# Properties

 Label 195.2.a.e.1.1 Level $195$ Weight $2$ Character 195.1 Self dual yes Analytic conductor $1.557$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.34292$$ of defining polynomial Character $$\chi$$ $$=$$ 195.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.48929 q^{2} -1.00000 q^{3} +4.19656 q^{4} -1.00000 q^{5} +2.48929 q^{6} -1.19656 q^{7} -5.46787 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.48929 q^{2} -1.00000 q^{3} +4.19656 q^{4} -1.00000 q^{5} +2.48929 q^{6} -1.19656 q^{7} -5.46787 q^{8} +1.00000 q^{9} +2.48929 q^{10} -1.19656 q^{11} -4.19656 q^{12} +1.00000 q^{13} +2.97858 q^{14} +1.00000 q^{15} +5.21798 q^{16} +6.17513 q^{17} -2.48929 q^{18} +6.97858 q^{19} -4.19656 q^{20} +1.19656 q^{21} +2.97858 q^{22} +4.17513 q^{23} +5.46787 q^{24} +1.00000 q^{25} -2.48929 q^{26} -1.00000 q^{27} -5.02142 q^{28} +6.00000 q^{29} -2.48929 q^{30} -2.97858 q^{31} -2.05333 q^{32} +1.19656 q^{33} -15.3717 q^{34} +1.19656 q^{35} +4.19656 q^{36} +7.78202 q^{37} -17.3717 q^{38} -1.00000 q^{39} +5.46787 q^{40} -6.17513 q^{41} -2.97858 q^{42} -9.95715 q^{43} -5.02142 q^{44} -1.00000 q^{45} -10.3931 q^{46} -1.02142 q^{47} -5.21798 q^{48} -5.56825 q^{49} -2.48929 q^{50} -6.17513 q^{51} +4.19656 q^{52} +10.1751 q^{53} +2.48929 q^{54} +1.19656 q^{55} +6.54262 q^{56} -6.97858 q^{57} -14.9357 q^{58} +5.37169 q^{59} +4.19656 q^{60} +12.5682 q^{61} +7.41454 q^{62} -1.19656 q^{63} -5.32464 q^{64} -1.00000 q^{65} -2.97858 q^{66} +9.37169 q^{67} +25.9143 q^{68} -4.17513 q^{69} -2.97858 q^{70} -5.19656 q^{71} -5.46787 q^{72} -11.9572 q^{73} -19.3717 q^{74} -1.00000 q^{75} +29.2860 q^{76} +1.43175 q^{77} +2.48929 q^{78} -1.78202 q^{79} -5.21798 q^{80} +1.00000 q^{81} +15.3717 q^{82} -5.37169 q^{83} +5.02142 q^{84} -6.17513 q^{85} +24.7862 q^{86} -6.00000 q^{87} +6.54262 q^{88} +10.1751 q^{89} +2.48929 q^{90} -1.19656 q^{91} +17.5212 q^{92} +2.97858 q^{93} +2.54262 q^{94} -6.97858 q^{95} +2.05333 q^{96} -1.82487 q^{97} +13.8610 q^{98} -1.19656 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 8 q^{4} - 3 q^{5} + q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 8 * q^4 - 3 * q^5 + q^7 + 6 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 8 q^{4} - 3 q^{5} + q^{7} + 6 q^{8} + 3 q^{9} + q^{11} - 8 q^{12} + 3 q^{13} - 6 q^{14} + 3 q^{15} + 26 q^{16} - q^{17} + 6 q^{19} - 8 q^{20} - q^{21} - 6 q^{22} - 7 q^{23} - 6 q^{24} + 3 q^{25} - 3 q^{27} - 30 q^{28} + 18 q^{29} + 6 q^{31} + 22 q^{32} - q^{33} - 22 q^{34} - q^{35} + 8 q^{36} + 13 q^{37} - 28 q^{38} - 3 q^{39} - 6 q^{40} + q^{41} + 6 q^{42} - 30 q^{44} - 3 q^{45} - 22 q^{46} - 18 q^{47} - 26 q^{48} + 12 q^{49} + q^{51} + 8 q^{52} + 11 q^{53} - q^{55} - 16 q^{56} - 6 q^{57} - 8 q^{59} + 8 q^{60} + 9 q^{61} + 28 q^{62} + q^{63} + 30 q^{64} - 3 q^{65} + 6 q^{66} + 4 q^{67} + 18 q^{68} + 7 q^{69} + 6 q^{70} - 11 q^{71} + 6 q^{72} - 6 q^{73} - 34 q^{74} - 3 q^{75} + 4 q^{76} + 33 q^{77} + 5 q^{79} - 26 q^{80} + 3 q^{81} + 22 q^{82} + 8 q^{83} + 30 q^{84} + q^{85} + 56 q^{86} - 18 q^{87} - 16 q^{88} + 11 q^{89} + q^{91} + 2 q^{92} - 6 q^{93} - 28 q^{94} - 6 q^{95} - 22 q^{96} - 25 q^{97} + 10 q^{98} + q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 8 * q^4 - 3 * q^5 + q^7 + 6 * q^8 + 3 * q^9 + q^11 - 8 * q^12 + 3 * q^13 - 6 * q^14 + 3 * q^15 + 26 * q^16 - q^17 + 6 * q^19 - 8 * q^20 - q^21 - 6 * q^22 - 7 * q^23 - 6 * q^24 + 3 * q^25 - 3 * q^27 - 30 * q^28 + 18 * q^29 + 6 * q^31 + 22 * q^32 - q^33 - 22 * q^34 - q^35 + 8 * q^36 + 13 * q^37 - 28 * q^38 - 3 * q^39 - 6 * q^40 + q^41 + 6 * q^42 - 30 * q^44 - 3 * q^45 - 22 * q^46 - 18 * q^47 - 26 * q^48 + 12 * q^49 + q^51 + 8 * q^52 + 11 * q^53 - q^55 - 16 * q^56 - 6 * q^57 - 8 * q^59 + 8 * q^60 + 9 * q^61 + 28 * q^62 + q^63 + 30 * q^64 - 3 * q^65 + 6 * q^66 + 4 * q^67 + 18 * q^68 + 7 * q^69 + 6 * q^70 - 11 * q^71 + 6 * q^72 - 6 * q^73 - 34 * q^74 - 3 * q^75 + 4 * q^76 + 33 * q^77 + 5 * q^79 - 26 * q^80 + 3 * q^81 + 22 * q^82 + 8 * q^83 + 30 * q^84 + q^85 + 56 * q^86 - 18 * q^87 - 16 * q^88 + 11 * q^89 + q^91 + 2 * q^92 - 6 * q^93 - 28 * q^94 - 6 * q^95 - 22 * q^96 - 25 * q^97 + 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.48929 −1.76019 −0.880096 0.474795i $$-0.842522\pi$$
−0.880096 + 0.474795i $$0.842522\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 4.19656 2.09828
$$5$$ −1.00000 −0.447214
$$6$$ 2.48929 1.01625
$$7$$ −1.19656 −0.452256 −0.226128 0.974098i $$-0.572607\pi$$
−0.226128 + 0.974098i $$0.572607\pi$$
$$8$$ −5.46787 −1.93318
$$9$$ 1.00000 0.333333
$$10$$ 2.48929 0.787182
$$11$$ −1.19656 −0.360776 −0.180388 0.983596i $$-0.557735\pi$$
−0.180388 + 0.983596i $$0.557735\pi$$
$$12$$ −4.19656 −1.21144
$$13$$ 1.00000 0.277350
$$14$$ 2.97858 0.796058
$$15$$ 1.00000 0.258199
$$16$$ 5.21798 1.30450
$$17$$ 6.17513 1.49769 0.748845 0.662745i $$-0.230609\pi$$
0.748845 + 0.662745i $$0.230609\pi$$
$$18$$ −2.48929 −0.586731
$$19$$ 6.97858 1.60100 0.800498 0.599336i $$-0.204569\pi$$
0.800498 + 0.599336i $$0.204569\pi$$
$$20$$ −4.19656 −0.938379
$$21$$ 1.19656 0.261110
$$22$$ 2.97858 0.635035
$$23$$ 4.17513 0.870576 0.435288 0.900291i $$-0.356647\pi$$
0.435288 + 0.900291i $$0.356647\pi$$
$$24$$ 5.46787 1.11612
$$25$$ 1.00000 0.200000
$$26$$ −2.48929 −0.488190
$$27$$ −1.00000 −0.192450
$$28$$ −5.02142 −0.948960
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ −2.48929 −0.454480
$$31$$ −2.97858 −0.534968 −0.267484 0.963562i $$-0.586192\pi$$
−0.267484 + 0.963562i $$0.586192\pi$$
$$32$$ −2.05333 −0.362980
$$33$$ 1.19656 0.208294
$$34$$ −15.3717 −2.63622
$$35$$ 1.19656 0.202255
$$36$$ 4.19656 0.699426
$$37$$ 7.78202 1.27936 0.639678 0.768643i $$-0.279068\pi$$
0.639678 + 0.768643i $$0.279068\pi$$
$$38$$ −17.3717 −2.81806
$$39$$ −1.00000 −0.160128
$$40$$ 5.46787 0.864545
$$41$$ −6.17513 −0.964394 −0.482197 0.876063i $$-0.660161\pi$$
−0.482197 + 0.876063i $$0.660161\pi$$
$$42$$ −2.97858 −0.459604
$$43$$ −9.95715 −1.51845 −0.759226 0.650827i $$-0.774422\pi$$
−0.759226 + 0.650827i $$0.774422\pi$$
$$44$$ −5.02142 −0.757008
$$45$$ −1.00000 −0.149071
$$46$$ −10.3931 −1.53238
$$47$$ −1.02142 −0.148990 −0.0744949 0.997221i $$-0.523734\pi$$
−0.0744949 + 0.997221i $$0.523734\pi$$
$$48$$ −5.21798 −0.753151
$$49$$ −5.56825 −0.795464
$$50$$ −2.48929 −0.352039
$$51$$ −6.17513 −0.864692
$$52$$ 4.19656 0.581958
$$53$$ 10.1751 1.39766 0.698831 0.715287i $$-0.253704\pi$$
0.698831 + 0.715287i $$0.253704\pi$$
$$54$$ 2.48929 0.338749
$$55$$ 1.19656 0.161344
$$56$$ 6.54262 0.874294
$$57$$ −6.97858 −0.924335
$$58$$ −14.9357 −1.96116
$$59$$ 5.37169 0.699335 0.349667 0.936874i $$-0.386295\pi$$
0.349667 + 0.936874i $$0.386295\pi$$
$$60$$ 4.19656 0.541773
$$61$$ 12.5682 1.60920 0.804600 0.593818i $$-0.202380\pi$$
0.804600 + 0.593818i $$0.202380\pi$$
$$62$$ 7.41454 0.941647
$$63$$ −1.19656 −0.150752
$$64$$ −5.32464 −0.665579
$$65$$ −1.00000 −0.124035
$$66$$ −2.97858 −0.366638
$$67$$ 9.37169 1.14493 0.572467 0.819928i $$-0.305986\pi$$
0.572467 + 0.819928i $$0.305986\pi$$
$$68$$ 25.9143 3.14257
$$69$$ −4.17513 −0.502627
$$70$$ −2.97858 −0.356008
$$71$$ −5.19656 −0.616718 −0.308359 0.951270i $$-0.599780\pi$$
−0.308359 + 0.951270i $$0.599780\pi$$
$$72$$ −5.46787 −0.644394
$$73$$ −11.9572 −1.39948 −0.699740 0.714398i $$-0.746701\pi$$
−0.699740 + 0.714398i $$0.746701\pi$$
$$74$$ −19.3717 −2.25191
$$75$$ −1.00000 −0.115470
$$76$$ 29.2860 3.35933
$$77$$ 1.43175 0.163163
$$78$$ 2.48929 0.281856
$$79$$ −1.78202 −0.200493 −0.100246 0.994963i $$-0.531963\pi$$
−0.100246 + 0.994963i $$0.531963\pi$$
$$80$$ −5.21798 −0.583388
$$81$$ 1.00000 0.111111
$$82$$ 15.3717 1.69752
$$83$$ −5.37169 −0.589620 −0.294810 0.955556i $$-0.595256\pi$$
−0.294810 + 0.955556i $$0.595256\pi$$
$$84$$ 5.02142 0.547882
$$85$$ −6.17513 −0.669787
$$86$$ 24.7862 2.67277
$$87$$ −6.00000 −0.643268
$$88$$ 6.54262 0.697445
$$89$$ 10.1751 1.07856 0.539281 0.842126i $$-0.318696\pi$$
0.539281 + 0.842126i $$0.318696\pi$$
$$90$$ 2.48929 0.262394
$$91$$ −1.19656 −0.125433
$$92$$ 17.5212 1.82671
$$93$$ 2.97858 0.308864
$$94$$ 2.54262 0.262251
$$95$$ −6.97858 −0.715987
$$96$$ 2.05333 0.209567
$$97$$ −1.82487 −0.185287 −0.0926435 0.995699i $$-0.529532\pi$$
−0.0926435 + 0.995699i $$0.529532\pi$$
$$98$$ 13.8610 1.40017
$$99$$ −1.19656 −0.120259
$$100$$ 4.19656 0.419656
$$101$$ −10.3503 −1.02989 −0.514945 0.857223i $$-0.672188\pi$$
−0.514945 + 0.857223i $$0.672188\pi$$
$$102$$ 15.3717 1.52202
$$103$$ 18.7434 1.84684 0.923420 0.383790i $$-0.125381\pi$$
0.923420 + 0.383790i $$0.125381\pi$$
$$104$$ −5.46787 −0.536168
$$105$$ −1.19656 −0.116772
$$106$$ −25.3288 −2.46016
$$107$$ −18.5682 −1.79506 −0.897530 0.440953i $$-0.854641\pi$$
−0.897530 + 0.440953i $$0.854641\pi$$
$$108$$ −4.19656 −0.403814
$$109$$ 8.39312 0.803915 0.401957 0.915658i $$-0.368330\pi$$
0.401957 + 0.915658i $$0.368330\pi$$
$$110$$ −2.97858 −0.283996
$$111$$ −7.78202 −0.738637
$$112$$ −6.24361 −0.589966
$$113$$ 7.95715 0.748546 0.374273 0.927319i $$-0.377892\pi$$
0.374273 + 0.927319i $$0.377892\pi$$
$$114$$ 17.3717 1.62701
$$115$$ −4.17513 −0.389333
$$116$$ 25.1793 2.33784
$$117$$ 1.00000 0.0924500
$$118$$ −13.3717 −1.23096
$$119$$ −7.38890 −0.677340
$$120$$ −5.46787 −0.499146
$$121$$ −9.56825 −0.869841
$$122$$ −31.2860 −2.83250
$$123$$ 6.17513 0.556793
$$124$$ −12.4998 −1.12251
$$125$$ −1.00000 −0.0894427
$$126$$ 2.97858 0.265353
$$127$$ −10.3931 −0.922240 −0.461120 0.887338i $$-0.652552\pi$$
−0.461120 + 0.887338i $$0.652552\pi$$
$$128$$ 17.3612 1.53453
$$129$$ 9.95715 0.876679
$$130$$ 2.48929 0.218325
$$131$$ −6.39312 −0.558569 −0.279285 0.960208i $$-0.590097\pi$$
−0.279285 + 0.960208i $$0.590097\pi$$
$$132$$ 5.02142 0.437059
$$133$$ −8.35027 −0.724060
$$134$$ −23.3288 −2.01531
$$135$$ 1.00000 0.0860663
$$136$$ −33.7648 −2.89531
$$137$$ 16.7434 1.43048 0.715242 0.698877i $$-0.246316\pi$$
0.715242 + 0.698877i $$0.246316\pi$$
$$138$$ 10.3931 0.884721
$$139$$ 5.78202 0.490424 0.245212 0.969469i $$-0.421142\pi$$
0.245212 + 0.969469i $$0.421142\pi$$
$$140$$ 5.02142 0.424388
$$141$$ 1.02142 0.0860193
$$142$$ 12.9357 1.08554
$$143$$ −1.19656 −0.100061
$$144$$ 5.21798 0.434832
$$145$$ −6.00000 −0.498273
$$146$$ 29.7648 2.46335
$$147$$ 5.56825 0.459262
$$148$$ 32.6577 2.68445
$$149$$ 15.3461 1.25720 0.628599 0.777730i $$-0.283629\pi$$
0.628599 + 0.777730i $$0.283629\pi$$
$$150$$ 2.48929 0.203250
$$151$$ −8.58546 −0.698675 −0.349337 0.936997i $$-0.613593\pi$$
−0.349337 + 0.936997i $$0.613593\pi$$
$$152$$ −38.1579 −3.09502
$$153$$ 6.17513 0.499230
$$154$$ −3.56404 −0.287198
$$155$$ 2.97858 0.239245
$$156$$ −4.19656 −0.335994
$$157$$ 2.78623 0.222365 0.111183 0.993800i $$-0.464536\pi$$
0.111183 + 0.993800i $$0.464536\pi$$
$$158$$ 4.43596 0.352906
$$159$$ −10.1751 −0.806941
$$160$$ 2.05333 0.162330
$$161$$ −4.99579 −0.393723
$$162$$ −2.48929 −0.195577
$$163$$ 8.76060 0.686183 0.343091 0.939302i $$-0.388526\pi$$
0.343091 + 0.939302i $$0.388526\pi$$
$$164$$ −25.9143 −2.02357
$$165$$ −1.19656 −0.0931519
$$166$$ 13.3717 1.03784
$$167$$ −17.3717 −1.34426 −0.672131 0.740432i $$-0.734621\pi$$
−0.672131 + 0.740432i $$0.734621\pi$$
$$168$$ −6.54262 −0.504774
$$169$$ 1.00000 0.0769231
$$170$$ 15.3717 1.17895
$$171$$ 6.97858 0.533665
$$172$$ −41.7858 −3.18614
$$173$$ −7.95715 −0.604971 −0.302486 0.953154i $$-0.597816\pi$$
−0.302486 + 0.953154i $$0.597816\pi$$
$$174$$ 14.9357 1.13227
$$175$$ −1.19656 −0.0904513
$$176$$ −6.24361 −0.470630
$$177$$ −5.37169 −0.403761
$$178$$ −25.3288 −1.89848
$$179$$ −15.5640 −1.16331 −0.581655 0.813435i $$-0.697595\pi$$
−0.581655 + 0.813435i $$0.697595\pi$$
$$180$$ −4.19656 −0.312793
$$181$$ 15.7820 1.17307 0.586534 0.809925i $$-0.300492\pi$$
0.586534 + 0.809925i $$0.300492\pi$$
$$182$$ 2.97858 0.220787
$$183$$ −12.5682 −0.929072
$$184$$ −22.8291 −1.68298
$$185$$ −7.78202 −0.572145
$$186$$ −7.41454 −0.543660
$$187$$ −7.38890 −0.540330
$$188$$ −4.28646 −0.312622
$$189$$ 1.19656 0.0870368
$$190$$ 17.3717 1.26028
$$191$$ 10.7434 0.777364 0.388682 0.921372i $$-0.372930\pi$$
0.388682 + 0.921372i $$0.372930\pi$$
$$192$$ 5.32464 0.384272
$$193$$ 9.73917 0.701041 0.350521 0.936555i $$-0.386005\pi$$
0.350521 + 0.936555i $$0.386005\pi$$
$$194$$ 4.54262 0.326141
$$195$$ 1.00000 0.0716115
$$196$$ −23.3675 −1.66911
$$197$$ −9.56404 −0.681410 −0.340705 0.940170i $$-0.610666\pi$$
−0.340705 + 0.940170i $$0.610666\pi$$
$$198$$ 2.97858 0.211678
$$199$$ 5.95715 0.422291 0.211146 0.977455i $$-0.432281\pi$$
0.211146 + 0.977455i $$0.432281\pi$$
$$200$$ −5.46787 −0.386636
$$201$$ −9.37169 −0.661028
$$202$$ 25.7648 1.81281
$$203$$ −7.17935 −0.503891
$$204$$ −25.9143 −1.81436
$$205$$ 6.17513 0.431290
$$206$$ −46.6577 −3.25080
$$207$$ 4.17513 0.290192
$$208$$ 5.21798 0.361802
$$209$$ −8.35027 −0.577600
$$210$$ 2.97858 0.205541
$$211$$ 23.9143 1.64633 0.823164 0.567803i $$-0.192206\pi$$
0.823164 + 0.567803i $$0.192206\pi$$
$$212$$ 42.7005 2.93269
$$213$$ 5.19656 0.356062
$$214$$ 46.2217 3.15965
$$215$$ 9.95715 0.679072
$$216$$ 5.46787 0.372041
$$217$$ 3.56404 0.241943
$$218$$ −20.8929 −1.41504
$$219$$ 11.9572 0.807990
$$220$$ 5.02142 0.338544
$$221$$ 6.17513 0.415385
$$222$$ 19.3717 1.30014
$$223$$ 2.62831 0.176004 0.0880022 0.996120i $$-0.471952\pi$$
0.0880022 + 0.996120i $$0.471952\pi$$
$$224$$ 2.45692 0.164160
$$225$$ 1.00000 0.0666667
$$226$$ −19.8077 −1.31759
$$227$$ 15.7648 1.04635 0.523174 0.852226i $$-0.324748\pi$$
0.523174 + 0.852226i $$0.324748\pi$$
$$228$$ −29.2860 −1.93951
$$229$$ 8.74338 0.577779 0.288890 0.957362i $$-0.406714\pi$$
0.288890 + 0.957362i $$0.406714\pi$$
$$230$$ 10.3931 0.685302
$$231$$ −1.43175 −0.0942022
$$232$$ −32.8072 −2.15390
$$233$$ −2.17513 −0.142498 −0.0712489 0.997459i $$-0.522698\pi$$
−0.0712489 + 0.997459i $$0.522698\pi$$
$$234$$ −2.48929 −0.162730
$$235$$ 1.02142 0.0666303
$$236$$ 22.5426 1.46740
$$237$$ 1.78202 0.115755
$$238$$ 18.3931 1.19225
$$239$$ 2.80344 0.181340 0.0906698 0.995881i $$-0.471099\pi$$
0.0906698 + 0.995881i $$0.471099\pi$$
$$240$$ 5.21798 0.336819
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ 23.8181 1.53109
$$243$$ −1.00000 −0.0641500
$$244$$ 52.7434 3.37655
$$245$$ 5.56825 0.355742
$$246$$ −15.3717 −0.980063
$$247$$ 6.97858 0.444036
$$248$$ 16.2865 1.03419
$$249$$ 5.37169 0.340417
$$250$$ 2.48929 0.157436
$$251$$ −23.9143 −1.50946 −0.754729 0.656037i $$-0.772232\pi$$
−0.754729 + 0.656037i $$0.772232\pi$$
$$252$$ −5.02142 −0.316320
$$253$$ −4.99579 −0.314083
$$254$$ 25.8715 1.62332
$$255$$ 6.17513 0.386702
$$256$$ −32.5678 −2.03549
$$257$$ −19.9572 −1.24489 −0.622447 0.782662i $$-0.713861\pi$$
−0.622447 + 0.782662i $$0.713861\pi$$
$$258$$ −24.7862 −1.54312
$$259$$ −9.31163 −0.578597
$$260$$ −4.19656 −0.260259
$$261$$ 6.00000 0.371391
$$262$$ 15.9143 0.983189
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ −6.54262 −0.402670
$$265$$ −10.1751 −0.625054
$$266$$ 20.7862 1.27449
$$267$$ −10.1751 −0.622708
$$268$$ 39.3288 2.40239
$$269$$ −2.35027 −0.143298 −0.0716492 0.997430i $$-0.522826\pi$$
−0.0716492 + 0.997430i $$0.522826\pi$$
$$270$$ −2.48929 −0.151493
$$271$$ 10.9786 0.666901 0.333451 0.942768i $$-0.391787\pi$$
0.333451 + 0.942768i $$0.391787\pi$$
$$272$$ 32.2217 1.95373
$$273$$ 1.19656 0.0724190
$$274$$ −41.6791 −2.51793
$$275$$ −1.19656 −0.0721551
$$276$$ −17.5212 −1.05465
$$277$$ 1.21377 0.0729283 0.0364642 0.999335i $$-0.488391\pi$$
0.0364642 + 0.999335i $$0.488391\pi$$
$$278$$ −14.3931 −0.863242
$$279$$ −2.97858 −0.178323
$$280$$ −6.54262 −0.390996
$$281$$ 11.9572 0.713304 0.356652 0.934237i $$-0.383918\pi$$
0.356652 + 0.934237i $$0.383918\pi$$
$$282$$ −2.54262 −0.151411
$$283$$ −29.8715 −1.77567 −0.887837 0.460158i $$-0.847793\pi$$
−0.887837 + 0.460158i $$0.847793\pi$$
$$284$$ −21.8077 −1.29405
$$285$$ 6.97858 0.413375
$$286$$ 2.97858 0.176127
$$287$$ 7.38890 0.436153
$$288$$ −2.05333 −0.120993
$$289$$ 21.1323 1.24308
$$290$$ 14.9357 0.877056
$$291$$ 1.82487 0.106975
$$292$$ −50.1789 −2.93650
$$293$$ 0.777809 0.0454401 0.0227200 0.999742i $$-0.492767\pi$$
0.0227200 + 0.999742i $$0.492767\pi$$
$$294$$ −13.8610 −0.808389
$$295$$ −5.37169 −0.312752
$$296$$ −42.5510 −2.47323
$$297$$ 1.19656 0.0694313
$$298$$ −38.2008 −2.21291
$$299$$ 4.17513 0.241454
$$300$$ −4.19656 −0.242288
$$301$$ 11.9143 0.686729
$$302$$ 21.3717 1.22980
$$303$$ 10.3503 0.594607
$$304$$ 36.4141 2.08849
$$305$$ −12.5682 −0.719656
$$306$$ −15.3717 −0.878741
$$307$$ 0.760597 0.0434095 0.0217048 0.999764i $$-0.493091\pi$$
0.0217048 + 0.999764i $$0.493091\pi$$
$$308$$ 6.00842 0.342362
$$309$$ −18.7434 −1.06627
$$310$$ −7.41454 −0.421117
$$311$$ −23.1281 −1.31147 −0.655736 0.754990i $$-0.727642\pi$$
−0.655736 + 0.754990i $$0.727642\pi$$
$$312$$ 5.46787 0.309557
$$313$$ −33.9143 −1.91695 −0.958475 0.285176i $$-0.907948\pi$$
−0.958475 + 0.285176i $$0.907948\pi$$
$$314$$ −6.93573 −0.391406
$$315$$ 1.19656 0.0674184
$$316$$ −7.47835 −0.420690
$$317$$ 9.64973 0.541983 0.270991 0.962582i $$-0.412648\pi$$
0.270991 + 0.962582i $$0.412648\pi$$
$$318$$ 25.3288 1.42037
$$319$$ −7.17935 −0.401966
$$320$$ 5.32464 0.297656
$$321$$ 18.5682 1.03638
$$322$$ 12.4360 0.693029
$$323$$ 43.0937 2.39780
$$324$$ 4.19656 0.233142
$$325$$ 1.00000 0.0554700
$$326$$ −21.8077 −1.20781
$$327$$ −8.39312 −0.464140
$$328$$ 33.7648 1.86435
$$329$$ 1.22219 0.0673816
$$330$$ 2.97858 0.163965
$$331$$ 15.3288 0.842550 0.421275 0.906933i $$-0.361583\pi$$
0.421275 + 0.906933i $$0.361583\pi$$
$$332$$ −22.5426 −1.23719
$$333$$ 7.78202 0.426452
$$334$$ 43.2432 2.36616
$$335$$ −9.37169 −0.512030
$$336$$ 6.24361 0.340617
$$337$$ −22.3503 −1.21750 −0.608748 0.793363i $$-0.708328\pi$$
−0.608748 + 0.793363i $$0.708328\pi$$
$$338$$ −2.48929 −0.135399
$$339$$ −7.95715 −0.432173
$$340$$ −25.9143 −1.40540
$$341$$ 3.56404 0.193004
$$342$$ −17.3717 −0.939354
$$343$$ 15.0386 0.812010
$$344$$ 54.4444 2.93544
$$345$$ 4.17513 0.224782
$$346$$ 19.8077 1.06487
$$347$$ −5.78202 −0.310395 −0.155198 0.987883i $$-0.549601\pi$$
−0.155198 + 0.987883i $$0.549601\pi$$
$$348$$ −25.1793 −1.34975
$$349$$ −27.5212 −1.47318 −0.736588 0.676342i $$-0.763564\pi$$
−0.736588 + 0.676342i $$0.763564\pi$$
$$350$$ 2.97858 0.159212
$$351$$ −1.00000 −0.0533761
$$352$$ 2.45692 0.130955
$$353$$ −28.7434 −1.52986 −0.764928 0.644116i $$-0.777225\pi$$
−0.764928 + 0.644116i $$0.777225\pi$$
$$354$$ 13.3717 0.710697
$$355$$ 5.19656 0.275805
$$356$$ 42.7005 2.26312
$$357$$ 7.38890 0.391062
$$358$$ 38.7434 2.04765
$$359$$ −12.5855 −0.664235 −0.332118 0.943238i $$-0.607763\pi$$
−0.332118 + 0.943238i $$0.607763\pi$$
$$360$$ 5.46787 0.288182
$$361$$ 29.7005 1.56319
$$362$$ −39.2860 −2.06483
$$363$$ 9.56825 0.502203
$$364$$ −5.02142 −0.263194
$$365$$ 11.9572 0.625866
$$366$$ 31.2860 1.63535
$$367$$ −27.9143 −1.45712 −0.728558 0.684985i $$-0.759809\pi$$
−0.728558 + 0.684985i $$0.759809\pi$$
$$368$$ 21.7858 1.13566
$$369$$ −6.17513 −0.321465
$$370$$ 19.3717 1.00709
$$371$$ −12.1751 −0.632102
$$372$$ 12.4998 0.648083
$$373$$ −2.35027 −0.121692 −0.0608462 0.998147i $$-0.519380\pi$$
−0.0608462 + 0.998147i $$0.519380\pi$$
$$374$$ 18.3931 0.951085
$$375$$ 1.00000 0.0516398
$$376$$ 5.58500 0.288025
$$377$$ 6.00000 0.309016
$$378$$ −2.97858 −0.153201
$$379$$ 24.5510 1.26110 0.630551 0.776148i $$-0.282829\pi$$
0.630551 + 0.776148i $$0.282829\pi$$
$$380$$ −29.2860 −1.50234
$$381$$ 10.3931 0.532455
$$382$$ −26.7434 −1.36831
$$383$$ 5.80765 0.296757 0.148379 0.988931i $$-0.452595\pi$$
0.148379 + 0.988931i $$0.452595\pi$$
$$384$$ −17.3612 −0.885961
$$385$$ −1.43175 −0.0729687
$$386$$ −24.2436 −1.23397
$$387$$ −9.95715 −0.506151
$$388$$ −7.65815 −0.388784
$$389$$ 13.6497 0.692069 0.346034 0.938222i $$-0.387528\pi$$
0.346034 + 0.938222i $$0.387528\pi$$
$$390$$ −2.48929 −0.126050
$$391$$ 25.7820 1.30385
$$392$$ 30.4464 1.53778
$$393$$ 6.39312 0.322490
$$394$$ 23.8077 1.19941
$$395$$ 1.78202 0.0896631
$$396$$ −5.02142 −0.252336
$$397$$ −12.1323 −0.608902 −0.304451 0.952528i $$-0.598473\pi$$
−0.304451 + 0.952528i $$0.598473\pi$$
$$398$$ −14.8291 −0.743314
$$399$$ 8.35027 0.418036
$$400$$ 5.21798 0.260899
$$401$$ −37.4439 −1.86986 −0.934930 0.354832i $$-0.884538\pi$$
−0.934930 + 0.354832i $$0.884538\pi$$
$$402$$ 23.3288 1.16354
$$403$$ −2.97858 −0.148373
$$404$$ −43.4355 −2.16100
$$405$$ −1.00000 −0.0496904
$$406$$ 17.8715 0.886946
$$407$$ −9.31163 −0.461561
$$408$$ 33.7648 1.67161
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ −15.3717 −0.759154
$$411$$ −16.7434 −0.825890
$$412$$ 78.6577 3.87519
$$413$$ −6.42754 −0.316279
$$414$$ −10.3931 −0.510794
$$415$$ 5.37169 0.263686
$$416$$ −2.05333 −0.100673
$$417$$ −5.78202 −0.283147
$$418$$ 20.7862 1.01669
$$419$$ 3.17935 0.155321 0.0776606 0.996980i $$-0.475255\pi$$
0.0776606 + 0.996980i $$0.475255\pi$$
$$420$$ −5.02142 −0.245020
$$421$$ −16.3074 −0.794775 −0.397388 0.917651i $$-0.630083\pi$$
−0.397388 + 0.917651i $$0.630083\pi$$
$$422$$ −59.5296 −2.89786
$$423$$ −1.02142 −0.0496633
$$424$$ −55.6363 −2.70194
$$425$$ 6.17513 0.299538
$$426$$ −12.9357 −0.626738
$$427$$ −15.0386 −0.727771
$$428$$ −77.9227 −3.76654
$$429$$ 1.19656 0.0577703
$$430$$ −24.7862 −1.19530
$$431$$ 4.58546 0.220874 0.110437 0.993883i $$-0.464775\pi$$
0.110437 + 0.993883i $$0.464775\pi$$
$$432$$ −5.21798 −0.251050
$$433$$ −38.3503 −1.84300 −0.921498 0.388383i $$-0.873034\pi$$
−0.921498 + 0.388383i $$0.873034\pi$$
$$434$$ −8.87192 −0.425866
$$435$$ 6.00000 0.287678
$$436$$ 35.2222 1.68684
$$437$$ 29.1365 1.39379
$$438$$ −29.7648 −1.42222
$$439$$ 7.73917 0.369371 0.184685 0.982798i $$-0.440873\pi$$
0.184685 + 0.982798i $$0.440873\pi$$
$$440$$ −6.54262 −0.311907
$$441$$ −5.56825 −0.265155
$$442$$ −15.3717 −0.731157
$$443$$ 34.9185 1.65903 0.829514 0.558485i $$-0.188617\pi$$
0.829514 + 0.558485i $$0.188617\pi$$
$$444$$ −32.6577 −1.54987
$$445$$ −10.1751 −0.482348
$$446$$ −6.54262 −0.309802
$$447$$ −15.3461 −0.725844
$$448$$ 6.37123 0.301012
$$449$$ −6.17513 −0.291423 −0.145711 0.989327i $$-0.546547\pi$$
−0.145711 + 0.989327i $$0.546547\pi$$
$$450$$ −2.48929 −0.117346
$$451$$ 7.38890 0.347930
$$452$$ 33.3927 1.57066
$$453$$ 8.58546 0.403380
$$454$$ −39.2432 −1.84177
$$455$$ 1.19656 0.0560955
$$456$$ 38.1579 1.78691
$$457$$ 1.38890 0.0649702 0.0324851 0.999472i $$-0.489658\pi$$
0.0324851 + 0.999472i $$0.489658\pi$$
$$458$$ −21.7648 −1.01700
$$459$$ −6.17513 −0.288231
$$460$$ −17.5212 −0.816930
$$461$$ 28.4826 1.32657 0.663283 0.748369i $$-0.269163\pi$$
0.663283 + 0.748369i $$0.269163\pi$$
$$462$$ 3.56404 0.165814
$$463$$ 16.3759 0.761053 0.380526 0.924770i $$-0.375743\pi$$
0.380526 + 0.924770i $$0.375743\pi$$
$$464$$ 31.3079 1.45343
$$465$$ −2.97858 −0.138128
$$466$$ 5.41454 0.250824
$$467$$ 25.7476 1.19146 0.595728 0.803186i $$-0.296864\pi$$
0.595728 + 0.803186i $$0.296864\pi$$
$$468$$ 4.19656 0.193986
$$469$$ −11.2138 −0.517804
$$470$$ −2.54262 −0.117282
$$471$$ −2.78623 −0.128383
$$472$$ −29.3717 −1.35194
$$473$$ 11.9143 0.547820
$$474$$ −4.43596 −0.203750
$$475$$ 6.97858 0.320199
$$476$$ −31.0080 −1.42125
$$477$$ 10.1751 0.465887
$$478$$ −6.97858 −0.319193
$$479$$ −7.58967 −0.346781 −0.173391 0.984853i $$-0.555472\pi$$
−0.173391 + 0.984853i $$0.555472\pi$$
$$480$$ −2.05333 −0.0937212
$$481$$ 7.78202 0.354830
$$482$$ 14.9357 0.680304
$$483$$ 4.99579 0.227316
$$484$$ −40.1537 −1.82517
$$485$$ 1.82487 0.0828629
$$486$$ 2.48929 0.112916
$$487$$ 4.41033 0.199851 0.0999255 0.994995i $$-0.468140\pi$$
0.0999255 + 0.994995i $$0.468140\pi$$
$$488$$ −68.7215 −3.11088
$$489$$ −8.76060 −0.396168
$$490$$ −13.8610 −0.626175
$$491$$ −0.0856914 −0.00386720 −0.00193360 0.999998i $$-0.500615\pi$$
−0.00193360 + 0.999998i $$0.500615\pi$$
$$492$$ 25.9143 1.16831
$$493$$ 37.0508 1.66868
$$494$$ −17.3717 −0.781589
$$495$$ 1.19656 0.0537813
$$496$$ −15.5422 −0.697863
$$497$$ 6.21798 0.278915
$$498$$ −13.3717 −0.599200
$$499$$ −17.7220 −0.793344 −0.396672 0.917960i $$-0.629835\pi$$
−0.396672 + 0.917960i $$0.629835\pi$$
$$500$$ −4.19656 −0.187676
$$501$$ 17.3717 0.776110
$$502$$ 59.5296 2.65694
$$503$$ −8.70054 −0.387938 −0.193969 0.981008i $$-0.562136\pi$$
−0.193969 + 0.981008i $$0.562136\pi$$
$$504$$ 6.54262 0.291431
$$505$$ 10.3503 0.460581
$$506$$ 12.4360 0.552846
$$507$$ −1.00000 −0.0444116
$$508$$ −43.6153 −1.93512
$$509$$ −33.3545 −1.47841 −0.739206 0.673480i $$-0.764799\pi$$
−0.739206 + 0.673480i $$0.764799\pi$$
$$510$$ −15.3717 −0.680670
$$511$$ 14.3074 0.632923
$$512$$ 46.3482 2.04832
$$513$$ −6.97858 −0.308112
$$514$$ 49.6791 2.19125
$$515$$ −18.7434 −0.825932
$$516$$ 41.7858 1.83952
$$517$$ 1.22219 0.0537519
$$518$$ 23.1793 1.01844
$$519$$ 7.95715 0.349280
$$520$$ 5.46787 0.239782
$$521$$ 18.7005 0.819285 0.409643 0.912246i $$-0.365653\pi$$
0.409643 + 0.912246i $$0.365653\pi$$
$$522$$ −14.9357 −0.653719
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −26.8291 −1.17203
$$525$$ 1.19656 0.0522221
$$526$$ −19.9143 −0.868305
$$527$$ −18.3931 −0.801217
$$528$$ 6.24361 0.271718
$$529$$ −5.56825 −0.242098
$$530$$ 25.3288 1.10021
$$531$$ 5.37169 0.233112
$$532$$ −35.0424 −1.51928
$$533$$ −6.17513 −0.267475
$$534$$ 25.3288 1.09609
$$535$$ 18.5682 0.802775
$$536$$ −51.2432 −2.21337
$$537$$ 15.5640 0.671638
$$538$$ 5.85050 0.252233
$$539$$ 6.66273 0.286984
$$540$$ 4.19656 0.180591
$$541$$ −41.5296 −1.78550 −0.892749 0.450555i $$-0.851226\pi$$
−0.892749 + 0.450555i $$0.851226\pi$$
$$542$$ −27.3288 −1.17387
$$543$$ −15.7820 −0.677271
$$544$$ −12.6796 −0.543632
$$545$$ −8.39312 −0.359522
$$546$$ −2.97858 −0.127471
$$547$$ −7.91431 −0.338391 −0.169196 0.985582i $$-0.554117\pi$$
−0.169196 + 0.985582i $$0.554117\pi$$
$$548$$ 70.2646 3.00155
$$549$$ 12.5682 0.536400
$$550$$ 2.97858 0.127007
$$551$$ 41.8715 1.78378
$$552$$ 22.8291 0.971670
$$553$$ 2.13229 0.0906742
$$554$$ −3.02142 −0.128368
$$555$$ 7.78202 0.330328
$$556$$ 24.2646 1.02905
$$557$$ 42.7005 1.80928 0.904640 0.426177i $$-0.140140\pi$$
0.904640 + 0.426177i $$0.140140\pi$$
$$558$$ 7.41454 0.313882
$$559$$ −9.95715 −0.421143
$$560$$ 6.24361 0.263841
$$561$$ 7.38890 0.311960
$$562$$ −29.7648 −1.25555
$$563$$ 1.04706 0.0441282 0.0220641 0.999757i $$-0.492976\pi$$
0.0220641 + 0.999757i $$0.492976\pi$$
$$564$$ 4.28646 0.180493
$$565$$ −7.95715 −0.334760
$$566$$ 74.3587 3.12553
$$567$$ −1.19656 −0.0502507
$$568$$ 28.4141 1.19223
$$569$$ 16.7778 0.703362 0.351681 0.936120i $$-0.385610\pi$$
0.351681 + 0.936120i $$0.385610\pi$$
$$570$$ −17.3717 −0.727620
$$571$$ −20.6111 −0.862548 −0.431274 0.902221i $$-0.641936\pi$$
−0.431274 + 0.902221i $$0.641936\pi$$
$$572$$ −5.02142 −0.209956
$$573$$ −10.7434 −0.448811
$$574$$ −18.3931 −0.767714
$$575$$ 4.17513 0.174115
$$576$$ −5.32464 −0.221860
$$577$$ 1.38890 0.0578208 0.0289104 0.999582i $$-0.490796\pi$$
0.0289104 + 0.999582i $$0.490796\pi$$
$$578$$ −52.6044 −2.18805
$$579$$ −9.73917 −0.404746
$$580$$ −25.1793 −1.04552
$$581$$ 6.42754 0.266659
$$582$$ −4.54262 −0.188298
$$583$$ −12.1751 −0.504243
$$584$$ 65.3801 2.70545
$$585$$ −1.00000 −0.0413449
$$586$$ −1.93619 −0.0799833
$$587$$ −0.935731 −0.0386218 −0.0193109 0.999814i $$-0.506147\pi$$
−0.0193109 + 0.999814i $$0.506147\pi$$
$$588$$ 23.3675 0.963659
$$589$$ −20.7862 −0.856482
$$590$$ 13.3717 0.550504
$$591$$ 9.56404 0.393412
$$592$$ 40.6064 1.66891
$$593$$ −0.478807 −0.0196622 −0.00983112 0.999952i $$-0.503129\pi$$
−0.00983112 + 0.999952i $$0.503129\pi$$
$$594$$ −2.97858 −0.122213
$$595$$ 7.38890 0.302916
$$596$$ 64.4006 2.63795
$$597$$ −5.95715 −0.243810
$$598$$ −10.3931 −0.425006
$$599$$ 29.0852 1.18839 0.594195 0.804321i $$-0.297471\pi$$
0.594195 + 0.804321i $$0.297471\pi$$
$$600$$ 5.46787 0.223225
$$601$$ 11.4318 0.466311 0.233155 0.972439i $$-0.425095\pi$$
0.233155 + 0.972439i $$0.425095\pi$$
$$602$$ −29.6582 −1.20878
$$603$$ 9.37169 0.381645
$$604$$ −36.0294 −1.46601
$$605$$ 9.56825 0.389005
$$606$$ −25.7648 −1.04662
$$607$$ 27.9143 1.13301 0.566503 0.824059i $$-0.308296\pi$$
0.566503 + 0.824059i $$0.308296\pi$$
$$608$$ −14.3293 −0.581130
$$609$$ 7.17935 0.290922
$$610$$ 31.2860 1.26673
$$611$$ −1.02142 −0.0413223
$$612$$ 25.9143 1.04752
$$613$$ −4.65394 −0.187971 −0.0939855 0.995574i $$-0.529961\pi$$
−0.0939855 + 0.995574i $$0.529961\pi$$
$$614$$ −1.89334 −0.0764092
$$615$$ −6.17513 −0.249005
$$616$$ −7.82862 −0.315424
$$617$$ −15.9572 −0.642411 −0.321205 0.947010i $$-0.604088\pi$$
−0.321205 + 0.947010i $$0.604088\pi$$
$$618$$ 46.6577 1.87685
$$619$$ 1.02142 0.0410545 0.0205272 0.999789i $$-0.493466\pi$$
0.0205272 + 0.999789i $$0.493466\pi$$
$$620$$ 12.4998 0.502003
$$621$$ −4.17513 −0.167542
$$622$$ 57.5725 2.30845
$$623$$ −12.1751 −0.487786
$$624$$ −5.21798 −0.208886
$$625$$ 1.00000 0.0400000
$$626$$ 84.4225 3.37420
$$627$$ 8.35027 0.333478
$$628$$ 11.6926 0.466585
$$629$$ 48.0550 1.91608
$$630$$ −2.97858 −0.118669
$$631$$ −20.4998 −0.816083 −0.408041 0.912963i $$-0.633788\pi$$
−0.408041 + 0.912963i $$0.633788\pi$$
$$632$$ 9.74384 0.387589
$$633$$ −23.9143 −0.950508
$$634$$ −24.0210 −0.953994
$$635$$ 10.3931 0.412438
$$636$$ −42.7005 −1.69319
$$637$$ −5.56825 −0.220622
$$638$$ 17.8715 0.707538
$$639$$ −5.19656 −0.205573
$$640$$ −17.3612 −0.686262
$$641$$ 38.2646 1.51136 0.755680 0.654941i $$-0.227307\pi$$
0.755680 + 0.654941i $$0.227307\pi$$
$$642$$ −46.2217 −1.82423
$$643$$ −33.1109 −1.30577 −0.652883 0.757459i $$-0.726440\pi$$
−0.652883 + 0.757459i $$0.726440\pi$$
$$644$$ −20.9651 −0.826141
$$645$$ −9.95715 −0.392063
$$646$$ −107.273 −4.22058
$$647$$ −16.9614 −0.666820 −0.333410 0.942782i $$-0.608199\pi$$
−0.333410 + 0.942782i $$0.608199\pi$$
$$648$$ −5.46787 −0.214798
$$649$$ −6.42754 −0.252303
$$650$$ −2.48929 −0.0976379
$$651$$ −3.56404 −0.139686
$$652$$ 36.7643 1.43980
$$653$$ −19.1709 −0.750216 −0.375108 0.926981i $$-0.622394\pi$$
−0.375108 + 0.926981i $$0.622394\pi$$
$$654$$ 20.8929 0.816976
$$655$$ 6.39312 0.249800
$$656$$ −32.2217 −1.25805
$$657$$ −11.9572 −0.466493
$$658$$ −3.04239 −0.118605
$$659$$ 37.8715 1.47526 0.737631 0.675204i $$-0.235944\pi$$
0.737631 + 0.675204i $$0.235944\pi$$
$$660$$ −5.02142 −0.195459
$$661$$ 24.3931 0.948782 0.474391 0.880314i $$-0.342668\pi$$
0.474391 + 0.880314i $$0.342668\pi$$
$$662$$ −38.1579 −1.48305
$$663$$ −6.17513 −0.239822
$$664$$ 29.3717 1.13984
$$665$$ 8.35027 0.323810
$$666$$ −19.3717 −0.750638
$$667$$ 25.0508 0.969971
$$668$$ −72.9013 −2.82064
$$669$$ −2.62831 −0.101616
$$670$$ 23.3288 0.901272
$$671$$ −15.0386 −0.580560
$$672$$ −2.45692 −0.0947779
$$673$$ −21.1281 −0.814428 −0.407214 0.913333i $$-0.633500\pi$$
−0.407214 + 0.913333i $$0.633500\pi$$
$$674$$ 55.6363 2.14303
$$675$$ −1.00000 −0.0384900
$$676$$ 4.19656 0.161406
$$677$$ −15.3973 −0.591767 −0.295884 0.955224i $$-0.595614\pi$$
−0.295884 + 0.955224i $$0.595614\pi$$
$$678$$ 19.8077 0.760708
$$679$$ 2.18356 0.0837972
$$680$$ 33.7648 1.29482
$$681$$ −15.7648 −0.604109
$$682$$ −8.87192 −0.339723
$$683$$ 30.0722 1.15068 0.575341 0.817914i $$-0.304869\pi$$
0.575341 + 0.817914i $$0.304869\pi$$
$$684$$ 29.2860 1.11978
$$685$$ −16.7434 −0.639732
$$686$$ −37.4355 −1.42929
$$687$$ −8.74338 −0.333581
$$688$$ −51.9562 −1.98081
$$689$$ 10.1751 0.387642
$$690$$ −10.3931 −0.395659
$$691$$ −8.14950 −0.310022 −0.155011 0.987913i $$-0.549541\pi$$
−0.155011 + 0.987913i $$0.549541\pi$$
$$692$$ −33.3927 −1.26940
$$693$$ 1.43175 0.0543877
$$694$$ 14.3931 0.546355
$$695$$ −5.78202 −0.219325
$$696$$ 32.8072 1.24355
$$697$$ −38.1323 −1.44436
$$698$$ 68.5082 2.59307
$$699$$ 2.17513 0.0822712
$$700$$ −5.02142 −0.189792
$$701$$ −28.6921 −1.08369 −0.541843 0.840480i $$-0.682273\pi$$
−0.541843 + 0.840480i $$0.682273\pi$$
$$702$$ 2.48929 0.0939521
$$703$$ 54.3074 2.04824
$$704$$ 6.37123 0.240125
$$705$$ −1.02142 −0.0384690
$$706$$ 71.5506 2.69284
$$707$$ 12.3847 0.465774
$$708$$ −22.5426 −0.847203
$$709$$ 12.3074 0.462215 0.231108 0.972928i $$-0.425765\pi$$
0.231108 + 0.972928i $$0.425765\pi$$
$$710$$ −12.9357 −0.485469
$$711$$ −1.78202 −0.0668310
$$712$$ −55.6363 −2.08506
$$713$$ −12.4360 −0.465730
$$714$$ −18.3931 −0.688345
$$715$$ 1.19656 0.0447487
$$716$$ −65.3154 −2.44095
$$717$$ −2.80344 −0.104696
$$718$$ 31.3288 1.16918
$$719$$ −28.7862 −1.07355 −0.536773 0.843727i $$-0.680357\pi$$
−0.536773 + 0.843727i $$0.680357\pi$$
$$720$$ −5.21798 −0.194463
$$721$$ −22.4275 −0.835245
$$722$$ −73.9332 −2.75151
$$723$$ 6.00000 0.223142
$$724$$ 66.2302 2.46142
$$725$$ 6.00000 0.222834
$$726$$ −23.8181 −0.883974
$$727$$ 34.3931 1.27557 0.637785 0.770214i $$-0.279851\pi$$
0.637785 + 0.770214i $$0.279851\pi$$
$$728$$ 6.54262 0.242485
$$729$$ 1.00000 0.0370370
$$730$$ −29.7648 −1.10164
$$731$$ −61.4868 −2.27417
$$732$$ −52.7434 −1.94945
$$733$$ −29.0042 −1.07129 −0.535647 0.844442i $$-0.679932\pi$$
−0.535647 + 0.844442i $$0.679932\pi$$
$$734$$ 69.4868 2.56480
$$735$$ −5.56825 −0.205388
$$736$$ −8.57292 −0.316002
$$737$$ −11.2138 −0.413065
$$738$$ 15.3717 0.565840
$$739$$ −6.27804 −0.230941 −0.115471 0.993311i $$-0.536838\pi$$
−0.115471 + 0.993311i $$0.536838\pi$$
$$740$$ −32.6577 −1.20052
$$741$$ −6.97858 −0.256364
$$742$$ 30.3074 1.11262
$$743$$ 12.2352 0.448866 0.224433 0.974490i $$-0.427947\pi$$
0.224433 + 0.974490i $$0.427947\pi$$
$$744$$ −16.2865 −0.597091
$$745$$ −15.3461 −0.562236
$$746$$ 5.85050 0.214202
$$747$$ −5.37169 −0.196540
$$748$$ −31.0080 −1.13376
$$749$$ 22.2180 0.811827
$$750$$ −2.48929 −0.0908960
$$751$$ −28.8757 −1.05369 −0.526844 0.849962i $$-0.676625\pi$$
−0.526844 + 0.849962i $$0.676625\pi$$
$$752$$ −5.32976 −0.194357
$$753$$ 23.9143 0.871486
$$754$$ −14.9357 −0.543927
$$755$$ 8.58546 0.312457
$$756$$ 5.02142 0.182627
$$757$$ 30.3503 1.10310 0.551550 0.834142i $$-0.314037\pi$$
0.551550 + 0.834142i $$0.314037\pi$$
$$758$$ −61.1146 −2.21978
$$759$$ 4.99579 0.181336
$$760$$ 38.1579 1.38413
$$761$$ −27.1709 −0.984945 −0.492473 0.870328i $$-0.663907\pi$$
−0.492473 + 0.870328i $$0.663907\pi$$
$$762$$ −25.8715 −0.937224
$$763$$ −10.0428 −0.363575
$$764$$ 45.0852 1.63113
$$765$$ −6.17513 −0.223262
$$766$$ −14.4569 −0.522350
$$767$$ 5.37169 0.193961
$$768$$ 32.5678 1.17519
$$769$$ −38.3503 −1.38295 −0.691473 0.722402i $$-0.743038\pi$$
−0.691473 + 0.722402i $$0.743038\pi$$
$$770$$ 3.56404 0.128439
$$771$$ 19.9572 0.718739
$$772$$ 40.8710 1.47098
$$773$$ −50.2646 −1.80789 −0.903946 0.427647i $$-0.859342\pi$$
−0.903946 + 0.427647i $$0.859342\pi$$
$$774$$ 24.7862 0.890923
$$775$$ −2.97858 −0.106994
$$776$$ 9.97812 0.358194
$$777$$ 9.31163 0.334053
$$778$$ −33.9781 −1.21817
$$779$$ −43.0937 −1.54399
$$780$$ 4.19656 0.150261
$$781$$ 6.21798 0.222497
$$782$$ −64.1789 −2.29503
$$783$$ −6.00000 −0.214423
$$784$$ −29.0550 −1.03768
$$785$$ −2.78623 −0.0994448
$$786$$ −15.9143 −0.567645
$$787$$ −13.2860 −0.473595 −0.236797 0.971559i $$-0.576098\pi$$
−0.236797 + 0.971559i $$0.576098\pi$$
$$788$$ −40.1360 −1.42979
$$789$$ −8.00000 −0.284808
$$790$$ −4.43596 −0.157824
$$791$$ −9.52119 −0.338535
$$792$$ 6.54262 0.232482
$$793$$ 12.5682 0.446312
$$794$$ 30.2008 1.07179
$$795$$ 10.1751 0.360875
$$796$$ 24.9995 0.886085
$$797$$ 9.82487 0.348015 0.174007 0.984744i $$-0.444328\pi$$
0.174007 + 0.984744i $$0.444328\pi$$
$$798$$ −20.7862 −0.735825
$$799$$ −6.30742 −0.223141
$$800$$ −2.05333 −0.0725961
$$801$$ 10.1751 0.359521
$$802$$ 93.2087 3.29131
$$803$$ 14.3074 0.504898
$$804$$ −39.3288 −1.38702
$$805$$ 4.99579 0.176078
$$806$$ 7.41454 0.261166
$$807$$ 2.35027 0.0827334
$$808$$ 56.5939 1.99097
$$809$$ −9.91431 −0.348569 −0.174284 0.984695i $$-0.555761\pi$$
−0.174284 + 0.984695i $$0.555761\pi$$
$$810$$ 2.48929 0.0874647
$$811$$ −36.5855 −1.28469 −0.642345 0.766416i $$-0.722038\pi$$
−0.642345 + 0.766416i $$0.722038\pi$$
$$812$$ −30.1285 −1.05730
$$813$$ −10.9786 −0.385036
$$814$$ 23.1793 0.812436
$$815$$ −8.76060 −0.306870
$$816$$ −32.2217 −1.12799
$$817$$ −69.4868 −2.43103
$$818$$ 34.8500 1.21850
$$819$$ −1.19656 −0.0418111
$$820$$ 25.9143 0.904967
$$821$$ −34.4741 −1.20316 −0.601578 0.798814i $$-0.705461\pi$$
−0.601578 + 0.798814i $$0.705461\pi$$
$$822$$ 41.6791 1.45373
$$823$$ 13.2566 0.462097 0.231048 0.972942i $$-0.425784\pi$$
0.231048 + 0.972942i $$0.425784\pi$$
$$824$$ −102.486 −3.57028
$$825$$ 1.19656 0.0416588
$$826$$ 16.0000 0.556711
$$827$$ 28.1495 0.978854 0.489427 0.872044i $$-0.337206\pi$$
0.489427 + 0.872044i $$0.337206\pi$$
$$828$$ 17.5212 0.608904
$$829$$ 16.3418 0.567576 0.283788 0.958887i $$-0.408409\pi$$
0.283788 + 0.958887i $$0.408409\pi$$
$$830$$ −13.3717 −0.464138
$$831$$ −1.21377 −0.0421052
$$832$$ −5.32464 −0.184599
$$833$$ −34.3847 −1.19136
$$834$$ 14.3931 0.498393
$$835$$ 17.3717 0.601172
$$836$$ −35.0424 −1.21197
$$837$$ 2.97858 0.102955
$$838$$ −7.91431 −0.273395
$$839$$ −30.3675 −1.04840 −0.524201 0.851595i $$-0.675636\pi$$
−0.524201 + 0.851595i $$0.675636\pi$$
$$840$$ 6.54262 0.225742
$$841$$ 7.00000 0.241379
$$842$$ 40.5939 1.39896
$$843$$ −11.9572 −0.411826
$$844$$ 100.358 3.45446
$$845$$ −1.00000 −0.0344010
$$846$$ 2.54262 0.0874169
$$847$$ 11.4490 0.393391
$$848$$ 53.0937 1.82324
$$849$$ 29.8715 1.02519
$$850$$ −15.3717 −0.527245
$$851$$ 32.4910 1.11378
$$852$$ 21.8077 0.747118
$$853$$ −42.1407 −1.44287 −0.721435 0.692482i $$-0.756517\pi$$
−0.721435 + 0.692482i $$0.756517\pi$$
$$854$$ 37.4355 1.28102
$$855$$ −6.97858 −0.238662
$$856$$ 101.529 3.47018
$$857$$ −2.17513 −0.0743012 −0.0371506 0.999310i $$-0.511828\pi$$
−0.0371506 + 0.999310i $$0.511828\pi$$
$$858$$ −2.97858 −0.101687
$$859$$ 18.5682 0.633541 0.316770 0.948502i $$-0.397402\pi$$
0.316770 + 0.948502i $$0.397402\pi$$
$$860$$ 41.7858 1.42488
$$861$$ −7.38890 −0.251813
$$862$$ −11.4145 −0.388781
$$863$$ 33.7220 1.14791 0.573954 0.818887i $$-0.305409\pi$$
0.573954 + 0.818887i $$0.305409\pi$$
$$864$$ 2.05333 0.0698556
$$865$$ 7.95715 0.270551
$$866$$ 95.4649 3.24403
$$867$$ −21.1323 −0.717690
$$868$$ 14.9567 0.507663
$$869$$ 2.13229 0.0723330
$$870$$ −14.9357 −0.506369
$$871$$ 9.37169 0.317548
$$872$$ −45.8924 −1.55411
$$873$$ −1.82487 −0.0617623
$$874$$ −72.5292 −2.45334
$$875$$ 1.19656 0.0404510
$$876$$ 50.1789 1.69539
$$877$$ 43.4868 1.46844 0.734222 0.678910i $$-0.237547\pi$$
0.734222 + 0.678910i $$0.237547\pi$$
$$878$$ −19.2650 −0.650164
$$879$$ −0.777809 −0.0262348
$$880$$ 6.24361 0.210472
$$881$$ 26.7005 0.899564 0.449782 0.893138i $$-0.351502\pi$$
0.449782 + 0.893138i $$0.351502\pi$$
$$882$$ 13.8610 0.466724
$$883$$ 20.2990 0.683116 0.341558 0.939861i $$-0.389045\pi$$
0.341558 + 0.939861i $$0.389045\pi$$
$$884$$ 25.9143 0.871593
$$885$$ 5.37169 0.180567
$$886$$ −86.9223 −2.92021
$$887$$ 36.0550 1.21061 0.605305 0.795994i $$-0.293051\pi$$
0.605305 + 0.795994i $$0.293051\pi$$
$$888$$ 42.5510 1.42792
$$889$$ 12.4360 0.417089
$$890$$ 25.3288 0.849025
$$891$$ −1.19656 −0.0400862
$$892$$ 11.0298 0.369307
$$893$$ −7.12808 −0.238532
$$894$$ 38.2008 1.27762
$$895$$ 15.5640 0.520248
$$896$$ −20.7737 −0.694000
$$897$$ −4.17513 −0.139404
$$898$$ 15.3717 0.512960
$$899$$ −17.8715 −0.596047
$$900$$ 4.19656 0.139885
$$901$$ 62.8328 2.09326
$$902$$ −18.3931 −0.612424
$$903$$ −11.9143 −0.396483
$$904$$ −43.5087 −1.44708
$$905$$ −15.7820 −0.524612
$$906$$ −21.3717 −0.710027
$$907$$ −7.26504 −0.241232 −0.120616 0.992699i $$-0.538487\pi$$
−0.120616 + 0.992699i $$0.538487\pi$$
$$908$$ 66.1579 2.19553
$$909$$ −10.3503 −0.343297
$$910$$ −2.97858 −0.0987389
$$911$$ −6.65769 −0.220579 −0.110290 0.993899i $$-0.535178\pi$$
−0.110290 + 0.993899i $$0.535178\pi$$
$$912$$ −36.4141 −1.20579
$$913$$ 6.42754 0.212721
$$914$$ −3.45738 −0.114360
$$915$$ 12.5682 0.415494
$$916$$ 36.6921 1.21234
$$917$$ 7.64973 0.252616
$$918$$ 15.3717 0.507341
$$919$$ −27.1831 −0.896688 −0.448344 0.893861i $$-0.647986\pi$$
−0.448344 + 0.893861i $$0.647986\pi$$
$$920$$ 22.8291 0.752652
$$921$$ −0.760597 −0.0250625
$$922$$ −70.9013 −2.33501
$$923$$ −5.19656 −0.171047
$$924$$ −6.00842 −0.197663
$$925$$ 7.78202 0.255871
$$926$$ −40.7643 −1.33960
$$927$$ 18.7434 0.615614
$$928$$ −12.3200 −0.404423
$$929$$ −15.3973 −0.505170 −0.252585 0.967575i $$-0.581281\pi$$
−0.252585 + 0.967575i $$0.581281\pi$$
$$930$$ 7.41454 0.243132
$$931$$ −38.8585 −1.27353
$$932$$ −9.12808 −0.299000
$$933$$ 23.1281 0.757179
$$934$$ −64.0932 −2.09719
$$935$$ 7.38890 0.241643
$$936$$ −5.46787 −0.178723
$$937$$ 1.12808 0.0368527 0.0184264 0.999830i $$-0.494134\pi$$
0.0184264 + 0.999830i $$0.494134\pi$$
$$938$$ 27.9143 0.911434
$$939$$ 33.9143 1.10675
$$940$$ 4.28646 0.139809
$$941$$ −30.1407 −0.982559 −0.491280 0.871002i $$-0.663471\pi$$
−0.491280 + 0.871002i $$0.663471\pi$$
$$942$$ 6.93573 0.225978
$$943$$ −25.7820 −0.839578
$$944$$ 28.0294 0.912279
$$945$$ −1.19656 −0.0389240
$$946$$ −29.6582 −0.964270
$$947$$ −20.0294 −0.650868 −0.325434 0.945565i $$-0.605510\pi$$
−0.325434 + 0.945565i $$0.605510\pi$$
$$948$$ 7.47835 0.242885
$$949$$ −11.9572 −0.388146
$$950$$ −17.3717 −0.563612
$$951$$ −9.64973 −0.312914
$$952$$ 40.4015 1.30942
$$953$$ −43.2259 −1.40023 −0.700113 0.714032i $$-0.746867\pi$$
−0.700113 + 0.714032i $$0.746867\pi$$
$$954$$ −25.3288 −0.820052
$$955$$ −10.7434 −0.347648
$$956$$ 11.7648 0.380501
$$957$$ 7.17935 0.232075
$$958$$ 18.8929 0.610401
$$959$$ −20.0344 −0.646945
$$960$$ −5.32464 −0.171852
$$961$$ −22.1281 −0.713809
$$962$$ −19.3717 −0.624568
$$963$$ −18.5682 −0.598353
$$964$$ −25.1793 −0.810972
$$965$$ −9.73917 −0.313515
$$966$$ −12.4360 −0.400120
$$967$$ 57.6875 1.85511 0.927553 0.373691i $$-0.121908\pi$$
0.927553 + 0.373691i $$0.121908\pi$$
$$968$$ 52.3179 1.68156
$$969$$ −43.0937 −1.38437
$$970$$ −4.54262 −0.145855
$$971$$ −19.5296 −0.626735 −0.313368 0.949632i $$-0.601457\pi$$
−0.313368 + 0.949632i $$0.601457\pi$$
$$972$$ −4.19656 −0.134605
$$973$$ −6.91852 −0.221798
$$974$$ −10.9786 −0.351776
$$975$$ −1.00000 −0.0320256
$$976$$ 65.5809 2.09919
$$977$$ −40.3074 −1.28955 −0.644774 0.764373i $$-0.723049\pi$$
−0.644774 + 0.764373i $$0.723049\pi$$
$$978$$ 21.8077 0.697332
$$979$$ −12.1751 −0.389119
$$980$$ 23.3675 0.746447
$$981$$ 8.39312 0.267972
$$982$$ 0.213311 0.00680702
$$983$$ 32.2008 1.02705 0.513523 0.858076i $$-0.328340\pi$$
0.513523 + 0.858076i $$0.328340\pi$$
$$984$$ −33.7648 −1.07638
$$985$$ 9.56404 0.304736
$$986$$ −92.2302 −2.93721
$$987$$ −1.22219 −0.0389028
$$988$$ 29.2860 0.931712
$$989$$ −41.5725 −1.32193
$$990$$ −2.97858 −0.0946654
$$991$$ 26.4826 0.841246 0.420623 0.907235i $$-0.361811\pi$$
0.420623 + 0.907235i $$0.361811\pi$$
$$992$$ 6.11599 0.194183
$$993$$ −15.3288 −0.486446
$$994$$ −15.4783 −0.490943
$$995$$ −5.95715 −0.188854
$$996$$ 22.5426 0.714290
$$997$$ 35.1365 1.11278 0.556392 0.830920i $$-0.312185\pi$$
0.556392 + 0.830920i $$0.312185\pi$$
$$998$$ 44.1151 1.39644
$$999$$ −7.78202 −0.246212
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 195.2.a.e.1.1 3
3.2 odd 2 585.2.a.n.1.3 3
4.3 odd 2 3120.2.a.bj.1.2 3
5.2 odd 4 975.2.c.i.274.2 6
5.3 odd 4 975.2.c.i.274.5 6
5.4 even 2 975.2.a.o.1.3 3
7.6 odd 2 9555.2.a.bq.1.1 3
12.11 even 2 9360.2.a.dd.1.2 3
13.12 even 2 2535.2.a.bc.1.3 3
15.2 even 4 2925.2.c.w.2224.5 6
15.8 even 4 2925.2.c.w.2224.2 6
15.14 odd 2 2925.2.a.bh.1.1 3
39.38 odd 2 7605.2.a.bx.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.1 3 1.1 even 1 trivial
585.2.a.n.1.3 3 3.2 odd 2
975.2.a.o.1.3 3 5.4 even 2
975.2.c.i.274.2 6 5.2 odd 4
975.2.c.i.274.5 6 5.3 odd 4
2535.2.a.bc.1.3 3 13.12 even 2
2925.2.a.bh.1.1 3 15.14 odd 2
2925.2.c.w.2224.2 6 15.8 even 4
2925.2.c.w.2224.5 6 15.2 even 4
3120.2.a.bj.1.2 3 4.3 odd 2
7605.2.a.bx.1.1 3 39.38 odd 2
9360.2.a.dd.1.2 3 12.11 even 2
9555.2.a.bq.1.1 3 7.6 odd 2