# Properties

 Label 100.6.c.a.49.2 Level $100$ Weight $6$ Character 100.49 Analytic conductor $16.038$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 100.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0383819813$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 100.49 Dual form 100.6.c.a.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+22.0000i q^{3} -218.000i q^{7} -241.000 q^{9} +O(q^{10})$$ $$q+22.0000i q^{3} -218.000i q^{7} -241.000 q^{9} -480.000 q^{11} -622.000i q^{13} -186.000i q^{17} +1204.00 q^{19} +4796.00 q^{21} -3186.00i q^{23} +44.0000i q^{27} -5526.00 q^{29} +9356.00 q^{31} -10560.0i q^{33} -5618.00i q^{37} +13684.0 q^{39} -14394.0 q^{41} -370.000i q^{43} -16146.0i q^{47} -30717.0 q^{49} +4092.00 q^{51} -4374.00i q^{53} +26488.0i q^{57} +11748.0 q^{59} +13202.0 q^{61} +52538.0i q^{63} +11542.0i q^{67} +70092.0 q^{69} -29532.0 q^{71} +33698.0i q^{73} +104640. i q^{77} -31208.0 q^{79} -59531.0 q^{81} -38466.0i q^{83} -121572. i q^{87} -119514. q^{89} -135596. q^{91} +205832. i q^{93} -94658.0i q^{97} +115680. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 482q^{9} + O(q^{10})$$ $$2q - 482q^{9} - 960q^{11} + 2408q^{19} + 9592q^{21} - 11052q^{29} + 18712q^{31} + 27368q^{39} - 28788q^{41} - 61434q^{49} + 8184q^{51} + 23496q^{59} + 26404q^{61} + 140184q^{69} - 59064q^{71} - 62416q^{79} - 119062q^{81} - 239028q^{89} - 271192q^{91} + 231360q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 22.0000i 1.41130i 0.708560 + 0.705650i $$0.249345\pi$$
−0.708560 + 0.705650i $$0.750655\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 218.000i − 1.68156i −0.541380 0.840778i $$-0.682098\pi$$
0.541380 0.840778i $$-0.317902\pi$$
$$8$$ 0 0
$$9$$ −241.000 −0.991770
$$10$$ 0 0
$$11$$ −480.000 −1.19608 −0.598039 0.801467i $$-0.704053\pi$$
−0.598039 + 0.801467i $$0.704053\pi$$
$$12$$ 0 0
$$13$$ − 622.000i − 1.02078i −0.859943 0.510390i $$-0.829501\pi$$
0.859943 0.510390i $$-0.170499\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 186.000i − 0.156096i −0.996950 0.0780478i $$-0.975131\pi$$
0.996950 0.0780478i $$-0.0248687\pi$$
$$18$$ 0 0
$$19$$ 1204.00 0.765143 0.382571 0.923926i $$-0.375039\pi$$
0.382571 + 0.923926i $$0.375039\pi$$
$$20$$ 0 0
$$21$$ 4796.00 2.37318
$$22$$ 0 0
$$23$$ − 3186.00i − 1.25582i −0.778287 0.627908i $$-0.783911\pi$$
0.778287 0.627908i $$-0.216089\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 44.0000i 0.0116156i
$$28$$ 0 0
$$29$$ −5526.00 −1.22016 −0.610079 0.792341i $$-0.708862\pi$$
−0.610079 + 0.792341i $$0.708862\pi$$
$$30$$ 0 0
$$31$$ 9356.00 1.74858 0.874291 0.485402i $$-0.161327\pi$$
0.874291 + 0.485402i $$0.161327\pi$$
$$32$$ 0 0
$$33$$ − 10560.0i − 1.68803i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 5618.00i − 0.674648i −0.941389 0.337324i $$-0.890478\pi$$
0.941389 0.337324i $$-0.109522\pi$$
$$38$$ 0 0
$$39$$ 13684.0 1.44063
$$40$$ 0 0
$$41$$ −14394.0 −1.33728 −0.668639 0.743587i $$-0.733123\pi$$
−0.668639 + 0.743587i $$0.733123\pi$$
$$42$$ 0 0
$$43$$ − 370.000i − 0.0305162i −0.999884 0.0152581i $$-0.995143\pi$$
0.999884 0.0152581i $$-0.00485699\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 16146.0i − 1.06615i −0.846066 0.533077i $$-0.821035\pi$$
0.846066 0.533077i $$-0.178965\pi$$
$$48$$ 0 0
$$49$$ −30717.0 −1.82763
$$50$$ 0 0
$$51$$ 4092.00 0.220298
$$52$$ 0 0
$$53$$ − 4374.00i − 0.213889i −0.994265 0.106945i $$-0.965893\pi$$
0.994265 0.106945i $$-0.0341068\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 26488.0i 1.07985i
$$58$$ 0 0
$$59$$ 11748.0 0.439374 0.219687 0.975570i $$-0.429496\pi$$
0.219687 + 0.975570i $$0.429496\pi$$
$$60$$ 0 0
$$61$$ 13202.0 0.454271 0.227136 0.973863i $$-0.427064\pi$$
0.227136 + 0.973863i $$0.427064\pi$$
$$62$$ 0 0
$$63$$ 52538.0i 1.66772i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11542.0i 0.314119i 0.987589 + 0.157059i $$0.0502014\pi$$
−0.987589 + 0.157059i $$0.949799\pi$$
$$68$$ 0 0
$$69$$ 70092.0 1.77233
$$70$$ 0 0
$$71$$ −29532.0 −0.695260 −0.347630 0.937632i $$-0.613013\pi$$
−0.347630 + 0.937632i $$0.613013\pi$$
$$72$$ 0 0
$$73$$ 33698.0i 0.740111i 0.929010 + 0.370056i $$0.120661\pi$$
−0.929010 + 0.370056i $$0.879339\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 104640.i 2.01127i
$$78$$ 0 0
$$79$$ −31208.0 −0.562598 −0.281299 0.959620i $$-0.590765\pi$$
−0.281299 + 0.959620i $$0.590765\pi$$
$$80$$ 0 0
$$81$$ −59531.0 −1.00816
$$82$$ 0 0
$$83$$ − 38466.0i − 0.612889i −0.951889 0.306444i $$-0.900861\pi$$
0.951889 0.306444i $$-0.0991394\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 121572.i − 1.72201i
$$88$$ 0 0
$$89$$ −119514. −1.59935 −0.799675 0.600432i $$-0.794995\pi$$
−0.799675 + 0.600432i $$0.794995\pi$$
$$90$$ 0 0
$$91$$ −135596. −1.71650
$$92$$ 0 0
$$93$$ 205832.i 2.46777i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 94658.0i − 1.02148i −0.859737 0.510738i $$-0.829372\pi$$
0.859737 0.510738i $$-0.170628\pi$$
$$98$$ 0 0
$$99$$ 115680. 1.18623
$$100$$ 0 0
$$101$$ 101046. 0.985634 0.492817 0.870133i $$-0.335967\pi$$
0.492817 + 0.870133i $$0.335967\pi$$
$$102$$ 0 0
$$103$$ − 143434.i − 1.33217i −0.745877 0.666084i $$-0.767969\pi$$
0.745877 0.666084i $$-0.232031\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 57054.0i 0.481755i 0.970555 + 0.240878i $$0.0774353\pi$$
−0.970555 + 0.240878i $$0.922565\pi$$
$$108$$ 0 0
$$109$$ 3118.00 0.0251368 0.0125684 0.999921i $$-0.495999\pi$$
0.0125684 + 0.999921i $$0.495999\pi$$
$$110$$ 0 0
$$111$$ 123596. 0.952132
$$112$$ 0 0
$$113$$ − 54534.0i − 0.401764i −0.979615 0.200882i $$-0.935619\pi$$
0.979615 0.200882i $$-0.0643808\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 149902.i 1.01238i
$$118$$ 0 0
$$119$$ −40548.0 −0.262484
$$120$$ 0 0
$$121$$ 69349.0 0.430603
$$122$$ 0 0
$$123$$ − 316668.i − 1.88730i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 24698.0i − 0.135879i −0.997689 0.0679395i $$-0.978358\pi$$
0.997689 0.0679395i $$-0.0216425\pi$$
$$128$$ 0 0
$$129$$ 8140.00 0.0430675
$$130$$ 0 0
$$131$$ 236640. 1.20479 0.602393 0.798200i $$-0.294214\pi$$
0.602393 + 0.798200i $$0.294214\pi$$
$$132$$ 0 0
$$133$$ − 262472.i − 1.28663i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 22158.0i 0.100862i 0.998728 + 0.0504312i $$0.0160596\pi$$
−0.998728 + 0.0504312i $$0.983940\pi$$
$$138$$ 0 0
$$139$$ 193204. 0.848163 0.424081 0.905624i $$-0.360597\pi$$
0.424081 + 0.905624i $$0.360597\pi$$
$$140$$ 0 0
$$141$$ 355212. 1.50467
$$142$$ 0 0
$$143$$ 298560.i 1.22093i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 675774.i − 2.57934i
$$148$$ 0 0
$$149$$ −448554. −1.65519 −0.827597 0.561322i $$-0.810293\pi$$
−0.827597 + 0.561322i $$0.810293\pi$$
$$150$$ 0 0
$$151$$ −140860. −0.502742 −0.251371 0.967891i $$-0.580881\pi$$
−0.251371 + 0.967891i $$0.580881\pi$$
$$152$$ 0 0
$$153$$ 44826.0i 0.154811i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 335878.i 1.08751i 0.839245 + 0.543754i $$0.182998\pi$$
−0.839245 + 0.543754i $$0.817002\pi$$
$$158$$ 0 0
$$159$$ 96228.0 0.301862
$$160$$ 0 0
$$161$$ −694548. −2.11173
$$162$$ 0 0
$$163$$ − 101650.i − 0.299667i −0.988711 0.149833i $$-0.952126\pi$$
0.988711 0.149833i $$-0.0478737\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 139242.i − 0.386348i −0.981164 0.193174i $$-0.938122\pi$$
0.981164 0.193174i $$-0.0618782\pi$$
$$168$$ 0 0
$$169$$ −15591.0 −0.0419911
$$170$$ 0 0
$$171$$ −290164. −0.758845
$$172$$ 0 0
$$173$$ − 265014.i − 0.673215i −0.941645 0.336607i $$-0.890721\pi$$
0.941645 0.336607i $$-0.109279\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 258456.i 0.620088i
$$178$$ 0 0
$$179$$ 142812. 0.333144 0.166572 0.986029i $$-0.446730\pi$$
0.166572 + 0.986029i $$0.446730\pi$$
$$180$$ 0 0
$$181$$ 109670. 0.248824 0.124412 0.992231i $$-0.460296\pi$$
0.124412 + 0.992231i $$0.460296\pi$$
$$182$$ 0 0
$$183$$ 290444.i 0.641113i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 89280.0i 0.186703i
$$188$$ 0 0
$$189$$ 9592.00 0.0195324
$$190$$ 0 0
$$191$$ 294948. 0.585008 0.292504 0.956264i $$-0.405511\pi$$
0.292504 + 0.956264i $$0.405511\pi$$
$$192$$ 0 0
$$193$$ 1.00303e6i 1.93831i 0.246459 + 0.969153i $$0.420733\pi$$
−0.246459 + 0.969153i $$0.579267\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 823998.i 1.51273i 0.654151 + 0.756364i $$0.273026\pi$$
−0.654151 + 0.756364i $$0.726974\pi$$
$$198$$ 0 0
$$199$$ 906712. 1.62307 0.811534 0.584305i $$-0.198633\pi$$
0.811534 + 0.584305i $$0.198633\pi$$
$$200$$ 0 0
$$201$$ −253924. −0.443316
$$202$$ 0 0
$$203$$ 1.20467e6i 2.05176i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 767826.i 1.24548i
$$208$$ 0 0
$$209$$ −577920. −0.915170
$$210$$ 0 0
$$211$$ 506384. 0.783022 0.391511 0.920173i $$-0.371953\pi$$
0.391511 + 0.920173i $$0.371953\pi$$
$$212$$ 0 0
$$213$$ − 649704.i − 0.981220i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 2.03961e6i − 2.94034i
$$218$$ 0 0
$$219$$ −741356. −1.04452
$$220$$ 0 0
$$221$$ −115692. −0.159339
$$222$$ 0 0
$$223$$ − 542050.i − 0.729923i −0.931023 0.364962i $$-0.881082\pi$$
0.931023 0.364962i $$-0.118918\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.44857e6i 1.86585i 0.360075 + 0.932924i $$0.382751\pi$$
−0.360075 + 0.932924i $$0.617249\pi$$
$$228$$ 0 0
$$229$$ 478786. 0.603327 0.301663 0.953414i $$-0.402458\pi$$
0.301663 + 0.953414i $$0.402458\pi$$
$$230$$ 0 0
$$231$$ −2.30208e6 −2.83851
$$232$$ 0 0
$$233$$ 374106.i 0.451445i 0.974192 + 0.225723i $$0.0724743\pi$$
−0.974192 + 0.225723i $$0.927526\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 686576.i − 0.793995i
$$238$$ 0 0
$$239$$ −169416. −0.191849 −0.0959245 0.995389i $$-0.530581\pi$$
−0.0959245 + 0.995389i $$0.530581\pi$$
$$240$$ 0 0
$$241$$ −353746. −0.392328 −0.196164 0.980571i $$-0.562848\pi$$
−0.196164 + 0.980571i $$0.562848\pi$$
$$242$$ 0 0
$$243$$ − 1.29899e6i − 1.41121i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 748888.i − 0.781042i
$$248$$ 0 0
$$249$$ 846252. 0.864971
$$250$$ 0 0
$$251$$ 1.25520e6 1.25756 0.628780 0.777583i $$-0.283555\pi$$
0.628780 + 0.777583i $$0.283555\pi$$
$$252$$ 0 0
$$253$$ 1.52928e6i 1.50205i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.12877e6i 1.06604i 0.846102 + 0.533021i $$0.178943\pi$$
−0.846102 + 0.533021i $$0.821057\pi$$
$$258$$ 0 0
$$259$$ −1.22472e6 −1.13446
$$260$$ 0 0
$$261$$ 1.33177e6 1.21012
$$262$$ 0 0
$$263$$ − 263082.i − 0.234532i −0.993101 0.117266i $$-0.962587\pi$$
0.993101 0.117266i $$-0.0374130\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 2.62931e6i − 2.25717i
$$268$$ 0 0
$$269$$ 1.18774e6 1.00079 0.500393 0.865798i $$-0.333189\pi$$
0.500393 + 0.865798i $$0.333189\pi$$
$$270$$ 0 0
$$271$$ 431300. 0.356744 0.178372 0.983963i $$-0.442917\pi$$
0.178372 + 0.983963i $$0.442917\pi$$
$$272$$ 0 0
$$273$$ − 2.98311e6i − 2.42250i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 743114.i − 0.581910i −0.956737 0.290955i $$-0.906027\pi$$
0.956737 0.290955i $$-0.0939730\pi$$
$$278$$ 0 0
$$279$$ −2.25480e6 −1.73419
$$280$$ 0 0
$$281$$ 1.92193e6 1.45201 0.726007 0.687687i $$-0.241374\pi$$
0.726007 + 0.687687i $$0.241374\pi$$
$$282$$ 0 0
$$283$$ − 1.63071e6i − 1.21035i −0.796092 0.605176i $$-0.793103\pi$$
0.796092 0.605176i $$-0.206897\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3.13789e6i 2.24871i
$$288$$ 0 0
$$289$$ 1.38526e6 0.975634
$$290$$ 0 0
$$291$$ 2.08248e6 1.44161
$$292$$ 0 0
$$293$$ 71250.0i 0.0484859i 0.999706 + 0.0242430i $$0.00771753\pi$$
−0.999706 + 0.0242430i $$0.992282\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 21120.0i − 0.0138932i
$$298$$ 0 0
$$299$$ −1.98169e6 −1.28191
$$300$$ 0 0
$$301$$ −80660.0 −0.0513147
$$302$$ 0 0
$$303$$ 2.22301e6i 1.39103i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.61762e6i 0.979560i 0.871846 + 0.489780i $$0.162923\pi$$
−0.871846 + 0.489780i $$0.837077\pi$$
$$308$$ 0 0
$$309$$ 3.15555e6 1.88009
$$310$$ 0 0
$$311$$ −682788. −0.400299 −0.200150 0.979765i $$-0.564143\pi$$
−0.200150 + 0.979765i $$0.564143\pi$$
$$312$$ 0 0
$$313$$ − 2.70444e6i − 1.56033i −0.625574 0.780165i $$-0.715135\pi$$
0.625574 0.780165i $$-0.284865\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.60347e6i − 1.45514i −0.686035 0.727568i $$-0.740650\pi$$
0.686035 0.727568i $$-0.259350\pi$$
$$318$$ 0 0
$$319$$ 2.65248e6 1.45940
$$320$$ 0 0
$$321$$ −1.25519e6 −0.679902
$$322$$ 0 0
$$323$$ − 223944.i − 0.119435i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 68596.0i 0.0354756i
$$328$$ 0 0
$$329$$ −3.51983e6 −1.79280
$$330$$ 0 0
$$331$$ −661432. −0.331830 −0.165915 0.986140i $$-0.553058\pi$$
−0.165915 + 0.986140i $$0.553058\pi$$
$$332$$ 0 0
$$333$$ 1.35394e6i 0.669096i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 1.71706e6i − 0.823588i −0.911277 0.411794i $$-0.864902\pi$$
0.911277 0.411794i $$-0.135098\pi$$
$$338$$ 0 0
$$339$$ 1.19975e6 0.567010
$$340$$ 0 0
$$341$$ −4.49088e6 −2.09144
$$342$$ 0 0
$$343$$ 3.03238e6i 1.39171i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 131370.i − 0.0585696i −0.999571 0.0292848i $$-0.990677\pi$$
0.999571 0.0292848i $$-0.00932298\pi$$
$$348$$ 0 0
$$349$$ −3.50951e6 −1.54235 −0.771175 0.636623i $$-0.780331\pi$$
−0.771175 + 0.636623i $$0.780331\pi$$
$$350$$ 0 0
$$351$$ 27368.0 0.0118570
$$352$$ 0 0
$$353$$ 2.21992e6i 0.948202i 0.880470 + 0.474101i $$0.157227\pi$$
−0.880470 + 0.474101i $$0.842773\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 892056.i − 0.370443i
$$358$$ 0 0
$$359$$ −4.39730e6 −1.80074 −0.900369 0.435128i $$-0.856703\pi$$
−0.900369 + 0.435128i $$0.856703\pi$$
$$360$$ 0 0
$$361$$ −1.02648e6 −0.414557
$$362$$ 0 0
$$363$$ 1.52568e6i 0.607710i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2.29824e6i 0.890697i 0.895357 + 0.445348i $$0.146920\pi$$
−0.895357 + 0.445348i $$0.853080\pi$$
$$368$$ 0 0
$$369$$ 3.46895e6 1.32627
$$370$$ 0 0
$$371$$ −953532. −0.359667
$$372$$ 0 0
$$373$$ − 1.73561e6i − 0.645920i −0.946413 0.322960i $$-0.895322\pi$$
0.946413 0.322960i $$-0.104678\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.43717e6i 1.24551i
$$378$$ 0 0
$$379$$ 5.39115e6 1.92789 0.963947 0.266094i $$-0.0857331\pi$$
0.963947 + 0.266094i $$0.0857331\pi$$
$$380$$ 0 0
$$381$$ 543356. 0.191766
$$382$$ 0 0
$$383$$ 3.27281e6i 1.14005i 0.821627 + 0.570026i $$0.193067\pi$$
−0.821627 + 0.570026i $$0.806933\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 89170.0i 0.0302650i
$$388$$ 0 0
$$389$$ −603114. −0.202081 −0.101040 0.994882i $$-0.532217\pi$$
−0.101040 + 0.994882i $$0.532217\pi$$
$$390$$ 0 0
$$391$$ −592596. −0.196027
$$392$$ 0 0
$$393$$ 5.20608e6i 1.70032i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 749422.i 0.238644i 0.992856 + 0.119322i $$0.0380721\pi$$
−0.992856 + 0.119322i $$0.961928\pi$$
$$398$$ 0 0
$$399$$ 5.77438e6 1.81582
$$400$$ 0 0
$$401$$ 5.31357e6 1.65016 0.825079 0.565018i $$-0.191131\pi$$
0.825079 + 0.565018i $$0.191131\pi$$
$$402$$ 0 0
$$403$$ − 5.81943e6i − 1.78492i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.69664e6i 0.806932i
$$408$$ 0 0
$$409$$ −999326. −0.295392 −0.147696 0.989033i $$-0.547186\pi$$
−0.147696 + 0.989033i $$0.547186\pi$$
$$410$$ 0 0
$$411$$ −487476. −0.142347
$$412$$ 0 0
$$413$$ − 2.56106e6i − 0.738831i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.25049e6i 1.19701i
$$418$$ 0 0
$$419$$ −2.03740e6 −0.566944 −0.283472 0.958980i $$-0.591486\pi$$
−0.283472 + 0.958980i $$0.591486\pi$$
$$420$$ 0 0
$$421$$ −5.11461e6 −1.40640 −0.703198 0.710994i $$-0.748245\pi$$
−0.703198 + 0.710994i $$0.748245\pi$$
$$422$$ 0 0
$$423$$ 3.89119e6i 1.05738i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 2.87804e6i − 0.763882i
$$428$$ 0 0
$$429$$ −6.56832e6 −1.72310
$$430$$ 0 0
$$431$$ −3.30404e6 −0.856747 −0.428374 0.903602i $$-0.640913\pi$$
−0.428374 + 0.903602i $$0.640913\pi$$
$$432$$ 0 0
$$433$$ − 2.01638e6i − 0.516836i −0.966033 0.258418i $$-0.916799\pi$$
0.966033 0.258418i $$-0.0832012\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3.83594e6i − 0.960879i
$$438$$ 0 0
$$439$$ −6.58321e6 −1.63033 −0.815166 0.579227i $$-0.803355\pi$$
−0.815166 + 0.579227i $$0.803355\pi$$
$$440$$ 0 0
$$441$$ 7.40280e6 1.81259
$$442$$ 0 0
$$443$$ − 4.81783e6i − 1.16638i −0.812334 0.583192i $$-0.801803\pi$$
0.812334 0.583192i $$-0.198197\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 9.86819e6i − 2.33598i
$$448$$ 0 0
$$449$$ 6.20399e6 1.45230 0.726149 0.687538i $$-0.241308\pi$$
0.726149 + 0.687538i $$0.241308\pi$$
$$450$$ 0 0
$$451$$ 6.90912e6 1.59949
$$452$$ 0 0
$$453$$ − 3.09892e6i − 0.709520i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2.84383e6i − 0.636962i −0.947929 0.318481i $$-0.896827\pi$$
0.947929 0.318481i $$-0.103173\pi$$
$$458$$ 0 0
$$459$$ 8184.00 0.00181315
$$460$$ 0 0
$$461$$ −1.75605e6 −0.384844 −0.192422 0.981312i $$-0.561634\pi$$
−0.192422 + 0.981312i $$0.561634\pi$$
$$462$$ 0 0
$$463$$ 7.66857e6i 1.66250i 0.555899 + 0.831250i $$0.312374\pi$$
−0.555899 + 0.831250i $$0.687626\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.35903e6i 0.288361i 0.989551 + 0.144181i $$0.0460546\pi$$
−0.989551 + 0.144181i $$0.953945\pi$$
$$468$$ 0 0
$$469$$ 2.51616e6 0.528209
$$470$$ 0 0
$$471$$ −7.38932e6 −1.53480
$$472$$ 0 0
$$473$$ 177600.i 0.0364998i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.05413e6i 0.212129i
$$478$$ 0 0
$$479$$ 2.02706e6 0.403672 0.201836 0.979419i $$-0.435309\pi$$
0.201836 + 0.979419i $$0.435309\pi$$
$$480$$ 0 0
$$481$$ −3.49440e6 −0.688667
$$482$$ 0 0
$$483$$ − 1.52801e7i − 2.98028i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 2.46427e6i − 0.470833i −0.971895 0.235416i $$-0.924355\pi$$
0.971895 0.235416i $$-0.0756454\pi$$
$$488$$ 0 0
$$489$$ 2.23630e6 0.422920
$$490$$ 0 0
$$491$$ 1.03848e7 1.94399 0.971996 0.234998i $$-0.0755084\pi$$
0.971996 + 0.234998i $$0.0755084\pi$$
$$492$$ 0 0
$$493$$ 1.02784e6i 0.190461i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.43798e6i 1.16912i
$$498$$ 0 0
$$499$$ −6.49416e6 −1.16754 −0.583769 0.811919i $$-0.698423\pi$$
−0.583769 + 0.811919i $$0.698423\pi$$
$$500$$ 0 0
$$501$$ 3.06332e6 0.545254
$$502$$ 0 0
$$503$$ − 1.03565e7i − 1.82513i −0.408931 0.912565i $$-0.634098\pi$$
0.408931 0.912565i $$-0.365902\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 343002.i − 0.0592621i
$$508$$ 0 0
$$509$$ −5.87305e6 −1.00478 −0.502388 0.864643i $$-0.667545\pi$$
−0.502388 + 0.864643i $$0.667545\pi$$
$$510$$ 0 0
$$511$$ 7.34616e6 1.24454
$$512$$ 0 0
$$513$$ 52976.0i 0.00888763i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 7.75008e6i 1.27520i
$$518$$ 0 0
$$519$$ 5.83031e6 0.950108
$$520$$ 0 0
$$521$$ 2.17295e6 0.350717 0.175358 0.984505i $$-0.443892\pi$$
0.175358 + 0.984505i $$0.443892\pi$$
$$522$$ 0 0
$$523$$ 1.07361e6i 0.171629i 0.996311 + 0.0858145i $$0.0273492\pi$$
−0.996311 + 0.0858145i $$0.972651\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1.74022e6i − 0.272946i
$$528$$ 0 0
$$529$$ −3.71425e6 −0.577075
$$530$$ 0 0
$$531$$ −2.83127e6 −0.435757
$$532$$ 0 0
$$533$$ 8.95307e6i 1.36507i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 3.14186e6i 0.470166i
$$538$$ 0 0
$$539$$ 1.47442e7 2.18599
$$540$$ 0 0
$$541$$ 7.09033e6 1.04153 0.520767 0.853699i $$-0.325646\pi$$
0.520767 + 0.853699i $$0.325646\pi$$
$$542$$ 0 0
$$543$$ 2.41274e6i 0.351165i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.69763e6i − 0.957091i −0.878063 0.478545i $$-0.841164\pi$$
0.878063 0.478545i $$-0.158836\pi$$
$$548$$ 0 0
$$549$$ −3.18168e6 −0.450532
$$550$$ 0 0
$$551$$ −6.65330e6 −0.933595
$$552$$ 0 0
$$553$$ 6.80334e6i 0.946040i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 1.19008e7i − 1.62532i −0.582735 0.812662i $$-0.698018\pi$$
0.582735 0.812662i $$-0.301982\pi$$
$$558$$ 0 0
$$559$$ −230140. −0.0311503
$$560$$ 0 0
$$561$$ −1.96416e6 −0.263493
$$562$$ 0 0
$$563$$ 8.75636e6i 1.16427i 0.813093 + 0.582133i $$0.197782\pi$$
−0.813093 + 0.582133i $$0.802218\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.29778e7i 1.69528i
$$568$$ 0 0
$$569$$ 1.15677e6 0.149784 0.0748922 0.997192i $$-0.476139\pi$$
0.0748922 + 0.997192i $$0.476139\pi$$
$$570$$ 0 0
$$571$$ −7.07807e6 −0.908500 −0.454250 0.890874i $$-0.650093\pi$$
−0.454250 + 0.890874i $$0.650093\pi$$
$$572$$ 0 0
$$573$$ 6.48886e6i 0.825623i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 3.13404e6i 0.391890i 0.980615 + 0.195945i $$0.0627775\pi$$
−0.980615 + 0.195945i $$0.937223\pi$$
$$578$$ 0 0
$$579$$ −2.20667e7 −2.73553
$$580$$ 0 0
$$581$$ −8.38559e6 −1.03061
$$582$$ 0 0
$$583$$ 2.09952e6i 0.255828i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 1.13833e7i − 1.36355i −0.731561 0.681776i $$-0.761208\pi$$
0.731561 0.681776i $$-0.238792\pi$$
$$588$$ 0 0
$$589$$ 1.12646e7 1.33791
$$590$$ 0 0
$$591$$ −1.81280e7 −2.13491
$$592$$ 0 0
$$593$$ − 1.58655e7i − 1.85275i −0.376599 0.926376i $$-0.622906\pi$$
0.376599 0.926376i $$-0.377094\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.99477e7i 2.29064i
$$598$$ 0 0
$$599$$ 1.50998e7 1.71951 0.859756 0.510705i $$-0.170615\pi$$
0.859756 + 0.510705i $$0.170615\pi$$
$$600$$ 0 0
$$601$$ −8.08705e6 −0.913280 −0.456640 0.889652i $$-0.650947\pi$$
−0.456640 + 0.889652i $$0.650947\pi$$
$$602$$ 0 0
$$603$$ − 2.78162e6i − 0.311534i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 710398.i 0.0782582i 0.999234 + 0.0391291i $$0.0124584\pi$$
−0.999234 + 0.0391291i $$0.987542\pi$$
$$608$$ 0 0
$$609$$ −2.65027e7 −2.89566
$$610$$ 0 0
$$611$$ −1.00428e7 −1.08831
$$612$$ 0 0
$$613$$ 5.96434e6i 0.641078i 0.947235 + 0.320539i $$0.103864\pi$$
−0.947235 + 0.320539i $$0.896136\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 1.48432e7i − 1.56970i −0.619689 0.784848i $$-0.712741\pi$$
0.619689 0.784848i $$-0.287259\pi$$
$$618$$ 0 0
$$619$$ 1.82042e7 1.90961 0.954807 0.297227i $$-0.0960620\pi$$
0.954807 + 0.297227i $$0.0960620\pi$$
$$620$$ 0 0
$$621$$ 140184. 0.0145871
$$622$$ 0 0
$$623$$ 2.60541e7i 2.68940i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 1.27142e7i − 1.29158i
$$628$$ 0 0
$$629$$ −1.04495e6 −0.105310
$$630$$ 0 0
$$631$$ 1.09461e6 0.109443 0.0547214 0.998502i $$-0.482573\pi$$
0.0547214 + 0.998502i $$0.482573\pi$$
$$632$$ 0 0
$$633$$ 1.11404e7i 1.10508i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.91060e7i 1.86561i
$$638$$ 0 0
$$639$$ 7.11721e6 0.689537
$$640$$ 0 0
$$641$$ 7.44046e6 0.715245 0.357622 0.933866i $$-0.383587\pi$$
0.357622 + 0.933866i $$0.383587\pi$$
$$642$$ 0 0
$$643$$ − 1.07915e7i − 1.02933i −0.857391 0.514665i $$-0.827916\pi$$
0.857391 0.514665i $$-0.172084\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.62998e6i 0.904409i 0.891914 + 0.452204i $$0.149362\pi$$
−0.891914 + 0.452204i $$0.850638\pi$$
$$648$$ 0 0
$$649$$ −5.63904e6 −0.525525
$$650$$ 0 0
$$651$$ 4.48714e7 4.14970
$$652$$ 0 0
$$653$$ − 1.00019e7i − 0.917905i −0.888461 0.458953i $$-0.848225\pi$$
0.888461 0.458953i $$-0.151775\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 8.12122e6i − 0.734020i
$$658$$ 0 0
$$659$$ 4.01060e6 0.359746 0.179873 0.983690i $$-0.442431\pi$$
0.179873 + 0.983690i $$0.442431\pi$$
$$660$$ 0 0
$$661$$ 1.20338e7 1.07127 0.535636 0.844449i $$-0.320072\pi$$
0.535636 + 0.844449i $$0.320072\pi$$
$$662$$ 0 0
$$663$$ − 2.54522e6i − 0.224876i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.76058e7i 1.53229i
$$668$$ 0 0
$$669$$ 1.19251e7 1.03014
$$670$$ 0 0
$$671$$ −6.33696e6 −0.543344
$$672$$ 0 0
$$673$$ 2.01231e6i 0.171260i 0.996327 + 0.0856301i $$0.0272903\pi$$
−0.996327 + 0.0856301i $$0.972710\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 1.62410e7i − 1.36188i −0.732337 0.680942i $$-0.761571\pi$$
0.732337 0.680942i $$-0.238429\pi$$
$$678$$ 0 0
$$679$$ −2.06354e7 −1.71767
$$680$$ 0 0
$$681$$ −3.18686e7 −2.63327
$$682$$ 0 0
$$683$$ 4.62910e6i 0.379704i 0.981813 + 0.189852i $$0.0608008\pi$$
−0.981813 + 0.189852i $$0.939199\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 1.05333e7i 0.851476i
$$688$$ 0 0
$$689$$ −2.72063e6 −0.218334
$$690$$ 0 0
$$691$$ 1.16794e7 0.930517 0.465258 0.885175i $$-0.345961\pi$$
0.465258 + 0.885175i $$0.345961\pi$$
$$692$$ 0 0
$$693$$ − 2.52182e7i − 1.99472i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.67728e6i 0.208743i
$$698$$ 0 0
$$699$$ −8.23033e6 −0.637125
$$700$$ 0 0
$$701$$ 1.99543e7 1.53370 0.766851 0.641825i $$-0.221822\pi$$
0.766851 + 0.641825i $$0.221822\pi$$
$$702$$ 0 0
$$703$$ − 6.76407e6i − 0.516202i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 2.20280e7i − 1.65740i
$$708$$ 0 0
$$709$$ 4.88331e6 0.364837 0.182419 0.983221i $$-0.441607\pi$$
0.182419 + 0.983221i $$0.441607\pi$$
$$710$$ 0 0
$$711$$ 7.52113e6 0.557968
$$712$$ 0 0
$$713$$ − 2.98082e7i − 2.19590i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 3.72715e6i − 0.270757i
$$718$$ 0 0
$$719$$ 1.35778e7 0.979505 0.489753 0.871861i $$-0.337087\pi$$
0.489753 + 0.871861i $$0.337087\pi$$
$$720$$ 0 0
$$721$$ −3.12686e7 −2.24012
$$722$$ 0 0
$$723$$ − 7.78241e6i − 0.553692i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 6.42411e6i − 0.450792i −0.974267 0.225396i $$-0.927632\pi$$
0.974267 0.225396i $$-0.0723677\pi$$
$$728$$ 0 0
$$729$$ 1.41117e7 0.983472
$$730$$ 0 0
$$731$$ −68820.0 −0.00476345
$$732$$ 0 0
$$733$$ 9.08556e6i 0.624585i 0.949986 + 0.312293i $$0.101097\pi$$
−0.949986 + 0.312293i $$0.898903\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 5.54016e6i − 0.375711i
$$738$$ 0 0
$$739$$ −2.02457e7 −1.36371 −0.681854 0.731488i $$-0.738826\pi$$
−0.681854 + 0.731488i $$0.738826\pi$$
$$740$$ 0 0
$$741$$ 1.64755e7 1.10229
$$742$$ 0 0
$$743$$ − 5.44831e6i − 0.362067i −0.983477 0.181034i $$-0.942056\pi$$
0.983477 0.181034i $$-0.0579443\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 9.27031e6i 0.607845i
$$748$$ 0 0
$$749$$ 1.24378e7 0.810099
$$750$$ 0 0
$$751$$ −1.14072e6 −0.0738041 −0.0369021 0.999319i $$-0.511749\pi$$
−0.0369021 + 0.999319i $$0.511749\pi$$
$$752$$ 0 0
$$753$$ 2.76144e7i 1.77479i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 1.90153e7i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$758$$ 0 0
$$759$$ −3.36442e7 −2.11985
$$760$$ 0 0
$$761$$ 2.23551e7 1.39931 0.699656 0.714480i $$-0.253337\pi$$
0.699656 + 0.714480i $$0.253337\pi$$
$$762$$ 0 0
$$763$$ − 679724.i − 0.0422689i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 7.30726e6i − 0.448504i
$$768$$ 0 0
$$769$$ 1.00704e7 0.614088 0.307044 0.951695i $$-0.400660\pi$$
0.307044 + 0.951695i $$0.400660\pi$$
$$770$$ 0 0
$$771$$ −2.48330e7 −1.50451
$$772$$ 0 0
$$773$$ − 4.05963e6i − 0.244364i −0.992508 0.122182i $$-0.961011\pi$$
0.992508 0.122182i $$-0.0389892\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 2.69439e7i − 1.60106i
$$778$$ 0 0
$$779$$ −1.73304e7 −1.02321
$$780$$ 0 0
$$781$$ 1.41754e7 0.831585
$$782$$ 0 0
$$783$$ − 243144.i − 0.0141729i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.72256e7i 0.991372i 0.868502 + 0.495686i $$0.165083\pi$$
−0.868502 + 0.495686i $$0.834917\pi$$
$$788$$ 0 0
$$789$$ 5.78780e6 0.330995
$$790$$ 0 0
$$791$$ −1.18884e7 −0.675589
$$792$$ 0 0
$$793$$ − 8.21164e6i − 0.463711i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.10793e7i 1.17547i 0.809055 + 0.587733i $$0.199980\pi$$
−0.809055 + 0.587733i $$0.800020\pi$$
$$798$$ 0 0
$$799$$ −3.00316e6 −0.166422
$$800$$ 0 0
$$801$$ 2.88029e7 1.58619
$$802$$ 0 0
$$803$$ − 1.61750e7i − 0.885231i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 2.61303e7i 1.41241i
$$808$$ 0 0
$$809$$ 1.87877e7 1.00926 0.504629 0.863336i $$-0.331629\pi$$
0.504629 + 0.863336i $$0.331629\pi$$
$$810$$ 0 0
$$811$$ −1.32456e7 −0.707164 −0.353582 0.935404i $$-0.615036\pi$$
−0.353582 + 0.935404i $$0.615036\pi$$
$$812$$ 0 0
$$813$$ 9.48860e6i 0.503473i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 445480.i − 0.0233493i
$$818$$ 0 0
$$819$$ 3.26786e7 1.70237
$$820$$ 0 0
$$821$$ −7.66925e6 −0.397096 −0.198548 0.980091i $$-0.563623\pi$$
−0.198548 + 0.980091i $$0.563623\pi$$
$$822$$ 0 0
$$823$$ − 8.82786e6i − 0.454314i −0.973858 0.227157i $$-0.927057\pi$$
0.973858 0.227157i $$-0.0729430\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 3.06923e7i 1.56051i 0.625463 + 0.780254i $$0.284910\pi$$
−0.625463 + 0.780254i $$0.715090\pi$$
$$828$$ 0 0
$$829$$ −3.28414e7 −1.65972 −0.829860 0.557972i $$-0.811580\pi$$
−0.829860 + 0.557972i $$0.811580\pi$$
$$830$$ 0 0
$$831$$ 1.63485e7 0.821250
$$832$$ 0 0
$$833$$ 5.71336e6i 0.285285i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 411664.i 0.0203109i
$$838$$ 0 0
$$839$$ 8.42117e6 0.413017 0.206508 0.978445i $$-0.433790\pi$$
0.206508 + 0.978445i $$0.433790\pi$$
$$840$$ 0 0
$$841$$ 1.00255e7 0.488784
$$842$$ 0 0
$$843$$ 4.22824e7i 2.04923i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1.51181e7i − 0.724083i
$$848$$ 0 0
$$849$$ 3.58757e7 1.70817
$$850$$ 0 0
$$851$$ −1.78989e7 −0.847234
$$852$$ 0 0
$$853$$ 2.35126e7i 1.10644i 0.833035 + 0.553221i $$0.186601\pi$$
−0.833035 + 0.553221i $$0.813399\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1.13050e7i 0.525799i 0.964823 + 0.262900i $$0.0846788\pi$$
−0.964823 + 0.262900i $$0.915321\pi$$
$$858$$ 0 0
$$859$$ −1.00078e7 −0.462758 −0.231379 0.972864i $$-0.574324\pi$$
−0.231379 + 0.972864i $$0.574324\pi$$
$$860$$ 0 0
$$861$$ −6.90336e7 −3.17360
$$862$$ 0 0
$$863$$ 2.61429e7i 1.19489i 0.801911 + 0.597443i $$0.203817\pi$$
−0.801911 + 0.597443i $$0.796183\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.04757e7i 1.37691i
$$868$$ 0 0
$$869$$ 1.49798e7 0.672911
$$870$$ 0 0
$$871$$ 7.17912e6 0.320646
$$872$$ 0 0
$$873$$ 2.28126e7i 1.01307i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.92041e6i 0.0843129i 0.999111 + 0.0421565i $$0.0134228\pi$$
−0.999111 + 0.0421565i $$0.986577\pi$$
$$878$$ 0 0
$$879$$ −1.56750e6 −0.0684282
$$880$$ 0 0
$$881$$ −2.56594e7 −1.11380 −0.556899 0.830580i $$-0.688009\pi$$
−0.556899 + 0.830580i $$0.688009\pi$$
$$882$$ 0 0
$$883$$ − 2.05643e7i − 0.887590i −0.896128 0.443795i $$-0.853632\pi$$
0.896128 0.443795i $$-0.146368\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 3.16868e7i − 1.35229i −0.736770 0.676143i $$-0.763650\pi$$
0.736770 0.676143i $$-0.236350\pi$$
$$888$$ 0 0
$$889$$ −5.38416e6 −0.228488
$$890$$ 0 0
$$891$$ 2.85749e7 1.20584
$$892$$ 0 0
$$893$$ − 1.94398e7i − 0.815761i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 4.35972e7i − 1.80916i
$$898$$ 0 0
$$899$$ −5.17013e7 −2.13355
$$900$$ 0 0
$$901$$ −813564. −0.0333872
$$902$$ 0 0
$$903$$ − 1.77452e6i − 0.0724205i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 3.96963e6i − 0.160225i −0.996786 0.0801127i $$-0.974472\pi$$
0.996786 0.0801127i $$-0.0255280\pi$$
$$908$$ 0 0
$$909$$ −2.43521e7 −0.977522
$$910$$ 0 0
$$911$$ −1.37945e7 −0.550692 −0.275346 0.961345i $$-0.588792\pi$$
−0.275346 + 0.961345i $$0.588792\pi$$
$$912$$ 0 0
$$913$$ 1.84637e7i 0.733063i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 5.15875e7i − 2.02592i
$$918$$ 0 0
$$919$$ −8.08126e6 −0.315639 −0.157819 0.987468i $$-0.550446\pi$$
−0.157819 + 0.987468i $$0.550446\pi$$
$$920$$ 0 0
$$921$$ −3.55877e7 −1.38245
$$922$$ 0 0
$$923$$ 1.83689e7i 0.709707i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 3.45676e7i 1.32120i
$$928$$ 0 0
$$929$$ −2.99956e7 −1.14030 −0.570150 0.821541i $$-0.693115\pi$$
−0.570150 + 0.821541i $$0.693115\pi$$
$$930$$ 0 0
$$931$$ −3.69833e7 −1.39840
$$932$$ 0 0
$$933$$ − 1.50213e7i − 0.564943i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.07620e7i 0.772540i 0.922386 + 0.386270i $$0.126237\pi$$
−0.922386 + 0.386270i $$0.873763\pi$$
$$938$$ 0 0
$$939$$ 5.94976e7 2.20209
$$940$$ 0 0
$$941$$ −3.47642e6 −0.127985 −0.0639923 0.997950i $$-0.520383\pi$$
−0.0639923 + 0.997950i $$0.520383\pi$$
$$942$$ 0 0
$$943$$ 4.58593e7i 1.67938i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1.86700e6i − 0.0676503i −0.999428 0.0338252i $$-0.989231\pi$$
0.999428 0.0338252i $$-0.0107689\pi$$
$$948$$ 0 0
$$949$$ 2.09602e7 0.755490
$$950$$ 0 0
$$951$$ 5.72763e7 2.05364
$$952$$ 0 0
$$953$$ − 3.85501e7i − 1.37497i −0.726199 0.687484i $$-0.758715\pi$$
0.726199 0.687484i $$-0.241285\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5.83546e7i 2.05966i
$$958$$ 0 0
$$959$$ 4.83044e6 0.169606
$$960$$ 0 0
$$961$$ 5.89056e7 2.05754
$$962$$ 0 0
$$963$$ − 1.37500e7i − 0.477790i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.64875e7i − 0.567008i −0.958971 0.283504i $$-0.908503\pi$$
0.958971 0.283504i $$-0.0914969\pi$$
$$968$$ 0 0
$$969$$ 4.92677e6 0.168559
$$970$$ 0 0
$$971$$ −2.36976e7 −0.806597 −0.403299 0.915068i $$-0.632136\pi$$
−0.403299 + 0.915068i $$0.632136\pi$$
$$972$$ 0 0
$$973$$ − 4.21185e7i − 1.42623i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 5.77590e7i 1.93590i 0.251143 + 0.967950i $$0.419194\pi$$
−0.251143 + 0.967950i $$0.580806\pi$$
$$978$$ 0 0
$$979$$ 5.73667e7 1.91295
$$980$$ 0 0
$$981$$ −751438. −0.0249299
$$982$$ 0 0
$$983$$ − 1.10103e7i − 0.363425i −0.983352 0.181712i $$-0.941836\pi$$
0.983352 0.181712i $$-0.0581640\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 7.74362e7i − 2.53018i
$$988$$ 0 0
$$989$$ −1.17882e6 −0.0383228
$$990$$ 0 0
$$991$$ 3.70807e7 1.19940 0.599700 0.800225i $$-0.295287\pi$$
0.599700 + 0.800225i $$0.295287\pi$$
$$992$$ 0 0
$$993$$ − 1.45515e7i − 0.468311i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 4.52935e6i − 0.144311i −0.997393 0.0721553i $$-0.977012\pi$$
0.997393 0.0721553i $$-0.0229877\pi$$
$$998$$ 0 0
$$999$$ 247192. 0.00783647
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 100.6.c.a.49.2 2
3.2 odd 2 900.6.d.h.649.1 2
4.3 odd 2 400.6.c.c.49.1 2
5.2 odd 4 20.6.a.a.1.1 1
5.3 odd 4 100.6.a.a.1.1 1
5.4 even 2 inner 100.6.c.a.49.1 2
15.2 even 4 180.6.a.e.1.1 1
15.8 even 4 900.6.a.b.1.1 1
15.14 odd 2 900.6.d.h.649.2 2
20.3 even 4 400.6.a.m.1.1 1
20.7 even 4 80.6.a.b.1.1 1
20.19 odd 2 400.6.c.c.49.2 2
35.27 even 4 980.6.a.b.1.1 1
40.27 even 4 320.6.a.n.1.1 1
40.37 odd 4 320.6.a.c.1.1 1
60.47 odd 4 720.6.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
20.6.a.a.1.1 1 5.2 odd 4
80.6.a.b.1.1 1 20.7 even 4
100.6.a.a.1.1 1 5.3 odd 4
100.6.c.a.49.1 2 5.4 even 2 inner
100.6.c.a.49.2 2 1.1 even 1 trivial
180.6.a.e.1.1 1 15.2 even 4
320.6.a.c.1.1 1 40.37 odd 4
320.6.a.n.1.1 1 40.27 even 4
400.6.a.m.1.1 1 20.3 even 4
400.6.c.c.49.1 2 4.3 odd 2
400.6.c.c.49.2 2 20.19 odd 2
720.6.a.l.1.1 1 60.47 odd 4
900.6.a.b.1.1 1 15.8 even 4
900.6.d.h.649.1 2 3.2 odd 2
900.6.d.h.649.2 2 15.14 odd 2
980.6.a.b.1.1 1 35.27 even 4