# Properties

 Label 2450.2.c.v.99.3 Level $2450$ Weight $2$ Character 2450.99 Analytic conductor $19.563$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 98) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.3 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.99 Dual form 2450.2.c.v.99.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.41421i q^{3} -1.00000 q^{4} +1.41421 q^{6} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.41421i q^{3} -1.00000 q^{4} +1.41421 q^{6} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.41421i q^{12} +1.00000 q^{16} +1.41421i q^{17} +1.00000i q^{18} -7.07107 q^{19} -2.00000i q^{22} +4.00000i q^{23} -1.41421 q^{24} -5.65685i q^{27} -2.00000 q^{29} -8.48528 q^{31} +1.00000i q^{32} +2.82843i q^{33} -1.41421 q^{34} -1.00000 q^{36} +10.0000i q^{37} -7.07107i q^{38} +9.89949 q^{41} -2.00000i q^{43} +2.00000 q^{44} -4.00000 q^{46} -2.82843i q^{47} -1.41421i q^{48} +2.00000 q^{51} +2.00000i q^{53} +5.65685 q^{54} +10.0000i q^{57} -2.00000i q^{58} -1.41421 q^{59} -2.82843 q^{61} -8.48528i q^{62} -1.00000 q^{64} -2.82843 q^{66} +12.0000i q^{67} -1.41421i q^{68} +5.65685 q^{69} -12.0000 q^{71} -1.00000i q^{72} -1.41421i q^{73} -10.0000 q^{74} +7.07107 q^{76} +4.00000 q^{79} -5.00000 q^{81} +9.89949i q^{82} +9.89949i q^{83} +2.00000 q^{86} +2.82843i q^{87} +2.00000i q^{88} -7.07107 q^{89} -4.00000i q^{92} +12.0000i q^{93} +2.82843 q^{94} +1.41421 q^{96} -9.89949i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{9} - 8 q^{11} + 4 q^{16} - 8 q^{29} - 4 q^{36} + 8 q^{44} - 16 q^{46} + 8 q^{51} - 4 q^{64} - 48 q^{71} - 40 q^{74} + 16 q^{79} - 20 q^{81} + 8 q^{86} - 8 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.41421i − 0.816497i −0.912871 0.408248i $$-0.866140\pi$$
0.912871 0.408248i $$-0.133860\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.41421 0.577350
$$7$$ 0 0
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.41421i 0.408248i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.41421i 0.342997i 0.985184 + 0.171499i $$0.0548609\pi$$
−0.985184 + 0.171499i $$0.945139\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −7.07107 −1.62221 −0.811107 0.584898i $$-0.801135\pi$$
−0.811107 + 0.584898i $$0.801135\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 2.00000i − 0.426401i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −1.41421 −0.288675
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.65685i − 1.08866i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.48528 −1.52400 −0.762001 0.647576i $$-0.775783\pi$$
−0.762001 + 0.647576i $$0.775783\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 2.82843i 0.492366i
$$34$$ −1.41421 −0.242536
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 7.07107i − 1.14708i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.89949 1.54604 0.773021 0.634381i $$-0.218745\pi$$
0.773021 + 0.634381i $$0.218745\pi$$
$$42$$ 0 0
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ − 2.82843i − 0.412568i −0.978492 0.206284i $$-0.933863\pi$$
0.978492 0.206284i $$-0.0661372\pi$$
$$48$$ − 1.41421i − 0.204124i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 5.65685 0.769800
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 10.0000i 1.32453i
$$58$$ − 2.00000i − 0.262613i
$$59$$ −1.41421 −0.184115 −0.0920575 0.995754i $$-0.529344\pi$$
−0.0920575 + 0.995754i $$0.529344\pi$$
$$60$$ 0 0
$$61$$ −2.82843 −0.362143 −0.181071 0.983470i $$-0.557957\pi$$
−0.181071 + 0.983470i $$0.557957\pi$$
$$62$$ − 8.48528i − 1.07763i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.82843 −0.348155
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 1.41421i − 0.171499i
$$69$$ 5.65685 0.681005
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 1.41421i − 0.165521i −0.996569 0.0827606i $$-0.973626\pi$$
0.996569 0.0827606i $$-0.0263737\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 7.07107 0.811107
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 9.89949i 1.09322i
$$83$$ 9.89949i 1.08661i 0.839535 + 0.543305i $$0.182827\pi$$
−0.839535 + 0.543305i $$0.817173\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 2.82843i 0.303239i
$$88$$ 2.00000i 0.213201i
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ 12.0000i 1.24434i
$$94$$ 2.82843 0.291730
$$95$$ 0 0
$$96$$ 1.41421 0.144338
$$97$$ − 9.89949i − 1.00514i −0.864536 0.502571i $$-0.832388\pi$$
0.864536 0.502571i $$-0.167612\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −8.48528 −0.844317 −0.422159 0.906522i $$-0.638727\pi$$
−0.422159 + 0.906522i $$0.638727\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ 2.82843i 0.278693i 0.990244 + 0.139347i $$0.0445002\pi$$
−0.990244 + 0.139347i $$0.955500\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 5.65685i 0.544331i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 14.1421 1.34231
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ −10.0000 −0.936586
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ − 1.41421i − 0.130189i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 2.82843i − 0.256074i
$$123$$ − 14.0000i − 1.26234i
$$124$$ 8.48528 0.762001
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −2.82843 −0.249029
$$130$$ 0 0
$$131$$ −12.7279 −1.11204 −0.556022 0.831168i $$-0.687673\pi$$
−0.556022 + 0.831168i $$0.687673\pi$$
$$132$$ − 2.82843i − 0.246183i
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 1.41421 0.121268
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 5.65685i 0.481543i
$$139$$ 9.89949 0.839664 0.419832 0.907602i $$-0.362089\pi$$
0.419832 + 0.907602i $$0.362089\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ − 12.0000i − 1.00702i
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 1.41421 0.117041
$$147$$ 0 0
$$148$$ − 10.0000i − 0.821995i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 7.07107i 0.573539i
$$153$$ 1.41421i 0.114332i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.3137i 0.902932i 0.892288 + 0.451466i $$0.149099\pi$$
−0.892288 + 0.451466i $$0.850901\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ 2.82843 0.224309
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 5.00000i − 0.392837i
$$163$$ − 10.0000i − 0.783260i −0.920123 0.391630i $$-0.871911\pi$$
0.920123 0.391630i $$-0.128089\pi$$
$$164$$ −9.89949 −0.773021
$$165$$ 0 0
$$166$$ −9.89949 −0.768350
$$167$$ 19.7990i 1.53209i 0.642786 + 0.766046i $$0.277779\pi$$
−0.642786 + 0.766046i $$0.722221\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −7.07107 −0.540738
$$172$$ 2.00000i 0.152499i
$$173$$ − 16.9706i − 1.29025i −0.764078 0.645124i $$-0.776806\pi$$
0.764078 0.645124i $$-0.223194\pi$$
$$174$$ −2.82843 −0.214423
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 2.00000i 0.150329i
$$178$$ − 7.07107i − 0.529999i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 4.00000i 0.295689i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −12.0000 −0.879883
$$187$$ − 2.82843i − 0.206835i
$$188$$ 2.82843i 0.206284i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 1.41421i 0.102062i
$$193$$ 16.0000i 1.15171i 0.817554 + 0.575853i $$0.195330\pi$$
−0.817554 + 0.575853i $$0.804670\pi$$
$$194$$ 9.89949 0.710742
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ − 2.00000i − 0.142134i
$$199$$ 8.48528 0.601506 0.300753 0.953702i $$-0.402762\pi$$
0.300753 + 0.953702i $$0.402762\pi$$
$$200$$ 0 0
$$201$$ 16.9706 1.19701
$$202$$ − 8.48528i − 0.597022i
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −2.82843 −0.197066
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ 14.1421 0.978232
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ 16.9706i 1.16280i
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ −5.65685 −0.384900
$$217$$ 0 0
$$218$$ 2.00000i 0.135457i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 14.1421i 0.949158i
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ 21.2132i 1.40797i 0.710215 + 0.703985i $$0.248598\pi$$
−0.710215 + 0.703985i $$0.751402\pi$$
$$228$$ − 10.0000i − 0.662266i
$$229$$ −16.9706 −1.12145 −0.560723 0.828003i $$-0.689477\pi$$
−0.560723 + 0.828003i $$0.689477\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ − 24.0000i − 1.57229i −0.618041 0.786146i $$-0.712073\pi$$
0.618041 0.786146i $$-0.287927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1.41421 0.0920575
$$237$$ − 5.65685i − 0.367452i
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 21.2132 1.36646 0.683231 0.730202i $$-0.260574\pi$$
0.683231 + 0.730202i $$0.260574\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ − 9.89949i − 0.635053i
$$244$$ 2.82843 0.181071
$$245$$ 0 0
$$246$$ 14.0000 0.892607
$$247$$ 0 0
$$248$$ 8.48528i 0.538816i
$$249$$ 14.0000 0.887214
$$250$$ 0 0
$$251$$ −9.89949 −0.624851 −0.312425 0.949942i $$-0.601141\pi$$
−0.312425 + 0.949942i $$0.601141\pi$$
$$252$$ 0 0
$$253$$ − 8.00000i − 0.502956i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 12.7279i − 0.793946i −0.917830 0.396973i $$-0.870061\pi$$
0.917830 0.396973i $$-0.129939\pi$$
$$258$$ − 2.82843i − 0.176090i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ − 12.7279i − 0.786334i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 2.82843 0.174078
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000i 0.611990i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −11.3137 −0.689809 −0.344904 0.938638i $$-0.612089\pi$$
−0.344904 + 0.938638i $$0.612089\pi$$
$$270$$ 0 0
$$271$$ −22.6274 −1.37452 −0.687259 0.726413i $$-0.741186\pi$$
−0.687259 + 0.726413i $$0.741186\pi$$
$$272$$ 1.41421i 0.0857493i
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ −5.65685 −0.340503
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 9.89949i 0.593732i
$$279$$ −8.48528 −0.508001
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ − 1.41421i − 0.0840663i −0.999116 0.0420331i $$-0.986616\pi$$
0.999116 0.0420331i $$-0.0133835\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 15.0000 0.882353
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 1.41421i 0.0827606i
$$293$$ − 19.7990i − 1.15667i −0.815800 0.578335i $$-0.803703\pi$$
0.815800 0.578335i $$-0.196297\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ 11.3137i 0.656488i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 16.0000i − 0.920697i
$$303$$ 12.0000i 0.689382i
$$304$$ −7.07107 −0.405554
$$305$$ 0 0
$$306$$ −1.41421 −0.0808452
$$307$$ 9.89949i 0.564994i 0.959268 + 0.282497i $$0.0911627\pi$$
−0.959268 + 0.282497i $$0.908837\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 11.3137 0.641542 0.320771 0.947157i $$-0.396058\pi$$
0.320771 + 0.947157i $$0.396058\pi$$
$$312$$ 0 0
$$313$$ 12.7279i 0.719425i 0.933063 + 0.359712i $$0.117125\pi$$
−0.933063 + 0.359712i $$0.882875\pi$$
$$314$$ −11.3137 −0.638470
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 10.0000i 0.561656i 0.959758 + 0.280828i $$0.0906090\pi$$
−0.959758 + 0.280828i $$0.909391\pi$$
$$318$$ 2.82843i 0.158610i
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ −5.65685 −0.315735
$$322$$ 0 0
$$323$$ − 10.0000i − 0.556415i
$$324$$ 5.00000 0.277778
$$325$$ 0 0
$$326$$ 10.0000 0.553849
$$327$$ − 2.82843i − 0.156412i
$$328$$ − 9.89949i − 0.546608i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ − 9.89949i − 0.543305i
$$333$$ 10.0000i 0.547997i
$$334$$ −19.7990 −1.08335
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 13.0000i 0.707107i
$$339$$ 16.9706 0.921714
$$340$$ 0 0
$$341$$ 16.9706 0.919007
$$342$$ − 7.07107i − 0.382360i
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 16.9706 0.912343
$$347$$ − 30.0000i − 1.61048i −0.592946 0.805242i $$-0.702035\pi$$
0.592946 0.805242i $$-0.297965\pi$$
$$348$$ − 2.82843i − 0.151620i
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 1.41421i − 0.0752710i −0.999292 0.0376355i $$-0.988017\pi$$
0.999292 0.0376355i $$-0.0119826\pi$$
$$354$$ −2.00000 −0.106299
$$355$$ 0 0
$$356$$ 7.07107 0.374766
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 31.0000 1.63158
$$362$$ 0 0
$$363$$ 9.89949i 0.519589i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −4.00000 −0.209083
$$367$$ − 28.2843i − 1.47643i −0.674567 0.738213i $$-0.735670\pi$$
0.674567 0.738213i $$-0.264330\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 9.89949 0.515347
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 12.0000i − 0.622171i
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ 2.82843 0.146254
$$375$$ 0 0
$$376$$ −2.82843 −0.145865
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 26.0000 1.33553 0.667765 0.744372i $$-0.267251\pi$$
0.667765 + 0.744372i $$0.267251\pi$$
$$380$$ 0 0
$$381$$ 22.6274 1.15924
$$382$$ − 4.00000i − 0.204658i
$$383$$ − 36.7696i − 1.87884i −0.342773 0.939418i $$-0.611366\pi$$
0.342773 0.939418i $$-0.388634\pi$$
$$384$$ −1.41421 −0.0721688
$$385$$ 0 0
$$386$$ −16.0000 −0.814379
$$387$$ − 2.00000i − 0.101666i
$$388$$ 9.89949i 0.502571i
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −5.65685 −0.286079
$$392$$ 0 0
$$393$$ 18.0000i 0.907980i
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 22.6274i − 1.13564i −0.823154 0.567819i $$-0.807787\pi$$
0.823154 0.567819i $$-0.192213\pi$$
$$398$$ 8.48528i 0.425329i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 16.9706i 0.846415i
$$403$$ 0 0
$$404$$ 8.48528 0.422159
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.0000i − 0.991363i
$$408$$ − 2.00000i − 0.0990148i
$$409$$ 38.1838 1.88807 0.944033 0.329851i $$-0.106999\pi$$
0.944033 + 0.329851i $$0.106999\pi$$
$$410$$ 0 0
$$411$$ 16.9706 0.837096
$$412$$ − 2.82843i − 0.139347i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 14.0000i − 0.685583i
$$418$$ 14.1421i 0.691714i
$$419$$ −9.89949 −0.483622 −0.241811 0.970323i $$-0.577741\pi$$
−0.241811 + 0.970323i $$0.577741\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ − 2.82843i − 0.137523i
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ −16.9706 −0.822226
$$427$$ 0 0
$$428$$ 4.00000i 0.193347i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ − 5.65685i − 0.272166i
$$433$$ − 29.6985i − 1.42722i −0.700544 0.713609i $$-0.747059\pi$$
0.700544 0.713609i $$-0.252941\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ − 28.2843i − 1.35302i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ −16.9706 −0.809961 −0.404980 0.914325i $$-0.632722\pi$$
−0.404980 + 0.914325i $$0.632722\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ −14.1421 −0.671156
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.1421i 0.668900i
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ −19.7990 −0.932298
$$452$$ − 12.0000i − 0.564433i
$$453$$ 22.6274i 1.06313i
$$454$$ −21.2132 −0.995585
$$455$$ 0 0
$$456$$ 10.0000 0.468293
$$457$$ 24.0000i 1.12267i 0.827588 + 0.561336i $$0.189713\pi$$
−0.827588 + 0.561336i $$0.810287\pi$$
$$458$$ − 16.9706i − 0.792982i
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −39.5980 −1.84426 −0.922131 0.386878i $$-0.873553\pi$$
−0.922131 + 0.386878i $$0.873553\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ − 32.5269i − 1.50517i −0.658497 0.752583i $$-0.728808\pi$$
0.658497 0.752583i $$-0.271192\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 16.0000 0.737241
$$472$$ 1.41421i 0.0650945i
$$473$$ 4.00000i 0.183920i
$$474$$ 5.65685 0.259828
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ 12.0000i 0.548867i
$$479$$ −31.1127 −1.42158 −0.710788 0.703407i $$-0.751661\pi$$
−0.710788 + 0.703407i $$0.751661\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 21.2132i 0.966235i
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 9.89949 0.449050
$$487$$ 12.0000i 0.543772i 0.962329 + 0.271886i $$0.0876473\pi$$
−0.962329 + 0.271886i $$0.912353\pi$$
$$488$$ 2.82843i 0.128037i
$$489$$ −14.1421 −0.639529
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 14.0000i 0.631169i
$$493$$ − 2.82843i − 0.127386i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.48528 −0.381000
$$497$$ 0 0
$$498$$ 14.0000i 0.627355i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 28.0000 1.25095
$$502$$ − 9.89949i − 0.441836i
$$503$$ 39.5980i 1.76559i 0.469762 + 0.882793i $$0.344340\pi$$
−0.469762 + 0.882793i $$0.655660\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ − 18.3848i − 0.816497i
$$508$$ − 16.0000i − 0.709885i
$$509$$ 22.6274 1.00294 0.501471 0.865174i $$-0.332792\pi$$
0.501471 + 0.865174i $$0.332792\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 40.0000i 1.76604i
$$514$$ 12.7279 0.561405
$$515$$ 0 0
$$516$$ 2.82843 0.124515
$$517$$ 5.65685i 0.248788i
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ 1.41421 0.0619578 0.0309789 0.999520i $$-0.490138\pi$$
0.0309789 + 0.999520i $$0.490138\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 12.7279i 0.556553i 0.960501 + 0.278277i $$0.0897632\pi$$
−0.960501 + 0.278277i $$0.910237\pi$$
$$524$$ 12.7279 0.556022
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ − 12.0000i − 0.522728i
$$528$$ 2.82843i 0.123091i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −1.41421 −0.0613716
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 16.9706i 0.732334i
$$538$$ − 11.3137i − 0.487769i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ − 22.6274i − 0.971931i
$$543$$ 0 0
$$544$$ −1.41421 −0.0606339
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 26.0000i − 1.11168i −0.831289 0.555840i $$-0.812397\pi$$
0.831289 0.555840i $$-0.187603\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ −2.82843 −0.120714
$$550$$ 0 0
$$551$$ 14.1421 0.602475
$$552$$ − 5.65685i − 0.240772i
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −9.89949 −0.419832
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ − 8.48528i − 0.359211i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 16.0000i 0.674919i
$$563$$ − 1.41421i − 0.0596020i −0.999556 0.0298010i $$-0.990513\pi$$
0.999556 0.0298010i $$-0.00948736\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ 1.41421 0.0594438
$$567$$ 0 0
$$568$$ 12.0000i 0.503509i
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −2.00000 −0.0836974 −0.0418487 0.999124i $$-0.513325\pi$$
−0.0418487 + 0.999124i $$0.513325\pi$$
$$572$$ 0 0
$$573$$ 5.65685i 0.236318i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ 21.2132i 0.883117i 0.897232 + 0.441559i $$0.145574\pi$$
−0.897232 + 0.441559i $$0.854426\pi$$
$$578$$ 15.0000i 0.623918i
$$579$$ 22.6274 0.940363
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 14.0000i − 0.580319i
$$583$$ − 4.00000i − 0.165663i
$$584$$ −1.41421 −0.0585206
$$585$$ 0 0
$$586$$ 19.7990 0.817889
$$587$$ 29.6985i 1.22579i 0.790165 + 0.612894i $$0.209995\pi$$
−0.790165 + 0.612894i $$0.790005\pi$$
$$588$$ 0 0
$$589$$ 60.0000 2.47226
$$590$$ 0 0
$$591$$ 2.82843 0.116346
$$592$$ 10.0000i 0.410997i
$$593$$ − 7.07107i − 0.290374i −0.989404 0.145187i $$-0.953622\pi$$
0.989404 0.145187i $$-0.0463784\pi$$
$$594$$ −11.3137 −0.464207
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ − 12.0000i − 0.491127i
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −29.6985 −1.21143 −0.605713 0.795683i $$-0.707112\pi$$
−0.605713 + 0.795683i $$0.707112\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ −12.0000 −0.487467
$$607$$ 16.9706i 0.688814i 0.938820 + 0.344407i $$0.111920\pi$$
−0.938820 + 0.344407i $$0.888080\pi$$
$$608$$ − 7.07107i − 0.286770i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 1.41421i − 0.0571662i
$$613$$ 30.0000i 1.21169i 0.795583 + 0.605844i $$0.207165\pi$$
−0.795583 + 0.605844i $$0.792835\pi$$
$$614$$ −9.89949 −0.399511
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 26.0000i − 1.04672i −0.852111 0.523360i $$-0.824678\pi$$
0.852111 0.523360i $$-0.175322\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 18.3848 0.738947 0.369473 0.929241i $$-0.379538\pi$$
0.369473 + 0.929241i $$0.379538\pi$$
$$620$$ 0 0
$$621$$ 22.6274 0.908007
$$622$$ 11.3137i 0.453638i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −12.7279 −0.508710
$$627$$ − 20.0000i − 0.798723i
$$628$$ − 11.3137i − 0.451466i
$$629$$ −14.1421 −0.563884
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ 16.9706i 0.674519i
$$634$$ −10.0000 −0.397151
$$635$$ 0 0
$$636$$ −2.82843 −0.112154
$$637$$ 0 0
$$638$$ 4.00000i 0.158362i
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ − 5.65685i − 0.223258i
$$643$$ 9.89949i 0.390398i 0.980764 + 0.195199i $$0.0625353\pi$$
−0.980764 + 0.195199i $$0.937465\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 10.0000 0.393445
$$647$$ − 8.48528i − 0.333591i −0.985992 0.166795i $$-0.946658\pi$$
0.985992 0.166795i $$-0.0533419\pi$$
$$648$$ 5.00000i 0.196419i
$$649$$ 2.82843 0.111025
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 10.0000i 0.391630i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 2.82843 0.110600
$$655$$ 0 0
$$656$$ 9.89949 0.386510
$$657$$ − 1.41421i − 0.0551737i
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ −8.48528 −0.330039 −0.165020 0.986290i $$-0.552769\pi$$
−0.165020 + 0.986290i $$0.552769\pi$$
$$662$$ 10.0000i 0.388661i
$$663$$ 0 0
$$664$$ 9.89949 0.384175
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ − 8.00000i − 0.309761i
$$668$$ − 19.7990i − 0.766046i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 5.65685 0.218380
$$672$$ 0 0
$$673$$ 12.0000i 0.462566i 0.972887 + 0.231283i $$0.0742923\pi$$
−0.972887 + 0.231283i $$0.925708\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 16.9706i 0.652232i 0.945330 + 0.326116i $$0.105740\pi$$
−0.945330 + 0.326116i $$0.894260\pi$$
$$678$$ 16.9706i 0.651751i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 30.0000 1.14960
$$682$$ 16.9706i 0.649836i
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 7.07107 0.270369
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 24.0000i 0.915657i
$$688$$ − 2.00000i − 0.0762493i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −12.7279 −0.484193 −0.242096 0.970252i $$-0.577835\pi$$
−0.242096 + 0.970252i $$0.577835\pi$$
$$692$$ 16.9706i 0.645124i
$$693$$ 0 0
$$694$$ 30.0000 1.13878
$$695$$ 0 0
$$696$$ 2.82843 0.107211
$$697$$ 14.0000i 0.530288i
$$698$$ 0 0
$$699$$ −33.9411 −1.28377
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ − 70.7107i − 2.66690i
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 1.41421 0.0532246
$$707$$ 0 0
$$708$$ − 2.00000i − 0.0751646i
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 7.07107i 0.264999i
$$713$$ − 33.9411i − 1.27111i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ − 16.9706i − 0.633777i
$$718$$ 32.0000i 1.19423i
$$719$$ 2.82843 0.105483 0.0527413 0.998608i $$-0.483204\pi$$
0.0527413 + 0.998608i $$0.483204\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 31.0000i 1.15370i
$$723$$ − 30.0000i − 1.11571i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ −9.89949 −0.367405
$$727$$ 19.7990i 0.734304i 0.930161 + 0.367152i $$0.119667\pi$$
−0.930161 + 0.367152i $$0.880333\pi$$
$$728$$ 0 0
$$729$$ −29.0000 −1.07407
$$730$$ 0 0
$$731$$ 2.82843 0.104613
$$732$$ − 4.00000i − 0.147844i
$$733$$ 42.4264i 1.56706i 0.621357 + 0.783528i $$0.286582\pi$$
−0.621357 + 0.783528i $$0.713418\pi$$
$$734$$ 28.2843 1.04399
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ − 24.0000i − 0.884051i
$$738$$ 9.89949i 0.364405i
$$739$$ 30.0000 1.10357 0.551784 0.833987i $$-0.313947\pi$$
0.551784 + 0.833987i $$0.313947\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 16.0000i − 0.586983i −0.955962 0.293492i $$-0.905183\pi$$
0.955962 0.293492i $$-0.0948173\pi$$
$$744$$ 12.0000 0.439941
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 9.89949i 0.362204i
$$748$$ 2.82843i 0.103418i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ − 2.82843i − 0.103142i
$$753$$ 14.0000i 0.510188i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 26.0000i 0.944363i
$$759$$ −11.3137 −0.410662
$$760$$ 0 0
$$761$$ 7.07107 0.256326 0.128163 0.991753i $$-0.459092\pi$$
0.128163 + 0.991753i $$0.459092\pi$$
$$762$$ 22.6274i 0.819705i
$$763$$ 0 0
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ 36.7696 1.32854
$$767$$ 0 0
$$768$$ − 1.41421i − 0.0510310i
$$769$$ 29.6985 1.07095 0.535477 0.844550i $$-0.320132\pi$$
0.535477 + 0.844550i $$0.320132\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 16.0000i − 0.575853i
$$773$$ 48.0833i 1.72943i 0.502259 + 0.864717i $$0.332502\pi$$
−0.502259 + 0.864717i $$0.667498\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ −9.89949 −0.355371
$$777$$ 0 0
$$778$$ − 26.0000i − 0.932145i
$$779$$ −70.0000 −2.50801
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ − 5.65685i − 0.202289i
$$783$$ 11.3137i 0.404319i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 1.41421i 0.0504113i 0.999682 + 0.0252056i $$0.00802405\pi$$
−0.999682 + 0.0252056i $$0.991976\pi$$
$$788$$ − 2.00000i − 0.0712470i
$$789$$ −16.9706 −0.604168
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 2.00000i 0.0710669i
$$793$$ 0 0
$$794$$ 22.6274 0.803017
$$795$$ 0 0
$$796$$ −8.48528 −0.300753
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −7.07107 −0.249844
$$802$$ − 18.0000i − 0.635602i
$$803$$ 2.82843i 0.0998130i
$$804$$ −16.9706 −0.598506
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 16.0000i 0.563227i
$$808$$ 8.48528i 0.298511i
$$809$$ 16.0000 0.562530 0.281265 0.959630i $$-0.409246\pi$$
0.281265 + 0.959630i $$0.409246\pi$$
$$810$$ 0 0
$$811$$ 29.6985 1.04285 0.521427 0.853296i $$-0.325400\pi$$
0.521427 + 0.853296i $$0.325400\pi$$
$$812$$ 0 0
$$813$$ 32.0000i 1.12229i
$$814$$ 20.0000 0.701000
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 14.1421i 0.494771i
$$818$$ 38.1838i 1.33506i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 16.9706i 0.591916i
$$823$$ − 40.0000i − 1.39431i −0.716919 0.697156i $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ 2.82843 0.0985329
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −31.1127 −1.08059 −0.540294 0.841476i $$-0.681687\pi$$
−0.540294 + 0.841476i $$0.681687\pi$$
$$830$$ 0 0
$$831$$ −2.82843 −0.0981170
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 14.0000 0.484780
$$835$$ 0 0
$$836$$ −14.1421 −0.489116
$$837$$ 48.0000i 1.65912i
$$838$$ − 9.89949i − 0.341972i
$$839$$ 19.7990 0.683537 0.341769 0.939784i $$-0.388974\pi$$
0.341769 + 0.939784i $$0.388974\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 30.0000i 1.03387i
$$843$$ − 22.6274i − 0.779330i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 2.82843 0.0972433
$$847$$ 0 0
$$848$$ 2.00000i 0.0686803i
$$849$$ −2.00000 −0.0686398
$$850$$ 0 0
$$851$$ −40.0000 −1.37118
$$852$$ − 16.9706i − 0.581402i
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 18.3848i − 0.628012i −0.949421 0.314006i $$-0.898329\pi$$
0.949421 0.314006i $$-0.101671\pi$$
$$858$$ 0 0
$$859$$ −26.8701 −0.916795 −0.458397 0.888747i $$-0.651576\pi$$
−0.458397 + 0.888747i $$0.651576\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 12.0000i 0.408722i
$$863$$ 4.00000i 0.136162i 0.997680 + 0.0680808i $$0.0216876\pi$$
−0.997680 + 0.0680808i $$0.978312\pi$$
$$864$$ 5.65685 0.192450
$$865$$ 0 0
$$866$$ 29.6985 1.00920
$$867$$ − 21.2132i − 0.720438i
$$868$$ 0 0
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 2.00000i − 0.0677285i
$$873$$ − 9.89949i − 0.335047i
$$874$$ 28.2843 0.956730
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ − 16.9706i − 0.572729i
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −29.6985 −1.00057 −0.500284 0.865862i $$-0.666771\pi$$
−0.500284 + 0.865862i $$0.666771\pi$$
$$882$$ 0 0
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 36.7696i 1.23460i 0.786728 + 0.617300i $$0.211774\pi$$
−0.786728 + 0.617300i $$0.788226\pi$$
$$888$$ − 14.1421i − 0.474579i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 10.0000 0.335013
$$892$$ 0 0
$$893$$ 20.0000i 0.669274i
$$894$$ −14.1421 −0.472984
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 30.0000i − 1.00111i
$$899$$ 16.9706 0.566000
$$900$$ 0 0
$$901$$ −2.82843 −0.0942286
$$902$$ − 19.7990i − 0.659234i
$$903$$ 0 0
$$904$$ 12.0000 0.399114
$$905$$ 0 0
$$906$$ −22.6274 −0.751746
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ − 21.2132i − 0.703985i
$$909$$ −8.48528 −0.281439
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 10.0000i 0.331133i
$$913$$ − 19.7990i − 0.655251i
$$914$$ −24.0000 −0.793849
$$915$$ 0 0
$$916$$ 16.9706 0.560723
$$917$$ 0 0
$$918$$ 8.00000i 0.264039i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 14.0000 0.461316
$$922$$ − 39.5980i − 1.30409i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.0000 0.525793
$$927$$ 2.82843i 0.0928977i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 32.5269 1.06717 0.533587 0.845745i $$-0.320844\pi$$
0.533587 + 0.845745i $$0.320844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 24.0000i 0.786146i
$$933$$ − 16.0000i − 0.523816i
$$934$$ 32.5269 1.06431
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 9.89949i − 0.323402i −0.986840 0.161701i $$-0.948302\pi$$
0.986840 0.161701i $$-0.0516981\pi$$
$$938$$ 0 0
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ 31.1127 1.01424 0.507122 0.861874i $$-0.330709\pi$$
0.507122 + 0.861874i $$0.330709\pi$$
$$942$$ 16.0000i 0.521308i
$$943$$ 39.5980i 1.28949i
$$944$$ −1.41421 −0.0460287
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ − 18.0000i − 0.584921i −0.956278 0.292461i $$-0.905526\pi$$
0.956278 0.292461i $$-0.0944741\pi$$
$$948$$ 5.65685i 0.183726i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 14.1421 0.458590
$$952$$ 0 0
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ − 5.65685i − 0.182860i
$$958$$ − 31.1127i − 1.00521i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ − 4.00000i − 0.128898i
$$964$$ −21.2132 −0.683231
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 12.0000i − 0.385894i −0.981209 0.192947i $$-0.938195\pi$$
0.981209 0.192947i $$-0.0618045\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ −14.1421 −0.454311
$$970$$ 0 0
$$971$$ −32.5269 −1.04384 −0.521919 0.852995i $$-0.674784\pi$$
−0.521919 + 0.852995i $$0.674784\pi$$
$$972$$ 9.89949i 0.317526i
$$973$$ 0 0
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −2.82843 −0.0905357
$$977$$ 12.0000i 0.383914i 0.981403 + 0.191957i $$0.0614834\pi$$
−0.981403 + 0.191957i $$0.938517\pi$$
$$978$$ − 14.1421i − 0.452216i
$$979$$ 14.1421 0.451985
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ − 12.0000i − 0.382935i
$$983$$ 48.0833i 1.53362i 0.641875 + 0.766809i $$0.278157\pi$$
−0.641875 + 0.766809i $$0.721843\pi$$
$$984$$ −14.0000 −0.446304
$$985$$ 0 0
$$986$$ 2.82843 0.0900755
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ − 8.48528i − 0.269408i
$$993$$ − 14.1421i − 0.448787i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ 31.1127i 0.985349i 0.870214 + 0.492675i $$0.163981\pi$$
−0.870214 + 0.492675i $$0.836019\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 56.5685 1.78975
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 2450.2.c.v.99.3 4
5.2 odd 4 2450.2.a.bj.1.1 2
5.3 odd 4 98.2.a.b.1.2 yes 2
5.4 even 2 inner 2450.2.c.v.99.2 4
7.6 odd 2 inner 2450.2.c.v.99.4 4
15.8 even 4 882.2.a.n.1.2 2
20.3 even 4 784.2.a.l.1.1 2
35.3 even 12 98.2.c.c.79.2 4
35.13 even 4 98.2.a.b.1.1 2
35.18 odd 12 98.2.c.c.79.1 4
35.23 odd 12 98.2.c.c.67.1 4
35.27 even 4 2450.2.a.bj.1.2 2
35.33 even 12 98.2.c.c.67.2 4
35.34 odd 2 inner 2450.2.c.v.99.1 4
40.3 even 4 3136.2.a.bm.1.2 2
40.13 odd 4 3136.2.a.bn.1.1 2
60.23 odd 4 7056.2.a.cl.1.2 2
105.23 even 12 882.2.g.l.361.1 4
105.38 odd 12 882.2.g.l.667.2 4
105.53 even 12 882.2.g.l.667.1 4
105.68 odd 12 882.2.g.l.361.2 4
105.83 odd 4 882.2.a.n.1.1 2
140.3 odd 12 784.2.i.m.177.1 4
140.23 even 12 784.2.i.m.753.2 4
140.83 odd 4 784.2.a.l.1.2 2
140.103 odd 12 784.2.i.m.753.1 4
140.123 even 12 784.2.i.m.177.2 4
280.13 even 4 3136.2.a.bn.1.2 2
280.83 odd 4 3136.2.a.bm.1.1 2
420.83 even 4 7056.2.a.cl.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 35.13 even 4
98.2.a.b.1.2 yes 2 5.3 odd 4
98.2.c.c.67.1 4 35.23 odd 12
98.2.c.c.67.2 4 35.33 even 12
98.2.c.c.79.1 4 35.18 odd 12
98.2.c.c.79.2 4 35.3 even 12
784.2.a.l.1.1 2 20.3 even 4
784.2.a.l.1.2 2 140.83 odd 4
784.2.i.m.177.1 4 140.3 odd 12
784.2.i.m.177.2 4 140.123 even 12
784.2.i.m.753.1 4 140.103 odd 12
784.2.i.m.753.2 4 140.23 even 12
882.2.a.n.1.1 2 105.83 odd 4
882.2.a.n.1.2 2 15.8 even 4
882.2.g.l.361.1 4 105.23 even 12
882.2.g.l.361.2 4 105.68 odd 12
882.2.g.l.667.1 4 105.53 even 12
882.2.g.l.667.2 4 105.38 odd 12
2450.2.a.bj.1.1 2 5.2 odd 4
2450.2.a.bj.1.2 2 35.27 even 4
2450.2.c.v.99.1 4 35.34 odd 2 inner
2450.2.c.v.99.2 4 5.4 even 2 inner
2450.2.c.v.99.3 4 1.1 even 1 trivial
2450.2.c.v.99.4 4 7.6 odd 2 inner
3136.2.a.bm.1.1 2 280.83 odd 4
3136.2.a.bm.1.2 2 40.3 even 4
3136.2.a.bn.1.1 2 40.13 odd 4
3136.2.a.bn.1.2 2 280.13 even 4
7056.2.a.cl.1.1 2 420.83 even 4
7056.2.a.cl.1.2 2 60.23 odd 4