# Properties

 Label 5445.2.a.be.1.2 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.737640$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} +3.28684 q^{7} -4.98884 q^{8} +O(q^{10})$$ $$q-2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} +3.28684 q^{7} -4.98884 q^{8} -2.45589 q^{10} -0.313133 q^{13} -8.07211 q^{14} +4.18926 q^{16} -5.00000 q^{17} -7.45408 q^{19} +4.03138 q^{20} -1.07392 q^{23} +1.00000 q^{25} +0.769020 q^{26} +13.2505 q^{28} +5.03647 q^{29} +3.44899 q^{31} -0.310680 q^{32} +12.2794 q^{34} +3.28684 q^{35} +2.63428 q^{37} +18.3064 q^{38} -4.98884 q^{40} -10.8472 q^{41} -5.51468 q^{43} +2.63743 q^{46} +11.9982 q^{47} +3.80333 q^{49} -2.45589 q^{50} -1.26236 q^{52} -4.93543 q^{53} -16.3975 q^{56} -12.3690 q^{58} +9.16409 q^{59} +9.18431 q^{61} -8.47033 q^{62} -7.61553 q^{64} -0.313133 q^{65} -15.2739 q^{67} -20.1569 q^{68} -8.07211 q^{70} -3.07211 q^{71} -8.65269 q^{73} -6.46950 q^{74} -30.0502 q^{76} +5.41446 q^{79} +4.18926 q^{80} +26.6395 q^{82} -16.2454 q^{83} -5.00000 q^{85} +13.5434 q^{86} -1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} -29.4662 q^{94} -7.45408 q^{95} +0.224082 q^{97} -9.34054 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8} + O(q^{10})$$ $$4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8} - 5 q^{10} + 3 q^{13} + 5 q^{14} + 15 q^{16} - 20 q^{17} + 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} + 3 q^{28} - 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} - 2 q^{35} - 7 q^{37} - q^{38} - 15 q^{40} - 20 q^{41} - 2 q^{43} + 7 q^{46} + 20 q^{47} + 8 q^{49} - 5 q^{50} - 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} - 7 q^{61} + 12 q^{62} + 49 q^{64} + 3 q^{65} - 13 q^{67} - 45 q^{68} + 5 q^{70} + 25 q^{71} + 23 q^{73} + 7 q^{74} - 7 q^{76} + 15 q^{80} + 11 q^{82} - 33 q^{83} - 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} - 17 q^{94} + 3 q^{95} - 25 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.45589 −1.73657 −0.868287 0.496062i $$-0.834779\pi$$
−0.868287 + 0.496062i $$0.834779\pi$$
$$3$$ 0 0
$$4$$ 4.03138 2.01569
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.28684 1.24231 0.621155 0.783688i $$-0.286664\pi$$
0.621155 + 0.783688i $$0.286664\pi$$
$$8$$ −4.98884 −1.76382
$$9$$ 0 0
$$10$$ −2.45589 −0.776620
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −0.313133 −0.0868476 −0.0434238 0.999057i $$-0.513827\pi$$
−0.0434238 + 0.999057i $$0.513827\pi$$
$$14$$ −8.07211 −2.15736
$$15$$ 0 0
$$16$$ 4.18926 1.04732
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ −7.45408 −1.71008 −0.855041 0.518560i $$-0.826468\pi$$
−0.855041 + 0.518560i $$0.826468\pi$$
$$20$$ 4.03138 0.901444
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.07392 −0.223928 −0.111964 0.993712i $$-0.535714\pi$$
−0.111964 + 0.993712i $$0.535714\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.769020 0.150817
$$27$$ 0 0
$$28$$ 13.2505 2.50411
$$29$$ 5.03647 0.935249 0.467624 0.883927i $$-0.345110\pi$$
0.467624 + 0.883927i $$0.345110\pi$$
$$30$$ 0 0
$$31$$ 3.44899 0.619457 0.309728 0.950825i $$-0.399762\pi$$
0.309728 + 0.950825i $$0.399762\pi$$
$$32$$ −0.310680 −0.0549210
$$33$$ 0 0
$$34$$ 12.2794 2.10591
$$35$$ 3.28684 0.555578
$$36$$ 0 0
$$37$$ 2.63428 0.433073 0.216537 0.976274i $$-0.430524\pi$$
0.216537 + 0.976274i $$0.430524\pi$$
$$38$$ 18.3064 2.96969
$$39$$ 0 0
$$40$$ −4.98884 −0.788805
$$41$$ −10.8472 −1.69405 −0.847024 0.531554i $$-0.821608\pi$$
−0.847024 + 0.531554i $$0.821608\pi$$
$$42$$ 0 0
$$43$$ −5.51468 −0.840980 −0.420490 0.907297i $$-0.638142\pi$$
−0.420490 + 0.907297i $$0.638142\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 2.63743 0.388868
$$47$$ 11.9982 1.75012 0.875058 0.484018i $$-0.160823\pi$$
0.875058 + 0.484018i $$0.160823\pi$$
$$48$$ 0 0
$$49$$ 3.80333 0.543333
$$50$$ −2.45589 −0.347315
$$51$$ 0 0
$$52$$ −1.26236 −0.175058
$$53$$ −4.93543 −0.677934 −0.338967 0.940798i $$-0.610077\pi$$
−0.338967 + 0.940798i $$0.610077\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −16.3975 −2.19121
$$57$$ 0 0
$$58$$ −12.3690 −1.62413
$$59$$ 9.16409 1.19306 0.596531 0.802590i $$-0.296545\pi$$
0.596531 + 0.802590i $$0.296545\pi$$
$$60$$ 0 0
$$61$$ 9.18431 1.17593 0.587965 0.808886i $$-0.299929\pi$$
0.587965 + 0.808886i $$0.299929\pi$$
$$62$$ −8.47033 −1.07573
$$63$$ 0 0
$$64$$ −7.61553 −0.951942
$$65$$ −0.313133 −0.0388394
$$66$$ 0 0
$$67$$ −15.2739 −1.86600 −0.933000 0.359876i $$-0.882819\pi$$
−0.933000 + 0.359876i $$0.882819\pi$$
$$68$$ −20.1569 −2.44438
$$69$$ 0 0
$$70$$ −8.07211 −0.964802
$$71$$ −3.07211 −0.364593 −0.182296 0.983244i $$-0.558353\pi$$
−0.182296 + 0.983244i $$0.558353\pi$$
$$72$$ 0 0
$$73$$ −8.65269 −1.01272 −0.506361 0.862322i $$-0.669009\pi$$
−0.506361 + 0.862322i $$0.669009\pi$$
$$74$$ −6.46950 −0.752064
$$75$$ 0 0
$$76$$ −30.0502 −3.44700
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.41446 0.609175 0.304587 0.952484i $$-0.401481\pi$$
0.304587 + 0.952484i $$0.401481\pi$$
$$80$$ 4.18926 0.468374
$$81$$ 0 0
$$82$$ 26.6395 2.94184
$$83$$ −16.2454 −1.78317 −0.891583 0.452857i $$-0.850405\pi$$
−0.891583 + 0.452857i $$0.850405\pi$$
$$84$$ 0 0
$$85$$ −5.00000 −0.542326
$$86$$ 13.5434 1.46042
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1.62118 −0.171845 −0.0859223 0.996302i $$-0.527384\pi$$
−0.0859223 + 0.996302i $$0.527384\pi$$
$$90$$ 0 0
$$91$$ −1.02922 −0.107892
$$92$$ −4.32938 −0.451369
$$93$$ 0 0
$$94$$ −29.4662 −3.03921
$$95$$ −7.45408 −0.764772
$$96$$ 0 0
$$97$$ 0.224082 0.0227521 0.0113760 0.999935i $$-0.496379\pi$$
0.0113760 + 0.999935i $$0.496379\pi$$
$$98$$ −9.34054 −0.943537
$$99$$ 0 0
$$100$$ 4.03138 0.403138
$$101$$ 0.505326 0.0502818 0.0251409 0.999684i $$-0.491997\pi$$
0.0251409 + 0.999684i $$0.491997\pi$$
$$102$$ 0 0
$$103$$ −6.40197 −0.630805 −0.315402 0.948958i $$-0.602139\pi$$
−0.315402 + 0.948958i $$0.602139\pi$$
$$104$$ 1.56217 0.153184
$$105$$ 0 0
$$106$$ 12.1209 1.17728
$$107$$ −2.09249 −0.202289 −0.101144 0.994872i $$-0.532250\pi$$
−0.101144 + 0.994872i $$0.532250\pi$$
$$108$$ 0 0
$$109$$ −6.69278 −0.641052 −0.320526 0.947240i $$-0.603860\pi$$
−0.320526 + 0.947240i $$0.603860\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 13.7694 1.30109
$$113$$ 10.7941 1.01542 0.507712 0.861527i $$-0.330491\pi$$
0.507712 + 0.861527i $$0.330491\pi$$
$$114$$ 0 0
$$115$$ −1.07392 −0.100144
$$116$$ 20.3039 1.88517
$$117$$ 0 0
$$118$$ −22.5060 −2.07184
$$119$$ −16.4342 −1.50652
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −22.5556 −2.04209
$$123$$ 0 0
$$124$$ 13.9042 1.24863
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −17.0033 −1.50880 −0.754398 0.656417i $$-0.772071\pi$$
−0.754398 + 0.656417i $$0.772071\pi$$
$$128$$ 19.3242 1.70804
$$129$$ 0 0
$$130$$ 0.769020 0.0674475
$$131$$ 0.0430508 0.00376136 0.00188068 0.999998i $$-0.499401\pi$$
0.00188068 + 0.999998i $$0.499401\pi$$
$$132$$ 0 0
$$133$$ −24.5004 −2.12445
$$134$$ 37.5109 3.24045
$$135$$ 0 0
$$136$$ 24.9442 2.13895
$$137$$ −7.36257 −0.629027 −0.314513 0.949253i $$-0.601841\pi$$
−0.314513 + 0.949253i $$0.601841\pi$$
$$138$$ 0 0
$$139$$ 13.2393 1.12295 0.561473 0.827495i $$-0.310235\pi$$
0.561473 + 0.827495i $$0.310235\pi$$
$$140$$ 13.2505 1.11987
$$141$$ 0 0
$$142$$ 7.54476 0.633142
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.03647 0.418256
$$146$$ 21.2500 1.75867
$$147$$ 0 0
$$148$$ 10.6198 0.872942
$$149$$ 5.87858 0.481592 0.240796 0.970576i $$-0.422591\pi$$
0.240796 + 0.970576i $$0.422591\pi$$
$$150$$ 0 0
$$151$$ 7.62821 0.620775 0.310387 0.950610i $$-0.399541\pi$$
0.310387 + 0.950610i $$0.399541\pi$$
$$152$$ 37.1872 3.01628
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3.44899 0.277029
$$156$$ 0 0
$$157$$ −10.1332 −0.808719 −0.404360 0.914600i $$-0.632506\pi$$
−0.404360 + 0.914600i $$0.632506\pi$$
$$158$$ −13.2973 −1.05788
$$159$$ 0 0
$$160$$ −0.310680 −0.0245614
$$161$$ −3.52981 −0.278188
$$162$$ 0 0
$$163$$ −5.02906 −0.393906 −0.196953 0.980413i $$-0.563105\pi$$
−0.196953 + 0.980413i $$0.563105\pi$$
$$164$$ −43.7292 −3.41468
$$165$$ 0 0
$$166$$ 39.8969 3.09660
$$167$$ −5.79105 −0.448125 −0.224062 0.974575i $$-0.571932\pi$$
−0.224062 + 0.974575i $$0.571932\pi$$
$$168$$ 0 0
$$169$$ −12.9019 −0.992457
$$170$$ 12.2794 0.941790
$$171$$ 0 0
$$172$$ −22.2318 −1.69516
$$173$$ −16.0652 −1.22142 −0.610708 0.791856i $$-0.709115\pi$$
−0.610708 + 0.791856i $$0.709115\pi$$
$$174$$ 0 0
$$175$$ 3.28684 0.248462
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 3.98143 0.298421
$$179$$ 8.30309 0.620602 0.310301 0.950638i $$-0.399570\pi$$
0.310301 + 0.950638i $$0.399570\pi$$
$$180$$ 0 0
$$181$$ −6.46425 −0.480484 −0.240242 0.970713i $$-0.577227\pi$$
−0.240242 + 0.970713i $$0.577227\pi$$
$$182$$ 2.52765 0.187362
$$183$$ 0 0
$$184$$ 5.35762 0.394969
$$185$$ 2.63428 0.193676
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 48.3693 3.52769
$$189$$ 0 0
$$190$$ 18.3064 1.32808
$$191$$ −15.3693 −1.11208 −0.556041 0.831155i $$-0.687680\pi$$
−0.556041 + 0.831155i $$0.687680\pi$$
$$192$$ 0 0
$$193$$ 15.7518 1.13384 0.566919 0.823773i $$-0.308135\pi$$
0.566919 + 0.823773i $$0.308135\pi$$
$$194$$ −0.550320 −0.0395107
$$195$$ 0 0
$$196$$ 15.3327 1.09519
$$197$$ −16.3940 −1.16802 −0.584010 0.811746i $$-0.698517\pi$$
−0.584010 + 0.811746i $$0.698517\pi$$
$$198$$ 0 0
$$199$$ 6.96500 0.493736 0.246868 0.969049i $$-0.420599\pi$$
0.246868 + 0.969049i $$0.420599\pi$$
$$200$$ −4.98884 −0.352764
$$201$$ 0 0
$$202$$ −1.24102 −0.0873180
$$203$$ 16.5541 1.16187
$$204$$ 0 0
$$205$$ −10.8472 −0.757602
$$206$$ 15.7225 1.09544
$$207$$ 0 0
$$208$$ −1.31180 −0.0909569
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −19.9531 −1.37363 −0.686814 0.726833i $$-0.740992\pi$$
−0.686814 + 0.726833i $$0.740992\pi$$
$$212$$ −19.8966 −1.36650
$$213$$ 0 0
$$214$$ 5.13892 0.351289
$$215$$ −5.51468 −0.376098
$$216$$ 0 0
$$217$$ 11.3363 0.769557
$$218$$ 16.4367 1.11323
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.56567 0.105318
$$222$$ 0 0
$$223$$ −20.1466 −1.34912 −0.674559 0.738221i $$-0.735666\pi$$
−0.674559 + 0.738221i $$0.735666\pi$$
$$224$$ −1.02116 −0.0682289
$$225$$ 0 0
$$226$$ −26.5091 −1.76336
$$227$$ −0.533937 −0.0354386 −0.0177193 0.999843i $$-0.505641\pi$$
−0.0177193 + 0.999843i $$0.505641\pi$$
$$228$$ 0 0
$$229$$ −22.1931 −1.46656 −0.733279 0.679928i $$-0.762011\pi$$
−0.733279 + 0.679928i $$0.762011\pi$$
$$230$$ 2.63743 0.173907
$$231$$ 0 0
$$232$$ −25.1261 −1.64961
$$233$$ 4.56567 0.299107 0.149553 0.988754i $$-0.452216\pi$$
0.149553 + 0.988754i $$0.452216\pi$$
$$234$$ 0 0
$$235$$ 11.9982 0.782676
$$236$$ 36.9439 2.40485
$$237$$ 0 0
$$238$$ 40.3606 2.61619
$$239$$ 5.86053 0.379086 0.189543 0.981872i $$-0.439299\pi$$
0.189543 + 0.981872i $$0.439299\pi$$
$$240$$ 0 0
$$241$$ 9.96074 0.641628 0.320814 0.947142i $$-0.396044\pi$$
0.320814 + 0.947142i $$0.396044\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 37.0254 2.37031
$$245$$ 3.80333 0.242986
$$246$$ 0 0
$$247$$ 2.33412 0.148517
$$248$$ −17.2065 −1.09261
$$249$$ 0 0
$$250$$ −2.45589 −0.155324
$$251$$ 16.8788 1.06538 0.532690 0.846310i $$-0.321181\pi$$
0.532690 + 0.846310i $$0.321181\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 41.7581 2.62014
$$255$$ 0 0
$$256$$ −32.2271 −2.01419
$$257$$ −11.1436 −0.695117 −0.347559 0.937658i $$-0.612989\pi$$
−0.347559 + 0.937658i $$0.612989\pi$$
$$258$$ 0 0
$$259$$ 8.65847 0.538011
$$260$$ −1.26236 −0.0782882
$$261$$ 0 0
$$262$$ −0.105728 −0.00653189
$$263$$ 26.8726 1.65704 0.828519 0.559961i $$-0.189184\pi$$
0.828519 + 0.559961i $$0.189184\pi$$
$$264$$ 0 0
$$265$$ −4.93543 −0.303181
$$266$$ 60.1701 3.68927
$$267$$ 0 0
$$268$$ −61.5748 −3.76128
$$269$$ 10.0629 0.613545 0.306773 0.951783i $$-0.400751\pi$$
0.306773 + 0.951783i $$0.400751\pi$$
$$270$$ 0 0
$$271$$ 10.5441 0.640509 0.320255 0.947331i $$-0.396232\pi$$
0.320255 + 0.947331i $$0.396232\pi$$
$$272$$ −20.9463 −1.27006
$$273$$ 0 0
$$274$$ 18.0816 1.09235
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −17.9376 −1.07777 −0.538883 0.842381i $$-0.681153\pi$$
−0.538883 + 0.842381i $$0.681153\pi$$
$$278$$ −32.5143 −1.95008
$$279$$ 0 0
$$280$$ −16.3975 −0.979939
$$281$$ −7.41103 −0.442105 −0.221052 0.975262i $$-0.570949\pi$$
−0.221052 + 0.975262i $$0.570949\pi$$
$$282$$ 0 0
$$283$$ 5.38684 0.320214 0.160107 0.987100i $$-0.448816\pi$$
0.160107 + 0.987100i $$0.448816\pi$$
$$284$$ −12.3848 −0.734905
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −35.6530 −2.10453
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ −12.3690 −0.726332
$$291$$ 0 0
$$292$$ −34.8823 −2.04133
$$293$$ 1.74006 0.101655 0.0508277 0.998707i $$-0.483814\pi$$
0.0508277 + 0.998707i $$0.483814\pi$$
$$294$$ 0 0
$$295$$ 9.16409 0.533554
$$296$$ −13.1420 −0.763864
$$297$$ 0 0
$$298$$ −14.4371 −0.836321
$$299$$ 0.336280 0.0194476
$$300$$ 0 0
$$301$$ −18.1259 −1.04476
$$302$$ −18.7340 −1.07802
$$303$$ 0 0
$$304$$ −31.2271 −1.79100
$$305$$ 9.18431 0.525892
$$306$$ 0 0
$$307$$ 21.3566 1.21889 0.609444 0.792829i $$-0.291393\pi$$
0.609444 + 0.792829i $$0.291393\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −8.47033 −0.481082
$$311$$ −32.8096 −1.86046 −0.930231 0.366975i $$-0.880393\pi$$
−0.930231 + 0.366975i $$0.880393\pi$$
$$312$$ 0 0
$$313$$ −3.45852 −0.195487 −0.0977436 0.995212i $$-0.531163\pi$$
−0.0977436 + 0.995212i $$0.531163\pi$$
$$314$$ 24.8860 1.40440
$$315$$ 0 0
$$316$$ 21.8278 1.22791
$$317$$ 2.87566 0.161513 0.0807565 0.996734i $$-0.474266\pi$$
0.0807565 + 0.996734i $$0.474266\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −7.61553 −0.425721
$$321$$ 0 0
$$322$$ 8.66881 0.483094
$$323$$ 37.2704 2.07378
$$324$$ 0 0
$$325$$ −0.313133 −0.0173695
$$326$$ 12.3508 0.684048
$$327$$ 0 0
$$328$$ 54.1150 2.98800
$$329$$ 39.4362 2.17419
$$330$$ 0 0
$$331$$ −14.1221 −0.776219 −0.388109 0.921613i $$-0.626872\pi$$
−0.388109 + 0.921613i $$0.626872\pi$$
$$332$$ −65.4915 −3.59431
$$333$$ 0 0
$$334$$ 14.2222 0.778202
$$335$$ −15.2739 −0.834501
$$336$$ 0 0
$$337$$ 15.9490 0.868796 0.434398 0.900721i $$-0.356961\pi$$
0.434398 + 0.900721i $$0.356961\pi$$
$$338$$ 31.6857 1.72348
$$339$$ 0 0
$$340$$ −20.1569 −1.09316
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −10.5070 −0.567322
$$344$$ 27.5118 1.48334
$$345$$ 0 0
$$346$$ 39.4543 2.12108
$$347$$ −29.6801 −1.59331 −0.796656 0.604433i $$-0.793400\pi$$
−0.796656 + 0.604433i $$0.793400\pi$$
$$348$$ 0 0
$$349$$ 31.6937 1.69653 0.848263 0.529574i $$-0.177648\pi$$
0.848263 + 0.529574i $$0.177648\pi$$
$$350$$ −8.07211 −0.431472
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1.20189 −0.0639703 −0.0319852 0.999488i $$-0.510183\pi$$
−0.0319852 + 0.999488i $$0.510183\pi$$
$$354$$ 0 0
$$355$$ −3.07211 −0.163051
$$356$$ −6.53559 −0.346385
$$357$$ 0 0
$$358$$ −20.3915 −1.07772
$$359$$ 11.6591 0.615343 0.307671 0.951493i $$-0.400450\pi$$
0.307671 + 0.951493i $$0.400450\pi$$
$$360$$ 0 0
$$361$$ 36.5633 1.92438
$$362$$ 15.8755 0.834396
$$363$$ 0 0
$$364$$ −4.14918 −0.217476
$$365$$ −8.65269 −0.452903
$$366$$ 0 0
$$367$$ 15.9860 0.834465 0.417232 0.908800i $$-0.363000\pi$$
0.417232 + 0.908800i $$0.363000\pi$$
$$368$$ −4.49894 −0.234523
$$369$$ 0 0
$$370$$ −6.46950 −0.336333
$$371$$ −16.2220 −0.842203
$$372$$ 0 0
$$373$$ 0.321975 0.0166712 0.00833561 0.999965i $$-0.497347\pi$$
0.00833561 + 0.999965i $$0.497347\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −59.8570 −3.08689
$$377$$ −1.57709 −0.0812241
$$378$$ 0 0
$$379$$ 11.4174 0.586475 0.293237 0.956040i $$-0.405267\pi$$
0.293237 + 0.956040i $$0.405267\pi$$
$$380$$ −30.0502 −1.54154
$$381$$ 0 0
$$382$$ 37.7452 1.93121
$$383$$ −28.3673 −1.44950 −0.724750 0.689012i $$-0.758045\pi$$
−0.724750 + 0.689012i $$0.758045\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −38.6846 −1.96899
$$387$$ 0 0
$$388$$ 0.903359 0.0458611
$$389$$ −15.1802 −0.769666 −0.384833 0.922986i $$-0.625741\pi$$
−0.384833 + 0.922986i $$0.625741\pi$$
$$390$$ 0 0
$$391$$ 5.36960 0.271553
$$392$$ −18.9742 −0.958341
$$393$$ 0 0
$$394$$ 40.2617 2.02835
$$395$$ 5.41446 0.272431
$$396$$ 0 0
$$397$$ 5.22461 0.262216 0.131108 0.991368i $$-0.458147\pi$$
0.131108 + 0.991368i $$0.458147\pi$$
$$398$$ −17.1053 −0.857409
$$399$$ 0 0
$$400$$ 4.18926 0.209463
$$401$$ −14.0007 −0.699160 −0.349580 0.936907i $$-0.613676\pi$$
−0.349580 + 0.936907i $$0.613676\pi$$
$$402$$ 0 0
$$403$$ −1.07999 −0.0537983
$$404$$ 2.03716 0.101352
$$405$$ 0 0
$$406$$ −40.6549 −2.01767
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 33.3112 1.64713 0.823567 0.567218i $$-0.191980\pi$$
0.823567 + 0.567218i $$0.191980\pi$$
$$410$$ 26.6395 1.31563
$$411$$ 0 0
$$412$$ −25.8088 −1.27151
$$413$$ 30.1209 1.48215
$$414$$ 0 0
$$415$$ −16.2454 −0.797456
$$416$$ 0.0972843 0.00476975
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 5.28460 0.258170 0.129085 0.991634i $$-0.458796\pi$$
0.129085 + 0.991634i $$0.458796\pi$$
$$420$$ 0 0
$$421$$ −30.7810 −1.50017 −0.750087 0.661340i $$-0.769988\pi$$
−0.750087 + 0.661340i $$0.769988\pi$$
$$422$$ 49.0026 2.38541
$$423$$ 0 0
$$424$$ 24.6221 1.19575
$$425$$ −5.00000 −0.242536
$$426$$ 0 0
$$427$$ 30.1874 1.46087
$$428$$ −8.43562 −0.407751
$$429$$ 0 0
$$430$$ 13.5434 0.653122
$$431$$ −12.3506 −0.594910 −0.297455 0.954736i $$-0.596138\pi$$
−0.297455 + 0.954736i $$0.596138\pi$$
$$432$$ 0 0
$$433$$ 1.41287 0.0678983 0.0339491 0.999424i $$-0.489192\pi$$
0.0339491 + 0.999424i $$0.489192\pi$$
$$434$$ −27.8406 −1.33639
$$435$$ 0 0
$$436$$ −26.9811 −1.29216
$$437$$ 8.00509 0.382935
$$438$$ 0 0
$$439$$ −7.58532 −0.362028 −0.181014 0.983481i $$-0.557938\pi$$
−0.181014 + 0.983481i $$0.557938\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.84510 −0.182893
$$443$$ −11.0662 −0.525771 −0.262885 0.964827i $$-0.584674\pi$$
−0.262885 + 0.964827i $$0.584674\pi$$
$$444$$ 0 0
$$445$$ −1.62118 −0.0768512
$$446$$ 49.4779 2.34284
$$447$$ 0 0
$$448$$ −25.0311 −1.18261
$$449$$ −6.32856 −0.298663 −0.149332 0.988787i $$-0.547712\pi$$
−0.149332 + 0.988787i $$0.547712\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 43.5152 2.04678
$$453$$ 0 0
$$454$$ 1.31129 0.0615418
$$455$$ −1.02922 −0.0482506
$$456$$ 0 0
$$457$$ −0.189579 −0.00886814 −0.00443407 0.999990i $$-0.501411\pi$$
−0.00443407 + 0.999990i $$0.501411\pi$$
$$458$$ 54.5036 2.54679
$$459$$ 0 0
$$460$$ −4.32938 −0.201859
$$461$$ 26.6198 1.23981 0.619904 0.784678i $$-0.287172\pi$$
0.619904 + 0.784678i $$0.287172\pi$$
$$462$$ 0 0
$$463$$ 20.9935 0.975652 0.487826 0.872941i $$-0.337790\pi$$
0.487826 + 0.872941i $$0.337790\pi$$
$$464$$ 21.0991 0.979501
$$465$$ 0 0
$$466$$ −11.2128 −0.519421
$$467$$ −7.89989 −0.365563 −0.182782 0.983154i $$-0.558510\pi$$
−0.182782 + 0.983154i $$0.558510\pi$$
$$468$$ 0 0
$$469$$ −50.2028 −2.31815
$$470$$ −29.4662 −1.35917
$$471$$ 0 0
$$472$$ −45.7182 −2.10435
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −7.45408 −0.342017
$$476$$ −66.2525 −3.03668
$$477$$ 0 0
$$478$$ −14.3928 −0.658311
$$479$$ −39.9728 −1.82640 −0.913201 0.407509i $$-0.866398\pi$$
−0.913201 + 0.407509i $$0.866398\pi$$
$$480$$ 0 0
$$481$$ −0.824882 −0.0376114
$$482$$ −24.4624 −1.11423
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0.224082 0.0101750
$$486$$ 0 0
$$487$$ −9.93556 −0.450223 −0.225112 0.974333i $$-0.572275\pi$$
−0.225112 + 0.974333i $$0.572275\pi$$
$$488$$ −45.8190 −2.07413
$$489$$ 0 0
$$490$$ −9.34054 −0.421963
$$491$$ −4.97349 −0.224451 −0.112225 0.993683i $$-0.535798\pi$$
−0.112225 + 0.993683i $$0.535798\pi$$
$$492$$ 0 0
$$493$$ −25.1823 −1.13416
$$494$$ −5.73234 −0.257910
$$495$$ 0 0
$$496$$ 14.4487 0.648767
$$497$$ −10.0975 −0.452937
$$498$$ 0 0
$$499$$ 43.7757 1.95967 0.979834 0.199812i $$-0.0640332\pi$$
0.979834 + 0.199812i $$0.0640332\pi$$
$$500$$ 4.03138 0.180289
$$501$$ 0 0
$$502$$ −41.4524 −1.85011
$$503$$ −6.28236 −0.280117 −0.140058 0.990143i $$-0.544729\pi$$
−0.140058 + 0.990143i $$0.544729\pi$$
$$504$$ 0 0
$$505$$ 0.505326 0.0224867
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −68.5467 −3.04127
$$509$$ −24.8381 −1.10093 −0.550465 0.834858i $$-0.685550\pi$$
−0.550465 + 0.834858i $$0.685550\pi$$
$$510$$ 0 0
$$511$$ −28.4400 −1.25811
$$512$$ 40.4976 1.78976
$$513$$ 0 0
$$514$$ 27.3674 1.20712
$$515$$ −6.40197 −0.282104
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −21.2642 −0.934296
$$519$$ 0 0
$$520$$ 1.56217 0.0685058
$$521$$ 6.94869 0.304428 0.152214 0.988348i $$-0.451360\pi$$
0.152214 + 0.988348i $$0.451360\pi$$
$$522$$ 0 0
$$523$$ −26.7510 −1.16974 −0.584869 0.811128i $$-0.698854\pi$$
−0.584869 + 0.811128i $$0.698854\pi$$
$$524$$ 0.173554 0.00758175
$$525$$ 0 0
$$526$$ −65.9962 −2.87757
$$527$$ −17.2449 −0.751202
$$528$$ 0 0
$$529$$ −21.8467 −0.949856
$$530$$ 12.1209 0.526496
$$531$$ 0 0
$$532$$ −98.7703 −4.28224
$$533$$ 3.39662 0.147124
$$534$$ 0 0
$$535$$ −2.09249 −0.0904662
$$536$$ 76.1989 3.29129
$$537$$ 0 0
$$538$$ −24.7133 −1.06547
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.5084 0.623767 0.311883 0.950120i $$-0.399040\pi$$
0.311883 + 0.950120i $$0.399040\pi$$
$$542$$ −25.8951 −1.11229
$$543$$ 0 0
$$544$$ 1.55340 0.0666015
$$545$$ −6.69278 −0.286687
$$546$$ 0 0
$$547$$ −26.7346 −1.14309 −0.571543 0.820572i $$-0.693655\pi$$
−0.571543 + 0.820572i $$0.693655\pi$$
$$548$$ −29.6813 −1.26792
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −37.5422 −1.59935
$$552$$ 0 0
$$553$$ 17.7965 0.756784
$$554$$ 44.0527 1.87162
$$555$$ 0 0
$$556$$ 53.3728 2.26351
$$557$$ −17.2444 −0.730670 −0.365335 0.930876i $$-0.619046\pi$$
−0.365335 + 0.930876i $$0.619046\pi$$
$$558$$ 0 0
$$559$$ 1.72683 0.0730371
$$560$$ 13.7694 0.581865
$$561$$ 0 0
$$562$$ 18.2006 0.767748
$$563$$ 0.831914 0.0350610 0.0175305 0.999846i $$-0.494420\pi$$
0.0175305 + 0.999846i $$0.494420\pi$$
$$564$$ 0 0
$$565$$ 10.7941 0.454112
$$566$$ −13.2295 −0.556076
$$567$$ 0 0
$$568$$ 15.3263 0.643076
$$569$$ −11.5961 −0.486132 −0.243066 0.970010i $$-0.578153\pi$$
−0.243066 + 0.970010i $$0.578153\pi$$
$$570$$ 0 0
$$571$$ 21.8414 0.914034 0.457017 0.889458i $$-0.348918\pi$$
0.457017 + 0.889458i $$0.348918\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 87.5598 3.65468
$$575$$ −1.07392 −0.0447856
$$576$$ 0 0
$$577$$ 9.74587 0.405726 0.202863 0.979207i $$-0.434975\pi$$
0.202863 + 0.979207i $$0.434975\pi$$
$$578$$ −19.6471 −0.817211
$$579$$ 0 0
$$580$$ 20.3039 0.843074
$$581$$ −53.3961 −2.21524
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 43.1669 1.78626
$$585$$ 0 0
$$586$$ −4.27339 −0.176532
$$587$$ −22.9441 −0.947005 −0.473502 0.880793i $$-0.657010\pi$$
−0.473502 + 0.880793i $$0.657010\pi$$
$$588$$ 0 0
$$589$$ −25.7090 −1.05932
$$590$$ −22.5060 −0.926556
$$591$$ 0 0
$$592$$ 11.0357 0.453565
$$593$$ −28.7819 −1.18193 −0.590965 0.806697i $$-0.701253\pi$$
−0.590965 + 0.806697i $$0.701253\pi$$
$$594$$ 0 0
$$595$$ −16.4342 −0.673737
$$596$$ 23.6988 0.970741
$$597$$ 0 0
$$598$$ −0.825867 −0.0337722
$$599$$ 29.1951 1.19288 0.596440 0.802657i $$-0.296581\pi$$
0.596440 + 0.802657i $$0.296581\pi$$
$$600$$ 0 0
$$601$$ −6.68087 −0.272518 −0.136259 0.990673i $$-0.543508\pi$$
−0.136259 + 0.990673i $$0.543508\pi$$
$$602$$ 44.5151 1.81430
$$603$$ 0 0
$$604$$ 30.7522 1.25129
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −42.6108 −1.72952 −0.864759 0.502188i $$-0.832529\pi$$
−0.864759 + 0.502188i $$0.832529\pi$$
$$608$$ 2.31583 0.0939194
$$609$$ 0 0
$$610$$ −22.5556 −0.913251
$$611$$ −3.75703 −0.151993
$$612$$ 0 0
$$613$$ −5.83156 −0.235535 −0.117767 0.993041i $$-0.537574\pi$$
−0.117767 + 0.993041i $$0.537574\pi$$
$$614$$ −52.4495 −2.11669
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 33.6386 1.35424 0.677119 0.735874i $$-0.263228\pi$$
0.677119 + 0.735874i $$0.263228\pi$$
$$618$$ 0 0
$$619$$ 42.5616 1.71070 0.855349 0.518053i $$-0.173343\pi$$
0.855349 + 0.518053i $$0.173343\pi$$
$$620$$ 13.9042 0.558405
$$621$$ 0 0
$$622$$ 80.5766 3.23083
$$623$$ −5.32856 −0.213484
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 8.49374 0.339478
$$627$$ 0 0
$$628$$ −40.8509 −1.63013
$$629$$ −13.1714 −0.525179
$$630$$ 0 0
$$631$$ −8.89989 −0.354299 −0.177149 0.984184i $$-0.556688\pi$$
−0.177149 + 0.984184i $$0.556688\pi$$
$$632$$ −27.0119 −1.07448
$$633$$ 0 0
$$634$$ −7.06228 −0.280479
$$635$$ −17.0033 −0.674755
$$636$$ 0 0
$$637$$ −1.19095 −0.0471871
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 19.3242 0.763858
$$641$$ 6.16806 0.243624 0.121812 0.992553i $$-0.461130\pi$$
0.121812 + 0.992553i $$0.461130\pi$$
$$642$$ 0 0
$$643$$ 4.35335 0.171680 0.0858398 0.996309i $$-0.472643\pi$$
0.0858398 + 0.996309i $$0.472643\pi$$
$$644$$ −14.2300 −0.560740
$$645$$ 0 0
$$646$$ −91.5318 −3.60127
$$647$$ −13.4933 −0.530478 −0.265239 0.964183i $$-0.585451\pi$$
−0.265239 + 0.964183i $$0.585451\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0.769020 0.0301635
$$651$$ 0 0
$$652$$ −20.2741 −0.793993
$$653$$ −27.4481 −1.07413 −0.537064 0.843541i $$-0.680467\pi$$
−0.537064 + 0.843541i $$0.680467\pi$$
$$654$$ 0 0
$$655$$ 0.0430508 0.00168213
$$656$$ −45.4418 −1.77420
$$657$$ 0 0
$$658$$ −96.8507 −3.77563
$$659$$ 18.7768 0.731441 0.365721 0.930725i $$-0.380823\pi$$
0.365721 + 0.930725i $$0.380823\pi$$
$$660$$ 0 0
$$661$$ −21.6525 −0.842184 −0.421092 0.907018i $$-0.638353\pi$$
−0.421092 + 0.907018i $$0.638353\pi$$
$$662$$ 34.6822 1.34796
$$663$$ 0 0
$$664$$ 81.0458 3.14519
$$665$$ −24.5004 −0.950084
$$666$$ 0 0
$$667$$ −5.40877 −0.209428
$$668$$ −23.3459 −0.903281
$$669$$ 0 0
$$670$$ 37.5109 1.44917
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 23.3021 0.898232 0.449116 0.893474i $$-0.351739\pi$$
0.449116 + 0.893474i $$0.351739\pi$$
$$674$$ −39.1689 −1.50873
$$675$$ 0 0
$$676$$ −52.0127 −2.00049
$$677$$ −33.2808 −1.27909 −0.639543 0.768756i $$-0.720876\pi$$
−0.639543 + 0.768756i $$0.720876\pi$$
$$678$$ 0 0
$$679$$ 0.736522 0.0282651
$$680$$ 24.9442 0.956566
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −16.9244 −0.647593 −0.323796 0.946127i $$-0.604959\pi$$
−0.323796 + 0.946127i $$0.604959\pi$$
$$684$$ 0 0
$$685$$ −7.36257 −0.281309
$$686$$ 25.8039 0.985197
$$687$$ 0 0
$$688$$ −23.1024 −0.880772
$$689$$ 1.54545 0.0588769
$$690$$ 0 0
$$691$$ −48.4335 −1.84250 −0.921249 0.388973i $$-0.872830\pi$$
−0.921249 + 0.388973i $$0.872830\pi$$
$$692$$ −64.7650 −2.46200
$$693$$ 0 0
$$694$$ 72.8910 2.76690
$$695$$ 13.2393 0.502197
$$696$$ 0 0
$$697$$ 54.2360 2.05434
$$698$$ −77.8362 −2.94614
$$699$$ 0 0
$$700$$ 13.2505 0.500822
$$701$$ 45.4161 1.71534 0.857672 0.514197i $$-0.171910\pi$$
0.857672 + 0.514197i $$0.171910\pi$$
$$702$$ 0 0
$$703$$ −19.6361 −0.740591
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 2.95171 0.111089
$$707$$ 1.66093 0.0624655
$$708$$ 0 0
$$709$$ −1.76497 −0.0662848 −0.0331424 0.999451i $$-0.510551\pi$$
−0.0331424 + 0.999451i $$0.510551\pi$$
$$710$$ 7.54476 0.283150
$$711$$ 0 0
$$712$$ 8.08780 0.303103
$$713$$ −3.70394 −0.138714
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 33.4729 1.25094
$$717$$ 0 0
$$718$$ −28.6334 −1.06859
$$719$$ 3.29998 0.123069 0.0615343 0.998105i $$-0.480401\pi$$
0.0615343 + 0.998105i $$0.480401\pi$$
$$720$$ 0 0
$$721$$ −21.0423 −0.783655
$$722$$ −89.7952 −3.34183
$$723$$ 0 0
$$724$$ −26.0599 −0.968507
$$725$$ 5.03647 0.187050
$$726$$ 0 0
$$727$$ 11.7838 0.437037 0.218519 0.975833i $$-0.429878\pi$$
0.218519 + 0.975833i $$0.429878\pi$$
$$728$$ 5.13461 0.190301
$$729$$ 0 0
$$730$$ 21.2500 0.786499
$$731$$ 27.5734 1.01984
$$732$$ 0 0
$$733$$ 5.73108 0.211682 0.105841 0.994383i $$-0.466246\pi$$
0.105841 + 0.994383i $$0.466246\pi$$
$$734$$ −39.2599 −1.44911
$$735$$ 0 0
$$736$$ 0.333646 0.0122983
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −21.0551 −0.774524 −0.387262 0.921970i $$-0.626579\pi$$
−0.387262 + 0.921970i $$0.626579\pi$$
$$740$$ 10.6198 0.390391
$$741$$ 0 0
$$742$$ 39.8393 1.46255
$$743$$ −13.1283 −0.481630 −0.240815 0.970571i $$-0.577415\pi$$
−0.240815 + 0.970571i $$0.577415\pi$$
$$744$$ 0 0
$$745$$ 5.87858 0.215375
$$746$$ −0.790734 −0.0289508
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −6.87768 −0.251305
$$750$$ 0 0
$$751$$ 25.6251 0.935073 0.467537 0.883974i $$-0.345142\pi$$
0.467537 + 0.883974i $$0.345142\pi$$
$$752$$ 50.2636 1.83292
$$753$$ 0 0
$$754$$ 3.87315 0.141052
$$755$$ 7.62821 0.277619
$$756$$ 0 0
$$757$$ 31.1970 1.13387 0.566936 0.823762i $$-0.308129\pi$$
0.566936 + 0.823762i $$0.308129\pi$$
$$758$$ −28.0400 −1.01846
$$759$$ 0 0
$$760$$ 37.1872 1.34892
$$761$$ −11.3761 −0.412382 −0.206191 0.978512i $$-0.566107\pi$$
−0.206191 + 0.978512i $$0.566107\pi$$
$$762$$ 0 0
$$763$$ −21.9981 −0.796385
$$764$$ −61.9594 −2.24161
$$765$$ 0 0
$$766$$ 69.6668 2.51717
$$767$$ −2.86958 −0.103615
$$768$$ 0 0
$$769$$ 10.3938 0.374811 0.187405 0.982283i $$-0.439992\pi$$
0.187405 + 0.982283i $$0.439992\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 63.5014 2.28547
$$773$$ −14.0348 −0.504796 −0.252398 0.967623i $$-0.581219\pi$$
−0.252398 + 0.967623i $$0.581219\pi$$
$$774$$ 0 0
$$775$$ 3.44899 0.123891
$$776$$ −1.11791 −0.0401306
$$777$$ 0 0
$$778$$ 37.2808 1.33658
$$779$$ 80.8559 2.89696
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −13.1871 −0.471571
$$783$$ 0 0
$$784$$ 15.9331 0.569041
$$785$$ −10.1332 −0.361670
$$786$$ 0 0
$$787$$ −13.8176 −0.492545 −0.246273 0.969201i $$-0.579206\pi$$
−0.246273 + 0.969201i $$0.579206\pi$$
$$788$$ −66.0902 −2.35437
$$789$$ 0 0
$$790$$ −13.2973 −0.473097
$$791$$ 35.4785 1.26147
$$792$$ 0 0
$$793$$ −2.87591 −0.102127
$$794$$ −12.8311 −0.455357
$$795$$ 0 0
$$796$$ 28.0786 0.995218
$$797$$ 5.38594 0.190780 0.0953898 0.995440i $$-0.469590\pi$$
0.0953898 + 0.995440i $$0.469590\pi$$
$$798$$ 0 0
$$799$$ −59.9910 −2.12233
$$800$$ −0.310680 −0.0109842
$$801$$ 0 0
$$802$$ 34.3840 1.21414
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −3.52981 −0.124409
$$806$$ 2.65234 0.0934248
$$807$$ 0 0
$$808$$ −2.52099 −0.0886880
$$809$$ −23.7748 −0.835876 −0.417938 0.908476i $$-0.637247\pi$$
−0.417938 + 0.908476i $$0.637247\pi$$
$$810$$ 0 0
$$811$$ 9.46335 0.332303 0.166152 0.986100i $$-0.446866\pi$$
0.166152 + 0.986100i $$0.446866\pi$$
$$812$$ 66.7358 2.34197
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −5.02906 −0.176160
$$816$$ 0 0
$$817$$ 41.1068 1.43815
$$818$$ −81.8086 −2.86037
$$819$$ 0 0
$$820$$ −43.7292 −1.52709
$$821$$ 10.4189 0.363622 0.181811 0.983334i $$-0.441804\pi$$
0.181811 + 0.983334i $$0.441804\pi$$
$$822$$ 0 0
$$823$$ 24.3540 0.848928 0.424464 0.905445i $$-0.360463\pi$$
0.424464 + 0.905445i $$0.360463\pi$$
$$824$$ 31.9384 1.11263
$$825$$ 0 0
$$826$$ −73.9736 −2.57387
$$827$$ 22.0006 0.765037 0.382519 0.923948i $$-0.375057\pi$$
0.382519 + 0.923948i $$0.375057\pi$$
$$828$$ 0 0
$$829$$ 3.03289 0.105337 0.0526683 0.998612i $$-0.483227\pi$$
0.0526683 + 0.998612i $$0.483227\pi$$
$$830$$ 39.8969 1.38484
$$831$$ 0 0
$$832$$ 2.38468 0.0826738
$$833$$ −19.0166 −0.658888
$$834$$ 0 0
$$835$$ −5.79105 −0.200408
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −12.9784 −0.448331
$$839$$ 48.5383 1.67573 0.837864 0.545879i $$-0.183804\pi$$
0.837864 + 0.545879i $$0.183804\pi$$
$$840$$ 0 0
$$841$$ −3.63399 −0.125310
$$842$$ 75.5946 2.60516
$$843$$ 0 0
$$844$$ −80.4386 −2.76881
$$845$$ −12.9019 −0.443840
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −20.6758 −0.710011
$$849$$ 0 0
$$850$$ 12.2794 0.421181
$$851$$ −2.82901 −0.0969773
$$852$$ 0 0
$$853$$ −11.7632 −0.402766 −0.201383 0.979513i $$-0.564544\pi$$
−0.201383 + 0.979513i $$0.564544\pi$$
$$854$$ −74.1368 −2.53691
$$855$$ 0 0
$$856$$ 10.4391 0.356801
$$857$$ −1.61311 −0.0551026 −0.0275513 0.999620i $$-0.508771\pi$$
−0.0275513 + 0.999620i $$0.508771\pi$$
$$858$$ 0 0
$$859$$ 47.3263 1.61475 0.807376 0.590038i $$-0.200887\pi$$
0.807376 + 0.590038i $$0.200887\pi$$
$$860$$ −22.2318 −0.758097
$$861$$ 0 0
$$862$$ 30.3318 1.03310
$$863$$ 9.97233 0.339462 0.169731 0.985490i $$-0.445710\pi$$
0.169731 + 0.985490i $$0.445710\pi$$
$$864$$ 0 0
$$865$$ −16.0652 −0.546234
$$866$$ −3.46985 −0.117910
$$867$$ 0 0
$$868$$ 45.7009 1.55119
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.78276 0.162058
$$872$$ 33.3892 1.13070
$$873$$ 0 0
$$874$$ −19.6596 −0.664996
$$875$$ 3.28684 0.111116
$$876$$ 0 0
$$877$$ 26.0057 0.878151 0.439076 0.898450i $$-0.355306\pi$$
0.439076 + 0.898450i $$0.355306\pi$$
$$878$$ 18.6287 0.628688
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −10.4081 −0.350657 −0.175329 0.984510i $$-0.556099\pi$$
−0.175329 + 0.984510i $$0.556099\pi$$
$$882$$ 0 0
$$883$$ −53.7283 −1.80810 −0.904051 0.427424i $$-0.859421\pi$$
−0.904051 + 0.427424i $$0.859421\pi$$
$$884$$ 6.31180 0.212289
$$885$$ 0 0
$$886$$ 27.1773 0.913040
$$887$$ 40.6246 1.36404 0.682021 0.731333i $$-0.261101\pi$$
0.682021 + 0.731333i $$0.261101\pi$$
$$888$$ 0 0
$$889$$ −55.8871 −1.87439
$$890$$ 3.98143 0.133458
$$891$$ 0 0
$$892$$ −81.2188 −2.71941
$$893$$ −89.4354 −2.99284
$$894$$ 0 0
$$895$$ 8.30309 0.277542
$$896$$ 63.5157 2.12191
$$897$$ 0 0
$$898$$ 15.5422 0.518651
$$899$$ 17.3707 0.579346
$$900$$ 0 0
$$901$$ 24.6772 0.822115
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −53.8501 −1.79103
$$905$$ −6.46425 −0.214879
$$906$$ 0 0
$$907$$ −19.4070 −0.644398 −0.322199 0.946672i $$-0.604422\pi$$
−0.322199 + 0.946672i $$0.604422\pi$$
$$908$$ −2.15250 −0.0714333
$$909$$ 0 0
$$910$$ 2.52765 0.0837907
$$911$$ −10.7208 −0.355195 −0.177597 0.984103i $$-0.556832\pi$$
−0.177597 + 0.984103i $$0.556832\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0.465585 0.0154002
$$915$$ 0 0
$$916$$ −89.4686 −2.95613
$$917$$ 0.141501 0.00467278
$$918$$ 0 0
$$919$$ −23.1310 −0.763021 −0.381511 0.924364i $$-0.624596\pi$$
−0.381511 + 0.924364i $$0.624596\pi$$
$$920$$ 5.35762 0.176635
$$921$$ 0 0
$$922$$ −65.3752 −2.15302
$$923$$ 0.961981 0.0316640
$$924$$ 0 0
$$925$$ 2.63428 0.0866147
$$926$$ −51.5577 −1.69429
$$927$$ 0 0
$$928$$ −1.56473 −0.0513648
$$929$$ −26.2273 −0.860489 −0.430245 0.902712i $$-0.641573\pi$$
−0.430245 + 0.902712i $$0.641573\pi$$
$$930$$ 0 0
$$931$$ −28.3503 −0.929144
$$932$$ 18.4059 0.602907
$$933$$ 0 0
$$934$$ 19.4012 0.634828
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.4011 0.764480 0.382240 0.924063i $$-0.375153\pi$$
0.382240 + 0.924063i $$0.375153\pi$$
$$938$$ 123.292 4.02564
$$939$$ 0 0
$$940$$ 48.3693 1.57763
$$941$$ 10.9687 0.357570 0.178785 0.983888i $$-0.442783\pi$$
0.178785 + 0.983888i $$0.442783\pi$$
$$942$$ 0 0
$$943$$ 11.6490 0.379345
$$944$$ 38.3908 1.24951
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 13.3652 0.434310 0.217155 0.976137i $$-0.430322\pi$$
0.217155 + 0.976137i $$0.430322\pi$$
$$948$$ 0 0
$$949$$ 2.70945 0.0879524
$$950$$ 18.3064 0.593937
$$951$$ 0 0
$$952$$ 81.9876 2.65723
$$953$$ 21.8242 0.706956 0.353478 0.935443i $$-0.384999\pi$$
0.353478 + 0.935443i $$0.384999\pi$$
$$954$$ 0 0
$$955$$ −15.3693 −0.497339
$$956$$ 23.6260 0.764120
$$957$$ 0 0
$$958$$ 98.1686 3.17168
$$959$$ −24.1996 −0.781446
$$960$$ 0 0
$$961$$ −19.1045 −0.616273
$$962$$ 2.02582 0.0653150
$$963$$ 0 0
$$964$$ 40.1555 1.29332
$$965$$ 15.7518 0.507068
$$966$$ 0 0
$$967$$ 16.6600 0.535750 0.267875 0.963454i $$-0.413679\pi$$
0.267875 + 0.963454i $$0.413679\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −0.550320 −0.0176697
$$971$$ −11.1032 −0.356320 −0.178160 0.984002i $$-0.557014\pi$$
−0.178160 + 0.984002i $$0.557014\pi$$
$$972$$ 0 0
$$973$$ 43.5156 1.39505
$$974$$ 24.4006 0.781846
$$975$$ 0 0
$$976$$ 38.4755 1.23157
$$977$$ −18.8144 −0.601926 −0.300963 0.953636i $$-0.597308\pi$$
−0.300963 + 0.953636i $$0.597308\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 15.3327 0.489784
$$981$$ 0 0
$$982$$ 12.2143 0.389775
$$983$$ 1.37848 0.0439667 0.0219833 0.999758i $$-0.493002\pi$$
0.0219833 + 0.999758i $$0.493002\pi$$
$$984$$ 0 0
$$985$$ −16.3940 −0.522355
$$986$$ 61.8450 1.96955
$$987$$ 0 0
$$988$$ 9.40973 0.299363
$$989$$ 5.92233 0.188319
$$990$$ 0 0
$$991$$ −46.3186 −1.47136 −0.735680 0.677329i $$-0.763137\pi$$
−0.735680 + 0.677329i $$0.763137\pi$$
$$992$$ −1.07153 −0.0340212
$$993$$ 0 0
$$994$$ 24.7984 0.786558
$$995$$ 6.96500 0.220805
$$996$$ 0 0
$$997$$ 14.5470 0.460709 0.230355 0.973107i $$-0.426011\pi$$
0.230355 + 0.973107i $$0.426011\pi$$
$$998$$ −107.508 −3.40311
$$999$$ 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 5445.2.a.be.1.2 4
3.2 odd 2 1815.2.a.x.1.3 4
11.3 even 5 495.2.n.d.361.1 8
11.4 even 5 495.2.n.d.181.1 8
11.10 odd 2 5445.2.a.bv.1.3 4
15.14 odd 2 9075.2.a.cl.1.2 4
33.14 odd 10 165.2.m.a.31.2 yes 8
33.26 odd 10 165.2.m.a.16.2 8
33.32 even 2 1815.2.a.o.1.2 4
165.14 odd 10 825.2.n.k.526.1 8
165.47 even 20 825.2.bx.h.724.4 16
165.59 odd 10 825.2.n.k.676.1 8
165.92 even 20 825.2.bx.h.49.1 16
165.113 even 20 825.2.bx.h.724.1 16
165.158 even 20 825.2.bx.h.49.4 16
165.164 even 2 9075.2.a.dj.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.16.2 8 33.26 odd 10
165.2.m.a.31.2 yes 8 33.14 odd 10
495.2.n.d.181.1 8 11.4 even 5
495.2.n.d.361.1 8 11.3 even 5
825.2.n.k.526.1 8 165.14 odd 10
825.2.n.k.676.1 8 165.59 odd 10
825.2.bx.h.49.1 16 165.92 even 20
825.2.bx.h.49.4 16 165.158 even 20
825.2.bx.h.724.1 16 165.113 even 20
825.2.bx.h.724.4 16 165.47 even 20
1815.2.a.o.1.2 4 33.32 even 2
1815.2.a.x.1.3 4 3.2 odd 2
5445.2.a.be.1.2 4 1.1 even 1 trivial
5445.2.a.bv.1.3 4 11.10 odd 2
9075.2.a.cl.1.2 4 15.14 odd 2
9075.2.a.dj.1.3 4 165.164 even 2