# Properties

 Label 7098.2.a.bw.1.1 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.56155 q^{10} +2.56155 q^{11} +1.00000 q^{12} -1.00000 q^{14} -3.56155 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +7.68466 q^{19} -3.56155 q^{20} -1.00000 q^{21} +2.56155 q^{22} -9.12311 q^{23} +1.00000 q^{24} +7.68466 q^{25} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} -3.56155 q^{30} +5.12311 q^{31} +1.00000 q^{32} +2.56155 q^{33} -5.00000 q^{34} +3.56155 q^{35} +1.00000 q^{36} -9.80776 q^{37} +7.68466 q^{38} -3.56155 q^{40} +4.12311 q^{41} -1.00000 q^{42} -2.87689 q^{43} +2.56155 q^{44} -3.56155 q^{45} -9.12311 q^{46} -7.68466 q^{47} +1.00000 q^{48} +1.00000 q^{49} +7.68466 q^{50} -5.00000 q^{51} -3.87689 q^{53} +1.00000 q^{54} -9.12311 q^{55} -1.00000 q^{56} +7.68466 q^{57} -1.00000 q^{58} +13.1231 q^{59} -3.56155 q^{60} -1.00000 q^{61} +5.12311 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.56155 q^{66} +1.12311 q^{67} -5.00000 q^{68} -9.12311 q^{69} +3.56155 q^{70} -4.00000 q^{71} +1.00000 q^{72} -2.43845 q^{73} -9.80776 q^{74} +7.68466 q^{75} +7.68466 q^{76} -2.56155 q^{77} +1.43845 q^{79} -3.56155 q^{80} +1.00000 q^{81} +4.12311 q^{82} -10.2462 q^{83} -1.00000 q^{84} +17.8078 q^{85} -2.87689 q^{86} -1.00000 q^{87} +2.56155 q^{88} -9.68466 q^{89} -3.56155 q^{90} -9.12311 q^{92} +5.12311 q^{93} -7.68466 q^{94} -27.3693 q^{95} +1.00000 q^{96} +12.2462 q^{97} +1.00000 q^{98} +2.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 3q^{10} + q^{11} + 2q^{12} - 2q^{14} - 3q^{15} + 2q^{16} - 10q^{17} + 2q^{18} + 3q^{19} - 3q^{20} - 2q^{21} + q^{22} - 10q^{23} + 2q^{24} + 3q^{25} + 2q^{27} - 2q^{28} - 2q^{29} - 3q^{30} + 2q^{31} + 2q^{32} + q^{33} - 10q^{34} + 3q^{35} + 2q^{36} + q^{37} + 3q^{38} - 3q^{40} - 2q^{42} - 14q^{43} + q^{44} - 3q^{45} - 10q^{46} - 3q^{47} + 2q^{48} + 2q^{49} + 3q^{50} - 10q^{51} - 16q^{53} + 2q^{54} - 10q^{55} - 2q^{56} + 3q^{57} - 2q^{58} + 18q^{59} - 3q^{60} - 2q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + q^{66} - 6q^{67} - 10q^{68} - 10q^{69} + 3q^{70} - 8q^{71} + 2q^{72} - 9q^{73} + q^{74} + 3q^{75} + 3q^{76} - q^{77} + 7q^{79} - 3q^{80} + 2q^{81} - 4q^{83} - 2q^{84} + 15q^{85} - 14q^{86} - 2q^{87} + q^{88} - 7q^{89} - 3q^{90} - 10q^{92} + 2q^{93} - 3q^{94} - 30q^{95} + 2q^{96} + 8q^{97} + 2q^{98} + q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.56155 −1.59277 −0.796387 0.604787i $$-0.793258\pi$$
−0.796387 + 0.604787i $$0.793258\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −3.56155 −1.12626
$$11$$ 2.56155 0.772337 0.386169 0.922428i $$-0.373798\pi$$
0.386169 + 0.922428i $$0.373798\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −1.00000 −0.267261
$$15$$ −3.56155 −0.919589
$$16$$ 1.00000 0.250000
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 7.68466 1.76298 0.881491 0.472201i $$-0.156540\pi$$
0.881491 + 0.472201i $$0.156540\pi$$
$$20$$ −3.56155 −0.796387
$$21$$ −1.00000 −0.218218
$$22$$ 2.56155 0.546125
$$23$$ −9.12311 −1.90230 −0.951150 0.308731i $$-0.900096\pi$$
−0.951150 + 0.308731i $$0.900096\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 7.68466 1.53693
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ −3.56155 −0.650248
$$31$$ 5.12311 0.920137 0.460068 0.887883i $$-0.347825\pi$$
0.460068 + 0.887883i $$0.347825\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.56155 0.445909
$$34$$ −5.00000 −0.857493
$$35$$ 3.56155 0.602012
$$36$$ 1.00000 0.166667
$$37$$ −9.80776 −1.61239 −0.806193 0.591652i $$-0.798476\pi$$
−0.806193 + 0.591652i $$0.798476\pi$$
$$38$$ 7.68466 1.24662
$$39$$ 0 0
$$40$$ −3.56155 −0.563131
$$41$$ 4.12311 0.643921 0.321960 0.946753i $$-0.395658\pi$$
0.321960 + 0.946753i $$0.395658\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −2.87689 −0.438722 −0.219361 0.975644i $$-0.570397\pi$$
−0.219361 + 0.975644i $$0.570397\pi$$
$$44$$ 2.56155 0.386169
$$45$$ −3.56155 −0.530925
$$46$$ −9.12311 −1.34513
$$47$$ −7.68466 −1.12092 −0.560461 0.828181i $$-0.689376\pi$$
−0.560461 + 0.828181i $$0.689376\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 7.68466 1.08677
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ −3.87689 −0.532532 −0.266266 0.963900i $$-0.585790\pi$$
−0.266266 + 0.963900i $$0.585790\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −9.12311 −1.23016
$$56$$ −1.00000 −0.133631
$$57$$ 7.68466 1.01786
$$58$$ −1.00000 −0.131306
$$59$$ 13.1231 1.70848 0.854241 0.519877i $$-0.174022\pi$$
0.854241 + 0.519877i $$0.174022\pi$$
$$60$$ −3.56155 −0.459794
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 5.12311 0.650635
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.56155 0.315305
$$67$$ 1.12311 0.137209 0.0686046 0.997644i $$-0.478145\pi$$
0.0686046 + 0.997644i $$0.478145\pi$$
$$68$$ −5.00000 −0.606339
$$69$$ −9.12311 −1.09829
$$70$$ 3.56155 0.425687
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −2.43845 −0.285399 −0.142699 0.989766i $$-0.545578\pi$$
−0.142699 + 0.989766i $$0.545578\pi$$
$$74$$ −9.80776 −1.14013
$$75$$ 7.68466 0.887348
$$76$$ 7.68466 0.881491
$$77$$ −2.56155 −0.291916
$$78$$ 0 0
$$79$$ 1.43845 0.161838 0.0809190 0.996721i $$-0.474214\pi$$
0.0809190 + 0.996721i $$0.474214\pi$$
$$80$$ −3.56155 −0.398194
$$81$$ 1.00000 0.111111
$$82$$ 4.12311 0.455321
$$83$$ −10.2462 −1.12467 −0.562334 0.826910i $$-0.690096\pi$$
−0.562334 + 0.826910i $$0.690096\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 17.8078 1.93152
$$86$$ −2.87689 −0.310223
$$87$$ −1.00000 −0.107211
$$88$$ 2.56155 0.273062
$$89$$ −9.68466 −1.02657 −0.513286 0.858218i $$-0.671572\pi$$
−0.513286 + 0.858218i $$0.671572\pi$$
$$90$$ −3.56155 −0.375421
$$91$$ 0 0
$$92$$ −9.12311 −0.951150
$$93$$ 5.12311 0.531241
$$94$$ −7.68466 −0.792612
$$95$$ −27.3693 −2.80803
$$96$$ 1.00000 0.102062
$$97$$ 12.2462 1.24341 0.621707 0.783250i $$-0.286439\pi$$
0.621707 + 0.783250i $$0.286439\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 2.56155 0.257446
$$100$$ 7.68466 0.768466
$$101$$ −19.5616 −1.94645 −0.973224 0.229860i $$-0.926173\pi$$
−0.973224 + 0.229860i $$0.926173\pi$$
$$102$$ −5.00000 −0.495074
$$103$$ −11.3693 −1.12025 −0.560126 0.828407i $$-0.689247\pi$$
−0.560126 + 0.828407i $$0.689247\pi$$
$$104$$ 0 0
$$105$$ 3.56155 0.347572
$$106$$ −3.87689 −0.376557
$$107$$ −9.43845 −0.912449 −0.456225 0.889865i $$-0.650799\pi$$
−0.456225 + 0.889865i $$0.650799\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 0.876894 0.0839912 0.0419956 0.999118i $$-0.486628\pi$$
0.0419956 + 0.999118i $$0.486628\pi$$
$$110$$ −9.12311 −0.869854
$$111$$ −9.80776 −0.930912
$$112$$ −1.00000 −0.0944911
$$113$$ 16.9309 1.59272 0.796361 0.604821i $$-0.206756\pi$$
0.796361 + 0.604821i $$0.206756\pi$$
$$114$$ 7.68466 0.719734
$$115$$ 32.4924 3.02993
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ 13.1231 1.20808
$$119$$ 5.00000 0.458349
$$120$$ −3.56155 −0.325124
$$121$$ −4.43845 −0.403495
$$122$$ −1.00000 −0.0905357
$$123$$ 4.12311 0.371768
$$124$$ 5.12311 0.460068
$$125$$ −9.56155 −0.855211
$$126$$ −1.00000 −0.0890871
$$127$$ 2.24621 0.199319 0.0996595 0.995022i $$-0.468225\pi$$
0.0996595 + 0.995022i $$0.468225\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −2.87689 −0.253296
$$130$$ 0 0
$$131$$ −16.4924 −1.44095 −0.720475 0.693481i $$-0.756076\pi$$
−0.720475 + 0.693481i $$0.756076\pi$$
$$132$$ 2.56155 0.222955
$$133$$ −7.68466 −0.666344
$$134$$ 1.12311 0.0970215
$$135$$ −3.56155 −0.306530
$$136$$ −5.00000 −0.428746
$$137$$ −5.80776 −0.496191 −0.248095 0.968736i $$-0.579805\pi$$
−0.248095 + 0.968736i $$0.579805\pi$$
$$138$$ −9.12311 −0.776610
$$139$$ −7.68466 −0.651804 −0.325902 0.945404i $$-0.605668\pi$$
−0.325902 + 0.945404i $$0.605668\pi$$
$$140$$ 3.56155 0.301006
$$141$$ −7.68466 −0.647165
$$142$$ −4.00000 −0.335673
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 3.56155 0.295771
$$146$$ −2.43845 −0.201807
$$147$$ 1.00000 0.0824786
$$148$$ −9.80776 −0.806193
$$149$$ −17.8078 −1.45887 −0.729434 0.684051i $$-0.760217\pi$$
−0.729434 + 0.684051i $$0.760217\pi$$
$$150$$ 7.68466 0.627450
$$151$$ −17.4384 −1.41912 −0.709560 0.704645i $$-0.751106\pi$$
−0.709560 + 0.704645i $$0.751106\pi$$
$$152$$ 7.68466 0.623308
$$153$$ −5.00000 −0.404226
$$154$$ −2.56155 −0.206416
$$155$$ −18.2462 −1.46557
$$156$$ 0 0
$$157$$ −5.80776 −0.463510 −0.231755 0.972774i $$-0.574447\pi$$
−0.231755 + 0.972774i $$0.574447\pi$$
$$158$$ 1.43845 0.114437
$$159$$ −3.87689 −0.307458
$$160$$ −3.56155 −0.281565
$$161$$ 9.12311 0.719001
$$162$$ 1.00000 0.0785674
$$163$$ −1.12311 −0.0879684 −0.0439842 0.999032i $$-0.514005\pi$$
−0.0439842 + 0.999032i $$0.514005\pi$$
$$164$$ 4.12311 0.321960
$$165$$ −9.12311 −0.710233
$$166$$ −10.2462 −0.795260
$$167$$ 20.4924 1.58575 0.792876 0.609383i $$-0.208583\pi$$
0.792876 + 0.609383i $$0.208583\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 0 0
$$170$$ 17.8078 1.36579
$$171$$ 7.68466 0.587661
$$172$$ −2.87689 −0.219361
$$173$$ −4.87689 −0.370783 −0.185392 0.982665i $$-0.559355\pi$$
−0.185392 + 0.982665i $$0.559355\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ −7.68466 −0.580906
$$176$$ 2.56155 0.193084
$$177$$ 13.1231 0.986393
$$178$$ −9.68466 −0.725896
$$179$$ −16.4924 −1.23270 −0.616351 0.787472i $$-0.711390\pi$$
−0.616351 + 0.787472i $$0.711390\pi$$
$$180$$ −3.56155 −0.265462
$$181$$ 13.2462 0.984583 0.492292 0.870430i $$-0.336159\pi$$
0.492292 + 0.870430i $$0.336159\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ −9.12311 −0.672564
$$185$$ 34.9309 2.56817
$$186$$ 5.12311 0.375644
$$187$$ −12.8078 −0.936596
$$188$$ −7.68466 −0.560461
$$189$$ −1.00000 −0.0727393
$$190$$ −27.3693 −1.98558
$$191$$ 5.75379 0.416330 0.208165 0.978094i $$-0.433251\pi$$
0.208165 + 0.978094i $$0.433251\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −20.3693 −1.46622 −0.733108 0.680113i $$-0.761931\pi$$
−0.733108 + 0.680113i $$0.761931\pi$$
$$194$$ 12.2462 0.879227
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −0.561553 −0.0400090 −0.0200045 0.999800i $$-0.506368\pi$$
−0.0200045 + 0.999800i $$0.506368\pi$$
$$198$$ 2.56155 0.182042
$$199$$ −17.6155 −1.24873 −0.624366 0.781132i $$-0.714643\pi$$
−0.624366 + 0.781132i $$0.714643\pi$$
$$200$$ 7.68466 0.543387
$$201$$ 1.12311 0.0792178
$$202$$ −19.5616 −1.37635
$$203$$ 1.00000 0.0701862
$$204$$ −5.00000 −0.350070
$$205$$ −14.6847 −1.02562
$$206$$ −11.3693 −0.792138
$$207$$ −9.12311 −0.634100
$$208$$ 0 0
$$209$$ 19.6847 1.36162
$$210$$ 3.56155 0.245770
$$211$$ −2.87689 −0.198054 −0.0990268 0.995085i $$-0.531573\pi$$
−0.0990268 + 0.995085i $$0.531573\pi$$
$$212$$ −3.87689 −0.266266
$$213$$ −4.00000 −0.274075
$$214$$ −9.43845 −0.645199
$$215$$ 10.2462 0.698786
$$216$$ 1.00000 0.0680414
$$217$$ −5.12311 −0.347779
$$218$$ 0.876894 0.0593908
$$219$$ −2.43845 −0.164775
$$220$$ −9.12311 −0.615080
$$221$$ 0 0
$$222$$ −9.80776 −0.658254
$$223$$ −16.4924 −1.10441 −0.552207 0.833707i $$-0.686214\pi$$
−0.552207 + 0.833707i $$0.686214\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 7.68466 0.512311
$$226$$ 16.9309 1.12622
$$227$$ 13.1231 0.871011 0.435506 0.900186i $$-0.356570\pi$$
0.435506 + 0.900186i $$0.356570\pi$$
$$228$$ 7.68466 0.508929
$$229$$ 22.8078 1.50718 0.753590 0.657345i $$-0.228321\pi$$
0.753590 + 0.657345i $$0.228321\pi$$
$$230$$ 32.4924 2.14249
$$231$$ −2.56155 −0.168538
$$232$$ −1.00000 −0.0656532
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 27.3693 1.78538
$$236$$ 13.1231 0.854241
$$237$$ 1.43845 0.0934372
$$238$$ 5.00000 0.324102
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ −3.56155 −0.229897
$$241$$ 11.3153 0.728885 0.364443 0.931226i $$-0.381260\pi$$
0.364443 + 0.931226i $$0.381260\pi$$
$$242$$ −4.43845 −0.285314
$$243$$ 1.00000 0.0641500
$$244$$ −1.00000 −0.0640184
$$245$$ −3.56155 −0.227539
$$246$$ 4.12311 0.262880
$$247$$ 0 0
$$248$$ 5.12311 0.325318
$$249$$ −10.2462 −0.649327
$$250$$ −9.56155 −0.604726
$$251$$ 8.49242 0.536037 0.268018 0.963414i $$-0.413631\pi$$
0.268018 + 0.963414i $$0.413631\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ −23.3693 −1.46922
$$254$$ 2.24621 0.140940
$$255$$ 17.8078 1.11517
$$256$$ 1.00000 0.0625000
$$257$$ −14.7538 −0.920316 −0.460158 0.887837i $$-0.652207\pi$$
−0.460158 + 0.887837i $$0.652207\pi$$
$$258$$ −2.87689 −0.179108
$$259$$ 9.80776 0.609425
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ −16.4924 −1.01891
$$263$$ −20.4924 −1.26362 −0.631808 0.775125i $$-0.717687\pi$$
−0.631808 + 0.775125i $$0.717687\pi$$
$$264$$ 2.56155 0.157653
$$265$$ 13.8078 0.848204
$$266$$ −7.68466 −0.471177
$$267$$ −9.68466 −0.592691
$$268$$ 1.12311 0.0686046
$$269$$ −4.87689 −0.297349 −0.148675 0.988886i $$-0.547501\pi$$
−0.148675 + 0.988886i $$0.547501\pi$$
$$270$$ −3.56155 −0.216749
$$271$$ 13.1231 0.797172 0.398586 0.917131i $$-0.369501\pi$$
0.398586 + 0.917131i $$0.369501\pi$$
$$272$$ −5.00000 −0.303170
$$273$$ 0 0
$$274$$ −5.80776 −0.350860
$$275$$ 19.6847 1.18703
$$276$$ −9.12311 −0.549146
$$277$$ −3.56155 −0.213993 −0.106996 0.994259i $$-0.534123\pi$$
−0.106996 + 0.994259i $$0.534123\pi$$
$$278$$ −7.68466 −0.460895
$$279$$ 5.12311 0.306712
$$280$$ 3.56155 0.212843
$$281$$ 7.80776 0.465772 0.232886 0.972504i $$-0.425183\pi$$
0.232886 + 0.972504i $$0.425183\pi$$
$$282$$ −7.68466 −0.457615
$$283$$ 8.49242 0.504822 0.252411 0.967620i $$-0.418776\pi$$
0.252411 + 0.967620i $$0.418776\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ −27.3693 −1.62122
$$286$$ 0 0
$$287$$ −4.12311 −0.243379
$$288$$ 1.00000 0.0589256
$$289$$ 8.00000 0.470588
$$290$$ 3.56155 0.209142
$$291$$ 12.2462 0.717886
$$292$$ −2.43845 −0.142699
$$293$$ −4.68466 −0.273681 −0.136840 0.990593i $$-0.543695\pi$$
−0.136840 + 0.990593i $$0.543695\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −46.7386 −2.72123
$$296$$ −9.80776 −0.570065
$$297$$ 2.56155 0.148636
$$298$$ −17.8078 −1.03158
$$299$$ 0 0
$$300$$ 7.68466 0.443674
$$301$$ 2.87689 0.165821
$$302$$ −17.4384 −1.00347
$$303$$ −19.5616 −1.12378
$$304$$ 7.68466 0.440745
$$305$$ 3.56155 0.203934
$$306$$ −5.00000 −0.285831
$$307$$ −33.9309 −1.93654 −0.968269 0.249912i $$-0.919598\pi$$
−0.968269 + 0.249912i $$0.919598\pi$$
$$308$$ −2.56155 −0.145958
$$309$$ −11.3693 −0.646778
$$310$$ −18.2462 −1.03632
$$311$$ 22.5616 1.27935 0.639674 0.768646i $$-0.279069\pi$$
0.639674 + 0.768646i $$0.279069\pi$$
$$312$$ 0 0
$$313$$ 27.6155 1.56092 0.780461 0.625205i $$-0.214984\pi$$
0.780461 + 0.625205i $$0.214984\pi$$
$$314$$ −5.80776 −0.327751
$$315$$ 3.56155 0.200671
$$316$$ 1.43845 0.0809190
$$317$$ −11.5616 −0.649362 −0.324681 0.945824i $$-0.605257\pi$$
−0.324681 + 0.945824i $$0.605257\pi$$
$$318$$ −3.87689 −0.217405
$$319$$ −2.56155 −0.143419
$$320$$ −3.56155 −0.199097
$$321$$ −9.43845 −0.526803
$$322$$ 9.12311 0.508411
$$323$$ −38.4233 −2.13793
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −1.12311 −0.0622031
$$327$$ 0.876894 0.0484924
$$328$$ 4.12311 0.227660
$$329$$ 7.68466 0.423669
$$330$$ −9.12311 −0.502210
$$331$$ −35.3693 −1.94407 −0.972037 0.234829i $$-0.924547\pi$$
−0.972037 + 0.234829i $$0.924547\pi$$
$$332$$ −10.2462 −0.562334
$$333$$ −9.80776 −0.537462
$$334$$ 20.4924 1.12130
$$335$$ −4.00000 −0.218543
$$336$$ −1.00000 −0.0545545
$$337$$ −29.4924 −1.60655 −0.803277 0.595605i $$-0.796912\pi$$
−0.803277 + 0.595605i $$0.796912\pi$$
$$338$$ 0 0
$$339$$ 16.9309 0.919559
$$340$$ 17.8078 0.965762
$$341$$ 13.1231 0.710656
$$342$$ 7.68466 0.415539
$$343$$ −1.00000 −0.0539949
$$344$$ −2.87689 −0.155112
$$345$$ 32.4924 1.74933
$$346$$ −4.87689 −0.262183
$$347$$ −0.315342 −0.0169284 −0.00846421 0.999964i $$-0.502694\pi$$
−0.00846421 + 0.999964i $$0.502694\pi$$
$$348$$ −1.00000 −0.0536056
$$349$$ −6.49242 −0.347531 −0.173766 0.984787i $$-0.555594\pi$$
−0.173766 + 0.984787i $$0.555594\pi$$
$$350$$ −7.68466 −0.410762
$$351$$ 0 0
$$352$$ 2.56155 0.136531
$$353$$ −9.80776 −0.522015 −0.261007 0.965337i $$-0.584055\pi$$
−0.261007 + 0.965337i $$0.584055\pi$$
$$354$$ 13.1231 0.697485
$$355$$ 14.2462 0.756110
$$356$$ −9.68466 −0.513286
$$357$$ 5.00000 0.264628
$$358$$ −16.4924 −0.871652
$$359$$ −4.63068 −0.244398 −0.122199 0.992506i $$-0.538995\pi$$
−0.122199 + 0.992506i $$0.538995\pi$$
$$360$$ −3.56155 −0.187710
$$361$$ 40.0540 2.10810
$$362$$ 13.2462 0.696205
$$363$$ −4.43845 −0.232958
$$364$$ 0 0
$$365$$ 8.68466 0.454576
$$366$$ −1.00000 −0.0522708
$$367$$ 11.3693 0.593474 0.296737 0.954959i $$-0.404102\pi$$
0.296737 + 0.954959i $$0.404102\pi$$
$$368$$ −9.12311 −0.475575
$$369$$ 4.12311 0.214640
$$370$$ 34.9309 1.81597
$$371$$ 3.87689 0.201278
$$372$$ 5.12311 0.265621
$$373$$ 21.5616 1.11641 0.558207 0.829701i $$-0.311489\pi$$
0.558207 + 0.829701i $$0.311489\pi$$
$$374$$ −12.8078 −0.662274
$$375$$ −9.56155 −0.493756
$$376$$ −7.68466 −0.396306
$$377$$ 0 0
$$378$$ −1.00000 −0.0514344
$$379$$ 0.492423 0.0252940 0.0126470 0.999920i $$-0.495974\pi$$
0.0126470 + 0.999920i $$0.495974\pi$$
$$380$$ −27.3693 −1.40402
$$381$$ 2.24621 0.115077
$$382$$ 5.75379 0.294389
$$383$$ 8.31534 0.424894 0.212447 0.977173i $$-0.431857\pi$$
0.212447 + 0.977173i $$0.431857\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 9.12311 0.464957
$$386$$ −20.3693 −1.03677
$$387$$ −2.87689 −0.146241
$$388$$ 12.2462 0.621707
$$389$$ 18.6847 0.947350 0.473675 0.880700i $$-0.342927\pi$$
0.473675 + 0.880700i $$0.342927\pi$$
$$390$$ 0 0
$$391$$ 45.6155 2.30688
$$392$$ 1.00000 0.0505076
$$393$$ −16.4924 −0.831933
$$394$$ −0.561553 −0.0282906
$$395$$ −5.12311 −0.257771
$$396$$ 2.56155 0.128723
$$397$$ 1.68466 0.0845506 0.0422753 0.999106i $$-0.486539\pi$$
0.0422753 + 0.999106i $$0.486539\pi$$
$$398$$ −17.6155 −0.882987
$$399$$ −7.68466 −0.384714
$$400$$ 7.68466 0.384233
$$401$$ 15.1771 0.757907 0.378954 0.925416i $$-0.376284\pi$$
0.378954 + 0.925416i $$0.376284\pi$$
$$402$$ 1.12311 0.0560154
$$403$$ 0 0
$$404$$ −19.5616 −0.973224
$$405$$ −3.56155 −0.176975
$$406$$ 1.00000 0.0496292
$$407$$ −25.1231 −1.24531
$$408$$ −5.00000 −0.247537
$$409$$ −30.4384 −1.50508 −0.752542 0.658544i $$-0.771173\pi$$
−0.752542 + 0.658544i $$0.771173\pi$$
$$410$$ −14.6847 −0.725224
$$411$$ −5.80776 −0.286476
$$412$$ −11.3693 −0.560126
$$413$$ −13.1231 −0.645746
$$414$$ −9.12311 −0.448376
$$415$$ 36.4924 1.79134
$$416$$ 0 0
$$417$$ −7.68466 −0.376319
$$418$$ 19.6847 0.962808
$$419$$ 22.7386 1.11085 0.555427 0.831565i $$-0.312555\pi$$
0.555427 + 0.831565i $$0.312555\pi$$
$$420$$ 3.56155 0.173786
$$421$$ 8.43845 0.411265 0.205632 0.978629i $$-0.434075\pi$$
0.205632 + 0.978629i $$0.434075\pi$$
$$422$$ −2.87689 −0.140045
$$423$$ −7.68466 −0.373641
$$424$$ −3.87689 −0.188279
$$425$$ −38.4233 −1.86380
$$426$$ −4.00000 −0.193801
$$427$$ 1.00000 0.0483934
$$428$$ −9.43845 −0.456225
$$429$$ 0 0
$$430$$ 10.2462 0.494116
$$431$$ −18.7386 −0.902608 −0.451304 0.892370i $$-0.649041\pi$$
−0.451304 + 0.892370i $$0.649041\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −5.31534 −0.255439 −0.127720 0.991810i $$-0.540766\pi$$
−0.127720 + 0.991810i $$0.540766\pi$$
$$434$$ −5.12311 −0.245917
$$435$$ 3.56155 0.170763
$$436$$ 0.876894 0.0419956
$$437$$ −70.1080 −3.35372
$$438$$ −2.43845 −0.116514
$$439$$ −28.4924 −1.35987 −0.679935 0.733273i $$-0.737992\pi$$
−0.679935 + 0.733273i $$0.737992\pi$$
$$440$$ −9.12311 −0.434927
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −34.4233 −1.63550 −0.817750 0.575574i $$-0.804779\pi$$
−0.817750 + 0.575574i $$0.804779\pi$$
$$444$$ −9.80776 −0.465456
$$445$$ 34.4924 1.63510
$$446$$ −16.4924 −0.780939
$$447$$ −17.8078 −0.842278
$$448$$ −1.00000 −0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 7.68466 0.362258
$$451$$ 10.5616 0.497324
$$452$$ 16.9309 0.796361
$$453$$ −17.4384 −0.819330
$$454$$ 13.1231 0.615898
$$455$$ 0 0
$$456$$ 7.68466 0.359867
$$457$$ −13.3153 −0.622865 −0.311433 0.950268i $$-0.600809\pi$$
−0.311433 + 0.950268i $$0.600809\pi$$
$$458$$ 22.8078 1.06574
$$459$$ −5.00000 −0.233380
$$460$$ 32.4924 1.51497
$$461$$ 30.0540 1.39975 0.699877 0.714264i $$-0.253238\pi$$
0.699877 + 0.714264i $$0.253238\pi$$
$$462$$ −2.56155 −0.119174
$$463$$ −11.6847 −0.543032 −0.271516 0.962434i $$-0.587525\pi$$
−0.271516 + 0.962434i $$0.587525\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ −18.2462 −0.846148
$$466$$ −6.00000 −0.277945
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 0 0
$$469$$ −1.12311 −0.0518602
$$470$$ 27.3693 1.26245
$$471$$ −5.80776 −0.267608
$$472$$ 13.1231 0.604040
$$473$$ −7.36932 −0.338842
$$474$$ 1.43845 0.0660701
$$475$$ 59.0540 2.70958
$$476$$ 5.00000 0.229175
$$477$$ −3.87689 −0.177511
$$478$$ 4.00000 0.182956
$$479$$ −2.56155 −0.117040 −0.0585202 0.998286i $$-0.518638\pi$$
−0.0585202 + 0.998286i $$0.518638\pi$$
$$480$$ −3.56155 −0.162562
$$481$$ 0 0
$$482$$ 11.3153 0.515400
$$483$$ 9.12311 0.415116
$$484$$ −4.43845 −0.201748
$$485$$ −43.6155 −1.98048
$$486$$ 1.00000 0.0453609
$$487$$ 23.6847 1.07325 0.536627 0.843819i $$-0.319698\pi$$
0.536627 + 0.843819i $$0.319698\pi$$
$$488$$ −1.00000 −0.0452679
$$489$$ −1.12311 −0.0507886
$$490$$ −3.56155 −0.160895
$$491$$ −1.75379 −0.0791474 −0.0395737 0.999217i $$-0.512600\pi$$
−0.0395737 + 0.999217i $$0.512600\pi$$
$$492$$ 4.12311 0.185884
$$493$$ 5.00000 0.225189
$$494$$ 0 0
$$495$$ −9.12311 −0.410053
$$496$$ 5.12311 0.230034
$$497$$ 4.00000 0.179425
$$498$$ −10.2462 −0.459144
$$499$$ −27.3693 −1.22522 −0.612609 0.790386i $$-0.709880\pi$$
−0.612609 + 0.790386i $$0.709880\pi$$
$$500$$ −9.56155 −0.427606
$$501$$ 20.4924 0.915534
$$502$$ 8.49242 0.379035
$$503$$ 20.4924 0.913712 0.456856 0.889541i $$-0.348975\pi$$
0.456856 + 0.889541i $$0.348975\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 69.6695 3.10025
$$506$$ −23.3693 −1.03889
$$507$$ 0 0
$$508$$ 2.24621 0.0996595
$$509$$ 27.3153 1.21073 0.605366 0.795948i $$-0.293027\pi$$
0.605366 + 0.795948i $$0.293027\pi$$
$$510$$ 17.8078 0.788541
$$511$$ 2.43845 0.107871
$$512$$ 1.00000 0.0441942
$$513$$ 7.68466 0.339286
$$514$$ −14.7538 −0.650762
$$515$$ 40.4924 1.78431
$$516$$ −2.87689 −0.126648
$$517$$ −19.6847 −0.865730
$$518$$ 9.80776 0.430928
$$519$$ −4.87689 −0.214072
$$520$$ 0 0
$$521$$ −17.0000 −0.744784 −0.372392 0.928076i $$-0.621462\pi$$
−0.372392 + 0.928076i $$0.621462\pi$$
$$522$$ −1.00000 −0.0437688
$$523$$ 15.1922 0.664310 0.332155 0.943225i $$-0.392224\pi$$
0.332155 + 0.943225i $$0.392224\pi$$
$$524$$ −16.4924 −0.720475
$$525$$ −7.68466 −0.335386
$$526$$ −20.4924 −0.893512
$$527$$ −25.6155 −1.11583
$$528$$ 2.56155 0.111477
$$529$$ 60.2311 2.61874
$$530$$ 13.8078 0.599771
$$531$$ 13.1231 0.569494
$$532$$ −7.68466 −0.333172
$$533$$ 0 0
$$534$$ −9.68466 −0.419096
$$535$$ 33.6155 1.45333
$$536$$ 1.12311 0.0485108
$$537$$ −16.4924 −0.711701
$$538$$ −4.87689 −0.210258
$$539$$ 2.56155 0.110334
$$540$$ −3.56155 −0.153265
$$541$$ 1.56155 0.0671364 0.0335682 0.999436i $$-0.489313\pi$$
0.0335682 + 0.999436i $$0.489313\pi$$
$$542$$ 13.1231 0.563686
$$543$$ 13.2462 0.568449
$$544$$ −5.00000 −0.214373
$$545$$ −3.12311 −0.133779
$$546$$ 0 0
$$547$$ −22.7386 −0.972234 −0.486117 0.873894i $$-0.661587\pi$$
−0.486117 + 0.873894i $$0.661587\pi$$
$$548$$ −5.80776 −0.248095
$$549$$ −1.00000 −0.0426790
$$550$$ 19.6847 0.839357
$$551$$ −7.68466 −0.327377
$$552$$ −9.12311 −0.388305
$$553$$ −1.43845 −0.0611690
$$554$$ −3.56155 −0.151316
$$555$$ 34.9309 1.48273
$$556$$ −7.68466 −0.325902
$$557$$ 21.7386 0.921095 0.460548 0.887635i $$-0.347653\pi$$
0.460548 + 0.887635i $$0.347653\pi$$
$$558$$ 5.12311 0.216878
$$559$$ 0 0
$$560$$ 3.56155 0.150503
$$561$$ −12.8078 −0.540744
$$562$$ 7.80776 0.329351
$$563$$ −0.630683 −0.0265801 −0.0132901 0.999912i $$-0.504230\pi$$
−0.0132901 + 0.999912i $$0.504230\pi$$
$$564$$ −7.68466 −0.323582
$$565$$ −60.3002 −2.53685
$$566$$ 8.49242 0.356963
$$567$$ −1.00000 −0.0419961
$$568$$ −4.00000 −0.167836
$$569$$ 11.6155 0.486948 0.243474 0.969907i $$-0.421713\pi$$
0.243474 + 0.969907i $$0.421713\pi$$
$$570$$ −27.3693 −1.14637
$$571$$ 13.7538 0.575578 0.287789 0.957694i $$-0.407080\pi$$
0.287789 + 0.957694i $$0.407080\pi$$
$$572$$ 0 0
$$573$$ 5.75379 0.240368
$$574$$ −4.12311 −0.172095
$$575$$ −70.1080 −2.92370
$$576$$ 1.00000 0.0416667
$$577$$ 31.3153 1.30367 0.651837 0.758359i $$-0.273998\pi$$
0.651837 + 0.758359i $$0.273998\pi$$
$$578$$ 8.00000 0.332756
$$579$$ −20.3693 −0.846520
$$580$$ 3.56155 0.147885
$$581$$ 10.2462 0.425084
$$582$$ 12.2462 0.507622
$$583$$ −9.93087 −0.411295
$$584$$ −2.43845 −0.100904
$$585$$ 0 0
$$586$$ −4.68466 −0.193521
$$587$$ 31.3693 1.29475 0.647375 0.762172i $$-0.275867\pi$$
0.647375 + 0.762172i $$0.275867\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 39.3693 1.62218
$$590$$ −46.7386 −1.92420
$$591$$ −0.561553 −0.0230992
$$592$$ −9.80776 −0.403097
$$593$$ 32.6155 1.33936 0.669680 0.742650i $$-0.266431\pi$$
0.669680 + 0.742650i $$0.266431\pi$$
$$594$$ 2.56155 0.105102
$$595$$ −17.8078 −0.730047
$$596$$ −17.8078 −0.729434
$$597$$ −17.6155 −0.720956
$$598$$ 0 0
$$599$$ −22.8769 −0.934725 −0.467362 0.884066i $$-0.654796\pi$$
−0.467362 + 0.884066i $$0.654796\pi$$
$$600$$ 7.68466 0.313725
$$601$$ 36.9309 1.50644 0.753221 0.657768i $$-0.228499\pi$$
0.753221 + 0.657768i $$0.228499\pi$$
$$602$$ 2.87689 0.117253
$$603$$ 1.12311 0.0457364
$$604$$ −17.4384 −0.709560
$$605$$ 15.8078 0.642677
$$606$$ −19.5616 −0.794634
$$607$$ 21.1231 0.857360 0.428680 0.903456i $$-0.358979\pi$$
0.428680 + 0.903456i $$0.358979\pi$$
$$608$$ 7.68466 0.311654
$$609$$ 1.00000 0.0405220
$$610$$ 3.56155 0.144203
$$611$$ 0 0
$$612$$ −5.00000 −0.202113
$$613$$ 36.3002 1.46615 0.733075 0.680147i $$-0.238084\pi$$
0.733075 + 0.680147i $$0.238084\pi$$
$$614$$ −33.9309 −1.36934
$$615$$ −14.6847 −0.592143
$$616$$ −2.56155 −0.103208
$$617$$ −27.4233 −1.10402 −0.552010 0.833837i $$-0.686139\pi$$
−0.552010 + 0.833837i $$0.686139\pi$$
$$618$$ −11.3693 −0.457341
$$619$$ 4.31534 0.173448 0.0867241 0.996232i $$-0.472360\pi$$
0.0867241 + 0.996232i $$0.472360\pi$$
$$620$$ −18.2462 −0.732785
$$621$$ −9.12311 −0.366098
$$622$$ 22.5616 0.904636
$$623$$ 9.68466 0.388008
$$624$$ 0 0
$$625$$ −4.36932 −0.174773
$$626$$ 27.6155 1.10374
$$627$$ 19.6847 0.786130
$$628$$ −5.80776 −0.231755
$$629$$ 49.0388 1.95531
$$630$$ 3.56155 0.141896
$$631$$ 23.1922 0.923268 0.461634 0.887070i $$-0.347263\pi$$
0.461634 + 0.887070i $$0.347263\pi$$
$$632$$ 1.43845 0.0572184
$$633$$ −2.87689 −0.114346
$$634$$ −11.5616 −0.459168
$$635$$ −8.00000 −0.317470
$$636$$ −3.87689 −0.153729
$$637$$ 0 0
$$638$$ −2.56155 −0.101413
$$639$$ −4.00000 −0.158238
$$640$$ −3.56155 −0.140783
$$641$$ 36.9309 1.45868 0.729341 0.684151i $$-0.239827\pi$$
0.729341 + 0.684151i $$0.239827\pi$$
$$642$$ −9.43845 −0.372506
$$643$$ −26.5616 −1.04749 −0.523743 0.851877i $$-0.675465\pi$$
−0.523743 + 0.851877i $$0.675465\pi$$
$$644$$ 9.12311 0.359501
$$645$$ 10.2462 0.403444
$$646$$ −38.4233 −1.51174
$$647$$ −20.8078 −0.818038 −0.409019 0.912526i $$-0.634129\pi$$
−0.409019 + 0.912526i $$0.634129\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 33.6155 1.31952
$$650$$ 0 0
$$651$$ −5.12311 −0.200790
$$652$$ −1.12311 −0.0439842
$$653$$ −18.8078 −0.736005 −0.368002 0.929825i $$-0.619958\pi$$
−0.368002 + 0.929825i $$0.619958\pi$$
$$654$$ 0.876894 0.0342893
$$655$$ 58.7386 2.29511
$$656$$ 4.12311 0.160980
$$657$$ −2.43845 −0.0951329
$$658$$ 7.68466 0.299579
$$659$$ −28.3153 −1.10301 −0.551505 0.834172i $$-0.685946\pi$$
−0.551505 + 0.834172i $$0.685946\pi$$
$$660$$ −9.12311 −0.355116
$$661$$ −9.80776 −0.381478 −0.190739 0.981641i $$-0.561088\pi$$
−0.190739 + 0.981641i $$0.561088\pi$$
$$662$$ −35.3693 −1.37467
$$663$$ 0 0
$$664$$ −10.2462 −0.397630
$$665$$ 27.3693 1.06134
$$666$$ −9.80776 −0.380043
$$667$$ 9.12311 0.353248
$$668$$ 20.4924 0.792876
$$669$$ −16.4924 −0.637634
$$670$$ −4.00000 −0.154533
$$671$$ −2.56155 −0.0988876
$$672$$ −1.00000 −0.0385758
$$673$$ 8.75379 0.337434 0.168717 0.985665i $$-0.446038\pi$$
0.168717 + 0.985665i $$0.446038\pi$$
$$674$$ −29.4924 −1.13601
$$675$$ 7.68466 0.295783
$$676$$ 0 0
$$677$$ −8.38447 −0.322241 −0.161121 0.986935i $$-0.551511\pi$$
−0.161121 + 0.986935i $$0.551511\pi$$
$$678$$ 16.9309 0.650226
$$679$$ −12.2462 −0.469966
$$680$$ 17.8078 0.682897
$$681$$ 13.1231 0.502879
$$682$$ 13.1231 0.502510
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 7.68466 0.293830
$$685$$ 20.6847 0.790320
$$686$$ −1.00000 −0.0381802
$$687$$ 22.8078 0.870170
$$688$$ −2.87689 −0.109681
$$689$$ 0 0
$$690$$ 32.4924 1.23697
$$691$$ −28.4924 −1.08390 −0.541951 0.840410i $$-0.682314\pi$$
−0.541951 + 0.840410i $$0.682314\pi$$
$$692$$ −4.87689 −0.185392
$$693$$ −2.56155 −0.0973053
$$694$$ −0.315342 −0.0119702
$$695$$ 27.3693 1.03818
$$696$$ −1.00000 −0.0379049
$$697$$ −20.6155 −0.780869
$$698$$ −6.49242 −0.245742
$$699$$ −6.00000 −0.226941
$$700$$ −7.68466 −0.290453
$$701$$ 9.05398 0.341964 0.170982 0.985274i $$-0.445306\pi$$
0.170982 + 0.985274i $$0.445306\pi$$
$$702$$ 0 0
$$703$$ −75.3693 −2.84261
$$704$$ 2.56155 0.0965422
$$705$$ 27.3693 1.03079
$$706$$ −9.80776 −0.369120
$$707$$ 19.5616 0.735688
$$708$$ 13.1231 0.493197
$$709$$ 20.3002 0.762390 0.381195 0.924495i $$-0.375513\pi$$
0.381195 + 0.924495i $$0.375513\pi$$
$$710$$ 14.2462 0.534651
$$711$$ 1.43845 0.0539460
$$712$$ −9.68466 −0.362948
$$713$$ −46.7386 −1.75038
$$714$$ 5.00000 0.187120
$$715$$ 0 0
$$716$$ −16.4924 −0.616351
$$717$$ 4.00000 0.149383
$$718$$ −4.63068 −0.172816
$$719$$ 21.9309 0.817883 0.408942 0.912561i $$-0.365898\pi$$
0.408942 + 0.912561i $$0.365898\pi$$
$$720$$ −3.56155 −0.132731
$$721$$ 11.3693 0.423415
$$722$$ 40.0540 1.49065
$$723$$ 11.3153 0.420822
$$724$$ 13.2462 0.492292
$$725$$ −7.68466 −0.285401
$$726$$ −4.43845 −0.164726
$$727$$ 50.2462 1.86353 0.931764 0.363063i $$-0.118269\pi$$
0.931764 + 0.363063i $$0.118269\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.68466 0.321434
$$731$$ 14.3845 0.532029
$$732$$ −1.00000 −0.0369611
$$733$$ 32.6155 1.20468 0.602341 0.798239i $$-0.294235\pi$$
0.602341 + 0.798239i $$0.294235\pi$$
$$734$$ 11.3693 0.419649
$$735$$ −3.56155 −0.131370
$$736$$ −9.12311 −0.336282
$$737$$ 2.87689 0.105972
$$738$$ 4.12311 0.151774
$$739$$ 47.3693 1.74251 0.871254 0.490832i $$-0.163307\pi$$
0.871254 + 0.490832i $$0.163307\pi$$
$$740$$ 34.9309 1.28408
$$741$$ 0 0
$$742$$ 3.87689 0.142325
$$743$$ 8.63068 0.316629 0.158315 0.987389i $$-0.449394\pi$$
0.158315 + 0.987389i $$0.449394\pi$$
$$744$$ 5.12311 0.187822
$$745$$ 63.4233 2.32365
$$746$$ 21.5616 0.789425
$$747$$ −10.2462 −0.374889
$$748$$ −12.8078 −0.468298
$$749$$ 9.43845 0.344873
$$750$$ −9.56155 −0.349139
$$751$$ −0.807764 −0.0294757 −0.0147379 0.999891i $$-0.504691\pi$$
−0.0147379 + 0.999891i $$0.504691\pi$$
$$752$$ −7.68466 −0.280231
$$753$$ 8.49242 0.309481
$$754$$ 0 0
$$755$$ 62.1080 2.26034
$$756$$ −1.00000 −0.0363696
$$757$$ −7.12311 −0.258894 −0.129447 0.991586i $$-0.541320\pi$$
−0.129447 + 0.991586i $$0.541320\pi$$
$$758$$ 0.492423 0.0178856
$$759$$ −23.3693 −0.848252
$$760$$ −27.3693 −0.992789
$$761$$ −42.4924 −1.54035 −0.770175 0.637833i $$-0.779831\pi$$
−0.770175 + 0.637833i $$0.779831\pi$$
$$762$$ 2.24621 0.0813716
$$763$$ −0.876894 −0.0317457
$$764$$ 5.75379 0.208165
$$765$$ 17.8078 0.643841
$$766$$ 8.31534 0.300446
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −22.9848 −0.828855 −0.414427 0.910082i $$-0.636018\pi$$
−0.414427 + 0.910082i $$0.636018\pi$$
$$770$$ 9.12311 0.328774
$$771$$ −14.7538 −0.531345
$$772$$ −20.3693 −0.733108
$$773$$ 26.4924 0.952866 0.476433 0.879211i $$-0.341929\pi$$
0.476433 + 0.879211i $$0.341929\pi$$
$$774$$ −2.87689 −0.103408
$$775$$ 39.3693 1.41419
$$776$$ 12.2462 0.439613
$$777$$ 9.80776 0.351852
$$778$$ 18.6847 0.669877
$$779$$ 31.6847 1.13522
$$780$$ 0 0
$$781$$ −10.2462 −0.366638
$$782$$ 45.6155 1.63121
$$783$$ −1.00000 −0.0357371
$$784$$ 1.00000 0.0357143
$$785$$ 20.6847 0.738267
$$786$$ −16.4924 −0.588265
$$787$$ 0.807764 0.0287937 0.0143968 0.999896i $$-0.495417\pi$$
0.0143968 + 0.999896i $$0.495417\pi$$
$$788$$ −0.561553 −0.0200045
$$789$$ −20.4924 −0.729550
$$790$$ −5.12311 −0.182272
$$791$$ −16.9309 −0.601992
$$792$$ 2.56155 0.0910208
$$793$$ 0 0
$$794$$ 1.68466 0.0597863
$$795$$ 13.8078 0.489711
$$796$$ −17.6155 −0.624366
$$797$$ −24.1080 −0.853947 −0.426974 0.904264i $$-0.640420\pi$$
−0.426974 + 0.904264i $$0.640420\pi$$
$$798$$ −7.68466 −0.272034
$$799$$ 38.4233 1.35932
$$800$$ 7.68466 0.271694
$$801$$ −9.68466 −0.342191
$$802$$ 15.1771 0.535921
$$803$$ −6.24621 −0.220424
$$804$$ 1.12311 0.0396089
$$805$$ −32.4924 −1.14521
$$806$$ 0 0
$$807$$ −4.87689 −0.171675
$$808$$ −19.5616 −0.688173
$$809$$ −10.9309 −0.384309 −0.192154 0.981365i $$-0.561547\pi$$
−0.192154 + 0.981365i $$0.561547\pi$$
$$810$$ −3.56155 −0.125140
$$811$$ 5.75379 0.202043 0.101021 0.994884i $$-0.467789\pi$$
0.101021 + 0.994884i $$0.467789\pi$$
$$812$$ 1.00000 0.0350931
$$813$$ 13.1231 0.460247
$$814$$ −25.1231 −0.880564
$$815$$ 4.00000 0.140114
$$816$$ −5.00000 −0.175035
$$817$$ −22.1080 −0.773459
$$818$$ −30.4384 −1.06426
$$819$$ 0 0
$$820$$ −14.6847 −0.512811
$$821$$ −12.0691 −0.421216 −0.210608 0.977571i $$-0.567544\pi$$
−0.210608 + 0.977571i $$0.567544\pi$$
$$822$$ −5.80776 −0.202569
$$823$$ 14.2462 0.496592 0.248296 0.968684i $$-0.420129\pi$$
0.248296 + 0.968684i $$0.420129\pi$$
$$824$$ −11.3693 −0.396069
$$825$$ 19.6847 0.685332
$$826$$ −13.1231 −0.456611
$$827$$ −11.5076 −0.400158 −0.200079 0.979780i $$-0.564120\pi$$
−0.200079 + 0.979780i $$0.564120\pi$$
$$828$$ −9.12311 −0.317050
$$829$$ 29.1080 1.01096 0.505480 0.862838i $$-0.331315\pi$$
0.505480 + 0.862838i $$0.331315\pi$$
$$830$$ 36.4924 1.26667
$$831$$ −3.56155 −0.123549
$$832$$ 0 0
$$833$$ −5.00000 −0.173240
$$834$$ −7.68466 −0.266098
$$835$$ −72.9848 −2.52574
$$836$$ 19.6847 0.680808
$$837$$ 5.12311 0.177080
$$838$$ 22.7386 0.785493
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 3.56155 0.122885
$$841$$ −28.0000 −0.965517
$$842$$ 8.43845 0.290808
$$843$$ 7.80776 0.268914
$$844$$ −2.87689 −0.0990268
$$845$$ 0 0
$$846$$ −7.68466 −0.264204
$$847$$ 4.43845 0.152507
$$848$$ −3.87689 −0.133133
$$849$$ 8.49242 0.291459
$$850$$ −38.4233 −1.31791
$$851$$ 89.4773 3.06724
$$852$$ −4.00000 −0.137038
$$853$$ −0.369317 −0.0126452 −0.00632258 0.999980i $$-0.502013\pi$$
−0.00632258 + 0.999980i $$0.502013\pi$$
$$854$$ 1.00000 0.0342193
$$855$$ −27.3693 −0.936011
$$856$$ −9.43845 −0.322599
$$857$$ −22.3002 −0.761760 −0.380880 0.924625i $$-0.624379\pi$$
−0.380880 + 0.924625i $$0.624379\pi$$
$$858$$ 0 0
$$859$$ −36.8078 −1.25586 −0.627932 0.778268i $$-0.716099\pi$$
−0.627932 + 0.778268i $$0.716099\pi$$
$$860$$ 10.2462 0.349393
$$861$$ −4.12311 −0.140515
$$862$$ −18.7386 −0.638240
$$863$$ 26.1080 0.888725 0.444362 0.895847i $$-0.353430\pi$$
0.444362 + 0.895847i $$0.353430\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 17.3693 0.590574
$$866$$ −5.31534 −0.180623
$$867$$ 8.00000 0.271694
$$868$$ −5.12311 −0.173890
$$869$$ 3.68466 0.124993
$$870$$ 3.56155 0.120748
$$871$$ 0 0
$$872$$ 0.876894 0.0296954
$$873$$ 12.2462 0.414471
$$874$$ −70.1080 −2.37144
$$875$$ 9.56155 0.323239
$$876$$ −2.43845 −0.0823875
$$877$$ 8.30019 0.280277 0.140139 0.990132i $$-0.455245\pi$$
0.140139 + 0.990132i $$0.455245\pi$$
$$878$$ −28.4924 −0.961573
$$879$$ −4.68466 −0.158010
$$880$$ −9.12311 −0.307540
$$881$$ 47.1771 1.58944 0.794718 0.606979i $$-0.207619\pi$$
0.794718 + 0.606979i $$0.207619\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 17.1231 0.576238 0.288119 0.957595i $$-0.406970\pi$$
0.288119 + 0.957595i $$0.406970\pi$$
$$884$$ 0 0
$$885$$ −46.7386 −1.57110
$$886$$ −34.4233 −1.15647
$$887$$ 5.43845 0.182605 0.0913026 0.995823i $$-0.470897\pi$$
0.0913026 + 0.995823i $$0.470897\pi$$
$$888$$ −9.80776 −0.329127
$$889$$ −2.24621 −0.0753355
$$890$$ 34.4924 1.15619
$$891$$ 2.56155 0.0858152
$$892$$ −16.4924 −0.552207
$$893$$ −59.0540 −1.97617
$$894$$ −17.8078 −0.595581
$$895$$ 58.7386 1.96342
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ −5.12311 −0.170865
$$900$$ 7.68466 0.256155
$$901$$ 19.3845 0.645790
$$902$$ 10.5616 0.351661
$$903$$ 2.87689 0.0957371
$$904$$ 16.9309 0.563112
$$905$$ −47.1771 −1.56822
$$906$$ −17.4384 −0.579354
$$907$$ 22.8769 0.759615 0.379807 0.925066i $$-0.375990\pi$$
0.379807 + 0.925066i $$0.375990\pi$$
$$908$$ 13.1231 0.435506
$$909$$ −19.5616 −0.648816
$$910$$ 0 0
$$911$$ −5.26137 −0.174317 −0.0871584 0.996194i $$-0.527779\pi$$
−0.0871584 + 0.996194i $$0.527779\pi$$
$$912$$ 7.68466 0.254464
$$913$$ −26.2462 −0.868623
$$914$$ −13.3153 −0.440432
$$915$$ 3.56155 0.117741
$$916$$ 22.8078 0.753590
$$917$$ 16.4924 0.544628
$$918$$ −5.00000 −0.165025
$$919$$ 6.56155 0.216446 0.108223 0.994127i $$-0.465484\pi$$
0.108223 + 0.994127i $$0.465484\pi$$
$$920$$ 32.4924 1.07124
$$921$$ −33.9309 −1.11806
$$922$$ 30.0540 0.989775
$$923$$ 0 0
$$924$$ −2.56155 −0.0842689
$$925$$ −75.3693 −2.47813
$$926$$ −11.6847 −0.383982
$$927$$ −11.3693 −0.373417
$$928$$ −1.00000 −0.0328266
$$929$$ −6.61553 −0.217048 −0.108524 0.994094i $$-0.534613\pi$$
−0.108524 + 0.994094i $$0.534613\pi$$
$$930$$ −18.2462 −0.598317
$$931$$ 7.68466 0.251855
$$932$$ −6.00000 −0.196537
$$933$$ 22.5616 0.738632
$$934$$ 28.0000 0.916188
$$935$$ 45.6155 1.49179
$$936$$ 0 0
$$937$$ 13.5616 0.443037 0.221518 0.975156i $$-0.428899\pi$$
0.221518 + 0.975156i $$0.428899\pi$$
$$938$$ −1.12311 −0.0366707
$$939$$ 27.6155 0.901199
$$940$$ 27.3693 0.892689
$$941$$ 53.3693 1.73979 0.869895 0.493237i $$-0.164186\pi$$
0.869895 + 0.493237i $$0.164186\pi$$
$$942$$ −5.80776 −0.189227
$$943$$ −37.6155 −1.22493
$$944$$ 13.1231 0.427121
$$945$$ 3.56155 0.115857
$$946$$ −7.36932 −0.239597
$$947$$ −0.315342 −0.0102472 −0.00512361 0.999987i $$-0.501631\pi$$
−0.00512361 + 0.999987i $$0.501631\pi$$
$$948$$ 1.43845 0.0467186
$$949$$ 0 0
$$950$$ 59.0540 1.91596
$$951$$ −11.5616 −0.374909
$$952$$ 5.00000 0.162051
$$953$$ −52.7386 −1.70837 −0.854186 0.519968i $$-0.825944\pi$$
−0.854186 + 0.519968i $$0.825944\pi$$
$$954$$ −3.87689 −0.125519
$$955$$ −20.4924 −0.663119
$$956$$ 4.00000 0.129369
$$957$$ −2.56155 −0.0828032
$$958$$ −2.56155 −0.0827600
$$959$$ 5.80776 0.187542
$$960$$ −3.56155 −0.114949
$$961$$ −4.75379 −0.153348
$$962$$ 0 0
$$963$$ −9.43845 −0.304150
$$964$$ 11.3153 0.364443
$$965$$ 72.5464 2.33535
$$966$$ 9.12311 0.293531
$$967$$ −4.49242 −0.144467 −0.0722333 0.997388i $$-0.523013\pi$$
−0.0722333 + 0.997388i $$0.523013\pi$$
$$968$$ −4.43845 −0.142657
$$969$$ −38.4233 −1.23433
$$970$$ −43.6155 −1.40041
$$971$$ −10.8769 −0.349056 −0.174528 0.984652i $$-0.555840\pi$$
−0.174528 + 0.984652i $$0.555840\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 7.68466 0.246359
$$974$$ 23.6847 0.758905
$$975$$ 0 0
$$976$$ −1.00000 −0.0320092
$$977$$ 2.05398 0.0657125 0.0328562 0.999460i $$-0.489540\pi$$
0.0328562 + 0.999460i $$0.489540\pi$$
$$978$$ −1.12311 −0.0359130
$$979$$ −24.8078 −0.792860
$$980$$ −3.56155 −0.113770
$$981$$ 0.876894 0.0279971
$$982$$ −1.75379 −0.0559656
$$983$$ 55.2311 1.76160 0.880799 0.473491i $$-0.157006\pi$$
0.880799 + 0.473491i $$0.157006\pi$$
$$984$$ 4.12311 0.131440
$$985$$ 2.00000 0.0637253
$$986$$ 5.00000 0.159232
$$987$$ 7.68466 0.244605
$$988$$ 0 0
$$989$$ 26.2462 0.834581
$$990$$ −9.12311 −0.289951
$$991$$ −1.43845 −0.0456938 −0.0228469 0.999739i $$-0.507273\pi$$
−0.0228469 + 0.999739i $$0.507273\pi$$
$$992$$ 5.12311 0.162659
$$993$$ −35.3693 −1.12241
$$994$$ 4.00000 0.126872
$$995$$ 62.7386 1.98895
$$996$$ −10.2462 −0.324664
$$997$$ −17.0000 −0.538395 −0.269198 0.963085i $$-0.586759\pi$$
−0.269198 + 0.963085i $$0.586759\pi$$
$$998$$ −27.3693 −0.866361
$$999$$ −9.80776 −0.310304
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 7098.2.a.bw.1.1 2
13.3 even 3 546.2.l.i.295.1 yes 4
13.9 even 3 546.2.l.i.211.1 4
13.12 even 2 7098.2.a.bq.1.2 2
39.29 odd 6 1638.2.r.z.1387.2 4
39.35 odd 6 1638.2.r.z.757.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.i.211.1 4 13.9 even 3
546.2.l.i.295.1 yes 4 13.3 even 3
1638.2.r.z.757.2 4 39.35 odd 6
1638.2.r.z.1387.2 4 39.29 odd 6
7098.2.a.bq.1.2 2 13.12 even 2
7098.2.a.bw.1.1 2 1.1 even 1 trivial