# Properties

 Label 1850.2.a.y.1.2 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1524.1 Defining polynomial: $$x^{3} - x^{2} - 7x + 1$$ x^3 - x^2 - 7*x + 1 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 $$0.140435$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -0.140435 q^{3} +1.00000 q^{4} +0.140435 q^{6} -1.14044 q^{7} -1.00000 q^{8} -2.98028 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -0.140435 q^{3} +1.00000 q^{4} +0.140435 q^{6} -1.14044 q^{7} -1.00000 q^{8} -2.98028 q^{9} +3.12071 q^{11} -0.140435 q^{12} -2.85956 q^{13} +1.14044 q^{14} +1.00000 q^{16} +2.14044 q^{17} +2.98028 q^{18} +4.26115 q^{19} +0.160157 q^{21} -3.12071 q^{22} -1.71913 q^{23} +0.140435 q^{24} +2.85956 q^{26} +0.839843 q^{27} -1.14044 q^{28} -4.28087 q^{29} +6.40158 q^{31} -1.00000 q^{32} -0.438259 q^{33} -2.14044 q^{34} -2.98028 q^{36} -1.00000 q^{37} -4.26115 q^{38} +0.401584 q^{39} -3.24143 q^{41} -0.160157 q^{42} -10.6994 q^{43} +3.12071 q^{44} +1.71913 q^{46} -0.140435 q^{48} -5.69941 q^{49} -0.300593 q^{51} -2.85956 q^{52} -7.96056 q^{53} -0.839843 q^{54} +1.14044 q^{56} -0.598416 q^{57} +4.28087 q^{58} +2.69941 q^{59} +4.57869 q^{61} -6.40158 q^{62} +3.39881 q^{63} +1.00000 q^{64} +0.438259 q^{66} -10.3819 q^{67} +2.14044 q^{68} +0.241427 q^{69} -7.10099 q^{71} +2.98028 q^{72} +0.261149 q^{73} +1.00000 q^{74} +4.26115 q^{76} -3.55897 q^{77} -0.401584 q^{78} -8.52230 q^{79} +8.82289 q^{81} +3.24143 q^{82} +1.16016 q^{83} +0.160157 q^{84} +10.6994 q^{86} +0.601186 q^{87} -3.12071 q^{88} -13.6430 q^{89} +3.26115 q^{91} -1.71913 q^{92} -0.899009 q^{93} +0.140435 q^{96} -14.2414 q^{97} +5.69941 q^{98} -9.30059 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 6 * q^9 $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} - 5 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} + 3 q^{16} + 7 q^{17} - 6 q^{18} - q^{19} + 16 q^{21} + 5 q^{22} - 4 q^{23} + q^{24} + 8 q^{26} - 13 q^{27} - 4 q^{28} - 14 q^{29} + 6 q^{31} - 3 q^{32} + q^{33} - 7 q^{34} + 6 q^{36} - 3 q^{37} + q^{38} - 12 q^{39} + 19 q^{41} - 16 q^{42} - 16 q^{43} - 5 q^{44} + 4 q^{46} - q^{48} - q^{49} - 17 q^{51} - 8 q^{52} + 6 q^{53} + 13 q^{54} + 4 q^{56} - 15 q^{57} + 14 q^{58} - 8 q^{59} + 12 q^{61} - 6 q^{62} - 22 q^{63} + 3 q^{64} - q^{66} - 3 q^{67} + 7 q^{68} - 28 q^{69} + 8 q^{71} - 6 q^{72} - 13 q^{73} + 3 q^{74} - q^{76} + 6 q^{77} + 12 q^{78} + 2 q^{79} + 15 q^{81} - 19 q^{82} + 19 q^{83} + 16 q^{84} + 16 q^{86} + 34 q^{87} + 5 q^{88} + q^{89} - 4 q^{91} - 4 q^{92} - 32 q^{93} + q^{96} - 14 q^{97} + q^{98} - 44 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 6 * q^9 - 5 * q^11 - q^12 - 8 * q^13 + 4 * q^14 + 3 * q^16 + 7 * q^17 - 6 * q^18 - q^19 + 16 * q^21 + 5 * q^22 - 4 * q^23 + q^24 + 8 * q^26 - 13 * q^27 - 4 * q^28 - 14 * q^29 + 6 * q^31 - 3 * q^32 + q^33 - 7 * q^34 + 6 * q^36 - 3 * q^37 + q^38 - 12 * q^39 + 19 * q^41 - 16 * q^42 - 16 * q^43 - 5 * q^44 + 4 * q^46 - q^48 - q^49 - 17 * q^51 - 8 * q^52 + 6 * q^53 + 13 * q^54 + 4 * q^56 - 15 * q^57 + 14 * q^58 - 8 * q^59 + 12 * q^61 - 6 * q^62 - 22 * q^63 + 3 * q^64 - q^66 - 3 * q^67 + 7 * q^68 - 28 * q^69 + 8 * q^71 - 6 * q^72 - 13 * q^73 + 3 * q^74 - q^76 + 6 * q^77 + 12 * q^78 + 2 * q^79 + 15 * q^81 - 19 * q^82 + 19 * q^83 + 16 * q^84 + 16 * q^86 + 34 * q^87 + 5 * q^88 + q^89 - 4 * q^91 - 4 * q^92 - 32 * q^93 + q^96 - 14 * q^97 + q^98 - 44 * 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
$$3$$ −0.140435 −0.0810804 −0.0405402 0.999178i $$-0.512908\pi$$
−0.0405402 + 0.999178i $$0.512908\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0.140435 0.0573325
$$7$$ −1.14044 −0.431044 −0.215522 0.976499i $$-0.569145\pi$$
−0.215522 + 0.976499i $$0.569145\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.98028 −0.993426
$$10$$ 0 0
$$11$$ 3.12071 0.940930 0.470465 0.882419i $$-0.344086\pi$$
0.470465 + 0.882419i $$0.344086\pi$$
$$12$$ −0.140435 −0.0405402
$$13$$ −2.85956 −0.793101 −0.396550 0.918013i $$-0.629793\pi$$
−0.396550 + 0.918013i $$0.629793\pi$$
$$14$$ 1.14044 0.304794
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.14044 0.519132 0.259566 0.965725i $$-0.416421\pi$$
0.259566 + 0.965725i $$0.416421\pi$$
$$18$$ 2.98028 0.702458
$$19$$ 4.26115 0.977575 0.488787 0.872403i $$-0.337439\pi$$
0.488787 + 0.872403i $$0.337439\pi$$
$$20$$ 0 0
$$21$$ 0.160157 0.0349492
$$22$$ −3.12071 −0.665338
$$23$$ −1.71913 −0.358463 −0.179232 0.983807i $$-0.557361\pi$$
−0.179232 + 0.983807i $$0.557361\pi$$
$$24$$ 0.140435 0.0286662
$$25$$ 0 0
$$26$$ 2.85956 0.560807
$$27$$ 0.839843 0.161628
$$28$$ −1.14044 −0.215522
$$29$$ −4.28087 −0.794938 −0.397469 0.917616i $$-0.630111\pi$$
−0.397469 + 0.917616i $$0.630111\pi$$
$$30$$ 0 0
$$31$$ 6.40158 1.14976 0.574879 0.818238i $$-0.305049\pi$$
0.574879 + 0.818238i $$0.305049\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −0.438259 −0.0762910
$$34$$ −2.14044 −0.367082
$$35$$ 0 0
$$36$$ −2.98028 −0.496713
$$37$$ −1.00000 −0.164399
$$38$$ −4.26115 −0.691250
$$39$$ 0.401584 0.0643049
$$40$$ 0 0
$$41$$ −3.24143 −0.506226 −0.253113 0.967437i $$-0.581454\pi$$
−0.253113 + 0.967437i $$0.581454\pi$$
$$42$$ −0.160157 −0.0247128
$$43$$ −10.6994 −1.63164 −0.815822 0.578303i $$-0.803715\pi$$
−0.815822 + 0.578303i $$0.803715\pi$$
$$44$$ 3.12071 0.470465
$$45$$ 0 0
$$46$$ 1.71913 0.253472
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −0.140435 −0.0202701
$$49$$ −5.69941 −0.814201
$$50$$ 0 0
$$51$$ −0.300593 −0.0420914
$$52$$ −2.85956 −0.396550
$$53$$ −7.96056 −1.09347 −0.546733 0.837307i $$-0.684129\pi$$
−0.546733 + 0.837307i $$0.684129\pi$$
$$54$$ −0.839843 −0.114288
$$55$$ 0 0
$$56$$ 1.14044 0.152397
$$57$$ −0.598416 −0.0792621
$$58$$ 4.28087 0.562106
$$59$$ 2.69941 0.351433 0.175716 0.984441i $$-0.443776\pi$$
0.175716 + 0.984441i $$0.443776\pi$$
$$60$$ 0 0
$$61$$ 4.57869 0.586242 0.293121 0.956075i $$-0.405306\pi$$
0.293121 + 0.956075i $$0.405306\pi$$
$$62$$ −6.40158 −0.813002
$$63$$ 3.39881 0.428210
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.438259 0.0539459
$$67$$ −10.3819 −1.26835 −0.634173 0.773191i $$-0.718659\pi$$
−0.634173 + 0.773191i $$0.718659\pi$$
$$68$$ 2.14044 0.259566
$$69$$ 0.241427 0.0290643
$$70$$ 0 0
$$71$$ −7.10099 −0.842733 −0.421366 0.906891i $$-0.638449\pi$$
−0.421366 + 0.906891i $$0.638449\pi$$
$$72$$ 2.98028 0.351229
$$73$$ 0.261149 0.0305651 0.0152826 0.999883i $$-0.495135\pi$$
0.0152826 + 0.999883i $$0.495135\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 4.26115 0.488787
$$77$$ −3.55897 −0.405582
$$78$$ −0.401584 −0.0454704
$$79$$ −8.52230 −0.958833 −0.479417 0.877587i $$-0.659152\pi$$
−0.479417 + 0.877587i $$0.659152\pi$$
$$80$$ 0 0
$$81$$ 8.82289 0.980321
$$82$$ 3.24143 0.357956
$$83$$ 1.16016 0.127344 0.0636719 0.997971i $$-0.479719\pi$$
0.0636719 + 0.997971i $$0.479719\pi$$
$$84$$ 0.160157 0.0174746
$$85$$ 0 0
$$86$$ 10.6994 1.15375
$$87$$ 0.601186 0.0644539
$$88$$ −3.12071 −0.332669
$$89$$ −13.6430 −1.44616 −0.723078 0.690766i $$-0.757273\pi$$
−0.723078 + 0.690766i $$0.757273\pi$$
$$90$$ 0 0
$$91$$ 3.26115 0.341861
$$92$$ −1.71913 −0.179232
$$93$$ −0.899009 −0.0932229
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0.140435 0.0143331
$$97$$ −14.2414 −1.44600 −0.722999 0.690849i $$-0.757237\pi$$
−0.722999 + 0.690849i $$0.757237\pi$$
$$98$$ 5.69941 0.575727
$$99$$ −9.30059 −0.934745
$$100$$ 0 0
$$101$$ 13.9606 1.38913 0.694564 0.719431i $$-0.255597\pi$$
0.694564 + 0.719431i $$0.255597\pi$$
$$102$$ 0.300593 0.0297631
$$103$$ 10.3621 1.02101 0.510506 0.859874i $$-0.329458\pi$$
0.510506 + 0.859874i $$0.329458\pi$$
$$104$$ 2.85956 0.280403
$$105$$ 0 0
$$106$$ 7.96056 0.773198
$$107$$ −4.70218 −0.454577 −0.227288 0.973828i $$-0.572986\pi$$
−0.227288 + 0.973828i $$0.572986\pi$$
$$108$$ 0.839843 0.0808139
$$109$$ −5.96056 −0.570918 −0.285459 0.958391i $$-0.592146\pi$$
−0.285459 + 0.958391i $$0.592146\pi$$
$$110$$ 0 0
$$111$$ 0.140435 0.0133295
$$112$$ −1.14044 −0.107761
$$113$$ 4.83984 0.455294 0.227647 0.973744i $$-0.426897\pi$$
0.227647 + 0.973744i $$0.426897\pi$$
$$114$$ 0.598416 0.0560468
$$115$$ 0 0
$$116$$ −4.28087 −0.397469
$$117$$ 8.52230 0.787887
$$118$$ −2.69941 −0.248501
$$119$$ −2.44103 −0.223769
$$120$$ 0 0
$$121$$ −1.26115 −0.114650
$$122$$ −4.57869 −0.414535
$$123$$ 0.455211 0.0410450
$$124$$ 6.40158 0.574879
$$125$$ 0 0
$$126$$ −3.39881 −0.302790
$$127$$ −0.899009 −0.0797741 −0.0398871 0.999204i $$-0.512700\pi$$
−0.0398871 + 0.999204i $$0.512700\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.50258 0.132294
$$130$$ 0 0
$$131$$ −0.980278 −0.0856473 −0.0428236 0.999083i $$-0.513635\pi$$
−0.0428236 + 0.999083i $$0.513635\pi$$
$$132$$ −0.438259 −0.0381455
$$133$$ −4.85956 −0.421378
$$134$$ 10.3819 0.896856
$$135$$ 0 0
$$136$$ −2.14044 −0.183541
$$137$$ 6.67969 0.570684 0.285342 0.958426i $$-0.407893\pi$$
0.285342 + 0.958426i $$0.407893\pi$$
$$138$$ −0.241427 −0.0205516
$$139$$ −9.68245 −0.821255 −0.410628 0.911803i $$-0.634690\pi$$
−0.410628 + 0.911803i $$0.634690\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 7.10099 0.595902
$$143$$ −8.92388 −0.746252
$$144$$ −2.98028 −0.248356
$$145$$ 0 0
$$146$$ −0.261149 −0.0216128
$$147$$ 0.800398 0.0660157
$$148$$ −1.00000 −0.0821995
$$149$$ −1.59842 −0.130947 −0.0654737 0.997854i $$-0.520856\pi$$
−0.0654737 + 0.997854i $$0.520856\pi$$
$$150$$ 0 0
$$151$$ −16.4829 −1.34136 −0.670678 0.741749i $$-0.733997\pi$$
−0.670678 + 0.741749i $$0.733997\pi$$
$$152$$ −4.26115 −0.345625
$$153$$ −6.37909 −0.515719
$$154$$ 3.55897 0.286790
$$155$$ 0 0
$$156$$ 0.401584 0.0321525
$$157$$ 10.1207 0.807721 0.403860 0.914821i $$-0.367668\pi$$
0.403860 + 0.914821i $$0.367668\pi$$
$$158$$ 8.52230 0.677998
$$159$$ 1.11794 0.0886587
$$160$$ 0 0
$$161$$ 1.96056 0.154513
$$162$$ −8.82289 −0.693192
$$163$$ −11.7389 −0.919458 −0.459729 0.888059i $$-0.652053\pi$$
−0.459729 + 0.888059i $$0.652053\pi$$
$$164$$ −3.24143 −0.253113
$$165$$ 0 0
$$166$$ −1.16016 −0.0900457
$$167$$ −10.6430 −0.823581 −0.411790 0.911279i $$-0.635096\pi$$
−0.411790 + 0.911279i $$0.635096\pi$$
$$168$$ −0.160157 −0.0123564
$$169$$ −4.82289 −0.370992
$$170$$ 0 0
$$171$$ −12.6994 −0.971148
$$172$$ −10.6994 −0.815822
$$173$$ 11.3988 0.866636 0.433318 0.901241i $$-0.357343\pi$$
0.433318 + 0.901241i $$0.357343\pi$$
$$174$$ −0.601186 −0.0455758
$$175$$ 0 0
$$176$$ 3.12071 0.235233
$$177$$ −0.379092 −0.0284943
$$178$$ 13.6430 1.02259
$$179$$ −18.9211 −1.41423 −0.707115 0.707098i $$-0.750004\pi$$
−0.707115 + 0.707098i $$0.750004\pi$$
$$180$$ 0 0
$$181$$ −18.8032 −1.39763 −0.698814 0.715303i $$-0.746289\pi$$
−0.698814 + 0.715303i $$0.746289\pi$$
$$182$$ −3.26115 −0.241732
$$183$$ −0.643011 −0.0475327
$$184$$ 1.71913 0.126736
$$185$$ 0 0
$$186$$ 0.899009 0.0659185
$$187$$ 6.67969 0.488467
$$188$$ 0 0
$$189$$ −0.957786 −0.0696687
$$190$$ 0 0
$$191$$ 15.0446 1.08859 0.544294 0.838894i $$-0.316797\pi$$
0.544294 + 0.838894i $$0.316797\pi$$
$$192$$ −0.140435 −0.0101350
$$193$$ −25.3227 −1.82277 −0.911384 0.411558i $$-0.864985\pi$$
−0.911384 + 0.411558i $$0.864985\pi$$
$$194$$ 14.2414 1.02247
$$195$$ 0 0
$$196$$ −5.69941 −0.407101
$$197$$ 2.68245 0.191117 0.0955585 0.995424i $$-0.469536\pi$$
0.0955585 + 0.995424i $$0.469536\pi$$
$$198$$ 9.30059 0.660964
$$199$$ 2.24143 0.158891 0.0794453 0.996839i $$-0.474685\pi$$
0.0794453 + 0.996839i $$0.474685\pi$$
$$200$$ 0 0
$$201$$ 1.45798 0.102838
$$202$$ −13.9606 −0.982261
$$203$$ 4.88206 0.342653
$$204$$ −0.300593 −0.0210457
$$205$$ 0 0
$$206$$ −10.3621 −0.721964
$$207$$ 5.12348 0.356107
$$208$$ −2.85956 −0.198275
$$209$$ 13.2978 0.919830
$$210$$ 0 0
$$211$$ −1.61814 −0.111397 −0.0556986 0.998448i $$-0.517739\pi$$
−0.0556986 + 0.998448i $$0.517739\pi$$
$$212$$ −7.96056 −0.546733
$$213$$ 0.997230 0.0683291
$$214$$ 4.70218 0.321434
$$215$$ 0 0
$$216$$ −0.839843 −0.0571440
$$217$$ −7.30059 −0.495597
$$218$$ 5.96056 0.403700
$$219$$ −0.0366745 −0.00247823
$$220$$ 0 0
$$221$$ −6.12071 −0.411724
$$222$$ −0.140435 −0.00942540
$$223$$ 3.39881 0.227601 0.113801 0.993504i $$-0.463697\pi$$
0.113801 + 0.993504i $$0.463697\pi$$
$$224$$ 1.14044 0.0761985
$$225$$ 0 0
$$226$$ −4.83984 −0.321942
$$227$$ −2.45798 −0.163142 −0.0815710 0.996668i $$-0.525994\pi$$
−0.0815710 + 0.996668i $$0.525994\pi$$
$$228$$ −0.598416 −0.0396311
$$229$$ 12.7637 0.843451 0.421725 0.906724i $$-0.361425\pi$$
0.421725 + 0.906724i $$0.361425\pi$$
$$230$$ 0 0
$$231$$ 0.499806 0.0328848
$$232$$ 4.28087 0.281053
$$233$$ 0.103761 0.00679760 0.00339880 0.999994i $$-0.498918\pi$$
0.00339880 + 0.999994i $$0.498918\pi$$
$$234$$ −8.52230 −0.557120
$$235$$ 0 0
$$236$$ 2.69941 0.175716
$$237$$ 1.19683 0.0777426
$$238$$ 2.44103 0.158228
$$239$$ −3.67969 −0.238019 −0.119010 0.992893i $$-0.537972\pi$$
−0.119010 + 0.992893i $$0.537972\pi$$
$$240$$ 0 0
$$241$$ −6.14044 −0.395540 −0.197770 0.980248i $$-0.563370\pi$$
−0.197770 + 0.980248i $$0.563370\pi$$
$$242$$ 1.26115 0.0810697
$$243$$ −3.75857 −0.241113
$$244$$ 4.57869 0.293121
$$245$$ 0 0
$$246$$ −0.455211 −0.0290232
$$247$$ −12.1850 −0.775315
$$248$$ −6.40158 −0.406501
$$249$$ −0.162927 −0.0103251
$$250$$ 0 0
$$251$$ −9.00000 −0.568075 −0.284037 0.958813i $$-0.591674\pi$$
−0.284037 + 0.958813i $$0.591674\pi$$
$$252$$ 3.39881 0.214105
$$253$$ −5.36491 −0.337289
$$254$$ 0.899009 0.0564088
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 16.5223 1.03063 0.515316 0.857000i $$-0.327674\pi$$
0.515316 + 0.857000i $$0.327674\pi$$
$$258$$ −1.50258 −0.0935462
$$259$$ 1.14044 0.0708632
$$260$$ 0 0
$$261$$ 12.7582 0.789712
$$262$$ 0.980278 0.0605618
$$263$$ 22.5223 1.38878 0.694392 0.719597i $$-0.255673\pi$$
0.694392 + 0.719597i $$0.255673\pi$$
$$264$$ 0.438259 0.0269729
$$265$$ 0 0
$$266$$ 4.85956 0.297959
$$267$$ 1.91596 0.117255
$$268$$ −10.3819 −0.634173
$$269$$ 27.2860 1.66366 0.831829 0.555032i $$-0.187294\pi$$
0.831829 + 0.555032i $$0.187294\pi$$
$$270$$ 0 0
$$271$$ −26.1456 −1.58823 −0.794116 0.607767i $$-0.792066\pi$$
−0.794116 + 0.607767i $$0.792066\pi$$
$$272$$ 2.14044 0.129783
$$273$$ −0.457981 −0.0277182
$$274$$ −6.67969 −0.403535
$$275$$ 0 0
$$276$$ 0.241427 0.0145322
$$277$$ 20.5223 1.23307 0.616533 0.787329i $$-0.288537\pi$$
0.616533 + 0.787329i $$0.288537\pi$$
$$278$$ 9.68245 0.580715
$$279$$ −19.0785 −1.14220
$$280$$ 0 0
$$281$$ −14.4580 −0.862491 −0.431245 0.902235i $$-0.641926\pi$$
−0.431245 + 0.902235i $$0.641926\pi$$
$$282$$ 0 0
$$283$$ −6.85679 −0.407594 −0.203797 0.979013i $$-0.565328\pi$$
−0.203797 + 0.979013i $$0.565328\pi$$
$$284$$ −7.10099 −0.421366
$$285$$ 0 0
$$286$$ 8.92388 0.527680
$$287$$ 3.69664 0.218206
$$288$$ 2.98028 0.175615
$$289$$ −12.4185 −0.730502
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0.261149 0.0152826
$$293$$ 19.2048 1.12195 0.560977 0.827832i $$-0.310426\pi$$
0.560977 + 0.827832i $$0.310426\pi$$
$$294$$ −0.800398 −0.0466802
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 2.62091 0.152080
$$298$$ 1.59842 0.0925938
$$299$$ 4.91596 0.284297
$$300$$ 0 0
$$301$$ 12.2020 0.703311
$$302$$ 16.4829 0.948482
$$303$$ −1.96056 −0.112631
$$304$$ 4.26115 0.244394
$$305$$ 0 0
$$306$$ 6.37909 0.364668
$$307$$ −16.2781 −0.929040 −0.464520 0.885563i $$-0.653773\pi$$
−0.464520 + 0.885563i $$0.653773\pi$$
$$308$$ −3.55897 −0.202791
$$309$$ −1.45521 −0.0827841
$$310$$ 0 0
$$311$$ −25.4801 −1.44484 −0.722421 0.691453i $$-0.756971\pi$$
−0.722421 + 0.691453i $$0.756971\pi$$
$$312$$ −0.401584 −0.0227352
$$313$$ −19.0197 −1.07506 −0.537529 0.843245i $$-0.680642\pi$$
−0.537529 + 0.843245i $$0.680642\pi$$
$$314$$ −10.1207 −0.571145
$$315$$ 0 0
$$316$$ −8.52230 −0.479417
$$317$$ −15.1653 −0.851769 −0.425884 0.904778i $$-0.640037\pi$$
−0.425884 + 0.904778i $$0.640037\pi$$
$$318$$ −1.11794 −0.0626912
$$319$$ −13.3594 −0.747981
$$320$$ 0 0
$$321$$ 0.660352 0.0368573
$$322$$ −1.96056 −0.109258
$$323$$ 9.12071 0.507490
$$324$$ 8.82289 0.490161
$$325$$ 0 0
$$326$$ 11.7389 0.650155
$$327$$ 0.837073 0.0462902
$$328$$ 3.24143 0.178978
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −23.4185 −1.28720 −0.643600 0.765362i $$-0.722560\pi$$
−0.643600 + 0.765362i $$0.722560\pi$$
$$332$$ 1.16016 0.0636719
$$333$$ 2.98028 0.163318
$$334$$ 10.6430 0.582360
$$335$$ 0 0
$$336$$ 0.160157 0.00873731
$$337$$ 28.9211 1.57543 0.787717 0.616038i $$-0.211263\pi$$
0.787717 + 0.616038i $$0.211263\pi$$
$$338$$ 4.82289 0.262331
$$339$$ −0.679685 −0.0369154
$$340$$ 0 0
$$341$$ 19.9775 1.08184
$$342$$ 12.6994 0.686705
$$343$$ 14.4829 0.782001
$$344$$ 10.6994 0.576873
$$345$$ 0 0
$$346$$ −11.3988 −0.612804
$$347$$ 32.8817 1.76518 0.882590 0.470143i $$-0.155798\pi$$
0.882590 + 0.470143i $$0.155798\pi$$
$$348$$ 0.601186 0.0322269
$$349$$ 15.2048 0.813892 0.406946 0.913452i $$-0.366594\pi$$
0.406946 + 0.913452i $$0.366594\pi$$
$$350$$ 0 0
$$351$$ −2.40158 −0.128187
$$352$$ −3.12071 −0.166335
$$353$$ 18.4829 0.983743 0.491872 0.870668i $$-0.336313\pi$$
0.491872 + 0.870668i $$0.336313\pi$$
$$354$$ 0.379092 0.0201485
$$355$$ 0 0
$$356$$ −13.6430 −0.723078
$$357$$ 0.342807 0.0181433
$$358$$ 18.9211 1.00001
$$359$$ 22.5223 1.18868 0.594341 0.804213i $$-0.297413\pi$$
0.594341 + 0.804213i $$0.297413\pi$$
$$360$$ 0 0
$$361$$ −0.842612 −0.0443480
$$362$$ 18.8032 0.988273
$$363$$ 0.177110 0.00929586
$$364$$ 3.26115 0.170931
$$365$$ 0 0
$$366$$ 0.643011 0.0336107
$$367$$ −15.3424 −0.800868 −0.400434 0.916326i $$-0.631141\pi$$
−0.400434 + 0.916326i $$0.631141\pi$$
$$368$$ −1.71913 −0.0896158
$$369$$ 9.66035 0.502898
$$370$$ 0 0
$$371$$ 9.07850 0.471332
$$372$$ −0.899009 −0.0466114
$$373$$ −12.6430 −0.654630 −0.327315 0.944915i $$-0.606144\pi$$
−0.327315 + 0.944915i $$0.606144\pi$$
$$374$$ −6.67969 −0.345398
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.2414 0.630466
$$378$$ 0.957786 0.0492632
$$379$$ 15.7807 0.810599 0.405299 0.914184i $$-0.367167\pi$$
0.405299 + 0.914184i $$0.367167\pi$$
$$380$$ 0 0
$$381$$ 0.126253 0.00646812
$$382$$ −15.0446 −0.769748
$$383$$ 27.8004 1.42053 0.710267 0.703932i $$-0.248574\pi$$
0.710267 + 0.703932i $$0.248574\pi$$
$$384$$ 0.140435 0.00716656
$$385$$ 0 0
$$386$$ 25.3227 1.28889
$$387$$ 31.8872 1.62092
$$388$$ −14.2414 −0.722999
$$389$$ 9.66273 0.489920 0.244960 0.969533i $$-0.421225\pi$$
0.244960 + 0.969533i $$0.421225\pi$$
$$390$$ 0 0
$$391$$ −3.67969 −0.186090
$$392$$ 5.69941 0.287864
$$393$$ 0.137666 0.00694432
$$394$$ −2.68245 −0.135140
$$395$$ 0 0
$$396$$ −9.30059 −0.467372
$$397$$ 23.4801 1.17843 0.589216 0.807976i $$-0.299437\pi$$
0.589216 + 0.807976i $$0.299437\pi$$
$$398$$ −2.24143 −0.112353
$$399$$ 0.682455 0.0341655
$$400$$ 0 0
$$401$$ 20.2781 1.01264 0.506320 0.862346i $$-0.331005\pi$$
0.506320 + 0.862346i $$0.331005\pi$$
$$402$$ −1.45798 −0.0727175
$$403$$ −18.3057 −0.911874
$$404$$ 13.9606 0.694564
$$405$$ 0 0
$$406$$ −4.88206 −0.242292
$$407$$ −3.12071 −0.154688
$$408$$ 0.300593 0.0148816
$$409$$ 4.17434 0.206408 0.103204 0.994660i $$-0.467091\pi$$
0.103204 + 0.994660i $$0.467091\pi$$
$$410$$ 0 0
$$411$$ −0.938064 −0.0462713
$$412$$ 10.3621 0.510506
$$413$$ −3.07850 −0.151483
$$414$$ −5.12348 −0.251805
$$415$$ 0 0
$$416$$ 2.85956 0.140202
$$417$$ 1.35976 0.0665877
$$418$$ −13.2978 −0.650418
$$419$$ −12.3175 −0.601751 −0.300876 0.953663i $$-0.597279\pi$$
−0.300876 + 0.953663i $$0.597279\pi$$
$$420$$ 0 0
$$421$$ 23.8872 1.16419 0.582096 0.813120i $$-0.302233\pi$$
0.582096 + 0.813120i $$0.302233\pi$$
$$422$$ 1.61814 0.0787697
$$423$$ 0 0
$$424$$ 7.96056 0.386599
$$425$$ 0 0
$$426$$ −0.997230 −0.0483160
$$427$$ −5.22170 −0.252696
$$428$$ −4.70218 −0.227288
$$429$$ 1.25323 0.0605064
$$430$$ 0 0
$$431$$ −2.80317 −0.135024 −0.0675119 0.997718i $$-0.521506\pi$$
−0.0675119 + 0.997718i $$0.521506\pi$$
$$432$$ 0.839843 0.0404069
$$433$$ −5.00000 −0.240285 −0.120142 0.992757i $$-0.538335\pi$$
−0.120142 + 0.992757i $$0.538335\pi$$
$$434$$ 7.30059 0.350440
$$435$$ 0 0
$$436$$ −5.96056 −0.285459
$$437$$ −7.32547 −0.350425
$$438$$ 0.0366745 0.00175238
$$439$$ 30.7976 1.46989 0.734945 0.678126i $$-0.237208\pi$$
0.734945 + 0.678126i $$0.237208\pi$$
$$440$$ 0 0
$$441$$ 16.9858 0.808848
$$442$$ 6.12071 0.291133
$$443$$ 16.5984 0.788615 0.394307 0.918979i $$-0.370985\pi$$
0.394307 + 0.918979i $$0.370985\pi$$
$$444$$ 0.140435 0.00666477
$$445$$ 0 0
$$446$$ −3.39881 −0.160939
$$447$$ 0.224474 0.0106173
$$448$$ −1.14044 −0.0538805
$$449$$ 9.75580 0.460405 0.230202 0.973143i $$-0.426061\pi$$
0.230202 + 0.973143i $$0.426061\pi$$
$$450$$ 0 0
$$451$$ −10.1156 −0.476323
$$452$$ 4.83984 0.227647
$$453$$ 2.31478 0.108758
$$454$$ 2.45798 0.115359
$$455$$ 0 0
$$456$$ 0.598416 0.0280234
$$457$$ 7.99723 0.374095 0.187047 0.982351i $$-0.440108\pi$$
0.187047 + 0.982351i $$0.440108\pi$$
$$458$$ −12.7637 −0.596410
$$459$$ 1.79763 0.0839061
$$460$$ 0 0
$$461$$ −20.6848 −0.963389 −0.481694 0.876339i $$-0.659978\pi$$
−0.481694 + 0.876339i $$0.659978\pi$$
$$462$$ −0.499806 −0.0232531
$$463$$ 7.52507 0.349720 0.174860 0.984593i $$-0.444053\pi$$
0.174860 + 0.984593i $$0.444053\pi$$
$$464$$ −4.28087 −0.198734
$$465$$ 0 0
$$466$$ −0.103761 −0.00480663
$$467$$ −6.84261 −0.316638 −0.158319 0.987388i $$-0.550607\pi$$
−0.158319 + 0.987388i $$0.550607\pi$$
$$468$$ 8.52230 0.393943
$$469$$ 11.8398 0.546713
$$470$$ 0 0
$$471$$ −1.42131 −0.0654903
$$472$$ −2.69941 −0.124250
$$473$$ −33.3898 −1.53526
$$474$$ −1.19683 −0.0549723
$$475$$ 0 0
$$476$$ −2.44103 −0.111884
$$477$$ 23.7247 1.08628
$$478$$ 3.67969 0.168305
$$479$$ −16.4016 −0.749408 −0.374704 0.927145i $$-0.622256\pi$$
−0.374704 + 0.927145i $$0.622256\pi$$
$$480$$ 0 0
$$481$$ 2.85956 0.130385
$$482$$ 6.14044 0.279689
$$483$$ −0.275331 −0.0125280
$$484$$ −1.26115 −0.0573249
$$485$$ 0 0
$$486$$ 3.75857 0.170492
$$487$$ −13.7270 −0.622032 −0.311016 0.950405i $$-0.600669\pi$$
−0.311016 + 0.950405i $$0.600669\pi$$
$$488$$ −4.57869 −0.207268
$$489$$ 1.64855 0.0745500
$$490$$ 0 0
$$491$$ 3.40435 0.153636 0.0768182 0.997045i $$-0.475524\pi$$
0.0768182 + 0.997045i $$0.475524\pi$$
$$492$$ 0.455211 0.0205225
$$493$$ −9.16293 −0.412677
$$494$$ 12.1850 0.548230
$$495$$ 0 0
$$496$$ 6.40158 0.287440
$$497$$ 8.09822 0.363255
$$498$$ 0.162927 0.00730094
$$499$$ −29.4631 −1.31895 −0.659475 0.751726i $$-0.729222\pi$$
−0.659475 + 0.751726i $$0.729222\pi$$
$$500$$ 0 0
$$501$$ 1.49466 0.0667763
$$502$$ 9.00000 0.401690
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −3.39881 −0.151395
$$505$$ 0 0
$$506$$ 5.36491 0.238499
$$507$$ 0.677304 0.0300801
$$508$$ −0.899009 −0.0398871
$$509$$ 12.9633 0.574589 0.287295 0.957842i $$-0.407244\pi$$
0.287295 + 0.957842i $$0.407244\pi$$
$$510$$ 0 0
$$511$$ −0.297823 −0.0131749
$$512$$ −1.00000 −0.0441942
$$513$$ 3.57869 0.158003
$$514$$ −16.5223 −0.728767
$$515$$ 0 0
$$516$$ 1.50258 0.0661472
$$517$$ 0 0
$$518$$ −1.14044 −0.0501079
$$519$$ −1.60080 −0.0702672
$$520$$ 0 0
$$521$$ −2.01972 −0.0884856 −0.0442428 0.999021i $$-0.514088\pi$$
−0.0442428 + 0.999021i $$0.514088\pi$$
$$522$$ −12.7582 −0.558411
$$523$$ −15.5562 −0.680225 −0.340113 0.940385i $$-0.610465\pi$$
−0.340113 + 0.940385i $$0.610465\pi$$
$$524$$ −0.980278 −0.0428236
$$525$$ 0 0
$$526$$ −22.5223 −0.982019
$$527$$ 13.7022 0.596876
$$528$$ −0.438259 −0.0190728
$$529$$ −20.0446 −0.871504
$$530$$ 0 0
$$531$$ −8.04498 −0.349123
$$532$$ −4.85956 −0.210689
$$533$$ 9.26907 0.401488
$$534$$ −1.91596 −0.0829118
$$535$$ 0 0
$$536$$ 10.3819 0.448428
$$537$$ 2.65719 0.114666
$$538$$ −27.2860 −1.17638
$$539$$ −17.7862 −0.766107
$$540$$ 0 0
$$541$$ −0.544789 −0.0234223 −0.0117112 0.999931i $$-0.503728\pi$$
−0.0117112 + 0.999931i $$0.503728\pi$$
$$542$$ 26.1456 1.12305
$$543$$ 2.64063 0.113320
$$544$$ −2.14044 −0.0917704
$$545$$ 0 0
$$546$$ 0.457981 0.0195998
$$547$$ 35.6261 1.52326 0.761630 0.648012i $$-0.224399\pi$$
0.761630 + 0.648012i $$0.224399\pi$$
$$548$$ 6.67969 0.285342
$$549$$ −13.6458 −0.582388
$$550$$ 0 0
$$551$$ −18.2414 −0.777111
$$552$$ −0.241427 −0.0102758
$$553$$ 9.71913 0.413299
$$554$$ −20.5223 −0.871909
$$555$$ 0 0
$$556$$ −9.68245 −0.410628
$$557$$ −31.9606 −1.35421 −0.677106 0.735885i $$-0.736766\pi$$
−0.677106 + 0.735885i $$0.736766\pi$$
$$558$$ 19.0785 0.807657
$$559$$ 30.5956 1.29406
$$560$$ 0 0
$$561$$ −0.938064 −0.0396051
$$562$$ 14.4580 0.609873
$$563$$ −14.6655 −0.618077 −0.309039 0.951049i $$-0.600007\pi$$
−0.309039 + 0.951049i $$0.600007\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 6.85679 0.288213
$$567$$ −10.0619 −0.422562
$$568$$ 7.10099 0.297951
$$569$$ 9.99723 0.419106 0.209553 0.977797i $$-0.432799\pi$$
0.209553 + 0.977797i $$0.432799\pi$$
$$570$$ 0 0
$$571$$ 27.6008 1.15506 0.577529 0.816370i $$-0.304017\pi$$
0.577529 + 0.816370i $$0.304017\pi$$
$$572$$ −8.92388 −0.373126
$$573$$ −2.11279 −0.0882632
$$574$$ −3.69664 −0.154295
$$575$$ 0 0
$$576$$ −2.98028 −0.124178
$$577$$ 3.78622 0.157622 0.0788111 0.996890i $$-0.474888\pi$$
0.0788111 + 0.996890i $$0.474888\pi$$
$$578$$ 12.4185 0.516543
$$579$$ 3.55620 0.147791
$$580$$ 0 0
$$581$$ −1.32308 −0.0548908
$$582$$ −2.00000 −0.0829027
$$583$$ −24.8426 −1.02888
$$584$$ −0.261149 −0.0108064
$$585$$ 0 0
$$586$$ −19.2048 −0.793341
$$587$$ −19.1771 −0.791524 −0.395762 0.918353i $$-0.629519\pi$$
−0.395762 + 0.918353i $$0.629519\pi$$
$$588$$ 0.800398 0.0330079
$$589$$ 27.2781 1.12397
$$590$$ 0 0
$$591$$ −0.376712 −0.0154958
$$592$$ −1.00000 −0.0410997
$$593$$ 12.4383 0.510778 0.255389 0.966838i $$-0.417796\pi$$
0.255389 + 0.966838i $$0.417796\pi$$
$$594$$ −2.62091 −0.107537
$$595$$ 0 0
$$596$$ −1.59842 −0.0654737
$$597$$ −0.314776 −0.0128829
$$598$$ −4.91596 −0.201029
$$599$$ −35.3819 −1.44566 −0.722832 0.691024i $$-0.757160\pi$$
−0.722832 + 0.691024i $$0.757160\pi$$
$$600$$ 0 0
$$601$$ 22.2951 0.909434 0.454717 0.890636i $$-0.349740\pi$$
0.454717 + 0.890636i $$0.349740\pi$$
$$602$$ −12.2020 −0.497316
$$603$$ 30.9408 1.26001
$$604$$ −16.4829 −0.670678
$$605$$ 0 0
$$606$$ 1.96056 0.0796421
$$607$$ 9.72705 0.394809 0.197404 0.980322i $$-0.436749\pi$$
0.197404 + 0.980322i $$0.436749\pi$$
$$608$$ −4.26115 −0.172812
$$609$$ −0.685613 −0.0277825
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −6.37909 −0.257860
$$613$$ −10.3203 −0.416834 −0.208417 0.978040i $$-0.566831\pi$$
−0.208417 + 0.978040i $$0.566831\pi$$
$$614$$ 16.2781 0.656931
$$615$$ 0 0
$$616$$ 3.55897 0.143395
$$617$$ 3.78345 0.152316 0.0761579 0.997096i $$-0.475735\pi$$
0.0761579 + 0.997096i $$0.475735\pi$$
$$618$$ 1.45521 0.0585372
$$619$$ 27.1179 1.08996 0.544981 0.838448i $$-0.316537\pi$$
0.544981 + 0.838448i $$0.316537\pi$$
$$620$$ 0 0
$$621$$ −1.44380 −0.0579376
$$622$$ 25.4801 1.02166
$$623$$ 15.5590 0.623357
$$624$$ 0.401584 0.0160762
$$625$$ 0 0
$$626$$ 19.0197 0.760181
$$627$$ −1.86748 −0.0745802
$$628$$ 10.1207 0.403860
$$629$$ −2.14044 −0.0853447
$$630$$ 0 0
$$631$$ −11.1179 −0.442598 −0.221299 0.975206i $$-0.571030\pi$$
−0.221299 + 0.975206i $$0.571030\pi$$
$$632$$ 8.52230 0.338999
$$633$$ 0.227244 0.00903213
$$634$$ 15.1653 0.602291
$$635$$ 0 0
$$636$$ 1.11794 0.0443293
$$637$$ 16.2978 0.645743
$$638$$ 13.3594 0.528903
$$639$$ 21.1629 0.837192
$$640$$ 0 0
$$641$$ 27.0446 1.06820 0.534099 0.845422i $$-0.320651\pi$$
0.534099 + 0.845422i $$0.320651\pi$$
$$642$$ −0.660352 −0.0260620
$$643$$ −29.7834 −1.17454 −0.587272 0.809389i $$-0.699798\pi$$
−0.587272 + 0.809389i $$0.699798\pi$$
$$644$$ 1.96056 0.0772567
$$645$$ 0 0
$$646$$ −9.12071 −0.358850
$$647$$ −19.3570 −0.761002 −0.380501 0.924781i $$-0.624248\pi$$
−0.380501 + 0.924781i $$0.624248\pi$$
$$648$$ −8.82289 −0.346596
$$649$$ 8.42408 0.330674
$$650$$ 0 0
$$651$$ 1.02526 0.0401832
$$652$$ −11.7389 −0.459729
$$653$$ 10.5392 0.412433 0.206216 0.978506i $$-0.433885\pi$$
0.206216 + 0.978506i $$0.433885\pi$$
$$654$$ −0.837073 −0.0327321
$$655$$ 0 0
$$656$$ −3.24143 −0.126556
$$657$$ −0.778296 −0.0303642
$$658$$ 0 0
$$659$$ 9.22447 0.359334 0.179667 0.983727i $$-0.442498\pi$$
0.179667 + 0.983727i $$0.442498\pi$$
$$660$$ 0 0
$$661$$ 27.3594 1.06416 0.532078 0.846695i $$-0.321411\pi$$
0.532078 + 0.846695i $$0.321411\pi$$
$$662$$ 23.4185 0.910187
$$663$$ 0.859565 0.0333827
$$664$$ −1.16016 −0.0450228
$$665$$ 0 0
$$666$$ −2.98028 −0.115483
$$667$$ 7.35937 0.284956
$$668$$ −10.6430 −0.411790
$$669$$ −0.477314 −0.0184540
$$670$$ 0 0
$$671$$ 14.2888 0.551613
$$672$$ −0.160157 −0.00617821
$$673$$ 13.1484 0.506832 0.253416 0.967357i $$-0.418446\pi$$
0.253416 + 0.967357i $$0.418446\pi$$
$$674$$ −28.9211 −1.11400
$$675$$ 0 0
$$676$$ −4.82289 −0.185496
$$677$$ −29.8817 −1.14845 −0.574223 0.818699i $$-0.694696\pi$$
−0.574223 + 0.818699i $$0.694696\pi$$
$$678$$ 0.679685 0.0261031
$$679$$ 16.2414 0.623289
$$680$$ 0 0
$$681$$ 0.345187 0.0132276
$$682$$ −19.9775 −0.764978
$$683$$ 26.9854 1.03257 0.516284 0.856417i $$-0.327315\pi$$
0.516284 + 0.856417i $$0.327315\pi$$
$$684$$ −12.6994 −0.485574
$$685$$ 0 0
$$686$$ −14.4829 −0.552958
$$687$$ −1.79248 −0.0683873
$$688$$ −10.6994 −0.407911
$$689$$ 22.7637 0.867229
$$690$$ 0 0
$$691$$ −30.7270 −1.16891 −0.584456 0.811425i $$-0.698692\pi$$
−0.584456 + 0.811425i $$0.698692\pi$$
$$692$$ 11.3988 0.433318
$$693$$ 10.6067 0.402916
$$694$$ −32.8817 −1.24817
$$695$$ 0 0
$$696$$ −0.601186 −0.0227879
$$697$$ −6.93806 −0.262798
$$698$$ −15.2048 −0.575508
$$699$$ −0.0145717 −0.000551152 0
$$700$$ 0 0
$$701$$ −27.7858 −1.04946 −0.524728 0.851270i $$-0.675833\pi$$
−0.524728 + 0.851270i $$0.675833\pi$$
$$702$$ 2.40158 0.0906419
$$703$$ −4.26115 −0.160712
$$704$$ 3.12071 0.117616
$$705$$ 0 0
$$706$$ −18.4829 −0.695611
$$707$$ −15.9211 −0.598775
$$708$$ −0.379092 −0.0142472
$$709$$ 9.71913 0.365010 0.182505 0.983205i $$-0.441579\pi$$
0.182505 + 0.983205i $$0.441579\pi$$
$$710$$ 0 0
$$711$$ 25.3988 0.952530
$$712$$ 13.6430 0.511293
$$713$$ −11.0052 −0.412146
$$714$$ −0.342807 −0.0128292
$$715$$ 0 0
$$716$$ −18.9211 −0.707115
$$717$$ 0.516758 0.0192987
$$718$$ −22.5223 −0.840525
$$719$$ 5.41577 0.201974 0.100987 0.994888i $$-0.467800\pi$$
0.100987 + 0.994888i $$0.467800\pi$$
$$720$$ 0 0
$$721$$ −11.8174 −0.440101
$$722$$ 0.842612 0.0313588
$$723$$ 0.862334 0.0320706
$$724$$ −18.8032 −0.698814
$$725$$ 0 0
$$726$$ −0.177110 −0.00657316
$$727$$ −13.7586 −0.510277 −0.255139 0.966904i $$-0.582121\pi$$
−0.255139 + 0.966904i $$0.582121\pi$$
$$728$$ −3.26115 −0.120866
$$729$$ −25.9408 −0.960772
$$730$$ 0 0
$$731$$ −22.9014 −0.847038
$$732$$ −0.643011 −0.0237664
$$733$$ −52.4095 −1.93579 −0.967895 0.251356i $$-0.919123\pi$$
−0.967895 + 0.251356i $$0.919123\pi$$
$$734$$ 15.3424 0.566299
$$735$$ 0 0
$$736$$ 1.71913 0.0633679
$$737$$ −32.3988 −1.19343
$$738$$ −9.66035 −0.355602
$$739$$ 50.0892 1.84256 0.921280 0.388899i $$-0.127145\pi$$
0.921280 + 0.388899i $$0.127145\pi$$
$$740$$ 0 0
$$741$$ 1.71121 0.0628628
$$742$$ −9.07850 −0.333282
$$743$$ 40.7976 1.49672 0.748360 0.663293i $$-0.230842\pi$$
0.748360 + 0.663293i $$0.230842\pi$$
$$744$$ 0.899009 0.0329593
$$745$$ 0 0
$$746$$ 12.6430 0.462894
$$747$$ −3.45759 −0.126507
$$748$$ 6.67969 0.244233
$$749$$ 5.36253 0.195943
$$750$$ 0 0
$$751$$ 2.84261 0.103728 0.0518642 0.998654i $$-0.483484\pi$$
0.0518642 + 0.998654i $$0.483484\pi$$
$$752$$ 0 0
$$753$$ 1.26392 0.0460597
$$754$$ −12.2414 −0.445806
$$755$$ 0 0
$$756$$ −0.957786 −0.0348343
$$757$$ −11.4383 −0.415731 −0.207865 0.978157i $$-0.566652\pi$$
−0.207865 + 0.978157i $$0.566652\pi$$
$$758$$ −15.7807 −0.573180
$$759$$ 0.753423 0.0273475
$$760$$ 0 0
$$761$$ 43.2663 1.56840 0.784201 0.620507i $$-0.213073\pi$$
0.784201 + 0.620507i $$0.213073\pi$$
$$762$$ −0.126253 −0.00457365
$$763$$ 6.79763 0.246091
$$764$$ 15.0446 0.544294
$$765$$ 0 0
$$766$$ −27.8004 −1.00447
$$767$$ −7.71913 −0.278722
$$768$$ −0.140435 −0.00506752
$$769$$ 26.5590 0.957741 0.478871 0.877886i $$-0.341046\pi$$
0.478871 + 0.877886i $$0.341046\pi$$
$$770$$ 0 0
$$771$$ −2.32031 −0.0835641
$$772$$ −25.3227 −0.911384
$$773$$ −12.3621 −0.444635 −0.222318 0.974974i $$-0.571362\pi$$
−0.222318 + 0.974974i $$0.571362\pi$$
$$774$$ −31.8872 −1.14616
$$775$$ 0 0
$$776$$ 14.2414 0.511237
$$777$$ −0.160157 −0.00574562
$$778$$ −9.66273 −0.346426
$$779$$ −13.8122 −0.494873
$$780$$ 0 0
$$781$$ −22.1602 −0.792953
$$782$$ 3.67969 0.131585
$$783$$ −3.59526 −0.128484
$$784$$ −5.69941 −0.203550
$$785$$ 0 0
$$786$$ −0.137666 −0.00491037
$$787$$ −6.03944 −0.215283 −0.107641 0.994190i $$-0.534330\pi$$
−0.107641 + 0.994190i $$0.534330\pi$$
$$788$$ 2.68245 0.0955585
$$789$$ −3.16293 −0.112603
$$790$$ 0 0
$$791$$ −5.51953 −0.196252
$$792$$ 9.30059 0.330482
$$793$$ −13.0931 −0.464949
$$794$$ −23.4801 −0.833277
$$795$$ 0 0
$$796$$ 2.24143 0.0794453
$$797$$ −35.6233 −1.26184 −0.630921 0.775847i $$-0.717323\pi$$
−0.630921 + 0.775847i $$0.717323\pi$$
$$798$$ −0.682455 −0.0241586
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 40.6600 1.43665
$$802$$ −20.2781 −0.716045
$$803$$ 0.814970 0.0287597
$$804$$ 1.45798 0.0514190
$$805$$ 0 0
$$806$$ 18.3057 0.644792
$$807$$ −3.83192 −0.134890
$$808$$ −13.9606 −0.491131
$$809$$ −23.6402 −0.831147 −0.415573 0.909560i $$-0.636419\pi$$
−0.415573 + 0.909560i $$0.636419\pi$$
$$810$$ 0 0
$$811$$ −3.27572 −0.115026 −0.0575130 0.998345i $$-0.518317\pi$$
−0.0575130 + 0.998345i $$0.518317\pi$$
$$812$$ 4.88206 0.171327
$$813$$ 3.67176 0.128774
$$814$$ 3.12071 0.109381
$$815$$ 0 0
$$816$$ −0.300593 −0.0105229
$$817$$ −45.5918 −1.59505
$$818$$ −4.17434 −0.145952
$$819$$ −9.71913 −0.339614
$$820$$ 0 0
$$821$$ −30.3621 −1.05965 −0.529823 0.848108i $$-0.677742\pi$$
−0.529823 + 0.848108i $$0.677742\pi$$
$$822$$ 0.938064 0.0327187
$$823$$ 14.2978 0.498391 0.249195 0.968453i $$-0.419834\pi$$
0.249195 + 0.968453i $$0.419834\pi$$
$$824$$ −10.3621 −0.360982
$$825$$ 0 0
$$826$$ 3.07850 0.107115
$$827$$ −7.90139 −0.274758 −0.137379 0.990519i $$-0.543868\pi$$
−0.137379 + 0.990519i $$0.543868\pi$$
$$828$$ 5.12348 0.178053
$$829$$ −43.3030 −1.50397 −0.751987 0.659178i $$-0.770905\pi$$
−0.751987 + 0.659178i $$0.770905\pi$$
$$830$$ 0 0
$$831$$ −2.88206 −0.0999774
$$832$$ −2.85956 −0.0991376
$$833$$ −12.1992 −0.422678
$$834$$ −1.35976 −0.0470846
$$835$$ 0 0
$$836$$ 13.2978 0.459915
$$837$$ 5.37632 0.185833
$$838$$ 12.3175 0.425503
$$839$$ 11.3819 0.392946 0.196473 0.980509i $$-0.437051\pi$$
0.196473 + 0.980509i $$0.437051\pi$$
$$840$$ 0 0
$$841$$ −10.6741 −0.368074
$$842$$ −23.8872 −0.823208
$$843$$ 2.03041 0.0699311
$$844$$ −1.61814 −0.0556986
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.43826 0.0494191
$$848$$ −7.96056 −0.273367
$$849$$ 0.962937 0.0330479
$$850$$ 0 0
$$851$$ 1.71913 0.0589310
$$852$$ 0.997230 0.0341645
$$853$$ −10.7862 −0.369313 −0.184656 0.982803i $$-0.559117\pi$$
−0.184656 + 0.982803i $$0.559117\pi$$
$$854$$ 5.22170 0.178683
$$855$$ 0 0
$$856$$ 4.70218 0.160717
$$857$$ 2.38186 0.0813629 0.0406814 0.999172i $$-0.487047\pi$$
0.0406814 + 0.999172i $$0.487047\pi$$
$$858$$ −1.25323 −0.0427845
$$859$$ −48.1428 −1.64261 −0.821306 0.570488i $$-0.806754\pi$$
−0.821306 + 0.570488i $$0.806754\pi$$
$$860$$ 0 0
$$861$$ −0.519139 −0.0176922
$$862$$ 2.80317 0.0954763
$$863$$ −29.8308 −1.01545 −0.507726 0.861518i $$-0.669514\pi$$
−0.507726 + 0.861518i $$0.669514\pi$$
$$864$$ −0.839843 −0.0285720
$$865$$ 0 0
$$866$$ 5.00000 0.169907
$$867$$ 1.74400 0.0592294
$$868$$ −7.30059 −0.247798
$$869$$ −26.5956 −0.902196
$$870$$ 0 0
$$871$$ 29.6876 1.00593
$$872$$ 5.96056 0.201850
$$873$$ 42.4434 1.43649
$$874$$ 7.32547 0.247788
$$875$$ 0 0
$$876$$ −0.0366745 −0.00123912
$$877$$ 48.9578 1.65319 0.826593 0.562799i $$-0.190276\pi$$
0.826593 + 0.562799i $$0.190276\pi$$
$$878$$ −30.7976 −1.03937
$$879$$ −2.69703 −0.0909684
$$880$$ 0 0
$$881$$ 3.50811 0.118191 0.0590957 0.998252i $$-0.481178\pi$$
0.0590957 + 0.998252i $$0.481178\pi$$
$$882$$ −16.9858 −0.571942
$$883$$ 48.8817 1.64500 0.822500 0.568766i $$-0.192579\pi$$
0.822500 + 0.568766i $$0.192579\pi$$
$$884$$ −6.12071 −0.205862
$$885$$ 0 0
$$886$$ −16.5984 −0.557635
$$887$$ −46.3701 −1.55695 −0.778477 0.627673i $$-0.784008\pi$$
−0.778477 + 0.627673i $$0.784008\pi$$
$$888$$ −0.140435 −0.00471270
$$889$$ 1.02526 0.0343862
$$890$$ 0 0
$$891$$ 27.5337 0.922414
$$892$$ 3.39881 0.113801
$$893$$ 0 0
$$894$$ −0.224474 −0.00750754
$$895$$ 0 0
$$896$$ 1.14044 0.0380993
$$897$$ −0.690375 −0.0230509
$$898$$ −9.75580 −0.325555
$$899$$ −27.4044 −0.913986
$$900$$ 0 0
$$901$$ −17.0391 −0.567653
$$902$$ 10.1156 0.336811
$$903$$ −1.71359 −0.0570247
$$904$$ −4.83984 −0.160971
$$905$$ 0 0
$$906$$ −2.31478 −0.0769033
$$907$$ −45.4631 −1.50958 −0.754789 0.655967i $$-0.772261\pi$$
−0.754789 + 0.655967i $$0.772261\pi$$
$$908$$ −2.45798 −0.0815710
$$909$$ −41.6063 −1.38000
$$910$$ 0 0
$$911$$ 17.7610 0.588447 0.294223 0.955737i $$-0.404939\pi$$
0.294223 + 0.955737i $$0.404939\pi$$
$$912$$ −0.598416 −0.0198155
$$913$$ 3.62052 0.119822
$$914$$ −7.99723 −0.264525
$$915$$ 0 0
$$916$$ 12.7637 0.421725
$$917$$ 1.11794 0.0369178
$$918$$ −1.79763 −0.0593306
$$919$$ 26.4829 0.873589 0.436794 0.899561i $$-0.356114\pi$$
0.436794 + 0.899561i $$0.356114\pi$$
$$920$$ 0 0
$$921$$ 2.28602 0.0753270
$$922$$ 20.6848 0.681219
$$923$$ 20.3057 0.668372
$$924$$ 0.499806 0.0164424
$$925$$ 0 0
$$926$$ −7.52507 −0.247289
$$927$$ −30.8821 −1.01430
$$928$$ 4.28087 0.140526
$$929$$ 59.3842 1.94833 0.974167 0.225829i $$-0.0725091\pi$$
0.974167 + 0.225829i $$0.0725091\pi$$
$$930$$ 0 0
$$931$$ −24.2860 −0.795942
$$932$$ 0.103761 0.00339880
$$933$$ 3.57830 0.117148
$$934$$ 6.84261 0.223897
$$935$$ 0 0
$$936$$ −8.52230 −0.278560
$$937$$ −15.1771 −0.495815 −0.247907 0.968784i $$-0.579743\pi$$
−0.247907 + 0.968784i $$0.579743\pi$$
$$938$$ −11.8398 −0.386585
$$939$$ 2.67104 0.0871662
$$940$$ 0 0
$$941$$ 12.8426 0.418657 0.209329 0.977845i $$-0.432872\pi$$
0.209329 + 0.977845i $$0.432872\pi$$
$$942$$ 1.42131 0.0463087
$$943$$ 5.57243 0.181463
$$944$$ 2.69941 0.0878582
$$945$$ 0 0
$$946$$ 33.3898 1.08560
$$947$$ 9.04459 0.293910 0.146955 0.989143i $$-0.453053\pi$$
0.146955 + 0.989143i $$0.453053\pi$$
$$948$$ 1.19683 0.0388713
$$949$$ −0.746771 −0.0242412
$$950$$ 0 0
$$951$$ 2.12975 0.0690617
$$952$$ 2.44103 0.0791142
$$953$$ 13.7783 0.446323 0.223161 0.974782i $$-0.428362\pi$$
0.223161 + 0.974782i $$0.428362\pi$$
$$954$$ −23.7247 −0.768115
$$955$$ 0 0
$$956$$ −3.67969 −0.119010
$$957$$ 1.87613 0.0606466
$$958$$ 16.4016 0.529911
$$959$$ −7.61775 −0.245990
$$960$$ 0 0
$$961$$ 9.98028 0.321944
$$962$$ −2.85956 −0.0921961
$$963$$ 14.0138 0.451588
$$964$$ −6.14044 −0.197770
$$965$$ 0 0
$$966$$ 0.275331 0.00885864
$$967$$ −60.1964 −1.93579 −0.967894 0.251360i $$-0.919122\pi$$
−0.967894 + 0.251360i $$0.919122\pi$$
$$968$$ 1.26115 0.0405349
$$969$$ −1.28087 −0.0411475
$$970$$ 0 0
$$971$$ −28.4123 −0.911793 −0.455897 0.890033i $$-0.650681\pi$$
−0.455897 + 0.890033i $$0.650681\pi$$
$$972$$ −3.75857 −0.120556
$$973$$ 11.0422 0.353997
$$974$$ 13.7270 0.439843
$$975$$ 0 0
$$976$$ 4.57869 0.146560
$$977$$ 55.7468 1.78350 0.891749 0.452531i $$-0.149479\pi$$
0.891749 + 0.452531i $$0.149479\pi$$
$$978$$ −1.64855 −0.0527148
$$979$$ −42.5759 −1.36073
$$980$$ 0 0
$$981$$ 17.7641 0.567164
$$982$$ −3.40435 −0.108637
$$983$$ −37.1179 −1.18388 −0.591939 0.805983i $$-0.701637\pi$$
−0.591939 + 0.805983i $$0.701637\pi$$
$$984$$ −0.455211 −0.0145116
$$985$$ 0 0
$$986$$ 9.16293 0.291807
$$987$$ 0 0
$$988$$ −12.1850 −0.387657
$$989$$ 18.3937 0.584884
$$990$$ 0 0
$$991$$ −38.0418 −1.20844 −0.604219 0.796818i $$-0.706515\pi$$
−0.604219 + 0.796818i $$0.706515\pi$$
$$992$$ −6.40158 −0.203250
$$993$$ 3.28879 0.104367
$$994$$ −8.09822 −0.256860
$$995$$ 0 0
$$996$$ −0.162927 −0.00516254
$$997$$ −11.6288 −0.368289 −0.184144 0.982899i $$-0.558951\pi$$
−0.184144 + 0.982899i $$0.558951\pi$$
$$998$$ 29.4631 0.932639
$$999$$ −0.839843 −0.0265714
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.y.1.2 3
5.2 odd 4 1850.2.b.p.149.2 6
5.3 odd 4 1850.2.b.p.149.5 6
5.4 even 2 1850.2.a.bc.1.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.y.1.2 3 1.1 even 1 trivial
1850.2.a.bc.1.2 yes 3 5.4 even 2
1850.2.b.p.149.2 6 5.2 odd 4
1850.2.b.p.149.5 6 5.3 odd 4