# Properties

 Label 1850.2.a.r.1.2 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.44949 q^{3} +1.00000 q^{4} -1.44949 q^{6} -2.44949 q^{7} -1.00000 q^{8} -0.898979 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.44949 q^{3} +1.00000 q^{4} -1.44949 q^{6} -2.44949 q^{7} -1.00000 q^{8} -0.898979 q^{9} +3.44949 q^{11} +1.44949 q^{12} +0.449490 q^{13} +2.44949 q^{14} +1.00000 q^{16} -3.44949 q^{17} +0.898979 q^{18} -5.00000 q^{19} -3.55051 q^{21} -3.44949 q^{22} -2.00000 q^{23} -1.44949 q^{24} -0.449490 q^{26} -5.65153 q^{27} -2.44949 q^{28} -0.898979 q^{29} +4.44949 q^{31} -1.00000 q^{32} +5.00000 q^{33} +3.44949 q^{34} -0.898979 q^{36} +1.00000 q^{37} +5.00000 q^{38} +0.651531 q^{39} +1.00000 q^{41} +3.55051 q^{42} +1.10102 q^{43} +3.44949 q^{44} +2.00000 q^{46} -9.79796 q^{47} +1.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} +0.449490 q^{52} -6.00000 q^{53} +5.65153 q^{54} +2.44949 q^{56} -7.24745 q^{57} +0.898979 q^{58} -2.00000 q^{59} -6.44949 q^{61} -4.44949 q^{62} +2.20204 q^{63} +1.00000 q^{64} -5.00000 q^{66} -4.55051 q^{67} -3.44949 q^{68} -2.89898 q^{69} -7.55051 q^{71} +0.898979 q^{72} -12.7980 q^{73} -1.00000 q^{74} -5.00000 q^{76} -8.44949 q^{77} -0.651531 q^{78} +7.79796 q^{79} -5.49490 q^{81} -1.00000 q^{82} -3.44949 q^{83} -3.55051 q^{84} -1.10102 q^{86} -1.30306 q^{87} -3.44949 q^{88} +14.3485 q^{89} -1.10102 q^{91} -2.00000 q^{92} +6.44949 q^{93} +9.79796 q^{94} -1.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -3.10102 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27} + 8 q^{29} + 4 q^{31} - 2 q^{32} + 10 q^{33} + 2 q^{34} + 8 q^{36} + 2 q^{37} + 10 q^{38} + 16 q^{39} + 2 q^{41} + 12 q^{42} + 12 q^{43} + 2 q^{44} + 4 q^{46} - 2 q^{48} - 2 q^{49} - 10 q^{51} - 4 q^{52} - 12 q^{53} + 26 q^{54} + 10 q^{57} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 4 q^{62} + 24 q^{63} + 2 q^{64} - 10 q^{66} - 14 q^{67} - 2 q^{68} + 4 q^{69} - 20 q^{71} - 8 q^{72} - 6 q^{73} - 2 q^{74} - 10 q^{76} - 12 q^{77} - 16 q^{78} - 4 q^{79} + 38 q^{81} - 2 q^{82} - 2 q^{83} - 12 q^{84} - 12 q^{86} - 32 q^{87} - 2 q^{88} + 14 q^{89} - 12 q^{91} - 4 q^{92} + 8 q^{93} + 2 q^{96} - 28 q^{97} + 2 q^{98} - 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.44949 0.836863 0.418432 0.908248i $$-0.362580\pi$$
0.418432 + 0.908248i $$0.362580\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.44949 −0.591752
$$7$$ −2.44949 −0.925820 −0.462910 0.886405i $$-0.653195\pi$$
−0.462910 + 0.886405i $$0.653195\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −0.898979 −0.299660
$$10$$ 0 0
$$11$$ 3.44949 1.04006 0.520030 0.854148i $$-0.325921\pi$$
0.520030 + 0.854148i $$0.325921\pi$$
$$12$$ 1.44949 0.418432
$$13$$ 0.449490 0.124666 0.0623330 0.998055i $$-0.480146\pi$$
0.0623330 + 0.998055i $$0.480146\pi$$
$$14$$ 2.44949 0.654654
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.44949 −0.836624 −0.418312 0.908303i $$-0.637378\pi$$
−0.418312 + 0.908303i $$0.637378\pi$$
$$18$$ 0.898979 0.211891
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ −3.55051 −0.774785
$$22$$ −3.44949 −0.735434
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ −1.44949 −0.295876
$$25$$ 0 0
$$26$$ −0.449490 −0.0881522
$$27$$ −5.65153 −1.08764
$$28$$ −2.44949 −0.462910
$$29$$ −0.898979 −0.166936 −0.0834681 0.996510i $$-0.526600\pi$$
−0.0834681 + 0.996510i $$0.526600\pi$$
$$30$$ 0 0
$$31$$ 4.44949 0.799152 0.399576 0.916700i $$-0.369157\pi$$
0.399576 + 0.916700i $$0.369157\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.00000 0.870388
$$34$$ 3.44949 0.591583
$$35$$ 0 0
$$36$$ −0.898979 −0.149830
$$37$$ 1.00000 0.164399
$$38$$ 5.00000 0.811107
$$39$$ 0.651531 0.104328
$$40$$ 0 0
$$41$$ 1.00000 0.156174 0.0780869 0.996947i $$-0.475119\pi$$
0.0780869 + 0.996947i $$0.475119\pi$$
$$42$$ 3.55051 0.547856
$$43$$ 1.10102 0.167904 0.0839520 0.996470i $$-0.473246\pi$$
0.0839520 + 0.996470i $$0.473246\pi$$
$$44$$ 3.44949 0.520030
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ −9.79796 −1.42918 −0.714590 0.699544i $$-0.753387\pi$$
−0.714590 + 0.699544i $$0.753387\pi$$
$$48$$ 1.44949 0.209216
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ 0.449490 0.0623330
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 5.65153 0.769076
$$55$$ 0 0
$$56$$ 2.44949 0.327327
$$57$$ −7.24745 −0.959948
$$58$$ 0.898979 0.118042
$$59$$ −2.00000 −0.260378 −0.130189 0.991489i $$-0.541558\pi$$
−0.130189 + 0.991489i $$0.541558\pi$$
$$60$$ 0 0
$$61$$ −6.44949 −0.825773 −0.412886 0.910783i $$-0.635479\pi$$
−0.412886 + 0.910783i $$0.635479\pi$$
$$62$$ −4.44949 −0.565086
$$63$$ 2.20204 0.277431
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.00000 −0.615457
$$67$$ −4.55051 −0.555933 −0.277967 0.960591i $$-0.589660\pi$$
−0.277967 + 0.960591i $$0.589660\pi$$
$$68$$ −3.44949 −0.418312
$$69$$ −2.89898 −0.348996
$$70$$ 0 0
$$71$$ −7.55051 −0.896081 −0.448040 0.894013i $$-0.647878\pi$$
−0.448040 + 0.894013i $$0.647878\pi$$
$$72$$ 0.898979 0.105946
$$73$$ −12.7980 −1.49789 −0.748944 0.662633i $$-0.769439\pi$$
−0.748944 + 0.662633i $$0.769439\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −5.00000 −0.573539
$$77$$ −8.44949 −0.962909
$$78$$ −0.651531 −0.0737713
$$79$$ 7.79796 0.877339 0.438669 0.898648i $$-0.355450\pi$$
0.438669 + 0.898648i $$0.355450\pi$$
$$80$$ 0 0
$$81$$ −5.49490 −0.610544
$$82$$ −1.00000 −0.110432
$$83$$ −3.44949 −0.378631 −0.189315 0.981916i $$-0.560627\pi$$
−0.189315 + 0.981916i $$0.560627\pi$$
$$84$$ −3.55051 −0.387392
$$85$$ 0 0
$$86$$ −1.10102 −0.118726
$$87$$ −1.30306 −0.139703
$$88$$ −3.44949 −0.367717
$$89$$ 14.3485 1.52093 0.760467 0.649376i $$-0.224970\pi$$
0.760467 + 0.649376i $$0.224970\pi$$
$$90$$ 0 0
$$91$$ −1.10102 −0.115418
$$92$$ −2.00000 −0.208514
$$93$$ 6.44949 0.668781
$$94$$ 9.79796 1.01058
$$95$$ 0 0
$$96$$ −1.44949 −0.147938
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −3.10102 −0.311664
$$100$$ 0 0
$$101$$ 1.10102 0.109556 0.0547778 0.998499i $$-0.482555\pi$$
0.0547778 + 0.998499i $$0.482555\pi$$
$$102$$ 5.00000 0.495074
$$103$$ −6.44949 −0.635487 −0.317744 0.948177i $$-0.602925\pi$$
−0.317744 + 0.948177i $$0.602925\pi$$
$$104$$ −0.449490 −0.0440761
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 8.55051 0.826609 0.413305 0.910593i $$-0.364375\pi$$
0.413305 + 0.910593i $$0.364375\pi$$
$$108$$ −5.65153 −0.543819
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 1.44949 0.137579
$$112$$ −2.44949 −0.231455
$$113$$ 4.55051 0.428076 0.214038 0.976825i $$-0.431338\pi$$
0.214038 + 0.976825i $$0.431338\pi$$
$$114$$ 7.24745 0.678786
$$115$$ 0 0
$$116$$ −0.898979 −0.0834681
$$117$$ −0.404082 −0.0373574
$$118$$ 2.00000 0.184115
$$119$$ 8.44949 0.774563
$$120$$ 0 0
$$121$$ 0.898979 0.0817254
$$122$$ 6.44949 0.583909
$$123$$ 1.44949 0.130696
$$124$$ 4.44949 0.399576
$$125$$ 0 0
$$126$$ −2.20204 −0.196173
$$127$$ 16.2474 1.44173 0.720864 0.693077i $$-0.243745\pi$$
0.720864 + 0.693077i $$0.243745\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.59592 0.140513
$$130$$ 0 0
$$131$$ 8.69694 0.759855 0.379928 0.925016i $$-0.375949\pi$$
0.379928 + 0.925016i $$0.375949\pi$$
$$132$$ 5.00000 0.435194
$$133$$ 12.2474 1.06199
$$134$$ 4.55051 0.393104
$$135$$ 0 0
$$136$$ 3.44949 0.295791
$$137$$ −9.69694 −0.828465 −0.414233 0.910171i $$-0.635950\pi$$
−0.414233 + 0.910171i $$0.635950\pi$$
$$138$$ 2.89898 0.246778
$$139$$ −12.5505 −1.06452 −0.532260 0.846581i $$-0.678657\pi$$
−0.532260 + 0.846581i $$0.678657\pi$$
$$140$$ 0 0
$$141$$ −14.2020 −1.19603
$$142$$ 7.55051 0.633625
$$143$$ 1.55051 0.129660
$$144$$ −0.898979 −0.0749150
$$145$$ 0 0
$$146$$ 12.7980 1.05917
$$147$$ −1.44949 −0.119552
$$148$$ 1.00000 0.0821995
$$149$$ −4.65153 −0.381068 −0.190534 0.981681i $$-0.561022\pi$$
−0.190534 + 0.981681i $$0.561022\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 3.10102 0.250703
$$154$$ 8.44949 0.680879
$$155$$ 0 0
$$156$$ 0.651531 0.0521642
$$157$$ −16.4495 −1.31281 −0.656406 0.754408i $$-0.727924\pi$$
−0.656406 + 0.754408i $$0.727924\pi$$
$$158$$ −7.79796 −0.620372
$$159$$ −8.69694 −0.689712
$$160$$ 0 0
$$161$$ 4.89898 0.386094
$$162$$ 5.49490 0.431720
$$163$$ −10.1010 −0.791173 −0.395586 0.918429i $$-0.629459\pi$$
−0.395586 + 0.918429i $$0.629459\pi$$
$$164$$ 1.00000 0.0780869
$$165$$ 0 0
$$166$$ 3.44949 0.267732
$$167$$ 10.4495 0.808606 0.404303 0.914625i $$-0.367514\pi$$
0.404303 + 0.914625i $$0.367514\pi$$
$$168$$ 3.55051 0.273928
$$169$$ −12.7980 −0.984458
$$170$$ 0 0
$$171$$ 4.49490 0.343733
$$172$$ 1.10102 0.0839520
$$173$$ 6.69694 0.509159 0.254579 0.967052i $$-0.418063\pi$$
0.254579 + 0.967052i $$0.418063\pi$$
$$174$$ 1.30306 0.0987848
$$175$$ 0 0
$$176$$ 3.44949 0.260015
$$177$$ −2.89898 −0.217901
$$178$$ −14.3485 −1.07546
$$179$$ −0.797959 −0.0596423 −0.0298211 0.999555i $$-0.509494\pi$$
−0.0298211 + 0.999555i $$0.509494\pi$$
$$180$$ 0 0
$$181$$ 6.89898 0.512797 0.256399 0.966571i $$-0.417464\pi$$
0.256399 + 0.966571i $$0.417464\pi$$
$$182$$ 1.10102 0.0816131
$$183$$ −9.34847 −0.691059
$$184$$ 2.00000 0.147442
$$185$$ 0 0
$$186$$ −6.44949 −0.472900
$$187$$ −11.8990 −0.870140
$$188$$ −9.79796 −0.714590
$$189$$ 13.8434 1.00696
$$190$$ 0 0
$$191$$ −7.79796 −0.564241 −0.282120 0.959379i $$-0.591038\pi$$
−0.282120 + 0.959379i $$0.591038\pi$$
$$192$$ 1.44949 0.104608
$$193$$ 2.55051 0.183590 0.0917949 0.995778i $$-0.470740\pi$$
0.0917949 + 0.995778i $$0.470740\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ 17.3485 1.23603 0.618014 0.786167i $$-0.287938\pi$$
0.618014 + 0.786167i $$0.287938\pi$$
$$198$$ 3.10102 0.220380
$$199$$ −15.5959 −1.10557 −0.552783 0.833325i $$-0.686434\pi$$
−0.552783 + 0.833325i $$0.686434\pi$$
$$200$$ 0 0
$$201$$ −6.59592 −0.465240
$$202$$ −1.10102 −0.0774675
$$203$$ 2.20204 0.154553
$$204$$ −5.00000 −0.350070
$$205$$ 0 0
$$206$$ 6.44949 0.449357
$$207$$ 1.79796 0.124967
$$208$$ 0.449490 0.0311665
$$209$$ −17.2474 −1.19303
$$210$$ 0 0
$$211$$ −4.55051 −0.313270 −0.156635 0.987657i $$-0.550065\pi$$
−0.156635 + 0.987657i $$0.550065\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −10.9444 −0.749897
$$214$$ −8.55051 −0.584501
$$215$$ 0 0
$$216$$ 5.65153 0.384538
$$217$$ −10.8990 −0.739871
$$218$$ −14.0000 −0.948200
$$219$$ −18.5505 −1.25353
$$220$$ 0 0
$$221$$ −1.55051 −0.104299
$$222$$ −1.44949 −0.0972834
$$223$$ −21.7980 −1.45970 −0.729850 0.683608i $$-0.760410\pi$$
−0.729850 + 0.683608i $$0.760410\pi$$
$$224$$ 2.44949 0.163663
$$225$$ 0 0
$$226$$ −4.55051 −0.302695
$$227$$ 16.6969 1.10821 0.554107 0.832445i $$-0.313060\pi$$
0.554107 + 0.832445i $$0.313060\pi$$
$$228$$ −7.24745 −0.479974
$$229$$ 5.79796 0.383140 0.191570 0.981479i $$-0.438642\pi$$
0.191570 + 0.981479i $$0.438642\pi$$
$$230$$ 0 0
$$231$$ −12.2474 −0.805823
$$232$$ 0.898979 0.0590209
$$233$$ 28.4949 1.86676 0.933381 0.358886i $$-0.116843\pi$$
0.933381 + 0.358886i $$0.116843\pi$$
$$234$$ 0.404082 0.0264157
$$235$$ 0 0
$$236$$ −2.00000 −0.130189
$$237$$ 11.3031 0.734213
$$238$$ −8.44949 −0.547699
$$239$$ 0.898979 0.0581501 0.0290751 0.999577i $$-0.490744\pi$$
0.0290751 + 0.999577i $$0.490744\pi$$
$$240$$ 0 0
$$241$$ −2.55051 −0.164293 −0.0821464 0.996620i $$-0.526178\pi$$
−0.0821464 + 0.996620i $$0.526178\pi$$
$$242$$ −0.898979 −0.0577886
$$243$$ 8.98979 0.576696
$$244$$ −6.44949 −0.412886
$$245$$ 0 0
$$246$$ −1.44949 −0.0924161
$$247$$ −2.24745 −0.143002
$$248$$ −4.44949 −0.282543
$$249$$ −5.00000 −0.316862
$$250$$ 0 0
$$251$$ −30.7980 −1.94395 −0.971975 0.235084i $$-0.924463\pi$$
−0.971975 + 0.235084i $$0.924463\pi$$
$$252$$ 2.20204 0.138716
$$253$$ −6.89898 −0.433735
$$254$$ −16.2474 −1.01946
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.89898 0.430347 0.215173 0.976576i $$-0.430968\pi$$
0.215173 + 0.976576i $$0.430968\pi$$
$$258$$ −1.59592 −0.0993575
$$259$$ −2.44949 −0.152204
$$260$$ 0 0
$$261$$ 0.808164 0.0500241
$$262$$ −8.69694 −0.537299
$$263$$ −0.202041 −0.0124584 −0.00622919 0.999981i $$-0.501983\pi$$
−0.00622919 + 0.999981i $$0.501983\pi$$
$$264$$ −5.00000 −0.307729
$$265$$ 0 0
$$266$$ −12.2474 −0.750939
$$267$$ 20.7980 1.27281
$$268$$ −4.55051 −0.277967
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ 30.0454 1.82513 0.912564 0.408933i $$-0.134099\pi$$
0.912564 + 0.408933i $$0.134099\pi$$
$$272$$ −3.44949 −0.209156
$$273$$ −1.59592 −0.0965893
$$274$$ 9.69694 0.585813
$$275$$ 0 0
$$276$$ −2.89898 −0.174498
$$277$$ −5.79796 −0.348366 −0.174183 0.984713i $$-0.555728\pi$$
−0.174183 + 0.984713i $$0.555728\pi$$
$$278$$ 12.5505 0.752730
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 8.89898 0.530869 0.265434 0.964129i $$-0.414485\pi$$
0.265434 + 0.964129i $$0.414485\pi$$
$$282$$ 14.2020 0.845719
$$283$$ 11.6969 0.695311 0.347655 0.937622i $$-0.386978\pi$$
0.347655 + 0.937622i $$0.386978\pi$$
$$284$$ −7.55051 −0.448040
$$285$$ 0 0
$$286$$ −1.55051 −0.0916836
$$287$$ −2.44949 −0.144589
$$288$$ 0.898979 0.0529729
$$289$$ −5.10102 −0.300060
$$290$$ 0 0
$$291$$ −20.2929 −1.18959
$$292$$ −12.7980 −0.748944
$$293$$ 28.0454 1.63843 0.819215 0.573486i $$-0.194409\pi$$
0.819215 + 0.573486i $$0.194409\pi$$
$$294$$ 1.44949 0.0845360
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ −19.4949 −1.13121
$$298$$ 4.65153 0.269456
$$299$$ −0.898979 −0.0519893
$$300$$ 0 0
$$301$$ −2.69694 −0.155449
$$302$$ 14.0000 0.805609
$$303$$ 1.59592 0.0916831
$$304$$ −5.00000 −0.286770
$$305$$ 0 0
$$306$$ −3.10102 −0.177274
$$307$$ −7.65153 −0.436696 −0.218348 0.975871i $$-0.570067\pi$$
−0.218348 + 0.975871i $$0.570067\pi$$
$$308$$ −8.44949 −0.481454
$$309$$ −9.34847 −0.531816
$$310$$ 0 0
$$311$$ −8.44949 −0.479127 −0.239563 0.970881i $$-0.577004\pi$$
−0.239563 + 0.970881i $$0.577004\pi$$
$$312$$ −0.651531 −0.0368857
$$313$$ −8.89898 −0.503000 −0.251500 0.967857i $$-0.580924\pi$$
−0.251500 + 0.967857i $$0.580924\pi$$
$$314$$ 16.4495 0.928298
$$315$$ 0 0
$$316$$ 7.79796 0.438669
$$317$$ 9.55051 0.536410 0.268205 0.963362i $$-0.413570\pi$$
0.268205 + 0.963362i $$0.413570\pi$$
$$318$$ 8.69694 0.487700
$$319$$ −3.10102 −0.173624
$$320$$ 0 0
$$321$$ 12.3939 0.691759
$$322$$ −4.89898 −0.273009
$$323$$ 17.2474 0.959674
$$324$$ −5.49490 −0.305272
$$325$$ 0 0
$$326$$ 10.1010 0.559444
$$327$$ 20.2929 1.12220
$$328$$ −1.00000 −0.0552158
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 19.8990 1.09375 0.546873 0.837215i $$-0.315818\pi$$
0.546873 + 0.837215i $$0.315818\pi$$
$$332$$ −3.44949 −0.189315
$$333$$ −0.898979 −0.0492638
$$334$$ −10.4495 −0.571771
$$335$$ 0 0
$$336$$ −3.55051 −0.193696
$$337$$ −15.6969 −0.855067 −0.427533 0.904000i $$-0.640617\pi$$
−0.427533 + 0.904000i $$0.640617\pi$$
$$338$$ 12.7980 0.696117
$$339$$ 6.59592 0.358241
$$340$$ 0 0
$$341$$ 15.3485 0.831166
$$342$$ −4.49490 −0.243056
$$343$$ 19.5959 1.05808
$$344$$ −1.10102 −0.0593630
$$345$$ 0 0
$$346$$ −6.69694 −0.360030
$$347$$ 18.7980 1.00913 0.504564 0.863374i $$-0.331653\pi$$
0.504564 + 0.863374i $$0.331653\pi$$
$$348$$ −1.30306 −0.0698514
$$349$$ −12.4495 −0.666406 −0.333203 0.942855i $$-0.608129\pi$$
−0.333203 + 0.942855i $$0.608129\pi$$
$$350$$ 0 0
$$351$$ −2.54031 −0.135591
$$352$$ −3.44949 −0.183858
$$353$$ −14.8990 −0.792993 −0.396496 0.918036i $$-0.629774\pi$$
−0.396496 + 0.918036i $$0.629774\pi$$
$$354$$ 2.89898 0.154079
$$355$$ 0 0
$$356$$ 14.3485 0.760467
$$357$$ 12.2474 0.648204
$$358$$ 0.797959 0.0421734
$$359$$ −31.3939 −1.65691 −0.828453 0.560059i $$-0.810778\pi$$
−0.828453 + 0.560059i $$0.810778\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −6.89898 −0.362602
$$363$$ 1.30306 0.0683930
$$364$$ −1.10102 −0.0577092
$$365$$ 0 0
$$366$$ 9.34847 0.488652
$$367$$ 14.2474 0.743711 0.371855 0.928291i $$-0.378722\pi$$
0.371855 + 0.928291i $$0.378722\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ −0.898979 −0.0467990
$$370$$ 0 0
$$371$$ 14.6969 0.763027
$$372$$ 6.44949 0.334390
$$373$$ 20.0454 1.03791 0.518956 0.854801i $$-0.326321\pi$$
0.518956 + 0.854801i $$0.326321\pi$$
$$374$$ 11.8990 0.615282
$$375$$ 0 0
$$376$$ 9.79796 0.505291
$$377$$ −0.404082 −0.0208113
$$378$$ −13.8434 −0.712026
$$379$$ −21.0454 −1.08103 −0.540515 0.841334i $$-0.681771\pi$$
−0.540515 + 0.841334i $$0.681771\pi$$
$$380$$ 0 0
$$381$$ 23.5505 1.20653
$$382$$ 7.79796 0.398978
$$383$$ 5.34847 0.273294 0.136647 0.990620i $$-0.456367\pi$$
0.136647 + 0.990620i $$0.456367\pi$$
$$384$$ −1.44949 −0.0739690
$$385$$ 0 0
$$386$$ −2.55051 −0.129818
$$387$$ −0.989795 −0.0503141
$$388$$ −14.0000 −0.710742
$$389$$ 13.1464 0.666550 0.333275 0.942830i $$-0.391846\pi$$
0.333275 + 0.942830i $$0.391846\pi$$
$$390$$ 0 0
$$391$$ 6.89898 0.348896
$$392$$ 1.00000 0.0505076
$$393$$ 12.6061 0.635895
$$394$$ −17.3485 −0.874003
$$395$$ 0 0
$$396$$ −3.10102 −0.155832
$$397$$ 15.3485 0.770318 0.385159 0.922850i $$-0.374147\pi$$
0.385159 + 0.922850i $$0.374147\pi$$
$$398$$ 15.5959 0.781753
$$399$$ 17.7526 0.888739
$$400$$ 0 0
$$401$$ 15.9444 0.796225 0.398112 0.917337i $$-0.369665\pi$$
0.398112 + 0.917337i $$0.369665\pi$$
$$402$$ 6.59592 0.328974
$$403$$ 2.00000 0.0996271
$$404$$ 1.10102 0.0547778
$$405$$ 0 0
$$406$$ −2.20204 −0.109285
$$407$$ 3.44949 0.170985
$$408$$ 5.00000 0.247537
$$409$$ −34.1464 −1.68843 −0.844216 0.536003i $$-0.819934\pi$$
−0.844216 + 0.536003i $$0.819934\pi$$
$$410$$ 0 0
$$411$$ −14.0556 −0.693312
$$412$$ −6.44949 −0.317744
$$413$$ 4.89898 0.241063
$$414$$ −1.79796 −0.0883649
$$415$$ 0 0
$$416$$ −0.449490 −0.0220380
$$417$$ −18.1918 −0.890858
$$418$$ 17.2474 0.843600
$$419$$ −21.4495 −1.04788 −0.523938 0.851756i $$-0.675538\pi$$
−0.523938 + 0.851756i $$0.675538\pi$$
$$420$$ 0 0
$$421$$ 32.8990 1.60340 0.801699 0.597728i $$-0.203930\pi$$
0.801699 + 0.597728i $$0.203930\pi$$
$$422$$ 4.55051 0.221515
$$423$$ 8.80816 0.428268
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 10.9444 0.530257
$$427$$ 15.7980 0.764517
$$428$$ 8.55051 0.413305
$$429$$ 2.24745 0.108508
$$430$$ 0 0
$$431$$ −20.6969 −0.996936 −0.498468 0.866908i $$-0.666104\pi$$
−0.498468 + 0.866908i $$0.666104\pi$$
$$432$$ −5.65153 −0.271909
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ 10.8990 0.523168
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 10.0000 0.478365
$$438$$ 18.5505 0.886378
$$439$$ −19.5959 −0.935262 −0.467631 0.883924i $$-0.654892\pi$$
−0.467631 + 0.883924i $$0.654892\pi$$
$$440$$ 0 0
$$441$$ 0.898979 0.0428085
$$442$$ 1.55051 0.0737503
$$443$$ 17.4495 0.829050 0.414525 0.910038i $$-0.363948\pi$$
0.414525 + 0.910038i $$0.363948\pi$$
$$444$$ 1.44949 0.0687897
$$445$$ 0 0
$$446$$ 21.7980 1.03216
$$447$$ −6.74235 −0.318902
$$448$$ −2.44949 −0.115728
$$449$$ 33.2474 1.56904 0.784522 0.620101i $$-0.212908\pi$$
0.784522 + 0.620101i $$0.212908\pi$$
$$450$$ 0 0
$$451$$ 3.44949 0.162430
$$452$$ 4.55051 0.214038
$$453$$ −20.2929 −0.953442
$$454$$ −16.6969 −0.783626
$$455$$ 0 0
$$456$$ 7.24745 0.339393
$$457$$ −15.2474 −0.713246 −0.356623 0.934248i $$-0.616072\pi$$
−0.356623 + 0.934248i $$0.616072\pi$$
$$458$$ −5.79796 −0.270921
$$459$$ 19.4949 0.909944
$$460$$ 0 0
$$461$$ 9.30306 0.433287 0.216643 0.976251i $$-0.430489\pi$$
0.216643 + 0.976251i $$0.430489\pi$$
$$462$$ 12.2474 0.569803
$$463$$ −9.55051 −0.443850 −0.221925 0.975064i $$-0.571234\pi$$
−0.221925 + 0.975064i $$0.571234\pi$$
$$464$$ −0.898979 −0.0417341
$$465$$ 0 0
$$466$$ −28.4949 −1.32000
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −0.404082 −0.0186787
$$469$$ 11.1464 0.514694
$$470$$ 0 0
$$471$$ −23.8434 −1.09864
$$472$$ 2.00000 0.0920575
$$473$$ 3.79796 0.174630
$$474$$ −11.3031 −0.519167
$$475$$ 0 0
$$476$$ 8.44949 0.387282
$$477$$ 5.39388 0.246969
$$478$$ −0.898979 −0.0411184
$$479$$ 6.24745 0.285453 0.142727 0.989762i $$-0.454413\pi$$
0.142727 + 0.989762i $$0.454413\pi$$
$$480$$ 0 0
$$481$$ 0.449490 0.0204950
$$482$$ 2.55051 0.116173
$$483$$ 7.10102 0.323108
$$484$$ 0.898979 0.0408627
$$485$$ 0 0
$$486$$ −8.98979 −0.407785
$$487$$ 30.7423 1.39307 0.696534 0.717523i $$-0.254724\pi$$
0.696534 + 0.717523i $$0.254724\pi$$
$$488$$ 6.44949 0.291955
$$489$$ −14.6413 −0.662104
$$490$$ 0 0
$$491$$ −41.7980 −1.88632 −0.943158 0.332345i $$-0.892160\pi$$
−0.943158 + 0.332345i $$0.892160\pi$$
$$492$$ 1.44949 0.0653480
$$493$$ 3.10102 0.139663
$$494$$ 2.24745 0.101117
$$495$$ 0 0
$$496$$ 4.44949 0.199788
$$497$$ 18.4949 0.829610
$$498$$ 5.00000 0.224055
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ 15.1464 0.676693
$$502$$ 30.7980 1.37458
$$503$$ −9.79796 −0.436869 −0.218435 0.975852i $$-0.570095\pi$$
−0.218435 + 0.975852i $$0.570095\pi$$
$$504$$ −2.20204 −0.0980867
$$505$$ 0 0
$$506$$ 6.89898 0.306697
$$507$$ −18.5505 −0.823857
$$508$$ 16.2474 0.720864
$$509$$ 29.3485 1.30085 0.650424 0.759571i $$-0.274591\pi$$
0.650424 + 0.759571i $$0.274591\pi$$
$$510$$ 0 0
$$511$$ 31.3485 1.38677
$$512$$ −1.00000 −0.0441942
$$513$$ 28.2577 1.24761
$$514$$ −6.89898 −0.304301
$$515$$ 0 0
$$516$$ 1.59592 0.0702564
$$517$$ −33.7980 −1.48643
$$518$$ 2.44949 0.107624
$$519$$ 9.70714 0.426096
$$520$$ 0 0
$$521$$ 10.1010 0.442534 0.221267 0.975213i $$-0.428981\pi$$
0.221267 + 0.975213i $$0.428981\pi$$
$$522$$ −0.808164 −0.0353724
$$523$$ 18.3939 0.804308 0.402154 0.915572i $$-0.368262\pi$$
0.402154 + 0.915572i $$0.368262\pi$$
$$524$$ 8.69694 0.379928
$$525$$ 0 0
$$526$$ 0.202041 0.00880941
$$527$$ −15.3485 −0.668590
$$528$$ 5.00000 0.217597
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 1.79796 0.0780248
$$532$$ 12.2474 0.530994
$$533$$ 0.449490 0.0194696
$$534$$ −20.7980 −0.900016
$$535$$ 0 0
$$536$$ 4.55051 0.196552
$$537$$ −1.15663 −0.0499124
$$538$$ −4.00000 −0.172452
$$539$$ −3.44949 −0.148580
$$540$$ 0 0
$$541$$ 25.3485 1.08982 0.544908 0.838496i $$-0.316565\pi$$
0.544908 + 0.838496i $$0.316565\pi$$
$$542$$ −30.0454 −1.29056
$$543$$ 10.0000 0.429141
$$544$$ 3.44949 0.147896
$$545$$ 0 0
$$546$$ 1.59592 0.0682990
$$547$$ −3.69694 −0.158070 −0.0790348 0.996872i $$-0.525184\pi$$
−0.0790348 + 0.996872i $$0.525184\pi$$
$$548$$ −9.69694 −0.414233
$$549$$ 5.79796 0.247451
$$550$$ 0 0
$$551$$ 4.49490 0.191489
$$552$$ 2.89898 0.123389
$$553$$ −19.1010 −0.812258
$$554$$ 5.79796 0.246332
$$555$$ 0 0
$$556$$ −12.5505 −0.532260
$$557$$ 21.7980 0.923609 0.461805 0.886982i $$-0.347202\pi$$
0.461805 + 0.886982i $$0.347202\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 0.494897 0.0209319
$$560$$ 0 0
$$561$$ −17.2474 −0.728188
$$562$$ −8.89898 −0.375381
$$563$$ −37.5959 −1.58448 −0.792240 0.610210i $$-0.791085\pi$$
−0.792240 + 0.610210i $$0.791085\pi$$
$$564$$ −14.2020 −0.598014
$$565$$ 0 0
$$566$$ −11.6969 −0.491659
$$567$$ 13.4597 0.565254
$$568$$ 7.55051 0.316812
$$569$$ 27.0454 1.13380 0.566901 0.823786i $$-0.308142\pi$$
0.566901 + 0.823786i $$0.308142\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 1.55051 0.0648301
$$573$$ −11.3031 −0.472192
$$574$$ 2.44949 0.102240
$$575$$ 0 0
$$576$$ −0.898979 −0.0374575
$$577$$ −21.0454 −0.876132 −0.438066 0.898943i $$-0.644336\pi$$
−0.438066 + 0.898943i $$0.644336\pi$$
$$578$$ 5.10102 0.212174
$$579$$ 3.69694 0.153640
$$580$$ 0 0
$$581$$ 8.44949 0.350544
$$582$$ 20.2929 0.841166
$$583$$ −20.6969 −0.857180
$$584$$ 12.7980 0.529583
$$585$$ 0 0
$$586$$ −28.0454 −1.15855
$$587$$ 6.30306 0.260155 0.130078 0.991504i $$-0.458477\pi$$
0.130078 + 0.991504i $$0.458477\pi$$
$$588$$ −1.44949 −0.0597759
$$589$$ −22.2474 −0.916690
$$590$$ 0 0
$$591$$ 25.1464 1.03439
$$592$$ 1.00000 0.0410997
$$593$$ −46.3939 −1.90517 −0.952584 0.304275i $$-0.901586\pi$$
−0.952584 + 0.304275i $$0.901586\pi$$
$$594$$ 19.4949 0.799885
$$595$$ 0 0
$$596$$ −4.65153 −0.190534
$$597$$ −22.6061 −0.925207
$$598$$ 0.898979 0.0367620
$$599$$ 38.9444 1.59122 0.795612 0.605806i $$-0.207149\pi$$
0.795612 + 0.605806i $$0.207149\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ 2.69694 0.109919
$$603$$ 4.09082 0.166591
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ −1.59592 −0.0648297
$$607$$ 28.7423 1.16662 0.583308 0.812251i $$-0.301758\pi$$
0.583308 + 0.812251i $$0.301758\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 3.19184 0.129340
$$610$$ 0 0
$$611$$ −4.40408 −0.178170
$$612$$ 3.10102 0.125351
$$613$$ 34.4949 1.39324 0.696618 0.717442i $$-0.254687\pi$$
0.696618 + 0.717442i $$0.254687\pi$$
$$614$$ 7.65153 0.308791
$$615$$ 0 0
$$616$$ 8.44949 0.340440
$$617$$ 4.89898 0.197225 0.0986127 0.995126i $$-0.468559\pi$$
0.0986127 + 0.995126i $$0.468559\pi$$
$$618$$ 9.34847 0.376051
$$619$$ −32.8990 −1.32232 −0.661161 0.750244i $$-0.729936\pi$$
−0.661161 + 0.750244i $$0.729936\pi$$
$$620$$ 0 0
$$621$$ 11.3031 0.453576
$$622$$ 8.44949 0.338794
$$623$$ −35.1464 −1.40811
$$624$$ 0.651531 0.0260821
$$625$$ 0 0
$$626$$ 8.89898 0.355675
$$627$$ −25.0000 −0.998404
$$628$$ −16.4495 −0.656406
$$629$$ −3.44949 −0.137540
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ −7.79796 −0.310186
$$633$$ −6.59592 −0.262164
$$634$$ −9.55051 −0.379299
$$635$$ 0 0
$$636$$ −8.69694 −0.344856
$$637$$ −0.449490 −0.0178094
$$638$$ 3.10102 0.122771
$$639$$ 6.78775 0.268519
$$640$$ 0 0
$$641$$ 21.5959 0.852987 0.426494 0.904491i $$-0.359749\pi$$
0.426494 + 0.904491i $$0.359749\pi$$
$$642$$ −12.3939 −0.489147
$$643$$ −32.2929 −1.27351 −0.636753 0.771068i $$-0.719723\pi$$
−0.636753 + 0.771068i $$0.719723\pi$$
$$644$$ 4.89898 0.193047
$$645$$ 0 0
$$646$$ −17.2474 −0.678592
$$647$$ 40.2474 1.58229 0.791145 0.611628i $$-0.209485\pi$$
0.791145 + 0.611628i $$0.209485\pi$$
$$648$$ 5.49490 0.215860
$$649$$ −6.89898 −0.270809
$$650$$ 0 0
$$651$$ −15.7980 −0.619171
$$652$$ −10.1010 −0.395586
$$653$$ 10.0454 0.393107 0.196554 0.980493i $$-0.437025\pi$$
0.196554 + 0.980493i $$0.437025\pi$$
$$654$$ −20.2929 −0.793513
$$655$$ 0 0
$$656$$ 1.00000 0.0390434
$$657$$ 11.5051 0.448857
$$658$$ −24.0000 −0.935617
$$659$$ 6.55051 0.255172 0.127586 0.991828i $$-0.459277\pi$$
0.127586 + 0.991828i $$0.459277\pi$$
$$660$$ 0 0
$$661$$ 25.7980 1.00342 0.501712 0.865035i $$-0.332704\pi$$
0.501712 + 0.865035i $$0.332704\pi$$
$$662$$ −19.8990 −0.773396
$$663$$ −2.24745 −0.0872837
$$664$$ 3.44949 0.133866
$$665$$ 0 0
$$666$$ 0.898979 0.0348347
$$667$$ 1.79796 0.0696172
$$668$$ 10.4495 0.404303
$$669$$ −31.5959 −1.22157
$$670$$ 0 0
$$671$$ −22.2474 −0.858853
$$672$$ 3.55051 0.136964
$$673$$ −5.79796 −0.223495 −0.111747 0.993737i $$-0.535645\pi$$
−0.111747 + 0.993737i $$0.535645\pi$$
$$674$$ 15.6969 0.604623
$$675$$ 0 0
$$676$$ −12.7980 −0.492229
$$677$$ 43.5959 1.67553 0.837764 0.546033i $$-0.183863\pi$$
0.837764 + 0.546033i $$0.183863\pi$$
$$678$$ −6.59592 −0.253315
$$679$$ 34.2929 1.31604
$$680$$ 0 0
$$681$$ 24.2020 0.927424
$$682$$ −15.3485 −0.587723
$$683$$ 7.00000 0.267848 0.133924 0.990992i $$-0.457242\pi$$
0.133924 + 0.990992i $$0.457242\pi$$
$$684$$ 4.49490 0.171867
$$685$$ 0 0
$$686$$ −19.5959 −0.748176
$$687$$ 8.40408 0.320636
$$688$$ 1.10102 0.0419760
$$689$$ −2.69694 −0.102745
$$690$$ 0 0
$$691$$ 29.6515 1.12800 0.563999 0.825776i $$-0.309262\pi$$
0.563999 + 0.825776i $$0.309262\pi$$
$$692$$ 6.69694 0.254579
$$693$$ 7.59592 0.288545
$$694$$ −18.7980 −0.713561
$$695$$ 0 0
$$696$$ 1.30306 0.0493924
$$697$$ −3.44949 −0.130659
$$698$$ 12.4495 0.471220
$$699$$ 41.3031 1.56223
$$700$$ 0 0
$$701$$ −34.2474 −1.29351 −0.646754 0.762699i $$-0.723874\pi$$
−0.646754 + 0.762699i $$0.723874\pi$$
$$702$$ 2.54031 0.0958776
$$703$$ −5.00000 −0.188579
$$704$$ 3.44949 0.130008
$$705$$ 0 0
$$706$$ 14.8990 0.560730
$$707$$ −2.69694 −0.101429
$$708$$ −2.89898 −0.108950
$$709$$ 16.8990 0.634654 0.317327 0.948316i $$-0.397215\pi$$
0.317327 + 0.948316i $$0.397215\pi$$
$$710$$ 0 0
$$711$$ −7.01021 −0.262903
$$712$$ −14.3485 −0.537732
$$713$$ −8.89898 −0.333269
$$714$$ −12.2474 −0.458349
$$715$$ 0 0
$$716$$ −0.797959 −0.0298211
$$717$$ 1.30306 0.0486637
$$718$$ 31.3939 1.17161
$$719$$ −0.651531 −0.0242980 −0.0121490 0.999926i $$-0.503867\pi$$
−0.0121490 + 0.999926i $$0.503867\pi$$
$$720$$ 0 0
$$721$$ 15.7980 0.588347
$$722$$ −6.00000 −0.223297
$$723$$ −3.69694 −0.137491
$$724$$ 6.89898 0.256399
$$725$$ 0 0
$$726$$ −1.30306 −0.0483611
$$727$$ −36.0000 −1.33517 −0.667583 0.744535i $$-0.732671\pi$$
−0.667583 + 0.744535i $$0.732671\pi$$
$$728$$ 1.10102 0.0408065
$$729$$ 29.5153 1.09316
$$730$$ 0 0
$$731$$ −3.79796 −0.140473
$$732$$ −9.34847 −0.345529
$$733$$ 15.5959 0.576048 0.288024 0.957623i $$-0.407002\pi$$
0.288024 + 0.957623i $$0.407002\pi$$
$$734$$ −14.2474 −0.525883
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ −15.6969 −0.578204
$$738$$ 0.898979 0.0330919
$$739$$ −31.5959 −1.16227 −0.581137 0.813806i $$-0.697392\pi$$
−0.581137 + 0.813806i $$0.697392\pi$$
$$740$$ 0 0
$$741$$ −3.25765 −0.119673
$$742$$ −14.6969 −0.539542
$$743$$ 6.40408 0.234943 0.117471 0.993076i $$-0.462521\pi$$
0.117471 + 0.993076i $$0.462521\pi$$
$$744$$ −6.44949 −0.236450
$$745$$ 0 0
$$746$$ −20.0454 −0.733915
$$747$$ 3.10102 0.113460
$$748$$ −11.8990 −0.435070
$$749$$ −20.9444 −0.765291
$$750$$ 0 0
$$751$$ −30.6969 −1.12015 −0.560074 0.828443i $$-0.689227\pi$$
−0.560074 + 0.828443i $$0.689227\pi$$
$$752$$ −9.79796 −0.357295
$$753$$ −44.6413 −1.62682
$$754$$ 0.404082 0.0147158
$$755$$ 0 0
$$756$$ 13.8434 0.503478
$$757$$ 13.7980 0.501495 0.250748 0.968052i $$-0.419324\pi$$
0.250748 + 0.968052i $$0.419324\pi$$
$$758$$ 21.0454 0.764404
$$759$$ −10.0000 −0.362977
$$760$$ 0 0
$$761$$ −14.3031 −0.518486 −0.259243 0.965812i $$-0.583473\pi$$
−0.259243 + 0.965812i $$0.583473\pi$$
$$762$$ −23.5505 −0.853145
$$763$$ −34.2929 −1.24148
$$764$$ −7.79796 −0.282120
$$765$$ 0 0
$$766$$ −5.34847 −0.193248
$$767$$ −0.898979 −0.0324603
$$768$$ 1.44949 0.0523040
$$769$$ 53.2474 1.92015 0.960076 0.279739i $$-0.0902480\pi$$
0.960076 + 0.279739i $$0.0902480\pi$$
$$770$$ 0 0
$$771$$ 10.0000 0.360141
$$772$$ 2.55051 0.0917949
$$773$$ −22.0454 −0.792918 −0.396459 0.918052i $$-0.629761\pi$$
−0.396459 + 0.918052i $$0.629761\pi$$
$$774$$ 0.989795 0.0355774
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ −3.55051 −0.127374
$$778$$ −13.1464 −0.471322
$$779$$ −5.00000 −0.179144
$$780$$ 0 0
$$781$$ −26.0454 −0.931978
$$782$$ −6.89898 −0.246707
$$783$$ 5.08061 0.181566
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ −12.6061 −0.449646
$$787$$ −30.6969 −1.09423 −0.547114 0.837058i $$-0.684274\pi$$
−0.547114 + 0.837058i $$0.684274\pi$$
$$788$$ 17.3485 0.618014
$$789$$ −0.292856 −0.0104260
$$790$$ 0 0
$$791$$ −11.1464 −0.396321
$$792$$ 3.10102 0.110190
$$793$$ −2.89898 −0.102946
$$794$$ −15.3485 −0.544697
$$795$$ 0 0
$$796$$ −15.5959 −0.552783
$$797$$ −4.24745 −0.150452 −0.0752262 0.997166i $$-0.523968\pi$$
−0.0752262 + 0.997166i $$0.523968\pi$$
$$798$$ −17.7526 −0.628434
$$799$$ 33.7980 1.19569
$$800$$ 0 0
$$801$$ −12.8990 −0.455763
$$802$$ −15.9444 −0.563016
$$803$$ −44.1464 −1.55789
$$804$$ −6.59592 −0.232620
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ 5.79796 0.204098
$$808$$ −1.10102 −0.0387338
$$809$$ −39.3939 −1.38501 −0.692507 0.721411i $$-0.743494\pi$$
−0.692507 + 0.721411i $$0.743494\pi$$
$$810$$ 0 0
$$811$$ 5.79796 0.203594 0.101797 0.994805i $$-0.467541\pi$$
0.101797 + 0.994805i $$0.467541\pi$$
$$812$$ 2.20204 0.0772765
$$813$$ 43.5505 1.52738
$$814$$ −3.44949 −0.120905
$$815$$ 0 0
$$816$$ −5.00000 −0.175035
$$817$$ −5.50510 −0.192599
$$818$$ 34.1464 1.19390
$$819$$ 0.989795 0.0345862
$$820$$ 0 0
$$821$$ 4.04541 0.141186 0.0705929 0.997505i $$-0.477511\pi$$
0.0705929 + 0.997505i $$0.477511\pi$$
$$822$$ 14.0556 0.490246
$$823$$ −33.8434 −1.17971 −0.589853 0.807511i $$-0.700814\pi$$
−0.589853 + 0.807511i $$0.700814\pi$$
$$824$$ 6.44949 0.224679
$$825$$ 0 0
$$826$$ −4.89898 −0.170457
$$827$$ −27.0000 −0.938882 −0.469441 0.882964i $$-0.655545\pi$$
−0.469441 + 0.882964i $$0.655545\pi$$
$$828$$ 1.79796 0.0624834
$$829$$ 21.3485 0.741463 0.370731 0.928740i $$-0.379107\pi$$
0.370731 + 0.928740i $$0.379107\pi$$
$$830$$ 0 0
$$831$$ −8.40408 −0.291534
$$832$$ 0.449490 0.0155833
$$833$$ 3.44949 0.119518
$$834$$ 18.1918 0.629932
$$835$$ 0 0
$$836$$ −17.2474 −0.596515
$$837$$ −25.1464 −0.869188
$$838$$ 21.4495 0.740960
$$839$$ −1.34847 −0.0465543 −0.0232772 0.999729i $$-0.507410\pi$$
−0.0232772 + 0.999729i $$0.507410\pi$$
$$840$$ 0 0
$$841$$ −28.1918 −0.972132
$$842$$ −32.8990 −1.13377
$$843$$ 12.8990 0.444264
$$844$$ −4.55051 −0.156635
$$845$$ 0 0
$$846$$ −8.80816 −0.302831
$$847$$ −2.20204 −0.0756630
$$848$$ −6.00000 −0.206041
$$849$$ 16.9546 0.581880
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ −10.9444 −0.374949
$$853$$ 19.3485 0.662479 0.331239 0.943547i $$-0.392533\pi$$
0.331239 + 0.943547i $$0.392533\pi$$
$$854$$ −15.7980 −0.540595
$$855$$ 0 0
$$856$$ −8.55051 −0.292250
$$857$$ 32.1464 1.09810 0.549051 0.835789i $$-0.314989\pi$$
0.549051 + 0.835789i $$0.314989\pi$$
$$858$$ −2.24745 −0.0767266
$$859$$ −9.49490 −0.323962 −0.161981 0.986794i $$-0.551788\pi$$
−0.161981 + 0.986794i $$0.551788\pi$$
$$860$$ 0 0
$$861$$ −3.55051 −0.121001
$$862$$ 20.6969 0.704941
$$863$$ −18.2474 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$864$$ 5.65153 0.192269
$$865$$ 0 0
$$866$$ 19.0000 0.645646
$$867$$ −7.39388 −0.251109
$$868$$ −10.8990 −0.369935
$$869$$ 26.8990 0.912485
$$870$$ 0 0
$$871$$ −2.04541 −0.0693060
$$872$$ −14.0000 −0.474100
$$873$$ 12.5857 0.425962
$$874$$ −10.0000 −0.338255
$$875$$ 0 0
$$876$$ −18.5505 −0.626764
$$877$$ −4.65153 −0.157071 −0.0785355 0.996911i $$-0.525024\pi$$
−0.0785355 + 0.996911i $$0.525024\pi$$
$$878$$ 19.5959 0.661330
$$879$$ 40.6515 1.37114
$$880$$ 0 0
$$881$$ 34.2929 1.15536 0.577678 0.816265i $$-0.303959\pi$$
0.577678 + 0.816265i $$0.303959\pi$$
$$882$$ −0.898979 −0.0302702
$$883$$ −26.1010 −0.878369 −0.439185 0.898397i $$-0.644733\pi$$
−0.439185 + 0.898397i $$0.644733\pi$$
$$884$$ −1.55051 −0.0521493
$$885$$ 0 0
$$886$$ −17.4495 −0.586227
$$887$$ −50.6969 −1.70224 −0.851118 0.524974i $$-0.824075\pi$$
−0.851118 + 0.524974i $$0.824075\pi$$
$$888$$ −1.44949 −0.0486417
$$889$$ −39.7980 −1.33478
$$890$$ 0 0
$$891$$ −18.9546 −0.635003
$$892$$ −21.7980 −0.729850
$$893$$ 48.9898 1.63938
$$894$$ 6.74235 0.225498
$$895$$ 0 0
$$896$$ 2.44949 0.0818317
$$897$$ −1.30306 −0.0435080
$$898$$ −33.2474 −1.10948
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 20.6969 0.689515
$$902$$ −3.44949 −0.114855
$$903$$ −3.90918 −0.130090
$$904$$ −4.55051 −0.151348
$$905$$ 0 0
$$906$$ 20.2929 0.674185
$$907$$ 34.8990 1.15880 0.579401 0.815043i $$-0.303287\pi$$
0.579401 + 0.815043i $$0.303287\pi$$
$$908$$ 16.6969 0.554107
$$909$$ −0.989795 −0.0328294
$$910$$ 0 0
$$911$$ −16.2474 −0.538302 −0.269151 0.963098i $$-0.586743\pi$$
−0.269151 + 0.963098i $$0.586743\pi$$
$$912$$ −7.24745 −0.239987
$$913$$ −11.8990 −0.393799
$$914$$ 15.2474 0.504341
$$915$$ 0 0
$$916$$ 5.79796 0.191570
$$917$$ −21.3031 −0.703489
$$918$$ −19.4949 −0.643427
$$919$$ −51.7980 −1.70866 −0.854329 0.519733i $$-0.826031\pi$$
−0.854329 + 0.519733i $$0.826031\pi$$
$$920$$ 0 0
$$921$$ −11.0908 −0.365455
$$922$$ −9.30306 −0.306380
$$923$$ −3.39388 −0.111711
$$924$$ −12.2474 −0.402911
$$925$$ 0 0
$$926$$ 9.55051 0.313849
$$927$$ 5.79796 0.190430
$$928$$ 0.898979 0.0295104
$$929$$ −5.79796 −0.190225 −0.0951124 0.995467i $$-0.530321\pi$$
−0.0951124 + 0.995467i $$0.530321\pi$$
$$930$$ 0 0
$$931$$ 5.00000 0.163868
$$932$$ 28.4949 0.933381
$$933$$ −12.2474 −0.400963
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0.404082 0.0132078
$$937$$ −14.5959 −0.476828 −0.238414 0.971164i $$-0.576627\pi$$
−0.238414 + 0.971164i $$0.576627\pi$$
$$938$$ −11.1464 −0.363944
$$939$$ −12.8990 −0.420942
$$940$$ 0 0
$$941$$ −41.3939 −1.34940 −0.674701 0.738091i $$-0.735727\pi$$
−0.674701 + 0.738091i $$0.735727\pi$$
$$942$$ 23.8434 0.776859
$$943$$ −2.00000 −0.0651290
$$944$$ −2.00000 −0.0650945
$$945$$ 0 0
$$946$$ −3.79796 −0.123482
$$947$$ −32.0000 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$948$$ 11.3031 0.367106
$$949$$ −5.75255 −0.186736
$$950$$ 0 0
$$951$$ 13.8434 0.448902
$$952$$ −8.44949 −0.273850
$$953$$ −25.4949 −0.825861 −0.412930 0.910763i $$-0.635495\pi$$
−0.412930 + 0.910763i $$0.635495\pi$$
$$954$$ −5.39388 −0.174633
$$955$$ 0 0
$$956$$ 0.898979 0.0290751
$$957$$ −4.49490 −0.145299
$$958$$ −6.24745 −0.201846
$$959$$ 23.7526 0.767010
$$960$$ 0 0
$$961$$ −11.2020 −0.361356
$$962$$ −0.449490 −0.0144921
$$963$$ −7.68673 −0.247702
$$964$$ −2.55051 −0.0821464
$$965$$ 0 0
$$966$$ −7.10102 −0.228472
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ −0.898979 −0.0288943
$$969$$ 25.0000 0.803116
$$970$$ 0 0
$$971$$ −53.5403 −1.71819 −0.859095 0.511816i $$-0.828973\pi$$
−0.859095 + 0.511816i $$0.828973\pi$$
$$972$$ 8.98979 0.288348
$$973$$ 30.7423 0.985554
$$974$$ −30.7423 −0.985048
$$975$$ 0 0
$$976$$ −6.44949 −0.206443
$$977$$ 23.7423 0.759585 0.379792 0.925072i $$-0.375995\pi$$
0.379792 + 0.925072i $$0.375995\pi$$
$$978$$ 14.6413 0.468178
$$979$$ 49.4949 1.58186
$$980$$ 0 0
$$981$$ −12.5857 −0.401831
$$982$$ 41.7980 1.33383
$$983$$ −18.4949 −0.589896 −0.294948 0.955513i $$-0.595302\pi$$
−0.294948 + 0.955513i $$0.595302\pi$$
$$984$$ −1.44949 −0.0462080
$$985$$ 0 0
$$986$$ −3.10102 −0.0987566
$$987$$ 34.7878 1.10731
$$988$$ −2.24745 −0.0715009
$$989$$ −2.20204 −0.0700208
$$990$$ 0 0
$$991$$ −16.4495 −0.522535 −0.261268 0.965266i $$-0.584141\pi$$
−0.261268 + 0.965266i $$0.584141\pi$$
$$992$$ −4.44949 −0.141271
$$993$$ 28.8434 0.915317
$$994$$ −18.4949 −0.586623
$$995$$ 0 0
$$996$$ −5.00000 −0.158431
$$997$$ −39.5505 −1.25258 −0.626289 0.779591i $$-0.715427\pi$$
−0.626289 + 0.779591i $$0.715427\pi$$
$$998$$ −22.0000 −0.696398
$$999$$ −5.65153 −0.178807
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 1850.2.a.r.1.2 2
5.2 odd 4 1850.2.b.k.149.1 4
5.3 odd 4 1850.2.b.k.149.4 4
5.4 even 2 1850.2.a.w.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.r.1.2 2 1.1 even 1 trivial
1850.2.a.w.1.1 yes 2 5.4 even 2
1850.2.b.k.149.1 4 5.2 odd 4
1850.2.b.k.149.4 4 5.3 odd 4