# Properties

 Label 3150.2.g.g.2899.2 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: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1050) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{11} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -4.00000 q^{19} -2.00000i q^{22} -7.00000i q^{23} -1.00000 q^{26} -1.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} +6.00000i q^{37} -4.00000i q^{38} +3.00000 q^{41} -1.00000i q^{43} +2.00000 q^{44} +7.00000 q^{46} -12.0000i q^{47} -1.00000 q^{49} -1.00000i q^{52} -11.0000i q^{53} +1.00000 q^{56} +1.00000i q^{58} -3.00000 q^{59} +5.00000 q^{61} +3.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} +1.00000i q^{68} -4.00000 q^{71} -14.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} -2.00000i q^{77} +2.00000 q^{79} +3.00000i q^{82} +3.00000i q^{83} +1.00000 q^{86} +2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} +7.00000i q^{92} +12.0000 q^{94} +10.0000i q^{97} -1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{11} - 2q^{14} + 2q^{16} - 8q^{19} - 2q^{26} + 2q^{29} + 6q^{31} + 2q^{34} + 6q^{41} + 4q^{44} + 14q^{46} - 2q^{49} + 2q^{56} - 6q^{59} + 10q^{61} - 2q^{64} - 8q^{71} - 12q^{74} + 8q^{76} + 4q^{79} + 2q^{86} + 20q^{89} - 2q^{91} + 24q^{94} + O(q^{100})$$

## 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
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i −0.992620 0.121268i $$-0.961304\pi$$
0.992620 0.121268i $$-0.0386960\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 2.00000i − 0.426401i
$$23$$ − 7.00000i − 1.45960i −0.683660 0.729800i $$-0.739613\pi$$
0.683660 0.729800i $$-0.260387\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 7.00000 1.03209
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 1.00000i − 0.138675i
$$53$$ − 11.0000i − 1.51097i −0.655168 0.755483i $$-0.727402\pi$$
0.655168 0.755483i $$-0.272598\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 1.00000i 0.131306i
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 3.00000i 0.381000i
$$63$$ 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 $$0.738112\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ − 2.00000i − 0.227921i
$$78$$ 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
$$82$$ 3.00000i 0.331295i
$$83$$ 3.00000i 0.329293i 0.986353 + 0.164646i $$0.0526483\pi$$
−0.986353 + 0.164646i $$0.947352\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 0 0
$$88$$ 2.00000i 0.213201i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 7.00000i 0.729800i
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 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
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 17.0000i − 1.67506i −0.546392 0.837530i $$-0.683999\pi$$
0.546392 0.837530i $$-0.316001\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ − 18.0000i − 1.74013i −0.492941 0.870063i $$-0.664078\pi$$
0.492941 0.870063i $$-0.335922\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ − 3.00000i − 0.276172i
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 5.00000i 0.452679i
$$123$$ 0 0
$$124$$ −3.00000 −0.269408
$$125$$ 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
$$133$$ − 4.00000i − 0.346844i
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 4.00000i 0.341743i 0.985293 + 0.170872i $$0.0546583\pi$$
−0.985293 + 0.170872i $$0.945342\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 4.00000i − 0.335673i
$$143$$ − 2.00000i − 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ − 6.00000i − 0.493197i
$$149$$ 5.00000 0.409616 0.204808 0.978802i $$-0.434343\pi$$
0.204808 + 0.978802i $$0.434343\pi$$
$$150$$ 0 0
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 0 0
$$154$$ 2.00000 0.161165
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 2.00000i 0.159111i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 7.00000 0.551677
$$162$$ 0 0
$$163$$ − 19.0000i − 1.48819i −0.668071 0.744097i $$-0.732880\pi$$
0.668071 0.744097i $$-0.267120\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ −3.00000 −0.232845
$$167$$ − 2.00000i − 0.154765i −0.997001 0.0773823i $$-0.975344\pi$$
0.997001 0.0773823i $$-0.0246562\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.00000i 0.0762493i
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 10.0000i 0.749532i
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ − 1.00000i − 0.0741249i
$$183$$ 0 0
$$184$$ −7.00000 −0.516047
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.00000i 0.146254i
$$188$$ 12.0000i 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.0000 0.940647 0.470323 0.882494i $$-0.344137\pi$$
0.470323 + 0.882494i $$0.344137\pi$$
$$192$$ 0 0
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 27.0000i 1.92367i 0.273629 + 0.961835i $$0.411776\pi$$
−0.273629 + 0.961835i $$0.588224\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.00000i 0.0701862i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 17.0000 1.18445
$$207$$ 0 0
$$208$$ 1.00000i 0.0693375i
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ 11.0000i 0.755483i
$$213$$ 0 0
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.00000i 0.203653i
$$218$$ 4.00000i 0.270914i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.00000 0.0672673
$$222$$ 0 0
$$223$$ − 13.0000i − 0.870544i −0.900299 0.435272i $$-0.856652\pi$$
0.900299 0.435272i $$-0.143348\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 7.00000i 0.464606i 0.972643 + 0.232303i $$0.0746261\pi$$
−0.972643 + 0.232303i $$0.925374\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 1.00000i − 0.0656532i
$$233$$ − 20.0000i − 1.31024i −0.755523 0.655122i $$-0.772617\pi$$
0.755523 0.655122i $$-0.227383\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 3.00000 0.195283
$$237$$ 0 0
$$238$$ 1.00000i 0.0648204i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 0 0
$$244$$ −5.00000 −0.320092
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.00000i − 0.254514i
$$248$$ − 3.00000i − 0.190500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 0 0
$$253$$ 14.0000i 0.880172i
$$254$$ 14.0000 0.878438
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 5.00000i − 0.311891i −0.987766 0.155946i $$-0.950158\pi$$
0.987766 0.155946i $$-0.0498425\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 8.00000i − 0.494242i
$$263$$ − 11.0000i − 0.678289i −0.940734 0.339145i $$-0.889862\pi$$
0.940734 0.339145i $$-0.110138\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 0 0
$$274$$ −4.00000 −0.241649
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 3.00000i 0.177084i
$$288$$ 0 0
$$289$$ 16.0000 0.941176
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 14.0000i 0.819288i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 5.00000i 0.289642i
$$299$$ 7.00000 0.404820
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 22.0000i 1.26596i
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.0000i 1.25561i 0.778372 + 0.627803i $$0.216046\pi$$
−0.778372 + 0.627803i $$0.783954\pi$$
$$308$$ 2.00000i 0.113961i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −34.0000 −1.92796 −0.963982 0.265969i $$-0.914308\pi$$
−0.963982 + 0.265969i $$0.914308\pi$$
$$312$$ 0 0
$$313$$ 18.0000i 1.01742i 0.860938 + 0.508710i $$0.169877\pi$$
−0.860938 + 0.508710i $$0.830123\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ − 17.0000i − 0.954815i −0.878682 0.477408i $$-0.841577\pi$$
0.878682 0.477408i $$-0.158423\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 7.00000i 0.390095i
$$323$$ 4.00000i 0.222566i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.0000 1.05231
$$327$$ 0 0
$$328$$ − 3.00000i − 0.165647i
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ − 3.00000i − 0.164646i
$$333$$ 0 0
$$334$$ 2.00000 0.109435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 27.0000i − 1.47078i −0.677642 0.735392i $$-0.736998\pi$$
0.677642 0.735392i $$-0.263002\pi$$
$$338$$ 12.0000i 0.652714i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 26.0000i 1.39575i 0.716218 + 0.697877i $$0.245872\pi$$
−0.716218 + 0.697877i $$0.754128\pi$$
$$348$$ 0 0
$$349$$ 1.00000 0.0535288 0.0267644 0.999642i $$-0.491480\pi$$
0.0267644 + 0.999642i $$0.491480\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 10.0000i − 0.532246i −0.963939 0.266123i $$-0.914257\pi$$
0.963939 0.266123i $$-0.0857428\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 10.0000i 0.528516i
$$359$$ −9.00000 −0.475002 −0.237501 0.971387i $$-0.576328\pi$$
−0.237501 + 0.971387i $$0.576328\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 18.0000i − 0.946059i
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 13.0000i − 0.678594i −0.940679 0.339297i $$-0.889811\pi$$
0.940679 0.339297i $$-0.110189\pi$$
$$368$$ − 7.00000i − 0.364900i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 11.0000 0.571092
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 1.00000i 0.0515026i
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 13.0000i 0.665138i
$$383$$ 20.0000i 1.02195i 0.859595 + 0.510976i $$0.170716\pi$$
−0.859595 + 0.510976i $$0.829284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 0 0
$$388$$ − 10.0000i − 0.507673i
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ 0 0
$$391$$ −7.00000 −0.354005
$$392$$ 1.00000i 0.0505076i
$$393$$ 0 0
$$394$$ −27.0000 −1.36024
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.00000i − 0.150566i −0.997162 0.0752828i $$-0.976014\pi$$
0.997162 0.0752828i $$-0.0239860\pi$$
$$398$$ 24.0000i 1.20301i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −16.0000 −0.799002 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$402$$ 0 0
$$403$$ 3.00000i 0.149441i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −1.00000 −0.0496292
$$407$$ − 12.0000i − 0.594818i
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 17.0000i 0.837530i
$$413$$ − 3.00000i − 0.147620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 8.00000i 0.391293i
$$419$$ −25.0000 −1.22133 −0.610665 0.791889i $$-0.709098\pi$$
−0.610665 + 0.791889i $$0.709098\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 15.0000i 0.730189i
$$423$$ 0 0
$$424$$ −11.0000 −0.534207
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 5.00000i 0.241967i
$$428$$ 18.0000i 0.870063i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −27.0000 −1.30054 −0.650272 0.759701i $$-0.725345\pi$$
−0.650272 + 0.759701i $$0.725345\pi$$
$$432$$ 0 0
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 28.0000i 1.33942i
$$438$$ 0 0
$$439$$ −7.00000 −0.334092 −0.167046 0.985949i $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1.00000i 0.0475651i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 13.0000 0.615568
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −16.0000 −0.755087 −0.377543 0.925992i $$-0.623231\pi$$
−0.377543 + 0.925992i $$0.623231\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 6.00000i 0.282216i
$$453$$ 0 0
$$454$$ −7.00000 −0.328526
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.0000i 0.795226i 0.917553 + 0.397613i $$0.130161\pi$$
−0.917553 + 0.397613i $$0.869839\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −32.0000 −1.49039 −0.745194 0.666847i $$-0.767643\pi$$
−0.745194 + 0.666847i $$0.767643\pi$$
$$462$$ 0 0
$$463$$ 36.0000i 1.67306i 0.547920 + 0.836531i $$0.315420\pi$$
−0.547920 + 0.836531i $$0.684580\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ 20.0000 0.926482
$$467$$ − 39.0000i − 1.80470i −0.430999 0.902352i $$-0.641839\pi$$
0.430999 0.902352i $$-0.358161\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3.00000i 0.138086i
$$473$$ 2.00000i 0.0919601i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.00000 −0.0458349
$$477$$ 0 0
$$478$$ − 24.0000i − 1.09773i
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ − 26.0000i − 1.18427i
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ − 5.00000i − 0.226339i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 0 0
$$493$$ − 1.00000i − 0.0450377i
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 3.00000 0.134704
$$497$$ − 4.00000i − 0.179425i
$$498$$ 0 0
$$499$$ 27.0000 1.20869 0.604343 0.796724i $$-0.293436\pi$$
0.604343 + 0.796724i $$0.293436\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 15.0000i 0.669483i
$$503$$ 2.00000i 0.0891756i 0.999005 + 0.0445878i $$0.0141974\pi$$
−0.999005 + 0.0445878i $$0.985803\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −14.0000 −0.622376
$$507$$ 0 0
$$508$$ 14.0000i 0.621150i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 5.00000 0.220541
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000i 1.05552i
$$518$$ − 6.00000i − 0.263625i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ − 8.00000i − 0.349816i −0.984585 0.174908i $$-0.944037\pi$$
0.984585 0.174908i $$-0.0559627\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ 11.0000 0.479623
$$527$$ − 3.00000i − 0.130682i
$$528$$ 0 0
$$529$$ −26.0000 −1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000i 0.173422i
$$533$$ 3.00000i 0.129944i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 18.0000i 0.776035i
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ − 20.0000i − 0.859074i
$$543$$ 0 0
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 15.0000i 0.641354i 0.947189 + 0.320677i $$0.103910\pi$$
−0.947189 + 0.320677i $$0.896090\pi$$
$$548$$ − 4.00000i − 0.170872i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 2.00000i 0.0850487i
$$554$$ −2.00000 −0.0849719
$$555$$ 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
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 41.0000i − 1.72794i −0.503540 0.863972i $$-0.667969\pi$$
0.503540 0.863972i $$-0.332031\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ 4.00000i 0.167836i
$$569$$ 14.0000 0.586911 0.293455 0.955973i $$-0.405195\pi$$
0.293455 + 0.955973i $$0.405195\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 0 0
$$574$$ −3.00000 −0.125218
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ 16.0000i 0.665512i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ 0 0
$$583$$ 22.0000i 0.911147i
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 27.0000i − 1.11441i −0.830375 0.557205i $$-0.811874\pi$$
0.830375 0.557205i $$-0.188126\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000i 0.246598i
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5.00000 −0.204808
$$597$$ 0 0
$$598$$ 7.00000i 0.286251i
$$599$$ −45.0000 −1.83865 −0.919325 0.393499i $$-0.871265\pi$$
−0.919325 + 0.393499i $$0.871265\pi$$
$$600$$ 0 0
$$601$$ 28.0000 1.14214 0.571072 0.820900i $$-0.306528\pi$$
0.571072 + 0.820900i $$0.306528\pi$$
$$602$$ 1.00000i 0.0407570i
$$603$$ 0 0
$$604$$ −22.0000 −0.895167
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 18.0000i 0.727013i 0.931592 + 0.363507i $$0.118421\pi$$
−0.931592 + 0.363507i $$0.881579\pi$$
$$614$$ −22.0000 −0.887848
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ 14.0000i 0.563619i 0.959470 + 0.281809i $$0.0909346\pi$$
−0.959470 + 0.281809i $$0.909065\pi$$
$$618$$ 0 0
$$619$$ −38.0000 −1.52735 −0.763674 0.645601i $$-0.776607\pi$$
−0.763674 + 0.645601i $$0.776607\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 34.0000i − 1.36328i
$$623$$ 10.0000i 0.400642i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.0000 −0.719425
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 26.0000 1.03504 0.517522 0.855670i $$-0.326855\pi$$
0.517522 + 0.855670i $$0.326855\pi$$
$$632$$ − 2.00000i − 0.0795557i
$$633$$ 0 0
$$634$$ 17.0000 0.675156
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 1.00000i − 0.0396214i
$$638$$ − 2.00000i − 0.0791808i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ 2.00000i 0.0788723i 0.999222 + 0.0394362i $$0.0125562\pi$$
−0.999222 + 0.0394362i $$0.987444\pi$$
$$644$$ −7.00000 −0.275839
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 6.00000i 0.235884i 0.993020 + 0.117942i $$0.0376297\pi$$
−0.993020 + 0.117942i $$0.962370\pi$$
$$648$$ 0 0
$$649$$ 6.00000 0.235521
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 19.0000i 0.744097i
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 3.00000 0.117130
$$657$$ 0 0
$$658$$ 12.0000i 0.467809i
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ − 3.00000i − 0.116598i
$$663$$ 0 0
$$664$$ 3.00000 0.116423
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 7.00000i − 0.271041i
$$668$$ 2.00000i 0.0773823i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ 27.0000i 1.04077i 0.853931 + 0.520387i $$0.174212\pi$$
−0.853931 + 0.520387i $$0.825788\pi$$
$$674$$ 27.0000 1.04000
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ − 20.0000i − 0.768662i −0.923195 0.384331i $$-0.874432\pi$$
0.923195 0.384331i $$-0.125568\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 6.00000i − 0.229752i
$$683$$ 6.00000i 0.229584i 0.993390 + 0.114792i $$0.0366201\pi$$
−0.993390 + 0.114792i $$0.963380\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ − 1.00000i − 0.0381246i
$$689$$ 11.0000 0.419067
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ 0 0
$$694$$ −26.0000 −0.986947
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 3.00000i − 0.113633i
$$698$$ 1.00000i 0.0378506i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 39.0000 1.47301 0.736505 0.676432i $$-0.236475\pi$$
0.736505 + 0.676432i $$0.236475\pi$$
$$702$$ 0 0
$$703$$ − 24.0000i − 0.905177i
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 10.0000 0.376355
$$707$$ 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
$$712$$ − 10.0000i − 0.374766i
$$713$$ − 21.0000i − 0.786456i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ − 9.00000i − 0.335877i
$$719$$ −34.0000 −1.26799 −0.633993 0.773339i $$-0.718585\pi$$
−0.633993 + 0.773339i $$0.718585\pi$$
$$720$$ 0 0
$$721$$ 17.0000 0.633113
$$722$$ − 3.00000i − 0.111648i
$$723$$ 0 0
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.00000i 0.0370879i 0.999828 + 0.0185440i $$0.00590307\pi$$
−0.999828 + 0.0185440i $$0.994097\pi$$
$$728$$ 1.00000i 0.0370625i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1.00000 −0.0369863
$$732$$ 0 0
$$733$$ 19.0000i 0.701781i 0.936416 + 0.350891i $$0.114121\pi$$
−0.936416 + 0.350891i $$0.885879\pi$$
$$734$$ 13.0000 0.479839
$$735$$ 0 0
$$736$$ 7.00000 0.258023
$$737$$ − 24.0000i − 0.884051i
$$738$$ 0 0
$$739$$ −11.0000 −0.404642 −0.202321 0.979319i $$-0.564848\pi$$
−0.202321 + 0.979319i $$0.564848\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 11.0000i 0.403823i
$$743$$ 3.00000i 0.110059i 0.998485 + 0.0550297i $$0.0175253\pi$$
−0.998485 + 0.0550297i $$0.982475\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ − 2.00000i − 0.0731272i
$$749$$ 18.0000 0.657706
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 0 0
$$754$$ −1.00000 −0.0364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 5.00000i 0.181608i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34.0000 −1.23250 −0.616250 0.787551i $$-0.711349\pi$$
−0.616250 + 0.787551i $$0.711349\pi$$
$$762$$ 0 0
$$763$$ 4.00000i 0.144810i
$$764$$ −13.0000 −0.470323
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ − 3.00000i − 0.108324i
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 2.00000i − 0.0719816i
$$773$$ 20.0000i 0.719350i 0.933078 + 0.359675i $$0.117112\pi$$
−0.933078 + 0.359675i $$0.882888\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 22.0000i 0.788738i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ − 7.00000i − 0.250319i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 38.0000i − 1.35455i −0.735728 0.677277i $$-0.763160\pi$$
0.735728 0.677277i $$-0.236840\pi$$
$$788$$ − 27.0000i − 0.961835i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 5.00000i 0.177555i
$$794$$ 3.00000 0.106466
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 42.0000i 1.48772i 0.668338 + 0.743858i $$0.267006\pi$$
−0.668338 + 0.743858i $$0.732994\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 16.0000i − 0.564980i
$$803$$ 28.0000i 0.988099i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3.00000 −0.105670
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 20.0000 0.703163 0.351581 0.936157i $$-0.385644\pi$$
0.351581 + 0.936157i $$0.385644\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ − 1.00000i − 0.0350931i
$$813$$ 0 0
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.00000i 0.139942i
$$818$$ 14.0000i 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ − 18.0000i − 0.627441i −0.949515 0.313720i $$-0.898425\pi$$
0.949515 0.313720i $$-0.101575\pi$$
$$824$$ −17.0000 −0.592223
$$825$$ 0 0
$$826$$ 3.00000 0.104383
$$827$$ − 50.0000i − 1.73867i −0.494223 0.869335i $$-0.664547\pi$$
0.494223 0.869335i $$-0.335453\pi$$
$$828$$ 0 0
$$829$$ −3.00000 −0.104194 −0.0520972 0.998642i $$-0.516591\pi$$
−0.0520972 + 0.998642i $$0.516591\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 1.00000i − 0.0346688i
$$833$$ 1.00000i 0.0346479i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 0 0
$$838$$ − 25.0000i − 0.863611i
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 34.0000i 1.17172i
$$843$$ 0 0
$$844$$ −15.0000 −0.516321
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 7.00000i − 0.240523i
$$848$$ − 11.0000i − 0.377742i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 42.0000 1.43974
$$852$$ 0 0
$$853$$ 43.0000i 1.47229i 0.676823 + 0.736146i $$0.263356\pi$$
−0.676823 + 0.736146i $$0.736644\pi$$
$$854$$ −5.00000 −0.171096
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ − 46.0000i − 1.57133i −0.618652 0.785665i $$-0.712321\pi$$
0.618652 0.785665i $$-0.287679\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 27.0000i − 0.919624i
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ − 3.00000i − 0.101827i
$$869$$ −4.00000 −0.135691
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ − 4.00000i − 0.135457i
$$873$$ 0 0
$$874$$ −28.0000 −0.947114
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 32.0000i − 1.08056i −0.841484 0.540282i $$-0.818318\pi$$
0.841484 0.540282i $$-0.181682\pi$$
$$878$$ − 7.00000i − 0.236239i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −5.00000 −0.168454 −0.0842271 0.996447i $$-0.526842\pi$$
−0.0842271 + 0.996447i $$0.526842\pi$$
$$882$$ 0 0
$$883$$ 29.0000i 0.975928i 0.872864 + 0.487964i $$0.162260\pi$$
−0.872864 + 0.487964i $$0.837740\pi$$
$$884$$ −1.00000 −0.0336336
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 50.0000i 1.67884i 0.543487 + 0.839418i $$0.317104\pi$$
−0.543487 + 0.839418i $$0.682896\pi$$
$$888$$ 0 0
$$889$$ 14.0000 0.469545
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 13.0000i 0.435272i
$$893$$ 48.0000i 1.60626i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 16.0000i − 0.533927i
$$899$$ 3.00000 0.100056
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ − 6.00000i − 0.199778i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11.0000i 0.365249i 0.983183 + 0.182625i $$0.0584593\pi$$
−0.983183 + 0.182625i $$0.941541\pi$$
$$908$$ − 7.00000i − 0.232303i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 19.0000 0.629498 0.314749 0.949175i $$-0.398080\pi$$
0.314749 + 0.949175i $$0.398080\pi$$
$$912$$ 0 0
$$913$$ − 6.00000i − 0.198571i
$$914$$ −17.0000 −0.562310
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ − 8.00000i − 0.264183i
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 32.0000i − 1.05386i
$$923$$ − 4.00000i − 0.131662i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 0 0
$$928$$ 1.00000i 0.0328266i
$$929$$ 25.0000 0.820223 0.410112 0.912035i $$-0.365490\pi$$
0.410112 + 0.912035i $$0.365490\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 20.0000i 0.655122i
$$933$$ 0 0
$$934$$ 39.0000 1.27612
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 14.0000i 0.457360i 0.973502 + 0.228680i $$0.0734410\pi$$
−0.973502 + 0.228680i $$0.926559\pi$$
$$938$$ − 12.0000i − 0.391814i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ − 21.0000i − 0.683854i
$$944$$ −3.00000 −0.0976417
$$945$$ 0 0
$$946$$ −2.00000 −0.0650256
$$947$$ − 32.0000i − 1.03986i −0.854209 0.519930i $$-0.825958\pi$$
0.854209 0.519930i $$-0.174042\pi$$
$$948$$ 0 0
$$949$$ 14.0000 0.454459
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 1.00000i − 0.0324102i
$$953$$ 40.0000i 1.29573i 0.761756 + 0.647864i $$0.224337\pi$$
−0.761756 + 0.647864i $$0.775663\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 8.00000i 0.258468i
$$959$$ −4.00000 −0.129167
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ − 6.00000i − 0.193448i
$$963$$ 0 0
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 44.0000 1.41203 0.706014 0.708198i $$-0.250492\pi$$
0.706014 + 0.708198i $$0.250492\pi$$
$$972$$ 0 0
$$973$$ 4.00000i 0.128234i
$$974$$ 22.0000 0.704925
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 0 0
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 30.0000i − 0.957338i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 1.00000 0.0318465
$$987$$ 0 0
$$988$$ 4.00000i 0.127257i
$$989$$ −7.00000 −0.222587
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 3.00000i 0.0952501i
$$993$$ 0 0
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.00000i 0.0633406i 0.999498 + 0.0316703i $$0.0100827\pi$$
−0.999498 + 0.0316703i $$0.989917\pi$$
$$998$$ 27.0000i 0.854670i
$$999$$ 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.g.2899.2 2
3.2 odd 2 1050.2.g.j.799.1 2
5.2 odd 4 3150.2.a.c.1.1 1
5.3 odd 4 3150.2.a.bl.1.1 1
5.4 even 2 inner 3150.2.g.g.2899.1 2
15.2 even 4 1050.2.a.o.1.1 yes 1
15.8 even 4 1050.2.a.e.1.1 1
15.14 odd 2 1050.2.g.j.799.2 2
60.23 odd 4 8400.2.a.bt.1.1 1
60.47 odd 4 8400.2.a.t.1.1 1
105.62 odd 4 7350.2.a.ca.1.1 1
105.83 odd 4 7350.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.e.1.1 1 15.8 even 4
1050.2.a.o.1.1 yes 1 15.2 even 4
1050.2.g.j.799.1 2 3.2 odd 2
1050.2.g.j.799.2 2 15.14 odd 2
3150.2.a.c.1.1 1 5.2 odd 4
3150.2.a.bl.1.1 1 5.3 odd 4
3150.2.g.g.2899.1 2 5.4 even 2 inner
3150.2.g.g.2899.2 2 1.1 even 1 trivial
7350.2.a.bj.1.1 1 105.83 odd 4
7350.2.a.ca.1.1 1 105.62 odd 4
8400.2.a.t.1.1 1 60.47 odd 4
8400.2.a.bt.1.1 1 60.23 odd 4