# Properties

 Label 1850.2.a.x.1.2 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} +1.26795 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} +1.26795 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.46410 q^{11} +2.73205 q^{12} -1.46410 q^{13} +1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} +4.46410 q^{18} -4.19615 q^{19} +3.46410 q^{21} +1.46410 q^{22} +8.00000 q^{23} +2.73205 q^{24} -1.46410 q^{26} +4.00000 q^{27} +1.26795 q^{28} -8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.46410 q^{34} +4.46410 q^{36} -1.00000 q^{37} -4.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} +3.46410 q^{42} +6.92820 q^{43} +1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} +2.73205 q^{48} -5.39230 q^{49} +4.00000 q^{51} -1.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +1.26795 q^{56} -11.4641 q^{57} -8.92820 q^{58} +0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +5.66025 q^{63} +1.00000 q^{64} +4.00000 q^{66} -13.6603 q^{67} +1.46410 q^{68} +21.8564 q^{69} -10.9282 q^{71} +4.46410 q^{72} -12.9282 q^{73} -1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} -4.00000 q^{78} +5.26795 q^{79} -2.46410 q^{81} -2.00000 q^{82} +5.26795 q^{83} +3.46410 q^{84} +6.92820 q^{86} -24.3923 q^{87} +1.46410 q^{88} -2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} -7.46410 q^{93} +1.26795 q^{94} +2.73205 q^{96} +2.00000 q^{97} -5.39230 q^{98} +6.53590 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26} + 8 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{33} - 4 q^{34} + 2 q^{36} - 2 q^{37} + 2 q^{38} - 8 q^{39} - 4 q^{41} - 4 q^{44} + 16 q^{46} + 6 q^{47} + 2 q^{48} + 10 q^{49} + 8 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 6 q^{56} - 16 q^{57} - 4 q^{58} - 10 q^{59} + 4 q^{61} - 2 q^{62} - 6 q^{63} + 2 q^{64} + 8 q^{66} - 10 q^{67} - 4 q^{68} + 16 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 2 q^{74} + 2 q^{76} - 24 q^{77} - 8 q^{78} + 14 q^{79} + 2 q^{81} - 4 q^{82} + 14 q^{83} - 28 q^{87} - 4 q^{88} - 4 q^{89} + 24 q^{91} + 16 q^{92} - 8 q^{93} + 6 q^{94} + 2 q^{96} + 4 q^{97} + 10 q^{98} + 20 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 4 * q^13 + 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^22 + 16 * q^23 + 2 * q^24 + 4 * q^26 + 8 * q^27 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^33 - 4 * q^34 + 2 * q^36 - 2 * q^37 + 2 * q^38 - 8 * q^39 - 4 * q^41 - 4 * q^44 + 16 * q^46 + 6 * q^47 + 2 * q^48 + 10 * q^49 + 8 * q^51 + 4 * q^52 + 12 * q^53 + 8 * q^54 + 6 * q^56 - 16 * q^57 - 4 * q^58 - 10 * q^59 + 4 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 8 * q^66 - 10 * q^67 - 4 * q^68 + 16 * q^69 - 8 * q^71 + 2 * q^72 - 12 * q^73 - 2 * q^74 + 2 * q^76 - 24 * q^77 - 8 * q^78 + 14 * q^79 + 2 * q^81 - 4 * q^82 + 14 * q^83 - 28 * q^87 - 4 * q^88 - 4 * q^89 + 24 * q^91 + 16 * q^92 - 8 * q^93 + 6 * q^94 + 2 * q^96 + 4 * q^97 + 10 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.73205 1.57735 0.788675 0.614810i $$-0.210767\pi$$
0.788675 + 0.614810i $$0.210767\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.73205 1.11536
$$7$$ 1.26795 0.479240 0.239620 0.970867i $$-0.422977\pi$$
0.239620 + 0.970867i $$0.422977\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.46410 1.48803
$$10$$ 0 0
$$11$$ 1.46410 0.441443 0.220722 0.975337i $$-0.429159\pi$$
0.220722 + 0.975337i $$0.429159\pi$$
$$12$$ 2.73205 0.788675
$$13$$ −1.46410 −0.406069 −0.203034 0.979172i $$-0.565080\pi$$
−0.203034 + 0.979172i $$0.565080\pi$$
$$14$$ 1.26795 0.338874
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.46410 0.355097 0.177548 0.984112i $$-0.443183\pi$$
0.177548 + 0.984112i $$0.443183\pi$$
$$18$$ 4.46410 1.05220
$$19$$ −4.19615 −0.962663 −0.481332 0.876539i $$-0.659847\pi$$
−0.481332 + 0.876539i $$0.659847\pi$$
$$20$$ 0 0
$$21$$ 3.46410 0.755929
$$22$$ 1.46410 0.312148
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 2.73205 0.557678
$$25$$ 0 0
$$26$$ −1.46410 −0.287134
$$27$$ 4.00000 0.769800
$$28$$ 1.26795 0.239620
$$29$$ −8.92820 −1.65793 −0.828963 0.559304i $$-0.811069\pi$$
−0.828963 + 0.559304i $$0.811069\pi$$
$$30$$ 0 0
$$31$$ −2.73205 −0.490691 −0.245345 0.969436i $$-0.578901\pi$$
−0.245345 + 0.969436i $$0.578901\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ 1.46410 0.251091
$$35$$ 0 0
$$36$$ 4.46410 0.744017
$$37$$ −1.00000 −0.164399
$$38$$ −4.19615 −0.680706
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 3.46410 0.534522
$$43$$ 6.92820 1.05654 0.528271 0.849076i $$-0.322841\pi$$
0.528271 + 0.849076i $$0.322841\pi$$
$$44$$ 1.46410 0.220722
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 1.26795 0.184949 0.0924747 0.995715i $$-0.470522\pi$$
0.0924747 + 0.995715i $$0.470522\pi$$
$$48$$ 2.73205 0.394338
$$49$$ −5.39230 −0.770329
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ −1.46410 −0.203034
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 1.26795 0.169437
$$57$$ −11.4641 −1.51846
$$58$$ −8.92820 −1.17233
$$59$$ 0.196152 0.0255369 0.0127684 0.999918i $$-0.495936\pi$$
0.0127684 + 0.999918i $$0.495936\pi$$
$$60$$ 0 0
$$61$$ 8.92820 1.14314 0.571570 0.820554i $$-0.306335\pi$$
0.571570 + 0.820554i $$0.306335\pi$$
$$62$$ −2.73205 −0.346971
$$63$$ 5.66025 0.713125
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ −13.6603 −1.66887 −0.834433 0.551110i $$-0.814205\pi$$
−0.834433 + 0.551110i $$0.814205\pi$$
$$68$$ 1.46410 0.177548
$$69$$ 21.8564 2.63120
$$70$$ 0 0
$$71$$ −10.9282 −1.29694 −0.648470 0.761241i $$-0.724591\pi$$
−0.648470 + 0.761241i $$0.724591\pi$$
$$72$$ 4.46410 0.526099
$$73$$ −12.9282 −1.51313 −0.756566 0.653917i $$-0.773124\pi$$
−0.756566 + 0.653917i $$0.773124\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −4.19615 −0.481332
$$77$$ 1.85641 0.211557
$$78$$ −4.00000 −0.452911
$$79$$ 5.26795 0.592691 0.296345 0.955081i $$-0.404232\pi$$
0.296345 + 0.955081i $$0.404232\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ −2.00000 −0.220863
$$83$$ 5.26795 0.578233 0.289116 0.957294i $$-0.406639\pi$$
0.289116 + 0.957294i $$0.406639\pi$$
$$84$$ 3.46410 0.377964
$$85$$ 0 0
$$86$$ 6.92820 0.747087
$$87$$ −24.3923 −2.61513
$$88$$ 1.46410 0.156074
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −1.85641 −0.194604
$$92$$ 8.00000 0.834058
$$93$$ −7.46410 −0.773991
$$94$$ 1.26795 0.130779
$$95$$ 0 0
$$96$$ 2.73205 0.278839
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −5.39230 −0.544705
$$99$$ 6.53590 0.656883
$$100$$ 0 0
$$101$$ −2.53590 −0.252331 −0.126166 0.992009i $$-0.540267\pi$$
−0.126166 + 0.992009i $$0.540267\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 13.4641 1.32666 0.663329 0.748328i $$-0.269143\pi$$
0.663329 + 0.748328i $$0.269143\pi$$
$$104$$ −1.46410 −0.143567
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −6.73205 −0.650812 −0.325406 0.945574i $$-0.605501\pi$$
−0.325406 + 0.945574i $$0.605501\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −2.73205 −0.259315
$$112$$ 1.26795 0.119810
$$113$$ 17.4641 1.64288 0.821442 0.570292i $$-0.193170\pi$$
0.821442 + 0.570292i $$0.193170\pi$$
$$114$$ −11.4641 −1.07371
$$115$$ 0 0
$$116$$ −8.92820 −0.828963
$$117$$ −6.53590 −0.604244
$$118$$ 0.196152 0.0180573
$$119$$ 1.85641 0.170177
$$120$$ 0 0
$$121$$ −8.85641 −0.805128
$$122$$ 8.92820 0.808322
$$123$$ −5.46410 −0.492681
$$124$$ −2.73205 −0.245345
$$125$$ 0 0
$$126$$ 5.66025 0.504256
$$127$$ 13.6603 1.21215 0.606076 0.795407i $$-0.292743\pi$$
0.606076 + 0.795407i $$0.292743\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 18.9282 1.66654
$$130$$ 0 0
$$131$$ 12.5885 1.09986 0.549929 0.835211i $$-0.314655\pi$$
0.549929 + 0.835211i $$0.314655\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −5.32051 −0.461347
$$134$$ −13.6603 −1.18007
$$135$$ 0 0
$$136$$ 1.46410 0.125546
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 21.8564 1.86054
$$139$$ −6.92820 −0.587643 −0.293821 0.955860i $$-0.594927\pi$$
−0.293821 + 0.955860i $$0.594927\pi$$
$$140$$ 0 0
$$141$$ 3.46410 0.291730
$$142$$ −10.9282 −0.917074
$$143$$ −2.14359 −0.179256
$$144$$ 4.46410 0.372008
$$145$$ 0 0
$$146$$ −12.9282 −1.06995
$$147$$ −14.7321 −1.21508
$$148$$ −1.00000 −0.0821995
$$149$$ −16.3923 −1.34291 −0.671455 0.741045i $$-0.734330\pi$$
−0.671455 + 0.741045i $$0.734330\pi$$
$$150$$ 0 0
$$151$$ 8.39230 0.682956 0.341478 0.939890i $$-0.389073\pi$$
0.341478 + 0.939890i $$0.389073\pi$$
$$152$$ −4.19615 −0.340353
$$153$$ 6.53590 0.528396
$$154$$ 1.85641 0.149593
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ −16.9282 −1.35102 −0.675509 0.737352i $$-0.736076\pi$$
−0.675509 + 0.737352i $$0.736076\pi$$
$$158$$ 5.26795 0.419096
$$159$$ 16.3923 1.29999
$$160$$ 0 0
$$161$$ 10.1436 0.799427
$$162$$ −2.46410 −0.193598
$$163$$ −23.3205 −1.82660 −0.913302 0.407284i $$-0.866476\pi$$
−0.913302 + 0.407284i $$0.866476\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 5.26795 0.408872
$$167$$ −5.46410 −0.422825 −0.211412 0.977397i $$-0.567806\pi$$
−0.211412 + 0.977397i $$0.567806\pi$$
$$168$$ 3.46410 0.267261
$$169$$ −10.8564 −0.835108
$$170$$ 0 0
$$171$$ −18.7321 −1.43248
$$172$$ 6.92820 0.528271
$$173$$ −10.0000 −0.760286 −0.380143 0.924928i $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ −24.3923 −1.84918
$$175$$ 0 0
$$176$$ 1.46410 0.110361
$$177$$ 0.535898 0.0402806
$$178$$ −2.00000 −0.149906
$$179$$ −17.6603 −1.31999 −0.659995 0.751270i $$-0.729441\pi$$
−0.659995 + 0.751270i $$0.729441\pi$$
$$180$$ 0 0
$$181$$ −1.46410 −0.108826 −0.0544129 0.998519i $$-0.517329\pi$$
−0.0544129 + 0.998519i $$0.517329\pi$$
$$182$$ −1.85641 −0.137606
$$183$$ 24.3923 1.80313
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ −7.46410 −0.547294
$$187$$ 2.14359 0.156755
$$188$$ 1.26795 0.0924747
$$189$$ 5.07180 0.368919
$$190$$ 0 0
$$191$$ −5.26795 −0.381175 −0.190588 0.981670i $$-0.561039\pi$$
−0.190588 + 0.981670i $$0.561039\pi$$
$$192$$ 2.73205 0.197169
$$193$$ 11.8564 0.853443 0.426721 0.904383i $$-0.359668\pi$$
0.426721 + 0.904383i $$0.359668\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −5.39230 −0.385165
$$197$$ −18.7846 −1.33835 −0.669174 0.743106i $$-0.733352\pi$$
−0.669174 + 0.743106i $$0.733352\pi$$
$$198$$ 6.53590 0.464486
$$199$$ 26.0526 1.84682 0.923408 0.383819i $$-0.125391\pi$$
0.923408 + 0.383819i $$0.125391\pi$$
$$200$$ 0 0
$$201$$ −37.3205 −2.63239
$$202$$ −2.53590 −0.178425
$$203$$ −11.3205 −0.794544
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 13.4641 0.938088
$$207$$ 35.7128 2.48221
$$208$$ −1.46410 −0.101517
$$209$$ −6.14359 −0.424961
$$210$$ 0 0
$$211$$ 9.85641 0.678543 0.339272 0.940688i $$-0.389819\pi$$
0.339272 + 0.940688i $$0.389819\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −29.8564 −2.04573
$$214$$ −6.73205 −0.460194
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ −3.46410 −0.235159
$$218$$ 2.00000 0.135457
$$219$$ −35.3205 −2.38674
$$220$$ 0 0
$$221$$ −2.14359 −0.144194
$$222$$ −2.73205 −0.183363
$$223$$ 22.0526 1.47675 0.738374 0.674391i $$-0.235594\pi$$
0.738374 + 0.674391i $$0.235594\pi$$
$$224$$ 1.26795 0.0847184
$$225$$ 0 0
$$226$$ 17.4641 1.16169
$$227$$ −3.60770 −0.239451 −0.119726 0.992807i $$-0.538201\pi$$
−0.119726 + 0.992807i $$0.538201\pi$$
$$228$$ −11.4641 −0.759229
$$229$$ 15.8564 1.04782 0.523910 0.851773i $$-0.324473\pi$$
0.523910 + 0.851773i $$0.324473\pi$$
$$230$$ 0 0
$$231$$ 5.07180 0.333700
$$232$$ −8.92820 −0.586165
$$233$$ −15.0718 −0.987386 −0.493693 0.869636i $$-0.664353\pi$$
−0.493693 + 0.869636i $$0.664353\pi$$
$$234$$ −6.53590 −0.427265
$$235$$ 0 0
$$236$$ 0.196152 0.0127684
$$237$$ 14.3923 0.934881
$$238$$ 1.85641 0.120333
$$239$$ −17.2679 −1.11697 −0.558485 0.829514i $$-0.688617\pi$$
−0.558485 + 0.829514i $$0.688617\pi$$
$$240$$ 0 0
$$241$$ −8.92820 −0.575116 −0.287558 0.957763i $$-0.592843\pi$$
−0.287558 + 0.957763i $$0.592843\pi$$
$$242$$ −8.85641 −0.569311
$$243$$ −18.7321 −1.20166
$$244$$ 8.92820 0.571570
$$245$$ 0 0
$$246$$ −5.46410 −0.348378
$$247$$ 6.14359 0.390907
$$248$$ −2.73205 −0.173485
$$249$$ 14.3923 0.912075
$$250$$ 0 0
$$251$$ 22.7321 1.43483 0.717417 0.696644i $$-0.245324\pi$$
0.717417 + 0.696644i $$0.245324\pi$$
$$252$$ 5.66025 0.356562
$$253$$ 11.7128 0.736378
$$254$$ 13.6603 0.857121
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 25.4641 1.58841 0.794204 0.607652i $$-0.207888\pi$$
0.794204 + 0.607652i $$0.207888\pi$$
$$258$$ 18.9282 1.17842
$$259$$ −1.26795 −0.0787865
$$260$$ 0 0
$$261$$ −39.8564 −2.46705
$$262$$ 12.5885 0.777717
$$263$$ −30.0526 −1.85312 −0.926560 0.376147i $$-0.877249\pi$$
−0.926560 + 0.376147i $$0.877249\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ −5.32051 −0.326221
$$267$$ −5.46410 −0.334398
$$268$$ −13.6603 −0.834433
$$269$$ −0.392305 −0.0239192 −0.0119596 0.999928i $$-0.503807\pi$$
−0.0119596 + 0.999928i $$0.503807\pi$$
$$270$$ 0 0
$$271$$ 16.7846 1.01959 0.509796 0.860295i $$-0.329721\pi$$
0.509796 + 0.860295i $$0.329721\pi$$
$$272$$ 1.46410 0.0887742
$$273$$ −5.07180 −0.306959
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 21.8564 1.31560
$$277$$ 26.2487 1.57713 0.788566 0.614950i $$-0.210824\pi$$
0.788566 + 0.614950i $$0.210824\pi$$
$$278$$ −6.92820 −0.415526
$$279$$ −12.1962 −0.730165
$$280$$ 0 0
$$281$$ 4.92820 0.293992 0.146996 0.989137i $$-0.453040\pi$$
0.146996 + 0.989137i $$0.453040\pi$$
$$282$$ 3.46410 0.206284
$$283$$ 4.39230 0.261095 0.130548 0.991442i $$-0.458326\pi$$
0.130548 + 0.991442i $$0.458326\pi$$
$$284$$ −10.9282 −0.648470
$$285$$ 0 0
$$286$$ −2.14359 −0.126753
$$287$$ −2.53590 −0.149689
$$288$$ 4.46410 0.263050
$$289$$ −14.8564 −0.873906
$$290$$ 0 0
$$291$$ 5.46410 0.320311
$$292$$ −12.9282 −0.756566
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −14.7321 −0.859191
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 5.85641 0.339823
$$298$$ −16.3923 −0.949581
$$299$$ −11.7128 −0.677369
$$300$$ 0 0
$$301$$ 8.78461 0.506336
$$302$$ 8.39230 0.482923
$$303$$ −6.92820 −0.398015
$$304$$ −4.19615 −0.240666
$$305$$ 0 0
$$306$$ 6.53590 0.373632
$$307$$ 12.5885 0.718461 0.359231 0.933249i $$-0.383039\pi$$
0.359231 + 0.933249i $$0.383039\pi$$
$$308$$ 1.85641 0.105779
$$309$$ 36.7846 2.09260
$$310$$ 0 0
$$311$$ 27.1244 1.53808 0.769041 0.639200i $$-0.220734\pi$$
0.769041 + 0.639200i $$0.220734\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −3.85641 −0.217977 −0.108988 0.994043i $$-0.534761\pi$$
−0.108988 + 0.994043i $$0.534761\pi$$
$$314$$ −16.9282 −0.955314
$$315$$ 0 0
$$316$$ 5.26795 0.296345
$$317$$ 31.8564 1.78923 0.894617 0.446834i $$-0.147448\pi$$
0.894617 + 0.446834i $$0.147448\pi$$
$$318$$ 16.3923 0.919235
$$319$$ −13.0718 −0.731880
$$320$$ 0 0
$$321$$ −18.3923 −1.02656
$$322$$ 10.1436 0.565280
$$323$$ −6.14359 −0.341839
$$324$$ −2.46410 −0.136895
$$325$$ 0 0
$$326$$ −23.3205 −1.29160
$$327$$ 5.46410 0.302166
$$328$$ −2.00000 −0.110432
$$329$$ 1.60770 0.0886351
$$330$$ 0 0
$$331$$ −8.87564 −0.487850 −0.243925 0.969794i $$-0.578435\pi$$
−0.243925 + 0.969794i $$0.578435\pi$$
$$332$$ 5.26795 0.289116
$$333$$ −4.46410 −0.244631
$$334$$ −5.46410 −0.298982
$$335$$ 0 0
$$336$$ 3.46410 0.188982
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ −10.8564 −0.590511
$$339$$ 47.7128 2.59140
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ −18.7321 −1.01291
$$343$$ −15.7128 −0.848412
$$344$$ 6.92820 0.373544
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ −17.0718 −0.916462 −0.458231 0.888833i $$-0.651517\pi$$
−0.458231 + 0.888833i $$0.651517\pi$$
$$348$$ −24.3923 −1.30756
$$349$$ 19.3205 1.03420 0.517102 0.855924i $$-0.327011\pi$$
0.517102 + 0.855924i $$0.327011\pi$$
$$350$$ 0 0
$$351$$ −5.85641 −0.312592
$$352$$ 1.46410 0.0780369
$$353$$ −15.8564 −0.843951 −0.421976 0.906607i $$-0.638663\pi$$
−0.421976 + 0.906607i $$0.638663\pi$$
$$354$$ 0.535898 0.0284827
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 5.07180 0.268428
$$358$$ −17.6603 −0.933373
$$359$$ 8.39230 0.442929 0.221464 0.975168i $$-0.428916\pi$$
0.221464 + 0.975168i $$0.428916\pi$$
$$360$$ 0 0
$$361$$ −1.39230 −0.0732792
$$362$$ −1.46410 −0.0769515
$$363$$ −24.1962 −1.26997
$$364$$ −1.85641 −0.0973021
$$365$$ 0 0
$$366$$ 24.3923 1.27501
$$367$$ 21.6603 1.13066 0.565328 0.824866i $$-0.308750\pi$$
0.565328 + 0.824866i $$0.308750\pi$$
$$368$$ 8.00000 0.417029
$$369$$ −8.92820 −0.464784
$$370$$ 0 0
$$371$$ 7.60770 0.394972
$$372$$ −7.46410 −0.386996
$$373$$ −30.7846 −1.59397 −0.796983 0.604001i $$-0.793572\pi$$
−0.796983 + 0.604001i $$0.793572\pi$$
$$374$$ 2.14359 0.110843
$$375$$ 0 0
$$376$$ 1.26795 0.0653895
$$377$$ 13.0718 0.673232
$$378$$ 5.07180 0.260865
$$379$$ 11.6077 0.596247 0.298124 0.954527i $$-0.403639\pi$$
0.298124 + 0.954527i $$0.403639\pi$$
$$380$$ 0 0
$$381$$ 37.3205 1.91199
$$382$$ −5.26795 −0.269532
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 2.73205 0.139419
$$385$$ 0 0
$$386$$ 11.8564 0.603475
$$387$$ 30.9282 1.57217
$$388$$ 2.00000 0.101535
$$389$$ 15.8564 0.803952 0.401976 0.915650i $$-0.368324\pi$$
0.401976 + 0.915650i $$0.368324\pi$$
$$390$$ 0 0
$$391$$ 11.7128 0.592342
$$392$$ −5.39230 −0.272353
$$393$$ 34.3923 1.73486
$$394$$ −18.7846 −0.946355
$$395$$ 0 0
$$396$$ 6.53590 0.328441
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 26.0526 1.30590
$$399$$ −14.5359 −0.727705
$$400$$ 0 0
$$401$$ −19.0718 −0.952400 −0.476200 0.879337i $$-0.657986\pi$$
−0.476200 + 0.879337i $$0.657986\pi$$
$$402$$ −37.3205 −1.86138
$$403$$ 4.00000 0.199254
$$404$$ −2.53590 −0.126166
$$405$$ 0 0
$$406$$ −11.3205 −0.561827
$$407$$ −1.46410 −0.0725728
$$408$$ 4.00000 0.198030
$$409$$ 16.9282 0.837046 0.418523 0.908206i $$-0.362548\pi$$
0.418523 + 0.908206i $$0.362548\pi$$
$$410$$ 0 0
$$411$$ −5.46410 −0.269524
$$412$$ 13.4641 0.663329
$$413$$ 0.248711 0.0122383
$$414$$ 35.7128 1.75519
$$415$$ 0 0
$$416$$ −1.46410 −0.0717835
$$417$$ −18.9282 −0.926918
$$418$$ −6.14359 −0.300493
$$419$$ −34.2487 −1.67316 −0.836580 0.547846i $$-0.815448\pi$$
−0.836580 + 0.547846i $$0.815448\pi$$
$$420$$ 0 0
$$421$$ −27.8564 −1.35764 −0.678819 0.734306i $$-0.737508\pi$$
−0.678819 + 0.734306i $$0.737508\pi$$
$$422$$ 9.85641 0.479802
$$423$$ 5.66025 0.275211
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −29.8564 −1.44655
$$427$$ 11.3205 0.547838
$$428$$ −6.73205 −0.325406
$$429$$ −5.85641 −0.282750
$$430$$ 0 0
$$431$$ 8.19615 0.394795 0.197397 0.980324i $$-0.436751\pi$$
0.197397 + 0.980324i $$0.436751\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −3.46410 −0.166282
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ −33.5692 −1.60583
$$438$$ −35.3205 −1.68768
$$439$$ −31.5167 −1.50421 −0.752104 0.659044i $$-0.770961\pi$$
−0.752104 + 0.659044i $$0.770961\pi$$
$$440$$ 0 0
$$441$$ −24.0718 −1.14628
$$442$$ −2.14359 −0.101960
$$443$$ 39.1244 1.85885 0.929427 0.369006i $$-0.120302\pi$$
0.929427 + 0.369006i $$0.120302\pi$$
$$444$$ −2.73205 −0.129657
$$445$$ 0 0
$$446$$ 22.0526 1.04422
$$447$$ −44.7846 −2.11824
$$448$$ 1.26795 0.0599050
$$449$$ 33.7128 1.59101 0.795503 0.605950i $$-0.207207\pi$$
0.795503 + 0.605950i $$0.207207\pi$$
$$450$$ 0 0
$$451$$ −2.92820 −0.137884
$$452$$ 17.4641 0.821442
$$453$$ 22.9282 1.07726
$$454$$ −3.60770 −0.169318
$$455$$ 0 0
$$456$$ −11.4641 −0.536856
$$457$$ 4.14359 0.193829 0.0969146 0.995293i $$-0.469103\pi$$
0.0969146 + 0.995293i $$0.469103\pi$$
$$458$$ 15.8564 0.740921
$$459$$ 5.85641 0.273354
$$460$$ 0 0
$$461$$ −26.7846 −1.24748 −0.623742 0.781630i $$-0.714388\pi$$
−0.623742 + 0.781630i $$0.714388\pi$$
$$462$$ 5.07180 0.235961
$$463$$ −5.07180 −0.235706 −0.117853 0.993031i $$-0.537601\pi$$
−0.117853 + 0.993031i $$0.537601\pi$$
$$464$$ −8.92820 −0.414481
$$465$$ 0 0
$$466$$ −15.0718 −0.698188
$$467$$ −37.1769 −1.72034 −0.860171 0.510005i $$-0.829643\pi$$
−0.860171 + 0.510005i $$0.829643\pi$$
$$468$$ −6.53590 −0.302122
$$469$$ −17.3205 −0.799787
$$470$$ 0 0
$$471$$ −46.2487 −2.13103
$$472$$ 0.196152 0.00902865
$$473$$ 10.1436 0.466403
$$474$$ 14.3923 0.661060
$$475$$ 0 0
$$476$$ 1.85641 0.0850883
$$477$$ 26.7846 1.22638
$$478$$ −17.2679 −0.789818
$$479$$ −34.0526 −1.55590 −0.777951 0.628325i $$-0.783741\pi$$
−0.777951 + 0.628325i $$0.783741\pi$$
$$480$$ 0 0
$$481$$ 1.46410 0.0667573
$$482$$ −8.92820 −0.406669
$$483$$ 27.7128 1.26098
$$484$$ −8.85641 −0.402564
$$485$$ 0 0
$$486$$ −18.7321 −0.849703
$$487$$ 16.3923 0.742806 0.371403 0.928472i $$-0.378877\pi$$
0.371403 + 0.928472i $$0.378877\pi$$
$$488$$ 8.92820 0.404161
$$489$$ −63.7128 −2.88119
$$490$$ 0 0
$$491$$ −1.07180 −0.0483695 −0.0241848 0.999708i $$-0.507699\pi$$
−0.0241848 + 0.999708i $$0.507699\pi$$
$$492$$ −5.46410 −0.246341
$$493$$ −13.0718 −0.588724
$$494$$ 6.14359 0.276413
$$495$$ 0 0
$$496$$ −2.73205 −0.122673
$$497$$ −13.8564 −0.621545
$$498$$ 14.3923 0.644935
$$499$$ 40.5885 1.81699 0.908494 0.417897i $$-0.137233\pi$$
0.908494 + 0.417897i $$0.137233\pi$$
$$500$$ 0 0
$$501$$ −14.9282 −0.666943
$$502$$ 22.7321 1.01458
$$503$$ 5.07180 0.226140 0.113070 0.993587i $$-0.463932\pi$$
0.113070 + 0.993587i $$0.463932\pi$$
$$504$$ 5.66025 0.252128
$$505$$ 0 0
$$506$$ 11.7128 0.520698
$$507$$ −29.6603 −1.31726
$$508$$ 13.6603 0.606076
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ −16.3923 −0.725153
$$512$$ 1.00000 0.0441942
$$513$$ −16.7846 −0.741059
$$514$$ 25.4641 1.12317
$$515$$ 0 0
$$516$$ 18.9282 0.833268
$$517$$ 1.85641 0.0816447
$$518$$ −1.26795 −0.0557105
$$519$$ −27.3205 −1.19924
$$520$$ 0 0
$$521$$ 33.4641 1.46609 0.733044 0.680181i $$-0.238099\pi$$
0.733044 + 0.680181i $$0.238099\pi$$
$$522$$ −39.8564 −1.74447
$$523$$ 4.78461 0.209216 0.104608 0.994514i $$-0.466641\pi$$
0.104608 + 0.994514i $$0.466641\pi$$
$$524$$ 12.5885 0.549929
$$525$$ 0 0
$$526$$ −30.0526 −1.31035
$$527$$ −4.00000 −0.174243
$$528$$ 4.00000 0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0.875644 0.0379997
$$532$$ −5.32051 −0.230673
$$533$$ 2.92820 0.126835
$$534$$ −5.46410 −0.236455
$$535$$ 0 0
$$536$$ −13.6603 −0.590033
$$537$$ −48.2487 −2.08209
$$538$$ −0.392305 −0.0169135
$$539$$ −7.89488 −0.340057
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 16.7846 0.720961
$$543$$ −4.00000 −0.171656
$$544$$ 1.46410 0.0627728
$$545$$ 0 0
$$546$$ −5.07180 −0.217053
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 39.8564 1.70103
$$550$$ 0 0
$$551$$ 37.4641 1.59602
$$552$$ 21.8564 0.930270
$$553$$ 6.67949 0.284041
$$554$$ 26.2487 1.11520
$$555$$ 0 0
$$556$$ −6.92820 −0.293821
$$557$$ 32.1051 1.36034 0.680169 0.733056i $$-0.261906\pi$$
0.680169 + 0.733056i $$0.261906\pi$$
$$558$$ −12.1962 −0.516304
$$559$$ −10.1436 −0.429028
$$560$$ 0 0
$$561$$ 5.85641 0.247258
$$562$$ 4.92820 0.207884
$$563$$ −17.0718 −0.719490 −0.359745 0.933051i $$-0.617136\pi$$
−0.359745 + 0.933051i $$0.617136\pi$$
$$564$$ 3.46410 0.145865
$$565$$ 0 0
$$566$$ 4.39230 0.184622
$$567$$ −3.12436 −0.131211
$$568$$ −10.9282 −0.458537
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −2.14359 −0.0896281
$$573$$ −14.3923 −0.601247
$$574$$ −2.53590 −0.105846
$$575$$ 0 0
$$576$$ 4.46410 0.186004
$$577$$ 3.60770 0.150190 0.0750952 0.997176i $$-0.476074\pi$$
0.0750952 + 0.997176i $$0.476074\pi$$
$$578$$ −14.8564 −0.617945
$$579$$ 32.3923 1.34618
$$580$$ 0 0
$$581$$ 6.67949 0.277112
$$582$$ 5.46410 0.226494
$$583$$ 8.78461 0.363821
$$584$$ −12.9282 −0.534973
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −3.21539 −0.132713 −0.0663567 0.997796i $$-0.521138\pi$$
−0.0663567 + 0.997796i $$0.521138\pi$$
$$588$$ −14.7321 −0.607540
$$589$$ 11.4641 0.472370
$$590$$ 0 0
$$591$$ −51.3205 −2.11104
$$592$$ −1.00000 −0.0410997
$$593$$ −6.78461 −0.278611 −0.139305 0.990249i $$-0.544487\pi$$
−0.139305 + 0.990249i $$0.544487\pi$$
$$594$$ 5.85641 0.240291
$$595$$ 0 0
$$596$$ −16.3923 −0.671455
$$597$$ 71.1769 2.91308
$$598$$ −11.7128 −0.478973
$$599$$ 2.53590 0.103614 0.0518070 0.998657i $$-0.483502\pi$$
0.0518070 + 0.998657i $$0.483502\pi$$
$$600$$ 0 0
$$601$$ 48.3923 1.97396 0.986982 0.160833i $$-0.0514180\pi$$
0.986982 + 0.160833i $$0.0514180\pi$$
$$602$$ 8.78461 0.358034
$$603$$ −60.9808 −2.48333
$$604$$ 8.39230 0.341478
$$605$$ 0 0
$$606$$ −6.92820 −0.281439
$$607$$ 40.7846 1.65540 0.827698 0.561174i $$-0.189650\pi$$
0.827698 + 0.561174i $$0.189650\pi$$
$$608$$ −4.19615 −0.170176
$$609$$ −30.9282 −1.25327
$$610$$ 0 0
$$611$$ −1.85641 −0.0751022
$$612$$ 6.53590 0.264198
$$613$$ −3.07180 −0.124069 −0.0620344 0.998074i $$-0.519759\pi$$
−0.0620344 + 0.998074i $$0.519759\pi$$
$$614$$ 12.5885 0.508029
$$615$$ 0 0
$$616$$ 1.85641 0.0747967
$$617$$ −12.9282 −0.520470 −0.260235 0.965545i $$-0.583800\pi$$
−0.260235 + 0.965545i $$0.583800\pi$$
$$618$$ 36.7846 1.47969
$$619$$ 9.85641 0.396162 0.198081 0.980186i $$-0.436529\pi$$
0.198081 + 0.980186i $$0.436529\pi$$
$$620$$ 0 0
$$621$$ 32.0000 1.28412
$$622$$ 27.1244 1.08759
$$623$$ −2.53590 −0.101599
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ −3.85641 −0.154133
$$627$$ −16.7846 −0.670313
$$628$$ −16.9282 −0.675509
$$629$$ −1.46410 −0.0583776
$$630$$ 0 0
$$631$$ −20.9808 −0.835231 −0.417615 0.908624i $$-0.637134\pi$$
−0.417615 + 0.908624i $$0.637134\pi$$
$$632$$ 5.26795 0.209548
$$633$$ 26.9282 1.07030
$$634$$ 31.8564 1.26518
$$635$$ 0 0
$$636$$ 16.3923 0.649997
$$637$$ 7.89488 0.312807
$$638$$ −13.0718 −0.517517
$$639$$ −48.7846 −1.92989
$$640$$ 0 0
$$641$$ −13.4641 −0.531800 −0.265900 0.964001i $$-0.585669\pi$$
−0.265900 + 0.964001i $$0.585669\pi$$
$$642$$ −18.3923 −0.725886
$$643$$ −41.4641 −1.63518 −0.817592 0.575798i $$-0.804692\pi$$
−0.817592 + 0.575798i $$0.804692\pi$$
$$644$$ 10.1436 0.399714
$$645$$ 0 0
$$646$$ −6.14359 −0.241716
$$647$$ −19.7128 −0.774991 −0.387495 0.921872i $$-0.626660\pi$$
−0.387495 + 0.921872i $$0.626660\pi$$
$$648$$ −2.46410 −0.0967991
$$649$$ 0.287187 0.0112731
$$650$$ 0 0
$$651$$ −9.46410 −0.370927
$$652$$ −23.3205 −0.913302
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 5.46410 0.213663
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ −57.7128 −2.25159
$$658$$ 1.60770 0.0626745
$$659$$ 6.92820 0.269884 0.134942 0.990853i $$-0.456915\pi$$
0.134942 + 0.990853i $$0.456915\pi$$
$$660$$ 0 0
$$661$$ −44.9282 −1.74750 −0.873752 0.486371i $$-0.838320\pi$$
−0.873752 + 0.486371i $$0.838320\pi$$
$$662$$ −8.87564 −0.344962
$$663$$ −5.85641 −0.227444
$$664$$ 5.26795 0.204436
$$665$$ 0 0
$$666$$ −4.46410 −0.172980
$$667$$ −71.4256 −2.76561
$$668$$ −5.46410 −0.211412
$$669$$ 60.2487 2.32935
$$670$$ 0 0
$$671$$ 13.0718 0.504631
$$672$$ 3.46410 0.133631
$$673$$ 19.0718 0.735164 0.367582 0.929991i $$-0.380186\pi$$
0.367582 + 0.929991i $$0.380186\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ −10.8564 −0.417554
$$677$$ 31.8564 1.22434 0.612171 0.790726i $$-0.290297\pi$$
0.612171 + 0.790726i $$0.290297\pi$$
$$678$$ 47.7128 1.83240
$$679$$ 2.53590 0.0973188
$$680$$ 0 0
$$681$$ −9.85641 −0.377698
$$682$$ −4.00000 −0.153168
$$683$$ −36.7846 −1.40752 −0.703762 0.710436i $$-0.748498\pi$$
−0.703762 + 0.710436i $$0.748498\pi$$
$$684$$ −18.7321 −0.716238
$$685$$ 0 0
$$686$$ −15.7128 −0.599918
$$687$$ 43.3205 1.65278
$$688$$ 6.92820 0.264135
$$689$$ −8.78461 −0.334667
$$690$$ 0 0
$$691$$ 20.3923 0.775760 0.387880 0.921710i $$-0.373208\pi$$
0.387880 + 0.921710i $$0.373208\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ 8.28719 0.314804
$$694$$ −17.0718 −0.648037
$$695$$ 0 0
$$696$$ −24.3923 −0.924588
$$697$$ −2.92820 −0.110914
$$698$$ 19.3205 0.731292
$$699$$ −41.1769 −1.55745
$$700$$ 0 0
$$701$$ 15.8564 0.598888 0.299444 0.954114i $$-0.403199\pi$$
0.299444 + 0.954114i $$0.403199\pi$$
$$702$$ −5.85641 −0.221036
$$703$$ 4.19615 0.158261
$$704$$ 1.46410 0.0551804
$$705$$ 0 0
$$706$$ −15.8564 −0.596764
$$707$$ −3.21539 −0.120927
$$708$$ 0.535898 0.0201403
$$709$$ 16.1436 0.606285 0.303143 0.952945i $$-0.401964\pi$$
0.303143 + 0.952945i $$0.401964\pi$$
$$710$$ 0 0
$$711$$ 23.5167 0.881944
$$712$$ −2.00000 −0.0749532
$$713$$ −21.8564 −0.818529
$$714$$ 5.07180 0.189807
$$715$$ 0 0
$$716$$ −17.6603 −0.659995
$$717$$ −47.1769 −1.76185
$$718$$ 8.39230 0.313198
$$719$$ 8.39230 0.312980 0.156490 0.987680i $$-0.449982\pi$$
0.156490 + 0.987680i $$0.449982\pi$$
$$720$$ 0 0
$$721$$ 17.0718 0.635787
$$722$$ −1.39230 −0.0518162
$$723$$ −24.3923 −0.907160
$$724$$ −1.46410 −0.0544129
$$725$$ 0 0
$$726$$ −24.1962 −0.898003
$$727$$ 32.7846 1.21591 0.607957 0.793970i $$-0.291989\pi$$
0.607957 + 0.793970i $$0.291989\pi$$
$$728$$ −1.85641 −0.0688030
$$729$$ −43.7846 −1.62165
$$730$$ 0 0
$$731$$ 10.1436 0.375174
$$732$$ 24.3923 0.901566
$$733$$ −0.143594 −0.00530375 −0.00265187 0.999996i $$-0.500844\pi$$
−0.00265187 + 0.999996i $$0.500844\pi$$
$$734$$ 21.6603 0.799495
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ −20.0000 −0.736709
$$738$$ −8.92820 −0.328652
$$739$$ −6.92820 −0.254858 −0.127429 0.991848i $$-0.540673\pi$$
−0.127429 + 0.991848i $$0.540673\pi$$
$$740$$ 0 0
$$741$$ 16.7846 0.616598
$$742$$ 7.60770 0.279287
$$743$$ 7.12436 0.261367 0.130684 0.991424i $$-0.458283\pi$$
0.130684 + 0.991424i $$0.458283\pi$$
$$744$$ −7.46410 −0.273647
$$745$$ 0 0
$$746$$ −30.7846 −1.12710
$$747$$ 23.5167 0.860430
$$748$$ 2.14359 0.0783775
$$749$$ −8.53590 −0.311895
$$750$$ 0 0
$$751$$ 0.392305 0.0143154 0.00715770 0.999974i $$-0.497722\pi$$
0.00715770 + 0.999974i $$0.497722\pi$$
$$752$$ 1.26795 0.0462373
$$753$$ 62.1051 2.26324
$$754$$ 13.0718 0.476047
$$755$$ 0 0
$$756$$ 5.07180 0.184459
$$757$$ 53.7128 1.95223 0.976113 0.217265i $$-0.0697135\pi$$
0.976113 + 0.217265i $$0.0697135\pi$$
$$758$$ 11.6077 0.421610
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 37.3205 1.35198
$$763$$ 2.53590 0.0918057
$$764$$ −5.26795 −0.190588
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ −0.287187 −0.0103697
$$768$$ 2.73205 0.0985844
$$769$$ 20.9282 0.754690 0.377345 0.926073i $$-0.376837\pi$$
0.377345 + 0.926073i $$0.376837\pi$$
$$770$$ 0 0
$$771$$ 69.5692 2.50547
$$772$$ 11.8564 0.426721
$$773$$ 38.7846 1.39499 0.697493 0.716592i $$-0.254299\pi$$
0.697493 + 0.716592i $$0.254299\pi$$
$$774$$ 30.9282 1.11169
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ −3.46410 −0.124274
$$778$$ 15.8564 0.568480
$$779$$ 8.39230 0.300686
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 11.7128 0.418849
$$783$$ −35.7128 −1.27627
$$784$$ −5.39230 −0.192582
$$785$$ 0 0
$$786$$ 34.3923 1.22673
$$787$$ −11.8038 −0.420762 −0.210381 0.977620i $$-0.567470\pi$$
−0.210381 + 0.977620i $$0.567470\pi$$
$$788$$ −18.7846 −0.669174
$$789$$ −82.1051 −2.92302
$$790$$ 0 0
$$791$$ 22.1436 0.787336
$$792$$ 6.53590 0.232243
$$793$$ −13.0718 −0.464193
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 26.0526 0.923408
$$797$$ −17.4641 −0.618610 −0.309305 0.950963i $$-0.600096\pi$$
−0.309305 + 0.950963i $$0.600096\pi$$
$$798$$ −14.5359 −0.514565
$$799$$ 1.85641 0.0656749
$$800$$ 0 0
$$801$$ −8.92820 −0.315463
$$802$$ −19.0718 −0.673449
$$803$$ −18.9282 −0.667962
$$804$$ −37.3205 −1.31619
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ −1.07180 −0.0377290
$$808$$ −2.53590 −0.0892126
$$809$$ −39.5692 −1.39118 −0.695590 0.718439i $$-0.744857\pi$$
−0.695590 + 0.718439i $$0.744857\pi$$
$$810$$ 0 0
$$811$$ −14.1436 −0.496649 −0.248324 0.968677i $$-0.579880\pi$$
−0.248324 + 0.968677i $$0.579880\pi$$
$$812$$ −11.3205 −0.397272
$$813$$ 45.8564 1.60825
$$814$$ −1.46410 −0.0513167
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ −29.0718 −1.01709
$$818$$ 16.9282 0.591881
$$819$$ −8.28719 −0.289578
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ −5.46410 −0.190582
$$823$$ 10.4449 0.364085 0.182043 0.983291i $$-0.441729\pi$$
0.182043 + 0.983291i $$0.441729\pi$$
$$824$$ 13.4641 0.469044
$$825$$ 0 0
$$826$$ 0.248711 0.00865377
$$827$$ −4.39230 −0.152735 −0.0763677 0.997080i $$-0.524332\pi$$
−0.0763677 + 0.997080i $$0.524332\pi$$
$$828$$ 35.7128 1.24111
$$829$$ 31.5692 1.09644 0.548222 0.836333i $$-0.315305\pi$$
0.548222 + 0.836333i $$0.315305\pi$$
$$830$$ 0 0
$$831$$ 71.7128 2.48769
$$832$$ −1.46410 −0.0507586
$$833$$ −7.89488 −0.273541
$$834$$ −18.9282 −0.655430
$$835$$ 0 0
$$836$$ −6.14359 −0.212481
$$837$$ −10.9282 −0.377734
$$838$$ −34.2487 −1.18310
$$839$$ 32.7846 1.13185 0.565925 0.824457i $$-0.308519\pi$$
0.565925 + 0.824457i $$0.308519\pi$$
$$840$$ 0 0
$$841$$ 50.7128 1.74872
$$842$$ −27.8564 −0.959995
$$843$$ 13.4641 0.463728
$$844$$ 9.85641 0.339272
$$845$$ 0 0
$$846$$ 5.66025 0.194604
$$847$$ −11.2295 −0.385849
$$848$$ 6.00000 0.206041
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ −29.8564 −1.02286
$$853$$ 30.0000 1.02718 0.513590 0.858036i $$-0.328315\pi$$
0.513590 + 0.858036i $$0.328315\pi$$
$$854$$ 11.3205 0.387380
$$855$$ 0 0
$$856$$ −6.73205 −0.230097
$$857$$ 4.14359 0.141542 0.0707712 0.997493i $$-0.477454\pi$$
0.0707712 + 0.997493i $$0.477454\pi$$
$$858$$ −5.85641 −0.199934
$$859$$ −14.4449 −0.492852 −0.246426 0.969162i $$-0.579256\pi$$
−0.246426 + 0.969162i $$0.579256\pi$$
$$860$$ 0 0
$$861$$ −6.92820 −0.236113
$$862$$ 8.19615 0.279162
$$863$$ −38.4449 −1.30868 −0.654339 0.756201i $$-0.727053\pi$$
−0.654339 + 0.756201i $$0.727053\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ −40.5885 −1.37846
$$868$$ −3.46410 −0.117579
$$869$$ 7.71281 0.261639
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 2.00000 0.0677285
$$873$$ 8.92820 0.302174
$$874$$ −33.5692 −1.13550
$$875$$ 0 0
$$876$$ −35.3205 −1.19337
$$877$$ 50.7846 1.71487 0.857437 0.514589i $$-0.172055\pi$$
0.857437 + 0.514589i $$0.172055\pi$$
$$878$$ −31.5167 −1.06364
$$879$$ −16.3923 −0.552899
$$880$$ 0 0
$$881$$ 47.3205 1.59427 0.797134 0.603802i $$-0.206348\pi$$
0.797134 + 0.603802i $$0.206348\pi$$
$$882$$ −24.0718 −0.810540
$$883$$ 34.2487 1.15256 0.576280 0.817252i $$-0.304504\pi$$
0.576280 + 0.817252i $$0.304504\pi$$
$$884$$ −2.14359 −0.0720969
$$885$$ 0 0
$$886$$ 39.1244 1.31441
$$887$$ −20.1962 −0.678120 −0.339060 0.940765i $$-0.610109\pi$$
−0.339060 + 0.940765i $$0.610109\pi$$
$$888$$ −2.73205 −0.0916816
$$889$$ 17.3205 0.580911
$$890$$ 0 0
$$891$$ −3.60770 −0.120862
$$892$$ 22.0526 0.738374
$$893$$ −5.32051 −0.178044
$$894$$ −44.7846 −1.49782
$$895$$ 0 0
$$896$$ 1.26795 0.0423592
$$897$$ −32.0000 −1.06845
$$898$$ 33.7128 1.12501
$$899$$ 24.3923 0.813529
$$900$$ 0 0
$$901$$ 8.78461 0.292658
$$902$$ −2.92820 −0.0974985
$$903$$ 24.0000 0.798670
$$904$$ 17.4641 0.580847
$$905$$ 0 0
$$906$$ 22.9282 0.761739
$$907$$ −5.75129 −0.190968 −0.0954842 0.995431i $$-0.530440\pi$$
−0.0954842 + 0.995431i $$0.530440\pi$$
$$908$$ −3.60770 −0.119726
$$909$$ −11.3205 −0.375478
$$910$$ 0 0
$$911$$ −27.9090 −0.924665 −0.462333 0.886707i $$-0.652987\pi$$
−0.462333 + 0.886707i $$0.652987\pi$$
$$912$$ −11.4641 −0.379614
$$913$$ 7.71281 0.255257
$$914$$ 4.14359 0.137058
$$915$$ 0 0
$$916$$ 15.8564 0.523910
$$917$$ 15.9615 0.527096
$$918$$ 5.85641 0.193290
$$919$$ −13.2679 −0.437669 −0.218835 0.975762i $$-0.570226\pi$$
−0.218835 + 0.975762i $$0.570226\pi$$
$$920$$ 0 0
$$921$$ 34.3923 1.13326
$$922$$ −26.7846 −0.882104
$$923$$ 16.0000 0.526646
$$924$$ 5.07180 0.166850
$$925$$ 0 0
$$926$$ −5.07180 −0.166670
$$927$$ 60.1051 1.97411
$$928$$ −8.92820 −0.293083
$$929$$ −15.8564 −0.520232 −0.260116 0.965577i $$-0.583761\pi$$
−0.260116 + 0.965577i $$0.583761\pi$$
$$930$$ 0 0
$$931$$ 22.6269 0.741568
$$932$$ −15.0718 −0.493693
$$933$$ 74.1051 2.42609
$$934$$ −37.1769 −1.21647
$$935$$ 0 0
$$936$$ −6.53590 −0.213633
$$937$$ −3.85641 −0.125983 −0.0629917 0.998014i $$-0.520064\pi$$
−0.0629917 + 0.998014i $$0.520064\pi$$
$$938$$ −17.3205 −0.565535
$$939$$ −10.5359 −0.343826
$$940$$ 0 0
$$941$$ 28.3923 0.925563 0.462781 0.886472i $$-0.346852\pi$$
0.462781 + 0.886472i $$0.346852\pi$$
$$942$$ −46.2487 −1.50686
$$943$$ −16.0000 −0.521032
$$944$$ 0.196152 0.00638422
$$945$$ 0 0
$$946$$ 10.1436 0.329797
$$947$$ 45.1769 1.46805 0.734026 0.679121i $$-0.237639\pi$$
0.734026 + 0.679121i $$0.237639\pi$$
$$948$$ 14.3923 0.467440
$$949$$ 18.9282 0.614435
$$950$$ 0 0
$$951$$ 87.0333 2.82225
$$952$$ 1.85641 0.0601665
$$953$$ 11.8564 0.384067 0.192033 0.981388i $$-0.438492\pi$$
0.192033 + 0.981388i $$0.438492\pi$$
$$954$$ 26.7846 0.867184
$$955$$ 0 0
$$956$$ −17.2679 −0.558485
$$957$$ −35.7128 −1.15443
$$958$$ −34.0526 −1.10019
$$959$$ −2.53590 −0.0818884
$$960$$ 0 0
$$961$$ −23.5359 −0.759223
$$962$$ 1.46410 0.0472045
$$963$$ −30.0526 −0.968430
$$964$$ −8.92820 −0.287558
$$965$$ 0 0
$$966$$ 27.7128 0.891645
$$967$$ 23.6077 0.759172 0.379586 0.925156i $$-0.376066\pi$$
0.379586 + 0.925156i $$0.376066\pi$$
$$968$$ −8.85641 −0.284656
$$969$$ −16.7846 −0.539199
$$970$$ 0 0
$$971$$ 14.9282 0.479069 0.239534 0.970888i $$-0.423005\pi$$
0.239534 + 0.970888i $$0.423005\pi$$
$$972$$ −18.7321 −0.600831
$$973$$ −8.78461 −0.281622
$$974$$ 16.3923 0.525243
$$975$$ 0 0
$$976$$ 8.92820 0.285785
$$977$$ 34.0000 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$978$$ −63.7128 −2.03731
$$979$$ −2.92820 −0.0935858
$$980$$ 0 0
$$981$$ 8.92820 0.285056
$$982$$ −1.07180 −0.0342024
$$983$$ 38.4449 1.22620 0.613100 0.790005i $$-0.289922\pi$$
0.613100 + 0.790005i $$0.289922\pi$$
$$984$$ −5.46410 −0.174189
$$985$$ 0 0
$$986$$ −13.0718 −0.416291
$$987$$ 4.39230 0.139809
$$988$$ 6.14359 0.195454
$$989$$ 55.4256 1.76243
$$990$$ 0 0
$$991$$ 5.94744 0.188927 0.0944633 0.995528i $$-0.469886\pi$$
0.0944633 + 0.995528i $$0.469886\pi$$
$$992$$ −2.73205 −0.0867427
$$993$$ −24.2487 −0.769510
$$994$$ −13.8564 −0.439499
$$995$$ 0 0
$$996$$ 14.3923 0.456038
$$997$$ 4.14359 0.131229 0.0656145 0.997845i $$-0.479099\pi$$
0.0656145 + 0.997845i $$0.479099\pi$$
$$998$$ 40.5885 1.28481
$$999$$ −4.00000 −0.126554
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.x.1.2 2
5.2 odd 4 1850.2.b.l.149.3 4
5.3 odd 4 1850.2.b.l.149.2 4
5.4 even 2 370.2.a.e.1.1 2
15.14 odd 2 3330.2.a.bd.1.2 2
20.19 odd 2 2960.2.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.1 2 5.4 even 2
1850.2.a.x.1.2 2 1.1 even 1 trivial
1850.2.b.l.149.2 4 5.3 odd 4
1850.2.b.l.149.3 4 5.2 odd 4
2960.2.a.q.1.2 2 20.19 odd 2
3330.2.a.bd.1.2 2 15.14 odd 2