# Properties

 Label 370.2.a.f.1.1 Level $370$ Weight $2$ Character 370.1 Self dual yes Analytic conductor $2.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 370.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -4.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -4.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.37228 q^{11} +2.00000 q^{12} +6.74456 q^{13} -4.37228 q^{14} +2.00000 q^{15} +1.00000 q^{16} +0.372281 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} -8.74456 q^{21} +2.37228 q^{22} -4.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} +6.74456 q^{26} -4.00000 q^{27} -4.37228 q^{28} -9.11684 q^{29} +2.00000 q^{30} -8.37228 q^{31} +1.00000 q^{32} +4.74456 q^{33} +0.372281 q^{34} -4.37228 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +13.4891 q^{39} +1.00000 q^{40} -0.372281 q^{41} -8.74456 q^{42} +1.62772 q^{43} +2.37228 q^{44} +1.00000 q^{45} -4.74456 q^{46} +2.74456 q^{47} +2.00000 q^{48} +12.1168 q^{49} +1.00000 q^{50} +0.744563 q^{51} +6.74456 q^{52} -4.37228 q^{53} -4.00000 q^{54} +2.37228 q^{55} -4.37228 q^{56} -4.00000 q^{57} -9.11684 q^{58} +1.25544 q^{59} +2.00000 q^{60} +0.372281 q^{61} -8.37228 q^{62} -4.37228 q^{63} +1.00000 q^{64} +6.74456 q^{65} +4.74456 q^{66} -6.74456 q^{67} +0.372281 q^{68} -9.48913 q^{69} -4.37228 q^{70} +4.74456 q^{71} +1.00000 q^{72} -2.74456 q^{73} +1.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} -10.3723 q^{77} +13.4891 q^{78} +6.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} -0.372281 q^{82} +10.7446 q^{83} -8.74456 q^{84} +0.372281 q^{85} +1.62772 q^{86} -18.2337 q^{87} +2.37228 q^{88} +10.0000 q^{89} +1.00000 q^{90} -29.4891 q^{91} -4.74456 q^{92} -16.7446 q^{93} +2.74456 q^{94} -2.00000 q^{95} +2.00000 q^{96} +17.1168 q^{97} +12.1168 q^{98} +2.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 4 q^{12} + 2 q^{13} - 3 q^{14} + 4 q^{15} + 2 q^{16} - 5 q^{17} + 2 q^{18} - 4 q^{19} + 2 q^{20} - 6 q^{21} - q^{22} + 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} - 3 q^{28} - q^{29} + 4 q^{30} - 11 q^{31} + 2 q^{32} - 2 q^{33} - 5 q^{34} - 3 q^{35} + 2 q^{36} + 2 q^{37} - 4 q^{38} + 4 q^{39} + 2 q^{40} + 5 q^{41} - 6 q^{42} + 9 q^{43} - q^{44} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 7 q^{49} + 2 q^{50} - 10 q^{51} + 2 q^{52} - 3 q^{53} - 8 q^{54} - q^{55} - 3 q^{56} - 8 q^{57} - q^{58} + 14 q^{59} + 4 q^{60} - 5 q^{61} - 11 q^{62} - 3 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} - 2 q^{67} - 5 q^{68} + 4 q^{69} - 3 q^{70} - 2 q^{71} + 2 q^{72} + 6 q^{73} + 2 q^{74} + 4 q^{75} - 4 q^{76} - 15 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} + 5 q^{82} + 10 q^{83} - 6 q^{84} - 5 q^{85} + 9 q^{86} - 2 q^{87} - q^{88} + 20 q^{89} + 2 q^{90} - 36 q^{91} + 2 q^{92} - 22 q^{93} - 6 q^{94} - 4 q^{95} + 4 q^{96} + 17 q^{97} + 7 q^{98} - q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 - q^11 + 4 * q^12 + 2 * q^13 - 3 * q^14 + 4 * q^15 + 2 * q^16 - 5 * q^17 + 2 * q^18 - 4 * q^19 + 2 * q^20 - 6 * q^21 - q^22 + 2 * q^23 + 4 * q^24 + 2 * q^25 + 2 * q^26 - 8 * q^27 - 3 * q^28 - q^29 + 4 * q^30 - 11 * q^31 + 2 * q^32 - 2 * q^33 - 5 * q^34 - 3 * q^35 + 2 * q^36 + 2 * q^37 - 4 * q^38 + 4 * q^39 + 2 * q^40 + 5 * q^41 - 6 * q^42 + 9 * q^43 - q^44 + 2 * q^45 + 2 * q^46 - 6 * q^47 + 4 * q^48 + 7 * q^49 + 2 * q^50 - 10 * q^51 + 2 * q^52 - 3 * q^53 - 8 * q^54 - q^55 - 3 * q^56 - 8 * q^57 - q^58 + 14 * q^59 + 4 * q^60 - 5 * q^61 - 11 * q^62 - 3 * q^63 + 2 * q^64 + 2 * q^65 - 2 * q^66 - 2 * q^67 - 5 * q^68 + 4 * q^69 - 3 * q^70 - 2 * q^71 + 2 * q^72 + 6 * q^73 + 2 * q^74 + 4 * q^75 - 4 * q^76 - 15 * q^77 + 4 * q^78 + 2 * q^79 + 2 * q^80 - 22 * q^81 + 5 * q^82 + 10 * q^83 - 6 * q^84 - 5 * q^85 + 9 * q^86 - 2 * q^87 - q^88 + 20 * q^89 + 2 * q^90 - 36 * q^91 + 2 * q^92 - 22 * q^93 - 6 * q^94 - 4 * q^95 + 4 * q^96 + 17 * q^97 + 7 * q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 2.00000 0.816497
$$7$$ −4.37228 −1.65257 −0.826284 0.563254i $$-0.809549\pi$$
−0.826284 + 0.563254i $$0.809549\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 2.37228 0.715270 0.357635 0.933862i $$-0.383583\pi$$
0.357635 + 0.933862i $$0.383583\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 6.74456 1.87061 0.935303 0.353849i $$-0.115127\pi$$
0.935303 + 0.353849i $$0.115127\pi$$
$$14$$ −4.37228 −1.16854
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ 0.372281 0.0902915 0.0451457 0.998980i $$-0.485625\pi$$
0.0451457 + 0.998980i $$0.485625\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −8.74456 −1.90822
$$22$$ 2.37228 0.505772
$$23$$ −4.74456 −0.989310 −0.494655 0.869090i $$-0.664706\pi$$
−0.494655 + 0.869090i $$0.664706\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 1.00000 0.200000
$$26$$ 6.74456 1.32272
$$27$$ −4.00000 −0.769800
$$28$$ −4.37228 −0.826284
$$29$$ −9.11684 −1.69296 −0.846478 0.532424i $$-0.821281\pi$$
−0.846478 + 0.532424i $$0.821281\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −8.37228 −1.50371 −0.751853 0.659331i $$-0.770840\pi$$
−0.751853 + 0.659331i $$0.770840\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.74456 0.825922
$$34$$ 0.372281 0.0638457
$$35$$ −4.37228 −0.739050
$$36$$ 1.00000 0.166667
$$37$$ 1.00000 0.164399
$$38$$ −2.00000 −0.324443
$$39$$ 13.4891 2.15999
$$40$$ 1.00000 0.158114
$$41$$ −0.372281 −0.0581406 −0.0290703 0.999577i $$-0.509255\pi$$
−0.0290703 + 0.999577i $$0.509255\pi$$
$$42$$ −8.74456 −1.34932
$$43$$ 1.62772 0.248225 0.124112 0.992268i $$-0.460392\pi$$
0.124112 + 0.992268i $$0.460392\pi$$
$$44$$ 2.37228 0.357635
$$45$$ 1.00000 0.149071
$$46$$ −4.74456 −0.699548
$$47$$ 2.74456 0.400336 0.200168 0.979762i $$-0.435851\pi$$
0.200168 + 0.979762i $$0.435851\pi$$
$$48$$ 2.00000 0.288675
$$49$$ 12.1168 1.73098
$$50$$ 1.00000 0.141421
$$51$$ 0.744563 0.104260
$$52$$ 6.74456 0.935303
$$53$$ −4.37228 −0.600579 −0.300290 0.953848i $$-0.597083\pi$$
−0.300290 + 0.953848i $$0.597083\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 2.37228 0.319878
$$56$$ −4.37228 −0.584271
$$57$$ −4.00000 −0.529813
$$58$$ −9.11684 −1.19710
$$59$$ 1.25544 0.163444 0.0817220 0.996655i $$-0.473958\pi$$
0.0817220 + 0.996655i $$0.473958\pi$$
$$60$$ 2.00000 0.258199
$$61$$ 0.372281 0.0476657 0.0238329 0.999716i $$-0.492413\pi$$
0.0238329 + 0.999716i $$0.492413\pi$$
$$62$$ −8.37228 −1.06328
$$63$$ −4.37228 −0.550856
$$64$$ 1.00000 0.125000
$$65$$ 6.74456 0.836560
$$66$$ 4.74456 0.584015
$$67$$ −6.74456 −0.823979 −0.411990 0.911188i $$-0.635166\pi$$
−0.411990 + 0.911188i $$0.635166\pi$$
$$68$$ 0.372281 0.0451457
$$69$$ −9.48913 −1.14236
$$70$$ −4.37228 −0.522588
$$71$$ 4.74456 0.563076 0.281538 0.959550i $$-0.409155\pi$$
0.281538 + 0.959550i $$0.409155\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −2.74456 −0.321227 −0.160613 0.987017i $$-0.551347\pi$$
−0.160613 + 0.987017i $$0.551347\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 2.00000 0.230940
$$76$$ −2.00000 −0.229416
$$77$$ −10.3723 −1.18203
$$78$$ 13.4891 1.52734
$$79$$ 6.74456 0.758823 0.379411 0.925228i $$-0.376127\pi$$
0.379411 + 0.925228i $$0.376127\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.0000 −1.22222
$$82$$ −0.372281 −0.0411116
$$83$$ 10.7446 1.17937 0.589684 0.807634i $$-0.299252\pi$$
0.589684 + 0.807634i $$0.299252\pi$$
$$84$$ −8.74456 −0.954110
$$85$$ 0.372281 0.0403796
$$86$$ 1.62772 0.175521
$$87$$ −18.2337 −1.95486
$$88$$ 2.37228 0.252886
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −29.4891 −3.09130
$$92$$ −4.74456 −0.494655
$$93$$ −16.7446 −1.73633
$$94$$ 2.74456 0.283080
$$95$$ −2.00000 −0.205196
$$96$$ 2.00000 0.204124
$$97$$ 17.1168 1.73795 0.868976 0.494854i $$-0.164778\pi$$
0.868976 + 0.494854i $$0.164778\pi$$
$$98$$ 12.1168 1.22399
$$99$$ 2.37228 0.238423
$$100$$ 1.00000 0.100000
$$101$$ 11.4891 1.14321 0.571605 0.820529i $$-0.306321\pi$$
0.571605 + 0.820529i $$0.306321\pi$$
$$102$$ 0.744563 0.0737227
$$103$$ 13.4891 1.32912 0.664562 0.747234i $$-0.268618\pi$$
0.664562 + 0.747234i $$0.268618\pi$$
$$104$$ 6.74456 0.661359
$$105$$ −8.74456 −0.853382
$$106$$ −4.37228 −0.424674
$$107$$ 19.4891 1.88408 0.942042 0.335494i $$-0.108903\pi$$
0.942042 + 0.335494i $$0.108903\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ −17.1168 −1.63950 −0.819748 0.572724i $$-0.805887\pi$$
−0.819748 + 0.572724i $$0.805887\pi$$
$$110$$ 2.37228 0.226188
$$111$$ 2.00000 0.189832
$$112$$ −4.37228 −0.413142
$$113$$ 11.6277 1.09384 0.546922 0.837184i $$-0.315799\pi$$
0.546922 + 0.837184i $$0.315799\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ −4.74456 −0.442433
$$116$$ −9.11684 −0.846478
$$117$$ 6.74456 0.623535
$$118$$ 1.25544 0.115572
$$119$$ −1.62772 −0.149213
$$120$$ 2.00000 0.182574
$$121$$ −5.37228 −0.488389
$$122$$ 0.372281 0.0337048
$$123$$ −0.744563 −0.0671350
$$124$$ −8.37228 −0.751853
$$125$$ 1.00000 0.0894427
$$126$$ −4.37228 −0.389514
$$127$$ −5.25544 −0.466345 −0.233172 0.972435i $$-0.574911\pi$$
−0.233172 + 0.972435i $$0.574911\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 3.25544 0.286625
$$130$$ 6.74456 0.591537
$$131$$ −14.7446 −1.28824 −0.644119 0.764925i $$-0.722776\pi$$
−0.644119 + 0.764925i $$0.722776\pi$$
$$132$$ 4.74456 0.412961
$$133$$ 8.74456 0.758250
$$134$$ −6.74456 −0.582641
$$135$$ −4.00000 −0.344265
$$136$$ 0.372281 0.0319229
$$137$$ 5.25544 0.449002 0.224501 0.974474i $$-0.427925\pi$$
0.224501 + 0.974474i $$0.427925\pi$$
$$138$$ −9.48913 −0.807768
$$139$$ −19.1168 −1.62147 −0.810735 0.585414i $$-0.800932\pi$$
−0.810735 + 0.585414i $$0.800932\pi$$
$$140$$ −4.37228 −0.369525
$$141$$ 5.48913 0.462268
$$142$$ 4.74456 0.398155
$$143$$ 16.0000 1.33799
$$144$$ 1.00000 0.0833333
$$145$$ −9.11684 −0.757113
$$146$$ −2.74456 −0.227142
$$147$$ 24.2337 1.99876
$$148$$ 1.00000 0.0821995
$$149$$ 11.4891 0.941226 0.470613 0.882340i $$-0.344033\pi$$
0.470613 + 0.882340i $$0.344033\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0.372281 0.0300972
$$154$$ −10.3723 −0.835822
$$155$$ −8.37228 −0.672478
$$156$$ 13.4891 1.07999
$$157$$ 11.6277 0.927993 0.463996 0.885837i $$-0.346415\pi$$
0.463996 + 0.885837i $$0.346415\pi$$
$$158$$ 6.74456 0.536569
$$159$$ −8.74456 −0.693489
$$160$$ 1.00000 0.0790569
$$161$$ 20.7446 1.63490
$$162$$ −11.0000 −0.864242
$$163$$ 13.6277 1.06741 0.533703 0.845672i $$-0.320800\pi$$
0.533703 + 0.845672i $$0.320800\pi$$
$$164$$ −0.372281 −0.0290703
$$165$$ 4.74456 0.369364
$$166$$ 10.7446 0.833940
$$167$$ −21.4891 −1.66288 −0.831439 0.555616i $$-0.812483\pi$$
−0.831439 + 0.555616i $$0.812483\pi$$
$$168$$ −8.74456 −0.674658
$$169$$ 32.4891 2.49916
$$170$$ 0.372281 0.0285527
$$171$$ −2.00000 −0.152944
$$172$$ 1.62772 0.124112
$$173$$ −5.86141 −0.445634 −0.222817 0.974860i $$-0.571525\pi$$
−0.222817 + 0.974860i $$0.571525\pi$$
$$174$$ −18.2337 −1.38229
$$175$$ −4.37228 −0.330513
$$176$$ 2.37228 0.178817
$$177$$ 2.51087 0.188729
$$178$$ 10.0000 0.749532
$$179$$ 8.23369 0.615415 0.307707 0.951481i $$-0.400438\pi$$
0.307707 + 0.951481i $$0.400438\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −15.4891 −1.15130 −0.575649 0.817697i $$-0.695250\pi$$
−0.575649 + 0.817697i $$0.695250\pi$$
$$182$$ −29.4891 −2.18588
$$183$$ 0.744563 0.0550397
$$184$$ −4.74456 −0.349774
$$185$$ 1.00000 0.0735215
$$186$$ −16.7446 −1.22777
$$187$$ 0.883156 0.0645828
$$188$$ 2.74456 0.200168
$$189$$ 17.4891 1.27215
$$190$$ −2.00000 −0.145095
$$191$$ 24.3723 1.76352 0.881758 0.471702i $$-0.156360\pi$$
0.881758 + 0.471702i $$0.156360\pi$$
$$192$$ 2.00000 0.144338
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 17.1168 1.22892
$$195$$ 13.4891 0.965976
$$196$$ 12.1168 0.865489
$$197$$ 4.51087 0.321387 0.160693 0.987004i $$-0.448627\pi$$
0.160693 + 0.987004i $$0.448627\pi$$
$$198$$ 2.37228 0.168591
$$199$$ −22.7446 −1.61232 −0.806160 0.591698i $$-0.798458\pi$$
−0.806160 + 0.591698i $$0.798458\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −13.4891 −0.951450
$$202$$ 11.4891 0.808372
$$203$$ 39.8614 2.79772
$$204$$ 0.744563 0.0521298
$$205$$ −0.372281 −0.0260013
$$206$$ 13.4891 0.939832
$$207$$ −4.74456 −0.329770
$$208$$ 6.74456 0.467651
$$209$$ −4.74456 −0.328188
$$210$$ −8.74456 −0.603432
$$211$$ 3.11684 0.214572 0.107286 0.994228i $$-0.465784\pi$$
0.107286 + 0.994228i $$0.465784\pi$$
$$212$$ −4.37228 −0.300290
$$213$$ 9.48913 0.650184
$$214$$ 19.4891 1.33225
$$215$$ 1.62772 0.111009
$$216$$ −4.00000 −0.272166
$$217$$ 36.6060 2.48498
$$218$$ −17.1168 −1.15930
$$219$$ −5.48913 −0.370921
$$220$$ 2.37228 0.159939
$$221$$ 2.51087 0.168900
$$222$$ 2.00000 0.134231
$$223$$ 4.37228 0.292790 0.146395 0.989226i $$-0.453233\pi$$
0.146395 + 0.989226i $$0.453233\pi$$
$$224$$ −4.37228 −0.292135
$$225$$ 1.00000 0.0666667
$$226$$ 11.6277 0.773464
$$227$$ −5.62772 −0.373525 −0.186762 0.982405i $$-0.559799\pi$$
−0.186762 + 0.982405i $$0.559799\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −4.74456 −0.312847
$$231$$ −20.7446 −1.36489
$$232$$ −9.11684 −0.598550
$$233$$ −28.2337 −1.84965 −0.924825 0.380392i $$-0.875789\pi$$
−0.924825 + 0.380392i $$0.875789\pi$$
$$234$$ 6.74456 0.440906
$$235$$ 2.74456 0.179036
$$236$$ 1.25544 0.0817220
$$237$$ 13.4891 0.876213
$$238$$ −1.62772 −0.105509
$$239$$ −22.6060 −1.46226 −0.731129 0.682239i $$-0.761006\pi$$
−0.731129 + 0.682239i $$0.761006\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 24.2337 1.56103 0.780515 0.625138i $$-0.214957\pi$$
0.780515 + 0.625138i $$0.214957\pi$$
$$242$$ −5.37228 −0.345343
$$243$$ −10.0000 −0.641500
$$244$$ 0.372281 0.0238329
$$245$$ 12.1168 0.774117
$$246$$ −0.744563 −0.0474716
$$247$$ −13.4891 −0.858292
$$248$$ −8.37228 −0.531640
$$249$$ 21.4891 1.36182
$$250$$ 1.00000 0.0632456
$$251$$ −11.4891 −0.725187 −0.362594 0.931947i $$-0.618109\pi$$
−0.362594 + 0.931947i $$0.618109\pi$$
$$252$$ −4.37228 −0.275428
$$253$$ −11.2554 −0.707623
$$254$$ −5.25544 −0.329755
$$255$$ 0.744563 0.0466263
$$256$$ 1.00000 0.0625000
$$257$$ 24.9783 1.55810 0.779050 0.626962i $$-0.215702\pi$$
0.779050 + 0.626962i $$0.215702\pi$$
$$258$$ 3.25544 0.202675
$$259$$ −4.37228 −0.271680
$$260$$ 6.74456 0.418280
$$261$$ −9.11684 −0.564318
$$262$$ −14.7446 −0.910922
$$263$$ −17.1168 −1.05547 −0.527735 0.849409i $$-0.676959\pi$$
−0.527735 + 0.849409i $$0.676959\pi$$
$$264$$ 4.74456 0.292008
$$265$$ −4.37228 −0.268587
$$266$$ 8.74456 0.536164
$$267$$ 20.0000 1.22398
$$268$$ −6.74456 −0.411990
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ 4.74456 0.288212 0.144106 0.989562i $$-0.453969\pi$$
0.144106 + 0.989562i $$0.453969\pi$$
$$272$$ 0.372281 0.0225729
$$273$$ −58.9783 −3.56953
$$274$$ 5.25544 0.317493
$$275$$ 2.37228 0.143054
$$276$$ −9.48913 −0.571178
$$277$$ −10.7446 −0.645578 −0.322789 0.946471i $$-0.604620\pi$$
−0.322789 + 0.946471i $$0.604620\pi$$
$$278$$ −19.1168 −1.14655
$$279$$ −8.37228 −0.501235
$$280$$ −4.37228 −0.261294
$$281$$ 13.2554 0.790753 0.395377 0.918519i $$-0.370614\pi$$
0.395377 + 0.918519i $$0.370614\pi$$
$$282$$ 5.48913 0.326873
$$283$$ −17.4891 −1.03962 −0.519810 0.854282i $$-0.673997\pi$$
−0.519810 + 0.854282i $$0.673997\pi$$
$$284$$ 4.74456 0.281538
$$285$$ −4.00000 −0.236940
$$286$$ 16.0000 0.946100
$$287$$ 1.62772 0.0960812
$$288$$ 1.00000 0.0589256
$$289$$ −16.8614 −0.991847
$$290$$ −9.11684 −0.535360
$$291$$ 34.2337 2.00681
$$292$$ −2.74456 −0.160613
$$293$$ 9.86141 0.576110 0.288055 0.957614i $$-0.406991\pi$$
0.288055 + 0.957614i $$0.406991\pi$$
$$294$$ 24.2337 1.41334
$$295$$ 1.25544 0.0730944
$$296$$ 1.00000 0.0581238
$$297$$ −9.48913 −0.550615
$$298$$ 11.4891 0.665547
$$299$$ −32.0000 −1.85061
$$300$$ 2.00000 0.115470
$$301$$ −7.11684 −0.410208
$$302$$ −20.0000 −1.15087
$$303$$ 22.9783 1.32007
$$304$$ −2.00000 −0.114708
$$305$$ 0.372281 0.0213168
$$306$$ 0.372281 0.0212819
$$307$$ −0.510875 −0.0291572 −0.0145786 0.999894i $$-0.504641\pi$$
−0.0145786 + 0.999894i $$0.504641\pi$$
$$308$$ −10.3723 −0.591016
$$309$$ 26.9783 1.53474
$$310$$ −8.37228 −0.475514
$$311$$ 4.37228 0.247929 0.123965 0.992287i $$-0.460439\pi$$
0.123965 + 0.992287i $$0.460439\pi$$
$$312$$ 13.4891 0.763671
$$313$$ 19.4891 1.10159 0.550795 0.834640i $$-0.314325\pi$$
0.550795 + 0.834640i $$0.314325\pi$$
$$314$$ 11.6277 0.656190
$$315$$ −4.37228 −0.246350
$$316$$ 6.74456 0.379411
$$317$$ 19.6277 1.10240 0.551201 0.834372i $$-0.314170\pi$$
0.551201 + 0.834372i $$0.314170\pi$$
$$318$$ −8.74456 −0.490371
$$319$$ −21.6277 −1.21092
$$320$$ 1.00000 0.0559017
$$321$$ 38.9783 2.17555
$$322$$ 20.7446 1.15605
$$323$$ −0.744563 −0.0414286
$$324$$ −11.0000 −0.611111
$$325$$ 6.74456 0.374121
$$326$$ 13.6277 0.754770
$$327$$ −34.2337 −1.89313
$$328$$ −0.372281 −0.0205558
$$329$$ −12.0000 −0.661581
$$330$$ 4.74456 0.261180
$$331$$ 0.510875 0.0280802 0.0140401 0.999901i $$-0.495531\pi$$
0.0140401 + 0.999901i $$0.495531\pi$$
$$332$$ 10.7446 0.589684
$$333$$ 1.00000 0.0547997
$$334$$ −21.4891 −1.17583
$$335$$ −6.74456 −0.368495
$$336$$ −8.74456 −0.477055
$$337$$ −18.7446 −1.02108 −0.510541 0.859854i $$-0.670555\pi$$
−0.510541 + 0.859854i $$0.670555\pi$$
$$338$$ 32.4891 1.76718
$$339$$ 23.2554 1.26306
$$340$$ 0.372281 0.0201898
$$341$$ −19.8614 −1.07556
$$342$$ −2.00000 −0.108148
$$343$$ −22.3723 −1.20799
$$344$$ 1.62772 0.0877607
$$345$$ −9.48913 −0.510877
$$346$$ −5.86141 −0.315111
$$347$$ 22.9783 1.23354 0.616769 0.787145i $$-0.288441\pi$$
0.616769 + 0.787145i $$0.288441\pi$$
$$348$$ −18.2337 −0.977428
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ −4.37228 −0.233708
$$351$$ −26.9783 −1.43999
$$352$$ 2.37228 0.126443
$$353$$ 5.86141 0.311971 0.155986 0.987759i $$-0.450145\pi$$
0.155986 + 0.987759i $$0.450145\pi$$
$$354$$ 2.51087 0.133451
$$355$$ 4.74456 0.251815
$$356$$ 10.0000 0.529999
$$357$$ −3.25544 −0.172296
$$358$$ 8.23369 0.435164
$$359$$ 14.9783 0.790522 0.395261 0.918569i $$-0.370654\pi$$
0.395261 + 0.918569i $$0.370654\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ −15.4891 −0.814090
$$363$$ −10.7446 −0.563943
$$364$$ −29.4891 −1.54565
$$365$$ −2.74456 −0.143657
$$366$$ 0.744563 0.0389189
$$367$$ 21.1168 1.10229 0.551145 0.834409i $$-0.314191\pi$$
0.551145 + 0.834409i $$0.314191\pi$$
$$368$$ −4.74456 −0.247327
$$369$$ −0.372281 −0.0193802
$$370$$ 1.00000 0.0519875
$$371$$ 19.1168 0.992497
$$372$$ −16.7446 −0.868165
$$373$$ −8.51087 −0.440676 −0.220338 0.975424i $$-0.570716\pi$$
−0.220338 + 0.975424i $$0.570716\pi$$
$$374$$ 0.883156 0.0456669
$$375$$ 2.00000 0.103280
$$376$$ 2.74456 0.141540
$$377$$ −61.4891 −3.16685
$$378$$ 17.4891 0.899544
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −10.5109 −0.538488
$$382$$ 24.3723 1.24699
$$383$$ 9.48913 0.484872 0.242436 0.970167i $$-0.422054\pi$$
0.242436 + 0.970167i $$0.422054\pi$$
$$384$$ 2.00000 0.102062
$$385$$ −10.3723 −0.528620
$$386$$ 2.00000 0.101797
$$387$$ 1.62772 0.0827416
$$388$$ 17.1168 0.868976
$$389$$ −13.8614 −0.702801 −0.351401 0.936225i $$-0.614294\pi$$
−0.351401 + 0.936225i $$0.614294\pi$$
$$390$$ 13.4891 0.683048
$$391$$ −1.76631 −0.0893262
$$392$$ 12.1168 0.611993
$$393$$ −29.4891 −1.48753
$$394$$ 4.51087 0.227255
$$395$$ 6.74456 0.339356
$$396$$ 2.37228 0.119212
$$397$$ −20.9783 −1.05287 −0.526434 0.850216i $$-0.676471\pi$$
−0.526434 + 0.850216i $$0.676471\pi$$
$$398$$ −22.7446 −1.14008
$$399$$ 17.4891 0.875551
$$400$$ 1.00000 0.0500000
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ −13.4891 −0.672776
$$403$$ −56.4674 −2.81284
$$404$$ 11.4891 0.571605
$$405$$ −11.0000 −0.546594
$$406$$ 39.8614 1.97829
$$407$$ 2.37228 0.117590
$$408$$ 0.744563 0.0368613
$$409$$ −28.2337 −1.39607 −0.698033 0.716066i $$-0.745941\pi$$
−0.698033 + 0.716066i $$0.745941\pi$$
$$410$$ −0.372281 −0.0183857
$$411$$ 10.5109 0.518463
$$412$$ 13.4891 0.664562
$$413$$ −5.48913 −0.270102
$$414$$ −4.74456 −0.233183
$$415$$ 10.7446 0.527430
$$416$$ 6.74456 0.330679
$$417$$ −38.2337 −1.87231
$$418$$ −4.74456 −0.232064
$$419$$ −9.48913 −0.463574 −0.231787 0.972767i $$-0.574457\pi$$
−0.231787 + 0.972767i $$0.574457\pi$$
$$420$$ −8.74456 −0.426691
$$421$$ 15.4891 0.754894 0.377447 0.926031i $$-0.376802\pi$$
0.377447 + 0.926031i $$0.376802\pi$$
$$422$$ 3.11684 0.151726
$$423$$ 2.74456 0.133445
$$424$$ −4.37228 −0.212337
$$425$$ 0.372281 0.0180583
$$426$$ 9.48913 0.459750
$$427$$ −1.62772 −0.0787708
$$428$$ 19.4891 0.942042
$$429$$ 32.0000 1.54497
$$430$$ 1.62772 0.0784956
$$431$$ 4.37228 0.210605 0.105303 0.994440i $$-0.466419\pi$$
0.105303 + 0.994440i $$0.466419\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −32.9783 −1.58483 −0.792417 0.609980i $$-0.791177\pi$$
−0.792417 + 0.609980i $$0.791177\pi$$
$$434$$ 36.6060 1.75714
$$435$$ −18.2337 −0.874238
$$436$$ −17.1168 −0.819748
$$437$$ 9.48913 0.453926
$$438$$ −5.48913 −0.262281
$$439$$ −7.62772 −0.364051 −0.182026 0.983294i $$-0.558265\pi$$
−0.182026 + 0.983294i $$0.558265\pi$$
$$440$$ 2.37228 0.113094
$$441$$ 12.1168 0.576993
$$442$$ 2.51087 0.119430
$$443$$ 28.9783 1.37680 0.688399 0.725332i $$-0.258314\pi$$
0.688399 + 0.725332i $$0.258314\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 10.0000 0.474045
$$446$$ 4.37228 0.207034
$$447$$ 22.9783 1.08683
$$448$$ −4.37228 −0.206571
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −0.883156 −0.0415862
$$452$$ 11.6277 0.546922
$$453$$ −40.0000 −1.87936
$$454$$ −5.62772 −0.264122
$$455$$ −29.4891 −1.38247
$$456$$ −4.00000 −0.187317
$$457$$ −9.86141 −0.461297 −0.230649 0.973037i $$-0.574085\pi$$
−0.230649 + 0.973037i $$0.574085\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −1.48913 −0.0695064
$$460$$ −4.74456 −0.221216
$$461$$ 24.0951 1.12222 0.561110 0.827741i $$-0.310374\pi$$
0.561110 + 0.827741i $$0.310374\pi$$
$$462$$ −20.7446 −0.965124
$$463$$ 17.4891 0.812789 0.406394 0.913698i $$-0.366786\pi$$
0.406394 + 0.913698i $$0.366786\pi$$
$$464$$ −9.11684 −0.423239
$$465$$ −16.7446 −0.776510
$$466$$ −28.2337 −1.30790
$$467$$ −21.3505 −0.987985 −0.493992 0.869466i $$-0.664463\pi$$
−0.493992 + 0.869466i $$0.664463\pi$$
$$468$$ 6.74456 0.311768
$$469$$ 29.4891 1.36168
$$470$$ 2.74456 0.126597
$$471$$ 23.2554 1.07155
$$472$$ 1.25544 0.0577862
$$473$$ 3.86141 0.177548
$$474$$ 13.4891 0.619576
$$475$$ −2.00000 −0.0917663
$$476$$ −1.62772 −0.0746064
$$477$$ −4.37228 −0.200193
$$478$$ −22.6060 −1.03397
$$479$$ 14.7446 0.673696 0.336848 0.941559i $$-0.390639\pi$$
0.336848 + 0.941559i $$0.390639\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 6.74456 0.307526
$$482$$ 24.2337 1.10381
$$483$$ 41.4891 1.88782
$$484$$ −5.37228 −0.244195
$$485$$ 17.1168 0.777236
$$486$$ −10.0000 −0.453609
$$487$$ 19.7228 0.893726 0.446863 0.894602i $$-0.352541\pi$$
0.446863 + 0.894602i $$0.352541\pi$$
$$488$$ 0.372281 0.0168524
$$489$$ 27.2554 1.23253
$$490$$ 12.1168 0.547383
$$491$$ −30.9783 −1.39803 −0.699014 0.715108i $$-0.746378\pi$$
−0.699014 + 0.715108i $$0.746378\pi$$
$$492$$ −0.744563 −0.0335675
$$493$$ −3.39403 −0.152859
$$494$$ −13.4891 −0.606904
$$495$$ 2.37228 0.106626
$$496$$ −8.37228 −0.375927
$$497$$ −20.7446 −0.930521
$$498$$ 21.4891 0.962951
$$499$$ 20.9783 0.939115 0.469558 0.882902i $$-0.344413\pi$$
0.469558 + 0.882902i $$0.344413\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −42.9783 −1.92013
$$502$$ −11.4891 −0.512785
$$503$$ 6.51087 0.290306 0.145153 0.989409i $$-0.453633\pi$$
0.145153 + 0.989409i $$0.453633\pi$$
$$504$$ −4.37228 −0.194757
$$505$$ 11.4891 0.511259
$$506$$ −11.2554 −0.500365
$$507$$ 64.9783 2.88579
$$508$$ −5.25544 −0.233172
$$509$$ −22.7446 −1.00814 −0.504068 0.863664i $$-0.668164\pi$$
−0.504068 + 0.863664i $$0.668164\pi$$
$$510$$ 0.744563 0.0329698
$$511$$ 12.0000 0.530849
$$512$$ 1.00000 0.0441942
$$513$$ 8.00000 0.353209
$$514$$ 24.9783 1.10174
$$515$$ 13.4891 0.594402
$$516$$ 3.25544 0.143313
$$517$$ 6.51087 0.286348
$$518$$ −4.37228 −0.192107
$$519$$ −11.7228 −0.514574
$$520$$ 6.74456 0.295769
$$521$$ −36.0951 −1.58135 −0.790677 0.612233i $$-0.790271\pi$$
−0.790677 + 0.612233i $$0.790271\pi$$
$$522$$ −9.11684 −0.399033
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ −14.7446 −0.644119
$$525$$ −8.74456 −0.381644
$$526$$ −17.1168 −0.746330
$$527$$ −3.11684 −0.135772
$$528$$ 4.74456 0.206481
$$529$$ −0.489125 −0.0212663
$$530$$ −4.37228 −0.189920
$$531$$ 1.25544 0.0544813
$$532$$ 8.74456 0.379125
$$533$$ −2.51087 −0.108758
$$534$$ 20.0000 0.865485
$$535$$ 19.4891 0.842588
$$536$$ −6.74456 −0.291321
$$537$$ 16.4674 0.710620
$$538$$ 10.0000 0.431131
$$539$$ 28.7446 1.23812
$$540$$ −4.00000 −0.172133
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 4.74456 0.203796
$$543$$ −30.9783 −1.32940
$$544$$ 0.372281 0.0159614
$$545$$ −17.1168 −0.733205
$$546$$ −58.9783 −2.52404
$$547$$ 33.6277 1.43782 0.718909 0.695104i $$-0.244642\pi$$
0.718909 + 0.695104i $$0.244642\pi$$
$$548$$ 5.25544 0.224501
$$549$$ 0.372281 0.0158886
$$550$$ 2.37228 0.101154
$$551$$ 18.2337 0.776781
$$552$$ −9.48913 −0.403884
$$553$$ −29.4891 −1.25401
$$554$$ −10.7446 −0.456493
$$555$$ 2.00000 0.0848953
$$556$$ −19.1168 −0.810735
$$557$$ −2.74456 −0.116291 −0.0581454 0.998308i $$-0.518519\pi$$
−0.0581454 + 0.998308i $$0.518519\pi$$
$$558$$ −8.37228 −0.354427
$$559$$ 10.9783 0.464331
$$560$$ −4.37228 −0.184763
$$561$$ 1.76631 0.0745738
$$562$$ 13.2554 0.559147
$$563$$ −30.0951 −1.26836 −0.634179 0.773187i $$-0.718662\pi$$
−0.634179 + 0.773187i $$0.718662\pi$$
$$564$$ 5.48913 0.231134
$$565$$ 11.6277 0.489182
$$566$$ −17.4891 −0.735123
$$567$$ 48.0951 2.01980
$$568$$ 4.74456 0.199077
$$569$$ 12.9783 0.544077 0.272038 0.962286i $$-0.412302\pi$$
0.272038 + 0.962286i $$0.412302\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 12.6060 0.527543 0.263772 0.964585i $$-0.415033\pi$$
0.263772 + 0.964585i $$0.415033\pi$$
$$572$$ 16.0000 0.668994
$$573$$ 48.7446 2.03633
$$574$$ 1.62772 0.0679397
$$575$$ −4.74456 −0.197862
$$576$$ 1.00000 0.0416667
$$577$$ 15.4891 0.644821 0.322410 0.946600i $$-0.395507\pi$$
0.322410 + 0.946600i $$0.395507\pi$$
$$578$$ −16.8614 −0.701342
$$579$$ 4.00000 0.166234
$$580$$ −9.11684 −0.378556
$$581$$ −46.9783 −1.94899
$$582$$ 34.2337 1.41903
$$583$$ −10.3723 −0.429576
$$584$$ −2.74456 −0.113571
$$585$$ 6.74456 0.278853
$$586$$ 9.86141 0.407371
$$587$$ −26.8397 −1.10779 −0.553896 0.832586i $$-0.686859\pi$$
−0.553896 + 0.832586i $$0.686859\pi$$
$$588$$ 24.2337 0.999380
$$589$$ 16.7446 0.689948
$$590$$ 1.25544 0.0516855
$$591$$ 9.02175 0.371105
$$592$$ 1.00000 0.0410997
$$593$$ 24.2337 0.995158 0.497579 0.867419i $$-0.334222\pi$$
0.497579 + 0.867419i $$0.334222\pi$$
$$594$$ −9.48913 −0.389344
$$595$$ −1.62772 −0.0667300
$$596$$ 11.4891 0.470613
$$597$$ −45.4891 −1.86175
$$598$$ −32.0000 −1.30858
$$599$$ −5.48913 −0.224280 −0.112140 0.993692i $$-0.535770\pi$$
−0.112140 + 0.993692i $$0.535770\pi$$
$$600$$ 2.00000 0.0816497
$$601$$ 0.372281 0.0151857 0.00759284 0.999971i $$-0.497583\pi$$
0.00759284 + 0.999971i $$0.497583\pi$$
$$602$$ −7.11684 −0.290061
$$603$$ −6.74456 −0.274660
$$604$$ −20.0000 −0.813788
$$605$$ −5.37228 −0.218414
$$606$$ 22.9783 0.933428
$$607$$ −39.7228 −1.61230 −0.806150 0.591712i $$-0.798452\pi$$
−0.806150 + 0.591712i $$0.798452\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 79.7228 3.23053
$$610$$ 0.372281 0.0150732
$$611$$ 18.5109 0.748870
$$612$$ 0.372281 0.0150486
$$613$$ −20.0951 −0.811633 −0.405817 0.913955i $$-0.633013\pi$$
−0.405817 + 0.913955i $$0.633013\pi$$
$$614$$ −0.510875 −0.0206172
$$615$$ −0.744563 −0.0300237
$$616$$ −10.3723 −0.417911
$$617$$ 25.7228 1.03556 0.517781 0.855513i $$-0.326758\pi$$
0.517781 + 0.855513i $$0.326758\pi$$
$$618$$ 26.9783 1.08522
$$619$$ 12.8832 0.517818 0.258909 0.965902i $$-0.416637\pi$$
0.258909 + 0.965902i $$0.416637\pi$$
$$620$$ −8.37228 −0.336239
$$621$$ 18.9783 0.761571
$$622$$ 4.37228 0.175313
$$623$$ −43.7228 −1.75172
$$624$$ 13.4891 0.539997
$$625$$ 1.00000 0.0400000
$$626$$ 19.4891 0.778942
$$627$$ −9.48913 −0.378959
$$628$$ 11.6277 0.463996
$$629$$ 0.372281 0.0148438
$$630$$ −4.37228 −0.174196
$$631$$ 44.0951 1.75540 0.877699 0.479212i $$-0.159078\pi$$
0.877699 + 0.479212i $$0.159078\pi$$
$$632$$ 6.74456 0.268284
$$633$$ 6.23369 0.247767
$$634$$ 19.6277 0.779516
$$635$$ −5.25544 −0.208556
$$636$$ −8.74456 −0.346744
$$637$$ 81.7228 3.23798
$$638$$ −21.6277 −0.856250
$$639$$ 4.74456 0.187692
$$640$$ 1.00000 0.0395285
$$641$$ −9.39403 −0.371042 −0.185521 0.982640i $$-0.559397\pi$$
−0.185521 + 0.982640i $$0.559397\pi$$
$$642$$ 38.9783 1.53835
$$643$$ −2.37228 −0.0935536 −0.0467768 0.998905i $$-0.514895\pi$$
−0.0467768 + 0.998905i $$0.514895\pi$$
$$644$$ 20.7446 0.817450
$$645$$ 3.25544 0.128183
$$646$$ −0.744563 −0.0292944
$$647$$ 9.48913 0.373056 0.186528 0.982450i $$-0.440276\pi$$
0.186528 + 0.982450i $$0.440276\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 2.97825 0.116907
$$650$$ 6.74456 0.264544
$$651$$ 73.2119 2.86940
$$652$$ 13.6277 0.533703
$$653$$ 23.4891 0.919201 0.459600 0.888126i $$-0.347993\pi$$
0.459600 + 0.888126i $$0.347993\pi$$
$$654$$ −34.2337 −1.33864
$$655$$ −14.7446 −0.576118
$$656$$ −0.372281 −0.0145351
$$657$$ −2.74456 −0.107076
$$658$$ −12.0000 −0.467809
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 4.74456 0.184682
$$661$$ −47.3505 −1.84172 −0.920861 0.389891i $$-0.872513\pi$$
−0.920861 + 0.389891i $$0.872513\pi$$
$$662$$ 0.510875 0.0198557
$$663$$ 5.02175 0.195029
$$664$$ 10.7446 0.416970
$$665$$ 8.74456 0.339100
$$666$$ 1.00000 0.0387492
$$667$$ 43.2554 1.67486
$$668$$ −21.4891 −0.831439
$$669$$ 8.74456 0.338084
$$670$$ −6.74456 −0.260565
$$671$$ 0.883156 0.0340939
$$672$$ −8.74456 −0.337329
$$673$$ −31.4891 −1.21382 −0.606908 0.794772i $$-0.707590\pi$$
−0.606908 + 0.794772i $$0.707590\pi$$
$$674$$ −18.7446 −0.722014
$$675$$ −4.00000 −0.153960
$$676$$ 32.4891 1.24958
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 23.2554 0.893120
$$679$$ −74.8397 −2.87208
$$680$$ 0.372281 0.0142763
$$681$$ −11.2554 −0.431309
$$682$$ −19.8614 −0.760533
$$683$$ 14.3723 0.549940 0.274970 0.961453i $$-0.411332\pi$$
0.274970 + 0.961453i $$0.411332\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 5.25544 0.200800
$$686$$ −22.3723 −0.854178
$$687$$ −20.0000 −0.763048
$$688$$ 1.62772 0.0620562
$$689$$ −29.4891 −1.12345
$$690$$ −9.48913 −0.361245
$$691$$ −29.3505 −1.11655 −0.558273 0.829657i $$-0.688536\pi$$
−0.558273 + 0.829657i $$0.688536\pi$$
$$692$$ −5.86141 −0.222817
$$693$$ −10.3723 −0.394010
$$694$$ 22.9783 0.872242
$$695$$ −19.1168 −0.725143
$$696$$ −18.2337 −0.691146
$$697$$ −0.138593 −0.00524960
$$698$$ −22.0000 −0.832712
$$699$$ −56.4674 −2.13579
$$700$$ −4.37228 −0.165257
$$701$$ 42.4674 1.60397 0.801985 0.597344i $$-0.203777\pi$$
0.801985 + 0.597344i $$0.203777\pi$$
$$702$$ −26.9783 −1.01823
$$703$$ −2.00000 −0.0754314
$$704$$ 2.37228 0.0894087
$$705$$ 5.48913 0.206732
$$706$$ 5.86141 0.220597
$$707$$ −50.2337 −1.88923
$$708$$ 2.51087 0.0943645
$$709$$ 22.8832 0.859395 0.429697 0.902973i $$-0.358620\pi$$
0.429697 + 0.902973i $$0.358620\pi$$
$$710$$ 4.74456 0.178060
$$711$$ 6.74456 0.252941
$$712$$ 10.0000 0.374766
$$713$$ 39.7228 1.48763
$$714$$ −3.25544 −0.121832
$$715$$ 16.0000 0.598366
$$716$$ 8.23369 0.307707
$$717$$ −45.2119 −1.68847
$$718$$ 14.9783 0.558983
$$719$$ −23.2554 −0.867281 −0.433641 0.901086i $$-0.642771\pi$$
−0.433641 + 0.901086i $$0.642771\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −58.9783 −2.19646
$$722$$ −15.0000 −0.558242
$$723$$ 48.4674 1.80252
$$724$$ −15.4891 −0.575649
$$725$$ −9.11684 −0.338591
$$726$$ −10.7446 −0.398768
$$727$$ −48.0000 −1.78022 −0.890111 0.455744i $$-0.849373\pi$$
−0.890111 + 0.455744i $$0.849373\pi$$
$$728$$ −29.4891 −1.09294
$$729$$ 13.0000 0.481481
$$730$$ −2.74456 −0.101581
$$731$$ 0.605969 0.0224126
$$732$$ 0.744563 0.0275198
$$733$$ −23.6277 −0.872710 −0.436355 0.899775i $$-0.643731\pi$$
−0.436355 + 0.899775i $$0.643731\pi$$
$$734$$ 21.1168 0.779437
$$735$$ 24.2337 0.893873
$$736$$ −4.74456 −0.174887
$$737$$ −16.0000 −0.589368
$$738$$ −0.372281 −0.0137039
$$739$$ −8.13859 −0.299383 −0.149691 0.988733i $$-0.547828\pi$$
−0.149691 + 0.988733i $$0.547828\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ −26.9783 −0.991071
$$742$$ 19.1168 0.701801
$$743$$ 0.372281 0.0136577 0.00682884 0.999977i $$-0.497826\pi$$
0.00682884 + 0.999977i $$0.497826\pi$$
$$744$$ −16.7446 −0.613885
$$745$$ 11.4891 0.420929
$$746$$ −8.51087 −0.311605
$$747$$ 10.7446 0.393123
$$748$$ 0.883156 0.0322914
$$749$$ −85.2119 −3.11358
$$750$$ 2.00000 0.0730297
$$751$$ −0.744563 −0.0271695 −0.0135847 0.999908i $$-0.504324\pi$$
−0.0135847 + 0.999908i $$0.504324\pi$$
$$752$$ 2.74456 0.100084
$$753$$ −22.9783 −0.837374
$$754$$ −61.4891 −2.23930
$$755$$ −20.0000 −0.727875
$$756$$ 17.4891 0.636073
$$757$$ −47.9565 −1.74301 −0.871504 0.490388i $$-0.836855\pi$$
−0.871504 + 0.490388i $$0.836855\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ −22.5109 −0.817093
$$760$$ −2.00000 −0.0725476
$$761$$ −0.372281 −0.0134952 −0.00674759 0.999977i $$-0.502148\pi$$
−0.00674759 + 0.999977i $$0.502148\pi$$
$$762$$ −10.5109 −0.380769
$$763$$ 74.8397 2.70938
$$764$$ 24.3723 0.881758
$$765$$ 0.372281 0.0134599
$$766$$ 9.48913 0.342856
$$767$$ 8.46738 0.305739
$$768$$ 2.00000 0.0721688
$$769$$ 11.7663 0.424304 0.212152 0.977237i $$-0.431953\pi$$
0.212152 + 0.977237i $$0.431953\pi$$
$$770$$ −10.3723 −0.373791
$$771$$ 49.9565 1.79914
$$772$$ 2.00000 0.0719816
$$773$$ 27.3505 0.983730 0.491865 0.870671i $$-0.336315\pi$$
0.491865 + 0.870671i $$0.336315\pi$$
$$774$$ 1.62772 0.0585071
$$775$$ −8.37228 −0.300741
$$776$$ 17.1168 0.614459
$$777$$ −8.74456 −0.313709
$$778$$ −13.8614 −0.496956
$$779$$ 0.744563 0.0266767
$$780$$ 13.4891 0.482988
$$781$$ 11.2554 0.402751
$$782$$ −1.76631 −0.0631632
$$783$$ 36.4674 1.30324
$$784$$ 12.1168 0.432744
$$785$$ 11.6277 0.415011
$$786$$ −29.4891 −1.05184
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 4.51087 0.160693
$$789$$ −34.2337 −1.21875
$$790$$ 6.74456 0.239961
$$791$$ −50.8397 −1.80765
$$792$$ 2.37228 0.0842953
$$793$$ 2.51087 0.0891638
$$794$$ −20.9783 −0.744490
$$795$$ −8.74456 −0.310138
$$796$$ −22.7446 −0.806160
$$797$$ −4.51087 −0.159783 −0.0798917 0.996804i $$-0.525457\pi$$
−0.0798917 + 0.996804i $$0.525457\pi$$
$$798$$ 17.4891 0.619108
$$799$$ 1.02175 0.0361469
$$800$$ 1.00000 0.0353553
$$801$$ 10.0000 0.353333
$$802$$ 10.0000 0.353112
$$803$$ −6.51087 −0.229764
$$804$$ −13.4891 −0.475725
$$805$$ 20.7446 0.731150
$$806$$ −56.4674 −1.98898
$$807$$ 20.0000 0.704033
$$808$$ 11.4891 0.404186
$$809$$ −28.5109 −1.00239 −0.501194 0.865335i $$-0.667106\pi$$
−0.501194 + 0.865335i $$0.667106\pi$$
$$810$$ −11.0000 −0.386501
$$811$$ 26.5109 0.930923 0.465461 0.885068i $$-0.345888\pi$$
0.465461 + 0.885068i $$0.345888\pi$$
$$812$$ 39.8614 1.39886
$$813$$ 9.48913 0.332798
$$814$$ 2.37228 0.0831484
$$815$$ 13.6277 0.477358
$$816$$ 0.744563 0.0260649
$$817$$ −3.25544 −0.113893
$$818$$ −28.2337 −0.987168
$$819$$ −29.4891 −1.03043
$$820$$ −0.372281 −0.0130006
$$821$$ −21.2554 −0.741820 −0.370910 0.928669i $$-0.620954\pi$$
−0.370910 + 0.928669i $$0.620954\pi$$
$$822$$ 10.5109 0.366609
$$823$$ 17.7228 0.617778 0.308889 0.951098i $$-0.400043\pi$$
0.308889 + 0.951098i $$0.400043\pi$$
$$824$$ 13.4891 0.469916
$$825$$ 4.74456 0.165184
$$826$$ −5.48913 −0.190991
$$827$$ −2.64947 −0.0921310 −0.0460655 0.998938i $$-0.514668\pi$$
−0.0460655 + 0.998938i $$0.514668\pi$$
$$828$$ −4.74456 −0.164885
$$829$$ 11.3505 0.394220 0.197110 0.980381i $$-0.436844\pi$$
0.197110 + 0.980381i $$0.436844\pi$$
$$830$$ 10.7446 0.372949
$$831$$ −21.4891 −0.745449
$$832$$ 6.74456 0.233826
$$833$$ 4.51087 0.156293
$$834$$ −38.2337 −1.32392
$$835$$ −21.4891 −0.743662
$$836$$ −4.74456 −0.164094
$$837$$ 33.4891 1.15755
$$838$$ −9.48913 −0.327796
$$839$$ −6.51087 −0.224780 −0.112390 0.993664i $$-0.535851\pi$$
−0.112390 + 0.993664i $$0.535851\pi$$
$$840$$ −8.74456 −0.301716
$$841$$ 54.1168 1.86610
$$842$$ 15.4891 0.533791
$$843$$ 26.5109 0.913083
$$844$$ 3.11684 0.107286
$$845$$ 32.4891 1.11766
$$846$$ 2.74456 0.0943600
$$847$$ 23.4891 0.807096
$$848$$ −4.37228 −0.150145
$$849$$ −34.9783 −1.20045
$$850$$ 0.372281 0.0127691
$$851$$ −4.74456 −0.162642
$$852$$ 9.48913 0.325092
$$853$$ 12.5109 0.428364 0.214182 0.976794i $$-0.431291\pi$$
0.214182 + 0.976794i $$0.431291\pi$$
$$854$$ −1.62772 −0.0556994
$$855$$ −2.00000 −0.0683986
$$856$$ 19.4891 0.666125
$$857$$ −57.8614 −1.97651 −0.988254 0.152820i $$-0.951164\pi$$
−0.988254 + 0.152820i $$0.951164\pi$$
$$858$$ 32.0000 1.09246
$$859$$ 37.7228 1.28709 0.643543 0.765410i $$-0.277464\pi$$
0.643543 + 0.765410i $$0.277464\pi$$
$$860$$ 1.62772 0.0555047
$$861$$ 3.25544 0.110945
$$862$$ 4.37228 0.148920
$$863$$ −20.8397 −0.709390 −0.354695 0.934982i $$-0.615415\pi$$
−0.354695 + 0.934982i $$0.615415\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ −5.86141 −0.199294
$$866$$ −32.9783 −1.12065
$$867$$ −33.7228 −1.14529
$$868$$ 36.6060 1.24249
$$869$$ 16.0000 0.542763
$$870$$ −18.2337 −0.618180
$$871$$ −45.4891 −1.54134
$$872$$ −17.1168 −0.579649
$$873$$ 17.1168 0.579317
$$874$$ 9.48913 0.320974
$$875$$ −4.37228 −0.147810
$$876$$ −5.48913 −0.185460
$$877$$ 43.6277 1.47320 0.736602 0.676327i $$-0.236429\pi$$
0.736602 + 0.676327i $$0.236429\pi$$
$$878$$ −7.62772 −0.257423
$$879$$ 19.7228 0.665234
$$880$$ 2.37228 0.0799696
$$881$$ −4.37228 −0.147306 −0.0736530 0.997284i $$-0.523466\pi$$
−0.0736530 + 0.997284i $$0.523466\pi$$
$$882$$ 12.1168 0.407995
$$883$$ −5.62772 −0.189388 −0.0946939 0.995506i $$-0.530187\pi$$
−0.0946939 + 0.995506i $$0.530187\pi$$
$$884$$ 2.51087 0.0844499
$$885$$ 2.51087 0.0844021
$$886$$ 28.9783 0.973543
$$887$$ −19.6277 −0.659034 −0.329517 0.944150i $$-0.606886\pi$$
−0.329517 + 0.944150i $$0.606886\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 22.9783 0.770666
$$890$$ 10.0000 0.335201
$$891$$ −26.0951 −0.874219
$$892$$ 4.37228 0.146395
$$893$$ −5.48913 −0.183687
$$894$$ 22.9783 0.768508
$$895$$ 8.23369 0.275222
$$896$$ −4.37228 −0.146068
$$897$$ −64.0000 −2.13690
$$898$$ 18.0000 0.600668
$$899$$ 76.3288 2.54571
$$900$$ 1.00000 0.0333333
$$901$$ −1.62772 −0.0542272
$$902$$ −0.883156 −0.0294059
$$903$$ −14.2337 −0.473667
$$904$$ 11.6277 0.386732
$$905$$ −15.4891 −0.514876
$$906$$ −40.0000 −1.32891
$$907$$ −28.4674 −0.945244 −0.472622 0.881265i $$-0.656692\pi$$
−0.472622 + 0.881265i $$0.656692\pi$$
$$908$$ −5.62772 −0.186762
$$909$$ 11.4891 0.381070
$$910$$ −29.4891 −0.977555
$$911$$ 52.2337 1.73058 0.865290 0.501272i $$-0.167134\pi$$
0.865290 + 0.501272i $$0.167134\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 25.4891 0.843567
$$914$$ −9.86141 −0.326186
$$915$$ 0.744563 0.0246145
$$916$$ −10.0000 −0.330409
$$917$$ 64.4674 2.12890
$$918$$ −1.48913 −0.0491485
$$919$$ −49.7228 −1.64020 −0.820102 0.572217i $$-0.806083\pi$$
−0.820102 + 0.572217i $$0.806083\pi$$
$$920$$ −4.74456 −0.156424
$$921$$ −1.02175 −0.0336678
$$922$$ 24.0951 0.793530
$$923$$ 32.0000 1.05329
$$924$$ −20.7446 −0.682446
$$925$$ 1.00000 0.0328798
$$926$$ 17.4891 0.574728
$$927$$ 13.4891 0.443041
$$928$$ −9.11684 −0.299275
$$929$$ −32.0951 −1.05301 −0.526503 0.850173i $$-0.676497\pi$$
−0.526503 + 0.850173i $$0.676497\pi$$
$$930$$ −16.7446 −0.549076
$$931$$ −24.2337 −0.794227
$$932$$ −28.2337 −0.924825
$$933$$ 8.74456 0.286284
$$934$$ −21.3505 −0.698611
$$935$$ 0.883156 0.0288823
$$936$$ 6.74456 0.220453
$$937$$ −8.97825 −0.293307 −0.146653 0.989188i $$-0.546850\pi$$
−0.146653 + 0.989188i $$0.546850\pi$$
$$938$$ 29.4891 0.962854
$$939$$ 38.9783 1.27201
$$940$$ 2.74456 0.0895178
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 23.2554 0.757703
$$943$$ 1.76631 0.0575190
$$944$$ 1.25544 0.0408610
$$945$$ 17.4891 0.568921
$$946$$ 3.86141 0.125545
$$947$$ −26.0951 −0.847977 −0.423988 0.905668i $$-0.639370\pi$$
−0.423988 + 0.905668i $$0.639370\pi$$
$$948$$ 13.4891 0.438106
$$949$$ −18.5109 −0.600888
$$950$$ −2.00000 −0.0648886
$$951$$ 39.2554 1.27294
$$952$$ −1.62772 −0.0527547
$$953$$ 11.7663 0.381148 0.190574 0.981673i $$-0.438965\pi$$
0.190574 + 0.981673i $$0.438965\pi$$
$$954$$ −4.37228 −0.141558
$$955$$ 24.3723 0.788669
$$956$$ −22.6060 −0.731129
$$957$$ −43.2554 −1.39825
$$958$$ 14.7446 0.476375
$$959$$ −22.9783 −0.742006
$$960$$ 2.00000 0.0645497
$$961$$ 39.0951 1.26113
$$962$$ 6.74456 0.217453
$$963$$ 19.4891 0.628028
$$964$$ 24.2337 0.780515
$$965$$ 2.00000 0.0643823
$$966$$ 41.4891 1.33489
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ −5.37228 −0.172672
$$969$$ −1.48913 −0.0478376
$$970$$ 17.1168 0.549589
$$971$$ 20.6060 0.661277 0.330639 0.943757i $$-0.392736\pi$$
0.330639 + 0.943757i $$0.392736\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ 83.5842 2.67959
$$974$$ 19.7228 0.631960
$$975$$ 13.4891 0.431998
$$976$$ 0.372281 0.0119164
$$977$$ −29.1168 −0.931530 −0.465765 0.884908i $$-0.654221\pi$$
−0.465765 + 0.884908i $$0.654221\pi$$
$$978$$ 27.2554 0.871533
$$979$$ 23.7228 0.758184
$$980$$ 12.1168 0.387058
$$981$$ −17.1168 −0.546499
$$982$$ −30.9783 −0.988556
$$983$$ 2.13859 0.0682105 0.0341053 0.999418i $$-0.489142\pi$$
0.0341053 + 0.999418i $$0.489142\pi$$
$$984$$ −0.744563 −0.0237358
$$985$$ 4.51087 0.143728
$$986$$ −3.39403 −0.108088
$$987$$ −24.0000 −0.763928
$$988$$ −13.4891 −0.429146
$$989$$ −7.72281 −0.245571
$$990$$ 2.37228 0.0753960
$$991$$ −14.6060 −0.463974 −0.231987 0.972719i $$-0.574523\pi$$
−0.231987 + 0.972719i $$0.574523\pi$$
$$992$$ −8.37228 −0.265820
$$993$$ 1.02175 0.0324242
$$994$$ −20.7446 −0.657978
$$995$$ −22.7446 −0.721051
$$996$$ 21.4891 0.680909
$$997$$ −8.23369 −0.260764 −0.130382 0.991464i $$-0.541620\pi$$
−0.130382 + 0.991464i $$0.541620\pi$$
$$998$$ 20.9783 0.664055
$$999$$ −4.00000 −0.126554
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 370.2.a.f.1.1 2
3.2 odd 2 3330.2.a.bb.1.1 2
4.3 odd 2 2960.2.a.o.1.2 2
5.2 odd 4 1850.2.b.m.149.3 4
5.3 odd 4 1850.2.b.m.149.2 4
5.4 even 2 1850.2.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.1 2 1.1 even 1 trivial
1850.2.a.q.1.2 2 5.4 even 2
1850.2.b.m.149.2 4 5.3 odd 4
1850.2.b.m.149.3 4 5.2 odd 4
2960.2.a.o.1.2 2 4.3 odd 2
3330.2.a.bb.1.1 2 3.2 odd 2