# Properties

 Label 1900.2.a.j.1.4 Level $1900$ Weight $2$ Character 1900.1 Self dual yes Analytic conductor $15.172$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.1715763840$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6x^{2} + 4$$ x^4 - 6*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.28825$$ of defining polynomial Character $$\chi$$ $$=$$ 1900.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} +O(q^{10})$$ $$q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} -5.23607 q^{11} -4.03631 q^{13} +1.08036 q^{17} -1.00000 q^{19} -6.47214 q^{21} +7.40492 q^{23} -1.74806 q^{27} +4.47214 q^{29} -4.00000 q^{31} -11.9814 q^{33} -6.86474 q^{37} -9.23607 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} +2.47214 q^{51} +6.86474 q^{53} -2.28825 q^{57} -10.4721 q^{59} +1.70820 q^{61} -6.32456 q^{63} -1.62054 q^{67} +16.9443 q^{69} -1.52786 q^{71} +13.7295 q^{73} +14.8098 q^{77} -11.4164 q^{79} -10.7082 q^{81} +5.24419 q^{83} +10.2333 q^{87} -14.9443 q^{89} +11.4164 q^{91} -9.15298 q^{93} -4.44897 q^{97} -11.7082 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{11} - 4 q^{19} - 8 q^{21} - 16 q^{31} - 28 q^{39} - 24 q^{41} + 4 q^{49} - 8 q^{51} - 24 q^{59} - 20 q^{61} + 32 q^{69} - 24 q^{71} + 8 q^{79} - 16 q^{81} - 24 q^{89} - 8 q^{91} - 20 q^{99}+O(q^{100})$$ 4 * q - 12 * q^11 - 4 * q^19 - 8 * q^21 - 16 * q^31 - 28 * q^39 - 24 * q^41 + 4 * q^49 - 8 * q^51 - 24 * q^59 - 20 * q^61 + 32 * q^69 - 24 * q^71 + 8 * q^79 - 16 * q^81 - 24 * q^89 - 8 * q^91 - 20 * 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
$$3$$ 2.28825 1.32112 0.660560 0.750774i $$-0.270319\pi$$
0.660560 + 0.750774i $$0.270319\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ 0 0
$$9$$ 2.23607 0.745356
$$10$$ 0 0
$$11$$ −5.23607 −1.57873 −0.789367 0.613922i $$-0.789591\pi$$
−0.789367 + 0.613922i $$0.789591\pi$$
$$12$$ 0 0
$$13$$ −4.03631 −1.11947 −0.559735 0.828671i $$-0.689097\pi$$
−0.559735 + 0.828671i $$0.689097\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.08036 0.262027 0.131013 0.991381i $$-0.458177\pi$$
0.131013 + 0.991381i $$0.458177\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −6.47214 −1.41234
$$22$$ 0 0
$$23$$ 7.40492 1.54403 0.772016 0.635603i $$-0.219248\pi$$
0.772016 + 0.635603i $$0.219248\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.74806 −0.336415
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −11.9814 −2.08570
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.86474 −1.12856 −0.564278 0.825585i $$-0.690845\pi$$
−0.564278 + 0.825585i $$0.690845\pi$$
$$38$$ 0 0
$$39$$ −9.23607 −1.47895
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −8.48528 −1.29399 −0.646997 0.762493i $$-0.723975\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.48528 −1.23771 −0.618853 0.785507i $$-0.712402\pi$$
−0.618853 + 0.785507i $$0.712402\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.47214 0.346168
$$52$$ 0 0
$$53$$ 6.86474 0.942944 0.471472 0.881881i $$-0.343723\pi$$
0.471472 + 0.881881i $$0.343723\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.28825 −0.303086
$$58$$ 0 0
$$59$$ −10.4721 −1.36336 −0.681678 0.731652i $$-0.738749\pi$$
−0.681678 + 0.731652i $$0.738749\pi$$
$$60$$ 0 0
$$61$$ 1.70820 0.218713 0.109357 0.994003i $$-0.465121\pi$$
0.109357 + 0.994003i $$0.465121\pi$$
$$62$$ 0 0
$$63$$ −6.32456 −0.796819
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −1.62054 −0.197981 −0.0989905 0.995088i $$-0.531561\pi$$
−0.0989905 + 0.995088i $$0.531561\pi$$
$$68$$ 0 0
$$69$$ 16.9443 2.03985
$$70$$ 0 0
$$71$$ −1.52786 −0.181324 −0.0906621 0.995882i $$-0.528898\pi$$
−0.0906621 + 0.995882i $$0.528898\pi$$
$$72$$ 0 0
$$73$$ 13.7295 1.60691 0.803457 0.595363i $$-0.202992\pi$$
0.803457 + 0.595363i $$0.202992\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 14.8098 1.68774
$$78$$ 0 0
$$79$$ −11.4164 −1.28445 −0.642223 0.766518i $$-0.721988\pi$$
−0.642223 + 0.766518i $$0.721988\pi$$
$$80$$ 0 0
$$81$$ −10.7082 −1.18980
$$82$$ 0 0
$$83$$ 5.24419 0.575625 0.287812 0.957687i $$-0.407072\pi$$
0.287812 + 0.957687i $$0.407072\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.2333 1.09713
$$88$$ 0 0
$$89$$ −14.9443 −1.58409 −0.792045 0.610463i $$-0.790983\pi$$
−0.792045 + 0.610463i $$0.790983\pi$$
$$90$$ 0 0
$$91$$ 11.4164 1.19676
$$92$$ 0 0
$$93$$ −9.15298 −0.949120
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −4.44897 −0.451725 −0.225862 0.974159i $$-0.572520\pi$$
−0.225862 + 0.974159i $$0.572520\pi$$
$$98$$ 0 0
$$99$$ −11.7082 −1.17672
$$100$$ 0 0
$$101$$ 0.763932 0.0760141 0.0380070 0.999277i $$-0.487899\pi$$
0.0380070 + 0.999277i $$0.487899\pi$$
$$102$$ 0 0
$$103$$ 15.3500 1.51248 0.756241 0.654293i $$-0.227034\pi$$
0.756241 + 0.654293i $$0.227034\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.4304 1.58838 0.794192 0.607666i $$-0.207894\pi$$
0.794192 + 0.607666i $$0.207894\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −15.7082 −1.49096
$$112$$ 0 0
$$113$$ −12.1089 −1.13911 −0.569556 0.821952i $$-0.692885\pi$$
−0.569556 + 0.821952i $$0.692885\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −9.02546 −0.834404
$$118$$ 0 0
$$119$$ −3.05573 −0.280118
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ 0 0
$$123$$ −13.7295 −1.23794
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.1058 0.896747 0.448374 0.893846i $$-0.352003\pi$$
0.448374 + 0.893846i $$0.352003\pi$$
$$128$$ 0 0
$$129$$ −19.4164 −1.70952
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 2.82843 0.245256
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.08036 0.0923016 0.0461508 0.998934i $$-0.485305\pi$$
0.0461508 + 0.998934i $$0.485305\pi$$
$$138$$ 0 0
$$139$$ 15.7082 1.33235 0.666176 0.745794i $$-0.267930\pi$$
0.666176 + 0.745794i $$0.267930\pi$$
$$140$$ 0 0
$$141$$ −19.4164 −1.63516
$$142$$ 0 0
$$143$$ 21.1344 1.76735
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.28825 0.188731
$$148$$ 0 0
$$149$$ 18.6525 1.52807 0.764035 0.645175i $$-0.223215\pi$$
0.764035 + 0.645175i $$0.223215\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 2.41577 0.195303
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 19.3863 1.54720 0.773599 0.633676i $$-0.218455\pi$$
0.773599 + 0.633676i $$0.218455\pi$$
$$158$$ 0 0
$$159$$ 15.7082 1.24574
$$160$$ 0 0
$$161$$ −20.9443 −1.65064
$$162$$ 0 0
$$163$$ 8.48528 0.664619 0.332309 0.943170i $$-0.392172\pi$$
0.332309 + 0.943170i $$0.392172\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4.70401 0.364007 0.182004 0.983298i $$-0.441742\pi$$
0.182004 + 0.983298i $$0.441742\pi$$
$$168$$ 0 0
$$169$$ 3.29180 0.253215
$$170$$ 0 0
$$171$$ −2.23607 −0.170996
$$172$$ 0 0
$$173$$ 13.1893 1.00276 0.501382 0.865226i $$-0.332825\pi$$
0.501382 + 0.865226i $$0.332825\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −23.9628 −1.80116
$$178$$ 0 0
$$179$$ −13.5279 −1.01112 −0.505560 0.862791i $$-0.668714\pi$$
−0.505560 + 0.862791i $$0.668714\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ 3.90879 0.288946
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.65685 −0.413670
$$188$$ 0 0
$$189$$ 4.94427 0.359643
$$190$$ 0 0
$$191$$ −20.9443 −1.51547 −0.757737 0.652560i $$-0.773695\pi$$
−0.757737 + 0.652560i $$0.773695\pi$$
$$192$$ 0 0
$$193$$ −4.44897 −0.320244 −0.160122 0.987097i $$-0.551189\pi$$
−0.160122 + 0.987097i $$0.551189\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.6491 0.901212 0.450606 0.892723i $$-0.351208\pi$$
0.450606 + 0.892723i $$0.351208\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −3.70820 −0.261557
$$202$$ 0 0
$$203$$ −12.6491 −0.887794
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 16.5579 1.15085
$$208$$ 0 0
$$209$$ 5.23607 0.362186
$$210$$ 0 0
$$211$$ −11.4164 −0.785938 −0.392969 0.919552i $$-0.628552\pi$$
−0.392969 + 0.919552i $$0.628552\pi$$
$$212$$ 0 0
$$213$$ −3.49613 −0.239551
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 11.3137 0.768025
$$218$$ 0 0
$$219$$ 31.4164 2.12292
$$220$$ 0 0
$$221$$ −4.36068 −0.293331
$$222$$ 0 0
$$223$$ 28.6668 1.91967 0.959836 0.280560i $$-0.0905202\pi$$
0.959836 + 0.280560i $$0.0905202\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −15.3500 −1.01882 −0.509408 0.860525i $$-0.670136\pi$$
−0.509408 + 0.860525i $$0.670136\pi$$
$$228$$ 0 0
$$229$$ −5.70820 −0.377209 −0.188604 0.982053i $$-0.560396\pi$$
−0.188604 + 0.982053i $$0.560396\pi$$
$$230$$ 0 0
$$231$$ 33.8885 2.22970
$$232$$ 0 0
$$233$$ −12.6491 −0.828671 −0.414335 0.910124i $$-0.635986\pi$$
−0.414335 + 0.910124i $$0.635986\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −26.1235 −1.69691
$$238$$ 0 0
$$239$$ 5.88854 0.380898 0.190449 0.981697i $$-0.439006\pi$$
0.190449 + 0.981697i $$0.439006\pi$$
$$240$$ 0 0
$$241$$ 24.8328 1.59962 0.799811 0.600252i $$-0.204933\pi$$
0.799811 + 0.600252i $$0.204933\pi$$
$$242$$ 0 0
$$243$$ −19.2588 −1.23545
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.03631 0.256824
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 29.8885 1.88655 0.943274 0.332015i $$-0.107728\pi$$
0.943274 + 0.332015i $$0.107728\pi$$
$$252$$ 0 0
$$253$$ −38.7727 −2.43762
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4.70401 −0.293428 −0.146714 0.989179i $$-0.546870\pi$$
−0.146714 + 0.989179i $$0.546870\pi$$
$$258$$ 0 0
$$259$$ 19.4164 1.20648
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ −9.56564 −0.589843 −0.294921 0.955522i $$-0.595293\pi$$
−0.294921 + 0.955522i $$0.595293\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −34.1962 −2.09277
$$268$$ 0 0
$$269$$ −2.94427 −0.179515 −0.0897577 0.995964i $$-0.528609\pi$$
−0.0897577 + 0.995964i $$0.528609\pi$$
$$270$$ 0 0
$$271$$ −19.1246 −1.16174 −0.580869 0.813997i $$-0.697287\pi$$
−0.580869 + 0.813997i $$0.697287\pi$$
$$272$$ 0 0
$$273$$ 26.1235 1.58107
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.24109 0.194738 0.0973691 0.995248i $$-0.468957\pi$$
0.0973691 + 0.995248i $$0.468957\pi$$
$$278$$ 0 0
$$279$$ −8.94427 −0.535480
$$280$$ 0 0
$$281$$ 1.41641 0.0844958 0.0422479 0.999107i $$-0.486548\pi$$
0.0422479 + 0.999107i $$0.486548\pi$$
$$282$$ 0 0
$$283$$ −16.5579 −0.984265 −0.492133 0.870520i $$-0.663782\pi$$
−0.492133 + 0.870520i $$0.663782\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 16.9706 1.00174
$$288$$ 0 0
$$289$$ −15.8328 −0.931342
$$290$$ 0 0
$$291$$ −10.1803 −0.596782
$$292$$ 0 0
$$293$$ 6.86474 0.401042 0.200521 0.979689i $$-0.435736\pi$$
0.200521 + 0.979689i $$0.435736\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 9.15298 0.531110
$$298$$ 0 0
$$299$$ −29.8885 −1.72850
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ 1.74806 0.100424
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.03631 −0.230364 −0.115182 0.993344i $$-0.536745\pi$$
−0.115182 + 0.993344i $$0.536745\pi$$
$$308$$ 0 0
$$309$$ 35.1246 1.99817
$$310$$ 0 0
$$311$$ 3.70820 0.210273 0.105136 0.994458i $$-0.466472\pi$$
0.105136 + 0.994458i $$0.466472\pi$$
$$312$$ 0 0
$$313$$ −27.4589 −1.55207 −0.776036 0.630689i $$-0.782772\pi$$
−0.776036 + 0.630689i $$0.782772\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −13.1893 −0.740784 −0.370392 0.928875i $$-0.620777\pi$$
−0.370392 + 0.928875i $$0.620777\pi$$
$$318$$ 0 0
$$319$$ −23.4164 −1.31107
$$320$$ 0 0
$$321$$ 37.5967 2.09845
$$322$$ 0 0
$$323$$ −1.08036 −0.0601130
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4.57649 0.253081
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −19.4164 −1.06722 −0.533611 0.845730i $$-0.679165\pi$$
−0.533611 + 0.845730i $$0.679165\pi$$
$$332$$ 0 0
$$333$$ −15.3500 −0.841176
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −7.27740 −0.396425 −0.198213 0.980159i $$-0.563514\pi$$
−0.198213 + 0.980159i $$0.563514\pi$$
$$338$$ 0 0
$$339$$ −27.7082 −1.50490
$$340$$ 0 0
$$341$$ 20.9443 1.13420
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −34.8639 −1.87159 −0.935795 0.352544i $$-0.885317\pi$$
−0.935795 + 0.352544i $$0.885317\pi$$
$$348$$ 0 0
$$349$$ −1.41641 −0.0758186 −0.0379093 0.999281i $$-0.512070\pi$$
−0.0379093 + 0.999281i $$0.512070\pi$$
$$350$$ 0 0
$$351$$ 7.05573 0.376607
$$352$$ 0 0
$$353$$ −15.8902 −0.845750 −0.422875 0.906188i $$-0.638979\pi$$
−0.422875 + 0.906188i $$0.638979\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.99226 −0.370069
$$358$$ 0 0
$$359$$ −11.1246 −0.587135 −0.293567 0.955938i $$-0.594842\pi$$
−0.293567 + 0.955938i $$0.594842\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 37.5648 1.97164
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 0 0
$$369$$ −13.4164 −0.698430
$$370$$ 0 0
$$371$$ −19.4164 −1.00805
$$372$$ 0 0
$$373$$ 34.3237 1.77721 0.888606 0.458670i $$-0.151674\pi$$
0.888606 + 0.458670i $$0.151674\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −18.0509 −0.929670
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 23.1246 1.18471
$$382$$ 0 0
$$383$$ −23.6777 −1.20987 −0.604936 0.796274i $$-0.706801\pi$$
−0.604936 + 0.796274i $$0.706801\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −18.9737 −0.964486
$$388$$ 0 0
$$389$$ 2.94427 0.149281 0.0746403 0.997211i $$-0.476219\pi$$
0.0746403 + 0.997211i $$0.476219\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −30.7000 −1.54079 −0.770395 0.637566i $$-0.779941\pi$$
−0.770395 + 0.637566i $$0.779941\pi$$
$$398$$ 0 0
$$399$$ 6.47214 0.324012
$$400$$ 0 0
$$401$$ 4.47214 0.223328 0.111664 0.993746i $$-0.464382\pi$$
0.111664 + 0.993746i $$0.464382\pi$$
$$402$$ 0 0
$$403$$ 16.1452 0.804252
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 35.9442 1.78169
$$408$$ 0 0
$$409$$ −9.41641 −0.465611 −0.232806 0.972523i $$-0.574791\pi$$
−0.232806 + 0.972523i $$0.574791\pi$$
$$410$$ 0 0
$$411$$ 2.47214 0.121941
$$412$$ 0 0
$$413$$ 29.6197 1.45749
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 35.9442 1.76020
$$418$$ 0 0
$$419$$ −3.05573 −0.149282 −0.0746410 0.997210i $$-0.523781\pi$$
−0.0746410 + 0.997210i $$0.523781\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 0 0
$$423$$ −18.9737 −0.922531
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −4.83153 −0.233814
$$428$$ 0 0
$$429$$ 48.3607 2.33488
$$430$$ 0 0
$$431$$ 10.4721 0.504425 0.252213 0.967672i $$-0.418842\pi$$
0.252213 + 0.967672i $$0.418842\pi$$
$$432$$ 0 0
$$433$$ 37.9774 1.82508 0.912540 0.408989i $$-0.134118\pi$$
0.912540 + 0.408989i $$0.134118\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −7.40492 −0.354225
$$438$$ 0 0
$$439$$ 34.8328 1.66248 0.831240 0.555914i $$-0.187632\pi$$
0.831240 + 0.555914i $$0.187632\pi$$
$$440$$ 0 0
$$441$$ 2.23607 0.106479
$$442$$ 0 0
$$443$$ −5.24419 −0.249159 −0.124580 0.992210i $$-0.539758\pi$$
−0.124580 + 0.992210i $$0.539758\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 42.6814 2.01876
$$448$$ 0 0
$$449$$ −7.52786 −0.355262 −0.177631 0.984097i $$-0.556843\pi$$
−0.177631 + 0.984097i $$0.556843\pi$$
$$450$$ 0 0
$$451$$ 31.4164 1.47934
$$452$$ 0 0
$$453$$ −18.3060 −0.860089
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.9473 1.77510 0.887551 0.460710i $$-0.152405\pi$$
0.887551 + 0.460710i $$0.152405\pi$$
$$458$$ 0 0
$$459$$ −1.88854 −0.0881497
$$460$$ 0 0
$$461$$ −11.8885 −0.553705 −0.276852 0.960912i $$-0.589291\pi$$
−0.276852 + 0.960912i $$0.589291\pi$$
$$462$$ 0 0
$$463$$ −33.5285 −1.55820 −0.779100 0.626900i $$-0.784324\pi$$
−0.779100 + 0.626900i $$0.784324\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −11.7264 −0.542632 −0.271316 0.962490i $$-0.587459\pi$$
−0.271316 + 0.962490i $$0.587459\pi$$
$$468$$ 0 0
$$469$$ 4.58359 0.211651
$$470$$ 0 0
$$471$$ 44.3607 2.04403
$$472$$ 0 0
$$473$$ 44.4295 2.04287
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 15.3500 0.702829
$$478$$ 0 0
$$479$$ 6.76393 0.309052 0.154526 0.987989i $$-0.450615\pi$$
0.154526 + 0.987989i $$0.450615\pi$$
$$480$$ 0 0
$$481$$ 27.7082 1.26339
$$482$$ 0 0
$$483$$ −47.9256 −2.18069
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −17.7658 −0.805044 −0.402522 0.915410i $$-0.631866\pi$$
−0.402522 + 0.915410i $$0.631866\pi$$
$$488$$ 0 0
$$489$$ 19.4164 0.878040
$$490$$ 0 0
$$491$$ −29.8885 −1.34885 −0.674426 0.738343i $$-0.735609\pi$$
−0.674426 + 0.738343i $$0.735609\pi$$
$$492$$ 0 0
$$493$$ 4.83153 0.217601
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.32145 0.193844
$$498$$ 0 0
$$499$$ 23.1246 1.03520 0.517600 0.855623i $$-0.326826\pi$$
0.517600 + 0.855623i $$0.326826\pi$$
$$500$$ 0 0
$$501$$ 10.7639 0.480897
$$502$$ 0 0
$$503$$ −39.1853 −1.74719 −0.873593 0.486656i $$-0.838216\pi$$
−0.873593 + 0.486656i $$0.838216\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 7.53244 0.334527
$$508$$ 0 0
$$509$$ −37.4164 −1.65845 −0.829227 0.558913i $$-0.811219\pi$$
−0.829227 + 0.558913i $$0.811219\pi$$
$$510$$ 0 0
$$511$$ −38.8328 −1.71786
$$512$$ 0 0
$$513$$ 1.74806 0.0771789
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 44.4295 1.95401
$$518$$ 0 0
$$519$$ 30.1803 1.32477
$$520$$ 0 0
$$521$$ 35.8885 1.57231 0.786153 0.618032i $$-0.212070\pi$$
0.786153 + 0.618032i $$0.212070\pi$$
$$522$$ 0 0
$$523$$ −3.62365 −0.158451 −0.0792255 0.996857i $$-0.525245\pi$$
−0.0792255 + 0.996857i $$0.525245\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.32145 −0.188245
$$528$$ 0 0
$$529$$ 31.8328 1.38404
$$530$$ 0 0
$$531$$ −23.4164 −1.01619
$$532$$ 0 0
$$533$$ 24.2179 1.04899
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −30.9551 −1.33581
$$538$$ 0 0
$$539$$ −5.23607 −0.225533
$$540$$ 0 0
$$541$$ −1.70820 −0.0734414 −0.0367207 0.999326i $$-0.511691\pi$$
−0.0367207 + 0.999326i $$0.511691\pi$$
$$542$$ 0 0
$$543$$ 13.7295 0.589188
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.20788 −0.0516453 −0.0258227 0.999667i $$-0.508221\pi$$
−0.0258227 + 0.999667i $$0.508221\pi$$
$$548$$ 0 0
$$549$$ 3.81966 0.163019
$$550$$ 0 0
$$551$$ −4.47214 −0.190519
$$552$$ 0 0
$$553$$ 32.2905 1.37313
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 11.5687 0.490184 0.245092 0.969500i $$-0.421182\pi$$
0.245092 + 0.969500i $$0.421182\pi$$
$$558$$ 0 0
$$559$$ 34.2492 1.44859
$$560$$ 0 0
$$561$$ −12.9443 −0.546508
$$562$$ 0 0
$$563$$ 17.3531 0.731347 0.365673 0.930743i $$-0.380839\pi$$
0.365673 + 0.930743i $$0.380839\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 30.2874 1.27195
$$568$$ 0 0
$$569$$ −21.0557 −0.882702 −0.441351 0.897335i $$-0.645501\pi$$
−0.441351 + 0.897335i $$0.645501\pi$$
$$570$$ 0 0
$$571$$ −15.1246 −0.632945 −0.316473 0.948602i $$-0.602499\pi$$
−0.316473 + 0.948602i $$0.602499\pi$$
$$572$$ 0 0
$$573$$ −47.9256 −2.00212
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 5.65685 0.235498 0.117749 0.993043i $$-0.462432\pi$$
0.117749 + 0.993043i $$0.462432\pi$$
$$578$$ 0 0
$$579$$ −10.1803 −0.423080
$$580$$ 0 0
$$581$$ −14.8328 −0.615369
$$582$$ 0 0
$$583$$ −35.9442 −1.48866
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −37.0246 −1.52817 −0.764084 0.645117i $$-0.776809\pi$$
−0.764084 + 0.645117i $$0.776809\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 28.9443 1.19061
$$592$$ 0 0
$$593$$ 14.8098 0.608167 0.304084 0.952645i $$-0.401650\pi$$
0.304084 + 0.952645i $$0.401650\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −36.6119 −1.49843
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −36.8328 −1.50244 −0.751221 0.660051i $$-0.770535\pi$$
−0.751221 + 0.660051i $$0.770535\pi$$
$$602$$ 0 0
$$603$$ −3.62365 −0.147566
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −12.9343 −0.524985 −0.262493 0.964934i $$-0.584545\pi$$
−0.262493 + 0.964934i $$0.584545\pi$$
$$608$$ 0 0
$$609$$ −28.9443 −1.17288
$$610$$ 0 0
$$611$$ 34.2492 1.38558
$$612$$ 0 0
$$613$$ 20.2117 0.816341 0.408170 0.912906i $$-0.366167\pi$$
0.408170 + 0.912906i $$0.366167\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 28.5393 1.14895 0.574475 0.818522i $$-0.305206\pi$$
0.574475 + 0.818522i $$0.305206\pi$$
$$618$$ 0 0
$$619$$ 0.875388 0.0351848 0.0175924 0.999845i $$-0.494400\pi$$
0.0175924 + 0.999845i $$0.494400\pi$$
$$620$$ 0 0
$$621$$ −12.9443 −0.519436
$$622$$ 0 0
$$623$$ 42.2688 1.69346
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 11.9814 0.478491
$$628$$ 0 0
$$629$$ −7.41641 −0.295712
$$630$$ 0 0
$$631$$ 11.7082 0.466096 0.233048 0.972465i $$-0.425130\pi$$
0.233048 + 0.972465i $$0.425130\pi$$
$$632$$ 0 0
$$633$$ −26.1235 −1.03832
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4.03631 −0.159924
$$638$$ 0 0
$$639$$ −3.41641 −0.135151
$$640$$ 0 0
$$641$$ −47.8885 −1.89148 −0.945742 0.324919i $$-0.894663\pi$$
−0.945742 + 0.324919i $$0.894663\pi$$
$$642$$ 0 0
$$643$$ 11.7264 0.462443 0.231221 0.972901i $$-0.425728\pi$$
0.231221 + 0.972901i $$0.425728\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −34.8639 −1.37064 −0.685320 0.728242i $$-0.740338\pi$$
−0.685320 + 0.728242i $$0.740338\pi$$
$$648$$ 0 0
$$649$$ 54.8328 2.15238
$$650$$ 0 0
$$651$$ 25.8885 1.01465
$$652$$ 0 0
$$653$$ 3.24109 0.126834 0.0634168 0.997987i $$-0.479800\pi$$
0.0634168 + 0.997987i $$0.479800\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 30.7000 1.19772
$$658$$ 0 0
$$659$$ −27.0557 −1.05394 −0.526971 0.849883i $$-0.676672\pi$$
−0.526971 + 0.849883i $$0.676672\pi$$
$$660$$ 0 0
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ 0 0
$$663$$ −9.97831 −0.387525
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 33.1158 1.28225
$$668$$ 0 0
$$669$$ 65.5967 2.53612
$$670$$ 0 0
$$671$$ −8.94427 −0.345290
$$672$$ 0 0
$$673$$ −42.8090 −1.65016 −0.825082 0.565013i $$-0.808871\pi$$
−0.825082 + 0.565013i $$0.808871\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 30.0022 1.15308 0.576540 0.817069i $$-0.304403\pi$$
0.576540 + 0.817069i $$0.304403\pi$$
$$678$$ 0 0
$$679$$ 12.5836 0.482914
$$680$$ 0 0
$$681$$ −35.1246 −1.34598
$$682$$ 0 0
$$683$$ −10.1058 −0.386689 −0.193344 0.981131i $$-0.561933\pi$$
−0.193344 + 0.981131i $$0.561933\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −13.0618 −0.498338
$$688$$ 0 0
$$689$$ −27.7082 −1.05560
$$690$$ 0 0
$$691$$ −15.1246 −0.575367 −0.287684 0.957725i $$-0.592885\pi$$
−0.287684 + 0.957725i $$0.592885\pi$$
$$692$$ 0 0
$$693$$ 33.1158 1.25797
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.48218 −0.245530
$$698$$ 0 0
$$699$$ −28.9443 −1.09477
$$700$$ 0 0
$$701$$ 41.1246 1.55326 0.776628 0.629960i $$-0.216929\pi$$
0.776628 + 0.629960i $$0.216929\pi$$
$$702$$ 0 0
$$703$$ 6.86474 0.258908
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.16073 −0.0812625
$$708$$ 0 0
$$709$$ 6.58359 0.247252 0.123626 0.992329i $$-0.460548\pi$$
0.123626 + 0.992329i $$0.460548\pi$$
$$710$$ 0 0
$$711$$ −25.5279 −0.957370
$$712$$ 0 0
$$713$$ −29.6197 −1.10927
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 13.4744 0.503212
$$718$$ 0 0
$$719$$ −21.5967 −0.805423 −0.402711 0.915327i $$-0.631932\pi$$
−0.402711 + 0.915327i $$0.631932\pi$$
$$720$$ 0 0
$$721$$ −43.4164 −1.61691
$$722$$ 0 0
$$723$$ 56.8236 2.11329
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −8.48528 −0.314702 −0.157351 0.987543i $$-0.550295\pi$$
−0.157351 + 0.987543i $$0.550295\pi$$
$$728$$ 0 0
$$729$$ −11.9443 −0.442380
$$730$$ 0 0
$$731$$ −9.16718 −0.339061
$$732$$ 0 0
$$733$$ 33.9411 1.25364 0.626822 0.779162i $$-0.284355\pi$$
0.626822 + 0.779162i $$0.284355\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.48528 0.312559
$$738$$ 0 0
$$739$$ 34.8328 1.28135 0.640673 0.767814i $$-0.278655\pi$$
0.640673 + 0.767814i $$0.278655\pi$$
$$740$$ 0 0
$$741$$ 9.23607 0.339295
$$742$$ 0 0
$$743$$ −6.86474 −0.251843 −0.125921 0.992040i $$-0.540189\pi$$
−0.125921 + 0.992040i $$0.540189\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 11.7264 0.429045
$$748$$ 0 0
$$749$$ −46.4721 −1.69805
$$750$$ 0 0
$$751$$ 47.4164 1.73025 0.865125 0.501557i $$-0.167239\pi$$
0.865125 + 0.501557i $$0.167239\pi$$
$$752$$ 0 0
$$753$$ 68.3923 2.49236
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0.825324 0.0299969 0.0149985 0.999888i $$-0.495226\pi$$
0.0149985 + 0.999888i $$0.495226\pi$$
$$758$$ 0 0
$$759$$ −88.7214 −3.22038
$$760$$ 0 0
$$761$$ 24.5410 0.889611 0.444806 0.895627i $$-0.353273\pi$$
0.444806 + 0.895627i $$0.353273\pi$$
$$762$$ 0 0
$$763$$ −5.65685 −0.204792
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 42.2688 1.52624
$$768$$ 0 0
$$769$$ 28.5410 1.02922 0.514608 0.857426i $$-0.327938\pi$$
0.514608 + 0.857426i $$0.327938\pi$$
$$770$$ 0 0
$$771$$ −10.7639 −0.387654
$$772$$ 0 0
$$773$$ −15.3500 −0.552102 −0.276051 0.961143i $$-0.589026\pi$$
−0.276051 + 0.961143i $$0.589026\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 44.4295 1.59390
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ −7.81758 −0.279378
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.62365 0.129169 0.0645845 0.997912i $$-0.479428\pi$$
0.0645845 + 0.997912i $$0.479428\pi$$
$$788$$ 0 0
$$789$$ −21.8885 −0.779253
$$790$$ 0 0
$$791$$ 34.2492 1.21776
$$792$$ 0 0
$$793$$ −6.89484 −0.244843
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −14.1120 −0.499874 −0.249937 0.968262i $$-0.580410\pi$$
−0.249937 + 0.968262i $$0.580410\pi$$
$$798$$ 0 0
$$799$$ −9.16718 −0.324312
$$800$$ 0 0
$$801$$ −33.4164 −1.18071
$$802$$ 0 0
$$803$$ −71.8885 −2.53689
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.73722 −0.237161
$$808$$ 0 0
$$809$$ −16.4721 −0.579129 −0.289565 0.957158i $$-0.593511\pi$$
−0.289565 + 0.957158i $$0.593511\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 0 0
$$813$$ −43.7618 −1.53479
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.48528 0.296862
$$818$$ 0 0
$$819$$ 25.5279 0.892016
$$820$$ 0 0
$$821$$ 21.0557 0.734850 0.367425 0.930053i $$-0.380239\pi$$
0.367425 + 0.930053i $$0.380239\pi$$
$$822$$ 0 0
$$823$$ −2.00310 −0.0698238 −0.0349119 0.999390i $$-0.511115\pi$$
−0.0349119 + 0.999390i $$0.511115\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20.5942 −0.716131 −0.358065 0.933696i $$-0.616564\pi$$
−0.358065 + 0.933696i $$0.616564\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ 7.41641 0.257272
$$832$$ 0 0
$$833$$ 1.08036 0.0374324
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 6.99226 0.241688
$$838$$ 0 0
$$839$$ −13.3050 −0.459338 −0.229669 0.973269i $$-0.573764\pi$$
−0.229669 + 0.973269i $$0.573764\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0 0
$$843$$ 3.24109 0.111629
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −46.4326 −1.59544
$$848$$ 0 0
$$849$$ −37.8885 −1.30033
$$850$$ 0 0
$$851$$ −50.8328 −1.74253
$$852$$ 0 0
$$853$$ −10.4884 −0.359115 −0.179558 0.983747i $$-0.557467\pi$$
−0.179558 + 0.983747i $$0.557467\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −40.8059 −1.39390 −0.696951 0.717119i $$-0.745460\pi$$
−0.696951 + 0.717119i $$0.745460\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 38.8328 1.32342
$$862$$ 0 0
$$863$$ 20.5942 0.701035 0.350518 0.936556i $$-0.386006\pi$$
0.350518 + 0.936556i $$0.386006\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −36.2294 −1.23041
$$868$$ 0 0
$$869$$ 59.7771 2.02780
$$870$$ 0 0
$$871$$ 6.54102 0.221634
$$872$$ 0 0
$$873$$ −9.94820 −0.336696
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −4.44897 −0.150231 −0.0751155 0.997175i $$-0.523933\pi$$
−0.0751155 + 0.997175i $$0.523933\pi$$
$$878$$ 0 0
$$879$$ 15.7082 0.529825
$$880$$ 0 0
$$881$$ −30.6525 −1.03271 −0.516354 0.856375i $$-0.672711\pi$$
−0.516354 + 0.856375i $$0.672711\pi$$
$$882$$ 0 0
$$883$$ 14.9675 0.503695 0.251848 0.967767i $$-0.418962\pi$$
0.251848 + 0.967767i $$0.418962\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36.3268 −1.21973 −0.609867 0.792504i $$-0.708777\pi$$
−0.609867 + 0.792504i $$0.708777\pi$$
$$888$$ 0 0
$$889$$ −28.5836 −0.958663
$$890$$ 0 0
$$891$$ 56.0689 1.87838
$$892$$ 0 0
$$893$$ 8.48528 0.283949
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −68.3923 −2.28355
$$898$$ 0 0
$$899$$ −17.8885 −0.596616
$$900$$ 0 0
$$901$$ 7.41641 0.247076
$$902$$ 0 0
$$903$$ 54.9179 1.82755
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −8.86784 −0.294452 −0.147226 0.989103i $$-0.547034\pi$$
−0.147226 + 0.989103i $$0.547034\pi$$
$$908$$ 0 0
$$909$$ 1.70820 0.0566575
$$910$$ 0 0
$$911$$ −37.5279 −1.24335 −0.621677 0.783274i $$-0.713548\pi$$
−0.621677 + 0.783274i $$0.713548\pi$$
$$912$$ 0 0
$$913$$ −27.4589 −0.908759
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −26.8328 −0.885133 −0.442566 0.896736i $$-0.645932\pi$$
−0.442566 + 0.896736i $$0.645932\pi$$
$$920$$ 0 0
$$921$$ −9.23607 −0.304339
$$922$$ 0 0
$$923$$ 6.16693 0.202987
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 34.3237 1.12734
$$928$$ 0 0
$$929$$ 16.4721 0.540433 0.270217 0.962800i $$-0.412905\pi$$
0.270217 + 0.962800i $$0.412905\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 8.48528 0.277796
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −51.6768 −1.68821 −0.844104 0.536180i $$-0.819867\pi$$
−0.844104 + 0.536180i $$0.819867\pi$$
$$938$$ 0 0
$$939$$ −62.8328 −2.05047
$$940$$ 0 0
$$941$$ −21.0557 −0.686397 −0.343199 0.939263i $$-0.611510\pi$$
−0.343199 + 0.939263i $$0.611510\pi$$
$$942$$ 0 0
$$943$$ −44.4295 −1.44682
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.6969 0.932525 0.466263 0.884646i $$-0.345600\pi$$
0.466263 + 0.884646i $$0.345600\pi$$
$$948$$ 0 0
$$949$$ −55.4164 −1.79889
$$950$$ 0 0
$$951$$ −30.1803 −0.978665
$$952$$ 0 0
$$953$$ 27.0764 0.877090 0.438545 0.898709i $$-0.355494\pi$$
0.438545 + 0.898709i $$0.355494\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −53.5825 −1.73208
$$958$$ 0 0
$$959$$ −3.05573 −0.0986746
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 36.7394 1.18391
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −16.5579 −0.532466 −0.266233 0.963909i $$-0.585779\pi$$
−0.266233 + 0.963909i $$0.585779\pi$$
$$968$$ 0 0
$$969$$ −2.47214 −0.0794164
$$970$$ 0 0
$$971$$ 4.36068 0.139941 0.0699704 0.997549i $$-0.477710\pi$$
0.0699704 + 0.997549i $$0.477710\pi$$
$$972$$ 0 0
$$973$$ −44.4295 −1.42434
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −47.1304 −1.50784 −0.753918 0.656969i $$-0.771838\pi$$
−0.753918 + 0.656969i $$0.771838\pi$$
$$978$$ 0 0
$$979$$ 78.2492 2.50086
$$980$$ 0 0
$$981$$ 4.47214 0.142784
$$982$$ 0 0
$$983$$ 33.4009 1.06532 0.532662 0.846328i $$-0.321192\pi$$
0.532662 + 0.846328i $$0.321192\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 54.9179 1.74806
$$988$$ 0 0
$$989$$ −62.8328 −1.99797
$$990$$ 0 0
$$991$$ −14.8328 −0.471180 −0.235590 0.971853i $$-0.575702\pi$$
−0.235590 + 0.971853i $$0.575702\pi$$
$$992$$ 0 0
$$993$$ −44.4295 −1.40993
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 30.7000 0.972280 0.486140 0.873881i $$-0.338405\pi$$
0.486140 + 0.873881i $$0.338405\pi$$
$$998$$ 0 0
$$999$$ 12.0000 0.379663
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 1900.2.a.j.1.4 4
4.3 odd 2 7600.2.a.ce.1.1 4
5.2 odd 4 380.2.c.a.229.1 4
5.3 odd 4 380.2.c.a.229.4 yes 4
5.4 even 2 inner 1900.2.a.j.1.1 4
15.2 even 4 3420.2.f.a.1369.3 4
15.8 even 4 3420.2.f.a.1369.4 4
20.3 even 4 1520.2.d.f.609.1 4
20.7 even 4 1520.2.d.f.609.4 4
20.19 odd 2 7600.2.a.ce.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.1 4 5.2 odd 4
380.2.c.a.229.4 yes 4 5.3 odd 4
1520.2.d.f.609.1 4 20.3 even 4
1520.2.d.f.609.4 4 20.7 even 4
1900.2.a.j.1.1 4 5.4 even 2 inner
1900.2.a.j.1.4 4 1.1 even 1 trivial
3420.2.f.a.1369.3 4 15.2 even 4
3420.2.f.a.1369.4 4 15.8 even 4
7600.2.a.ce.1.1 4 4.3 odd 2
7600.2.a.ce.1.4 4 20.19 odd 2