# Properties

 Label 950.2.b.h.799.3 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 799.3 Root $$2.25342i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.h.799.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +3.25342i q^{3} -1.00000 q^{4} +3.25342 q^{6} +0.0778929i q^{7} +1.00000i q^{8} -7.58473 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +3.25342i q^{3} -1.00000 q^{4} +3.25342 q^{6} +0.0778929i q^{7} +1.00000i q^{8} -7.58473 q^{9} -4.50684 q^{11} -3.25342i q^{12} -5.33131i q^{13} +0.0778929 q^{14} +1.00000 q^{16} -7.33131i q^{17} +7.58473i q^{18} -1.00000 q^{19} -0.253418 q^{21} +4.50684i q^{22} +3.40920i q^{23} -3.25342 q^{24} -5.33131 q^{26} -14.9160i q^{27} -0.0778929i q^{28} +1.33131 q^{29} -2.50684 q^{31} -1.00000i q^{32} -14.6626i q^{33} -7.33131 q^{34} +7.58473 q^{36} +5.50684i q^{37} +1.00000i q^{38} +17.3450 q^{39} +0.253418i q^{42} +0.506836i q^{43} +4.50684 q^{44} +3.40920 q^{46} +5.66262i q^{47} +3.25342i q^{48} +6.99393 q^{49} +23.8518 q^{51} +5.33131i q^{52} -12.9358i q^{53} -14.9160 q^{54} -0.0778929 q^{56} -3.25342i q^{57} -1.33131i q^{58} -7.56499 q^{59} -2.15579 q^{61} +2.50684i q^{62} -0.590796i q^{63} -1.00000 q^{64} -14.6626 q^{66} +4.58473i q^{67} +7.33131i q^{68} -11.0916 q^{69} -10.8579 q^{71} -7.58473i q^{72} -5.09763i q^{73} +5.50684 q^{74} +1.00000 q^{76} -0.351050i q^{77} -17.3450i q^{78} -17.0137 q^{79} +25.7739 q^{81} -13.1695i q^{83} +0.253418 q^{84} +0.506836 q^{86} +4.33131i q^{87} -4.50684i q^{88} -15.0137 q^{89} +0.415271 q^{91} -3.40920i q^{92} -8.15579i q^{93} +5.66262 q^{94} +3.25342 q^{96} -7.67629i q^{97} -6.99393i q^{98} +34.1831 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} + 4q^{6} - 10q^{9} + O(q^{10})$$ $$6q - 6q^{4} + 4q^{6} - 10q^{9} + 4q^{11} - 4q^{14} + 6q^{16} - 6q^{19} + 14q^{21} - 4q^{24} - 12q^{26} - 12q^{29} + 16q^{31} - 24q^{34} + 10q^{36} + 22q^{39} - 4q^{44} - 4q^{46} - 18q^{49} + 30q^{51} - 34q^{54} + 4q^{56} - 12q^{59} - 4q^{61} - 6q^{64} - 48q^{66} - 12q^{71} + 2q^{74} + 6q^{76} - 40q^{79} + 46q^{81} - 14q^{84} - 28q^{86} - 28q^{89} + 38q^{91} - 6q^{94} + 4q^{96} + 72q^{99} + O(q^{100})$$

## 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
$$3$$ 3.25342i 1.87836i 0.343423 + 0.939181i $$0.388414\pi$$
−0.343423 + 0.939181i $$0.611586\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 3.25342 1.32820
$$7$$ 0.0778929i 0.0294407i 0.999892 + 0.0147204i $$0.00468581\pi$$
−0.999892 + 0.0147204i $$0.995314\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −7.58473 −2.52824
$$10$$ 0 0
$$11$$ −4.50684 −1.35886 −0.679431 0.733739i $$-0.737773\pi$$
−0.679431 + 0.733739i $$0.737773\pi$$
$$12$$ − 3.25342i − 0.939181i
$$13$$ − 5.33131i − 1.47864i −0.673354 0.739320i $$-0.735147\pi$$
0.673354 0.739320i $$-0.264853\pi$$
$$14$$ 0.0778929 0.0208177
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 7.33131i − 1.77810i −0.457806 0.889052i $$-0.651365\pi$$
0.457806 0.889052i $$-0.348635\pi$$
$$18$$ 7.58473i 1.78774i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −0.253418 −0.0553003
$$22$$ 4.50684i 0.960861i
$$23$$ 3.40920i 0.710868i 0.934701 + 0.355434i $$0.115667\pi$$
−0.934701 + 0.355434i $$0.884333\pi$$
$$24$$ −3.25342 −0.664101
$$25$$ 0 0
$$26$$ −5.33131 −1.04556
$$27$$ − 14.9160i − 2.87059i
$$28$$ − 0.0778929i − 0.0147204i
$$29$$ 1.33131 0.247218 0.123609 0.992331i $$-0.460553\pi$$
0.123609 + 0.992331i $$0.460553\pi$$
$$30$$ 0 0
$$31$$ −2.50684 −0.450241 −0.225121 0.974331i $$-0.572278\pi$$
−0.225121 + 0.974331i $$0.572278\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 14.6626i − 2.55243i
$$34$$ −7.33131 −1.25731
$$35$$ 0 0
$$36$$ 7.58473 1.26412
$$37$$ 5.50684i 0.905318i 0.891684 + 0.452659i $$0.149525\pi$$
−0.891684 + 0.452659i $$0.850475\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 17.3450 2.77742
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0.253418i 0.0391033i
$$43$$ 0.506836i 0.0772918i 0.999253 + 0.0386459i $$0.0123044\pi$$
−0.999253 + 0.0386459i $$0.987696\pi$$
$$44$$ 4.50684 0.679431
$$45$$ 0 0
$$46$$ 3.40920 0.502660
$$47$$ 5.66262i 0.825978i 0.910736 + 0.412989i $$0.135515\pi$$
−0.910736 + 0.412989i $$0.864485\pi$$
$$48$$ 3.25342i 0.469590i
$$49$$ 6.99393 0.999133
$$50$$ 0 0
$$51$$ 23.8518 3.33992
$$52$$ 5.33131i 0.739320i
$$53$$ − 12.9358i − 1.77687i −0.459006 0.888433i $$-0.651794\pi$$
0.459006 0.888433i $$-0.348206\pi$$
$$54$$ −14.9160 −2.02982
$$55$$ 0 0
$$56$$ −0.0778929 −0.0104089
$$57$$ − 3.25342i − 0.430926i
$$58$$ − 1.33131i − 0.174810i
$$59$$ −7.56499 −0.984878 −0.492439 0.870347i $$-0.663894\pi$$
−0.492439 + 0.870347i $$0.663894\pi$$
$$60$$ 0 0
$$61$$ −2.15579 −0.276020 −0.138010 0.990431i $$-0.544071\pi$$
−0.138010 + 0.990431i $$0.544071\pi$$
$$62$$ 2.50684i 0.318369i
$$63$$ − 0.590796i − 0.0744333i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −14.6626 −1.80484
$$67$$ 4.58473i 0.560114i 0.959983 + 0.280057i $$0.0903533\pi$$
−0.959983 + 0.280057i $$0.909647\pi$$
$$68$$ 7.33131i 0.889052i
$$69$$ −11.0916 −1.33527
$$70$$ 0 0
$$71$$ −10.8579 −1.28859 −0.644297 0.764775i $$-0.722850\pi$$
−0.644297 + 0.764775i $$0.722850\pi$$
$$72$$ − 7.58473i − 0.893869i
$$73$$ − 5.09763i − 0.596633i −0.954467 0.298316i $$-0.903575\pi$$
0.954467 0.298316i $$-0.0964250\pi$$
$$74$$ 5.50684 0.640157
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ − 0.351050i − 0.0400059i
$$78$$ − 17.3450i − 1.96393i
$$79$$ −17.0137 −1.91419 −0.957094 0.289778i $$-0.906418\pi$$
−0.957094 + 0.289778i $$0.906418\pi$$
$$80$$ 0 0
$$81$$ 25.7739 2.86377
$$82$$ 0 0
$$83$$ − 13.1695i − 1.44554i −0.691091 0.722768i $$-0.742870\pi$$
0.691091 0.722768i $$-0.257130\pi$$
$$84$$ 0.253418 0.0276502
$$85$$ 0 0
$$86$$ 0.506836 0.0546535
$$87$$ 4.33131i 0.464365i
$$88$$ − 4.50684i − 0.480430i
$$89$$ −15.0137 −1.59145 −0.795723 0.605661i $$-0.792909\pi$$
−0.795723 + 0.605661i $$0.792909\pi$$
$$90$$ 0 0
$$91$$ 0.415271 0.0435322
$$92$$ − 3.40920i − 0.355434i
$$93$$ − 8.15579i − 0.845716i
$$94$$ 5.66262 0.584055
$$95$$ 0 0
$$96$$ 3.25342 0.332051
$$97$$ − 7.67629i − 0.779410i −0.920940 0.389705i $$-0.872577\pi$$
0.920940 0.389705i $$-0.127423\pi$$
$$98$$ − 6.99393i − 0.706494i
$$99$$ 34.1831 3.43553
$$100$$ 0 0
$$101$$ −4.15579 −0.413516 −0.206758 0.978392i $$-0.566291\pi$$
−0.206758 + 0.978392i $$0.566291\pi$$
$$102$$ − 23.8518i − 2.36168i
$$103$$ 2.35105i 0.231656i 0.993269 + 0.115828i $$0.0369521\pi$$
−0.993269 + 0.115828i $$0.963048\pi$$
$$104$$ 5.33131 0.522778
$$105$$ 0 0
$$106$$ −12.9358 −1.25643
$$107$$ 14.0334i 1.35666i 0.734757 + 0.678331i $$0.237296\pi$$
−0.734757 + 0.678331i $$0.762704\pi$$
$$108$$ 14.9160i 1.43530i
$$109$$ 0.0778929 0.00746078 0.00373039 0.999993i $$-0.498813\pi$$
0.00373039 + 0.999993i $$0.498813\pi$$
$$110$$ 0 0
$$111$$ −17.9160 −1.70052
$$112$$ 0.0778929i 0.00736018i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −3.25342 −0.304711
$$115$$ 0 0
$$116$$ −1.33131 −0.123609
$$117$$ 40.4365i 3.73836i
$$118$$ 7.56499i 0.696414i
$$119$$ 0.571057 0.0523487
$$120$$ 0 0
$$121$$ 9.31157 0.846506
$$122$$ 2.15579i 0.195176i
$$123$$ 0 0
$$124$$ 2.50684 0.225121
$$125$$ 0 0
$$126$$ −0.590796 −0.0526323
$$127$$ − 17.8321i − 1.58234i −0.611596 0.791171i $$-0.709472\pi$$
0.611596 0.791171i $$-0.290528\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −1.64895 −0.145182
$$130$$ 0 0
$$131$$ −1.49316 −0.130458 −0.0652292 0.997870i $$-0.520778\pi$$
−0.0652292 + 0.997870i $$0.520778\pi$$
$$132$$ 14.6626i 1.27622i
$$133$$ − 0.0778929i − 0.00675417i
$$134$$ 4.58473 0.396060
$$135$$ 0 0
$$136$$ 7.33131 0.628655
$$137$$ 8.42894i 0.720133i 0.932927 + 0.360067i $$0.117246\pi$$
−0.932927 + 0.360067i $$0.882754\pi$$
$$138$$ 11.0916i 0.944177i
$$139$$ −8.81841 −0.747968 −0.373984 0.927435i $$-0.622008\pi$$
−0.373984 + 0.927435i $$0.622008\pi$$
$$140$$ 0 0
$$141$$ −18.4229 −1.55149
$$142$$ 10.8579i 0.911174i
$$143$$ 24.0273i 2.00927i
$$144$$ −7.58473 −0.632061
$$145$$ 0 0
$$146$$ −5.09763 −0.421883
$$147$$ 22.7542i 1.87673i
$$148$$ − 5.50684i − 0.452659i
$$149$$ −17.6763 −1.44810 −0.724049 0.689748i $$-0.757721\pi$$
−0.724049 + 0.689748i $$0.757721\pi$$
$$150$$ 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
$$153$$ 55.6060i 4.49548i
$$154$$ −0.351050 −0.0282884
$$155$$ 0 0
$$156$$ −17.3450 −1.38871
$$157$$ − 0.506836i − 0.0404499i −0.999795 0.0202250i $$-0.993562\pi$$
0.999795 0.0202250i $$-0.00643824\pi$$
$$158$$ 17.0137i 1.35354i
$$159$$ 42.0855 3.33760
$$160$$ 0 0
$$161$$ −0.265553 −0.0209285
$$162$$ − 25.7739i − 2.02499i
$$163$$ 0.830542i 0.0650531i 0.999471 + 0.0325265i $$0.0103553\pi$$
−0.999471 + 0.0325265i $$0.989645\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −13.1695 −1.02215
$$167$$ − 16.5068i − 1.27734i −0.769482 0.638669i $$-0.779485\pi$$
0.769482 0.638669i $$-0.220515\pi$$
$$168$$ − 0.253418i − 0.0195516i
$$169$$ −15.4229 −1.18638
$$170$$ 0 0
$$171$$ 7.58473 0.580019
$$172$$ − 0.506836i − 0.0386459i
$$173$$ 18.8321i 1.43178i 0.698215 + 0.715888i $$0.253978\pi$$
−0.698215 + 0.715888i $$0.746022\pi$$
$$174$$ 4.33131 0.328356
$$175$$ 0 0
$$176$$ −4.50684 −0.339716
$$177$$ − 24.6121i − 1.84996i
$$178$$ 15.0137i 1.12532i
$$179$$ 7.15579 0.534849 0.267424 0.963579i $$-0.413827\pi$$
0.267424 + 0.963579i $$0.413827\pi$$
$$180$$ 0 0
$$181$$ 12.5205 0.930642 0.465321 0.885142i $$-0.345939\pi$$
0.465321 + 0.885142i $$0.345939\pi$$
$$182$$ − 0.415271i − 0.0307819i
$$183$$ − 7.01367i − 0.518466i
$$184$$ −3.40920 −0.251330
$$185$$ 0 0
$$186$$ −8.15579 −0.598011
$$187$$ 33.0410i 2.41620i
$$188$$ − 5.66262i − 0.412989i
$$189$$ 1.16185 0.0845124
$$190$$ 0 0
$$191$$ −4.90237 −0.354723 −0.177361 0.984146i $$-0.556756\pi$$
−0.177361 + 0.984146i $$0.556756\pi$$
$$192$$ − 3.25342i − 0.234795i
$$193$$ 18.1558i 1.30688i 0.756977 + 0.653441i $$0.226675\pi$$
−0.756977 + 0.653441i $$0.773325\pi$$
$$194$$ −7.67629 −0.551126
$$195$$ 0 0
$$196$$ −6.99393 −0.499567
$$197$$ 2.98633i 0.212767i 0.994325 + 0.106384i $$0.0339271\pi$$
−0.994325 + 0.106384i $$0.966073\pi$$
$$198$$ − 34.1831i − 2.42929i
$$199$$ 3.06422 0.217217 0.108608 0.994085i $$-0.465361\pi$$
0.108608 + 0.994085i $$0.465361\pi$$
$$200$$ 0 0
$$201$$ −14.9160 −1.05210
$$202$$ 4.15579i 0.292400i
$$203$$ 0.103700i 0.00727829i
$$204$$ −23.8518 −1.66996
$$205$$ 0 0
$$206$$ 2.35105 0.163805
$$207$$ − 25.8579i − 1.79725i
$$208$$ − 5.33131i − 0.369660i
$$209$$ 4.50684 0.311744
$$210$$ 0 0
$$211$$ 19.2534 1.32546 0.662730 0.748858i $$-0.269398\pi$$
0.662730 + 0.748858i $$0.269398\pi$$
$$212$$ 12.9358i 0.888433i
$$213$$ − 35.3252i − 2.42045i
$$214$$ 14.0334 0.959304
$$215$$ 0 0
$$216$$ 14.9160 1.01491
$$217$$ − 0.195265i − 0.0132554i
$$218$$ − 0.0778929i − 0.00527557i
$$219$$ 16.5847 1.12069
$$220$$ 0 0
$$221$$ −39.0855 −2.62918
$$222$$ 17.9160i 1.20245i
$$223$$ − 14.5068i − 0.971450i −0.874112 0.485725i $$-0.838556\pi$$
0.874112 0.485725i $$-0.161444\pi$$
$$224$$ 0.0778929 0.00520444
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 21.5984i 1.43354i 0.697312 + 0.716768i $$0.254379\pi$$
−0.697312 + 0.716768i $$0.745621\pi$$
$$228$$ 3.25342i 0.215463i
$$229$$ 19.0137 1.25646 0.628229 0.778028i $$-0.283780\pi$$
0.628229 + 0.778028i $$0.283780\pi$$
$$230$$ 0 0
$$231$$ 1.14211 0.0751456
$$232$$ 1.33131i 0.0874048i
$$233$$ − 6.01367i − 0.393969i −0.980407 0.196984i $$-0.936885\pi$$
0.980407 0.196984i $$-0.0631148\pi$$
$$234$$ 40.4365 2.64342
$$235$$ 0 0
$$236$$ 7.56499 0.492439
$$237$$ − 55.3526i − 3.59554i
$$238$$ − 0.571057i − 0.0370161i
$$239$$ −15.4092 −0.996739 −0.498369 0.866965i $$-0.666068\pi$$
−0.498369 + 0.866965i $$0.666068\pi$$
$$240$$ 0 0
$$241$$ −4.81841 −0.310381 −0.155190 0.987885i $$-0.549599\pi$$
−0.155190 + 0.987885i $$0.549599\pi$$
$$242$$ − 9.31157i − 0.598570i
$$243$$ 39.1052i 2.50860i
$$244$$ 2.15579 0.138010
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.33131i 0.339223i
$$248$$ − 2.50684i − 0.159184i
$$249$$ 42.8458 2.71524
$$250$$ 0 0
$$251$$ 1.52051 0.0959736 0.0479868 0.998848i $$-0.484719\pi$$
0.0479868 + 0.998848i $$0.484719\pi$$
$$252$$ 0.590796i 0.0372167i
$$253$$ − 15.3647i − 0.965972i
$$254$$ −17.8321 −1.11888
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 16.5068i − 1.02967i −0.857290 0.514834i $$-0.827854\pi$$
0.857290 0.514834i $$-0.172146\pi$$
$$258$$ 1.64895i 0.102659i
$$259$$ −0.428943 −0.0266532
$$260$$ 0 0
$$261$$ −10.0976 −0.625028
$$262$$ 1.49316i 0.0922480i
$$263$$ − 18.0273i − 1.11161i −0.831311 0.555807i $$-0.812409\pi$$
0.831311 0.555807i $$-0.187591\pi$$
$$264$$ 14.6626 0.902422
$$265$$ 0 0
$$266$$ −0.0778929 −0.00477592
$$267$$ − 48.8458i − 2.98931i
$$268$$ − 4.58473i − 0.280057i
$$269$$ −20.2089 −1.23216 −0.616080 0.787683i $$-0.711280\pi$$
−0.616080 + 0.787683i $$0.711280\pi$$
$$270$$ 0 0
$$271$$ 6.08396 0.369574 0.184787 0.982779i $$-0.440840\pi$$
0.184787 + 0.982779i $$0.440840\pi$$
$$272$$ − 7.33131i − 0.444526i
$$273$$ 1.35105i 0.0817693i
$$274$$ 8.42894 0.509211
$$275$$ 0 0
$$276$$ 11.0916 0.667634
$$277$$ 31.3647i 1.88452i 0.334877 + 0.942262i $$0.391305\pi$$
−0.334877 + 0.942262i $$0.608695\pi$$
$$278$$ 8.81841i 0.528893i
$$279$$ 19.0137 1.13832
$$280$$ 0 0
$$281$$ 11.3252 0.675607 0.337804 0.941217i $$-0.390316\pi$$
0.337804 + 0.941217i $$0.390316\pi$$
$$282$$ 18.4229i 1.09707i
$$283$$ 26.1437i 1.55408i 0.629452 + 0.777039i $$0.283279\pi$$
−0.629452 + 0.777039i $$0.716721\pi$$
$$284$$ 10.8579 0.644297
$$285$$ 0 0
$$286$$ 24.0273 1.42077
$$287$$ 0 0
$$288$$ 7.58473i 0.446934i
$$289$$ −36.7481 −2.16165
$$290$$ 0 0
$$291$$ 24.9742 1.46401
$$292$$ 5.09763i 0.298316i
$$293$$ 1.33131i 0.0777760i 0.999244 + 0.0388880i $$0.0123816\pi$$
−0.999244 + 0.0388880i $$0.987618\pi$$
$$294$$ 22.7542 1.32705
$$295$$ 0 0
$$296$$ −5.50684 −0.320078
$$297$$ 67.2241i 3.90074i
$$298$$ 17.6763i 1.02396i
$$299$$ 18.1755 1.05112
$$300$$ 0 0
$$301$$ −0.0394789 −0.00227553
$$302$$ − 20.0000i − 1.15087i
$$303$$ − 13.5205i − 0.776733i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 55.6060 3.17878
$$307$$ 2.84421i 0.162328i 0.996701 + 0.0811639i $$0.0258637\pi$$
−0.996701 + 0.0811639i $$0.974136\pi$$
$$308$$ 0.351050i 0.0200030i
$$309$$ −7.64895 −0.435134
$$310$$ 0 0
$$311$$ −16.3895 −0.929361 −0.464681 0.885478i $$-0.653831\pi$$
−0.464681 + 0.885478i $$0.653831\pi$$
$$312$$ 17.3450i 0.981966i
$$313$$ − 21.0471i − 1.18965i −0.803855 0.594826i $$-0.797221\pi$$
0.803855 0.594826i $$-0.202779\pi$$
$$314$$ −0.506836 −0.0286024
$$315$$ 0 0
$$316$$ 17.0137 0.957094
$$317$$ 30.8902i 1.73497i 0.497465 + 0.867484i $$0.334264\pi$$
−0.497465 + 0.867484i $$0.665736\pi$$
$$318$$ − 42.0855i − 2.36004i
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −45.6566 −2.54830
$$322$$ 0.265553i 0.0147987i
$$323$$ 7.33131i 0.407925i
$$324$$ −25.7739 −1.43188
$$325$$ 0 0
$$326$$ 0.830542 0.0459995
$$327$$ 0.253418i 0.0140140i
$$328$$ 0 0
$$329$$ −0.441078 −0.0243174
$$330$$ 0 0
$$331$$ 28.3845 1.56015 0.780076 0.625685i $$-0.215181\pi$$
0.780076 + 0.625685i $$0.215181\pi$$
$$332$$ 13.1695i 0.722768i
$$333$$ − 41.7679i − 2.28886i
$$334$$ −16.5068 −0.903214
$$335$$ 0 0
$$336$$ −0.253418 −0.0138251
$$337$$ 17.8716i 0.973526i 0.873534 + 0.486763i $$0.161822\pi$$
−0.873534 + 0.486763i $$0.838178\pi$$
$$338$$ 15.4229i 0.838894i
$$339$$ 19.5205 1.06021
$$340$$ 0 0
$$341$$ 11.2979 0.611816
$$342$$ − 7.58473i − 0.410135i
$$343$$ 1.09003i 0.0588560i
$$344$$ −0.506836 −0.0273268
$$345$$ 0 0
$$346$$ 18.8321 1.01242
$$347$$ − 33.0410i − 1.77373i −0.462024 0.886867i $$-0.652877\pi$$
0.462024 0.886867i $$-0.347123\pi$$
$$348$$ − 4.33131i − 0.232183i
$$349$$ 19.7158 1.05536 0.527681 0.849443i $$-0.323062\pi$$
0.527681 + 0.849443i $$0.323062\pi$$
$$350$$ 0 0
$$351$$ −79.5220 −4.24457
$$352$$ 4.50684i 0.240215i
$$353$$ − 5.90997i − 0.314556i −0.987554 0.157278i $$-0.949728\pi$$
0.987554 0.157278i $$-0.0502719\pi$$
$$354$$ −24.6121 −1.30812
$$355$$ 0 0
$$356$$ 15.0137 0.795723
$$357$$ 1.85789i 0.0983298i
$$358$$ − 7.15579i − 0.378195i
$$359$$ 7.00760 0.369847 0.184924 0.982753i $$-0.440796\pi$$
0.184924 + 0.982753i $$0.440796\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 12.5205i − 0.658063i
$$363$$ 30.2944i 1.59005i
$$364$$ −0.415271 −0.0217661
$$365$$ 0 0
$$366$$ −7.01367 −0.366611
$$367$$ − 17.7158i − 0.924756i −0.886683 0.462378i $$-0.846996\pi$$
0.886683 0.462378i $$-0.153004\pi$$
$$368$$ 3.40920i 0.177717i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.00760 0.0523122
$$372$$ 8.15579i 0.422858i
$$373$$ 15.4487i 0.799902i 0.916536 + 0.399951i $$0.130973\pi$$
−0.916536 + 0.399951i $$0.869027\pi$$
$$374$$ 33.0410 1.70851
$$375$$ 0 0
$$376$$ −5.66262 −0.292027
$$377$$ − 7.09763i − 0.365547i
$$378$$ − 1.16185i − 0.0597593i
$$379$$ −34.4563 −1.76990 −0.884950 0.465685i $$-0.845808\pi$$
−0.884950 + 0.465685i $$0.845808\pi$$
$$380$$ 0 0
$$381$$ 58.0152 2.97221
$$382$$ 4.90237i 0.250827i
$$383$$ − 12.0273i − 0.614569i −0.951618 0.307284i $$-0.900580\pi$$
0.951618 0.307284i $$-0.0994203\pi$$
$$384$$ −3.25342 −0.166025
$$385$$ 0 0
$$386$$ 18.1558 0.924105
$$387$$ − 3.84421i − 0.195412i
$$388$$ 7.67629i 0.389705i
$$389$$ 15.3647 0.779022 0.389511 0.921022i $$-0.372644\pi$$
0.389511 + 0.921022i $$0.372644\pi$$
$$390$$ 0 0
$$391$$ 24.9939 1.26400
$$392$$ 6.99393i 0.353247i
$$393$$ − 4.85789i − 0.245048i
$$394$$ 2.98633 0.150449
$$395$$ 0 0
$$396$$ −34.1831 −1.71777
$$397$$ − 1.32524i − 0.0665121i −0.999447 0.0332560i $$-0.989412\pi$$
0.999447 0.0332560i $$-0.0105877\pi$$
$$398$$ − 3.06422i − 0.153596i
$$399$$ 0.253418 0.0126868
$$400$$ 0 0
$$401$$ −6.46736 −0.322964 −0.161482 0.986876i $$-0.551627\pi$$
−0.161482 + 0.986876i $$0.551627\pi$$
$$402$$ 14.9160i 0.743944i
$$403$$ 13.3647i 0.665744i
$$404$$ 4.15579 0.206758
$$405$$ 0 0
$$406$$ 0.103700 0.00514653
$$407$$ − 24.8184i − 1.23020i
$$408$$ 23.8518i 1.18084i
$$409$$ 1.36472 0.0674812 0.0337406 0.999431i $$-0.489258\pi$$
0.0337406 + 0.999431i $$0.489258\pi$$
$$410$$ 0 0
$$411$$ −27.4229 −1.35267
$$412$$ − 2.35105i − 0.115828i
$$413$$ − 0.589259i − 0.0289955i
$$414$$ −25.8579 −1.27085
$$415$$ 0 0
$$416$$ −5.33131 −0.261389
$$417$$ − 28.6900i − 1.40495i
$$418$$ − 4.50684i − 0.220437i
$$419$$ −16.6231 −0.812094 −0.406047 0.913852i $$-0.633093\pi$$
−0.406047 + 0.913852i $$0.633093\pi$$
$$420$$ 0 0
$$421$$ −31.1128 −1.51635 −0.758174 0.652053i $$-0.773908\pi$$
−0.758174 + 0.652053i $$0.773908\pi$$
$$422$$ − 19.2534i − 0.937242i
$$423$$ − 42.9495i − 2.08827i
$$424$$ 12.9358 0.628217
$$425$$ 0 0
$$426$$ −35.3252 −1.71151
$$427$$ − 0.167920i − 0.00812623i
$$428$$ − 14.0334i − 0.678331i
$$429$$ −78.1710 −3.77413
$$430$$ 0 0
$$431$$ −10.1831 −0.490504 −0.245252 0.969459i $$-0.578871\pi$$
−0.245252 + 0.969459i $$0.578871\pi$$
$$432$$ − 14.9160i − 0.717648i
$$433$$ 22.1953i 1.06664i 0.845915 + 0.533318i $$0.179055\pi$$
−0.845915 + 0.533318i $$0.820945\pi$$
$$434$$ −0.195265 −0.00937300
$$435$$ 0 0
$$436$$ −0.0778929 −0.00373039
$$437$$ − 3.40920i − 0.163084i
$$438$$ − 16.5847i − 0.792449i
$$439$$ 9.32524 0.445070 0.222535 0.974925i $$-0.428567\pi$$
0.222535 + 0.974925i $$0.428567\pi$$
$$440$$ 0 0
$$441$$ −53.0471 −2.52605
$$442$$ 39.0855i 1.85911i
$$443$$ − 13.9879i − 0.664584i −0.943177 0.332292i $$-0.892178\pi$$
0.943177 0.332292i $$-0.107822\pi$$
$$444$$ 17.9160 0.850258
$$445$$ 0 0
$$446$$ −14.5068 −0.686919
$$447$$ − 57.5084i − 2.72005i
$$448$$ − 0.0778929i − 0.00368009i
$$449$$ −13.4932 −0.636782 −0.318391 0.947960i $$-0.603142\pi$$
−0.318391 + 0.947960i $$0.603142\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000i 0.282216i
$$453$$ 65.0684i 3.05718i
$$454$$ 21.5984 1.01366
$$455$$ 0 0
$$456$$ 3.25342 0.152355
$$457$$ − 9.68236i − 0.452922i −0.974020 0.226461i $$-0.927284\pi$$
0.974020 0.226461i $$-0.0727155\pi$$
$$458$$ − 19.0137i − 0.888451i
$$459$$ −109.354 −5.10421
$$460$$ 0 0
$$461$$ 8.66262 0.403459 0.201729 0.979441i $$-0.435344\pi$$
0.201729 + 0.979441i $$0.435344\pi$$
$$462$$ − 1.14211i − 0.0531359i
$$463$$ 28.0015i 1.30134i 0.759360 + 0.650671i $$0.225512\pi$$
−0.759360 + 0.650671i $$0.774488\pi$$
$$464$$ 1.33131 0.0618046
$$465$$ 0 0
$$466$$ −6.01367 −0.278578
$$467$$ − 16.9742i − 0.785472i −0.919651 0.392736i $$-0.871529\pi$$
0.919651 0.392736i $$-0.128471\pi$$
$$468$$ − 40.4365i − 1.86918i
$$469$$ −0.357118 −0.0164902
$$470$$ 0 0
$$471$$ 1.64895 0.0759796
$$472$$ − 7.56499i − 0.348207i
$$473$$ − 2.28423i − 0.105029i
$$474$$ −55.3526 −2.54243
$$475$$ 0 0
$$476$$ −0.571057 −0.0261743
$$477$$ 98.1144i 4.49235i
$$478$$ 15.4092i 0.704801i
$$479$$ −10.0532 −0.459340 −0.229670 0.973269i $$-0.573765\pi$$
−0.229670 + 0.973269i $$0.573765\pi$$
$$480$$ 0 0
$$481$$ 29.3587 1.33864
$$482$$ 4.81841i 0.219472i
$$483$$ − 0.863954i − 0.0393113i
$$484$$ −9.31157 −0.423253
$$485$$ 0 0
$$486$$ 39.1052 1.77385
$$487$$ − 19.2089i − 0.870440i −0.900324 0.435220i $$-0.856671\pi$$
0.900324 0.435220i $$-0.143329\pi$$
$$488$$ − 2.15579i − 0.0975878i
$$489$$ −2.70210 −0.122193
$$490$$ 0 0
$$491$$ 5.32524 0.240325 0.120162 0.992754i $$-0.461658\pi$$
0.120162 + 0.992754i $$0.461658\pi$$
$$492$$ 0 0
$$493$$ − 9.76025i − 0.439580i
$$494$$ 5.33131 0.239867
$$495$$ 0 0
$$496$$ −2.50684 −0.112560
$$497$$ − 0.845752i − 0.0379372i
$$498$$ − 42.8458i − 1.91996i
$$499$$ 11.9605 0.535426 0.267713 0.963499i $$-0.413732\pi$$
0.267713 + 0.963499i $$0.413732\pi$$
$$500$$ 0 0
$$501$$ 53.7036 2.39930
$$502$$ − 1.52051i − 0.0678636i
$$503$$ − 19.3313i − 0.861941i −0.902366 0.430970i $$-0.858171\pi$$
0.902366 0.430970i $$-0.141829\pi$$
$$504$$ 0.590796 0.0263162
$$505$$ 0 0
$$506$$ −15.3647 −0.683045
$$507$$ − 50.1771i − 2.22844i
$$508$$ 17.8321i 0.791171i
$$509$$ 36.1573 1.60265 0.801323 0.598232i $$-0.204130\pi$$
0.801323 + 0.598232i $$0.204130\pi$$
$$510$$ 0 0
$$511$$ 0.397069 0.0175653
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 14.9160i 0.658559i
$$514$$ −16.5068 −0.728085
$$515$$ 0 0
$$516$$ 1.64895 0.0725910
$$517$$ − 25.5205i − 1.12239i
$$518$$ 0.428943i 0.0188467i
$$519$$ −61.2686 −2.68939
$$520$$ 0 0
$$521$$ −31.5205 −1.38094 −0.690469 0.723362i $$-0.742596\pi$$
−0.690469 + 0.723362i $$0.742596\pi$$
$$522$$ 10.0976i 0.441961i
$$523$$ 5.59840i 0.244801i 0.992481 + 0.122400i $$0.0390592\pi$$
−0.992481 + 0.122400i $$0.960941\pi$$
$$524$$ 1.49316 0.0652292
$$525$$ 0 0
$$526$$ −18.0273 −0.786030
$$527$$ 18.3784i 0.800575i
$$528$$ − 14.6626i − 0.638109i
$$529$$ 11.3773 0.494667
$$530$$ 0 0
$$531$$ 57.3784 2.49001
$$532$$ 0.0778929i 0.00337708i
$$533$$ 0 0
$$534$$ −48.8458 −2.11376
$$535$$ 0 0
$$536$$ −4.58473 −0.198030
$$537$$ 23.2808i 1.00464i
$$538$$ 20.2089i 0.871269i
$$539$$ −31.5205 −1.35768
$$540$$ 0 0
$$541$$ 15.1968 0.653362 0.326681 0.945135i $$-0.394070\pi$$
0.326681 + 0.945135i $$0.394070\pi$$
$$542$$ − 6.08396i − 0.261328i
$$543$$ 40.7344i 1.74808i
$$544$$ −7.33131 −0.314327
$$545$$ 0 0
$$546$$ 1.35105 0.0578196
$$547$$ 26.6505i 1.13949i 0.821821 + 0.569746i $$0.192959\pi$$
−0.821821 + 0.569746i $$0.807041\pi$$
$$548$$ − 8.42894i − 0.360067i
$$549$$ 16.3511 0.697846
$$550$$ 0 0
$$551$$ −1.33131 −0.0567158
$$552$$ − 11.0916i − 0.472088i
$$553$$ − 1.32524i − 0.0563551i
$$554$$ 31.3647 1.33256
$$555$$ 0 0
$$556$$ 8.81841 0.373984
$$557$$ − 12.8458i − 0.544292i −0.962256 0.272146i $$-0.912267\pi$$
0.962256 0.272146i $$-0.0877334\pi$$
$$558$$ − 19.0137i − 0.804913i
$$559$$ 2.70210 0.114287
$$560$$ 0 0
$$561$$ −107.496 −4.53849
$$562$$ − 11.3252i − 0.477727i
$$563$$ 10.1968i 0.429744i 0.976642 + 0.214872i $$0.0689334\pi$$
−0.976642 + 0.214872i $$0.931067\pi$$
$$564$$ 18.4229 0.775743
$$565$$ 0 0
$$566$$ 26.1437 1.09890
$$567$$ 2.00760i 0.0843115i
$$568$$ − 10.8579i − 0.455587i
$$569$$ 24.3784 1.02200 0.510998 0.859582i $$-0.329276\pi$$
0.510998 + 0.859582i $$0.329276\pi$$
$$570$$ 0 0
$$571$$ 8.32371 0.348336 0.174168 0.984716i $$-0.444276\pi$$
0.174168 + 0.984716i $$0.444276\pi$$
$$572$$ − 24.0273i − 1.00463i
$$573$$ − 15.9495i − 0.666298i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.58473 0.316030
$$577$$ 7.29290i 0.303607i 0.988411 + 0.151804i $$0.0485081\pi$$
−0.988411 + 0.151804i $$0.951492\pi$$
$$578$$ 36.7481i 1.52852i
$$579$$ −59.0684 −2.45480
$$580$$ 0 0
$$581$$ 1.02581 0.0425576
$$582$$ − 24.9742i − 1.03521i
$$583$$ 58.2994i 2.41452i
$$584$$ 5.09763 0.210942
$$585$$ 0 0
$$586$$ 1.33131 0.0549959
$$587$$ − 5.53264i − 0.228357i −0.993460 0.114178i $$-0.963576\pi$$
0.993460 0.114178i $$-0.0364235\pi$$
$$588$$ − 22.7542i − 0.938367i
$$589$$ 2.50684 0.103292
$$590$$ 0 0
$$591$$ −9.71577 −0.399653
$$592$$ 5.50684i 0.226330i
$$593$$ − 26.3252i − 1.08105i −0.841329 0.540524i $$-0.818226\pi$$
0.841329 0.540524i $$-0.181774\pi$$
$$594$$ 67.2241 2.75824
$$595$$ 0 0
$$596$$ 17.6763 0.724049
$$597$$ 9.96919i 0.408012i
$$598$$ − 18.1755i − 0.743252i
$$599$$ −22.2994 −0.911130 −0.455565 0.890202i $$-0.650563\pi$$
−0.455565 + 0.890202i $$0.650563\pi$$
$$600$$ 0 0
$$601$$ 32.4947 1.32549 0.662743 0.748847i $$-0.269392\pi$$
0.662743 + 0.748847i $$0.269392\pi$$
$$602$$ 0.0394789i 0.00160904i
$$603$$ − 34.7739i − 1.41610i
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ −13.5205 −0.549233
$$607$$ − 8.35105i − 0.338959i −0.985534 0.169479i $$-0.945791\pi$$
0.985534 0.169479i $$-0.0542086\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ −0.337378 −0.0136713
$$610$$ 0 0
$$611$$ 30.1892 1.22132
$$612$$ − 55.6060i − 2.24774i
$$613$$ − 38.2994i − 1.54690i −0.633858 0.773450i $$-0.718529\pi$$
0.633858 0.773450i $$-0.281471\pi$$
$$614$$ 2.84421 0.114783
$$615$$ 0 0
$$616$$ 0.351050 0.0141442
$$617$$ 14.3526i 0.577813i 0.957357 + 0.288907i $$0.0932917\pi$$
−0.957357 + 0.288907i $$0.906708\pi$$
$$618$$ 7.64895i 0.307686i
$$619$$ 8.62314 0.346593 0.173297 0.984870i $$-0.444558\pi$$
0.173297 + 0.984870i $$0.444558\pi$$
$$620$$ 0 0
$$621$$ 50.8518 2.04061
$$622$$ 16.3895i 0.657158i
$$623$$ − 1.16946i − 0.0468533i
$$624$$ 17.3450 0.694355
$$625$$ 0 0
$$626$$ −21.0471 −0.841211
$$627$$ 14.6626i 0.585569i
$$628$$ 0.506836i 0.0202250i
$$629$$ 40.3723 1.60975
$$630$$ 0 0
$$631$$ −10.3374 −0.411525 −0.205762 0.978602i $$-0.565967\pi$$
−0.205762 + 0.978602i $$0.565967\pi$$
$$632$$ − 17.0137i − 0.676768i
$$633$$ 62.6394i 2.48969i
$$634$$ 30.8902 1.22681
$$635$$ 0 0
$$636$$ −42.0855 −1.66880
$$637$$ − 37.2868i − 1.47736i
$$638$$ 6.00000i 0.237542i
$$639$$ 82.3541 3.25788
$$640$$ 0 0
$$641$$ 5.88369 0.232392 0.116196 0.993226i $$-0.462930\pi$$
0.116196 + 0.993226i $$0.462930\pi$$
$$642$$ 45.6566i 1.80192i
$$643$$ − 25.5084i − 1.00595i −0.864300 0.502976i $$-0.832238\pi$$
0.864300 0.502976i $$-0.167762\pi$$
$$644$$ 0.265553 0.0104642
$$645$$ 0 0
$$646$$ 7.33131 0.288447
$$647$$ 5.09157i 0.200170i 0.994979 + 0.100085i $$0.0319115\pi$$
−0.994979 + 0.100085i $$0.968089\pi$$
$$648$$ 25.7739i 1.01250i
$$649$$ 34.0942 1.33831
$$650$$ 0 0
$$651$$ 0.635277 0.0248985
$$652$$ − 0.830542i − 0.0325265i
$$653$$ − 11.1816i − 0.437570i −0.975773 0.218785i $$-0.929791\pi$$
0.975773 0.218785i $$-0.0702093\pi$$
$$654$$ 0.253418 0.00990943
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 38.6642i 1.50843i
$$658$$ 0.441078i 0.0171950i
$$659$$ 13.7542 0.535787 0.267894 0.963449i $$-0.413672\pi$$
0.267894 + 0.963449i $$0.413672\pi$$
$$660$$ 0 0
$$661$$ −12.6171 −0.490747 −0.245374 0.969429i $$-0.578911\pi$$
−0.245374 + 0.969429i $$0.578911\pi$$
$$662$$ − 28.3845i − 1.10319i
$$663$$ − 127.161i − 4.93854i
$$664$$ 13.1695 0.511074
$$665$$ 0 0
$$666$$ −41.7679 −1.61847
$$667$$ 4.53871i 0.175740i
$$668$$ 16.5068i 0.638669i
$$669$$ 47.1968 1.82473
$$670$$ 0 0
$$671$$ 9.71577 0.375073
$$672$$ 0.253418i 0.00977581i
$$673$$ 2.35105i 0.0906263i 0.998973 + 0.0453132i $$0.0144286\pi$$
−0.998973 + 0.0453132i $$0.985571\pi$$
$$674$$ 17.8716 0.688387
$$675$$ 0 0
$$676$$ 15.4229 0.593188
$$677$$ 23.7663i 0.913414i 0.889617 + 0.456707i $$0.150971\pi$$
−0.889617 + 0.456707i $$0.849029\pi$$
$$678$$ − 19.5205i − 0.749681i
$$679$$ 0.597928 0.0229464
$$680$$ 0 0
$$681$$ −70.2686 −2.69270
$$682$$ − 11.2979i − 0.432619i
$$683$$ 28.1695i 1.07787i 0.842346 + 0.538937i $$0.181174\pi$$
−0.842346 + 0.538937i $$0.818826\pi$$
$$684$$ −7.58473 −0.290009
$$685$$ 0 0
$$686$$ 1.09003 0.0416174
$$687$$ 61.8594i 2.36008i
$$688$$ 0.506836i 0.0193229i
$$689$$ −68.9647 −2.62734
$$690$$ 0 0
$$691$$ 2.32371 0.0883979 0.0441990 0.999023i $$-0.485926\pi$$
0.0441990 + 0.999023i $$0.485926\pi$$
$$692$$ − 18.8321i − 0.715888i
$$693$$ 2.66262i 0.101145i
$$694$$ −33.0410 −1.25422
$$695$$ 0 0
$$696$$ −4.33131 −0.164178
$$697$$ 0 0
$$698$$ − 19.7158i − 0.746253i
$$699$$ 19.5650 0.740016
$$700$$ 0 0
$$701$$ 23.7036 0.895274 0.447637 0.894215i $$-0.352266\pi$$
0.447637 + 0.894215i $$0.352266\pi$$
$$702$$ 79.5220i 3.00137i
$$703$$ − 5.50684i − 0.207694i
$$704$$ 4.50684 0.169858
$$705$$ 0 0
$$706$$ −5.90997 −0.222425
$$707$$ − 0.323706i − 0.0121742i
$$708$$ 24.6121i 0.924978i
$$709$$ −9.05315 −0.339998 −0.169999 0.985444i $$-0.554376\pi$$
−0.169999 + 0.985444i $$0.554376\pi$$
$$710$$ 0 0
$$711$$ 129.044 4.83953
$$712$$ − 15.0137i − 0.562661i
$$713$$ − 8.54631i − 0.320062i
$$714$$ 1.85789 0.0695297
$$715$$ 0 0
$$716$$ −7.15579 −0.267424
$$717$$ − 50.1326i − 1.87224i
$$718$$ − 7.00760i − 0.261521i
$$719$$ −35.0734 −1.30802 −0.654008 0.756488i $$-0.726914\pi$$
−0.654008 + 0.756488i $$0.726914\pi$$
$$720$$ 0 0
$$721$$ −0.183130 −0.00682012
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 15.6763i − 0.583008i
$$724$$ −12.5205 −0.465321
$$725$$ 0 0
$$726$$ 30.2944 1.12433
$$727$$ − 30.1386i − 1.11778i −0.829242 0.558890i $$-0.811227\pi$$
0.829242 0.558890i $$-0.188773\pi$$
$$728$$ 0.415271i 0.0153910i
$$729$$ −49.9039 −1.84829
$$730$$ 0 0
$$731$$ 3.71577 0.137433
$$732$$ 7.01367i 0.259233i
$$733$$ 47.8321i 1.76672i 0.468697 + 0.883359i $$0.344724\pi$$
−0.468697 + 0.883359i $$0.655276\pi$$
$$734$$ −17.7158 −0.653901
$$735$$ 0 0
$$736$$ 3.40920 0.125665
$$737$$ − 20.6626i − 0.761117i
$$738$$ 0 0
$$739$$ 22.4674 0.826475 0.413238 0.910623i $$-0.364398\pi$$
0.413238 + 0.910623i $$0.364398\pi$$
$$740$$ 0 0
$$741$$ −17.3450 −0.637184
$$742$$ − 1.00760i − 0.0369903i
$$743$$ − 11.7926i − 0.432629i −0.976324 0.216314i $$-0.930596\pi$$
0.976324 0.216314i $$-0.0694036\pi$$
$$744$$ 8.15579 0.299006
$$745$$ 0 0
$$746$$ 15.4487 0.565616
$$747$$ 99.8868i 3.65467i
$$748$$ − 33.0410i − 1.20810i
$$749$$ −1.09310 −0.0399411
$$750$$ 0 0
$$751$$ −2.03948 −0.0744216 −0.0372108 0.999307i $$-0.511847\pi$$
−0.0372108 + 0.999307i $$0.511847\pi$$
$$752$$ 5.66262i 0.206495i
$$753$$ 4.94685i 0.180273i
$$754$$ −7.09763 −0.258481
$$755$$ 0 0
$$756$$ −1.16185 −0.0422562
$$757$$ − 43.3526i − 1.57568i −0.615882 0.787838i $$-0.711200\pi$$
0.615882 0.787838i $$-0.288800\pi$$
$$758$$ 34.4563i 1.25151i
$$759$$ 49.9879 1.81444
$$760$$ 0 0
$$761$$ 6.62921 0.240309 0.120154 0.992755i $$-0.461661\pi$$
0.120154 + 0.992755i $$0.461661\pi$$
$$762$$ − 58.0152i − 2.10167i
$$763$$ 0.00606730i 0 0.000219651i
$$764$$ 4.90237 0.177361
$$765$$ 0 0
$$766$$ −12.0273 −0.434566
$$767$$ 40.3313i 1.45628i
$$768$$ 3.25342i 0.117398i
$$769$$ 19.6429 0.708340 0.354170 0.935181i $$-0.384763\pi$$
0.354170 + 0.935181i $$0.384763\pi$$
$$770$$ 0 0
$$771$$ 53.7036 1.93409
$$772$$ − 18.1558i − 0.653441i
$$773$$ 14.4107i 0.518318i 0.965835 + 0.259159i $$0.0834454\pi$$
−0.965835 + 0.259159i $$0.916555\pi$$
$$774$$ −3.84421 −0.138177
$$775$$ 0 0
$$776$$ 7.67629 0.275563
$$777$$ − 1.39553i − 0.0500644i
$$778$$ − 15.3647i − 0.550852i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 48.9347 1.75102
$$782$$ − 24.9939i − 0.893781i
$$783$$ − 19.8579i − 0.709663i
$$784$$ 6.99393 0.249783
$$785$$ 0 0
$$786$$ −4.85789 −0.173275
$$787$$ − 24.3176i − 0.866830i −0.901194 0.433415i $$-0.857308\pi$$
0.901194 0.433415i $$-0.142692\pi$$
$$788$$ − 2.98633i − 0.106384i
$$789$$ 58.6505 2.08801
$$790$$ 0 0
$$791$$ 0.467357 0.0166173
$$792$$ 34.1831i 1.21464i
$$793$$ 11.4932i 0.408134i
$$794$$ −1.32524 −0.0470311
$$795$$ 0 0
$$796$$ −3.06422 −0.108608
$$797$$ 7.30397i 0.258720i 0.991598 + 0.129360i $$0.0412922\pi$$
−0.991598 + 0.129360i $$0.958708\pi$$
$$798$$ − 0.253418i − 0.00897090i
$$799$$ 41.5144 1.46868
$$800$$ 0 0
$$801$$ 113.875 4.02356
$$802$$ 6.46736i 0.228370i
$$803$$ 22.9742i 0.810742i
$$804$$ 14.9160 0.526048
$$805$$ 0 0
$$806$$ 13.3647 0.470752
$$807$$ − 65.7481i − 2.31444i
$$808$$ − 4.15579i − 0.146200i
$$809$$ 0.584729 0.0205580 0.0102790 0.999947i $$-0.496728\pi$$
0.0102790 + 0.999947i $$0.496728\pi$$
$$810$$ 0 0
$$811$$ 1.90997 0.0670682 0.0335341 0.999438i $$-0.489324\pi$$
0.0335341 + 0.999438i $$0.489324\pi$$
$$812$$ − 0.103700i − 0.00363914i
$$813$$ 19.7937i 0.694194i
$$814$$ −24.8184 −0.869885
$$815$$ 0 0
$$816$$ 23.8518 0.834981
$$817$$ − 0.506836i − 0.0177319i
$$818$$ − 1.36472i − 0.0477164i
$$819$$ −3.14972 −0.110060
$$820$$ 0 0
$$821$$ −38.2226 −1.33398 −0.666989 0.745067i $$-0.732417\pi$$
−0.666989 + 0.745067i $$0.732417\pi$$
$$822$$ 27.4229i 0.956483i
$$823$$ − 45.7481i − 1.59468i −0.603531 0.797340i $$-0.706240\pi$$
0.603531 0.797340i $$-0.293760\pi$$
$$824$$ −2.35105 −0.0819027
$$825$$ 0 0
$$826$$ −0.589259 −0.0205029
$$827$$ 22.6687i 0.788268i 0.919053 + 0.394134i $$0.128955\pi$$
−0.919053 + 0.394134i $$0.871045\pi$$
$$828$$ 25.8579i 0.898624i
$$829$$ 4.63028 0.160816 0.0804081 0.996762i $$-0.474378\pi$$
0.0804081 + 0.996762i $$0.474378\pi$$
$$830$$ 0 0
$$831$$ −102.043 −3.53982
$$832$$ 5.33131i 0.184830i
$$833$$ − 51.2747i − 1.77656i
$$834$$ −28.6900 −0.993452
$$835$$ 0 0
$$836$$ −4.50684 −0.155872
$$837$$ 37.3921i 1.29246i
$$838$$ 16.6231i 0.574237i
$$839$$ 50.4826 1.74285 0.871426 0.490527i $$-0.163196\pi$$
0.871426 + 0.490527i $$0.163196\pi$$
$$840$$ 0 0
$$841$$ −27.2276 −0.938883
$$842$$ 31.1128i 1.07222i
$$843$$ 36.8458i 1.26904i
$$844$$ −19.2534 −0.662730
$$845$$ 0 0
$$846$$ −42.9495 −1.47663
$$847$$ 0.725305i 0.0249218i
$$848$$ − 12.9358i − 0.444216i
$$849$$ −85.0562 −2.91912
$$850$$ 0 0
$$851$$ −18.7739 −0.643562
$$852$$ 35.3252i 1.21022i
$$853$$ 36.2105i 1.23982i 0.784672 + 0.619912i $$0.212832\pi$$
−0.784672 + 0.619912i $$0.787168\pi$$
$$854$$ −0.167920 −0.00574611
$$855$$ 0 0
$$856$$ −14.0334 −0.479652
$$857$$ − 25.1968i − 0.860706i −0.902661 0.430353i $$-0.858389\pi$$
0.902661 0.430353i $$-0.141611\pi$$
$$858$$ 78.1710i 2.66871i
$$859$$ −18.8579 −0.643423 −0.321711 0.946838i $$-0.604258\pi$$
−0.321711 + 0.946838i $$0.604258\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10.1831i 0.346839i
$$863$$ 15.0137i 0.511071i 0.966800 + 0.255536i $$0.0822518\pi$$
−0.966800 + 0.255536i $$0.917748\pi$$
$$864$$ −14.9160 −0.507454
$$865$$ 0 0
$$866$$ 22.1953 0.754226
$$867$$ − 119.557i − 4.06037i
$$868$$ 0.195265i 0.00662771i
$$869$$ 76.6778 2.60112
$$870$$ 0 0
$$871$$ 24.4426 0.828206
$$872$$ 0.0778929i 0.00263779i
$$873$$ 58.2226i 1.97054i
$$874$$ −3.40920 −0.115318
$$875$$ 0 0
$$876$$ −16.5847 −0.560346
$$877$$ − 35.1128i − 1.18568i −0.805322 0.592838i $$-0.798007\pi$$
0.805322 0.592838i $$-0.201993\pi$$
$$878$$ − 9.32524i − 0.314712i
$$879$$ −4.33131 −0.146091
$$880$$ 0 0
$$881$$ −41.3389 −1.39274 −0.696372 0.717681i $$-0.745203\pi$$
−0.696372 + 0.717681i $$0.745203\pi$$
$$882$$ 53.0471i 1.78619i
$$883$$ − 12.6353i − 0.425211i −0.977138 0.212605i $$-0.931805\pi$$
0.977138 0.212605i $$-0.0681949\pi$$
$$884$$ 39.0855 1.31459
$$885$$ 0 0
$$886$$ −13.9879 −0.469932
$$887$$ − 4.15579i − 0.139538i −0.997563 0.0697688i $$-0.977774\pi$$
0.997563 0.0697688i $$-0.0222262\pi$$
$$888$$ − 17.9160i − 0.601223i
$$889$$ 1.38899 0.0465853
$$890$$ 0 0
$$891$$ −116.159 −3.89147
$$892$$ 14.5068i 0.485725i
$$893$$ − 5.66262i − 0.189492i
$$894$$ −57.5084 −1.92337
$$895$$ 0 0
$$896$$ −0.0778929 −0.00260222
$$897$$ 59.1326i 1.97438i
$$898$$ 13.4932i 0.450273i
$$899$$ −3.33738 −0.111308
$$900$$ 0 0
$$901$$ −94.8362 −3.15945
$$902$$ 0 0
$$903$$ − 0.128441i − 0.00427426i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 65.0684 2.16175
$$907$$ 41.7542i 1.38643i 0.720733 + 0.693213i $$0.243805\pi$$
−0.720733 + 0.693213i $$0.756195\pi$$
$$908$$ − 21.5984i − 0.716768i
$$909$$ 31.5205 1.04547
$$910$$ 0 0
$$911$$ −9.12998 −0.302490 −0.151245 0.988496i $$-0.548328\pi$$
−0.151245 + 0.988496i $$0.548328\pi$$
$$912$$ − 3.25342i − 0.107731i
$$913$$ 59.3526i 1.96428i
$$914$$ −9.68236 −0.320264
$$915$$ 0 0
$$916$$ −19.0137 −0.628229
$$917$$ − 0.116307i − 0.00384079i
$$918$$ 109.354i 3.60922i
$$919$$ −19.0197 −0.627403 −0.313702 0.949522i $$-0.601569\pi$$
−0.313702 + 0.949522i $$0.601569\pi$$
$$920$$ 0 0
$$921$$ −9.25342 −0.304910
$$922$$ − 8.66262i − 0.285288i
$$923$$ 57.8868i 1.90537i
$$924$$ −1.14211 −0.0375728
$$925$$ 0 0
$$926$$ 28.0015 0.920188
$$927$$ − 17.8321i − 0.585682i
$$928$$ − 1.33131i − 0.0437024i
$$929$$ 7.77239 0.255004 0.127502 0.991838i $$-0.459304\pi$$
0.127502 + 0.991838i $$0.459304\pi$$
$$930$$ 0 0
$$931$$ −6.99393 −0.229217
$$932$$ 6.01367i 0.196984i
$$933$$ − 53.3218i − 1.74568i
$$934$$ −16.9742 −0.555413
$$935$$ 0 0
$$936$$ −40.4365 −1.32171
$$937$$ 45.9818i 1.50216i 0.660211 + 0.751080i $$0.270467\pi$$
−0.660211 + 0.751080i $$0.729533\pi$$
$$938$$ 0.357118i 0.0116603i
$$939$$ 68.4750 2.23460
$$940$$ 0 0
$$941$$ −40.6242 −1.32431 −0.662156 0.749366i $$-0.730358\pi$$
−0.662156 + 0.749366i $$0.730358\pi$$
$$942$$ − 1.64895i − 0.0537257i
$$943$$ 0 0
$$944$$ −7.56499 −0.246219
$$945$$ 0 0
$$946$$ −2.28423 −0.0742666
$$947$$ 2.28423i 0.0742274i 0.999311 + 0.0371137i $$0.0118164\pi$$
−0.999311 + 0.0371137i $$0.988184\pi$$
$$948$$ 55.3526i 1.79777i
$$949$$ −27.1771 −0.882205
$$950$$ 0 0
$$951$$ −100.499 −3.25890
$$952$$ 0.571057i 0.0185081i
$$953$$ − 57.5084i − 1.86288i −0.363896 0.931439i $$-0.618554\pi$$
0.363896 0.931439i $$-0.381446\pi$$
$$954$$ 98.1144 3.17657
$$955$$ 0 0
$$956$$ 15.4092 0.498369
$$957$$ − 19.5205i − 0.631008i
$$958$$ 10.0532i 0.324803i
$$959$$ −0.656555 −0.0212013
$$960$$ 0 0
$$961$$ −24.7158 −0.797283
$$962$$ − 29.3587i − 0.946561i
$$963$$ − 106.440i − 3.42997i
$$964$$ 4.81841 0.155190
$$965$$ 0 0
$$966$$ −0.863954 −0.0277973
$$967$$ 4.70210i 0.151209i 0.997138 + 0.0756047i $$0.0240887\pi$$
−0.997138 + 0.0756047i $$0.975911\pi$$
$$968$$ 9.31157i 0.299285i
$$969$$ −23.8518 −0.766231
$$970$$ 0 0
$$971$$ 14.9863 0.480934 0.240467 0.970657i $$-0.422699\pi$$
0.240467 + 0.970657i $$0.422699\pi$$
$$972$$ − 39.1052i − 1.25430i
$$973$$ − 0.686891i − 0.0220207i
$$974$$ −19.2089 −0.615494
$$975$$ 0 0
$$976$$ −2.15579 −0.0690050
$$977$$ − 8.89737i − 0.284652i −0.989820 0.142326i $$-0.954542\pi$$
0.989820 0.142326i $$-0.0454581\pi$$
$$978$$ 2.70210i 0.0864037i
$$979$$ 67.6642 2.16256
$$980$$ 0 0
$$981$$ −0.590796 −0.0188627
$$982$$ − 5.32524i − 0.169935i
$$983$$ − 1.60947i − 0.0513341i −0.999671 0.0256671i $$-0.991829\pi$$
0.999671 0.0256671i $$-0.00817098\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −9.76025 −0.310830
$$987$$ − 1.43501i − 0.0456769i
$$988$$ − 5.33131i − 0.169612i
$$989$$ −1.72791 −0.0549443
$$990$$ 0 0
$$991$$ 16.8974 0.536763 0.268381 0.963313i $$-0.413511\pi$$
0.268381 + 0.963313i $$0.413511\pi$$
$$992$$ 2.50684i 0.0795921i
$$993$$ 92.3465i 2.93053i
$$994$$ −0.845752 −0.0268256
$$995$$ 0 0
$$996$$ −42.8458 −1.35762
$$997$$ − 24.1558i − 0.765021i −0.923951 0.382511i $$-0.875060\pi$$
0.923951 0.382511i $$-0.124940\pi$$
$$998$$ − 11.9605i − 0.378604i
$$999$$ 82.1402 2.59880
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.799.3 6
5.2 odd 4 950.2.a.l.1.3 yes 3
5.3 odd 4 950.2.a.j.1.1 3
5.4 even 2 inner 950.2.b.h.799.4 6
15.2 even 4 8550.2.a.ci.1.2 3
15.8 even 4 8550.2.a.cp.1.2 3
20.3 even 4 7600.2.a.bz.1.3 3
20.7 even 4 7600.2.a.bk.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.1 3 5.3 odd 4
950.2.a.l.1.3 yes 3 5.2 odd 4
950.2.b.h.799.3 6 1.1 even 1 trivial
950.2.b.h.799.4 6 5.4 even 2 inner
7600.2.a.bk.1.1 3 20.7 even 4
7600.2.a.bz.1.3 3 20.3 even 4
8550.2.a.ci.1.2 3 15.2 even 4
8550.2.a.cp.1.2 3 15.8 even 4