# Properties

 Label 3450.2.a.bq.1.3 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.363328$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.50466 q^{11} +1.00000 q^{12} +2.72666 q^{13} -2.00000 q^{14} +1.00000 q^{16} -0.726656 q^{17} -1.00000 q^{18} -7.78734 q^{19} +2.00000 q^{21} -3.50466 q^{22} +1.00000 q^{23} -1.00000 q^{24} -2.72666 q^{26} +1.00000 q^{27} +2.00000 q^{28} -2.00000 q^{29} +8.28267 q^{31} -1.00000 q^{32} +3.50466 q^{33} +0.726656 q^{34} +1.00000 q^{36} +9.50466 q^{37} +7.78734 q^{38} +2.72666 q^{39} +0.726656 q^{41} -2.00000 q^{42} +5.50466 q^{43} +3.50466 q^{44} -1.00000 q^{46} +2.72666 q^{47} +1.00000 q^{48} -3.00000 q^{49} -0.726656 q^{51} +2.72666 q^{52} -0.231321 q^{53} -1.00000 q^{54} -2.00000 q^{56} -7.78734 q^{57} +2.00000 q^{58} +9.29200 q^{59} +4.05135 q^{61} -8.28267 q^{62} +2.00000 q^{63} +1.00000 q^{64} -3.50466 q^{66} +4.05135 q^{67} -0.726656 q^{68} +1.00000 q^{69} -11.7360 q^{71} -1.00000 q^{72} +3.71733 q^{73} -9.50466 q^{74} -7.78734 q^{76} +7.00933 q^{77} -2.72666 q^{78} -9.73599 q^{79} +1.00000 q^{81} -0.726656 q^{82} -13.5233 q^{83} +2.00000 q^{84} -5.50466 q^{86} -2.00000 q^{87} -3.50466 q^{88} -9.55602 q^{89} +5.45331 q^{91} +1.00000 q^{92} +8.28267 q^{93} -2.72666 q^{94} -1.00000 q^{96} +4.54669 q^{97} +3.00000 q^{98} +3.50466 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + 3q^{12} + 4q^{13} - 6q^{14} + 3q^{16} + 2q^{17} - 3q^{18} + 4q^{19} + 6q^{21} + 3q^{23} - 3q^{24} - 4q^{26} + 3q^{27} + 6q^{28} - 6q^{29} + 8q^{31} - 3q^{32} - 2q^{34} + 3q^{36} + 18q^{37} - 4q^{38} + 4q^{39} - 2q^{41} - 6q^{42} + 6q^{43} - 3q^{46} + 4q^{47} + 3q^{48} - 9q^{49} + 2q^{51} + 4q^{52} + 14q^{53} - 3q^{54} - 6q^{56} + 4q^{57} + 6q^{58} - 10q^{59} + 10q^{61} - 8q^{62} + 6q^{63} + 3q^{64} + 10q^{67} + 2q^{68} + 3q^{69} - 10q^{71} - 3q^{72} + 28q^{73} - 18q^{74} + 4q^{76} - 4q^{78} - 4q^{79} + 3q^{81} + 2q^{82} + 12q^{83} + 6q^{84} - 6q^{86} - 6q^{87} - 16q^{89} + 8q^{91} + 3q^{92} + 8q^{93} - 4q^{94} - 3q^{96} + 22q^{97} + 9q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.50466 1.05670 0.528348 0.849028i $$-0.322812\pi$$
0.528348 + 0.849028i $$0.322812\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.72666 0.756238 0.378119 0.925757i $$-0.376571\pi$$
0.378119 + 0.925757i $$0.376571\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.726656 −0.176240 −0.0881200 0.996110i $$-0.528086\pi$$
−0.0881200 + 0.996110i $$0.528086\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −7.78734 −1.78654 −0.893269 0.449523i $$-0.851594\pi$$
−0.893269 + 0.449523i $$0.851594\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ −3.50466 −0.747197
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.72666 −0.534741
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 8.28267 1.48761 0.743806 0.668396i $$-0.233019\pi$$
0.743806 + 0.668396i $$0.233019\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 3.50466 0.610084
$$34$$ 0.726656 0.124621
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 9.50466 1.56256 0.781279 0.624182i $$-0.214568\pi$$
0.781279 + 0.624182i $$0.214568\pi$$
$$38$$ 7.78734 1.26327
$$39$$ 2.72666 0.436614
$$40$$ 0 0
$$41$$ 0.726656 0.113485 0.0567423 0.998389i $$-0.481929\pi$$
0.0567423 + 0.998389i $$0.481929\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 5.50466 0.839453 0.419727 0.907651i $$-0.362126\pi$$
0.419727 + 0.907651i $$0.362126\pi$$
$$44$$ 3.50466 0.528348
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 2.72666 0.397724 0.198862 0.980028i $$-0.436275\pi$$
0.198862 + 0.980028i $$0.436275\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −0.726656 −0.101752
$$52$$ 2.72666 0.378119
$$53$$ −0.231321 −0.0317744 −0.0158872 0.999874i $$-0.505057\pi$$
−0.0158872 + 0.999874i $$0.505057\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ −7.78734 −1.03146
$$58$$ 2.00000 0.262613
$$59$$ 9.29200 1.20972 0.604858 0.796334i $$-0.293230\pi$$
0.604858 + 0.796334i $$0.293230\pi$$
$$60$$ 0 0
$$61$$ 4.05135 0.518722 0.259361 0.965780i $$-0.416488\pi$$
0.259361 + 0.965780i $$0.416488\pi$$
$$62$$ −8.28267 −1.05190
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −3.50466 −0.431394
$$67$$ 4.05135 0.494951 0.247476 0.968894i $$-0.420399\pi$$
0.247476 + 0.968894i $$0.420399\pi$$
$$68$$ −0.726656 −0.0881200
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −11.7360 −1.39281 −0.696403 0.717651i $$-0.745217\pi$$
−0.696403 + 0.717651i $$0.745217\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 3.71733 0.435080 0.217540 0.976051i $$-0.430197\pi$$
0.217540 + 0.976051i $$0.430197\pi$$
$$74$$ −9.50466 −1.10489
$$75$$ 0 0
$$76$$ −7.78734 −0.893269
$$77$$ 7.00933 0.798787
$$78$$ −2.72666 −0.308733
$$79$$ −9.73599 −1.09538 −0.547692 0.836680i $$-0.684493\pi$$
−0.547692 + 0.836680i $$0.684493\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −0.726656 −0.0802458
$$83$$ −13.5233 −1.48438 −0.742189 0.670191i $$-0.766212\pi$$
−0.742189 + 0.670191i $$0.766212\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −5.50466 −0.593583
$$87$$ −2.00000 −0.214423
$$88$$ −3.50466 −0.373598
$$89$$ −9.55602 −1.01294 −0.506468 0.862259i $$-0.669049\pi$$
−0.506468 + 0.862259i $$0.669049\pi$$
$$90$$ 0 0
$$91$$ 5.45331 0.571663
$$92$$ 1.00000 0.104257
$$93$$ 8.28267 0.858873
$$94$$ −2.72666 −0.281233
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 4.54669 0.461646 0.230823 0.972996i $$-0.425858\pi$$
0.230823 + 0.972996i $$0.425858\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 3.50466 0.352232
$$100$$ 0 0
$$101$$ 3.45331 0.343617 0.171809 0.985130i $$-0.445039\pi$$
0.171809 + 0.985130i $$0.445039\pi$$
$$102$$ 0.726656 0.0719497
$$103$$ 14.4626 1.42505 0.712523 0.701649i $$-0.247552\pi$$
0.712523 + 0.701649i $$0.247552\pi$$
$$104$$ −2.72666 −0.267371
$$105$$ 0 0
$$106$$ 0.231321 0.0224679
$$107$$ −13.0607 −1.26262 −0.631312 0.775529i $$-0.717483\pi$$
−0.631312 + 0.775529i $$0.717483\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.9580 −1.04958 −0.524792 0.851231i $$-0.675857\pi$$
−0.524792 + 0.851231i $$0.675857\pi$$
$$110$$ 0 0
$$111$$ 9.50466 0.902143
$$112$$ 2.00000 0.188982
$$113$$ 11.7360 1.10403 0.552014 0.833835i $$-0.313859\pi$$
0.552014 + 0.833835i $$0.313859\pi$$
$$114$$ 7.78734 0.729351
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 2.72666 0.252079
$$118$$ −9.29200 −0.855398
$$119$$ −1.45331 −0.133225
$$120$$ 0 0
$$121$$ 1.28267 0.116607
$$122$$ −4.05135 −0.366792
$$123$$ 0.726656 0.0655204
$$124$$ 8.28267 0.743806
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 12.2827 1.08991 0.544955 0.838465i $$-0.316547\pi$$
0.544955 + 0.838465i $$0.316547\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 5.50466 0.484659
$$130$$ 0 0
$$131$$ −14.8480 −1.29728 −0.648639 0.761097i $$-0.724661\pi$$
−0.648639 + 0.761097i $$0.724661\pi$$
$$132$$ 3.50466 0.305042
$$133$$ −15.5747 −1.35050
$$134$$ −4.05135 −0.349983
$$135$$ 0 0
$$136$$ 0.726656 0.0623103
$$137$$ 3.27334 0.279661 0.139830 0.990175i $$-0.455344\pi$$
0.139830 + 0.990175i $$0.455344\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 14.0187 1.18905 0.594524 0.804078i $$-0.297341\pi$$
0.594524 + 0.804078i $$0.297341\pi$$
$$140$$ 0 0
$$141$$ 2.72666 0.229626
$$142$$ 11.7360 0.984862
$$143$$ 9.55602 0.799114
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −3.71733 −0.307648
$$147$$ −3.00000 −0.247436
$$148$$ 9.50466 0.781279
$$149$$ −4.49534 −0.368272 −0.184136 0.982901i $$-0.558949\pi$$
−0.184136 + 0.982901i $$0.558949\pi$$
$$150$$ 0 0
$$151$$ 22.3013 1.81486 0.907428 0.420207i $$-0.138043\pi$$
0.907428 + 0.420207i $$0.138043\pi$$
$$152$$ 7.78734 0.631636
$$153$$ −0.726656 −0.0587467
$$154$$ −7.00933 −0.564828
$$155$$ 0 0
$$156$$ 2.72666 0.218307
$$157$$ 9.60737 0.766751 0.383376 0.923592i $$-0.374761\pi$$
0.383376 + 0.923592i $$0.374761\pi$$
$$158$$ 9.73599 0.774553
$$159$$ −0.231321 −0.0183449
$$160$$ 0 0
$$161$$ 2.00000 0.157622
$$162$$ −1.00000 −0.0785674
$$163$$ 14.1214 1.10607 0.553035 0.833158i $$-0.313470\pi$$
0.553035 + 0.833158i $$0.313470\pi$$
$$164$$ 0.726656 0.0567423
$$165$$ 0 0
$$166$$ 13.5233 1.04961
$$167$$ 8.74531 0.676733 0.338366 0.941014i $$-0.390126\pi$$
0.338366 + 0.941014i $$0.390126\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ −5.56534 −0.428103
$$170$$ 0 0
$$171$$ −7.78734 −0.595513
$$172$$ 5.50466 0.419727
$$173$$ 5.89730 0.448363 0.224182 0.974547i $$-0.428029\pi$$
0.224182 + 0.974547i $$0.428029\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 3.50466 0.264174
$$177$$ 9.29200 0.698430
$$178$$ 9.55602 0.716254
$$179$$ −0.623954 −0.0466365 −0.0233182 0.999728i $$-0.507423\pi$$
−0.0233182 + 0.999728i $$0.507423\pi$$
$$180$$ 0 0
$$181$$ −14.0700 −1.04582 −0.522908 0.852389i $$-0.675153\pi$$
−0.522908 + 0.852389i $$0.675153\pi$$
$$182$$ −5.45331 −0.404226
$$183$$ 4.05135 0.299485
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −8.28267 −0.607315
$$187$$ −2.54669 −0.186232
$$188$$ 2.72666 0.198862
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −19.5747 −1.41637 −0.708187 0.706025i $$-0.750487\pi$$
−0.708187 + 0.706025i $$0.750487\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −10.0187 −0.721159 −0.360579 0.932729i $$-0.617421\pi$$
−0.360579 + 0.932729i $$0.617421\pi$$
$$194$$ −4.54669 −0.326433
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ −3.50466 −0.249066
$$199$$ 23.1893 1.64385 0.821923 0.569599i $$-0.192901\pi$$
0.821923 + 0.569599i $$0.192901\pi$$
$$200$$ 0 0
$$201$$ 4.05135 0.285760
$$202$$ −3.45331 −0.242974
$$203$$ −4.00000 −0.280745
$$204$$ −0.726656 −0.0508761
$$205$$ 0 0
$$206$$ −14.4626 −1.00766
$$207$$ 1.00000 0.0695048
$$208$$ 2.72666 0.189060
$$209$$ −27.2920 −1.88783
$$210$$ 0 0
$$211$$ −7.57467 −0.521462 −0.260731 0.965411i $$-0.583964\pi$$
−0.260731 + 0.965411i $$0.583964\pi$$
$$212$$ −0.231321 −0.0158872
$$213$$ −11.7360 −0.804136
$$214$$ 13.0607 0.892810
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 16.5653 1.12453
$$218$$ 10.9580 0.742168
$$219$$ 3.71733 0.251194
$$220$$ 0 0
$$221$$ −1.98134 −0.133280
$$222$$ −9.50466 −0.637911
$$223$$ −26.5840 −1.78020 −0.890098 0.455769i $$-0.849364\pi$$
−0.890098 + 0.455769i $$0.849364\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −11.7360 −0.780666
$$227$$ −11.9673 −0.794298 −0.397149 0.917754i $$-0.630000\pi$$
−0.397149 + 0.917754i $$0.630000\pi$$
$$228$$ −7.78734 −0.515729
$$229$$ 26.5327 1.75333 0.876663 0.481104i $$-0.159764\pi$$
0.876663 + 0.481104i $$0.159764\pi$$
$$230$$ 0 0
$$231$$ 7.00933 0.461180
$$232$$ 2.00000 0.131306
$$233$$ 24.4040 1.59876 0.799381 0.600825i $$-0.205161\pi$$
0.799381 + 0.600825i $$0.205161\pi$$
$$234$$ −2.72666 −0.178247
$$235$$ 0 0
$$236$$ 9.29200 0.604858
$$237$$ −9.73599 −0.632420
$$238$$ 1.45331 0.0942043
$$239$$ −3.55602 −0.230020 −0.115010 0.993364i $$-0.536690\pi$$
−0.115010 + 0.993364i $$0.536690\pi$$
$$240$$ 0 0
$$241$$ 3.55602 0.229063 0.114532 0.993420i $$-0.463463\pi$$
0.114532 + 0.993420i $$0.463463\pi$$
$$242$$ −1.28267 −0.0824533
$$243$$ 1.00000 0.0641500
$$244$$ 4.05135 0.259361
$$245$$ 0 0
$$246$$ −0.726656 −0.0463299
$$247$$ −21.2334 −1.35105
$$248$$ −8.28267 −0.525950
$$249$$ −13.5233 −0.857006
$$250$$ 0 0
$$251$$ 6.05135 0.381958 0.190979 0.981594i $$-0.438834\pi$$
0.190979 + 0.981594i $$0.438834\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 3.50466 0.220336
$$254$$ −12.2827 −0.770683
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 16.9066 1.05461 0.527303 0.849677i $$-0.323203\pi$$
0.527303 + 0.849677i $$0.323203\pi$$
$$258$$ −5.50466 −0.342705
$$259$$ 19.0093 1.18118
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 14.8480 0.917314
$$263$$ 0.887968 0.0547545 0.0273772 0.999625i $$-0.491284\pi$$
0.0273772 + 0.999625i $$0.491284\pi$$
$$264$$ −3.50466 −0.215697
$$265$$ 0 0
$$266$$ 15.5747 0.954944
$$267$$ −9.55602 −0.584819
$$268$$ 4.05135 0.247476
$$269$$ 20.1214 1.22682 0.613410 0.789764i $$-0.289797\pi$$
0.613410 + 0.789764i $$0.289797\pi$$
$$270$$ 0 0
$$271$$ −30.5840 −1.85785 −0.928923 0.370273i $$-0.879264\pi$$
−0.928923 + 0.370273i $$0.879264\pi$$
$$272$$ −0.726656 −0.0440600
$$273$$ 5.45331 0.330050
$$274$$ −3.27334 −0.197750
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ −4.38538 −0.263492 −0.131746 0.991284i $$-0.542058\pi$$
−0.131746 + 0.991284i $$0.542058\pi$$
$$278$$ −14.0187 −0.840783
$$279$$ 8.28267 0.495871
$$280$$ 0 0
$$281$$ 10.1214 0.603790 0.301895 0.953341i $$-0.402381\pi$$
0.301895 + 0.953341i $$0.402381\pi$$
$$282$$ −2.72666 −0.162370
$$283$$ 25.1820 1.49692 0.748458 0.663182i $$-0.230794\pi$$
0.748458 + 0.663182i $$0.230794\pi$$
$$284$$ −11.7360 −0.696403
$$285$$ 0 0
$$286$$ −9.55602 −0.565059
$$287$$ 1.45331 0.0857864
$$288$$ −1.00000 −0.0589256
$$289$$ −16.4720 −0.968939
$$290$$ 0 0
$$291$$ 4.54669 0.266532
$$292$$ 3.71733 0.217540
$$293$$ 0.759350 0.0443617 0.0221809 0.999754i $$-0.492939\pi$$
0.0221809 + 0.999754i $$0.492939\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −9.50466 −0.552447
$$297$$ 3.50466 0.203361
$$298$$ 4.49534 0.260408
$$299$$ 2.72666 0.157687
$$300$$ 0 0
$$301$$ 11.0093 0.634567
$$302$$ −22.3013 −1.28330
$$303$$ 3.45331 0.198388
$$304$$ −7.78734 −0.446634
$$305$$ 0 0
$$306$$ 0.726656 0.0415402
$$307$$ −27.5747 −1.57377 −0.786885 0.617100i $$-0.788307\pi$$
−0.786885 + 0.617100i $$0.788307\pi$$
$$308$$ 7.00933 0.399394
$$309$$ 14.4626 0.822751
$$310$$ 0 0
$$311$$ 24.1214 1.36780 0.683898 0.729577i $$-0.260283\pi$$
0.683898 + 0.729577i $$0.260283\pi$$
$$312$$ −2.72666 −0.154367
$$313$$ 8.58400 0.485196 0.242598 0.970127i $$-0.422000\pi$$
0.242598 + 0.970127i $$0.422000\pi$$
$$314$$ −9.60737 −0.542175
$$315$$ 0 0
$$316$$ −9.73599 −0.547692
$$317$$ 24.1214 1.35479 0.677395 0.735619i $$-0.263109\pi$$
0.677395 + 0.735619i $$0.263109\pi$$
$$318$$ 0.231321 0.0129718
$$319$$ −7.00933 −0.392447
$$320$$ 0 0
$$321$$ −13.0607 −0.728976
$$322$$ −2.00000 −0.111456
$$323$$ 5.65872 0.314860
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −14.1214 −0.782110
$$327$$ −10.9580 −0.605978
$$328$$ −0.726656 −0.0401229
$$329$$ 5.45331 0.300651
$$330$$ 0 0
$$331$$ 8.10270 0.445365 0.222682 0.974891i $$-0.428519\pi$$
0.222682 + 0.974891i $$0.428519\pi$$
$$332$$ −13.5233 −0.742189
$$333$$ 9.50466 0.520852
$$334$$ −8.74531 −0.478522
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ −24.1214 −1.31397 −0.656987 0.753902i $$-0.728169\pi$$
−0.656987 + 0.753902i $$0.728169\pi$$
$$338$$ 5.56534 0.302715
$$339$$ 11.7360 0.637411
$$340$$ 0 0
$$341$$ 29.0280 1.57195
$$342$$ 7.78734 0.421091
$$343$$ −20.0000 −1.07990
$$344$$ −5.50466 −0.296792
$$345$$ 0 0
$$346$$ −5.89730 −0.317041
$$347$$ −21.9160 −1.17651 −0.588255 0.808675i $$-0.700185\pi$$
−0.588255 + 0.808675i $$0.700185\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 26.5653 1.42201 0.711005 0.703187i $$-0.248240\pi$$
0.711005 + 0.703187i $$0.248240\pi$$
$$350$$ 0 0
$$351$$ 2.72666 0.145538
$$352$$ −3.50466 −0.186799
$$353$$ −22.9507 −1.22154 −0.610772 0.791807i $$-0.709141\pi$$
−0.610772 + 0.791807i $$0.709141\pi$$
$$354$$ −9.29200 −0.493864
$$355$$ 0 0
$$356$$ −9.55602 −0.506468
$$357$$ −1.45331 −0.0769175
$$358$$ 0.623954 0.0329770
$$359$$ 5.02799 0.265367 0.132683 0.991158i $$-0.457641\pi$$
0.132683 + 0.991158i $$0.457641\pi$$
$$360$$ 0 0
$$361$$ 41.6426 2.19172
$$362$$ 14.0700 0.739503
$$363$$ 1.28267 0.0673228
$$364$$ 5.45331 0.285831
$$365$$ 0 0
$$366$$ −4.05135 −0.211768
$$367$$ 7.45331 0.389060 0.194530 0.980897i $$-0.437682\pi$$
0.194530 + 0.980897i $$0.437682\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0.726656 0.0378282
$$370$$ 0 0
$$371$$ −0.462642 −0.0240192
$$372$$ 8.28267 0.429437
$$373$$ 14.4953 0.750540 0.375270 0.926916i $$-0.377550\pi$$
0.375270 + 0.926916i $$0.377550\pi$$
$$374$$ 2.54669 0.131686
$$375$$ 0 0
$$376$$ −2.72666 −0.140617
$$377$$ −5.45331 −0.280860
$$378$$ −2.00000 −0.102869
$$379$$ 17.1379 0.880317 0.440159 0.897920i $$-0.354922\pi$$
0.440159 + 0.897920i $$0.354922\pi$$
$$380$$ 0 0
$$381$$ 12.2827 0.629260
$$382$$ 19.5747 1.00153
$$383$$ 0.565344 0.0288878 0.0144439 0.999896i $$-0.495402\pi$$
0.0144439 + 0.999896i $$0.495402\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 10.0187 0.509936
$$387$$ 5.50466 0.279818
$$388$$ 4.54669 0.230823
$$389$$ −29.5233 −1.49689 −0.748446 0.663196i $$-0.769200\pi$$
−0.748446 + 0.663196i $$0.769200\pi$$
$$390$$ 0 0
$$391$$ −0.726656 −0.0367486
$$392$$ 3.00000 0.151523
$$393$$ −14.8480 −0.748983
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 3.50466 0.176116
$$397$$ −37.7360 −1.89391 −0.946957 0.321359i $$-0.895860\pi$$
−0.946957 + 0.321359i $$0.895860\pi$$
$$398$$ −23.1893 −1.16237
$$399$$ −15.5747 −0.779709
$$400$$ 0 0
$$401$$ −6.90663 −0.344900 −0.172450 0.985018i $$-0.555168\pi$$
−0.172450 + 0.985018i $$0.555168\pi$$
$$402$$ −4.05135 −0.202063
$$403$$ 22.5840 1.12499
$$404$$ 3.45331 0.171809
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 33.3107 1.65115
$$408$$ 0.726656 0.0359749
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 3.27334 0.161462
$$412$$ 14.4626 0.712523
$$413$$ 18.5840 0.914459
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ −2.72666 −0.133685
$$417$$ 14.0187 0.686497
$$418$$ 27.2920 1.33490
$$419$$ 27.9673 1.36629 0.683146 0.730282i $$-0.260611\pi$$
0.683146 + 0.730282i $$0.260611\pi$$
$$420$$ 0 0
$$421$$ 2.59804 0.126621 0.0633103 0.997994i $$-0.479834\pi$$
0.0633103 + 0.997994i $$0.479834\pi$$
$$422$$ 7.57467 0.368729
$$423$$ 2.72666 0.132575
$$424$$ 0.231321 0.0112339
$$425$$ 0 0
$$426$$ 11.7360 0.568610
$$427$$ 8.10270 0.392117
$$428$$ −13.0607 −0.631312
$$429$$ 9.55602 0.461369
$$430$$ 0 0
$$431$$ −15.2147 −0.732868 −0.366434 0.930444i $$-0.619421\pi$$
−0.366434 + 0.930444i $$0.619421\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −27.1307 −1.30382 −0.651909 0.758297i $$-0.726032\pi$$
−0.651909 + 0.758297i $$0.726032\pi$$
$$434$$ −16.5653 −0.795162
$$435$$ 0 0
$$436$$ −10.9580 −0.524792
$$437$$ −7.78734 −0.372519
$$438$$ −3.71733 −0.177621
$$439$$ −1.65872 −0.0791663 −0.0395832 0.999216i $$-0.512603\pi$$
−0.0395832 + 0.999216i $$0.512603\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 1.98134 0.0942429
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 9.50466 0.451071
$$445$$ 0 0
$$446$$ 26.5840 1.25879
$$447$$ −4.49534 −0.212622
$$448$$ 2.00000 0.0944911
$$449$$ −14.9507 −0.705568 −0.352784 0.935705i $$-0.614765\pi$$
−0.352784 + 0.935705i $$0.614765\pi$$
$$450$$ 0 0
$$451$$ 2.54669 0.119919
$$452$$ 11.7360 0.552014
$$453$$ 22.3013 1.04781
$$454$$ 11.9673 0.561654
$$455$$ 0 0
$$456$$ 7.78734 0.364675
$$457$$ 1.43466 0.0671104 0.0335552 0.999437i $$-0.489317\pi$$
0.0335552 + 0.999437i $$0.489317\pi$$
$$458$$ −26.5327 −1.23979
$$459$$ −0.726656 −0.0339174
$$460$$ 0 0
$$461$$ −13.5747 −0.632236 −0.316118 0.948720i $$-0.602379\pi$$
−0.316118 + 0.948720i $$0.602379\pi$$
$$462$$ −7.00933 −0.326103
$$463$$ 1.73599 0.0806781 0.0403390 0.999186i $$-0.487156\pi$$
0.0403390 + 0.999186i $$0.487156\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −24.4040 −1.13049
$$467$$ −28.1727 −1.30368 −0.651839 0.758358i $$-0.726002\pi$$
−0.651839 + 0.758358i $$0.726002\pi$$
$$468$$ 2.72666 0.126040
$$469$$ 8.10270 0.374148
$$470$$ 0 0
$$471$$ 9.60737 0.442684
$$472$$ −9.29200 −0.427699
$$473$$ 19.2920 0.887047
$$474$$ 9.73599 0.447189
$$475$$ 0 0
$$476$$ −1.45331 −0.0666125
$$477$$ −0.231321 −0.0105915
$$478$$ 3.55602 0.162648
$$479$$ −26.3786 −1.20527 −0.602634 0.798017i $$-0.705882\pi$$
−0.602634 + 0.798017i $$0.705882\pi$$
$$480$$ 0 0
$$481$$ 25.9160 1.18167
$$482$$ −3.55602 −0.161972
$$483$$ 2.00000 0.0910032
$$484$$ 1.28267 0.0583033
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 20.2827 0.919096 0.459548 0.888153i $$-0.348011\pi$$
0.459548 + 0.888153i $$0.348011\pi$$
$$488$$ −4.05135 −0.183396
$$489$$ 14.1214 0.638590
$$490$$ 0 0
$$491$$ 18.6426 0.841329 0.420665 0.907216i $$-0.361797\pi$$
0.420665 + 0.907216i $$0.361797\pi$$
$$492$$ 0.726656 0.0327602
$$493$$ 1.45331 0.0654539
$$494$$ 21.2334 0.955335
$$495$$ 0 0
$$496$$ 8.28267 0.371903
$$497$$ −23.4720 −1.05286
$$498$$ 13.5233 0.605995
$$499$$ 20.9907 0.939671 0.469836 0.882754i $$-0.344313\pi$$
0.469836 + 0.882754i $$0.344313\pi$$
$$500$$ 0 0
$$501$$ 8.74531 0.390712
$$502$$ −6.05135 −0.270085
$$503$$ −13.5560 −0.604433 −0.302216 0.953239i $$-0.597727\pi$$
−0.302216 + 0.953239i $$0.597727\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ −3.50466 −0.155801
$$507$$ −5.56534 −0.247166
$$508$$ 12.2827 0.544955
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 7.43466 0.328890
$$512$$ −1.00000 −0.0441942
$$513$$ −7.78734 −0.343819
$$514$$ −16.9066 −0.745719
$$515$$ 0 0
$$516$$ 5.50466 0.242329
$$517$$ 9.55602 0.420273
$$518$$ −19.0093 −0.835222
$$519$$ 5.89730 0.258863
$$520$$ 0 0
$$521$$ −44.6027 −1.95408 −0.977039 0.213061i $$-0.931657\pi$$
−0.977039 + 0.213061i $$0.931657\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 19.7287 0.862677 0.431339 0.902190i $$-0.358041\pi$$
0.431339 + 0.902190i $$0.358041\pi$$
$$524$$ −14.8480 −0.648639
$$525$$ 0 0
$$526$$ −0.887968 −0.0387173
$$527$$ −6.01866 −0.262177
$$528$$ 3.50466 0.152521
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 9.29200 0.403238
$$532$$ −15.5747 −0.675248
$$533$$ 1.98134 0.0858215
$$534$$ 9.55602 0.413529
$$535$$ 0 0
$$536$$ −4.05135 −0.174992
$$537$$ −0.623954 −0.0269256
$$538$$ −20.1214 −0.867493
$$539$$ −10.5140 −0.452870
$$540$$ 0 0
$$541$$ −13.4720 −0.579205 −0.289603 0.957147i $$-0.593523\pi$$
−0.289603 + 0.957147i $$0.593523\pi$$
$$542$$ 30.5840 1.31370
$$543$$ −14.0700 −0.603802
$$544$$ 0.726656 0.0311551
$$545$$ 0 0
$$546$$ −5.45331 −0.233380
$$547$$ 23.0466 0.985403 0.492702 0.870198i $$-0.336009\pi$$
0.492702 + 0.870198i $$0.336009\pi$$
$$548$$ 3.27334 0.139830
$$549$$ 4.05135 0.172907
$$550$$ 0 0
$$551$$ 15.5747 0.663503
$$552$$ −1.00000 −0.0425628
$$553$$ −19.4720 −0.828032
$$554$$ 4.38538 0.186317
$$555$$ 0 0
$$556$$ 14.0187 0.594524
$$557$$ 37.2593 1.57873 0.789364 0.613926i $$-0.210411\pi$$
0.789364 + 0.613926i $$0.210411\pi$$
$$558$$ −8.28267 −0.350633
$$559$$ 15.0093 0.634827
$$560$$ 0 0
$$561$$ −2.54669 −0.107521
$$562$$ −10.1214 −0.426944
$$563$$ −46.5513 −1.96190 −0.980952 0.194251i $$-0.937772\pi$$
−0.980952 + 0.194251i $$0.937772\pi$$
$$564$$ 2.72666 0.114813
$$565$$ 0 0
$$566$$ −25.1820 −1.05848
$$567$$ 2.00000 0.0839921
$$568$$ 11.7360 0.492431
$$569$$ −20.0000 −0.838444 −0.419222 0.907884i $$-0.637697\pi$$
−0.419222 + 0.907884i $$0.637697\pi$$
$$570$$ 0 0
$$571$$ −13.6660 −0.571903 −0.285952 0.958244i $$-0.592310\pi$$
−0.285952 + 0.958244i $$0.592310\pi$$
$$572$$ 9.55602 0.399557
$$573$$ −19.5747 −0.817744
$$574$$ −1.45331 −0.0606601
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 6.30133 0.262328 0.131164 0.991361i $$-0.458129\pi$$
0.131164 + 0.991361i $$0.458129\pi$$
$$578$$ 16.4720 0.685144
$$579$$ −10.0187 −0.416361
$$580$$ 0 0
$$581$$ −27.0466 −1.12208
$$582$$ −4.54669 −0.188466
$$583$$ −0.810702 −0.0335758
$$584$$ −3.71733 −0.153824
$$585$$ 0 0
$$586$$ −0.759350 −0.0313685
$$587$$ 35.5747 1.46832 0.734162 0.678974i $$-0.237575\pi$$
0.734162 + 0.678974i $$0.237575\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −64.5000 −2.65767
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 9.50466 0.390639
$$593$$ −4.54669 −0.186710 −0.0933550 0.995633i $$-0.529759\pi$$
−0.0933550 + 0.995633i $$0.529759\pi$$
$$594$$ −3.50466 −0.143798
$$595$$ 0 0
$$596$$ −4.49534 −0.184136
$$597$$ 23.1893 0.949075
$$598$$ −2.72666 −0.111501
$$599$$ 16.1214 0.658701 0.329350 0.944208i $$-0.393170\pi$$
0.329350 + 0.944208i $$0.393170\pi$$
$$600$$ 0 0
$$601$$ −9.39470 −0.383218 −0.191609 0.981471i $$-0.561371\pi$$
−0.191609 + 0.981471i $$0.561371\pi$$
$$602$$ −11.0093 −0.448707
$$603$$ 4.05135 0.164984
$$604$$ 22.3013 0.907428
$$605$$ 0 0
$$606$$ −3.45331 −0.140281
$$607$$ −10.9066 −0.442686 −0.221343 0.975196i $$-0.571044\pi$$
−0.221343 + 0.975196i $$0.571044\pi$$
$$608$$ 7.78734 0.315818
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 7.43466 0.300774
$$612$$ −0.726656 −0.0293733
$$613$$ −43.4206 −1.75374 −0.876871 0.480725i $$-0.840373\pi$$
−0.876871 + 0.480725i $$0.840373\pi$$
$$614$$ 27.5747 1.11282
$$615$$ 0 0
$$616$$ −7.00933 −0.282414
$$617$$ −20.0959 −0.809031 −0.404516 0.914531i $$-0.632560\pi$$
−0.404516 + 0.914531i $$0.632560\pi$$
$$618$$ −14.4626 −0.581773
$$619$$ 27.3247 1.09827 0.549136 0.835733i $$-0.314957\pi$$
0.549136 + 0.835733i $$0.314957\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ −24.1214 −0.967178
$$623$$ −19.1120 −0.765707
$$624$$ 2.72666 0.109154
$$625$$ 0 0
$$626$$ −8.58400 −0.343086
$$627$$ −27.2920 −1.08994
$$628$$ 9.60737 0.383376
$$629$$ −6.90663 −0.275385
$$630$$ 0 0
$$631$$ 39.8947 1.58818 0.794091 0.607799i $$-0.207947\pi$$
0.794091 + 0.607799i $$0.207947\pi$$
$$632$$ 9.73599 0.387277
$$633$$ −7.57467 −0.301066
$$634$$ −24.1214 −0.957982
$$635$$ 0 0
$$636$$ −0.231321 −0.00917247
$$637$$ −8.17997 −0.324102
$$638$$ 7.00933 0.277502
$$639$$ −11.7360 −0.464268
$$640$$ 0 0
$$641$$ 49.1680 1.94202 0.971010 0.239040i $$-0.0768327\pi$$
0.971010 + 0.239040i $$0.0768327\pi$$
$$642$$ 13.0607 0.515464
$$643$$ −20.9766 −0.827238 −0.413619 0.910450i $$-0.635735\pi$$
−0.413619 + 0.910450i $$0.635735\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 0 0
$$646$$ −5.65872 −0.222639
$$647$$ −25.9813 −1.02143 −0.510716 0.859749i $$-0.670620\pi$$
−0.510716 + 0.859749i $$0.670620\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 32.5653 1.27830
$$650$$ 0 0
$$651$$ 16.5653 0.649247
$$652$$ 14.1214 0.553035
$$653$$ −0.0840454 −0.00328895 −0.00164448 0.999999i $$-0.500523\pi$$
−0.00164448 + 0.999999i $$0.500523\pi$$
$$654$$ 10.9580 0.428491
$$655$$ 0 0
$$656$$ 0.726656 0.0283712
$$657$$ 3.71733 0.145027
$$658$$ −5.45331 −0.212592
$$659$$ 10.6167 0.413568 0.206784 0.978387i $$-0.433700\pi$$
0.206784 + 0.978387i $$0.433700\pi$$
$$660$$ 0 0
$$661$$ −39.9860 −1.55527 −0.777637 0.628714i $$-0.783582\pi$$
−0.777637 + 0.628714i $$0.783582\pi$$
$$662$$ −8.10270 −0.314920
$$663$$ −1.98134 −0.0769490
$$664$$ 13.5233 0.524807
$$665$$ 0 0
$$666$$ −9.50466 −0.368298
$$667$$ −2.00000 −0.0774403
$$668$$ 8.74531 0.338366
$$669$$ −26.5840 −1.02780
$$670$$ 0 0
$$671$$ 14.1986 0.548132
$$672$$ −2.00000 −0.0771517
$$673$$ 2.86667 0.110502 0.0552511 0.998472i $$-0.482404\pi$$
0.0552511 + 0.998472i $$0.482404\pi$$
$$674$$ 24.1214 0.929120
$$675$$ 0 0
$$676$$ −5.56534 −0.214052
$$677$$ −3.76868 −0.144842 −0.0724211 0.997374i $$-0.523073\pi$$
−0.0724211 + 0.997374i $$0.523073\pi$$
$$678$$ −11.7360 −0.450718
$$679$$ 9.09337 0.348972
$$680$$ 0 0
$$681$$ −11.9673 −0.458588
$$682$$ −29.0280 −1.11154
$$683$$ −39.6120 −1.51571 −0.757855 0.652423i $$-0.773753\pi$$
−0.757855 + 0.652423i $$0.773753\pi$$
$$684$$ −7.78734 −0.297756
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 26.5327 1.01228
$$688$$ 5.50466 0.209863
$$689$$ −0.630732 −0.0240290
$$690$$ 0 0
$$691$$ −49.0653 −1.86653 −0.933266 0.359186i $$-0.883054\pi$$
−0.933266 + 0.359186i $$0.883054\pi$$
$$692$$ 5.89730 0.224182
$$693$$ 7.00933 0.266262
$$694$$ 21.9160 0.831918
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ −0.528030 −0.0200005
$$698$$ −26.5653 −1.00551
$$699$$ 24.4040 0.923045
$$700$$ 0 0
$$701$$ −10.5140 −0.397108 −0.198554 0.980090i $$-0.563625\pi$$
−0.198554 + 0.980090i $$0.563625\pi$$
$$702$$ −2.72666 −0.102911
$$703$$ −74.0160 −2.79157
$$704$$ 3.50466 0.132087
$$705$$ 0 0
$$706$$ 22.9507 0.863762
$$707$$ 6.90663 0.259750
$$708$$ 9.29200 0.349215
$$709$$ −17.1447 −0.643884 −0.321942 0.946759i $$-0.604336\pi$$
−0.321942 + 0.946759i $$0.604336\pi$$
$$710$$ 0 0
$$711$$ −9.73599 −0.365128
$$712$$ 9.55602 0.358127
$$713$$ 8.28267 0.310189
$$714$$ 1.45331 0.0543889
$$715$$ 0 0
$$716$$ −0.623954 −0.0233182
$$717$$ −3.55602 −0.132802
$$718$$ −5.02799 −0.187643
$$719$$ 29.0093 1.08187 0.540933 0.841066i $$-0.318071\pi$$
0.540933 + 0.841066i $$0.318071\pi$$
$$720$$ 0 0
$$721$$ 28.9253 1.07723
$$722$$ −41.6426 −1.54978
$$723$$ 3.55602 0.132250
$$724$$ −14.0700 −0.522908
$$725$$ 0 0
$$726$$ −1.28267 −0.0476044
$$727$$ −38.9253 −1.44366 −0.721829 0.692071i $$-0.756698\pi$$
−0.721829 + 0.692071i $$0.756698\pi$$
$$728$$ −5.45331 −0.202113
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 4.05135 0.149742
$$733$$ −5.50466 −0.203319 −0.101660 0.994819i $$-0.532415\pi$$
−0.101660 + 0.994819i $$0.532415\pi$$
$$734$$ −7.45331 −0.275107
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 14.1986 0.523013
$$738$$ −0.726656 −0.0267486
$$739$$ 28.6027 1.05217 0.526083 0.850433i $$-0.323660\pi$$
0.526083 + 0.850433i $$0.323660\pi$$
$$740$$ 0 0
$$741$$ −21.2334 −0.780028
$$742$$ 0.462642 0.0169841
$$743$$ −9.98134 −0.366180 −0.183090 0.983096i $$-0.558610\pi$$
−0.183090 + 0.983096i $$0.558610\pi$$
$$744$$ −8.28267 −0.303657
$$745$$ 0 0
$$746$$ −14.4953 −0.530712
$$747$$ −13.5233 −0.494792
$$748$$ −2.54669 −0.0931161
$$749$$ −26.1214 −0.954454
$$750$$ 0 0
$$751$$ −21.8387 −0.796905 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$752$$ 2.72666 0.0994309
$$753$$ 6.05135 0.220524
$$754$$ 5.45331 0.198598
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 16.5513 0.601568 0.300784 0.953692i $$-0.402752\pi$$
0.300784 + 0.953692i $$0.402752\pi$$
$$758$$ −17.1379 −0.622478
$$759$$ 3.50466 0.127211
$$760$$ 0 0
$$761$$ 20.5840 0.746169 0.373085 0.927797i $$-0.378300\pi$$
0.373085 + 0.927797i $$0.378300\pi$$
$$762$$ −12.2827 −0.444954
$$763$$ −21.9160 −0.793411
$$764$$ −19.5747 −0.708187
$$765$$ 0 0
$$766$$ −0.565344 −0.0204267
$$767$$ 25.3361 0.914833
$$768$$ 1.00000 0.0360844
$$769$$ 7.76142 0.279884 0.139942 0.990160i $$-0.455308\pi$$
0.139942 + 0.990160i $$0.455308\pi$$
$$770$$ 0 0
$$771$$ 16.9066 0.608877
$$772$$ −10.0187 −0.360579
$$773$$ 21.8247 0.784978 0.392489 0.919757i $$-0.371614\pi$$
0.392489 + 0.919757i $$0.371614\pi$$
$$774$$ −5.50466 −0.197861
$$775$$ 0 0
$$776$$ −4.54669 −0.163217
$$777$$ 19.0093 0.681956
$$778$$ 29.5233 1.05846
$$779$$ −5.65872 −0.202745
$$780$$ 0 0
$$781$$ −41.1307 −1.47177
$$782$$ 0.726656 0.0259852
$$783$$ −2.00000 −0.0714742
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 14.8480 0.529611
$$787$$ 46.4299 1.65505 0.827524 0.561430i $$-0.189748\pi$$
0.827524 + 0.561430i $$0.189748\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0.887968 0.0316125
$$790$$ 0 0
$$791$$ 23.4720 0.834567
$$792$$ −3.50466 −0.124533
$$793$$ 11.0466 0.392278
$$794$$ 37.7360 1.33920
$$795$$ 0 0
$$796$$ 23.1893 0.821923
$$797$$ −44.3340 −1.57039 −0.785196 0.619248i $$-0.787438\pi$$
−0.785196 + 0.619248i $$0.787438\pi$$
$$798$$ 15.5747 0.551337
$$799$$ −1.98134 −0.0700949
$$800$$ 0 0
$$801$$ −9.55602 −0.337645
$$802$$ 6.90663 0.243881
$$803$$ 13.0280 0.459748
$$804$$ 4.05135 0.142880
$$805$$ 0 0
$$806$$ −22.5840 −0.795488
$$807$$ 20.1214 0.708305
$$808$$ −3.45331 −0.121487
$$809$$ −1.96269 −0.0690043 −0.0345022 0.999405i $$-0.510985\pi$$
−0.0345022 + 0.999405i $$0.510985\pi$$
$$810$$ 0 0
$$811$$ 31.8973 1.12007 0.560033 0.828470i $$-0.310789\pi$$
0.560033 + 0.828470i $$0.310789\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ −30.5840 −1.07263
$$814$$ −33.3107 −1.16754
$$815$$ 0 0
$$816$$ −0.726656 −0.0254381
$$817$$ −42.8667 −1.49972
$$818$$ −2.00000 −0.0699284
$$819$$ 5.45331 0.190554
$$820$$ 0 0
$$821$$ 11.4906 0.401026 0.200513 0.979691i $$-0.435739\pi$$
0.200513 + 0.979691i $$0.435739\pi$$
$$822$$ −3.27334 −0.114171
$$823$$ −37.1307 −1.29429 −0.647147 0.762365i $$-0.724038\pi$$
−0.647147 + 0.762365i $$0.724038\pi$$
$$824$$ −14.4626 −0.503830
$$825$$ 0 0
$$826$$ −18.5840 −0.646620
$$827$$ −40.2100 −1.39824 −0.699120 0.715005i $$-0.746425\pi$$
−0.699120 + 0.715005i $$0.746425\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ −6.88797 −0.239229 −0.119615 0.992820i $$-0.538166\pi$$
−0.119615 + 0.992820i $$0.538166\pi$$
$$830$$ 0 0
$$831$$ −4.38538 −0.152127
$$832$$ 2.72666 0.0945298
$$833$$ 2.17997 0.0755315
$$834$$ −14.0187 −0.485426
$$835$$ 0 0
$$836$$ −27.2920 −0.943914
$$837$$ 8.28267 0.286291
$$838$$ −27.9673 −0.966115
$$839$$ −26.3786 −0.910690 −0.455345 0.890315i $$-0.650484\pi$$
−0.455345 + 0.890315i $$0.650484\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −2.59804 −0.0895343
$$843$$ 10.1214 0.348598
$$844$$ −7.57467 −0.260731
$$845$$ 0 0
$$846$$ −2.72666 −0.0937444
$$847$$ 2.56534 0.0881463
$$848$$ −0.231321 −0.00794359
$$849$$ 25.1820 0.864245
$$850$$ 0 0
$$851$$ 9.50466 0.325816
$$852$$ −11.7360 −0.402068
$$853$$ 11.7546 0.402471 0.201236 0.979543i $$-0.435504\pi$$
0.201236 + 0.979543i $$0.435504\pi$$
$$854$$ −8.10270 −0.277269
$$855$$ 0 0
$$856$$ 13.0607 0.446405
$$857$$ −53.5347 −1.82871 −0.914356 0.404912i $$-0.867302\pi$$
−0.914356 + 0.404912i $$0.867302\pi$$
$$858$$ −9.55602 −0.326237
$$859$$ −7.11203 −0.242659 −0.121330 0.992612i $$-0.538716\pi$$
−0.121330 + 0.992612i $$0.538716\pi$$
$$860$$ 0 0
$$861$$ 1.45331 0.0495288
$$862$$ 15.2147 0.518216
$$863$$ 28.0373 0.954401 0.477201 0.878794i $$-0.341651\pi$$
0.477201 + 0.878794i $$0.341651\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 27.1307 0.921938
$$867$$ −16.4720 −0.559417
$$868$$ 16.5653 0.562264
$$869$$ −34.1214 −1.15749
$$870$$ 0 0
$$871$$ 11.0466 0.374301
$$872$$ 10.9580 0.371084
$$873$$ 4.54669 0.153882
$$874$$ 7.78734 0.263411
$$875$$ 0 0
$$876$$ 3.71733 0.125597
$$877$$ −46.4040 −1.56695 −0.783476 0.621422i $$-0.786555\pi$$
−0.783476 + 0.621422i $$0.786555\pi$$
$$878$$ 1.65872 0.0559790
$$879$$ 0.759350 0.0256123
$$880$$ 0 0
$$881$$ 14.6494 0.493550 0.246775 0.969073i $$-0.420629\pi$$
0.246775 + 0.969073i $$0.420629\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −13.7614 −0.463109 −0.231554 0.972822i $$-0.574381\pi$$
−0.231554 + 0.972822i $$0.574381\pi$$
$$884$$ −1.98134 −0.0666398
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 35.8573 1.20397 0.601986 0.798507i $$-0.294376\pi$$
0.601986 + 0.798507i $$0.294376\pi$$
$$888$$ −9.50466 −0.318956
$$889$$ 24.5653 0.823895
$$890$$ 0 0
$$891$$ 3.50466 0.117411
$$892$$ −26.5840 −0.890098
$$893$$ −21.2334 −0.710548
$$894$$ 4.49534 0.150347
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 2.72666 0.0910404
$$898$$ 14.9507 0.498912
$$899$$ −16.5653 −0.552485
$$900$$ 0 0
$$901$$ 0.168091 0.00559992
$$902$$ −2.54669 −0.0847954
$$903$$ 11.0093 0.366368
$$904$$ −11.7360 −0.390333
$$905$$ 0 0
$$906$$ −22.3013 −0.740912
$$907$$ 1.60737 0.0533718 0.0266859 0.999644i $$-0.491505\pi$$
0.0266859 + 0.999644i $$0.491505\pi$$
$$908$$ −11.9673 −0.397149
$$909$$ 3.45331 0.114539
$$910$$ 0 0
$$911$$ −15.8973 −0.526701 −0.263350 0.964700i $$-0.584828\pi$$
−0.263350 + 0.964700i $$0.584828\pi$$
$$912$$ −7.78734 −0.257864
$$913$$ −47.3947 −1.56854
$$914$$ −1.43466 −0.0474542
$$915$$ 0 0
$$916$$ 26.5327 0.876663
$$917$$ −29.6960 −0.980649
$$918$$ 0.726656 0.0239832
$$919$$ −42.3013 −1.39539 −0.697696 0.716394i $$-0.745791\pi$$
−0.697696 + 0.716394i $$0.745791\pi$$
$$920$$ 0 0
$$921$$ −27.5747 −0.908616
$$922$$ 13.5747 0.447058
$$923$$ −32.0000 −1.05329
$$924$$ 7.00933 0.230590
$$925$$ 0 0
$$926$$ −1.73599 −0.0570480
$$927$$ 14.4626 0.475015
$$928$$ 2.00000 0.0656532
$$929$$ −42.0628 −1.38003 −0.690017 0.723793i $$-0.742397\pi$$
−0.690017 + 0.723793i $$0.742397\pi$$
$$930$$ 0 0
$$931$$ 23.3620 0.765659
$$932$$ 24.4040 0.799381
$$933$$ 24.1214 0.789698
$$934$$ 28.1727 0.921839
$$935$$ 0 0
$$936$$ −2.72666 −0.0891236
$$937$$ 8.44398 0.275853 0.137926 0.990442i $$-0.455956\pi$$
0.137926 + 0.990442i $$0.455956\pi$$
$$938$$ −8.10270 −0.264563
$$939$$ 8.58400 0.280128
$$940$$ 0 0
$$941$$ −19.2993 −0.629138 −0.314569 0.949235i $$-0.601860\pi$$
−0.314569 + 0.949235i $$0.601860\pi$$
$$942$$ −9.60737 −0.313025
$$943$$ 0.726656 0.0236632
$$944$$ 9.29200 0.302429
$$945$$ 0 0
$$946$$ −19.2920 −0.627237
$$947$$ −3.36927 −0.109486 −0.0547432 0.998500i $$-0.517434\pi$$
−0.0547432 + 0.998500i $$0.517434\pi$$
$$948$$ −9.73599 −0.316210
$$949$$ 10.1359 0.329024
$$950$$ 0 0
$$951$$ 24.1214 0.782189
$$952$$ 1.45331 0.0471021
$$953$$ 24.0441 0.778865 0.389432 0.921055i $$-0.372671\pi$$
0.389432 + 0.921055i $$0.372671\pi$$
$$954$$ 0.231321 0.00748929
$$955$$ 0 0
$$956$$ −3.55602 −0.115010
$$957$$ −7.00933 −0.226579
$$958$$ 26.3786 0.852254
$$959$$ 6.54669 0.211404
$$960$$ 0 0
$$961$$ 37.6027 1.21299
$$962$$ −25.9160 −0.835564
$$963$$ −13.0607 −0.420875
$$964$$ 3.55602 0.114532
$$965$$ 0 0
$$966$$ −2.00000 −0.0643489
$$967$$ −26.9439 −0.866459 −0.433229 0.901284i $$-0.642626\pi$$
−0.433229 + 0.901284i $$0.642626\pi$$
$$968$$ −1.28267 −0.0412266
$$969$$ 5.65872 0.181784
$$970$$ 0 0
$$971$$ −5.84595 −0.187605 −0.0938027 0.995591i $$-0.529902\pi$$
−0.0938027 + 0.995591i $$0.529902\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 28.0373 0.898835
$$974$$ −20.2827 −0.649899
$$975$$ 0 0
$$976$$ 4.05135 0.129681
$$977$$ 23.8387 0.762667 0.381334 0.924437i $$-0.375465\pi$$
0.381334 + 0.924437i $$0.375465\pi$$
$$978$$ −14.1214 −0.451551
$$979$$ −33.4906 −1.07037
$$980$$ 0 0
$$981$$ −10.9580 −0.349861
$$982$$ −18.6426 −0.594910
$$983$$ 44.6680 1.42469 0.712345 0.701830i $$-0.247633\pi$$
0.712345 + 0.701830i $$0.247633\pi$$
$$984$$ −0.726656 −0.0231650
$$985$$ 0 0
$$986$$ −1.45331 −0.0462829
$$987$$ 5.45331 0.173581
$$988$$ −21.2334 −0.675524
$$989$$ 5.50466 0.175038
$$990$$ 0 0
$$991$$ −10.0586 −0.319522 −0.159761 0.987156i $$-0.551072\pi$$
−0.159761 + 0.987156i $$0.551072\pi$$
$$992$$ −8.28267 −0.262975
$$993$$ 8.10270 0.257132
$$994$$ 23.4720 0.744486
$$995$$ 0 0
$$996$$ −13.5233 −0.428503
$$997$$ −38.2640 −1.21183 −0.605917 0.795528i $$-0.707194\pi$$
−0.605917 + 0.795528i $$0.707194\pi$$
$$998$$ −20.9907 −0.664448
$$999$$ 9.50466 0.300714
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 3450.2.a.bq.1.3 3
5.2 odd 4 690.2.d.d.139.1 6
5.3 odd 4 690.2.d.d.139.4 yes 6
5.4 even 2 3450.2.a.br.1.3 3
15.2 even 4 2070.2.d.d.829.6 6
15.8 even 4 2070.2.d.d.829.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.d.139.1 6 5.2 odd 4
690.2.d.d.139.4 yes 6 5.3 odd 4
2070.2.d.d.829.3 6 15.8 even 4
2070.2.d.d.829.6 6 15.2 even 4
3450.2.a.bq.1.3 3 1.1 even 1 trivial
3450.2.a.br.1.3 3 5.4 even 2