# Properties

 Label 475.2.a.j.1.1 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.66064384.1 Defining polynomial: $$x^{6} - 9x^{4} + 13x^{2} - 1$$ x^6 - 9*x^4 + 13*x^2 - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30397$$ of defining polynomial Character $$\chi$$ $$=$$ 475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41987 q^{2} +0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} -3.18676 q^{7} -4.49073 q^{8} -2.71155 q^{9} +O(q^{10})$$ $$q-2.41987 q^{2} +0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} -3.18676 q^{7} -4.49073 q^{8} -2.71155 q^{9} +4.15544 q^{11} +2.07086 q^{12} +2.07086 q^{13} +7.71155 q^{14} +3.15544 q^{16} +5.79470 q^{17} +6.56159 q^{18} -1.00000 q^{19} -1.71155 q^{21} -10.0556 q^{22} -2.60794 q^{23} -2.41188 q^{24} -5.01121 q^{26} -3.06756 q^{27} -12.2874 q^{28} +6.00000 q^{29} +2.59933 q^{31} +1.34571 q^{32} +2.23180 q^{33} -14.0224 q^{34} -10.4551 q^{36} +4.30266 q^{37} +2.41987 q^{38} +1.11222 q^{39} -0.599328 q^{41} +4.14172 q^{42} +3.18676 q^{43} +16.0224 q^{44} +6.31087 q^{46} +11.7086 q^{47} +1.69472 q^{48} +3.15544 q^{49} +3.11222 q^{51} +7.98476 q^{52} +11.7503 q^{53} +7.42309 q^{54} +14.3109 q^{56} -0.537080 q^{57} -14.5192 q^{58} -1.71155 q^{59} -8.75476 q^{61} -6.29004 q^{62} +8.64104 q^{63} -9.56732 q^{64} -5.40067 q^{66} +4.76228 q^{67} +22.3430 q^{68} -1.40067 q^{69} +13.7115 q^{71} +12.1768 q^{72} -2.72714 q^{73} -10.4119 q^{74} -3.85577 q^{76} -13.2424 q^{77} -2.69142 q^{78} -1.40067 q^{79} +6.48711 q^{81} +1.45030 q^{82} -7.07154 q^{83} -6.59933 q^{84} -7.71155 q^{86} +3.22248 q^{87} -18.6609 q^{88} +16.5353 q^{89} -6.59933 q^{91} -10.0556 q^{92} +1.39605 q^{93} -28.3333 q^{94} +0.722754 q^{96} +2.07086 q^{97} -7.63575 q^{98} -11.2677 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4} + 14 q^{9}+O(q^{10})$$ 6 * q + 8 * q^4 + 14 * q^9 $$6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100})$$ 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.41987 −1.71111 −0.855553 0.517715i $$-0.826783\pi$$
−0.855553 + 0.517715i $$0.826783\pi$$
$$3$$ 0.537080 0.310083 0.155042 0.987908i $$-0.450449\pi$$
0.155042 + 0.987908i $$0.450449\pi$$
$$4$$ 3.85577 1.92789
$$5$$ 0 0
$$6$$ −1.29966 −0.530586
$$7$$ −3.18676 −1.20448 −0.602241 0.798314i $$-0.705725\pi$$
−0.602241 + 0.798314i $$0.705725\pi$$
$$8$$ −4.49073 −1.58771
$$9$$ −2.71155 −0.903848
$$10$$ 0 0
$$11$$ 4.15544 1.25291 0.626456 0.779457i $$-0.284505\pi$$
0.626456 + 0.779457i $$0.284505\pi$$
$$12$$ 2.07086 0.597805
$$13$$ 2.07086 0.574353 0.287176 0.957878i $$-0.407283\pi$$
0.287176 + 0.957878i $$0.407283\pi$$
$$14$$ 7.71155 2.06100
$$15$$ 0 0
$$16$$ 3.15544 0.788859
$$17$$ 5.79470 1.40542 0.702710 0.711476i $$-0.251973\pi$$
0.702710 + 0.711476i $$0.251973\pi$$
$$18$$ 6.56159 1.54658
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.71155 −0.373490
$$22$$ −10.0556 −2.14386
$$23$$ −2.60794 −0.543793 −0.271896 0.962327i $$-0.587651\pi$$
−0.271896 + 0.962327i $$0.587651\pi$$
$$24$$ −2.41188 −0.492323
$$25$$ 0 0
$$26$$ −5.01121 −0.982779
$$27$$ −3.06756 −0.590352
$$28$$ −12.2874 −2.32210
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 2.59933 0.466853 0.233427 0.972374i $$-0.425006\pi$$
0.233427 + 0.972374i $$0.425006\pi$$
$$32$$ 1.34571 0.237890
$$33$$ 2.23180 0.388507
$$34$$ −14.0224 −2.40482
$$35$$ 0 0
$$36$$ −10.4551 −1.74252
$$37$$ 4.30266 0.707353 0.353677 0.935368i $$-0.384931\pi$$
0.353677 + 0.935368i $$0.384931\pi$$
$$38$$ 2.41987 0.392555
$$39$$ 1.11222 0.178097
$$40$$ 0 0
$$41$$ −0.599328 −0.0935993 −0.0467997 0.998904i $$-0.514902\pi$$
−0.0467997 + 0.998904i $$0.514902\pi$$
$$42$$ 4.14172 0.639081
$$43$$ 3.18676 0.485976 0.242988 0.970029i $$-0.421872\pi$$
0.242988 + 0.970029i $$0.421872\pi$$
$$44$$ 16.0224 2.41547
$$45$$ 0 0
$$46$$ 6.31087 0.930487
$$47$$ 11.7086 1.70787 0.853937 0.520376i $$-0.174208\pi$$
0.853937 + 0.520376i $$0.174208\pi$$
$$48$$ 1.69472 0.244612
$$49$$ 3.15544 0.450777
$$50$$ 0 0
$$51$$ 3.11222 0.435798
$$52$$ 7.98476 1.10729
$$53$$ 11.7503 1.61403 0.807017 0.590529i $$-0.201081\pi$$
0.807017 + 0.590529i $$0.201081\pi$$
$$54$$ 7.42309 1.01015
$$55$$ 0 0
$$56$$ 14.3109 1.91237
$$57$$ −0.537080 −0.0711380
$$58$$ −14.5192 −1.90647
$$59$$ −1.71155 −0.222824 −0.111412 0.993774i $$-0.535537\pi$$
−0.111412 + 0.993774i $$0.535537\pi$$
$$60$$ 0 0
$$61$$ −8.75476 −1.12093 −0.560466 0.828177i $$-0.689378\pi$$
−0.560466 + 0.828177i $$0.689378\pi$$
$$62$$ −6.29004 −0.798836
$$63$$ 8.64104 1.08867
$$64$$ −9.56732 −1.19591
$$65$$ 0 0
$$66$$ −5.40067 −0.664777
$$67$$ 4.76228 0.581805 0.290902 0.956753i $$-0.406044\pi$$
0.290902 + 0.956753i $$0.406044\pi$$
$$68$$ 22.3430 2.70949
$$69$$ −1.40067 −0.168621
$$70$$ 0 0
$$71$$ 13.7115 1.62726 0.813631 0.581382i $$-0.197488\pi$$
0.813631 + 0.581382i $$0.197488\pi$$
$$72$$ 12.1768 1.43505
$$73$$ −2.72714 −0.319188 −0.159594 0.987183i $$-0.551018\pi$$
−0.159594 + 0.987183i $$0.551018\pi$$
$$74$$ −10.4119 −1.21036
$$75$$ 0 0
$$76$$ −3.85577 −0.442287
$$77$$ −13.2424 −1.50911
$$78$$ −2.69142 −0.304743
$$79$$ −1.40067 −0.157588 −0.0787939 0.996891i $$-0.525107\pi$$
−0.0787939 + 0.996891i $$0.525107\pi$$
$$80$$ 0 0
$$81$$ 6.48711 0.720790
$$82$$ 1.45030 0.160158
$$83$$ −7.07154 −0.776203 −0.388101 0.921617i $$-0.626869\pi$$
−0.388101 + 0.921617i $$0.626869\pi$$
$$84$$ −6.59933 −0.720046
$$85$$ 0 0
$$86$$ −7.71155 −0.831557
$$87$$ 3.22248 0.345486
$$88$$ −18.6609 −1.98926
$$89$$ 16.5353 1.75274 0.876370 0.481639i $$-0.159958\pi$$
0.876370 + 0.481639i $$0.159958\pi$$
$$90$$ 0 0
$$91$$ −6.59933 −0.691798
$$92$$ −10.0556 −1.04837
$$93$$ 1.39605 0.144763
$$94$$ −28.3333 −2.92236
$$95$$ 0 0
$$96$$ 0.722754 0.0737658
$$97$$ 2.07086 0.210264 0.105132 0.994458i $$-0.466474\pi$$
0.105132 + 0.994458i $$0.466474\pi$$
$$98$$ −7.63575 −0.771327
$$99$$ −11.2677 −1.13244
$$100$$ 0 0
$$101$$ −1.71155 −0.170305 −0.0851525 0.996368i $$-0.527138\pi$$
−0.0851525 + 0.996368i $$0.527138\pi$$
$$102$$ −7.53116 −0.745696
$$103$$ 5.75296 0.566856 0.283428 0.958994i $$-0.408528\pi$$
0.283428 + 0.958994i $$0.408528\pi$$
$$104$$ −9.29966 −0.911907
$$105$$ 0 0
$$106$$ −28.4343 −2.76178
$$107$$ −15.4324 −1.49191 −0.745955 0.665996i $$-0.768007\pi$$
−0.745955 + 0.665996i $$0.768007\pi$$
$$108$$ −11.8278 −1.13813
$$109$$ 11.7115 1.12176 0.560881 0.827896i $$-0.310462\pi$$
0.560881 + 0.827896i $$0.310462\pi$$
$$110$$ 0 0
$$111$$ 2.31087 0.219338
$$112$$ −10.0556 −0.950167
$$113$$ −10.5927 −0.996477 −0.498239 0.867040i $$-0.666020\pi$$
−0.498239 + 0.867040i $$0.666020\pi$$
$$114$$ 1.29966 0.121725
$$115$$ 0 0
$$116$$ 23.1346 2.14800
$$117$$ −5.61523 −0.519128
$$118$$ 4.14172 0.381276
$$119$$ −18.4663 −1.69280
$$120$$ 0 0
$$121$$ 6.26765 0.569787
$$122$$ 21.1854 1.91804
$$123$$ −0.321887 −0.0290236
$$124$$ 10.0224 0.900040
$$125$$ 0 0
$$126$$ −20.9102 −1.86283
$$127$$ −6.07484 −0.539055 −0.269528 0.962993i $$-0.586868\pi$$
−0.269528 + 0.962993i $$0.586868\pi$$
$$128$$ 20.4602 1.80845
$$129$$ 1.71155 0.150693
$$130$$ 0 0
$$131$$ −13.5785 −1.18636 −0.593181 0.805069i $$-0.702128\pi$$
−0.593181 + 0.805069i $$0.702128\pi$$
$$132$$ 8.60532 0.748997
$$133$$ 3.18676 0.276327
$$134$$ −11.5241 −0.995530
$$135$$ 0 0
$$136$$ −26.0224 −2.23140
$$137$$ −7.94302 −0.678618 −0.339309 0.940675i $$-0.610193\pi$$
−0.339309 + 0.940675i $$0.610193\pi$$
$$138$$ 3.38944 0.288529
$$139$$ 3.26765 0.277159 0.138579 0.990351i $$-0.455746\pi$$
0.138579 + 0.990351i $$0.455746\pi$$
$$140$$ 0 0
$$141$$ 6.28845 0.529583
$$142$$ −33.1802 −2.78442
$$143$$ 8.60532 0.719613
$$144$$ −8.55611 −0.713009
$$145$$ 0 0
$$146$$ 6.59933 0.546164
$$147$$ 1.69472 0.139778
$$148$$ 16.5901 1.36370
$$149$$ 8.44389 0.691751 0.345875 0.938280i $$-0.387582\pi$$
0.345875 + 0.938280i $$0.387582\pi$$
$$150$$ 0 0
$$151$$ 0.887783 0.0722468 0.0361234 0.999347i $$-0.488499\pi$$
0.0361234 + 0.999347i $$0.488499\pi$$
$$152$$ 4.49073 0.364246
$$153$$ −15.7126 −1.27029
$$154$$ 32.0448 2.58225
$$155$$ 0 0
$$156$$ 4.28845 0.343351
$$157$$ 4.14172 0.330545 0.165273 0.986248i $$-0.447150\pi$$
0.165273 + 0.986248i $$0.447150\pi$$
$$158$$ 3.38944 0.269650
$$159$$ 6.31087 0.500485
$$160$$ 0 0
$$161$$ 8.31087 0.654989
$$162$$ −15.6980 −1.23335
$$163$$ −24.7126 −1.93564 −0.967819 0.251647i $$-0.919028\pi$$
−0.967819 + 0.251647i $$0.919028\pi$$
$$164$$ −2.31087 −0.180449
$$165$$ 0 0
$$166$$ 17.1122 1.32817
$$167$$ −3.60464 −0.278935 −0.139468 0.990227i $$-0.544539\pi$$
−0.139468 + 0.990227i $$0.544539\pi$$
$$168$$ 7.68608 0.592994
$$169$$ −8.71155 −0.670119
$$170$$ 0 0
$$171$$ 2.71155 0.207357
$$172$$ 12.2874 0.936907
$$173$$ −22.4205 −1.70460 −0.852300 0.523054i $$-0.824793\pi$$
−0.852300 + 0.523054i $$0.824793\pi$$
$$174$$ −7.79798 −0.591164
$$175$$ 0 0
$$176$$ 13.1122 0.988371
$$177$$ −0.919237 −0.0690941
$$178$$ −40.0133 −2.99912
$$179$$ −5.13464 −0.383781 −0.191890 0.981416i $$-0.561462\pi$$
−0.191890 + 0.981416i $$0.561462\pi$$
$$180$$ 0 0
$$181$$ 20.8462 1.54948 0.774742 0.632277i $$-0.217880\pi$$
0.774742 + 0.632277i $$0.217880\pi$$
$$182$$ 15.9695 1.18374
$$183$$ −4.70201 −0.347583
$$184$$ 11.7115 0.863387
$$185$$ 0 0
$$186$$ −3.37825 −0.247706
$$187$$ 24.0795 1.76087
$$188$$ 45.1457 3.29259
$$189$$ 9.77557 0.711068
$$190$$ 0 0
$$191$$ −5.26765 −0.381154 −0.190577 0.981672i $$-0.561036\pi$$
−0.190577 + 0.981672i $$0.561036\pi$$
$$192$$ −5.13842 −0.370833
$$193$$ 2.07086 0.149064 0.0745318 0.997219i $$-0.476254\pi$$
0.0745318 + 0.997219i $$0.476254\pi$$
$$194$$ −5.01121 −0.359784
$$195$$ 0 0
$$196$$ 12.1666 0.869046
$$197$$ 10.4318 0.743232 0.371616 0.928387i $$-0.378804\pi$$
0.371616 + 0.928387i $$0.378804\pi$$
$$198$$ 27.2663 1.93773
$$199$$ −2.73235 −0.193691 −0.0968455 0.995299i $$-0.530875\pi$$
−0.0968455 + 0.995299i $$0.530875\pi$$
$$200$$ 0 0
$$201$$ 2.55773 0.180408
$$202$$ 4.14172 0.291410
$$203$$ −19.1206 −1.34200
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −13.9214 −0.969951
$$207$$ 7.07154 0.491506
$$208$$ 6.53446 0.453083
$$209$$ −4.15544 −0.287438
$$210$$ 0 0
$$211$$ −15.7340 −1.08317 −0.541585 0.840646i $$-0.682176\pi$$
−0.541585 + 0.840646i $$0.682176\pi$$
$$212$$ 45.3066 3.11167
$$213$$ 7.36420 0.504586
$$214$$ 37.3445 2.55282
$$215$$ 0 0
$$216$$ 13.7756 0.937309
$$217$$ −8.28343 −0.562316
$$218$$ −28.3404 −1.91946
$$219$$ −1.46469 −0.0989748
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ −5.59201 −0.375311
$$223$$ 18.8219 1.26041 0.630203 0.776430i $$-0.282972\pi$$
0.630203 + 0.776430i $$0.282972\pi$$
$$224$$ −4.28845 −0.286534
$$225$$ 0 0
$$226$$ 25.6330 1.70508
$$227$$ −14.4418 −0.958533 −0.479267 0.877669i $$-0.659097\pi$$
−0.479267 + 0.877669i $$0.659097\pi$$
$$228$$ −2.07086 −0.137146
$$229$$ −4.17785 −0.276080 −0.138040 0.990427i $$-0.544080\pi$$
−0.138040 + 0.990427i $$0.544080\pi$$
$$230$$ 0 0
$$231$$ −7.11222 −0.467950
$$232$$ −26.9444 −1.76898
$$233$$ 12.0847 0.791697 0.395849 0.918316i $$-0.370450\pi$$
0.395849 + 0.918316i $$0.370450\pi$$
$$234$$ 13.5881 0.888283
$$235$$ 0 0
$$236$$ −6.59933 −0.429580
$$237$$ −0.752273 −0.0488654
$$238$$ 44.6861 2.89657
$$239$$ −11.3541 −0.734435 −0.367218 0.930135i $$-0.619690\pi$$
−0.367218 + 0.930135i $$0.619690\pi$$
$$240$$ 0 0
$$241$$ −3.40067 −0.219057 −0.109528 0.993984i $$-0.534934\pi$$
−0.109528 + 0.993984i $$0.534934\pi$$
$$242$$ −15.1669 −0.974966
$$243$$ 12.6868 0.813857
$$244$$ −33.7564 −2.16103
$$245$$ 0 0
$$246$$ 0.778925 0.0496625
$$247$$ −2.07086 −0.131766
$$248$$ −11.6729 −0.741229
$$249$$ −3.79798 −0.240687
$$250$$ 0 0
$$251$$ 3.04322 0.192086 0.0960432 0.995377i $$-0.469381\pi$$
0.0960432 + 0.995377i $$0.469381\pi$$
$$252$$ 33.3179 2.09883
$$253$$ −10.8371 −0.681324
$$254$$ 14.7003 0.922381
$$255$$ 0 0
$$256$$ −30.3765 −1.89853
$$257$$ 17.2881 1.07840 0.539201 0.842177i $$-0.318726\pi$$
0.539201 + 0.842177i $$0.318726\pi$$
$$258$$ −4.14172 −0.257852
$$259$$ −13.7115 −0.851994
$$260$$ 0 0
$$261$$ −16.2693 −1.00704
$$262$$ 32.8583 2.02999
$$263$$ −1.19336 −0.0735859 −0.0367930 0.999323i $$-0.511714\pi$$
−0.0367930 + 0.999323i $$0.511714\pi$$
$$264$$ −10.0224 −0.616837
$$265$$ 0 0
$$266$$ −7.71155 −0.472825
$$267$$ 8.88078 0.543495
$$268$$ 18.3623 1.12165
$$269$$ 22.1089 1.34800 0.674000 0.738731i $$-0.264575\pi$$
0.674000 + 0.738731i $$0.264575\pi$$
$$270$$ 0 0
$$271$$ −4.08644 −0.248234 −0.124117 0.992268i $$-0.539610\pi$$
−0.124117 + 0.992268i $$0.539610\pi$$
$$272$$ 18.2848 1.10868
$$273$$ −3.54437 −0.214515
$$274$$ 19.2211 1.16119
$$275$$ 0 0
$$276$$ −5.40067 −0.325082
$$277$$ −10.0199 −0.602037 −0.301019 0.953618i $$-0.597327\pi$$
−0.301019 + 0.953618i $$0.597327\pi$$
$$278$$ −7.90730 −0.474248
$$279$$ −7.04820 −0.421964
$$280$$ 0 0
$$281$$ 0.599328 0.0357529 0.0178765 0.999840i $$-0.494309\pi$$
0.0178765 + 0.999840i $$0.494309\pi$$
$$282$$ −15.2172 −0.906174
$$283$$ 5.41856 0.322100 0.161050 0.986946i $$-0.448512\pi$$
0.161050 + 0.986946i $$0.448512\pi$$
$$284$$ 52.8686 3.13717
$$285$$ 0 0
$$286$$ −20.8238 −1.23133
$$287$$ 1.90991 0.112739
$$288$$ −3.64895 −0.215017
$$289$$ 16.5785 0.975207
$$290$$ 0 0
$$291$$ 1.11222 0.0651993
$$292$$ −10.5152 −0.615358
$$293$$ 3.46691 0.202539 0.101269 0.994859i $$-0.467710\pi$$
0.101269 + 0.994859i $$0.467710\pi$$
$$294$$ −4.10101 −0.239176
$$295$$ 0 0
$$296$$ −19.3221 −1.12307
$$297$$ −12.7470 −0.739658
$$298$$ −20.4331 −1.18366
$$299$$ −5.40067 −0.312329
$$300$$ 0 0
$$301$$ −10.1554 −0.585350
$$302$$ −2.14832 −0.123622
$$303$$ −0.919237 −0.0528088
$$304$$ −3.15544 −0.180977
$$305$$ 0 0
$$306$$ 38.0224 2.17360
$$307$$ −16.5901 −0.946846 −0.473423 0.880835i $$-0.656982\pi$$
−0.473423 + 0.880835i $$0.656982\pi$$
$$308$$ −51.0596 −2.90939
$$309$$ 3.08980 0.175772
$$310$$ 0 0
$$311$$ −4.15544 −0.235633 −0.117817 0.993035i $$-0.537589\pi$$
−0.117817 + 0.993035i $$0.537589\pi$$
$$312$$ −4.99466 −0.282767
$$313$$ −0.919237 −0.0519583 −0.0259792 0.999662i $$-0.508270\pi$$
−0.0259792 + 0.999662i $$0.508270\pi$$
$$314$$ −10.0224 −0.565598
$$315$$ 0 0
$$316$$ −5.40067 −0.303812
$$317$$ −26.7292 −1.50126 −0.750630 0.660723i $$-0.770250\pi$$
−0.750630 + 0.660723i $$0.770250\pi$$
$$318$$ −15.2715 −0.856383
$$319$$ 24.9326 1.39596
$$320$$ 0 0
$$321$$ −8.28845 −0.462616
$$322$$ −20.1112 −1.12076
$$323$$ −5.79470 −0.322426
$$324$$ 25.0128 1.38960
$$325$$ 0 0
$$326$$ 59.8012 3.31208
$$327$$ 6.29004 0.347840
$$328$$ 2.69142 0.148609
$$329$$ −37.3125 −2.05710
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ −27.2663 −1.49643
$$333$$ −11.6669 −0.639340
$$334$$ 8.72275 0.477288
$$335$$ 0 0
$$336$$ −5.40067 −0.294631
$$337$$ 22.5040 1.22587 0.612935 0.790133i $$-0.289989\pi$$
0.612935 + 0.790133i $$0.289989\pi$$
$$338$$ 21.0808 1.14664
$$339$$ −5.68913 −0.308991
$$340$$ 0 0
$$341$$ 10.8013 0.584926
$$342$$ −6.56159 −0.354810
$$343$$ 12.2517 0.661530
$$344$$ −14.3109 −0.771591
$$345$$ 0 0
$$346$$ 54.2547 2.91675
$$347$$ −14.4543 −0.775946 −0.387973 0.921671i $$-0.626825\pi$$
−0.387973 + 0.921671i $$0.626825\pi$$
$$348$$ 12.4252 0.666058
$$349$$ −13.3541 −0.714828 −0.357414 0.933946i $$-0.616342\pi$$
−0.357414 + 0.933946i $$0.616342\pi$$
$$350$$ 0 0
$$351$$ −6.35248 −0.339070
$$352$$ 5.59201 0.298055
$$353$$ 17.6410 0.938937 0.469469 0.882949i $$-0.344446\pi$$
0.469469 + 0.882949i $$0.344446\pi$$
$$354$$ 2.22443 0.118227
$$355$$ 0 0
$$356$$ 63.7564 3.37908
$$357$$ −9.91789 −0.524910
$$358$$ 12.4252 0.656690
$$359$$ 12.4663 0.657947 0.328973 0.944339i $$-0.393297\pi$$
0.328973 + 0.944339i $$0.393297\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −50.4451 −2.65133
$$363$$ 3.36623 0.176681
$$364$$ −25.4455 −1.33371
$$365$$ 0 0
$$366$$ 11.3783 0.594751
$$367$$ −16.4291 −0.857594 −0.428797 0.903401i $$-0.641062\pi$$
−0.428797 + 0.903401i $$0.641062\pi$$
$$368$$ −8.22918 −0.428976
$$369$$ 1.62511 0.0845996
$$370$$ 0 0
$$371$$ −37.4455 −1.94407
$$372$$ 5.38284 0.279087
$$373$$ −5.29334 −0.274079 −0.137039 0.990566i $$-0.543759\pi$$
−0.137039 + 0.990566i $$0.543759\pi$$
$$374$$ −58.2693 −3.01303
$$375$$ 0 0
$$376$$ −52.5801 −2.71161
$$377$$ 12.4252 0.639928
$$378$$ −23.6556 −1.21671
$$379$$ 14.5353 0.746629 0.373314 0.927705i $$-0.378221\pi$$
0.373314 + 0.927705i $$0.378221\pi$$
$$380$$ 0 0
$$381$$ −3.26268 −0.167152
$$382$$ 12.7470 0.652195
$$383$$ 0.453598 0.0231778 0.0115889 0.999933i $$-0.496311\pi$$
0.0115889 + 0.999933i $$0.496311\pi$$
$$384$$ 10.9888 0.560769
$$385$$ 0 0
$$386$$ −5.01121 −0.255064
$$387$$ −8.64104 −0.439249
$$388$$ 7.98476 0.405365
$$389$$ 16.1554 0.819113 0.409557 0.912285i $$-0.365683\pi$$
0.409557 + 0.912285i $$0.365683\pi$$
$$390$$ 0 0
$$391$$ −15.1122 −0.764258
$$392$$ −14.1702 −0.715704
$$393$$ −7.29276 −0.367871
$$394$$ −25.2435 −1.27175
$$395$$ 0 0
$$396$$ −43.4455 −2.18322
$$397$$ 32.7563 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$398$$ 6.61192 0.331426
$$399$$ 1.71155 0.0856844
$$400$$ 0 0
$$401$$ 12.0864 0.603568 0.301784 0.953376i $$-0.402418\pi$$
0.301784 + 0.953376i $$0.402418\pi$$
$$402$$ −6.18936 −0.308697
$$403$$ 5.38284 0.268138
$$404$$ −6.59933 −0.328329
$$405$$ 0 0
$$406$$ 46.2693 2.29631
$$407$$ 17.8794 0.886251
$$408$$ −13.9761 −0.691921
$$409$$ −19.1346 −0.946147 −0.473073 0.881023i $$-0.656855\pi$$
−0.473073 + 0.881023i $$0.656855\pi$$
$$410$$ 0 0
$$411$$ −4.26604 −0.210428
$$412$$ 22.1821 1.09283
$$413$$ 5.45428 0.268388
$$414$$ −17.1122 −0.841020
$$415$$ 0 0
$$416$$ 2.78678 0.136633
$$417$$ 1.75499 0.0859423
$$418$$ 10.0556 0.491836
$$419$$ 8.04484 0.393016 0.196508 0.980502i $$-0.437040\pi$$
0.196508 + 0.980502i $$0.437040\pi$$
$$420$$ 0 0
$$421$$ −29.3591 −1.43087 −0.715437 0.698678i $$-0.753772\pi$$
−0.715437 + 0.698678i $$0.753772\pi$$
$$422$$ 38.0742 1.85342
$$423$$ −31.7484 −1.54366
$$424$$ −52.7676 −2.56262
$$425$$ 0 0
$$426$$ −17.8204 −0.863401
$$427$$ 27.8993 1.35014
$$428$$ −59.5040 −2.87623
$$429$$ 4.62175 0.223140
$$430$$ 0 0
$$431$$ −32.0448 −1.54355 −0.771773 0.635898i $$-0.780630\pi$$
−0.771773 + 0.635898i $$0.780630\pi$$
$$432$$ −9.67948 −0.465704
$$433$$ 0.482831 0.0232034 0.0116017 0.999933i $$-0.496307\pi$$
0.0116017 + 0.999933i $$0.496307\pi$$
$$434$$ 20.0448 0.962183
$$435$$ 0 0
$$436$$ 45.1571 2.16263
$$437$$ 2.60794 0.124755
$$438$$ 3.54437 0.169356
$$439$$ −27.3591 −1.30578 −0.652889 0.757454i $$-0.726443\pi$$
−0.652889 + 0.757454i $$0.726443\pi$$
$$440$$ 0 0
$$441$$ −8.55611 −0.407434
$$442$$ −29.0384 −1.38122
$$443$$ 23.3815 1.11089 0.555444 0.831554i $$-0.312548\pi$$
0.555444 + 0.831554i $$0.312548\pi$$
$$444$$ 8.91020 0.422859
$$445$$ 0 0
$$446$$ −45.5465 −2.15669
$$447$$ 4.53505 0.214500
$$448$$ 30.4887 1.44046
$$449$$ 23.1346 1.09179 0.545895 0.837853i $$-0.316190\pi$$
0.545895 + 0.837853i $$0.316190\pi$$
$$450$$ 0 0
$$451$$ −2.49047 −0.117272
$$452$$ −40.8430 −1.92109
$$453$$ 0.476811 0.0224025
$$454$$ 34.9472 1.64015
$$455$$ 0 0
$$456$$ 2.41188 0.112947
$$457$$ 21.2503 0.994049 0.497025 0.867736i $$-0.334426\pi$$
0.497025 + 0.867736i $$0.334426\pi$$
$$458$$ 10.1099 0.472403
$$459$$ −17.7756 −0.829692
$$460$$ 0 0
$$461$$ 31.5785 1.47076 0.735379 0.677656i $$-0.237004\pi$$
0.735379 + 0.677656i $$0.237004\pi$$
$$462$$ 17.2106 0.800712
$$463$$ −15.6119 −0.725547 −0.362774 0.931877i $$-0.618170\pi$$
−0.362774 + 0.931877i $$0.618170\pi$$
$$464$$ 18.9326 0.878925
$$465$$ 0 0
$$466$$ −29.2435 −1.35468
$$467$$ −29.8264 −1.38020 −0.690101 0.723713i $$-0.742434\pi$$
−0.690101 + 0.723713i $$0.742434\pi$$
$$468$$ −21.6510 −1.00082
$$469$$ −15.1762 −0.700774
$$470$$ 0 0
$$471$$ 2.22443 0.102496
$$472$$ 7.68608 0.353781
$$473$$ 13.2424 0.608885
$$474$$ 1.82040 0.0836139
$$475$$ 0 0
$$476$$ −71.2019 −3.26353
$$477$$ −31.8616 −1.45884
$$478$$ 27.4754 1.25670
$$479$$ 4.53531 0.207223 0.103612 0.994618i $$-0.466960\pi$$
0.103612 + 0.994618i $$0.466960\pi$$
$$480$$ 0 0
$$481$$ 8.91020 0.406270
$$482$$ 8.22918 0.374829
$$483$$ 4.46360 0.203101
$$484$$ 24.1666 1.09848
$$485$$ 0 0
$$486$$ −30.7003 −1.39260
$$487$$ 39.2550 1.77881 0.889407 0.457116i $$-0.151118\pi$$
0.889407 + 0.457116i $$0.151118\pi$$
$$488$$ 39.3153 1.77972
$$489$$ −13.2726 −0.600209
$$490$$ 0 0
$$491$$ 38.8910 1.75513 0.877564 0.479460i $$-0.159168\pi$$
0.877564 + 0.479460i $$0.159168\pi$$
$$492$$ −1.24112 −0.0559542
$$493$$ 34.7682 1.56588
$$494$$ 5.01121 0.225465
$$495$$ 0 0
$$496$$ 8.20202 0.368281
$$497$$ −43.6954 −1.96001
$$498$$ 9.19063 0.411842
$$499$$ 4.73235 0.211849 0.105924 0.994374i $$-0.466220\pi$$
0.105924 + 0.994374i $$0.466220\pi$$
$$500$$ 0 0
$$501$$ −1.93598 −0.0864931
$$502$$ −7.36420 −0.328680
$$503$$ 1.85567 0.0827400 0.0413700 0.999144i $$-0.486828\pi$$
0.0413700 + 0.999144i $$0.486828\pi$$
$$504$$ −38.8046 −1.72849
$$505$$ 0 0
$$506$$ 26.2244 1.16582
$$507$$ −4.67880 −0.207793
$$508$$ −23.4232 −1.03924
$$509$$ 22.8878 1.01448 0.507242 0.861804i $$-0.330665\pi$$
0.507242 + 0.861804i $$0.330665\pi$$
$$510$$ 0 0
$$511$$ 8.69074 0.384456
$$512$$ 32.5867 1.44014
$$513$$ 3.06756 0.135436
$$514$$ −41.8350 −1.84526
$$515$$ 0 0
$$516$$ 6.59933 0.290519
$$517$$ 48.6543 2.13982
$$518$$ 33.1802 1.45785
$$519$$ −12.0416 −0.528568
$$520$$ 0 0
$$521$$ −29.7340 −1.30267 −0.651334 0.758791i $$-0.725790\pi$$
−0.651334 + 0.758791i $$0.725790\pi$$
$$522$$ 39.3695 1.72316
$$523$$ 30.6497 1.34022 0.670109 0.742263i $$-0.266248\pi$$
0.670109 + 0.742263i $$0.266248\pi$$
$$524$$ −52.3557 −2.28717
$$525$$ 0 0
$$526$$ 2.88778 0.125913
$$527$$ 15.0623 0.656125
$$528$$ 7.04231 0.306477
$$529$$ −16.1987 −0.704289
$$530$$ 0 0
$$531$$ 4.64093 0.201399
$$532$$ 12.2874 0.532727
$$533$$ −1.24112 −0.0537590
$$534$$ −21.4903 −0.929978
$$535$$ 0 0
$$536$$ −21.3861 −0.923739
$$537$$ −2.75771 −0.119004
$$538$$ −53.5006 −2.30657
$$539$$ 13.1122 0.564783
$$540$$ 0 0
$$541$$ 2.21946 0.0954220 0.0477110 0.998861i $$-0.484807\pi$$
0.0477110 + 0.998861i $$0.484807\pi$$
$$542$$ 9.88865 0.424754
$$543$$ 11.1961 0.480469
$$544$$ 7.79798 0.334336
$$545$$ 0 0
$$546$$ 8.57691 0.367058
$$547$$ −14.4297 −0.616970 −0.308485 0.951229i $$-0.599822\pi$$
−0.308485 + 0.951229i $$0.599822\pi$$
$$548$$ −30.6265 −1.30830
$$549$$ 23.7389 1.01315
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 6.29004 0.267722
$$553$$ 4.46360 0.189812
$$554$$ 24.2469 1.03015
$$555$$ 0 0
$$556$$ 12.5993 0.534331
$$557$$ 19.4610 0.824588 0.412294 0.911051i $$-0.364728\pi$$
0.412294 + 0.911051i $$0.364728\pi$$
$$558$$ 17.0557 0.722026
$$559$$ 6.59933 0.279122
$$560$$ 0 0
$$561$$ 12.9326 0.546016
$$562$$ −1.45030 −0.0611771
$$563$$ −12.3649 −0.521118 −0.260559 0.965458i $$-0.583907\pi$$
−0.260559 + 0.965458i $$0.583907\pi$$
$$564$$ 24.2469 1.02098
$$565$$ 0 0
$$566$$ −13.1122 −0.551148
$$567$$ −20.6729 −0.868179
$$568$$ −61.5748 −2.58362
$$569$$ 15.0898 0.632597 0.316299 0.948660i $$-0.397560\pi$$
0.316299 + 0.948660i $$0.397560\pi$$
$$570$$ 0 0
$$571$$ 17.1571 0.718000 0.359000 0.933337i $$-0.383118\pi$$
0.359000 + 0.933337i $$0.383118\pi$$
$$572$$ 33.1802 1.38733
$$573$$ −2.82915 −0.118190
$$574$$ −4.62175 −0.192908
$$575$$ 0 0
$$576$$ 25.9422 1.08093
$$577$$ −22.5165 −0.937374 −0.468687 0.883364i $$-0.655273\pi$$
−0.468687 + 0.883364i $$0.655273\pi$$
$$578$$ −40.1179 −1.66868
$$579$$ 1.11222 0.0462222
$$580$$ 0 0
$$581$$ 22.5353 0.934922
$$582$$ −2.69142 −0.111563
$$583$$ 48.8278 2.02224
$$584$$ 12.2469 0.506778
$$585$$ 0 0
$$586$$ −8.38946 −0.346566
$$587$$ 7.49544 0.309370 0.154685 0.987964i $$-0.450564\pi$$
0.154685 + 0.987964i $$0.450564\pi$$
$$588$$ 6.53446 0.269477
$$589$$ −2.59933 −0.107103
$$590$$ 0 0
$$591$$ 5.60269 0.230464
$$592$$ 13.5768 0.558002
$$593$$ 27.8094 1.14199 0.570997 0.820952i $$-0.306557\pi$$
0.570997 + 0.820952i $$0.306557\pi$$
$$594$$ 30.8462 1.26563
$$595$$ 0 0
$$596$$ 32.5577 1.33362
$$597$$ −1.46749 −0.0600603
$$598$$ 13.0689 0.534428
$$599$$ −45.4903 −1.85869 −0.929343 0.369219i $$-0.879625\pi$$
−0.929343 + 0.369219i $$0.879625\pi$$
$$600$$ 0 0
$$601$$ −16.5993 −0.677101 −0.338550 0.940948i $$-0.609937\pi$$
−0.338550 + 0.940948i $$0.609937\pi$$
$$602$$ 24.5748 1.00160
$$603$$ −12.9131 −0.525863
$$604$$ 3.42309 0.139284
$$605$$ 0 0
$$606$$ 2.22443 0.0903614
$$607$$ −5.08417 −0.206360 −0.103180 0.994663i $$-0.532902\pi$$
−0.103180 + 0.994663i $$0.532902\pi$$
$$608$$ −1.34571 −0.0545758
$$609$$ −10.2693 −0.416132
$$610$$ 0 0
$$611$$ 24.2469 0.980923
$$612$$ −60.5842 −2.44897
$$613$$ 4.63706 0.187289 0.0936445 0.995606i $$-0.470148\pi$$
0.0936445 + 0.995606i $$0.470148\pi$$
$$614$$ 40.1458 1.62015
$$615$$ 0 0
$$616$$ 59.4679 2.39603
$$617$$ 40.2874 1.62191 0.810955 0.585108i $$-0.198948\pi$$
0.810955 + 0.585108i $$0.198948\pi$$
$$618$$ −7.47691 −0.300766
$$619$$ −43.3815 −1.74365 −0.871825 0.489818i $$-0.837063\pi$$
−0.871825 + 0.489818i $$0.837063\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 10.0556 0.403194
$$623$$ −52.6940 −2.11114
$$624$$ 3.50953 0.140494
$$625$$ 0 0
$$626$$ 2.22443 0.0889062
$$627$$ −2.23180 −0.0891296
$$628$$ 15.9695 0.637253
$$629$$ 24.9326 0.994129
$$630$$ 0 0
$$631$$ 7.53369 0.299911 0.149956 0.988693i $$-0.452087\pi$$
0.149956 + 0.988693i $$0.452087\pi$$
$$632$$ 6.29004 0.250204
$$633$$ −8.45040 −0.335873
$$634$$ 64.6812 2.56882
$$635$$ 0 0
$$636$$ 24.3333 0.964878
$$637$$ 6.53446 0.258905
$$638$$ −60.3337 −2.38863
$$639$$ −37.1795 −1.47080
$$640$$ 0 0
$$641$$ −23.3075 −0.920591 −0.460296 0.887766i $$-0.652257\pi$$
−0.460296 + 0.887766i $$0.652257\pi$$
$$642$$ 20.0570 0.791586
$$643$$ 1.87419 0.0739110 0.0369555 0.999317i $$-0.488234\pi$$
0.0369555 + 0.999317i $$0.488234\pi$$
$$644$$ 32.0448 1.26274
$$645$$ 0 0
$$646$$ 14.0224 0.551705
$$647$$ −47.0371 −1.84922 −0.924609 0.380917i $$-0.875608\pi$$
−0.924609 + 0.380917i $$0.875608\pi$$
$$648$$ −29.1319 −1.14441
$$649$$ −7.11222 −0.279179
$$650$$ 0 0
$$651$$ −4.44887 −0.174365
$$652$$ −95.2861 −3.73169
$$653$$ 24.1630 0.945571 0.472785 0.881178i $$-0.343249\pi$$
0.472785 + 0.881178i $$0.343249\pi$$
$$654$$ −15.2211 −0.595191
$$655$$ 0 0
$$656$$ −1.89114 −0.0738367
$$657$$ 7.39477 0.288497
$$658$$ 90.2914 3.51992
$$659$$ 10.2885 0.400781 0.200391 0.979716i $$-0.435779\pi$$
0.200391 + 0.979716i $$0.435779\pi$$
$$660$$ 0 0
$$661$$ 5.93598 0.230883 0.115441 0.993314i $$-0.463172\pi$$
0.115441 + 0.993314i $$0.463172\pi$$
$$662$$ −19.3590 −0.752407
$$663$$ 6.44496 0.250302
$$664$$ 31.7564 1.23239
$$665$$ 0 0
$$666$$ 28.2323 1.09398
$$667$$ −15.6476 −0.605879
$$668$$ −13.8987 −0.537755
$$669$$ 10.1089 0.390831
$$670$$ 0 0
$$671$$ −36.3799 −1.40443
$$672$$ −2.30324 −0.0888496
$$673$$ −40.7053 −1.56907 −0.784537 0.620082i $$-0.787099\pi$$
−0.784537 + 0.620082i $$0.787099\pi$$
$$674$$ −54.4567 −2.09759
$$675$$ 0 0
$$676$$ −33.5897 −1.29191
$$677$$ 18.8761 0.725469 0.362734 0.931893i $$-0.381843\pi$$
0.362734 + 0.931893i $$0.381843\pi$$
$$678$$ 13.7669 0.528716
$$679$$ −6.59933 −0.253259
$$680$$ 0 0
$$681$$ −7.75638 −0.297225
$$682$$ −26.1379 −1.00087
$$683$$ −19.6576 −0.752179 −0.376089 0.926583i $$-0.622731\pi$$
−0.376089 + 0.926583i $$0.622731\pi$$
$$684$$ 10.4551 0.399761
$$685$$ 0 0
$$686$$ −29.6475 −1.13195
$$687$$ −2.24384 −0.0856079
$$688$$ 10.0556 0.383367
$$689$$ 24.3333 0.927025
$$690$$ 0 0
$$691$$ −48.2451 −1.83533 −0.917665 0.397354i $$-0.869928\pi$$
−0.917665 + 0.397354i $$0.869928\pi$$
$$692$$ −86.4483 −3.28627
$$693$$ 35.9073 1.36401
$$694$$ 34.9775 1.32773
$$695$$ 0 0
$$696$$ −14.4713 −0.548533
$$697$$ −3.47293 −0.131546
$$698$$ 32.3152 1.22315
$$699$$ 6.49047 0.245492
$$700$$ 0 0
$$701$$ 0.512889 0.0193715 0.00968577 0.999953i $$-0.496917\pi$$
0.00968577 + 0.999953i $$0.496917\pi$$
$$702$$ 15.3722 0.580185
$$703$$ −4.30266 −0.162278
$$704$$ −39.7564 −1.49838
$$705$$ 0 0
$$706$$ −42.6890 −1.60662
$$707$$ 5.45428 0.205129
$$708$$ −3.54437 −0.133205
$$709$$ −8.84618 −0.332225 −0.166113 0.986107i $$-0.553122\pi$$
−0.166113 + 0.986107i $$0.553122\pi$$
$$710$$ 0 0
$$711$$ 3.79798 0.142436
$$712$$ −74.2556 −2.78285
$$713$$ −6.77889 −0.253871
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ −19.7980 −0.739885
$$717$$ −6.09806 −0.227736
$$718$$ −30.1669 −1.12582
$$719$$ 7.84456 0.292553 0.146276 0.989244i $$-0.453271\pi$$
0.146276 + 0.989244i $$0.453271\pi$$
$$720$$ 0 0
$$721$$ −18.3333 −0.682767
$$722$$ −2.41987 −0.0900582
$$723$$ −1.82643 −0.0679258
$$724$$ 80.3781 2.98723
$$725$$ 0 0
$$726$$ −8.14584 −0.302321
$$727$$ −2.91130 −0.107974 −0.0539870 0.998542i $$-0.517193\pi$$
−0.0539870 + 0.998542i $$0.517193\pi$$
$$728$$ 29.6358 1.09838
$$729$$ −12.6475 −0.468427
$$730$$ 0 0
$$731$$ 18.4663 0.683001
$$732$$ −18.1299 −0.670100
$$733$$ −33.7775 −1.24760 −0.623800 0.781584i $$-0.714412\pi$$
−0.623800 + 0.781584i $$0.714412\pi$$
$$734$$ 39.7564 1.46743
$$735$$ 0 0
$$736$$ −3.50953 −0.129363
$$737$$ 19.7893 0.728950
$$738$$ −3.93254 −0.144759
$$739$$ −35.8030 −1.31703 −0.658517 0.752566i $$-0.728816\pi$$
−0.658517 + 0.752566i $$0.728816\pi$$
$$740$$ 0 0
$$741$$ −1.11222 −0.0408583
$$742$$ 90.6133 3.32652
$$743$$ −5.66948 −0.207993 −0.103996 0.994578i $$-0.533163\pi$$
−0.103996 + 0.994578i $$0.533163\pi$$
$$744$$ −6.26927 −0.229843
$$745$$ 0 0
$$746$$ 12.8092 0.468978
$$747$$ 19.1748 0.701569
$$748$$ 92.8451 3.39475
$$749$$ 49.1795 1.79698
$$750$$ 0 0
$$751$$ −27.4679 −1.00232 −0.501159 0.865355i $$-0.667093\pi$$
−0.501159 + 0.865355i $$0.667093\pi$$
$$752$$ 36.9457 1.34727
$$753$$ 1.63445 0.0595628
$$754$$ −30.0673 −1.09498
$$755$$ 0 0
$$756$$ 37.6924 1.37086
$$757$$ −41.6370 −1.51332 −0.756662 0.653806i $$-0.773171\pi$$
−0.756662 + 0.653806i $$0.773171\pi$$
$$758$$ −35.1736 −1.27756
$$759$$ −5.82040 −0.211267
$$760$$ 0 0
$$761$$ −9.46967 −0.343275 −0.171638 0.985160i $$-0.554906\pi$$
−0.171638 + 0.985160i $$0.554906\pi$$
$$762$$ 7.89526 0.286015
$$763$$ −37.3219 −1.35114
$$764$$ −20.3109 −0.734822
$$765$$ 0 0
$$766$$ −1.09765 −0.0396597
$$767$$ −3.54437 −0.127980
$$768$$ −16.3146 −0.588703
$$769$$ 1.90858 0.0688253 0.0344127 0.999408i $$-0.489044\pi$$
0.0344127 + 0.999408i $$0.489044\pi$$
$$770$$ 0 0
$$771$$ 9.28510 0.334395
$$772$$ 7.98476 0.287378
$$773$$ −28.4007 −1.02150 −0.510751 0.859729i $$-0.670633\pi$$
−0.510751 + 0.859729i $$0.670633\pi$$
$$774$$ 20.9102 0.751602
$$775$$ 0 0
$$776$$ −9.29966 −0.333838
$$777$$ −7.36420 −0.264189
$$778$$ −39.0941 −1.40159
$$779$$ 0.599328 0.0214732
$$780$$ 0 0
$$781$$ 56.9775 2.03881
$$782$$ 36.5696 1.30773
$$783$$ −18.4053 −0.657753
$$784$$ 9.95678 0.355599
$$785$$ 0 0
$$786$$ 17.6475 0.629466
$$787$$ 15.6708 0.558605 0.279303 0.960203i $$-0.409897\pi$$
0.279303 + 0.960203i $$0.409897\pi$$
$$788$$ 40.2225 1.43287
$$789$$ −0.640931 −0.0228178
$$790$$ 0 0
$$791$$ 33.7564 1.20024
$$792$$ 50.6000 1.79799
$$793$$ −18.1299 −0.643811
$$794$$ −79.2659 −2.81304
$$795$$ 0 0
$$796$$ −10.5353 −0.373414
$$797$$ −36.9225 −1.30786 −0.653932 0.756554i $$-0.726882\pi$$
−0.653932 + 0.756554i $$0.726882\pi$$
$$798$$ −4.14172 −0.146615
$$799$$ 67.8478 2.40028
$$800$$ 0 0
$$801$$ −44.8362 −1.58421
$$802$$ −29.2476 −1.03277
$$803$$ −11.3325 −0.399914
$$804$$ 9.86201 0.347806
$$805$$ 0 0
$$806$$ −13.0258 −0.458813
$$807$$ 11.8742 0.417993
$$808$$ 7.68608 0.270396
$$809$$ 43.0465 1.51343 0.756716 0.653743i $$-0.226802\pi$$
0.756716 + 0.653743i $$0.226802\pi$$
$$810$$ 0 0
$$811$$ 32.2469 1.13234 0.566170 0.824288i $$-0.308425\pi$$
0.566170 + 0.824288i $$0.308425\pi$$
$$812$$ −73.7245 −2.58722
$$813$$ −2.19475 −0.0769731
$$814$$ −43.2659 −1.51647
$$815$$ 0 0
$$816$$ 9.82040 0.343783
$$817$$ −3.18676 −0.111491
$$818$$ 46.3033 1.61896
$$819$$ 17.8944 0.625280
$$820$$ 0 0
$$821$$ 5.18121 0.180826 0.0904128 0.995904i $$-0.471181\pi$$
0.0904128 + 0.995904i $$0.471181\pi$$
$$822$$ 10.3233 0.360065
$$823$$ −25.8517 −0.901133 −0.450567 0.892743i $$-0.648778\pi$$
−0.450567 + 0.892743i $$0.648778\pi$$
$$824$$ −25.8350 −0.900004
$$825$$ 0 0
$$826$$ −13.1987 −0.459240
$$827$$ −9.97816 −0.346974 −0.173487 0.984836i $$-0.555504\pi$$
−0.173487 + 0.984836i $$0.555504\pi$$
$$828$$ 27.2663 0.947568
$$829$$ 21.9808 0.763425 0.381713 0.924281i $$-0.375334\pi$$
0.381713 + 0.924281i $$0.375334\pi$$
$$830$$ 0 0
$$831$$ −5.38149 −0.186682
$$832$$ −19.8126 −0.686877
$$833$$ 18.2848 0.633531
$$834$$ −4.24685 −0.147056
$$835$$ 0 0
$$836$$ −16.0224 −0.554147
$$837$$ −7.97359 −0.275607
$$838$$ −19.4675 −0.672492
$$839$$ −16.1089 −0.556140 −0.278070 0.960561i $$-0.589695\pi$$
−0.278070 + 0.960561i $$0.589695\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 71.0451 2.44838
$$843$$ 0.321887 0.0110864
$$844$$ −60.6666 −2.08823
$$845$$ 0 0
$$846$$ 76.8270 2.64137
$$847$$ −19.9735 −0.686298
$$848$$ 37.0775 1.27324
$$849$$ 2.91020 0.0998779
$$850$$ 0 0
$$851$$ −11.2211 −0.384653
$$852$$ 28.3947 0.972785
$$853$$ 44.9615 1.53945 0.769727 0.638373i $$-0.220392\pi$$
0.769727 + 0.638373i $$0.220392\pi$$
$$854$$ −67.5128 −2.31024
$$855$$ 0 0
$$856$$ 69.3029 2.36872
$$857$$ −46.2431 −1.57963 −0.789817 0.613343i $$-0.789824\pi$$
−0.789817 + 0.613343i $$0.789824\pi$$
$$858$$ −11.1840 −0.381816
$$859$$ −10.7772 −0.367713 −0.183856 0.982953i $$-0.558858\pi$$
−0.183856 + 0.982953i $$0.558858\pi$$
$$860$$ 0 0
$$861$$ 1.02578 0.0349584
$$862$$ 77.5443 2.64117
$$863$$ 22.7966 0.776007 0.388003 0.921658i $$-0.373165\pi$$
0.388003 + 0.921658i $$0.373165\pi$$
$$864$$ −4.12804 −0.140439
$$865$$ 0 0
$$866$$ −1.16839 −0.0397034
$$867$$ 8.90400 0.302396
$$868$$ −31.9390 −1.08408
$$869$$ −5.82040 −0.197444
$$870$$ 0 0
$$871$$ 9.86201 0.334161
$$872$$ −52.5934 −1.78104
$$873$$ −5.61523 −0.190047
$$874$$ −6.31087 −0.213468
$$875$$ 0 0
$$876$$ −5.64752 −0.190812
$$877$$ −4.94644 −0.167029 −0.0835146 0.996507i $$-0.526615\pi$$
−0.0835146 + 0.996507i $$0.526615\pi$$
$$878$$ 66.2054 2.23432
$$879$$ 1.86201 0.0628039
$$880$$ 0 0
$$881$$ 2.53033 0.0852490 0.0426245 0.999091i $$-0.486428\pi$$
0.0426245 + 0.999091i $$0.486428\pi$$
$$882$$ 20.7047 0.697163
$$883$$ 29.7430 1.00093 0.500465 0.865757i $$-0.333162\pi$$
0.500465 + 0.865757i $$0.333162\pi$$
$$884$$ 46.2693 1.55620
$$885$$ 0 0
$$886$$ −56.5801 −1.90085
$$887$$ 45.5450 1.52925 0.764626 0.644474i $$-0.222924\pi$$
0.764626 + 0.644474i $$0.222924\pi$$
$$888$$ −10.3775 −0.348246
$$889$$ 19.3591 0.649282
$$890$$ 0 0
$$891$$ 26.9568 0.903086
$$892$$ 72.5729 2.42992
$$893$$ −11.7086 −0.391813
$$894$$ −10.9742 −0.367033
$$895$$ 0 0
$$896$$ −65.2019 −2.17824
$$897$$ −2.90059 −0.0968480
$$898$$ −55.9828 −1.86817
$$899$$ 15.5960 0.520155
$$900$$ 0 0
$$901$$ 68.0897 2.26840
$$902$$ 6.02662 0.200664
$$903$$ −5.45428 −0.181507
$$904$$ 47.5689 1.58212
$$905$$ 0 0
$$906$$ −1.15382 −0.0383331
$$907$$ 33.2034 1.10250 0.551250 0.834340i $$-0.314151\pi$$
0.551250 + 0.834340i $$0.314151\pi$$
$$908$$ −55.6841 −1.84794
$$909$$ 4.64093 0.153930
$$910$$ 0 0
$$911$$ 55.7788 1.84803 0.924017 0.382351i $$-0.124886\pi$$
0.924017 + 0.382351i $$0.124886\pi$$
$$912$$ −1.69472 −0.0561179
$$913$$ −29.3853 −0.972513
$$914$$ −51.4231 −1.70092
$$915$$ 0 0
$$916$$ −16.1089 −0.532252
$$917$$ 43.2715 1.42895
$$918$$ 43.0146 1.41969
$$919$$ −28.5769 −0.942665 −0.471333 0.881956i $$-0.656227\pi$$
−0.471333 + 0.881956i $$0.656227\pi$$
$$920$$ 0 0
$$921$$ −8.91020 −0.293601
$$922$$ −76.4159 −2.51662
$$923$$ 28.3947 0.934622
$$924$$ −27.4231 −0.902153
$$925$$ 0 0
$$926$$ 37.7788 1.24149
$$927$$ −15.5994 −0.512352
$$928$$ 8.07426 0.265051
$$929$$ −54.4937 −1.78788 −0.893940 0.448186i $$-0.852070\pi$$
−0.893940 + 0.448186i $$0.852070\pi$$
$$930$$ 0 0
$$931$$ −3.15544 −0.103415
$$932$$ 46.5960 1.52630
$$933$$ −2.23180 −0.0730659
$$934$$ 72.1761 2.36167
$$935$$ 0 0
$$936$$ 25.2165 0.824226
$$937$$ −37.1484 −1.21359 −0.606793 0.794860i $$-0.707544\pi$$
−0.606793 + 0.794860i $$0.707544\pi$$
$$938$$ 36.7245 1.19910
$$939$$ −0.493704 −0.0161114
$$940$$ 0 0
$$941$$ 32.3973 1.05612 0.528061 0.849206i $$-0.322919\pi$$
0.528061 + 0.849206i $$0.322919\pi$$
$$942$$ −5.38284 −0.175382
$$943$$ 1.56301 0.0508986
$$944$$ −5.40067 −0.175777
$$945$$ 0 0
$$946$$ −32.0448 −1.04187
$$947$$ 35.4662 1.15250 0.576249 0.817275i $$-0.304516\pi$$
0.576249 + 0.817275i $$0.304516\pi$$
$$948$$ −2.90059 −0.0942069
$$949$$ −5.64752 −0.183326
$$950$$ 0 0
$$951$$ −14.3557 −0.465516
$$952$$ 82.9272 2.68769
$$953$$ −14.3411 −0.464553 −0.232277 0.972650i $$-0.574617\pi$$
−0.232277 + 0.972650i $$0.574617\pi$$
$$954$$ 77.1009 2.49623
$$955$$ 0 0
$$956$$ −43.7788 −1.41591
$$957$$ 13.3908 0.432864
$$958$$ −10.9749 −0.354581
$$959$$ 25.3125 0.817383
$$960$$ 0 0
$$961$$ −24.2435 −0.782048
$$962$$ −21.5615 −0.695172
$$963$$ 41.8458 1.34846
$$964$$ −13.1122 −0.422316
$$965$$ 0 0
$$966$$ −10.8013 −0.347528
$$967$$ −27.9351 −0.898331 −0.449165 0.893449i $$-0.648279\pi$$
−0.449165 + 0.893449i $$0.648279\pi$$
$$968$$ −28.1463 −0.904657
$$969$$ −3.11222 −0.0999788
$$970$$ 0 0
$$971$$ −42.5993 −1.36708 −0.683539 0.729914i $$-0.739560\pi$$
−0.683539 + 0.729914i $$0.739560\pi$$
$$972$$ 48.9173 1.56902
$$973$$ −10.4132 −0.333833
$$974$$ −94.9920 −3.04374
$$975$$ 0 0
$$976$$ −27.6251 −0.884258
$$977$$ 39.5597 1.26563 0.632813 0.774304i $$-0.281900\pi$$
0.632813 + 0.774304i $$0.281900\pi$$
$$978$$ 32.1180 1.02702
$$979$$ 68.7114 2.19603
$$980$$ 0 0
$$981$$ −31.7564 −1.01390
$$982$$ −94.1112 −3.00321
$$983$$ −39.7689 −1.26843 −0.634215 0.773157i $$-0.718677\pi$$
−0.634215 + 0.773157i $$0.718677\pi$$
$$984$$ 1.44551 0.0460811
$$985$$ 0 0
$$986$$ −84.1345 −2.67939
$$987$$ −20.0398 −0.637874
$$988$$ −7.98476 −0.254029
$$989$$ −8.31087 −0.264270
$$990$$ 0 0
$$991$$ 55.2019 1.75355 0.876773 0.480905i $$-0.159692\pi$$
0.876773 + 0.480905i $$0.159692\pi$$
$$992$$ 3.49794 0.111060
$$993$$ 4.29664 0.136350
$$994$$ 105.737 3.35378
$$995$$ 0 0
$$996$$ −14.6442 −0.464018
$$997$$ −11.6543 −0.369097 −0.184548 0.982823i $$-0.559082\pi$$
−0.184548 + 0.982823i $$0.559082\pi$$
$$998$$ −11.4517 −0.362496
$$999$$ −13.1987 −0.417587
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 475.2.a.j.1.1 6
3.2 odd 2 4275.2.a.br.1.6 6
4.3 odd 2 7600.2.a.ck.1.3 6
5.2 odd 4 95.2.b.b.39.1 6
5.3 odd 4 95.2.b.b.39.6 yes 6
5.4 even 2 inner 475.2.a.j.1.6 6
15.2 even 4 855.2.c.d.514.6 6
15.8 even 4 855.2.c.d.514.1 6
15.14 odd 2 4275.2.a.br.1.1 6
19.18 odd 2 9025.2.a.bx.1.6 6
20.3 even 4 1520.2.d.h.609.3 6
20.7 even 4 1520.2.d.h.609.4 6
20.19 odd 2 7600.2.a.ck.1.4 6
95.18 even 4 1805.2.b.e.1084.1 6
95.37 even 4 1805.2.b.e.1084.6 6
95.94 odd 2 9025.2.a.bx.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.1 6 5.2 odd 4
95.2.b.b.39.6 yes 6 5.3 odd 4
475.2.a.j.1.1 6 1.1 even 1 trivial
475.2.a.j.1.6 6 5.4 even 2 inner
855.2.c.d.514.1 6 15.8 even 4
855.2.c.d.514.6 6 15.2 even 4
1520.2.d.h.609.3 6 20.3 even 4
1520.2.d.h.609.4 6 20.7 even 4
1805.2.b.e.1084.1 6 95.18 even 4
1805.2.b.e.1084.6 6 95.37 even 4
4275.2.a.br.1.1 6 15.14 odd 2
4275.2.a.br.1.6 6 3.2 odd 2
7600.2.a.ck.1.3 6 4.3 odd 2
7600.2.a.ck.1.4 6 20.19 odd 2
9025.2.a.bx.1.1 6 95.94 odd 2
9025.2.a.bx.1.6 6 19.18 odd 2