# Properties

 Label 1850.2.b.i.149.4 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.4 Root $$2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.i.149.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +2.60555i q^{7} -1.00000i q^{8} -7.90833 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +2.60555i q^{7} -1.00000i q^{8} -7.90833 q^{9} -2.30278 q^{11} -3.30278i q^{12} +1.30278i q^{13} -2.60555 q^{14} +1.00000 q^{16} +6.00000i q^{17} -7.90833i q^{18} -2.00000 q^{19} -8.60555 q^{21} -2.30278i q^{22} +3.90833i q^{23} +3.30278 q^{24} -1.30278 q^{26} -16.2111i q^{27} -2.60555i q^{28} +3.90833 q^{29} -0.302776 q^{31} +1.00000i q^{32} -7.60555i q^{33} -6.00000 q^{34} +7.90833 q^{36} -1.00000i q^{37} -2.00000i q^{38} -4.30278 q^{39} +9.90833 q^{41} -8.60555i q^{42} +0.605551i q^{43} +2.30278 q^{44} -3.90833 q^{46} -4.60555i q^{47} +3.30278i q^{48} +0.211103 q^{49} -19.8167 q^{51} -1.30278i q^{52} -6.00000i q^{53} +16.2111 q^{54} +2.60555 q^{56} -6.60555i q^{57} +3.90833i q^{58} -10.6056 q^{59} +7.51388 q^{61} -0.302776i q^{62} -20.6056i q^{63} -1.00000 q^{64} +7.60555 q^{66} +3.51388i q^{67} -6.00000i q^{68} -12.9083 q^{69} +6.00000 q^{71} +7.90833i q^{72} -12.3028i q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -4.30278i q^{78} -9.11943 q^{79} +29.8167 q^{81} +9.90833i q^{82} +2.78890i q^{83} +8.60555 q^{84} -0.605551 q^{86} +12.9083i q^{87} +2.30278i q^{88} +9.21110 q^{89} -3.39445 q^{91} -3.90833i q^{92} -1.00000i q^{93} +4.60555 q^{94} -3.30278 q^{96} +16.4222i q^{97} +0.211103i q^{98} +18.2111 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 $$4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 - 2 * q^11 + 4 * q^14 + 4 * q^16 - 8 * q^19 - 20 * q^21 + 6 * q^24 + 2 * q^26 - 6 * q^29 + 6 * q^31 - 24 * q^34 + 10 * q^36 - 10 * q^39 + 18 * q^41 + 2 * q^44 + 6 * q^46 - 28 * q^49 - 36 * q^51 + 36 * q^54 - 4 * q^56 - 28 * q^59 - 6 * q^61 - 4 * q^64 + 16 * q^66 - 30 * q^69 + 24 * q^71 + 4 * q^74 + 8 * q^76 + 14 * q^79 + 76 * q^81 + 20 * q^84 + 12 * q^86 + 8 * q^89 - 28 * q^91 + 4 * q^94 - 6 * q^96 + 44 * 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
$$3$$ 3.30278i 1.90686i 0.301617 + 0.953429i $$0.402474\pi$$
−0.301617 + 0.953429i $$0.597526\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −3.30278 −1.34835
$$7$$ 2.60555i 0.984806i 0.870367 + 0.492403i $$0.163881\pi$$
−0.870367 + 0.492403i $$0.836119\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −7.90833 −2.63611
$$10$$ 0 0
$$11$$ −2.30278 −0.694313 −0.347156 0.937807i $$-0.612853\pi$$
−0.347156 + 0.937807i $$0.612853\pi$$
$$12$$ − 3.30278i − 0.953429i
$$13$$ 1.30278i 0.361325i 0.983545 + 0.180662i $$0.0578242\pi$$
−0.983545 + 0.180662i $$0.942176\pi$$
$$14$$ −2.60555 −0.696363
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 7.90833i − 1.86401i
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −8.60555 −1.87789
$$22$$ − 2.30278i − 0.490953i
$$23$$ 3.90833i 0.814942i 0.913218 + 0.407471i $$0.133589\pi$$
−0.913218 + 0.407471i $$0.866411\pi$$
$$24$$ 3.30278 0.674176
$$25$$ 0 0
$$26$$ −1.30278 −0.255495
$$27$$ − 16.2111i − 3.11983i
$$28$$ − 2.60555i − 0.492403i
$$29$$ 3.90833 0.725758 0.362879 0.931836i $$-0.381794\pi$$
0.362879 + 0.931836i $$0.381794\pi$$
$$30$$ 0 0
$$31$$ −0.302776 −0.0543801 −0.0271901 0.999630i $$-0.508656\pi$$
−0.0271901 + 0.999630i $$0.508656\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 7.60555i − 1.32396i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 7.90833 1.31805
$$37$$ − 1.00000i − 0.164399i
$$38$$ − 2.00000i − 0.324443i
$$39$$ −4.30278 −0.688996
$$40$$ 0 0
$$41$$ 9.90833 1.54742 0.773710 0.633540i $$-0.218399\pi$$
0.773710 + 0.633540i $$0.218399\pi$$
$$42$$ − 8.60555i − 1.32787i
$$43$$ 0.605551i 0.0923457i 0.998933 + 0.0461729i $$0.0147025\pi$$
−0.998933 + 0.0461729i $$0.985297\pi$$
$$44$$ 2.30278 0.347156
$$45$$ 0 0
$$46$$ −3.90833 −0.576251
$$47$$ − 4.60555i − 0.671789i −0.941900 0.335894i $$-0.890961\pi$$
0.941900 0.335894i $$-0.109039\pi$$
$$48$$ 3.30278i 0.476715i
$$49$$ 0.211103 0.0301575
$$50$$ 0 0
$$51$$ −19.8167 −2.77489
$$52$$ − 1.30278i − 0.180662i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 16.2111 2.20605
$$55$$ 0 0
$$56$$ 2.60555 0.348181
$$57$$ − 6.60555i − 0.874927i
$$58$$ 3.90833i 0.513188i
$$59$$ −10.6056 −1.38073 −0.690363 0.723464i $$-0.742549\pi$$
−0.690363 + 0.723464i $$0.742549\pi$$
$$60$$ 0 0
$$61$$ 7.51388 0.962054 0.481027 0.876706i $$-0.340264\pi$$
0.481027 + 0.876706i $$0.340264\pi$$
$$62$$ − 0.302776i − 0.0384525i
$$63$$ − 20.6056i − 2.59606i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 7.60555 0.936179
$$67$$ 3.51388i 0.429289i 0.976692 + 0.214644i $$0.0688592\pi$$
−0.976692 + 0.214644i $$0.931141\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −12.9083 −1.55398
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 7.90833i 0.932005i
$$73$$ − 12.3028i − 1.43993i −0.694010 0.719965i $$-0.744158\pi$$
0.694010 0.719965i $$-0.255842\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ − 6.00000i − 0.683763i
$$78$$ − 4.30278i − 0.487193i
$$79$$ −9.11943 −1.02602 −0.513008 0.858384i $$-0.671469\pi$$
−0.513008 + 0.858384i $$0.671469\pi$$
$$80$$ 0 0
$$81$$ 29.8167 3.31296
$$82$$ 9.90833i 1.09419i
$$83$$ 2.78890i 0.306121i 0.988217 + 0.153061i $$0.0489130\pi$$
−0.988217 + 0.153061i $$0.951087\pi$$
$$84$$ 8.60555 0.938943
$$85$$ 0 0
$$86$$ −0.605551 −0.0652983
$$87$$ 12.9083i 1.38392i
$$88$$ 2.30278i 0.245477i
$$89$$ 9.21110 0.976375 0.488187 0.872739i $$-0.337658\pi$$
0.488187 + 0.872739i $$0.337658\pi$$
$$90$$ 0 0
$$91$$ −3.39445 −0.355835
$$92$$ − 3.90833i − 0.407471i
$$93$$ − 1.00000i − 0.103695i
$$94$$ 4.60555 0.475026
$$95$$ 0 0
$$96$$ −3.30278 −0.337088
$$97$$ 16.4222i 1.66742i 0.552201 + 0.833711i $$0.313788\pi$$
−0.552201 + 0.833711i $$0.686212\pi$$
$$98$$ 0.211103i 0.0213246i
$$99$$ 18.2111 1.83028
$$100$$ 0 0
$$101$$ −12.4222 −1.23606 −0.618028 0.786156i $$-0.712068\pi$$
−0.618028 + 0.786156i $$0.712068\pi$$
$$102$$ − 19.8167i − 1.96214i
$$103$$ − 0.302776i − 0.0298334i −0.999889 0.0149167i $$-0.995252\pi$$
0.999889 0.0149167i $$-0.00474830\pi$$
$$104$$ 1.30278 0.127748
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 0.697224i − 0.0674032i −0.999432 0.0337016i $$-0.989270\pi$$
0.999432 0.0337016i $$-0.0107296\pi$$
$$108$$ 16.2111i 1.55991i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 3.30278 0.313486
$$112$$ 2.60555i 0.246201i
$$113$$ − 3.21110i − 0.302075i −0.988528 0.151038i $$-0.951739\pi$$
0.988528 0.151038i $$-0.0482614\pi$$
$$114$$ 6.60555 0.618667
$$115$$ 0 0
$$116$$ −3.90833 −0.362879
$$117$$ − 10.3028i − 0.952492i
$$118$$ − 10.6056i − 0.976320i
$$119$$ −15.6333 −1.43310
$$120$$ 0 0
$$121$$ −5.69722 −0.517929
$$122$$ 7.51388i 0.680275i
$$123$$ 32.7250i 2.95071i
$$124$$ 0.302776 0.0271901
$$125$$ 0 0
$$126$$ 20.6056 1.83569
$$127$$ 19.2111i 1.70471i 0.522964 + 0.852355i $$0.324826\pi$$
−0.522964 + 0.852355i $$0.675174\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ 10.6056 0.926611 0.463306 0.886199i $$-0.346663\pi$$
0.463306 + 0.886199i $$0.346663\pi$$
$$132$$ 7.60555i 0.661978i
$$133$$ − 5.21110i − 0.451860i
$$134$$ −3.51388 −0.303553
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 0.908327i − 0.0776036i −0.999247 0.0388018i $$-0.987646\pi$$
0.999247 0.0388018i $$-0.0123541\pi$$
$$138$$ − 12.9083i − 1.09883i
$$139$$ 1.90833 0.161862 0.0809311 0.996720i $$-0.474211\pi$$
0.0809311 + 0.996720i $$0.474211\pi$$
$$140$$ 0 0
$$141$$ 15.2111 1.28101
$$142$$ 6.00000i 0.503509i
$$143$$ − 3.00000i − 0.250873i
$$144$$ −7.90833 −0.659027
$$145$$ 0 0
$$146$$ 12.3028 1.01818
$$147$$ 0.697224i 0.0575061i
$$148$$ 1.00000i 0.0821995i
$$149$$ −19.8167 −1.62344 −0.811722 0.584044i $$-0.801469\pi$$
−0.811722 + 0.584044i $$0.801469\pi$$
$$150$$ 0 0
$$151$$ −20.6056 −1.67686 −0.838428 0.545012i $$-0.816525\pi$$
−0.838428 + 0.545012i $$0.816525\pi$$
$$152$$ 2.00000i 0.162221i
$$153$$ − 47.4500i − 3.83610i
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 4.30278 0.344498
$$157$$ 7.21110i 0.575509i 0.957704 + 0.287754i $$0.0929087\pi$$
−0.957704 + 0.287754i $$0.907091\pi$$
$$158$$ − 9.11943i − 0.725503i
$$159$$ 19.8167 1.57156
$$160$$ 0 0
$$161$$ −10.1833 −0.802560
$$162$$ 29.8167i 2.34262i
$$163$$ 8.42221i 0.659678i 0.944037 + 0.329839i $$0.106994\pi$$
−0.944037 + 0.329839i $$0.893006\pi$$
$$164$$ −9.90833 −0.773710
$$165$$ 0 0
$$166$$ −2.78890 −0.216460
$$167$$ 5.51388i 0.426677i 0.976978 + 0.213338i $$0.0684337\pi$$
−0.976978 + 0.213338i $$0.931566\pi$$
$$168$$ 8.60555i 0.663933i
$$169$$ 11.3028 0.869444
$$170$$ 0 0
$$171$$ 15.8167 1.20953
$$172$$ − 0.605551i − 0.0461729i
$$173$$ − 8.78890i − 0.668207i −0.942536 0.334104i $$-0.891566\pi$$
0.942536 0.334104i $$-0.108434\pi$$
$$174$$ −12.9083 −0.978578
$$175$$ 0 0
$$176$$ −2.30278 −0.173578
$$177$$ − 35.0278i − 2.63285i
$$178$$ 9.21110i 0.690401i
$$179$$ 13.8167 1.03271 0.516353 0.856376i $$-0.327289\pi$$
0.516353 + 0.856376i $$0.327289\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ − 3.39445i − 0.251613i
$$183$$ 24.8167i 1.83450i
$$184$$ 3.90833 0.288126
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ − 13.8167i − 1.01037i
$$188$$ 4.60555i 0.335894i
$$189$$ 42.2389 3.07242
$$190$$ 0 0
$$191$$ −5.51388 −0.398970 −0.199485 0.979901i $$-0.563927\pi$$
−0.199485 + 0.979901i $$0.563927\pi$$
$$192$$ − 3.30278i − 0.238357i
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ −16.4222 −1.17905
$$195$$ 0 0
$$196$$ −0.211103 −0.0150788
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 18.2111i 1.29421i
$$199$$ −26.4222 −1.87302 −0.936510 0.350640i $$-0.885964\pi$$
−0.936510 + 0.350640i $$0.885964\pi$$
$$200$$ 0 0
$$201$$ −11.6056 −0.818592
$$202$$ − 12.4222i − 0.874023i
$$203$$ 10.1833i 0.714731i
$$204$$ 19.8167 1.38744
$$205$$ 0 0
$$206$$ 0.302776 0.0210954
$$207$$ − 30.9083i − 2.14828i
$$208$$ 1.30278i 0.0903312i
$$209$$ 4.60555 0.318573
$$210$$ 0 0
$$211$$ 10.3028 0.709272 0.354636 0.935004i $$-0.384605\pi$$
0.354636 + 0.935004i $$0.384605\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 19.8167i 1.35781i
$$214$$ 0.697224 0.0476613
$$215$$ 0 0
$$216$$ −16.2111 −1.10303
$$217$$ − 0.788897i − 0.0535538i
$$218$$ − 2.00000i − 0.135457i
$$219$$ 40.6333 2.74574
$$220$$ 0 0
$$221$$ −7.81665 −0.525805
$$222$$ 3.30278i 0.221668i
$$223$$ − 5.81665i − 0.389512i −0.980852 0.194756i $$-0.937609\pi$$
0.980852 0.194756i $$-0.0623915\pi$$
$$224$$ −2.60555 −0.174091
$$225$$ 0 0
$$226$$ 3.21110 0.213599
$$227$$ − 13.8167i − 0.917044i −0.888683 0.458522i $$-0.848379\pi$$
0.888683 0.458522i $$-0.151621\pi$$
$$228$$ 6.60555i 0.437463i
$$229$$ −24.6056 −1.62598 −0.812990 0.582277i $$-0.802162\pi$$
−0.812990 + 0.582277i $$0.802162\pi$$
$$230$$ 0 0
$$231$$ 19.8167 1.30384
$$232$$ − 3.90833i − 0.256594i
$$233$$ 8.51388i 0.557763i 0.960326 + 0.278881i $$0.0899636\pi$$
−0.960326 + 0.278881i $$0.910036\pi$$
$$234$$ 10.3028 0.673514
$$235$$ 0 0
$$236$$ 10.6056 0.690363
$$237$$ − 30.1194i − 1.95647i
$$238$$ − 15.6333i − 1.01336i
$$239$$ 17.5139 1.13288 0.566439 0.824103i $$-0.308321\pi$$
0.566439 + 0.824103i $$0.308321\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ − 5.69722i − 0.366231i
$$243$$ 49.8444i 3.19752i
$$244$$ −7.51388 −0.481027
$$245$$ 0 0
$$246$$ −32.7250 −2.08647
$$247$$ − 2.60555i − 0.165787i
$$248$$ 0.302776i 0.0192263i
$$249$$ −9.21110 −0.583730
$$250$$ 0 0
$$251$$ −21.2111 −1.33883 −0.669416 0.742887i $$-0.733456\pi$$
−0.669416 + 0.742887i $$0.733456\pi$$
$$252$$ 20.6056i 1.29803i
$$253$$ − 9.00000i − 0.565825i
$$254$$ −19.2111 −1.20541
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 3.21110i − 0.200303i −0.994972 0.100152i $$-0.968067\pi$$
0.994972 0.100152i $$-0.0319328\pi$$
$$258$$ − 2.00000i − 0.124515i
$$259$$ 2.60555 0.161901
$$260$$ 0 0
$$261$$ −30.9083 −1.91318
$$262$$ 10.6056i 0.655213i
$$263$$ 13.8167i 0.851971i 0.904730 + 0.425986i $$0.140073\pi$$
−0.904730 + 0.425986i $$0.859927\pi$$
$$264$$ −7.60555 −0.468089
$$265$$ 0 0
$$266$$ 5.21110 0.319513
$$267$$ 30.4222i 1.86181i
$$268$$ − 3.51388i − 0.214644i
$$269$$ 21.2111 1.29326 0.646632 0.762802i $$-0.276177\pi$$
0.646632 + 0.762802i $$0.276177\pi$$
$$270$$ 0 0
$$271$$ −22.4222 −1.36205 −0.681026 0.732259i $$-0.738466\pi$$
−0.681026 + 0.732259i $$0.738466\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ − 11.2111i − 0.678527i
$$274$$ 0.908327 0.0548740
$$275$$ 0 0
$$276$$ 12.9083 0.776990
$$277$$ − 0.119429i − 0.00717582i −0.999994 0.00358791i $$-0.998858\pi$$
0.999994 0.00358791i $$-0.00114207\pi$$
$$278$$ 1.90833i 0.114454i
$$279$$ 2.39445 0.143352
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 15.2111i 0.905808i
$$283$$ 24.6056i 1.46265i 0.682030 + 0.731324i $$0.261097\pi$$
−0.682030 + 0.731324i $$0.738903\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 25.8167i 1.52391i
$$288$$ − 7.90833i − 0.466003i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −54.2389 −3.17954
$$292$$ 12.3028i 0.719965i
$$293$$ 11.0278i 0.644248i 0.946697 + 0.322124i $$0.104397\pi$$
−0.946697 + 0.322124i $$0.895603\pi$$
$$294$$ −0.697224 −0.0406630
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 37.3305i 2.16614i
$$298$$ − 19.8167i − 1.14795i
$$299$$ −5.09167 −0.294459
$$300$$ 0 0
$$301$$ −1.57779 −0.0909426
$$302$$ − 20.6056i − 1.18572i
$$303$$ − 41.0278i − 2.35698i
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 47.4500 2.71253
$$307$$ − 17.9083i − 1.02208i −0.859556 0.511041i $$-0.829260\pi$$
0.859556 0.511041i $$-0.170740\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 15.9083 0.902078 0.451039 0.892504i $$-0.351053\pi$$
0.451039 + 0.892504i $$0.351053\pi$$
$$312$$ 4.30278i 0.243597i
$$313$$ − 9.02776i − 0.510279i −0.966904 0.255139i $$-0.917879\pi$$
0.966904 0.255139i $$-0.0821214\pi$$
$$314$$ −7.21110 −0.406946
$$315$$ 0 0
$$316$$ 9.11943 0.513008
$$317$$ − 9.21110i − 0.517347i −0.965965 0.258674i $$-0.916715\pi$$
0.965965 0.258674i $$-0.0832854\pi$$
$$318$$ 19.8167i 1.11126i
$$319$$ −9.00000 −0.503903
$$320$$ 0 0
$$321$$ 2.30278 0.128528
$$322$$ − 10.1833i − 0.567496i
$$323$$ − 12.0000i − 0.667698i
$$324$$ −29.8167 −1.65648
$$325$$ 0 0
$$326$$ −8.42221 −0.466463
$$327$$ − 6.60555i − 0.365288i
$$328$$ − 9.90833i − 0.547096i
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −13.2111 −0.726148 −0.363074 0.931760i $$-0.618273\pi$$
−0.363074 + 0.931760i $$0.618273\pi$$
$$332$$ − 2.78890i − 0.153061i
$$333$$ 7.90833i 0.433374i
$$334$$ −5.51388 −0.301706
$$335$$ 0 0
$$336$$ −8.60555 −0.469471
$$337$$ − 6.11943i − 0.333347i −0.986012 0.166673i $$-0.946697\pi$$
0.986012 0.166673i $$-0.0533025\pi$$
$$338$$ 11.3028i 0.614790i
$$339$$ 10.6056 0.576014
$$340$$ 0 0
$$341$$ 0.697224 0.0377568
$$342$$ 15.8167i 0.855267i
$$343$$ 18.7889i 1.01451i
$$344$$ 0.605551 0.0326491
$$345$$ 0 0
$$346$$ 8.78890 0.472494
$$347$$ − 10.1833i − 0.546671i −0.961919 0.273335i $$-0.911873\pi$$
0.961919 0.273335i $$-0.0881269\pi$$
$$348$$ − 12.9083i − 0.691959i
$$349$$ −28.2389 −1.51159 −0.755796 0.654807i $$-0.772750\pi$$
−0.755796 + 0.654807i $$0.772750\pi$$
$$350$$ 0 0
$$351$$ 21.1194 1.12727
$$352$$ − 2.30278i − 0.122738i
$$353$$ 10.1833i 0.542005i 0.962579 + 0.271002i $$0.0873551\pi$$
−0.962579 + 0.271002i $$0.912645\pi$$
$$354$$ 35.0278 1.86170
$$355$$ 0 0
$$356$$ −9.21110 −0.488187
$$357$$ − 51.6333i − 2.73272i
$$358$$ 13.8167i 0.730233i
$$359$$ −3.21110 −0.169476 −0.0847378 0.996403i $$-0.527005\pi$$
−0.0847378 + 0.996403i $$0.527005\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000i 1.05118i
$$363$$ − 18.8167i − 0.987618i
$$364$$ 3.39445 0.177917
$$365$$ 0 0
$$366$$ −24.8167 −1.29719
$$367$$ − 3.81665i − 0.199228i −0.995026 0.0996139i $$-0.968239\pi$$
0.995026 0.0996139i $$-0.0317607\pi$$
$$368$$ 3.90833i 0.203736i
$$369$$ −78.3583 −4.07917
$$370$$ 0 0
$$371$$ 15.6333 0.811641
$$372$$ 1.00000i 0.0518476i
$$373$$ − 17.8167i − 0.922511i −0.887267 0.461256i $$-0.847399\pi$$
0.887267 0.461256i $$-0.152601\pi$$
$$374$$ 13.8167 0.714442
$$375$$ 0 0
$$376$$ −4.60555 −0.237513
$$377$$ 5.09167i 0.262235i
$$378$$ 42.2389i 2.17253i
$$379$$ −24.3305 −1.24978 −0.624888 0.780715i $$-0.714855\pi$$
−0.624888 + 0.780715i $$0.714855\pi$$
$$380$$ 0 0
$$381$$ −63.4500 −3.25064
$$382$$ − 5.51388i − 0.282115i
$$383$$ − 36.8444i − 1.88266i −0.337486 0.941331i $$-0.609576\pi$$
0.337486 0.941331i $$-0.390424\pi$$
$$384$$ 3.30278 0.168544
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ − 4.78890i − 0.243433i
$$388$$ − 16.4222i − 0.833711i
$$389$$ 37.1194 1.88203 0.941015 0.338365i $$-0.109874\pi$$
0.941015 + 0.338365i $$0.109874\pi$$
$$390$$ 0 0
$$391$$ −23.4500 −1.18592
$$392$$ − 0.211103i − 0.0106623i
$$393$$ 35.0278i 1.76692i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −18.2111 −0.915142
$$397$$ − 6.18335i − 0.310333i −0.987888 0.155167i $$-0.950409\pi$$
0.987888 0.155167i $$-0.0495914\pi$$
$$398$$ − 26.4222i − 1.32443i
$$399$$ 17.2111 0.861633
$$400$$ 0 0
$$401$$ −7.81665 −0.390345 −0.195173 0.980769i $$-0.562527\pi$$
−0.195173 + 0.980769i $$0.562527\pi$$
$$402$$ − 11.6056i − 0.578832i
$$403$$ − 0.394449i − 0.0196489i
$$404$$ 12.4222 0.618028
$$405$$ 0 0
$$406$$ −10.1833 −0.505391
$$407$$ 2.30278i 0.114144i
$$408$$ 19.8167i 0.981071i
$$409$$ −31.0278 −1.53422 −0.767112 0.641513i $$-0.778307\pi$$
−0.767112 + 0.641513i $$0.778307\pi$$
$$410$$ 0 0
$$411$$ 3.00000 0.147979
$$412$$ 0.302776i 0.0149167i
$$413$$ − 27.6333i − 1.35975i
$$414$$ 30.9083 1.51906
$$415$$ 0 0
$$416$$ −1.30278 −0.0638738
$$417$$ 6.30278i 0.308648i
$$418$$ 4.60555i 0.225265i
$$419$$ −36.1472 −1.76591 −0.882953 0.469462i $$-0.844448\pi$$
−0.882953 + 0.469462i $$0.844448\pi$$
$$420$$ 0 0
$$421$$ −3.72498 −0.181544 −0.0907722 0.995872i $$-0.528934\pi$$
−0.0907722 + 0.995872i $$0.528934\pi$$
$$422$$ 10.3028i 0.501531i
$$423$$ 36.4222i 1.77091i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −19.8167 −0.960120
$$427$$ 19.5778i 0.947436i
$$428$$ 0.697224i 0.0337016i
$$429$$ 9.90833 0.478379
$$430$$ 0 0
$$431$$ 9.21110 0.443683 0.221842 0.975083i $$-0.428793\pi$$
0.221842 + 0.975083i $$0.428793\pi$$
$$432$$ − 16.2111i − 0.779957i
$$433$$ 34.9361i 1.67892i 0.543421 + 0.839461i $$0.317129\pi$$
−0.543421 + 0.839461i $$0.682871\pi$$
$$434$$ 0.788897 0.0378683
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 7.81665i − 0.373921i
$$438$$ 40.6333i 1.94153i
$$439$$ −30.3305 −1.44760 −0.723799 0.690011i $$-0.757606\pi$$
−0.723799 + 0.690011i $$0.757606\pi$$
$$440$$ 0 0
$$441$$ −1.66947 −0.0794985
$$442$$ − 7.81665i − 0.371800i
$$443$$ 32.7250i 1.55481i 0.629000 + 0.777405i $$0.283465\pi$$
−0.629000 + 0.777405i $$0.716535\pi$$
$$444$$ −3.30278 −0.156743
$$445$$ 0 0
$$446$$ 5.81665 0.275427
$$447$$ − 65.4500i − 3.09568i
$$448$$ − 2.60555i − 0.123101i
$$449$$ 15.2111 0.717856 0.358928 0.933365i $$-0.383142\pi$$
0.358928 + 0.933365i $$0.383142\pi$$
$$450$$ 0 0
$$451$$ −22.8167 −1.07439
$$452$$ 3.21110i 0.151038i
$$453$$ − 68.0555i − 3.19753i
$$454$$ 13.8167 0.648448
$$455$$ 0 0
$$456$$ −6.60555 −0.309333
$$457$$ 2.60555i 0.121883i 0.998141 + 0.0609413i $$0.0194102\pi$$
−0.998141 + 0.0609413i $$0.980590\pi$$
$$458$$ − 24.6056i − 1.14974i
$$459$$ 97.2666 4.54002
$$460$$ 0 0
$$461$$ 12.4222 0.578560 0.289280 0.957245i $$-0.406584\pi$$
0.289280 + 0.957245i $$0.406584\pi$$
$$462$$ 19.8167i 0.921954i
$$463$$ 26.6972i 1.24073i 0.784315 + 0.620363i $$0.213015\pi$$
−0.784315 + 0.620363i $$0.786985\pi$$
$$464$$ 3.90833 0.181440
$$465$$ 0 0
$$466$$ −8.51388 −0.394398
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 10.3028i 0.476246i
$$469$$ −9.15559 −0.422766
$$470$$ 0 0
$$471$$ −23.8167 −1.09741
$$472$$ 10.6056i 0.488160i
$$473$$ − 1.39445i − 0.0641168i
$$474$$ 30.1194 1.38343
$$475$$ 0 0
$$476$$ 15.6333 0.716551
$$477$$ 47.4500i 2.17258i
$$478$$ 17.5139i 0.801066i
$$479$$ 13.1194 0.599442 0.299721 0.954027i $$-0.403106\pi$$
0.299721 + 0.954027i $$0.403106\pi$$
$$480$$ 0 0
$$481$$ 1.30278 0.0594015
$$482$$ 8.00000i 0.364390i
$$483$$ − 33.6333i − 1.53037i
$$484$$ 5.69722 0.258965
$$485$$ 0 0
$$486$$ −49.8444 −2.26099
$$487$$ 37.2111i 1.68620i 0.537760 + 0.843098i $$0.319271\pi$$
−0.537760 + 0.843098i $$0.680729\pi$$
$$488$$ − 7.51388i − 0.340137i
$$489$$ −27.8167 −1.25791
$$490$$ 0 0
$$491$$ 17.7250 0.799917 0.399959 0.916533i $$-0.369025\pi$$
0.399959 + 0.916533i $$0.369025\pi$$
$$492$$ − 32.7250i − 1.47536i
$$493$$ 23.4500i 1.05613i
$$494$$ 2.60555 0.117229
$$495$$ 0 0
$$496$$ −0.302776 −0.0135950
$$497$$ 15.6333i 0.701250i
$$498$$ − 9.21110i − 0.412759i
$$499$$ 42.2389 1.89087 0.945436 0.325809i $$-0.105637\pi$$
0.945436 + 0.325809i $$0.105637\pi$$
$$500$$ 0 0
$$501$$ −18.2111 −0.813612
$$502$$ − 21.2111i − 0.946698i
$$503$$ − 6.48612i − 0.289202i −0.989490 0.144601i $$-0.953810\pi$$
0.989490 0.144601i $$-0.0461898\pi$$
$$504$$ −20.6056 −0.917844
$$505$$ 0 0
$$506$$ 9.00000 0.400099
$$507$$ 37.3305i 1.65791i
$$508$$ − 19.2111i − 0.852355i
$$509$$ 4.18335 0.185424 0.0927118 0.995693i $$-0.470446\pi$$
0.0927118 + 0.995693i $$0.470446\pi$$
$$510$$ 0 0
$$511$$ 32.0555 1.41805
$$512$$ 1.00000i 0.0441942i
$$513$$ 32.4222i 1.43148i
$$514$$ 3.21110 0.141636
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ 10.6056i 0.466432i
$$518$$ 2.60555i 0.114481i
$$519$$ 29.0278 1.27418
$$520$$ 0 0
$$521$$ 33.6333 1.47350 0.736751 0.676164i $$-0.236359\pi$$
0.736751 + 0.676164i $$0.236359\pi$$
$$522$$ − 30.9083i − 1.35282i
$$523$$ − 18.2389i − 0.797530i −0.917053 0.398765i $$-0.869439\pi$$
0.917053 0.398765i $$-0.130561\pi$$
$$524$$ −10.6056 −0.463306
$$525$$ 0 0
$$526$$ −13.8167 −0.602435
$$527$$ − 1.81665i − 0.0791347i
$$528$$ − 7.60555i − 0.330989i
$$529$$ 7.72498 0.335869
$$530$$ 0 0
$$531$$ 83.8722 3.63974
$$532$$ 5.21110i 0.225930i
$$533$$ 12.9083i 0.559122i
$$534$$ −30.4222 −1.31650
$$535$$ 0 0
$$536$$ 3.51388 0.151776
$$537$$ 45.6333i 1.96922i
$$538$$ 21.2111i 0.914476i
$$539$$ −0.486122 −0.0209387
$$540$$ 0 0
$$541$$ 25.9361 1.11508 0.557540 0.830150i $$-0.311745\pi$$
0.557540 + 0.830150i $$0.311745\pi$$
$$542$$ − 22.4222i − 0.963116i
$$543$$ 66.0555i 2.83471i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 11.2111 0.479791
$$547$$ 20.6056i 0.881030i 0.897745 + 0.440515i $$0.145204\pi$$
−0.897745 + 0.440515i $$0.854796\pi$$
$$548$$ 0.908327i 0.0388018i
$$549$$ −59.4222 −2.53608
$$550$$ 0 0
$$551$$ −7.81665 −0.333001
$$552$$ 12.9083i 0.549415i
$$553$$ − 23.7611i − 1.01043i
$$554$$ 0.119429 0.00507407
$$555$$ 0 0
$$556$$ −1.90833 −0.0809311
$$557$$ − 11.5139i − 0.487859i −0.969793 0.243929i $$-0.921564\pi$$
0.969793 0.243929i $$-0.0784365\pi$$
$$558$$ 2.39445i 0.101365i
$$559$$ −0.788897 −0.0333668
$$560$$ 0 0
$$561$$ 45.6333 1.92664
$$562$$ − 12.0000i − 0.506189i
$$563$$ − 28.0555i − 1.18240i −0.806525 0.591199i $$-0.798655\pi$$
0.806525 0.591199i $$-0.201345\pi$$
$$564$$ −15.2111 −0.640503
$$565$$ 0 0
$$566$$ −24.6056 −1.03425
$$567$$ 77.6888i 3.26262i
$$568$$ − 6.00000i − 0.251754i
$$569$$ −18.4222 −0.772299 −0.386150 0.922436i $$-0.626195\pi$$
−0.386150 + 0.922436i $$0.626195\pi$$
$$570$$ 0 0
$$571$$ −16.6972 −0.698757 −0.349379 0.936982i $$-0.613607\pi$$
−0.349379 + 0.936982i $$0.613607\pi$$
$$572$$ 3.00000i 0.125436i
$$573$$ − 18.2111i − 0.760780i
$$574$$ −25.8167 −1.07757
$$575$$ 0 0
$$576$$ 7.90833 0.329514
$$577$$ − 22.2389i − 0.925816i −0.886406 0.462908i $$-0.846806\pi$$
0.886406 0.462908i $$-0.153194\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 13.2111 0.549035
$$580$$ 0 0
$$581$$ −7.26662 −0.301470
$$582$$ − 54.2389i − 2.24827i
$$583$$ 13.8167i 0.572227i
$$584$$ −12.3028 −0.509092
$$585$$ 0 0
$$586$$ −11.0278 −0.455552
$$587$$ 45.6333i 1.88349i 0.336330 + 0.941744i $$0.390814\pi$$
−0.336330 + 0.941744i $$0.609186\pi$$
$$588$$ − 0.697224i − 0.0287530i
$$589$$ 0.605551 0.0249513
$$590$$ 0 0
$$591$$ −19.8167 −0.815148
$$592$$ − 1.00000i − 0.0410997i
$$593$$ 18.4861i 0.759134i 0.925164 + 0.379567i $$0.123927\pi$$
−0.925164 + 0.379567i $$0.876073\pi$$
$$594$$ −37.3305 −1.53169
$$595$$ 0 0
$$596$$ 19.8167 0.811722
$$597$$ − 87.2666i − 3.57158i
$$598$$ − 5.09167i − 0.208214i
$$599$$ 20.7889 0.849411 0.424706 0.905331i $$-0.360378\pi$$
0.424706 + 0.905331i $$0.360378\pi$$
$$600$$ 0 0
$$601$$ −24.3028 −0.991331 −0.495665 0.868514i $$-0.665076\pi$$
−0.495665 + 0.868514i $$0.665076\pi$$
$$602$$ − 1.57779i − 0.0643061i
$$603$$ − 27.7889i − 1.13165i
$$604$$ 20.6056 0.838428
$$605$$ 0 0
$$606$$ 41.0278 1.66664
$$607$$ 13.4861i 0.547385i 0.961817 + 0.273692i $$0.0882450\pi$$
−0.961817 + 0.273692i $$0.911755\pi$$
$$608$$ − 2.00000i − 0.0811107i
$$609$$ −33.6333 −1.36289
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 47.4500i 1.91805i
$$613$$ − 29.8167i − 1.20428i −0.798389 0.602142i $$-0.794314\pi$$
0.798389 0.602142i $$-0.205686\pi$$
$$614$$ 17.9083 0.722721
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ 42.5694i 1.71378i 0.515500 + 0.856890i $$0.327606\pi$$
−0.515500 + 0.856890i $$0.672394\pi$$
$$618$$ 1.00000i 0.0402259i
$$619$$ 6.30278 0.253330 0.126665 0.991946i $$-0.459573\pi$$
0.126665 + 0.991946i $$0.459573\pi$$
$$620$$ 0 0
$$621$$ 63.3583 2.54248
$$622$$ 15.9083i 0.637866i
$$623$$ 24.0000i 0.961540i
$$624$$ −4.30278 −0.172249
$$625$$ 0 0
$$626$$ 9.02776 0.360822
$$627$$ 15.2111i 0.607473i
$$628$$ − 7.21110i − 0.287754i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 14.6972 0.585087 0.292544 0.956252i $$-0.405498\pi$$
0.292544 + 0.956252i $$0.405498\pi$$
$$632$$ 9.11943i 0.362751i
$$633$$ 34.0278i 1.35248i
$$634$$ 9.21110 0.365820
$$635$$ 0 0
$$636$$ −19.8167 −0.785781
$$637$$ 0.275019i 0.0108967i
$$638$$ − 9.00000i − 0.356313i
$$639$$ −47.4500 −1.87709
$$640$$ 0 0
$$641$$ −20.5139 −0.810249 −0.405125 0.914261i $$-0.632772\pi$$
−0.405125 + 0.914261i $$0.632772\pi$$
$$642$$ 2.30278i 0.0908833i
$$643$$ − 8.18335i − 0.322720i −0.986896 0.161360i $$-0.948412\pi$$
0.986896 0.161360i $$-0.0515880\pi$$
$$644$$ 10.1833 0.401280
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ − 20.9361i − 0.823082i −0.911391 0.411541i $$-0.864991\pi$$
0.911391 0.411541i $$-0.135009\pi$$
$$648$$ − 29.8167i − 1.17131i
$$649$$ 24.4222 0.958655
$$650$$ 0 0
$$651$$ 2.60555 0.102120
$$652$$ − 8.42221i − 0.329839i
$$653$$ − 3.90833i − 0.152945i −0.997072 0.0764723i $$-0.975634\pi$$
0.997072 0.0764723i $$-0.0243657\pi$$
$$654$$ 6.60555 0.258297
$$655$$ 0 0
$$656$$ 9.90833 0.386855
$$657$$ 97.2944i 3.79581i
$$658$$ 12.0000i 0.467809i
$$659$$ 16.8806 0.657574 0.328787 0.944404i $$-0.393360\pi$$
0.328787 + 0.944404i $$0.393360\pi$$
$$660$$ 0 0
$$661$$ −30.5139 −1.18685 −0.593426 0.804888i $$-0.702225\pi$$
−0.593426 + 0.804888i $$0.702225\pi$$
$$662$$ − 13.2111i − 0.513464i
$$663$$ − 25.8167i − 1.00264i
$$664$$ 2.78890 0.108230
$$665$$ 0 0
$$666$$ −7.90833 −0.306441
$$667$$ 15.2750i 0.591451i
$$668$$ − 5.51388i − 0.213338i
$$669$$ 19.2111 0.742744
$$670$$ 0 0
$$671$$ −17.3028 −0.667966
$$672$$ − 8.60555i − 0.331966i
$$673$$ 20.6972i 0.797819i 0.916990 + 0.398910i $$0.130611\pi$$
−0.916990 + 0.398910i $$0.869389\pi$$
$$674$$ 6.11943 0.235712
$$675$$ 0 0
$$676$$ −11.3028 −0.434722
$$677$$ − 14.2389i − 0.547244i −0.961837 0.273622i $$-0.911778\pi$$
0.961837 0.273622i $$-0.0882217\pi$$
$$678$$ 10.6056i 0.407304i
$$679$$ −42.7889 −1.64209
$$680$$ 0 0
$$681$$ 45.6333 1.74867
$$682$$ 0.697224i 0.0266981i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −15.8167 −0.604765
$$685$$ 0 0
$$686$$ −18.7889 −0.717363
$$687$$ − 81.2666i − 3.10051i
$$688$$ 0.605551i 0.0230864i
$$689$$ 7.81665 0.297791
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 8.78890i 0.334104i
$$693$$ 47.4500i 1.80247i
$$694$$ 10.1833 0.386555
$$695$$ 0 0
$$696$$ 12.9083 0.489289
$$697$$ 59.4500i 2.25183i
$$698$$ − 28.2389i − 1.06886i
$$699$$ −28.1194 −1.06357
$$700$$ 0 0
$$701$$ 40.1194 1.51529 0.757645 0.652667i $$-0.226350\pi$$
0.757645 + 0.652667i $$0.226350\pi$$
$$702$$ 21.1194i 0.797101i
$$703$$ 2.00000i 0.0754314i
$$704$$ 2.30278 0.0867891
$$705$$ 0 0
$$706$$ −10.1833 −0.383255
$$707$$ − 32.3667i − 1.21727i
$$708$$ 35.0278i 1.31642i
$$709$$ 41.3305 1.55220 0.776100 0.630609i $$-0.217195\pi$$
0.776100 + 0.630609i $$0.217195\pi$$
$$710$$ 0 0
$$711$$ 72.1194 2.70469
$$712$$ − 9.21110i − 0.345201i
$$713$$ − 1.18335i − 0.0443167i
$$714$$ 51.6333 1.93233
$$715$$ 0 0
$$716$$ −13.8167 −0.516353
$$717$$ 57.8444i 2.16024i
$$718$$ − 3.21110i − 0.119837i
$$719$$ 51.6333 1.92560 0.962799 0.270220i $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 0 0
$$721$$ 0.788897 0.0293801
$$722$$ − 15.0000i − 0.558242i
$$723$$ 26.4222i 0.982652i
$$724$$ −20.0000 −0.743294
$$725$$ 0 0
$$726$$ 18.8167 0.698352
$$727$$ − 19.0917i − 0.708071i −0.935232 0.354035i $$-0.884809\pi$$
0.935232 0.354035i $$-0.115191\pi$$
$$728$$ 3.39445i 0.125807i
$$729$$ −75.1749 −2.78426
$$730$$ 0 0
$$731$$ −3.63331 −0.134383
$$732$$ − 24.8167i − 0.917250i
$$733$$ − 13.6333i − 0.503558i −0.967785 0.251779i $$-0.918984\pi$$
0.967785 0.251779i $$-0.0810156\pi$$
$$734$$ 3.81665 0.140875
$$735$$ 0 0
$$736$$ −3.90833 −0.144063
$$737$$ − 8.09167i − 0.298061i
$$738$$ − 78.3583i − 2.88441i
$$739$$ 2.66947 0.0981980 0.0490990 0.998794i $$-0.484365\pi$$
0.0490990 + 0.998794i $$0.484365\pi$$
$$740$$ 0 0
$$741$$ 8.60555 0.316133
$$742$$ 15.6333i 0.573917i
$$743$$ − 29.4500i − 1.08041i −0.841532 0.540207i $$-0.818346\pi$$
0.841532 0.540207i $$-0.181654\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 17.8167 0.652314
$$747$$ − 22.0555i − 0.806969i
$$748$$ 13.8167i 0.505187i
$$749$$ 1.81665 0.0663791
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ − 4.60555i − 0.167947i
$$753$$ − 70.0555i − 2.55296i
$$754$$ −5.09167 −0.185428
$$755$$ 0 0
$$756$$ −42.2389 −1.53621
$$757$$ − 5.69722i − 0.207069i −0.994626 0.103535i $$-0.966985\pi$$
0.994626 0.103535i $$-0.0330152\pi$$
$$758$$ − 24.3305i − 0.883725i
$$759$$ 29.7250 1.07895
$$760$$ 0 0
$$761$$ 16.8806 0.611920 0.305960 0.952044i $$-0.401023\pi$$
0.305960 + 0.952044i $$0.401023\pi$$
$$762$$ − 63.4500i − 2.29855i
$$763$$ − 5.21110i − 0.188655i
$$764$$ 5.51388 0.199485
$$765$$ 0 0
$$766$$ 36.8444 1.33124
$$767$$ − 13.8167i − 0.498890i
$$768$$ 3.30278i 0.119179i
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 10.6056 0.381950
$$772$$ 4.00000i 0.143963i
$$773$$ 22.0555i 0.793282i 0.917974 + 0.396641i $$0.129824\pi$$
−0.917974 + 0.396641i $$0.870176\pi$$
$$774$$ 4.78890 0.172133
$$775$$ 0 0
$$776$$ 16.4222 0.589523
$$777$$ 8.60555i 0.308722i
$$778$$ 37.1194i 1.33080i
$$779$$ −19.8167 −0.710005
$$780$$ 0 0
$$781$$ −13.8167 −0.494399
$$782$$ − 23.4500i − 0.838569i
$$783$$ − 63.3583i − 2.26424i
$$784$$ 0.211103 0.00753938
$$785$$ 0 0
$$786$$ −35.0278 −1.24940
$$787$$ − 10.7889i − 0.384583i −0.981338 0.192291i $$-0.938408\pi$$
0.981338 0.192291i $$-0.0615919\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ −45.6333 −1.62459
$$790$$ 0 0
$$791$$ 8.36669 0.297485
$$792$$ − 18.2111i − 0.647103i
$$793$$ 9.78890i 0.347614i
$$794$$ 6.18335 0.219439
$$795$$ 0 0
$$796$$ 26.4222 0.936510
$$797$$ − 22.3305i − 0.790988i −0.918469 0.395494i $$-0.870573\pi$$
0.918469 0.395494i $$-0.129427\pi$$
$$798$$ 17.2111i 0.609266i
$$799$$ 27.6333 0.977596
$$800$$ 0 0
$$801$$ −72.8444 −2.57383
$$802$$ − 7.81665i − 0.276016i
$$803$$ 28.3305i 0.999763i
$$804$$ 11.6056 0.409296
$$805$$ 0 0
$$806$$ 0.394449 0.0138939
$$807$$ 70.0555i 2.46607i
$$808$$ 12.4222i 0.437012i
$$809$$ 35.4500 1.24635 0.623177 0.782081i $$-0.285842\pi$$
0.623177 + 0.782081i $$0.285842\pi$$
$$810$$ 0 0
$$811$$ −7.14719 −0.250972 −0.125486 0.992095i $$-0.540049\pi$$
−0.125486 + 0.992095i $$0.540049\pi$$
$$812$$ − 10.1833i − 0.357365i
$$813$$ − 74.0555i − 2.59724i
$$814$$ −2.30278 −0.0807122
$$815$$ 0 0
$$816$$ −19.8167 −0.693722
$$817$$ − 1.21110i − 0.0423711i
$$818$$ − 31.0278i − 1.08486i
$$819$$ 26.8444 0.938020
$$820$$ 0 0
$$821$$ −3.21110 −0.112068 −0.0560341 0.998429i $$-0.517846\pi$$
−0.0560341 + 0.998429i $$0.517846\pi$$
$$822$$ 3.00000i 0.104637i
$$823$$ 44.8444i 1.56318i 0.623794 + 0.781589i $$0.285590\pi$$
−0.623794 + 0.781589i $$0.714410\pi$$
$$824$$ −0.302776 −0.0105477
$$825$$ 0 0
$$826$$ 27.6333 0.961486
$$827$$ − 34.6056i − 1.20335i −0.798740 0.601676i $$-0.794500\pi$$
0.798740 0.601676i $$-0.205500\pi$$
$$828$$ 30.9083i 1.07414i
$$829$$ 27.7250 0.962928 0.481464 0.876466i $$-0.340105\pi$$
0.481464 + 0.876466i $$0.340105\pi$$
$$830$$ 0 0
$$831$$ 0.394449 0.0136833
$$832$$ − 1.30278i − 0.0451656i
$$833$$ 1.26662i 0.0438856i
$$834$$ −6.30278 −0.218247
$$835$$ 0 0
$$836$$ −4.60555 −0.159286
$$837$$ 4.90833i 0.169657i
$$838$$ − 36.1472i − 1.24868i
$$839$$ 12.9722 0.447852 0.223926 0.974606i $$-0.428113\pi$$
0.223926 + 0.974606i $$0.428113\pi$$
$$840$$ 0 0
$$841$$ −13.7250 −0.473275
$$842$$ − 3.72498i − 0.128371i
$$843$$ − 39.6333i − 1.36504i
$$844$$ −10.3028 −0.354636
$$845$$ 0 0
$$846$$ −36.4222 −1.25222
$$847$$ − 14.8444i − 0.510060i
$$848$$ − 6.00000i − 0.206041i
$$849$$ −81.2666 −2.78906
$$850$$ 0 0
$$851$$ 3.90833 0.133976
$$852$$ − 19.8167i − 0.678907i
$$853$$ 42.5416i 1.45660i 0.685260 + 0.728299i $$0.259689\pi$$
−0.685260 + 0.728299i $$0.740311\pi$$
$$854$$ −19.5778 −0.669938
$$855$$ 0 0
$$856$$ −0.697224 −0.0238306
$$857$$ − 42.8444i − 1.46354i −0.681553 0.731769i $$-0.738695\pi$$
0.681553 0.731769i $$-0.261305\pi$$
$$858$$ 9.90833i 0.338265i
$$859$$ −48.0555 −1.63963 −0.819816 0.572626i $$-0.805925\pi$$
−0.819816 + 0.572626i $$0.805925\pi$$
$$860$$ 0 0
$$861$$ −85.2666 −2.90588
$$862$$ 9.21110i 0.313731i
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ 16.2111 0.551513
$$865$$ 0 0
$$866$$ −34.9361 −1.18718
$$867$$ − 62.7527i − 2.13119i
$$868$$ 0.788897i 0.0267769i
$$869$$ 21.0000 0.712376
$$870$$ 0 0
$$871$$ −4.57779 −0.155113
$$872$$ 2.00000i 0.0677285i
$$873$$ − 129.872i − 4.39551i
$$874$$ 7.81665 0.264402
$$875$$ 0 0
$$876$$ −40.6333 −1.37287
$$877$$ 7.21110i 0.243502i 0.992561 + 0.121751i $$0.0388509\pi$$
−0.992561 + 0.121751i $$0.961149\pi$$
$$878$$ − 30.3305i − 1.02361i
$$879$$ −36.4222 −1.22849
$$880$$ 0 0
$$881$$ −28.5416 −0.961592 −0.480796 0.876832i $$-0.659652\pi$$
−0.480796 + 0.876832i $$0.659652\pi$$
$$882$$ − 1.66947i − 0.0562139i
$$883$$ 26.4222i 0.889178i 0.895735 + 0.444589i $$0.146650\pi$$
−0.895735 + 0.444589i $$0.853350\pi$$
$$884$$ 7.81665 0.262903
$$885$$ 0 0
$$886$$ −32.7250 −1.09942
$$887$$ 0.422205i 0.0141763i 0.999975 + 0.00708813i $$0.00225624\pi$$
−0.999975 + 0.00708813i $$0.997744\pi$$
$$888$$ − 3.30278i − 0.110834i
$$889$$ −50.0555 −1.67881
$$890$$ 0 0
$$891$$ −68.6611 −2.30023
$$892$$ 5.81665i 0.194756i
$$893$$ 9.21110i 0.308238i
$$894$$ 65.4500 2.18897
$$895$$ 0 0
$$896$$ 2.60555 0.0870454
$$897$$ − 16.8167i − 0.561492i
$$898$$ 15.2111i 0.507601i
$$899$$ −1.18335 −0.0394668
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ − 22.8167i − 0.759711i
$$903$$ − 5.21110i − 0.173415i
$$904$$ −3.21110 −0.106800
$$905$$ 0 0
$$906$$ 68.0555 2.26099
$$907$$ − 26.0000i − 0.863316i −0.902037 0.431658i $$-0.857929\pi$$
0.902037 0.431658i $$-0.142071\pi$$
$$908$$ 13.8167i 0.458522i
$$909$$ 98.2389 3.25838
$$910$$ 0 0
$$911$$ −17.5778 −0.582378 −0.291189 0.956665i $$-0.594051\pi$$
−0.291189 + 0.956665i $$0.594051\pi$$
$$912$$ − 6.60555i − 0.218732i
$$913$$ − 6.42221i − 0.212544i
$$914$$ −2.60555 −0.0861840
$$915$$ 0 0
$$916$$ 24.6056 0.812990
$$917$$ 27.6333i 0.912532i
$$918$$ 97.2666i 3.21028i
$$919$$ 9.57779 0.315942 0.157971 0.987444i $$-0.449505\pi$$
0.157971 + 0.987444i $$0.449505\pi$$
$$920$$ 0 0
$$921$$ 59.1472 1.94897
$$922$$ 12.4222i 0.409104i
$$923$$ 7.81665i 0.257288i
$$924$$ −19.8167 −0.651920
$$925$$ 0 0
$$926$$ −26.6972 −0.877325
$$927$$ 2.39445i 0.0786440i
$$928$$ 3.90833i 0.128297i
$$929$$ 18.4861 0.606510 0.303255 0.952909i $$-0.401927\pi$$
0.303255 + 0.952909i $$0.401927\pi$$
$$930$$ 0 0
$$931$$ −0.422205 −0.0138372
$$932$$ − 8.51388i − 0.278881i
$$933$$ 52.5416i 1.72014i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ −10.3028 −0.336757
$$937$$ 18.0917i 0.591029i 0.955338 + 0.295515i $$0.0954911\pi$$
−0.955338 + 0.295515i $$0.904509\pi$$
$$938$$ − 9.15559i − 0.298941i
$$939$$ 29.8167 0.973030
$$940$$ 0 0
$$941$$ 13.8167 0.450410 0.225205 0.974311i $$-0.427695\pi$$
0.225205 + 0.974311i $$0.427695\pi$$
$$942$$ − 23.8167i − 0.775989i
$$943$$ 38.7250i 1.26106i
$$944$$ −10.6056 −0.345181
$$945$$ 0 0
$$946$$ 1.39445 0.0453374
$$947$$ 3.63331i 0.118067i 0.998256 + 0.0590333i $$0.0188018\pi$$
−0.998256 + 0.0590333i $$0.981198\pi$$
$$948$$ 30.1194i 0.978234i
$$949$$ 16.0278 0.520283
$$950$$ 0 0
$$951$$ 30.4222 0.986508
$$952$$ 15.6333i 0.506678i
$$953$$ 49.7527i 1.61165i 0.592154 + 0.805825i $$0.298278\pi$$
−0.592154 + 0.805825i $$0.701722\pi$$
$$954$$ −47.4500 −1.53625
$$955$$ 0 0
$$956$$ −17.5139 −0.566439
$$957$$ − 29.7250i − 0.960872i
$$958$$ 13.1194i 0.423870i
$$959$$ 2.36669 0.0764245
$$960$$ 0 0
$$961$$ −30.9083 −0.997043
$$962$$ 1.30278i 0.0420032i
$$963$$ 5.51388i 0.177682i
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ 33.6333 1.08213
$$967$$ 6.72498i 0.216261i 0.994137 + 0.108130i $$0.0344864\pi$$
−0.994137 + 0.108130i $$0.965514\pi$$
$$968$$ 5.69722i 0.183116i
$$969$$ 39.6333 1.27321
$$970$$ 0 0
$$971$$ −22.5416 −0.723395 −0.361698 0.932295i $$-0.617803\pi$$
−0.361698 + 0.932295i $$0.617803\pi$$
$$972$$ − 49.8444i − 1.59876i
$$973$$ 4.97224i 0.159403i
$$974$$ −37.2111 −1.19232
$$975$$ 0 0
$$976$$ 7.51388 0.240513
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 27.8167i − 0.889479i
$$979$$ −21.2111 −0.677910
$$980$$ 0 0
$$981$$ 15.8167 0.504987
$$982$$ 17.7250i 0.565627i
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 32.7250 1.04323
$$985$$ 0 0
$$986$$ −23.4500 −0.746799
$$987$$ 39.6333i 1.26154i
$$988$$ 2.60555i 0.0828936i
$$989$$ −2.36669 −0.0752564
$$990$$ 0 0
$$991$$ 50.6972 1.61045 0.805225 0.592969i $$-0.202044\pi$$
0.805225 + 0.592969i $$0.202044\pi$$
$$992$$ − 0.302776i − 0.00961314i
$$993$$ − 43.6333i − 1.38466i
$$994$$ −15.6333 −0.495858
$$995$$ 0 0
$$996$$ 9.21110 0.291865
$$997$$ 52.4222i 1.66023i 0.557594 + 0.830114i $$0.311725\pi$$
−0.557594 + 0.830114i $$0.688275\pi$$
$$998$$ 42.2389i 1.33705i
$$999$$ −16.2111 −0.512897
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.i.149.4 4
5.2 odd 4 74.2.a.a.1.2 2
5.3 odd 4 1850.2.a.u.1.1 2
5.4 even 2 inner 1850.2.b.i.149.1 4
15.2 even 4 666.2.a.j.1.2 2
20.7 even 4 592.2.a.f.1.1 2
35.27 even 4 3626.2.a.a.1.1 2
40.27 even 4 2368.2.a.ba.1.2 2
40.37 odd 4 2368.2.a.s.1.1 2
55.32 even 4 8954.2.a.p.1.2 2
60.47 odd 4 5328.2.a.bf.1.2 2
185.147 odd 4 2738.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 5.2 odd 4
592.2.a.f.1.1 2 20.7 even 4
666.2.a.j.1.2 2 15.2 even 4
1850.2.a.u.1.1 2 5.3 odd 4
1850.2.b.i.149.1 4 5.4 even 2 inner
1850.2.b.i.149.4 4 1.1 even 1 trivial
2368.2.a.s.1.1 2 40.37 odd 4
2368.2.a.ba.1.2 2 40.27 even 4
2738.2.a.l.1.2 2 185.147 odd 4
3626.2.a.a.1.1 2 35.27 even 4
5328.2.a.bf.1.2 2 60.47 odd 4
8954.2.a.p.1.2 2 55.32 even 4