# Properties

 Label 825.2.a.m.1.3 Level $825$ Weight $2$ Character 825.1 Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.76156 q^{2} +1.00000 q^{3} +5.62620 q^{4} +2.76156 q^{6} -1.86464 q^{7} +10.0140 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.76156 q^{2} +1.00000 q^{3} +5.62620 q^{4} +2.76156 q^{6} -1.86464 q^{7} +10.0140 q^{8} +1.00000 q^{9} +1.00000 q^{11} +5.62620 q^{12} -4.62620 q^{13} -5.14931 q^{14} +16.4017 q^{16} -2.49084 q^{17} +2.76156 q^{18} -5.38776 q^{19} -1.86464 q^{21} +2.76156 q^{22} -7.14931 q^{23} +10.0140 q^{24} -12.7755 q^{26} +1.00000 q^{27} -10.4908 q^{28} -3.52311 q^{29} +8.62620 q^{31} +25.2663 q^{32} +1.00000 q^{33} -6.87859 q^{34} +5.62620 q^{36} +8.87859 q^{37} -14.8786 q^{38} -4.62620 q^{39} -0.761557 q^{41} -5.14931 q^{42} +7.40171 q^{43} +5.62620 q^{44} -19.7432 q^{46} +0.373802 q^{47} +16.4017 q^{48} -3.52311 q^{49} -2.49084 q^{51} -26.0279 q^{52} +5.45856 q^{53} +2.76156 q^{54} -18.6724 q^{56} -5.38776 q^{57} -9.72928 q^{58} +5.14931 q^{59} +4.42003 q^{61} +23.8217 q^{62} -1.86464 q^{63} +36.9711 q^{64} +2.76156 q^{66} -11.9431 q^{67} -14.0140 q^{68} -7.14931 q^{69} +11.6262 q^{71} +10.0140 q^{72} -6.77551 q^{73} +24.5187 q^{74} -30.3126 q^{76} -1.86464 q^{77} -12.7755 q^{78} -6.01395 q^{79} +1.00000 q^{81} -2.10308 q^{82} -14.5693 q^{83} -10.4908 q^{84} +20.4402 q^{86} -3.52311 q^{87} +10.0140 q^{88} +9.04623 q^{89} +8.62620 q^{91} -40.2234 q^{92} +8.62620 q^{93} +1.03228 q^{94} +25.2663 q^{96} -16.3169 q^{97} -9.72928 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 3 * q^3 + 8 * q^4 + 2 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 $$3 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} + 3 q^{11} + 8 q^{12} - 5 q^{13} + 6 q^{14} + 10 q^{16} + 4 q^{17} + 2 q^{18} - q^{19} - 3 q^{21} + 2 q^{22} + 6 q^{24} - 8 q^{26} + 3 q^{27} - 20 q^{28} + 2 q^{29} + 17 q^{31} + 34 q^{32} + 3 q^{33} + 6 q^{34} + 8 q^{36} - 18 q^{38} - 5 q^{39} + 4 q^{41} + 6 q^{42} - 17 q^{43} + 8 q^{44} - 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} + 4 q^{51} - 30 q^{52} + 6 q^{53} + 2 q^{54} - 22 q^{56} - q^{57} - 24 q^{58} - 6 q^{59} - 3 q^{61} + 16 q^{62} - 3 q^{63} + 34 q^{64} + 2 q^{66} - 7 q^{67} - 18 q^{68} + 26 q^{71} + 6 q^{72} + 10 q^{73} + 14 q^{74} - 24 q^{76} - 3 q^{77} - 8 q^{78} + 6 q^{79} + 3 q^{81} - 10 q^{82} - 6 q^{83} - 20 q^{84} + 28 q^{86} + 2 q^{87} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} + 2 q^{94} + 34 q^{96} - 29 q^{97} - 24 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 + 3 * q^3 + 8 * q^4 + 2 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 + 3 * q^11 + 8 * q^12 - 5 * q^13 + 6 * q^14 + 10 * q^16 + 4 * q^17 + 2 * q^18 - q^19 - 3 * q^21 + 2 * q^22 + 6 * q^24 - 8 * q^26 + 3 * q^27 - 20 * q^28 + 2 * q^29 + 17 * q^31 + 34 * q^32 + 3 * q^33 + 6 * q^34 + 8 * q^36 - 18 * q^38 - 5 * q^39 + 4 * q^41 + 6 * q^42 - 17 * q^43 + 8 * q^44 - 30 * q^46 + 10 * q^47 + 10 * q^48 + 2 * q^49 + 4 * q^51 - 30 * q^52 + 6 * q^53 + 2 * q^54 - 22 * q^56 - q^57 - 24 * q^58 - 6 * q^59 - 3 * q^61 + 16 * q^62 - 3 * q^63 + 34 * q^64 + 2 * q^66 - 7 * q^67 - 18 * q^68 + 26 * q^71 + 6 * q^72 + 10 * q^73 + 14 * q^74 - 24 * q^76 - 3 * q^77 - 8 * q^78 + 6 * q^79 + 3 * q^81 - 10 * q^82 - 6 * q^83 - 20 * q^84 + 28 * q^86 + 2 * q^87 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 26 * q^92 + 17 * q^93 + 2 * q^94 + 34 * q^96 - 29 * q^97 - 24 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.76156 1.95272 0.976358 0.216160i $$-0.0693534\pi$$
0.976358 + 0.216160i $$0.0693534\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 5.62620 2.81310
$$5$$ 0 0
$$6$$ 2.76156 1.12740
$$7$$ −1.86464 −0.704768 −0.352384 0.935855i $$-0.614629\pi$$
−0.352384 + 0.935855i $$0.614629\pi$$
$$8$$ 10.0140 3.54047
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 5.62620 1.62414
$$13$$ −4.62620 −1.28308 −0.641538 0.767091i $$-0.721703\pi$$
−0.641538 + 0.767091i $$0.721703\pi$$
$$14$$ −5.14931 −1.37621
$$15$$ 0 0
$$16$$ 16.4017 4.10043
$$17$$ −2.49084 −0.604117 −0.302059 0.953289i $$-0.597674\pi$$
−0.302059 + 0.953289i $$0.597674\pi$$
$$18$$ 2.76156 0.650905
$$19$$ −5.38776 −1.23604 −0.618018 0.786164i $$-0.712064\pi$$
−0.618018 + 0.786164i $$0.712064\pi$$
$$20$$ 0 0
$$21$$ −1.86464 −0.406898
$$22$$ 2.76156 0.588766
$$23$$ −7.14931 −1.49073 −0.745367 0.666654i $$-0.767726\pi$$
−0.745367 + 0.666654i $$0.767726\pi$$
$$24$$ 10.0140 2.04409
$$25$$ 0 0
$$26$$ −12.7755 −2.50548
$$27$$ 1.00000 0.192450
$$28$$ −10.4908 −1.98258
$$29$$ −3.52311 −0.654226 −0.327113 0.944985i $$-0.606076\pi$$
−0.327113 + 0.944985i $$0.606076\pi$$
$$30$$ 0 0
$$31$$ 8.62620 1.54931 0.774655 0.632384i $$-0.217923\pi$$
0.774655 + 0.632384i $$0.217923\pi$$
$$32$$ 25.2663 4.46650
$$33$$ 1.00000 0.174078
$$34$$ −6.87859 −1.17967
$$35$$ 0 0
$$36$$ 5.62620 0.937700
$$37$$ 8.87859 1.45963 0.729816 0.683644i $$-0.239606\pi$$
0.729816 + 0.683644i $$0.239606\pi$$
$$38$$ −14.8786 −2.41363
$$39$$ −4.62620 −0.740785
$$40$$ 0 0
$$41$$ −0.761557 −0.118935 −0.0594676 0.998230i $$-0.518940\pi$$
−0.0594676 + 0.998230i $$0.518940\pi$$
$$42$$ −5.14931 −0.794556
$$43$$ 7.40171 1.12875 0.564375 0.825519i $$-0.309117\pi$$
0.564375 + 0.825519i $$0.309117\pi$$
$$44$$ 5.62620 0.848181
$$45$$ 0 0
$$46$$ −19.7432 −2.91098
$$47$$ 0.373802 0.0545246 0.0272623 0.999628i $$-0.491321\pi$$
0.0272623 + 0.999628i $$0.491321\pi$$
$$48$$ 16.4017 2.36738
$$49$$ −3.52311 −0.503302
$$50$$ 0 0
$$51$$ −2.49084 −0.348787
$$52$$ −26.0279 −3.60942
$$53$$ 5.45856 0.749791 0.374896 0.927067i $$-0.377679\pi$$
0.374896 + 0.927067i $$0.377679\pi$$
$$54$$ 2.76156 0.375800
$$55$$ 0 0
$$56$$ −18.6724 −2.49521
$$57$$ −5.38776 −0.713626
$$58$$ −9.72928 −1.27752
$$59$$ 5.14931 0.670383 0.335192 0.942150i $$-0.391199\pi$$
0.335192 + 0.942150i $$0.391199\pi$$
$$60$$ 0 0
$$61$$ 4.42003 0.565927 0.282963 0.959131i $$-0.408682\pi$$
0.282963 + 0.959131i $$0.408682\pi$$
$$62$$ 23.8217 3.02536
$$63$$ −1.86464 −0.234923
$$64$$ 36.9711 4.62138
$$65$$ 0 0
$$66$$ 2.76156 0.339924
$$67$$ −11.9431 −1.45909 −0.729544 0.683934i $$-0.760268\pi$$
−0.729544 + 0.683934i $$0.760268\pi$$
$$68$$ −14.0140 −1.69944
$$69$$ −7.14931 −0.860676
$$70$$ 0 0
$$71$$ 11.6262 1.37978 0.689888 0.723916i $$-0.257660\pi$$
0.689888 + 0.723916i $$0.257660\pi$$
$$72$$ 10.0140 1.18016
$$73$$ −6.77551 −0.793014 −0.396507 0.918032i $$-0.629778\pi$$
−0.396507 + 0.918032i $$0.629778\pi$$
$$74$$ 24.5187 2.85025
$$75$$ 0 0
$$76$$ −30.3126 −3.47709
$$77$$ −1.86464 −0.212496
$$78$$ −12.7755 −1.44654
$$79$$ −6.01395 −0.676623 −0.338311 0.941034i $$-0.609856\pi$$
−0.338311 + 0.941034i $$0.609856\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.10308 −0.232247
$$83$$ −14.5693 −1.59919 −0.799597 0.600538i $$-0.794953\pi$$
−0.799597 + 0.600538i $$0.794953\pi$$
$$84$$ −10.4908 −1.14464
$$85$$ 0 0
$$86$$ 20.4402 2.20413
$$87$$ −3.52311 −0.377718
$$88$$ 10.0140 1.06749
$$89$$ 9.04623 0.958898 0.479449 0.877570i $$-0.340836\pi$$
0.479449 + 0.877570i $$0.340836\pi$$
$$90$$ 0 0
$$91$$ 8.62620 0.904271
$$92$$ −40.2234 −4.19358
$$93$$ 8.62620 0.894495
$$94$$ 1.03228 0.106471
$$95$$ 0 0
$$96$$ 25.2663 2.57874
$$97$$ −16.3169 −1.65673 −0.828367 0.560185i $$-0.810730\pi$$
−0.828367 + 0.560185i $$0.810730\pi$$
$$98$$ −9.72928 −0.982806
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −1.03228 −0.102715 −0.0513576 0.998680i $$-0.516355\pi$$
−0.0513576 + 0.998680i $$0.516355\pi$$
$$102$$ −6.87859 −0.681082
$$103$$ −3.04623 −0.300154 −0.150077 0.988674i $$-0.547952\pi$$
−0.150077 + 0.988674i $$0.547952\pi$$
$$104$$ −46.3265 −4.54269
$$105$$ 0 0
$$106$$ 15.0741 1.46413
$$107$$ 8.50479 0.822189 0.411095 0.911593i $$-0.365147\pi$$
0.411095 + 0.911593i $$0.365147\pi$$
$$108$$ 5.62620 0.541381
$$109$$ −6.14931 −0.588997 −0.294499 0.955652i $$-0.595153\pi$$
−0.294499 + 0.955652i $$0.595153\pi$$
$$110$$ 0 0
$$111$$ 8.87859 0.842719
$$112$$ −30.5833 −2.88985
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −14.8786 −1.39351
$$115$$ 0 0
$$116$$ −19.8217 −1.84040
$$117$$ −4.62620 −0.427692
$$118$$ 14.2201 1.30907
$$119$$ 4.64452 0.425762
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 12.2062 1.10509
$$123$$ −0.761557 −0.0686673
$$124$$ 48.5327 4.35837
$$125$$ 0 0
$$126$$ −5.14931 −0.458737
$$127$$ 9.26635 0.822256 0.411128 0.911578i $$-0.365135\pi$$
0.411128 + 0.911578i $$0.365135\pi$$
$$128$$ 51.5650 4.55774
$$129$$ 7.40171 0.651684
$$130$$ 0 0
$$131$$ 4.06455 0.355121 0.177561 0.984110i $$-0.443179\pi$$
0.177561 + 0.984110i $$0.443179\pi$$
$$132$$ 5.62620 0.489698
$$133$$ 10.0462 0.871119
$$134$$ −32.9817 −2.84918
$$135$$ 0 0
$$136$$ −24.9431 −2.13886
$$137$$ −5.93545 −0.507100 −0.253550 0.967322i $$-0.581598\pi$$
−0.253550 + 0.967322i $$0.581598\pi$$
$$138$$ −19.7432 −1.68066
$$139$$ 6.71096 0.569216 0.284608 0.958644i $$-0.408137\pi$$
0.284608 + 0.958644i $$0.408137\pi$$
$$140$$ 0 0
$$141$$ 0.373802 0.0314798
$$142$$ 32.1064 2.69431
$$143$$ −4.62620 −0.386862
$$144$$ 16.4017 1.36681
$$145$$ 0 0
$$146$$ −18.7110 −1.54853
$$147$$ −3.52311 −0.290582
$$148$$ 49.9527 4.10609
$$149$$ 5.74324 0.470504 0.235252 0.971934i $$-0.424408\pi$$
0.235252 + 0.971934i $$0.424408\pi$$
$$150$$ 0 0
$$151$$ 10.5616 0.859495 0.429747 0.902949i $$-0.358603\pi$$
0.429747 + 0.902949i $$0.358603\pi$$
$$152$$ −53.9527 −4.37614
$$153$$ −2.49084 −0.201372
$$154$$ −5.14931 −0.414943
$$155$$ 0 0
$$156$$ −26.0279 −2.08390
$$157$$ −3.10308 −0.247653 −0.123827 0.992304i $$-0.539517\pi$$
−0.123827 + 0.992304i $$0.539517\pi$$
$$158$$ −16.6079 −1.32125
$$159$$ 5.45856 0.432892
$$160$$ 0 0
$$161$$ 13.3309 1.05062
$$162$$ 2.76156 0.216968
$$163$$ −15.6724 −1.22756 −0.613780 0.789477i $$-0.710352\pi$$
−0.613780 + 0.789477i $$0.710352\pi$$
$$164$$ −4.28467 −0.334577
$$165$$ 0 0
$$166$$ −40.2341 −3.12277
$$167$$ 8.98168 0.695023 0.347512 0.937676i $$-0.387027\pi$$
0.347512 + 0.937676i $$0.387027\pi$$
$$168$$ −18.6724 −1.44061
$$169$$ 8.40171 0.646285
$$170$$ 0 0
$$171$$ −5.38776 −0.412012
$$172$$ 41.6435 3.17529
$$173$$ 11.5092 0.875025 0.437513 0.899212i $$-0.355860\pi$$
0.437513 + 0.899212i $$0.355860\pi$$
$$174$$ −9.72928 −0.737575
$$175$$ 0 0
$$176$$ 16.4017 1.23633
$$177$$ 5.14931 0.387046
$$178$$ 24.9817 1.87246
$$179$$ 10.3738 0.775374 0.387687 0.921791i $$-0.373274\pi$$
0.387687 + 0.921791i $$0.373274\pi$$
$$180$$ 0 0
$$181$$ −2.66473 −0.198068 −0.0990339 0.995084i $$-0.531575\pi$$
−0.0990339 + 0.995084i $$0.531575\pi$$
$$182$$ 23.8217 1.76578
$$183$$ 4.42003 0.326738
$$184$$ −71.5929 −5.27790
$$185$$ 0 0
$$186$$ 23.8217 1.74669
$$187$$ −2.49084 −0.182148
$$188$$ 2.10308 0.153383
$$189$$ −1.86464 −0.135633
$$190$$ 0 0
$$191$$ 14.6724 1.06166 0.530830 0.847478i $$-0.321880\pi$$
0.530830 + 0.847478i $$0.321880\pi$$
$$192$$ 36.9711 2.66816
$$193$$ −5.10308 −0.367328 −0.183664 0.982989i $$-0.558796\pi$$
−0.183664 + 0.982989i $$0.558796\pi$$
$$194$$ −45.0602 −3.23513
$$195$$ 0 0
$$196$$ −19.8217 −1.41584
$$197$$ −3.74324 −0.266694 −0.133347 0.991069i $$-0.542573\pi$$
−0.133347 + 0.991069i $$0.542573\pi$$
$$198$$ 2.76156 0.196255
$$199$$ −8.08476 −0.573114 −0.286557 0.958063i $$-0.592511\pi$$
−0.286557 + 0.958063i $$0.592511\pi$$
$$200$$ 0 0
$$201$$ −11.9431 −0.842404
$$202$$ −2.85069 −0.200574
$$203$$ 6.56934 0.461077
$$204$$ −14.0140 −0.981173
$$205$$ 0 0
$$206$$ −8.41233 −0.586115
$$207$$ −7.14931 −0.496912
$$208$$ −75.8776 −5.26116
$$209$$ −5.38776 −0.372679
$$210$$ 0 0
$$211$$ −15.9431 −1.09757 −0.548786 0.835963i $$-0.684910\pi$$
−0.548786 + 0.835963i $$0.684910\pi$$
$$212$$ 30.7110 2.10924
$$213$$ 11.6262 0.796614
$$214$$ 23.4865 1.60550
$$215$$ 0 0
$$216$$ 10.0140 0.681363
$$217$$ −16.0848 −1.09190
$$218$$ −16.9817 −1.15014
$$219$$ −6.77551 −0.457847
$$220$$ 0 0
$$221$$ 11.5231 0.775129
$$222$$ 24.5187 1.64559
$$223$$ 22.1772 1.48510 0.742548 0.669793i $$-0.233617\pi$$
0.742548 + 0.669793i $$0.233617\pi$$
$$224$$ −47.1127 −3.14785
$$225$$ 0 0
$$226$$ 16.5693 1.10218
$$227$$ −5.93545 −0.393950 −0.196975 0.980409i $$-0.563112\pi$$
−0.196975 + 0.980409i $$0.563112\pi$$
$$228$$ −30.3126 −2.00750
$$229$$ −8.31695 −0.549599 −0.274800 0.961502i $$-0.588612\pi$$
−0.274800 + 0.961502i $$0.588612\pi$$
$$230$$ 0 0
$$231$$ −1.86464 −0.122684
$$232$$ −35.2803 −2.31627
$$233$$ −28.5187 −1.86833 −0.934163 0.356848i $$-0.883852\pi$$
−0.934163 + 0.356848i $$0.883852\pi$$
$$234$$ −12.7755 −0.835161
$$235$$ 0 0
$$236$$ 28.9711 1.88585
$$237$$ −6.01395 −0.390648
$$238$$ 12.8261 0.831393
$$239$$ 24.0925 1.55841 0.779206 0.626768i $$-0.215623\pi$$
0.779206 + 0.626768i $$0.215623\pi$$
$$240$$ 0 0
$$241$$ −5.16763 −0.332877 −0.166438 0.986052i $$-0.553227\pi$$
−0.166438 + 0.986052i $$0.553227\pi$$
$$242$$ 2.76156 0.177520
$$243$$ 1.00000 0.0641500
$$244$$ 24.8680 1.59201
$$245$$ 0 0
$$246$$ −2.10308 −0.134088
$$247$$ 24.9248 1.58593
$$248$$ 86.3823 5.48528
$$249$$ −14.5693 −0.923295
$$250$$ 0 0
$$251$$ −9.25240 −0.584006 −0.292003 0.956417i $$-0.594322\pi$$
−0.292003 + 0.956417i $$0.594322\pi$$
$$252$$ −10.4908 −0.660861
$$253$$ −7.14931 −0.449473
$$254$$ 25.5896 1.60563
$$255$$ 0 0
$$256$$ 68.4575 4.27860
$$257$$ 25.2158 1.57292 0.786458 0.617644i $$-0.211913\pi$$
0.786458 + 0.617644i $$0.211913\pi$$
$$258$$ 20.4402 1.27255
$$259$$ −16.5554 −1.02870
$$260$$ 0 0
$$261$$ −3.52311 −0.218075
$$262$$ 11.2245 0.693451
$$263$$ −5.45856 −0.336589 −0.168295 0.985737i $$-0.553826\pi$$
−0.168295 + 0.985737i $$0.553826\pi$$
$$264$$ 10.0140 0.616316
$$265$$ 0 0
$$266$$ 27.7432 1.70105
$$267$$ 9.04623 0.553620
$$268$$ −67.1945 −4.10456
$$269$$ 17.0741 1.04103 0.520514 0.853853i $$-0.325740\pi$$
0.520514 + 0.853853i $$0.325740\pi$$
$$270$$ 0 0
$$271$$ 9.77988 0.594085 0.297043 0.954864i $$-0.404000\pi$$
0.297043 + 0.954864i $$0.404000\pi$$
$$272$$ −40.8540 −2.47714
$$273$$ 8.62620 0.522081
$$274$$ −16.3911 −0.990221
$$275$$ 0 0
$$276$$ −40.2234 −2.42117
$$277$$ −5.91524 −0.355412 −0.177706 0.984084i $$-0.556868\pi$$
−0.177706 + 0.984084i $$0.556868\pi$$
$$278$$ 18.5327 1.11152
$$279$$ 8.62620 0.516437
$$280$$ 0 0
$$281$$ −6.76156 −0.403361 −0.201680 0.979451i $$-0.564640\pi$$
−0.201680 + 0.979451i $$0.564640\pi$$
$$282$$ 1.03228 0.0614711
$$283$$ −8.81841 −0.524200 −0.262100 0.965041i $$-0.584415\pi$$
−0.262100 + 0.965041i $$0.584415\pi$$
$$284$$ 65.4113 3.88145
$$285$$ 0 0
$$286$$ −12.7755 −0.755432
$$287$$ 1.42003 0.0838218
$$288$$ 25.2663 1.48883
$$289$$ −10.7957 −0.635042
$$290$$ 0 0
$$291$$ −16.3169 −0.956516
$$292$$ −38.1204 −2.23083
$$293$$ 30.4908 1.78129 0.890647 0.454696i $$-0.150252\pi$$
0.890647 + 0.454696i $$0.150252\pi$$
$$294$$ −9.72928 −0.567423
$$295$$ 0 0
$$296$$ 88.9098 5.16778
$$297$$ 1.00000 0.0580259
$$298$$ 15.8603 0.918761
$$299$$ 33.0741 1.91273
$$300$$ 0 0
$$301$$ −13.8015 −0.795507
$$302$$ 29.1666 1.67835
$$303$$ −1.03228 −0.0593027
$$304$$ −88.3684 −5.06827
$$305$$ 0 0
$$306$$ −6.87859 −0.393223
$$307$$ −0.767815 −0.0438215 −0.0219107 0.999760i $$-0.506975\pi$$
−0.0219107 + 0.999760i $$0.506975\pi$$
$$308$$ −10.4908 −0.597771
$$309$$ −3.04623 −0.173294
$$310$$ 0 0
$$311$$ −33.8496 −1.91944 −0.959719 0.280963i $$-0.909346\pi$$
−0.959719 + 0.280963i $$0.909346\pi$$
$$312$$ −46.3265 −2.62272
$$313$$ 8.11078 0.458448 0.229224 0.973374i $$-0.426381\pi$$
0.229224 + 0.973374i $$0.426381\pi$$
$$314$$ −8.56934 −0.483596
$$315$$ 0 0
$$316$$ −33.8357 −1.90341
$$317$$ −8.06455 −0.452950 −0.226475 0.974017i $$-0.572720\pi$$
−0.226475 + 0.974017i $$0.572720\pi$$
$$318$$ 15.0741 0.845316
$$319$$ −3.52311 −0.197257
$$320$$ 0 0
$$321$$ 8.50479 0.474691
$$322$$ 36.8140 2.05157
$$323$$ 13.4200 0.746710
$$324$$ 5.62620 0.312567
$$325$$ 0 0
$$326$$ −43.2803 −2.39707
$$327$$ −6.14931 −0.340058
$$328$$ −7.62620 −0.421086
$$329$$ −0.697006 −0.0384272
$$330$$ 0 0
$$331$$ 22.4769 1.23544 0.617721 0.786398i $$-0.288056\pi$$
0.617721 + 0.786398i $$0.288056\pi$$
$$332$$ −81.9700 −4.49869
$$333$$ 8.87859 0.486544
$$334$$ 24.8034 1.35718
$$335$$ 0 0
$$336$$ −30.5833 −1.66846
$$337$$ −6.32757 −0.344685 −0.172342 0.985037i $$-0.555134\pi$$
−0.172342 + 0.985037i $$0.555134\pi$$
$$338$$ 23.2018 1.26201
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 8.62620 0.467135
$$342$$ −14.8786 −0.804542
$$343$$ 19.6218 1.05948
$$344$$ 74.1204 3.99630
$$345$$ 0 0
$$346$$ 31.7832 1.70868
$$347$$ 0.178261 0.00956954 0.00478477 0.999989i $$-0.498477\pi$$
0.00478477 + 0.999989i $$0.498477\pi$$
$$348$$ −19.8217 −1.06256
$$349$$ −19.7293 −1.05608 −0.528042 0.849218i $$-0.677074\pi$$
−0.528042 + 0.849218i $$0.677074\pi$$
$$350$$ 0 0
$$351$$ −4.62620 −0.246928
$$352$$ 25.2663 1.34670
$$353$$ 19.5231 1.03911 0.519555 0.854437i $$-0.326098\pi$$
0.519555 + 0.854437i $$0.326098\pi$$
$$354$$ 14.2201 0.755791
$$355$$ 0 0
$$356$$ 50.8959 2.69748
$$357$$ 4.64452 0.245814
$$358$$ 28.6478 1.51409
$$359$$ 35.6926 1.88379 0.941893 0.335914i $$-0.109045\pi$$
0.941893 + 0.335914i $$0.109045\pi$$
$$360$$ 0 0
$$361$$ 10.0279 0.527785
$$362$$ −7.35881 −0.386770
$$363$$ 1.00000 0.0524864
$$364$$ 48.5327 2.54380
$$365$$ 0 0
$$366$$ 12.2062 0.638027
$$367$$ −20.1127 −1.04987 −0.524936 0.851141i $$-0.675911\pi$$
−0.524936 + 0.851141i $$0.675911\pi$$
$$368$$ −117.261 −6.11265
$$369$$ −0.761557 −0.0396451
$$370$$ 0 0
$$371$$ −10.1783 −0.528429
$$372$$ 48.5327 2.51630
$$373$$ 2.56165 0.132637 0.0663185 0.997799i $$-0.478875\pi$$
0.0663185 + 0.997799i $$0.478875\pi$$
$$374$$ −6.87859 −0.355684
$$375$$ 0 0
$$376$$ 3.74324 0.193043
$$377$$ 16.2986 0.839422
$$378$$ −5.14931 −0.264852
$$379$$ 23.2601 1.19479 0.597395 0.801947i $$-0.296202\pi$$
0.597395 + 0.801947i $$0.296202\pi$$
$$380$$ 0 0
$$381$$ 9.26635 0.474729
$$382$$ 40.5187 2.07312
$$383$$ −25.7938 −1.31800 −0.659002 0.752142i $$-0.729021\pi$$
−0.659002 + 0.752142i $$0.729021\pi$$
$$384$$ 51.5650 2.63141
$$385$$ 0 0
$$386$$ −14.0925 −0.717287
$$387$$ 7.40171 0.376250
$$388$$ −91.8024 −4.66056
$$389$$ 2.33527 0.118403 0.0592014 0.998246i $$-0.481145\pi$$
0.0592014 + 0.998246i $$0.481145\pi$$
$$390$$ 0 0
$$391$$ 17.8078 0.900578
$$392$$ −35.2803 −1.78192
$$393$$ 4.06455 0.205029
$$394$$ −10.3372 −0.520778
$$395$$ 0 0
$$396$$ 5.62620 0.282727
$$397$$ −17.1955 −0.863019 −0.431510 0.902108i $$-0.642019\pi$$
−0.431510 + 0.902108i $$0.642019\pi$$
$$398$$ −22.3265 −1.11913
$$399$$ 10.0462 0.502941
$$400$$ 0 0
$$401$$ −26.5693 −1.32681 −0.663405 0.748261i $$-0.730889\pi$$
−0.663405 + 0.748261i $$0.730889\pi$$
$$402$$ −32.9817 −1.64498
$$403$$ −39.9065 −1.98788
$$404$$ −5.80779 −0.288948
$$405$$ 0 0
$$406$$ 18.1416 0.900353
$$407$$ 8.87859 0.440096
$$408$$ −24.9431 −1.23487
$$409$$ −20.4200 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$410$$ 0 0
$$411$$ −5.93545 −0.292774
$$412$$ −17.1387 −0.844362
$$413$$ −9.60162 −0.472465
$$414$$ −19.7432 −0.970327
$$415$$ 0 0
$$416$$ −116.887 −5.73086
$$417$$ 6.71096 0.328637
$$418$$ −14.8786 −0.727736
$$419$$ −13.5896 −0.663893 −0.331947 0.943298i $$-0.607705\pi$$
−0.331947 + 0.943298i $$0.607705\pi$$
$$420$$ 0 0
$$421$$ −39.6175 −1.93084 −0.965418 0.260705i $$-0.916045\pi$$
−0.965418 + 0.260705i $$0.916045\pi$$
$$422$$ −44.0279 −2.14324
$$423$$ 0.373802 0.0181749
$$424$$ 54.6618 2.65461
$$425$$ 0 0
$$426$$ 32.1064 1.55556
$$427$$ −8.24177 −0.398847
$$428$$ 47.8496 2.31290
$$429$$ −4.62620 −0.223355
$$430$$ 0 0
$$431$$ −6.56934 −0.316434 −0.158217 0.987404i $$-0.550575\pi$$
−0.158217 + 0.987404i $$0.550575\pi$$
$$432$$ 16.4017 0.789128
$$433$$ 36.4113 1.74982 0.874908 0.484290i $$-0.160922\pi$$
0.874908 + 0.484290i $$0.160922\pi$$
$$434$$ −44.4190 −2.13218
$$435$$ 0 0
$$436$$ −34.5972 −1.65691
$$437$$ 38.5187 1.84260
$$438$$ −18.7110 −0.894044
$$439$$ 25.3878 1.21169 0.605846 0.795582i $$-0.292835\pi$$
0.605846 + 0.795582i $$0.292835\pi$$
$$440$$ 0 0
$$441$$ −3.52311 −0.167767
$$442$$ 31.8217 1.51361
$$443$$ 20.9065 0.993298 0.496649 0.867952i $$-0.334564\pi$$
0.496649 + 0.867952i $$0.334564\pi$$
$$444$$ 49.9527 2.37065
$$445$$ 0 0
$$446$$ 61.2437 2.89997
$$447$$ 5.74324 0.271646
$$448$$ −68.9377 −3.25700
$$449$$ −34.9450 −1.64916 −0.824579 0.565747i $$-0.808588\pi$$
−0.824579 + 0.565747i $$0.808588\pi$$
$$450$$ 0 0
$$451$$ −0.761557 −0.0358603
$$452$$ 33.7572 1.58780
$$453$$ 10.5616 0.496229
$$454$$ −16.3911 −0.769272
$$455$$ 0 0
$$456$$ −53.9527 −2.52657
$$457$$ −7.31695 −0.342272 −0.171136 0.985247i $$-0.554744\pi$$
−0.171136 + 0.985247i $$0.554744\pi$$
$$458$$ −22.9677 −1.07321
$$459$$ −2.49084 −0.116262
$$460$$ 0 0
$$461$$ −14.0279 −0.653345 −0.326672 0.945138i $$-0.605927\pi$$
−0.326672 + 0.945138i $$0.605927\pi$$
$$462$$ −5.14931 −0.239568
$$463$$ 4.33527 0.201477 0.100739 0.994913i $$-0.467879\pi$$
0.100739 + 0.994913i $$0.467879\pi$$
$$464$$ −57.7851 −2.68261
$$465$$ 0 0
$$466$$ −78.7561 −3.64831
$$467$$ 3.70138 0.171279 0.0856396 0.996326i $$-0.472707\pi$$
0.0856396 + 0.996326i $$0.472707\pi$$
$$468$$ −26.0279 −1.20314
$$469$$ 22.2697 1.02832
$$470$$ 0 0
$$471$$ −3.10308 −0.142983
$$472$$ 51.5650 2.37347
$$473$$ 7.40171 0.340331
$$474$$ −16.6079 −0.762825
$$475$$ 0 0
$$476$$ 26.1310 1.19771
$$477$$ 5.45856 0.249930
$$478$$ 66.5327 3.04313
$$479$$ −22.8680 −1.04486 −0.522432 0.852681i $$-0.674975\pi$$
−0.522432 + 0.852681i $$0.674975\pi$$
$$480$$ 0 0
$$481$$ −41.0741 −1.87282
$$482$$ −14.2707 −0.650013
$$483$$ 13.3309 0.606577
$$484$$ 5.62620 0.255736
$$485$$ 0 0
$$486$$ 2.76156 0.125267
$$487$$ −5.42962 −0.246039 −0.123020 0.992404i $$-0.539258\pi$$
−0.123020 + 0.992404i $$0.539258\pi$$
$$488$$ 44.2620 2.00365
$$489$$ −15.6724 −0.708732
$$490$$ 0 0
$$491$$ 21.2803 0.960367 0.480183 0.877168i $$-0.340570\pi$$
0.480183 + 0.877168i $$0.340570\pi$$
$$492$$ −4.28467 −0.193168
$$493$$ 8.77551 0.395229
$$494$$ 68.8313 3.09687
$$495$$ 0 0
$$496$$ 141.484 6.35284
$$497$$ −21.6787 −0.972422
$$498$$ −40.2341 −1.80293
$$499$$ −16.0077 −0.716603 −0.358301 0.933606i $$-0.616644\pi$$
−0.358301 + 0.933606i $$0.616644\pi$$
$$500$$ 0 0
$$501$$ 8.98168 0.401272
$$502$$ −25.5510 −1.14040
$$503$$ 12.0925 0.539176 0.269588 0.962976i $$-0.413112\pi$$
0.269588 + 0.962976i $$0.413112\pi$$
$$504$$ −18.6724 −0.831736
$$505$$ 0 0
$$506$$ −19.7432 −0.877694
$$507$$ 8.40171 0.373133
$$508$$ 52.1343 2.31309
$$509$$ 14.7110 0.652052 0.326026 0.945361i $$-0.394290\pi$$
0.326026 + 0.945361i $$0.394290\pi$$
$$510$$ 0 0
$$511$$ 12.6339 0.558891
$$512$$ 85.9194 3.79714
$$513$$ −5.38776 −0.237875
$$514$$ 69.6347 3.07146
$$515$$ 0 0
$$516$$ 41.6435 1.83325
$$517$$ 0.373802 0.0164398
$$518$$ −45.7187 −2.00876
$$519$$ 11.5092 0.505196
$$520$$ 0 0
$$521$$ −17.2803 −0.757064 −0.378532 0.925588i $$-0.623571\pi$$
−0.378532 + 0.925588i $$0.623571\pi$$
$$522$$ −9.72928 −0.425839
$$523$$ −42.7326 −1.86857 −0.934283 0.356532i $$-0.883959\pi$$
−0.934283 + 0.356532i $$0.883959\pi$$
$$524$$ 22.8680 0.998992
$$525$$ 0 0
$$526$$ −15.0741 −0.657264
$$527$$ −21.4865 −0.935965
$$528$$ 16.4017 0.713793
$$529$$ 28.1127 1.22229
$$530$$ 0 0
$$531$$ 5.14931 0.223461
$$532$$ 56.5221 2.45054
$$533$$ 3.52311 0.152603
$$534$$ 24.9817 1.08106
$$535$$ 0 0
$$536$$ −119.598 −5.16585
$$537$$ 10.3738 0.447663
$$538$$ 47.1512 2.03283
$$539$$ −3.52311 −0.151751
$$540$$ 0 0
$$541$$ 8.69075 0.373644 0.186822 0.982394i $$-0.440181\pi$$
0.186822 + 0.982394i $$0.440181\pi$$
$$542$$ 27.0077 1.16008
$$543$$ −2.66473 −0.114355
$$544$$ −62.9344 −2.69829
$$545$$ 0 0
$$546$$ 23.8217 1.01948
$$547$$ 44.1064 1.88585 0.942927 0.333000i $$-0.108061\pi$$
0.942927 + 0.333000i $$0.108061\pi$$
$$548$$ −33.3940 −1.42652
$$549$$ 4.42003 0.188642
$$550$$ 0 0
$$551$$ 18.9817 0.808647
$$552$$ −71.5929 −3.04720
$$553$$ 11.2139 0.476862
$$554$$ −16.3353 −0.694019
$$555$$ 0 0
$$556$$ 37.7572 1.60126
$$557$$ 7.69264 0.325948 0.162974 0.986630i $$-0.447891\pi$$
0.162974 + 0.986630i $$0.447891\pi$$
$$558$$ 23.8217 1.00845
$$559$$ −34.2418 −1.44827
$$560$$ 0 0
$$561$$ −2.49084 −0.105163
$$562$$ −18.6724 −0.787649
$$563$$ −9.25240 −0.389942 −0.194971 0.980809i $$-0.562461\pi$$
−0.194971 + 0.980809i $$0.562461\pi$$
$$564$$ 2.10308 0.0885558
$$565$$ 0 0
$$566$$ −24.3525 −1.02361
$$567$$ −1.86464 −0.0783076
$$568$$ 116.424 4.88505
$$569$$ 4.96772 0.208258 0.104129 0.994564i $$-0.466795\pi$$
0.104129 + 0.994564i $$0.466795\pi$$
$$570$$ 0 0
$$571$$ 6.02021 0.251938 0.125969 0.992034i $$-0.459796\pi$$
0.125969 + 0.992034i $$0.459796\pi$$
$$572$$ −26.0279 −1.08828
$$573$$ 14.6724 0.612949
$$574$$ 3.92150 0.163680
$$575$$ 0 0
$$576$$ 36.9711 1.54046
$$577$$ −8.25240 −0.343552 −0.171776 0.985136i $$-0.554950\pi$$
−0.171776 + 0.985136i $$0.554950\pi$$
$$578$$ −29.8130 −1.24006
$$579$$ −5.10308 −0.212077
$$580$$ 0 0
$$581$$ 27.1666 1.12706
$$582$$ −45.0602 −1.86780
$$583$$ 5.45856 0.226071
$$584$$ −67.8496 −2.80764
$$585$$ 0 0
$$586$$ 84.2022 3.47836
$$587$$ 11.4846 0.474019 0.237010 0.971507i $$-0.423833\pi$$
0.237010 + 0.971507i $$0.423833\pi$$
$$588$$ −19.8217 −0.817435
$$589$$ −46.4758 −1.91500
$$590$$ 0 0
$$591$$ −3.74324 −0.153976
$$592$$ 145.624 5.98511
$$593$$ 39.0375 1.60308 0.801539 0.597943i $$-0.204015\pi$$
0.801539 + 0.597943i $$0.204015\pi$$
$$594$$ 2.76156 0.113308
$$595$$ 0 0
$$596$$ 32.3126 1.32357
$$597$$ −8.08476 −0.330887
$$598$$ 91.3361 3.73501
$$599$$ −30.1589 −1.23226 −0.616130 0.787645i $$-0.711300\pi$$
−0.616130 + 0.787645i $$0.711300\pi$$
$$600$$ 0 0
$$601$$ 19.1310 0.780369 0.390185 0.920737i $$-0.372411\pi$$
0.390185 + 0.920737i $$0.372411\pi$$
$$602$$ −38.1137 −1.55340
$$603$$ −11.9431 −0.486362
$$604$$ 59.4219 2.41784
$$605$$ 0 0
$$606$$ −2.85069 −0.115801
$$607$$ −4.20617 −0.170723 −0.0853615 0.996350i $$-0.527205\pi$$
−0.0853615 + 0.996350i $$0.527205\pi$$
$$608$$ −136.129 −5.52076
$$609$$ 6.56934 0.266203
$$610$$ 0 0
$$611$$ −1.72928 −0.0699593
$$612$$ −14.0140 −0.566480
$$613$$ −5.45856 −0.220469 −0.110235 0.993906i $$-0.535160\pi$$
−0.110235 + 0.993906i $$0.535160\pi$$
$$614$$ −2.12036 −0.0855709
$$615$$ 0 0
$$616$$ −18.6724 −0.752334
$$617$$ −30.1974 −1.21570 −0.607851 0.794051i $$-0.707968\pi$$
−0.607851 + 0.794051i $$0.707968\pi$$
$$618$$ −8.41233 −0.338394
$$619$$ 23.1955 0.932308 0.466154 0.884704i $$-0.345639\pi$$
0.466154 + 0.884704i $$0.345639\pi$$
$$620$$ 0 0
$$621$$ −7.14931 −0.286892
$$622$$ −93.4777 −3.74812
$$623$$ −16.8680 −0.675801
$$624$$ −75.8776 −3.03753
$$625$$ 0 0
$$626$$ 22.3984 0.895219
$$627$$ −5.38776 −0.215166
$$628$$ −17.4586 −0.696673
$$629$$ −22.1151 −0.881789
$$630$$ 0 0
$$631$$ −3.19554 −0.127212 −0.0636062 0.997975i $$-0.520260\pi$$
−0.0636062 + 0.997975i $$0.520260\pi$$
$$632$$ −60.2234 −2.39556
$$633$$ −15.9431 −0.633683
$$634$$ −22.2707 −0.884483
$$635$$ 0 0
$$636$$ 30.7110 1.21777
$$637$$ 16.2986 0.645775
$$638$$ −9.72928 −0.385186
$$639$$ 11.6262 0.459925
$$640$$ 0 0
$$641$$ 28.1974 1.11373 0.556866 0.830603i $$-0.312004\pi$$
0.556866 + 0.830603i $$0.312004\pi$$
$$642$$ 23.4865 0.926937
$$643$$ −17.9634 −0.708406 −0.354203 0.935169i $$-0.615248\pi$$
−0.354203 + 0.935169i $$0.615248\pi$$
$$644$$ 75.0023 2.95550
$$645$$ 0 0
$$646$$ 37.0602 1.45811
$$647$$ 40.9990 1.61184 0.805918 0.592028i $$-0.201672\pi$$
0.805918 + 0.592028i $$0.201672\pi$$
$$648$$ 10.0140 0.393385
$$649$$ 5.14931 0.202128
$$650$$ 0 0
$$651$$ −16.0848 −0.630412
$$652$$ −88.1762 −3.45325
$$653$$ −11.6926 −0.457568 −0.228784 0.973477i $$-0.573475\pi$$
−0.228784 + 0.973477i $$0.573475\pi$$
$$654$$ −16.9817 −0.664036
$$655$$ 0 0
$$656$$ −12.4908 −0.487685
$$657$$ −6.77551 −0.264338
$$658$$ −1.92482 −0.0750374
$$659$$ −1.72928 −0.0673633 −0.0336816 0.999433i $$-0.510723\pi$$
−0.0336816 + 0.999433i $$0.510723\pi$$
$$660$$ 0 0
$$661$$ 0.232185 0.00903096 0.00451548 0.999990i $$-0.498563\pi$$
0.00451548 + 0.999990i $$0.498563\pi$$
$$662$$ 62.0712 2.41247
$$663$$ 11.5231 0.447521
$$664$$ −145.897 −5.66189
$$665$$ 0 0
$$666$$ 24.5187 0.950082
$$667$$ 25.1878 0.975277
$$668$$ 50.5327 1.95517
$$669$$ 22.1772 0.857421
$$670$$ 0 0
$$671$$ 4.42003 0.170633
$$672$$ −47.1127 −1.81741
$$673$$ −33.5144 −1.29188 −0.645942 0.763386i $$-0.723535\pi$$
−0.645942 + 0.763386i $$0.723535\pi$$
$$674$$ −17.4740 −0.673072
$$675$$ 0 0
$$676$$ 47.2697 1.81806
$$677$$ 13.0183 0.500335 0.250167 0.968203i $$-0.419514\pi$$
0.250167 + 0.968203i $$0.419514\pi$$
$$678$$ 16.5693 0.636342
$$679$$ 30.4252 1.16761
$$680$$ 0 0
$$681$$ −5.93545 −0.227447
$$682$$ 23.8217 0.912182
$$683$$ 4.54333 0.173846 0.0869228 0.996215i $$-0.472297\pi$$
0.0869228 + 0.996215i $$0.472297\pi$$
$$684$$ −30.3126 −1.15903
$$685$$ 0 0
$$686$$ 54.1868 2.06886
$$687$$ −8.31695 −0.317311
$$688$$ 121.401 4.62836
$$689$$ −25.2524 −0.962040
$$690$$ 0 0
$$691$$ −19.8988 −0.756986 −0.378493 0.925604i $$-0.623558\pi$$
−0.378493 + 0.925604i $$0.623558\pi$$
$$692$$ 64.7528 2.46153
$$693$$ −1.86464 −0.0708319
$$694$$ 0.492277 0.0186866
$$695$$ 0 0
$$696$$ −35.2803 −1.33730
$$697$$ 1.89692 0.0718508
$$698$$ −54.4835 −2.06223
$$699$$ −28.5187 −1.07868
$$700$$ 0 0
$$701$$ −15.8444 −0.598436 −0.299218 0.954185i $$-0.596726\pi$$
−0.299218 + 0.954185i $$0.596726\pi$$
$$702$$ −12.7755 −0.482181
$$703$$ −47.8357 −1.80416
$$704$$ 36.9711 1.39340
$$705$$ 0 0
$$706$$ 53.9142 2.02909
$$707$$ 1.92482 0.0723904
$$708$$ 28.9711 1.08880
$$709$$ 8.82174 0.331307 0.165654 0.986184i $$-0.447027\pi$$
0.165654 + 0.986184i $$0.447027\pi$$
$$710$$ 0 0
$$711$$ −6.01395 −0.225541
$$712$$ 90.5885 3.39495
$$713$$ −61.6714 −2.30961
$$714$$ 12.8261 0.480005
$$715$$ 0 0
$$716$$ 58.3651 2.18120
$$717$$ 24.0925 0.899749
$$718$$ 98.5673 3.67850
$$719$$ 36.7668 1.37117 0.685585 0.727993i $$-0.259547\pi$$
0.685585 + 0.727993i $$0.259547\pi$$
$$720$$ 0 0
$$721$$ 5.68012 0.211539
$$722$$ 27.6926 1.03061
$$723$$ −5.16763 −0.192186
$$724$$ −14.9923 −0.557185
$$725$$ 0 0
$$726$$ 2.76156 0.102491
$$727$$ 5.27261 0.195550 0.0977751 0.995209i $$-0.468827\pi$$
0.0977751 + 0.995209i $$0.468827\pi$$
$$728$$ 86.3823 3.20154
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −18.4365 −0.681897
$$732$$ 24.8680 0.919147
$$733$$ −37.9508 −1.40175 −0.700873 0.713286i $$-0.747206\pi$$
−0.700873 + 0.713286i $$0.747206\pi$$
$$734$$ −55.5423 −2.05010
$$735$$ 0 0
$$736$$ −180.637 −6.65837
$$737$$ −11.9431 −0.439931
$$738$$ −2.10308 −0.0774156
$$739$$ 37.2943 1.37189 0.685946 0.727653i $$-0.259389\pi$$
0.685946 + 0.727653i $$0.259389\pi$$
$$740$$ 0 0
$$741$$ 24.9248 0.915636
$$742$$ −28.1078 −1.03187
$$743$$ −22.5693 −0.827989 −0.413994 0.910279i $$-0.635867\pi$$
−0.413994 + 0.910279i $$0.635867\pi$$
$$744$$ 86.3823 3.16693
$$745$$ 0 0
$$746$$ 7.07414 0.259002
$$747$$ −14.5693 −0.533064
$$748$$ −14.0140 −0.512401
$$749$$ −15.8584 −0.579453
$$750$$ 0 0
$$751$$ −9.55102 −0.348522 −0.174261 0.984700i $$-0.555754\pi$$
−0.174261 + 0.984700i $$0.555754\pi$$
$$752$$ 6.13099 0.223574
$$753$$ −9.25240 −0.337176
$$754$$ 45.0096 1.63915
$$755$$ 0 0
$$756$$ −10.4908 −0.381548
$$757$$ −30.3188 −1.10196 −0.550978 0.834519i $$-0.685745\pi$$
−0.550978 + 0.834519i $$0.685745\pi$$
$$758$$ 64.2341 2.33309
$$759$$ −7.14931 −0.259504
$$760$$ 0 0
$$761$$ −1.43066 −0.0518613 −0.0259306 0.999664i $$-0.508255\pi$$
−0.0259306 + 0.999664i $$0.508255\pi$$
$$762$$ 25.5896 0.927012
$$763$$ 11.4663 0.415106
$$764$$ 82.5500 2.98655
$$765$$ 0 0
$$766$$ −71.2311 −2.57369
$$767$$ −23.8217 −0.860153
$$768$$ 68.4575 2.47025
$$769$$ −35.4017 −1.27662 −0.638309 0.769780i $$-0.720366\pi$$
−0.638309 + 0.769780i $$0.720366\pi$$
$$770$$ 0 0
$$771$$ 25.2158 0.908123
$$772$$ −28.7110 −1.03333
$$773$$ 13.4094 0.482303 0.241151 0.970488i $$-0.422475\pi$$
0.241151 + 0.970488i $$0.422475\pi$$
$$774$$ 20.4402 0.734709
$$775$$ 0 0
$$776$$ −163.397 −5.86562
$$777$$ −16.5554 −0.593921
$$778$$ 6.44898 0.231207
$$779$$ 4.10308 0.147008
$$780$$ 0 0
$$781$$ 11.6262 0.416018
$$782$$ 49.1772 1.75857
$$783$$ −3.52311 −0.125906
$$784$$ −57.7851 −2.06375
$$785$$ 0 0
$$786$$ 11.2245 0.400364
$$787$$ 37.3232 1.33043 0.665214 0.746653i $$-0.268340\pi$$
0.665214 + 0.746653i $$0.268340\pi$$
$$788$$ −21.0602 −0.750238
$$789$$ −5.45856 −0.194330
$$790$$ 0 0
$$791$$ −11.1878 −0.397794
$$792$$ 10.0140 0.355830
$$793$$ −20.4479 −0.726128
$$794$$ −47.4865 −1.68523
$$795$$ 0 0
$$796$$ −45.4865 −1.61223
$$797$$ 8.05581 0.285352 0.142676 0.989769i $$-0.454429\pi$$
0.142676 + 0.989769i $$0.454429\pi$$
$$798$$ 27.7432 0.982100
$$799$$ −0.931080 −0.0329393
$$800$$ 0 0
$$801$$ 9.04623 0.319633
$$802$$ −73.3728 −2.59088
$$803$$ −6.77551 −0.239103
$$804$$ −67.1945 −2.36977
$$805$$ 0 0
$$806$$ −110.204 −3.88177
$$807$$ 17.0741 0.601038
$$808$$ −10.3372 −0.363660
$$809$$ −42.3771 −1.48990 −0.744950 0.667120i $$-0.767527\pi$$
−0.744950 + 0.667120i $$0.767527\pi$$
$$810$$ 0 0
$$811$$ −21.3878 −0.751026 −0.375513 0.926817i $$-0.622533\pi$$
−0.375513 + 0.926817i $$0.622533\pi$$
$$812$$ 36.9604 1.29706
$$813$$ 9.77988 0.342995
$$814$$ 24.5187 0.859382
$$815$$ 0 0
$$816$$ −40.8540 −1.43018
$$817$$ −39.8786 −1.39518
$$818$$ −56.3911 −1.97167
$$819$$ 8.62620 0.301424
$$820$$ 0 0
$$821$$ −47.8776 −1.67094 −0.835469 0.549537i $$-0.814804\pi$$
−0.835469 + 0.549537i $$0.814804\pi$$
$$822$$ −16.3911 −0.571705
$$823$$ −16.3555 −0.570116 −0.285058 0.958510i $$-0.592013\pi$$
−0.285058 + 0.958510i $$0.592013\pi$$
$$824$$ −30.5048 −1.06268
$$825$$ 0 0
$$826$$ −26.5154 −0.922589
$$827$$ 44.7110 1.55475 0.777376 0.629036i $$-0.216550\pi$$
0.777376 + 0.629036i $$0.216550\pi$$
$$828$$ −40.2234 −1.39786
$$829$$ −38.5972 −1.34054 −0.670269 0.742118i $$-0.733821\pi$$
−0.670269 + 0.742118i $$0.733821\pi$$
$$830$$ 0 0
$$831$$ −5.91524 −0.205197
$$832$$ −171.035 −5.92959
$$833$$ 8.77551 0.304053
$$834$$ 18.5327 0.641735
$$835$$ 0 0
$$836$$ −30.3126 −1.04838
$$837$$ 8.62620 0.298165
$$838$$ −37.5283 −1.29639
$$839$$ −0.710960 −0.0245451 −0.0122725 0.999925i $$-0.503907\pi$$
−0.0122725 + 0.999925i $$0.503907\pi$$
$$840$$ 0 0
$$841$$ −16.5877 −0.571988
$$842$$ −109.406 −3.77038
$$843$$ −6.76156 −0.232880
$$844$$ −89.6993 −3.08758
$$845$$ 0 0
$$846$$ 1.03228 0.0354904
$$847$$ −1.86464 −0.0640698
$$848$$ 89.5298 3.07446
$$849$$ −8.81841 −0.302647
$$850$$ 0 0
$$851$$ −63.4758 −2.17592
$$852$$ 65.4113 2.24095
$$853$$ −52.7187 −1.80505 −0.902526 0.430635i $$-0.858290\pi$$
−0.902526 + 0.430635i $$0.858290\pi$$
$$854$$ −22.7601 −0.778835
$$855$$ 0 0
$$856$$ 85.1666 2.91093
$$857$$ −16.7124 −0.570885 −0.285442 0.958396i $$-0.592141\pi$$
−0.285442 + 0.958396i $$0.592141\pi$$
$$858$$ −12.7755 −0.436149
$$859$$ −41.9142 −1.43009 −0.715047 0.699076i $$-0.753595\pi$$
−0.715047 + 0.699076i $$0.753595\pi$$
$$860$$ 0 0
$$861$$ 1.42003 0.0483945
$$862$$ −18.1416 −0.617906
$$863$$ 41.4219 1.41002 0.705009 0.709198i $$-0.250943\pi$$
0.705009 + 0.709198i $$0.250943\pi$$
$$864$$ 25.2663 0.859579
$$865$$ 0 0
$$866$$ 100.552 3.41689
$$867$$ −10.7957 −0.366642
$$868$$ −90.4961 −3.07164
$$869$$ −6.01395 −0.204009
$$870$$ 0 0
$$871$$ 55.2514 1.87212
$$872$$ −61.5789 −2.08533
$$873$$ −16.3169 −0.552245
$$874$$ 106.372 3.59808
$$875$$ 0 0
$$876$$ −38.1204 −1.28797
$$877$$ 12.0848 0.408073 0.204037 0.978963i $$-0.434594\pi$$
0.204037 + 0.978963i $$0.434594\pi$$
$$878$$ 70.1097 2.36609
$$879$$ 30.4908 1.02843
$$880$$ 0 0
$$881$$ −31.8130 −1.07181 −0.535904 0.844279i $$-0.680029\pi$$
−0.535904 + 0.844279i $$0.680029\pi$$
$$882$$ −9.72928 −0.327602
$$883$$ 20.9894 0.706349 0.353174 0.935558i $$-0.385102\pi$$
0.353174 + 0.935558i $$0.385102\pi$$
$$884$$ 64.8313 2.18051
$$885$$ 0 0
$$886$$ 57.7345 1.93963
$$887$$ −49.2032 −1.65208 −0.826042 0.563609i $$-0.809412\pi$$
−0.826042 + 0.563609i $$0.809412\pi$$
$$888$$ 88.9098 2.98362
$$889$$ −17.2784 −0.579499
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 124.773 4.17772
$$893$$ −2.01395 −0.0673944
$$894$$ 15.8603 0.530447
$$895$$ 0 0
$$896$$ −96.1502 −3.21215
$$897$$ 33.0741 1.10431
$$898$$ −96.5027 −3.22034
$$899$$ −30.3911 −1.01360
$$900$$ 0 0
$$901$$ −13.5964 −0.452962
$$902$$ −2.10308 −0.0700250
$$903$$ −13.8015 −0.459286
$$904$$ 60.0837 1.99835
$$905$$ 0 0
$$906$$ 29.1666 0.968995
$$907$$ 24.8526 0.825216 0.412608 0.910909i $$-0.364618\pi$$
0.412608 + 0.910909i $$0.364618\pi$$
$$908$$ −33.3940 −1.10822
$$909$$ −1.03228 −0.0342384
$$910$$ 0 0
$$911$$ 45.4758 1.50668 0.753341 0.657630i $$-0.228441\pi$$
0.753341 + 0.657630i $$0.228441\pi$$
$$912$$ −88.3684 −2.92617
$$913$$ −14.5693 −0.482175
$$914$$ −20.2062 −0.668361
$$915$$ 0 0
$$916$$ −46.7928 −1.54608
$$917$$ −7.57893 −0.250278
$$918$$ −6.87859 −0.227027
$$919$$ −15.7355 −0.519068 −0.259534 0.965734i $$-0.583569\pi$$
−0.259534 + 0.965734i $$0.583569\pi$$
$$920$$ 0 0
$$921$$ −0.767815 −0.0253004
$$922$$ −38.7389 −1.27580
$$923$$ −53.7851 −1.77036
$$924$$ −10.4908 −0.345123
$$925$$ 0 0
$$926$$ 11.9721 0.393427
$$927$$ −3.04623 −0.100051
$$928$$ −89.0162 −2.92210
$$929$$ −2.06455 −0.0677357 −0.0338679 0.999426i $$-0.510783\pi$$
−0.0338679 + 0.999426i $$0.510783\pi$$
$$930$$ 0 0
$$931$$ 18.9817 0.622099
$$932$$ −160.452 −5.25578
$$933$$ −33.8496 −1.10819
$$934$$ 10.2216 0.334460
$$935$$ 0 0
$$936$$ −46.3265 −1.51423
$$937$$ −40.1493 −1.31162 −0.655810 0.754926i $$-0.727673\pi$$
−0.655810 + 0.754926i $$0.727673\pi$$
$$938$$ 61.4990 2.00801
$$939$$ 8.11078 0.264685
$$940$$ 0 0
$$941$$ 26.4050 0.860780 0.430390 0.902643i $$-0.358376\pi$$
0.430390 + 0.902643i $$0.358376\pi$$
$$942$$ −8.56934 −0.279204
$$943$$ 5.44461 0.177301
$$944$$ 84.4575 2.74886
$$945$$ 0 0
$$946$$ 20.4402 0.664570
$$947$$ −8.67243 −0.281816 −0.140908 0.990023i $$-0.545002\pi$$
−0.140908 + 0.990023i $$0.545002\pi$$
$$948$$ −33.8357 −1.09893
$$949$$ 31.3449 1.01750
$$950$$ 0 0
$$951$$ −8.06455 −0.261511
$$952$$ 46.5100 1.50740
$$953$$ 26.8540 0.869887 0.434943 0.900458i $$-0.356768\pi$$
0.434943 + 0.900458i $$0.356768\pi$$
$$954$$ 15.0741 0.488043
$$955$$ 0 0
$$956$$ 135.549 4.38397
$$957$$ −3.52311 −0.113886
$$958$$ −63.1512 −2.04032
$$959$$ 11.0675 0.357388
$$960$$ 0 0
$$961$$ 43.4113 1.40036
$$962$$ −113.429 −3.65708
$$963$$ 8.50479 0.274063
$$964$$ −29.0741 −0.936415
$$965$$ 0 0
$$966$$ 36.8140 1.18447
$$967$$ 55.8496 1.79600 0.898002 0.439992i $$-0.145019\pi$$
0.898002 + 0.439992i $$0.145019\pi$$
$$968$$ 10.0140 0.321861
$$969$$ 13.4200 0.431113
$$970$$ 0 0
$$971$$ 30.0664 0.964878 0.482439 0.875930i $$-0.339751\pi$$
0.482439 + 0.875930i $$0.339751\pi$$
$$972$$ 5.62620 0.180460
$$973$$ −12.5135 −0.401165
$$974$$ −14.9942 −0.480445
$$975$$ 0 0
$$976$$ 72.4961 2.32054
$$977$$ −15.7014 −0.502331 −0.251166 0.967944i $$-0.580814\pi$$
−0.251166 + 0.967944i $$0.580814\pi$$
$$978$$ −43.2803 −1.38395
$$979$$ 9.04623 0.289119
$$980$$ 0 0
$$981$$ −6.14931 −0.196332
$$982$$ 58.7668 1.87532
$$983$$ 51.9894 1.65820 0.829102 0.559098i $$-0.188852\pi$$
0.829102 + 0.559098i $$0.188852\pi$$
$$984$$ −7.62620 −0.243114
$$985$$ 0 0
$$986$$ 24.2341 0.771770
$$987$$ −0.697006 −0.0221860
$$988$$ 140.232 4.46137
$$989$$ −52.9171 −1.68267
$$990$$ 0 0
$$991$$ 27.6445 0.878157 0.439079 0.898449i $$-0.355305\pi$$
0.439079 + 0.898449i $$0.355305\pi$$
$$992$$ 217.953 6.92000
$$993$$ 22.4769 0.713282
$$994$$ −59.8669 −1.89886
$$995$$ 0 0
$$996$$ −81.9700 −2.59732
$$997$$ −40.5048 −1.28280 −0.641400 0.767207i $$-0.721646\pi$$
−0.641400 + 0.767207i $$0.721646\pi$$
$$998$$ −44.2062 −1.39932
$$999$$ 8.87859 0.280906
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 825.2.a.m.1.3 yes 3
3.2 odd 2 2475.2.a.z.1.1 3
5.2 odd 4 825.2.c.f.199.6 6
5.3 odd 4 825.2.c.f.199.1 6
5.4 even 2 825.2.a.i.1.1 3
11.10 odd 2 9075.2.a.cd.1.1 3
15.2 even 4 2475.2.c.q.199.1 6
15.8 even 4 2475.2.c.q.199.6 6
15.14 odd 2 2475.2.a.bd.1.3 3
55.54 odd 2 9075.2.a.cj.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.1 3 5.4 even 2
825.2.a.m.1.3 yes 3 1.1 even 1 trivial
825.2.c.f.199.1 6 5.3 odd 4
825.2.c.f.199.6 6 5.2 odd 4
2475.2.a.z.1.1 3 3.2 odd 2
2475.2.a.bd.1.3 3 15.14 odd 2
2475.2.c.q.199.1 6 15.2 even 4
2475.2.c.q.199.6 6 15.8 even 4
9075.2.a.cd.1.1 3 11.10 odd 2
9075.2.a.cj.1.3 3 55.54 odd 2