# Properties

 Label 9200.2.a.ct.1.1 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.521397.1 Defining polynomial: $$x^{5} - 9x^{3} - 3x^{2} + 18x + 12$$ x^5 - 9*x^3 - 3*x^2 + 18*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.61696$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.61696 q^{3} -3.83744 q^{7} +3.84849 q^{9} +O(q^{10})$$ $$q-2.61696 q^{3} -3.83744 q^{7} +3.84849 q^{9} +0.508005 q^{11} +1.01106 q^{13} +1.44705 q^{17} +0.508005 q^{19} +10.0424 q^{21} +1.00000 q^{23} -2.22047 q^{27} +7.51040 q^{29} +0.439038 q^{31} -1.32943 q^{33} -7.02642 q^{37} -2.64590 q^{39} +5.47041 q^{41} -6.72592 q^{43} +2.64098 q^{47} +7.72592 q^{49} -3.78688 q^{51} +4.77648 q^{53} -1.32943 q^{57} -3.85345 q^{59} -9.05844 q^{61} -14.7683 q^{63} -3.45696 q^{67} -2.61696 q^{69} +2.73649 q^{71} -9.21300 q^{73} -1.94944 q^{77} -10.5504 q^{79} -5.73458 q^{81} +1.40211 q^{83} -19.6544 q^{87} +6.77086 q^{89} -3.87986 q^{91} -1.14895 q^{93} -0.313420 q^{97} +1.95506 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 5 * q - 4 * q^7 + 3 * q^9 $$5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57} + q^{59} + 10 q^{61} - 8 q^{67} - 8 q^{71} + 6 q^{73} - 27 q^{81} + 2 q^{83} - 39 q^{87} + 14 q^{89} + 46 q^{91} - 3 q^{93} + 6 q^{97} + 6 q^{99}+O(q^{100})$$ 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * q^99

## 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
$$3$$ −2.61696 −1.51090 −0.755452 0.655204i $$-0.772583\pi$$
−0.755452 + 0.655204i $$0.772583\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.83744 −1.45041 −0.725207 0.688531i $$-0.758256\pi$$
−0.725207 + 0.688531i $$0.758256\pi$$
$$8$$ 0 0
$$9$$ 3.84849 1.28283
$$10$$ 0 0
$$11$$ 0.508005 0.153169 0.0765847 0.997063i $$-0.475598\pi$$
0.0765847 + 0.997063i $$0.475598\pi$$
$$12$$ 0 0
$$13$$ 1.01106 0.280417 0.140208 0.990122i $$-0.455223\pi$$
0.140208 + 0.990122i $$0.455223\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.44705 0.350961 0.175481 0.984483i $$-0.443852\pi$$
0.175481 + 0.984483i $$0.443852\pi$$
$$18$$ 0 0
$$19$$ 0.508005 0.116544 0.0582722 0.998301i $$-0.481441\pi$$
0.0582722 + 0.998301i $$0.481441\pi$$
$$20$$ 0 0
$$21$$ 10.0424 2.19144
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −2.22047 −0.427330
$$28$$ 0 0
$$29$$ 7.51040 1.39465 0.697323 0.716757i $$-0.254374\pi$$
0.697323 + 0.716757i $$0.254374\pi$$
$$30$$ 0 0
$$31$$ 0.439038 0.0788536 0.0394268 0.999222i $$-0.487447\pi$$
0.0394268 + 0.999222i $$0.487447\pi$$
$$32$$ 0 0
$$33$$ −1.32943 −0.231424
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.02642 −1.15514 −0.577568 0.816343i $$-0.695998\pi$$
−0.577568 + 0.816343i $$0.695998\pi$$
$$38$$ 0 0
$$39$$ −2.64590 −0.423683
$$40$$ 0 0
$$41$$ 5.47041 0.854335 0.427167 0.904173i $$-0.359512\pi$$
0.427167 + 0.904173i $$0.359512\pi$$
$$42$$ 0 0
$$43$$ −6.72592 −1.02569 −0.512847 0.858480i $$-0.671409\pi$$
−0.512847 + 0.858480i $$0.671409\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.64098 0.385227 0.192614 0.981275i $$-0.438304\pi$$
0.192614 + 0.981275i $$0.438304\pi$$
$$48$$ 0 0
$$49$$ 7.72592 1.10370
$$50$$ 0 0
$$51$$ −3.78688 −0.530269
$$52$$ 0 0
$$53$$ 4.77648 0.656100 0.328050 0.944660i $$-0.393609\pi$$
0.328050 + 0.944660i $$0.393609\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.32943 −0.176087
$$58$$ 0 0
$$59$$ −3.85345 −0.501676 −0.250838 0.968029i $$-0.580706\pi$$
−0.250838 + 0.968029i $$0.580706\pi$$
$$60$$ 0 0
$$61$$ −9.05844 −1.15981 −0.579907 0.814683i $$-0.696911\pi$$
−0.579907 + 0.814683i $$0.696911\pi$$
$$62$$ 0 0
$$63$$ −14.7683 −1.86064
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.45696 −0.422335 −0.211167 0.977450i $$-0.567727\pi$$
−0.211167 + 0.977450i $$0.567727\pi$$
$$68$$ 0 0
$$69$$ −2.61696 −0.315045
$$70$$ 0 0
$$71$$ 2.73649 0.324762 0.162381 0.986728i $$-0.448083\pi$$
0.162381 + 0.986728i $$0.448083\pi$$
$$72$$ 0 0
$$73$$ −9.21300 −1.07830 −0.539150 0.842210i $$-0.681254\pi$$
−0.539150 + 0.842210i $$0.681254\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.94944 −0.222159
$$78$$ 0 0
$$79$$ −10.5504 −1.18702 −0.593508 0.804828i $$-0.702258\pi$$
−0.593508 + 0.804828i $$0.702258\pi$$
$$80$$ 0 0
$$81$$ −5.73458 −0.637176
$$82$$ 0 0
$$83$$ 1.40211 0.153901 0.0769505 0.997035i $$-0.475482\pi$$
0.0769505 + 0.997035i $$0.475482\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −19.6544 −2.10718
$$88$$ 0 0
$$89$$ 6.77086 0.717710 0.358855 0.933393i $$-0.383167\pi$$
0.358855 + 0.933393i $$0.383167\pi$$
$$90$$ 0 0
$$91$$ −3.87986 −0.406720
$$92$$ 0 0
$$93$$ −1.14895 −0.119140
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.313420 −0.0318230 −0.0159115 0.999873i $$-0.505065\pi$$
−0.0159115 + 0.999873i $$0.505065\pi$$
$$98$$ 0 0
$$99$$ 1.95506 0.196490
$$100$$ 0 0
$$101$$ −4.83182 −0.480784 −0.240392 0.970676i $$-0.577276\pi$$
−0.240392 + 0.970676i $$0.577276\pi$$
$$102$$ 0 0
$$103$$ −8.07746 −0.795896 −0.397948 0.917408i $$-0.630278\pi$$
−0.397948 + 0.917408i $$0.630278\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.1587 1.07876 0.539378 0.842064i $$-0.318659\pi$$
0.539378 + 0.842064i $$0.318659\pi$$
$$108$$ 0 0
$$109$$ 17.6015 1.68592 0.842958 0.537979i $$-0.180812\pi$$
0.842958 + 0.537979i $$0.180812\pi$$
$$110$$ 0 0
$$111$$ 18.3879 1.74530
$$112$$ 0 0
$$113$$ −4.06454 −0.382360 −0.191180 0.981555i $$-0.561231\pi$$
−0.191180 + 0.981555i $$0.561231\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.89104 0.359727
$$118$$ 0 0
$$119$$ −5.55296 −0.509039
$$120$$ 0 0
$$121$$ −10.7419 −0.976539
$$122$$ 0 0
$$123$$ −14.3159 −1.29082
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.75246 −0.421713 −0.210856 0.977517i $$-0.567625\pi$$
−0.210856 + 0.977517i $$0.567625\pi$$
$$128$$ 0 0
$$129$$ 17.6015 1.54972
$$130$$ 0 0
$$131$$ −1.96851 −0.171989 −0.0859946 0.996296i $$-0.527407\pi$$
−0.0859946 + 0.996296i $$0.527407\pi$$
$$132$$ 0 0
$$133$$ −1.94944 −0.169038
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −5.64285 −0.482101 −0.241051 0.970513i $$-0.577492\pi$$
−0.241051 + 0.970513i $$0.577492\pi$$
$$138$$ 0 0
$$139$$ 7.59724 0.644390 0.322195 0.946673i $$-0.395579\pi$$
0.322195 + 0.946673i $$0.395579\pi$$
$$140$$ 0 0
$$141$$ −6.91136 −0.582041
$$142$$ 0 0
$$143$$ 0.513622 0.0429512
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −20.2184 −1.66759
$$148$$ 0 0
$$149$$ 10.7709 0.882384 0.441192 0.897413i $$-0.354556\pi$$
0.441192 + 0.897413i $$0.354556\pi$$
$$150$$ 0 0
$$151$$ 16.5333 1.34546 0.672729 0.739889i $$-0.265122\pi$$
0.672729 + 0.739889i $$0.265122\pi$$
$$152$$ 0 0
$$153$$ 5.56896 0.450224
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −11.5436 −0.921281 −0.460640 0.887587i $$-0.652380\pi$$
−0.460640 + 0.887587i $$0.652380\pi$$
$$158$$ 0 0
$$159$$ −12.4999 −0.991304
$$160$$ 0 0
$$161$$ −3.83744 −0.302432
$$162$$ 0 0
$$163$$ −3.32143 −0.260155 −0.130077 0.991504i $$-0.541523\pi$$
−0.130077 + 0.991504i $$0.541523\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.6742 −0.980755 −0.490378 0.871510i $$-0.663141\pi$$
−0.490378 + 0.871510i $$0.663141\pi$$
$$168$$ 0 0
$$169$$ −11.9778 −0.921367
$$170$$ 0 0
$$171$$ 1.95506 0.149507
$$172$$ 0 0
$$173$$ 17.5733 1.33607 0.668035 0.744130i $$-0.267136\pi$$
0.668035 + 0.744130i $$0.267136\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.0843 0.757984
$$178$$ 0 0
$$179$$ 14.8952 1.11332 0.556659 0.830741i $$-0.312083\pi$$
0.556659 + 0.830741i $$0.312083\pi$$
$$180$$ 0 0
$$181$$ −23.2868 −1.73089 −0.865446 0.501003i $$-0.832965\pi$$
−0.865446 + 0.501003i $$0.832965\pi$$
$$182$$ 0 0
$$183$$ 23.7056 1.75237
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.735109 0.0537565
$$188$$ 0 0
$$189$$ 8.52093 0.619806
$$190$$ 0 0
$$191$$ −6.92360 −0.500974 −0.250487 0.968120i $$-0.580591\pi$$
−0.250487 + 0.968120i $$0.580591\pi$$
$$192$$ 0 0
$$193$$ −7.74318 −0.557366 −0.278683 0.960383i $$-0.589898\pi$$
−0.278683 + 0.960383i $$0.589898\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −23.8908 −1.70215 −0.851074 0.525045i $$-0.824048\pi$$
−0.851074 + 0.525045i $$0.824048\pi$$
$$198$$ 0 0
$$199$$ 23.9444 1.69737 0.848687 0.528896i $$-0.177394\pi$$
0.848687 + 0.528896i $$0.177394\pi$$
$$200$$ 0 0
$$201$$ 9.04673 0.638107
$$202$$ 0 0
$$203$$ −28.8207 −2.02282
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 3.84849 0.267489
$$208$$ 0 0
$$209$$ 0.258070 0.0178510
$$210$$ 0 0
$$211$$ −14.4709 −0.996220 −0.498110 0.867114i $$-0.665973\pi$$
−0.498110 + 0.867114i $$0.665973\pi$$
$$212$$ 0 0
$$213$$ −7.16129 −0.490684
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.68478 −0.114370
$$218$$ 0 0
$$219$$ 24.1101 1.62921
$$220$$ 0 0
$$221$$ 1.46305 0.0984153
$$222$$ 0 0
$$223$$ 4.74755 0.317919 0.158960 0.987285i $$-0.449186\pi$$
0.158960 + 0.987285i $$0.449186\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −8.98399 −0.596288 −0.298144 0.954521i $$-0.596368\pi$$
−0.298144 + 0.954521i $$0.596368\pi$$
$$228$$ 0 0
$$229$$ 20.7290 1.36981 0.684906 0.728631i $$-0.259843\pi$$
0.684906 + 0.728631i $$0.259843\pi$$
$$230$$ 0 0
$$231$$ 5.10161 0.335661
$$232$$ 0 0
$$233$$ 26.0634 1.70747 0.853735 0.520707i $$-0.174332\pi$$
0.853735 + 0.520707i $$0.174332\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 27.6101 1.79347
$$238$$ 0 0
$$239$$ 2.94209 0.190308 0.0951540 0.995463i $$-0.469666\pi$$
0.0951540 + 0.995463i $$0.469666\pi$$
$$240$$ 0 0
$$241$$ −24.9413 −1.60661 −0.803306 0.595567i $$-0.796927\pi$$
−0.803306 + 0.595567i $$0.796927\pi$$
$$242$$ 0 0
$$243$$ 21.6686 1.39004
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.513622 0.0326810
$$248$$ 0 0
$$249$$ −3.66926 −0.232530
$$250$$ 0 0
$$251$$ −21.1908 −1.33755 −0.668774 0.743465i $$-0.733181\pi$$
−0.668774 + 0.743465i $$0.733181\pi$$
$$252$$ 0 0
$$253$$ 0.508005 0.0319380
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 26.2110 1.63500 0.817500 0.575929i $$-0.195360\pi$$
0.817500 + 0.575929i $$0.195360\pi$$
$$258$$ 0 0
$$259$$ 26.9634 1.67543
$$260$$ 0 0
$$261$$ 28.9037 1.78910
$$262$$ 0 0
$$263$$ 5.65276 0.348564 0.174282 0.984696i $$-0.444240\pi$$
0.174282 + 0.984696i $$0.444240\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −17.7191 −1.08439
$$268$$ 0 0
$$269$$ 9.65772 0.588841 0.294421 0.955676i $$-0.404873\pi$$
0.294421 + 0.955676i $$0.404873\pi$$
$$270$$ 0 0
$$271$$ −8.39040 −0.509680 −0.254840 0.966983i $$-0.582023\pi$$
−0.254840 + 0.966983i $$0.582023\pi$$
$$272$$ 0 0
$$273$$ 10.1535 0.614515
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 30.3317 1.82246 0.911229 0.411900i $$-0.135135\pi$$
0.911229 + 0.411900i $$0.135135\pi$$
$$278$$ 0 0
$$279$$ 1.68964 0.101156
$$280$$ 0 0
$$281$$ −25.3132 −1.51006 −0.755028 0.655692i $$-0.772377\pi$$
−0.755028 + 0.655692i $$0.772377\pi$$
$$282$$ 0 0
$$283$$ 12.9629 0.770566 0.385283 0.922798i $$-0.374104\pi$$
0.385283 + 0.922798i $$0.374104\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.9924 −1.23914
$$288$$ 0 0
$$289$$ −14.9060 −0.876826
$$290$$ 0 0
$$291$$ 0.820209 0.0480815
$$292$$ 0 0
$$293$$ 25.9469 1.51584 0.757918 0.652350i $$-0.226217\pi$$
0.757918 + 0.652350i $$0.226217\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1.12801 −0.0654539
$$298$$ 0 0
$$299$$ 1.01106 0.0584709
$$300$$ 0 0
$$301$$ 25.8103 1.48768
$$302$$ 0 0
$$303$$ 12.6447 0.726419
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −12.0640 −0.688531 −0.344266 0.938872i $$-0.611872\pi$$
−0.344266 + 0.938872i $$0.611872\pi$$
$$308$$ 0 0
$$309$$ 21.1384 1.20252
$$310$$ 0 0
$$311$$ 31.8125 1.80392 0.901959 0.431821i $$-0.142129\pi$$
0.901959 + 0.431821i $$0.142129\pi$$
$$312$$ 0 0
$$313$$ −4.30122 −0.243119 −0.121560 0.992584i $$-0.538790\pi$$
−0.121560 + 0.992584i $$0.538790\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.28317 −0.352898 −0.176449 0.984310i $$-0.556461\pi$$
−0.176449 + 0.984310i $$0.556461\pi$$
$$318$$ 0 0
$$319$$ 3.81532 0.213617
$$320$$ 0 0
$$321$$ −29.2020 −1.62990
$$322$$ 0 0
$$323$$ 0.735109 0.0409026
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −46.0624 −2.54726
$$328$$ 0 0
$$329$$ −10.1346 −0.558739
$$330$$ 0 0
$$331$$ 8.00053 0.439749 0.219874 0.975528i $$-0.429435\pi$$
0.219874 + 0.975528i $$0.429435\pi$$
$$332$$ 0 0
$$333$$ −27.0411 −1.48184
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.1436 0.552554 0.276277 0.961078i $$-0.410899\pi$$
0.276277 + 0.961078i $$0.410899\pi$$
$$338$$ 0 0
$$339$$ 10.6367 0.577709
$$340$$ 0 0
$$341$$ 0.223034 0.0120780
$$342$$ 0 0
$$343$$ −2.78567 −0.150412
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.8747 1.01325 0.506625 0.862167i $$-0.330893\pi$$
0.506625 + 0.862167i $$0.330893\pi$$
$$348$$ 0 0
$$349$$ 9.26611 0.496004 0.248002 0.968760i $$-0.420226\pi$$
0.248002 + 0.968760i $$0.420226\pi$$
$$350$$ 0 0
$$351$$ −2.24502 −0.119831
$$352$$ 0 0
$$353$$ 8.04442 0.428161 0.214081 0.976816i $$-0.431324\pi$$
0.214081 + 0.976816i $$0.431324\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 14.5319 0.769109
$$358$$ 0 0
$$359$$ −20.5443 −1.08429 −0.542144 0.840285i $$-0.682387\pi$$
−0.542144 + 0.840285i $$0.682387\pi$$
$$360$$ 0 0
$$361$$ −18.7419 −0.986417
$$362$$ 0 0
$$363$$ 28.1112 1.47546
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.64032 0.451021 0.225511 0.974241i $$-0.427595\pi$$
0.225511 + 0.974241i $$0.427595\pi$$
$$368$$ 0 0
$$369$$ 21.0528 1.09597
$$370$$ 0 0
$$371$$ −18.3294 −0.951617
$$372$$ 0 0
$$373$$ −19.0442 −0.986073 −0.493037 0.870009i $$-0.664113\pi$$
−0.493037 + 0.870009i $$0.664113\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.59344 0.391082
$$378$$ 0 0
$$379$$ −0.926121 −0.0475717 −0.0237858 0.999717i $$-0.507572\pi$$
−0.0237858 + 0.999717i $$0.507572\pi$$
$$380$$ 0 0
$$381$$ 12.4370 0.637167
$$382$$ 0 0
$$383$$ 9.38526 0.479564 0.239782 0.970827i $$-0.422924\pi$$
0.239782 + 0.970827i $$0.422924\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −25.8847 −1.31579
$$388$$ 0 0
$$389$$ 2.22865 0.112997 0.0564985 0.998403i $$-0.482006\pi$$
0.0564985 + 0.998403i $$0.482006\pi$$
$$390$$ 0 0
$$391$$ 1.44705 0.0731805
$$392$$ 0 0
$$393$$ 5.15151 0.259859
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13.0448 0.654701 0.327350 0.944903i $$-0.393844\pi$$
0.327350 + 0.944903i $$0.393844\pi$$
$$398$$ 0 0
$$399$$ 5.10161 0.255400
$$400$$ 0 0
$$401$$ −2.22221 −0.110972 −0.0554859 0.998459i $$-0.517671\pi$$
−0.0554859 + 0.998459i $$0.517671\pi$$
$$402$$ 0 0
$$403$$ 0.443893 0.0221119
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.56946 −0.176931
$$408$$ 0 0
$$409$$ −19.5232 −0.965362 −0.482681 0.875796i $$-0.660337\pi$$
−0.482681 + 0.875796i $$0.660337\pi$$
$$410$$ 0 0
$$411$$ 14.7671 0.728409
$$412$$ 0 0
$$413$$ 14.7874 0.727638
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −19.8817 −0.973611
$$418$$ 0 0
$$419$$ 24.9180 1.21732 0.608662 0.793430i $$-0.291707\pi$$
0.608662 + 0.793430i $$0.291707\pi$$
$$420$$ 0 0
$$421$$ 6.10721 0.297647 0.148824 0.988864i $$-0.452451\pi$$
0.148824 + 0.988864i $$0.452451\pi$$
$$422$$ 0 0
$$423$$ 10.1638 0.494181
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 34.7612 1.68221
$$428$$ 0 0
$$429$$ −1.34413 −0.0648952
$$430$$ 0 0
$$431$$ −4.44095 −0.213913 −0.106956 0.994264i $$-0.534111\pi$$
−0.106956 + 0.994264i $$0.534111\pi$$
$$432$$ 0 0
$$433$$ 3.25554 0.156451 0.0782257 0.996936i $$-0.475075\pi$$
0.0782257 + 0.996936i $$0.475075\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.508005 0.0243012
$$438$$ 0 0
$$439$$ 26.9611 1.28678 0.643392 0.765537i $$-0.277527\pi$$
0.643392 + 0.765537i $$0.277527\pi$$
$$440$$ 0 0
$$441$$ 29.7331 1.41586
$$442$$ 0 0
$$443$$ 16.0072 0.760526 0.380263 0.924878i $$-0.375833\pi$$
0.380263 + 0.924878i $$0.375833\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −28.1869 −1.33320
$$448$$ 0 0
$$449$$ −37.4278 −1.76633 −0.883163 0.469066i $$-0.844591\pi$$
−0.883163 + 0.469066i $$0.844591\pi$$
$$450$$ 0 0
$$451$$ 2.77900 0.130858
$$452$$ 0 0
$$453$$ −43.2670 −2.03286
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.75949 −0.409752 −0.204876 0.978788i $$-0.565679\pi$$
−0.204876 + 0.978788i $$0.565679\pi$$
$$458$$ 0 0
$$459$$ −3.21314 −0.149976
$$460$$ 0 0
$$461$$ 11.4459 0.533089 0.266545 0.963823i $$-0.414118\pi$$
0.266545 + 0.963823i $$0.414118\pi$$
$$462$$ 0 0
$$463$$ −8.23570 −0.382745 −0.191373 0.981517i $$-0.561294\pi$$
−0.191373 + 0.981517i $$0.561294\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.9436 1.33935 0.669675 0.742654i $$-0.266433\pi$$
0.669675 + 0.742654i $$0.266433\pi$$
$$468$$ 0 0
$$469$$ 13.2659 0.612561
$$470$$ 0 0
$$471$$ 30.2092 1.39197
$$472$$ 0 0
$$473$$ −3.41680 −0.157105
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 18.3823 0.841666
$$478$$ 0 0
$$479$$ −31.1031 −1.42114 −0.710569 0.703627i $$-0.751563\pi$$
−0.710569 + 0.703627i $$0.751563\pi$$
$$480$$ 0 0
$$481$$ −7.10410 −0.323919
$$482$$ 0 0
$$483$$ 10.0424 0.456946
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1.90588 −0.0863635 −0.0431817 0.999067i $$-0.513749\pi$$
−0.0431817 + 0.999067i $$0.513749\pi$$
$$488$$ 0 0
$$489$$ 8.69206 0.393069
$$490$$ 0 0
$$491$$ 27.2222 1.22852 0.614259 0.789104i $$-0.289455\pi$$
0.614259 + 0.789104i $$0.289455\pi$$
$$492$$ 0 0
$$493$$ 10.8679 0.489467
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10.5011 −0.471039
$$498$$ 0 0
$$499$$ 9.38913 0.420315 0.210158 0.977668i $$-0.432602\pi$$
0.210158 + 0.977668i $$0.432602\pi$$
$$500$$ 0 0
$$501$$ 33.1678 1.48183
$$502$$ 0 0
$$503$$ −29.5613 −1.31807 −0.659037 0.752110i $$-0.729036\pi$$
−0.659037 + 0.752110i $$0.729036\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 31.3454 1.39210
$$508$$ 0 0
$$509$$ 0.448890 0.0198967 0.00994835 0.999951i $$-0.496833\pi$$
0.00994835 + 0.999951i $$0.496833\pi$$
$$510$$ 0 0
$$511$$ 35.3543 1.56398
$$512$$ 0 0
$$513$$ −1.12801 −0.0498030
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.34163 0.0590050
$$518$$ 0 0
$$519$$ −45.9886 −2.01867
$$520$$ 0 0
$$521$$ 26.8712 1.17725 0.588623 0.808407i $$-0.299670\pi$$
0.588623 + 0.808407i $$0.299670\pi$$
$$522$$ 0 0
$$523$$ −2.90473 −0.127015 −0.0635075 0.997981i $$-0.520229\pi$$
−0.0635075 + 0.997981i $$0.520229\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0.635311 0.0276746
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −14.8300 −0.643566
$$532$$ 0 0
$$533$$ 5.53089 0.239570
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −38.9801 −1.68212
$$538$$ 0 0
$$539$$ 3.92481 0.169053
$$540$$ 0 0
$$541$$ 34.0386 1.46343 0.731716 0.681610i $$-0.238720\pi$$
0.731716 + 0.681610i $$0.238720\pi$$
$$542$$ 0 0
$$543$$ 60.9406 2.61521
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 23.8364 1.01917 0.509586 0.860420i $$-0.329798\pi$$
0.509586 + 0.860420i $$0.329798\pi$$
$$548$$ 0 0
$$549$$ −34.8613 −1.48785
$$550$$ 0 0
$$551$$ 3.81532 0.162538
$$552$$ 0 0
$$553$$ 40.4866 1.72167
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 40.4201 1.71265 0.856326 0.516435i $$-0.172741\pi$$
0.856326 + 0.516435i $$0.172741\pi$$
$$558$$ 0 0
$$559$$ −6.80028 −0.287621
$$560$$ 0 0
$$561$$ −1.92375 −0.0812209
$$562$$ 0 0
$$563$$ 6.54758 0.275948 0.137974 0.990436i $$-0.455941\pi$$
0.137974 + 0.990436i $$0.455941\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 22.0061 0.924169
$$568$$ 0 0
$$569$$ 20.4590 0.857686 0.428843 0.903379i $$-0.358921\pi$$
0.428843 + 0.903379i $$0.358921\pi$$
$$570$$ 0 0
$$571$$ −4.71300 −0.197233 −0.0986164 0.995126i $$-0.531442\pi$$
−0.0986164 + 0.995126i $$0.531442\pi$$
$$572$$ 0 0
$$573$$ 18.1188 0.756924
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −25.9331 −1.07961 −0.539805 0.841790i $$-0.681502\pi$$
−0.539805 + 0.841790i $$0.681502\pi$$
$$578$$ 0 0
$$579$$ 20.2636 0.842127
$$580$$ 0 0
$$581$$ −5.38049 −0.223220
$$582$$ 0 0
$$583$$ 2.42648 0.100494
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10.4711 −0.432189 −0.216095 0.976372i $$-0.569332\pi$$
−0.216095 + 0.976372i $$0.569332\pi$$
$$588$$ 0 0
$$589$$ 0.223034 0.00918995
$$590$$ 0 0
$$591$$ 62.5213 2.57178
$$592$$ 0 0
$$593$$ 20.8158 0.854803 0.427401 0.904062i $$-0.359429\pi$$
0.427401 + 0.904062i $$0.359429\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −62.6616 −2.56457
$$598$$ 0 0
$$599$$ 14.7951 0.604511 0.302256 0.953227i $$-0.402260\pi$$
0.302256 + 0.953227i $$0.402260\pi$$
$$600$$ 0 0
$$601$$ −33.8995 −1.38279 −0.691394 0.722477i $$-0.743003\pi$$
−0.691394 + 0.722477i $$0.743003\pi$$
$$602$$ 0 0
$$603$$ −13.3041 −0.541784
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.9350 −1.33679 −0.668394 0.743807i $$-0.733018\pi$$
−0.668394 + 0.743807i $$0.733018\pi$$
$$608$$ 0 0
$$609$$ 75.4227 3.05628
$$610$$ 0 0
$$611$$ 2.67018 0.108024
$$612$$ 0 0
$$613$$ −24.8664 −1.00434 −0.502172 0.864768i $$-0.667465\pi$$
−0.502172 + 0.864768i $$0.667465\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −20.1280 −0.810324 −0.405162 0.914245i $$-0.632785\pi$$
−0.405162 + 0.914245i $$0.632785\pi$$
$$618$$ 0 0
$$619$$ 15.3719 0.617847 0.308924 0.951087i $$-0.400031\pi$$
0.308924 + 0.951087i $$0.400031\pi$$
$$620$$ 0 0
$$621$$ −2.22047 −0.0891046
$$622$$ 0 0
$$623$$ −25.9828 −1.04098
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −0.675358 −0.0269712
$$628$$ 0 0
$$629$$ −10.1676 −0.405408
$$630$$ 0 0
$$631$$ 25.7115 1.02356 0.511779 0.859117i $$-0.328987\pi$$
0.511779 + 0.859117i $$0.328987\pi$$
$$632$$ 0 0
$$633$$ 37.8699 1.50519
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.81134 0.309497
$$638$$ 0 0
$$639$$ 10.5314 0.416614
$$640$$ 0 0
$$641$$ 2.95684 0.116788 0.0583941 0.998294i $$-0.481402\pi$$
0.0583941 + 0.998294i $$0.481402\pi$$
$$642$$ 0 0
$$643$$ 16.7764 0.661596 0.330798 0.943702i $$-0.392682\pi$$
0.330798 + 0.943702i $$0.392682\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1.99247 0.0783322 0.0391661 0.999233i $$-0.487530\pi$$
0.0391661 + 0.999233i $$0.487530\pi$$
$$648$$ 0 0
$$649$$ −1.95757 −0.0768414
$$650$$ 0 0
$$651$$ 4.40901 0.172803
$$652$$ 0 0
$$653$$ 46.4989 1.81964 0.909822 0.414999i $$-0.136218\pi$$
0.909822 + 0.414999i $$0.136218\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −35.4562 −1.38328
$$658$$ 0 0
$$659$$ 37.1468 1.44703 0.723517 0.690307i $$-0.242524\pi$$
0.723517 + 0.690307i $$0.242524\pi$$
$$660$$ 0 0
$$661$$ −7.01756 −0.272952 −0.136476 0.990643i $$-0.543578\pi$$
−0.136476 + 0.990643i $$0.543578\pi$$
$$662$$ 0 0
$$663$$ −3.82874 −0.148696
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7.51040 0.290804
$$668$$ 0 0
$$669$$ −12.4242 −0.480346
$$670$$ 0 0
$$671$$ −4.60174 −0.177648
$$672$$ 0 0
$$673$$ −6.81155 −0.262566 −0.131283 0.991345i $$-0.541910\pi$$
−0.131283 + 0.991345i $$0.541910\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0.522812 0.0200933 0.0100466 0.999950i $$-0.496802\pi$$
0.0100466 + 0.999950i $$0.496802\pi$$
$$678$$ 0 0
$$679$$ 1.20273 0.0461566
$$680$$ 0 0
$$681$$ 23.5108 0.900934
$$682$$ 0 0
$$683$$ −2.93412 −0.112271 −0.0561355 0.998423i $$-0.517878\pi$$
−0.0561355 + 0.998423i $$0.517878\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −54.2471 −2.06965
$$688$$ 0 0
$$689$$ 4.82929 0.183981
$$690$$ 0 0
$$691$$ 31.8456 1.21146 0.605731 0.795669i $$-0.292881\pi$$
0.605731 + 0.795669i $$0.292881\pi$$
$$692$$ 0 0
$$693$$ −7.50240 −0.284993
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 7.91596 0.299838
$$698$$ 0 0
$$699$$ −68.2070 −2.57982
$$700$$ 0 0
$$701$$ 11.2070 0.423283 0.211642 0.977347i $$-0.432119\pi$$
0.211642 + 0.977347i $$0.432119\pi$$
$$702$$ 0 0
$$703$$ −3.56946 −0.134625
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.5418 0.697336
$$708$$ 0 0
$$709$$ 23.7300 0.891198 0.445599 0.895233i $$-0.352991\pi$$
0.445599 + 0.895233i $$0.352991\pi$$
$$710$$ 0 0
$$711$$ −40.6033 −1.52274
$$712$$ 0 0
$$713$$ 0.439038 0.0164421
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −7.69934 −0.287537
$$718$$ 0 0
$$719$$ 27.8687 1.03933 0.519664 0.854371i $$-0.326057\pi$$
0.519664 + 0.854371i $$0.326057\pi$$
$$720$$ 0 0
$$721$$ 30.9968 1.15438
$$722$$ 0 0
$$723$$ 65.2705 2.42744
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 47.5773 1.76455 0.882273 0.470739i $$-0.156013\pi$$
0.882273 + 0.470739i $$0.156013\pi$$
$$728$$ 0 0
$$729$$ −39.5022 −1.46304
$$730$$ 0 0
$$731$$ −9.73274 −0.359978
$$732$$ 0 0
$$733$$ 15.4086 0.569129 0.284565 0.958657i $$-0.408151\pi$$
0.284565 + 0.958657i $$0.408151\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.75615 −0.0646888
$$738$$ 0 0
$$739$$ −53.0212 −1.95042 −0.975208 0.221292i $$-0.928973\pi$$
−0.975208 + 0.221292i $$0.928973\pi$$
$$740$$ 0 0
$$741$$ −1.34413 −0.0493778
$$742$$ 0 0
$$743$$ −15.6611 −0.574551 −0.287275 0.957848i $$-0.592750\pi$$
−0.287275 + 0.957848i $$0.592750\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 5.39599 0.197429
$$748$$ 0 0
$$749$$ −42.8209 −1.56464
$$750$$ 0 0
$$751$$ −34.6216 −1.26336 −0.631679 0.775230i $$-0.717634\pi$$
−0.631679 + 0.775230i $$0.717634\pi$$
$$752$$ 0 0
$$753$$ 55.4554 2.02091
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −38.4389 −1.39709 −0.698543 0.715568i $$-0.746168\pi$$
−0.698543 + 0.715568i $$0.746168\pi$$
$$758$$ 0 0
$$759$$ −1.32943 −0.0482553
$$760$$ 0 0
$$761$$ 42.0932 1.52588 0.762939 0.646470i $$-0.223755\pi$$
0.762939 + 0.646470i $$0.223755\pi$$
$$762$$ 0 0
$$763$$ −67.5446 −2.44528
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −3.89605 −0.140678
$$768$$ 0 0
$$769$$ 34.2859 1.23638 0.618191 0.786028i $$-0.287866\pi$$
0.618191 + 0.786028i $$0.287866\pi$$
$$770$$ 0 0
$$771$$ −68.5933 −2.47033
$$772$$ 0 0
$$773$$ −12.7422 −0.458304 −0.229152 0.973391i $$-0.573595\pi$$
−0.229152 + 0.973391i $$0.573595\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −70.5623 −2.53141
$$778$$ 0 0
$$779$$ 2.77900 0.0995679
$$780$$ 0 0
$$781$$ 1.39015 0.0497436
$$782$$ 0 0
$$783$$ −16.6766 −0.595975
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −28.5534 −1.01782 −0.508910 0.860820i $$-0.669951\pi$$
−0.508910 + 0.860820i $$0.669951\pi$$
$$788$$ 0 0
$$789$$ −14.7931 −0.526647
$$790$$ 0 0
$$791$$ 15.5974 0.554580
$$792$$ 0 0
$$793$$ −9.15859 −0.325231
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −8.39925 −0.297517 −0.148758 0.988874i $$-0.547528\pi$$
−0.148758 + 0.988874i $$0.547528\pi$$
$$798$$ 0 0
$$799$$ 3.82164 0.135200
$$800$$ 0 0
$$801$$ 26.0576 0.920701
$$802$$ 0 0
$$803$$ −4.68026 −0.165163
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −25.2739 −0.889683
$$808$$ 0 0
$$809$$ −28.8816 −1.01542 −0.507712 0.861527i $$-0.669509\pi$$
−0.507712 + 0.861527i $$0.669509\pi$$
$$810$$ 0 0
$$811$$ 46.5292 1.63386 0.816931 0.576736i $$-0.195674\pi$$
0.816931 + 0.576736i $$0.195674\pi$$
$$812$$ 0 0
$$813$$ 21.9574 0.770078
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3.41680 −0.119539
$$818$$ 0 0
$$819$$ −14.9316 −0.521753
$$820$$ 0 0
$$821$$ −2.99065 −0.104374 −0.0521872 0.998637i $$-0.516619\pi$$
−0.0521872 + 0.998637i $$0.516619\pi$$
$$822$$ 0 0
$$823$$ −5.14357 −0.179294 −0.0896468 0.995974i $$-0.528574\pi$$
−0.0896468 + 0.995974i $$0.528574\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 53.4288 1.85790 0.928950 0.370205i $$-0.120713\pi$$
0.928950 + 0.370205i $$0.120713\pi$$
$$828$$ 0 0
$$829$$ 11.5290 0.400420 0.200210 0.979753i $$-0.435838\pi$$
0.200210 + 0.979753i $$0.435838\pi$$
$$830$$ 0 0
$$831$$ −79.3770 −2.75356
$$832$$ 0 0
$$833$$ 11.1798 0.387357
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −0.974874 −0.0336966
$$838$$ 0 0
$$839$$ −12.0983 −0.417678 −0.208839 0.977950i $$-0.566968\pi$$
−0.208839 + 0.977950i $$0.566968\pi$$
$$840$$ 0 0
$$841$$ 27.4061 0.945038
$$842$$ 0 0
$$843$$ 66.2436 2.28155
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 41.2215 1.41639
$$848$$ 0 0
$$849$$ −33.9235 −1.16425
$$850$$ 0 0
$$851$$ −7.02642 −0.240862
$$852$$ 0 0
$$853$$ 3.10103 0.106177 0.0530886 0.998590i $$-0.483093\pi$$
0.0530886 + 0.998590i $$0.483093\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.39457 0.150116 0.0750578 0.997179i $$-0.476086\pi$$
0.0750578 + 0.997179i $$0.476086\pi$$
$$858$$ 0 0
$$859$$ 29.0453 0.991014 0.495507 0.868604i $$-0.334982\pi$$
0.495507 + 0.868604i $$0.334982\pi$$
$$860$$ 0 0
$$861$$ 54.9362 1.87222
$$862$$ 0 0
$$863$$ 40.1020 1.36509 0.682543 0.730845i $$-0.260874\pi$$
0.682543 + 0.730845i $$0.260874\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 39.0086 1.32480
$$868$$ 0 0
$$869$$ −5.35968 −0.181815
$$870$$ 0 0
$$871$$ −3.49518 −0.118430
$$872$$ 0 0
$$873$$ −1.20620 −0.0408235
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37.7503 1.27474 0.637368 0.770559i $$-0.280023\pi$$
0.637368 + 0.770559i $$0.280023\pi$$
$$878$$ 0 0
$$879$$ −67.9021 −2.29028
$$880$$ 0 0
$$881$$ −29.1541 −0.982227 −0.491113 0.871096i $$-0.663410\pi$$
−0.491113 + 0.871096i $$0.663410\pi$$
$$882$$ 0 0
$$883$$ 43.3971 1.46043 0.730214 0.683218i $$-0.239420\pi$$
0.730214 + 0.683218i $$0.239420\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 25.4091 0.853155 0.426578 0.904451i $$-0.359719\pi$$
0.426578 + 0.904451i $$0.359719\pi$$
$$888$$ 0 0
$$889$$ 18.2373 0.611658
$$890$$ 0 0
$$891$$ −2.91320 −0.0975958
$$892$$ 0 0
$$893$$ 1.34163 0.0448961
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −2.64590 −0.0883439
$$898$$ 0 0
$$899$$ 3.29735 0.109973
$$900$$ 0 0
$$901$$ 6.91181 0.230266
$$902$$ 0 0
$$903$$ −67.5446 −2.24774
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −2.00778 −0.0666672 −0.0333336 0.999444i $$-0.510612\pi$$
−0.0333336 + 0.999444i $$0.510612\pi$$
$$908$$ 0 0
$$909$$ −18.5952 −0.616765
$$910$$ 0 0
$$911$$ 8.45792 0.280223 0.140112 0.990136i $$-0.455254\pi$$
0.140112 + 0.990136i $$0.455254\pi$$
$$912$$ 0 0
$$913$$ 0.712277 0.0235729
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 7.55402 0.249456
$$918$$ 0 0
$$919$$ −0.840028 −0.0277100 −0.0138550 0.999904i $$-0.504410\pi$$
−0.0138550 + 0.999904i $$0.504410\pi$$
$$920$$ 0 0
$$921$$ 31.5711 1.04030
$$922$$ 0 0
$$923$$ 2.76675 0.0910686
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −31.0861 −1.02100
$$928$$ 0 0
$$929$$ 15.7877 0.517979 0.258989 0.965880i $$-0.416611\pi$$
0.258989 + 0.965880i $$0.416611\pi$$
$$930$$ 0 0
$$931$$ 3.92481 0.128630
$$932$$ 0 0
$$933$$ −83.2520 −2.72555
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 31.6568 1.03418 0.517091 0.855930i $$-0.327015\pi$$
0.517091 + 0.855930i $$0.327015\pi$$
$$938$$ 0 0
$$939$$ 11.2561 0.367330
$$940$$ 0 0
$$941$$ −38.3932 −1.25158 −0.625792 0.779990i $$-0.715224\pi$$
−0.625792 + 0.779990i $$0.715224\pi$$
$$942$$ 0 0
$$943$$ 5.47041 0.178141
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 34.3783 1.11714 0.558572 0.829456i $$-0.311349\pi$$
0.558572 + 0.829456i $$0.311349\pi$$
$$948$$ 0 0
$$949$$ −9.31486 −0.302373
$$950$$ 0 0
$$951$$ 16.4428 0.533195
$$952$$ 0 0
$$953$$ 48.3782 1.56712 0.783562 0.621314i $$-0.213401\pi$$
0.783562 + 0.621314i $$0.213401\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −9.98456 −0.322755
$$958$$ 0 0
$$959$$ 21.6541 0.699247
$$960$$ 0 0
$$961$$ −30.8072 −0.993782
$$962$$ 0 0
$$963$$ 42.9443 1.38386
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 51.3705 1.65196 0.825982 0.563697i $$-0.190621\pi$$
0.825982 + 0.563697i $$0.190621\pi$$
$$968$$ 0 0
$$969$$ −1.92375 −0.0617999
$$970$$ 0 0
$$971$$ −50.4587 −1.61930 −0.809649 0.586914i $$-0.800343\pi$$
−0.809649 + 0.586914i $$0.800343\pi$$
$$972$$ 0 0
$$973$$ −29.1539 −0.934633
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 32.1406 1.02827 0.514134 0.857710i $$-0.328113\pi$$
0.514134 + 0.857710i $$0.328113\pi$$
$$978$$ 0 0
$$979$$ 3.43964 0.109931
$$980$$ 0 0
$$981$$ 67.7392 2.16275
$$982$$ 0 0
$$983$$ −24.1306 −0.769647 −0.384824 0.922990i $$-0.625738\pi$$
−0.384824 + 0.922990i $$0.625738\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 26.5219 0.844201
$$988$$ 0 0
$$989$$ −6.72592 −0.213872
$$990$$ 0 0
$$991$$ −40.1267 −1.27467 −0.637333 0.770588i $$-0.719962\pi$$
−0.637333 + 0.770588i $$0.719962\pi$$
$$992$$ 0 0
$$993$$ −20.9371 −0.664418
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −48.9890 −1.55150 −0.775749 0.631042i $$-0.782628\pi$$
−0.775749 + 0.631042i $$0.782628\pi$$
$$998$$ 0 0
$$999$$ 15.6020 0.493625
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 9200.2.a.ct.1.1 5
4.3 odd 2 4600.2.a.bf.1.5 yes 5
5.4 even 2 9200.2.a.cv.1.5 5
20.3 even 4 4600.2.e.w.4049.10 10
20.7 even 4 4600.2.e.w.4049.1 10
20.19 odd 2 4600.2.a.bd.1.1 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.1 5 20.19 odd 2
4600.2.a.bf.1.5 yes 5 4.3 odd 2
4600.2.e.w.4049.1 10 20.7 even 4
4600.2.e.w.4049.10 10 20.3 even 4
9200.2.a.ct.1.1 5 1.1 even 1 trivial
9200.2.a.cv.1.5 5 5.4 even 2