 + 0.270599i $$0.912778\pi$$
$$578$$ − 16.0000i − 0.665512i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 11.0000 0.456357
$$582$$ 0 0
$$583$$ 20.0000i 0.828315i
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 13.0000i 0.536567i 0.963340 + 0.268284i $$0.0864565\pi$$
−0.963340 + 0.268284i $$0.913544\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000i 0.328798i
$$593$$ − 39.0000i − 1.60154i −0.598973 0.800769i $$-0.704424\pi$$
0.598973 0.800769i $$-0.295576\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.0000 −0.491539
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 21.0000 0.856608 0.428304 0.903635i $$-0.359111\pi$$
0.428304 + 0.903635i $$0.359111\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ − 3.00000i − 0.121666i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ − 18.0000i − 0.727013i −0.931592 0.363507i $$-0.881579\pi$$
0.931592 0.363507i $$-0.118421\pi$$
$$614$$ 13.0000 0.524637
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ − 14.0000i − 0.563619i −0.959470 0.281809i $$-0.909065\pi$$
0.959470 0.281809i $$-0.0909346\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 6.00000i 0.240578i
$$623$$ 11.0000i 0.440706i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ − 4.00000i − 0.159617i
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 2.00000i 0.0795557i
$$633$$ 0 0
$$634$$ −4.00000 −0.158860
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ 30.0000i 1.18771i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ − 16.0000i − 0.630978i −0.948929 0.315489i $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 3.00000 0.118033
$$647$$ 8.00000i 0.314512i 0.987558 + 0.157256i $$0.0502649\pi$$
−0.987558 + 0.157256i $$0.949735\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 19.0000i − 0.744097i
$$653$$ − 28.0000i − 1.09572i −0.836569 0.547862i $$-0.815442\pi$$
0.836569 0.547862i $$-0.184558\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −11.0000 −0.429478
$$657$$ 0 0
$$658$$ 2.00000i 0.0779681i
$$659$$ 1.00000 0.0389545 0.0194772 0.999810i $$-0.493800\pi$$
0.0194772 + 0.999810i $$0.493800\pi$$
$$660$$ 0 0
$$661$$ −50.0000 −1.94477 −0.972387 0.233373i $$-0.925024\pi$$
−0.972387 + 0.233373i $$0.925024\pi$$
$$662$$ 17.0000i 0.660724i
$$663$$ 0 0
$$664$$ −11.0000 −0.426883
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 12.0000i 0.464294i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ −29.0000 −1.11704
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 48.0000i 1.84479i 0.386248 + 0.922395i $$0.373771\pi$$
−0.386248 + 0.922395i $$0.626229\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 20.0000i 0.765840i
$$683$$ − 13.0000i − 0.497431i −0.968577 0.248716i $$-0.919992\pi$$
0.968577 0.248716i $$-0.0800084\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 49.0000 1.86405 0.932024 0.362397i $$-0.118041\pi$$
0.932024 + 0.362397i $$0.118041\pi$$
$$692$$ − 2.00000i − 0.0760286i
$$693$$ 0 0
$$694$$ −19.0000 −0.721230
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 11.0000i − 0.416655i
$$698$$ − 8.00000i − 0.302804i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 32.0000 1.20862 0.604312 0.796748i $$-0.293448\pi$$
0.604312 + 0.796748i $$0.293448\pi$$
$$702$$ 0 0
$$703$$ 24.0000i 0.905177i
$$704$$ −5.00000 −0.188445
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 11.0000i − 0.412242i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3.00000 −0.112115
$$717$$ 0 0
$$718$$ − 26.0000i − 0.970311i
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 10.0000i 0.372161i
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 6.00000i 0.222528i 0.993791 + 0.111264i $$0.0354899\pi$$
−0.993791 + 0.111264i $$0.964510\pi$$
$$728$$ 6.00000i 0.222375i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ − 40.0000i − 1.47743i −0.674016 0.738717i $$-0.735432\pi$$
0.674016 0.738717i $$-0.264568\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 45.0000i 1.65760i
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 4.00000i − 0.146845i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ − 5.00000i − 0.182818i
$$749$$ −3.00000 −0.109618
$$750$$ 0 0
$$751$$ 50.0000 1.82453 0.912263 0.409605i $$-0.134333\pi$$
0.912263 + 0.409605i $$0.134333\pi$$
$$752$$ − 2.00000i − 0.0729325i
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 9.00000i 0.326895i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ − 18.0000i − 0.651644i
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ 24.0000i 0.866590i
$$768$$ 0 0
$$769$$ −19.0000 −0.685158 −0.342579 0.939489i $$-0.611300\pi$$
−0.342579 + 0.939489i $$0.611300\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 19.0000i − 0.683825i
$$773$$ 36.0000i 1.29483i 0.762138 + 0.647415i $$0.224150\pi$$
−0.762138 + 0.647415i $$0.775850\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ − 8.00000i − 0.286814i
$$779$$ −33.0000 −1.18235
$$780$$ 0 0
$$781$$ 50.0000 1.78914
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 52.0000i 1.85360i 0.375555 + 0.926800i $$0.377452\pi$$
−0.375555 + 0.926800i $$0.622548\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −1.00000 −0.0355559
$$792$$ 0 0
$$793$$ − 12.0000i − 0.426132i
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ 2.00000 0.0707549
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 37.0000i 1.30652i
$$803$$ 35.0000i 1.23512i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −24.0000 −0.845364
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ 0 0
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24.0000i 0.839654i
$$818$$ − 21.0000i − 0.734248i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ 0 0
$$823$$ − 10.0000i − 0.348578i −0.984695 0.174289i $$-0.944237\pi$$
0.984695 0.174289i $$-0.0557627\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ −4.00000 −0.139178
$$827$$ − 41.0000i − 1.42571i −0.701312 0.712855i $$-0.747402\pi$$
0.701312 0.712855i $$-0.252598\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 1.00000i − 0.0346479i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −15.0000 −0.518786
$$837$$ 0 0
$$838$$ 39.0000i 1.34723i
$$839$$ 2.00000 0.0690477 0.0345238 0.999404i $$-0.489009\pi$$
0.0345238 + 0.999404i $$0.489009\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 20.0000i − 0.689246i
$$843$$ 0 0
$$844$$ −1.00000 −0.0344214
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 14.0000i − 0.481046i
$$848$$ 4.00000i 0.137361i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 34.0000i 1.16414i 0.813139 + 0.582069i $$0.197757\pi$$
−0.813139 + 0.582069i $$0.802243\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ − 3.00000i − 0.102478i −0.998686 0.0512390i $$-0.983683\pi$$
0.998686 0.0512390i $$-0.0163170\pi$$
$$858$$ 0 0
$$859$$ 51.0000 1.74010 0.870049 0.492966i $$-0.164087\pi$$
0.870049 + 0.492966i $$0.164087\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 36.0000i − 1.22616i
$$863$$ 4.00000i 0.136162i 0.997680 + 0.0680808i $$0.0216876\pi$$
−0.997680 + 0.0680808i $$0.978312\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 1.00000 0.0339814
$$867$$ 0 0
$$868$$ − 4.00000i − 0.135769i
$$869$$ 10.0000 0.339227
$$870$$ 0 0
$$871$$ −54.0000 −1.82972
$$872$$ 18.0000i 0.609557i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000i 1.08056i 0.841484 + 0.540282i $$0.181682\pi$$
−0.841484 + 0.540282i $$0.818318\pi$$
$$878$$ 28.0000i 0.944954i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −26.0000 −0.875962 −0.437981 0.898984i $$-0.644306\pi$$
−0.437981 + 0.898984i $$0.644306\pi$$
$$882$$ 0 0
$$883$$ − 15.0000i − 0.504790i −0.967624 0.252395i $$-0.918782\pi$$
0.967624 0.252395i $$-0.0812183\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 37.0000 1.24304
$$887$$ 34.0000i 1.14161i 0.821086 + 0.570804i $$0.193368\pi$$
−0.821086 + 0.570804i $$0.806632\pi$$
$$888$$ 0 0
$$889$$ −14.0000 −0.469545
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 22.0000i 0.736614i
$$893$$ − 6.00000i − 0.200782i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 33.0000i − 1.10122i
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 55.0000i 1.83130i
$$903$$ 0 0
$$904$$ 1.00000 0.0332595
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ − 28.0000i − 0.929213i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 55.0000i 1.82023i
$$914$$ 25.0000 0.826927
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 8.00000i 0.264183i
$$918$$ 0 0
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 38.0000i − 1.25146i
$$923$$ 60.0000i 1.97492i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 0 0
$$928$$ 6.00000i 0.196960i
$$929$$ 46.0000 1.50921 0.754606 0.656179i $$-0.227828\pi$$
0.754606 + 0.656179i $$0.227828\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ − 6.00000i − 0.196537i
$$933$$ 0 0
$$934$$ 4.00000 0.130884
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 7.00000i − 0.228680i −0.993442 0.114340i $$-0.963525\pi$$
0.993442 0.114340i $$-0.0364753\pi$$
$$938$$ − 9.00000i − 0.293860i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −56.0000 −1.82555 −0.912774 0.408465i $$-0.866064\pi$$
−0.912774 + 0.408465i $$0.866064\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 40.0000 1.30051
$$947$$ 4.00000i 0.129983i 0.997886 + 0.0649913i $$0.0207020\pi$$
−0.997886 + 0.0649913i $$0.979298\pi$$
$$948$$ 0 0
$$949$$ −42.0000 −1.36338
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.00000i 0.0324102i
$$953$$ 9.00000i 0.291539i 0.989319 + 0.145769i $$0.0465657\pi$$
−0.989319 + 0.145769i $$0.953434\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −4.00000 −0.129369
$$957$$ 0 0
$$958$$ 6.00000i 0.193851i
$$959$$ 3.00000 0.0968751
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 48.0000i 1.54758i
$$963$$ 0 0
$$964$$ 5.00000 0.161039
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 2.00000i − 0.0643157i −0.999483 0.0321578i $$-0.989762\pi$$
0.999483 0.0321578i $$-0.0102379\pi$$
$$968$$ 14.0000i 0.449977i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 51.0000 1.63667 0.818334 0.574743i $$-0.194898\pi$$
0.818334 + 0.574743i $$0.194898\pi$$
$$972$$ 0 0
$$973$$ − 11.0000i − 0.352644i
$$974$$ −34.0000 −1.08943
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 21.0000i − 0.671850i −0.941889 0.335925i $$-0.890951\pi$$
0.941889 0.335925i $$-0.109049\pi$$
$$978$$ 0 0
$$979$$ −55.0000 −1.75781
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 12.0000i − 0.382935i
$$983$$ − 4.00000i − 0.127580i −0.997963 0.0637901i $$-0.979681\pi$$
0.997963 0.0637901i $$-0.0203188\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −6.00000 −0.191079
$$987$$ 0 0
$$988$$ − 18.0000i − 0.572656i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 0 0
$$994$$ −10.0000 −0.317181
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ 36.0000i 1.13956i
$$999$$ 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 3150.2.g.v.2899.1 2
3.2 odd 2 350.2.c.a.99.2 2
5.2 odd 4 3150.2.a.bq.1.1 1
5.3 odd 4 3150.2.a.j.1.1 1
5.4 even 2 inner 3150.2.g.v.2899.2 2
12.11 even 2 2800.2.g.a.449.1 2
15.2 even 4 350.2.a.c.1.1 1
15.8 even 4 350.2.a.d.1.1 yes 1
15.14 odd 2 350.2.c.a.99.1 2
21.20 even 2 2450.2.c.r.99.2 2
60.23 odd 4 2800.2.a.bg.1.1 1
60.47 odd 4 2800.2.a.b.1.1 1
60.59 even 2 2800.2.g.a.449.2 2
105.62 odd 4 2450.2.a.a.1.1 1
105.83 odd 4 2450.2.a.bg.1.1 1
105.104 even 2 2450.2.c.r.99.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.c.1.1 1 15.2 even 4
350.2.a.d.1.1 yes 1 15.8 even 4
350.2.c.a.99.1 2 15.14 odd 2
350.2.c.a.99.2 2 3.2 odd 2
2450.2.a.a.1.1 1 105.62 odd 4
2450.2.a.bg.1.1 1 105.83 odd 4
2450.2.c.r.99.1 2 105.104 even 2
2450.2.c.r.99.2 2 21.20 even 2
2800.2.a.b.1.1 1 60.47 odd 4
2800.2.a.bg.1.1 1 60.23 odd 4
2800.2.g.a.449.1 2 12.11 even 2
2800.2.g.a.449.2 2 60.59 even 2
3150.2.a.j.1.1 1 5.3 odd 4
3150.2.a.bq.1.1 1 5.2 odd 4
3150.2.g.v.2899.1 2 1.1 even 1 trivial
3150.2.g.v.2899.2 2 5.4 even 2 inner