# Properties

 Label 8085.2.a.bk.1.2 Level $8085$ Weight $2$ Character 8085.1 Self dual yes Analytic conductor $64.559$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8085.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.5590500342$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 8085.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -1.00000 q^{5} +0.193937 q^{6} +0.768452 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -1.00000 q^{5} +0.193937 q^{6} +0.768452 q^{8} +1.00000 q^{9} +0.193937 q^{10} +1.00000 q^{11} +1.96239 q^{12} -2.96239 q^{13} +1.00000 q^{15} +3.77575 q^{16} +4.57452 q^{17} -0.193937 q^{18} +4.31265 q^{19} +1.96239 q^{20} -0.193937 q^{22} -6.70052 q^{23} -0.768452 q^{24} +1.00000 q^{25} +0.574515 q^{26} -1.00000 q^{27} -3.61213 q^{29} -0.193937 q^{30} -9.92478 q^{31} -2.26916 q^{32} -1.00000 q^{33} -0.887166 q^{34} -1.96239 q^{36} -2.00000 q^{37} -0.836381 q^{38} +2.96239 q^{39} -0.768452 q^{40} +4.38787 q^{41} -9.27504 q^{43} -1.96239 q^{44} -1.00000 q^{45} +1.29948 q^{46} +9.92478 q^{47} -3.77575 q^{48} -0.193937 q^{50} -4.57452 q^{51} +5.81336 q^{52} +4.70052 q^{53} +0.193937 q^{54} -1.00000 q^{55} -4.31265 q^{57} +0.700523 q^{58} -10.7005 q^{59} -1.96239 q^{60} +8.70052 q^{61} +1.92478 q^{62} -7.11142 q^{64} +2.96239 q^{65} +0.193937 q^{66} +5.92478 q^{67} -8.97698 q^{68} +6.70052 q^{69} +9.92478 q^{71} +0.768452 q^{72} +7.73813 q^{73} +0.387873 q^{74} -1.00000 q^{75} -8.46310 q^{76} -0.574515 q^{78} +11.5369 q^{79} -3.77575 q^{80} +1.00000 q^{81} -0.850969 q^{82} -10.8872 q^{83} -4.57452 q^{85} +1.79877 q^{86} +3.61213 q^{87} +0.768452 q^{88} +2.77575 q^{89} +0.193937 q^{90} +13.1490 q^{92} +9.92478 q^{93} -1.92478 q^{94} -4.31265 q^{95} +2.26916 q^{96} -0.0752228 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9 $$3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9 + q^10 + 3 * q^11 - 5 * q^12 + 2 * q^13 + 3 * q^15 + 13 * q^16 + 2 * q^17 - q^18 - 8 * q^19 - 5 * q^20 - q^22 + 9 * q^24 + 3 * q^25 - 10 * q^26 - 3 * q^27 - 10 * q^29 - q^30 - 8 * q^31 - 29 * q^32 - 3 * q^33 + 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 + 9 * q^40 + 14 * q^41 + 4 * q^43 + 5 * q^44 - 3 * q^45 + 24 * q^46 + 8 * q^47 - 13 * q^48 - q^50 - 2 * q^51 + 30 * q^52 - 6 * q^53 + q^54 - 3 * q^55 + 8 * q^57 - 18 * q^58 - 12 * q^59 + 5 * q^60 + 6 * q^61 - 16 * q^62 + 13 * q^64 - 2 * q^65 + q^66 - 4 * q^67 - 42 * q^68 + 8 * q^71 - 9 * q^72 + 14 * q^73 + 2 * q^74 - 3 * q^75 - 48 * q^76 + 10 * q^78 + 12 * q^79 - 13 * q^80 + 3 * q^81 - 26 * q^82 - 2 * q^85 - 8 * q^86 + 10 * q^87 - 9 * q^88 + 10 * q^89 + q^90 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + 29 * q^96 - 22 * q^97 + 3 * q^99

## 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 $$-0.521843\pi$$
−0.0685669 + 0.997647i $$0.521843\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.96239 −0.981194
$$5$$ −1.00000 −0.447214
$$6$$ 0.193937 0.0791743
$$7$$ 0 0
$$8$$ 0.768452 0.271689
$$9$$ 1.00000 0.333333
$$10$$ 0.193937 0.0613281
$$11$$ 1.00000 0.301511
$$12$$ 1.96239 0.566493
$$13$$ −2.96239 −0.821619 −0.410809 0.911721i $$-0.634754\pi$$
−0.410809 + 0.911721i $$0.634754\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 3.77575 0.943937
$$17$$ 4.57452 1.10948 0.554741 0.832023i $$-0.312817\pi$$
0.554741 + 0.832023i $$0.312817\pi$$
$$18$$ −0.193937 −0.0457113
$$19$$ 4.31265 0.989390 0.494695 0.869067i $$-0.335280\pi$$
0.494695 + 0.869067i $$0.335280\pi$$
$$20$$ 1.96239 0.438803
$$21$$ 0 0
$$22$$ −0.193937 −0.0413474
$$23$$ −6.70052 −1.39716 −0.698578 0.715534i $$-0.746183\pi$$
−0.698578 + 0.715534i $$0.746183\pi$$
$$24$$ −0.768452 −0.156860
$$25$$ 1.00000 0.200000
$$26$$ 0.574515 0.112672
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −3.61213 −0.670755 −0.335378 0.942084i $$-0.608864\pi$$
−0.335378 + 0.942084i $$0.608864\pi$$
$$30$$ −0.193937 −0.0354078
$$31$$ −9.92478 −1.78254 −0.891271 0.453470i $$-0.850186\pi$$
−0.891271 + 0.453470i $$0.850186\pi$$
$$32$$ −2.26916 −0.401134
$$33$$ −1.00000 −0.174078
$$34$$ −0.887166 −0.152148
$$35$$ 0 0
$$36$$ −1.96239 −0.327065
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −0.836381 −0.135679
$$39$$ 2.96239 0.474362
$$40$$ −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
$$45$$ −1.00000 −0.149071
$$46$$ 1.29948 0.191597
$$47$$ 9.92478 1.44768 0.723839 0.689969i $$-0.242376\pi$$
0.723839 + 0.689969i $$0.242376\pi$$
$$48$$ −3.77575 −0.544982
$$49$$ 0 0
$$50$$ −0.193937 −0.0274268
$$51$$ −4.57452 −0.640560
$$52$$ 5.81336 0.806168
$$53$$ 4.70052 0.645667 0.322833 0.946456i $$-0.395365\pi$$
0.322833 + 0.946456i $$0.395365\pi$$
$$54$$ 0.193937 0.0263914
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ −4.31265 −0.571224
$$58$$ 0.700523 0.0919832
$$59$$ −10.7005 −1.39309 −0.696545 0.717513i $$-0.745280\pi$$
−0.696545 + 0.717513i $$0.745280\pi$$
$$60$$ −1.96239 −0.253343
$$61$$ 8.70052 1.11399 0.556994 0.830517i $$-0.311955\pi$$
0.556994 + 0.830517i $$0.311955\pi$$
$$62$$ 1.92478 0.244447
$$63$$ 0 0
$$64$$ −7.11142 −0.888927
$$65$$ 2.96239 0.367439
$$66$$ 0.193937 0.0238719
$$67$$ 5.92478 0.723827 0.361913 0.932212i $$-0.382124\pi$$
0.361913 + 0.932212i $$0.382124\pi$$
$$68$$ −8.97698 −1.08862
$$69$$ 6.70052 0.806648
$$70$$ 0 0
$$71$$ 9.92478 1.17785 0.588927 0.808186i $$-0.299550\pi$$
0.588927 + 0.808186i $$0.299550\pi$$
$$72$$ 0.768452 0.0905629
$$73$$ 7.73813 0.905680 0.452840 0.891592i $$-0.350411\pi$$
0.452840 + 0.891592i $$0.350411\pi$$
$$74$$ 0.387873 0.0450893
$$75$$ −1.00000 −0.115470
$$76$$ −8.46310 −0.970784
$$77$$ 0 0
$$78$$ −0.574515 −0.0650511
$$79$$ 11.5369 1.29800 0.649002 0.760787i $$-0.275187\pi$$
0.649002 + 0.760787i $$0.275187\pi$$
$$80$$ −3.77575 −0.422141
$$81$$ 1.00000 0.111111
$$82$$ −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
$$87$$ 3.61213 0.387261
$$88$$ 0.768452 0.0819173
$$89$$ 2.77575 0.294229 0.147114 0.989120i $$-0.453001\pi$$
0.147114 + 0.989120i $$0.453001\pi$$
$$90$$ 0.193937 0.0204427
$$91$$ 0 0
$$92$$ 13.1490 1.37088
$$93$$ 9.92478 1.02915
$$94$$ −1.92478 −0.198526
$$95$$ −4.31265 −0.442469
$$96$$ 2.26916 0.231595
$$97$$ −0.0752228 −0.00763772 −0.00381886 0.999993i $$-0.501216\pi$$
−0.00381886 + 0.999993i $$0.501216\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −1.96239 −0.196239
$$101$$ 15.0884 1.50135 0.750676 0.660671i $$-0.229728\pi$$
0.750676 + 0.660671i $$0.229728\pi$$
$$102$$ 0.887166 0.0878425
$$103$$ 3.22425 0.317695 0.158848 0.987303i $$-0.449222\pi$$
0.158848 + 0.987303i $$0.449222\pi$$
$$104$$ −2.27645 −0.223225
$$105$$ 0 0
$$106$$ −0.911603 −0.0885427
$$107$$ −0.962389 −0.0930376 −0.0465188 0.998917i $$-0.514813\pi$$
−0.0465188 + 0.998917i $$0.514813\pi$$
$$108$$ 1.96239 0.188831
$$109$$ 11.4010 1.09202 0.546011 0.837778i $$-0.316146\pi$$
0.546011 + 0.837778i $$0.316146\pi$$
$$110$$ 0.193937 0.0184911
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0.836381 0.0783342
$$115$$ 6.70052 0.624827
$$116$$ 7.08840 0.658141
$$117$$ −2.96239 −0.273873
$$118$$ 2.07522 0.191040
$$119$$ 0 0
$$120$$ 0.768452 0.0701498
$$121$$ 1.00000 0.0909091
$$122$$ −1.68735 −0.152765
$$123$$ −4.38787 −0.395641
$$124$$ 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
$$129$$ 9.27504 0.816622
$$130$$ −0.574515 −0.0503883
$$131$$ 5.92478 0.517650 0.258825 0.965924i $$-0.416665\pi$$
0.258825 + 0.965924i $$0.416665\pi$$
$$132$$ 1.96239 0.170804
$$133$$ 0 0
$$134$$ −1.14903 −0.0992612
$$135$$ 1.00000 0.0860663
$$136$$ 3.51530 0.301434
$$137$$ 13.8496 1.18325 0.591624 0.806214i $$-0.298487\pi$$
0.591624 + 0.806214i $$0.298487\pi$$
$$138$$ −1.29948 −0.110619
$$139$$ −13.6121 −1.15457 −0.577283 0.816544i $$-0.695887\pi$$
−0.577283 + 0.816544i $$0.695887\pi$$
$$140$$ 0 0
$$141$$ −9.92478 −0.835817
$$142$$ −1.92478 −0.161524
$$143$$ −2.96239 −0.247727
$$144$$ 3.77575 0.314646
$$145$$ 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 $$-0.479948\pi$$
0.0629540 + 0.998016i $$0.479948\pi$$
$$150$$ 0.193937 0.0158349
$$151$$ −6.76116 −0.550215 −0.275108 0.961413i $$-0.588713\pi$$
−0.275108 + 0.961413i $$0.588713\pi$$
$$152$$ 3.31406 0.268806
$$153$$ 4.57452 0.369828
$$154$$ 0 0
$$155$$ 9.92478 0.797177
$$156$$ −5.81336 −0.465441
$$157$$ 5.47627 0.437054 0.218527 0.975831i $$-0.429875\pi$$
0.218527 + 0.975831i $$0.429875\pi$$
$$158$$ −2.23743 −0.178000
$$159$$ −4.70052 −0.372776
$$160$$ 2.26916 0.179393
$$161$$ 0 0
$$162$$ −0.193937 −0.0152371
$$163$$ 12.6253 0.988890 0.494445 0.869209i $$-0.335371\pi$$
0.494445 + 0.869209i $$0.335371\pi$$
$$164$$ −8.61071 −0.672384
$$165$$ 1.00000 0.0778499
$$166$$ 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
$$171$$ 4.31265 0.329797
$$172$$ 18.2012 1.38783
$$173$$ 8.57452 0.651908 0.325954 0.945386i $$-0.394314\pi$$
0.325954 + 0.945386i $$0.394314\pi$$
$$174$$ −0.700523 −0.0531065
$$175$$ 0 0
$$176$$ 3.77575 0.284608
$$177$$ 10.7005 0.804301
$$178$$ −0.538319 −0.0403487
$$179$$ 14.1768 1.05962 0.529812 0.848115i $$-0.322263\pi$$
0.529812 + 0.848115i $$0.322263\pi$$
$$180$$ 1.96239 0.146268
$$181$$ 5.22425 0.388316 0.194158 0.980970i $$-0.437803\pi$$
0.194158 + 0.980970i $$0.437803\pi$$
$$182$$ 0 0
$$183$$ −8.70052 −0.643161
$$184$$ −5.14903 −0.379592
$$185$$ 2.00000 0.147043
$$186$$ −1.92478 −0.141132
$$187$$ 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 $$-0.705423\pi$$
−0.601482 + 0.798886i $$0.705423\pi$$
$$192$$ 7.11142 0.513222
$$193$$ −16.3634 −1.17787 −0.588933 0.808182i $$-0.700452\pi$$
−0.588933 + 0.808182i $$0.700452\pi$$
$$194$$ 0.0145884 0.00104739
$$195$$ −2.96239 −0.212141
$$196$$ 0 0
$$197$$ −20.4241 −1.45515 −0.727577 0.686026i $$-0.759354\pi$$
−0.727577 + 0.686026i $$0.759354\pi$$
$$198$$ −0.193937 −0.0137825
$$199$$ 8.62530 0.611431 0.305716 0.952123i $$-0.401104\pi$$
0.305716 + 0.952123i $$0.401104\pi$$
$$200$$ 0.768452 0.0543378
$$201$$ −5.92478 −0.417902
$$202$$ −2.92619 −0.205886
$$203$$ 0 0
$$204$$ 8.97698 0.628514
$$205$$ −4.38787 −0.306462
$$206$$ −0.625301 −0.0435668
$$207$$ −6.70052 −0.465719
$$208$$ −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
$$213$$ −9.92478 −0.680035
$$214$$ 0.186642 0.0127586
$$215$$ 9.27504 0.632552
$$216$$ −0.768452 −0.0522865
$$217$$ 0 0
$$218$$ −2.21108 −0.149753
$$219$$ −7.73813 −0.522895
$$220$$ 1.96239 0.132304
$$221$$ −13.5515 −0.911572
$$222$$ −0.387873 −0.0260323
$$223$$ 6.70052 0.448700 0.224350 0.974509i $$-0.427974\pi$$
0.224350 + 0.974509i $$0.427974\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 1.16362 0.0774028
$$227$$ −16.9624 −1.12583 −0.562917 0.826514i $$-0.690321\pi$$
−0.562917 + 0.826514i $$0.690321\pi$$
$$228$$ 8.46310 0.560482
$$229$$ −25.8496 −1.70819 −0.854093 0.520120i $$-0.825887\pi$$
−0.854093 + 0.520120i $$0.825887\pi$$
$$230$$ −1.29948 −0.0856849
$$231$$ 0 0
$$232$$ −2.77575 −0.182237
$$233$$ −19.2750 −1.26275 −0.631375 0.775478i $$-0.717509\pi$$
−0.631375 + 0.775478i $$0.717509\pi$$
$$234$$ 0.574515 0.0375573
$$235$$ −9.92478 −0.647421
$$236$$ 20.9986 1.36689
$$237$$ −11.5369 −0.749402
$$238$$ 0 0
$$239$$ 26.5501 1.71738 0.858691 0.512494i $$-0.171278\pi$$
0.858691 + 0.512494i $$0.171278\pi$$
$$240$$ 3.77575 0.243723
$$241$$ −28.5501 −1.83907 −0.919536 0.393006i $$-0.871435\pi$$
−0.919536 + 0.393006i $$0.871435\pi$$
$$242$$ −0.193937 −0.0124667
$$243$$ −1.00000 −0.0641500
$$244$$ −17.0738 −1.09304
$$245$$ 0 0
$$246$$ 0.850969 0.0542558
$$247$$ −12.7757 −0.812901
$$248$$ −7.62672 −0.484297
$$249$$ 10.8872 0.689946
$$250$$ 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
$$255$$ 4.57452 0.286467
$$256$$ 13.0752 0.817201
$$257$$ −8.70052 −0.542724 −0.271362 0.962477i $$-0.587474\pi$$
−0.271362 + 0.962477i $$0.587474\pi$$
$$258$$ −1.79877 −0.111986
$$259$$ 0 0
$$260$$ −5.81336 −0.360529
$$261$$ −3.61213 −0.223585
$$262$$ −1.14903 −0.0709874
$$263$$ 12.2882 0.757724 0.378862 0.925453i $$-0.376316\pi$$
0.378862 + 0.925453i $$0.376316\pi$$
$$264$$ −0.768452 −0.0472950
$$265$$ −4.70052 −0.288751
$$266$$ 0 0
$$267$$ −2.77575 −0.169873
$$268$$ −11.6267 −0.710215
$$269$$ 5.84955 0.356654 0.178327 0.983971i $$-0.442932\pi$$
0.178327 + 0.983971i $$0.442932\pi$$
$$270$$ −0.193937 −0.0118026
$$271$$ 5.08840 0.309098 0.154549 0.987985i $$-0.450608\pi$$
0.154549 + 0.987985i $$0.450608\pi$$
$$272$$ 17.2722 1.04728
$$273$$ 0 0
$$274$$ −2.68594 −0.162263
$$275$$ 1.00000 0.0603023
$$276$$ −13.1490 −0.791479
$$277$$ 1.41090 0.0847725 0.0423863 0.999101i $$-0.486504\pi$$
0.0423863 + 0.999101i $$0.486504\pi$$
$$278$$ 2.63989 0.158330
$$279$$ −9.92478 −0.594181
$$280$$ 0 0
$$281$$ −4.38787 −0.261759 −0.130879 0.991398i $$-0.541780\pi$$
−0.130879 + 0.991398i $$0.541780\pi$$
$$282$$ 1.92478 0.114619
$$283$$ −26.5745 −1.57969 −0.789845 0.613306i $$-0.789839\pi$$
−0.789845 + 0.613306i $$0.789839\pi$$
$$284$$ −19.4763 −1.15570
$$285$$ 4.31265 0.255459
$$286$$ 0.574515 0.0339718
$$287$$ 0 0
$$288$$ −2.26916 −0.133711
$$289$$ 3.92619 0.230952
$$290$$ −0.700523 −0.0411362
$$291$$ 0.0752228 0.00440964
$$292$$ −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
$$297$$ −1.00000 −0.0580259
$$298$$ −0.298062 −0.0172663
$$299$$ 19.8496 1.14793
$$300$$ 1.96239 0.113299
$$301$$ 0 0
$$302$$ 1.31124 0.0754531
$$303$$ −15.0884 −0.866806
$$304$$ 16.2835 0.933921
$$305$$ −8.70052 −0.498191
$$306$$ −0.887166 −0.0507159
$$307$$ 16.6497 0.950251 0.475125 0.879918i $$-0.342403\pi$$
0.475125 + 0.879918i $$0.342403\pi$$
$$308$$ 0 0
$$309$$ −3.22425 −0.183421
$$310$$ −1.92478 −0.109320
$$311$$ −32.9986 −1.87118 −0.935589 0.353091i $$-0.885131\pi$$
−0.935589 + 0.353091i $$0.885131\pi$$
$$312$$ 2.27645 0.128879
$$313$$ −15.4010 −0.870519 −0.435259 0.900305i $$-0.643343\pi$$
−0.435259 + 0.900305i $$0.643343\pi$$
$$314$$ −1.06205 −0.0599349
$$315$$ 0 0
$$316$$ −22.6399 −1.27359
$$317$$ 2.15045 0.120781 0.0603905 0.998175i $$-0.480765\pi$$
0.0603905 + 0.998175i $$0.480765\pi$$
$$318$$ 0.911603 0.0511202
$$319$$ −3.61213 −0.202240
$$320$$ 7.11142 0.397540
$$321$$ 0.962389 0.0537153
$$322$$ 0 0
$$323$$ 19.7283 1.09771
$$324$$ −1.96239 −0.109022
$$325$$ −2.96239 −0.164324
$$326$$ −2.44851 −0.135610
$$327$$ −11.4010 −0.630479
$$328$$ 3.37187 0.186180
$$329$$ 0 0
$$330$$ −0.193937 −0.0106759
$$331$$ −14.5501 −0.799745 −0.399872 0.916571i $$-0.630946\pi$$
−0.399872 + 0.916571i $$0.630946\pi$$
$$332$$ 21.3649 1.17255
$$333$$ −2.00000 −0.109599
$$334$$ 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
$$339$$ 6.00000 0.325875
$$340$$ 8.97698 0.486845
$$341$$ −9.92478 −0.537457
$$342$$ −0.836381 −0.0452263
$$343$$ 0 0
$$344$$ −7.12742 −0.384285
$$345$$ −6.70052 −0.360744
$$346$$ −1.66291 −0.0893987
$$347$$ −0.962389 −0.0516637 −0.0258319 0.999666i $$-0.508223\pi$$
−0.0258319 + 0.999666i $$0.508223\pi$$
$$348$$ −7.08840 −0.379978
$$349$$ −20.7005 −1.10807 −0.554037 0.832492i $$-0.686913\pi$$
−0.554037 + 0.832492i $$0.686913\pi$$
$$350$$ 0 0
$$351$$ 2.96239 0.158121
$$352$$ −2.26916 −0.120947
$$353$$ −20.5501 −1.09377 −0.546885 0.837208i $$-0.684187\pi$$
−0.546885 + 0.837208i $$0.684187\pi$$
$$354$$ −2.07522 −0.110297
$$355$$ −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 $$-0.343165\pi$$
0.473017 + 0.881053i $$0.343165\pi$$
$$360$$ −0.768452 −0.0405010
$$361$$ −0.401047 −0.0211077
$$362$$ −1.01317 −0.0532512
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −7.73813 −0.405032
$$366$$ 1.68735 0.0881992
$$367$$ 29.6531 1.54788 0.773939 0.633261i $$-0.218284\pi$$
0.773939 + 0.633261i $$0.218284\pi$$
$$368$$ −25.2995 −1.31883
$$369$$ 4.38787 0.228424
$$370$$ −0.387873 −0.0201646
$$371$$ 0 0
$$372$$ −19.4763 −1.00980
$$373$$ −9.13918 −0.473209 −0.236604 0.971606i $$-0.576035\pi$$
−0.236604 + 0.971606i $$0.576035\pi$$
$$374$$ −0.887166 −0.0458743
$$375$$ 1.00000 0.0516398
$$376$$ 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
$$381$$ 14.5745 0.746675
$$382$$ 3.22425 0.164967
$$383$$ 34.9234 1.78450 0.892250 0.451541i $$-0.149126\pi$$
0.892250 + 0.451541i $$0.149126\pi$$
$$384$$ −5.91748 −0.301975
$$385$$ 0 0
$$386$$ 3.17347 0.161525
$$387$$ −9.27504 −0.471477
$$388$$ 0.147616 0.00749408
$$389$$ 2.77575 0.140736 0.0703680 0.997521i $$-0.477583\pi$$
0.0703680 + 0.997521i $$0.477583\pi$$
$$390$$ 0.574515 0.0290917
$$391$$ −30.6516 −1.55012
$$392$$ 0 0
$$393$$ −5.92478 −0.298865
$$394$$ 3.96097 0.199551
$$395$$ −11.5369 −0.580485
$$396$$ −1.96239 −0.0986137
$$397$$ 19.9248 0.999996 0.499998 0.866027i $$-0.333334\pi$$
0.499998 + 0.866027i $$0.333334\pi$$
$$398$$ −1.67276 −0.0838479
$$399$$ 0 0
$$400$$ 3.77575 0.188787
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 1.14903 0.0573085
$$403$$ 29.4010 1.46457
$$404$$ −29.6093 −1.47312
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ −3.51530 −0.174033
$$409$$ 13.0738 0.646458 0.323229 0.946321i $$-0.395232\pi$$
0.323229 + 0.946321i $$0.395232\pi$$
$$410$$ 0.850969 0.0420264
$$411$$ −13.8496 −0.683148
$$412$$ −6.32724 −0.311721
$$413$$ 0 0
$$414$$ 1.29948 0.0638658
$$415$$ 10.8872 0.534430
$$416$$ 6.72213 0.329580
$$417$$ 13.6121 0.666589
$$418$$ −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
$$423$$ 9.92478 0.482559
$$424$$ 3.61213 0.175420
$$425$$ 4.57452 0.221897
$$426$$ 1.92478 0.0932558
$$427$$ 0 0
$$428$$ 1.88858 0.0912880
$$429$$ 2.96239 0.143025
$$430$$ −1.79877 −0.0867444
$$431$$ −33.8759 −1.63174 −0.815872 0.578232i $$-0.803743\pi$$
−0.815872 + 0.578232i $$0.803743\pi$$
$$432$$ −3.77575 −0.181661
$$433$$ 9.47627 0.455400 0.227700 0.973731i $$-0.426879\pi$$
0.227700 + 0.973731i $$0.426879\pi$$
$$434$$ 0 0
$$435$$ −3.61213 −0.173188
$$436$$ −22.3733 −1.07149
$$437$$ −28.8970 −1.38233
$$438$$ 1.50071 0.0717066
$$439$$ 29.4617 1.40613 0.703065 0.711126i $$-0.251814\pi$$
0.703065 + 0.711126i $$0.251814\pi$$
$$440$$ −0.768452 −0.0366345
$$441$$ 0 0
$$442$$ 2.62813 0.125007
$$443$$ −19.0738 −0.906224 −0.453112 0.891454i $$-0.649686\pi$$
−0.453112 + 0.891454i $$0.649686\pi$$
$$444$$ −3.92478 −0.186262
$$445$$ −2.77575 −0.131583
$$446$$ −1.29948 −0.0615320
$$447$$ −1.53690 −0.0726931
$$448$$ 0 0
$$449$$ 35.8759 1.69309 0.846544 0.532318i $$-0.178679\pi$$
0.846544 + 0.532318i $$0.178679\pi$$
$$450$$ −0.193937 −0.00914226
$$451$$ 4.38787 0.206617
$$452$$ 11.7743 0.553818
$$453$$ 6.76116 0.317667
$$454$$ 3.28963 0.154390
$$455$$ 0 0
$$456$$ −3.31406 −0.155195
$$457$$ 5.28963 0.247438 0.123719 0.992317i $$-0.460518\pi$$
0.123719 + 0.992317i $$0.460518\pi$$
$$458$$ 5.01317 0.234250
$$459$$ −4.57452 −0.213520
$$460$$ −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
$$465$$ −9.92478 −0.460251
$$466$$ 3.73813 0.173166
$$467$$ −18.7005 −0.865357 −0.432679 0.901548i $$-0.642431\pi$$
−0.432679 + 0.901548i $$0.642431\pi$$
$$468$$ 5.81336 0.268723
$$469$$ 0 0
$$470$$ 1.92478 0.0887834
$$471$$ −5.47627 −0.252333
$$472$$ −8.22284 −0.378487
$$473$$ −9.27504 −0.426467
$$474$$ 2.23743 0.102768
$$475$$ 4.31265 0.197878
$$476$$ 0 0
$$477$$ 4.70052 0.215222
$$478$$ −5.14903 −0.235511
$$479$$ 9.29948 0.424904 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$480$$ −2.26916 −0.103572
$$481$$ 5.92478 0.270147
$$482$$ 5.53690 0.252199
$$483$$ 0 0
$$484$$ −1.96239 −0.0891995
$$485$$ 0.0752228 0.00341569
$$486$$ 0.193937 0.00879714
$$487$$ −35.4763 −1.60758 −0.803792 0.594911i $$-0.797187\pi$$
−0.803792 + 0.594911i $$0.797187\pi$$
$$488$$ 6.68594 0.302658
$$489$$ −12.6253 −0.570936
$$490$$ 0 0
$$491$$ 24.7757 1.11811 0.559057 0.829129i $$-0.311163\pi$$
0.559057 + 0.829129i $$0.311163\pi$$
$$492$$ 8.61071 0.388201
$$493$$ −16.5237 −0.744191
$$494$$ 2.47768 0.111476
$$495$$ −1.00000 −0.0449467
$$496$$ −37.4734 −1.68261
$$497$$ 0 0
$$498$$ −2.11142 −0.0946150
$$499$$ 14.1768 0.634640 0.317320 0.948318i $$-0.397217\pi$$
0.317320 + 0.948318i $$0.397217\pi$$
$$500$$ 1.96239 0.0877607
$$501$$ 18.3634 0.820418
$$502$$ 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
$$507$$ 4.22425 0.187606
$$508$$ 28.6009 1.26896
$$509$$ −1.10299 −0.0488890 −0.0244445 0.999701i $$-0.507782\pi$$
−0.0244445 + 0.999701i $$0.507782\pi$$
$$510$$ −0.887166 −0.0392844
$$511$$ 0 0
$$512$$ −14.3707 −0.635103
$$513$$ −4.31265 −0.190408
$$514$$ 1.68735 0.0744258
$$515$$ −3.22425 −0.142078
$$516$$ −18.2012 −0.801265
$$517$$ 9.92478 0.436491
$$518$$ 0 0
$$519$$ −8.57452 −0.376379
$$520$$ 2.27645 0.0998291
$$521$$ 12.4485 0.545379 0.272690 0.962102i $$-0.412087\pi$$
0.272690 + 0.962102i $$0.412087\pi$$
$$522$$ 0.700523 0.0306611
$$523$$ −30.0508 −1.31403 −0.657015 0.753878i $$-0.728181\pi$$
−0.657015 + 0.753878i $$0.728181\pi$$
$$524$$ −11.6267 −0.507915
$$525$$ 0 0
$$526$$ −2.38313 −0.103910
$$527$$ −45.4010 −1.97770
$$528$$ −3.77575 −0.164318
$$529$$ 21.8970 0.952044
$$530$$ 0.911603 0.0395975
$$531$$ −10.7005 −0.464363
$$532$$ 0 0
$$533$$ −12.9986 −0.563031
$$534$$ 0.538319 0.0232953
$$535$$ 0.962389 0.0416077
$$536$$ 4.55291 0.196656
$$537$$ −14.1768 −0.611774
$$538$$ −1.13444 −0.0489093
$$539$$ 0 0
$$540$$ −1.96239 −0.0844478
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −0.986826 −0.0423878
$$543$$ −5.22425 −0.224194
$$544$$ −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
$$549$$ 8.70052 0.371329
$$550$$ −0.193937 −0.00826948
$$551$$ −15.5778 −0.663638
$$552$$ 5.14903 0.219157
$$553$$ 0 0
$$554$$ −0.273624 −0.0116252
$$555$$ −2.00000 −0.0848953
$$556$$ 26.7123 1.13285
$$557$$ −11.7988 −0.499930 −0.249965 0.968255i $$-0.580419\pi$$
−0.249965 + 0.968255i $$0.580419\pi$$
$$558$$ 1.92478 0.0814823
$$559$$ 27.4763 1.16212
$$560$$ 0 0
$$561$$ −4.57452 −0.193136
$$562$$ 0.850969 0.0358960
$$563$$ −30.4847 −1.28478 −0.642389 0.766379i $$-0.722056\pi$$
−0.642389 + 0.766379i $$0.722056\pi$$
$$564$$ 19.4763 0.820099
$$565$$ 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 $$-0.692206\pi$$
−0.567802 + 0.823165i $$0.692206\pi$$
$$570$$ −0.836381 −0.0350321
$$571$$ 7.28489 0.304863 0.152432 0.988314i $$-0.451290\pi$$
0.152432 + 0.988314i $$0.451290\pi$$
$$572$$ 5.81336 0.243069
$$573$$ 16.6253 0.694532
$$574$$ 0 0
$$575$$ −6.70052 −0.279431
$$576$$ −7.11142 −0.296309
$$577$$ 31.6239 1.31652 0.658260 0.752791i $$-0.271293\pi$$
0.658260 + 0.752791i $$0.271293\pi$$
$$578$$ −0.761432 −0.0316714
$$579$$ 16.3634 0.680041
$$580$$ −7.08840 −0.294330
$$581$$ 0 0
$$582$$ −0.0145884 −0.000604711 0
$$583$$ 4.70052 0.194676
$$584$$ 5.94639 0.246063
$$585$$ 2.96239 0.122480
$$586$$ −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
$$591$$ 20.4241 0.840134
$$592$$ −7.55149 −0.310364
$$593$$ −34.4993 −1.41672 −0.708358 0.705853i $$-0.750564\pi$$
−0.708358 + 0.705853i $$0.750564\pi$$
$$594$$ 0.193937 0.00795731
$$595$$ 0 0
$$596$$ −3.01600 −0.123540
$$597$$ −8.62530 −0.353010
$$598$$ −3.84955 −0.157420
$$599$$ −14.4485 −0.590350 −0.295175 0.955443i $$-0.595378\pi$$
−0.295175 + 0.955443i $$0.595378\pi$$
$$600$$ −0.768452 −0.0313719
$$601$$ 15.9248 0.649585 0.324793 0.945785i $$-0.394705\pi$$
0.324793 + 0.945785i $$0.394705\pi$$
$$602$$ 0 0
$$603$$ 5.92478 0.241276
$$604$$ 13.2680 0.539868
$$605$$ −1.00000 −0.0406558
$$606$$ 2.92619 0.118868
$$607$$ 14.5745 0.591561 0.295781 0.955256i $$-0.404420\pi$$
0.295781 + 0.955256i $$0.404420\pi$$
$$608$$ −9.78609 −0.396878
$$609$$ 0 0
$$610$$ 1.68735 0.0683188
$$611$$ −29.4010 −1.18944
$$612$$ −8.97698 −0.362873
$$613$$ 16.4123 0.662887 0.331443 0.943475i $$-0.392464\pi$$
0.331443 + 0.943475i $$0.392464\pi$$
$$614$$ −3.22899 −0.130312
$$615$$ 4.38787 0.176936
$$616$$ 0 0
$$617$$ −17.8496 −0.718596 −0.359298 0.933223i $$-0.616984\pi$$
−0.359298 + 0.933223i $$0.616984\pi$$
$$618$$ 0.625301 0.0251533
$$619$$ 0.402462 0.0161763 0.00808815 0.999967i $$-0.497425\pi$$
0.00808815 + 0.999967i $$0.497425\pi$$
$$620$$ −19.4763 −0.782186
$$621$$ 6.70052 0.268883
$$622$$ 6.39963 0.256602
$$623$$ 0 0
$$624$$ 11.1852 0.447767
$$625$$ 1.00000 0.0400000
$$626$$ 2.98683 0.119378
$$627$$ −4.31265 −0.172231
$$628$$ −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
$$633$$ −9.08840 −0.361231
$$634$$ −0.417050 −0.0165632
$$635$$ 14.5745 0.578372
$$636$$ 9.22425 0.365765
$$637$$ 0 0
$$638$$ 0.700523 0.0277340
$$639$$ 9.92478 0.392618
$$640$$ −5.91748 −0.233909
$$641$$ −28.0263 −1.10697 −0.553487 0.832858i $$-0.686703\pi$$
−0.553487 + 0.832858i $$0.686703\pi$$
$$642$$ −0.186642 −0.00736619
$$643$$ 4.62530 0.182404 0.0912020 0.995832i $$-0.470929\pi$$
0.0912020 + 0.995832i $$0.470929\pi$$
$$644$$ 0 0
$$645$$ −9.27504 −0.365204
$$646$$ −3.82604 −0.150533
$$647$$ −23.5778 −0.926941 −0.463470 0.886112i $$-0.653396\pi$$
−0.463470 + 0.886112i $$0.653396\pi$$
$$648$$ 0.768452 0.0301876
$$649$$ −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 $$-0.514031\pi$$
−0.0440641 + 0.999029i $$0.514031\pi$$
$$654$$ 2.21108 0.0864601
$$655$$ −5.92478 −0.231500
$$656$$ 16.5675 0.646852
$$657$$ 7.73813 0.301893
$$658$$ 0 0
$$659$$ −41.4010 −1.61276 −0.806378 0.591401i $$-0.798575\pi$$
−0.806378 + 0.591401i $$0.798575\pi$$
$$660$$ −1.96239 −0.0763859
$$661$$ −3.40105 −0.132285 −0.0661427 0.997810i $$-0.521069\pi$$
−0.0661427 + 0.997810i $$0.521069\pi$$
$$662$$ 2.82179 0.109672
$$663$$ 13.5515 0.526296
$$664$$ −8.36626 −0.324674
$$665$$ 0 0
$$666$$ 0.387873 0.0150298
$$667$$ 24.2031 0.937149
$$668$$ 36.0362 1.39428
$$669$$ −6.70052 −0.259057
$$670$$ 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
$$675$$ −1.00000 −0.0384900
$$676$$ 8.28963 0.318832
$$677$$ −18.9018 −0.726453 −0.363227 0.931701i $$-0.618325\pi$$
−0.363227 + 0.931701i $$0.618325\pi$$
$$678$$ −1.16362 −0.0446885
$$679$$ 0 0
$$680$$ −3.51530 −0.134805
$$681$$ 16.9624 0.650000
$$682$$ 1.92478 0.0737035
$$683$$ −20.8773 −0.798848 −0.399424 0.916766i $$-0.630790\pi$$
−0.399424 + 0.916766i $$0.630790\pi$$
$$684$$ −8.46310 −0.323595
$$685$$ −13.8496 −0.529164
$$686$$ 0 0
$$687$$ 25.8496 0.986222
$$688$$ −35.0202 −1.33513
$$689$$ −13.9248 −0.530492
$$690$$ 1.29948 0.0494702
$$691$$ 2.44851 0.0931456 0.0465728 0.998915i $$-0.485170\pi$$
0.0465728 + 0.998915i $$0.485170\pi$$
$$692$$ −16.8265 −0.639649
$$693$$ 0 0
$$694$$ 0.186642 0.00708485
$$695$$ 13.6121 0.516337
$$696$$ 2.77575 0.105214
$$697$$ 20.0724 0.760296
$$698$$ 4.01459 0.151954
$$699$$ 19.2750 0.729049
$$700$$ 0 0
$$701$$ −2.98683 −0.112811 −0.0564054 0.998408i $$-0.517964\pi$$
−0.0564054 + 0.998408i $$0.517964\pi$$
$$702$$ −0.574515 −0.0216837
$$703$$ −8.62530 −0.325309
$$704$$ −7.11142 −0.268022
$$705$$ 9.92478 0.373789
$$706$$ 3.98541 0.149993
$$707$$ 0 0
$$708$$ −20.9986 −0.789175
$$709$$ 24.1768 0.907979 0.453989 0.891007i $$-0.350000\pi$$
0.453989 + 0.891007i $$0.350000\pi$$
$$710$$ 1.92478 0.0722356
$$711$$ 11.5369 0.432668
$$712$$ 2.13303 0.0799386
$$713$$ 66.5012 2.49049
$$714$$ 0 0
$$715$$ 2.96239 0.110787
$$716$$ −27.8204 −1.03970
$$717$$ −26.5501 −0.991531
$$718$$ −3.47627 −0.129733
$$719$$ 30.0263 1.11979 0.559897 0.828562i $$-0.310841\pi$$
0.559897 + 0.828562i $$0.310841\pi$$
$$720$$ −3.77575 −0.140714
$$721$$ 0 0
$$722$$ 0.0777777 0.00289459
$$723$$ 28.5501 1.06179
$$724$$ −10.2520 −0.381013
$$725$$ −3.61213 −0.134151
$$726$$ 0.193937 0.00719766
$$727$$ −14.9525 −0.554559 −0.277279 0.960789i $$-0.589433\pi$$
−0.277279 + 0.960789i $$0.589433\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 1.50071 0.0555437
$$731$$ −42.4288 −1.56929
$$732$$ 17.0738 0.631066
$$733$$ 19.1128 0.705949 0.352974 0.935633i $$-0.385170\pi$$
0.352974 + 0.935633i $$0.385170\pi$$
$$734$$ −5.75081 −0.212266
$$735$$ 0 0
$$736$$ 15.2046 0.560447
$$737$$ 5.92478 0.218242
$$738$$ −0.850969 −0.0313246
$$739$$ −3.31406 −0.121910 −0.0609549 0.998141i $$-0.519415\pi$$
−0.0609549 + 0.998141i $$0.519415\pi$$
$$740$$ −3.92478 −0.144278
$$741$$ 12.7757 0.469329
$$742$$ 0 0
$$743$$ −34.9887 −1.28361 −0.641806 0.766867i $$-0.721815\pi$$
−0.641806 + 0.766867i $$0.721815\pi$$
$$744$$ 7.62672 0.279609
$$745$$ −1.53690 −0.0563078
$$746$$ 1.77242 0.0648930
$$747$$ −10.8872 −0.398341
$$748$$ −8.97698 −0.328231
$$749$$ 0 0
$$750$$ −0.193937 −0.00708156
$$751$$ −26.9234 −0.982447 −0.491224 0.871033i $$-0.663450\pi$$
−0.491224 + 0.871033i $$0.663450\pi$$
$$752$$ 37.4734 1.36652
$$753$$ 29.9248 1.09052
$$754$$ −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
$$759$$ 6.70052 0.243214
$$760$$ −3.31406 −0.120214
$$761$$ 30.9380 1.12150 0.560750 0.827985i $$-0.310513\pi$$
0.560750 + 0.827985i $$0.310513\pi$$
$$762$$ −2.82653 −0.102394
$$763$$ 0 0
$$764$$ 32.6253 1.18034
$$765$$ −4.57452 −0.165392
$$766$$ −6.77292 −0.244715
$$767$$ 31.6991 1.14459
$$768$$ −13.0752 −0.471811
$$769$$ −9.32582 −0.336298 −0.168149 0.985762i $$-0.553779\pi$$
−0.168149 + 0.985762i $$0.553779\pi$$
$$770$$ 0 0
$$771$$ 8.70052 0.313342
$$772$$ 32.1114 1.15572
$$773$$ −44.7005 −1.60777 −0.803883 0.594787i $$-0.797236\pi$$
−0.803883 + 0.594787i $$0.797236\pi$$
$$774$$ 1.79877 0.0646554
$$775$$ −9.92478 −0.356509
$$776$$ −0.0578051 −0.00207508
$$777$$ 0 0
$$778$$ −0.538319 −0.0192997
$$779$$ 18.9234 0.678000
$$780$$ 5.81336 0.208152
$$781$$ 9.92478 0.355136
$$782$$ 5.94448 0.212574
$$783$$ 3.61213 0.129087
$$784$$ 0 0
$$785$$ −5.47627 −0.195456
$$786$$ 1.14903 0.0409846
$$787$$ −21.6775 −0.772719 −0.386360 0.922348i $$-0.626268\pi$$
−0.386360 + 0.922348i $$0.626268\pi$$
$$788$$ 40.0800 1.42779
$$789$$ −12.2882 −0.437472
$$790$$ 2.23743 0.0796041
$$791$$ 0 0
$$792$$ 0.768452 0.0273058
$$793$$ −25.7743 −0.915273
$$794$$ −3.86414 −0.137133
$$795$$ 4.70052 0.166710
$$796$$ −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
$$801$$ 2.77575 0.0980762
$$802$$ −0.387873 −0.0136963
$$803$$ 7.73813 0.273073
$$804$$ 11.6267 0.410043
$$805$$ 0 0
$$806$$ −5.70194 −0.200842
$$807$$ −5.84955 −0.205914
$$808$$ 11.5947 0.407900
$$809$$ −23.6121 −0.830158 −0.415079 0.909785i $$-0.636246\pi$$
−0.415079 + 0.909785i $$0.636246\pi$$
$$810$$ 0.193937 0.00681424
$$811$$ 26.0870 0.916038 0.458019 0.888942i $$-0.348559\pi$$
0.458019 + 0.888942i $$0.348559\pi$$
$$812$$ 0 0
$$813$$ −5.08840 −0.178458
$$814$$ 0.387873 0.0135949
$$815$$ −12.6253 −0.442245
$$816$$ −17.2722 −0.604648
$$817$$ −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 $$-0.898444\pi$$
−0.949535 + 0.313662i $$0.898444\pi$$
$$822$$ 2.68594 0.0936827
$$823$$ 0.121269 0.00422716 0.00211358 0.999998i $$-0.499327\pi$$
0.00211358 + 0.999998i $$0.499327\pi$$
$$824$$ 2.47768 0.0863142
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ −18.2130 −0.633328 −0.316664 0.948538i $$-0.602563\pi$$
−0.316664 + 0.948538i $$0.602563\pi$$
$$828$$ 13.1490 0.456960
$$829$$ −13.0738 −0.454072 −0.227036 0.973886i $$-0.572904\pi$$
−0.227036 + 0.973886i $$0.572904\pi$$
$$830$$ −2.11142 −0.0732884
$$831$$ −1.41090 −0.0489434
$$832$$ 21.0668 0.730359
$$833$$ 0 0
$$834$$ −2.63989 −0.0914119
$$835$$ 18.3634 0.635493
$$836$$ −8.46310 −0.292702
$$837$$ 9.92478 0.343050
$$838$$ 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
$$843$$ 4.38787 0.151126
$$844$$ −17.8350 −0.613905
$$845$$ 4.22425 0.145319
$$846$$ −1.92478 −0.0661752
$$847$$ 0 0
$$848$$ 17.7480 0.609468
$$849$$ 26.5745 0.912035
$$850$$ −0.887166 −0.0304295
$$851$$ 13.4010 0.459382
$$852$$ 19.4763 0.667246
$$853$$ −40.6155 −1.39065 −0.695323 0.718697i $$-0.744739\pi$$
−0.695323 + 0.718697i $$0.744739\pi$$
$$854$$ 0 0
$$855$$ −4.31265 −0.147490
$$856$$ −0.739549 −0.0252773
$$857$$ −20.1721 −0.689064 −0.344532 0.938775i $$-0.611962\pi$$
−0.344532 + 0.938775i $$0.611962\pi$$
$$858$$ −0.574515 −0.0196136
$$859$$ −21.8035 −0.743926 −0.371963 0.928248i $$-0.621315\pi$$
−0.371963 + 0.928248i $$0.621315\pi$$
$$860$$ −18.2012 −0.620657
$$861$$ 0 0
$$862$$ 6.56978 0.223767
$$863$$ 35.4274 1.20596 0.602981 0.797755i $$-0.293979\pi$$
0.602981 + 0.797755i $$0.293979\pi$$
$$864$$ 2.26916 0.0771984
$$865$$ −8.57452 −0.291542
$$866$$ −1.83780 −0.0624508
$$867$$ −3.92619 −0.133340
$$868$$ 0 0
$$869$$ 11.5369 0.391363
$$870$$ 0.700523 0.0237500
$$871$$ −17.5515 −0.594710
$$872$$ 8.76116 0.296690
$$873$$ −0.0752228 −0.00254591
$$874$$ 5.60419 0.189564
$$875$$ 0 0
$$876$$ 15.1852 0.513061
$$877$$ −14.0362 −0.473969 −0.236984 0.971513i $$-0.576159\pi$$
−0.236984 + 0.971513i $$0.576159\pi$$
$$878$$ −5.71370 −0.192828
$$879$$ −3.42548 −0.115539
$$880$$ −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
$$885$$ −10.7005 −0.359694
$$886$$ 3.69911 0.124274
$$887$$ −6.93604 −0.232889 −0.116445 0.993197i $$-0.537150\pi$$
−0.116445 + 0.993197i $$0.537150\pi$$
$$888$$ 1.53690 0.0515751
$$889$$ 0 0
$$890$$ 0.538319 0.0180445
$$891$$ 1.00000 0.0335013
$$892$$ −13.1490 −0.440262
$$893$$ 42.8021 1.43232
$$894$$ 0.298062 0.00996868
$$895$$ −14.1768 −0.473878
$$896$$ 0 0
$$897$$ −19.8496 −0.662757
$$898$$ −6.95765 −0.232180
$$899$$ 35.8496 1.19565
$$900$$ −1.96239 −0.0654130
$$901$$ 21.5026 0.716356
$$902$$ −0.850969 −0.0283342
$$903$$ 0 0
$$904$$ −4.61071 −0.153350
$$905$$ −5.22425 −0.173660
$$906$$ −1.31124 −0.0435629
$$907$$ 53.2017 1.76653 0.883267 0.468870i $$-0.155339\pi$$
0.883267 + 0.468870i $$0.155339\pi$$
$$908$$ 33.2868 1.10466
$$909$$ 15.0884 0.500451
$$910$$ 0 0
$$911$$ 36.4749 1.20847 0.604233 0.796808i $$-0.293480\pi$$
0.604233 + 0.796808i $$0.293480\pi$$
$$912$$ −16.2835 −0.539200
$$913$$ −10.8872 −0.360313
$$914$$ −1.02585 −0.0339322
$$915$$ 8.70052 0.287630
$$916$$ 50.7269 1.67606
$$917$$ 0 0
$$918$$ 0.887166 0.0292808
$$919$$ 9.73340 0.321075 0.160538 0.987030i $$-0.448677\pi$$
0.160538 + 0.987030i $$0.448677\pi$$
$$920$$ 5.14903 0.169759
$$921$$ −16.6497 −0.548628
$$922$$ 7.04746 0.232096
$$923$$ −29.4010 −0.967747
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −2.04605 −0.0672372
$$927$$ 3.22425 0.105898
$$928$$ 8.19649 0.269063
$$929$$ 24.1768 0.793215 0.396607 0.917988i $$-0.370187\pi$$
0.396607 + 0.917988i $$0.370187\pi$$
$$930$$ 1.92478 0.0631159
$$931$$ 0 0
$$932$$ 37.8251 1.23900
$$933$$ 32.9986 1.08033
$$934$$ 3.62672 0.118670
$$935$$ −4.57452 −0.149603
$$936$$ −2.27645 −0.0744082
$$937$$ 7.48612 0.244561 0.122280 0.992496i $$-0.460979\pi$$
0.122280 + 0.992496i $$0.460979\pi$$
$$938$$ 0 0
$$939$$ 15.4010 0.502594
$$940$$ 19.4763 0.635246
$$941$$ 21.2360 0.692274 0.346137 0.938184i $$-0.387493\pi$$
0.346137 + 0.938184i $$0.387493\pi$$
$$942$$ 1.06205 0.0346034
$$943$$ −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 $$-0.580909\pi$$
−0.251456 + 0.967869i $$0.580909\pi$$
$$948$$ 22.6399 0.735309
$$949$$ −22.9234 −0.744124
$$950$$ −0.836381 −0.0271358
$$951$$ −2.15045 −0.0697330
$$952$$ 0 0
$$953$$ −32.0508 −1.03823 −0.519113 0.854705i $$-0.673738\pi$$
−0.519113 + 0.854705i $$0.673738\pi$$
$$954$$ −0.911603 −0.0295142
$$955$$ 16.6253 0.537982
$$956$$ −52.1016 −1.68509
$$957$$ 3.61213 0.116763
$$958$$ −1.80351 −0.0582687
$$959$$ 0 0
$$960$$ −7.11142 −0.229520
$$961$$ 67.5012 2.17746
$$962$$ −1.14903 −0.0370462
$$963$$ −0.962389 −0.0310125
$$964$$ 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
$$969$$ −19.7283 −0.633764
$$970$$ −0.0145884 −0.000468407 0
$$971$$ −36.2031 −1.16181 −0.580907 0.813970i $$-0.697302\pi$$
−0.580907 + 0.813970i $$0.697302\pi$$
$$972$$ 1.96239 0.0629436
$$973$$ 0 0
$$974$$ 6.88015 0.220454
$$975$$ 2.96239 0.0948724
$$976$$ 32.8510 1.05153
$$977$$ −28.1476 −0.900522 −0.450261 0.892897i $$-0.648669\pi$$
−0.450261 + 0.892897i $$0.648669\pi$$
$$978$$ 2.44851 0.0782946
$$979$$ 2.77575 0.0887132
$$980$$ 0 0
$$981$$ 11.4010 0.364007
$$982$$ −4.80492 −0.153331
$$983$$ −7.07381 −0.225619 −0.112810 0.993617i $$-0.535985\pi$$
−0.112810 + 0.993617i $$0.535985\pi$$
$$984$$ −3.37187 −0.107491
$$985$$ 20.4241 0.650765
$$986$$ 3.20456 0.102054
$$987$$ 0 0
$$988$$ 25.0710 0.797614
$$989$$ 62.1476 1.97618
$$990$$ 0.193937 0.00616371
$$991$$ 44.4260 1.41124 0.705619 0.708592i $$-0.250669\pi$$
0.705619 + 0.708592i $$0.250669\pi$$
$$992$$ 22.5209 0.715039
$$993$$ 14.5501 0.461733
$$994$$ 0 0
$$995$$ −8.62530 −0.273440
$$996$$ −21.3649 −0.676971
$$997$$ 28.4847 0.902120 0.451060 0.892494i $$-0.351046\pi$$
0.451060 + 0.892494i $$0.351046\pi$$
$$998$$ −2.74940 −0.0870307
$$999$$ 2.00000 0.0632772
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 8085.2.a.bk.1.2 3
7.6 odd 2 165.2.a.c.1.2 3
21.20 even 2 495.2.a.e.1.2 3
28.27 even 2 2640.2.a.be.1.1 3
35.13 even 4 825.2.c.g.199.4 6
35.27 even 4 825.2.c.g.199.3 6
35.34 odd 2 825.2.a.k.1.2 3
77.76 even 2 1815.2.a.m.1.2 3
84.83 odd 2 7920.2.a.cj.1.1 3
105.62 odd 4 2475.2.c.r.199.4 6
105.83 odd 4 2475.2.c.r.199.3 6
105.104 even 2 2475.2.a.bb.1.2 3
231.230 odd 2 5445.2.a.z.1.2 3
385.384 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 7.6 odd 2
495.2.a.e.1.2 3 21.20 even 2
825.2.a.k.1.2 3 35.34 odd 2
825.2.c.g.199.3 6 35.27 even 4
825.2.c.g.199.4 6 35.13 even 4
1815.2.a.m.1.2 3 77.76 even 2
2475.2.a.bb.1.2 3 105.104 even 2
2475.2.c.r.199.3 6 105.83 odd 4
2475.2.c.r.199.4 6 105.62 odd 4
2640.2.a.be.1.1 3 28.27 even 2
5445.2.a.z.1.2 3 231.230 odd 2
7920.2.a.cj.1.1 3 84.83 odd 2
8085.2.a.bk.1.2 3 1.1 even 1 trivial
9075.2.a.cf.1.2 3 385.384 even 2