# Properties

 Label 2800.2.g.t.449.2 Level $2800$ Weight $2$ Character 2800.449 Analytic conductor $22.358$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$-1.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.449 Dual form 2800.2.g.t.449.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155i q^{3} -1.00000i q^{7} +0.561553 q^{9} +O(q^{10})$$ $$q-1.56155i q^{3} -1.00000i q^{7} +0.561553 q^{9} +1.56155 q^{11} +0.438447i q^{13} +0.438447i q^{17} -7.12311 q^{19} -1.56155 q^{21} -3.12311i q^{23} -5.56155i q^{27} -6.68466 q^{29} -2.43845i q^{33} -6.00000i q^{37} +0.684658 q^{39} +5.12311 q^{41} -0.876894i q^{43} -8.68466i q^{47} -1.00000 q^{49} +0.684658 q^{51} -5.12311i q^{53} +11.1231i q^{57} -4.00000 q^{59} +15.3693 q^{61} -0.561553i q^{63} +10.2462i q^{67} -4.87689 q^{69} -8.00000 q^{71} -12.2462i q^{73} -1.56155i q^{77} -2.43845 q^{79} -7.00000 q^{81} -4.00000i q^{83} +10.4384i q^{87} +1.12311 q^{89} +0.438447 q^{91} -5.80776i q^{97} +0.876894 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{9} + O(q^{10})$$ $$4q - 6q^{9} - 2q^{11} - 12q^{19} + 2q^{21} - 2q^{29} - 22q^{39} + 4q^{41} - 4q^{49} - 22q^{51} - 16q^{59} + 12q^{61} - 36q^{69} - 32q^{71} - 18q^{79} - 28q^{81} - 12q^{89} + 10q^{91} + 20q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.56155i − 0.901563i −0.892634 0.450781i $$-0.851145\pi$$
0.892634 0.450781i $$-0.148855\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 0 0
$$9$$ 0.561553 0.187184
$$10$$ 0 0
$$11$$ 1.56155 0.470826 0.235413 0.971895i $$-0.424356\pi$$
0.235413 + 0.971895i $$0.424356\pi$$
$$12$$ 0 0
$$13$$ 0.438447i 0.121603i 0.998150 + 0.0608017i $$0.0193657\pi$$
−0.998150 + 0.0608017i $$0.980634\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.438447i 0.106339i 0.998586 + 0.0531695i $$0.0169324\pi$$
−0.998586 + 0.0531695i $$0.983068\pi$$
$$18$$ 0 0
$$19$$ −7.12311 −1.63415 −0.817076 0.576530i $$-0.804407\pi$$
−0.817076 + 0.576530i $$0.804407\pi$$
$$20$$ 0 0
$$21$$ −1.56155 −0.340759
$$22$$ 0 0
$$23$$ − 3.12311i − 0.651213i −0.945505 0.325606i $$-0.894432\pi$$
0.945505 0.325606i $$-0.105568\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.56155i − 1.07032i
$$28$$ 0 0
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ − 2.43845i − 0.424479i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 0.684658 0.109633
$$40$$ 0 0
$$41$$ 5.12311 0.800095 0.400047 0.916494i $$-0.368994\pi$$
0.400047 + 0.916494i $$0.368994\pi$$
$$42$$ 0 0
$$43$$ − 0.876894i − 0.133725i −0.997762 0.0668626i $$-0.978701\pi$$
0.997762 0.0668626i $$-0.0212989\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 8.68466i − 1.26679i −0.773830 0.633394i $$-0.781661\pi$$
0.773830 0.633394i $$-0.218339\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0.684658 0.0958714
$$52$$ 0 0
$$53$$ − 5.12311i − 0.703713i −0.936054 0.351856i $$-0.885551\pi$$
0.936054 0.351856i $$-0.114449\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 11.1231i 1.47329i
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 15.3693 1.96784 0.983920 0.178611i $$-0.0571605\pi$$
0.983920 + 0.178611i $$0.0571605\pi$$
$$62$$ 0 0
$$63$$ − 0.561553i − 0.0707490i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.2462i 1.25177i 0.779914 + 0.625887i $$0.215263\pi$$
−0.779914 + 0.625887i $$0.784737\pi$$
$$68$$ 0 0
$$69$$ −4.87689 −0.587109
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ − 12.2462i − 1.43331i −0.697428 0.716655i $$-0.745672\pi$$
0.697428 0.716655i $$-0.254328\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1.56155i − 0.177955i
$$78$$ 0 0
$$79$$ −2.43845 −0.274347 −0.137173 0.990547i $$-0.543802\pi$$
−0.137173 + 0.990547i $$0.543802\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.4384i 1.11912i
$$88$$ 0 0
$$89$$ 1.12311 0.119049 0.0595245 0.998227i $$-0.481042\pi$$
0.0595245 + 0.998227i $$0.481042\pi$$
$$90$$ 0 0
$$91$$ 0.438447 0.0459618
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 5.80776i − 0.589689i −0.955545 0.294845i $$-0.904732\pi$$
0.955545 0.294845i $$-0.0952679\pi$$
$$98$$ 0 0
$$99$$ 0.876894 0.0881312
$$100$$ 0 0
$$101$$ −16.2462 −1.61656 −0.808279 0.588799i $$-0.799601\pi$$
−0.808279 + 0.588799i $$0.799601\pi$$
$$102$$ 0 0
$$103$$ − 5.56155i − 0.547996i −0.961730 0.273998i $$-0.911654\pi$$
0.961730 0.273998i $$-0.0883462\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.3693i 1.29246i 0.763142 + 0.646230i $$0.223655\pi$$
−0.763142 + 0.646230i $$0.776345\pi$$
$$108$$ 0 0
$$109$$ −5.31534 −0.509117 −0.254559 0.967057i $$-0.581930\pi$$
−0.254559 + 0.967057i $$0.581930\pi$$
$$110$$ 0 0
$$111$$ −9.36932 −0.889296
$$112$$ 0 0
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.246211i 0.0227622i
$$118$$ 0 0
$$119$$ 0.438447 0.0401924
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 6.24621i − 0.554262i −0.960832 0.277131i $$-0.910616\pi$$
0.960832 0.277131i $$-0.0893835\pi$$
$$128$$ 0 0
$$129$$ −1.36932 −0.120562
$$130$$ 0 0
$$131$$ 0.876894 0.0766146 0.0383073 0.999266i $$-0.487803\pi$$
0.0383073 + 0.999266i $$0.487803\pi$$
$$132$$ 0 0
$$133$$ 7.12311i 0.617652i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 17.1231i 1.46293i 0.681881 + 0.731463i $$0.261162\pi$$
−0.681881 + 0.731463i $$0.738838\pi$$
$$138$$ 0 0
$$139$$ −15.1231 −1.28273 −0.641363 0.767238i $$-0.721631\pi$$
−0.641363 + 0.767238i $$0.721631\pi$$
$$140$$ 0 0
$$141$$ −13.5616 −1.14209
$$142$$ 0 0
$$143$$ 0.684658i 0.0572540i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.56155i 0.128795i
$$148$$ 0 0
$$149$$ −12.2462 −1.00325 −0.501624 0.865086i $$-0.667264\pi$$
−0.501624 + 0.865086i $$0.667264\pi$$
$$150$$ 0 0
$$151$$ 6.93087 0.564026 0.282013 0.959411i $$-0.408998\pi$$
0.282013 + 0.959411i $$0.408998\pi$$
$$152$$ 0 0
$$153$$ 0.246211i 0.0199050i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 20.2462i − 1.61582i −0.589303 0.807912i $$-0.700598\pi$$
0.589303 0.807912i $$-0.299402\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ −3.12311 −0.246135
$$162$$ 0 0
$$163$$ 7.12311i 0.557925i 0.960302 + 0.278962i $$0.0899905\pi$$
−0.960302 + 0.278962i $$0.910010\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 6.93087i − 0.536327i −0.963373 0.268163i $$-0.913583\pi$$
0.963373 0.268163i $$-0.0864167\pi$$
$$168$$ 0 0
$$169$$ 12.8078 0.985213
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ − 4.43845i − 0.337449i −0.985663 0.168724i $$-0.946035\pi$$
0.985663 0.168724i $$-0.0539648\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.24621i 0.469494i
$$178$$ 0 0
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −17.6155 −1.30935 −0.654676 0.755910i $$-0.727195\pi$$
−0.654676 + 0.755910i $$0.727195\pi$$
$$182$$ 0 0
$$183$$ − 24.0000i − 1.77413i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.684658i 0.0500672i
$$188$$ 0 0
$$189$$ −5.56155 −0.404543
$$190$$ 0 0
$$191$$ 13.5616 0.981280 0.490640 0.871363i $$-0.336763\pi$$
0.490640 + 0.871363i $$0.336763\pi$$
$$192$$ 0 0
$$193$$ 19.3693i 1.39423i 0.716957 + 0.697117i $$0.245534\pi$$
−0.716957 + 0.697117i $$0.754466\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 1.12311i − 0.0800180i −0.999199 0.0400090i $$-0.987261\pi$$
0.999199 0.0400090i $$-0.0127387\pi$$
$$198$$ 0 0
$$199$$ −1.75379 −0.124323 −0.0621614 0.998066i $$-0.519799\pi$$
−0.0621614 + 0.998066i $$0.519799\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ 6.68466i 0.469171i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 1.75379i − 0.121897i
$$208$$ 0 0
$$209$$ −11.1231 −0.769401
$$210$$ 0 0
$$211$$ −14.0540 −0.967516 −0.483758 0.875202i $$-0.660728\pi$$
−0.483758 + 0.875202i $$0.660728\pi$$
$$212$$ 0 0
$$213$$ 12.4924i 0.855967i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −19.1231 −1.29222
$$220$$ 0 0
$$221$$ −0.192236 −0.0129312
$$222$$ 0 0
$$223$$ 2.43845i 0.163291i 0.996661 + 0.0816453i $$0.0260175\pi$$
−0.996661 + 0.0816453i $$0.973983\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 11.3153i 0.751026i 0.926817 + 0.375513i $$0.122533\pi$$
−0.926817 + 0.375513i $$0.877467\pi$$
$$228$$ 0 0
$$229$$ −10.8769 −0.718765 −0.359383 0.933190i $$-0.617013\pi$$
−0.359383 + 0.933190i $$0.617013\pi$$
$$230$$ 0 0
$$231$$ −2.43845 −0.160438
$$232$$ 0 0
$$233$$ 5.12311i 0.335626i 0.985819 + 0.167813i $$0.0536704\pi$$
−0.985819 + 0.167813i $$0.946330\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3.80776i 0.247341i
$$238$$ 0 0
$$239$$ 19.8078 1.28126 0.640629 0.767851i $$-0.278674\pi$$
0.640629 + 0.767851i $$0.278674\pi$$
$$240$$ 0 0
$$241$$ −4.24621 −0.273523 −0.136761 0.990604i $$-0.543669\pi$$
−0.136761 + 0.990604i $$0.543669\pi$$
$$242$$ 0 0
$$243$$ − 5.75379i − 0.369106i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3.12311i − 0.198718i
$$248$$ 0 0
$$249$$ −6.24621 −0.395838
$$250$$ 0 0
$$251$$ 8.87689 0.560305 0.280152 0.959956i $$-0.409615\pi$$
0.280152 + 0.959956i $$0.409615\pi$$
$$252$$ 0 0
$$253$$ − 4.87689i − 0.306608i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 10.4924i 0.654499i 0.944938 + 0.327250i $$0.106122\pi$$
−0.944938 + 0.327250i $$0.893878\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −3.75379 −0.232354
$$262$$ 0 0
$$263$$ 12.8769i 0.794023i 0.917814 + 0.397012i $$0.129953\pi$$
−0.917814 + 0.397012i $$0.870047\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 1.75379i − 0.107330i
$$268$$ 0 0
$$269$$ 20.7386 1.26446 0.632228 0.774782i $$-0.282140\pi$$
0.632228 + 0.774782i $$0.282140\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ − 0.684658i − 0.0414374i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.246211i 0.0147934i 0.999973 + 0.00739670i $$0.00235446\pi$$
−0.999973 + 0.00739670i $$0.997646\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.4384 0.742016 0.371008 0.928630i $$-0.379012\pi$$
0.371008 + 0.928630i $$0.379012\pi$$
$$282$$ 0 0
$$283$$ 11.3153i 0.672627i 0.941750 + 0.336314i $$0.109180\pi$$
−0.941750 + 0.336314i $$0.890820\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 5.12311i − 0.302407i
$$288$$ 0 0
$$289$$ 16.8078 0.988692
$$290$$ 0 0
$$291$$ −9.06913 −0.531642
$$292$$ 0 0
$$293$$ − 2.68466i − 0.156839i −0.996920 0.0784197i $$-0.975013\pi$$
0.996920 0.0784197i $$-0.0249874\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 8.68466i − 0.503935i
$$298$$ 0 0
$$299$$ 1.36932 0.0791896
$$300$$ 0 0
$$301$$ −0.876894 −0.0505434
$$302$$ 0 0
$$303$$ 25.3693i 1.45743i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 19.3153i − 1.10238i −0.834378 0.551192i $$-0.814173\pi$$
0.834378 0.551192i $$-0.185827\pi$$
$$308$$ 0 0
$$309$$ −8.68466 −0.494053
$$310$$ 0 0
$$311$$ −31.6155 −1.79275 −0.896376 0.443294i $$-0.853810\pi$$
−0.896376 + 0.443294i $$0.853810\pi$$
$$312$$ 0 0
$$313$$ − 22.3002i − 1.26048i −0.776400 0.630241i $$-0.782956\pi$$
0.776400 0.630241i $$-0.217044\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 10.4924i − 0.589313i −0.955603 0.294657i $$-0.904795\pi$$
0.955603 0.294657i $$-0.0952053\pi$$
$$318$$ 0 0
$$319$$ −10.4384 −0.584441
$$320$$ 0 0
$$321$$ 20.8769 1.16523
$$322$$ 0 0
$$323$$ − 3.12311i − 0.173774i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 8.30019i 0.459001i
$$328$$ 0 0
$$329$$ −8.68466 −0.478801
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ − 3.36932i − 0.184637i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.50758i 0.0821230i 0.999157 + 0.0410615i $$0.0130740\pi$$
−0.999157 + 0.0410615i $$0.986926\pi$$
$$338$$ 0 0
$$339$$ −21.8617 −1.18737
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.12311i 0.382388i 0.981552 + 0.191194i $$0.0612360\pi$$
−0.981552 + 0.191194i $$0.938764\pi$$
$$348$$ 0 0
$$349$$ −10.4924 −0.561646 −0.280823 0.959760i $$-0.590607\pi$$
−0.280823 + 0.959760i $$0.590607\pi$$
$$350$$ 0 0
$$351$$ 2.43845 0.130155
$$352$$ 0 0
$$353$$ 5.80776i 0.309116i 0.987984 + 0.154558i $$0.0493954\pi$$
−0.987984 + 0.154558i $$0.950605\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 0.684658i − 0.0362360i
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 0 0
$$363$$ 13.3693i 0.701707i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.68466i − 0.453335i −0.973972 0.226668i $$-0.927217\pi$$
0.973972 0.226668i $$-0.0727831\pi$$
$$368$$ 0 0
$$369$$ 2.87689 0.149765
$$370$$ 0 0
$$371$$ −5.12311 −0.265978
$$372$$ 0 0
$$373$$ 4.63068i 0.239768i 0.992788 + 0.119884i $$0.0382522\pi$$
−0.992788 + 0.119884i $$0.961748\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 2.93087i − 0.150947i
$$378$$ 0 0
$$379$$ −16.4924 −0.847159 −0.423579 0.905859i $$-0.639227\pi$$
−0.423579 + 0.905859i $$0.639227\pi$$
$$380$$ 0 0
$$381$$ −9.75379 −0.499702
$$382$$ 0 0
$$383$$ − 6.24621i − 0.319166i −0.987184 0.159583i $$-0.948985\pi$$
0.987184 0.159583i $$-0.0510150\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 0.492423i − 0.0250312i
$$388$$ 0 0
$$389$$ 24.9309 1.26405 0.632023 0.774950i $$-0.282225\pi$$
0.632023 + 0.774950i $$0.282225\pi$$
$$390$$ 0 0
$$391$$ 1.36932 0.0692493
$$392$$ 0 0
$$393$$ − 1.36932i − 0.0690729i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 27.5616i − 1.38327i −0.722245 0.691637i $$-0.756890\pi$$
0.722245 0.691637i $$-0.243110\pi$$
$$398$$ 0 0
$$399$$ 11.1231 0.556852
$$400$$ 0 0
$$401$$ 31.5616 1.57611 0.788054 0.615606i $$-0.211089\pi$$
0.788054 + 0.615606i $$0.211089\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 9.36932i − 0.464420i
$$408$$ 0 0
$$409$$ −6.49242 −0.321030 −0.160515 0.987033i $$-0.551315\pi$$
−0.160515 + 0.987033i $$0.551315\pi$$
$$410$$ 0 0
$$411$$ 26.7386 1.31892
$$412$$ 0 0
$$413$$ 4.00000i 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 23.6155i 1.15646i
$$418$$ 0 0
$$419$$ 26.2462 1.28221 0.641106 0.767453i $$-0.278476\pi$$
0.641106 + 0.767453i $$0.278476\pi$$
$$420$$ 0 0
$$421$$ −2.68466 −0.130842 −0.0654211 0.997858i $$-0.520839\pi$$
−0.0654211 + 0.997858i $$0.520839\pi$$
$$422$$ 0 0
$$423$$ − 4.87689i − 0.237123i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 15.3693i − 0.743773i
$$428$$ 0 0
$$429$$ 1.06913 0.0516181
$$430$$ 0 0
$$431$$ 19.8078 0.954106 0.477053 0.878874i $$-0.341705\pi$$
0.477053 + 0.878874i $$0.341705\pi$$
$$432$$ 0 0
$$433$$ 8.24621i 0.396288i 0.980173 + 0.198144i $$0.0634913\pi$$
−0.980173 + 0.198144i $$0.936509\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 22.2462i 1.06418i
$$438$$ 0 0
$$439$$ 9.36932 0.447173 0.223587 0.974684i $$-0.428223\pi$$
0.223587 + 0.974684i $$0.428223\pi$$
$$440$$ 0 0
$$441$$ −0.561553 −0.0267406
$$442$$ 0 0
$$443$$ 2.63068i 0.124988i 0.998045 + 0.0624938i $$0.0199054\pi$$
−0.998045 + 0.0624938i $$0.980095\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 19.1231i 0.904492i
$$448$$ 0 0
$$449$$ 1.80776 0.0853137 0.0426568 0.999090i $$-0.486418\pi$$
0.0426568 + 0.999090i $$0.486418\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 0 0
$$453$$ − 10.8229i − 0.508505i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.1231i 0.800985i 0.916300 + 0.400493i $$0.131161\pi$$
−0.916300 + 0.400493i $$0.868839\pi$$
$$458$$ 0 0
$$459$$ 2.43845 0.113817
$$460$$ 0 0
$$461$$ −13.1231 −0.611204 −0.305602 0.952159i $$-0.598858\pi$$
−0.305602 + 0.952159i $$0.598858\pi$$
$$462$$ 0 0
$$463$$ − 12.4924i − 0.580572i −0.956940 0.290286i $$-0.906250\pi$$
0.956940 0.290286i $$-0.0937505\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.4384i 1.03833i 0.854675 + 0.519164i $$0.173757\pi$$
−0.854675 + 0.519164i $$0.826243\pi$$
$$468$$ 0 0
$$469$$ 10.2462 0.473126
$$470$$ 0 0
$$471$$ −31.6155 −1.45677
$$472$$ 0 0
$$473$$ − 1.36932i − 0.0629613i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 2.87689i − 0.131724i
$$478$$ 0 0
$$479$$ 4.87689 0.222831 0.111415 0.993774i $$-0.464462\pi$$
0.111415 + 0.993774i $$0.464462\pi$$
$$480$$ 0 0
$$481$$ 2.63068 0.119949
$$482$$ 0 0
$$483$$ 4.87689i 0.221906i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 3.12311i − 0.141521i −0.997493 0.0707607i $$-0.977457\pi$$
0.997493 0.0707607i $$-0.0225427\pi$$
$$488$$ 0 0
$$489$$ 11.1231 0.503004
$$490$$ 0 0
$$491$$ 41.1771 1.85830 0.929148 0.369708i $$-0.120542\pi$$
0.929148 + 0.369708i $$0.120542\pi$$
$$492$$ 0 0
$$493$$ − 2.93087i − 0.132000i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000i 0.358849i
$$498$$ 0 0
$$499$$ 41.1771 1.84334 0.921670 0.387976i $$-0.126826\pi$$
0.921670 + 0.387976i $$0.126826\pi$$
$$500$$ 0 0
$$501$$ −10.8229 −0.483532
$$502$$ 0 0
$$503$$ − 38.9309i − 1.73584i −0.496703 0.867921i $$-0.665456\pi$$
0.496703 0.867921i $$-0.334544\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 20.0000i − 0.888231i
$$508$$ 0 0
$$509$$ 11.7538 0.520978 0.260489 0.965477i $$-0.416116\pi$$
0.260489 + 0.965477i $$0.416116\pi$$
$$510$$ 0 0
$$511$$ −12.2462 −0.541740
$$512$$ 0 0
$$513$$ 39.6155i 1.74907i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 13.5616i − 0.596436i
$$518$$ 0 0
$$519$$ −6.93087 −0.304231
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ − 40.4924i − 1.77061i −0.465011 0.885305i $$-0.653950\pi$$
0.465011 0.885305i $$-0.346050\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.2462 0.575922
$$530$$ 0 0
$$531$$ −2.24621 −0.0974773
$$532$$ 0 0
$$533$$ 2.24621i 0.0972942i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 31.2311i − 1.34772i
$$538$$ 0 0
$$539$$ −1.56155 −0.0672608
$$540$$ 0 0
$$541$$ −37.8078 −1.62548 −0.812741 0.582625i $$-0.802026\pi$$
−0.812741 + 0.582625i $$0.802026\pi$$
$$542$$ 0 0
$$543$$ 27.5076i 1.18046i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 2.24621i − 0.0960411i −0.998846 0.0480205i $$-0.984709\pi$$
0.998846 0.0480205i $$-0.0152913\pi$$
$$548$$ 0 0
$$549$$ 8.63068 0.368349
$$550$$ 0 0
$$551$$ 47.6155 2.02849
$$552$$ 0 0
$$553$$ 2.43845i 0.103693i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13.1231i 0.556044i 0.960575 + 0.278022i $$0.0896788\pi$$
−0.960575 + 0.278022i $$0.910321\pi$$
$$558$$ 0 0
$$559$$ 0.384472 0.0162614
$$560$$ 0 0
$$561$$ 1.06913 0.0451387
$$562$$ 0 0
$$563$$ 28.0000i 1.18006i 0.807382 + 0.590030i $$0.200884\pi$$
−0.807382 + 0.590030i $$0.799116\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 7.00000i 0.293972i
$$568$$ 0 0
$$569$$ 30.9848 1.29895 0.649476 0.760382i $$-0.274988\pi$$
0.649476 + 0.760382i $$0.274988\pi$$
$$570$$ 0 0
$$571$$ −40.4924 −1.69456 −0.847278 0.531150i $$-0.821760\pi$$
−0.847278 + 0.531150i $$0.821760\pi$$
$$572$$ 0 0
$$573$$ − 21.1771i − 0.884685i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 24.0540i 1.00138i 0.865627 + 0.500690i $$0.166920\pi$$
−0.865627 + 0.500690i $$0.833080\pi$$
$$578$$ 0 0
$$579$$ 30.2462 1.25699
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ − 8.00000i − 0.331326i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 26.2462i − 1.08330i −0.840605 0.541649i $$-0.817800\pi$$
0.840605 0.541649i $$-0.182200\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −1.75379 −0.0721412
$$592$$ 0 0
$$593$$ − 27.5616i − 1.13182i −0.824468 0.565909i $$-0.808525\pi$$
0.824468 0.565909i $$-0.191475\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.73863i 0.112085i
$$598$$ 0 0
$$599$$ −11.8078 −0.482452 −0.241226 0.970469i $$-0.577550\pi$$
−0.241226 + 0.970469i $$0.577550\pi$$
$$600$$ 0 0
$$601$$ 6.49242 0.264831 0.132416 0.991194i $$-0.457727\pi$$
0.132416 + 0.991194i $$0.457727\pi$$
$$602$$ 0 0
$$603$$ 5.75379i 0.234312i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 42.0540i − 1.70692i −0.521160 0.853459i $$-0.674500\pi$$
0.521160 0.853459i $$-0.325500\pi$$
$$608$$ 0 0
$$609$$ 10.4384 0.422987
$$610$$ 0 0
$$611$$ 3.80776 0.154046
$$612$$ 0 0
$$613$$ 40.7386i 1.64542i 0.568463 + 0.822709i $$0.307538\pi$$
−0.568463 + 0.822709i $$0.692462\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 32.2462i − 1.29818i −0.760710 0.649092i $$-0.775149\pi$$
0.760710 0.649092i $$-0.224851\pi$$
$$618$$ 0 0
$$619$$ 32.1080 1.29053 0.645264 0.763960i $$-0.276747\pi$$
0.645264 + 0.763960i $$0.276747\pi$$
$$620$$ 0 0
$$621$$ −17.3693 −0.697007
$$622$$ 0 0
$$623$$ − 1.12311i − 0.0449963i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 17.3693i 0.693664i
$$628$$ 0 0
$$629$$ 2.63068 0.104892
$$630$$ 0 0
$$631$$ 11.8078 0.470060 0.235030 0.971988i $$-0.424481\pi$$
0.235030 + 0.971988i $$0.424481\pi$$
$$632$$ 0 0
$$633$$ 21.9460i 0.872276i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 0.438447i − 0.0173719i
$$638$$ 0 0
$$639$$ −4.49242 −0.177717
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ − 1.56155i − 0.0615816i −0.999526 0.0307908i $$-0.990197\pi$$
0.999526 0.0307908i $$-0.00980257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.4924i 1.43467i 0.696731 + 0.717333i $$0.254637\pi$$
−0.696731 + 0.717333i $$0.745363\pi$$
$$648$$ 0 0
$$649$$ −6.24621 −0.245185
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 33.2311i − 1.30043i −0.759750 0.650216i $$-0.774678\pi$$
0.759750 0.650216i $$-0.225322\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 6.87689i − 0.268293i
$$658$$ 0 0
$$659$$ 9.17708 0.357488 0.178744 0.983896i $$-0.442797\pi$$
0.178744 + 0.983896i $$0.442797\pi$$
$$660$$ 0 0
$$661$$ −5.12311 −0.199266 −0.0996329 0.995024i $$-0.531767\pi$$
−0.0996329 + 0.995024i $$0.531767\pi$$
$$662$$ 0 0
$$663$$ 0.300187i 0.0116583i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.8769i 0.808357i
$$668$$ 0 0
$$669$$ 3.80776 0.147217
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ 31.8617i 1.22818i 0.789236 + 0.614090i $$0.210477\pi$$
−0.789236 + 0.614090i $$0.789523\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 4.93087i − 0.189509i −0.995501 0.0947544i $$-0.969793\pi$$
0.995501 0.0947544i $$-0.0302066\pi$$
$$678$$ 0 0
$$679$$ −5.80776 −0.222882
$$680$$ 0 0
$$681$$ 17.6695 0.677097
$$682$$ 0 0
$$683$$ 6.73863i 0.257847i 0.991655 + 0.128923i $$0.0411521\pi$$
−0.991655 + 0.128923i $$0.958848\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16.9848i 0.648012i
$$688$$ 0 0
$$689$$ 2.24621 0.0855738
$$690$$ 0 0
$$691$$ 24.4924 0.931736 0.465868 0.884854i $$-0.345742\pi$$
0.465868 + 0.884854i $$0.345742\pi$$
$$692$$ 0 0
$$693$$ − 0.876894i − 0.0333105i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.24621i 0.0850813i
$$698$$ 0 0
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ 28.9309 1.09270 0.546352 0.837556i $$-0.316016\pi$$
0.546352 + 0.837556i $$0.316016\pi$$
$$702$$ 0 0
$$703$$ 42.7386i 1.61192i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.2462i 0.611002i
$$708$$ 0 0
$$709$$ −27.1771 −1.02066 −0.510328 0.859980i $$-0.670476\pi$$
−0.510328 + 0.859980i $$0.670476\pi$$
$$710$$ 0 0
$$711$$ −1.36932 −0.0513534
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 30.9309i − 1.15513i
$$718$$ 0 0
$$719$$ 8.38447 0.312688 0.156344 0.987703i $$-0.450029\pi$$
0.156344 + 0.987703i $$0.450029\pi$$
$$720$$ 0 0
$$721$$ −5.56155 −0.207123
$$722$$ 0 0
$$723$$ 6.63068i 0.246598i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.4924i 1.94684i 0.229035 + 0.973418i $$0.426443\pi$$
−0.229035 + 0.973418i $$0.573557\pi$$
$$728$$ 0 0
$$729$$ −29.9848 −1.11055
$$730$$ 0 0
$$731$$ 0.384472 0.0142202
$$732$$ 0 0
$$733$$ 6.68466i 0.246903i 0.992351 + 0.123452i $$0.0393964\pi$$
−0.992351 + 0.123452i $$0.960604\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 16.0000i 0.589368i
$$738$$ 0 0
$$739$$ 34.9309 1.28495 0.642476 0.766305i $$-0.277907\pi$$
0.642476 + 0.766305i $$0.277907\pi$$
$$740$$ 0 0
$$741$$ −4.87689 −0.179157
$$742$$ 0 0
$$743$$ 32.9848i 1.21010i 0.796189 + 0.605048i $$0.206846\pi$$
−0.796189 + 0.605048i $$0.793154\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 2.24621i − 0.0821846i
$$748$$ 0 0
$$749$$ 13.3693 0.488504
$$750$$ 0 0
$$751$$ −17.0691 −0.622861 −0.311431 0.950269i $$-0.600808\pi$$
−0.311431 + 0.950269i $$0.600808\pi$$
$$752$$ 0 0
$$753$$ − 13.8617i − 0.505150i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 39.3693i − 1.43090i −0.698663 0.715451i $$-0.746221\pi$$
0.698663 0.715451i $$-0.253779\pi$$
$$758$$ 0 0
$$759$$ −7.61553 −0.276426
$$760$$ 0 0
$$761$$ 48.2462 1.74892 0.874462 0.485094i $$-0.161215\pi$$
0.874462 + 0.485094i $$0.161215\pi$$
$$762$$ 0 0
$$763$$ 5.31534i 0.192428i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1.75379i − 0.0633256i
$$768$$ 0 0
$$769$$ 42.4924 1.53232 0.766158 0.642652i $$-0.222166\pi$$
0.766158 + 0.642652i $$0.222166\pi$$
$$770$$ 0 0
$$771$$ 16.3845 0.590072
$$772$$ 0 0
$$773$$ 36.9309i 1.32831i 0.747594 + 0.664156i $$0.231209\pi$$
−0.747594 + 0.664156i $$0.768791\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 9.36932i 0.336122i
$$778$$ 0 0
$$779$$ −36.4924 −1.30748
$$780$$ 0 0
$$781$$ −12.4924 −0.447014
$$782$$ 0 0
$$783$$ 37.1771i 1.32860i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 49.1771i − 1.75297i −0.481426 0.876487i $$-0.659881\pi$$
0.481426 0.876487i $$-0.340119\pi$$
$$788$$ 0 0
$$789$$ 20.1080 0.715862
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 6.73863i 0.239296i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 24.0540i − 0.852036i −0.904715 0.426018i $$-0.859916\pi$$
0.904715 0.426018i $$-0.140084\pi$$
$$798$$ 0 0
$$799$$ 3.80776 0.134709
$$800$$ 0 0
$$801$$ 0.630683 0.0222841
$$802$$ 0 0
$$803$$ − 19.1231i − 0.674840i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 32.3845i − 1.13999i
$$808$$ 0 0
$$809$$ −16.5464 −0.581740 −0.290870 0.956763i $$-0.593945\pi$$
−0.290870 + 0.956763i $$0.593945\pi$$
$$810$$ 0 0
$$811$$ −19.6155 −0.688794 −0.344397 0.938824i $$-0.611917\pi$$
−0.344397 + 0.938824i $$0.611917\pi$$
$$812$$ 0 0
$$813$$ − 24.9848i − 0.876257i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 6.24621i 0.218527i
$$818$$ 0 0
$$819$$ 0.246211 0.00860332
$$820$$ 0 0
$$821$$ −21.4233 −0.747678 −0.373839 0.927494i $$-0.621959\pi$$
−0.373839 + 0.927494i $$0.621959\pi$$
$$822$$ 0 0
$$823$$ 36.4924i 1.27205i 0.771670 + 0.636023i $$0.219422\pi$$
−0.771670 + 0.636023i $$0.780578\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 5.36932i − 0.186709i −0.995633 0.0933547i $$-0.970241\pi$$
0.995633 0.0933547i $$-0.0297591\pi$$
$$828$$ 0 0
$$829$$ −34.8769 −1.21132 −0.605662 0.795722i $$-0.707092\pi$$
−0.605662 + 0.795722i $$0.707092\pi$$
$$830$$ 0 0
$$831$$ 0.384472 0.0133372
$$832$$ 0 0
$$833$$ − 0.438447i − 0.0151913i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −28.8769 −0.996941 −0.498471 0.866907i $$-0.666105\pi$$
−0.498471 + 0.866907i $$0.666105\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ 0 0
$$843$$ − 19.4233i − 0.668974i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8.56155i 0.294178i
$$848$$ 0 0
$$849$$ 17.6695 0.606416
$$850$$ 0 0
$$851$$ −18.7386 −0.642352
$$852$$ 0 0
$$853$$ − 7.26137i − 0.248624i −0.992243 0.124312i $$-0.960328\pi$$
0.992243 0.124312i $$-0.0396724\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15.7538i 0.538139i 0.963121 + 0.269070i $$0.0867162\pi$$
−0.963121 + 0.269070i $$0.913284\pi$$
$$858$$ 0 0
$$859$$ −16.4924 −0.562714 −0.281357 0.959603i $$-0.590785\pi$$
−0.281357 + 0.959603i $$0.590785\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 25.7538i 0.876669i 0.898812 + 0.438335i $$0.144431\pi$$
−0.898812 + 0.438335i $$0.855569\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 26.2462i − 0.891368i
$$868$$ 0 0
$$869$$ −3.80776 −0.129170
$$870$$ 0 0
$$871$$ −4.49242 −0.152220
$$872$$ 0 0
$$873$$ − 3.26137i − 0.110381i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 40.2462i 1.35902i 0.733667 + 0.679509i $$0.237807\pi$$
−0.733667 + 0.679509i $$0.762193\pi$$
$$878$$ 0 0
$$879$$ −4.19224 −0.141401
$$880$$ 0 0
$$881$$ −11.8617 −0.399632 −0.199816 0.979833i $$-0.564034\pi$$
−0.199816 + 0.979833i $$0.564034\pi$$
$$882$$ 0 0
$$883$$ 8.49242i 0.285793i 0.989738 + 0.142896i $$0.0456416\pi$$
−0.989738 + 0.142896i $$0.954358\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 20.4924i 0.688068i 0.938957 + 0.344034i $$0.111794\pi$$
−0.938957 + 0.344034i $$0.888206\pi$$
$$888$$ 0 0
$$889$$ −6.24621 −0.209491
$$890$$ 0 0
$$891$$ −10.9309 −0.366198
$$892$$ 0 0
$$893$$ 61.8617i 2.07012i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 2.13826i − 0.0713944i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 2.24621 0.0748321
$$902$$ 0 0
$$903$$ 1.36932i 0.0455680i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 24.1080i − 0.800491i −0.916408 0.400246i $$-0.868925\pi$$
0.916408 0.400246i $$-0.131075\pi$$
$$908$$ 0 0
$$909$$ −9.12311 −0.302594
$$910$$ 0 0
$$911$$ 28.4924 0.943996 0.471998 0.881600i $$-0.343533\pi$$
0.471998 + 0.881600i $$0.343533\pi$$
$$912$$ 0 0
$$913$$ − 6.24621i − 0.206719i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 0.876894i − 0.0289576i
$$918$$ 0 0
$$919$$ 40.3002 1.32938 0.664690 0.747119i $$-0.268564\pi$$
0.664690 + 0.747119i $$0.268564\pi$$
$$920$$ 0 0
$$921$$ −30.1619 −0.993869
$$922$$ 0 0
$$923$$ − 3.50758i − 0.115453i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 3.12311i − 0.102576i
$$928$$ 0 0
$$929$$ −22.1080 −0.725338 −0.362669 0.931918i $$-0.618134\pi$$
−0.362669 + 0.931918i $$0.618134\pi$$
$$930$$ 0 0
$$931$$ 7.12311 0.233450
$$932$$ 0 0
$$933$$ 49.3693i 1.61628i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 55.6695i 1.81864i 0.416094 + 0.909322i $$0.363399\pi$$
−0.416094 + 0.909322i $$0.636601\pi$$
$$938$$ 0 0
$$939$$ −34.8229 −1.13640
$$940$$ 0 0
$$941$$ 43.8617 1.42985 0.714926 0.699200i $$-0.246460\pi$$
0.714926 + 0.699200i $$0.246460\pi$$
$$942$$ 0 0
$$943$$ − 16.0000i − 0.521032i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.00000i 0.129983i 0.997886 + 0.0649913i $$0.0207020\pi$$
−0.997886 + 0.0649913i $$0.979298\pi$$
$$948$$ 0 0
$$949$$ 5.36932 0.174295
$$950$$ 0 0
$$951$$ −16.3845 −0.531303
$$952$$ 0 0
$$953$$ − 33.1231i − 1.07296i −0.843912 0.536481i $$-0.819753\pi$$
0.843912 0.536481i $$-0.180247\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 16.3002i 0.526910i
$$958$$ 0 0
$$959$$ 17.1231 0.552934
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 7.50758i 0.241928i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 35.1231i − 1.12948i −0.825268 0.564741i $$-0.808976\pi$$
0.825268 0.564741i $$-0.191024\pi$$
$$968$$ 0 0
$$969$$ −4.87689 −0.156668
$$970$$ 0 0
$$971$$ −49.4773 −1.58780 −0.793901 0.608048i $$-0.791953\pi$$
−0.793901 + 0.608048i $$0.791953\pi$$
$$972$$ 0 0
$$973$$ 15.1231i 0.484825i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 33.2311i − 1.06316i −0.847009 0.531578i $$-0.821599\pi$$
0.847009 0.531578i $$-0.178401\pi$$
$$978$$ 0 0
$$979$$ 1.75379 0.0560513
$$980$$ 0 0
$$981$$ −2.98485 −0.0952988
$$982$$ 0 0
$$983$$ − 51.4233i − 1.64015i −0.572257 0.820074i $$-0.693932\pi$$
0.572257 0.820074i $$-0.306068\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 13.5616i 0.431669i
$$988$$ 0 0
$$989$$ −2.73863 −0.0870835
$$990$$ 0 0
$$991$$ −12.4924 −0.396835 −0.198417 0.980118i $$-0.563580\pi$$
−0.198417 + 0.980118i $$0.563580\pi$$
$$992$$ 0 0
$$993$$ 18.7386i 0.594653i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.68466i 0.0850240i 0.999096 + 0.0425120i $$0.0135361\pi$$
−0.999096 + 0.0425120i $$0.986464\pi$$
$$998$$ 0 0
$$999$$ −33.3693 −1.05576
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 2800.2.g.t.449.2 4
4.3 odd 2 175.2.b.b.99.4 4
5.2 odd 4 560.2.a.i.1.1 2
5.3 odd 4 2800.2.a.bi.1.2 2
5.4 even 2 inner 2800.2.g.t.449.3 4
12.11 even 2 1575.2.d.e.1324.1 4
15.2 even 4 5040.2.a.bt.1.1 2
20.3 even 4 175.2.a.f.1.2 2
20.7 even 4 35.2.a.b.1.1 2
20.19 odd 2 175.2.b.b.99.1 4
28.27 even 2 1225.2.b.f.99.4 4
35.27 even 4 3920.2.a.bs.1.2 2
40.27 even 4 2240.2.a.bh.1.1 2
40.37 odd 4 2240.2.a.bd.1.2 2
60.23 odd 4 1575.2.a.p.1.1 2
60.47 odd 4 315.2.a.e.1.2 2
60.59 even 2 1575.2.d.e.1324.4 4
140.27 odd 4 245.2.a.d.1.1 2
140.47 odd 12 245.2.e.h.116.2 4
140.67 even 12 245.2.e.i.226.2 4
140.83 odd 4 1225.2.a.s.1.2 2
140.87 odd 12 245.2.e.h.226.2 4
140.107 even 12 245.2.e.i.116.2 4
140.139 even 2 1225.2.b.f.99.1 4
220.87 odd 4 4235.2.a.m.1.2 2
260.207 even 4 5915.2.a.l.1.2 2
420.167 even 4 2205.2.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 20.7 even 4
175.2.a.f.1.2 2 20.3 even 4
175.2.b.b.99.1 4 20.19 odd 2
175.2.b.b.99.4 4 4.3 odd 2
245.2.a.d.1.1 2 140.27 odd 4
245.2.e.h.116.2 4 140.47 odd 12
245.2.e.h.226.2 4 140.87 odd 12
245.2.e.i.116.2 4 140.107 even 12
245.2.e.i.226.2 4 140.67 even 12
315.2.a.e.1.2 2 60.47 odd 4
560.2.a.i.1.1 2 5.2 odd 4
1225.2.a.s.1.2 2 140.83 odd 4
1225.2.b.f.99.1 4 140.139 even 2
1225.2.b.f.99.4 4 28.27 even 2
1575.2.a.p.1.1 2 60.23 odd 4
1575.2.d.e.1324.1 4 12.11 even 2
1575.2.d.e.1324.4 4 60.59 even 2
2205.2.a.x.1.2 2 420.167 even 4
2240.2.a.bd.1.2 2 40.37 odd 4
2240.2.a.bh.1.1 2 40.27 even 4
2800.2.a.bi.1.2 2 5.3 odd 4
2800.2.g.t.449.2 4 1.1 even 1 trivial
2800.2.g.t.449.3 4 5.4 even 2 inner
3920.2.a.bs.1.2 2 35.27 even 4
4235.2.a.m.1.2 2 220.87 odd 4
5040.2.a.bt.1.1 2 15.2 even 4
5915.2.a.l.1.2 2 260.207 even 4