# Properties

 Label 637.2.a.i.1.1 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*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 1.1 Root $$-1.65544$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.65544 q^{2} +2.39593 q^{3} +5.05137 q^{4} +3.65544 q^{5} -6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} +O(q^{10})$$ $$q-2.65544 q^{2} +2.39593 q^{3} +5.05137 q^{4} +3.65544 q^{5} -6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} -9.70682 q^{10} +0.655442 q^{11} +12.1027 q^{12} -1.00000 q^{13} +8.75819 q^{15} +11.4136 q^{16} +2.39593 q^{17} -7.27721 q^{18} -2.70682 q^{19} +18.4650 q^{20} -1.74049 q^{22} +7.36226 q^{23} -19.4136 q^{24} +8.36226 q^{25} +2.65544 q^{26} -0.621770 q^{27} -0.208136 q^{29} -23.2569 q^{30} -1.13642 q^{31} -14.1027 q^{32} +1.57040 q^{33} -6.36226 q^{34} +13.8432 q^{36} -7.44731 q^{37} +7.18780 q^{38} -2.39593 q^{39} -29.6191 q^{40} -10.2055 q^{41} -3.10275 q^{43} +3.31088 q^{44} +10.0177 q^{45} -19.5501 q^{46} +4.60407 q^{47} +27.3463 q^{48} -22.2055 q^{50} +5.74049 q^{51} -5.05137 q^{52} +5.25951 q^{53} +1.65107 q^{54} +2.39593 q^{55} -6.48535 q^{57} +0.552694 q^{58} +8.25951 q^{59} +44.2409 q^{60} -1.89725 q^{61} +3.01770 q^{62} +14.6218 q^{64} -3.65544 q^{65} -4.17009 q^{66} -12.8946 q^{67} +12.1027 q^{68} +17.6395 q^{69} -6.75819 q^{71} -22.2055 q^{72} +12.5367 q^{73} +19.7759 q^{74} +20.0354 q^{75} -13.6731 q^{76} +6.36226 q^{78} -1.51902 q^{79} +41.7219 q^{80} -9.71119 q^{81} +27.1001 q^{82} +15.7582 q^{83} +8.75819 q^{85} +8.23917 q^{86} -0.498680 q^{87} -5.31088 q^{88} -14.8096 q^{89} -26.6014 q^{90} +37.1895 q^{92} -2.72279 q^{93} -12.2258 q^{94} -9.89461 q^{95} -33.7892 q^{96} -10.0177 q^{97} +1.79623 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} - 6 q^{8} + 11 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 4 * q^3 + 6 * q^4 + 5 * q^5 + 2 * q^6 - 6 * q^8 + 11 * q^9 $$3 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} - 6 q^{8} + 11 q^{9} - 14 q^{10} - 4 q^{11} + 18 q^{12} - 3 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 8 q^{18} + 7 q^{19} + 16 q^{20} - 8 q^{22} + q^{23} - 28 q^{24} + 4 q^{25} + 2 q^{26} + 22 q^{27} - 7 q^{29} - 24 q^{30} - 3 q^{31} - 24 q^{32} - 10 q^{33} + 2 q^{34} + 26 q^{36} - 10 q^{37} + 12 q^{38} - 4 q^{39} - 22 q^{40} + 6 q^{41} + 9 q^{43} - 2 q^{44} + 3 q^{45} - 28 q^{46} + 17 q^{47} + 16 q^{48} - 30 q^{50} + 20 q^{51} - 6 q^{52} + 13 q^{53} + 28 q^{54} + 4 q^{55} + 4 q^{57} + 14 q^{58} + 22 q^{59} + 42 q^{60} - 24 q^{61} - 18 q^{62} + 20 q^{64} - 5 q^{65} - 30 q^{66} - 14 q^{67} + 18 q^{68} + 2 q^{69} + 4 q^{71} - 30 q^{72} + 5 q^{73} + 8 q^{74} + 6 q^{75} - 8 q^{76} - 2 q^{78} + q^{79} + 40 q^{80} + 15 q^{81} + 20 q^{82} + 23 q^{83} + 2 q^{85} + 6 q^{86} + 20 q^{87} - 4 q^{88} - 11 q^{89} - 40 q^{90} + 30 q^{92} - 38 q^{93} - 16 q^{94} - 5 q^{95} - 52 q^{96} - 3 q^{97} - 30 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 4 * q^3 + 6 * q^4 + 5 * q^5 + 2 * q^6 - 6 * q^8 + 11 * q^9 - 14 * q^10 - 4 * q^11 + 18 * q^12 - 3 * q^13 + 2 * q^15 + 4 * q^16 + 4 * q^17 + 8 * q^18 + 7 * q^19 + 16 * q^20 - 8 * q^22 + q^23 - 28 * q^24 + 4 * q^25 + 2 * q^26 + 22 * q^27 - 7 * q^29 - 24 * q^30 - 3 * q^31 - 24 * q^32 - 10 * q^33 + 2 * q^34 + 26 * q^36 - 10 * q^37 + 12 * q^38 - 4 * q^39 - 22 * q^40 + 6 * q^41 + 9 * q^43 - 2 * q^44 + 3 * q^45 - 28 * q^46 + 17 * q^47 + 16 * q^48 - 30 * q^50 + 20 * q^51 - 6 * q^52 + 13 * q^53 + 28 * q^54 + 4 * q^55 + 4 * q^57 + 14 * q^58 + 22 * q^59 + 42 * q^60 - 24 * q^61 - 18 * q^62 + 20 * q^64 - 5 * q^65 - 30 * q^66 - 14 * q^67 + 18 * q^68 + 2 * q^69 + 4 * q^71 - 30 * q^72 + 5 * q^73 + 8 * q^74 + 6 * q^75 - 8 * q^76 - 2 * q^78 + q^79 + 40 * q^80 + 15 * q^81 + 20 * q^82 + 23 * q^83 + 2 * q^85 + 6 * q^86 + 20 * q^87 - 4 * q^88 - 11 * q^89 - 40 * q^90 + 30 * q^92 - 38 * q^93 - 16 * q^94 - 5 * q^95 - 52 * q^96 - 3 * q^97 - 30 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.65544 −1.87768 −0.938841 0.344352i $$-0.888099\pi$$
−0.938841 + 0.344352i $$0.888099\pi$$
$$3$$ 2.39593 1.38329 0.691646 0.722237i $$-0.256886\pi$$
0.691646 + 0.722237i $$0.256886\pi$$
$$4$$ 5.05137 2.52569
$$5$$ 3.65544 1.63476 0.817382 0.576096i $$-0.195425\pi$$
0.817382 + 0.576096i $$0.195425\pi$$
$$6$$ −6.36226 −2.59738
$$7$$ 0 0
$$8$$ −8.10275 −2.86475
$$9$$ 2.74049 0.913496
$$10$$ −9.70682 −3.06956
$$11$$ 0.655442 0.197623 0.0988117 0.995106i $$-0.468496\pi$$
0.0988117 + 0.995106i $$0.468496\pi$$
$$12$$ 12.1027 3.49376
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 8.75819 2.26136
$$16$$ 11.4136 2.85341
$$17$$ 2.39593 0.581099 0.290549 0.956860i $$-0.406162\pi$$
0.290549 + 0.956860i $$0.406162\pi$$
$$18$$ −7.27721 −1.71526
$$19$$ −2.70682 −0.620986 −0.310493 0.950576i $$-0.600494\pi$$
−0.310493 + 0.950576i $$0.600494\pi$$
$$20$$ 18.4650 4.12890
$$21$$ 0 0
$$22$$ −1.74049 −0.371074
$$23$$ 7.36226 1.53514 0.767569 0.640967i $$-0.221466\pi$$
0.767569 + 0.640967i $$0.221466\pi$$
$$24$$ −19.4136 −3.96279
$$25$$ 8.36226 1.67245
$$26$$ 2.65544 0.520775
$$27$$ −0.621770 −0.119660
$$28$$ 0 0
$$29$$ −0.208136 −0.0386499 −0.0193250 0.999813i $$-0.506152\pi$$
−0.0193250 + 0.999813i $$0.506152\pi$$
$$30$$ −23.2569 −4.24610
$$31$$ −1.13642 −0.204107 −0.102054 0.994779i $$-0.532541\pi$$
−0.102054 + 0.994779i $$0.532541\pi$$
$$32$$ −14.1027 −2.49304
$$33$$ 1.57040 0.273371
$$34$$ −6.36226 −1.09112
$$35$$ 0 0
$$36$$ 13.8432 2.30721
$$37$$ −7.44731 −1.22433 −0.612165 0.790730i $$-0.709701\pi$$
−0.612165 + 0.790730i $$0.709701\pi$$
$$38$$ 7.18780 1.16601
$$39$$ −2.39593 −0.383656
$$40$$ −29.6191 −4.68320
$$41$$ −10.2055 −1.59383 −0.796915 0.604091i $$-0.793536\pi$$
−0.796915 + 0.604091i $$0.793536\pi$$
$$42$$ 0 0
$$43$$ −3.10275 −0.473165 −0.236582 0.971611i $$-0.576027\pi$$
−0.236582 + 0.971611i $$0.576027\pi$$
$$44$$ 3.31088 0.499135
$$45$$ 10.0177 1.49335
$$46$$ −19.5501 −2.88250
$$47$$ 4.60407 0.671572 0.335786 0.941938i $$-0.390998\pi$$
0.335786 + 0.941938i $$0.390998\pi$$
$$48$$ 27.3463 3.94710
$$49$$ 0 0
$$50$$ −22.2055 −3.14033
$$51$$ 5.74049 0.803829
$$52$$ −5.05137 −0.700500
$$53$$ 5.25951 0.722449 0.361225 0.932479i $$-0.382359\pi$$
0.361225 + 0.932479i $$0.382359\pi$$
$$54$$ 1.65107 0.224683
$$55$$ 2.39593 0.323067
$$56$$ 0 0
$$57$$ −6.48535 −0.859005
$$58$$ 0.552694 0.0725723
$$59$$ 8.25951 1.07530 0.537648 0.843169i $$-0.319313\pi$$
0.537648 + 0.843169i $$0.319313\pi$$
$$60$$ 44.2409 5.71148
$$61$$ −1.89725 −0.242918 −0.121459 0.992596i $$-0.538757\pi$$
−0.121459 + 0.992596i $$0.538757\pi$$
$$62$$ 3.01770 0.383248
$$63$$ 0 0
$$64$$ 14.6218 1.82772
$$65$$ −3.65544 −0.453402
$$66$$ −4.17009 −0.513303
$$67$$ −12.8946 −1.57533 −0.787664 0.616105i $$-0.788710\pi$$
−0.787664 + 0.616105i $$0.788710\pi$$
$$68$$ 12.1027 1.46767
$$69$$ 17.6395 2.12354
$$70$$ 0 0
$$71$$ −6.75819 −0.802050 −0.401025 0.916067i $$-0.631346\pi$$
−0.401025 + 0.916067i $$0.631346\pi$$
$$72$$ −22.2055 −2.61694
$$73$$ 12.5367 1.46731 0.733656 0.679521i $$-0.237812\pi$$
0.733656 + 0.679521i $$0.237812\pi$$
$$74$$ 19.7759 2.29890
$$75$$ 20.0354 2.31349
$$76$$ −13.6731 −1.56842
$$77$$ 0 0
$$78$$ 6.36226 0.720384
$$79$$ −1.51902 −0.170903 −0.0854516 0.996342i $$-0.527233\pi$$
−0.0854516 + 0.996342i $$0.527233\pi$$
$$80$$ 41.7219 4.66465
$$81$$ −9.71119 −1.07902
$$82$$ 27.1001 2.99271
$$83$$ 15.7582 1.72969 0.864843 0.502042i $$-0.167418\pi$$
0.864843 + 0.502042i $$0.167418\pi$$
$$84$$ 0 0
$$85$$ 8.75819 0.949959
$$86$$ 8.23917 0.888453
$$87$$ −0.498680 −0.0534641
$$88$$ −5.31088 −0.566142
$$89$$ −14.8096 −1.56981 −0.784905 0.619616i $$-0.787288\pi$$
−0.784905 + 0.619616i $$0.787288\pi$$
$$90$$ −26.6014 −2.80404
$$91$$ 0 0
$$92$$ 37.1895 3.87728
$$93$$ −2.72279 −0.282340
$$94$$ −12.2258 −1.26100
$$95$$ −9.89461 −1.01517
$$96$$ −33.7892 −3.44860
$$97$$ −10.0177 −1.01714 −0.508572 0.861020i $$-0.669826\pi$$
−0.508572 + 0.861020i $$0.669826\pi$$
$$98$$ 0 0
$$99$$ 1.79623 0.180528
$$100$$ 42.2409 4.22409
$$101$$ −3.01770 −0.300273 −0.150136 0.988665i $$-0.547971\pi$$
−0.150136 + 0.988665i $$0.547971\pi$$
$$102$$ −15.2435 −1.50934
$$103$$ −5.03804 −0.496413 −0.248207 0.968707i $$-0.579841\pi$$
−0.248207 + 0.968707i $$0.579841\pi$$
$$104$$ 8.10275 0.794540
$$105$$ 0 0
$$106$$ −13.9663 −1.35653
$$107$$ 11.8432 1.14493 0.572465 0.819929i $$-0.305987\pi$$
0.572465 + 0.819929i $$0.305987\pi$$
$$108$$ −3.14079 −0.302223
$$109$$ 3.55005 0.340034 0.170017 0.985441i $$-0.445618\pi$$
0.170017 + 0.985441i $$0.445618\pi$$
$$110$$ −6.36226 −0.606618
$$111$$ −17.8432 −1.69361
$$112$$ 0 0
$$113$$ −9.46501 −0.890393 −0.445197 0.895433i $$-0.646866\pi$$
−0.445197 + 0.895433i $$0.646866\pi$$
$$114$$ 17.2215 1.61294
$$115$$ 26.9123 2.50959
$$116$$ −1.05137 −0.0976176
$$117$$ −2.74049 −0.253358
$$118$$ −21.9327 −2.01906
$$119$$ 0 0
$$120$$ −70.9654 −6.47823
$$121$$ −10.5704 −0.960945
$$122$$ 5.03804 0.456123
$$123$$ −24.4517 −2.20473
$$124$$ −5.74049 −0.515511
$$125$$ 12.2905 1.09930
$$126$$ 0 0
$$127$$ 5.46765 0.485175 0.242588 0.970130i $$-0.422004\pi$$
0.242588 + 0.970130i $$0.422004\pi$$
$$128$$ −10.6218 −0.938841
$$129$$ −7.43397 −0.654525
$$130$$ 9.70682 0.851344
$$131$$ 9.82991 0.858843 0.429421 0.903104i $$-0.358718\pi$$
0.429421 + 0.903104i $$0.358718\pi$$
$$132$$ 7.93265 0.690449
$$133$$ 0 0
$$134$$ 34.2409 2.95796
$$135$$ −2.27284 −0.195615
$$136$$ −19.4136 −1.66471
$$137$$ 17.5501 1.49940 0.749701 0.661777i $$-0.230197\pi$$
0.749701 + 0.661777i $$0.230197\pi$$
$$138$$ −46.8406 −3.98734
$$139$$ −4.91495 −0.416881 −0.208440 0.978035i $$-0.566839\pi$$
−0.208440 + 0.978035i $$0.566839\pi$$
$$140$$ 0 0
$$141$$ 11.0310 0.928981
$$142$$ 17.9460 1.50599
$$143$$ −0.655442 −0.0548108
$$144$$ 31.2789 2.60658
$$145$$ −0.760830 −0.0631835
$$146$$ −33.2905 −2.75515
$$147$$ 0 0
$$148$$ −37.6191 −3.09227
$$149$$ −10.3419 −0.847243 −0.423621 0.905839i $$-0.639241\pi$$
−0.423621 + 0.905839i $$0.639241\pi$$
$$150$$ −53.2029 −4.34400
$$151$$ 5.07171 0.412730 0.206365 0.978475i $$-0.433837\pi$$
0.206365 + 0.978475i $$0.433837\pi$$
$$152$$ 21.9327 1.77897
$$153$$ 6.56603 0.530832
$$154$$ 0 0
$$155$$ −4.15412 −0.333667
$$156$$ −12.1027 −0.968995
$$157$$ −12.6014 −1.00570 −0.502852 0.864373i $$-0.667716\pi$$
−0.502852 + 0.864373i $$0.667716\pi$$
$$158$$ 4.03367 0.320902
$$159$$ 12.6014 0.999358
$$160$$ −51.5518 −4.07553
$$161$$ 0 0
$$162$$ 25.7875 2.02606
$$163$$ 3.20814 0.251281 0.125640 0.992076i $$-0.459901\pi$$
0.125640 + 0.992076i $$0.459901\pi$$
$$164$$ −51.5518 −4.02552
$$165$$ 5.74049 0.446896
$$166$$ −41.8450 −3.24780
$$167$$ 8.12045 0.628379 0.314190 0.949360i $$-0.398267\pi$$
0.314190 + 0.949360i $$0.398267\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −23.2569 −1.78372
$$171$$ −7.41800 −0.567269
$$172$$ −15.6731 −1.19507
$$173$$ −10.3286 −0.785268 −0.392634 0.919695i $$-0.628436\pi$$
−0.392634 + 0.919695i $$0.628436\pi$$
$$174$$ 1.32422 0.100389
$$175$$ 0 0
$$176$$ 7.48098 0.563900
$$177$$ 19.7892 1.48745
$$178$$ 39.3259 2.94760
$$179$$ −2.37823 −0.177757 −0.0888786 0.996042i $$-0.528328\pi$$
−0.0888786 + 0.996042i $$0.528328\pi$$
$$180$$ 50.6032 3.77174
$$181$$ 21.8096 1.62109 0.810546 0.585675i $$-0.199170\pi$$
0.810546 + 0.585675i $$0.199170\pi$$
$$182$$ 0 0
$$183$$ −4.54569 −0.336027
$$184$$ −59.6545 −4.39779
$$185$$ −27.2232 −2.00149
$$186$$ 7.23021 0.530144
$$187$$ 1.57040 0.114839
$$188$$ 23.2569 1.69618
$$189$$ 0 0
$$190$$ 26.2746 1.90616
$$191$$ −25.0868 −1.81522 −0.907608 0.419819i $$-0.862093\pi$$
−0.907608 + 0.419819i $$0.862093\pi$$
$$192$$ 35.0328 2.52827
$$193$$ 11.3109 0.814175 0.407088 0.913389i $$-0.366544\pi$$
0.407088 + 0.913389i $$0.366544\pi$$
$$194$$ 26.6014 1.90987
$$195$$ −8.75819 −0.627187
$$196$$ 0 0
$$197$$ −16.7919 −1.19637 −0.598185 0.801358i $$-0.704111\pi$$
−0.598185 + 0.801358i $$0.704111\pi$$
$$198$$ −4.76979 −0.338974
$$199$$ 20.5341 1.45562 0.727811 0.685777i $$-0.240538\pi$$
0.727811 + 0.685777i $$0.240538\pi$$
$$200$$ −67.7573 −4.79116
$$201$$ −30.8946 −2.17914
$$202$$ 8.01333 0.563816
$$203$$ 0 0
$$204$$ 28.9974 2.03022
$$205$$ −37.3056 −2.60554
$$206$$ 13.3782 0.932105
$$207$$ 20.1762 1.40234
$$208$$ −11.4136 −0.791393
$$209$$ −1.77416 −0.122721
$$210$$ 0 0
$$211$$ −15.7785 −1.08624 −0.543119 0.839655i $$-0.682757\pi$$
−0.543119 + 0.839655i $$0.682757\pi$$
$$212$$ 26.5678 1.82468
$$213$$ −16.1922 −1.10947
$$214$$ −31.4490 −2.14981
$$215$$ −11.3419 −0.773512
$$216$$ 5.03804 0.342795
$$217$$ 0 0
$$218$$ −9.42697 −0.638475
$$219$$ 30.0371 2.02972
$$220$$ 12.1027 0.815967
$$221$$ −2.39593 −0.161168
$$222$$ 47.3817 3.18005
$$223$$ −8.44731 −0.565673 −0.282837 0.959168i $$-0.591275\pi$$
−0.282837 + 0.959168i $$0.591275\pi$$
$$224$$ 0 0
$$225$$ 22.9167 1.52778
$$226$$ 25.1338 1.67187
$$227$$ 6.68912 0.443972 0.221986 0.975050i $$-0.428746\pi$$
0.221986 + 0.975050i $$0.428746\pi$$
$$228$$ −32.7599 −2.16958
$$229$$ 8.63510 0.570624 0.285312 0.958435i $$-0.407903\pi$$
0.285312 + 0.958435i $$0.407903\pi$$
$$230$$ −71.4641 −4.71220
$$231$$ 0 0
$$232$$ 1.68648 0.110723
$$233$$ −4.16745 −0.273019 −0.136510 0.990639i $$-0.543588\pi$$
−0.136510 + 0.990639i $$0.543588\pi$$
$$234$$ 7.27721 0.475726
$$235$$ 16.8299 1.09786
$$236$$ 41.7219 2.71586
$$237$$ −3.63947 −0.236409
$$238$$ 0 0
$$239$$ −1.79450 −0.116077 −0.0580384 0.998314i $$-0.518485\pi$$
−0.0580384 + 0.998314i $$0.518485\pi$$
$$240$$ 99.9628 6.45257
$$241$$ 13.8609 0.892862 0.446431 0.894818i $$-0.352695\pi$$
0.446431 + 0.894818i $$0.352695\pi$$
$$242$$ 28.0691 1.80435
$$243$$ −21.4020 −1.37294
$$244$$ −9.58373 −0.613535
$$245$$ 0 0
$$246$$ 64.9300 4.13979
$$247$$ 2.70682 0.172231
$$248$$ 9.20814 0.584717
$$249$$ 37.7556 2.39266
$$250$$ −32.6368 −2.06413
$$251$$ 14.7449 0.930687 0.465344 0.885130i $$-0.345931\pi$$
0.465344 + 0.885130i $$0.345931\pi$$
$$252$$ 0 0
$$253$$ 4.82554 0.303379
$$254$$ −14.5190 −0.911004
$$255$$ 20.9840 1.31407
$$256$$ −1.03804 −0.0648776
$$257$$ 23.7068 1.47879 0.739395 0.673272i $$-0.235112\pi$$
0.739395 + 0.673272i $$0.235112\pi$$
$$258$$ 19.7405 1.22899
$$259$$ 0 0
$$260$$ −18.4650 −1.14515
$$261$$ −0.570395 −0.0353066
$$262$$ −26.1027 −1.61263
$$263$$ 11.3756 0.701449 0.350725 0.936479i $$-0.385935\pi$$
0.350725 + 0.936479i $$0.385935\pi$$
$$264$$ −12.7245 −0.783140
$$265$$ 19.2258 1.18103
$$266$$ 0 0
$$267$$ −35.4827 −2.17151
$$268$$ −65.1355 −3.97878
$$269$$ −11.1054 −0.677107 −0.338554 0.940947i $$-0.609938\pi$$
−0.338554 + 0.940947i $$0.609938\pi$$
$$270$$ 6.03540 0.367303
$$271$$ −12.7245 −0.772959 −0.386480 0.922298i $$-0.626309\pi$$
−0.386480 + 0.922298i $$0.626309\pi$$
$$272$$ 27.3463 1.65811
$$273$$ 0 0
$$274$$ −46.6032 −2.81540
$$275$$ 5.48098 0.330515
$$276$$ 89.1036 5.36340
$$277$$ −3.00000 −0.180253 −0.0901263 0.995930i $$-0.528727\pi$$
−0.0901263 + 0.995930i $$0.528727\pi$$
$$278$$ 13.0514 0.782769
$$279$$ −3.11435 −0.186451
$$280$$ 0 0
$$281$$ 3.44731 0.205649 0.102825 0.994700i $$-0.467212\pi$$
0.102825 + 0.994700i $$0.467212\pi$$
$$282$$ −29.2923 −1.74433
$$283$$ −12.4783 −0.741760 −0.370880 0.928681i $$-0.620944\pi$$
−0.370880 + 0.928681i $$0.620944\pi$$
$$284$$ −34.1382 −2.02573
$$285$$ −23.7068 −1.40427
$$286$$ 1.74049 0.102917
$$287$$ 0 0
$$288$$ −38.6484 −2.27738
$$289$$ −11.2595 −0.662324
$$290$$ 2.02034 0.118638
$$291$$ −24.0017 −1.40701
$$292$$ 63.3277 3.70597
$$293$$ −13.5341 −0.790670 −0.395335 0.918537i $$-0.629371\pi$$
−0.395335 + 0.918537i $$0.629371\pi$$
$$294$$ 0 0
$$295$$ 30.1922 1.75786
$$296$$ 60.3436 3.50740
$$297$$ −0.407534 −0.0236475
$$298$$ 27.4624 1.59085
$$299$$ −7.36226 −0.425770
$$300$$ 101.206 5.84315
$$301$$ 0 0
$$302$$ −13.4676 −0.774976
$$303$$ −7.23021 −0.415365
$$304$$ −30.8946 −1.77193
$$305$$ −6.93529 −0.397114
$$306$$ −17.4357 −0.996733
$$307$$ 28.2365 1.61154 0.805772 0.592226i $$-0.201751\pi$$
0.805772 + 0.592226i $$0.201751\pi$$
$$308$$ 0 0
$$309$$ −12.0708 −0.686684
$$310$$ 11.0310 0.626521
$$311$$ −24.7042 −1.40085 −0.700423 0.713728i $$-0.747005\pi$$
−0.700423 + 0.713728i $$0.747005\pi$$
$$312$$ 19.4136 1.09908
$$313$$ −20.8122 −1.17638 −0.588188 0.808724i $$-0.700158\pi$$
−0.588188 + 0.808724i $$0.700158\pi$$
$$314$$ 33.4624 1.88839
$$315$$ 0 0
$$316$$ −7.67314 −0.431648
$$317$$ 26.1382 1.46806 0.734032 0.679114i $$-0.237636\pi$$
0.734032 + 0.679114i $$0.237636\pi$$
$$318$$ −33.4624 −1.87648
$$319$$ −0.136421 −0.00763813
$$320$$ 53.4490 2.98789
$$321$$ 28.3756 1.58377
$$322$$ 0 0
$$323$$ −6.48535 −0.360854
$$324$$ −49.0548 −2.72527
$$325$$ −8.36226 −0.463855
$$326$$ −8.51902 −0.471825
$$327$$ 8.50569 0.470366
$$328$$ 82.6926 4.56593
$$329$$ 0 0
$$330$$ −15.2435 −0.839129
$$331$$ −24.1382 −1.32675 −0.663376 0.748286i $$-0.730877\pi$$
−0.663376 + 0.748286i $$0.730877\pi$$
$$332$$ 79.6005 4.36865
$$333$$ −20.4093 −1.11842
$$334$$ −21.5634 −1.17990
$$335$$ −47.1355 −2.57529
$$336$$ 0 0
$$337$$ −30.5297 −1.66306 −0.831530 0.555480i $$-0.812534\pi$$
−0.831530 + 0.555480i $$0.812534\pi$$
$$338$$ −2.65544 −0.144437
$$339$$ −22.6775 −1.23167
$$340$$ 44.2409 2.39930
$$341$$ −0.744859 −0.0403364
$$342$$ 19.6981 1.06515
$$343$$ 0 0
$$344$$ 25.1408 1.35550
$$345$$ 64.4801 3.47149
$$346$$ 27.4270 1.47448
$$347$$ −24.9974 −1.34193 −0.670964 0.741490i $$-0.734120\pi$$
−0.670964 + 0.741490i $$0.734120\pi$$
$$348$$ −2.51902 −0.135034
$$349$$ −1.83887 −0.0984324 −0.0492162 0.998788i $$-0.515672\pi$$
−0.0492162 + 0.998788i $$0.515672\pi$$
$$350$$ 0 0
$$351$$ 0.621770 0.0331876
$$352$$ −9.24354 −0.492682
$$353$$ −23.2569 −1.23784 −0.618919 0.785455i $$-0.712429\pi$$
−0.618919 + 0.785455i $$0.712429\pi$$
$$354$$ −52.5491 −2.79296
$$355$$ −24.7042 −1.31116
$$356$$ −74.8087 −3.96485
$$357$$ 0 0
$$358$$ 6.31525 0.333772
$$359$$ −21.4473 −1.13195 −0.565973 0.824424i $$-0.691499\pi$$
−0.565973 + 0.824424i $$0.691499\pi$$
$$360$$ −81.1709 −4.27808
$$361$$ −11.6731 −0.614376
$$362$$ −57.9140 −3.04389
$$363$$ −25.3259 −1.32927
$$364$$ 0 0
$$365$$ 45.8273 2.39871
$$366$$ 12.0708 0.630951
$$367$$ −1.12045 −0.0584870 −0.0292435 0.999572i $$-0.509310\pi$$
−0.0292435 + 0.999572i $$0.509310\pi$$
$$368$$ 84.0301 4.38037
$$369$$ −27.9681 −1.45596
$$370$$ 72.2896 3.75816
$$371$$ 0 0
$$372$$ −13.7538 −0.713102
$$373$$ 15.6058 0.808038 0.404019 0.914751i $$-0.367613\pi$$
0.404019 + 0.914751i $$0.367613\pi$$
$$374$$ −4.17009 −0.215630
$$375$$ 29.4473 1.52065
$$376$$ −37.3056 −1.92389
$$377$$ 0.208136 0.0107196
$$378$$ 0 0
$$379$$ 12.7849 0.656714 0.328357 0.944554i $$-0.393505\pi$$
0.328357 + 0.944554i $$0.393505\pi$$
$$380$$ −49.9814 −2.56399
$$381$$ 13.1001 0.671139
$$382$$ 66.6165 3.40840
$$383$$ 34.0354 1.73913 0.869564 0.493820i $$-0.164400\pi$$
0.869564 + 0.493820i $$0.164400\pi$$
$$384$$ −25.4490 −1.29869
$$385$$ 0 0
$$386$$ −30.0354 −1.52876
$$387$$ −8.50305 −0.432234
$$388$$ −50.6032 −2.56899
$$389$$ −24.6705 −1.25084 −0.625422 0.780287i $$-0.715073\pi$$
−0.625422 + 0.780287i $$0.715073\pi$$
$$390$$ 23.2569 1.17766
$$391$$ 17.6395 0.892066
$$392$$ 0 0
$$393$$ 23.5518 1.18803
$$394$$ 44.5898 2.24640
$$395$$ −5.55269 −0.279386
$$396$$ 9.07344 0.455958
$$397$$ −4.97966 −0.249922 −0.124961 0.992162i $$-0.539881\pi$$
−0.124961 + 0.992162i $$0.539881\pi$$
$$398$$ −54.5271 −2.73320
$$399$$ 0 0
$$400$$ 95.4438 4.77219
$$401$$ 0.689115 0.0344128 0.0172064 0.999852i $$-0.494523\pi$$
0.0172064 + 0.999852i $$0.494523\pi$$
$$402$$ 82.0389 4.09173
$$403$$ 1.13642 0.0566092
$$404$$ −15.2435 −0.758394
$$405$$ −35.4987 −1.76394
$$406$$ 0 0
$$407$$ −4.88128 −0.241956
$$408$$ −46.5137 −2.30277
$$409$$ 19.9770 0.987800 0.493900 0.869519i $$-0.335571\pi$$
0.493900 + 0.869519i $$0.335571\pi$$
$$410$$ 99.0629 4.89237
$$411$$ 42.0487 2.07411
$$412$$ −25.4490 −1.25378
$$413$$ 0 0
$$414$$ −53.5767 −2.63315
$$415$$ 57.6032 2.82763
$$416$$ 14.1027 0.691444
$$417$$ −11.7759 −0.576668
$$418$$ 4.71119 0.230432
$$419$$ 19.6661 0.960754 0.480377 0.877062i $$-0.340500\pi$$
0.480377 + 0.877062i $$0.340500\pi$$
$$420$$ 0 0
$$421$$ 14.9283 0.727560 0.363780 0.931485i $$-0.381486\pi$$
0.363780 + 0.931485i $$0.381486\pi$$
$$422$$ 41.8990 2.03961
$$423$$ 12.6174 0.613479
$$424$$ −42.6165 −2.06964
$$425$$ 20.0354 0.971860
$$426$$ 42.9974 2.08323
$$427$$ 0 0
$$428$$ 59.8246 2.89173
$$429$$ −1.57040 −0.0758194
$$430$$ 30.1178 1.45241
$$431$$ −32.6838 −1.57433 −0.787163 0.616746i $$-0.788451\pi$$
−0.787163 + 0.616746i $$0.788451\pi$$
$$432$$ −7.09665 −0.341438
$$433$$ −8.96196 −0.430684 −0.215342 0.976539i $$-0.569087\pi$$
−0.215342 + 0.976539i $$0.569087\pi$$
$$434$$ 0 0
$$435$$ −1.82290 −0.0874012
$$436$$ 17.9327 0.858818
$$437$$ −19.9283 −0.953299
$$438$$ −79.7619 −3.81117
$$439$$ −7.93265 −0.378605 −0.189302 0.981919i $$-0.560623\pi$$
−0.189302 + 0.981919i $$0.560623\pi$$
$$440$$ −19.4136 −0.925509
$$441$$ 0 0
$$442$$ 6.36226 0.302622
$$443$$ −8.91058 −0.423355 −0.211677 0.977340i $$-0.567893\pi$$
−0.211677 + 0.977340i $$0.567893\pi$$
$$444$$ −90.1329 −4.27752
$$445$$ −54.1355 −2.56627
$$446$$ 22.4313 1.06215
$$447$$ −24.7785 −1.17198
$$448$$ 0 0
$$449$$ −8.45168 −0.398859 −0.199430 0.979912i $$-0.563909\pi$$
−0.199430 + 0.979912i $$0.563909\pi$$
$$450$$ −60.8539 −2.86868
$$451$$ −6.68912 −0.314978
$$452$$ −47.8113 −2.24885
$$453$$ 12.1515 0.570926
$$454$$ −17.7626 −0.833638
$$455$$ 0 0
$$456$$ 52.5491 2.46084
$$457$$ 23.1692 1.08381 0.541904 0.840440i $$-0.317703\pi$$
0.541904 + 0.840440i $$0.317703\pi$$
$$458$$ −22.9300 −1.07145
$$459$$ −1.48972 −0.0695341
$$460$$ 135.944 6.33843
$$461$$ 2.27284 0.105857 0.0529284 0.998598i $$-0.483144\pi$$
0.0529284 + 0.998598i $$0.483144\pi$$
$$462$$ 0 0
$$463$$ −4.10976 −0.190997 −0.0954983 0.995430i $$-0.530444\pi$$
−0.0954983 + 0.995430i $$0.530444\pi$$
$$464$$ −2.37559 −0.110284
$$465$$ −9.95299 −0.461559
$$466$$ 11.0664 0.512643
$$467$$ 32.9150 1.52312 0.761561 0.648093i $$-0.224433\pi$$
0.761561 + 0.648093i $$0.224433\pi$$
$$468$$ −13.8432 −0.639904
$$469$$ 0 0
$$470$$ −44.6908 −2.06143
$$471$$ −30.1922 −1.39118
$$472$$ −66.9247 −3.08046
$$473$$ −2.03367 −0.0935084
$$474$$ 9.66441 0.443901
$$475$$ −22.6351 −1.03857
$$476$$ 0 0
$$477$$ 14.4136 0.659955
$$478$$ 4.76520 0.217955
$$479$$ −9.31525 −0.425625 −0.212812 0.977093i $$-0.568262\pi$$
−0.212812 + 0.977093i $$0.568262\pi$$
$$480$$ −123.515 −5.63764
$$481$$ 7.44731 0.339568
$$482$$ −36.8069 −1.67651
$$483$$ 0 0
$$484$$ −53.3950 −2.42705
$$485$$ −36.6191 −1.66279
$$486$$ 56.8319 2.57795
$$487$$ 20.2409 0.917203 0.458601 0.888642i $$-0.348351\pi$$
0.458601 + 0.888642i $$0.348351\pi$$
$$488$$ 15.3730 0.695901
$$489$$ 7.68648 0.347594
$$490$$ 0 0
$$491$$ −4.36226 −0.196866 −0.0984330 0.995144i $$-0.531383\pi$$
−0.0984330 + 0.995144i $$0.531383\pi$$
$$492$$ −123.515 −5.56847
$$493$$ −0.498680 −0.0224594
$$494$$ −7.18780 −0.323394
$$495$$ 6.56603 0.295121
$$496$$ −12.9707 −0.582401
$$497$$ 0 0
$$498$$ −100.258 −4.49265
$$499$$ 9.69348 0.433940 0.216970 0.976178i $$-0.430383\pi$$
0.216970 + 0.976178i $$0.430383\pi$$
$$500$$ 62.0841 2.77649
$$501$$ 19.4560 0.869232
$$502$$ −39.1541 −1.74753
$$503$$ 2.64843 0.118088 0.0590439 0.998255i $$-0.481195\pi$$
0.0590439 + 0.998255i $$0.481195\pi$$
$$504$$ 0 0
$$505$$ −11.0310 −0.490875
$$506$$ −12.8139 −0.569649
$$507$$ 2.39593 0.106407
$$508$$ 27.6191 1.22540
$$509$$ −13.9416 −0.617951 −0.308976 0.951070i $$-0.599986\pi$$
−0.308976 + 0.951070i $$0.599986\pi$$
$$510$$ −55.7219 −2.46741
$$511$$ 0 0
$$512$$ 24.0000 1.06066
$$513$$ 1.68302 0.0743070
$$514$$ −62.9521 −2.77670
$$515$$ −18.4163 −0.811518
$$516$$ −37.5518 −1.65313
$$517$$ 3.01770 0.132718
$$518$$ 0 0
$$519$$ −24.7466 −1.08625
$$520$$ 29.6191 1.29888
$$521$$ −14.6218 −0.640591 −0.320296 0.947318i $$-0.603782\pi$$
−0.320296 + 0.947318i $$0.603782\pi$$
$$522$$ 1.51465 0.0662945
$$523$$ 16.5190 0.722326 0.361163 0.932503i $$-0.382380\pi$$
0.361163 + 0.932503i $$0.382380\pi$$
$$524$$ 49.6545 2.16917
$$525$$ 0 0
$$526$$ −30.2072 −1.31710
$$527$$ −2.72279 −0.118607
$$528$$ 17.9239 0.780038
$$529$$ 31.2029 1.35665
$$530$$ −51.0531 −2.21761
$$531$$ 22.6351 0.982280
$$532$$ 0 0
$$533$$ 10.2055 0.442049
$$534$$ 94.2223 4.07740
$$535$$ 43.2923 1.87169
$$536$$ 104.482 4.51293
$$537$$ −5.69808 −0.245890
$$538$$ 29.4897 1.27139
$$539$$ 0 0
$$540$$ −11.4810 −0.494063
$$541$$ −43.1018 −1.85309 −0.926546 0.376181i $$-0.877237\pi$$
−0.926546 + 0.376181i $$0.877237\pi$$
$$542$$ 33.7892 1.45137
$$543$$ 52.2542 2.24244
$$544$$ −33.7892 −1.44870
$$545$$ 12.9770 0.555874
$$546$$ 0 0
$$547$$ −13.5057 −0.577462 −0.288731 0.957410i $$-0.593233\pi$$
−0.288731 + 0.957410i $$0.593233\pi$$
$$548$$ 88.6519 3.78702
$$549$$ −5.19940 −0.221905
$$550$$ −14.5544 −0.620603
$$551$$ 0.563387 0.0240011
$$552$$ −142.928 −6.08343
$$553$$ 0 0
$$554$$ 7.96633 0.338457
$$555$$ −65.2249 −2.76864
$$556$$ −24.8273 −1.05291
$$557$$ 1.35157 0.0572677 0.0286338 0.999590i $$-0.490884\pi$$
0.0286338 + 0.999590i $$0.490884\pi$$
$$558$$ 8.26998 0.350096
$$559$$ 3.10275 0.131232
$$560$$ 0 0
$$561$$ 3.76256 0.158855
$$562$$ −9.15412 −0.386143
$$563$$ 31.7626 1.33863 0.669316 0.742978i $$-0.266587\pi$$
0.669316 + 0.742978i $$0.266587\pi$$
$$564$$ 55.7219 2.34631
$$565$$ −34.5988 −1.45558
$$566$$ 33.1355 1.39279
$$567$$ 0 0
$$568$$ 54.7599 2.29768
$$569$$ 30.3730 1.27330 0.636650 0.771153i $$-0.280320\pi$$
0.636650 + 0.771153i $$0.280320\pi$$
$$570$$ 62.9521 2.63677
$$571$$ −0.432244 −0.0180888 −0.00904442 0.999959i $$-0.502879\pi$$
−0.00904442 + 0.999959i $$0.502879\pi$$
$$572$$ −3.31088 −0.138435
$$573$$ −60.1062 −2.51097
$$574$$ 0 0
$$575$$ 61.5651 2.56744
$$576$$ 40.0708 1.66962
$$577$$ 18.1382 0.755101 0.377551 0.925989i $$-0.376766\pi$$
0.377551 + 0.925989i $$0.376766\pi$$
$$578$$ 29.8990 1.24363
$$579$$ 27.1001 1.12624
$$580$$ −3.84324 −0.159582
$$581$$ 0 0
$$582$$ 63.7352 2.64191
$$583$$ 3.44731 0.142773
$$584$$ −101.582 −4.20349
$$585$$ −10.0177 −0.414181
$$586$$ 35.9390 1.48463
$$587$$ 19.3065 0.796865 0.398433 0.917198i $$-0.369554\pi$$
0.398433 + 0.917198i $$0.369554\pi$$
$$588$$ 0 0
$$589$$ 3.07608 0.126748
$$590$$ −80.1736 −3.30069
$$591$$ −40.2322 −1.65493
$$592$$ −85.0008 −3.49351
$$593$$ 8.20113 0.336780 0.168390 0.985720i $$-0.446143\pi$$
0.168390 + 0.985720i $$0.446143\pi$$
$$594$$ 1.08218 0.0444025
$$595$$ 0 0
$$596$$ −52.2409 −2.13987
$$597$$ 49.1983 2.01355
$$598$$ 19.5501 0.799461
$$599$$ 23.4783 0.959299 0.479649 0.877460i $$-0.340764\pi$$
0.479649 + 0.877460i $$0.340764\pi$$
$$600$$ −162.342 −6.62758
$$601$$ −8.96196 −0.365566 −0.182783 0.983153i $$-0.558511\pi$$
−0.182783 + 0.983153i $$0.558511\pi$$
$$602$$ 0 0
$$603$$ −35.3375 −1.43906
$$604$$ 25.6191 1.04243
$$605$$ −38.6395 −1.57092
$$606$$ 19.1994 0.779922
$$607$$ 43.4641 1.76415 0.882077 0.471106i $$-0.156145\pi$$
0.882077 + 0.471106i $$0.156145\pi$$
$$608$$ 38.1736 1.54814
$$609$$ 0 0
$$610$$ 18.4163 0.745653
$$611$$ −4.60407 −0.186261
$$612$$ 33.1675 1.34071
$$613$$ −21.4490 −0.866318 −0.433159 0.901317i $$-0.642601\pi$$
−0.433159 + 0.901317i $$0.642601\pi$$
$$614$$ −74.9805 −3.02597
$$615$$ −89.3817 −3.60422
$$616$$ 0 0
$$617$$ −12.1294 −0.488312 −0.244156 0.969736i $$-0.578511\pi$$
−0.244156 + 0.969736i $$0.578511\pi$$
$$618$$ 32.0533 1.28937
$$619$$ 12.7245 0.511442 0.255721 0.966751i $$-0.417687\pi$$
0.255721 + 0.966751i $$0.417687\pi$$
$$620$$ −20.9840 −0.842739
$$621$$ −4.57763 −0.183694
$$622$$ 65.6005 2.63034
$$623$$ 0 0
$$624$$ −27.3463 −1.09473
$$625$$ 3.11608 0.124643
$$626$$ 55.2656 2.20886
$$627$$ −4.25077 −0.169759
$$628$$ −63.6545 −2.54009
$$629$$ −17.8432 −0.711456
$$630$$ 0 0
$$631$$ −11.7538 −0.467912 −0.233956 0.972247i $$-0.575167\pi$$
−0.233956 + 0.972247i $$0.575167\pi$$
$$632$$ 12.3082 0.489596
$$633$$ −37.8043 −1.50259
$$634$$ −69.4084 −2.75656
$$635$$ 19.9867 0.793147
$$636$$ 63.6545 2.52407
$$637$$ 0 0
$$638$$ 0.362259 0.0143420
$$639$$ −18.5208 −0.732670
$$640$$ −38.8273 −1.53478
$$641$$ 4.68648 0.185105 0.0925523 0.995708i $$-0.470497\pi$$
0.0925523 + 0.995708i $$0.470497\pi$$
$$642$$ −75.3497 −2.97382
$$643$$ −0.751182 −0.0296237 −0.0148119 0.999890i $$-0.504715\pi$$
−0.0148119 + 0.999890i $$0.504715\pi$$
$$644$$ 0 0
$$645$$ −27.1745 −1.06999
$$646$$ 17.2215 0.677570
$$647$$ 40.8476 1.60589 0.802943 0.596056i $$-0.203267\pi$$
0.802943 + 0.596056i $$0.203267\pi$$
$$648$$ 78.6873 3.09113
$$649$$ 5.41363 0.212504
$$650$$ 22.2055 0.870971
$$651$$ 0 0
$$652$$ 16.2055 0.634656
$$653$$ −46.4783 −1.81884 −0.909419 0.415881i $$-0.863473\pi$$
−0.909419 + 0.415881i $$0.863473\pi$$
$$654$$ −22.5864 −0.883197
$$655$$ 35.9327 1.40400
$$656$$ −116.482 −4.54785
$$657$$ 34.3568 1.34038
$$658$$ 0 0
$$659$$ 30.3596 1.18264 0.591321 0.806436i $$-0.298606\pi$$
0.591321 + 0.806436i $$0.298606\pi$$
$$660$$ 28.9974 1.12872
$$661$$ −0.107118 −0.00416640 −0.00208320 0.999998i $$-0.500663\pi$$
−0.00208320 + 0.999998i $$0.500663\pi$$
$$662$$ 64.0975 2.49122
$$663$$ −5.74049 −0.222942
$$664$$ −127.685 −4.95513
$$665$$ 0 0
$$666$$ 54.1956 2.10004
$$667$$ −1.53235 −0.0593330
$$668$$ 41.0194 1.58709
$$669$$ −20.2392 −0.782492
$$670$$ 125.166 4.83557
$$671$$ −1.24354 −0.0480063
$$672$$ 0 0
$$673$$ 38.5385 1.48555 0.742774 0.669542i $$-0.233510\pi$$
0.742774 + 0.669542i $$0.233510\pi$$
$$674$$ 81.0699 3.12270
$$675$$ −5.19940 −0.200125
$$676$$ 5.05137 0.194284
$$677$$ −20.7803 −0.798650 −0.399325 0.916809i $$-0.630756\pi$$
−0.399325 + 0.916809i $$0.630756\pi$$
$$678$$ 60.2188 2.31269
$$679$$ 0 0
$$680$$ −70.9654 −2.72140
$$681$$ 16.0267 0.614143
$$682$$ 1.97793 0.0757388
$$683$$ 27.5837 1.05546 0.527731 0.849412i $$-0.323043\pi$$
0.527731 + 0.849412i $$0.323043\pi$$
$$684$$ −37.4711 −1.43274
$$685$$ 64.1532 2.45117
$$686$$ 0 0
$$687$$ 20.6891 0.789339
$$688$$ −35.4136 −1.35013
$$689$$ −5.25951 −0.200371
$$690$$ −171.223 −6.51835
$$691$$ 43.6775 1.66157 0.830785 0.556593i $$-0.187892\pi$$
0.830785 + 0.556593i $$0.187892\pi$$
$$692$$ −52.1736 −1.98334
$$693$$ 0 0
$$694$$ 66.3791 2.51971
$$695$$ −17.9663 −0.681502
$$696$$ 4.04068 0.153162
$$697$$ −24.4517 −0.926173
$$698$$ 4.88301 0.184825
$$699$$ −9.98494 −0.377665
$$700$$ 0 0
$$701$$ −20.8973 −0.789278 −0.394639 0.918836i $$-0.629130\pi$$
−0.394639 + 0.918836i $$0.629130\pi$$
$$702$$ −1.65107 −0.0623157
$$703$$ 20.1585 0.760292
$$704$$ 9.58373 0.361200
$$705$$ 40.3233 1.51866
$$706$$ 61.7573 2.32427
$$707$$ 0 0
$$708$$ 99.9628 3.75683
$$709$$ 9.93966 0.373292 0.186646 0.982427i $$-0.440238\pi$$
0.186646 + 0.982427i $$0.440238\pi$$
$$710$$ 65.6005 2.46194
$$711$$ −4.16286 −0.156119
$$712$$ 119.998 4.49712
$$713$$ −8.36663 −0.313333
$$714$$ 0 0
$$715$$ −2.39593 −0.0896028
$$716$$ −12.0133 −0.448959
$$717$$ −4.29951 −0.160568
$$718$$ 56.9521 2.12543
$$719$$ −11.9797 −0.446766 −0.223383 0.974731i $$-0.571710\pi$$
−0.223383 + 0.974731i $$0.571710\pi$$
$$720$$ 114.338 4.26114
$$721$$ 0 0
$$722$$ 30.9974 1.15360
$$723$$ 33.2099 1.23509
$$724$$ 110.168 4.09437
$$725$$ −1.74049 −0.0646402
$$726$$ 67.2516 2.49594
$$727$$ 24.1736 0.896547 0.448274 0.893896i $$-0.352039\pi$$
0.448274 + 0.893896i $$0.352039\pi$$
$$728$$ 0 0
$$729$$ −22.1443 −0.820157
$$730$$ −121.692 −4.50401
$$731$$ −7.43397 −0.274955
$$732$$ −22.9620 −0.848698
$$733$$ −36.4473 −1.34621 −0.673106 0.739546i $$-0.735040\pi$$
−0.673106 + 0.739546i $$0.735040\pi$$
$$734$$ 2.97529 0.109820
$$735$$ 0 0
$$736$$ −103.828 −3.82715
$$737$$ −8.45168 −0.311321
$$738$$ 74.2676 2.73383
$$739$$ 43.2772 1.59198 0.795989 0.605311i $$-0.206951\pi$$
0.795989 + 0.605311i $$0.206951\pi$$
$$740$$ −137.515 −5.05514
$$741$$ 6.48535 0.238245
$$742$$ 0 0
$$743$$ 22.6572 0.831211 0.415606 0.909545i $$-0.363570\pi$$
0.415606 + 0.909545i $$0.363570\pi$$
$$744$$ 22.0621 0.808835
$$745$$ −37.8043 −1.38504
$$746$$ −41.4403 −1.51724
$$747$$ 43.1852 1.58006
$$748$$ 7.93265 0.290047
$$749$$ 0 0
$$750$$ −78.1956 −2.85530
$$751$$ 29.8679 1.08990 0.544948 0.838470i $$-0.316549\pi$$
0.544948 + 0.838470i $$0.316549\pi$$
$$752$$ 52.5491 1.91627
$$753$$ 35.3277 1.28741
$$754$$ −0.552694 −0.0201279
$$755$$ 18.5394 0.674716
$$756$$ 0 0
$$757$$ 2.55706 0.0929380 0.0464690 0.998920i $$-0.485203\pi$$
0.0464690 + 0.998920i $$0.485203\pi$$
$$758$$ −33.9494 −1.23310
$$759$$ 11.5617 0.419662
$$760$$ 80.1736 2.90820
$$761$$ 14.7289 0.533922 0.266961 0.963707i $$-0.413981\pi$$
0.266961 + 0.963707i $$0.413981\pi$$
$$762$$ −34.7866 −1.26019
$$763$$ 0 0
$$764$$ −126.723 −4.58467
$$765$$ 24.0017 0.867784
$$766$$ −90.3791 −3.26553
$$767$$ −8.25951 −0.298234
$$768$$ −2.48708 −0.0897447
$$769$$ −36.1692 −1.30429 −0.652147 0.758092i $$-0.726132\pi$$
−0.652147 + 0.758092i $$0.726132\pi$$
$$770$$ 0 0
$$771$$ 56.7999 2.04560
$$772$$ 57.1355 2.05635
$$773$$ 38.9567 1.40117 0.700587 0.713567i $$-0.252921\pi$$
0.700587 + 0.713567i $$0.252921\pi$$
$$774$$ 22.5794 0.811598
$$775$$ −9.50305 −0.341360
$$776$$ 81.1709 2.91387
$$777$$ 0 0
$$778$$ 65.5111 2.34869
$$779$$ 27.6244 0.989747
$$780$$ −44.2409 −1.58408
$$781$$ −4.42960 −0.158504
$$782$$ −46.8406 −1.67502
$$783$$ 0.129413 0.00462484
$$784$$ 0 0
$$785$$ −46.0638 −1.64409
$$786$$ −62.5404 −2.23074
$$787$$ 29.1045 1.03746 0.518731 0.854937i $$-0.326405\pi$$
0.518731 + 0.854937i $$0.326405\pi$$
$$788$$ −84.8220 −3.02166
$$789$$ 27.2551 0.970309
$$790$$ 14.7449 0.524599
$$791$$ 0 0
$$792$$ −14.5544 −0.517169
$$793$$ 1.89725 0.0673734
$$794$$ 13.2232 0.469274
$$795$$ 46.0638 1.63371
$$796$$ 103.725 3.67645
$$797$$ −17.8920 −0.633766 −0.316883 0.948465i $$-0.602636\pi$$
−0.316883 + 0.948465i $$0.602636\pi$$
$$798$$ 0 0
$$799$$ 11.0310 0.390250
$$800$$ −117.931 −4.16948
$$801$$ −40.5855 −1.43402
$$802$$ −1.82991 −0.0646162
$$803$$ 8.21710 0.289975
$$804$$ −156.060 −5.50382
$$805$$ 0 0
$$806$$ −3.01770 −0.106294
$$807$$ −26.6078 −0.936637
$$808$$ 24.4517 0.860207
$$809$$ 20.4543 0.719135 0.359568 0.933119i $$-0.382924\pi$$
0.359568 + 0.933119i $$0.382924\pi$$
$$810$$ 94.2647 3.31212
$$811$$ −31.6458 −1.11123 −0.555617 0.831438i $$-0.687518\pi$$
−0.555617 + 0.831438i $$0.687518\pi$$
$$812$$ 0 0
$$813$$ −30.4871 −1.06923
$$814$$ 12.9620 0.454316
$$815$$ 11.7272 0.410784
$$816$$ 65.5198 2.29365
$$817$$ 8.39857 0.293829
$$818$$ −53.0478 −1.85477
$$819$$ 0 0
$$820$$ −188.445 −6.58077
$$821$$ 18.0761 0.630860 0.315430 0.948949i $$-0.397851\pi$$
0.315430 + 0.948949i $$0.397851\pi$$
$$822$$ −111.658 −3.89452
$$823$$ −23.0514 −0.803520 −0.401760 0.915745i $$-0.631601\pi$$
−0.401760 + 0.915745i $$0.631601\pi$$
$$824$$ 40.8220 1.42210
$$825$$ 13.1321 0.457199
$$826$$ 0 0
$$827$$ −26.3756 −0.917169 −0.458585 0.888651i $$-0.651643\pi$$
−0.458585 + 0.888651i $$0.651643\pi$$
$$828$$ 101.918 3.54188
$$829$$ 45.1152 1.56691 0.783457 0.621446i $$-0.213454\pi$$
0.783457 + 0.621446i $$0.213454\pi$$
$$830$$ −152.962 −5.30938
$$831$$ −7.18780 −0.249342
$$832$$ −14.6218 −0.506919
$$833$$ 0 0
$$834$$ 31.2702 1.08280
$$835$$ 29.6838 1.02725
$$836$$ −8.96196 −0.309956
$$837$$ 0.706592 0.0244234
$$838$$ −52.2223 −1.80399
$$839$$ −30.4871 −1.05253 −0.526265 0.850320i $$-0.676408\pi$$
−0.526265 + 0.850320i $$0.676408\pi$$
$$840$$ 0 0
$$841$$ −28.9567 −0.998506
$$842$$ −39.6412 −1.36613
$$843$$ 8.25951 0.284473
$$844$$ −79.7033 −2.74350
$$845$$ 3.65544 0.125751
$$846$$ −33.5048 −1.15192
$$847$$ 0 0
$$848$$ 60.0301 2.06144
$$849$$ −29.8973 −1.02607
$$850$$ −53.2029 −1.82484
$$851$$ −54.8290 −1.87951
$$852$$ −81.7927 −2.80217
$$853$$ −33.2746 −1.13930 −0.569650 0.821888i $$-0.692921\pi$$
−0.569650 + 0.821888i $$0.692921\pi$$
$$854$$ 0 0
$$855$$ −27.1161 −0.927350
$$856$$ −95.9628 −3.27994
$$857$$ 45.6139 1.55814 0.779070 0.626937i $$-0.215692\pi$$
0.779070 + 0.626937i $$0.215692\pi$$
$$858$$ 4.17009 0.142365
$$859$$ 20.9353 0.714303 0.357151 0.934046i $$-0.383748\pi$$
0.357151 + 0.934046i $$0.383748\pi$$
$$860$$ −57.2923 −1.95365
$$861$$ 0 0
$$862$$ 86.7900 2.95608
$$863$$ 2.96196 0.100826 0.0504131 0.998728i $$-0.483946\pi$$
0.0504131 + 0.998728i $$0.483946\pi$$
$$864$$ 8.76866 0.298316
$$865$$ −37.7556 −1.28373
$$866$$ 23.7980 0.808688
$$867$$ −26.9770 −0.916188
$$868$$ 0 0
$$869$$ −0.995631 −0.0337745
$$870$$ 4.84060 0.164112
$$871$$ 12.8946 0.436917
$$872$$ −28.7652 −0.974113
$$873$$ −27.4534 −0.929157
$$874$$ 52.9184 1.78999
$$875$$ 0 0
$$876$$ 151.729 5.12644
$$877$$ −36.9370 −1.24727 −0.623637 0.781714i $$-0.714346\pi$$
−0.623637 + 0.781714i $$0.714346\pi$$
$$878$$ 21.0647 0.710899
$$879$$ −32.4267 −1.09373
$$880$$ 27.3463 0.921843
$$881$$ 30.8946 1.04087 0.520433 0.853903i $$-0.325771\pi$$
0.520433 + 0.853903i $$0.325771\pi$$
$$882$$ 0 0
$$883$$ −9.64648 −0.324630 −0.162315 0.986739i $$-0.551896\pi$$
−0.162315 + 0.986739i $$0.551896\pi$$
$$884$$ −12.1027 −0.407059
$$885$$ 72.3384 2.43163
$$886$$ 23.6615 0.794925
$$887$$ −51.3056 −1.72267 −0.861337 0.508034i $$-0.830372\pi$$
−0.861337 + 0.508034i $$0.830372\pi$$
$$888$$ 144.579 4.85176
$$889$$ 0 0
$$890$$ 143.754 4.81864
$$891$$ −6.36512 −0.213240
$$892$$ −42.6705 −1.42871
$$893$$ −12.4624 −0.417037
$$894$$ 65.7980 2.20061
$$895$$ −8.69348 −0.290591
$$896$$ 0 0
$$897$$ −17.6395 −0.588965
$$898$$ 22.4429 0.748931
$$899$$ 0.236531 0.00788873
$$900$$ 115.761 3.85869
$$901$$ 12.6014 0.419814
$$902$$ 17.7626 0.591429
$$903$$ 0 0
$$904$$ 76.6926 2.55076
$$905$$ 79.7236 2.65010
$$906$$ −32.2676 −1.07202
$$907$$ −18.5385 −0.615559 −0.307780 0.951458i $$-0.599586\pi$$
−0.307780 + 0.951458i $$0.599586\pi$$
$$908$$ 33.7892 1.12133
$$909$$ −8.26998 −0.274298
$$910$$ 0 0
$$911$$ 20.7272 0.686721 0.343361 0.939204i $$-0.388435\pi$$
0.343361 + 0.939204i $$0.388435\pi$$
$$912$$ −74.0214 −2.45109
$$913$$ 10.3286 0.341826
$$914$$ −61.5244 −2.03505
$$915$$ −16.6165 −0.549324
$$916$$ 43.6191 1.44122
$$917$$ 0 0
$$918$$ 3.95586 0.130563
$$919$$ −5.13205 −0.169291 −0.0846454 0.996411i $$-0.526976\pi$$
−0.0846454 + 0.996411i $$0.526976\pi$$
$$920$$ −218.064 −7.18935
$$921$$ 67.6528 2.22924
$$922$$ −6.03540 −0.198765
$$923$$ 6.75819 0.222449
$$924$$ 0 0
$$925$$ −62.2763 −2.04763
$$926$$ 10.9132 0.358631
$$927$$ −13.8067 −0.453472
$$928$$ 2.93529 0.0963557
$$929$$ 19.9283 0.653826 0.326913 0.945054i $$-0.393992\pi$$
0.326913 + 0.945054i $$0.393992\pi$$
$$930$$ 26.4296 0.866661
$$931$$ 0 0
$$932$$ −21.0514 −0.689561
$$933$$ −59.1895 −1.93778
$$934$$ −87.4038 −2.85994
$$935$$ 5.74049 0.187734
$$936$$ 22.2055 0.725809
$$937$$ −1.64475 −0.0537316 −0.0268658 0.999639i $$-0.508553\pi$$
−0.0268658 + 0.999639i $$0.508553\pi$$
$$938$$ 0 0
$$939$$ −49.8646 −1.62727
$$940$$ 85.0142 2.77286
$$941$$ −17.7849 −0.579770 −0.289885 0.957062i $$-0.593617\pi$$
−0.289885 + 0.957062i $$0.593617\pi$$
$$942$$ 80.1736 2.61220
$$943$$ −75.1355 −2.44675
$$944$$ 94.2710 3.06826
$$945$$ 0 0
$$946$$ 5.40030 0.175579
$$947$$ 10.6484 0.346028 0.173014 0.984919i $$-0.444649\pi$$
0.173014 + 0.984919i $$0.444649\pi$$
$$948$$ −18.3843 −0.597095
$$949$$ −12.5367 −0.406959
$$950$$ 60.1062 1.95010
$$951$$ 62.6252 2.03076
$$952$$ 0 0
$$953$$ −41.5544 −1.34608 −0.673040 0.739606i $$-0.735012\pi$$
−0.673040 + 0.739606i $$0.735012\pi$$
$$954$$ −38.2746 −1.23919
$$955$$ −91.7033 −2.96745
$$956$$ −9.06471 −0.293174
$$957$$ −0.326856 −0.0105658
$$958$$ 24.7361 0.799188
$$959$$ 0 0
$$960$$ 128.060 4.13313
$$961$$ −29.7085 −0.958340
$$962$$ −19.7759 −0.637600
$$963$$ 32.4563 1.04589
$$964$$ 70.0168 2.25509
$$965$$ 41.3463 1.33098
$$966$$ 0 0
$$967$$ −5.51465 −0.177339 −0.0886696 0.996061i $$-0.528262\pi$$
−0.0886696 + 0.996061i $$0.528262\pi$$
$$968$$ 85.6493 2.75287
$$969$$ −15.5385 −0.499167
$$970$$ 97.2400 3.12219
$$971$$ 39.8246 1.27803 0.639017 0.769193i $$-0.279342\pi$$
0.639017 + 0.769193i $$0.279342\pi$$
$$972$$ −108.110 −3.46762
$$973$$ 0 0
$$974$$ −53.7485 −1.72221
$$975$$ −20.0354 −0.641646
$$976$$ −21.6545 −0.693145
$$977$$ 1.93092 0.0617757 0.0308879 0.999523i $$-0.490167\pi$$
0.0308879 + 0.999523i $$0.490167\pi$$
$$978$$ −20.4110 −0.652672
$$979$$ −9.70682 −0.310231
$$980$$ 0 0
$$981$$ 9.72889 0.310619
$$982$$ 11.5837 0.369652
$$983$$ −43.0238 −1.37225 −0.686123 0.727485i $$-0.740689\pi$$
−0.686123 + 0.727485i $$0.740689\pi$$
$$984$$ 198.126 6.31602
$$985$$ −61.3817 −1.95578
$$986$$ 1.32422 0.0421717
$$987$$ 0 0
$$988$$ 13.6731 0.435001
$$989$$ −22.8432 −0.726373
$$990$$ −17.4357 −0.554143
$$991$$ −17.6058 −0.559267 −0.279633 0.960107i $$-0.590213\pi$$
−0.279633 + 0.960107i $$0.590213\pi$$
$$992$$ 16.0267 0.508847
$$993$$ −57.8334 −1.83529
$$994$$ 0 0
$$995$$ 75.0612 2.37960
$$996$$ 190.717 6.04311
$$997$$ −12.9707 −0.410786 −0.205393 0.978680i $$-0.565847\pi$$
−0.205393 + 0.978680i $$0.565847\pi$$
$$998$$ −25.7405 −0.814801
$$999$$ 4.63051 0.146503
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 637.2.a.i.1.1 yes 3
3.2 odd 2 5733.2.a.bd.1.3 3
7.2 even 3 637.2.e.k.508.3 6
7.3 odd 6 637.2.e.l.79.3 6
7.4 even 3 637.2.e.k.79.3 6
7.5 odd 6 637.2.e.l.508.3 6
7.6 odd 2 637.2.a.h.1.1 3
13.12 even 2 8281.2.a.bk.1.3 3
21.20 even 2 5733.2.a.be.1.3 3
91.90 odd 2 8281.2.a.bh.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.1 3 7.6 odd 2
637.2.a.i.1.1 yes 3 1.1 even 1 trivial
637.2.e.k.79.3 6 7.4 even 3
637.2.e.k.508.3 6 7.2 even 3
637.2.e.l.79.3 6 7.3 odd 6
637.2.e.l.508.3 6 7.5 odd 6
5733.2.a.bd.1.3 3 3.2 odd 2
5733.2.a.be.1.3 3 21.20 even 2
8281.2.a.bh.1.3 3 91.90 odd 2
8281.2.a.bk.1.3 3 13.12 even 2