# Properties

 Label 1617.2.a.r.1.3 Level $1617$ Weight $2$ Character 1617.1 Self dual yes Analytic conductor $12.912$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9118100068$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 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 $$2.11491$$ of defining polynomial Character $$\chi$$ $$=$$ 1617.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.11491 q^{2} +1.00000 q^{3} +2.47283 q^{4} +1.64207 q^{5} +2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.11491 q^{2} +1.00000 q^{3} +2.47283 q^{4} +1.64207 q^{5} +2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.47283 q^{10} +1.00000 q^{11} +2.47283 q^{12} +4.58774 q^{13} +1.64207 q^{15} -2.83076 q^{16} +0.715853 q^{17} +2.11491 q^{18} +1.64207 q^{19} +4.06058 q^{20} +2.11491 q^{22} +1.00000 q^{24} -2.30359 q^{25} +9.70265 q^{26} +1.00000 q^{27} -5.53341 q^{29} +3.47283 q^{30} -5.17548 q^{31} -7.98680 q^{32} +1.00000 q^{33} +1.51396 q^{34} +2.47283 q^{36} -6.58774 q^{37} +3.47283 q^{38} +4.58774 q^{39} +1.64207 q^{40} +5.17548 q^{41} +5.17548 q^{43} +2.47283 q^{44} +1.64207 q^{45} +7.87189 q^{47} -2.83076 q^{48} -4.87189 q^{50} +0.715853 q^{51} +11.3447 q^{52} +2.71585 q^{53} +2.11491 q^{54} +1.64207 q^{55} +1.64207 q^{57} -11.7026 q^{58} -3.15604 q^{59} +4.06058 q^{60} +13.1755 q^{61} -10.9457 q^{62} -11.2298 q^{64} +7.53341 q^{65} +2.11491 q^{66} +4.21037 q^{67} +1.77018 q^{68} -1.85244 q^{71} +1.00000 q^{72} -10.4791 q^{73} -13.9325 q^{74} -2.30359 q^{75} +4.06058 q^{76} +9.70265 q^{78} -12.4596 q^{79} -4.64832 q^{80} +1.00000 q^{81} +10.9457 q^{82} -9.17548 q^{83} +1.17548 q^{85} +10.9457 q^{86} -5.53341 q^{87} +1.00000 q^{88} +4.71585 q^{89} +3.47283 q^{90} -5.17548 q^{93} +16.6483 q^{94} +2.69641 q^{95} -7.98680 q^{96} -13.6351 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 2 * q^4 + 4 * q^5 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} + 4 q^{19} - 5 q^{20} + 3 q^{24} + 3 q^{25} + 11 q^{26} + 3 q^{27} + 6 q^{29} + 5 q^{30} + 8 q^{31} - 4 q^{32} + 3 q^{33} - 10 q^{34} + 2 q^{36} - 8 q^{37} + 5 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 2 q^{44} + 4 q^{45} + 10 q^{47} - 4 q^{48} - q^{50} + 4 q^{51} + 15 q^{52} + 10 q^{53} + 4 q^{55} + 4 q^{57} - 17 q^{58} + 6 q^{59} - 5 q^{60} + 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} + 3 q^{72} + 2 q^{73} - 11 q^{74} + 3 q^{75} - 5 q^{76} + 11 q^{78} - 12 q^{79} + 15 q^{80} + 3 q^{81} + 22 q^{82} - 4 q^{83} - 20 q^{85} + 22 q^{86} + 6 q^{87} + 3 q^{88} + 16 q^{89} + 5 q^{90} + 8 q^{93} + 21 q^{94} + 18 q^{95} - 4 q^{96} + 8 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 2 * q^4 + 4 * q^5 + 3 * q^8 + 3 * q^9 + 5 * q^10 + 3 * q^11 + 2 * q^12 + 2 * q^13 + 4 * q^15 - 4 * q^16 + 4 * q^17 + 4 * q^19 - 5 * q^20 + 3 * q^24 + 3 * q^25 + 11 * q^26 + 3 * q^27 + 6 * q^29 + 5 * q^30 + 8 * q^31 - 4 * q^32 + 3 * q^33 - 10 * q^34 + 2 * q^36 - 8 * q^37 + 5 * q^38 + 2 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 2 * q^44 + 4 * q^45 + 10 * q^47 - 4 * q^48 - q^50 + 4 * q^51 + 15 * q^52 + 10 * q^53 + 4 * q^55 + 4 * q^57 - 17 * q^58 + 6 * q^59 - 5 * q^60 + 16 * q^61 - 22 * q^62 - 21 * q^64 + 8 * q^67 + 18 * q^68 + 3 * q^72 + 2 * q^73 - 11 * q^74 + 3 * q^75 - 5 * q^76 + 11 * q^78 - 12 * q^79 + 15 * q^80 + 3 * q^81 + 22 * q^82 - 4 * q^83 - 20 * q^85 + 22 * q^86 + 6 * q^87 + 3 * q^88 + 16 * q^89 + 5 * q^90 + 8 * q^93 + 21 * q^94 + 18 * q^95 - 4 * q^96 + 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.11491 1.49547 0.747733 0.664000i $$-0.231142\pi$$
0.747733 + 0.664000i $$0.231142\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.47283 1.23642
$$5$$ 1.64207 0.734358 0.367179 0.930150i $$-0.380324\pi$$
0.367179 + 0.930150i $$0.380324\pi$$
$$6$$ 2.11491 0.863407
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.47283 1.09821
$$11$$ 1.00000 0.301511
$$12$$ 2.47283 0.713846
$$13$$ 4.58774 1.27241 0.636205 0.771520i $$-0.280503\pi$$
0.636205 + 0.771520i $$0.280503\pi$$
$$14$$ 0 0
$$15$$ 1.64207 0.423982
$$16$$ −2.83076 −0.707690
$$17$$ 0.715853 0.173620 0.0868099 0.996225i $$-0.472333\pi$$
0.0868099 + 0.996225i $$0.472333\pi$$
$$18$$ 2.11491 0.498488
$$19$$ 1.64207 0.376718 0.188359 0.982100i $$-0.439683\pi$$
0.188359 + 0.982100i $$0.439683\pi$$
$$20$$ 4.06058 0.907972
$$21$$ 0 0
$$22$$ 2.11491 0.450900
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −2.30359 −0.460719
$$26$$ 9.70265 1.90285
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −5.53341 −1.02753 −0.513764 0.857931i $$-0.671749\pi$$
−0.513764 + 0.857931i $$0.671749\pi$$
$$30$$ 3.47283 0.634050
$$31$$ −5.17548 −0.929544 −0.464772 0.885430i $$-0.653864\pi$$
−0.464772 + 0.885430i $$0.653864\pi$$
$$32$$ −7.98680 −1.41188
$$33$$ 1.00000 0.174078
$$34$$ 1.51396 0.259642
$$35$$ 0 0
$$36$$ 2.47283 0.412139
$$37$$ −6.58774 −1.08302 −0.541509 0.840695i $$-0.682147\pi$$
−0.541509 + 0.840695i $$0.682147\pi$$
$$38$$ 3.47283 0.563368
$$39$$ 4.58774 0.734627
$$40$$ 1.64207 0.259635
$$41$$ 5.17548 0.808275 0.404137 0.914698i $$-0.367572\pi$$
0.404137 + 0.914698i $$0.367572\pi$$
$$42$$ 0 0
$$43$$ 5.17548 0.789254 0.394627 0.918841i $$-0.370874\pi$$
0.394627 + 0.918841i $$0.370874\pi$$
$$44$$ 2.47283 0.372794
$$45$$ 1.64207 0.244786
$$46$$ 0 0
$$47$$ 7.87189 1.14823 0.574116 0.818774i $$-0.305346\pi$$
0.574116 + 0.818774i $$0.305346\pi$$
$$48$$ −2.83076 −0.408585
$$49$$ 0 0
$$50$$ −4.87189 −0.688989
$$51$$ 0.715853 0.100239
$$52$$ 11.3447 1.57323
$$53$$ 2.71585 0.373051 0.186526 0.982450i $$-0.440277\pi$$
0.186526 + 0.982450i $$0.440277\pi$$
$$54$$ 2.11491 0.287802
$$55$$ 1.64207 0.221417
$$56$$ 0 0
$$57$$ 1.64207 0.217498
$$58$$ −11.7026 −1.53663
$$59$$ −3.15604 −0.410881 −0.205440 0.978670i $$-0.565863\pi$$
−0.205440 + 0.978670i $$0.565863\pi$$
$$60$$ 4.06058 0.524218
$$61$$ 13.1755 1.68695 0.843474 0.537170i $$-0.180507\pi$$
0.843474 + 0.537170i $$0.180507\pi$$
$$62$$ −10.9457 −1.39010
$$63$$ 0 0
$$64$$ −11.2298 −1.40373
$$65$$ 7.53341 0.934404
$$66$$ 2.11491 0.260327
$$67$$ 4.21037 0.514378 0.257189 0.966361i $$-0.417204\pi$$
0.257189 + 0.966361i $$0.417204\pi$$
$$68$$ 1.77018 0.214666
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.85244 −0.219844 −0.109922 0.993940i $$-0.535060\pi$$
−0.109922 + 0.993940i $$0.535060\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −10.4791 −1.22648 −0.613242 0.789895i $$-0.710135\pi$$
−0.613242 + 0.789895i $$0.710135\pi$$
$$74$$ −13.9325 −1.61962
$$75$$ −2.30359 −0.265996
$$76$$ 4.06058 0.465780
$$77$$ 0 0
$$78$$ 9.70265 1.09861
$$79$$ −12.4596 −1.40182 −0.700909 0.713251i $$-0.747222\pi$$
−0.700909 + 0.713251i $$0.747222\pi$$
$$80$$ −4.64832 −0.519698
$$81$$ 1.00000 0.111111
$$82$$ 10.9457 1.20875
$$83$$ −9.17548 −1.00714 −0.503570 0.863954i $$-0.667980\pi$$
−0.503570 + 0.863954i $$0.667980\pi$$
$$84$$ 0 0
$$85$$ 1.17548 0.127499
$$86$$ 10.9457 1.18030
$$87$$ −5.53341 −0.593244
$$88$$ 1.00000 0.106600
$$89$$ 4.71585 0.499879 0.249940 0.968261i $$-0.419589\pi$$
0.249940 + 0.968261i $$0.419589\pi$$
$$90$$ 3.47283 0.366069
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.17548 −0.536673
$$94$$ 16.6483 1.71714
$$95$$ 2.69641 0.276645
$$96$$ −7.98680 −0.815149
$$97$$ −13.6351 −1.38444 −0.692218 0.721688i $$-0.743366\pi$$
−0.692218 + 0.721688i $$0.743366\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −5.69641 −0.569641
$$101$$ −14.6072 −1.45347 −0.726735 0.686918i $$-0.758963\pi$$
−0.726735 + 0.686918i $$0.758963\pi$$
$$102$$ 1.51396 0.149905
$$103$$ 17.8913 1.76289 0.881443 0.472291i $$-0.156573\pi$$
0.881443 + 0.472291i $$0.156573\pi$$
$$104$$ 4.58774 0.449865
$$105$$ 0 0
$$106$$ 5.74378 0.557885
$$107$$ 11.1560 1.07849 0.539247 0.842147i $$-0.318709\pi$$
0.539247 + 0.842147i $$0.318709\pi$$
$$108$$ 2.47283 0.237949
$$109$$ −0.108664 −0.0104082 −0.00520408 0.999986i $$-0.501657\pi$$
−0.00520408 + 0.999986i $$0.501657\pi$$
$$110$$ 3.47283 0.331122
$$111$$ −6.58774 −0.625281
$$112$$ 0 0
$$113$$ −13.0668 −1.22922 −0.614611 0.788830i $$-0.710687\pi$$
−0.614611 + 0.788830i $$0.710687\pi$$
$$114$$ 3.47283 0.325261
$$115$$ 0 0
$$116$$ −13.6832 −1.27045
$$117$$ 4.58774 0.424137
$$118$$ −6.67472 −0.614458
$$119$$ 0 0
$$120$$ 1.64207 0.149900
$$121$$ 1.00000 0.0909091
$$122$$ 27.8649 2.52277
$$123$$ 5.17548 0.466658
$$124$$ −12.7981 −1.14930
$$125$$ −11.9930 −1.07269
$$126$$ 0 0
$$127$$ −20.4596 −1.81550 −0.907749 0.419513i $$-0.862201\pi$$
−0.907749 + 0.419513i $$0.862201\pi$$
$$128$$ −7.77643 −0.687346
$$129$$ 5.17548 0.455676
$$130$$ 15.9325 1.39737
$$131$$ −15.0668 −1.31639 −0.658197 0.752846i $$-0.728681\pi$$
−0.658197 + 0.752846i $$0.728681\pi$$
$$132$$ 2.47283 0.215233
$$133$$ 0 0
$$134$$ 8.90454 0.769235
$$135$$ 1.64207 0.141327
$$136$$ 0.715853 0.0613839
$$137$$ −14.4596 −1.23537 −0.617685 0.786426i $$-0.711929\pi$$
−0.617685 + 0.786426i $$0.711929\pi$$
$$138$$ 0 0
$$139$$ −5.13659 −0.435680 −0.217840 0.975985i $$-0.569901\pi$$
−0.217840 + 0.975985i $$0.569901\pi$$
$$140$$ 0 0
$$141$$ 7.87189 0.662933
$$142$$ −3.91774 −0.328770
$$143$$ 4.58774 0.383646
$$144$$ −2.83076 −0.235897
$$145$$ −9.08627 −0.754573
$$146$$ −22.1623 −1.83416
$$147$$ 0 0
$$148$$ −16.2904 −1.33906
$$149$$ 16.8176 1.37775 0.688874 0.724881i $$-0.258105\pi$$
0.688874 + 0.724881i $$0.258105\pi$$
$$150$$ −4.87189 −0.397788
$$151$$ 4.71585 0.383771 0.191885 0.981417i $$-0.438540\pi$$
0.191885 + 0.981417i $$0.438540\pi$$
$$152$$ 1.64207 0.133190
$$153$$ 0.715853 0.0578733
$$154$$ 0 0
$$155$$ −8.49852 −0.682618
$$156$$ 11.3447 0.908305
$$157$$ −11.0279 −0.880124 −0.440062 0.897967i $$-0.645044\pi$$
−0.440062 + 0.897967i $$0.645044\pi$$
$$158$$ −26.3510 −2.09637
$$159$$ 2.71585 0.215381
$$160$$ −13.1149 −1.03682
$$161$$ 0 0
$$162$$ 2.11491 0.166163
$$163$$ 5.64207 0.441921 0.220961 0.975283i $$-0.429081\pi$$
0.220961 + 0.975283i $$0.429081\pi$$
$$164$$ 12.7981 0.999364
$$165$$ 1.64207 0.127835
$$166$$ −19.4053 −1.50614
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ 8.04737 0.619029
$$170$$ 2.48604 0.190670
$$171$$ 1.64207 0.125573
$$172$$ 12.7981 0.975847
$$173$$ −15.0279 −1.14255 −0.571276 0.820758i $$-0.693551\pi$$
−0.571276 + 0.820758i $$0.693551\pi$$
$$174$$ −11.7026 −0.887176
$$175$$ 0 0
$$176$$ −2.83076 −0.213377
$$177$$ −3.15604 −0.237222
$$178$$ 9.97359 0.747552
$$179$$ 9.89134 0.739313 0.369657 0.929168i $$-0.379475\pi$$
0.369657 + 0.929168i $$0.379475\pi$$
$$180$$ 4.06058 0.302657
$$181$$ 6.82452 0.507262 0.253631 0.967301i $$-0.418375\pi$$
0.253631 + 0.967301i $$0.418375\pi$$
$$182$$ 0 0
$$183$$ 13.1755 0.973960
$$184$$ 0 0
$$185$$ −10.8176 −0.795323
$$186$$ −10.9457 −0.802575
$$187$$ 0.715853 0.0523483
$$188$$ 19.4659 1.41969
$$189$$ 0 0
$$190$$ 5.70265 0.413714
$$191$$ 11.0279 0.797953 0.398976 0.916961i $$-0.369366\pi$$
0.398976 + 0.916961i $$0.369366\pi$$
$$192$$ −11.2298 −0.810442
$$193$$ −7.85244 −0.565231 −0.282616 0.959233i $$-0.591202\pi$$
−0.282616 + 0.959233i $$0.591202\pi$$
$$194$$ −28.8370 −2.07038
$$195$$ 7.53341 0.539479
$$196$$ 0 0
$$197$$ 19.8913 1.41720 0.708599 0.705611i $$-0.249327\pi$$
0.708599 + 0.705611i $$0.249327\pi$$
$$198$$ 2.11491 0.150300
$$199$$ 23.5264 1.66775 0.833873 0.551957i $$-0.186119\pi$$
0.833873 + 0.551957i $$0.186119\pi$$
$$200$$ −2.30359 −0.162889
$$201$$ 4.21037 0.296976
$$202$$ −30.8929 −2.17361
$$203$$ 0 0
$$204$$ 1.77018 0.123938
$$205$$ 8.49852 0.593563
$$206$$ 37.8385 2.63633
$$207$$ 0 0
$$208$$ −12.9868 −0.900472
$$209$$ 1.64207 0.113585
$$210$$ 0 0
$$211$$ −4.25622 −0.293010 −0.146505 0.989210i $$-0.546803\pi$$
−0.146505 + 0.989210i $$0.546803\pi$$
$$212$$ 6.71585 0.461247
$$213$$ −1.85244 −0.126927
$$214$$ 23.5940 1.61285
$$215$$ 8.49852 0.579595
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −0.229815 −0.0155650
$$219$$ −10.4791 −0.708110
$$220$$ 4.06058 0.273764
$$221$$ 3.28415 0.220916
$$222$$ −13.9325 −0.935086
$$223$$ −3.74378 −0.250702 −0.125351 0.992112i $$-0.540006\pi$$
−0.125351 + 0.992112i $$0.540006\pi$$
$$224$$ 0 0
$$225$$ −2.30359 −0.153573
$$226$$ −27.6351 −1.83826
$$227$$ −15.4876 −1.02795 −0.513973 0.857807i $$-0.671827\pi$$
−0.513973 + 0.857807i $$0.671827\pi$$
$$228$$ 4.06058 0.268918
$$229$$ −8.00000 −0.528655 −0.264327 0.964433i $$-0.585150\pi$$
−0.264327 + 0.964433i $$0.585150\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −5.53341 −0.363286
$$233$$ −5.54037 −0.362962 −0.181481 0.983394i $$-0.558089\pi$$
−0.181481 + 0.983394i $$0.558089\pi$$
$$234$$ 9.70265 0.634282
$$235$$ 12.9262 0.843214
$$236$$ −7.80435 −0.508020
$$237$$ −12.4596 −0.809340
$$238$$ 0 0
$$239$$ −1.26470 −0.0818067 −0.0409033 0.999163i $$-0.513024\pi$$
−0.0409033 + 0.999163i $$0.513024\pi$$
$$240$$ −4.64832 −0.300048
$$241$$ −23.8719 −1.53772 −0.768862 0.639415i $$-0.779177\pi$$
−0.768862 + 0.639415i $$0.779177\pi$$
$$242$$ 2.11491 0.135951
$$243$$ 1.00000 0.0641500
$$244$$ 32.5808 2.08577
$$245$$ 0 0
$$246$$ 10.9457 0.697870
$$247$$ 7.53341 0.479339
$$248$$ −5.17548 −0.328643
$$249$$ −9.17548 −0.581473
$$250$$ −25.3642 −1.60417
$$251$$ 22.0194 1.38986 0.694928 0.719080i $$-0.255436\pi$$
0.694928 + 0.719080i $$0.255436\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −43.2702 −2.71502
$$255$$ 1.17548 0.0736116
$$256$$ 6.01320 0.375825
$$257$$ 27.3161 1.70393 0.851965 0.523599i $$-0.175411\pi$$
0.851965 + 0.523599i $$0.175411\pi$$
$$258$$ 10.9457 0.681448
$$259$$ 0 0
$$260$$ 18.6289 1.15531
$$261$$ −5.53341 −0.342509
$$262$$ −31.8649 −1.96862
$$263$$ −11.8719 −0.732052 −0.366026 0.930605i $$-0.619282\pi$$
−0.366026 + 0.930605i $$0.619282\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 4.45963 0.273953
$$266$$ 0 0
$$267$$ 4.71585 0.288606
$$268$$ 10.4115 0.635986
$$269$$ 5.63511 0.343579 0.171789 0.985134i $$-0.445045\pi$$
0.171789 + 0.985134i $$0.445045\pi$$
$$270$$ 3.47283 0.211350
$$271$$ −5.68097 −0.345094 −0.172547 0.985001i $$-0.555200\pi$$
−0.172547 + 0.985001i $$0.555200\pi$$
$$272$$ −2.02641 −0.122869
$$273$$ 0 0
$$274$$ −30.5808 −1.84745
$$275$$ −2.30359 −0.138912
$$276$$ 0 0
$$277$$ −25.4876 −1.53140 −0.765699 0.643199i $$-0.777607\pi$$
−0.765699 + 0.643199i $$0.777607\pi$$
$$278$$ −10.8634 −0.651544
$$279$$ −5.17548 −0.309848
$$280$$ 0 0
$$281$$ 17.5723 1.04828 0.524138 0.851633i $$-0.324388\pi$$
0.524138 + 0.851633i $$0.324388\pi$$
$$282$$ 16.6483 0.991393
$$283$$ 25.1296 1.49380 0.746901 0.664936i $$-0.231541\pi$$
0.746901 + 0.664936i $$0.231541\pi$$
$$284$$ −4.58078 −0.271819
$$285$$ 2.69641 0.159721
$$286$$ 9.70265 0.573730
$$287$$ 0 0
$$288$$ −7.98680 −0.470626
$$289$$ −16.4876 −0.969856
$$290$$ −19.2166 −1.12844
$$291$$ −13.6351 −0.799304
$$292$$ −25.9130 −1.51644
$$293$$ −14.3510 −0.838392 −0.419196 0.907896i $$-0.637688\pi$$
−0.419196 + 0.907896i $$0.637688\pi$$
$$294$$ 0 0
$$295$$ −5.18244 −0.301734
$$296$$ −6.58774 −0.382905
$$297$$ 1.00000 0.0580259
$$298$$ 35.5676 2.06037
$$299$$ 0 0
$$300$$ −5.69641 −0.328882
$$301$$ 0 0
$$302$$ 9.97359 0.573916
$$303$$ −14.6072 −0.839161
$$304$$ −4.64832 −0.266599
$$305$$ 21.6351 1.23882
$$306$$ 1.51396 0.0865475
$$307$$ 13.1366 0.749745 0.374872 0.927076i $$-0.377687\pi$$
0.374872 + 0.927076i $$0.377687\pi$$
$$308$$ 0 0
$$309$$ 17.8913 1.01780
$$310$$ −17.9736 −1.02083
$$311$$ −14.3510 −0.813769 −0.406884 0.913480i $$-0.633385\pi$$
−0.406884 + 0.913480i $$0.633385\pi$$
$$312$$ 4.58774 0.259730
$$313$$ 23.7438 1.34208 0.671039 0.741422i $$-0.265848\pi$$
0.671039 + 0.741422i $$0.265848\pi$$
$$314$$ −23.3230 −1.31620
$$315$$ 0 0
$$316$$ −30.8106 −1.73323
$$317$$ −6.45963 −0.362809 −0.181404 0.983409i $$-0.558064\pi$$
−0.181404 + 0.983409i $$0.558064\pi$$
$$318$$ 5.74378 0.322095
$$319$$ −5.53341 −0.309811
$$320$$ −18.4402 −1.03084
$$321$$ 11.1560 0.622669
$$322$$ 0 0
$$323$$ 1.17548 0.0654056
$$324$$ 2.47283 0.137380
$$325$$ −10.5683 −0.586224
$$326$$ 11.9325 0.660878
$$327$$ −0.108664 −0.00600915
$$328$$ 5.17548 0.285768
$$329$$ 0 0
$$330$$ 3.47283 0.191173
$$331$$ 31.2702 1.71877 0.859384 0.511332i $$-0.170848\pi$$
0.859384 + 0.511332i $$0.170848\pi$$
$$332$$ −22.6894 −1.24525
$$333$$ −6.58774 −0.361006
$$334$$ 33.8385 1.85156
$$335$$ 6.91373 0.377738
$$336$$ 0 0
$$337$$ −16.3121 −0.888575 −0.444288 0.895884i $$-0.646543\pi$$
−0.444288 + 0.895884i $$0.646543\pi$$
$$338$$ 17.0194 0.925736
$$339$$ −13.0668 −0.709692
$$340$$ 2.90677 0.157642
$$341$$ −5.17548 −0.280268
$$342$$ 3.47283 0.187789
$$343$$ 0 0
$$344$$ 5.17548 0.279043
$$345$$ 0 0
$$346$$ −31.7827 −1.70865
$$347$$ −6.86341 −0.368447 −0.184224 0.982884i $$-0.558977\pi$$
−0.184224 + 0.982884i $$0.558977\pi$$
$$348$$ −13.6832 −0.733497
$$349$$ 23.8580 1.27709 0.638544 0.769585i $$-0.279537\pi$$
0.638544 + 0.769585i $$0.279537\pi$$
$$350$$ 0 0
$$351$$ 4.58774 0.244876
$$352$$ −7.98680 −0.425698
$$353$$ 11.7368 0.624688 0.312344 0.949969i $$-0.398886\pi$$
0.312344 + 0.949969i $$0.398886\pi$$
$$354$$ −6.67472 −0.354758
$$355$$ −3.04185 −0.161444
$$356$$ 11.6615 0.618059
$$357$$ 0 0
$$358$$ 20.9193 1.10562
$$359$$ −27.7827 −1.46631 −0.733157 0.680060i $$-0.761954\pi$$
−0.733157 + 0.680060i $$0.761954\pi$$
$$360$$ 1.64207 0.0865449
$$361$$ −16.3036 −0.858084
$$362$$ 14.4332 0.758593
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ −17.2074 −0.900677
$$366$$ 27.8649 1.45652
$$367$$ −26.5544 −1.38613 −0.693064 0.720877i $$-0.743739\pi$$
−0.693064 + 0.720877i $$0.743739\pi$$
$$368$$ 0 0
$$369$$ 5.17548 0.269425
$$370$$ −22.8781 −1.18938
$$371$$ 0 0
$$372$$ −12.7981 −0.663551
$$373$$ 5.06682 0.262350 0.131175 0.991359i $$-0.458125\pi$$
0.131175 + 0.991359i $$0.458125\pi$$
$$374$$ 1.51396 0.0782851
$$375$$ −11.9930 −0.619318
$$376$$ 7.87189 0.405962
$$377$$ −25.3859 −1.30744
$$378$$ 0 0
$$379$$ −30.5613 −1.56983 −0.784915 0.619603i $$-0.787294\pi$$
−0.784915 + 0.619603i $$0.787294\pi$$
$$380$$ 6.66776 0.342049
$$381$$ −20.4596 −1.04818
$$382$$ 23.3230 1.19331
$$383$$ 9.13659 0.466858 0.233429 0.972374i $$-0.425005\pi$$
0.233429 + 0.972374i $$0.425005\pi$$
$$384$$ −7.77643 −0.396839
$$385$$ 0 0
$$386$$ −16.6072 −0.845284
$$387$$ 5.17548 0.263085
$$388$$ −33.7174 −1.71174
$$389$$ 21.2702 1.07844 0.539222 0.842164i $$-0.318719\pi$$
0.539222 + 0.842164i $$0.318719\pi$$
$$390$$ 15.9325 0.806772
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −15.0668 −0.760020
$$394$$ 42.0683 2.11937
$$395$$ −20.4596 −1.02944
$$396$$ 2.47283 0.124265
$$397$$ 28.9582 1.45337 0.726684 0.686972i $$-0.241060\pi$$
0.726684 + 0.686972i $$0.241060\pi$$
$$398$$ 49.7563 2.49406
$$399$$ 0 0
$$400$$ 6.52092 0.326046
$$401$$ 32.3510 1.61553 0.807765 0.589505i $$-0.200677\pi$$
0.807765 + 0.589505i $$0.200677\pi$$
$$402$$ 8.90454 0.444118
$$403$$ −23.7438 −1.18276
$$404$$ −36.1212 −1.79709
$$405$$ 1.64207 0.0815953
$$406$$ 0 0
$$407$$ −6.58774 −0.326542
$$408$$ 0.715853 0.0354400
$$409$$ 2.31207 0.114325 0.0571623 0.998365i $$-0.481795\pi$$
0.0571623 + 0.998365i $$0.481795\pi$$
$$410$$ 17.9736 0.887652
$$411$$ −14.4596 −0.713241
$$412$$ 44.2423 2.17966
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −15.0668 −0.739601
$$416$$ −36.6414 −1.79649
$$417$$ −5.13659 −0.251540
$$418$$ 3.47283 0.169862
$$419$$ 38.4263 1.87725 0.938623 0.344945i $$-0.112102\pi$$
0.938623 + 0.344945i $$0.112102\pi$$
$$420$$ 0 0
$$421$$ 15.5070 0.755765 0.377883 0.925854i $$-0.376652\pi$$
0.377883 + 0.925854i $$0.376652\pi$$
$$422$$ −9.00152 −0.438187
$$423$$ 7.87189 0.382744
$$424$$ 2.71585 0.131893
$$425$$ −1.64903 −0.0799899
$$426$$ −3.91774 −0.189815
$$427$$ 0 0
$$428$$ 27.5870 1.33347
$$429$$ 4.58774 0.221498
$$430$$ 17.9736 0.866764
$$431$$ 37.9666 1.82879 0.914394 0.404825i $$-0.132668\pi$$
0.914394 + 0.404825i $$0.132668\pi$$
$$432$$ −2.83076 −0.136195
$$433$$ 10.1087 0.485791 0.242896 0.970052i $$-0.421903\pi$$
0.242896 + 0.970052i $$0.421903\pi$$
$$434$$ 0 0
$$435$$ −9.08627 −0.435653
$$436$$ −0.268709 −0.0128688
$$437$$ 0 0
$$438$$ −22.1623 −1.05895
$$439$$ −22.5224 −1.07494 −0.537469 0.843284i $$-0.680619\pi$$
−0.537469 + 0.843284i $$0.680619\pi$$
$$440$$ 1.64207 0.0782828
$$441$$ 0 0
$$442$$ 6.94567 0.330372
$$443$$ 20.6630 0.981731 0.490865 0.871235i $$-0.336681\pi$$
0.490865 + 0.871235i $$0.336681\pi$$
$$444$$ −16.2904 −0.773108
$$445$$ 7.74378 0.367090
$$446$$ −7.91774 −0.374916
$$447$$ 16.8176 0.795443
$$448$$ 0 0
$$449$$ 40.8106 1.92597 0.962986 0.269553i $$-0.0868759\pi$$
0.962986 + 0.269553i $$0.0868759\pi$$
$$450$$ −4.87189 −0.229663
$$451$$ 5.17548 0.243704
$$452$$ −32.3121 −1.51983
$$453$$ 4.71585 0.221570
$$454$$ −32.7547 −1.53726
$$455$$ 0 0
$$456$$ 1.64207 0.0768971
$$457$$ −28.3899 −1.32802 −0.664011 0.747723i $$-0.731147\pi$$
−0.664011 + 0.747723i $$0.731147\pi$$
$$458$$ −16.9193 −0.790585
$$459$$ 0.715853 0.0334131
$$460$$ 0 0
$$461$$ 21.4178 0.997526 0.498763 0.866739i $$-0.333788\pi$$
0.498763 + 0.866739i $$0.333788\pi$$
$$462$$ 0 0
$$463$$ 20.7478 0.964231 0.482116 0.876108i $$-0.339868\pi$$
0.482116 + 0.876108i $$0.339868\pi$$
$$464$$ 15.6638 0.727172
$$465$$ −8.49852 −0.394110
$$466$$ −11.7174 −0.542797
$$467$$ 26.8998 1.24477 0.622387 0.782709i $$-0.286163\pi$$
0.622387 + 0.782709i $$0.286163\pi$$
$$468$$ 11.3447 0.524410
$$469$$ 0 0
$$470$$ 27.3378 1.26100
$$471$$ −11.0279 −0.508140
$$472$$ −3.15604 −0.145268
$$473$$ 5.17548 0.237969
$$474$$ −26.3510 −1.21034
$$475$$ −3.78267 −0.173561
$$476$$ 0 0
$$477$$ 2.71585 0.124350
$$478$$ −2.67472 −0.122339
$$479$$ 30.8106 1.40777 0.703886 0.710313i $$-0.251447\pi$$
0.703886 + 0.710313i $$0.251447\pi$$
$$480$$ −13.1149 −0.598611
$$481$$ −30.2229 −1.37804
$$482$$ −50.4868 −2.29961
$$483$$ 0 0
$$484$$ 2.47283 0.112402
$$485$$ −22.3899 −1.01667
$$486$$ 2.11491 0.0959342
$$487$$ −23.4876 −1.06432 −0.532161 0.846643i $$-0.678620\pi$$
−0.532161 + 0.846643i $$0.678620\pi$$
$$488$$ 13.1755 0.596426
$$489$$ 5.64207 0.255143
$$490$$ 0 0
$$491$$ −16.6266 −0.750350 −0.375175 0.926954i $$-0.622417\pi$$
−0.375175 + 0.926954i $$0.622417\pi$$
$$492$$ 12.7981 0.576983
$$493$$ −3.96111 −0.178399
$$494$$ 15.9325 0.716835
$$495$$ 1.64207 0.0738057
$$496$$ 14.6506 0.657829
$$497$$ 0 0
$$498$$ −19.4053 −0.869572
$$499$$ −7.99304 −0.357818 −0.178909 0.983866i $$-0.557257\pi$$
−0.178909 + 0.983866i $$0.557257\pi$$
$$500$$ −29.6568 −1.32629
$$501$$ 16.0000 0.714827
$$502$$ 46.5691 2.07848
$$503$$ 24.6630 1.09967 0.549835 0.835273i $$-0.314690\pi$$
0.549835 + 0.835273i $$0.314690\pi$$
$$504$$ 0 0
$$505$$ −23.9861 −1.06737
$$506$$ 0 0
$$507$$ 8.04737 0.357396
$$508$$ −50.5933 −2.24471
$$509$$ 15.0668 0.667825 0.333912 0.942604i $$-0.391631\pi$$
0.333912 + 0.942604i $$0.391631\pi$$
$$510$$ 2.48604 0.110084
$$511$$ 0 0
$$512$$ 28.2702 1.24938
$$513$$ 1.64207 0.0724993
$$514$$ 57.7710 2.54817
$$515$$ 29.3789 1.29459
$$516$$ 12.7981 0.563405
$$517$$ 7.87189 0.346205
$$518$$ 0 0
$$519$$ −15.0279 −0.659653
$$520$$ 7.53341 0.330362
$$521$$ −42.3051 −1.85342 −0.926710 0.375777i $$-0.877376\pi$$
−0.926710 + 0.375777i $$0.877376\pi$$
$$522$$ −11.7026 −0.512211
$$523$$ 5.84548 0.255605 0.127803 0.991800i $$-0.459208\pi$$
0.127803 + 0.991800i $$0.459208\pi$$
$$524$$ −37.2577 −1.62761
$$525$$ 0 0
$$526$$ −25.1079 −1.09476
$$527$$ −3.70488 −0.161387
$$528$$ −2.83076 −0.123193
$$529$$ −23.0000 −1.00000
$$530$$ 9.43171 0.409687
$$531$$ −3.15604 −0.136960
$$532$$ 0 0
$$533$$ 23.7438 1.02846
$$534$$ 9.97359 0.431600
$$535$$ 18.3190 0.792001
$$536$$ 4.21037 0.181860
$$537$$ 9.89134 0.426843
$$538$$ 11.9177 0.513810
$$539$$ 0 0
$$540$$ 4.06058 0.174739
$$541$$ −40.2982 −1.73255 −0.866276 0.499565i $$-0.833493\pi$$
−0.866276 + 0.499565i $$0.833493\pi$$
$$542$$ −12.0147 −0.516076
$$543$$ 6.82452 0.292868
$$544$$ −5.71737 −0.245130
$$545$$ −0.178435 −0.00764331
$$546$$ 0 0
$$547$$ 2.98903 0.127802 0.0639009 0.997956i $$-0.479646\pi$$
0.0639009 + 0.997956i $$0.479646\pi$$
$$548$$ −35.7563 −1.52743
$$549$$ 13.1755 0.562316
$$550$$ −4.87189 −0.207738
$$551$$ −9.08627 −0.387088
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −53.9038 −2.29015
$$555$$ −10.8176 −0.459180
$$556$$ −12.7019 −0.538682
$$557$$ 21.1127 0.894573 0.447286 0.894391i $$-0.352390\pi$$
0.447286 + 0.894391i $$0.352390\pi$$
$$558$$ −10.9457 −0.463367
$$559$$ 23.7438 1.00425
$$560$$ 0 0
$$561$$ 0.715853 0.0302233
$$562$$ 37.1638 1.56766
$$563$$ 29.8774 1.25918 0.629591 0.776926i $$-0.283222\pi$$
0.629591 + 0.776926i $$0.283222\pi$$
$$564$$ 19.4659 0.819661
$$565$$ −21.4567 −0.902689
$$566$$ 53.1468 2.23393
$$567$$ 0 0
$$568$$ −1.85244 −0.0777267
$$569$$ 0.186452 0.00781648 0.00390824 0.999992i $$-0.498756\pi$$
0.00390824 + 0.999992i $$0.498756\pi$$
$$570$$ 5.70265 0.238858
$$571$$ 3.06682 0.128342 0.0641712 0.997939i $$-0.479560\pi$$
0.0641712 + 0.997939i $$0.479560\pi$$
$$572$$ 11.3447 0.474347
$$573$$ 11.0279 0.460698
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −11.2298 −0.467909
$$577$$ 18.8495 0.784715 0.392357 0.919813i $$-0.371660\pi$$
0.392357 + 0.919813i $$0.371660\pi$$
$$578$$ −34.8697 −1.45039
$$579$$ −7.85244 −0.326336
$$580$$ −22.4688 −0.932967
$$581$$ 0 0
$$582$$ −28.8370 −1.19533
$$583$$ 2.71585 0.112479
$$584$$ −10.4791 −0.433627
$$585$$ 7.53341 0.311468
$$586$$ −30.3510 −1.25379
$$587$$ −8.11419 −0.334908 −0.167454 0.985880i $$-0.553555\pi$$
−0.167454 + 0.985880i $$0.553555\pi$$
$$588$$ 0 0
$$589$$ −8.49852 −0.350176
$$590$$ −10.9604 −0.451232
$$591$$ 19.8913 0.818220
$$592$$ 18.6483 0.766441
$$593$$ 6.36489 0.261375 0.130687 0.991424i $$-0.458282\pi$$
0.130687 + 0.991424i $$0.458282\pi$$
$$594$$ 2.11491 0.0867757
$$595$$ 0 0
$$596$$ 41.5870 1.70347
$$597$$ 23.5264 0.962873
$$598$$ 0 0
$$599$$ 0.676959 0.0276598 0.0138299 0.999904i $$-0.495598\pi$$
0.0138299 + 0.999904i $$0.495598\pi$$
$$600$$ −2.30359 −0.0940438
$$601$$ −22.6964 −0.925806 −0.462903 0.886409i $$-0.653192\pi$$
−0.462903 + 0.886409i $$0.653192\pi$$
$$602$$ 0 0
$$603$$ 4.21037 0.171459
$$604$$ 11.6615 0.474501
$$605$$ 1.64207 0.0667598
$$606$$ −30.8929 −1.25494
$$607$$ 14.6141 0.593170 0.296585 0.955006i $$-0.404152\pi$$
0.296585 + 0.955006i $$0.404152\pi$$
$$608$$ −13.1149 −0.531880
$$609$$ 0 0
$$610$$ 45.7563 1.85262
$$611$$ 36.1142 1.46102
$$612$$ 1.77018 0.0715555
$$613$$ −28.3899 −1.14666 −0.573328 0.819326i $$-0.694348\pi$$
−0.573328 + 0.819326i $$0.694348\pi$$
$$614$$ 27.7827 1.12122
$$615$$ 8.49852 0.342694
$$616$$ 0 0
$$617$$ −21.5404 −0.867183 −0.433591 0.901110i $$-0.642754\pi$$
−0.433591 + 0.901110i $$0.642754\pi$$
$$618$$ 37.8385 1.52209
$$619$$ 23.7827 0.955906 0.477953 0.878385i $$-0.341379\pi$$
0.477953 + 0.878385i $$0.341379\pi$$
$$620$$ −21.0154 −0.844000
$$621$$ 0 0
$$622$$ −30.3510 −1.21696
$$623$$ 0 0
$$624$$ −12.9868 −0.519888
$$625$$ −8.17548 −0.327019
$$626$$ 50.2159 2.00703
$$627$$ 1.64207 0.0655781
$$628$$ −27.2702 −1.08820
$$629$$ −4.71585 −0.188033
$$630$$ 0 0
$$631$$ −40.4068 −1.60857 −0.804285 0.594244i $$-0.797451\pi$$
−0.804285 + 0.594244i $$0.797451\pi$$
$$632$$ −12.4596 −0.495617
$$633$$ −4.25622 −0.169170
$$634$$ −13.6615 −0.542568
$$635$$ −33.5962 −1.33323
$$636$$ 6.71585 0.266301
$$637$$ 0 0
$$638$$ −11.7026 −0.463312
$$639$$ −1.85244 −0.0732815
$$640$$ −12.7695 −0.504758
$$641$$ 13.4876 0.532726 0.266363 0.963873i $$-0.414178\pi$$
0.266363 + 0.963873i $$0.414178\pi$$
$$642$$ 23.5940 0.931180
$$643$$ 26.8245 1.05786 0.528928 0.848667i $$-0.322594\pi$$
0.528928 + 0.848667i $$0.322594\pi$$
$$644$$ 0 0
$$645$$ 8.49852 0.334629
$$646$$ 2.48604 0.0978118
$$647$$ 42.4791 1.67002 0.835012 0.550231i $$-0.185460\pi$$
0.835012 + 0.550231i $$0.185460\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −3.15604 −0.123885
$$650$$ −22.3510 −0.876677
$$651$$ 0 0
$$652$$ 13.9519 0.546399
$$653$$ 7.39281 0.289303 0.144652 0.989483i $$-0.453794\pi$$
0.144652 + 0.989483i $$0.453794\pi$$
$$654$$ −0.229815 −0.00898648
$$655$$ −24.7408 −0.966704
$$656$$ −14.6506 −0.572008
$$657$$ −10.4791 −0.408828
$$658$$ 0 0
$$659$$ 18.2368 0.710404 0.355202 0.934790i $$-0.384412\pi$$
0.355202 + 0.934790i $$0.384412\pi$$
$$660$$ 4.06058 0.158058
$$661$$ 8.16451 0.317563 0.158781 0.987314i $$-0.449243\pi$$
0.158781 + 0.987314i $$0.449243\pi$$
$$662$$ 66.1336 2.57036
$$663$$ 3.28415 0.127546
$$664$$ −9.17548 −0.356078
$$665$$ 0 0
$$666$$ −13.9325 −0.539872
$$667$$ 0 0
$$668$$ 39.5653 1.53083
$$669$$ −3.74378 −0.144743
$$670$$ 14.6219 0.564894
$$671$$ 13.1755 0.508634
$$672$$ 0 0
$$673$$ 43.6212 1.68147 0.840737 0.541444i $$-0.182122\pi$$
0.840737 + 0.541444i $$0.182122\pi$$
$$674$$ −34.4985 −1.32883
$$675$$ −2.30359 −0.0886654
$$676$$ 19.8998 0.765377
$$677$$ −23.1057 −0.888025 −0.444012 0.896021i $$-0.646445\pi$$
−0.444012 + 0.896021i $$0.646445\pi$$
$$678$$ −27.6351 −1.06132
$$679$$ 0 0
$$680$$ 1.17548 0.0450777
$$681$$ −15.4876 −0.593484
$$682$$ −10.9457 −0.419131
$$683$$ 27.5653 1.05476 0.527379 0.849630i $$-0.323175\pi$$
0.527379 + 0.849630i $$0.323175\pi$$
$$684$$ 4.06058 0.155260
$$685$$ −23.7438 −0.907203
$$686$$ 0 0
$$687$$ −8.00000 −0.305219
$$688$$ −14.6506 −0.558547
$$689$$ 12.4596 0.474674
$$690$$ 0 0
$$691$$ −5.85244 −0.222637 −0.111319 0.993785i $$-0.535507\pi$$
−0.111319 + 0.993785i $$0.535507\pi$$
$$692$$ −37.1616 −1.41267
$$693$$ 0 0
$$694$$ −14.5155 −0.551000
$$695$$ −8.43466 −0.319945
$$696$$ −5.53341 −0.209743
$$697$$ 3.70488 0.140332
$$698$$ 50.4574 1.90984
$$699$$ −5.54037 −0.209556
$$700$$ 0 0
$$701$$ 9.32304 0.352126 0.176063 0.984379i $$-0.443664\pi$$
0.176063 + 0.984379i $$0.443664\pi$$
$$702$$ 9.70265 0.366203
$$703$$ −10.8176 −0.407992
$$704$$ −11.2298 −0.423240
$$705$$ 12.9262 0.486830
$$706$$ 24.8223 0.934199
$$707$$ 0 0
$$708$$ −7.80435 −0.293306
$$709$$ 41.5629 1.56093 0.780463 0.625202i $$-0.214983\pi$$
0.780463 + 0.625202i $$0.214983\pi$$
$$710$$ −6.43322 −0.241435
$$711$$ −12.4596 −0.467273
$$712$$ 4.71585 0.176734
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 7.53341 0.281734
$$716$$ 24.4596 0.914099
$$717$$ −1.26470 −0.0472311
$$718$$ −58.7578 −2.19282
$$719$$ −31.4372 −1.17241 −0.586205 0.810162i $$-0.699379\pi$$
−0.586205 + 0.810162i $$0.699379\pi$$
$$720$$ −4.64832 −0.173233
$$721$$ 0 0
$$722$$ −34.4806 −1.28323
$$723$$ −23.8719 −0.887805
$$724$$ 16.8759 0.627188
$$725$$ 12.7467 0.473402
$$726$$ 2.11491 0.0784916
$$727$$ −14.2732 −0.529363 −0.264681 0.964336i $$-0.585267\pi$$
−0.264681 + 0.964336i $$0.585267\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −36.3921 −1.34693
$$731$$ 3.70488 0.137030
$$732$$ 32.5808 1.20422
$$733$$ −2.31207 −0.0853983 −0.0426992 0.999088i $$-0.513596\pi$$
−0.0426992 + 0.999088i $$0.513596\pi$$
$$734$$ −56.1600 −2.07291
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.21037 0.155091
$$738$$ 10.9457 0.402916
$$739$$ −26.0558 −0.958480 −0.479240 0.877684i $$-0.659088\pi$$
−0.479240 + 0.877684i $$0.659088\pi$$
$$740$$ −26.7500 −0.983350
$$741$$ 7.53341 0.276747
$$742$$ 0 0
$$743$$ 5.04737 0.185170 0.0925851 0.995705i $$-0.470487\pi$$
0.0925851 + 0.995705i $$0.470487\pi$$
$$744$$ −5.17548 −0.189742
$$745$$ 27.6157 1.01176
$$746$$ 10.7159 0.392335
$$747$$ −9.17548 −0.335713
$$748$$ 1.77018 0.0647244
$$749$$ 0 0
$$750$$ −25.3642 −0.926169
$$751$$ −40.4387 −1.47563 −0.737815 0.675002i $$-0.764143\pi$$
−0.737815 + 0.675002i $$0.764143\pi$$
$$752$$ −22.2834 −0.812593
$$753$$ 22.0194 0.802433
$$754$$ −53.6887 −1.95523
$$755$$ 7.74378 0.281825
$$756$$ 0 0
$$757$$ −24.4263 −0.887788 −0.443894 0.896079i $$-0.646403\pi$$
−0.443894 + 0.896079i $$0.646403\pi$$
$$758$$ −64.6344 −2.34763
$$759$$ 0 0
$$760$$ 2.69641 0.0978089
$$761$$ 5.18940 0.188116 0.0940579 0.995567i $$-0.470016\pi$$
0.0940579 + 0.995567i $$0.470016\pi$$
$$762$$ −43.2702 −1.56751
$$763$$ 0 0
$$764$$ 27.2702 0.986602
$$765$$ 1.17548 0.0424997
$$766$$ 19.3230 0.698170
$$767$$ −14.4791 −0.522809
$$768$$ 6.01320 0.216983
$$769$$ −28.1670 −1.01573 −0.507864 0.861437i $$-0.669565\pi$$
−0.507864 + 0.861437i $$0.669565\pi$$
$$770$$ 0 0
$$771$$ 27.3161 0.983765
$$772$$ −19.4178 −0.698861
$$773$$ 38.2662 1.37634 0.688170 0.725549i $$-0.258414\pi$$
0.688170 + 0.725549i $$0.258414\pi$$
$$774$$ 10.9457 0.393434
$$775$$ 11.9222 0.428259
$$776$$ −13.6351 −0.489472
$$777$$ 0 0
$$778$$ 44.9846 1.61277
$$779$$ 8.49852 0.304491
$$780$$ 18.6289 0.667021
$$781$$ −1.85244 −0.0662856
$$782$$ 0 0
$$783$$ −5.53341 −0.197748
$$784$$ 0 0
$$785$$ −18.1087 −0.646326
$$786$$ −31.8649 −1.13658
$$787$$ −13.4247 −0.478540 −0.239270 0.970953i $$-0.576908\pi$$
−0.239270 + 0.970953i $$0.576908\pi$$
$$788$$ 49.1880 1.75225
$$789$$ −11.8719 −0.422650
$$790$$ −43.2702 −1.53949
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ 60.4457 2.14649
$$794$$ 61.2438 2.17346
$$795$$ 4.45963 0.158167
$$796$$ 58.1770 2.06203
$$797$$ −22.3440 −0.791465 −0.395733 0.918366i $$-0.629509\pi$$
−0.395733 + 0.918366i $$0.629509\pi$$
$$798$$ 0 0
$$799$$ 5.63511 0.199356
$$800$$ 18.3983 0.650479
$$801$$ 4.71585 0.166626
$$802$$ 68.4193 2.41597
$$803$$ −10.4791 −0.369799
$$804$$ 10.4115 0.367187
$$805$$ 0 0
$$806$$ −50.2159 −1.76878
$$807$$ 5.63511 0.198365
$$808$$ −14.6072 −0.513879
$$809$$ 31.8455 1.11963 0.559814 0.828618i $$-0.310873\pi$$
0.559814 + 0.828618i $$0.310873\pi$$
$$810$$ 3.47283 0.122023
$$811$$ 8.70889 0.305811 0.152905 0.988241i $$-0.451137\pi$$
0.152905 + 0.988241i $$0.451137\pi$$
$$812$$ 0 0
$$813$$ −5.68097 −0.199240
$$814$$ −13.9325 −0.488333
$$815$$ 9.26470 0.324528
$$816$$ −2.02641 −0.0709385
$$817$$ 8.49852 0.297326
$$818$$ 4.88982 0.170968
$$819$$ 0 0
$$820$$ 21.0154 0.733891
$$821$$ 6.45267 0.225200 0.112600 0.993640i $$-0.464082\pi$$
0.112600 + 0.993640i $$0.464082\pi$$
$$822$$ −30.5808 −1.06663
$$823$$ 25.3719 0.884410 0.442205 0.896914i $$-0.354196\pi$$
0.442205 + 0.896914i $$0.354196\pi$$
$$824$$ 17.8913 0.623274
$$825$$ −2.30359 −0.0802009
$$826$$ 0 0
$$827$$ 14.0194 0.487504 0.243752 0.969838i $$-0.421622\pi$$
0.243752 + 0.969838i $$0.421622\pi$$
$$828$$ 0 0
$$829$$ −48.9970 −1.70174 −0.850869 0.525378i $$-0.823924\pi$$
−0.850869 + 0.525378i $$0.823924\pi$$
$$830$$ −31.8649 −1.10605
$$831$$ −25.4876 −0.884153
$$832$$ −51.5195 −1.78612
$$833$$ 0 0
$$834$$ −10.8634 −0.376169
$$835$$ 26.2732 0.909221
$$836$$ 4.06058 0.140438
$$837$$ −5.17548 −0.178891
$$838$$ 81.2680 2.80736
$$839$$ −13.2647 −0.457948 −0.228974 0.973433i $$-0.573537\pi$$
−0.228974 + 0.973433i $$0.573537\pi$$
$$840$$ 0 0
$$841$$ 1.61862 0.0558144
$$842$$ 32.7959 1.13022
$$843$$ 17.5723 0.605222
$$844$$ −10.5249 −0.362283
$$845$$ 13.2144 0.454588
$$846$$ 16.6483 0.572381
$$847$$ 0 0
$$848$$ −7.68793 −0.264005
$$849$$ 25.1296 0.862447
$$850$$ −3.48755 −0.119622
$$851$$ 0 0
$$852$$ −4.58078 −0.156935
$$853$$ −4.25622 −0.145730 −0.0728651 0.997342i $$-0.523214\pi$$
−0.0728651 + 0.997342i $$0.523214\pi$$
$$854$$ 0 0
$$855$$ 2.69641 0.0922151
$$856$$ 11.1560 0.381305
$$857$$ −23.5264 −0.803648 −0.401824 0.915717i $$-0.631624\pi$$
−0.401824 + 0.915717i $$0.631624\pi$$
$$858$$ 9.70265 0.331243
$$859$$ −31.0279 −1.05866 −0.529330 0.848416i $$-0.677556\pi$$
−0.529330 + 0.848416i $$0.677556\pi$$
$$860$$ 21.0154 0.716620
$$861$$ 0 0
$$862$$ 80.2959 2.73489
$$863$$ −46.3121 −1.57648 −0.788241 0.615367i $$-0.789008\pi$$
−0.788241 + 0.615367i $$0.789008\pi$$
$$864$$ −7.98680 −0.271716
$$865$$ −24.6770 −0.839042
$$866$$ 21.3789 0.726484
$$867$$ −16.4876 −0.559947
$$868$$ 0 0
$$869$$ −12.4596 −0.422664
$$870$$ −19.2166 −0.651504
$$871$$ 19.3161 0.654500
$$872$$ −0.108664 −0.00367984
$$873$$ −13.6351 −0.461479
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −25.9130 −0.875520
$$877$$ −12.8245 −0.433053 −0.216527 0.976277i $$-0.569473\pi$$
−0.216527 + 0.976277i $$0.569473\pi$$
$$878$$ −47.6329 −1.60753
$$879$$ −14.3510 −0.484046
$$880$$ −4.64832 −0.156695
$$881$$ −56.9512 −1.91873 −0.959367 0.282160i $$-0.908949\pi$$
−0.959367 + 0.282160i $$0.908949\pi$$
$$882$$ 0 0
$$883$$ 8.00696 0.269456 0.134728 0.990883i $$-0.456984\pi$$
0.134728 + 0.990883i $$0.456984\pi$$
$$884$$ 8.12115 0.273144
$$885$$ −5.18244 −0.174206
$$886$$ 43.7004 1.46814
$$887$$ 35.7827 1.20146 0.600732 0.799450i $$-0.294876\pi$$
0.600732 + 0.799450i $$0.294876\pi$$
$$888$$ −6.58774 −0.221070
$$889$$ 0 0
$$890$$ 16.3774 0.548971
$$891$$ 1.00000 0.0335013
$$892$$ −9.25774 −0.309972
$$893$$ 12.9262 0.432559
$$894$$ 35.5676 1.18956
$$895$$ 16.2423 0.542920
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 86.3106 2.88022
$$899$$ 28.6381 0.955133
$$900$$ −5.69641 −0.189880
$$901$$ 1.94415 0.0647690
$$902$$ 10.9457 0.364451
$$903$$ 0 0
$$904$$ −13.0668 −0.434596
$$905$$ 11.2064 0.372512
$$906$$ 9.97359 0.331350
$$907$$ 5.43171 0.180357 0.0901784 0.995926i $$-0.471256\pi$$
0.0901784 + 0.995926i $$0.471256\pi$$
$$908$$ −38.2982 −1.27097
$$909$$ −14.6072 −0.484490
$$910$$ 0 0
$$911$$ 16.6770 0.552532 0.276266 0.961081i $$-0.410903\pi$$
0.276266 + 0.961081i $$0.410903\pi$$
$$912$$ −4.64832 −0.153921
$$913$$ −9.17548 −0.303664
$$914$$ −60.0419 −1.98601
$$915$$ 21.6351 0.715235
$$916$$ −19.7827 −0.653638
$$917$$ 0 0
$$918$$ 1.51396 0.0499682
$$919$$ 14.4038 0.475137 0.237568 0.971371i $$-0.423650\pi$$
0.237568 + 0.971371i $$0.423650\pi$$
$$920$$ 0 0
$$921$$ 13.1366 0.432865
$$922$$ 45.2966 1.49177
$$923$$ −8.49852 −0.279732
$$924$$ 0 0
$$925$$ 15.1755 0.498967
$$926$$ 43.8796 1.44197
$$927$$ 17.8913 0.587629
$$928$$ 44.1942 1.45075
$$929$$ 1.89830 0.0622811 0.0311405 0.999515i $$-0.490086\pi$$
0.0311405 + 0.999515i $$0.490086\pi$$
$$930$$ −17.9736 −0.589377
$$931$$ 0 0
$$932$$ −13.7004 −0.448772
$$933$$ −14.3510 −0.469830
$$934$$ 56.8906 1.86152
$$935$$ 1.17548 0.0384424
$$936$$ 4.58774 0.149955
$$937$$ −13.1755 −0.430424 −0.215212 0.976567i $$-0.569044\pi$$
−0.215212 + 0.976567i $$0.569044\pi$$
$$938$$ 0 0
$$939$$ 23.7438 0.774849
$$940$$ 31.9644 1.04256
$$941$$ 0.381842 0.0124477 0.00622385 0.999981i $$-0.498019\pi$$
0.00622385 + 0.999981i $$0.498019\pi$$
$$942$$ −23.3230 −0.759906
$$943$$ 0 0
$$944$$ 8.93398 0.290776
$$945$$ 0 0
$$946$$ 10.9457 0.355874
$$947$$ −52.2284 −1.69719 −0.848597 0.529040i $$-0.822552\pi$$
−0.848597 + 0.529040i $$0.822552\pi$$
$$948$$ −30.8106 −1.00068
$$949$$ −48.0753 −1.56059
$$950$$ −8.00000 −0.259554
$$951$$ −6.45963 −0.209468
$$952$$ 0 0
$$953$$ −45.7757 −1.48282 −0.741410 0.671052i $$-0.765843\pi$$
−0.741410 + 0.671052i $$0.765843\pi$$
$$954$$ 5.74378 0.185962
$$955$$ 18.1087 0.585983
$$956$$ −3.12739 −0.101147
$$957$$ −5.53341 −0.178870
$$958$$ 65.1616 2.10527
$$959$$ 0 0
$$960$$ −18.4402 −0.595154
$$961$$ −4.21438 −0.135948
$$962$$ −63.9185 −2.06082
$$963$$ 11.1560 0.359498
$$964$$ −59.0312 −1.90127
$$965$$ −12.8943 −0.415082
$$966$$ 0 0
$$967$$ −15.3230 −0.492756 −0.246378 0.969174i $$-0.579240\pi$$
−0.246378 + 0.969174i $$0.579240\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 1.17548 0.0377620
$$970$$ −47.3525 −1.52040
$$971$$ 12.0753 0.387515 0.193757 0.981049i $$-0.437933\pi$$
0.193757 + 0.981049i $$0.437933\pi$$
$$972$$ 2.47283 0.0793162
$$973$$ 0 0
$$974$$ −49.6740 −1.59166
$$975$$ −10.5683 −0.338456
$$976$$ −37.2966 −1.19384
$$977$$ −39.3650 −1.25940 −0.629698 0.776840i $$-0.716822\pi$$
−0.629698 + 0.776840i $$0.716822\pi$$
$$978$$ 11.9325 0.381558
$$979$$ 4.71585 0.150719
$$980$$ 0 0
$$981$$ −0.108664 −0.00346939
$$982$$ −35.1638 −1.12212
$$983$$ 10.5683 0.337076 0.168538 0.985695i $$-0.446095\pi$$
0.168538 + 0.985695i $$0.446095\pi$$
$$984$$ 5.17548 0.164988
$$985$$ 32.6630 1.04073
$$986$$ −8.37737 −0.266790
$$987$$ 0 0
$$988$$ 18.6289 0.592663
$$989$$ 0 0
$$990$$ 3.47283 0.110374
$$991$$ 42.8036 1.35970 0.679851 0.733350i $$-0.262044\pi$$
0.679851 + 0.733350i $$0.262044\pi$$
$$992$$ 41.3355 1.31240
$$993$$ 31.2702 0.992331
$$994$$ 0 0
$$995$$ 38.6322 1.22472
$$996$$ −22.6894 −0.718943
$$997$$ −10.3121 −0.326587 −0.163293 0.986578i $$-0.552212\pi$$
−0.163293 + 0.986578i $$0.552212\pi$$
$$998$$ −16.9045 −0.535104
$$999$$ −6.58774 −0.208427
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 1617.2.a.r.1.3 yes 3
3.2 odd 2 4851.2.a.bl.1.1 3
7.6 odd 2 1617.2.a.q.1.3 3
21.20 even 2 4851.2.a.bm.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.q.1.3 3 7.6 odd 2
1617.2.a.r.1.3 yes 3 1.1 even 1 trivial
4851.2.a.bl.1.1 3 3.2 odd 2
4851.2.a.bm.1.1 3 21.20 even 2