# Properties

 Label 1575.2.d.e.1324.1 Level $1575$ Weight $2$ Character 1575.1324 Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.5764383184$$ 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 1324.1 Root $$-2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1324 Dual form 1575.2.d.e.1324.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.56155i q^{2} -4.56155 q^{4} +1.00000i q^{7} +6.56155i q^{8} +O(q^{10})$$ $$q-2.56155i q^{2} -4.56155 q^{4} +1.00000i q^{7} +6.56155i q^{8} +1.56155 q^{11} +0.438447i q^{13} +2.56155 q^{14} +7.68466 q^{16} -0.438447i q^{17} +7.12311 q^{19} -4.00000i q^{22} -3.12311i q^{23} +1.12311 q^{26} -4.56155i q^{28} +6.68466 q^{29} -6.56155i q^{32} -1.12311 q^{34} -6.00000i q^{37} -18.2462i q^{38} -5.12311 q^{41} +0.876894i q^{43} -7.12311 q^{44} -8.00000 q^{46} -8.68466i q^{47} -1.00000 q^{49} -2.00000i q^{52} +5.12311i q^{53} -6.56155 q^{56} -17.1231i q^{58} -4.00000 q^{59} +15.3693 q^{61} -1.43845 q^{64} -10.2462i q^{67} +2.00000i q^{68} -8.00000 q^{71} -12.2462i q^{73} -15.3693 q^{74} -32.4924 q^{76} +1.56155i q^{77} +2.43845 q^{79} +13.1231i q^{82} -4.00000i q^{83} +2.24621 q^{86} +10.2462i q^{88} -1.12311 q^{89} -0.438447 q^{91} +14.2462i q^{92} -22.2462 q^{94} -5.80776i q^{97} +2.56155i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} + O(q^{10})$$ $$4q - 10q^{4} - 2q^{11} + 2q^{14} + 6q^{16} + 12q^{19} - 12q^{26} + 2q^{29} + 12q^{34} - 4q^{41} - 12q^{44} - 32q^{46} - 4q^{49} - 18q^{56} - 16q^{59} + 12q^{61} - 14q^{64} - 32q^{71} - 12q^{74} - 64q^{76} + 18q^{79} - 24q^{86} + 12q^{89} - 10q^{91} - 56q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.56155i − 1.81129i −0.424035 0.905646i $$-0.639387\pi$$
0.424035 0.905646i $$-0.360613\pi$$
$$3$$ 0 0
$$4$$ −4.56155 −2.28078
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 6.56155i 2.31986i
$$9$$ 0 0
$$10$$ 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 $$0.980634\pi$$
$$14$$ 2.56155 0.684604
$$15$$ 0 0
$$16$$ 7.68466 1.92116
$$17$$ − 0.438447i − 0.106339i −0.998586 0.0531695i $$-0.983068\pi$$
0.998586 0.0531695i $$-0.0169324\pi$$
$$18$$ 0 0
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 3.12311i − 0.651213i −0.945505 0.325606i $$-0.894432\pi$$
0.945505 0.325606i $$-0.105568\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.12311 0.220259
$$27$$ 0 0
$$28$$ − 4.56155i − 0.862052i
$$29$$ 6.68466 1.24131 0.620655 0.784084i $$-0.286867\pi$$
0.620655 + 0.784084i $$0.286867\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 6.56155i − 1.15993i
$$33$$ 0 0
$$34$$ −1.12311 −0.192611
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ − 18.2462i − 2.95993i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.12311 −0.800095 −0.400047 0.916494i $$-0.631006\pi$$
−0.400047 + 0.916494i $$0.631006\pi$$
$$42$$ 0 0
$$43$$ 0.876894i 0.133725i 0.997762 + 0.0668626i $$0.0212989\pi$$
−0.997762 + 0.0668626i $$0.978701\pi$$
$$44$$ −7.12311 −1.07385
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ − 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 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 5.12311i 0.703713i 0.936054 + 0.351856i $$0.114449\pi$$
−0.936054 + 0.351856i $$0.885551\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.56155 −0.876824
$$57$$ 0 0
$$58$$ − 17.1231i − 2.24837i
$$59$$ −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 0
$$64$$ −1.43845 −0.179806
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 10.2462i − 1.25177i −0.779914 0.625887i $$-0.784737\pi$$
0.779914 0.625887i $$-0.215263\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 0 0
$$70$$ 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 $$-0.254328\pi$$
$$74$$ −15.3693 −1.78665
$$75$$ 0 0
$$76$$ −32.4924 −3.72714
$$77$$ 1.56155i 0.177955i
$$78$$ 0 0
$$79$$ 2.43845 0.274347 0.137173 0.990547i $$-0.456198\pi$$
0.137173 + 0.990547i $$0.456198\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 13.1231i 1.44920i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.24621 0.242215
$$87$$ 0 0
$$88$$ 10.2462i 1.09225i
$$89$$ −1.12311 −0.119049 −0.0595245 0.998227i $$-0.518958\pi$$
−0.0595245 + 0.998227i $$0.518958\pi$$
$$90$$ 0 0
$$91$$ −0.438447 −0.0459618
$$92$$ 14.2462i 1.48527i
$$93$$ 0 0
$$94$$ −22.2462 −2.29452
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 5.80776i − 0.589689i −0.955545 0.294845i $$-0.904732\pi$$
0.955545 0.294845i $$-0.0952679\pi$$
$$98$$ 2.56155i 0.258756i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ 0 0
$$103$$ 5.56155i 0.547996i 0.961730 + 0.273998i $$0.0883462\pi$$
−0.961730 + 0.273998i $$0.911654\pi$$
$$104$$ −2.87689 −0.282103
$$105$$ 0 0
$$106$$ 13.1231 1.27463
$$107$$ 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
$$111$$ 0 0
$$112$$ 7.68466i 0.726132i
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −30.4924 −2.83115
$$117$$ 0 0
$$118$$ 10.2462i 0.943240i
$$119$$ 0.438447 0.0401924
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ − 39.3693i − 3.56433i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.24621i 0.554262i 0.960832 + 0.277131i $$0.0893835\pi$$
−0.960832 + 0.277131i $$0.910616\pi$$
$$128$$ − 9.43845i − 0.834249i
$$129$$ 0 0
$$130$$ 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
$$134$$ −26.2462 −2.26733
$$135$$ 0 0
$$136$$ 2.87689 0.246692
$$137$$ − 17.1231i − 1.46293i −0.681881 0.731463i $$-0.738838\pi$$
0.681881 0.731463i $$-0.261162\pi$$
$$138$$ 0 0
$$139$$ 15.1231 1.28273 0.641363 0.767238i $$-0.278369\pi$$
0.641363 + 0.767238i $$0.278369\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 20.4924i 1.71969i
$$143$$ 0.684658i 0.0572540i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −31.3693 −2.59614
$$147$$ 0 0
$$148$$ 27.3693i 2.24974i
$$149$$ 12.2462 1.00325 0.501624 0.865086i $$-0.332736\pi$$
0.501624 + 0.865086i $$0.332736\pi$$
$$150$$ 0 0
$$151$$ −6.93087 −0.564026 −0.282013 0.959411i $$-0.591002\pi$$
−0.282013 + 0.959411i $$0.591002\pi$$
$$152$$ 46.7386i 3.79100i
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 20.2462i − 1.61582i −0.589303 0.807912i $$-0.700598\pi$$
0.589303 0.807912i $$-0.299402\pi$$
$$158$$ − 6.24621i − 0.496922i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.12311 0.246135
$$162$$ 0 0
$$163$$ − 7.12311i − 0.557925i −0.960302 0.278962i $$-0.910010\pi$$
0.960302 0.278962i $$-0.0899905\pi$$
$$164$$ 23.3693 1.82484
$$165$$ 0 0
$$166$$ −10.2462 −0.795260
$$167$$ − 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
$$171$$ 0 0
$$172$$ − 4.00000i − 0.304997i
$$173$$ 4.43845i 0.337449i 0.985663 + 0.168724i $$0.0539648\pi$$
−0.985663 + 0.168724i $$0.946035\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ 2.87689i 0.215632i
$$179$$ 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 $$0.727195\pi$$
$$182$$ 1.12311i 0.0832501i
$$183$$ 0 0
$$184$$ 20.4924 1.51072
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 0.684658i − 0.0500672i
$$188$$ 39.6155i 2.88926i
$$189$$ 0 0
$$190$$ 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 $$0.754466\pi$$
$$194$$ −14.8769 −1.06810
$$195$$ 0 0
$$196$$ 4.56155 0.325825
$$197$$ 1.12311i 0.0800180i 0.999199 + 0.0400090i $$0.0127387\pi$$
−0.999199 + 0.0400090i $$0.987261\pi$$
$$198$$ 0 0
$$199$$ 1.75379 0.124323 0.0621614 0.998066i $$-0.480201\pi$$
0.0621614 + 0.998066i $$0.480201\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 41.6155i − 2.92806i
$$203$$ 6.68466i 0.469171i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 14.2462 0.992581
$$207$$ 0 0
$$208$$ 3.36932i 0.233620i
$$209$$ 11.1231 0.769401
$$210$$ 0 0
$$211$$ 14.0540 0.967516 0.483758 0.875202i $$-0.339272\pi$$
0.483758 + 0.875202i $$0.339272\pi$$
$$212$$ − 23.3693i − 1.60501i
$$213$$ 0 0
$$214$$ 34.2462 2.34102
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 13.6155i 0.922160i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.192236 0.0129312
$$222$$ 0 0
$$223$$ − 2.43845i − 0.163291i −0.996661 0.0816453i $$-0.973983\pi$$
0.996661 0.0816453i $$-0.0260175\pi$$
$$224$$ 6.56155 0.438412
$$225$$ 0 0
$$226$$ 35.8617 2.38549
$$227$$ 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
$$231$$ 0 0
$$232$$ 43.8617i 2.87966i
$$233$$ − 5.12311i − 0.335626i −0.985819 0.167813i $$-0.946330\pi$$
0.985819 0.167813i $$-0.0536704\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 18.2462 1.18773
$$237$$ 0 0
$$238$$ − 1.12311i − 0.0728001i
$$239$$ 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 $$0.543669\pi$$
$$242$$ 21.9309i 1.40977i
$$243$$ 0 0
$$244$$ −70.1080 −4.48820
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.12311i 0.198718i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ −27.0540 −1.69087
$$257$$ − 10.4924i − 0.654499i −0.944938 0.327250i $$-0.893878\pi$$
0.944938 0.327250i $$-0.106122\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 2.24621i − 0.138771i
$$263$$ 12.8769i 0.794023i 0.917814 + 0.397012i $$0.129953\pi$$
−0.917814 + 0.397012i $$0.870047\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 18.2462 1.11875
$$267$$ 0 0
$$268$$ 46.7386i 2.85502i
$$269$$ −20.7386 −1.26446 −0.632228 0.774782i $$-0.717860\pi$$
−0.632228 + 0.774782i $$0.717860\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ − 3.36932i − 0.204295i
$$273$$ 0 0
$$274$$ −43.8617 −2.64978
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.246211i 0.0147934i 0.999973 + 0.00739670i $$0.00235446\pi$$
−0.999973 + 0.00739670i $$0.997646\pi$$
$$278$$ − 38.7386i − 2.32339i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.4384 −0.742016 −0.371008 0.928630i $$-0.620988\pi$$
−0.371008 + 0.928630i $$0.620988\pi$$
$$282$$ 0 0
$$283$$ − 11.3153i − 0.672627i −0.941750 0.336314i $$-0.890820\pi$$
0.941750 0.336314i $$-0.109180\pi$$
$$284$$ 36.4924 2.16543
$$285$$ 0 0
$$286$$ 1.75379 0.103704
$$287$$ − 5.12311i − 0.302407i
$$288$$ 0 0
$$289$$ 16.8078 0.988692
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 55.8617i 3.26906i
$$293$$ 2.68466i 0.156839i 0.996920 + 0.0784197i $$0.0249874\pi$$
−0.996920 + 0.0784197i $$0.975013\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 39.3693 2.28830
$$297$$ 0 0
$$298$$ − 31.3693i − 1.81718i
$$299$$ 1.36932 0.0791896
$$300$$ 0 0
$$301$$ −0.876894 −0.0505434
$$302$$ 17.7538i 1.02162i
$$303$$ 0 0
$$304$$ 54.7386 3.13948
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 19.3153i 1.10238i 0.834378 + 0.551192i $$0.185827\pi$$
−0.834378 + 0.551192i $$0.814173\pi$$
$$308$$ − 7.12311i − 0.405877i
$$309$$ 0 0
$$310$$ 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 $$-0.217044\pi$$
$$314$$ −51.8617 −2.92673
$$315$$ 0 0
$$316$$ −11.1231 −0.625724
$$317$$ 10.4924i 0.589313i 0.955603 + 0.294657i $$0.0952053\pi$$
−0.955603 + 0.294657i $$0.904795\pi$$
$$318$$ 0 0
$$319$$ 10.4384 0.584441
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 8.00000i − 0.445823i
$$323$$ − 3.12311i − 0.173774i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −18.2462 −1.01056
$$327$$ 0 0
$$328$$ − 33.6155i − 1.85611i
$$329$$ 8.68466 0.478801
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 18.2462i 1.00139i
$$333$$ 0 0
$$334$$ −17.7538 −0.971444
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.50758i 0.0821230i 0.999157 + 0.0410615i $$0.0130740\pi$$
−0.999157 + 0.0410615i $$0.986926\pi$$
$$338$$ − 32.8078i − 1.78451i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −5.75379 −0.310223
$$345$$ 0 0
$$346$$ 11.3693 0.611218
$$347$$ 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
$$351$$ 0 0
$$352$$ − 10.2462i − 0.546125i
$$353$$ − 5.80776i − 0.309116i −0.987984 0.154558i $$-0.950605\pi$$
0.987984 0.154558i $$-0.0493954\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 5.12311 0.271524
$$357$$ 0 0
$$358$$ − 51.2311i − 2.70765i
$$359$$ 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
$$362$$ 45.1231i 2.37162i
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.68466i 0.453335i 0.973972 + 0.226668i $$0.0727831\pi$$
−0.973972 + 0.226668i $$0.927217\pi$$
$$368$$ − 24.0000i − 1.25109i
$$369$$ 0 0
$$370$$ 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 $$0.961748\pi$$
$$374$$ −1.75379 −0.0906863
$$375$$ 0 0
$$376$$ 56.9848 2.93877
$$377$$ 2.93087i 0.150947i
$$378$$ 0 0
$$379$$ 16.4924 0.847159 0.423579 0.905859i $$-0.360773\pi$$
0.423579 + 0.905859i $$0.360773\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 34.7386i − 1.77738i
$$383$$ − 6.24621i − 0.319166i −0.987184 0.159583i $$-0.948985\pi$$
0.987184 0.159583i $$-0.0510150\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 49.6155 2.52536
$$387$$ 0 0
$$388$$ 26.4924i 1.34495i
$$389$$ −24.9309 −1.26405 −0.632023 0.774950i $$-0.717775\pi$$
−0.632023 + 0.774950i $$0.717775\pi$$
$$390$$ 0 0
$$391$$ −1.36932 −0.0692493
$$392$$ − 6.56155i − 0.331408i
$$393$$ 0 0
$$394$$ 2.87689 0.144936
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 27.5616i − 1.38327i −0.722245 0.691637i $$-0.756890\pi$$
0.722245 0.691637i $$-0.243110\pi$$
$$398$$ − 4.49242i − 0.225185i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −31.5616 −1.57611 −0.788054 0.615606i $$-0.788911\pi$$
−0.788054 + 0.615606i $$0.788911\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −74.1080 −3.68701
$$405$$ 0 0
$$406$$ 17.1231 0.849805
$$407$$ − 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
$$411$$ 0 0
$$412$$ − 25.3693i − 1.24986i
$$413$$ − 4.00000i − 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.87689 0.141051
$$417$$ 0 0
$$418$$ − 28.4924i − 1.39361i
$$419$$ 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 $$0.520839\pi$$
$$422$$ − 36.0000i − 1.75245i
$$423$$ 0 0
$$424$$ −33.6155 −1.63251
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.3693i 0.743773i
$$428$$ − 60.9848i − 2.94781i
$$429$$ 0 0
$$430$$ 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
$$436$$ 24.2462 1.16118
$$437$$ − 22.2462i − 1.06418i
$$438$$ 0 0
$$439$$ −9.36932 −0.447173 −0.223587 0.974684i $$-0.571777\pi$$
−0.223587 + 0.974684i $$0.571777\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 0.492423i − 0.0234221i
$$443$$ 2.63068i 0.124988i 0.998045 + 0.0624938i $$0.0199054\pi$$
−0.998045 + 0.0624938i $$0.980095\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −6.24621 −0.295767
$$447$$ 0 0
$$448$$ − 1.43845i − 0.0679602i
$$449$$ −1.80776 −0.0853137 −0.0426568 0.999090i $$-0.513582\pi$$
−0.0426568 + 0.999090i $$0.513582\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ − 63.8617i − 3.00380i
$$453$$ 0 0
$$454$$ 28.9848 1.36033
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.1231i 0.800985i 0.916300 + 0.400493i $$0.131161\pi$$
−0.916300 + 0.400493i $$0.868839\pi$$
$$458$$ 27.8617i 1.30189i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.1231 0.611204 0.305602 0.952159i $$-0.401142\pi$$
0.305602 + 0.952159i $$0.401142\pi$$
$$462$$ 0 0
$$463$$ 12.4924i 0.580572i 0.956940 + 0.290286i $$0.0937505\pi$$
−0.956940 + 0.290286i $$0.906250\pi$$
$$464$$ 51.3693 2.38476
$$465$$ 0 0
$$466$$ −13.1231 −0.607916
$$467$$ 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
$$471$$ 0 0
$$472$$ − 26.2462i − 1.20808i
$$473$$ 1.36932i 0.0629613i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 0 0
$$478$$ − 50.7386i − 2.32073i
$$479$$ 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
$$482$$ 10.8769i 0.495429i
$$483$$ 0 0
$$484$$ 39.0540 1.77518
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3.12311i 0.141521i 0.997493 + 0.0707607i $$0.0225427\pi$$
−0.997493 + 0.0707607i $$0.977457\pi$$
$$488$$ 100.847i 4.56511i
$$489$$ 0 0
$$490$$ 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
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 8.00000i − 0.358849i
$$498$$ 0 0
$$499$$ −41.1771 −1.84334 −0.921670 0.387976i $$-0.873174\pi$$
−0.921670 + 0.387976i $$0.873174\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 22.7386i − 1.01487i
$$503$$ − 38.9309i − 1.73584i −0.496703 0.867921i $$-0.665456\pi$$
0.496703 0.867921i $$-0.334544\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −12.4924 −0.555356
$$507$$ 0 0
$$508$$ − 28.4924i − 1.26415i
$$509$$ −11.7538 −0.520978 −0.260489 0.965477i $$-0.583884\pi$$
−0.260489 + 0.965477i $$0.583884\pi$$
$$510$$ 0 0
$$511$$ 12.2462 0.541740
$$512$$ 50.4233i 2.22842i
$$513$$ 0 0
$$514$$ −26.8769 −1.18549
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 13.5616i − 0.596436i
$$518$$ − 15.3693i − 0.675289i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ 40.4924i 1.77061i 0.465011 + 0.885305i $$0.346050\pi$$
−0.465011 + 0.885305i $$0.653950\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 32.9848 1.43821
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.2462 0.575922
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 32.4924i − 1.40873i
$$533$$ − 2.24621i − 0.0972942i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 67.2311 2.90394
$$537$$ 0 0
$$538$$ 53.1231i 2.29030i
$$539$$ −1.56155 −0.0672608
$$540$$ 0 0
$$541$$ −37.8078 −1.62548 −0.812741 0.582625i $$-0.802026\pi$$
−0.812741 + 0.582625i $$0.802026\pi$$
$$542$$ 40.9848i 1.76045i
$$543$$ 0 0
$$544$$ −2.87689 −0.123346
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.24621i 0.0960411i 0.998846 + 0.0480205i $$0.0152913\pi$$
−0.998846 + 0.0480205i $$0.984709\pi$$
$$548$$ 78.1080i 3.33661i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 47.6155 2.02849
$$552$$ 0 0
$$553$$ 2.43845i 0.103693i
$$554$$ 0.630683 0.0267952
$$555$$ 0 0
$$556$$ −68.9848 −2.92561
$$557$$ − 13.1231i − 0.556044i −0.960575 0.278022i $$-0.910321\pi$$
0.960575 0.278022i $$-0.0896788\pi$$
$$558$$ 0 0
$$559$$ −0.384472 −0.0162614
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 31.8617i 1.34401i
$$563$$ 28.0000i 1.18006i 0.807382 + 0.590030i $$0.200884\pi$$
−0.807382 + 0.590030i $$0.799116\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −28.9848 −1.21832
$$567$$ 0 0
$$568$$ − 52.4924i − 2.20253i
$$569$$ −30.9848 −1.29895 −0.649476 0.760382i $$-0.725012\pi$$
−0.649476 + 0.760382i $$0.725012\pi$$
$$570$$ 0 0
$$571$$ 40.4924 1.69456 0.847278 0.531150i $$-0.178240\pi$$
0.847278 + 0.531150i $$0.178240\pi$$
$$572$$ − 3.12311i − 0.130584i
$$573$$ 0 0
$$574$$ −13.1231 −0.547748
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 24.0540i 1.00138i 0.865627 + 0.500690i $$0.166920\pi$$
−0.865627 + 0.500690i $$0.833080\pi$$
$$578$$ − 43.0540i − 1.79081i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ 8.00000i 0.331326i
$$584$$ 80.3542 3.32508
$$585$$ 0 0
$$586$$ 6.87689 0.284082
$$587$$ − 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
$$591$$ 0 0
$$592$$ − 46.1080i − 1.89503i
$$593$$ 27.5616i 1.13182i 0.824468 + 0.565909i $$0.191475\pi$$
−0.824468 + 0.565909i $$0.808525\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −55.8617 −2.28819
$$597$$ 0 0
$$598$$ − 3.50758i − 0.143436i
$$599$$ −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 $$0.457727\pi$$
$$602$$ 2.24621i 0.0915487i
$$603$$ 0 0
$$604$$ 31.6155 1.28642
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 42.0540i 1.70692i 0.521160 + 0.853459i $$0.325500\pi$$
−0.521160 + 0.853459i $$0.674500\pi$$
$$608$$ − 46.7386i − 1.89550i
$$609$$ 0 0
$$610$$ 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 $$0.692462\pi$$
$$614$$ 49.4773 1.99674
$$615$$ 0 0
$$616$$ −10.2462 −0.412832
$$617$$ 32.2462i 1.29818i 0.760710 + 0.649092i $$0.224851\pi$$
−0.760710 + 0.649092i $$0.775149\pi$$
$$618$$ 0 0
$$619$$ −32.1080 −1.29053 −0.645264 0.763960i $$-0.723253\pi$$
−0.645264 + 0.763960i $$0.723253\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 80.9848i 3.24720i
$$623$$ − 1.12311i − 0.0449963i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −57.1231 −2.28310
$$627$$ 0 0
$$628$$ 92.3542i 3.68533i
$$629$$ −2.63068 −0.104892
$$630$$ 0 0
$$631$$ −11.8078 −0.470060 −0.235030 0.971988i $$-0.575519\pi$$
−0.235030 + 0.971988i $$0.575519\pi$$
$$632$$ 16.0000i 0.636446i
$$633$$ 0 0
$$634$$ 26.8769 1.06742
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 0.438447i − 0.0173719i
$$638$$ − 26.7386i − 1.05859i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ 1.56155i 0.0615816i 0.999526 + 0.0307908i $$0.00980257\pi$$
−0.999526 + 0.0307908i $$0.990197\pi$$
$$644$$ −14.2462 −0.561379
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 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
$$652$$ 32.4924i 1.27250i
$$653$$ 33.2311i 1.30043i 0.759750 + 0.650216i $$0.225322\pi$$
−0.759750 + 0.650216i $$0.774678\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −39.3693 −1.53711
$$657$$ 0 0
$$658$$ − 22.2462i − 0.867248i
$$659$$ 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 $$0.531767\pi$$
$$662$$ − 30.7386i − 1.19469i
$$663$$ 0 0
$$664$$ 26.2462 1.01855
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 20.8769i − 0.808357i
$$668$$ 31.6155i 1.22324i
$$669$$ 0 0
$$670$$ 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 $$0.789523\pi$$
$$674$$ 3.86174 0.148749
$$675$$ 0 0
$$676$$ −58.4233 −2.24705
$$677$$ 4.93087i 0.189509i 0.995501 + 0.0947544i $$0.0302066\pi$$
−0.995501 + 0.0947544i $$0.969793\pi$$
$$678$$ 0 0
$$679$$ 5.80776 0.222882
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 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
$$686$$ −2.56155 −0.0978005
$$687$$ 0 0
$$688$$ 6.73863i 0.256908i
$$689$$ −2.24621 −0.0855738
$$690$$ 0 0
$$691$$ −24.4924 −0.931736 −0.465868 0.884854i $$-0.654258\pi$$
−0.465868 + 0.884854i $$0.654258\pi$$
$$692$$ − 20.2462i − 0.769645i
$$693$$ 0 0
$$694$$ 18.2462 0.692617
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.24621i 0.0850813i
$$698$$ 26.8769i 1.01731i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −28.9309 −1.09270 −0.546352 0.837556i $$-0.683984\pi$$
−0.546352 + 0.837556i $$0.683984\pi$$
$$702$$ 0 0
$$703$$ − 42.7386i − 1.61192i
$$704$$ −2.24621 −0.0846573
$$705$$ 0 0
$$706$$ −14.8769 −0.559899
$$707$$ 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
$$711$$ 0 0
$$712$$ − 7.36932i − 0.276177i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −91.2311 −3.40946
$$717$$ 0 0
$$718$$ − 20.4924i − 0.764770i
$$719$$ 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
$$722$$ − 81.3002i − 3.02568i
$$723$$ 0 0
$$724$$ 80.3542 2.98634
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 52.4924i − 1.94684i −0.229035 0.973418i $$-0.573557\pi$$
0.229035 0.973418i $$-0.426443\pi$$
$$728$$ − 2.87689i − 0.106625i
$$729$$ 0 0
$$730$$ 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 $$0.960604\pi$$
$$734$$ 22.2462 0.821123
$$735$$ 0 0
$$736$$ −20.4924 −0.755361
$$737$$ − 16.0000i − 0.589368i
$$738$$ 0 0
$$739$$ −34.9309 −1.28495 −0.642476 0.766305i $$-0.722093\pi$$
−0.642476 + 0.766305i $$0.722093\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 13.1231i 0.481764i
$$743$$ 32.9848i 1.21010i 0.796189 + 0.605048i $$0.206846\pi$$
−0.796189 + 0.605048i $$0.793154\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 11.8617 0.434289
$$747$$ 0 0
$$748$$ 3.12311i 0.114192i
$$749$$ −13.3693 −0.488504
$$750$$ 0 0
$$751$$ 17.0691 0.622861 0.311431 0.950269i $$-0.399192\pi$$
0.311431 + 0.950269i $$0.399192\pi$$
$$752$$ − 66.7386i − 2.43371i
$$753$$ 0 0
$$754$$ 7.50758 0.273410
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 39.3693i − 1.43090i −0.698663 0.715451i $$-0.746221\pi$$
0.698663 0.715451i $$-0.253779\pi$$
$$758$$ − 42.2462i − 1.53445i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −48.2462 −1.74892 −0.874462 0.485094i $$-0.838785\pi$$
−0.874462 + 0.485094i $$0.838785\pi$$
$$762$$ 0 0
$$763$$ − 5.31534i − 0.192428i
$$764$$ −61.8617 −2.23808
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ − 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
$$771$$ 0 0
$$772$$ − 88.3542i − 3.17994i
$$773$$ − 36.9309i − 1.32831i −0.747594 0.664156i $$-0.768791\pi$$
0.747594 0.664156i $$-0.231209\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 38.1080 1.36800
$$777$$ 0 0
$$778$$ 63.8617i 2.28955i
$$779$$ −36.4924 −1.30748
$$780$$ 0 0
$$781$$ −12.4924 −0.447014
$$782$$ 3.50758i 0.125431i
$$783$$ 0 0
$$784$$ −7.68466 −0.274452
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 49.1771i 1.75297i 0.481426 + 0.876487i $$0.340119\pi$$
−0.481426 + 0.876487i $$0.659881\pi$$
$$788$$ − 5.12311i − 0.182503i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 6.73863i 0.239296i
$$794$$ −70.6004 −2.50551
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 24.0540i 0.852036i 0.904715 + 0.426018i $$0.140084\pi$$
−0.904715 + 0.426018i $$0.859916\pi$$
$$798$$ 0 0
$$799$$ −3.80776 −0.134709
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 80.8466i 2.85479i
$$803$$ − 19.1231i − 0.674840i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 106.600i 3.75019i
$$809$$ 16.5464 0.581740 0.290870 0.956763i $$-0.406055\pi$$
0.290870 + 0.956763i $$0.406055\pi$$
$$810$$ 0 0
$$811$$ 19.6155 0.688794 0.344397 0.938824i $$-0.388083\pi$$
0.344397 + 0.938824i $$0.388083\pi$$
$$812$$ − 30.4924i − 1.07007i
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 6.24621i 0.218527i
$$818$$ 16.6307i 0.581478i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 21.4233 0.747678 0.373839 0.927494i $$-0.378041\pi$$
0.373839 + 0.927494i $$0.378041\pi$$
$$822$$ 0 0
$$823$$ − 36.4924i − 1.27205i −0.771670 0.636023i $$-0.780578\pi$$
0.771670 0.636023i $$-0.219422\pi$$
$$824$$ −36.4924 −1.27127
$$825$$ 0 0
$$826$$ −10.2462 −0.356511
$$827$$ − 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 0
$$832$$ − 0.630683i − 0.0218650i
$$833$$ 0.438447i 0.0151913i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −50.7386 −1.75483
$$837$$ 0 0
$$838$$ − 67.2311i − 2.32246i
$$839$$ −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
$$842$$ 6.87689i 0.236993i
$$843$$ 0 0
$$844$$ −64.1080 −2.20669
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 8.56155i − 0.294178i
$$848$$ 39.3693i 1.35195i
$$849$$ 0 0
$$850$$ 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 $$-0.0396724\pi$$
$$854$$ 39.3693 1.34719
$$855$$ 0 0
$$856$$ −87.7235 −2.99833
$$857$$ − 15.7538i − 0.538139i −0.963121 0.269070i $$-0.913284\pi$$
0.963121 0.269070i $$-0.0867162\pi$$
$$858$$ 0 0
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 50.7386i − 1.72816i
$$863$$ 25.7538i 0.876669i 0.898812 + 0.438335i $$0.144431\pi$$
−0.898812 + 0.438335i $$0.855569\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 21.1231 0.717792
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 3.80776 0.129170
$$870$$ 0 0
$$871$$ 4.49242 0.152220
$$872$$ − 34.8769i − 1.18108i
$$873$$ 0 0
$$874$$ −56.9848 −1.92754
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 40.2462i 1.35902i 0.733667 + 0.679509i $$0.237807\pi$$
−0.733667 + 0.679509i $$0.762193\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 11.8617 0.399632 0.199816 0.979833i $$-0.435966\pi$$
0.199816 + 0.979833i $$0.435966\pi$$
$$882$$ 0 0
$$883$$ − 8.49242i − 0.285793i −0.989738 0.142896i $$-0.954358\pi$$
0.989738 0.142896i $$-0.0456416\pi$$
$$884$$ −0.876894 −0.0294931
$$885$$ 0 0
$$886$$ 6.73863 0.226389
$$887$$ 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
$$891$$ 0 0
$$892$$ 11.1231i 0.372429i
$$893$$ − 61.8617i − 2.07012i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 9.43845 0.315316
$$897$$ 0 0
$$898$$ 4.63068i 0.154528i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 2.24621 0.0748321
$$902$$ 20.4924i 0.682323i
$$903$$ 0 0
$$904$$ −91.8617 −3.05528
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 24.1080i 0.800491i 0.916408 + 0.400246i $$0.131075\pi$$
−0.916408 + 0.400246i $$0.868925\pi$$
$$908$$ − 51.6155i − 1.71292i
$$909$$ 0 0
$$910$$ 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
$$914$$ 43.8617 1.45082
$$915$$ 0 0
$$916$$ 49.6155 1.63934
$$917$$ 0.876894i 0.0289576i
$$918$$ 0 0
$$919$$ −40.3002 −1.32938 −0.664690 0.747119i $$-0.731436\pi$$
−0.664690 + 0.747119i $$0.731436\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 33.6155i − 1.10707i
$$923$$ − 3.50758i − 0.115453i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 32.0000 1.05159
$$927$$ 0 0
$$928$$ − 43.8617i − 1.43983i
$$929$$ 22.1080 0.725338 0.362669 0.931918i $$-0.381866\pi$$
0.362669 + 0.931918i $$0.381866\pi$$
$$930$$ 0 0
$$931$$ −7.12311 −0.233450
$$932$$ 23.3693i 0.765487i
$$933$$ 0 0
$$934$$ 57.4773 1.88071
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 55.6695i 1.81864i 0.416094 + 0.909322i $$0.363399\pi$$
−0.416094 + 0.909322i $$0.636601\pi$$
$$938$$ − 26.2462i − 0.856969i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −43.8617 −1.42985 −0.714926 0.699200i $$-0.753540\pi$$
−0.714926 + 0.699200i $$0.753540\pi$$
$$942$$ 0 0
$$943$$ 16.0000i 0.521032i
$$944$$ −30.7386 −1.00046
$$945$$ 0 0
$$946$$ 3.50758 0.114041
$$947$$ 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
$$951$$ 0 0
$$952$$ 2.87689i 0.0932407i
$$953$$ 33.1231i 1.07296i 0.843912 + 0.536481i $$0.180247\pi$$
−0.843912 + 0.536481i $$0.819753\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −90.3542 −2.92226
$$957$$ 0 0
$$958$$ − 12.4924i − 0.403612i
$$959$$ 17.1231 0.552934
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 6.73863i − 0.217262i
$$963$$ 0 0
$$964$$ 19.3693 0.623844
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 35.1231i 1.12948i 0.825268 + 0.564741i $$0.191024\pi$$
−0.825268 + 0.564741i $$0.808976\pi$$
$$968$$ − 56.1771i − 1.80560i
$$969$$ 0 0
$$970$$ 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
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 118.108 3.78054
$$977$$ 33.2311i 1.06316i 0.847009 + 0.531578i $$0.178401\pi$$
−0.847009 + 0.531578i $$0.821599\pi$$
$$978$$ 0 0
$$979$$ −1.75379 −0.0560513
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 105.477i − 3.36591i
$$983$$ − 51.4233i − 1.64015i −0.572257 0.820074i $$-0.693932\pi$$
0.572257 0.820074i $$-0.306068\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −7.50758 −0.239090
$$987$$ 0 0
$$988$$ − 14.2462i − 0.453232i
$$989$$ 2.73863 0.0870835
$$990$$ 0 0
$$991$$ 12.4924 0.396835 0.198417 0.980118i $$-0.436420\pi$$
0.198417 + 0.980118i $$0.436420\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −20.4924 −0.649980
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.68466i 0.0850240i 0.999096 + 0.0425120i $$0.0135361\pi$$
−0.999096 + 0.0425120i $$0.986464\pi$$
$$998$$ 105.477i 3.33882i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1575.2.d.e.1324.1 4
3.2 odd 2 175.2.b.b.99.4 4
5.2 odd 4 315.2.a.e.1.2 2
5.3 odd 4 1575.2.a.p.1.1 2
5.4 even 2 inner 1575.2.d.e.1324.4 4
12.11 even 2 2800.2.g.t.449.2 4
15.2 even 4 35.2.a.b.1.1 2
15.8 even 4 175.2.a.f.1.2 2
15.14 odd 2 175.2.b.b.99.1 4
20.7 even 4 5040.2.a.bt.1.1 2
21.20 even 2 1225.2.b.f.99.4 4
35.27 even 4 2205.2.a.x.1.2 2
60.23 odd 4 2800.2.a.bi.1.2 2
60.47 odd 4 560.2.a.i.1.1 2
60.59 even 2 2800.2.g.t.449.3 4
105.2 even 12 245.2.e.i.116.2 4
105.17 odd 12 245.2.e.h.226.2 4
105.32 even 12 245.2.e.i.226.2 4
105.47 odd 12 245.2.e.h.116.2 4
105.62 odd 4 245.2.a.d.1.1 2
105.83 odd 4 1225.2.a.s.1.2 2
105.104 even 2 1225.2.b.f.99.1 4
120.77 even 4 2240.2.a.bh.1.1 2
120.107 odd 4 2240.2.a.bd.1.2 2
165.32 odd 4 4235.2.a.m.1.2 2
195.77 even 4 5915.2.a.l.1.2 2
420.167 even 4 3920.2.a.bs.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 15.2 even 4
175.2.a.f.1.2 2 15.8 even 4
175.2.b.b.99.1 4 15.14 odd 2
175.2.b.b.99.4 4 3.2 odd 2
245.2.a.d.1.1 2 105.62 odd 4
245.2.e.h.116.2 4 105.47 odd 12
245.2.e.h.226.2 4 105.17 odd 12
245.2.e.i.116.2 4 105.2 even 12
245.2.e.i.226.2 4 105.32 even 12
315.2.a.e.1.2 2 5.2 odd 4
560.2.a.i.1.1 2 60.47 odd 4
1225.2.a.s.1.2 2 105.83 odd 4
1225.2.b.f.99.1 4 105.104 even 2
1225.2.b.f.99.4 4 21.20 even 2
1575.2.a.p.1.1 2 5.3 odd 4
1575.2.d.e.1324.1 4 1.1 even 1 trivial
1575.2.d.e.1324.4 4 5.4 even 2 inner
2205.2.a.x.1.2 2 35.27 even 4
2240.2.a.bd.1.2 2 120.107 odd 4
2240.2.a.bh.1.1 2 120.77 even 4
2800.2.a.bi.1.2 2 60.23 odd 4
2800.2.g.t.449.2 4 12.11 even 2
2800.2.g.t.449.3 4 60.59 even 2
3920.2.a.bs.1.2 2 420.167 even 4
4235.2.a.m.1.2 2 165.32 odd 4
5040.2.a.bt.1.1 2 20.7 even 4
5915.2.a.l.1.2 2 195.77 even 4