# Properties

 Label 495.2.a.e.1.2 Level $495$ Weight $2$ Character 495.1 Self dual yes Analytic conductor $3.953$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 495.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.193937 q^{2} -1.96239 q^{4} -1.00000 q^{5} +3.35026 q^{7} -0.768452 q^{8} +O(q^{10})$$ $$q+0.193937 q^{2} -1.96239 q^{4} -1.00000 q^{5} +3.35026 q^{7} -0.768452 q^{8} -0.193937 q^{10} -1.00000 q^{11} +2.96239 q^{13} +0.649738 q^{14} +3.77575 q^{16} +4.57452 q^{17} -4.31265 q^{19} +1.96239 q^{20} -0.193937 q^{22} +6.70052 q^{23} +1.00000 q^{25} +0.574515 q^{26} -6.57452 q^{28} +3.61213 q^{29} +9.92478 q^{31} +2.26916 q^{32} +0.887166 q^{34} -3.35026 q^{35} -2.00000 q^{37} -0.836381 q^{38} +0.768452 q^{40} +4.38787 q^{41} -9.27504 q^{43} +1.96239 q^{44} +1.29948 q^{46} +9.92478 q^{47} +4.22425 q^{49} +0.193937 q^{50} -5.81336 q^{52} -4.70052 q^{53} +1.00000 q^{55} -2.57452 q^{56} +0.700523 q^{58} -10.7005 q^{59} -8.70052 q^{61} +1.92478 q^{62} -7.11142 q^{64} -2.96239 q^{65} +5.92478 q^{67} -8.97698 q^{68} -0.649738 q^{70} -9.92478 q^{71} -7.73813 q^{73} -0.387873 q^{74} +8.46310 q^{76} -3.35026 q^{77} +11.5369 q^{79} -3.77575 q^{80} +0.850969 q^{82} -10.8872 q^{83} -4.57452 q^{85} -1.79877 q^{86} +0.768452 q^{88} +2.77575 q^{89} +9.92478 q^{91} -13.1490 q^{92} +1.92478 q^{94} +4.31265 q^{95} +0.0752228 q^{97} +0.819237 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 5 q^{4} - 3 q^{5} + 9 q^{8}+O(q^{10})$$ 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 9 * q^8 $$3 q + q^{2} + 5 q^{4} - 3 q^{5} + 9 q^{8} - q^{10} - 3 q^{11} - 2 q^{13} + 12 q^{14} + 13 q^{16} + 2 q^{17} + 8 q^{19} - 5 q^{20} - q^{22} + 3 q^{25} - 10 q^{26} - 8 q^{28} + 10 q^{29} + 8 q^{31} + 29 q^{32} - 30 q^{34} - 6 q^{37} - 9 q^{40} + 14 q^{41} + 4 q^{43} - 5 q^{44} + 24 q^{46} + 8 q^{47} + 11 q^{49} + q^{50} - 30 q^{52} + 6 q^{53} + 3 q^{55} + 4 q^{56} - 18 q^{58} - 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} + 2 q^{65} - 4 q^{67} - 42 q^{68} - 12 q^{70} - 8 q^{71} - 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 13 q^{80} + 26 q^{82} - 2 q^{85} + 8 q^{86} - 9 q^{88} + 10 q^{89} + 8 q^{91} - 16 q^{92} - 16 q^{94} - 8 q^{95} + 22 q^{97} - 39 q^{98}+O(q^{100})$$ 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 9 * q^8 - q^10 - 3 * q^11 - 2 * q^13 + 12 * q^14 + 13 * q^16 + 2 * q^17 + 8 * q^19 - 5 * q^20 - q^22 + 3 * q^25 - 10 * q^26 - 8 * q^28 + 10 * q^29 + 8 * q^31 + 29 * q^32 - 30 * q^34 - 6 * q^37 - 9 * q^40 + 14 * q^41 + 4 * q^43 - 5 * q^44 + 24 * q^46 + 8 * q^47 + 11 * q^49 + q^50 - 30 * q^52 + 6 * q^53 + 3 * q^55 + 4 * q^56 - 18 * q^58 - 12 * q^59 - 6 * q^61 - 16 * q^62 + 13 * q^64 + 2 * q^65 - 4 * q^67 - 42 * q^68 - 12 * q^70 - 8 * q^71 - 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 13 * q^80 + 26 * q^82 - 2 * q^85 + 8 * q^86 - 9 * q^88 + 10 * q^89 + 8 * q^91 - 16 * q^92 - 16 * q^94 - 8 * q^95 + 22 * q^97 - 39 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.193937 0.137134 0.0685669 0.997647i $$-0.478157\pi$$
0.0685669 + 0.997647i $$0.478157\pi$$
$$3$$ 0 0
$$4$$ −1.96239 −0.981194
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 3.35026 1.26628 0.633140 0.774037i $$-0.281766\pi$$
0.633140 + 0.774037i $$0.281766\pi$$
$$8$$ −0.768452 −0.271689
$$9$$ 0 0
$$10$$ −0.193937 −0.0613281
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 2.96239 0.821619 0.410809 0.911721i $$-0.365246\pi$$
0.410809 + 0.911721i $$0.365246\pi$$
$$14$$ 0.649738 0.173650
$$15$$ 0 0
$$16$$ 3.77575 0.943937
$$17$$ 4.57452 1.10948 0.554741 0.832023i $$-0.312817\pi$$
0.554741 + 0.832023i $$0.312817\pi$$
$$18$$ 0 0
$$19$$ −4.31265 −0.989390 −0.494695 0.869067i $$-0.664720\pi$$
−0.494695 + 0.869067i $$0.664720\pi$$
$$20$$ 1.96239 0.438803
$$21$$ 0 0
$$22$$ −0.193937 −0.0413474
$$23$$ 6.70052 1.39716 0.698578 0.715534i $$-0.253817\pi$$
0.698578 + 0.715534i $$0.253817\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.574515 0.112672
$$27$$ 0 0
$$28$$ −6.57452 −1.24247
$$29$$ 3.61213 0.670755 0.335378 0.942084i $$-0.391136\pi$$
0.335378 + 0.942084i $$0.391136\pi$$
$$30$$ 0 0
$$31$$ 9.92478 1.78254 0.891271 0.453470i $$-0.149814\pi$$
0.891271 + 0.453470i $$0.149814\pi$$
$$32$$ 2.26916 0.401134
$$33$$ 0 0
$$34$$ 0.887166 0.152148
$$35$$ −3.35026 −0.566298
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −0.836381 −0.135679
$$39$$ 0 0
$$40$$ 0.768452 0.121503
$$41$$ 4.38787 0.685271 0.342635 0.939468i $$-0.388680\pi$$
0.342635 + 0.939468i $$0.388680\pi$$
$$42$$ 0 0
$$43$$ −9.27504 −1.41443 −0.707215 0.706998i $$-0.750049\pi$$
−0.707215 + 0.706998i $$0.750049\pi$$
$$44$$ 1.96239 0.295841
$$45$$ 0 0
$$46$$ 1.29948 0.191597
$$47$$ 9.92478 1.44768 0.723839 0.689969i $$-0.242376\pi$$
0.723839 + 0.689969i $$0.242376\pi$$
$$48$$ 0 0
$$49$$ 4.22425 0.603465
$$50$$ 0.193937 0.0274268
$$51$$ 0 0
$$52$$ −5.81336 −0.806168
$$53$$ −4.70052 −0.645667 −0.322833 0.946456i $$-0.604635\pi$$
−0.322833 + 0.946456i $$0.604635\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ −2.57452 −0.344034
$$57$$ 0 0
$$58$$ 0.700523 0.0919832
$$59$$ −10.7005 −1.39309 −0.696545 0.717513i $$-0.745280\pi$$
−0.696545 + 0.717513i $$0.745280\pi$$
$$60$$ 0 0
$$61$$ −8.70052 −1.11399 −0.556994 0.830517i $$-0.688045\pi$$
−0.556994 + 0.830517i $$0.688045\pi$$
$$62$$ 1.92478 0.244447
$$63$$ 0 0
$$64$$ −7.11142 −0.888927
$$65$$ −2.96239 −0.367439
$$66$$ 0 0
$$67$$ 5.92478 0.723827 0.361913 0.932212i $$-0.382124\pi$$
0.361913 + 0.932212i $$0.382124\pi$$
$$68$$ −8.97698 −1.08862
$$69$$ 0 0
$$70$$ −0.649738 −0.0776586
$$71$$ −9.92478 −1.17785 −0.588927 0.808186i $$-0.700450\pi$$
−0.588927 + 0.808186i $$0.700450\pi$$
$$72$$ 0 0
$$73$$ −7.73813 −0.905680 −0.452840 0.891592i $$-0.649589\pi$$
−0.452840 + 0.891592i $$0.649589\pi$$
$$74$$ −0.387873 −0.0450893
$$75$$ 0 0
$$76$$ 8.46310 0.970784
$$77$$ −3.35026 −0.381798
$$78$$ 0 0
$$79$$ 11.5369 1.29800 0.649002 0.760787i $$-0.275187\pi$$
0.649002 + 0.760787i $$0.275187\pi$$
$$80$$ −3.77575 −0.422141
$$81$$ 0 0
$$82$$ 0.850969 0.0939738
$$83$$ −10.8872 −1.19502 −0.597511 0.801861i $$-0.703844\pi$$
−0.597511 + 0.801861i $$0.703844\pi$$
$$84$$ 0 0
$$85$$ −4.57452 −0.496176
$$86$$ −1.79877 −0.193966
$$87$$ 0 0
$$88$$ 0.768452 0.0819173
$$89$$ 2.77575 0.294229 0.147114 0.989120i $$-0.453001\pi$$
0.147114 + 0.989120i $$0.453001\pi$$
$$90$$ 0 0
$$91$$ 9.92478 1.04040
$$92$$ −13.1490 −1.37088
$$93$$ 0 0
$$94$$ 1.92478 0.198526
$$95$$ 4.31265 0.442469
$$96$$ 0 0
$$97$$ 0.0752228 0.00763772 0.00381886 0.999993i $$-0.498784\pi$$
0.00381886 + 0.999993i $$0.498784\pi$$
$$98$$ 0.819237 0.0827555
$$99$$ 0 0
$$100$$ −1.96239 −0.196239
$$101$$ 15.0884 1.50135 0.750676 0.660671i $$-0.229728\pi$$
0.750676 + 0.660671i $$0.229728\pi$$
$$102$$ 0 0
$$103$$ −3.22425 −0.317695 −0.158848 0.987303i $$-0.550778\pi$$
−0.158848 + 0.987303i $$0.550778\pi$$
$$104$$ −2.27645 −0.223225
$$105$$ 0 0
$$106$$ −0.911603 −0.0885427
$$107$$ 0.962389 0.0930376 0.0465188 0.998917i $$-0.485187\pi$$
0.0465188 + 0.998917i $$0.485187\pi$$
$$108$$ 0 0
$$109$$ 11.4010 1.09202 0.546011 0.837778i $$-0.316146\pi$$
0.546011 + 0.837778i $$0.316146\pi$$
$$110$$ 0.193937 0.0184911
$$111$$ 0 0
$$112$$ 12.6497 1.19529
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −6.70052 −0.624827
$$116$$ −7.08840 −0.658141
$$117$$ 0 0
$$118$$ −2.07522 −0.191040
$$119$$ 15.3258 1.40492
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −1.68735 −0.152765
$$123$$ 0 0
$$124$$ −19.4763 −1.74902
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −14.5745 −1.29328 −0.646640 0.762796i $$-0.723826\pi$$
−0.646640 + 0.762796i $$0.723826\pi$$
$$128$$ −5.91748 −0.523037
$$129$$ 0 0
$$130$$ −0.574515 −0.0503883
$$131$$ 5.92478 0.517650 0.258825 0.965924i $$-0.416665\pi$$
0.258825 + 0.965924i $$0.416665\pi$$
$$132$$ 0 0
$$133$$ −14.4485 −1.25284
$$134$$ 1.14903 0.0992612
$$135$$ 0 0
$$136$$ −3.51530 −0.301434
$$137$$ −13.8496 −1.18325 −0.591624 0.806214i $$-0.701513\pi$$
−0.591624 + 0.806214i $$0.701513\pi$$
$$138$$ 0 0
$$139$$ 13.6121 1.15457 0.577283 0.816544i $$-0.304113\pi$$
0.577283 + 0.816544i $$0.304113\pi$$
$$140$$ 6.57452 0.555648
$$141$$ 0 0
$$142$$ −1.92478 −0.161524
$$143$$ −2.96239 −0.247727
$$144$$ 0 0
$$145$$ −3.61213 −0.299971
$$146$$ −1.50071 −0.124199
$$147$$ 0 0
$$148$$ 3.92478 0.322615
$$149$$ −1.53690 −0.125908 −0.0629540 0.998016i $$-0.520052\pi$$
−0.0629540 + 0.998016i $$0.520052\pi$$
$$150$$ 0 0
$$151$$ −6.76116 −0.550215 −0.275108 0.961413i $$-0.588713\pi$$
−0.275108 + 0.961413i $$0.588713\pi$$
$$152$$ 3.31406 0.268806
$$153$$ 0 0
$$154$$ −0.649738 −0.0523574
$$155$$ −9.92478 −0.797177
$$156$$ 0 0
$$157$$ −5.47627 −0.437054 −0.218527 0.975831i $$-0.570125\pi$$
−0.218527 + 0.975831i $$0.570125\pi$$
$$158$$ 2.23743 0.178000
$$159$$ 0 0
$$160$$ −2.26916 −0.179393
$$161$$ 22.4485 1.76919
$$162$$ 0 0
$$163$$ 12.6253 0.988890 0.494445 0.869209i $$-0.335371\pi$$
0.494445 + 0.869209i $$0.335371\pi$$
$$164$$ −8.61071 −0.672384
$$165$$ 0 0
$$166$$ −2.11142 −0.163878
$$167$$ −18.3634 −1.42101 −0.710503 0.703695i $$-0.751532\pi$$
−0.710503 + 0.703695i $$0.751532\pi$$
$$168$$ 0 0
$$169$$ −4.22425 −0.324943
$$170$$ −0.887166 −0.0680425
$$171$$ 0 0
$$172$$ 18.2012 1.38783
$$173$$ 8.57452 0.651908 0.325954 0.945386i $$-0.394314\pi$$
0.325954 + 0.945386i $$0.394314\pi$$
$$174$$ 0 0
$$175$$ 3.35026 0.253256
$$176$$ −3.77575 −0.284608
$$177$$ 0 0
$$178$$ 0.538319 0.0403487
$$179$$ −14.1768 −1.05962 −0.529812 0.848115i $$-0.677737\pi$$
−0.529812 + 0.848115i $$0.677737\pi$$
$$180$$ 0 0
$$181$$ −5.22425 −0.388316 −0.194158 0.980970i $$-0.562197\pi$$
−0.194158 + 0.980970i $$0.562197\pi$$
$$182$$ 1.92478 0.142674
$$183$$ 0 0
$$184$$ −5.14903 −0.379592
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −4.57452 −0.334522
$$188$$ −19.4763 −1.42045
$$189$$ 0 0
$$190$$ 0.836381 0.0606774
$$191$$ 16.6253 1.20296 0.601482 0.798886i $$-0.294577\pi$$
0.601482 + 0.798886i $$0.294577\pi$$
$$192$$ 0 0
$$193$$ −16.3634 −1.17787 −0.588933 0.808182i $$-0.700452\pi$$
−0.588933 + 0.808182i $$0.700452\pi$$
$$194$$ 0.0145884 0.00104739
$$195$$ 0 0
$$196$$ −8.28963 −0.592116
$$197$$ 20.4241 1.45515 0.727577 0.686026i $$-0.240646\pi$$
0.727577 + 0.686026i $$0.240646\pi$$
$$198$$ 0 0
$$199$$ −8.62530 −0.611431 −0.305716 0.952123i $$-0.598896\pi$$
−0.305716 + 0.952123i $$0.598896\pi$$
$$200$$ −0.768452 −0.0543378
$$201$$ 0 0
$$202$$ 2.92619 0.205886
$$203$$ 12.1016 0.849364
$$204$$ 0 0
$$205$$ −4.38787 −0.306462
$$206$$ −0.625301 −0.0435668
$$207$$ 0 0
$$208$$ 11.1852 0.775556
$$209$$ 4.31265 0.298312
$$210$$ 0 0
$$211$$ 9.08840 0.625671 0.312836 0.949807i $$-0.398721\pi$$
0.312836 + 0.949807i $$0.398721\pi$$
$$212$$ 9.22425 0.633524
$$213$$ 0 0
$$214$$ 0.186642 0.0127586
$$215$$ 9.27504 0.632552
$$216$$ 0 0
$$217$$ 33.2506 2.25720
$$218$$ 2.21108 0.149753
$$219$$ 0 0
$$220$$ −1.96239 −0.132304
$$221$$ 13.5515 0.911572
$$222$$ 0 0
$$223$$ −6.70052 −0.448700 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$224$$ 7.60228 0.507949
$$225$$ 0 0
$$226$$ 1.16362 0.0774028
$$227$$ −16.9624 −1.12583 −0.562917 0.826514i $$-0.690321\pi$$
−0.562917 + 0.826514i $$0.690321\pi$$
$$228$$ 0 0
$$229$$ 25.8496 1.70819 0.854093 0.520120i $$-0.174113\pi$$
0.854093 + 0.520120i $$0.174113\pi$$
$$230$$ −1.29948 −0.0856849
$$231$$ 0 0
$$232$$ −2.77575 −0.182237
$$233$$ 19.2750 1.26275 0.631375 0.775478i $$-0.282491\pi$$
0.631375 + 0.775478i $$0.282491\pi$$
$$234$$ 0 0
$$235$$ −9.92478 −0.647421
$$236$$ 20.9986 1.36689
$$237$$ 0 0
$$238$$ 2.97224 0.192662
$$239$$ −26.5501 −1.71738 −0.858691 0.512494i $$-0.828722\pi$$
−0.858691 + 0.512494i $$0.828722\pi$$
$$240$$ 0 0
$$241$$ 28.5501 1.83907 0.919536 0.393006i $$-0.128565\pi$$
0.919536 + 0.393006i $$0.128565\pi$$
$$242$$ 0.193937 0.0124667
$$243$$ 0 0
$$244$$ 17.0738 1.09304
$$245$$ −4.22425 −0.269878
$$246$$ 0 0
$$247$$ −12.7757 −0.812901
$$248$$ −7.62672 −0.484297
$$249$$ 0 0
$$250$$ −0.193937 −0.0122656
$$251$$ −29.9248 −1.88884 −0.944418 0.328748i $$-0.893373\pi$$
−0.944418 + 0.328748i $$0.893373\pi$$
$$252$$ 0 0
$$253$$ −6.70052 −0.421258
$$254$$ −2.82653 −0.177352
$$255$$ 0 0
$$256$$ 13.0752 0.817201
$$257$$ −8.70052 −0.542724 −0.271362 0.962477i $$-0.587474\pi$$
−0.271362 + 0.962477i $$0.587474\pi$$
$$258$$ 0 0
$$259$$ −6.70052 −0.416350
$$260$$ 5.81336 0.360529
$$261$$ 0 0
$$262$$ 1.14903 0.0709874
$$263$$ −12.2882 −0.757724 −0.378862 0.925453i $$-0.623684\pi$$
−0.378862 + 0.925453i $$0.623684\pi$$
$$264$$ 0 0
$$265$$ 4.70052 0.288751
$$266$$ −2.80209 −0.171807
$$267$$ 0 0
$$268$$ −11.6267 −0.710215
$$269$$ 5.84955 0.356654 0.178327 0.983971i $$-0.442932\pi$$
0.178327 + 0.983971i $$0.442932\pi$$
$$270$$ 0 0
$$271$$ −5.08840 −0.309098 −0.154549 0.987985i $$-0.549392\pi$$
−0.154549 + 0.987985i $$0.549392\pi$$
$$272$$ 17.2722 1.04728
$$273$$ 0 0
$$274$$ −2.68594 −0.162263
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 1.41090 0.0847725 0.0423863 0.999101i $$-0.486504\pi$$
0.0423863 + 0.999101i $$0.486504\pi$$
$$278$$ 2.63989 0.158330
$$279$$ 0 0
$$280$$ 2.57452 0.153857
$$281$$ 4.38787 0.261759 0.130879 0.991398i $$-0.458220\pi$$
0.130879 + 0.991398i $$0.458220\pi$$
$$282$$ 0 0
$$283$$ 26.5745 1.57969 0.789845 0.613306i $$-0.210161\pi$$
0.789845 + 0.613306i $$0.210161\pi$$
$$284$$ 19.4763 1.15570
$$285$$ 0 0
$$286$$ −0.574515 −0.0339718
$$287$$ 14.7005 0.867744
$$288$$ 0 0
$$289$$ 3.92619 0.230952
$$290$$ −0.700523 −0.0411362
$$291$$ 0 0
$$292$$ 15.1852 0.888648
$$293$$ 3.42548 0.200119 0.100059 0.994981i $$-0.468097\pi$$
0.100059 + 0.994981i $$0.468097\pi$$
$$294$$ 0 0
$$295$$ 10.7005 0.623009
$$296$$ 1.53690 0.0893307
$$297$$ 0 0
$$298$$ −0.298062 −0.0172663
$$299$$ 19.8496 1.14793
$$300$$ 0 0
$$301$$ −31.0738 −1.79106
$$302$$ −1.31124 −0.0754531
$$303$$ 0 0
$$304$$ −16.2835 −0.933921
$$305$$ 8.70052 0.498191
$$306$$ 0 0
$$307$$ −16.6497 −0.950251 −0.475125 0.879918i $$-0.657597\pi$$
−0.475125 + 0.879918i $$0.657597\pi$$
$$308$$ 6.57452 0.374618
$$309$$ 0 0
$$310$$ −1.92478 −0.109320
$$311$$ −32.9986 −1.87118 −0.935589 0.353091i $$-0.885131\pi$$
−0.935589 + 0.353091i $$0.885131\pi$$
$$312$$ 0 0
$$313$$ 15.4010 0.870519 0.435259 0.900305i $$-0.356657\pi$$
0.435259 + 0.900305i $$0.356657\pi$$
$$314$$ −1.06205 −0.0599349
$$315$$ 0 0
$$316$$ −22.6399 −1.27359
$$317$$ −2.15045 −0.120781 −0.0603905 0.998175i $$-0.519235\pi$$
−0.0603905 + 0.998175i $$0.519235\pi$$
$$318$$ 0 0
$$319$$ −3.61213 −0.202240
$$320$$ 7.11142 0.397540
$$321$$ 0 0
$$322$$ 4.35359 0.242616
$$323$$ −19.7283 −1.09771
$$324$$ 0 0
$$325$$ 2.96239 0.164324
$$326$$ 2.44851 0.135610
$$327$$ 0 0
$$328$$ −3.37187 −0.186180
$$329$$ 33.2506 1.83316
$$330$$ 0 0
$$331$$ −14.5501 −0.799745 −0.399872 0.916571i $$-0.630946\pi$$
−0.399872 + 0.916571i $$0.630946\pi$$
$$332$$ 21.3649 1.17255
$$333$$ 0 0
$$334$$ −3.56134 −0.194868
$$335$$ −5.92478 −0.323705
$$336$$ 0 0
$$337$$ 16.2619 0.885840 0.442920 0.896561i $$-0.353943\pi$$
0.442920 + 0.896561i $$0.353943\pi$$
$$338$$ −0.819237 −0.0445606
$$339$$ 0 0
$$340$$ 8.97698 0.486845
$$341$$ −9.92478 −0.537457
$$342$$ 0 0
$$343$$ −9.29948 −0.502125
$$344$$ 7.12742 0.384285
$$345$$ 0 0
$$346$$ 1.66291 0.0893987
$$347$$ 0.962389 0.0516637 0.0258319 0.999666i $$-0.491777\pi$$
0.0258319 + 0.999666i $$0.491777\pi$$
$$348$$ 0 0
$$349$$ 20.7005 1.10807 0.554037 0.832492i $$-0.313087\pi$$
0.554037 + 0.832492i $$0.313087\pi$$
$$350$$ 0.649738 0.0347300
$$351$$ 0 0
$$352$$ −2.26916 −0.120947
$$353$$ −20.5501 −1.09377 −0.546885 0.837208i $$-0.684187\pi$$
−0.546885 + 0.837208i $$0.684187\pi$$
$$354$$ 0 0
$$355$$ 9.92478 0.526752
$$356$$ −5.44709 −0.288695
$$357$$ 0 0
$$358$$ −2.74940 −0.145310
$$359$$ −17.9248 −0.946034 −0.473017 0.881053i $$-0.656835\pi$$
−0.473017 + 0.881053i $$0.656835\pi$$
$$360$$ 0 0
$$361$$ −0.401047 −0.0211077
$$362$$ −1.01317 −0.0532512
$$363$$ 0 0
$$364$$ −19.4763 −1.02083
$$365$$ 7.73813 0.405032
$$366$$ 0 0
$$367$$ −29.6531 −1.54788 −0.773939 0.633261i $$-0.781716\pi$$
−0.773939 + 0.633261i $$0.781716\pi$$
$$368$$ 25.2995 1.31883
$$369$$ 0 0
$$370$$ 0.387873 0.0201646
$$371$$ −15.7480 −0.817595
$$372$$ 0 0
$$373$$ −9.13918 −0.473209 −0.236604 0.971606i $$-0.576035\pi$$
−0.236604 + 0.971606i $$0.576035\pi$$
$$374$$ −0.887166 −0.0458743
$$375$$ 0 0
$$376$$ −7.62672 −0.393318
$$377$$ 10.7005 0.551105
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −8.46310 −0.434148
$$381$$ 0 0
$$382$$ 3.22425 0.164967
$$383$$ 34.9234 1.78450 0.892250 0.451541i $$-0.149126\pi$$
0.892250 + 0.451541i $$0.149126\pi$$
$$384$$ 0 0
$$385$$ 3.35026 0.170745
$$386$$ −3.17347 −0.161525
$$387$$ 0 0
$$388$$ −0.147616 −0.00749408
$$389$$ −2.77575 −0.140736 −0.0703680 0.997521i $$-0.522417\pi$$
−0.0703680 + 0.997521i $$0.522417\pi$$
$$390$$ 0 0
$$391$$ 30.6516 1.55012
$$392$$ −3.24614 −0.163955
$$393$$ 0 0
$$394$$ 3.96097 0.199551
$$395$$ −11.5369 −0.580485
$$396$$ 0 0
$$397$$ −19.9248 −0.999996 −0.499998 0.866027i $$-0.666666\pi$$
−0.499998 + 0.866027i $$0.666666\pi$$
$$398$$ −1.67276 −0.0838479
$$399$$ 0 0
$$400$$ 3.77575 0.188787
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ 29.4010 1.46457
$$404$$ −29.6093 −1.47312
$$405$$ 0 0
$$406$$ 2.34694 0.116477
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ −13.0738 −0.646458 −0.323229 0.946321i $$-0.604768\pi$$
−0.323229 + 0.946321i $$0.604768\pi$$
$$410$$ −0.850969 −0.0420264
$$411$$ 0 0
$$412$$ 6.32724 0.311721
$$413$$ −35.8496 −1.76404
$$414$$ 0 0
$$415$$ 10.8872 0.534430
$$416$$ 6.72213 0.329580
$$417$$ 0 0
$$418$$ 0.836381 0.0409087
$$419$$ −7.22425 −0.352928 −0.176464 0.984307i $$-0.556466\pi$$
−0.176464 + 0.984307i $$0.556466\pi$$
$$420$$ 0 0
$$421$$ 30.6253 1.49259 0.746293 0.665618i $$-0.231832\pi$$
0.746293 + 0.665618i $$0.231832\pi$$
$$422$$ 1.76257 0.0858007
$$423$$ 0 0
$$424$$ 3.61213 0.175420
$$425$$ 4.57452 0.221897
$$426$$ 0 0
$$427$$ −29.1490 −1.41062
$$428$$ −1.88858 −0.0912880
$$429$$ 0 0
$$430$$ 1.79877 0.0867444
$$431$$ 33.8759 1.63174 0.815872 0.578232i $$-0.196257\pi$$
0.815872 + 0.578232i $$0.196257\pi$$
$$432$$ 0 0
$$433$$ −9.47627 −0.455400 −0.227700 0.973731i $$-0.573121\pi$$
−0.227700 + 0.973731i $$0.573121\pi$$
$$434$$ 6.44851 0.309538
$$435$$ 0 0
$$436$$ −22.3733 −1.07149
$$437$$ −28.8970 −1.38233
$$438$$ 0 0
$$439$$ −29.4617 −1.40613 −0.703065 0.711126i $$-0.748186\pi$$
−0.703065 + 0.711126i $$0.748186\pi$$
$$440$$ −0.768452 −0.0366345
$$441$$ 0 0
$$442$$ 2.62813 0.125007
$$443$$ 19.0738 0.906224 0.453112 0.891454i $$-0.350314\pi$$
0.453112 + 0.891454i $$0.350314\pi$$
$$444$$ 0 0
$$445$$ −2.77575 −0.131583
$$446$$ −1.29948 −0.0615320
$$447$$ 0 0
$$448$$ −23.8251 −1.12563
$$449$$ −35.8759 −1.69309 −0.846544 0.532318i $$-0.821321\pi$$
−0.846544 + 0.532318i $$0.821321\pi$$
$$450$$ 0 0
$$451$$ −4.38787 −0.206617
$$452$$ −11.7743 −0.553818
$$453$$ 0 0
$$454$$ −3.28963 −0.154390
$$455$$ −9.92478 −0.465281
$$456$$ 0 0
$$457$$ 5.28963 0.247438 0.123719 0.992317i $$-0.460518\pi$$
0.123719 + 0.992317i $$0.460518\pi$$
$$458$$ 5.01317 0.234250
$$459$$ 0 0
$$460$$ 13.1490 0.613077
$$461$$ −36.3390 −1.69248 −0.846238 0.532805i $$-0.821138\pi$$
−0.846238 + 0.532805i $$0.821138\pi$$
$$462$$ 0 0
$$463$$ 10.5501 0.490304 0.245152 0.969485i $$-0.421162\pi$$
0.245152 + 0.969485i $$0.421162\pi$$
$$464$$ 13.6385 0.633150
$$465$$ 0 0
$$466$$ 3.73813 0.173166
$$467$$ −18.7005 −0.865357 −0.432679 0.901548i $$-0.642431\pi$$
−0.432679 + 0.901548i $$0.642431\pi$$
$$468$$ 0 0
$$469$$ 19.8496 0.916567
$$470$$ −1.92478 −0.0887834
$$471$$ 0 0
$$472$$ 8.22284 0.378487
$$473$$ 9.27504 0.426467
$$474$$ 0 0
$$475$$ −4.31265 −0.197878
$$476$$ −30.0752 −1.37850
$$477$$ 0 0
$$478$$ −5.14903 −0.235511
$$479$$ 9.29948 0.424904 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$480$$ 0 0
$$481$$ −5.92478 −0.270147
$$482$$ 5.53690 0.252199
$$483$$ 0 0
$$484$$ −1.96239 −0.0891995
$$485$$ −0.0752228 −0.00341569
$$486$$ 0 0
$$487$$ −35.4763 −1.60758 −0.803792 0.594911i $$-0.797187\pi$$
−0.803792 + 0.594911i $$0.797187\pi$$
$$488$$ 6.68594 0.302658
$$489$$ 0 0
$$490$$ −0.819237 −0.0370094
$$491$$ −24.7757 −1.11811 −0.559057 0.829129i $$-0.688837\pi$$
−0.559057 + 0.829129i $$0.688837\pi$$
$$492$$ 0 0
$$493$$ 16.5237 0.744191
$$494$$ −2.47768 −0.111476
$$495$$ 0 0
$$496$$ 37.4734 1.68261
$$497$$ −33.2506 −1.49149
$$498$$ 0 0
$$499$$ 14.1768 0.634640 0.317320 0.948318i $$-0.397217\pi$$
0.317320 + 0.948318i $$0.397217\pi$$
$$500$$ 1.96239 0.0877607
$$501$$ 0 0
$$502$$ −5.80351 −0.259023
$$503$$ 8.43866 0.376261 0.188131 0.982144i $$-0.439757\pi$$
0.188131 + 0.982144i $$0.439757\pi$$
$$504$$ 0 0
$$505$$ −15.0884 −0.671425
$$506$$ −1.29948 −0.0577688
$$507$$ 0 0
$$508$$ 28.6009 1.26896
$$509$$ −1.10299 −0.0488890 −0.0244445 0.999701i $$-0.507782\pi$$
−0.0244445 + 0.999701i $$0.507782\pi$$
$$510$$ 0 0
$$511$$ −25.9248 −1.14684
$$512$$ 14.3707 0.635103
$$513$$ 0 0
$$514$$ −1.68735 −0.0744258
$$515$$ 3.22425 0.142078
$$516$$ 0 0
$$517$$ −9.92478 −0.436491
$$518$$ −1.29948 −0.0570957
$$519$$ 0 0
$$520$$ 2.27645 0.0998291
$$521$$ 12.4485 0.545379 0.272690 0.962102i $$-0.412087\pi$$
0.272690 + 0.962102i $$0.412087\pi$$
$$522$$ 0 0
$$523$$ 30.0508 1.31403 0.657015 0.753878i $$-0.271819\pi$$
0.657015 + 0.753878i $$0.271819\pi$$
$$524$$ −11.6267 −0.507915
$$525$$ 0 0
$$526$$ −2.38313 −0.103910
$$527$$ 45.4010 1.97770
$$528$$ 0 0
$$529$$ 21.8970 0.952044
$$530$$ 0.911603 0.0395975
$$531$$ 0 0
$$532$$ 28.3536 1.22928
$$533$$ 12.9986 0.563031
$$534$$ 0 0
$$535$$ −0.962389 −0.0416077
$$536$$ −4.55291 −0.196656
$$537$$ 0 0
$$538$$ 1.13444 0.0489093
$$539$$ −4.22425 −0.181951
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −0.986826 −0.0423878
$$543$$ 0 0
$$544$$ 10.3803 0.445052
$$545$$ −11.4010 −0.488367
$$546$$ 0 0
$$547$$ −14.3028 −0.611544 −0.305772 0.952105i $$-0.598914\pi$$
−0.305772 + 0.952105i $$0.598914\pi$$
$$548$$ 27.1782 1.16100
$$549$$ 0 0
$$550$$ −0.193937 −0.00826948
$$551$$ −15.5778 −0.663638
$$552$$ 0 0
$$553$$ 38.6516 1.64364
$$554$$ 0.273624 0.0116252
$$555$$ 0 0
$$556$$ −26.7123 −1.13285
$$557$$ 11.7988 0.499930 0.249965 0.968255i $$-0.419581\pi$$
0.249965 + 0.968255i $$0.419581\pi$$
$$558$$ 0 0
$$559$$ −27.4763 −1.16212
$$560$$ −12.6497 −0.534549
$$561$$ 0 0
$$562$$ 0.850969 0.0358960
$$563$$ −30.4847 −1.28478 −0.642389 0.766379i $$-0.722056\pi$$
−0.642389 + 0.766379i $$0.722056\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 5.15377 0.216629
$$567$$ 0 0
$$568$$ 7.62672 0.320010
$$569$$ 27.0884 1.13560 0.567802 0.823165i $$-0.307794\pi$$
0.567802 + 0.823165i $$0.307794\pi$$
$$570$$ 0 0
$$571$$ 7.28489 0.304863 0.152432 0.988314i $$-0.451290\pi$$
0.152432 + 0.988314i $$0.451290\pi$$
$$572$$ 5.81336 0.243069
$$573$$ 0 0
$$574$$ 2.85097 0.118997
$$575$$ 6.70052 0.279431
$$576$$ 0 0
$$577$$ −31.6239 −1.31652 −0.658260 0.752791i $$-0.728707\pi$$
−0.658260 + 0.752791i $$0.728707\pi$$
$$578$$ 0.761432 0.0316714
$$579$$ 0 0
$$580$$ 7.08840 0.294330
$$581$$ −36.4749 −1.51323
$$582$$ 0 0
$$583$$ 4.70052 0.194676
$$584$$ 5.94639 0.246063
$$585$$ 0 0
$$586$$ 0.664327 0.0274431
$$587$$ −33.1490 −1.36821 −0.684103 0.729385i $$-0.739806\pi$$
−0.684103 + 0.729385i $$0.739806\pi$$
$$588$$ 0 0
$$589$$ −42.8021 −1.76363
$$590$$ 2.07522 0.0854356
$$591$$ 0 0
$$592$$ −7.55149 −0.310364
$$593$$ −34.4993 −1.41672 −0.708358 0.705853i $$-0.750564\pi$$
−0.708358 + 0.705853i $$0.750564\pi$$
$$594$$ 0 0
$$595$$ −15.3258 −0.628298
$$596$$ 3.01600 0.123540
$$597$$ 0 0
$$598$$ 3.84955 0.157420
$$599$$ 14.4485 0.590350 0.295175 0.955443i $$-0.404622\pi$$
0.295175 + 0.955443i $$0.404622\pi$$
$$600$$ 0 0
$$601$$ −15.9248 −0.649585 −0.324793 0.945785i $$-0.605295\pi$$
−0.324793 + 0.945785i $$0.605295\pi$$
$$602$$ −6.02635 −0.245616
$$603$$ 0 0
$$604$$ 13.2680 0.539868
$$605$$ −1.00000 −0.0406558
$$606$$ 0 0
$$607$$ −14.5745 −0.591561 −0.295781 0.955256i $$-0.595580\pi$$
−0.295781 + 0.955256i $$0.595580\pi$$
$$608$$ −9.78609 −0.396878
$$609$$ 0 0
$$610$$ 1.68735 0.0683188
$$611$$ 29.4010 1.18944
$$612$$ 0 0
$$613$$ 16.4123 0.662887 0.331443 0.943475i $$-0.392464\pi$$
0.331443 + 0.943475i $$0.392464\pi$$
$$614$$ −3.22899 −0.130312
$$615$$ 0 0
$$616$$ 2.57452 0.103730
$$617$$ 17.8496 0.718596 0.359298 0.933223i $$-0.383016\pi$$
0.359298 + 0.933223i $$0.383016\pi$$
$$618$$ 0 0
$$619$$ −0.402462 −0.0161763 −0.00808815 0.999967i $$-0.502575\pi$$
−0.00808815 + 0.999967i $$0.502575\pi$$
$$620$$ 19.4763 0.782186
$$621$$ 0 0
$$622$$ −6.39963 −0.256602
$$623$$ 9.29948 0.372576
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 2.98683 0.119378
$$627$$ 0 0
$$628$$ 10.7466 0.428835
$$629$$ −9.14903 −0.364796
$$630$$ 0 0
$$631$$ −38.0263 −1.51380 −0.756902 0.653528i $$-0.773288\pi$$
−0.756902 + 0.653528i $$0.773288\pi$$
$$632$$ −8.86556 −0.352653
$$633$$ 0 0
$$634$$ −0.417050 −0.0165632
$$635$$ 14.5745 0.578372
$$636$$ 0 0
$$637$$ 12.5139 0.495818
$$638$$ −0.700523 −0.0277340
$$639$$ 0 0
$$640$$ 5.91748 0.233909
$$641$$ 28.0263 1.10697 0.553487 0.832858i $$-0.313297\pi$$
0.553487 + 0.832858i $$0.313297\pi$$
$$642$$ 0 0
$$643$$ −4.62530 −0.182404 −0.0912020 0.995832i $$-0.529071\pi$$
−0.0912020 + 0.995832i $$0.529071\pi$$
$$644$$ −44.0527 −1.73592
$$645$$ 0 0
$$646$$ −3.82604 −0.150533
$$647$$ −23.5778 −0.926941 −0.463470 0.886112i $$-0.653396\pi$$
−0.463470 + 0.886112i $$0.653396\pi$$
$$648$$ 0 0
$$649$$ 10.7005 0.420032
$$650$$ 0.574515 0.0225344
$$651$$ 0 0
$$652$$ −24.7757 −0.970293
$$653$$ 2.25202 0.0881282 0.0440641 0.999029i $$-0.485969\pi$$
0.0440641 + 0.999029i $$0.485969\pi$$
$$654$$ 0 0
$$655$$ −5.92478 −0.231500
$$656$$ 16.5675 0.646852
$$657$$ 0 0
$$658$$ 6.44851 0.251389
$$659$$ 41.4010 1.61276 0.806378 0.591401i $$-0.201425\pi$$
0.806378 + 0.591401i $$0.201425\pi$$
$$660$$ 0 0
$$661$$ 3.40105 0.132285 0.0661427 0.997810i $$-0.478931\pi$$
0.0661427 + 0.997810i $$0.478931\pi$$
$$662$$ −2.82179 −0.109672
$$663$$ 0 0
$$664$$ 8.36626 0.324674
$$665$$ 14.4485 0.560289
$$666$$ 0 0
$$667$$ 24.2031 0.937149
$$668$$ 36.0362 1.39428
$$669$$ 0 0
$$670$$ −1.14903 −0.0443909
$$671$$ 8.70052 0.335880
$$672$$ 0 0
$$673$$ 0.887166 0.0341977 0.0170989 0.999854i $$-0.494557\pi$$
0.0170989 + 0.999854i $$0.494557\pi$$
$$674$$ 3.15377 0.121479
$$675$$ 0 0
$$676$$ 8.28963 0.318832
$$677$$ −18.9018 −0.726453 −0.363227 0.931701i $$-0.618325\pi$$
−0.363227 + 0.931701i $$0.618325\pi$$
$$678$$ 0 0
$$679$$ 0.252016 0.00967149
$$680$$ 3.51530 0.134805
$$681$$ 0 0
$$682$$ −1.92478 −0.0737035
$$683$$ 20.8773 0.798848 0.399424 0.916766i $$-0.369210\pi$$
0.399424 + 0.916766i $$0.369210\pi$$
$$684$$ 0 0
$$685$$ 13.8496 0.529164
$$686$$ −1.80351 −0.0688583
$$687$$ 0 0
$$688$$ −35.0202 −1.33513
$$689$$ −13.9248 −0.530492
$$690$$ 0 0
$$691$$ −2.44851 −0.0931456 −0.0465728 0.998915i $$-0.514830\pi$$
−0.0465728 + 0.998915i $$0.514830\pi$$
$$692$$ −16.8265 −0.639649
$$693$$ 0 0
$$694$$ 0.186642 0.00708485
$$695$$ −13.6121 −0.516337
$$696$$ 0 0
$$697$$ 20.0724 0.760296
$$698$$ 4.01459 0.151954
$$699$$ 0 0
$$700$$ −6.57452 −0.248493
$$701$$ 2.98683 0.112811 0.0564054 0.998408i $$-0.482036\pi$$
0.0564054 + 0.998408i $$0.482036\pi$$
$$702$$ 0 0
$$703$$ 8.62530 0.325309
$$704$$ 7.11142 0.268022
$$705$$ 0 0
$$706$$ −3.98541 −0.149993
$$707$$ 50.5501 1.90113
$$708$$ 0 0
$$709$$ 24.1768 0.907979 0.453989 0.891007i $$-0.350000\pi$$
0.453989 + 0.891007i $$0.350000\pi$$
$$710$$ 1.92478 0.0722356
$$711$$ 0 0
$$712$$ −2.13303 −0.0799386
$$713$$ 66.5012 2.49049
$$714$$ 0 0
$$715$$ 2.96239 0.110787
$$716$$ 27.8204 1.03970
$$717$$ 0 0
$$718$$ −3.47627 −0.129733
$$719$$ 30.0263 1.11979 0.559897 0.828562i $$-0.310841\pi$$
0.559897 + 0.828562i $$0.310841\pi$$
$$720$$ 0 0
$$721$$ −10.8021 −0.402291
$$722$$ −0.0777777 −0.00289459
$$723$$ 0 0
$$724$$ 10.2520 0.381013
$$725$$ 3.61213 0.134151
$$726$$ 0 0
$$727$$ 14.9525 0.554559 0.277279 0.960789i $$-0.410567\pi$$
0.277279 + 0.960789i $$0.410567\pi$$
$$728$$ −7.62672 −0.282665
$$729$$ 0 0
$$730$$ 1.50071 0.0555437
$$731$$ −42.4288 −1.56929
$$732$$ 0 0
$$733$$ −19.1128 −0.705949 −0.352974 0.935633i $$-0.614830\pi$$
−0.352974 + 0.935633i $$0.614830\pi$$
$$734$$ −5.75081 −0.212266
$$735$$ 0 0
$$736$$ 15.2046 0.560447
$$737$$ −5.92478 −0.218242
$$738$$ 0 0
$$739$$ −3.31406 −0.121910 −0.0609549 0.998141i $$-0.519415\pi$$
−0.0609549 + 0.998141i $$0.519415\pi$$
$$740$$ −3.92478 −0.144278
$$741$$ 0 0
$$742$$ −3.05411 −0.112120
$$743$$ 34.9887 1.28361 0.641806 0.766867i $$-0.278185\pi$$
0.641806 + 0.766867i $$0.278185\pi$$
$$744$$ 0 0
$$745$$ 1.53690 0.0563078
$$746$$ −1.77242 −0.0648930
$$747$$ 0 0
$$748$$ 8.97698 0.328231
$$749$$ 3.22425 0.117812
$$750$$ 0 0
$$751$$ −26.9234 −0.982447 −0.491224 0.871033i $$-0.663450\pi$$
−0.491224 + 0.871033i $$0.663450\pi$$
$$752$$ 37.4734 1.36652
$$753$$ 0 0
$$754$$ 2.07522 0.0755752
$$755$$ 6.76116 0.246064
$$756$$ 0 0
$$757$$ 15.9248 0.578796 0.289398 0.957209i $$-0.406545\pi$$
0.289398 + 0.957209i $$0.406545\pi$$
$$758$$ −3.87873 −0.140882
$$759$$ 0 0
$$760$$ −3.31406 −0.120214
$$761$$ 30.9380 1.12150 0.560750 0.827985i $$-0.310513\pi$$
0.560750 + 0.827985i $$0.310513\pi$$
$$762$$ 0 0
$$763$$ 38.1965 1.38281
$$764$$ −32.6253 −1.18034
$$765$$ 0 0
$$766$$ 6.77292 0.244715
$$767$$ −31.6991 −1.14459
$$768$$ 0 0
$$769$$ 9.32582 0.336298 0.168149 0.985762i $$-0.446221\pi$$
0.168149 + 0.985762i $$0.446221\pi$$
$$770$$ 0.649738 0.0234149
$$771$$ 0 0
$$772$$ 32.1114 1.15572
$$773$$ −44.7005 −1.60777 −0.803883 0.594787i $$-0.797236\pi$$
−0.803883 + 0.594787i $$0.797236\pi$$
$$774$$ 0 0
$$775$$ 9.92478 0.356509
$$776$$ −0.0578051 −0.00207508
$$777$$ 0 0
$$778$$ −0.538319 −0.0192997
$$779$$ −18.9234 −0.678000
$$780$$ 0 0
$$781$$ 9.92478 0.355136
$$782$$ 5.94448 0.212574
$$783$$ 0 0
$$784$$ 15.9497 0.569633
$$785$$ 5.47627 0.195456
$$786$$ 0 0
$$787$$ 21.6775 0.772719 0.386360 0.922348i $$-0.373732\pi$$
0.386360 + 0.922348i $$0.373732\pi$$
$$788$$ −40.0800 −1.42779
$$789$$ 0 0
$$790$$ −2.23743 −0.0796041
$$791$$ 20.1016 0.714730
$$792$$ 0 0
$$793$$ −25.7743 −0.915273
$$794$$ −3.86414 −0.137133
$$795$$ 0 0
$$796$$ 16.9262 0.599933
$$797$$ −22.7466 −0.805725 −0.402862 0.915261i $$-0.631985\pi$$
−0.402862 + 0.915261i $$0.631985\pi$$
$$798$$ 0 0
$$799$$ 45.4010 1.60617
$$800$$ 2.26916 0.0802269
$$801$$ 0 0
$$802$$ −0.387873 −0.0136963
$$803$$ 7.73813 0.273073
$$804$$ 0 0
$$805$$ −22.4485 −0.791206
$$806$$ 5.70194 0.200842
$$807$$ 0 0
$$808$$ −11.5947 −0.407900
$$809$$ 23.6121 0.830158 0.415079 0.909785i $$-0.363754\pi$$
0.415079 + 0.909785i $$0.363754\pi$$
$$810$$ 0 0
$$811$$ −26.0870 −0.916038 −0.458019 0.888942i $$-0.651441\pi$$
−0.458019 + 0.888942i $$0.651441\pi$$
$$812$$ −23.7480 −0.833391
$$813$$ 0 0
$$814$$ 0.387873 0.0135949
$$815$$ −12.6253 −0.442245
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ −2.53549 −0.0886513
$$819$$ 0 0
$$820$$ 8.61071 0.300699
$$821$$ 54.4142 1.89907 0.949535 0.313662i $$-0.101556\pi$$
0.949535 + 0.313662i $$0.101556\pi$$
$$822$$ 0 0
$$823$$ 0.121269 0.00422716 0.00211358 0.999998i $$-0.499327\pi$$
0.00211358 + 0.999998i $$0.499327\pi$$
$$824$$ 2.47768 0.0863142
$$825$$ 0 0
$$826$$ −6.95254 −0.241910
$$827$$ 18.2130 0.633328 0.316664 0.948538i $$-0.397437\pi$$
0.316664 + 0.948538i $$0.397437\pi$$
$$828$$ 0 0
$$829$$ 13.0738 0.454072 0.227036 0.973886i $$-0.427096\pi$$
0.227036 + 0.973886i $$0.427096\pi$$
$$830$$ 2.11142 0.0732884
$$831$$ 0 0
$$832$$ −21.0668 −0.730359
$$833$$ 19.3239 0.669534
$$834$$ 0 0
$$835$$ 18.3634 0.635493
$$836$$ −8.46310 −0.292702
$$837$$ 0 0
$$838$$ −1.40105 −0.0483984
$$839$$ 26.5501 0.916610 0.458305 0.888795i $$-0.348457\pi$$
0.458305 + 0.888795i $$0.348457\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ 5.93937 0.204684
$$843$$ 0 0
$$844$$ −17.8350 −0.613905
$$845$$ 4.22425 0.145319
$$846$$ 0 0
$$847$$ 3.35026 0.115116
$$848$$ −17.7480 −0.609468
$$849$$ 0 0
$$850$$ 0.887166 0.0304295
$$851$$ −13.4010 −0.459382
$$852$$ 0 0
$$853$$ 40.6155 1.39065 0.695323 0.718697i $$-0.255261\pi$$
0.695323 + 0.718697i $$0.255261\pi$$
$$854$$ −5.65306 −0.193444
$$855$$ 0 0
$$856$$ −0.739549 −0.0252773
$$857$$ −20.1721 −0.689064 −0.344532 0.938775i $$-0.611962\pi$$
−0.344532 + 0.938775i $$0.611962\pi$$
$$858$$ 0 0
$$859$$ 21.8035 0.743926 0.371963 0.928248i $$-0.378685\pi$$
0.371963 + 0.928248i $$0.378685\pi$$
$$860$$ −18.2012 −0.620657
$$861$$ 0 0
$$862$$ 6.56978 0.223767
$$863$$ −35.4274 −1.20596 −0.602981 0.797755i $$-0.706021\pi$$
−0.602981 + 0.797755i $$0.706021\pi$$
$$864$$ 0 0
$$865$$ −8.57452 −0.291542
$$866$$ −1.83780 −0.0624508
$$867$$ 0 0
$$868$$ −65.2506 −2.21475
$$869$$ −11.5369 −0.391363
$$870$$ 0 0
$$871$$ 17.5515 0.594710
$$872$$ −8.76116 −0.296690
$$873$$ 0 0
$$874$$ −5.60419 −0.189564
$$875$$ −3.35026 −0.113260
$$876$$ 0 0
$$877$$ −14.0362 −0.473969 −0.236984 0.971513i $$-0.576159\pi$$
−0.236984 + 0.971513i $$0.576159\pi$$
$$878$$ −5.71370 −0.192828
$$879$$ 0 0
$$880$$ 3.77575 0.127280
$$881$$ 21.0738 0.709995 0.354997 0.934867i $$-0.384482\pi$$
0.354997 + 0.934867i $$0.384482\pi$$
$$882$$ 0 0
$$883$$ −42.1476 −1.41838 −0.709190 0.705017i $$-0.750939\pi$$
−0.709190 + 0.705017i $$0.750939\pi$$
$$884$$ −26.5933 −0.894429
$$885$$ 0 0
$$886$$ 3.69911 0.124274
$$887$$ −6.93604 −0.232889 −0.116445 0.993197i $$-0.537150\pi$$
−0.116445 + 0.993197i $$0.537150\pi$$
$$888$$ 0 0
$$889$$ −48.8284 −1.63765
$$890$$ −0.538319 −0.0180445
$$891$$ 0 0
$$892$$ 13.1490 0.440262
$$893$$ −42.8021 −1.43232
$$894$$ 0 0
$$895$$ 14.1768 0.473878
$$896$$ −19.8251 −0.662311
$$897$$ 0 0
$$898$$ −6.95765 −0.232180
$$899$$ 35.8496 1.19565
$$900$$ 0 0
$$901$$ −21.5026 −0.716356
$$902$$ −0.850969 −0.0283342
$$903$$ 0 0
$$904$$ −4.61071 −0.153350
$$905$$ 5.22425 0.173660
$$906$$ 0 0
$$907$$ 53.2017 1.76653 0.883267 0.468870i $$-0.155339\pi$$
0.883267 + 0.468870i $$0.155339\pi$$
$$908$$ 33.2868 1.10466
$$909$$ 0 0
$$910$$ −1.92478 −0.0638057
$$911$$ −36.4749 −1.20847 −0.604233 0.796808i $$-0.706520\pi$$
−0.604233 + 0.796808i $$0.706520\pi$$
$$912$$ 0 0
$$913$$ 10.8872 0.360313
$$914$$ 1.02585 0.0339322
$$915$$ 0 0
$$916$$ −50.7269 −1.67606
$$917$$ 19.8496 0.655490
$$918$$ 0 0
$$919$$ 9.73340 0.321075 0.160538 0.987030i $$-0.448677\pi$$
0.160538 + 0.987030i $$0.448677\pi$$
$$920$$ 5.14903 0.169759
$$921$$ 0 0
$$922$$ −7.04746 −0.232096
$$923$$ −29.4010 −0.967747
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 2.04605 0.0672372
$$927$$ 0 0
$$928$$ 8.19649 0.269063
$$929$$ 24.1768 0.793215 0.396607 0.917988i $$-0.370187\pi$$
0.396607 + 0.917988i $$0.370187\pi$$
$$930$$ 0 0
$$931$$ −18.2177 −0.597062
$$932$$ −37.8251 −1.23900
$$933$$ 0 0
$$934$$ −3.62672 −0.118670
$$935$$ 4.57452 0.149603
$$936$$ 0 0
$$937$$ −7.48612 −0.244561 −0.122280 0.992496i $$-0.539021\pi$$
−0.122280 + 0.992496i $$0.539021\pi$$
$$938$$ 3.84955 0.125692
$$939$$ 0 0
$$940$$ 19.4763 0.635246
$$941$$ 21.2360 0.692274 0.346137 0.938184i $$-0.387493\pi$$
0.346137 + 0.938184i $$0.387493\pi$$
$$942$$ 0 0
$$943$$ 29.4010 0.957430
$$944$$ −40.4025 −1.31499
$$945$$ 0 0
$$946$$ 1.79877 0.0584830
$$947$$ 15.4763 0.502911 0.251456 0.967869i $$-0.419091\pi$$
0.251456 + 0.967869i $$0.419091\pi$$
$$948$$ 0 0
$$949$$ −22.9234 −0.744124
$$950$$ −0.836381 −0.0271358
$$951$$ 0 0
$$952$$ −11.7772 −0.381700
$$953$$ 32.0508 1.03823 0.519113 0.854705i $$-0.326262\pi$$
0.519113 + 0.854705i $$0.326262\pi$$
$$954$$ 0 0
$$955$$ −16.6253 −0.537982
$$956$$ 52.1016 1.68509
$$957$$ 0 0
$$958$$ 1.80351 0.0582687
$$959$$ −46.3996 −1.49832
$$960$$ 0 0
$$961$$ 67.5012 2.17746
$$962$$ −1.14903 −0.0370462
$$963$$ 0 0
$$964$$ −56.0263 −1.80449
$$965$$ 16.3634 0.526758
$$966$$ 0 0
$$967$$ −17.3766 −0.558794 −0.279397 0.960176i $$-0.590135\pi$$
−0.279397 + 0.960176i $$0.590135\pi$$
$$968$$ −0.768452 −0.0246990
$$969$$ 0 0
$$970$$ −0.0145884 −0.000468407 0
$$971$$ −36.2031 −1.16181 −0.580907 0.813970i $$-0.697302\pi$$
−0.580907 + 0.813970i $$0.697302\pi$$
$$972$$ 0 0
$$973$$ 45.6042 1.46200
$$974$$ −6.88015 −0.220454
$$975$$ 0 0
$$976$$ −32.8510 −1.05153
$$977$$ 28.1476 0.900522 0.450261 0.892897i $$-0.351331\pi$$
0.450261 + 0.892897i $$0.351331\pi$$
$$978$$ 0 0
$$979$$ −2.77575 −0.0887132
$$980$$ 8.28963 0.264802
$$981$$ 0 0
$$982$$ −4.80492 −0.153331
$$983$$ −7.07381 −0.225619 −0.112810 0.993617i $$-0.535985\pi$$
−0.112810 + 0.993617i $$0.535985\pi$$
$$984$$ 0 0
$$985$$ −20.4241 −0.650765
$$986$$ 3.20456 0.102054
$$987$$ 0 0
$$988$$ 25.0710 0.797614
$$989$$ −62.1476 −1.97618
$$990$$ 0 0
$$991$$ 44.4260 1.41124 0.705619 0.708592i $$-0.250669\pi$$
0.705619 + 0.708592i $$0.250669\pi$$
$$992$$ 22.5209 0.715039
$$993$$ 0 0
$$994$$ −6.44851 −0.204534
$$995$$ 8.62530 0.273440
$$996$$ 0 0
$$997$$ −28.4847 −0.902120 −0.451060 0.892494i $$-0.648954\pi$$
−0.451060 + 0.892494i $$0.648954\pi$$
$$998$$ 2.74940 0.0870307
$$999$$ 0 0
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 495.2.a.e.1.2 3
3.2 odd 2 165.2.a.c.1.2 3
4.3 odd 2 7920.2.a.cj.1.1 3
5.2 odd 4 2475.2.c.r.199.4 6
5.3 odd 4 2475.2.c.r.199.3 6
5.4 even 2 2475.2.a.bb.1.2 3
11.10 odd 2 5445.2.a.z.1.2 3
12.11 even 2 2640.2.a.be.1.1 3
15.2 even 4 825.2.c.g.199.3 6
15.8 even 4 825.2.c.g.199.4 6
15.14 odd 2 825.2.a.k.1.2 3
21.20 even 2 8085.2.a.bk.1.2 3
33.32 even 2 1815.2.a.m.1.2 3
165.164 even 2 9075.2.a.cf.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 3.2 odd 2
495.2.a.e.1.2 3 1.1 even 1 trivial
825.2.a.k.1.2 3 15.14 odd 2
825.2.c.g.199.3 6 15.2 even 4
825.2.c.g.199.4 6 15.8 even 4
1815.2.a.m.1.2 3 33.32 even 2
2475.2.a.bb.1.2 3 5.4 even 2
2475.2.c.r.199.3 6 5.3 odd 4
2475.2.c.r.199.4 6 5.2 odd 4
2640.2.a.be.1.1 3 12.11 even 2
5445.2.a.z.1.2 3 11.10 odd 2
7920.2.a.cj.1.1 3 4.3 odd 2
8085.2.a.bk.1.2 3 21.20 even 2
9075.2.a.cf.1.2 3 165.164 even 2