# Properties

 Label 2475.2.f.d.2276.2 Level $2475$ Weight $2$ Character 2475.2276 Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2276.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.2276 Dual form 2475.2.f.d.2276.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} -3.00000 q^{8} +(3.00000 + 1.41421i) q^{11} -2.82843i q^{13} +1.41421i q^{14} -1.00000 q^{16} +2.00000 q^{17} +4.24264i q^{19} +(3.00000 + 1.41421i) q^{22} -2.82843i q^{26} -1.41421i q^{28} -6.00000 q^{29} +5.00000 q^{32} +2.00000 q^{34} -6.00000 q^{37} +4.24264i q^{38} -6.00000 q^{41} +1.41421i q^{43} +(-3.00000 - 1.41421i) q^{44} +8.48528i q^{47} +5.00000 q^{49} +2.82843i q^{52} +4.24264i q^{53} -4.24264i q^{56} -6.00000 q^{58} +8.48528i q^{59} +8.48528i q^{61} +7.00000 q^{64} -2.00000 q^{68} +8.48528i q^{71} +11.3137i q^{73} -6.00000 q^{74} -4.24264i q^{76} +(-2.00000 + 4.24264i) q^{77} +12.7279i q^{79} -6.00000 q^{82} +14.0000 q^{83} +1.41421i q^{86} +(-9.00000 - 4.24264i) q^{88} -7.07107i q^{89} +4.00000 q^{91} +8.48528i q^{94} -12.0000 q^{97} +5.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{11} - 2 q^{16} + 4 q^{17} + 6 q^{22} - 12 q^{29} + 10 q^{32} + 4 q^{34} - 12 q^{37} - 12 q^{41} - 6 q^{44} + 10 q^{49} - 12 q^{58} + 14 q^{64} - 4 q^{68} - 12 q^{74} - 4 q^{77} - 12 q^{82} + 28 q^{83} - 18 q^{88} + 8 q^{91} - 24 q^{97} + 10 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.41421i 0.534522i 0.963624 + 0.267261i $$0.0861187\pi$$
−0.963624 + 0.267261i $$0.913881\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 + 1.41421i 0.904534 + 0.426401i
$$12$$ 0 0
$$13$$ 2.82843i 0.784465i −0.919866 0.392232i $$-0.871703\pi$$
0.919866 0.392232i $$-0.128297\pi$$
$$14$$ 1.41421i 0.377964i
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 4.24264i 0.973329i 0.873589 + 0.486664i $$0.161786\pi$$
−0.873589 + 0.486664i $$0.838214\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000 + 1.41421i 0.639602 + 0.301511i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.82843i 0.554700i
$$27$$ 0 0
$$28$$ 1.41421i 0.267261i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 4.24264i 0.688247i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 1.41421i 0.215666i 0.994169 + 0.107833i $$0.0343911\pi$$
−0.994169 + 0.107833i $$0.965609\pi$$
$$44$$ −3.00000 1.41421i −0.452267 0.213201i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.48528i 1.23771i 0.785507 + 0.618853i $$0.212402\pi$$
−0.785507 + 0.618853i $$0.787598\pi$$
$$48$$ 0 0
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.82843i 0.392232i
$$53$$ 4.24264i 0.582772i 0.956606 + 0.291386i $$0.0941163\pi$$
−0.956606 + 0.291386i $$0.905884\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.24264i 0.566947i
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 8.48528i 1.10469i 0.833616 + 0.552345i $$0.186267\pi$$
−0.833616 + 0.552345i $$0.813733\pi$$
$$60$$ 0 0
$$61$$ 8.48528i 1.08643i 0.839594 + 0.543214i $$0.182793\pi$$
−0.839594 + 0.543214i $$0.817207\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.48528i 1.00702i 0.863990 + 0.503509i $$0.167958\pi$$
−0.863990 + 0.503509i $$0.832042\pi$$
$$72$$ 0 0
$$73$$ 11.3137i 1.32417i 0.749429 + 0.662085i $$0.230328\pi$$
−0.749429 + 0.662085i $$0.769672\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 4.24264i 0.486664i
$$77$$ −2.00000 + 4.24264i −0.227921 + 0.483494i
$$78$$ 0 0
$$79$$ 12.7279i 1.43200i 0.698099 + 0.716002i $$0.254030\pi$$
−0.698099 + 0.716002i $$0.745970\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −6.00000 −0.662589
$$83$$ 14.0000 1.53670 0.768350 0.640030i $$-0.221078\pi$$
0.768350 + 0.640030i $$0.221078\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.41421i 0.152499i
$$87$$ 0 0
$$88$$ −9.00000 4.24264i −0.959403 0.452267i
$$89$$ 7.07107i 0.749532i −0.927119 0.374766i $$-0.877723\pi$$
0.927119 0.374766i $$-0.122277\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.48528i 0.875190i
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 5.00000 0.505076
$$99$$ 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
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 8.48528i 0.832050i
$$105$$ 0 0
$$106$$ 4.24264i 0.412082i
$$107$$ −14.0000 −1.35343 −0.676716 0.736245i $$-0.736597\pi$$
−0.676716 + 0.736245i $$0.736597\pi$$
$$108$$ 0 0
$$109$$ 16.9706i 1.62549i 0.582623 + 0.812743i $$0.302026\pi$$
−0.582623 + 0.812743i $$0.697974\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.41421i 0.133631i
$$113$$ 12.7279i 1.19734i −0.800995 0.598671i $$-0.795696\pi$$
0.800995 0.598671i $$-0.204304\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 8.48528i 0.781133i
$$119$$ 2.82843i 0.259281i
$$120$$ 0 0
$$121$$ 7.00000 + 8.48528i 0.636364 + 0.771389i
$$122$$ 8.48528i 0.768221i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 18.3848i 1.63139i −0.578486 0.815693i $$-0.696356\pi$$
0.578486 0.815693i $$-0.303644\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 4.24264i 0.362473i −0.983440 0.181237i $$-0.941990\pi$$
0.983440 0.181237i $$-0.0580100\pi$$
$$138$$ 0 0
$$139$$ 4.24264i 0.359856i 0.983680 + 0.179928i $$0.0575865\pi$$
−0.983680 + 0.179928i $$0.942414\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.48528i 0.712069i
$$143$$ 4.00000 8.48528i 0.334497 0.709575i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 11.3137i 0.936329i
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 4.24264i 0.345261i 0.984987 + 0.172631i $$0.0552267\pi$$
−0.984987 + 0.172631i $$0.944773\pi$$
$$152$$ 12.7279i 1.03237i
$$153$$ 0 0
$$154$$ −2.00000 + 4.24264i −0.161165 + 0.341882i
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 24.0000 1.91541 0.957704 0.287754i $$-0.0929087\pi$$
0.957704 + 0.287754i $$0.0929087\pi$$
$$158$$ 12.7279i 1.01258i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 24.0000 1.87983 0.939913 0.341415i $$-0.110906\pi$$
0.939913 + 0.341415i $$0.110906\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 14.0000 1.08661
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.41421i 0.107833i
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 1.41421i −0.226134 0.106600i
$$177$$ 0 0
$$178$$ 7.07107i 0.529999i
$$179$$ 5.65685i 0.422813i −0.977398 0.211407i $$-0.932196\pi$$
0.977398 0.211407i $$-0.0678044\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.00000 + 2.82843i 0.438763 + 0.206835i
$$188$$ 8.48528i 0.618853i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2.82843i 0.204658i 0.994751 + 0.102329i $$0.0326294\pi$$
−0.994751 + 0.102329i $$0.967371\pi$$
$$192$$ 0 0
$$193$$ 19.7990i 1.42516i −0.701590 0.712581i $$-0.747526\pi$$
0.701590 0.712581i $$-0.252474\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ −5.00000 −0.357143
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 8.48528i 0.595550i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ 0 0
$$208$$ 2.82843i 0.196116i
$$209$$ −6.00000 + 12.7279i −0.415029 + 0.880409i
$$210$$ 0 0
$$211$$ 4.24264i 0.292075i 0.989279 + 0.146038i $$0.0466521\pi$$
−0.989279 + 0.146038i $$0.953348\pi$$
$$212$$ 4.24264i 0.291386i
$$213$$ 0 0
$$214$$ −14.0000 −0.957020
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 16.9706i 1.14939i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.65685i 0.380521i
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 7.07107i 0.472456i
$$225$$ 0 0
$$226$$ 12.7279i 0.846649i
$$227$$ 2.00000 0.132745 0.0663723 0.997795i $$-0.478857\pi$$
0.0663723 + 0.997795i $$0.478857\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 18.0000 1.18176
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.48528i 0.552345i
$$237$$ 0 0
$$238$$ 2.82843i 0.183340i
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 8.48528i 0.546585i 0.961931 + 0.273293i $$0.0881127\pi$$
−0.961931 + 0.273293i $$0.911887\pi$$
$$242$$ 7.00000 + 8.48528i 0.449977 + 0.545455i
$$243$$ 0 0
$$244$$ 8.48528i 0.543214i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.9706i 1.07117i −0.844481 0.535586i $$-0.820091\pi$$
0.844481 0.535586i $$-0.179909\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 18.3848i 1.15356i
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 29.6985i 1.85254i −0.376860 0.926270i $$-0.622996\pi$$
0.376860 0.926270i $$-0.377004\pi$$
$$258$$ 0 0
$$259$$ 8.48528i 0.527250i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −18.0000 −1.11204
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 12.7279i 0.776035i 0.921652 + 0.388018i $$0.126840\pi$$
−0.921652 + 0.388018i $$0.873160\pi$$
$$270$$ 0 0
$$271$$ 12.7279i 0.773166i 0.922255 + 0.386583i $$0.126345\pi$$
−0.922255 + 0.386583i $$0.873655\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 4.24264i 0.256307i
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 31.1127i 1.86938i 0.355463 + 0.934690i $$0.384323\pi$$
−0.355463 + 0.934690i $$0.615677\pi$$
$$278$$ 4.24264i 0.254457i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 1.41421i 0.0840663i −0.999116 0.0420331i $$-0.986616\pi$$
0.999116 0.0420331i $$-0.0133835\pi$$
$$284$$ 8.48528i 0.503509i
$$285$$ 0 0
$$286$$ 4.00000 8.48528i 0.236525 0.501745i
$$287$$ 8.48528i 0.500870i
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.3137i 0.662085i
$$293$$ −10.0000 −0.584206 −0.292103 0.956387i $$-0.594355\pi$$
−0.292103 + 0.956387i $$0.594355\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −2.00000 −0.115278
$$302$$ 4.24264i 0.244137i
$$303$$ 0 0
$$304$$ 4.24264i 0.243332i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 24.0416i 1.37213i −0.727541 0.686064i $$-0.759337\pi$$
0.727541 0.686064i $$-0.240663\pi$$
$$308$$ 2.00000 4.24264i 0.113961 0.241747i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 16.9706i 0.962312i 0.876635 + 0.481156i $$0.159783\pi$$
−0.876635 + 0.481156i $$0.840217\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 24.0000 1.35440
$$315$$ 0 0
$$316$$ 12.7279i 0.716002i
$$317$$ 21.2132i 1.19145i −0.803188 0.595726i $$-0.796864\pi$$
0.803188 0.595726i $$-0.203136\pi$$
$$318$$ 0 0
$$319$$ −18.0000 8.48528i −1.00781 0.475085i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.48528i 0.472134i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ 0 0
$$328$$ 18.0000 0.993884
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ 0 0
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 11.3137i 0.616297i 0.951338 + 0.308148i $$0.0997094\pi$$
−0.951338 + 0.308148i $$0.900291\pi$$
$$338$$ 5.00000 0.271964
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 16.9706i 0.916324i
$$344$$ 4.24264i 0.228748i
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ 14.0000 0.751559 0.375780 0.926709i $$-0.377375\pi$$
0.375780 + 0.926709i $$0.377375\pi$$
$$348$$ 0 0
$$349$$ 8.48528i 0.454207i 0.973871 + 0.227103i $$0.0729255\pi$$
−0.973871 + 0.227103i $$0.927074\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 15.0000 + 7.07107i 0.799503 + 0.376889i
$$353$$ 4.24264i 0.225813i −0.993606 0.112906i $$-0.963984\pi$$
0.993606 0.112906i $$-0.0360161\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 7.07107i 0.374766i
$$357$$ 0 0
$$358$$ 5.65685i 0.298974i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −12.0000 −0.626395 −0.313197 0.949688i $$-0.601400\pi$$
−0.313197 + 0.949688i $$0.601400\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 22.6274i 1.17160i −0.810454 0.585802i $$-0.800780\pi$$
0.810454 0.585802i $$-0.199220\pi$$
$$374$$ 6.00000 + 2.82843i 0.310253 + 0.146254i
$$375$$ 0 0
$$376$$ 25.4558i 1.31278i
$$377$$ 16.9706i 0.874028i
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 2.82843i 0.144715i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 19.7990i 1.00774i
$$387$$ 0 0
$$388$$ 12.0000 0.609208
$$389$$ 21.2132i 1.07555i 0.843088 + 0.537776i $$0.180735\pi$$
−0.843088 + 0.537776i $$0.819265\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −15.0000 −0.757614
$$393$$ 0 0
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.8701i 1.34183i 0.741536 + 0.670913i $$0.234098\pi$$
−0.741536 + 0.670913i $$0.765902\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 8.48528i 0.421117i
$$407$$ −18.0000 8.48528i −0.892227 0.420600i
$$408$$ 0 0
$$409$$ 8.48528i 0.419570i −0.977748 0.209785i $$-0.932724\pi$$
0.977748 0.209785i $$-0.0672764\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −12.0000 −0.591198
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 14.1421i 0.693375i
$$417$$ 0 0
$$418$$ −6.00000 + 12.7279i −0.293470 + 0.622543i
$$419$$ 39.5980i 1.93449i −0.253849 0.967244i $$-0.581697\pi$$
0.253849 0.967244i $$-0.418303\pi$$
$$420$$ 0 0
$$421$$ 24.0000 1.16969 0.584844 0.811146i $$-0.301156\pi$$
0.584844 + 0.811146i $$0.301156\pi$$
$$422$$ 4.24264i 0.206529i
$$423$$ 0 0
$$424$$ 12.7279i 0.618123i
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −12.0000 −0.580721
$$428$$ 14.0000 0.676716
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 16.9706i 0.812743i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 38.1838i 1.82241i −0.411951 0.911206i $$-0.635153\pi$$
0.411951 0.911206i $$-0.364847\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 5.65685i 0.269069i
$$443$$ 25.4558i 1.20944i −0.796437 0.604722i $$-0.793284\pi$$
0.796437 0.604722i $$-0.206716\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ 0 0
$$448$$ 9.89949i 0.467707i
$$449$$ 32.5269i 1.53504i −0.641025 0.767520i $$-0.721491\pi$$
0.641025 0.767520i $$-0.278509\pi$$
$$450$$ 0 0
$$451$$ −18.0000 8.48528i −0.847587 0.399556i
$$452$$ 12.7279i 0.598671i
$$453$$ 0 0
$$454$$ 2.00000 0.0938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 28.2843i 1.32308i −0.749909 0.661541i $$-0.769903\pi$$
0.749909 0.661541i $$-0.230097\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ 8.48528i 0.392652i 0.980539 + 0.196326i $$0.0629011\pi$$
−0.980539 + 0.196326i $$0.937099\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 25.4558i 1.17170i
$$473$$ −2.00000 + 4.24264i −0.0919601 + 0.195077i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.82843i 0.129641i
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 16.9706i 0.773791i
$$482$$ 8.48528i 0.386494i
$$483$$ 0 0
$$484$$ −7.00000 8.48528i −0.318182 0.385695i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 25.4558i 1.15233i
$$489$$ 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
$$493$$ −12.0000 −0.540453
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 16.9706i 0.757433i
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 18.3848i 0.815693i
$$509$$ 4.24264i 0.188052i 0.995570 + 0.0940259i $$0.0299736\pi$$
−0.995570 + 0.0940259i $$0.970026\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 29.6985i 1.30994i
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −12.0000 + 25.4558i −0.527759 + 1.11955i
$$518$$ 8.48528i 0.372822i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7.07107i 0.309789i −0.987931 0.154895i $$-0.950496\pi$$
0.987931 0.154895i $$-0.0495038\pi$$
$$522$$ 0 0
$$523$$ 7.07107i 0.309196i 0.987977 + 0.154598i $$0.0494083\pi$$
−0.987977 + 0.154598i $$0.950592\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 6.00000 0.260133
$$533$$ 16.9706i 0.735077i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 12.7279i 0.548740i
$$539$$ 15.0000 + 7.07107i 0.646096 + 0.304572i
$$540$$ 0 0
$$541$$ 33.9411i 1.45924i 0.683851 + 0.729621i $$0.260304\pi$$
−0.683851 + 0.729621i $$0.739696\pi$$
$$542$$ 12.7279i 0.546711i
$$543$$ 0 0
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9.89949i 0.423272i 0.977349 + 0.211636i $$0.0678791\pi$$
−0.977349 + 0.211636i $$0.932121\pi$$
$$548$$ 4.24264i 0.181237i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 25.4558i 1.08446i
$$552$$ 0 0
$$553$$ −18.0000 −0.765438
$$554$$ 31.1127i 1.32185i
$$555$$ 0 0
$$556$$ 4.24264i 0.179928i
$$557$$ 34.0000 1.44063 0.720313 0.693649i $$-0.243998\pi$$
0.720313 + 0.693649i $$0.243998\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1.41421i 0.0594438i
$$567$$ 0 0
$$568$$ 25.4558i 1.06810i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 4.24264i 0.177549i 0.996052 + 0.0887745i $$0.0282950\pi$$
−0.996052 + 0.0887745i $$0.971705\pi$$
$$572$$ −4.00000 + 8.48528i −0.167248 + 0.354787i
$$573$$ 0 0
$$574$$ 8.48528i 0.354169i
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 19.7990i 0.821401i
$$582$$ 0 0
$$583$$ −6.00000 + 12.7279i −0.248495 + 0.527137i
$$584$$ 33.9411i 1.40449i
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ 42.4264i 1.75113i 0.483105 + 0.875563i $$0.339509\pi$$
−0.483105 + 0.875563i $$0.660491\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ 2.00000 0.0821302 0.0410651 0.999156i $$-0.486925\pi$$
0.0410651 + 0.999156i $$0.486925\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 11.3137i 0.462266i 0.972922 + 0.231133i $$0.0742432\pi$$
−0.972922 + 0.231133i $$0.925757\pi$$
$$600$$ 0 0
$$601$$ 33.9411i 1.38449i −0.721664 0.692244i $$-0.756622\pi$$
0.721664 0.692244i $$-0.243378\pi$$
$$602$$ −2.00000 −0.0815139
$$603$$ 0 0
$$604$$ 4.24264i 0.172631i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.5269i 1.32023i 0.751166 + 0.660113i $$0.229492\pi$$
−0.751166 + 0.660113i $$0.770508\pi$$
$$608$$ 21.2132i 0.860309i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ 28.2843i 1.14239i −0.820814 0.571195i $$-0.806480\pi$$
0.820814 0.571195i $$-0.193520\pi$$
$$614$$ 24.0416i 0.970241i
$$615$$ 0 0
$$616$$ 6.00000 12.7279i 0.241747 0.512823i
$$617$$ 4.24264i 0.170802i 0.996347 + 0.0854011i $$0.0272172\pi$$
−0.996347 + 0.0854011i $$0.972783\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16.9706i 0.680458i
$$623$$ 10.0000 0.400642
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ −24.0000 −0.957704
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 38.1838i 1.51887i
$$633$$ 0 0
$$634$$ 21.2132i 0.842484i
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14.1421i 0.560332i
$$638$$ −18.0000 8.48528i −0.712627 0.335936i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 15.5563i 0.614439i −0.951639 0.307219i $$-0.900601\pi$$
0.951639 0.307219i $$-0.0993986\pi$$
$$642$$ 0 0
$$643$$ −12.0000 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 8.48528i 0.333849i
$$647$$ 16.9706i 0.667182i 0.942718 + 0.333591i $$0.108260\pi$$
−0.942718 + 0.333591i $$0.891740\pi$$
$$648$$ 0 0
$$649$$ −12.0000 + 25.4558i −0.471041 + 0.999229i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −24.0000 −0.939913
$$653$$ 29.6985i 1.16219i 0.813835 + 0.581096i $$0.197376\pi$$
−0.813835 + 0.581096i $$0.802624\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 12.0000 0.466393
$$663$$ 0 0
$$664$$ −42.0000 −1.62992
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 8.00000 0.309529
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −12.0000 + 25.4558i −0.463255 + 0.982712i
$$672$$ 0 0
$$673$$ 36.7696i 1.41736i 0.705529 + 0.708681i $$0.250709\pi$$
−0.705529 + 0.708681i $$0.749291\pi$$
$$674$$ 11.3137i 0.435788i
$$675$$ 0 0
$$676$$ −5.00000 −0.192308
$$677$$ 46.0000 1.76792 0.883962 0.467559i $$-0.154866\pi$$
0.883962 + 0.467559i $$0.154866\pi$$
$$678$$ 0 0
$$679$$ 16.9706i 0.651270i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 33.9411i 1.29872i 0.760481 + 0.649361i $$0.224963\pi$$
−0.760481 + 0.649361i $$0.775037\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 16.9706i 0.647939i
$$687$$ 0 0
$$688$$ 1.41421i 0.0539164i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 0 0
$$694$$ 14.0000 0.531433
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 8.48528i 0.321173i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 25.4558i 0.960085i
$$704$$ 21.0000 + 9.89949i 0.791467 + 0.373101i
$$705$$ 0 0
$$706$$ 4.24264i 0.159674i
$$707$$ 8.48528i 0.319122i
$$708$$ 0 0
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 21.2132i 0.794998i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 5.65685i 0.211407i
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ 31.1127i 1.16031i 0.814507 + 0.580154i $$0.197008\pi$$
−0.814507 + 0.580154i $$0.802992\pi$$
$$720$$ 0 0
$$721$$ 16.9706i 0.632017i
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −48.0000 −1.78022 −0.890111 0.455744i $$-0.849373\pi$$
−0.890111 + 0.455744i $$0.849373\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2.82843i 0.104613i
$$732$$ 0 0
$$733$$ 31.1127i 1.14917i −0.818444 0.574587i $$-0.805163\pi$$
0.818444 0.574587i $$-0.194837\pi$$
$$734$$ −12.0000 −0.442928
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 46.6690i 1.71675i −0.513024 0.858374i $$-0.671475\pi$$
0.513024 0.858374i $$-0.328525\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −6.00000 −0.220267
$$743$$ 10.0000 0.366864 0.183432 0.983032i $$-0.441279\pi$$
0.183432 + 0.983032i $$0.441279\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22.6274i 0.828449i
$$747$$ 0 0
$$748$$ −6.00000 2.82843i −0.219382 0.103418i
$$749$$ 19.7990i 0.723439i
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 8.48528i 0.309426i
$$753$$ 0 0
$$754$$ 16.9706i 0.618031i
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 42.0000 1.52652 0.763258 0.646094i $$-0.223599\pi$$
0.763258 + 0.646094i $$0.223599\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 2.82843i 0.102329i
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 19.7990i 0.712581i
$$773$$ 4.24264i 0.152597i 0.997085 + 0.0762986i $$0.0243102\pi$$
−0.997085 + 0.0762986i $$0.975690\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 36.0000 1.29232
$$777$$ 0 0
$$778$$ 21.2132i 0.760530i
$$779$$ 25.4558i 0.912050i
$$780$$ 0 0
$$781$$ −12.0000 + 25.4558i −0.429394 + 0.910882i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −5.00000 −0.178571
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 9.89949i 0.352879i −0.984311 0.176439i $$-0.943542\pi$$
0.984311 0.176439i $$-0.0564580\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 24.0000 0.852265
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ 29.6985i 1.05197i 0.850493 + 0.525987i $$0.176304\pi$$
−0.850493 + 0.525987i $$0.823696\pi$$
$$798$$ 0 0
$$799$$ 16.9706i 0.600375i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 26.8701i 0.948815i
$$803$$ −16.0000 + 33.9411i −0.564628