# Properties

 Label 1850.2.b.p.149.2 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.37161216.1 Defining polynomial: $$x^{6} + 15x^{4} + 51x^{2} + 1$$ x^6 + 15*x^4 + 51*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.2 Root $$0.140435i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.p.149.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +0.140435i q^{3} -1.00000 q^{4} +0.140435 q^{6} -1.14044i q^{7} +1.00000i q^{8} +2.98028 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +0.140435i q^{3} -1.00000 q^{4} +0.140435 q^{6} -1.14044i q^{7} +1.00000i q^{8} +2.98028 q^{9} +3.12071 q^{11} -0.140435i q^{12} +2.85956i q^{13} -1.14044 q^{14} +1.00000 q^{16} +2.14044i q^{17} -2.98028i q^{18} -4.26115 q^{19} +0.160157 q^{21} -3.12071i q^{22} +1.71913i q^{23} -0.140435 q^{24} +2.85956 q^{26} +0.839843i q^{27} +1.14044i q^{28} +4.28087 q^{29} +6.40158 q^{31} -1.00000i q^{32} +0.438259i q^{33} +2.14044 q^{34} -2.98028 q^{36} -1.00000i q^{37} +4.26115i q^{38} -0.401584 q^{39} -3.24143 q^{41} -0.160157i q^{42} +10.6994i q^{43} -3.12071 q^{44} +1.71913 q^{46} +0.140435i q^{48} +5.69941 q^{49} -0.300593 q^{51} -2.85956i q^{52} +7.96056i q^{53} +0.839843 q^{54} +1.14044 q^{56} -0.598416i q^{57} -4.28087i q^{58} -2.69941 q^{59} +4.57869 q^{61} -6.40158i q^{62} -3.39881i q^{63} -1.00000 q^{64} +0.438259 q^{66} -10.3819i q^{67} -2.14044i q^{68} -0.241427 q^{69} -7.10099 q^{71} +2.98028i q^{72} -0.261149i q^{73} -1.00000 q^{74} +4.26115 q^{76} -3.55897i q^{77} +0.401584i q^{78} +8.52230 q^{79} +8.82289 q^{81} +3.24143i q^{82} -1.16016i q^{83} -0.160157 q^{84} +10.6994 q^{86} +0.601186i q^{87} +3.12071i q^{88} +13.6430 q^{89} +3.26115 q^{91} -1.71913i q^{92} +0.899009i q^{93} +0.140435 q^{96} -14.2414i q^{97} -5.69941i q^{98} +9.30059 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 2 q^{6} - 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 + 2 * q^6 - 12 * q^9 $$6 q - 6 q^{4} + 2 q^{6} - 12 q^{9} - 10 q^{11} - 8 q^{14} + 6 q^{16} + 2 q^{19} + 32 q^{21} - 2 q^{24} + 16 q^{26} + 28 q^{29} + 12 q^{31} + 14 q^{34} + 12 q^{36} + 24 q^{39} + 38 q^{41} + 10 q^{44} + 8 q^{46} + 2 q^{49} - 34 q^{51} - 26 q^{54} + 8 q^{56} + 16 q^{59} + 24 q^{61} - 6 q^{64} - 2 q^{66} + 56 q^{69} + 16 q^{71} - 6 q^{74} - 2 q^{76} - 4 q^{79} + 30 q^{81} - 32 q^{84} + 32 q^{86} - 2 q^{89} - 8 q^{91} + 2 q^{96} + 88 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 + 2 * q^6 - 12 * q^9 - 10 * q^11 - 8 * q^14 + 6 * q^16 + 2 * q^19 + 32 * q^21 - 2 * q^24 + 16 * q^26 + 28 * q^29 + 12 * q^31 + 14 * q^34 + 12 * q^36 + 24 * q^39 + 38 * q^41 + 10 * q^44 + 8 * q^46 + 2 * q^49 - 34 * q^51 - 26 * q^54 + 8 * q^56 + 16 * q^59 + 24 * q^61 - 6 * q^64 - 2 * q^66 + 56 * q^69 + 16 * q^71 - 6 * q^74 - 2 * q^76 - 4 * q^79 + 30 * q^81 - 32 * q^84 + 32 * q^86 - 2 * q^89 - 8 * q^91 + 2 * q^96 + 88 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0.140435i 0.0810804i 0.999178 + 0.0405402i $$0.0129079\pi$$
−0.999178 + 0.0405402i $$0.987092\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0.140435 0.0573325
$$7$$ − 1.14044i − 0.431044i −0.976499 0.215522i $$-0.930855\pi$$
0.976499 0.215522i $$-0.0691453\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 2.98028 0.993426
$$10$$ 0 0
$$11$$ 3.12071 0.940930 0.470465 0.882419i $$-0.344086\pi$$
0.470465 + 0.882419i $$0.344086\pi$$
$$12$$ − 0.140435i − 0.0405402i
$$13$$ 2.85956i 0.793101i 0.918013 + 0.396550i $$0.129793\pi$$
−0.918013 + 0.396550i $$0.870207\pi$$
$$14$$ −1.14044 −0.304794
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.14044i 0.519132i 0.965725 + 0.259566i $$0.0835795\pi$$
−0.965725 + 0.259566i $$0.916421\pi$$
$$18$$ − 2.98028i − 0.702458i
$$19$$ −4.26115 −0.977575 −0.488787 0.872403i $$-0.662561\pi$$
−0.488787 + 0.872403i $$0.662561\pi$$
$$20$$ 0 0
$$21$$ 0.160157 0.0349492
$$22$$ − 3.12071i − 0.665338i
$$23$$ 1.71913i 0.358463i 0.983807 + 0.179232i $$0.0573611\pi$$
−0.983807 + 0.179232i $$0.942639\pi$$
$$24$$ −0.140435 −0.0286662
$$25$$ 0 0
$$26$$ 2.85956 0.560807
$$27$$ 0.839843i 0.161628i
$$28$$ 1.14044i 0.215522i
$$29$$ 4.28087 0.794938 0.397469 0.917616i $$-0.369889\pi$$
0.397469 + 0.917616i $$0.369889\pi$$
$$30$$ 0 0
$$31$$ 6.40158 1.14976 0.574879 0.818238i $$-0.305049\pi$$
0.574879 + 0.818238i $$0.305049\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0.438259i 0.0762910i
$$34$$ 2.14044 0.367082
$$35$$ 0 0
$$36$$ −2.98028 −0.496713
$$37$$ − 1.00000i − 0.164399i
$$38$$ 4.26115i 0.691250i
$$39$$ −0.401584 −0.0643049
$$40$$ 0 0
$$41$$ −3.24143 −0.506226 −0.253113 0.967437i $$-0.581454\pi$$
−0.253113 + 0.967437i $$0.581454\pi$$
$$42$$ − 0.160157i − 0.0247128i
$$43$$ 10.6994i 1.63164i 0.578303 + 0.815822i $$0.303715\pi$$
−0.578303 + 0.815822i $$0.696285\pi$$
$$44$$ −3.12071 −0.470465
$$45$$ 0 0
$$46$$ 1.71913 0.253472
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0.140435i 0.0202701i
$$49$$ 5.69941 0.814201
$$50$$ 0 0
$$51$$ −0.300593 −0.0420914
$$52$$ − 2.85956i − 0.396550i
$$53$$ 7.96056i 1.09347i 0.837307 + 0.546733i $$0.184129\pi$$
−0.837307 + 0.546733i $$0.815871\pi$$
$$54$$ 0.839843 0.114288
$$55$$ 0 0
$$56$$ 1.14044 0.152397
$$57$$ − 0.598416i − 0.0792621i
$$58$$ − 4.28087i − 0.562106i
$$59$$ −2.69941 −0.351433 −0.175716 0.984441i $$-0.556224\pi$$
−0.175716 + 0.984441i $$0.556224\pi$$
$$60$$ 0 0
$$61$$ 4.57869 0.586242 0.293121 0.956075i $$-0.405306\pi$$
0.293121 + 0.956075i $$0.405306\pi$$
$$62$$ − 6.40158i − 0.813002i
$$63$$ − 3.39881i − 0.428210i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0.438259 0.0539459
$$67$$ − 10.3819i − 1.26835i −0.773191 0.634173i $$-0.781341\pi$$
0.773191 0.634173i $$-0.218659\pi$$
$$68$$ − 2.14044i − 0.259566i
$$69$$ −0.241427 −0.0290643
$$70$$ 0 0
$$71$$ −7.10099 −0.842733 −0.421366 0.906891i $$-0.638449\pi$$
−0.421366 + 0.906891i $$0.638449\pi$$
$$72$$ 2.98028i 0.351229i
$$73$$ − 0.261149i − 0.0305651i −0.999883 0.0152826i $$-0.995135\pi$$
0.999883 0.0152826i $$-0.00486478\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 4.26115 0.488787
$$77$$ − 3.55897i − 0.405582i
$$78$$ 0.401584i 0.0454704i
$$79$$ 8.52230 0.958833 0.479417 0.877587i $$-0.340848\pi$$
0.479417 + 0.877587i $$0.340848\pi$$
$$80$$ 0 0
$$81$$ 8.82289 0.980321
$$82$$ 3.24143i 0.357956i
$$83$$ − 1.16016i − 0.127344i −0.997971 0.0636719i $$-0.979719\pi$$
0.997971 0.0636719i $$-0.0202811\pi$$
$$84$$ −0.160157 −0.0174746
$$85$$ 0 0
$$86$$ 10.6994 1.15375
$$87$$ 0.601186i 0.0644539i
$$88$$ 3.12071i 0.332669i
$$89$$ 13.6430 1.44616 0.723078 0.690766i $$-0.242727\pi$$
0.723078 + 0.690766i $$0.242727\pi$$
$$90$$ 0 0
$$91$$ 3.26115 0.341861
$$92$$ − 1.71913i − 0.179232i
$$93$$ 0.899009i 0.0932229i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0.140435 0.0143331
$$97$$ − 14.2414i − 1.44600i −0.690849 0.722999i $$-0.742763\pi$$
0.690849 0.722999i $$-0.257237\pi$$
$$98$$ − 5.69941i − 0.575727i
$$99$$ 9.30059 0.934745
$$100$$ 0 0
$$101$$ 13.9606 1.38913 0.694564 0.719431i $$-0.255597\pi$$
0.694564 + 0.719431i $$0.255597\pi$$
$$102$$ 0.300593i 0.0297631i
$$103$$ − 10.3621i − 1.02101i −0.859874 0.510506i $$-0.829458\pi$$
0.859874 0.510506i $$-0.170542\pi$$
$$104$$ −2.85956 −0.280403
$$105$$ 0 0
$$106$$ 7.96056 0.773198
$$107$$ − 4.70218i − 0.454577i −0.973828 0.227288i $$-0.927014\pi$$
0.973828 0.227288i $$-0.0729860\pi$$
$$108$$ − 0.839843i − 0.0808139i
$$109$$ 5.96056 0.570918 0.285459 0.958391i $$-0.407854\pi$$
0.285459 + 0.958391i $$0.407854\pi$$
$$110$$ 0 0
$$111$$ 0.140435 0.0133295
$$112$$ − 1.14044i − 0.107761i
$$113$$ − 4.83984i − 0.455294i −0.973744 0.227647i $$-0.926897\pi$$
0.973744 0.227647i $$-0.0731032\pi$$
$$114$$ −0.598416 −0.0560468
$$115$$ 0 0
$$116$$ −4.28087 −0.397469
$$117$$ 8.52230i 0.787887i
$$118$$ 2.69941i 0.248501i
$$119$$ 2.44103 0.223769
$$120$$ 0 0
$$121$$ −1.26115 −0.114650
$$122$$ − 4.57869i − 0.414535i
$$123$$ − 0.455211i − 0.0410450i
$$124$$ −6.40158 −0.574879
$$125$$ 0 0
$$126$$ −3.39881 −0.302790
$$127$$ − 0.899009i − 0.0797741i −0.999204 0.0398871i $$-0.987300\pi$$
0.999204 0.0398871i $$-0.0126998\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −1.50258 −0.132294
$$130$$ 0 0
$$131$$ −0.980278 −0.0856473 −0.0428236 0.999083i $$-0.513635\pi$$
−0.0428236 + 0.999083i $$0.513635\pi$$
$$132$$ − 0.438259i − 0.0381455i
$$133$$ 4.85956i 0.421378i
$$134$$ −10.3819 −0.896856
$$135$$ 0 0
$$136$$ −2.14044 −0.183541
$$137$$ 6.67969i 0.570684i 0.958426 + 0.285342i $$0.0921072\pi$$
−0.958426 + 0.285342i $$0.907893\pi$$
$$138$$ 0.241427i 0.0205516i
$$139$$ 9.68245 0.821255 0.410628 0.911803i $$-0.365310\pi$$
0.410628 + 0.911803i $$0.365310\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 7.10099i 0.595902i
$$143$$ 8.92388i 0.746252i
$$144$$ 2.98028 0.248356
$$145$$ 0 0
$$146$$ −0.261149 −0.0216128
$$147$$ 0.800398i 0.0660157i
$$148$$ 1.00000i 0.0821995i
$$149$$ 1.59842 0.130947 0.0654737 0.997854i $$-0.479144\pi$$
0.0654737 + 0.997854i $$0.479144\pi$$
$$150$$ 0 0
$$151$$ −16.4829 −1.34136 −0.670678 0.741749i $$-0.733997\pi$$
−0.670678 + 0.741749i $$0.733997\pi$$
$$152$$ − 4.26115i − 0.345625i
$$153$$ 6.37909i 0.515719i
$$154$$ −3.55897 −0.286790
$$155$$ 0 0
$$156$$ 0.401584 0.0321525
$$157$$ 10.1207i 0.807721i 0.914821 + 0.403860i $$0.132332\pi$$
−0.914821 + 0.403860i $$0.867668\pi$$
$$158$$ − 8.52230i − 0.677998i
$$159$$ −1.11794 −0.0886587
$$160$$ 0 0
$$161$$ 1.96056 0.154513
$$162$$ − 8.82289i − 0.693192i
$$163$$ 11.7389i 0.919458i 0.888059 + 0.459729i $$0.152053\pi$$
−0.888059 + 0.459729i $$0.847947\pi$$
$$164$$ 3.24143 0.253113
$$165$$ 0 0
$$166$$ −1.16016 −0.0900457
$$167$$ − 10.6430i − 0.823581i −0.911279 0.411790i $$-0.864904\pi$$
0.911279 0.411790i $$-0.135096\pi$$
$$168$$ 0.160157i 0.0123564i
$$169$$ 4.82289 0.370992
$$170$$ 0 0
$$171$$ −12.6994 −0.971148
$$172$$ − 10.6994i − 0.815822i
$$173$$ − 11.3988i − 0.866636i −0.901241 0.433318i $$-0.857343\pi$$
0.901241 0.433318i $$-0.142657\pi$$
$$174$$ 0.601186 0.0455758
$$175$$ 0 0
$$176$$ 3.12071 0.235233
$$177$$ − 0.379092i − 0.0284943i
$$178$$ − 13.6430i − 1.02259i
$$179$$ 18.9211 1.41423 0.707115 0.707098i $$-0.249996\pi$$
0.707115 + 0.707098i $$0.249996\pi$$
$$180$$ 0 0
$$181$$ −18.8032 −1.39763 −0.698814 0.715303i $$-0.746289\pi$$
−0.698814 + 0.715303i $$0.746289\pi$$
$$182$$ − 3.26115i − 0.241732i
$$183$$ 0.643011i 0.0475327i
$$184$$ −1.71913 −0.126736
$$185$$ 0 0
$$186$$ 0.899009 0.0659185
$$187$$ 6.67969i 0.488467i
$$188$$ 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 $$0.316797\pi$$
$$192$$ − 0.140435i − 0.0101350i
$$193$$ 25.3227i 1.82277i 0.411558 + 0.911384i $$0.364985\pi$$
−0.411558 + 0.911384i $$0.635015\pi$$
$$194$$ −14.2414 −1.02247
$$195$$ 0 0
$$196$$ −5.69941 −0.407101
$$197$$ 2.68245i 0.191117i 0.995424 + 0.0955585i $$0.0304637\pi$$
−0.995424 + 0.0955585i $$0.969536\pi$$
$$198$$ − 9.30059i − 0.660964i
$$199$$ −2.24143 −0.158891 −0.0794453 0.996839i $$-0.525315\pi$$
−0.0794453 + 0.996839i $$0.525315\pi$$
$$200$$ 0 0
$$201$$ 1.45798 0.102838
$$202$$ − 13.9606i − 0.982261i
$$203$$ − 4.88206i − 0.342653i
$$204$$ 0.300593 0.0210457
$$205$$ 0 0
$$206$$ −10.3621 −0.721964
$$207$$ 5.12348i 0.356107i
$$208$$ 2.85956i 0.198275i
$$209$$ −13.2978 −0.919830
$$210$$ 0 0
$$211$$ −1.61814 −0.111397 −0.0556986 0.998448i $$-0.517739\pi$$
−0.0556986 + 0.998448i $$0.517739\pi$$
$$212$$ − 7.96056i − 0.546733i
$$213$$ − 0.997230i − 0.0683291i
$$214$$ −4.70218 −0.321434
$$215$$ 0 0
$$216$$ −0.839843 −0.0571440
$$217$$ − 7.30059i − 0.495597i
$$218$$ − 5.96056i − 0.403700i
$$219$$ 0.0366745 0.00247823
$$220$$ 0 0
$$221$$ −6.12071 −0.411724
$$222$$ − 0.140435i − 0.00942540i
$$223$$ − 3.39881i − 0.227601i −0.993504 0.113801i $$-0.963697\pi$$
0.993504 0.113801i $$-0.0363025\pi$$
$$224$$ −1.14044 −0.0761985
$$225$$ 0 0
$$226$$ −4.83984 −0.321942
$$227$$ − 2.45798i − 0.163142i −0.996668 0.0815710i $$-0.974006\pi$$
0.996668 0.0815710i $$-0.0259937\pi$$
$$228$$ 0.598416i 0.0396311i
$$229$$ −12.7637 −0.843451 −0.421725 0.906724i $$-0.638575\pi$$
−0.421725 + 0.906724i $$0.638575\pi$$
$$230$$ 0 0
$$231$$ 0.499806 0.0328848
$$232$$ 4.28087i 0.281053i
$$233$$ − 0.103761i − 0.00679760i −0.999994 0.00339880i $$-0.998918\pi$$
0.999994 0.00339880i $$-0.00108187\pi$$
$$234$$ 8.52230 0.557120
$$235$$ 0 0
$$236$$ 2.69941 0.175716
$$237$$ 1.19683i 0.0777426i
$$238$$ − 2.44103i − 0.158228i
$$239$$ 3.67969 0.238019 0.119010 0.992893i $$-0.462028\pi$$
0.119010 + 0.992893i $$0.462028\pi$$
$$240$$ 0 0
$$241$$ −6.14044 −0.395540 −0.197770 0.980248i $$-0.563370\pi$$
−0.197770 + 0.980248i $$0.563370\pi$$
$$242$$ 1.26115i 0.0810697i
$$243$$ 3.75857i 0.241113i
$$244$$ −4.57869 −0.293121
$$245$$ 0 0
$$246$$ −0.455211 −0.0290232
$$247$$ − 12.1850i − 0.775315i
$$248$$ 6.40158i 0.406501i
$$249$$ 0.162927 0.0103251
$$250$$ 0 0
$$251$$ −9.00000 −0.568075 −0.284037 0.958813i $$-0.591674\pi$$
−0.284037 + 0.958813i $$0.591674\pi$$
$$252$$ 3.39881i 0.214105i
$$253$$ 5.36491i 0.337289i
$$254$$ −0.899009 −0.0564088
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 16.5223i 1.03063i 0.857000 + 0.515316i $$0.172326\pi$$
−0.857000 + 0.515316i $$0.827674\pi$$
$$258$$ 1.50258i 0.0935462i
$$259$$ −1.14044 −0.0708632
$$260$$ 0 0
$$261$$ 12.7582 0.789712
$$262$$ 0.980278i 0.0605618i
$$263$$ − 22.5223i − 1.38878i −0.719597 0.694392i $$-0.755673\pi$$
0.719597 0.694392i $$-0.244327\pi$$
$$264$$ −0.438259 −0.0269729
$$265$$ 0 0
$$266$$ 4.85956 0.297959
$$267$$ 1.91596i 0.117255i
$$268$$ 10.3819i 0.634173i
$$269$$ −27.2860 −1.66366 −0.831829 0.555032i $$-0.812706\pi$$
−0.831829 + 0.555032i $$0.812706\pi$$
$$270$$ 0 0
$$271$$ −26.1456 −1.58823 −0.794116 0.607767i $$-0.792066\pi$$
−0.794116 + 0.607767i $$0.792066\pi$$
$$272$$ 2.14044i 0.129783i
$$273$$ 0.457981i 0.0277182i
$$274$$ 6.67969 0.403535
$$275$$ 0 0
$$276$$ 0.241427 0.0145322
$$277$$ 20.5223i 1.23307i 0.787329 + 0.616533i $$0.211463\pi$$
−0.787329 + 0.616533i $$0.788537\pi$$
$$278$$ − 9.68245i − 0.580715i
$$279$$ 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
$$283$$ 6.85679i 0.407594i 0.979013 + 0.203797i $$0.0653283\pi$$
−0.979013 + 0.203797i $$0.934672\pi$$
$$284$$ 7.10099 0.421366
$$285$$ 0 0
$$286$$ 8.92388 0.527680
$$287$$ 3.69664i 0.218206i
$$288$$ − 2.98028i − 0.175615i
$$289$$ 12.4185 0.730502
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0.261149i 0.0152826i
$$293$$ − 19.2048i − 1.12195i −0.827832 0.560977i $$-0.810426\pi$$
0.827832 0.560977i $$-0.189574\pi$$
$$294$$ 0.800398 0.0466802
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 2.62091i 0.152080i
$$298$$ − 1.59842i − 0.0925938i
$$299$$ −4.91596 −0.284297
$$300$$ 0 0
$$301$$ 12.2020 0.703311
$$302$$ 16.4829i 0.948482i
$$303$$ 1.96056i 0.112631i
$$304$$ −4.26115 −0.244394
$$305$$ 0 0
$$306$$ 6.37909 0.364668
$$307$$ − 16.2781i − 0.929040i −0.885563 0.464520i $$-0.846227\pi$$
0.885563 0.464520i $$-0.153773\pi$$
$$308$$ 3.55897i 0.202791i
$$309$$ 1.45521 0.0827841
$$310$$ 0 0
$$311$$ −25.4801 −1.44484 −0.722421 0.691453i $$-0.756971\pi$$
−0.722421 + 0.691453i $$0.756971\pi$$
$$312$$ − 0.401584i − 0.0227352i
$$313$$ 19.0197i 1.07506i 0.843245 + 0.537529i $$0.180642\pi$$
−0.843245 + 0.537529i $$0.819358\pi$$
$$314$$ 10.1207 0.571145
$$315$$ 0 0
$$316$$ −8.52230 −0.479417
$$317$$ − 15.1653i − 0.851769i −0.904778 0.425884i $$-0.859963\pi$$
0.904778 0.425884i $$-0.140037\pi$$
$$318$$ 1.11794i 0.0626912i
$$319$$ 13.3594 0.747981
$$320$$ 0 0
$$321$$ 0.660352 0.0368573
$$322$$ − 1.96056i − 0.109258i
$$323$$ − 9.12071i − 0.507490i
$$324$$ −8.82289 −0.490161
$$325$$ 0 0
$$326$$ 11.7389 0.650155
$$327$$ 0.837073i 0.0462902i
$$328$$ − 3.24143i − 0.178978i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −23.4185 −1.28720 −0.643600 0.765362i $$-0.722560\pi$$
−0.643600 + 0.765362i $$0.722560\pi$$
$$332$$ 1.16016i 0.0636719i
$$333$$ − 2.98028i − 0.163318i
$$334$$ −10.6430 −0.582360
$$335$$ 0 0
$$336$$ 0.160157 0.00873731
$$337$$ 28.9211i 1.57543i 0.616038 + 0.787717i $$0.288737\pi$$
−0.616038 + 0.787717i $$0.711263\pi$$
$$338$$ − 4.82289i − 0.262331i
$$339$$ 0.679685 0.0369154
$$340$$ 0 0
$$341$$ 19.9775 1.08184
$$342$$ 12.6994i 0.686705i
$$343$$ − 14.4829i − 0.782001i
$$344$$ −10.6994 −0.576873
$$345$$ 0 0
$$346$$ −11.3988 −0.612804
$$347$$ 32.8817i 1.76518i 0.470143 + 0.882590i $$0.344202\pi$$
−0.470143 + 0.882590i $$0.655798\pi$$
$$348$$ − 0.601186i − 0.0322269i
$$349$$ −15.2048 −0.813892 −0.406946 0.913452i $$-0.633406\pi$$
−0.406946 + 0.913452i $$0.633406\pi$$
$$350$$ 0 0
$$351$$ −2.40158 −0.128187
$$352$$ − 3.12071i − 0.166335i
$$353$$ − 18.4829i − 0.983743i −0.870668 0.491872i $$-0.836313\pi$$
0.870668 0.491872i $$-0.163687\pi$$
$$354$$ −0.379092 −0.0201485
$$355$$ 0 0
$$356$$ −13.6430 −0.723078
$$357$$ 0.342807i 0.0181433i
$$358$$ − 18.9211i − 1.00001i
$$359$$ −22.5223 −1.18868 −0.594341 0.804213i $$-0.702587\pi$$
−0.594341 + 0.804213i $$0.702587\pi$$
$$360$$ 0 0
$$361$$ −0.842612 −0.0443480
$$362$$ 18.8032i 0.988273i
$$363$$ − 0.177110i − 0.00929586i
$$364$$ −3.26115 −0.170931
$$365$$ 0 0
$$366$$ 0.643011 0.0336107
$$367$$ − 15.3424i − 0.800868i −0.916326 0.400434i $$-0.868859\pi$$
0.916326 0.400434i $$-0.131141\pi$$
$$368$$ 1.71913i 0.0896158i
$$369$$ −9.66035 −0.502898
$$370$$ 0 0
$$371$$ 9.07850 0.471332
$$372$$ − 0.899009i − 0.0466114i
$$373$$ 12.6430i 0.654630i 0.944915 + 0.327315i $$0.106144\pi$$
−0.944915 + 0.327315i $$0.893856\pi$$
$$374$$ 6.67969 0.345398
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.2414i 0.630466i
$$378$$ − 0.957786i − 0.0492632i
$$379$$ −15.7807 −0.810599 −0.405299 0.914184i $$-0.632833\pi$$
−0.405299 + 0.914184i $$0.632833\pi$$
$$380$$ 0 0
$$381$$ 0.126253 0.00646812
$$382$$ − 15.0446i − 0.769748i
$$383$$ − 27.8004i − 1.42053i −0.703932 0.710267i $$-0.748574\pi$$
0.703932 0.710267i $$-0.251426\pi$$
$$384$$ −0.140435 −0.00716656
$$385$$ 0 0
$$386$$ 25.3227 1.28889
$$387$$ 31.8872i 1.62092i
$$388$$ 14.2414i 0.722999i
$$389$$ −9.66273 −0.489920 −0.244960 0.969533i $$-0.578775\pi$$
−0.244960 + 0.969533i $$0.578775\pi$$
$$390$$ 0 0
$$391$$ −3.67969 −0.186090
$$392$$ 5.69941i 0.287864i
$$393$$ − 0.137666i − 0.00694432i
$$394$$ 2.68245 0.135140
$$395$$ 0 0
$$396$$ −9.30059 −0.467372
$$397$$ 23.4801i 1.17843i 0.807976 + 0.589216i $$0.200563\pi$$
−0.807976 + 0.589216i $$0.799437\pi$$
$$398$$ 2.24143i 0.112353i
$$399$$ −0.682455 −0.0341655
$$400$$ 0 0
$$401$$ 20.2781 1.01264 0.506320 0.862346i $$-0.331005\pi$$
0.506320 + 0.862346i $$0.331005\pi$$
$$402$$ − 1.45798i − 0.0727175i
$$403$$ 18.3057i 0.911874i
$$404$$ −13.9606 −0.694564
$$405$$ 0 0
$$406$$ −4.88206 −0.242292
$$407$$ − 3.12071i − 0.154688i
$$408$$ − 0.300593i − 0.0148816i
$$409$$ −4.17434 −0.206408 −0.103204 0.994660i $$-0.532909\pi$$
−0.103204 + 0.994660i $$0.532909\pi$$
$$410$$ 0 0
$$411$$ −0.938064 −0.0462713
$$412$$ 10.3621i 0.510506i
$$413$$ 3.07850i 0.151483i
$$414$$ 5.12348 0.251805
$$415$$ 0 0
$$416$$ 2.85956 0.140202
$$417$$ 1.35976i 0.0665877i
$$418$$ 13.2978i 0.650418i
$$419$$ 12.3175 0.601751 0.300876 0.953663i $$-0.402721\pi$$
0.300876 + 0.953663i $$0.402721\pi$$
$$420$$ 0 0
$$421$$ 23.8872 1.16419 0.582096 0.813120i $$-0.302233\pi$$
0.582096 + 0.813120i $$0.302233\pi$$
$$422$$ 1.61814i 0.0787697i
$$423$$ 0 0
$$424$$ −7.96056 −0.386599
$$425$$ 0 0
$$426$$ −0.997230 −0.0483160
$$427$$ − 5.22170i − 0.252696i
$$428$$ 4.70218i 0.227288i
$$429$$ −1.25323 −0.0605064
$$430$$ 0 0
$$431$$ −2.80317 −0.135024 −0.0675119 0.997718i $$-0.521506\pi$$
−0.0675119 + 0.997718i $$0.521506\pi$$
$$432$$ 0.839843i 0.0404069i
$$433$$ 5.00000i 0.240285i 0.992757 + 0.120142i $$0.0383351\pi$$
−0.992757 + 0.120142i $$0.961665\pi$$
$$434$$ −7.30059 −0.350440
$$435$$ 0 0
$$436$$ −5.96056 −0.285459
$$437$$ − 7.32547i − 0.350425i
$$438$$ − 0.0366745i − 0.00175238i
$$439$$ −30.7976 −1.46989 −0.734945 0.678126i $$-0.762792\pi$$
−0.734945 + 0.678126i $$0.762792\pi$$
$$440$$ 0 0
$$441$$ 16.9858 0.808848
$$442$$ 6.12071i 0.291133i
$$443$$ − 16.5984i − 0.788615i −0.918979 0.394307i $$-0.870985\pi$$
0.918979 0.394307i $$-0.129015\pi$$
$$444$$ −0.140435 −0.00666477
$$445$$ 0 0
$$446$$ −3.39881 −0.160939
$$447$$ 0.224474i 0.0106173i
$$448$$ 1.14044i 0.0538805i
$$449$$ −9.75580 −0.460405 −0.230202 0.973143i $$-0.573939\pi$$
−0.230202 + 0.973143i $$0.573939\pi$$
$$450$$ 0 0
$$451$$ −10.1156 −0.476323
$$452$$ 4.83984i 0.227647i
$$453$$ − 2.31478i − 0.108758i
$$454$$ −2.45798 −0.115359
$$455$$ 0 0
$$456$$ 0.598416 0.0280234
$$457$$ 7.99723i 0.374095i 0.982351 + 0.187047i $$0.0598918\pi$$
−0.982351 + 0.187047i $$0.940108\pi$$
$$458$$ 12.7637i 0.596410i
$$459$$ −1.79763 −0.0839061
$$460$$ 0 0
$$461$$ −20.6848 −0.963389 −0.481694 0.876339i $$-0.659978\pi$$
−0.481694 + 0.876339i $$0.659978\pi$$
$$462$$ − 0.499806i − 0.0232531i
$$463$$ − 7.52507i − 0.349720i −0.984593 0.174860i $$-0.944053\pi$$
0.984593 0.174860i $$-0.0559472\pi$$
$$464$$ 4.28087 0.198734
$$465$$ 0 0
$$466$$ −0.103761 −0.00480663
$$467$$ − 6.84261i − 0.316638i −0.987388 0.158319i $$-0.949393\pi$$
0.987388 0.158319i $$-0.0506075\pi$$
$$468$$ − 8.52230i − 0.393943i
$$469$$ −11.8398 −0.546713
$$470$$ 0 0
$$471$$ −1.42131 −0.0654903
$$472$$ − 2.69941i − 0.124250i
$$473$$ 33.3898i 1.53526i
$$474$$ 1.19683 0.0549723
$$475$$ 0 0
$$476$$ −2.44103 −0.111884
$$477$$ 23.7247i 1.08628i
$$478$$ − 3.67969i − 0.168305i
$$479$$ 16.4016 0.749408 0.374704 0.927145i $$-0.377744\pi$$
0.374704 + 0.927145i $$0.377744\pi$$
$$480$$ 0 0
$$481$$ 2.85956 0.130385
$$482$$ 6.14044i 0.279689i
$$483$$ 0.275331i 0.0125280i
$$484$$ 1.26115 0.0573249
$$485$$ 0 0
$$486$$ 3.75857 0.170492
$$487$$ − 13.7270i − 0.622032i −0.950405 0.311016i $$-0.899331\pi$$
0.950405 0.311016i $$-0.100669\pi$$
$$488$$ 4.57869i 0.207268i
$$489$$ −1.64855 −0.0745500
$$490$$ 0 0
$$491$$ 3.40435 0.153636 0.0768182 0.997045i $$-0.475524\pi$$
0.0768182 + 0.997045i $$0.475524\pi$$
$$492$$ 0.455211i 0.0205225i
$$493$$ 9.16293i 0.412677i
$$494$$ −12.1850 −0.548230
$$495$$ 0 0
$$496$$ 6.40158 0.287440
$$497$$ 8.09822i 0.363255i
$$498$$ − 0.162927i − 0.00730094i
$$499$$ 29.4631 1.31895 0.659475 0.751726i $$-0.270778\pi$$
0.659475 + 0.751726i $$0.270778\pi$$
$$500$$ 0 0
$$501$$ 1.49466 0.0667763
$$502$$ 9.00000i 0.401690i
$$503$$ − 24.0000i − 1.07011i −0.844818 0.535054i $$-0.820291\pi$$
0.844818 0.535054i $$-0.179709\pi$$
$$504$$ 3.39881 0.151395
$$505$$ 0 0
$$506$$ 5.36491 0.238499
$$507$$ 0.677304i 0.0300801i
$$508$$ 0.899009i 0.0398871i
$$509$$ −12.9633 −0.574589 −0.287295 0.957842i $$-0.592756\pi$$
−0.287295 + 0.957842i $$0.592756\pi$$
$$510$$ 0 0
$$511$$ −0.297823 −0.0131749
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 3.57869i − 0.158003i
$$514$$ 16.5223 0.728767
$$515$$ 0 0
$$516$$ 1.50258 0.0661472
$$517$$ 0 0
$$518$$ 1.14044i 0.0501079i
$$519$$ 1.60080 0.0702672
$$520$$ 0 0
$$521$$ −2.01972 −0.0884856 −0.0442428 0.999021i $$-0.514088\pi$$
−0.0442428 + 0.999021i $$0.514088\pi$$
$$522$$ − 12.7582i − 0.558411i
$$523$$ 15.5562i 0.680225i 0.940385 + 0.340113i $$0.110465\pi$$
−0.940385 + 0.340113i $$0.889535\pi$$
$$524$$ 0.980278 0.0428236
$$525$$ 0 0
$$526$$ −22.5223 −0.982019
$$527$$ 13.7022i 0.596876i
$$528$$ 0.438259i 0.0190728i
$$529$$ 20.0446 0.871504
$$530$$ 0 0
$$531$$ −8.04498 −0.349123
$$532$$ − 4.85956i − 0.210689i
$$533$$ − 9.26907i − 0.401488i
$$534$$ 1.91596 0.0829118
$$535$$ 0 0
$$536$$ 10.3819 0.448428
$$537$$ 2.65719i 0.114666i
$$538$$ 27.2860i 1.17638i
$$539$$ 17.7862 0.766107
$$540$$ 0 0
$$541$$ −0.544789 −0.0234223 −0.0117112 0.999931i $$-0.503728\pi$$
−0.0117112 + 0.999931i $$0.503728\pi$$
$$542$$ 26.1456i 1.12305i
$$543$$ − 2.64063i − 0.113320i
$$544$$ 2.14044 0.0917704
$$545$$ 0 0
$$546$$ 0.457981 0.0195998
$$547$$ 35.6261i 1.52326i 0.648012 + 0.761630i $$0.275601\pi$$
−0.648012 + 0.761630i $$0.724399\pi$$
$$548$$ − 6.67969i − 0.285342i
$$549$$ 13.6458 0.582388
$$550$$ 0 0
$$551$$ −18.2414 −0.777111
$$552$$ − 0.241427i − 0.0102758i
$$553$$ − 9.71913i − 0.413299i
$$554$$ 20.5223 0.871909
$$555$$ 0 0
$$556$$ −9.68245 −0.410628
$$557$$ − 31.9606i − 1.35421i −0.735885 0.677106i $$-0.763234\pi$$
0.735885 0.677106i $$-0.236766\pi$$
$$558$$ − 19.0785i − 0.807657i
$$559$$ −30.5956 −1.29406
$$560$$ 0 0
$$561$$ −0.938064 −0.0396051
$$562$$ 14.4580i 0.609873i
$$563$$ 14.6655i 0.618077i 0.951049 + 0.309039i $$0.100007\pi$$
−0.951049 + 0.309039i $$0.899993\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 6.85679 0.288213
$$567$$ − 10.0619i − 0.422562i
$$568$$ − 7.10099i − 0.297951i
$$569$$ −9.99723 −0.419106 −0.209553 0.977797i $$-0.567201\pi$$
−0.209553 + 0.977797i $$0.567201\pi$$
$$570$$ 0 0
$$571$$ 27.6008 1.15506 0.577529 0.816370i $$-0.304017\pi$$
0.577529 + 0.816370i $$0.304017\pi$$
$$572$$ − 8.92388i − 0.373126i
$$573$$ 2.11279i 0.0882632i
$$574$$ 3.69664 0.154295
$$575$$ 0 0
$$576$$ −2.98028 −0.124178
$$577$$ 3.78622i 0.157622i 0.996890 + 0.0788111i $$0.0251124\pi$$
−0.996890 + 0.0788111i $$0.974888\pi$$
$$578$$ − 12.4185i − 0.516543i
$$579$$ −3.55620 −0.147791
$$580$$ 0 0
$$581$$ −1.32308 −0.0548908
$$582$$ − 2.00000i − 0.0829027i
$$583$$ 24.8426i 1.02888i
$$584$$ 0.261149 0.0108064
$$585$$ 0 0
$$586$$ −19.2048 −0.793341
$$587$$ − 19.1771i − 0.791524i −0.918353 0.395762i $$-0.870481\pi$$
0.918353 0.395762i $$-0.129519\pi$$
$$588$$ − 0.800398i − 0.0330079i
$$589$$ −27.2781 −1.12397
$$590$$ 0 0
$$591$$ −0.376712 −0.0154958
$$592$$ − 1.00000i − 0.0410997i
$$593$$ − 12.4383i − 0.510778i −0.966838 0.255389i $$-0.917796\pi$$
0.966838 0.255389i $$-0.0822035\pi$$
$$594$$ 2.62091 0.107537
$$595$$ 0 0
$$596$$ −1.59842 −0.0654737
$$597$$ − 0.314776i − 0.0128829i
$$598$$ 4.91596i 0.201029i
$$599$$ 35.3819 1.44566 0.722832 0.691024i $$-0.242840\pi$$
0.722832 + 0.691024i $$0.242840\pi$$
$$600$$ 0 0
$$601$$ 22.2951 0.909434 0.454717 0.890636i $$-0.349740\pi$$
0.454717 + 0.890636i $$0.349740\pi$$
$$602$$ − 12.2020i − 0.497316i
$$603$$ − 30.9408i − 1.26001i
$$604$$ 16.4829 0.670678
$$605$$ 0 0
$$606$$ 1.96056 0.0796421
$$607$$ 9.72705i 0.394809i 0.980322 + 0.197404i $$0.0632512\pi$$
−0.980322 + 0.197404i $$0.936749\pi$$
$$608$$ 4.26115i 0.172812i
$$609$$ 0.685613 0.0277825
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.37909i − 0.257860i
$$613$$ 10.3203i 0.416834i 0.978040 + 0.208417i $$0.0668310\pi$$
−0.978040 + 0.208417i $$0.933169\pi$$
$$614$$ −16.2781 −0.656931
$$615$$ 0 0
$$616$$ 3.55897 0.143395
$$617$$ 3.78345i 0.152316i 0.997096 + 0.0761579i $$0.0242653\pi$$
−0.997096 + 0.0761579i $$0.975735\pi$$
$$618$$ − 1.45521i − 0.0585372i
$$619$$ −27.1179 −1.08996 −0.544981 0.838448i $$-0.683463\pi$$
−0.544981 + 0.838448i $$0.683463\pi$$
$$620$$ 0 0
$$621$$ −1.44380 −0.0579376
$$622$$ 25.4801i 1.02166i
$$623$$ − 15.5590i − 0.623357i
$$624$$ −0.401584 −0.0160762
$$625$$ 0 0
$$626$$ 19.0197 0.760181
$$627$$ − 1.86748i − 0.0745802i
$$628$$ − 10.1207i − 0.403860i
$$629$$ 2.14044 0.0853447
$$630$$ 0 0
$$631$$ −11.1179 −0.442598 −0.221299 0.975206i $$-0.571030\pi$$
−0.221299 + 0.975206i $$0.571030\pi$$
$$632$$ 8.52230i 0.338999i
$$633$$ − 0.227244i − 0.00903213i
$$634$$ −15.1653 −0.602291
$$635$$ 0 0
$$636$$ 1.11794 0.0443293
$$637$$ 16.2978i 0.645743i
$$638$$ − 13.3594i − 0.528903i
$$639$$ −21.1629 −0.837192
$$640$$ 0 0
$$641$$ 27.0446 1.06820 0.534099 0.845422i $$-0.320651\pi$$
0.534099 + 0.845422i $$0.320651\pi$$
$$642$$ − 0.660352i − 0.0260620i
$$643$$ 29.7834i 1.17454i 0.809389 + 0.587272i $$0.199798\pi$$
−0.809389 + 0.587272i $$0.800202\pi$$
$$644$$ −1.96056 −0.0772567
$$645$$ 0 0
$$646$$ −9.12071 −0.358850
$$647$$ − 19.3570i − 0.761002i −0.924781 0.380501i $$-0.875752\pi$$
0.924781 0.380501i $$-0.124248\pi$$
$$648$$ 8.82289i 0.346596i
$$649$$ −8.42408 −0.330674
$$650$$ 0 0
$$651$$ 1.02526 0.0401832
$$652$$ − 11.7389i − 0.459729i
$$653$$ − 10.5392i − 0.412433i −0.978506 0.206216i $$-0.933885\pi$$
0.978506 0.206216i $$-0.0661151\pi$$
$$654$$ 0.837073 0.0327321
$$655$$ 0 0
$$656$$ −3.24143 −0.126556
$$657$$ − 0.778296i − 0.0303642i
$$658$$ 0 0
$$659$$ −9.22447 −0.359334 −0.179667 0.983727i $$-0.557502\pi$$
−0.179667 + 0.983727i $$0.557502\pi$$
$$660$$ 0 0
$$661$$ 27.3594 1.06416 0.532078 0.846695i $$-0.321411\pi$$
0.532078 + 0.846695i $$0.321411\pi$$
$$662$$ 23.4185i 0.910187i
$$663$$ − 0.859565i − 0.0333827i
$$664$$ 1.16016 0.0450228
$$665$$ 0 0
$$666$$ −2.98028 −0.115483
$$667$$ 7.35937i 0.284956i
$$668$$ 10.6430i 0.411790i
$$669$$ 0.477314 0.0184540
$$670$$ 0 0
$$671$$ 14.2888 0.551613
$$672$$ − 0.160157i − 0.00617821i
$$673$$ − 13.1484i − 0.506832i −0.967357 0.253416i $$-0.918446\pi$$
0.967357 0.253416i $$-0.0815541\pi$$
$$674$$ 28.9211 1.11400
$$675$$ 0 0
$$676$$ −4.82289 −0.185496
$$677$$ − 29.8817i − 1.14845i −0.818699 0.574223i $$-0.805304\pi$$
0.818699 0.574223i $$-0.194696\pi$$
$$678$$ − 0.679685i − 0.0261031i
$$679$$ −16.2414 −0.623289
$$680$$ 0 0
$$681$$ 0.345187 0.0132276
$$682$$ − 19.9775i − 0.764978i
$$683$$ − 26.9854i − 1.03257i −0.856417 0.516284i $$-0.827315\pi$$
0.856417 0.516284i $$-0.172685\pi$$
$$684$$ 12.6994 0.485574
$$685$$ 0 0
$$686$$ −14.4829 −0.552958
$$687$$ − 1.79248i − 0.0683873i
$$688$$ 10.6994i 0.407911i
$$689$$ −22.7637 −0.867229
$$690$$ 0 0
$$691$$ −30.7270 −1.16891 −0.584456 0.811425i $$-0.698692\pi$$
−0.584456 + 0.811425i $$0.698692\pi$$
$$692$$ 11.3988i 0.433318i
$$693$$ − 10.6067i − 0.402916i
$$694$$ 32.8817 1.24817
$$695$$ 0 0
$$696$$ −0.601186 −0.0227879
$$697$$ − 6.93806i − 0.262798i
$$698$$ 15.2048i 0.575508i
$$699$$ 0.0145717 0.000551152 0
$$700$$ 0 0
$$701$$ −27.7858 −1.04946 −0.524728 0.851270i $$-0.675833\pi$$
−0.524728 + 0.851270i $$0.675833\pi$$
$$702$$ 2.40158i 0.0906419i
$$703$$ 4.26115i 0.160712i
$$704$$ −3.12071 −0.117616
$$705$$ 0 0
$$706$$ −18.4829 −0.695611
$$707$$ − 15.9211i − 0.598775i
$$708$$ 0.379092i 0.0142472i
$$709$$ −9.71913 −0.365010 −0.182505 0.983205i $$-0.558421\pi$$
−0.182505 + 0.983205i $$0.558421\pi$$
$$710$$ 0 0
$$711$$ 25.3988 0.952530
$$712$$ 13.6430i 0.511293i
$$713$$ 11.0052i 0.412146i
$$714$$ 0.342807 0.0128292
$$715$$ 0 0
$$716$$ −18.9211 −0.707115
$$717$$ 0.516758i 0.0192987i
$$718$$ 22.5223i 0.840525i
$$719$$ −5.41577 −0.201974 −0.100987 0.994888i $$-0.532200\pi$$
−0.100987 + 0.994888i $$0.532200\pi$$
$$720$$ 0 0
$$721$$ −11.8174 −0.440101
$$722$$ 0.842612i 0.0313588i
$$723$$ − 0.862334i − 0.0320706i
$$724$$ 18.8032 0.698814
$$725$$ 0 0
$$726$$ −0.177110 −0.00657316
$$727$$ − 13.7586i − 0.510277i −0.966904 0.255139i $$-0.917879\pi$$
0.966904 0.255139i $$-0.0821211\pi$$
$$728$$ 3.26115i 0.120866i
$$729$$ 25.9408 0.960772
$$730$$ 0 0
$$731$$ −22.9014 −0.847038
$$732$$ − 0.643011i − 0.0237664i
$$733$$ 52.4095i 1.93579i 0.251356 + 0.967895i $$0.419123\pi$$
−0.251356 + 0.967895i $$0.580877\pi$$
$$734$$ −15.3424 −0.566299
$$735$$ 0 0
$$736$$ 1.71913 0.0633679
$$737$$ − 32.3988i − 1.19343i
$$738$$ 9.66035i 0.355602i
$$739$$ −50.0892 −1.84256 −0.921280 0.388899i $$-0.872855\pi$$
−0.921280 + 0.388899i $$0.872855\pi$$
$$740$$ 0 0
$$741$$ 1.71121 0.0628628
$$742$$ − 9.07850i − 0.333282i
$$743$$ − 40.7976i − 1.49672i −0.663293 0.748360i $$-0.730842\pi$$
0.663293 0.748360i $$-0.269158\pi$$
$$744$$ −0.899009 −0.0329593
$$745$$ 0 0
$$746$$ 12.6430 0.462894
$$747$$ − 3.45759i − 0.126507i
$$748$$ − 6.67969i − 0.244233i
$$749$$ −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
$$753$$ − 1.26392i − 0.0460597i
$$754$$ 12.2414 0.445806
$$755$$ 0 0
$$756$$ −0.957786 −0.0348343
$$757$$ − 11.4383i − 0.415731i −0.978157 0.207865i $$-0.933348\pi$$
0.978157 0.207865i $$-0.0666516\pi$$
$$758$$ 15.7807i 0.573180i
$$759$$ −0.753423 −0.0273475
$$760$$ 0 0
$$761$$ 43.2663 1.56840 0.784201 0.620507i $$-0.213073\pi$$
0.784201 + 0.620507i $$0.213073\pi$$
$$762$$ − 0.126253i − 0.00457365i
$$763$$ − 6.79763i − 0.246091i
$$764$$ −15.0446 −0.544294
$$765$$ 0 0
$$766$$ −27.8004 −1.00447
$$767$$ − 7.71913i − 0.278722i
$$768$$ 0.140435i 0.00506752i
$$769$$ −26.5590 −0.957741 −0.478871 0.877886i $$-0.658954\pi$$
−0.478871 + 0.877886i $$0.658954\pi$$
$$770$$ 0 0
$$771$$ −2.32031 −0.0835641
$$772$$ − 25.3227i − 0.911384i
$$773$$ 12.3621i 0.444635i 0.974974 + 0.222318i $$0.0713622\pi$$
−0.974974 + 0.222318i $$0.928638\pi$$
$$774$$ 31.8872 1.14616
$$775$$ 0 0
$$776$$ 14.2414 0.511237
$$777$$ − 0.160157i − 0.00574562i
$$778$$ 9.66273i 0.346426i
$$779$$ 13.8122 0.494873
$$780$$ 0 0
$$781$$ −22.1602 −0.792953
$$782$$ 3.67969i 0.131585i
$$783$$ 3.59526i 0.128484i
$$784$$ 5.69941 0.203550
$$785$$ 0 0
$$786$$ −0.137666 −0.00491037
$$787$$ − 6.03944i − 0.215283i −0.994190 0.107641i $$-0.965670\pi$$
0.994190 0.107641i $$-0.0343299\pi$$
$$788$$ − 2.68245i − 0.0955585i
$$789$$ 3.16293 0.112603
$$790$$ 0 0
$$791$$ −5.51953 −0.196252
$$792$$ 9.30059i 0.330482i
$$793$$ 13.0931i 0.464949i
$$794$$ 23.4801 0.833277
$$795$$ 0 0
$$796$$ 2.24143 0.0794453
$$797$$ − 35.6233i − 1.26184i −0.775847 0.630921i $$-0.782677\pi$$
0.775847 0.630921i $$-0.217323\pi$$
$$798$$ 0.682455i 0.0241586i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 40.6600 1.43665
$$802$$ − 20.2781i − 0.716045i
$$803$$ − 0.814970i − 0.0287597i
$$804$$ −1.45798 −0.0514190
$$805$$ 0 0
$$806$$ 18.3057 0.644792
$$807$$ − 3.83192i − 0.134890i
$$808$$ 13.9606i 0.491131i
$$809$$ 23.6402 0.831147 0.415573 0.909560i $$-0.363581\pi$$
0.415573 + 0.909560i $$0.363581\pi$$
$$810$$ 0 0
$$811$$ −3.27572 −0.115026 −0.0575130 0.998345i $$-0.518317\pi$$
−0.0575130 + 0.998345i $$0.518317\pi$$
$$812$$ 4.88206i 0.171327i
$$813$$ − 3.67176i − 0.128774i
$$814$$ −3.12071 −0.109381
$$815$$ 0 0
$$816$$ −0.300593 −0.0105229
$$817$$ − 45.5918i − 1.59505i
$$818$$ 4.17434i 0.145952i
$$819$$ 9.71913 0.339614
$$820$$ 0 0
$$821$$ −30.3621 −1.05965 −0.529823 0.848108i $$-0.677742\pi$$
−0.529823 + 0.848108i $$0.677742\pi$$
$$822$$ 0.938064i 0.0327187i
$$823$$ − 14.2978i − 0.498391i −0.968453 0.249195i $$-0.919834\pi$$
0.968453 0.249195i $$-0.0801661\pi$$
$$824$$ 10.3621 0.360982
$$825$$ 0 0
$$826$$ 3.07850 0.107115
$$827$$ − 7.90139i − 0.274758i −0.990519 0.137379i $$-0.956132\pi$$
0.990519 0.137379i $$-0.0438679\pi$$
$$828$$ − 5.12348i − 0.178053i
$$829$$ 43.3030 1.50397 0.751987 0.659178i $$-0.229095\pi$$
0.751987 + 0.659178i $$0.229095\pi$$
$$830$$ 0 0
$$831$$ −2.88206 −0.0999774
$$832$$ − 2.85956i − 0.0991376i
$$833$$ 12.1992i 0.422678i
$$834$$ 1.35976 0.0470846
$$835$$ 0 0
$$836$$ 13.2978 0.459915
$$837$$ 5.37632i 0.185833i
$$838$$ − 12.3175i − 0.425503i
$$839$$ −11.3819 −0.392946 −0.196473 0.980509i $$-0.562949\pi$$
−0.196473 + 0.980509i $$0.562949\pi$$
$$840$$ 0 0
$$841$$ −10.6741 −0.368074
$$842$$ − 23.8872i − 0.823208i
$$843$$ − 2.03041i − 0.0699311i
$$844$$ 1.61814 0.0556986
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.43826i 0.0494191i
$$848$$ 7.96056i 0.273367i
$$849$$ −0.962937 −0.0330479
$$850$$ 0 0
$$851$$ 1.71913 0.0589310
$$852$$ 0.997230i 0.0341645i
$$853$$ 10.7862i 0.369313i 0.982803 + 0.184656i $$0.0591173\pi$$
−0.982803 + 0.184656i $$0.940883\pi$$
$$854$$ −5.22170 −0.178683
$$855$$ 0 0
$$856$$ 4.70218 0.160717
$$857$$ 2.38186i 0.0813629i 0.999172 + 0.0406814i $$0.0129529\pi$$
−0.999172 + 0.0406814i $$0.987047\pi$$
$$858$$ 1.25323i 0.0427845i
$$859$$ 48.1428 1.64261 0.821306 0.570488i $$-0.193246\pi$$
0.821306 + 0.570488i $$0.193246\pi$$
$$860$$ 0 0
$$861$$ −0.519139 −0.0176922
$$862$$ 2.80317i 0.0954763i
$$863$$ 29.8308i 1.01545i 0.861518 + 0.507726i $$0.169514\pi$$
−0.861518 + 0.507726i $$0.830486\pi$$
$$864$$ 0.839843 0.0285720
$$865$$ 0 0
$$866$$ 5.00000 0.169907
$$867$$ 1.74400i 0.0592294i
$$868$$ 7.30059i 0.247798i
$$869$$ 26.5956 0.902196
$$870$$ 0 0
$$871$$ 29.6876 1.00593
$$872$$ 5.96056i 0.201850i
$$873$$ − 42.4434i − 1.43649i
$$874$$ −7.32547 −0.247788
$$875$$ 0 0
$$876$$ −0.0366745 −0.00123912
$$877$$ 48.9578i 1.65319i 0.562799 + 0.826593i $$0.309724\pi$$
−0.562799 + 0.826593i $$0.690276\pi$$
$$878$$ 30.7976i 1.03937i
$$879$$ 2.69703 0.0909684
$$880$$ 0 0
$$881$$ 3.50811 0.118191 0.0590957 0.998252i $$-0.481178\pi$$
0.0590957 + 0.998252i $$0.481178\pi$$
$$882$$ − 16.9858i − 0.571942i
$$883$$ − 48.8817i − 1.64500i −0.568766 0.822500i $$-0.692579\pi$$
0.568766 0.822500i $$-0.307421\pi$$
$$884$$ 6.12071 0.205862
$$885$$ 0 0
$$886$$ −16.5984 −0.557635
$$887$$ − 46.3701i − 1.55695i −0.627673 0.778477i $$-0.715992\pi$$
0.627673 0.778477i $$-0.284008\pi$$
$$888$$ 0.140435i 0.00471270i
$$889$$ −1.02526 −0.0343862
$$890$$ 0 0
$$891$$ 27.5337 0.922414
$$892$$ 3.39881i 0.113801i
$$893$$ 0 0
$$894$$ 0.224474 0.00750754
$$895$$ 0 0
$$896$$ 1.14044 0.0380993
$$897$$ − 0.690375i − 0.0230509i
$$898$$ 9.75580i 0.325555i
$$899$$ 27.4044 0.913986
$$900$$ 0 0
$$901$$ −17.0391 −0.567653
$$902$$ 10.1156i 0.336811i
$$903$$ 1.71359i 0.0570247i
$$904$$ 4.83984 0.160971
$$905$$ 0 0
$$906$$ −2.31478 −0.0769033
$$907$$ − 45.4631i − 1.50958i −0.655967 0.754789i $$-0.727739\pi$$
0.655967 0.754789i $$-0.272261\pi$$
$$908$$ 2.45798i 0.0815710i
$$909$$ 41.6063 1.38000
$$910$$ 0 0
$$911$$ 17.7610 0.588447 0.294223 0.955737i $$-0.404939\pi$$
0.294223 + 0.955737i $$0.404939\pi$$
$$912$$ − 0.598416i − 0.0198155i
$$913$$ − 3.62052i − 0.119822i
$$914$$ 7.99723 0.264525
$$915$$ 0 0
$$916$$ 12.7637 0.421725
$$917$$ 1.11794i 0.0369178i
$$918$$ 1.79763i 0.0593306i
$$919$$ −26.4829 −0.873589 −0.436794 0.899561i $$-0.643886\pi$$
−0.436794 + 0.899561i $$0.643886\pi$$
$$920$$ 0 0
$$921$$ 2.28602 0.0753270
$$922$$ 20.6848i 0.681219i
$$923$$ − 20.3057i − 0.668372i
$$924$$ −0.499806 −0.0164424
$$925$$ 0 0
$$926$$ −7.52507 −0.247289
$$927$$ − 30.8821i − 1.01430i
$$928$$ − 4.28087i − 0.140526i
$$929$$ −59.3842 −1.94833 −0.974167 0.225829i $$-0.927491\pi$$
−0.974167 + 0.225829i $$0.927491\pi$$
$$930$$ 0 0
$$931$$ −24.2860 −0.795942
$$932$$ 0.103761i 0.00339880i
$$933$$ − 3.57830i − 0.117148i
$$934$$ −6.84261 −0.223897
$$935$$ 0 0
$$936$$ −8.52230 −0.278560
$$937$$ − 15.1771i − 0.495815i −0.968784 0.247907i $$-0.920257\pi$$
0.968784 0.247907i $$-0.0797428\pi$$
$$938$$ 11.8398i 0.386585i
$$939$$ −2.67104 −0.0871662
$$940$$ 0 0
$$941$$ 12.8426 0.418657 0.209329 0.977845i $$-0.432872\pi$$
0.209329 + 0.977845i $$0.432872\pi$$
$$942$$ 1.42131i 0.0463087i
$$943$$ − 5.57243i − 0.181463i
$$944$$ −2.69941 −0.0878582
$$945$$ 0 0
$$946$$ 33.3898 1.08560
$$947$$ 9.04459i 0.293910i 0.989143 + 0.146955i $$0.0469472\pi$$
−0.989143 + 0.146955i $$0.953053\pi$$
$$948$$ − 1.19683i − 0.0388713i
$$949$$ 0.746771 0.0242412
$$950$$ 0 0
$$951$$ 2.12975 0.0690617
$$952$$ 2.44103i 0.0791142i
$$953$$ − 13.7783i − 0.446323i −0.974782 0.223161i $$-0.928362\pi$$
0.974782 0.223161i $$-0.0716377\pi$$
$$954$$ 23.7247 0.768115
$$955$$ 0 0
$$956$$ −3.67969 −0.119010
$$957$$ 1.87613i 0.0606466i
$$958$$ − 16.4016i − 0.529911i
$$959$$ 7.61775 0.245990
$$960$$ 0 0
$$961$$ 9.98028 0.321944
$$962$$ − 2.85956i − 0.0921961i
$$963$$ − 14.0138i − 0.451588i
$$964$$ 6.14044 0.197770
$$965$$ 0 0
$$966$$ 0.275331 0.00885864
$$967$$ − 60.1964i − 1.93579i −0.251360 0.967894i $$-0.580878\pi$$
0.251360 0.967894i $$-0.419122\pi$$
$$968$$ − 1.26115i − 0.0405349i
$$969$$ 1.28087 0.0411475
$$970$$ 0 0
$$971$$ −28.4123 −0.911793 −0.455897 0.890033i $$-0.650681\pi$$
−0.455897 + 0.890033i $$0.650681\pi$$
$$972$$ − 3.75857i − 0.120556i
$$973$$ − 11.0422i − 0.353997i
$$974$$ −13.7270 −0.439843
$$975$$ 0 0
$$976$$ 4.57869 0.146560
$$977$$ 55.7468i 1.78350i 0.452531 + 0.891749i $$0.350521\pi$$
−0.452531 + 0.891749i $$0.649479\pi$$
$$978$$ 1.64855i 0.0527148i
$$979$$ 42.5759 1.36073
$$980$$ 0 0
$$981$$ 17.7641 0.567164
$$982$$ − 3.40435i − 0.108637i
$$983$$ 37.1179i 1.18388i 0.805983 + 0.591939i $$0.201637\pi$$
−0.805983 + 0.591939i $$0.798363\pi$$
$$984$$ 0.455211 0.0145116
$$985$$ 0 0
$$986$$ 9.16293 0.291807
$$987$$ 0 0
$$988$$ 12.1850i 0.387657i
$$989$$ −18.3937 −0.584884
$$990$$ 0 0
$$991$$ −38.0418 −1.20844 −0.604219 0.796818i $$-0.706515\pi$$
−0.604219 + 0.796818i $$0.706515\pi$$
$$992$$ − 6.40158i − 0.203250i
$$993$$ − 3.28879i − 0.104367i
$$994$$ 8.09822 0.256860
$$995$$ 0 0
$$996$$ −0.162927 −0.00516254
$$997$$ − 11.6288i − 0.368289i −0.982899 0.184144i $$-0.941049\pi$$
0.982899 0.184144i $$-0.0589514\pi$$
$$998$$ − 29.4631i − 0.932639i
$$999$$ 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.b.p.149.2 6
5.2 odd 4 1850.2.a.bc.1.2 yes 3
5.3 odd 4 1850.2.a.y.1.2 3
5.4 even 2 inner 1850.2.b.p.149.5 6

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