# Properties

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

# Related objects

## 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 210) 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.e.2899.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} -2.00000i q^{37} -4.00000i q^{38} -2.00000 q^{41} +12.0000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} +2.00000i q^{68} -8.00000 q^{71} +14.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} -4.00000i q^{77} +2.00000i q^{82} +12.0000i q^{83} +12.0000 q^{86} -4.00000i q^{88} +2.00000 q^{89} -2.00000 q^{91} +8.00000i q^{92} +8.00000 q^{94} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 8q^{11} + 2q^{14} + 2q^{16} + 8q^{19} + 4q^{26} + 12q^{29} - 16q^{31} - 4q^{34} - 4q^{41} + 8q^{44} - 16q^{46} - 2q^{49} - 2q^{56} + 8q^{59} - 4q^{61} - 2q^{64} - 16q^{71} - 4q^{74} - 8q^{76} + 24q^{86} + 4q^{89} - 4q^{91} + 16q^{94} + O(q^{100})$$

## 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
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 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 $$0.540868\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 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 $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 4.00000i − 0.455842i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.00000i 0.220863i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 0 0
$$88$$ − 4.00000i − 0.426401i
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 8.00000i 0.834058i
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 20.0000i 1.93347i 0.255774 + 0.966736i $$0.417670\pi$$
−0.255774 + 0.966736i $$0.582330\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ − 4.00000i − 0.368230i
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ − 8.00000i − 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ 2.00000i 0.164399i
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 0 0
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ − 12.0000i − 0.939913i −0.882690 0.469956i $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 12.0000i − 0.914991i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ − 2.00000i − 0.149906i
$$179$$ −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 $$0.621207\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ 0 0
$$184$$ 8.00000 0.589768
$$185$$ 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 $$0.303498\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000i 0.422159i
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 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 $$0.258301\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 0 0
$$214$$ 20.0000 1.36717
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 8.00000i − 0.543075i
$$218$$ − 2.00000i − 0.135457i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ − 2.00000i − 0.129641i
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ − 8.00000i − 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ 0 0
$$255$$ 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
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.0000i 0.741362i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.0000i 0.841178i 0.907251 + 0.420589i $$0.138177\pi$$
−0.907251 + 0.420589i $$0.861823\pi$$
$$278$$ 20.0000i 1.19952i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ − 2.00000i − 0.118056i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 14.0000i − 0.819288i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 18.0000i 1.04271i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ − 8.00000i − 0.460348i
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ − 34.0000i − 1.92179i −0.276907 0.960897i $$-0.589309\pi$$
0.276907 0.960897i $$-0.410691\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 8.00000i − 0.445823i
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 0 0
$$328$$ − 2.00000i − 0.110432i
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 0 0
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000i 0.213201i
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 20.0000i 1.05703i
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000i 0.525588i
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 32.0000i − 1.67039i −0.549957 0.835193i $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 16.0000i − 0.818631i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ − 10.0000i − 0.507673i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.797017i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 8.00000i 0.396545i
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000i 0.394132i
$$413$$ 4.00000i 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 16.0000i 0.782586i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 2.00000i − 0.0967868i
$$428$$ − 20.0000i − 0.966736i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 6.00000i 0.288342i 0.989553 + 0.144171i $$0.0460515\pi$$
−0.989553 + 0.144171i $$0.953949\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ − 32.0000i − 1.53077i
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 4.00000i − 0.190261i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 14.0000i 0.658505i
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 38.0000i − 1.77757i −0.458329 0.888783i $$-0.651552\pi$$
0.458329 0.888783i $$-0.348448\pi$$
$$458$$ − 26.0000i − 1.21490i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 34.0000 1.58354 0.791769 0.610821i $$-0.209160\pi$$
0.791769 + 0.610821i $$0.209160\pi$$
$$462$$ 0 0
$$463$$ 16.0000i 0.743583i 0.928316 + 0.371792i $$0.121256\pi$$
−0.928316 + 0.371792i $$0.878744\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000i 0.184115i
$$473$$ − 48.0000i − 2.20704i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 0 0
$$478$$ 16.0000i 0.731823i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ − 18.0000i − 0.819878i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 24.0000i − 1.08754i −0.839233 0.543772i $$-0.816996\pi$$
0.839233 0.543772i $$-0.183004\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 8.00000i − 0.358849i
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 4.00000i 0.178529i
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 32.0000i − 1.40736i
$$518$$ − 2.00000i − 0.0878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 16.0000i 0.696971i
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 4.00000i − 0.173259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ − 30.0000i − 1.29339i
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 24.0000i 1.03089i
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 14.0000 0.594803
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 10.0000i 0.421825i
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ − 8.00000i − 0.335673i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 26.0000i 1.08239i 0.840896 + 0.541197i $$0.182029\pi$$
−0.840896 + 0.541197i $$0.817971\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ − 24.0000i − 0.993978i
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ − 16.0000i − 0.654289i
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 12.0000i 0.489083i
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 0 0
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ − 10.0000i − 0.402585i −0.979531 0.201292i $$-0.935486\pi$$
0.979531 0.201292i $$-0.0645141\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 8.00000i 0.320771i
$$623$$ 2.00000i 0.0801283i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −34.0000 −1.35891
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −14.0000 −0.556011
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 24.0000i 0.950169i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 12.0000i 0.469956i
$$653$$ − 34.0000i − 1.33052i −0.746611 0.665261i $$-0.768320\pi$$
0.746611 0.665261i $$-0.231680\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ 8.00000i 0.311872i
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 48.0000i − 1.85857i
$$668$$ − 16.0000i − 0.619059i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 32.0000i − 1.22534i
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 12.0000i 0.457496i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ − 14.0000i − 0.532200i
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.00000i 0.151511i
$$698$$ − 34.0000i − 1.28692i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 2.00000i 0.0749532i
$$713$$ 64.0000i 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 8.00000i 0.298557i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 3.00000i 0.111648i
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 24.0000i − 0.890111i −0.895503 0.445055i $$-0.853184\pi$$
0.895503 0.445055i $$-0.146816\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ − 30.0000i − 1.10808i −0.832492 0.554038i $$-0.813086\pi$$
0.832492 0.554038i $$-0.186914\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ − 48.0000i − 1.76810i
$$738$$ 0 0
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 6.00000i 0.220267i
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ − 8.00000i − 0.292509i
$$749$$ −20.0000 −0.730784
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 34.0000i − 1.23575i −0.786276 0.617876i $$-0.787994\pi$$
0.786276 0.617876i $$-0.212006\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2.00000 −0.0724999 −0.0362500 0.999343i $$-0.511541\pi$$
−0.0362500 + 0.999343i $$0.511541\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 8.00000i 0.288863i
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 26.0000i − 0.935155i −0.883952 0.467578i $$-0.845127\pi$$
0.883952 0.467578i $$-0.154873\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 18.0000i 0.645331i
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 16.0000i 0.572159i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 0 0
$$793$$ − 4.00000i − 0.142044i
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 18.0000i 0.635602i
$$803$$ − 56.0000i − 1.97620i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 0 0
$$808$$ − 6.00000i − 0.211079i
$$809$$ −38.0000 −1.33601 −0.668004 0.744157i $$-0.732851\pi$$
−0.668004 + 0.744157i $$0.732851\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 48.0000i 1.67931i
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ 0 0
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 2.00000i 0.0692959i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ − 12.0000i − 0.414533i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 26.0000i 0.896019i
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.00000i 0.171802i
$$848$$ 6.00000i 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −16.0000 −0.548473
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ −20.0000 −0.683586
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 6.00000 0.203888
$$867$$ 0 0
$$868$$ 8.00000i 0.271538i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 2.00000i 0.0677285i
$$873$$ 0 0
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 10.0000i − 0.337676i −0.985644 0.168838i $$-0.945999\pi$$
0.985644 0.168838i $$-0.0540015\pi$$
$$878$$ 32.0000i 1.07995i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 0 0
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 16.0000i − 0.537227i −0.963248 0.268614i $$-0.913434\pi$$
0.963248 0.268614i $$-0.0865655\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 14.0000i 0.467186i
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 8.00000i − 0.266371i
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 12.0000i − 0.398453i −0.979953 0.199227i $$-0.936157\pi$$
0.979953 0.199227i $$-0.0638430\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ − 48.0000i − 1.58857i
$$914$$ −38.0000 −1.25693
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ − 12.0000i − 0.396275i
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 34.0000i − 1.11973i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.0000 0.525793
$$927$$ 0 0
$$928$$ − 6.00000i − 0.196960i
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ − 10.0000i − 0.327561i
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.0000i 0.588034i 0.955800 + 0.294017i $$0.0949923\pi$$
−0.955800 + 0.294017i $$0.905008\pi$$
$$938$$ 12.0000i 0.391814i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ 16.0000i 0.521032i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2.00000i 0.0648204i
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 4.00000i − 0.128965i
$$963$$ 0 0
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ − 20.0000i − 0.641171i
$$974$$ −24.0000 −0.769010
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 36.0000i 1.14881i
$$983$$ 32.0000i 1.02064i 0.859984 + 0.510321i $$0.170473\pi$$
−0.859984 + 0.510321i $$0.829527\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 96.0000 3.05262
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 38.0000i 1.20347i 0.798695 + 0.601736i $$0.205524\pi$$
−0.798695 + 0.601736i $$0.794476\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 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.e.2899.1 2
3.2 odd 2 1050.2.g.d.799.2 2
5.2 odd 4 3150.2.a.w.1.1 1
5.3 odd 4 630.2.a.b.1.1 1
5.4 even 2 inner 3150.2.g.e.2899.2 2
15.2 even 4 1050.2.a.h.1.1 1
15.8 even 4 210.2.a.c.1.1 1
15.14 odd 2 1050.2.g.d.799.1 2
20.3 even 4 5040.2.a.i.1.1 1
35.13 even 4 4410.2.a.l.1.1 1
60.23 odd 4 1680.2.a.q.1.1 1
60.47 odd 4 8400.2.a.p.1.1 1
105.23 even 12 1470.2.i.f.361.1 2
105.38 odd 12 1470.2.i.b.961.1 2
105.53 even 12 1470.2.i.f.961.1 2
105.62 odd 4 7350.2.a.p.1.1 1
105.68 odd 12 1470.2.i.b.361.1 2
105.83 odd 4 1470.2.a.q.1.1 1
120.53 even 4 6720.2.a.bp.1.1 1
120.83 odd 4 6720.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.c.1.1 1 15.8 even 4
630.2.a.b.1.1 1 5.3 odd 4
1050.2.a.h.1.1 1 15.2 even 4
1050.2.g.d.799.1 2 15.14 odd 2
1050.2.g.d.799.2 2 3.2 odd 2
1470.2.a.q.1.1 1 105.83 odd 4
1470.2.i.b.361.1 2 105.68 odd 12
1470.2.i.b.961.1 2 105.38 odd 12
1470.2.i.f.361.1 2 105.23 even 12
1470.2.i.f.961.1 2 105.53 even 12
1680.2.a.q.1.1 1 60.23 odd 4
3150.2.a.w.1.1 1 5.2 odd 4
3150.2.g.e.2899.1 2 1.1 even 1 trivial
3150.2.g.e.2899.2 2 5.4 even 2 inner
4410.2.a.l.1.1 1 35.13 even 4
5040.2.a.i.1.1 1 20.3 even 4
6720.2.a.k.1.1 1 120.83 odd 4
6720.2.a.bp.1.1 1 120.53 even 4
7350.2.a.p.1.1 1 105.62 odd 4
8400.2.a.p.1.1 1 60.47 odd 4