# Properties

 Label 3850.2.c.q.1849.2 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.2 Root $$0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.q.1849.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.47214 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.47214 q^{9} +1.00000 q^{11} -1.23607i q^{12} -3.23607i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.47214i q^{17} -1.47214i q^{18} +7.23607 q^{19} +1.23607 q^{21} -1.00000i q^{22} +4.00000i q^{23} -1.23607 q^{24} -3.23607 q^{26} +5.52786i q^{27} +1.00000i q^{28} -4.47214 q^{29} +2.00000 q^{31} -1.00000i q^{32} +1.23607i q^{33} -2.47214 q^{34} -1.47214 q^{36} -6.94427i q^{37} -7.23607i q^{38} +4.00000 q^{39} -2.47214 q^{41} -1.23607i q^{42} -10.4721i q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000i q^{47} +1.23607i q^{48} -1.00000 q^{49} +3.05573 q^{51} +3.23607i q^{52} +8.47214i q^{53} +5.52786 q^{54} +1.00000 q^{56} +8.94427i q^{57} +4.47214i q^{58} -2.76393 q^{59} -0.763932 q^{61} -2.00000i q^{62} -1.47214i q^{63} -1.00000 q^{64} +1.23607 q^{66} -11.4164i q^{67} +2.47214i q^{68} -4.94427 q^{69} +6.47214 q^{71} +1.47214i q^{72} +12.9443i q^{73} -6.94427 q^{74} -7.23607 q^{76} -1.00000i q^{77} -4.00000i q^{78} -2.41641 q^{81} +2.47214i q^{82} -12.1803i q^{83} -1.23607 q^{84} -10.4721 q^{86} -5.52786i q^{87} +1.00000i q^{88} -10.0000 q^{89} -3.23607 q^{91} -4.00000i q^{92} +2.47214i q^{93} +2.00000 q^{94} +1.23607 q^{96} -12.4721i q^{97} +1.00000i q^{98} +1.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9 $$4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9 + 4 * q^11 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 4 * q^21 + 4 * q^24 - 4 * q^26 + 8 * q^31 + 8 * q^34 + 12 * q^36 + 16 * q^39 + 8 * q^41 - 4 * q^44 + 16 * q^46 - 4 * q^49 + 48 * q^51 + 40 * q^54 + 4 * q^56 - 20 * q^59 - 12 * q^61 - 4 * q^64 - 4 * q^66 + 16 * q^69 + 8 * q^71 + 8 * q^74 - 20 * q^76 + 44 * q^81 + 4 * q^84 - 24 * q^86 - 40 * q^89 - 4 * q^91 + 8 * q^94 - 4 * q^96 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.23607i 0.713644i 0.934172 + 0.356822i $$0.116140\pi$$
−0.934172 + 0.356822i $$0.883860\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.23607 0.504623
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 1.47214 0.490712
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 1.23607i − 0.356822i
$$13$$ − 3.23607i − 0.897524i −0.893651 0.448762i $$-0.851865\pi$$
0.893651 0.448762i $$-0.148135\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.47214i − 0.599581i −0.954005 0.299791i $$-0.903083\pi$$
0.954005 0.299791i $$-0.0969168\pi$$
$$18$$ − 1.47214i − 0.346986i
$$19$$ 7.23607 1.66007 0.830034 0.557713i $$-0.188321\pi$$
0.830034 + 0.557713i $$0.188321\pi$$
$$20$$ 0 0
$$21$$ 1.23607 0.269732
$$22$$ − 1.00000i − 0.213201i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −1.23607 −0.252311
$$25$$ 0 0
$$26$$ −3.23607 −0.634645
$$27$$ 5.52786i 1.06384i
$$28$$ 1.00000i 0.188982i
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 1.23607i 0.215172i
$$34$$ −2.47214 −0.423968
$$35$$ 0 0
$$36$$ −1.47214 −0.245356
$$37$$ − 6.94427i − 1.14163i −0.821078 0.570816i $$-0.806627\pi$$
0.821078 0.570816i $$-0.193373\pi$$
$$38$$ − 7.23607i − 1.17385i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −2.47214 −0.386083 −0.193041 0.981191i $$-0.561835\pi$$
−0.193041 + 0.981191i $$0.561835\pi$$
$$42$$ − 1.23607i − 0.190729i
$$43$$ − 10.4721i − 1.59699i −0.602004 0.798493i $$-0.705631\pi$$
0.602004 0.798493i $$-0.294369\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ 1.23607i 0.178411i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 3.05573 0.427888
$$52$$ 3.23607i 0.448762i
$$53$$ 8.47214i 1.16374i 0.813283 + 0.581869i $$0.197678\pi$$
−0.813283 + 0.581869i $$0.802322\pi$$
$$54$$ 5.52786 0.752247
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 8.94427i 1.18470i
$$58$$ 4.47214i 0.587220i
$$59$$ −2.76393 −0.359833 −0.179917 0.983682i $$-0.557583\pi$$
−0.179917 + 0.983682i $$0.557583\pi$$
$$60$$ 0 0
$$61$$ −0.763932 −0.0978115 −0.0489057 0.998803i $$-0.515573\pi$$
−0.0489057 + 0.998803i $$0.515573\pi$$
$$62$$ − 2.00000i − 0.254000i
$$63$$ − 1.47214i − 0.185472i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 1.23607 0.152149
$$67$$ − 11.4164i − 1.39474i −0.716713 0.697368i $$-0.754354\pi$$
0.716713 0.697368i $$-0.245646\pi$$
$$68$$ 2.47214i 0.299791i
$$69$$ −4.94427 −0.595220
$$70$$ 0 0
$$71$$ 6.47214 0.768101 0.384051 0.923312i $$-0.374529\pi$$
0.384051 + 0.923312i $$0.374529\pi$$
$$72$$ 1.47214i 0.173493i
$$73$$ 12.9443i 1.51501i 0.652828 + 0.757506i $$0.273582\pi$$
−0.652828 + 0.757506i $$0.726418\pi$$
$$74$$ −6.94427 −0.807255
$$75$$ 0 0
$$76$$ −7.23607 −0.830034
$$77$$ − 1.00000i − 0.113961i
$$78$$ − 4.00000i − 0.452911i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ −2.41641 −0.268490
$$82$$ 2.47214i 0.273002i
$$83$$ − 12.1803i − 1.33697i −0.743727 0.668483i $$-0.766944\pi$$
0.743727 0.668483i $$-0.233056\pi$$
$$84$$ −1.23607 −0.134866
$$85$$ 0 0
$$86$$ −10.4721 −1.12924
$$87$$ − 5.52786i − 0.592649i
$$88$$ 1.00000i 0.106600i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −3.23607 −0.339232
$$92$$ − 4.00000i − 0.417029i
$$93$$ 2.47214i 0.256349i
$$94$$ 2.00000 0.206284
$$95$$ 0 0
$$96$$ 1.23607 0.126156
$$97$$ − 12.4721i − 1.26635i −0.774007 0.633177i $$-0.781751\pi$$
0.774007 0.633177i $$-0.218249\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 1.47214 0.147955
$$100$$ 0 0
$$101$$ 8.18034 0.813974 0.406987 0.913434i $$-0.366579\pi$$
0.406987 + 0.913434i $$0.366579\pi$$
$$102$$ − 3.05573i − 0.302562i
$$103$$ − 14.9443i − 1.47250i −0.676708 0.736251i $$-0.736594\pi$$
0.676708 0.736251i $$-0.263406\pi$$
$$104$$ 3.23607 0.317323
$$105$$ 0 0
$$106$$ 8.47214 0.822887
$$107$$ − 2.47214i − 0.238990i −0.992835 0.119495i $$-0.961872\pi$$
0.992835 0.119495i $$-0.0381276\pi$$
$$108$$ − 5.52786i − 0.531919i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 8.58359 0.814719
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 0.472136i − 0.0444148i −0.999753 0.0222074i $$-0.992931\pi$$
0.999753 0.0222074i $$-0.00706942\pi$$
$$114$$ 8.94427 0.837708
$$115$$ 0 0
$$116$$ 4.47214 0.415227
$$117$$ − 4.76393i − 0.440426i
$$118$$ 2.76393i 0.254441i
$$119$$ −2.47214 −0.226620
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0.763932i 0.0691632i
$$123$$ − 3.05573i − 0.275526i
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ −1.47214 −0.131148
$$127$$ 12.0000i 1.06483i 0.846484 + 0.532414i $$0.178715\pi$$
−0.846484 + 0.532414i $$0.821285\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 12.9443 1.13968
$$130$$ 0 0
$$131$$ 4.76393 0.416227 0.208113 0.978105i $$-0.433268\pi$$
0.208113 + 0.978105i $$0.433268\pi$$
$$132$$ − 1.23607i − 0.107586i
$$133$$ − 7.23607i − 0.627447i
$$134$$ −11.4164 −0.986227
$$135$$ 0 0
$$136$$ 2.47214 0.211984
$$137$$ 19.8885i 1.69919i 0.527433 + 0.849596i $$0.323154\pi$$
−0.527433 + 0.849596i $$0.676846\pi$$
$$138$$ 4.94427i 0.420884i
$$139$$ 21.7082 1.84127 0.920633 0.390429i $$-0.127673\pi$$
0.920633 + 0.390429i $$0.127673\pi$$
$$140$$ 0 0
$$141$$ −2.47214 −0.208191
$$142$$ − 6.47214i − 0.543130i
$$143$$ − 3.23607i − 0.270614i
$$144$$ 1.47214 0.122678
$$145$$ 0 0
$$146$$ 12.9443 1.07128
$$147$$ − 1.23607i − 0.101949i
$$148$$ 6.94427i 0.570816i
$$149$$ 22.3607 1.83186 0.915929 0.401340i $$-0.131455\pi$$
0.915929 + 0.401340i $$0.131455\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 7.23607i 0.586923i
$$153$$ − 3.63932i − 0.294222i
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 12.6525i 1.00978i 0.863184 + 0.504889i $$0.168466\pi$$
−0.863184 + 0.504889i $$0.831534\pi$$
$$158$$ 0 0
$$159$$ −10.4721 −0.830494
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 2.41641i 0.189851i
$$163$$ − 19.4164i − 1.52081i −0.649449 0.760405i $$-0.725000\pi$$
0.649449 0.760405i $$-0.275000\pi$$
$$164$$ 2.47214 0.193041
$$165$$ 0 0
$$166$$ −12.1803 −0.945378
$$167$$ − 11.4164i − 0.883428i −0.897156 0.441714i $$-0.854371\pi$$
0.897156 0.441714i $$-0.145629\pi$$
$$168$$ 1.23607i 0.0953647i
$$169$$ 2.52786 0.194451
$$170$$ 0 0
$$171$$ 10.6525 0.814615
$$172$$ 10.4721i 0.798493i
$$173$$ − 3.23607i − 0.246034i −0.992405 0.123017i $$-0.960743\pi$$
0.992405 0.123017i $$-0.0392569\pi$$
$$174$$ −5.52786 −0.419066
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ − 3.41641i − 0.256793i
$$178$$ 10.0000i 0.749532i
$$179$$ 8.94427 0.668526 0.334263 0.942480i $$-0.391513\pi$$
0.334263 + 0.942480i $$0.391513\pi$$
$$180$$ 0 0
$$181$$ 9.23607 0.686512 0.343256 0.939242i $$-0.388470\pi$$
0.343256 + 0.939242i $$0.388470\pi$$
$$182$$ 3.23607i 0.239873i
$$183$$ − 0.944272i − 0.0698026i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 2.47214 0.181266
$$187$$ − 2.47214i − 0.180780i
$$188$$ − 2.00000i − 0.145865i
$$189$$ 5.52786 0.402093
$$190$$ 0 0
$$191$$ −2.47214 −0.178877 −0.0894387 0.995992i $$-0.528507\pi$$
−0.0894387 + 0.995992i $$0.528507\pi$$
$$192$$ − 1.23607i − 0.0892055i
$$193$$ − 14.9443i − 1.07571i −0.843037 0.537856i $$-0.819234\pi$$
0.843037 0.537856i $$-0.180766\pi$$
$$194$$ −12.4721 −0.895447
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ − 1.47214i − 0.104620i
$$199$$ −18.9443 −1.34292 −0.671462 0.741039i $$-0.734333\pi$$
−0.671462 + 0.741039i $$0.734333\pi$$
$$200$$ 0 0
$$201$$ 14.1115 0.995345
$$202$$ − 8.18034i − 0.575567i
$$203$$ 4.47214i 0.313882i
$$204$$ −3.05573 −0.213944
$$205$$ 0 0
$$206$$ −14.9443 −1.04122
$$207$$ 5.88854i 0.409282i
$$208$$ − 3.23607i − 0.224381i
$$209$$ 7.23607 0.500529
$$210$$ 0 0
$$211$$ −13.5279 −0.931297 −0.465648 0.884970i $$-0.654179\pi$$
−0.465648 + 0.884970i $$0.654179\pi$$
$$212$$ − 8.47214i − 0.581869i
$$213$$ 8.00000i 0.548151i
$$214$$ −2.47214 −0.168992
$$215$$ 0 0
$$216$$ −5.52786 −0.376124
$$217$$ − 2.00000i − 0.135769i
$$218$$ − 10.0000i − 0.677285i
$$219$$ −16.0000 −1.08118
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ − 8.58359i − 0.576093i
$$223$$ − 0.472136i − 0.0316166i −0.999875 0.0158083i $$-0.994968\pi$$
0.999875 0.0158083i $$-0.00503214\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −0.472136 −0.0314060
$$227$$ 19.2361i 1.27674i 0.769729 + 0.638371i $$0.220392\pi$$
−0.769729 + 0.638371i $$0.779608\pi$$
$$228$$ − 8.94427i − 0.592349i
$$229$$ −17.2361 −1.13899 −0.569496 0.821994i $$-0.692861\pi$$
−0.569496 + 0.821994i $$0.692861\pi$$
$$230$$ 0 0
$$231$$ 1.23607 0.0813273
$$232$$ − 4.47214i − 0.293610i
$$233$$ − 14.9443i − 0.979032i −0.871994 0.489516i $$-0.837174\pi$$
0.871994 0.489516i $$-0.162826\pi$$
$$234$$ −4.76393 −0.311428
$$235$$ 0 0
$$236$$ 2.76393 0.179917
$$237$$ 0 0
$$238$$ 2.47214i 0.160245i
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ 15.4164 0.993058 0.496529 0.868020i $$-0.334608\pi$$
0.496529 + 0.868020i $$0.334608\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ 13.5967i 0.872232i
$$244$$ 0.763932 0.0489057
$$245$$ 0 0
$$246$$ −3.05573 −0.194826
$$247$$ − 23.4164i − 1.48995i
$$248$$ 2.00000i 0.127000i
$$249$$ 15.0557 0.954118
$$250$$ 0 0
$$251$$ 29.2361 1.84536 0.922682 0.385562i $$-0.125992\pi$$
0.922682 + 0.385562i $$0.125992\pi$$
$$252$$ 1.47214i 0.0927358i
$$253$$ 4.00000i 0.251478i
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 6.94427i − 0.433172i −0.976264 0.216586i $$-0.930508\pi$$
0.976264 0.216586i $$-0.0694921\pi$$
$$258$$ − 12.9443i − 0.805875i
$$259$$ −6.94427 −0.431496
$$260$$ 0 0
$$261$$ −6.58359 −0.407514
$$262$$ − 4.76393i − 0.294317i
$$263$$ − 4.94427i − 0.304877i −0.988313 0.152438i $$-0.951287\pi$$
0.988313 0.152438i $$-0.0487126\pi$$
$$264$$ −1.23607 −0.0760747
$$265$$ 0 0
$$266$$ −7.23607 −0.443672
$$267$$ − 12.3607i − 0.756461i
$$268$$ 11.4164i 0.697368i
$$269$$ 22.7639 1.38794 0.693971 0.720003i $$-0.255860\pi$$
0.693971 + 0.720003i $$0.255860\pi$$
$$270$$ 0 0
$$271$$ 0.944272 0.0573604 0.0286802 0.999589i $$-0.490870\pi$$
0.0286802 + 0.999589i $$0.490870\pi$$
$$272$$ − 2.47214i − 0.149895i
$$273$$ − 4.00000i − 0.242091i
$$274$$ 19.8885 1.20151
$$275$$ 0 0
$$276$$ 4.94427 0.297610
$$277$$ − 3.52786i − 0.211969i −0.994368 0.105984i $$-0.966201\pi$$
0.994368 0.105984i $$-0.0337994\pi$$
$$278$$ − 21.7082i − 1.30197i
$$279$$ 2.94427 0.176269
$$280$$ 0 0
$$281$$ 28.8328 1.72002 0.860011 0.510276i $$-0.170457\pi$$
0.860011 + 0.510276i $$0.170457\pi$$
$$282$$ 2.47214i 0.147214i
$$283$$ 14.6525i 0.870999i 0.900189 + 0.435500i $$0.143428\pi$$
−0.900189 + 0.435500i $$0.856572\pi$$
$$284$$ −6.47214 −0.384051
$$285$$ 0 0
$$286$$ −3.23607 −0.191353
$$287$$ 2.47214i 0.145926i
$$288$$ − 1.47214i − 0.0867464i
$$289$$ 10.8885 0.640503
$$290$$ 0 0
$$291$$ 15.4164 0.903726
$$292$$ − 12.9443i − 0.757506i
$$293$$ − 26.6525i − 1.55705i −0.627611 0.778527i $$-0.715967\pi$$
0.627611 0.778527i $$-0.284033\pi$$
$$294$$ −1.23607 −0.0720889
$$295$$ 0 0
$$296$$ 6.94427 0.403628
$$297$$ 5.52786i 0.320759i
$$298$$ − 22.3607i − 1.29532i
$$299$$ 12.9443 0.748587
$$300$$ 0 0
$$301$$ −10.4721 −0.603604
$$302$$ − 12.0000i − 0.690522i
$$303$$ 10.1115i 0.580888i
$$304$$ 7.23607 0.415017
$$305$$ 0 0
$$306$$ −3.63932 −0.208046
$$307$$ 26.0689i 1.48783i 0.668274 + 0.743915i $$0.267033\pi$$
−0.668274 + 0.743915i $$0.732967\pi$$
$$308$$ 1.00000i 0.0569803i
$$309$$ 18.4721 1.05084
$$310$$ 0 0
$$311$$ −21.4164 −1.21441 −0.607207 0.794544i $$-0.707710\pi$$
−0.607207 + 0.794544i $$0.707710\pi$$
$$312$$ 4.00000i 0.226455i
$$313$$ 19.5279i 1.10378i 0.833917 + 0.551890i $$0.186093\pi$$
−0.833917 + 0.551890i $$0.813907\pi$$
$$314$$ 12.6525 0.714021
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 30.9443i 1.73800i 0.494809 + 0.869002i $$0.335238\pi$$
−0.494809 + 0.869002i $$0.664762\pi$$
$$318$$ 10.4721i 0.587248i
$$319$$ −4.47214 −0.250392
$$320$$ 0 0
$$321$$ 3.05573 0.170554
$$322$$ − 4.00000i − 0.222911i
$$323$$ − 17.8885i − 0.995345i
$$324$$ 2.41641 0.134245
$$325$$ 0 0
$$326$$ −19.4164 −1.07538
$$327$$ 12.3607i 0.683547i
$$328$$ − 2.47214i − 0.136501i
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ −16.9443 −0.931341 −0.465671 0.884958i $$-0.654187\pi$$
−0.465671 + 0.884958i $$0.654187\pi$$
$$332$$ 12.1803i 0.668483i
$$333$$ − 10.2229i − 0.560212i
$$334$$ −11.4164 −0.624678
$$335$$ 0 0
$$336$$ 1.23607 0.0674330
$$337$$ − 18.0000i − 0.980522i −0.871576 0.490261i $$-0.836901\pi$$
0.871576 0.490261i $$-0.163099\pi$$
$$338$$ − 2.52786i − 0.137498i
$$339$$ 0.583592 0.0316964
$$340$$ 0 0
$$341$$ 2.00000 0.108306
$$342$$ − 10.6525i − 0.576020i
$$343$$ 1.00000i 0.0539949i
$$344$$ 10.4721 0.564620
$$345$$ 0 0
$$346$$ −3.23607 −0.173972
$$347$$ − 2.47214i − 0.132711i −0.997796 0.0663556i $$-0.978863\pi$$
0.997796 0.0663556i $$-0.0211372\pi$$
$$348$$ 5.52786i 0.296325i
$$349$$ −21.7082 −1.16201 −0.581007 0.813899i $$-0.697341\pi$$
−0.581007 + 0.813899i $$0.697341\pi$$
$$350$$ 0 0
$$351$$ 17.8885 0.954820
$$352$$ − 1.00000i − 0.0533002i
$$353$$ − 17.0557i − 0.907785i −0.891056 0.453892i $$-0.850035\pi$$
0.891056 0.453892i $$-0.149965\pi$$
$$354$$ −3.41641 −0.181580
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ − 3.05573i − 0.161726i
$$358$$ − 8.94427i − 0.472719i
$$359$$ −26.8328 −1.41618 −0.708091 0.706121i $$-0.750443\pi$$
−0.708091 + 0.706121i $$0.750443\pi$$
$$360$$ 0 0
$$361$$ 33.3607 1.75583
$$362$$ − 9.23607i − 0.485437i
$$363$$ 1.23607i 0.0648767i
$$364$$ 3.23607 0.169616
$$365$$ 0 0
$$366$$ −0.944272 −0.0493579
$$367$$ 5.41641i 0.282734i 0.989957 + 0.141367i $$0.0451498\pi$$
−0.989957 + 0.141367i $$0.954850\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −3.63932 −0.189455
$$370$$ 0 0
$$371$$ 8.47214 0.439851
$$372$$ − 2.47214i − 0.128174i
$$373$$ − 6.00000i − 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ −2.47214 −0.127831
$$375$$ 0 0
$$376$$ −2.00000 −0.103142
$$377$$ 14.4721i 0.745353i
$$378$$ − 5.52786i − 0.284323i
$$379$$ −14.4721 −0.743384 −0.371692 0.928356i $$-0.621222\pi$$
−0.371692 + 0.928356i $$0.621222\pi$$
$$380$$ 0 0
$$381$$ −14.8328 −0.759908
$$382$$ 2.47214i 0.126485i
$$383$$ − 23.8885i − 1.22065i −0.792152 0.610324i $$-0.791039\pi$$
0.792152 0.610324i $$-0.208961\pi$$
$$384$$ −1.23607 −0.0630778
$$385$$ 0 0
$$386$$ −14.9443 −0.760643
$$387$$ − 15.4164i − 0.783660i
$$388$$ 12.4721i 0.633177i
$$389$$ −33.4164 −1.69428 −0.847140 0.531370i $$-0.821677\pi$$
−0.847140 + 0.531370i $$0.821677\pi$$
$$390$$ 0 0
$$391$$ 9.88854 0.500085
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 5.88854i 0.297038i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ −1.47214 −0.0739776
$$397$$ 23.7082i 1.18988i 0.803770 + 0.594940i $$0.202824\pi$$
−0.803770 + 0.594940i $$0.797176\pi$$
$$398$$ 18.9443i 0.949591i
$$399$$ 8.94427 0.447774
$$400$$ 0 0
$$401$$ 14.3607 0.717138 0.358569 0.933503i $$-0.383265\pi$$
0.358569 + 0.933503i $$0.383265\pi$$
$$402$$ − 14.1115i − 0.703815i
$$403$$ − 6.47214i − 0.322400i
$$404$$ −8.18034 −0.406987
$$405$$ 0 0
$$406$$ 4.47214 0.221948
$$407$$ − 6.94427i − 0.344215i
$$408$$ 3.05573i 0.151281i
$$409$$ −3.41641 −0.168930 −0.0844652 0.996426i $$-0.526918\pi$$
−0.0844652 + 0.996426i $$0.526918\pi$$
$$410$$ 0 0
$$411$$ −24.5836 −1.21262
$$412$$ 14.9443i 0.736251i
$$413$$ 2.76393i 0.136004i
$$414$$ 5.88854 0.289406
$$415$$ 0 0
$$416$$ −3.23607 −0.158661
$$417$$ 26.8328i 1.31401i
$$418$$ − 7.23607i − 0.353928i
$$419$$ −17.2361 −0.842037 −0.421019 0.907052i $$-0.638327\pi$$
−0.421019 + 0.907052i $$0.638327\pi$$
$$420$$ 0 0
$$421$$ 16.4721 0.802803 0.401401 0.915902i $$-0.368523\pi$$
0.401401 + 0.915902i $$0.368523\pi$$
$$422$$ 13.5279i 0.658526i
$$423$$ 2.94427i 0.143155i
$$424$$ −8.47214 −0.411443
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 0.763932i 0.0369693i
$$428$$ 2.47214i 0.119495i
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ 23.0557 1.11056 0.555278 0.831665i $$-0.312612\pi$$
0.555278 + 0.831665i $$0.312612\pi$$
$$432$$ 5.52786i 0.265959i
$$433$$ 28.4721i 1.36828i 0.729349 + 0.684142i $$0.239823\pi$$
−0.729349 + 0.684142i $$0.760177\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 28.9443i 1.38459i
$$438$$ 16.0000i 0.764510i
$$439$$ −8.94427 −0.426887 −0.213443 0.976955i $$-0.568468\pi$$
−0.213443 + 0.976955i $$0.568468\pi$$
$$440$$ 0 0
$$441$$ −1.47214 −0.0701017
$$442$$ 8.00000i 0.380521i
$$443$$ − 24.9443i − 1.18514i −0.805520 0.592569i $$-0.798114\pi$$
0.805520 0.592569i $$-0.201886\pi$$
$$444$$ −8.58359 −0.407359
$$445$$ 0 0
$$446$$ −0.472136 −0.0223563
$$447$$ 27.6393i 1.30729i
$$448$$ 1.00000i 0.0472456i
$$449$$ −18.9443 −0.894035 −0.447018 0.894525i $$-0.647514\pi$$
−0.447018 + 0.894525i $$0.647514\pi$$
$$450$$ 0 0
$$451$$ −2.47214 −0.116408
$$452$$ 0.472136i 0.0222074i
$$453$$ 14.8328i 0.696906i
$$454$$ 19.2361 0.902793
$$455$$ 0 0
$$456$$ −8.94427 −0.418854
$$457$$ − 26.9443i − 1.26040i −0.776433 0.630200i $$-0.782973\pi$$
0.776433 0.630200i $$-0.217027\pi$$
$$458$$ 17.2361i 0.805389i
$$459$$ 13.6656 0.637857
$$460$$ 0 0
$$461$$ 24.7639 1.15337 0.576686 0.816966i $$-0.304346\pi$$
0.576686 + 0.816966i $$0.304346\pi$$
$$462$$ − 1.23607i − 0.0575071i
$$463$$ − 30.4721i − 1.41616i −0.706132 0.708080i $$-0.749562\pi$$
0.706132 0.708080i $$-0.250438\pi$$
$$464$$ −4.47214 −0.207614
$$465$$ 0 0
$$466$$ −14.9443 −0.692280
$$467$$ 27.1246i 1.25518i 0.778545 + 0.627589i $$0.215958\pi$$
−0.778545 + 0.627589i $$0.784042\pi$$
$$468$$ 4.76393i 0.220213i
$$469$$ −11.4164 −0.527161
$$470$$ 0 0
$$471$$ −15.6393 −0.720622
$$472$$ − 2.76393i − 0.127220i
$$473$$ − 10.4721i − 0.481509i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.47214 0.113310
$$477$$ 12.4721i 0.571060i
$$478$$ 20.0000i 0.914779i
$$479$$ −12.3607 −0.564774 −0.282387 0.959301i $$-0.591126\pi$$
−0.282387 + 0.959301i $$0.591126\pi$$
$$480$$ 0 0
$$481$$ −22.4721 −1.02464
$$482$$ − 15.4164i − 0.702198i
$$483$$ 4.94427i 0.224972i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 13.5967 0.616761
$$487$$ − 16.9443i − 0.767818i −0.923371 0.383909i $$-0.874578\pi$$
0.923371 0.383909i $$-0.125422\pi$$
$$488$$ − 0.763932i − 0.0345816i
$$489$$ 24.0000 1.08532
$$490$$ 0 0
$$491$$ −16.9443 −0.764684 −0.382342 0.924021i $$-0.624882\pi$$
−0.382342 + 0.924021i $$0.624882\pi$$
$$492$$ 3.05573i 0.137763i
$$493$$ 11.0557i 0.497925i
$$494$$ −23.4164 −1.05355
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ − 6.47214i − 0.290315i
$$498$$ − 15.0557i − 0.674663i
$$499$$ 32.3607 1.44866 0.724331 0.689452i $$-0.242149\pi$$
0.724331 + 0.689452i $$0.242149\pi$$
$$500$$ 0 0
$$501$$ 14.1115 0.630453
$$502$$ − 29.2361i − 1.30487i
$$503$$ 4.00000i 0.178351i 0.996016 + 0.0891756i $$0.0284232\pi$$
−0.996016 + 0.0891756i $$0.971577\pi$$
$$504$$ 1.47214 0.0655741
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ 3.12461i 0.138769i
$$508$$ − 12.0000i − 0.532414i
$$509$$ −24.0689 −1.06683 −0.533417 0.845852i $$-0.679092\pi$$
−0.533417 + 0.845852i $$0.679092\pi$$
$$510$$ 0 0
$$511$$ 12.9443 0.572621
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 40.0000i 1.76604i
$$514$$ −6.94427 −0.306299
$$515$$ 0 0
$$516$$ −12.9443 −0.569840
$$517$$ 2.00000i 0.0879599i
$$518$$ 6.94427i 0.305114i
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ −10.3607 −0.453910 −0.226955 0.973905i $$-0.572877\pi$$
−0.226955 + 0.973905i $$0.572877\pi$$
$$522$$ 6.58359i 0.288156i
$$523$$ − 14.2918i − 0.624937i −0.949928 0.312468i $$-0.898844\pi$$
0.949928 0.312468i $$-0.101156\pi$$
$$524$$ −4.76393 −0.208113
$$525$$ 0 0
$$526$$ −4.94427 −0.215580
$$527$$ − 4.94427i − 0.215376i
$$528$$ 1.23607i 0.0537930i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −4.06888 −0.176575
$$532$$ 7.23607i 0.313723i
$$533$$ 8.00000i 0.346518i
$$534$$ −12.3607 −0.534899
$$535$$ 0 0
$$536$$ 11.4164 0.493114
$$537$$ 11.0557i 0.477090i
$$538$$ − 22.7639i − 0.981423i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −26.9443 −1.15842 −0.579212 0.815177i $$-0.696640\pi$$
−0.579212 + 0.815177i $$0.696640\pi$$
$$542$$ − 0.944272i − 0.0405600i
$$543$$ 11.4164i 0.489925i
$$544$$ −2.47214 −0.105992
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ 0.944272i 0.0403742i 0.999796 + 0.0201871i $$0.00642618\pi$$
−0.999796 + 0.0201871i $$0.993574\pi$$
$$548$$ − 19.8885i − 0.849596i
$$549$$ −1.12461 −0.0479973
$$550$$ 0 0
$$551$$ −32.3607 −1.37861
$$552$$ − 4.94427i − 0.210442i
$$553$$ 0 0
$$554$$ −3.52786 −0.149885
$$555$$ 0 0
$$556$$ −21.7082 −0.920633
$$557$$ − 24.8328i − 1.05220i −0.850423 0.526100i $$-0.823654\pi$$
0.850423 0.526100i $$-0.176346\pi$$
$$558$$ − 2.94427i − 0.124641i
$$559$$ −33.8885 −1.43333
$$560$$ 0 0
$$561$$ 3.05573 0.129013
$$562$$ − 28.8328i − 1.21624i
$$563$$ 31.2361i 1.31644i 0.752824 + 0.658222i $$0.228691\pi$$
−0.752824 + 0.658222i $$0.771309\pi$$
$$564$$ 2.47214 0.104096
$$565$$ 0 0
$$566$$ 14.6525 0.615889
$$567$$ 2.41641i 0.101480i
$$568$$ 6.47214i 0.271565i
$$569$$ 36.8328 1.54411 0.772056 0.635555i $$-0.219228\pi$$
0.772056 + 0.635555i $$0.219228\pi$$
$$570$$ 0 0
$$571$$ −10.1115 −0.423151 −0.211576 0.977362i $$-0.567859\pi$$
−0.211576 + 0.977362i $$0.567859\pi$$
$$572$$ 3.23607i 0.135307i
$$573$$ − 3.05573i − 0.127655i
$$574$$ 2.47214 0.103185
$$575$$ 0 0
$$576$$ −1.47214 −0.0613390
$$577$$ − 26.9443i − 1.12170i −0.827916 0.560852i $$-0.810474\pi$$
0.827916 0.560852i $$-0.189526\pi$$
$$578$$ − 10.8885i − 0.452904i
$$579$$ 18.4721 0.767676
$$580$$ 0 0
$$581$$ −12.1803 −0.505326
$$582$$ − 15.4164i − 0.639031i
$$583$$ 8.47214i 0.350880i
$$584$$ −12.9443 −0.535638
$$585$$ 0 0
$$586$$ −26.6525 −1.10100
$$587$$ 5.81966i 0.240203i 0.992762 + 0.120102i $$0.0383220\pi$$
−0.992762 + 0.120102i $$0.961678\pi$$
$$588$$ 1.23607i 0.0509746i
$$589$$ 14.4721 0.596314
$$590$$ 0 0
$$591$$ 22.2492 0.915211
$$592$$ − 6.94427i − 0.285408i
$$593$$ 24.0000i 0.985562i 0.870153 + 0.492781i $$0.164020\pi$$
−0.870153 + 0.492781i $$0.835980\pi$$
$$594$$ 5.52786 0.226811
$$595$$ 0 0
$$596$$ −22.3607 −0.915929
$$597$$ − 23.4164i − 0.958370i
$$598$$ − 12.9443i − 0.529331i
$$599$$ 32.3607 1.32222 0.661111 0.750288i $$-0.270085\pi$$
0.661111 + 0.750288i $$0.270085\pi$$
$$600$$ 0 0
$$601$$ −34.8328 −1.42086 −0.710430 0.703768i $$-0.751500\pi$$
−0.710430 + 0.703768i $$0.751500\pi$$
$$602$$ 10.4721i 0.426812i
$$603$$ − 16.8065i − 0.684414i
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ 10.1115 0.410750
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ − 7.23607i − 0.293461i
$$609$$ −5.52786 −0.224000
$$610$$ 0 0
$$611$$ 6.47214 0.261835
$$612$$ 3.63932i 0.147111i
$$613$$ 28.4721i 1.14998i 0.818161 + 0.574989i $$0.194994\pi$$
−0.818161 + 0.574989i $$0.805006\pi$$
$$614$$ 26.0689 1.05205
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ − 21.4164i − 0.862192i −0.902306 0.431096i $$-0.858127\pi$$
0.902306 0.431096i $$-0.141873\pi$$
$$618$$ − 18.4721i − 0.743058i
$$619$$ −18.5410 −0.745227 −0.372613 0.927987i $$-0.621538\pi$$
−0.372613 + 0.927987i $$0.621538\pi$$
$$620$$ 0 0
$$621$$ −22.1115 −0.887302
$$622$$ 21.4164i 0.858720i
$$623$$ 10.0000i 0.400642i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ 19.5279 0.780490
$$627$$ 8.94427i 0.357200i
$$628$$ − 12.6525i − 0.504889i
$$629$$ −17.1672 −0.684500
$$630$$ 0 0
$$631$$ −31.4164 −1.25067 −0.625334 0.780357i $$-0.715037\pi$$
−0.625334 + 0.780357i $$0.715037\pi$$
$$632$$ 0 0
$$633$$ − 16.7214i − 0.664614i
$$634$$ 30.9443 1.22895
$$635$$ 0 0
$$636$$ 10.4721 0.415247
$$637$$ 3.23607i 0.128218i
$$638$$ 4.47214i 0.177054i
$$639$$ 9.52786 0.376916
$$640$$ 0 0
$$641$$ 27.5279 1.08729 0.543643 0.839317i $$-0.317045\pi$$
0.543643 + 0.839317i $$0.317045\pi$$
$$642$$ − 3.05573i − 0.120600i
$$643$$ − 18.7639i − 0.739977i −0.929036 0.369989i $$-0.879362\pi$$
0.929036 0.369989i $$-0.120638\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ −17.8885 −0.703815
$$647$$ 28.8328i 1.13353i 0.823878 + 0.566767i $$0.191806\pi$$
−0.823878 + 0.566767i $$0.808194\pi$$
$$648$$ − 2.41641i − 0.0949255i
$$649$$ −2.76393 −0.108494
$$650$$ 0 0
$$651$$ 2.47214 0.0968906
$$652$$ 19.4164i 0.760405i
$$653$$ 46.3607i 1.81423i 0.420879 + 0.907117i $$0.361722\pi$$
−0.420879 + 0.907117i $$0.638278\pi$$
$$654$$ 12.3607 0.483341
$$655$$ 0 0
$$656$$ −2.47214 −0.0965207
$$657$$ 19.0557i 0.743435i
$$658$$ − 2.00000i − 0.0779681i
$$659$$ −16.5836 −0.646005 −0.323003 0.946398i $$-0.604692\pi$$
−0.323003 + 0.946398i $$0.604692\pi$$
$$660$$ 0 0
$$661$$ −3.12461 −0.121533 −0.0607667 0.998152i $$-0.519355\pi$$
−0.0607667 + 0.998152i $$0.519355\pi$$
$$662$$ 16.9443i 0.658558i
$$663$$ − 9.88854i − 0.384039i
$$664$$ 12.1803 0.472689
$$665$$ 0 0
$$666$$ −10.2229 −0.396130
$$667$$ − 17.8885i − 0.692647i
$$668$$ 11.4164i 0.441714i
$$669$$ 0.583592 0.0225630
$$670$$ 0 0
$$671$$ −0.763932 −0.0294913
$$672$$ − 1.23607i − 0.0476824i
$$673$$ − 3.88854i − 0.149892i −0.997188 0.0749462i $$-0.976122\pi$$
0.997188 0.0749462i $$-0.0238785\pi$$
$$674$$ −18.0000 −0.693334
$$675$$ 0 0
$$676$$ −2.52786 −0.0972255
$$677$$ 26.0689i 1.00191i 0.865474 + 0.500954i $$0.167018\pi$$
−0.865474 + 0.500954i $$0.832982\pi$$
$$678$$ − 0.583592i − 0.0224127i
$$679$$ −12.4721 −0.478637
$$680$$ 0 0
$$681$$ −23.7771 −0.911140
$$682$$ − 2.00000i − 0.0765840i
$$683$$ 32.9443i 1.26058i 0.776361 + 0.630289i $$0.217064\pi$$
−0.776361 + 0.630289i $$0.782936\pi$$
$$684$$ −10.6525 −0.407308
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ − 21.3050i − 0.812835i
$$688$$ − 10.4721i − 0.399246i
$$689$$ 27.4164 1.04448
$$690$$ 0 0
$$691$$ 12.6525 0.481323 0.240661 0.970609i $$-0.422636\pi$$
0.240661 + 0.970609i $$0.422636\pi$$
$$692$$ 3.23607i 0.123017i
$$693$$ − 1.47214i − 0.0559218i
$$694$$ −2.47214 −0.0938410
$$695$$ 0 0
$$696$$ 5.52786 0.209533
$$697$$ 6.11146i 0.231488i
$$698$$ 21.7082i 0.821668i
$$699$$ 18.4721 0.698680
$$700$$ 0 0
$$701$$ −42.7214 −1.61356 −0.806782 0.590850i $$-0.798793\pi$$
−0.806782 + 0.590850i $$0.798793\pi$$
$$702$$ − 17.8885i − 0.675160i
$$703$$ − 50.2492i − 1.89519i
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −17.0557 −0.641901
$$707$$ − 8.18034i − 0.307653i
$$708$$ 3.41641i 0.128396i
$$709$$ −4.47214 −0.167955 −0.0839773 0.996468i $$-0.526762\pi$$
−0.0839773 + 0.996468i $$0.526762\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 10.0000i − 0.374766i
$$713$$ 8.00000i 0.299602i
$$714$$ −3.05573 −0.114358
$$715$$ 0 0
$$716$$ −8.94427 −0.334263
$$717$$ − 24.7214i − 0.923236i
$$718$$ 26.8328i 1.00139i
$$719$$ −16.8328 −0.627758 −0.313879 0.949463i $$-0.601629\pi$$
−0.313879 + 0.949463i $$0.601629\pi$$
$$720$$ 0 0
$$721$$ −14.9443 −0.556554
$$722$$ − 33.3607i − 1.24156i
$$723$$ 19.0557i 0.708690i
$$724$$ −9.23607 −0.343256
$$725$$ 0 0
$$726$$ 1.23607 0.0458748
$$727$$ − 18.0000i − 0.667583i −0.942647 0.333792i $$-0.891672\pi$$
0.942647 0.333792i $$-0.108328\pi$$
$$728$$ − 3.23607i − 0.119937i
$$729$$ −24.0557 −0.890953
$$730$$ 0 0
$$731$$ −25.8885 −0.957522
$$732$$ 0.944272i 0.0349013i
$$733$$ 49.1246i 1.81446i 0.420636 + 0.907229i $$0.361807\pi$$
−0.420636 + 0.907229i $$0.638193\pi$$
$$734$$ 5.41641 0.199923
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 11.4164i − 0.420529i
$$738$$ 3.63932i 0.133965i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 28.9443 1.06329
$$742$$ − 8.47214i − 0.311022i
$$743$$ 21.8885i 0.803013i 0.915856 + 0.401506i $$0.131513\pi$$
−0.915856 + 0.401506i $$0.868487\pi$$
$$744$$ −2.47214 −0.0906329
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ − 17.9311i − 0.656065i
$$748$$ 2.47214i 0.0903902i
$$749$$ −2.47214 −0.0903299
$$750$$ 0 0
$$751$$ −16.9443 −0.618305 −0.309153 0.951012i $$-0.600045\pi$$
−0.309153 + 0.951012i $$0.600045\pi$$
$$752$$ 2.00000i 0.0729325i
$$753$$ 36.1378i 1.31693i
$$754$$ 14.4721 0.527044
$$755$$ 0 0
$$756$$ −5.52786 −0.201046
$$757$$ 23.3050i 0.847033i 0.905888 + 0.423516i $$0.139204\pi$$
−0.905888 + 0.423516i $$0.860796\pi$$
$$758$$ 14.4721i 0.525652i
$$759$$ −4.94427 −0.179466
$$760$$ 0 0
$$761$$ −11.4164 −0.413844 −0.206922 0.978357i $$-0.566345\pi$$
−0.206922 + 0.978357i $$0.566345\pi$$
$$762$$ 14.8328i 0.537336i
$$763$$ − 10.0000i − 0.362024i
$$764$$ 2.47214 0.0894387
$$765$$ 0 0
$$766$$ −23.8885 −0.863128
$$767$$ 8.94427i 0.322959i
$$768$$ 1.23607i 0.0446028i
$$769$$ 43.4164 1.56564 0.782818 0.622251i $$-0.213782\pi$$
0.782818 + 0.622251i $$0.213782\pi$$
$$770$$ 0 0
$$771$$ 8.58359 0.309131
$$772$$ 14.9443i 0.537856i
$$773$$ 15.7082i 0.564985i 0.959270 + 0.282492i $$0.0911612\pi$$
−0.959270 + 0.282492i $$0.908839\pi$$
$$774$$ −15.4164 −0.554131
$$775$$ 0 0
$$776$$ 12.4721 0.447724
$$777$$ − 8.58359i − 0.307935i
$$778$$ 33.4164i 1.19804i
$$779$$ −17.8885 −0.640924
$$780$$ 0 0
$$781$$ 6.47214 0.231591
$$782$$ − 9.88854i − 0.353614i
$$783$$ − 24.7214i − 0.883469i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 5.88854 0.210037
$$787$$ 28.1803i 1.00452i 0.864716 + 0.502260i $$0.167498\pi$$
−0.864716 + 0.502260i $$0.832502\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 6.11146 0.217574
$$790$$ 0 0
$$791$$ −0.472136 −0.0167872
$$792$$ 1.47214i 0.0523101i
$$793$$ 2.47214i 0.0877881i
$$794$$ 23.7082 0.841373
$$795$$ 0 0
$$796$$ 18.9443 0.671462
$$797$$ 41.5967i 1.47343i 0.676202 + 0.736716i $$0.263625\pi$$
−0.676202 + 0.736716i $$0.736375\pi$$
$$798$$ − 8.94427i − 0.316624i
$$799$$ 4.94427 0.174916
$$800$$ 0 0
$$801$$ −14.7214 −0.520154
$$802$$ − 14.3607i − 0.507093i
$$803$$ 12.9443i 0.456793i
$$804$$ −14.1115 −0.497673
$$805$$ 0 0
$$806$$ −6.47214 −0.227971
$$807$$ 28.1378i 0.990496i
$$808$$ 8.18034i 0.287783i
$$809$$ 21.0557 0.740280 0.370140 0.928976i $$-0.379310\pi$$
0.370140 + 0.928976i $$0.379310\pi$$
$$810$$ 0 0
$$811$$ 4.76393 0.167284 0.0836421 0.996496i $$-0.473345\pi$$
0.0836421 + 0.996496i $$0.473345\pi$$
$$812$$ − 4.47214i − 0.156941i
$$813$$ 1.16718i 0.0409349i
$$814$$ −6.94427 −0.243397
$$815$$ 0 0
$$816$$ 3.05573 0.106972
$$817$$ − 75.7771i − 2.65110i
$$818$$ 3.41641i 0.119452i
$$819$$ −4.76393 −0.166465
$$820$$ 0 0
$$821$$ −1.41641 −0.0494330 −0.0247165 0.999695i $$-0.507868\pi$$
−0.0247165 + 0.999695i $$0.507868\pi$$
$$822$$ 24.5836i 0.857451i
$$823$$ − 46.2492i − 1.61215i −0.591816 0.806073i $$-0.701589\pi$$
0.591816 0.806073i $$-0.298411\pi$$
$$824$$ 14.9443 0.520608
$$825$$ 0 0
$$826$$ 2.76393 0.0961695
$$827$$ − 16.9443i − 0.589210i −0.955619 0.294605i $$-0.904812\pi$$
0.955619 0.294605i $$-0.0951881\pi$$
$$828$$ − 5.88854i − 0.204641i
$$829$$ −11.7082 −0.406643 −0.203321 0.979112i $$-0.565174\pi$$
−0.203321 + 0.979112i $$0.565174\pi$$
$$830$$ 0 0
$$831$$ 4.36068 0.151270
$$832$$ 3.23607i 0.112190i
$$833$$ 2.47214i 0.0856544i
$$834$$ 26.8328 0.929144
$$835$$ 0 0
$$836$$ −7.23607 −0.250265
$$837$$ 11.0557i 0.382142i
$$838$$ 17.2361i 0.595410i
$$839$$ −16.8328 −0.581133 −0.290567 0.956855i $$-0.593844\pi$$
−0.290567 + 0.956855i $$0.593844\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ − 16.4721i − 0.567667i
$$843$$ 35.6393i 1.22748i
$$844$$ 13.5279 0.465648
$$845$$ 0 0
$$846$$ 2.94427 0.101226
$$847$$ − 1.00000i − 0.0343604i
$$848$$ 8.47214i 0.290934i
$$849$$ −18.1115 −0.621584
$$850$$ 0 0
$$851$$ 27.7771 0.952186
$$852$$ − 8.00000i − 0.274075i
$$853$$ 32.5410i 1.11418i 0.830451 + 0.557092i $$0.188083\pi$$
−0.830451 + 0.557092i $$0.811917\pi$$
$$854$$ 0.763932 0.0261412
$$855$$ 0 0
$$856$$ 2.47214 0.0844959
$$857$$ 46.4721i 1.58746i 0.608272 + 0.793729i $$0.291863\pi$$
−0.608272 + 0.793729i $$0.708137\pi$$
$$858$$ − 4.00000i − 0.136558i
$$859$$ −15.1246 −0.516045 −0.258023 0.966139i $$-0.583071\pi$$
−0.258023 + 0.966139i $$0.583071\pi$$
$$860$$ 0 0
$$861$$ −3.05573 −0.104139
$$862$$ − 23.0557i − 0.785281i
$$863$$ 0.583592i 0.0198657i 0.999951 + 0.00993285i $$0.00316178\pi$$
−0.999951 + 0.00993285i $$0.996838\pi$$
$$864$$ 5.52786 0.188062
$$865$$ 0 0
$$866$$ 28.4721 0.967523
$$867$$ 13.4590i 0.457091i
$$868$$ 2.00000i 0.0678844i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −36.9443 −1.25181
$$872$$ 10.0000i 0.338643i
$$873$$ − 18.3607i − 0.621415i
$$874$$ 28.9443 0.979055
$$875$$ 0 0
$$876$$ 16.0000 0.540590
$$877$$ − 9.05573i − 0.305790i −0.988242 0.152895i $$-0.951140\pi$$
0.988242 0.152895i $$-0.0488597\pi$$
$$878$$ 8.94427i 0.301855i
$$879$$ 32.9443 1.11118
$$880$$ 0 0
$$881$$ 28.8328 0.971402 0.485701 0.874125i $$-0.338564\pi$$
0.485701 + 0.874125i $$0.338564\pi$$
$$882$$ 1.47214i 0.0495694i
$$883$$ − 2.83282i − 0.0953318i −0.998863 0.0476659i $$-0.984822\pi$$
0.998863 0.0476659i $$-0.0151783\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ −24.9443 −0.838019
$$887$$ 44.3607i 1.48949i 0.667351 + 0.744743i $$0.267428\pi$$
−0.667351 + 0.744743i $$0.732572\pi$$
$$888$$ 8.58359i 0.288046i
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ −2.41641 −0.0809527
$$892$$ 0.472136i 0.0158083i
$$893$$ 14.4721i 0.484292i
$$894$$ 27.6393 0.924397
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 16.0000i 0.534224i
$$898$$ 18.9443i 0.632179i
$$899$$ −8.94427 −0.298308
$$900$$ 0 0
$$901$$ 20.9443 0.697755
$$902$$ 2.47214i 0.0823131i
$$903$$ − 12.9443i − 0.430758i
$$904$$ 0.472136 0.0157030
$$905$$ 0 0
$$906$$ 14.8328 0.492787
$$907$$ 24.3607i 0.808883i 0.914564 + 0.404442i $$0.132534\pi$$
−0.914564 + 0.404442i $$0.867466\pi$$
$$908$$ − 19.2361i − 0.638371i
$$909$$ 12.0426 0.399427
$$910$$ 0 0
$$911$$ −28.0000 −0.927681 −0.463841 0.885919i $$-0.653529\pi$$
−0.463841 + 0.885919i $$0.653529\pi$$
$$912$$ 8.94427i 0.296174i
$$913$$ − 12.1803i − 0.403110i
$$914$$ −26.9443 −0.891237
$$915$$ 0 0
$$916$$ 17.2361 0.569496
$$917$$ − 4.76393i − 0.157319i
$$918$$ − 13.6656i − 0.451033i
$$919$$ 22.1115 0.729390 0.364695 0.931127i $$-0.381173\pi$$
0.364695 + 0.931127i $$0.381173\pi$$
$$920$$ 0 0
$$921$$ −32.2229 −1.06178
$$922$$ − 24.7639i − 0.815557i
$$923$$ − 20.9443i − 0.689389i
$$924$$ −1.23607 −0.0406637
$$925$$ 0 0
$$926$$ −30.4721 −1.00138
$$927$$ − 22.0000i − 0.722575i
$$928$$ 4.47214i 0.146805i
$$929$$ −40.2492 −1.32053 −0.660267 0.751031i $$-0.729557\pi$$
−0.660267 + 0.751031i $$0.729557\pi$$
$$930$$ 0 0
$$931$$ −7.23607 −0.237153
$$932$$ 14.9443i 0.489516i
$$933$$ − 26.4721i − 0.866659i
$$934$$ 27.1246 0.887544
$$935$$ 0 0
$$936$$ 4.76393 0.155714
$$937$$ 3.05573i 0.0998263i 0.998754 + 0.0499131i $$0.0158945\pi$$
−0.998754 + 0.0499131i $$0.984106\pi$$
$$938$$ 11.4164i 0.372759i
$$939$$ −24.1378 −0.787706
$$940$$ 0 0
$$941$$ −11.8197 −0.385310 −0.192655 0.981267i $$-0.561710\pi$$
−0.192655 + 0.981267i $$0.561710\pi$$
$$942$$ 15.6393i 0.509557i
$$943$$ − 9.88854i − 0.322015i
$$944$$ −2.76393 −0.0899583
$$945$$ 0 0
$$946$$ −10.4721 −0.340479
$$947$$ − 16.9443i − 0.550615i −0.961356 0.275307i $$-0.911220\pi$$
0.961356 0.275307i $$-0.0887796\pi$$
$$948$$ 0 0
$$949$$ 41.8885 1.35976
$$950$$ 0 0
$$951$$ −38.2492 −1.24032
$$952$$ − 2.47214i − 0.0801224i
$$953$$ 22.9443i 0.743238i 0.928385 + 0.371619i $$0.121197\pi$$
−0.928385 + 0.371619i $$0.878803\pi$$
$$954$$ 12.4721 0.403800
$$955$$ 0 0
$$956$$ 20.0000 0.646846
$$957$$ − 5.52786i − 0.178690i
$$958$$ 12.3607i 0.399355i
$$959$$ 19.8885 0.642235
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 22.4721i 0.724531i
$$963$$ − 3.63932i − 0.117275i
$$964$$ −15.4164 −0.496529
$$965$$ 0 0
$$966$$ 4.94427 0.159079
$$967$$ − 45.8885i − 1.47568i −0.674978 0.737838i $$-0.735847\pi$$
0.674978 0.737838i $$-0.264153\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ 22.1115 0.710322
$$970$$ 0 0
$$971$$ 50.5410 1.62194 0.810969 0.585089i $$-0.198940\pi$$
0.810969 + 0.585089i $$0.198940\pi$$
$$972$$ − 13.5967i − 0.436116i
$$973$$ − 21.7082i − 0.695933i
$$974$$ −16.9443 −0.542929
$$975$$ 0 0
$$976$$ −0.763932 −0.0244529
$$977$$ 28.8328i 0.922443i 0.887285 + 0.461222i $$0.152589\pi$$
−0.887285 + 0.461222i $$0.847411\pi$$
$$978$$ − 24.0000i − 0.767435i
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 14.7214 0.470017
$$982$$ 16.9443i 0.540713i
$$983$$ 14.0000i 0.446531i 0.974758 + 0.223265i $$0.0716716\pi$$
−0.974758 + 0.223265i $$0.928328\pi$$
$$984$$ 3.05573 0.0974131
$$985$$ 0 0
$$986$$ 11.0557 0.352086
$$987$$ 2.47214i 0.0786890i
$$988$$ 23.4164i 0.744975i
$$989$$ 41.8885 1.33198
$$990$$ 0 0
$$991$$ −0.360680 −0.0114574 −0.00572869 0.999984i $$-0.501824\pi$$
−0.00572869 + 0.999984i $$0.501824\pi$$
$$992$$ − 2.00000i − 0.0635001i
$$993$$ − 20.9443i − 0.664646i
$$994$$ −6.47214 −0.205284
$$995$$ 0 0
$$996$$ −15.0557 −0.477059
$$997$$ − 24.1803i − 0.765799i −0.923790 0.382900i $$-0.874926\pi$$
0.923790 0.382900i $$-0.125074\pi$$
$$998$$ − 32.3607i − 1.02436i
$$999$$ 38.3870 1.21451
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 3850.2.c.q.1849.2 4
5.2 odd 4 154.2.a.d.1.2 2
5.3 odd 4 3850.2.a.bj.1.1 2
5.4 even 2 inner 3850.2.c.q.1849.3 4
15.2 even 4 1386.2.a.m.1.2 2
20.7 even 4 1232.2.a.p.1.1 2
35.2 odd 12 1078.2.e.q.67.1 4
35.12 even 12 1078.2.e.n.67.2 4
35.17 even 12 1078.2.e.n.177.2 4
35.27 even 4 1078.2.a.w.1.1 2
35.32 odd 12 1078.2.e.q.177.1 4
40.27 even 4 4928.2.a.bk.1.2 2
40.37 odd 4 4928.2.a.bt.1.1 2
55.32 even 4 1694.2.a.l.1.2 2
105.62 odd 4 9702.2.a.cu.1.1 2
140.27 odd 4 8624.2.a.bf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.2 2 5.2 odd 4
1078.2.a.w.1.1 2 35.27 even 4
1078.2.e.n.67.2 4 35.12 even 12
1078.2.e.n.177.2 4 35.17 even 12
1078.2.e.q.67.1 4 35.2 odd 12
1078.2.e.q.177.1 4 35.32 odd 12
1232.2.a.p.1.1 2 20.7 even 4
1386.2.a.m.1.2 2 15.2 even 4
1694.2.a.l.1.2 2 55.32 even 4
3850.2.a.bj.1.1 2 5.3 odd 4
3850.2.c.q.1849.2 4 1.1 even 1 trivial
3850.2.c.q.1849.3 4 5.4 even 2 inner
4928.2.a.bk.1.2 2 40.27 even 4
4928.2.a.bt.1.1 2 40.37 odd 4
8624.2.a.bf.1.2 2 140.27 odd 4
9702.2.a.cu.1.1 2 105.62 odd 4