# Properties

 Label 75.2.b.b.49.2 Level $75$ Weight $2$ Character 75.49 Analytic conductor $0.599$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.49 Dual form 75.2.b.b.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +3.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -7.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{94} +5.00000 q^{96} -2.00000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} - 2q^{16} - 8q^{19} + 6q^{24} + 4q^{26} + 4q^{29} + 4q^{34} - 2q^{36} - 4q^{39} + 20q^{41} - 8q^{44} + 14q^{49} - 4q^{51} - 2q^{54} + 8q^{59} - 4q^{61} - 14q^{64} - 8q^{66} - 16q^{71} - 20q^{74} - 8q^{76} + 2q^{81} - 8q^{86} + 12q^{89} + 16q^{94} + 10q^{96} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 4.00000i 0.696311i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 2.00000i 0.262613i
$$59$$ 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 $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 2.00000i − 0.214423i
$$88$$ − 12.0000i − 1.27920i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 2.00000i 0.184900i
$$118$$ 4.00000i 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 10.0000i − 0.901670i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ − 8.00000i − 0.671345i
$$143$$ 8.00000i 0.668994i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ − 7.00000i − 0.577350i
$$148$$ 10.0000i 0.821995i
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ − 12.0000i − 0.973329i
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 4.00000i − 0.300658i
$$178$$ 6.00000i 0.449719i
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ − 8.00000i − 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 7.00000i 0.505181i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 8.00000i 0.548151i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ 0 0
$$218$$ − 14.0000i − 0.948200i
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 10.0000i 0.671156i
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 8.00000i 0.509028i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ − 12.0000i − 0.741362i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ − 12.0000i − 0.713326i −0.934233 0.356663i $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ − 5.00000i − 0.294628i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 10.0000i 0.585206i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −30.0000 −1.74371
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 22.0000i − 1.27443i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 8.00000i − 0.460348i
$$303$$ − 6.00000i − 0.344691i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 14.0000i 0.774202i
$$328$$ 30.0000i 1.65647i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ − 10.0000i − 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ − 2.00000i − 0.107211i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ − 20.0000i − 1.06600i
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ − 20.0000i − 1.05703i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 10.0000i − 0.525588i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 24.0000 1.23771
$$377$$ − 4.00000i − 0.206010i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 16.0000i 0.818631i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 2.00000i − 0.101535i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 21.0000i 1.06066i
$$393$$ 12.0000i 0.605320i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 40.0000i − 1.98273i
$$408$$ − 6.00000i − 0.297044i
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ − 16.0000i − 0.788263i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ − 4.00000i − 0.195881i
$$418$$ 16.0000i 0.782586i
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 8.00000i 0.388973i
$$424$$ 30.0000 1.45693
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ 10.0000i 0.477818i
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 4.00000i − 0.190261i
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 22.0000i 1.04056i
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 2.00000i 0.0940721i
$$453$$ 8.00000i 0.375873i
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −12.0000 −0.561951
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 24.0000i 1.11537i 0.830051 + 0.557687i $$0.188311\pi$$
−0.830051 + 0.557687i $$0.811689\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 12.0000i 0.552345i
$$473$$ − 16.0000i − 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10.0000i 0.457869i
$$478$$ 16.0000i 0.731823i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ − 14.0000i − 0.637683i
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 32.0000i − 1.45006i −0.688718 0.725029i $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ − 4.00000i − 0.180151i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000i 0.537733i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000i 0.535586i
$$503$$ − 32.0000i − 1.42681i −0.700752 0.713405i $$-0.747152\pi$$
0.700752 0.713405i $$-0.252848\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 8.00000i 0.354943i
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 11.0000i − 0.486136i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ − 4.00000i − 0.174078i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ − 20.0000i − 0.866296i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 36.0000 1.55496
$$537$$ 20.0000i 0.863064i
$$538$$ − 14.0000i − 0.603583i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ 10.0000i 0.429141i
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ − 6.00000i − 0.253095i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ − 24.0000i − 1.00702i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ − 16.0000i − 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ 2.00000 0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 2.00000i − 0.0829027i
$$583$$ 40.0000i 1.65663i
$$584$$ −30.0000 −1.24141
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 10.0000i − 0.410997i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −22.0000 −0.901155
$$597$$ − 8.00000i − 0.327418i
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 8.00000i 0.324710i 0.986732 + 0.162355i $$0.0519090\pi$$
−0.986732 + 0.162355i $$0.948091\pi$$
$$608$$ − 20.0000i − 0.811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 2.00000i 0.0808452i
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ − 16.0000i − 0.643614i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ − 16.0000i − 0.638978i
$$628$$ − 14.0000i − 0.558661i
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ − 20.0000i − 0.794929i
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ − 14.0000i − 0.554700i
$$638$$ − 8.00000i − 0.316723i
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 36.0000i − 1.41970i −0.704352 0.709851i $$-0.748762\pi$$
0.704352 0.709851i $$-0.251238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ − 32.0000i − 1.25805i −0.777385 0.629025i $$-0.783454\pi$$
0.777385 0.629025i $$-0.216546\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ 46.0000i 1.80012i 0.435767 + 0.900060i $$0.356477\pi$$
−0.435767 + 0.900060i $$0.643523\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ − 10.0000i − 0.390137i
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ 4.00000i 0.155347i
$$664$$ −36.0000 −1.39707
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ − 30.0000i − 1.15642i −0.815890 0.578208i $$-0.803752\pi$$
0.815890 0.578208i $$-0.196248\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.00000i 0.228914i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ − 20.0000i − 0.757554i
$$698$$ 2.00000i 0.0757011i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 40.0000i − 1.50863i
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ − 4.00000i − 0.150329i
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 18.0000i 0.674579i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ − 16.0000i − 0.597531i
$$718$$ 24.0000i 0.895672i
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 14.0000i 0.520666i
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ 16.0000i 0.593407i 0.954970 + 0.296704i $$0.0958873\pi$$
−0.954970 + 0.296704i $$0.904113\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 2.00000i 0.0739221i
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000i 1.76810i
$$738$$ − 10.0000i − 0.368105i
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ − 16.0000i − 0.586983i −0.955962 0.293492i $$-0.905183\pi$$
0.955962 0.293492i $$-0.0948173\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ − 12.0000i − 0.439057i
$$748$$ 8.00000i 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ − 12.0000i − 0.437304i
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000i 0.944986i 0.881334 + 0.472493i $$0.156646\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ 0 0
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ − 8.00000i − 0.288863i
$$768$$ 17.0000i 0.613435i
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 2.00000i 0.0719816i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ − 6.00000i − 0.215110i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 2.00000i 0.0714742i
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 12.0000i 0.426401i
$$793$$ 4.00000i 0.142044i
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 2.00000i 0.0708436i 0.999372 + 0.0354218i $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000i 0.635602i
$$803$$ − 40.0000i − 1.41157i
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.0000i 0.492823i
$$808$$ 18.0000i 0.633238i
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 48.0000 1.67216
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ −6.00000 −0.208138
$$832$$ 14.0000i 0.485363i
$$833$$ − 14.0000i − 0.485071i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ − 4.00000i − 0.138178i
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 26.0000i − 0.896019i
$$843$$ 6.00000i 0.206651i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ 0 0
$$848$$ 10.0000i 0.343401i
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 8.00000i 0.274075i
$$853$$ 6.00000i 0.205436i 0.994711 + 0.102718i $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 56.0000i − 1.90626i −0.302558 0.953131i $$-0.597840\pi$$
0.302558 0.953131i $$-0.402160\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ − 42.0000i − 1.42230i
$$873$$ 2.00000i 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ − 30.0000i − 1.01303i −0.862232 0.506514i $$-0.830934\pi$$
0.862232 0.506514i $$-0.169066\pi$$
$$878$$ − 40.0000i − 1.34993i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 30.0000i 1.00673i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 8.00000i 0.267860i
$$893$$ 32.0000i 1.07084i
$$894$$ −22.0000 −0.735790
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ − 40.0000i − 1.33185i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 48.0000i − 1.58857i
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ − 18.0000i − 0.592798i
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −24.0000 −0.788689
$$927$$ 16.0000i 0.525509i
$$928$$ 10.0000i 0.328266i
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 6.00000i − 0.196537i
$$933$$ 24.0000i 0.785725i
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 54.0000i 1.76410i 0.471153 + 0.882052i $$0.343838\pi$$
−0.471153 + 0.882052i $$0.656162\pi$$
$$938$$ 0 0
$$939$$ 26.0000 0.848478
$$940$$ 0 0
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ − 14.0000i − 0.456145i
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 0 0
$$953$$ − 22.0000i − 0.712650i −0.934362 0.356325i $$-0.884030\pi$$
0.934362 0.356325i $$-0.115970\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 8.00000i 0.258603i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 20.0000i 0.644826i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 15.0000i 0.482118i
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 2.00000i − 0.0639857i −0.999488 0.0319928i $$-0.989815\pi$$
0.999488 0.0319928i $$-0.0101854\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 28.0000i 0.893516i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 30.0000 0.956365
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ − 12.0000i − 0.380808i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 54.0000i − 1.71020i −0.518465 0.855099i $$-0.673497\pi$$
0.518465 0.855099i $$-0.326503\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ −10.0000 −0.316386
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.2.b.b.49.2 2
3.2 odd 2 225.2.b.b.199.1 2
4.3 odd 2 1200.2.f.h.49.2 2
5.2 odd 4 15.2.a.a.1.1 1
5.3 odd 4 75.2.a.b.1.1 1
5.4 even 2 inner 75.2.b.b.49.1 2
8.3 odd 2 4800.2.f.c.3649.1 2
8.5 even 2 4800.2.f.bf.3649.2 2
12.11 even 2 3600.2.f.e.2449.1 2
15.2 even 4 45.2.a.a.1.1 1
15.8 even 4 225.2.a.b.1.1 1
15.14 odd 2 225.2.b.b.199.2 2
20.3 even 4 1200.2.a.e.1.1 1
20.7 even 4 240.2.a.d.1.1 1
20.19 odd 2 1200.2.f.h.49.1 2
35.2 odd 12 735.2.i.e.361.1 2
35.12 even 12 735.2.i.d.361.1 2
35.13 even 4 3675.2.a.j.1.1 1
35.17 even 12 735.2.i.d.226.1 2
35.27 even 4 735.2.a.c.1.1 1
35.32 odd 12 735.2.i.e.226.1 2
40.3 even 4 4800.2.a.bz.1.1 1
40.13 odd 4 4800.2.a.t.1.1 1
40.19 odd 2 4800.2.f.c.3649.2 2
40.27 even 4 960.2.a.a.1.1 1
40.29 even 2 4800.2.f.bf.3649.1 2
40.37 odd 4 960.2.a.l.1.1 1
45.2 even 12 405.2.e.c.271.1 2
45.7 odd 12 405.2.e.f.271.1 2
45.22 odd 12 405.2.e.f.136.1 2
45.32 even 12 405.2.e.c.136.1 2
55.32 even 4 1815.2.a.d.1.1 1
55.43 even 4 9075.2.a.g.1.1 1
60.23 odd 4 3600.2.a.u.1.1 1
60.47 odd 4 720.2.a.c.1.1 1
60.59 even 2 3600.2.f.e.2449.2 2
65.12 odd 4 2535.2.a.j.1.1 1
80.27 even 4 3840.2.k.r.1921.1 2
80.37 odd 4 3840.2.k.m.1921.2 2
80.67 even 4 3840.2.k.r.1921.2 2
80.77 odd 4 3840.2.k.m.1921.1 2
85.67 odd 4 4335.2.a.c.1.1 1
95.37 even 4 5415.2.a.j.1.1 1
105.62 odd 4 2205.2.a.i.1.1 1
115.22 even 4 7935.2.a.d.1.1 1
120.77 even 4 2880.2.a.y.1.1 1
120.107 odd 4 2880.2.a.bc.1.1 1
165.32 odd 4 5445.2.a.c.1.1 1
195.77 even 4 7605.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 5.2 odd 4
45.2.a.a.1.1 1 15.2 even 4
75.2.a.b.1.1 1 5.3 odd 4
75.2.b.b.49.1 2 5.4 even 2 inner
75.2.b.b.49.2 2 1.1 even 1 trivial
225.2.a.b.1.1 1 15.8 even 4
225.2.b.b.199.1 2 3.2 odd 2
225.2.b.b.199.2 2 15.14 odd 2
240.2.a.d.1.1 1 20.7 even 4
405.2.e.c.136.1 2 45.32 even 12
405.2.e.c.271.1 2 45.2 even 12
405.2.e.f.136.1 2 45.22 odd 12
405.2.e.f.271.1 2 45.7 odd 12
720.2.a.c.1.1 1 60.47 odd 4
735.2.a.c.1.1 1 35.27 even 4
735.2.i.d.226.1 2 35.17 even 12
735.2.i.d.361.1 2 35.12 even 12
735.2.i.e.226.1 2 35.32 odd 12
735.2.i.e.361.1 2 35.2 odd 12
960.2.a.a.1.1 1 40.27 even 4
960.2.a.l.1.1 1 40.37 odd 4
1200.2.a.e.1.1 1 20.3 even 4
1200.2.f.h.49.1 2 20.19 odd 2
1200.2.f.h.49.2 2 4.3 odd 2
1815.2.a.d.1.1 1 55.32 even 4
2205.2.a.i.1.1 1 105.62 odd 4
2535.2.a.j.1.1 1 65.12 odd 4
2880.2.a.y.1.1 1 120.77 even 4
2880.2.a.bc.1.1 1 120.107 odd 4
3600.2.a.u.1.1 1 60.23 odd 4
3600.2.f.e.2449.1 2 12.11 even 2
3600.2.f.e.2449.2 2 60.59 even 2
3675.2.a.j.1.1 1 35.13 even 4
3840.2.k.m.1921.1 2 80.77 odd 4
3840.2.k.m.1921.2 2 80.37 odd 4
3840.2.k.r.1921.1 2 80.27 even 4
3840.2.k.r.1921.2 2 80.67 even 4
4335.2.a.c.1.1 1 85.67 odd 4
4800.2.a.t.1.1 1 40.13 odd 4
4800.2.a.bz.1.1 1 40.3 even 4
4800.2.f.c.3649.1 2 8.3 odd 2
4800.2.f.c.3649.2 2 40.19 odd 2
4800.2.f.bf.3649.1 2 40.29 even 2
4800.2.f.bf.3649.2 2 8.5 even 2
5415.2.a.j.1.1 1 95.37 even 4
5445.2.a.c.1.1 1 165.32 odd 4
7605.2.a.g.1.1 1 195.77 even 4
7935.2.a.d.1.1 1 115.22 even 4
9075.2.a.g.1.1 1 55.43 even 4