# Properties

 Label 950.2.b.h.799.6 Level $950$ Weight $2$ Character 950.799 Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.63107136.1 Defining polynomial: $$x^{6} + 13 x^{4} + 42 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.6 Root $$2.77339i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.h.799.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.77339i q^{3} -1.00000 q^{4} -1.77339 q^{6} -2.69168i q^{7} -1.00000i q^{8} -0.144903 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.77339i q^{3} -1.00000 q^{4} -1.77339 q^{6} -2.69168i q^{7} -1.00000i q^{8} -0.144903 q^{9} +5.54677 q^{11} -1.77339i q^{12} +2.91829i q^{13} +2.69168 q^{14} +1.00000 q^{16} +4.91829i q^{17} -0.144903i q^{18} -1.00000 q^{19} +4.77339 q^{21} +5.54677i q^{22} -3.60997i q^{23} +1.77339 q^{24} -2.91829 q^{26} +5.06319i q^{27} +2.69168i q^{28} -1.08171 q^{29} +7.54677 q^{31} +1.00000i q^{32} +9.83658i q^{33} -4.91829 q^{34} +0.144903 q^{36} +4.54677i q^{37} -1.00000i q^{38} -5.17526 q^{39} +4.77339i q^{42} +9.54677i q^{43} -5.54677 q^{44} +3.60997 q^{46} -0.836581i q^{47} +1.77339i q^{48} -0.245129 q^{49} -8.72203 q^{51} -2.91829i q^{52} -9.78523i q^{53} -5.06319 q^{54} -2.69168 q^{56} -1.77339i q^{57} -1.08171i q^{58} -12.9933 q^{59} -7.38336 q^{61} +7.54677i q^{62} +0.390032i q^{63} -1.00000 q^{64} -9.83658 q^{66} +2.85510i q^{67} -4.91829i q^{68} +6.40187 q^{69} +14.4769 q^{71} +0.144903i q^{72} -5.15674i q^{73} -4.54677 q^{74} +1.00000 q^{76} -14.9301i q^{77} -5.17526i q^{78} +3.09355 q^{79} -9.41371 q^{81} -1.71019i q^{83} -4.77339 q^{84} -9.54677 q^{86} -1.91829i q^{87} -5.54677i q^{88} +5.09355 q^{89} +7.85510 q^{91} +3.60997i q^{92} +13.3834i q^{93} +0.836581 q^{94} -1.77339 q^{96} -17.2570i q^{97} -0.245129i q^{98} -0.803744 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} + 4q^{6} - 10q^{9} + O(q^{10})$$ $$6q - 6q^{4} + 4q^{6} - 10q^{9} + 4q^{11} - 4q^{14} + 6q^{16} - 6q^{19} + 14q^{21} - 4q^{24} - 12q^{26} - 12q^{29} + 16q^{31} - 24q^{34} + 10q^{36} + 22q^{39} - 4q^{44} - 4q^{46} - 18q^{49} + 30q^{51} - 34q^{54} + 4q^{56} - 12q^{59} - 4q^{61} - 6q^{64} - 48q^{66} - 12q^{71} + 2q^{74} + 6q^{76} - 40q^{79} + 46q^{81} - 14q^{84} - 28q^{86} - 28q^{89} + 38q^{91} - 6q^{94} + 4q^{96} + 72q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$3$$ 1.77339i 1.02387i 0.859025 + 0.511933i $$0.171070\pi$$
−0.859025 + 0.511933i $$0.828930\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.77339 −0.723982
$$7$$ − 2.69168i − 1.01736i −0.860956 0.508679i $$-0.830134\pi$$
0.860956 0.508679i $$-0.169866\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −0.144903 −0.0483010
$$10$$ 0 0
$$11$$ 5.54677 1.67242 0.836208 0.548413i $$-0.184768\pi$$
0.836208 + 0.548413i $$0.184768\pi$$
$$12$$ − 1.77339i − 0.511933i
$$13$$ 2.91829i 0.809388i 0.914452 + 0.404694i $$0.132622\pi$$
−0.914452 + 0.404694i $$0.867378\pi$$
$$14$$ 2.69168 0.719381
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.91829i 1.19286i 0.802665 + 0.596430i $$0.203415\pi$$
−0.802665 + 0.596430i $$0.796585\pi$$
$$18$$ − 0.144903i − 0.0341539i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 4.77339 1.04164
$$22$$ 5.54677i 1.18258i
$$23$$ − 3.60997i − 0.752730i −0.926471 0.376365i $$-0.877174\pi$$
0.926471 0.376365i $$-0.122826\pi$$
$$24$$ 1.77339 0.361991
$$25$$ 0 0
$$26$$ −2.91829 −0.572324
$$27$$ 5.06319i 0.974412i
$$28$$ 2.69168i 0.508679i
$$29$$ −1.08171 −0.200868 −0.100434 0.994944i $$-0.532023\pi$$
−0.100434 + 0.994944i $$0.532023\pi$$
$$30$$ 0 0
$$31$$ 7.54677 1.35544 0.677720 0.735320i $$-0.262968\pi$$
0.677720 + 0.735320i $$0.262968\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 9.83658i 1.71233i
$$34$$ −4.91829 −0.843480
$$35$$ 0 0
$$36$$ 0.144903 0.0241505
$$37$$ 4.54677i 0.747485i 0.927532 + 0.373743i $$0.121926\pi$$
−0.927532 + 0.373743i $$0.878074\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ −5.17526 −0.828705
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 4.77339i 0.736550i
$$43$$ 9.54677i 1.45587i 0.685646 + 0.727935i $$0.259520\pi$$
−0.685646 + 0.727935i $$0.740480\pi$$
$$44$$ −5.54677 −0.836208
$$45$$ 0 0
$$46$$ 3.60997 0.532261
$$47$$ − 0.836581i − 0.122028i −0.998137 0.0610139i $$-0.980567\pi$$
0.998137 0.0610139i $$-0.0194334\pi$$
$$48$$ 1.77339i 0.255966i
$$49$$ −0.245129 −0.0350184
$$50$$ 0 0
$$51$$ −8.72203 −1.22133
$$52$$ − 2.91829i − 0.404694i
$$53$$ − 9.78523i − 1.34410i −0.740504 0.672052i $$-0.765413\pi$$
0.740504 0.672052i $$-0.234587\pi$$
$$54$$ −5.06319 −0.689013
$$55$$ 0 0
$$56$$ −2.69168 −0.359691
$$57$$ − 1.77339i − 0.234891i
$$58$$ − 1.08171i − 0.142035i
$$59$$ −12.9933 −1.69159 −0.845793 0.533511i $$-0.820872\pi$$
−0.845793 + 0.533511i $$0.820872\pi$$
$$60$$ 0 0
$$61$$ −7.38336 −0.945342 −0.472671 0.881239i $$-0.656710\pi$$
−0.472671 + 0.881239i $$0.656710\pi$$
$$62$$ 7.54677i 0.958441i
$$63$$ 0.390032i 0.0491394i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −9.83658 −1.21080
$$67$$ 2.85510i 0.348806i 0.984674 + 0.174403i $$0.0557995\pi$$
−0.984674 + 0.174403i $$0.944201\pi$$
$$68$$ − 4.91829i − 0.596430i
$$69$$ 6.40187 0.770695
$$70$$ 0 0
$$71$$ 14.4769 1.71809 0.859046 0.511898i $$-0.171057\pi$$
0.859046 + 0.511898i $$0.171057\pi$$
$$72$$ 0.144903i 0.0170770i
$$73$$ − 5.15674i − 0.603551i −0.953379 0.301776i $$-0.902421\pi$$
0.953379 0.301776i $$-0.0975793\pi$$
$$74$$ −4.54677 −0.528552
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ − 14.9301i − 1.70145i
$$78$$ − 5.17526i − 0.585983i
$$79$$ 3.09355 0.348052 0.174026 0.984741i $$-0.444322\pi$$
0.174026 + 0.984741i $$0.444322\pi$$
$$80$$ 0 0
$$81$$ −9.41371 −1.04597
$$82$$ 0 0
$$83$$ − 1.71019i − 0.187718i −0.995585 0.0938591i $$-0.970080\pi$$
0.995585 0.0938591i $$-0.0299203\pi$$
$$84$$ −4.77339 −0.520819
$$85$$ 0 0
$$86$$ −9.54677 −1.02946
$$87$$ − 1.91829i − 0.205662i
$$88$$ − 5.54677i − 0.591288i
$$89$$ 5.09355 0.539915 0.269958 0.962872i $$-0.412990\pi$$
0.269958 + 0.962872i $$0.412990\pi$$
$$90$$ 0 0
$$91$$ 7.85510 0.823438
$$92$$ 3.60997i 0.376365i
$$93$$ 13.3834i 1.38779i
$$94$$ 0.836581 0.0862867
$$95$$ 0 0
$$96$$ −1.77339 −0.180996
$$97$$ − 17.2570i − 1.75218i −0.482148 0.876090i $$-0.660143\pi$$
0.482148 0.876090i $$-0.339857\pi$$
$$98$$ − 0.245129i − 0.0247618i
$$99$$ −0.803744 −0.0807793
$$100$$ 0 0
$$101$$ −9.38336 −0.933679 −0.466839 0.884342i $$-0.654607\pi$$
−0.466839 + 0.884342i $$0.654607\pi$$
$$102$$ − 8.72203i − 0.863610i
$$103$$ 12.9301i 1.27404i 0.770846 + 0.637022i $$0.219834\pi$$
−0.770846 + 0.637022i $$0.780166\pi$$
$$104$$ 2.91829 0.286162
$$105$$ 0 0
$$106$$ 9.78523 0.950425
$$107$$ 18.9420i 1.83119i 0.402102 + 0.915595i $$0.368280\pi$$
−0.402102 + 0.915595i $$0.631720\pi$$
$$108$$ − 5.06319i − 0.487206i
$$109$$ 2.69168 0.257816 0.128908 0.991657i $$-0.458853\pi$$
0.128908 + 0.991657i $$0.458853\pi$$
$$110$$ 0 0
$$111$$ −8.06319 −0.765324
$$112$$ − 2.69168i − 0.254340i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 1.77339 0.166093
$$115$$ 0 0
$$116$$ 1.08171 0.100434
$$117$$ − 0.422869i − 0.0390942i
$$118$$ − 12.9933i − 1.19613i
$$119$$ 13.2385 1.21357
$$120$$ 0 0
$$121$$ 19.7667 1.79697
$$122$$ − 7.38336i − 0.668458i
$$123$$ 0 0
$$124$$ −7.54677 −0.677720
$$125$$ 0 0
$$126$$ −0.390032 −0.0347468
$$127$$ − 1.87361i − 0.166256i −0.996539 0.0831282i $$-0.973509\pi$$
0.996539 0.0831282i $$-0.0264911\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −16.9301 −1.49061
$$130$$ 0 0
$$131$$ −11.5468 −1.00885 −0.504423 0.863457i $$-0.668295\pi$$
−0.504423 + 0.863457i $$0.668295\pi$$
$$132$$ − 9.83658i − 0.856164i
$$133$$ 2.69168i 0.233398i
$$134$$ −2.85510 −0.246643
$$135$$ 0 0
$$136$$ 4.91829 0.421740
$$137$$ 4.23845i 0.362115i 0.983472 + 0.181058i $$0.0579521\pi$$
−0.983472 + 0.181058i $$0.942048\pi$$
$$138$$ 6.40187i 0.544964i
$$139$$ −9.21994 −0.782025 −0.391012 0.920385i $$-0.627875\pi$$
−0.391012 + 0.920385i $$0.627875\pi$$
$$140$$ 0 0
$$141$$ 1.48358 0.124940
$$142$$ 14.4769i 1.21487i
$$143$$ 16.1871i 1.35363i
$$144$$ −0.144903 −0.0120752
$$145$$ 0 0
$$146$$ 5.15674 0.426775
$$147$$ − 0.434709i − 0.0358542i
$$148$$ − 4.54677i − 0.373743i
$$149$$ 7.25697 0.594514 0.297257 0.954797i $$-0.403928\pi$$
0.297257 + 0.954797i $$0.403928\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 0.712675i − 0.0576163i
$$154$$ 14.9301 1.20310
$$155$$ 0 0
$$156$$ 5.17526 0.414352
$$157$$ − 9.54677i − 0.761916i −0.924592 0.380958i $$-0.875594\pi$$
0.924592 0.380958i $$-0.124406\pi$$
$$158$$ 3.09355i 0.246110i
$$159$$ 17.3530 1.37618
$$160$$ 0 0
$$161$$ −9.71687 −0.765797
$$162$$ − 9.41371i − 0.739611i
$$163$$ − 15.7102i − 1.23052i −0.788325 0.615259i $$-0.789052\pi$$
0.788325 0.615259i $$-0.210948\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 1.71019 0.132737
$$167$$ 6.45323i 0.499366i 0.968328 + 0.249683i $$0.0803263\pi$$
−0.968328 + 0.249683i $$0.919674\pi$$
$$168$$ − 4.77339i − 0.368275i
$$169$$ 4.48358 0.344891
$$170$$ 0 0
$$171$$ 0.144903 0.0110810
$$172$$ − 9.54677i − 0.727935i
$$173$$ 0.873614i 0.0664196i 0.999448 + 0.0332098i $$0.0105730\pi$$
−0.999448 + 0.0332098i $$0.989427\pi$$
$$174$$ 1.91829 0.145425
$$175$$ 0 0
$$176$$ 5.54677 0.418104
$$177$$ − 23.0422i − 1.73196i
$$178$$ 5.09355i 0.381778i
$$179$$ 12.3834 0.925575 0.462788 0.886469i $$-0.346849\pi$$
0.462788 + 0.886469i $$0.346849\pi$$
$$180$$ 0 0
$$181$$ −17.6403 −1.31119 −0.655597 0.755111i $$-0.727583\pi$$
−0.655597 + 0.755111i $$0.727583\pi$$
$$182$$ 7.85510i 0.582259i
$$183$$ − 13.0935i − 0.967903i
$$184$$ −3.60997 −0.266130
$$185$$ 0 0
$$186$$ −13.3834 −0.981315
$$187$$ 27.2806i 1.99496i
$$188$$ 0.836581i 0.0610139i
$$189$$ 13.6285 0.991326
$$190$$ 0 0
$$191$$ −15.1567 −1.09670 −0.548352 0.836248i $$-0.684744\pi$$
−0.548352 + 0.836248i $$0.684744\pi$$
$$192$$ − 1.77339i − 0.127983i
$$193$$ − 23.3834i − 1.68317i −0.540124 0.841585i $$-0.681623\pi$$
0.540124 0.841585i $$-0.318377\pi$$
$$194$$ 17.2570 1.23898
$$195$$ 0 0
$$196$$ 0.245129 0.0175092
$$197$$ − 23.0935i − 1.64535i −0.568514 0.822674i $$-0.692481\pi$$
0.568514 0.822674i $$-0.307519\pi$$
$$198$$ − 0.803744i − 0.0571196i
$$199$$ 25.7852 1.82787 0.913933 0.405865i $$-0.133030\pi$$
0.913933 + 0.405865i $$0.133030\pi$$
$$200$$ 0 0
$$201$$ −5.06319 −0.357130
$$202$$ − 9.38336i − 0.660211i
$$203$$ 2.91161i 0.204355i
$$204$$ 8.72203 0.610665
$$205$$ 0 0
$$206$$ −12.9301 −0.900885
$$207$$ 0.523095i 0.0363576i
$$208$$ 2.91829i 0.202347i
$$209$$ −5.54677 −0.383678
$$210$$ 0 0
$$211$$ 14.2266 0.979400 0.489700 0.871891i $$-0.337106\pi$$
0.489700 + 0.871891i $$0.337106\pi$$
$$212$$ 9.78523i 0.672052i
$$213$$ 25.6732i 1.75910i
$$214$$ −18.9420 −1.29485
$$215$$ 0 0
$$216$$ 5.06319 0.344507
$$217$$ − 20.3135i − 1.37897i
$$218$$ 2.69168i 0.182303i
$$219$$ 9.14490 0.617955
$$220$$ 0 0
$$221$$ −14.3530 −0.965487
$$222$$ − 8.06319i − 0.541166i
$$223$$ 4.45323i 0.298210i 0.988821 + 0.149105i $$0.0476392\pi$$
−0.988821 + 0.149105i $$0.952361\pi$$
$$224$$ 2.69168 0.179845
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 5.94865i 0.394826i 0.980320 + 0.197413i $$0.0632539\pi$$
−0.980320 + 0.197413i $$0.936746\pi$$
$$228$$ 1.77339i 0.117445i
$$229$$ −1.09355 −0.0722638 −0.0361319 0.999347i $$-0.511504\pi$$
−0.0361319 + 0.999347i $$0.511504\pi$$
$$230$$ 0 0
$$231$$ 26.4769 1.74205
$$232$$ 1.08171i 0.0710177i
$$233$$ − 14.0935i − 0.923299i −0.887062 0.461650i $$-0.847258\pi$$
0.887062 0.461650i $$-0.152742\pi$$
$$234$$ 0.422869 0.0276438
$$235$$ 0 0
$$236$$ 12.9933 0.845793
$$237$$ 5.48606i 0.356358i
$$238$$ 13.2385i 0.858121i
$$239$$ −15.6100 −1.00972 −0.504862 0.863200i $$-0.668457\pi$$
−0.504862 + 0.863200i $$0.668457\pi$$
$$240$$ 0 0
$$241$$ −5.21994 −0.336246 −0.168123 0.985766i $$-0.553771\pi$$
−0.168123 + 0.985766i $$0.553771\pi$$
$$242$$ 19.7667i 1.27065i
$$243$$ − 1.50458i − 0.0965188i
$$244$$ 7.38336 0.472671
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 2.91829i − 0.185686i
$$248$$ − 7.54677i − 0.479221i
$$249$$ 3.03284 0.192198
$$250$$ 0 0
$$251$$ −28.6403 −1.80776 −0.903881 0.427785i $$-0.859294\pi$$
−0.903881 + 0.427785i $$0.859294\pi$$
$$252$$ − 0.390032i − 0.0245697i
$$253$$ − 20.0237i − 1.25888i
$$254$$ 1.87361 0.117561
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.45323i 0.402541i 0.979536 + 0.201271i $$0.0645070\pi$$
−0.979536 + 0.201271i $$0.935493\pi$$
$$258$$ − 16.9301i − 1.05402i
$$259$$ 12.2385 0.760460
$$260$$ 0 0
$$261$$ 0.156743 0.00970214
$$262$$ − 11.5468i − 0.713362i
$$263$$ − 22.1871i − 1.36812i −0.729428 0.684058i $$-0.760214\pi$$
0.729428 0.684058i $$-0.239786\pi$$
$$264$$ 9.83658 0.605400
$$265$$ 0 0
$$266$$ −2.69168 −0.165037
$$267$$ 9.03284i 0.552801i
$$268$$ − 2.85510i − 0.174403i
$$269$$ 20.4070 1.24424 0.622119 0.782922i $$-0.286272\pi$$
0.622119 + 0.782922i $$0.286272\pi$$
$$270$$ 0 0
$$271$$ 15.9368 0.968092 0.484046 0.875043i $$-0.339167\pi$$
0.484046 + 0.875043i $$0.339167\pi$$
$$272$$ 4.91829i 0.298215i
$$273$$ 13.9301i 0.843090i
$$274$$ −4.23845 −0.256054
$$275$$ 0 0
$$276$$ −6.40187 −0.385347
$$277$$ 4.02368i 0.241759i 0.992667 + 0.120880i $$0.0385715\pi$$
−0.992667 + 0.120880i $$0.961428\pi$$
$$278$$ − 9.21994i − 0.552975i
$$279$$ −1.09355 −0.0654691
$$280$$ 0 0
$$281$$ 1.67316 0.0998124 0.0499062 0.998754i $$-0.484108\pi$$
0.0499062 + 0.998754i $$0.484108\pi$$
$$282$$ 1.48358i 0.0883460i
$$283$$ − 16.8931i − 1.00419i −0.864812 0.502095i $$-0.832563\pi$$
0.864812 0.502095i $$-0.167437\pi$$
$$284$$ −14.4769 −0.859046
$$285$$ 0 0
$$286$$ −16.1871 −0.957163
$$287$$ 0 0
$$288$$ − 0.144903i − 0.00853849i
$$289$$ −7.18958 −0.422916
$$290$$ 0 0
$$291$$ 30.6033 1.79400
$$292$$ 5.15674i 0.301776i
$$293$$ 1.08171i 0.0631942i 0.999501 + 0.0315971i $$0.0100593\pi$$
−0.999501 + 0.0315971i $$0.989941\pi$$
$$294$$ 0.434709 0.0253527
$$295$$ 0 0
$$296$$ 4.54677 0.264276
$$297$$ 28.0844i 1.62962i
$$298$$ 7.25697i 0.420385i
$$299$$ 10.5349 0.609251
$$300$$ 0 0
$$301$$ 25.6968 1.48114
$$302$$ 20.0000i 1.15087i
$$303$$ − 16.6403i − 0.955962i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0.712675 0.0407409
$$307$$ 2.38336i 0.136025i 0.997684 + 0.0680126i $$0.0216658\pi$$
−0.997684 + 0.0680126i $$0.978334\pi$$
$$308$$ 14.9301i 0.850723i
$$309$$ −22.9301 −1.30445
$$310$$ 0 0
$$311$$ −29.4584 −1.67043 −0.835216 0.549922i $$-0.814657\pi$$
−0.835216 + 0.549922i $$0.814657\pi$$
$$312$$ 5.17526i 0.292991i
$$313$$ − 32.0355i − 1.81075i −0.424608 0.905377i $$-0.639588\pi$$
0.424608 0.905377i $$-0.360412\pi$$
$$314$$ 9.54677 0.538756
$$315$$ 0 0
$$316$$ −3.09355 −0.174026
$$317$$ − 26.6665i − 1.49774i −0.662717 0.748870i $$-0.730597\pi$$
0.662717 0.748870i $$-0.269403\pi$$
$$318$$ 17.3530i 0.973108i
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −33.5915 −1.87489
$$322$$ − 9.71687i − 0.541500i
$$323$$ − 4.91829i − 0.273661i
$$324$$ 9.41371 0.522984
$$325$$ 0 0
$$326$$ 15.7102 0.870107
$$327$$ 4.77339i 0.263969i
$$328$$ 0 0
$$329$$ −2.25181 −0.124146
$$330$$ 0 0
$$331$$ −19.8721 −1.09227 −0.546135 0.837697i $$-0.683901\pi$$
−0.546135 + 0.837697i $$0.683901\pi$$
$$332$$ 1.71019i 0.0938591i
$$333$$ − 0.658841i − 0.0361043i
$$334$$ −6.45323 −0.353105
$$335$$ 0 0
$$336$$ 4.77339 0.260410
$$337$$ 27.5705i 1.50186i 0.660383 + 0.750929i $$0.270394\pi$$
−0.660383 + 0.750929i $$0.729606\pi$$
$$338$$ 4.48358i 0.243875i
$$339$$ −10.6403 −0.577903
$$340$$ 0 0
$$341$$ 41.8603 2.26686
$$342$$ 0.144903i 0.00783545i
$$343$$ − 18.1819i − 0.981732i
$$344$$ 9.54677 0.514728
$$345$$ 0 0
$$346$$ −0.873614 −0.0469658
$$347$$ − 27.2806i − 1.46450i −0.681035 0.732251i $$-0.738470\pi$$
0.681035 0.732251i $$-0.261530\pi$$
$$348$$ 1.91829i 0.102831i
$$349$$ −30.9538 −1.65692 −0.828460 0.560049i $$-0.810782\pi$$
−0.828460 + 0.560049i $$0.810782\pi$$
$$350$$ 0 0
$$351$$ −14.7759 −0.788677
$$352$$ 5.54677i 0.295644i
$$353$$ − 11.1819i − 0.595154i −0.954698 0.297577i $$-0.903821\pi$$
0.954698 0.297577i $$-0.0961786\pi$$
$$354$$ 23.0422 1.22468
$$355$$ 0 0
$$356$$ −5.09355 −0.269958
$$357$$ 23.4769i 1.24253i
$$358$$ 12.3834i 0.654481i
$$359$$ −20.3387 −1.07343 −0.536717 0.843762i $$-0.680336\pi$$
−0.536717 + 0.843762i $$0.680336\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 17.6403i − 0.927155i
$$363$$ 35.0540i 1.83986i
$$364$$ −7.85510 −0.411719
$$365$$ 0 0
$$366$$ 13.0935 0.684411
$$367$$ − 32.9538i − 1.72017i −0.510147 0.860087i $$-0.670409\pi$$
0.510147 0.860087i $$-0.329591\pi$$
$$368$$ − 3.60997i − 0.188183i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −26.3387 −1.36744
$$372$$ − 13.3834i − 0.693895i
$$373$$ 10.0869i 0.522278i 0.965301 + 0.261139i $$0.0840982\pi$$
−0.965301 + 0.261139i $$0.915902\pi$$
$$374$$ −27.2806 −1.41065
$$375$$ 0 0
$$376$$ −0.836581 −0.0431434
$$377$$ − 3.15674i − 0.162581i
$$378$$ 13.6285i 0.700974i
$$379$$ 18.4256 0.946457 0.473229 0.880940i $$-0.343088\pi$$
0.473229 + 0.880940i $$0.343088\pi$$
$$380$$ 0 0
$$381$$ 3.32264 0.170224
$$382$$ − 15.1567i − 0.775486i
$$383$$ − 28.1871i − 1.44029i −0.693822 0.720147i $$-0.744074\pi$$
0.693822 0.720147i $$-0.255926\pi$$
$$384$$ 1.77339 0.0904978
$$385$$ 0 0
$$386$$ 23.3834 1.19018
$$387$$ − 1.38336i − 0.0703199i
$$388$$ 17.2570i 0.876090i
$$389$$ −20.0237 −1.01524 −0.507620 0.861581i $$-0.669475\pi$$
−0.507620 + 0.861581i $$0.669475\pi$$
$$390$$ 0 0
$$391$$ 17.7549 0.897902
$$392$$ 0.245129i 0.0123809i
$$393$$ − 20.4769i − 1.03292i
$$394$$ 23.0935 1.16344
$$395$$ 0 0
$$396$$ 0.803744 0.0403896
$$397$$ − 8.32684i − 0.417912i −0.977925 0.208956i $$-0.932993\pi$$
0.977925 0.208956i $$-0.0670066\pi$$
$$398$$ 25.7852i 1.29250i
$$399$$ −4.77339 −0.238968
$$400$$ 0 0
$$401$$ −22.1501 −1.10612 −0.553061 0.833141i $$-0.686540\pi$$
−0.553061 + 0.833141i $$0.686540\pi$$
$$402$$ − 5.06319i − 0.252529i
$$403$$ 22.0237i 1.09708i
$$404$$ 9.38336 0.466839
$$405$$ 0 0
$$406$$ −2.91161 −0.144501
$$407$$ 25.2199i 1.25011i
$$408$$ 8.72203i 0.431805i
$$409$$ −34.0237 −1.68236 −0.841181 0.540753i $$-0.818139\pi$$
−0.841181 + 0.540753i $$0.818139\pi$$
$$410$$ 0 0
$$411$$ −7.51642 −0.370758
$$412$$ − 12.9301i − 0.637022i
$$413$$ 34.9738i 1.72095i
$$414$$ −0.523095 −0.0257087
$$415$$ 0 0
$$416$$ −2.91829 −0.143081
$$417$$ − 16.3505i − 0.800688i
$$418$$ − 5.54677i − 0.271302i
$$419$$ −37.5334 −1.83363 −0.916814 0.399315i $$-0.869248\pi$$
−0.916814 + 0.399315i $$0.869248\pi$$
$$420$$ 0 0
$$421$$ 33.8341 1.64897 0.824487 0.565882i $$-0.191464\pi$$
0.824487 + 0.565882i $$0.191464\pi$$
$$422$$ 14.2266i 0.692541i
$$423$$ 0.121223i 0.00589406i
$$424$$ −9.78523 −0.475213
$$425$$ 0 0
$$426$$ −25.6732 −1.24387
$$427$$ 19.8736i 0.961752i
$$428$$ − 18.9420i − 0.915595i
$$429$$ −28.7060 −1.38594
$$430$$ 0 0
$$431$$ 24.8037 1.19475 0.597377 0.801960i $$-0.296210\pi$$
0.597377 + 0.801960i $$0.296210\pi$$
$$432$$ 5.06319i 0.243603i
$$433$$ − 1.68651i − 0.0810487i −0.999179 0.0405244i $$-0.987097\pi$$
0.999179 0.0405244i $$-0.0129028\pi$$
$$434$$ 20.3135 0.975079
$$435$$ 0 0
$$436$$ −2.69168 −0.128908
$$437$$ 3.60997i 0.172688i
$$438$$ 9.14490i 0.436960i
$$439$$ −0.326839 −0.0155992 −0.00779958 0.999970i $$-0.502483\pi$$
−0.00779958 + 0.999970i $$0.502483\pi$$
$$440$$ 0 0
$$441$$ 0.0355199 0.00169142
$$442$$ − 14.3530i − 0.682703i
$$443$$ − 0.490258i − 0.0232929i −0.999932 0.0116464i $$-0.996293\pi$$
0.999932 0.0116464i $$-0.00370726\pi$$
$$444$$ 8.06319 0.382662
$$445$$ 0 0
$$446$$ −4.45323 −0.210866
$$447$$ 12.8694i 0.608703i
$$448$$ 2.69168i 0.127170i
$$449$$ −23.5468 −1.11124 −0.555621 0.831436i $$-0.687519\pi$$
−0.555621 + 0.831436i $$0.687519\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ 35.4677i 1.66642i
$$454$$ −5.94865 −0.279184
$$455$$ 0 0
$$456$$ −1.77339 −0.0830465
$$457$$ − 8.01184i − 0.374778i −0.982286 0.187389i $$-0.939997\pi$$
0.982286 0.187389i $$-0.0600025\pi$$
$$458$$ − 1.09355i − 0.0510982i
$$459$$ −24.9023 −1.16234
$$460$$ 0 0
$$461$$ 3.83658 0.178687 0.0893437 0.996001i $$-0.471523\pi$$
0.0893437 + 0.996001i $$0.471523\pi$$
$$462$$ 26.4769i 1.23182i
$$463$$ 6.58381i 0.305975i 0.988228 + 0.152988i $$0.0488895\pi$$
−0.988228 + 0.152988i $$0.951110\pi$$
$$464$$ −1.08171 −0.0502171
$$465$$ 0 0
$$466$$ 14.0935 0.652871
$$467$$ 22.6033i 1.04596i 0.852346 + 0.522978i $$0.175179\pi$$
−0.852346 + 0.522978i $$0.824821\pi$$
$$468$$ 0.422869i 0.0195471i
$$469$$ 7.68500 0.354860
$$470$$ 0 0
$$471$$ 16.9301 0.780099
$$472$$ 12.9933i 0.598066i
$$473$$ 52.9538i 2.43482i
$$474$$ −5.48606 −0.251983
$$475$$ 0 0
$$476$$ −13.2385 −0.606783
$$477$$ 1.41791i 0.0649215i
$$478$$ − 15.6100i − 0.713983i
$$479$$ 35.7904 1.63530 0.817652 0.575712i $$-0.195275\pi$$
0.817652 + 0.575712i $$0.195275\pi$$
$$480$$ 0 0
$$481$$ −13.2688 −0.605006
$$482$$ − 5.21994i − 0.237762i
$$483$$ − 17.2318i − 0.784073i
$$484$$ −19.7667 −0.898487
$$485$$ 0 0
$$486$$ 1.50458 0.0682491
$$487$$ − 21.4070i − 0.970045i −0.874502 0.485023i $$-0.838811\pi$$
0.874502 0.485023i $$-0.161189\pi$$
$$488$$ 7.38336i 0.334229i
$$489$$ 27.8603 1.25988
$$490$$ 0 0
$$491$$ −4.32684 −0.195267 −0.0976337 0.995222i $$-0.531127\pi$$
−0.0976337 + 0.995222i $$0.531127\pi$$
$$492$$ 0 0
$$493$$ − 5.32016i − 0.239608i
$$494$$ 2.91829 0.131300
$$495$$ 0 0
$$496$$ 7.54677 0.338860
$$497$$ − 38.9672i − 1.74792i
$$498$$ 3.03284i 0.135905i
$$499$$ 37.6968 1.68754 0.843771 0.536703i $$-0.180330\pi$$
0.843771 + 0.536703i $$0.180330\pi$$
$$500$$ 0 0
$$501$$ −11.4441 −0.511283
$$502$$ − 28.6403i − 1.27828i
$$503$$ 16.9183i 0.754349i 0.926142 + 0.377175i $$0.123104\pi$$
−0.926142 + 0.377175i $$0.876896\pi$$
$$504$$ 0.390032 0.0173734
$$505$$ 0 0
$$506$$ 20.0237 0.890161
$$507$$ 7.95113i 0.353122i
$$508$$ 1.87361i 0.0831282i
$$509$$ 6.79955 0.301385 0.150692 0.988581i $$-0.451850\pi$$
0.150692 + 0.988581i $$0.451850\pi$$
$$510$$ 0 0
$$511$$ −13.8803 −0.614028
$$512$$ 1.00000i 0.0441942i
$$513$$ − 5.06319i − 0.223545i
$$514$$ −6.45323 −0.284640
$$515$$ 0 0
$$516$$ 16.9301 0.745307
$$517$$ − 4.64032i − 0.204081i
$$518$$ 12.2385i 0.537727i
$$519$$ −1.54926 −0.0680048
$$520$$ 0 0
$$521$$ −1.35968 −0.0595685 −0.0297842 0.999556i $$-0.509482\pi$$
−0.0297842 + 0.999556i $$0.509482\pi$$
$$522$$ 0.156743i 0.00686045i
$$523$$ 21.9486i 0.959747i 0.877338 + 0.479874i $$0.159318\pi$$
−0.877338 + 0.479874i $$0.840682\pi$$
$$524$$ 11.5468 0.504423
$$525$$ 0 0
$$526$$ 22.1871 0.967404
$$527$$ 37.1172i 1.61685i
$$528$$ 9.83658i 0.428082i
$$529$$ 9.96813 0.433397
$$530$$ 0 0
$$531$$ 1.88277 0.0817053
$$532$$ − 2.69168i − 0.116699i
$$533$$ 0 0
$$534$$ −9.03284 −0.390889
$$535$$ 0 0
$$536$$ 2.85510 0.123321
$$537$$ 21.9605i 0.947665i
$$538$$ 20.4070i 0.879810i
$$539$$ −1.35968 −0.0585654
$$540$$ 0 0
$$541$$ −39.8973 −1.71532 −0.857659 0.514218i $$-0.828082\pi$$
−0.857659 + 0.514218i $$0.828082\pi$$
$$542$$ 15.9368i 0.684544i
$$543$$ − 31.2831i − 1.34249i
$$544$$ −4.91829 −0.210870
$$545$$ 0 0
$$546$$ −13.9301 −0.596155
$$547$$ − 7.34632i − 0.314106i −0.987590 0.157053i $$-0.949801\pi$$
0.987590 0.157053i $$-0.0501994\pi$$
$$548$$ − 4.23845i − 0.181058i
$$549$$ 1.06987 0.0456609
$$550$$ 0 0
$$551$$ 1.08171 0.0460824
$$552$$ − 6.40187i − 0.272482i
$$553$$ − 8.32684i − 0.354093i
$$554$$ −4.02368 −0.170950
$$555$$ 0 0
$$556$$ 9.21994 0.391012
$$557$$ − 26.9672i − 1.14264i −0.820729 0.571318i $$-0.806432\pi$$
0.820729 0.571318i $$-0.193568\pi$$
$$558$$ − 1.09355i − 0.0462936i
$$559$$ −27.8603 −1.17836
$$560$$ 0 0
$$561$$ −48.3792 −2.04257
$$562$$ 1.67316i 0.0705780i
$$563$$ 44.8973i 1.89220i 0.323881 + 0.946098i $$0.395012\pi$$
−0.323881 + 0.946098i $$0.604988\pi$$
$$564$$ −1.48358 −0.0624701
$$565$$ 0 0
$$566$$ 16.8931 0.710070
$$567$$ 25.3387i 1.06412i
$$568$$ − 14.4769i − 0.607437i
$$569$$ −31.1172 −1.30450 −0.652251 0.758003i $$-0.726175\pi$$
−0.652251 + 0.758003i $$0.726175\pi$$
$$570$$ 0 0
$$571$$ 33.2570 1.39176 0.695880 0.718158i $$-0.255014\pi$$
0.695880 + 0.718158i $$0.255014\pi$$
$$572$$ − 16.1871i − 0.676817i
$$573$$ − 26.8788i − 1.12288i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0.144903 0.00603762
$$577$$ 23.4702i 0.977078i 0.872542 + 0.488539i $$0.162470\pi$$
−0.872542 + 0.488539i $$0.837530\pi$$
$$578$$ − 7.18958i − 0.299047i
$$579$$ 41.4677 1.72334
$$580$$ 0 0
$$581$$ −4.60329 −0.190977
$$582$$ 30.6033i 1.26855i
$$583$$ − 54.2765i − 2.24790i
$$584$$ −5.15674 −0.213388
$$585$$ 0 0
$$586$$ −1.08171 −0.0446850
$$587$$ − 10.1501i − 0.418938i −0.977815 0.209469i $$-0.932826\pi$$
0.977815 0.209469i $$-0.0671735\pi$$
$$588$$ 0.434709i 0.0179271i
$$589$$ −7.54677 −0.310959
$$590$$ 0 0
$$591$$ 40.9538 1.68461
$$592$$ 4.54677i 0.186871i
$$593$$ 16.6732i 0.684685i 0.939575 + 0.342342i $$0.111220\pi$$
−0.939575 + 0.342342i $$0.888780\pi$$
$$594$$ −28.0844 −1.15232
$$595$$ 0 0
$$596$$ −7.25697 −0.297257
$$597$$ 45.7272i 1.87149i
$$598$$ 10.5349i 0.430806i
$$599$$ −18.2765 −0.746756 −0.373378 0.927679i $$-0.621800\pi$$
−0.373378 + 0.927679i $$0.621800\pi$$
$$600$$ 0 0
$$601$$ 7.96297 0.324816 0.162408 0.986724i $$-0.448074\pi$$
0.162408 + 0.986724i $$0.448074\pi$$
$$602$$ 25.6968i 1.04733i
$$603$$ − 0.413712i − 0.0168477i
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 16.6403 0.675967
$$607$$ − 6.93013i − 0.281285i −0.990060 0.140643i $$-0.955083\pi$$
0.990060 0.140643i $$-0.0449169\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ −5.16342 −0.209232
$$610$$ 0 0
$$611$$ 2.44139 0.0987679
$$612$$ 0.712675i 0.0288082i
$$613$$ 34.2765i 1.38441i 0.721700 + 0.692206i $$0.243361\pi$$
−0.721700 + 0.692206i $$0.756639\pi$$
$$614$$ −2.38336 −0.0961844
$$615$$ 0 0
$$616$$ −14.9301 −0.601552
$$617$$ 35.5139i 1.42974i 0.699259 + 0.714869i $$0.253514\pi$$
−0.699259 + 0.714869i $$0.746486\pi$$
$$618$$ − 22.9301i − 0.922385i
$$619$$ 29.5334 1.18705 0.593524 0.804816i $$-0.297736\pi$$
0.593524 + 0.804816i $$0.297736\pi$$
$$620$$ 0 0
$$621$$ 18.2780 0.733470
$$622$$ − 29.4584i − 1.18117i
$$623$$ − 13.7102i − 0.549287i
$$624$$ −5.17526 −0.207176
$$625$$ 0 0
$$626$$ 32.0355 1.28040
$$627$$ − 9.83658i − 0.392835i
$$628$$ 9.54677i 0.380958i
$$629$$ −22.3624 −0.891646
$$630$$ 0 0
$$631$$ −15.1634 −0.603646 −0.301823 0.953364i $$-0.597595\pi$$
−0.301823 + 0.953364i $$0.597595\pi$$
$$632$$ − 3.09355i − 0.123055i
$$633$$ 25.2293i 1.00277i
$$634$$ 26.6665 1.05906
$$635$$ 0 0
$$636$$ −17.3530 −0.688091
$$637$$ − 0.715358i − 0.0283435i
$$638$$ − 6.00000i − 0.237542i
$$639$$ −2.09775 −0.0829855
$$640$$ 0 0
$$641$$ −25.0802 −0.990608 −0.495304 0.868720i $$-0.664943\pi$$
−0.495304 + 0.868720i $$0.664943\pi$$
$$642$$ − 33.5915i − 1.32575i
$$643$$ − 19.1306i − 0.754437i −0.926124 0.377218i $$-0.876881\pi$$
0.926124 0.377218i $$-0.123119\pi$$
$$644$$ 9.71687 0.382898
$$645$$ 0 0
$$646$$ 4.91829 0.193508
$$647$$ 12.4019i 0.487568i 0.969830 + 0.243784i $$0.0783888\pi$$
−0.969830 + 0.243784i $$0.921611\pi$$
$$648$$ 9.41371i 0.369806i
$$649$$ −72.0710 −2.82904
$$650$$ 0 0
$$651$$ 36.0237 1.41188
$$652$$ 15.7102i 0.615259i
$$653$$ 10.7801i 0.421856i 0.977502 + 0.210928i $$0.0676486\pi$$
−0.977502 + 0.210928i $$0.932351\pi$$
$$654$$ −4.77339 −0.186654
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0.747227i 0.0291521i
$$658$$ − 2.25181i − 0.0877845i
$$659$$ −8.56529 −0.333656 −0.166828 0.985986i $$-0.553353\pi$$
−0.166828 + 0.985986i $$0.553353\pi$$
$$660$$ 0 0
$$661$$ −26.2883 −1.02250 −0.511248 0.859433i $$-0.670817\pi$$
−0.511248 + 0.859433i $$0.670817\pi$$
$$662$$ − 19.8721i − 0.772351i
$$663$$ − 25.4534i − 0.988529i
$$664$$ −1.71019 −0.0663684
$$665$$ 0 0
$$666$$ 0.658841 0.0255296
$$667$$ 3.90494i 0.151200i
$$668$$ − 6.45323i − 0.249683i
$$669$$ −7.89729 −0.305327
$$670$$ 0 0
$$671$$ −40.9538 −1.58100
$$672$$ 4.77339i 0.184137i
$$673$$ 12.9301i 0.498420i 0.968449 + 0.249210i $$0.0801709\pi$$
−0.968449 + 0.249210i $$0.919829\pi$$
$$674$$ −27.5705 −1.06197
$$675$$ 0 0
$$676$$ −4.48358 −0.172445
$$677$$ − 15.9250i − 0.612046i −0.952024 0.306023i $$-0.901002\pi$$
0.952024 0.306023i $$-0.0989985\pi$$
$$678$$ − 10.6403i − 0.408639i
$$679$$ −46.4502 −1.78260
$$680$$ 0 0
$$681$$ −10.5493 −0.404248
$$682$$ 41.8603i 1.60291i
$$683$$ − 13.2898i − 0.508520i −0.967136 0.254260i $$-0.918168\pi$$
0.967136 0.254260i $$-0.0818319\pi$$
$$684$$ −0.144903 −0.00554050
$$685$$ 0 0
$$686$$ 18.1819 0.694190
$$687$$ − 1.93929i − 0.0739884i
$$688$$ 9.54677i 0.363967i
$$689$$ 28.5561 1.08790
$$690$$ 0 0
$$691$$ 27.2570 1.03690 0.518452 0.855107i $$-0.326508\pi$$
0.518452 + 0.855107i $$0.326508\pi$$
$$692$$ − 0.873614i − 0.0332098i
$$693$$ 2.16342i 0.0821815i
$$694$$ 27.2806 1.03556
$$695$$ 0 0
$$696$$ −1.91829 −0.0727126
$$697$$ 0 0
$$698$$ − 30.9538i − 1.17162i
$$699$$ 24.9933 0.945334
$$700$$ 0 0
$$701$$ −41.4441 −1.56532 −0.782660 0.622449i $$-0.786138\pi$$
−0.782660 + 0.622449i $$0.786138\pi$$
$$702$$ − 14.7759i − 0.557679i
$$703$$ − 4.54677i − 0.171485i
$$704$$ −5.54677 −0.209052
$$705$$ 0 0
$$706$$ 11.1819 0.420838
$$707$$ 25.2570i 0.949886i
$$708$$ 23.0422i 0.865979i
$$709$$ 36.7904 1.38169 0.690846 0.723002i $$-0.257238\pi$$
0.690846 + 0.723002i $$0.257238\pi$$
$$710$$ 0 0
$$711$$ −0.448264 −0.0168112
$$712$$ − 5.09355i − 0.190889i
$$713$$ − 27.2436i − 1.02028i
$$714$$ −23.4769 −0.878601
$$715$$ 0 0
$$716$$ −12.3834 −0.462788
$$717$$ − 27.6825i − 1.03382i
$$718$$ − 20.3387i − 0.759033i
$$719$$ 4.13726 0.154294 0.0771469 0.997020i $$-0.475419\pi$$
0.0771469 + 0.997020i $$0.475419\pi$$
$$720$$ 0 0
$$721$$ 34.8037 1.29616
$$722$$ 1.00000i 0.0372161i
$$723$$ − 9.25697i − 0.344270i
$$724$$ 17.6403 0.655597
$$725$$ 0 0
$$726$$ −35.0540 −1.30098
$$727$$ − 40.4374i − 1.49974i −0.661585 0.749870i $$-0.730116\pi$$
0.661585 0.749870i $$-0.269884\pi$$
$$728$$ − 7.85510i − 0.291129i
$$729$$ −25.5729 −0.947146
$$730$$ 0 0
$$731$$ −46.9538 −1.73665
$$732$$ 13.0935i 0.483952i
$$733$$ − 28.1264i − 1.03887i −0.854509 0.519436i $$-0.826142\pi$$
0.854509 0.519436i $$-0.173858\pi$$
$$734$$ 32.9538 1.21635
$$735$$ 0 0
$$736$$ 3.60997 0.133065
$$737$$ 15.8366i 0.583348i
$$738$$ 0 0
$$739$$ 38.1501 1.40337 0.701686 0.712486i $$-0.252431\pi$$
0.701686 + 0.712486i $$0.252431\pi$$
$$740$$ 0 0
$$741$$ 5.17526 0.190118
$$742$$ − 26.3387i − 0.966923i
$$743$$ 17.8232i 0.653871i 0.945047 + 0.326935i $$0.106016\pi$$
−0.945047 + 0.326935i $$0.893984\pi$$
$$744$$ 13.3834 0.490658
$$745$$ 0 0
$$746$$ −10.0869 −0.369307
$$747$$ 0.247812i 0.00906697i
$$748$$ − 27.2806i − 0.997479i
$$749$$ 50.9857 1.86298
$$750$$ 0 0
$$751$$ 23.6968 0.864710 0.432355 0.901703i $$-0.357683\pi$$
0.432355 + 0.901703i $$0.357683\pi$$
$$752$$ − 0.836581i − 0.0305070i
$$753$$ − 50.7904i − 1.85090i
$$754$$ 3.15674 0.114962
$$755$$ 0 0
$$756$$ −13.6285 −0.495663
$$757$$ − 6.51394i − 0.236753i −0.992969 0.118377i $$-0.962231\pi$$
0.992969 0.118377i $$-0.0377690\pi$$
$$758$$ 18.4256i 0.669246i
$$759$$ 35.5097 1.28892
$$760$$ 0 0
$$761$$ 34.7786 1.26072 0.630361 0.776302i $$-0.282907\pi$$
0.630361 + 0.776302i $$0.282907\pi$$
$$762$$ 3.32264i 0.120367i
$$763$$ − 7.24513i − 0.262291i
$$764$$ 15.1567 0.548352
$$765$$ 0 0
$$766$$ 28.1871 1.01844
$$767$$ − 37.9183i − 1.36915i
$$768$$ 1.77339i 0.0639916i
$$769$$ 27.6850 0.998347 0.499173 0.866502i $$-0.333637\pi$$
0.499173 + 0.866502i $$0.333637\pi$$
$$770$$ 0 0
$$771$$ −11.4441 −0.412148
$$772$$ 23.3834i 0.841585i
$$773$$ 19.9738i 0.718409i 0.933259 + 0.359205i $$0.116952\pi$$
−0.933259 + 0.359205i $$0.883048\pi$$
$$774$$ 1.38336 0.0497237
$$775$$ 0 0
$$776$$ −17.2570 −0.619489
$$777$$ 21.7035i 0.778609i
$$778$$ − 20.0237i − 0.717884i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 80.3001 2.87336
$$782$$ 17.7549i 0.634913i
$$783$$ − 5.47691i − 0.195729i
$$784$$ −0.245129 −0.00875461
$$785$$ 0 0
$$786$$ 20.4769 0.730387
$$787$$ 42.0118i 1.49756i 0.662818 + 0.748780i $$0.269360\pi$$
−0.662818 + 0.748780i $$0.730640\pi$$
$$788$$ 23.0935i 0.822674i
$$789$$ 39.3463 1.40077
$$790$$ 0 0
$$791$$ 16.1501 0.574230
$$792$$ 0.803744i 0.0285598i
$$793$$ − 21.5468i − 0.765148i
$$794$$ 8.32684 0.295508
$$795$$ 0 0
$$796$$ −25.7852 −0.913933
$$797$$ − 45.1054i − 1.59771i −0.601520 0.798857i $$-0.705438\pi$$
0.601520 0.798857i $$-0.294562\pi$$
$$798$$ − 4.77339i − 0.168976i
$$799$$ 4.11455 0.145562
$$800$$ 0 0
$$801$$ −0.738070 −0.0260784
$$802$$ − 22.1501i − 0.782146i
$$803$$ − 28.6033i − 1.00939i
$$804$$ 5.06319 0.178565
$$805$$ 0 0
$$806$$ −22.0237 −0.775751
$$807$$ 36.1896i 1.27393i
$$808$$ 9.38336i 0.330105i
$$809$$ −6.85510 −0.241012 −0.120506 0.992713i $$-0.538452\pi$$
−0.120506 + 0.992713i $$0.538452\pi$$
$$810$$ 0 0
$$811$$ −15.1819 −0.533110 −0.266555 0.963820i $$-0.585885\pi$$
−0.266555 + 0.963820i $$0.585885\pi$$
$$812$$ − 2.91161i − 0.102178i
$$813$$ 28.2621i 0.991196i
$$814$$ −25.2199 −0.883958
$$815$$ 0 0
$$816$$ −8.72203 −0.305332
$$817$$ − 9.54677i − 0.333999i
$$818$$ − 34.0237i − 1.18961i
$$819$$ −1.13823 −0.0397729
$$820$$ 0 0
$$821$$ 22.5006 0.785276 0.392638 0.919693i $$-0.371563\pi$$
0.392638 + 0.919693i $$0.371563\pi$$
$$822$$ − 7.51642i − 0.262165i
$$823$$ 16.1896i 0.564333i 0.959365 + 0.282167i $$0.0910531\pi$$
−0.959365 + 0.282167i $$0.908947\pi$$
$$824$$ 12.9301 0.450442
$$825$$ 0 0
$$826$$ −34.9738 −1.21690
$$827$$ − 25.0817i − 0.872177i −0.899904 0.436088i $$-0.856364\pi$$
0.899904 0.436088i $$-0.143636\pi$$
$$828$$ − 0.523095i − 0.0181788i
$$829$$ −21.3068 −0.740016 −0.370008 0.929029i $$-0.620645\pi$$
−0.370008 + 0.929029i $$0.620645\pi$$
$$830$$ 0 0
$$831$$ −7.13554 −0.247529
$$832$$ − 2.91829i − 0.101174i
$$833$$ − 1.20562i − 0.0417721i
$$834$$ 16.3505 0.566172
$$835$$ 0 0
$$836$$ 5.54677 0.191839
$$837$$ 38.2108i 1.32076i
$$838$$ − 37.5334i − 1.29657i
$$839$$ 11.4727 0.396082 0.198041 0.980194i $$-0.436542\pi$$
0.198041 + 0.980194i $$0.436542\pi$$
$$840$$ 0 0
$$841$$ −27.8299 −0.959652
$$842$$ 33.8341i 1.16600i
$$843$$ 2.96716i 0.102195i
$$844$$ −14.2266 −0.489700
$$845$$ 0 0
$$846$$ −0.121223 −0.00416773
$$847$$ − 53.2056i − 1.82817i
$$848$$ − 9.78523i − 0.336026i
$$849$$ 29.9580 1.02816
$$850$$ 0 0
$$851$$ 16.4137 0.562655
$$852$$ − 25.6732i − 0.879548i
$$853$$ 38.9908i 1.33502i 0.744600 + 0.667511i $$0.232640\pi$$
−0.744600 + 0.667511i $$0.767360\pi$$
$$854$$ −19.8736 −0.680061
$$855$$ 0 0
$$856$$ 18.9420 0.647423
$$857$$ − 29.8973i − 1.02127i −0.859797 0.510636i $$-0.829410\pi$$
0.859797 0.510636i $$-0.170590\pi$$
$$858$$ − 28.7060i − 0.980007i
$$859$$ 6.47691 0.220989 0.110495 0.993877i $$-0.464757\pi$$
0.110495 + 0.993877i $$0.464757\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 24.8037i 0.844819i
$$863$$ 5.09355i 0.173386i 0.996235 + 0.0866932i $$0.0276300\pi$$
−0.996235 + 0.0866932i $$0.972370\pi$$
$$864$$ −5.06319 −0.172253
$$865$$ 0 0
$$866$$ 1.68651 0.0573101
$$867$$ − 12.7499i − 0.433010i
$$868$$ 20.3135i 0.689485i
$$869$$ 17.1592 0.582087
$$870$$ 0 0
$$871$$ −8.33200 −0.282319
$$872$$ − 2.69168i − 0.0911517i
$$873$$ 2.50059i 0.0846320i
$$874$$ −3.60997 −0.122109
$$875$$ 0 0
$$876$$ −9.14490 −0.308978
$$877$$ − 29.8341i − 1.00743i −0.863871 0.503713i $$-0.831967\pi$$
0.863871 0.503713i $$-0.168033\pi$$
$$878$$ − 0.326839i − 0.0110303i
$$879$$ −1.91829 −0.0647023
$$880$$ 0 0
$$881$$ −11.5796 −0.390127 −0.195064 0.980791i $$-0.562491\pi$$
−0.195064 + 0.980791i $$0.562491\pi$$
$$882$$ 0.0355199i 0.00119602i
$$883$$ 48.0237i 1.61613i 0.589096 + 0.808063i $$0.299484\pi$$
−0.589096 + 0.808063i $$0.700516\pi$$
$$884$$ 14.3530 0.482744
$$885$$ 0 0
$$886$$ 0.490258 0.0164705
$$887$$ 9.38336i 0.315062i 0.987514 + 0.157531i $$0.0503535\pi$$
−0.987514 + 0.157531i $$0.949647\pi$$
$$888$$ 8.06319i 0.270583i
$$889$$ −5.04316 −0.169142
$$890$$ 0 0
$$891$$ −52.2157 −1.74929
$$892$$ − 4.45323i − 0.149105i
$$893$$ 0.836581i 0.0279951i
$$894$$ −12.8694 −0.430418
$$895$$ 0 0
$$896$$ −2.69168 −0.0899226
$$897$$ 18.6825i 0.623791i
$$898$$ − 23.5468i − 0.785766i
$$899$$ −8.16342 −0.272265
$$900$$ 0 0
$$901$$ 48.1266 1.60333
$$902$$ 0 0
$$903$$ 45.5705i 1.51649i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −35.4677 −1.17834
$$907$$ − 19.4347i − 0.645319i −0.946515 0.322659i $$-0.895423\pi$$
0.946515 0.322659i $$-0.104577\pi$$
$$908$$ − 5.94865i − 0.197413i
$$909$$ 1.35968 0.0450976
$$910$$ 0 0
$$911$$ −19.9866 −0.662187 −0.331094 0.943598i $$-0.607418\pi$$
−0.331094 + 0.943598i $$0.607418\pi$$
$$912$$ − 1.77339i − 0.0587227i
$$913$$ − 9.48606i − 0.313943i
$$914$$ 8.01184 0.265008
$$915$$ 0 0
$$916$$ 1.09355 0.0361319
$$917$$ 31.0802i 1.02636i
$$918$$ − 24.9023i − 0.821897i
$$919$$ −6.15158 −0.202922 −0.101461 0.994840i $$-0.532352\pi$$
−0.101461 + 0.994840i $$0.532352\pi$$
$$920$$ 0 0
$$921$$ −4.22661 −0.139272
$$922$$ 3.83658i 0.126351i
$$923$$ 42.2478i 1.39060i
$$924$$ −26.4769 −0.871026
$$925$$ 0 0
$$926$$ −6.58381 −0.216357
$$927$$ − 1.87361i − 0.0615375i
$$928$$ − 1.08171i − 0.0355089i
$$929$$ 7.17010 0.235243 0.117622 0.993058i $$-0.462473\pi$$
0.117622 + 0.993058i $$0.462473\pi$$
$$930$$ 0 0
$$931$$ 0.245129 0.00803378
$$932$$ 14.0935i 0.461650i
$$933$$ − 52.2411i − 1.71030i
$$934$$ −22.6033 −0.739602
$$935$$ 0 0
$$936$$ −0.422869 −0.0138219
$$937$$ − 24.2646i − 0.792690i −0.918102 0.396345i $$-0.870278\pi$$
0.918102 0.396345i $$-0.129722\pi$$
$$938$$ 7.68500i 0.250924i
$$939$$ 56.8114 1.85397
$$940$$ 0 0
$$941$$ −7.44806 −0.242800 −0.121400 0.992604i $$-0.538738\pi$$
−0.121400 + 0.992604i $$0.538738\pi$$
$$942$$ 16.9301i 0.551613i
$$943$$ 0 0
$$944$$ −12.9933 −0.422897
$$945$$ 0 0
$$946$$ −52.9538 −1.72168
$$947$$ − 52.9538i − 1.72077i −0.509647 0.860384i $$-0.670224\pi$$
0.509647 0.860384i $$-0.329776\pi$$
$$948$$ − 5.48606i − 0.178179i
$$949$$ 15.0489 0.488507
$$950$$ 0 0
$$951$$ 47.2900 1.53348
$$952$$ − 13.2385i − 0.429061i
$$953$$ 12.8694i 0.416881i 0.978035 + 0.208441i $$0.0668388\pi$$
−0.978035 + 0.208441i $$0.933161\pi$$
$$954$$ −1.41791 −0.0459065
$$955$$ 0 0
$$956$$ 15.6100 0.504862
$$957$$ − 10.6403i − 0.343953i
$$958$$ 35.7904i 1.15634i
$$959$$ 11.4085 0.368401
$$960$$ 0 0
$$961$$ 25.9538 0.837220
$$962$$ − 13.2688i − 0.427804i
$$963$$ − 2.74475i − 0.0884482i
$$964$$ 5.21994 0.168123
$$965$$ 0 0
$$966$$ 17.2318 0.554423
$$967$$ 25.8603i 0.831610i 0.909454 + 0.415805i $$0.136500\pi$$
−0.909454 + 0.415805i $$0.863500\pi$$
$$968$$ − 19.7667i − 0.635326i
$$969$$ 8.72203 0.280192
$$970$$ 0 0
$$971$$ 35.0935 1.12621 0.563103 0.826387i $$-0.309608\pi$$
0.563103 + 0.826387i $$0.309608\pi$$
$$972$$ 1.50458i 0.0482594i
$$973$$ 24.8171i 0.795600i
$$974$$ 21.4070 0.685926
$$975$$ 0 0
$$976$$ −7.38336 −0.236335
$$977$$ − 42.1737i − 1.34926i −0.738157 0.674629i $$-0.764304\pi$$
0.738157 0.674629i $$-0.235696\pi$$
$$978$$ 27.8603i 0.890873i
$$979$$ 28.2528 0.902963
$$980$$ 0 0
$$981$$ −0.390032 −0.0124528
$$982$$ − 4.32684i − 0.138075i
$$983$$ 42.6270i 1.35959i 0.733403 + 0.679795i $$0.237931\pi$$
−0.733403 + 0.679795i $$0.762069\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 5.32016 0.169428
$$987$$ − 3.99332i − 0.127109i
$$988$$ 2.91829i 0.0928432i
$$989$$ 34.4636 1.09588
$$990$$ 0 0
$$991$$ −34.1737 −1.08556 −0.542782 0.839873i $$-0.682629\pi$$
−0.542782 + 0.839873i $$0.682629\pi$$
$$992$$ 7.54677i 0.239610i
$$993$$ − 35.2409i − 1.11834i
$$994$$ 38.9672 1.23596
$$995$$ 0 0
$$996$$ −3.03284 −0.0960991
$$997$$ 29.3834i 0.930580i 0.885158 + 0.465290i $$0.154050\pi$$
−0.885158 + 0.465290i $$0.845950\pi$$
$$998$$ 37.6968i 1.19327i
$$999$$ −23.0212 −0.728359
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 950.2.b.h.799.6 6
5.2 odd 4 950.2.a.j.1.3 3
5.3 odd 4 950.2.a.l.1.1 yes 3
5.4 even 2 inner 950.2.b.h.799.1 6
15.2 even 4 8550.2.a.cp.1.3 3
15.8 even 4 8550.2.a.ci.1.1 3
20.3 even 4 7600.2.a.bk.1.3 3
20.7 even 4 7600.2.a.bz.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.3 3 5.2 odd 4
950.2.a.l.1.1 yes 3 5.3 odd 4
950.2.b.h.799.1 6 5.4 even 2 inner
950.2.b.h.799.6 6 1.1 even 1 trivial
7600.2.a.bk.1.3 3 20.3 even 4
7600.2.a.bz.1.1 3 20.7 even 4
8550.2.a.ci.1.1 3 15.8 even 4
8550.2.a.cp.1.3 3 15.2 even 4