# Properties

 Label 1225.2.b.f.99.1 Level $1225$ Weight $2$ Character 1225.99 Analytic conductor $9.782$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.1 Root $$-2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 1225.99 Dual form 1225.2.b.f.99.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.56155i q^{2} +1.56155i q^{3} -4.56155 q^{4} +4.00000 q^{6} +6.56155i q^{8} +0.561553 q^{9} +O(q^{10})$$ $$q-2.56155i q^{2} +1.56155i q^{3} -4.56155 q^{4} +4.00000 q^{6} +6.56155i q^{8} +0.561553 q^{9} -1.56155 q^{11} -7.12311i q^{12} +0.438447i q^{13} +7.68466 q^{16} +0.438447i q^{17} -1.43845i q^{18} -7.12311 q^{19} +4.00000i q^{22} -3.12311i q^{23} -10.2462 q^{24} +1.12311 q^{26} +5.56155i q^{27} -6.68466 q^{29} -6.56155i q^{32} -2.43845i q^{33} +1.12311 q^{34} -2.56155 q^{36} +6.00000i q^{37} +18.2462i q^{38} -0.684658 q^{39} -5.12311 q^{41} -0.876894i q^{43} +7.12311 q^{44} -8.00000 q^{46} +8.68466i q^{47} +12.0000i q^{48} -0.684658 q^{51} -2.00000i q^{52} +5.12311i q^{53} +14.2462 q^{54} -11.1231i q^{57} +17.1231i q^{58} -4.00000 q^{59} -15.3693 q^{61} -1.43845 q^{64} -6.24621 q^{66} +10.2462i q^{67} -2.00000i q^{68} +4.87689 q^{69} +8.00000 q^{71} +3.68466i q^{72} -12.2462i q^{73} +15.3693 q^{74} +32.4924 q^{76} +1.75379i q^{78} +2.43845 q^{79} -7.00000 q^{81} +13.1231i q^{82} +4.00000i q^{83} -2.24621 q^{86} -10.4384i q^{87} -10.2462i q^{88} -1.12311 q^{89} +14.2462i q^{92} +22.2462 q^{94} +10.2462 q^{96} -5.80776i q^{97} -0.876894 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} + 16q^{6} - 6q^{9} + O(q^{10})$$ $$4q - 10q^{4} + 16q^{6} - 6q^{9} + 2q^{11} + 6q^{16} - 12q^{19} - 8q^{24} - 12q^{26} - 2q^{29} - 12q^{34} - 2q^{36} + 22q^{39} - 4q^{41} + 12q^{44} - 32q^{46} + 22q^{51} + 24q^{54} - 16q^{59} - 12q^{61} - 14q^{64} + 8q^{66} + 36q^{69} + 32q^{71} + 12q^{74} + 64q^{76} + 18q^{79} - 28q^{81} + 24q^{86} + 12q^{89} + 56q^{94} + 8q^{96} - 20q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

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

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.56155i − 1.81129i −0.424035 0.905646i $$-0.639387\pi$$
0.424035 0.905646i $$-0.360613\pi$$
$$3$$ 1.56155i 0.901563i 0.892634 + 0.450781i $$0.148855\pi$$
−0.892634 + 0.450781i $$0.851145\pi$$
$$4$$ −4.56155 −2.28078
$$5$$ 0 0
$$6$$ 4.00000 1.63299
$$7$$ 0 0
$$8$$ 6.56155i 2.31986i
$$9$$ 0.561553 0.187184
$$10$$ 0 0
$$11$$ −1.56155 −0.470826 −0.235413 0.971895i $$-0.575644\pi$$
−0.235413 + 0.971895i $$0.575644\pi$$
$$12$$ − 7.12311i − 2.05626i
$$13$$ 0.438447i 0.121603i 0.998150 + 0.0608017i $$0.0193657\pi$$
−0.998150 + 0.0608017i $$0.980634\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 7.68466 1.92116
$$17$$ 0.438447i 0.106339i 0.998586 + 0.0531695i $$0.0169324\pi$$
−0.998586 + 0.0531695i $$0.983068\pi$$
$$18$$ − 1.43845i − 0.339045i
$$19$$ −7.12311 −1.63415 −0.817076 0.576530i $$-0.804407\pi$$
−0.817076 + 0.576530i $$0.804407\pi$$
$$20$$ 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 $$-0.105568\pi$$
$$24$$ −10.2462 −2.09150
$$25$$ 0 0
$$26$$ 1.12311 0.220259
$$27$$ 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 $$0.5\pi$$
$$32$$ − 6.56155i − 1.15993i
$$33$$ − 2.43845i − 0.424479i
$$34$$ 1.12311 0.192611
$$35$$ 0 0
$$36$$ −2.56155 −0.426925
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 18.2462i 2.95993i
$$39$$ −0.684658 −0.109633
$$40$$ 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 $$-0.978701\pi$$
0.997762 0.0668626i $$-0.0212989\pi$$
$$44$$ 7.12311 1.07385
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 8.68466i 1.26679i 0.773830 + 0.633394i $$0.218339\pi$$
−0.773830 + 0.633394i $$0.781661\pi$$
$$48$$ 12.0000i 1.73205i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −0.684658 −0.0958714
$$52$$ − 2.00000i − 0.277350i
$$53$$ 5.12311i 0.703713i 0.936054 + 0.351856i $$0.114449\pi$$
−0.936054 + 0.351856i $$0.885551\pi$$
$$54$$ 14.2462 1.93866
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 11.1231i − 1.47329i
$$58$$ 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 $$-0.942839\pi$$
−0.983920 + 0.178611i $$0.942839\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −1.43845 −0.179806
$$65$$ 0 0
$$66$$ −6.24621 −0.768855
$$67$$ 10.2462i 1.25177i 0.779914 + 0.625887i $$0.215263\pi$$
−0.779914 + 0.625887i $$0.784737\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 4.87689 0.587109
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 3.68466i 0.434241i
$$73$$ − 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
$$77$$ 0 0
$$78$$ 1.75379i 0.198577i
$$79$$ 2.43845 0.274347 0.137173 0.990547i $$-0.456198\pi$$
0.137173 + 0.990547i $$0.456198\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 13.1231i 1.44920i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.24621 −0.242215
$$87$$ − 10.4384i − 1.11912i
$$88$$ − 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 0
$$92$$ 14.2462i 1.48527i
$$93$$ 0 0
$$94$$ 22.2462 2.29452
$$95$$ 0 0
$$96$$ 10.2462 1.04575
$$97$$ − 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 $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ 1.75379i 0.173651i
$$103$$ 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 $$0.776345\pi$$
$$108$$ − 25.3693i − 2.44116i
$$109$$ −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 $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ −28.4924 −2.66856
$$115$$ 0 0
$$116$$ 30.4924 2.83115
$$117$$ 0.246211i 0.0227622i
$$118$$ 10.2462i 0.943240i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ 39.3693i 3.56433i
$$123$$ − 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 $$-0.0893835\pi$$
$$128$$ − 9.43845i − 0.834249i
$$129$$ 1.36932 0.120562
$$130$$ 0 0
$$131$$ 0.876894 0.0766146 0.0383073 0.999266i $$-0.487803\pi$$
0.0383073 + 0.999266i $$0.487803\pi$$
$$132$$ 11.1231i 0.968142i
$$133$$ 0 0
$$134$$ 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 $$-0.261162\pi$$
$$138$$ − 12.4924i − 1.06343i
$$139$$ −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
$$142$$ − 20.4924i − 1.71969i
$$143$$ − 0.684658i − 0.0572540i
$$144$$ 4.31534 0.359612
$$145$$ 0 0
$$146$$ −31.3693 −2.59614
$$147$$ 0 0
$$148$$ − 27.3693i − 2.24974i
$$149$$ −12.2462 −1.00325 −0.501624 0.865086i $$-0.667264\pi$$
−0.501624 + 0.865086i $$0.667264\pi$$
$$150$$ 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
$$153$$ 0.246211i 0.0199050i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.12311 0.250049
$$157$$ − 20.2462i − 1.61582i −0.589303 0.807912i $$-0.700598\pi$$
0.589303 0.807912i $$-0.299402\pi$$
$$158$$ − 6.24621i − 0.496922i
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 17.9309i 1.40878i
$$163$$ 7.12311i 0.557925i 0.960302 + 0.278962i $$0.0899905\pi$$
−0.960302 + 0.278962i $$0.910010\pi$$
$$164$$ 23.3693 1.82484
$$165$$ 0 0
$$166$$ 10.2462 0.795260
$$167$$ 6.93087i 0.536327i 0.963373 + 0.268163i $$0.0864167\pi$$
−0.963373 + 0.268163i $$0.913583\pi$$
$$168$$ 0 0
$$169$$ 12.8078 0.985213
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ − 4.43845i − 0.337449i −0.985663 0.168724i $$-0.946035\pi$$
0.985663 0.168724i $$-0.0539648\pi$$
$$174$$ −26.7386 −2.02705
$$175$$ 0 0
$$176$$ −12.0000 −0.904534
$$177$$ − 6.24621i − 0.469494i
$$178$$ 2.87689i 0.215632i
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 17.6155 1.30935 0.654676 0.755910i $$-0.272805\pi$$
0.654676 + 0.755910i $$0.272805\pi$$
$$182$$ 0 0
$$183$$ − 24.0000i − 1.77413i
$$184$$ 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 $$-0.663237\pi$$
−0.490640 + 0.871363i $$0.663237\pi$$
$$192$$ − 2.24621i − 0.162106i
$$193$$ − 19.3693i − 1.39423i −0.716957 0.697117i $$-0.754466\pi$$
0.716957 0.697117i $$-0.245534\pi$$
$$194$$ −14.8769 −1.06810
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.12311i 0.0800180i 0.999199 + 0.0400090i $$0.0127387\pi$$
−0.999199 + 0.0400090i $$0.987261\pi$$
$$198$$ 2.24621i 0.159631i
$$199$$ −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
$$202$$ − 41.6155i − 2.92806i
$$203$$ 0 0
$$204$$ 3.12311 0.218661
$$205$$ 0 0
$$206$$ 14.2462 0.992581
$$207$$ − 1.75379i − 0.121897i
$$208$$ 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
$$213$$ 12.4924i 0.855967i
$$214$$ 34.2462 2.34102
$$215$$ 0 0
$$216$$ −36.4924 −2.48299
$$217$$ 0 0
$$218$$ 13.6155i 0.922160i
$$219$$ 19.1231 1.29222
$$220$$ 0 0
$$221$$ −0.192236 −0.0129312
$$222$$ 24.0000i 1.61077i
$$223$$ − 2.43845i − 0.163291i −0.996661 0.0816453i $$-0.973983\pi$$
0.996661 0.0816453i $$-0.0260175\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 35.8617 2.38549
$$227$$ − 11.3153i − 0.751026i −0.926817 0.375513i $$-0.877467\pi$$
0.926817 0.375513i $$-0.122533\pi$$
$$228$$ 50.7386i 3.36025i
$$229$$ 10.8769 0.718765 0.359383 0.933190i $$-0.382987\pi$$
0.359383 + 0.933190i $$0.382987\pi$$
$$230$$ 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 $$-0.0536704\pi$$
$$234$$ 0.630683 0.0412290
$$235$$ 0 0
$$236$$ 18.2462 1.18773
$$237$$ 3.80776i 0.247341i
$$238$$ 0 0
$$239$$ −19.8078 −1.28126 −0.640629 0.767851i $$-0.721326\pi$$
−0.640629 + 0.767851i $$0.721326\pi$$
$$240$$ 0 0
$$241$$ 4.24621 0.273523 0.136761 0.990604i $$-0.456331\pi$$
0.136761 + 0.990604i $$0.456331\pi$$
$$242$$ 21.9309i 1.40977i
$$243$$ 5.75379i 0.369106i
$$244$$ 70.1080 4.48820
$$245$$ 0 0
$$246$$ −20.4924 −1.30655
$$247$$ − 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
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −27.0540 −1.69087
$$257$$ 10.4924i 0.654499i 0.944938 + 0.327250i $$0.106122\pi$$
−0.944938 + 0.327250i $$0.893878\pi$$
$$258$$ − 3.50758i − 0.218372i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −3.75379 −0.232354
$$262$$ − 2.24621i − 0.138771i
$$263$$ 12.8769i 0.794023i 0.917814 + 0.397012i $$0.129953\pi$$
−0.917814 + 0.397012i $$0.870047\pi$$
$$264$$ 16.0000 0.984732
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 1.75379i − 0.107330i
$$268$$ − 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 $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 3.36932i 0.204295i
$$273$$ 0 0
$$274$$ −43.8617 −2.64978
$$275$$ 0 0
$$276$$ −22.2462 −1.33906
$$277$$ − 0.246211i − 0.0147934i −0.999973 0.00739670i $$-0.997646\pi$$
0.999973 0.00739670i $$-0.00235446\pi$$
$$278$$ 38.7386i 2.32339i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.4384 0.742016 0.371008 0.928630i $$-0.379012\pi$$
0.371008 + 0.928630i $$0.379012\pi$$
$$282$$ 34.7386i 2.06866i
$$283$$ − 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
$$287$$ 0 0
$$288$$ − 3.68466i − 0.217121i
$$289$$ 16.8078 0.988692
$$290$$ 0 0
$$291$$ 9.06913 0.531642
$$292$$ 55.8617i 3.26906i
$$293$$ − 2.68466i − 0.156839i −0.996920 0.0784197i $$-0.975013\pi$$
0.996920 0.0784197i $$-0.0249874\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −39.3693 −2.28830
$$297$$ − 8.68466i − 0.503935i
$$298$$ 31.3693i 1.81718i
$$299$$ 1.36932 0.0791896
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 17.7538i 1.02162i
$$303$$ 25.3693i 1.45743i
$$304$$ −54.7386 −3.13948
$$305$$ 0 0
$$306$$ 0.630683 0.0360538
$$307$$ 19.3153i 1.10238i 0.834378 + 0.551192i $$0.185827\pi$$
−0.834378 + 0.551192i $$0.814173\pi$$
$$308$$ 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 $$0.853810\pi$$
$$312$$ − 4.49242i − 0.254333i
$$313$$ − 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 $$0.904795\pi$$
$$318$$ 20.4924i 1.14916i
$$319$$ 10.4384 0.584441
$$320$$ 0 0
$$321$$ −20.8769 −1.16523
$$322$$ 0 0
$$323$$ − 3.12311i − 0.173774i
$$324$$ 31.9309 1.77394
$$325$$ 0 0
$$326$$ 18.2462 1.01056
$$327$$ − 8.30019i − 0.459001i
$$328$$ − 33.6155i − 1.85611i
$$329$$ 0 0
$$330$$ 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
$$333$$ 3.36932i 0.184637i
$$334$$ 17.7538 0.971444
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 1.50758i − 0.0821230i −0.999157 0.0410615i $$-0.986926\pi$$
0.999157 0.0410615i $$-0.0130740\pi$$
$$338$$ − 32.8078i − 1.78451i
$$339$$ −21.8617 −1.18737
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 10.2462i 0.554052i
$$343$$ 0 0
$$344$$ 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 $$0.938764\pi$$
$$348$$ 47.6155i 2.55246i
$$349$$ 10.4924 0.561646 0.280823 0.959760i $$-0.409393\pi$$
0.280823 + 0.959760i $$0.409393\pi$$
$$350$$ 0 0
$$351$$ −2.43845 −0.130155
$$352$$ 10.2462i 0.546125i
$$353$$ 5.80776i 0.309116i 0.987984 + 0.154558i $$0.0493954\pi$$
−0.987984 + 0.154558i $$0.950605\pi$$
$$354$$ −16.0000 −0.850390
$$355$$ 0 0
$$356$$ 5.12311 0.271524
$$357$$ 0 0
$$358$$ 51.2311i 2.70765i
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ − 45.1231i − 2.37162i
$$363$$ − 13.3693i − 0.701707i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −61.4773 −3.21347
$$367$$ 8.68466i 0.453335i 0.973972 + 0.226668i $$0.0727831\pi$$
−0.973972 + 0.226668i $$0.927217\pi$$
$$368$$ − 24.0000i − 1.25109i
$$369$$ −2.87689 −0.149765
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 4.63068i − 0.239768i −0.992788 0.119884i $$-0.961748\pi$$
0.992788 0.119884i $$-0.0382522\pi$$
$$374$$ −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
$$381$$ 9.75379 0.499702
$$382$$ 34.7386i 1.77738i
$$383$$ 6.24621i 0.319166i 0.987184 + 0.159583i $$0.0510150\pi$$
−0.987184 + 0.159583i $$0.948985\pi$$
$$384$$ 14.7386 0.752128
$$385$$ 0 0
$$386$$ −49.6155 −2.52536
$$387$$ − 0.492423i − 0.0250312i
$$388$$ 26.4924i 1.34495i
$$389$$ 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
$$394$$ 2.87689 0.144936
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ − 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 $$-0.211089\pi$$
0.788054 + 0.615606i $$0.211089\pi$$
$$402$$ 40.9848i 2.04414i
$$403$$ 0 0
$$404$$ −74.1080 −3.68701
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 9.36932i − 0.464420i
$$408$$ − 4.49242i − 0.222408i
$$409$$ 6.49242 0.321030 0.160515 0.987033i $$-0.448685\pi$$
0.160515 + 0.987033i $$0.448685\pi$$
$$410$$ 0 0
$$411$$ 26.7386 1.31892
$$412$$ − 25.3693i − 1.24986i
$$413$$ 0 0
$$414$$ −4.49242 −0.220791
$$415$$ 0 0
$$416$$ 2.87689 0.141051
$$417$$ − 23.6155i − 1.15646i
$$418$$ − 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
$$423$$ 4.87689i 0.237123i
$$424$$ −33.6155 −1.63251
$$425$$ 0 0
$$426$$ 32.0000 1.55041
$$427$$ 0 0
$$428$$ − 60.9848i − 2.94781i
$$429$$ 1.06913 0.0516181
$$430$$ 0 0
$$431$$ −19.8078 −0.954106 −0.477053 0.878874i $$-0.658295\pi$$
−0.477053 + 0.878874i $$0.658295\pi$$
$$432$$ 42.7386i 2.05626i
$$433$$ 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
$$438$$ − 48.9848i − 2.34059i
$$439$$ 9.36932 0.447173 0.223587 0.974684i $$-0.428223\pi$$
0.223587 + 0.974684i $$0.428223\pi$$
$$440$$ 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 $$0.980095\pi$$
$$444$$ 42.7386 2.02829
$$445$$ 0 0
$$446$$ −6.24621 −0.295767
$$447$$ − 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
$$452$$ − 63.8617i − 3.00380i
$$453$$ − 10.8229i − 0.508505i
$$454$$ −28.9848 −1.36033
$$455$$ 0 0
$$456$$ 72.9848 3.41783
$$457$$ − 17.1231i − 0.800985i −0.916300 0.400493i $$-0.868839\pi$$
0.916300 0.400493i $$-0.131161\pi$$
$$458$$ − 27.8617i − 1.30189i
$$459$$ −2.43845 −0.113817
$$460$$ 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 $$-0.906250\pi$$
0.956940 0.290286i $$-0.0937505\pi$$
$$464$$ −51.3693 −2.38476
$$465$$ 0 0
$$466$$ −13.1231 −0.607916
$$467$$ − 22.4384i − 1.03833i −0.854675 0.519164i $$-0.826243\pi$$
0.854675 0.519164i $$-0.173757\pi$$
$$468$$ − 1.12311i − 0.0519156i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 31.6155 1.45677
$$472$$ − 26.2462i − 1.20808i
$$473$$ 1.36932i 0.0629613i
$$474$$ 9.75379 0.448006
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.87689i 0.131724i
$$478$$ 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
$$486$$ 14.7386 0.668558
$$487$$ − 3.12311i − 0.141521i −0.997493 0.0707607i $$-0.977457\pi$$
0.997493 0.0707607i $$-0.0225427\pi$$
$$488$$ − 100.847i − 4.56511i
$$489$$ −11.1231 −0.503004
$$490$$ 0 0
$$491$$ −41.1771 −1.85830 −0.929148 0.369708i $$-0.879458\pi$$
−0.929148 + 0.369708i $$0.879458\pi$$
$$492$$ 36.4924i 1.64521i
$$493$$ − 2.93087i − 0.132000i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 16.0000i 0.716977i
$$499$$ −41.1771 −1.84334 −0.921670 0.387976i $$-0.873174\pi$$
−0.921670 + 0.387976i $$0.873174\pi$$
$$500$$ 0 0
$$501$$ −10.8229 −0.483532
$$502$$ − 22.7386i − 1.01487i
$$503$$ 38.9309i 1.73584i 0.496703 + 0.867921i $$0.334544\pi$$
−0.496703 + 0.867921i $$0.665456\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 12.4924 0.555356
$$507$$ 20.0000i 0.888231i
$$508$$ 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
$$511$$ 0 0
$$512$$ 50.4233i 2.22842i
$$513$$ − 39.6155i − 1.74907i
$$514$$ 26.8769 1.18549
$$515$$ 0 0
$$516$$ −6.24621 −0.274974
$$517$$ − 13.5616i − 0.596436i
$$518$$ 0 0
$$519$$ 6.93087 0.304231
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 9.61553i 0.420860i
$$523$$ 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
$$528$$ − 18.7386i − 0.815494i
$$529$$ 13.2462 0.575922
$$530$$ 0 0
$$531$$ −2.24621 −0.0974773
$$532$$ 0 0
$$533$$ − 2.24621i − 0.0972942i
$$534$$ −4.49242 −0.194406
$$535$$ 0 0
$$536$$ −67.2311 −2.90394
$$537$$ − 31.2311i − 1.34772i
$$538$$ 53.1231i 2.29030i
$$539$$ 0 0
$$540$$ 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
$$543$$ 27.5076i 1.18046i
$$544$$ 2.87689 0.123346
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 2.24621i − 0.0960411i −0.998846 0.0480205i $$-0.984709\pi$$
0.998846 0.0480205i $$-0.0152913\pi$$
$$548$$ 78.1080i 3.33661i
$$549$$ −8.63068 −0.368349
$$550$$ 0 0
$$551$$ 47.6155 2.02849
$$552$$ 32.0000i 1.36201i
$$553$$ 0 0
$$554$$ −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
$$561$$ 1.06913 0.0451387
$$562$$ − 31.8617i − 1.34401i
$$563$$ − 28.0000i − 1.18006i −0.807382 0.590030i $$-0.799116\pi$$
0.807382 0.590030i $$-0.200884\pi$$
$$564$$ 61.8617 2.60485
$$565$$ 0 0
$$566$$ −28.9848 −1.21832
$$567$$ 0 0
$$568$$ 52.4924i 2.20253i
$$569$$ 30.9848 1.29895 0.649476 0.760382i $$-0.274988\pi$$
0.649476 + 0.760382i $$0.274988\pi$$
$$570$$ 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
$$573$$ − 21.1771i − 0.884685i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −0.807764 −0.0336568
$$577$$ 24.0540i 1.00138i 0.865627 + 0.500690i $$0.166920\pi$$
−0.865627 + 0.500690i $$0.833080\pi$$
$$578$$ − 43.0540i − 1.79081i
$$579$$ 30.2462 1.25699
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 23.2311i − 0.962958i
$$583$$ − 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 $$0.182200\pi$$
−0.840605 + 0.541649i $$0.817800\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −1.75379 −0.0721412
$$592$$ 46.1080i 1.89503i
$$593$$ − 27.5616i − 1.13182i −0.824468 0.565909i $$-0.808525\pi$$
0.824468 0.565909i $$-0.191475\pi$$
$$594$$ −22.2462 −0.912773
$$595$$ 0 0
$$596$$ 55.8617 2.28819
$$597$$ − 2.73863i − 0.112085i
$$598$$ − 3.50758i − 0.143436i
$$599$$ 11.8078 0.482452 0.241226 0.970469i $$-0.422450\pi$$
0.241226 + 0.970469i $$0.422450\pi$$
$$600$$ 0 0
$$601$$ −6.49242 −0.264831 −0.132416 0.991194i $$-0.542273\pi$$
−0.132416 + 0.991194i $$0.542273\pi$$
$$602$$ 0 0
$$603$$ 5.75379i 0.234312i
$$604$$ 31.6155 1.28642
$$605$$ 0 0
$$606$$ 64.9848 2.63983
$$607$$ 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
$$612$$ − 1.12311i − 0.0453989i
$$613$$ − 40.7386i − 1.64542i −0.568463 0.822709i $$-0.692462\pi$$
0.568463 0.822709i $$-0.307538\pi$$
$$614$$ 49.4773 1.99674
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 32.2462i 1.29818i 0.760710 + 0.649092i $$0.224851\pi$$
−0.760710 + 0.649092i $$0.775149\pi$$
$$618$$ 22.2462i 0.894874i
$$619$$ 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
$$622$$ 80.9848i 3.24720i
$$623$$ 0 0
$$624$$ −5.26137 −0.210623
$$625$$ 0 0
$$626$$ −57.1231 −2.28310
$$627$$ 17.3693i 0.693664i
$$628$$ 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
$$633$$ 21.9460i 0.872276i
$$634$$ 26.8769 1.06742
$$635$$ 0 0
$$636$$ 36.4924 1.44702
$$637$$ 0 0
$$638$$ − 26.7386i − 1.05859i
$$639$$ 4.49242 0.177717
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 53.4773i 2.11058i
$$643$$ 1.56155i 0.0615816i 0.999526 + 0.0307908i $$0.00980257\pi$$
−0.999526 + 0.0307908i $$0.990197\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ − 36.4924i − 1.43467i −0.696731 0.717333i $$-0.745363\pi$$
0.696731 0.717333i $$-0.254637\pi$$
$$648$$ − 45.9309i − 1.80433i
$$649$$ 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 $$0.774678\pi$$
$$654$$ −21.2614 −0.831385
$$655$$ 0 0
$$656$$ −39.3693 −1.53711
$$657$$ − 6.87689i − 0.268293i
$$658$$ 0 0
$$659$$ −9.17708 −0.357488 −0.178744 0.983896i $$-0.557203\pi$$
−0.178744 + 0.983896i $$0.557203\pi$$
$$660$$ 0 0
$$661$$ 5.12311 0.199266 0.0996329 0.995024i $$-0.468233\pi$$
0.0996329 + 0.995024i $$0.468233\pi$$
$$662$$ − 30.7386i − 1.19469i
$$663$$ − 0.300187i − 0.0116583i
$$664$$ −26.2462 −1.01855
$$665$$ 0 0
$$666$$ 8.63068 0.334432
$$667$$ 20.8769i 0.808357i
$$668$$ − 31.6155i − 1.22324i
$$669$$ 3.80776 0.147217
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ − 31.8617i − 1.22818i −0.789236 0.614090i $$-0.789523\pi$$
0.789236 0.614090i $$-0.210477\pi$$
$$674$$ −3.86174 −0.148749
$$675$$ 0 0
$$676$$ −58.4233 −2.24705
$$677$$ − 4.93087i − 0.189509i −0.995501 0.0947544i $$-0.969793\pi$$
0.995501 0.0947544i $$-0.0302066\pi$$
$$678$$ 56.0000i 2.15067i
$$679$$ 0 0
$$680$$ 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 $$0.958848\pi$$
$$684$$ 18.2462 0.697661
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16.9848i 0.648012i
$$688$$ − 6.73863i − 0.256908i
$$689$$ −2.24621 −0.0855738
$$690$$ 0 0
$$691$$ 24.4924 0.931736 0.465868 0.884854i $$-0.345742\pi$$
0.465868 + 0.884854i $$0.345742\pi$$
$$692$$ 20.2462i 0.769645i
$$693$$ 0 0
$$694$$ 18.2462 0.692617
$$695$$ 0 0
$$696$$ 68.4924 2.59620
$$697$$ − 2.24621i − 0.0850813i
$$698$$ − 26.8769i − 1.01731i
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ 28.9309 1.09270 0.546352 0.837556i $$-0.316016\pi$$
0.546352 + 0.837556i $$0.316016\pi$$
$$702$$ 6.24621i 0.235748i
$$703$$ − 42.7386i − 1.61192i
$$704$$ 2.24621 0.0846573
$$705$$ 0 0
$$706$$ 14.8769 0.559899
$$707$$ 0 0
$$708$$ 28.4924i 1.07081i
$$709$$ −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
$$712$$ − 7.36932i − 0.276177i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 91.2311 3.40946
$$717$$ − 30.9309i − 1.15513i
$$718$$ 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
$$721$$ 0 0
$$722$$ − 81.3002i − 3.02568i
$$723$$ 6.63068i 0.246598i
$$724$$ −80.3542 −2.98634
$$725$$ 0 0
$$726$$ −34.2462 −1.27100
$$727$$ − 52.4924i − 1.94684i −0.229035 0.973418i $$-0.573557\pi$$
0.229035 0.973418i $$-0.426443\pi$$
$$728$$ 0 0
$$729$$ −29.9848 −1.11055
$$730$$ 0 0
$$731$$ 0.384472 0.0142202
$$732$$ 109.477i 4.04640i
$$733$$ 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
$$738$$ 7.36932i 0.271268i
$$739$$ −34.9309 −1.28495 −0.642476 0.766305i $$-0.722093\pi$$
−0.642476 + 0.766305i $$0.722093\pi$$
$$740$$ 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
$$746$$ −11.8617 −0.434289
$$747$$ 2.24621i 0.0821846i
$$748$$ 3.12311i 0.114192i
$$749$$ 0 0
$$750$$ 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
$$753$$ 13.8617i 0.505150i
$$754$$ −7.50758 −0.273410
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 39.3693i 1.43090i 0.698663 + 0.715451i $$0.253779\pi$$
−0.698663 + 0.715451i $$0.746221\pi$$
$$758$$ − 42.2462i − 1.53445i
$$759$$ −7.61553 −0.276426
$$760$$ 0 0
$$761$$ −48.2462 −1.74892 −0.874462 0.485094i $$-0.838785\pi$$
−0.874462 + 0.485094i $$0.838785\pi$$
$$762$$ − 24.9848i − 0.905105i
$$763$$ 0 0
$$764$$ 61.8617 2.23808
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ − 1.75379i − 0.0633256i
$$768$$ − 42.2462i − 1.52443i
$$769$$ −42.4924 −1.53232 −0.766158 0.642652i $$-0.777834\pi$$
−0.766158 + 0.642652i $$0.777834\pi$$
$$770$$ 0 0
$$771$$ −16.3845 −0.590072
$$772$$ 88.3542i 3.17994i
$$773$$ 36.9309i 1.32831i 0.747594 + 0.664156i $$0.231209\pi$$
−0.747594 + 0.664156i $$0.768791\pi$$
$$774$$ −1.26137 −0.0453389
$$775$$ 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
$$783$$ − 37.1771i − 1.32860i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 3.50758 0.125111
$$787$$ 49.1771i 1.75297i 0.481426 + 0.876487i $$0.340119\pi$$
−0.481426 + 0.876487i $$0.659881\pi$$
$$788$$ − 5.12311i − 0.182503i
$$789$$ −20.1080 −0.715862
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 5.75379i − 0.204452i
$$793$$ − 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 $$-0.859916\pi$$
0.904715 0.426018i $$-0.140084\pi$$
$$798$$ 0 0
$$799$$ −3.80776 −0.134709
$$800$$ 0 0
$$801$$ −0.630683 −0.0222841
$$802$$ − 80.8466i − 2.85479i
$$803$$ 19.1231i 0.674840i
$$804$$ 72.9848 2.57398
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 32.3845i − 1.13999i
$$808$$ 106.600i 3.75019i
$$809$$ −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
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ −5.26137 −0.184185
$$817$$ 6.24621i 0.218527i
$$818$$ − 16.6307i − 0.581478i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −21.4233 −0.747678 −0.373839 0.927494i $$-0.621959\pi$$
−0.373839 + 0.927494i $$0.621959\pi$$
$$822$$ − 68.4924i − 2.38895i
$$823$$ 36.4924i 1.27205i 0.771670 + 0.636023i $$0.219422\pi$$
−0.771670 + 0.636023i $$0.780578\pi$$
$$824$$ −36.4924 −1.27127
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 5.36932i − 0.186709i −0.995633 0.0933547i $$-0.970241\pi$$
0.995633 0.0933547i $$-0.0297591\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ 34.8769 1.21132 0.605662 0.795722i $$-0.292908\pi$$
0.605662 + 0.795722i $$0.292908\pi$$
$$830$$ 0 0
$$831$$ 0.384472 0.0133372
$$832$$ − 0.630683i − 0.0218650i
$$833$$ 0 0
$$834$$ −60.4924 −2.09468
$$835$$ 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
$$843$$ 19.4233i 0.668974i
$$844$$ −64.1080 −2.20669
$$845$$ 0 0
$$846$$ 12.4924 0.429498
$$847$$ 0 0
$$848$$ 39.3693i 1.35195i
$$849$$ 17.6695 0.606416
$$850$$ 0 0
$$851$$ 18.7386 0.642352
$$852$$ − 56.9848i − 1.95227i
$$853$$ − 7.26137i − 0.248624i −0.992243 0.124312i $$-0.960328\pi$$
0.992243 0.124312i $$-0.0396724\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −87.7235 −2.99833
$$857$$ 15.7538i 0.538139i 0.963121 + 0.269070i $$0.0867162\pi$$
−0.963121 + 0.269070i $$0.913284\pi$$
$$858$$ − 2.73863i − 0.0934954i
$$859$$ −16.4924 −0.562714 −0.281357 0.959603i $$-0.590785\pi$$
−0.281357 + 0.959603i $$0.590785\pi$$
$$860$$ 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 $$0.855569\pi$$
$$864$$ 36.4924 1.24150
$$865$$ 0 0
$$866$$ 21.1231 0.717792
$$867$$ 26.2462i 0.891368i
$$868$$ 0 0
$$869$$ −3.80776 −0.129170
$$870$$ 0 0
$$871$$ −4.49242 −0.152220
$$872$$ − 34.8769i − 1.18108i
$$873$$ − 3.26137i − 0.110381i
$$874$$ 56.9848 1.92754
$$875$$ 0 0
$$876$$ −87.2311 −2.94726
$$877$$ − 40.2462i − 1.35902i −0.733667 0.679509i $$-0.762193\pi$$
0.733667 0.679509i $$-0.237807\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ 4.19224 0.141401
$$880$$ 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 $$0.0456416\pi$$
−0.989738 + 0.142896i $$0.954358\pi$$
$$884$$ 0.876894 0.0294931
$$885$$ 0 0
$$886$$ 6.73863 0.226389
$$887$$ − 20.4924i − 0.688068i −0.938957 0.344034i $$-0.888206\pi$$
0.938957 0.344034i $$-0.111794\pi$$
$$888$$ − 61.4773i − 2.06304i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 10.9309 0.366198
$$892$$ 11.1231i 0.372429i
$$893$$ − 61.8617i − 2.07012i
$$894$$ −48.9848 −1.63830
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.13826i 0.0713944i
$$898$$ − 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
$$906$$ −27.7235 −0.921051
$$907$$ − 24.1080i − 0.800491i −0.916408 0.400246i $$-0.868925\pi$$
0.916408 0.400246i $$-0.131075\pi$$
$$908$$ 51.6155i 1.71292i
$$909$$ 9.12311 0.302594
$$910$$ 0 0
$$911$$ −28.4924 −0.943996 −0.471998 0.881600i $$-0.656467\pi$$
−0.471998 + 0.881600i $$0.656467\pi$$
$$912$$ − 85.4773i − 2.83044i
$$913$$ − 6.24621i − 0.206719i
$$914$$ −43.8617 −1.45082
$$915$$ 0 0
$$916$$ −49.6155 −1.63934
$$917$$ 0 0
$$918$$ 6.24621i 0.206156i
$$919$$ −40.3002 −1.32938 −0.664690 0.747119i $$-0.731436\pi$$
−0.664690 + 0.747119i $$0.731436\pi$$
$$920$$ 0 0
$$921$$ −30.1619 −0.993869
$$922$$ − 33.6155i − 1.10707i
$$923$$ 3.50758i 0.115453i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ 3.12311i 0.102576i
$$928$$ 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
$$931$$ 0 0
$$932$$ 23.3693i 0.765487i
$$933$$ − 49.3693i − 1.61628i
$$934$$ −57.4773 −1.88071
$$935$$ 0 0
$$936$$ −1.61553 −0.0528052
$$937$$ 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 $$-0.753540\pi$$
−0.714926 + 0.699200i $$0.753540\pi$$
$$942$$ − 80.9848i − 2.63863i
$$943$$ 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 $$0.979298\pi$$
$$948$$ − 17.3693i − 0.564129i
$$949$$ 5.36932 0.174295
$$950$$ 0 0
$$951$$ −16.3845 −0.531303
$$952$$ 0 0
$$953$$ 33.1231i 1.07296i 0.843912 + 0.536481i $$0.180247\pi$$
−0.843912 + 0.536481i $$0.819753\pi$$
$$954$$ 7.36932 0.238590
$$955$$ 0 0
$$956$$ 90.3542 2.92226
$$957$$ 16.3002i 0.526910i
$$958$$ − 12.4924i − 0.403612i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 6.73863i 0.217262i
$$963$$ 7.50758i 0.241928i
$$964$$ −19.3693 −0.623844
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 35.1231i − 1.12948i −0.825268 0.564741i $$-0.808976\pi$$
0.825268 0.564741i $$-0.191024\pi$$
$$968$$ − 56.1771i − 1.80560i
$$969$$ 4.87689 0.156668
$$970$$ 0 0
$$971$$ −49.4773 −1.58780 −0.793901 0.608048i $$-0.791953\pi$$
−0.793901 + 0.608048i $$0.791953\pi$$
$$972$$ − 26.2462i − 0.841848i
$$973$$ 0 0
$$974$$ −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 $$0.821599\pi$$
$$978$$ 28.4924i 0.911087i
$$979$$ 1.75379 0.0560513
$$980$$ 0 0
$$981$$ −2.98485 −0.0952988
$$982$$ 105.477i 3.36591i
$$983$$ 51.4233i 1.64015i 0.572257 + 0.820074i $$0.306068\pi$$
−0.572257 + 0.820074i $$0.693932\pi$$
$$984$$ 52.4924 1.67340
$$985$$ 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
$$993$$ 18.7386i 0.594653i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 28.4924 0.902817
$$997$$ 2.68466i 0.0850240i 0.999096 + 0.0425120i $$0.0135361\pi$$
−0.999096 + 0.0425120i $$0.986464\pi$$
$$998$$ 105.477i 3.33882i
$$999$$ −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 1225.2.b.f.99.1 4
5.2 odd 4 1225.2.a.s.1.2 2
5.3 odd 4 245.2.a.d.1.1 2
5.4 even 2 inner 1225.2.b.f.99.4 4
7.6 odd 2 175.2.b.b.99.1 4
15.8 even 4 2205.2.a.x.1.2 2
20.3 even 4 3920.2.a.bs.1.2 2
21.20 even 2 1575.2.d.e.1324.4 4
28.27 even 2 2800.2.g.t.449.3 4
35.3 even 12 245.2.e.i.226.2 4
35.13 even 4 35.2.a.b.1.1 2
35.18 odd 12 245.2.e.h.226.2 4
35.23 odd 12 245.2.e.h.116.2 4
35.27 even 4 175.2.a.f.1.2 2
35.33 even 12 245.2.e.i.116.2 4
35.34 odd 2 175.2.b.b.99.4 4
105.62 odd 4 1575.2.a.p.1.1 2
105.83 odd 4 315.2.a.e.1.2 2
105.104 even 2 1575.2.d.e.1324.1 4
140.27 odd 4 2800.2.a.bi.1.2 2
140.83 odd 4 560.2.a.i.1.1 2
140.139 even 2 2800.2.g.t.449.2 4
280.13 even 4 2240.2.a.bh.1.1 2
280.83 odd 4 2240.2.a.bd.1.2 2
385.153 odd 4 4235.2.a.m.1.2 2
420.83 even 4 5040.2.a.bt.1.1 2
455.363 even 4 5915.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 35.13 even 4
175.2.a.f.1.2 2 35.27 even 4
175.2.b.b.99.1 4 7.6 odd 2
175.2.b.b.99.4 4 35.34 odd 2
245.2.a.d.1.1 2 5.3 odd 4
245.2.e.h.116.2 4 35.23 odd 12
245.2.e.h.226.2 4 35.18 odd 12
245.2.e.i.116.2 4 35.33 even 12
245.2.e.i.226.2 4 35.3 even 12
315.2.a.e.1.2 2 105.83 odd 4
560.2.a.i.1.1 2 140.83 odd 4
1225.2.a.s.1.2 2 5.2 odd 4
1225.2.b.f.99.1 4 1.1 even 1 trivial
1225.2.b.f.99.4 4 5.4 even 2 inner
1575.2.a.p.1.1 2 105.62 odd 4
1575.2.d.e.1324.1 4 105.104 even 2
1575.2.d.e.1324.4 4 21.20 even 2
2205.2.a.x.1.2 2 15.8 even 4
2240.2.a.bd.1.2 2 280.83 odd 4
2240.2.a.bh.1.1 2 280.13 even 4
2800.2.a.bi.1.2 2 140.27 odd 4
2800.2.g.t.449.2 4 140.139 even 2
2800.2.g.t.449.3 4 28.27 even 2
3920.2.a.bs.1.2 2 20.3 even 4
4235.2.a.m.1.2 2 385.153 odd 4
5040.2.a.bt.1.1 2 420.83 even 4
5915.2.a.l.1.2 2 455.363 even 4