# Properties

 Label 2394.2.a.x.1.2 Level $2394$ Weight $2$ Character 2394.1 Self dual yes Analytic conductor $19.116$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 798) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 2394.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} +5.46410 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.46410 q^{17} +1.00000 q^{19} +3.46410 q^{20} +5.46410 q^{22} +5.46410 q^{23} +7.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} +6.00000 q^{29} +6.92820 q^{31} +1.00000 q^{32} -7.46410 q^{34} -3.46410 q^{35} +10.0000 q^{37} +1.00000 q^{38} +3.46410 q^{40} +2.00000 q^{41} -6.92820 q^{43} +5.46410 q^{44} +5.46410 q^{46} -10.9282 q^{47} +1.00000 q^{49} +7.00000 q^{50} -3.46410 q^{52} -2.00000 q^{53} +18.9282 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -4.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -12.0000 q^{65} +9.46410 q^{67} -7.46410 q^{68} -3.46410 q^{70} -2.92820 q^{71} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -5.46410 q^{77} -12.3923 q^{79} +3.46410 q^{80} +2.00000 q^{82} -8.00000 q^{83} -25.8564 q^{85} -6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +3.46410 q^{91} +5.46410 q^{92} -10.9282 q^{94} +3.46410 q^{95} +4.53590 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 14 * q^25 - 2 * q^28 + 12 * q^29 + 2 * q^32 - 8 * q^34 + 20 * q^37 + 2 * q^38 + 4 * q^41 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 14 * q^50 - 4 * q^53 + 24 * q^55 - 2 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 - 24 * q^65 + 12 * q^67 - 8 * q^68 + 8 * q^71 + 20 * q^73 + 20 * q^74 + 2 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^82 - 16 * q^83 - 24 * q^85 + 4 * q^88 + 4 * q^89 + 4 * q^92 - 8 * q^94 + 16 * q^97 + 2 * q^98

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 3.46410 1.54919 0.774597 0.632456i $$-0.217953\pi$$
0.774597 + 0.632456i $$0.217953\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 3.46410 1.09545
$$11$$ 5.46410 1.64749 0.823744 0.566961i $$-0.191881\pi$$
0.823744 + 0.566961i $$0.191881\pi$$
$$12$$ 0 0
$$13$$ −3.46410 −0.960769 −0.480384 0.877058i $$-0.659503\pi$$
−0.480384 + 0.877058i $$0.659503\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −7.46410 −1.81031 −0.905155 0.425081i $$-0.860246\pi$$
−0.905155 + 0.425081i $$0.860246\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 3.46410 0.774597
$$21$$ 0 0
$$22$$ 5.46410 1.16495
$$23$$ 5.46410 1.13934 0.569672 0.821872i $$-0.307070\pi$$
0.569672 + 0.821872i $$0.307070\pi$$
$$24$$ 0 0
$$25$$ 7.00000 1.40000
$$26$$ −3.46410 −0.679366
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 6.92820 1.24434 0.622171 0.782881i $$-0.286251\pi$$
0.622171 + 0.782881i $$0.286251\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −7.46410 −1.28008
$$35$$ −3.46410 −0.585540
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 3.46410 0.547723
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −6.92820 −1.05654 −0.528271 0.849076i $$-0.677159\pi$$
−0.528271 + 0.849076i $$0.677159\pi$$
$$44$$ 5.46410 0.823744
$$45$$ 0 0
$$46$$ 5.46410 0.805638
$$47$$ −10.9282 −1.59404 −0.797021 0.603951i $$-0.793592\pi$$
−0.797021 + 0.603951i $$0.793592\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 7.00000 0.989949
$$51$$ 0 0
$$52$$ −3.46410 −0.480384
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 18.9282 2.55228
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −4.92820 −0.630992 −0.315496 0.948927i $$-0.602171\pi$$
−0.315496 + 0.948927i $$0.602171\pi$$
$$62$$ 6.92820 0.879883
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ 9.46410 1.15622 0.578112 0.815957i $$-0.303790\pi$$
0.578112 + 0.815957i $$0.303790\pi$$
$$68$$ −7.46410 −0.905155
$$69$$ 0 0
$$70$$ −3.46410 −0.414039
$$71$$ −2.92820 −0.347514 −0.173757 0.984789i $$-0.555591\pi$$
−0.173757 + 0.984789i $$0.555591\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −5.46410 −0.622692
$$78$$ 0 0
$$79$$ −12.3923 −1.39424 −0.697122 0.716953i $$-0.745536\pi$$
−0.697122 + 0.716953i $$0.745536\pi$$
$$80$$ 3.46410 0.387298
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ −25.8564 −2.80452
$$86$$ −6.92820 −0.747087
$$87$$ 0 0
$$88$$ 5.46410 0.582475
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 3.46410 0.363137
$$92$$ 5.46410 0.569672
$$93$$ 0 0
$$94$$ −10.9282 −1.12716
$$95$$ 3.46410 0.355409
$$96$$ 0 0
$$97$$ 4.53590 0.460551 0.230275 0.973126i $$-0.426037\pi$$
0.230275 + 0.973126i $$0.426037\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 7.00000 0.700000
$$101$$ −4.53590 −0.451339 −0.225669 0.974204i $$-0.572457\pi$$
−0.225669 + 0.974204i $$0.572457\pi$$
$$102$$ 0 0
$$103$$ −6.92820 −0.682656 −0.341328 0.939944i $$-0.610877\pi$$
−0.341328 + 0.939944i $$0.610877\pi$$
$$104$$ −3.46410 −0.339683
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 18.9282 1.80473
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −11.8564 −1.11536 −0.557678 0.830057i $$-0.688308\pi$$
−0.557678 + 0.830057i $$0.688308\pi$$
$$114$$ 0 0
$$115$$ 18.9282 1.76506
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 7.46410 0.684233
$$120$$ 0 0
$$121$$ 18.8564 1.71422
$$122$$ −4.92820 −0.446179
$$123$$ 0 0
$$124$$ 6.92820 0.622171
$$125$$ 6.92820 0.619677
$$126$$ 0 0
$$127$$ −17.4641 −1.54969 −0.774844 0.632152i $$-0.782172\pi$$
−0.774844 + 0.632152i $$0.782172\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −12.0000 −1.05247
$$131$$ 5.07180 0.443125 0.221562 0.975146i $$-0.428884\pi$$
0.221562 + 0.975146i $$0.428884\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 9.46410 0.817574
$$135$$ 0 0
$$136$$ −7.46410 −0.640041
$$137$$ 0.928203 0.0793018 0.0396509 0.999214i $$-0.487375\pi$$
0.0396509 + 0.999214i $$0.487375\pi$$
$$138$$ 0 0
$$139$$ 9.85641 0.836009 0.418005 0.908445i $$-0.362730\pi$$
0.418005 + 0.908445i $$0.362730\pi$$
$$140$$ −3.46410 −0.292770
$$141$$ 0 0
$$142$$ −2.92820 −0.245729
$$143$$ −18.9282 −1.58286
$$144$$ 0 0
$$145$$ 20.7846 1.72607
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ 3.07180 0.251651 0.125826 0.992052i $$-0.459842\pi$$
0.125826 + 0.992052i $$0.459842\pi$$
$$150$$ 0 0
$$151$$ 1.46410 0.119147 0.0595734 0.998224i $$-0.481026\pi$$
0.0595734 + 0.998224i $$0.481026\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ −5.46410 −0.440310
$$155$$ 24.0000 1.92773
$$156$$ 0 0
$$157$$ 19.8564 1.58471 0.792357 0.610058i $$-0.208854\pi$$
0.792357 + 0.610058i $$0.208854\pi$$
$$158$$ −12.3923 −0.985879
$$159$$ 0 0
$$160$$ 3.46410 0.273861
$$161$$ −5.46410 −0.430632
$$162$$ 0 0
$$163$$ 1.07180 0.0839496 0.0419748 0.999119i $$-0.486635\pi$$
0.0419748 + 0.999119i $$0.486635\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 18.9282 1.46471 0.732354 0.680924i $$-0.238422\pi$$
0.732354 + 0.680924i $$0.238422\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ −25.8564 −1.98310
$$171$$ 0 0
$$172$$ −6.92820 −0.528271
$$173$$ 0.928203 0.0705700 0.0352850 0.999377i $$-0.488766\pi$$
0.0352850 + 0.999377i $$0.488766\pi$$
$$174$$ 0 0
$$175$$ −7.00000 −0.529150
$$176$$ 5.46410 0.411872
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 9.85641 0.736702 0.368351 0.929687i $$-0.379922\pi$$
0.368351 + 0.929687i $$0.379922\pi$$
$$180$$ 0 0
$$181$$ −8.53590 −0.634468 −0.317234 0.948347i $$-0.602754\pi$$
−0.317234 + 0.948347i $$0.602754\pi$$
$$182$$ 3.46410 0.256776
$$183$$ 0 0
$$184$$ 5.46410 0.402819
$$185$$ 34.6410 2.54686
$$186$$ 0 0
$$187$$ −40.7846 −2.98247
$$188$$ −10.9282 −0.797021
$$189$$ 0 0
$$190$$ 3.46410 0.251312
$$191$$ 5.46410 0.395369 0.197684 0.980266i $$-0.436658\pi$$
0.197684 + 0.980266i $$0.436658\pi$$
$$192$$ 0 0
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ 4.53590 0.325659
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −4.92820 −0.351120 −0.175560 0.984469i $$-0.556174\pi$$
−0.175560 + 0.984469i $$0.556174\pi$$
$$198$$ 0 0
$$199$$ −24.7846 −1.75693 −0.878467 0.477803i $$-0.841433\pi$$
−0.878467 + 0.477803i $$0.841433\pi$$
$$200$$ 7.00000 0.494975
$$201$$ 0 0
$$202$$ −4.53590 −0.319145
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 6.92820 0.483887
$$206$$ −6.92820 −0.482711
$$207$$ 0 0
$$208$$ −3.46410 −0.240192
$$209$$ 5.46410 0.377960
$$210$$ 0 0
$$211$$ 1.46410 0.100793 0.0503965 0.998729i $$-0.483952\pi$$
0.0503965 + 0.998729i $$0.483952\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ −24.0000 −1.63679
$$216$$ 0 0
$$217$$ −6.92820 −0.470317
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 18.9282 1.27614
$$221$$ 25.8564 1.73929
$$222$$ 0 0
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −11.8564 −0.788676
$$227$$ −14.9282 −0.990820 −0.495410 0.868659i $$-0.664982\pi$$
−0.495410 + 0.868659i $$0.664982\pi$$
$$228$$ 0 0
$$229$$ 3.07180 0.202990 0.101495 0.994836i $$-0.467637\pi$$
0.101495 + 0.994836i $$0.467637\pi$$
$$230$$ 18.9282 1.24809
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −26.7846 −1.75472 −0.877359 0.479834i $$-0.840697\pi$$
−0.877359 + 0.479834i $$0.840697\pi$$
$$234$$ 0 0
$$235$$ −37.8564 −2.46948
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 7.46410 0.483826
$$239$$ −30.2487 −1.95663 −0.978313 0.207131i $$-0.933587\pi$$
−0.978313 + 0.207131i $$0.933587\pi$$
$$240$$ 0 0
$$241$$ 21.3205 1.37337 0.686687 0.726953i $$-0.259064\pi$$
0.686687 + 0.726953i $$0.259064\pi$$
$$242$$ 18.8564 1.21214
$$243$$ 0 0
$$244$$ −4.92820 −0.315496
$$245$$ 3.46410 0.221313
$$246$$ 0 0
$$247$$ −3.46410 −0.220416
$$248$$ 6.92820 0.439941
$$249$$ 0 0
$$250$$ 6.92820 0.438178
$$251$$ −5.07180 −0.320129 −0.160064 0.987107i $$-0.551170\pi$$
−0.160064 + 0.987107i $$0.551170\pi$$
$$252$$ 0 0
$$253$$ 29.8564 1.87706
$$254$$ −17.4641 −1.09580
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.9282 −1.55498 −0.777489 0.628896i $$-0.783507\pi$$
−0.777489 + 0.628896i $$0.783507\pi$$
$$258$$ 0 0
$$259$$ −10.0000 −0.621370
$$260$$ −12.0000 −0.744208
$$261$$ 0 0
$$262$$ 5.07180 0.313337
$$263$$ −11.3205 −0.698052 −0.349026 0.937113i $$-0.613488\pi$$
−0.349026 + 0.937113i $$0.613488\pi$$
$$264$$ 0 0
$$265$$ −6.92820 −0.425596
$$266$$ −1.00000 −0.0613139
$$267$$ 0 0
$$268$$ 9.46410 0.578112
$$269$$ 16.9282 1.03213 0.516065 0.856549i $$-0.327396\pi$$
0.516065 + 0.856549i $$0.327396\pi$$
$$270$$ 0 0
$$271$$ −29.8564 −1.81365 −0.906824 0.421510i $$-0.861500\pi$$
−0.906824 + 0.421510i $$0.861500\pi$$
$$272$$ −7.46410 −0.452578
$$273$$ 0 0
$$274$$ 0.928203 0.0560748
$$275$$ 38.2487 2.30648
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 9.85641 0.591148
$$279$$ 0 0
$$280$$ −3.46410 −0.207020
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ −2.92820 −0.173757
$$285$$ 0 0
$$286$$ −18.9282 −1.11925
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ 38.7128 2.27722
$$290$$ 20.7846 1.22051
$$291$$ 0 0
$$292$$ 10.0000 0.585206
$$293$$ −7.07180 −0.413139 −0.206569 0.978432i $$-0.566230\pi$$
−0.206569 + 0.978432i $$0.566230\pi$$
$$294$$ 0 0
$$295$$ −13.8564 −0.806751
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ 3.07180 0.177944
$$299$$ −18.9282 −1.09465
$$300$$ 0 0
$$301$$ 6.92820 0.399335
$$302$$ 1.46410 0.0842496
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ −17.0718 −0.977528
$$306$$ 0 0
$$307$$ 20.7846 1.18624 0.593120 0.805114i $$-0.297896\pi$$
0.593120 + 0.805114i $$0.297896\pi$$
$$308$$ −5.46410 −0.311346
$$309$$ 0 0
$$310$$ 24.0000 1.36311
$$311$$ −2.14359 −0.121552 −0.0607760 0.998151i $$-0.519358\pi$$
−0.0607760 + 0.998151i $$0.519358\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 19.8564 1.12056
$$315$$ 0 0
$$316$$ −12.3923 −0.697122
$$317$$ −7.85641 −0.441260 −0.220630 0.975358i $$-0.570811\pi$$
−0.220630 + 0.975358i $$0.570811\pi$$
$$318$$ 0 0
$$319$$ 32.7846 1.83559
$$320$$ 3.46410 0.193649
$$321$$ 0 0
$$322$$ −5.46410 −0.304502
$$323$$ −7.46410 −0.415314
$$324$$ 0 0
$$325$$ −24.2487 −1.34508
$$326$$ 1.07180 0.0593613
$$327$$ 0 0
$$328$$ 2.00000 0.110432
$$329$$ 10.9282 0.602491
$$330$$ 0 0
$$331$$ 14.5359 0.798965 0.399483 0.916741i $$-0.369190\pi$$
0.399483 + 0.916741i $$0.369190\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 0 0
$$334$$ 18.9282 1.03571
$$335$$ 32.7846 1.79121
$$336$$ 0 0
$$337$$ 10.7846 0.587475 0.293738 0.955886i $$-0.405101\pi$$
0.293738 + 0.955886i $$0.405101\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ −25.8564 −1.40226
$$341$$ 37.8564 2.05004
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −6.92820 −0.373544
$$345$$ 0 0
$$346$$ 0.928203 0.0499005
$$347$$ −27.3205 −1.46664 −0.733321 0.679883i $$-0.762031\pi$$
−0.733321 + 0.679883i $$0.762031\pi$$
$$348$$ 0 0
$$349$$ 35.8564 1.91935 0.959675 0.281113i $$-0.0907035\pi$$
0.959675 + 0.281113i $$0.0907035\pi$$
$$350$$ −7.00000 −0.374166
$$351$$ 0 0
$$352$$ 5.46410 0.291238
$$353$$ −5.32051 −0.283182 −0.141591 0.989925i $$-0.545222\pi$$
−0.141591 + 0.989925i $$0.545222\pi$$
$$354$$ 0 0
$$355$$ −10.1436 −0.538366
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 9.85641 0.520927
$$359$$ −35.3205 −1.86415 −0.932073 0.362272i $$-0.882001\pi$$
−0.932073 + 0.362272i $$0.882001\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −8.53590 −0.448637
$$363$$ 0 0
$$364$$ 3.46410 0.181568
$$365$$ 34.6410 1.81319
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 5.46410 0.284836
$$369$$ 0 0
$$370$$ 34.6410 1.80090
$$371$$ 2.00000 0.103835
$$372$$ 0 0
$$373$$ 10.7846 0.558406 0.279203 0.960232i $$-0.409930\pi$$
0.279203 + 0.960232i $$0.409930\pi$$
$$374$$ −40.7846 −2.10892
$$375$$ 0 0
$$376$$ −10.9282 −0.563579
$$377$$ −20.7846 −1.07046
$$378$$ 0 0
$$379$$ −1.46410 −0.0752058 −0.0376029 0.999293i $$-0.511972\pi$$
−0.0376029 + 0.999293i $$0.511972\pi$$
$$380$$ 3.46410 0.177705
$$381$$ 0 0
$$382$$ 5.46410 0.279568
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ −18.9282 −0.964671
$$386$$ −22.0000 −1.11977
$$387$$ 0 0
$$388$$ 4.53590 0.230275
$$389$$ −29.7128 −1.50650 −0.753250 0.657735i $$-0.771515\pi$$
−0.753250 + 0.657735i $$0.771515\pi$$
$$390$$ 0 0
$$391$$ −40.7846 −2.06257
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ −4.92820 −0.248279
$$395$$ −42.9282 −2.15995
$$396$$ 0 0
$$397$$ 19.8564 0.996564 0.498282 0.867015i $$-0.333964\pi$$
0.498282 + 0.867015i $$0.333964\pi$$
$$398$$ −24.7846 −1.24234
$$399$$ 0 0
$$400$$ 7.00000 0.350000
$$401$$ −19.8564 −0.991582 −0.495791 0.868442i $$-0.665122\pi$$
−0.495791 + 0.868442i $$0.665122\pi$$
$$402$$ 0 0
$$403$$ −24.0000 −1.19553
$$404$$ −4.53590 −0.225669
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ 54.6410 2.70845
$$408$$ 0 0
$$409$$ −31.1769 −1.54160 −0.770800 0.637078i $$-0.780143\pi$$
−0.770800 + 0.637078i $$0.780143\pi$$
$$410$$ 6.92820 0.342160
$$411$$ 0 0
$$412$$ −6.92820 −0.341328
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ −27.7128 −1.36037
$$416$$ −3.46410 −0.169842
$$417$$ 0 0
$$418$$ 5.46410 0.267258
$$419$$ −18.9282 −0.924703 −0.462352 0.886697i $$-0.652994\pi$$
−0.462352 + 0.886697i $$0.652994\pi$$
$$420$$ 0 0
$$421$$ 18.7846 0.915506 0.457753 0.889079i $$-0.348654\pi$$
0.457753 + 0.889079i $$0.348654\pi$$
$$422$$ 1.46410 0.0712714
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ −52.2487 −2.53443
$$426$$ 0 0
$$427$$ 4.92820 0.238492
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −24.0000 −1.15738
$$431$$ 34.9282 1.68243 0.841216 0.540699i $$-0.181840\pi$$
0.841216 + 0.540699i $$0.181840\pi$$
$$432$$ 0 0
$$433$$ 6.67949 0.320996 0.160498 0.987036i $$-0.448690\pi$$
0.160498 + 0.987036i $$0.448690\pi$$
$$434$$ −6.92820 −0.332564
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 5.46410 0.261383
$$438$$ 0 0
$$439$$ −4.78461 −0.228357 −0.114178 0.993460i $$-0.536424\pi$$
−0.114178 + 0.993460i $$0.536424\pi$$
$$440$$ 18.9282 0.902367
$$441$$ 0 0
$$442$$ 25.8564 1.22986
$$443$$ 11.3205 0.537854 0.268927 0.963161i $$-0.413331\pi$$
0.268927 + 0.963161i $$0.413331\pi$$
$$444$$ 0 0
$$445$$ 6.92820 0.328428
$$446$$ −12.0000 −0.568216
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 7.85641 0.370767 0.185383 0.982666i $$-0.440647\pi$$
0.185383 + 0.982666i $$0.440647\pi$$
$$450$$ 0 0
$$451$$ 10.9282 0.514589
$$452$$ −11.8564 −0.557678
$$453$$ 0 0
$$454$$ −14.9282 −0.700615
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ 12.1436 0.568053 0.284027 0.958816i $$-0.408330\pi$$
0.284027 + 0.958816i $$0.408330\pi$$
$$458$$ 3.07180 0.143536
$$459$$ 0 0
$$460$$ 18.9282 0.882532
$$461$$ −38.1051 −1.77473 −0.887366 0.461065i $$-0.847467\pi$$
−0.887366 + 0.461065i $$0.847467\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −26.7846 −1.24077
$$467$$ −35.7128 −1.65259 −0.826296 0.563236i $$-0.809556\pi$$
−0.826296 + 0.563236i $$0.809556\pi$$
$$468$$ 0 0
$$469$$ −9.46410 −0.437012
$$470$$ −37.8564 −1.74619
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ −37.8564 −1.74064
$$474$$ 0 0
$$475$$ 7.00000 0.321182
$$476$$ 7.46410 0.342117
$$477$$ 0 0
$$478$$ −30.2487 −1.38354
$$479$$ −27.7128 −1.26623 −0.633115 0.774057i $$-0.718224\pi$$
−0.633115 + 0.774057i $$0.718224\pi$$
$$480$$ 0 0
$$481$$ −34.6410 −1.57949
$$482$$ 21.3205 0.971123
$$483$$ 0 0
$$484$$ 18.8564 0.857109
$$485$$ 15.7128 0.713482
$$486$$ 0 0
$$487$$ −39.3205 −1.78178 −0.890891 0.454217i $$-0.849919\pi$$
−0.890891 + 0.454217i $$0.849919\pi$$
$$488$$ −4.92820 −0.223089
$$489$$ 0 0
$$490$$ 3.46410 0.156492
$$491$$ 40.3923 1.82288 0.911440 0.411434i $$-0.134972\pi$$
0.911440 + 0.411434i $$0.134972\pi$$
$$492$$ 0 0
$$493$$ −44.7846 −2.01700
$$494$$ −3.46410 −0.155857
$$495$$ 0 0
$$496$$ 6.92820 0.311086
$$497$$ 2.92820 0.131348
$$498$$ 0 0
$$499$$ −22.9282 −1.02641 −0.513204 0.858267i $$-0.671541\pi$$
−0.513204 + 0.858267i $$0.671541\pi$$
$$500$$ 6.92820 0.309839
$$501$$ 0 0
$$502$$ −5.07180 −0.226365
$$503$$ 21.8564 0.974529 0.487264 0.873254i $$-0.337995\pi$$
0.487264 + 0.873254i $$0.337995\pi$$
$$504$$ 0 0
$$505$$ −15.7128 −0.699211
$$506$$ 29.8564 1.32728
$$507$$ 0 0
$$508$$ −17.4641 −0.774844
$$509$$ 44.6410 1.97868 0.989339 0.145630i $$-0.0465209\pi$$
0.989339 + 0.145630i $$0.0465209\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −24.9282 −1.09954
$$515$$ −24.0000 −1.05757
$$516$$ 0 0
$$517$$ −59.7128 −2.62617
$$518$$ −10.0000 −0.439375
$$519$$ 0 0
$$520$$ −12.0000 −0.526235
$$521$$ −25.7128 −1.12650 −0.563249 0.826287i $$-0.690449\pi$$
−0.563249 + 0.826287i $$0.690449\pi$$
$$522$$ 0 0
$$523$$ 26.6410 1.16493 0.582465 0.812856i $$-0.302088\pi$$
0.582465 + 0.812856i $$0.302088\pi$$
$$524$$ 5.07180 0.221562
$$525$$ 0 0
$$526$$ −11.3205 −0.493598
$$527$$ −51.7128 −2.25265
$$528$$ 0 0
$$529$$ 6.85641 0.298105
$$530$$ −6.92820 −0.300942
$$531$$ 0 0
$$532$$ −1.00000 −0.0433555
$$533$$ −6.92820 −0.300094
$$534$$ 0 0
$$535$$ 13.8564 0.599065
$$536$$ 9.46410 0.408787
$$537$$ 0 0
$$538$$ 16.9282 0.729827
$$539$$ 5.46410 0.235356
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ −29.8564 −1.28244
$$543$$ 0 0
$$544$$ −7.46410 −0.320021
$$545$$ 6.92820 0.296772
$$546$$ 0 0
$$547$$ −10.2487 −0.438203 −0.219102 0.975702i $$-0.570313\pi$$
−0.219102 + 0.975702i $$0.570313\pi$$
$$548$$ 0.928203 0.0396509
$$549$$ 0 0
$$550$$ 38.2487 1.63093
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 12.3923 0.526974
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ 9.85641 0.418005
$$557$$ −4.92820 −0.208815 −0.104407 0.994535i $$-0.533295\pi$$
−0.104407 + 0.994535i $$0.533295\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ −3.46410 −0.146385
$$561$$ 0 0
$$562$$ 26.0000 1.09674
$$563$$ −25.8564 −1.08972 −0.544859 0.838528i $$-0.683417\pi$$
−0.544859 + 0.838528i $$0.683417\pi$$
$$564$$ 0 0
$$565$$ −41.0718 −1.72790
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ −2.92820 −0.122865
$$569$$ 45.7128 1.91638 0.958190 0.286131i $$-0.0923694\pi$$
0.958190 + 0.286131i $$0.0923694\pi$$
$$570$$ 0 0
$$571$$ 37.5692 1.57222 0.786111 0.618085i $$-0.212091\pi$$
0.786111 + 0.618085i $$0.212091\pi$$
$$572$$ −18.9282 −0.791428
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 38.2487 1.59508
$$576$$ 0 0
$$577$$ −3.85641 −0.160544 −0.0802722 0.996773i $$-0.525579\pi$$
−0.0802722 + 0.996773i $$0.525579\pi$$
$$578$$ 38.7128 1.61024
$$579$$ 0 0
$$580$$ 20.7846 0.863034
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ −10.9282 −0.452600
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ −7.07180 −0.292133
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ 6.92820 0.285472
$$590$$ −13.8564 −0.570459
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ −12.5359 −0.514788 −0.257394 0.966307i $$-0.582864\pi$$
−0.257394 + 0.966307i $$0.582864\pi$$
$$594$$ 0 0
$$595$$ 25.8564 1.06001
$$596$$ 3.07180 0.125826
$$597$$ 0 0
$$598$$ −18.9282 −0.774032
$$599$$ −13.0718 −0.534099 −0.267050 0.963683i $$-0.586049\pi$$
−0.267050 + 0.963683i $$0.586049\pi$$
$$600$$ 0 0
$$601$$ 6.67949 0.272462 0.136231 0.990677i $$-0.456501\pi$$
0.136231 + 0.990677i $$0.456501\pi$$
$$602$$ 6.92820 0.282372
$$603$$ 0 0
$$604$$ 1.46410 0.0595734
$$605$$ 65.3205 2.65566
$$606$$ 0 0
$$607$$ −17.8564 −0.724769 −0.362385 0.932029i $$-0.618037\pi$$
−0.362385 + 0.932029i $$0.618037\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −17.0718 −0.691217
$$611$$ 37.8564 1.53151
$$612$$ 0 0
$$613$$ −23.8564 −0.963551 −0.481776 0.876295i $$-0.660008\pi$$
−0.481776 + 0.876295i $$0.660008\pi$$
$$614$$ 20.7846 0.838799
$$615$$ 0 0
$$616$$ −5.46410 −0.220155
$$617$$ −12.9282 −0.520470 −0.260235 0.965545i $$-0.583800\pi$$
−0.260235 + 0.965545i $$0.583800\pi$$
$$618$$ 0 0
$$619$$ 26.6410 1.07079 0.535396 0.844601i $$-0.320162\pi$$
0.535396 + 0.844601i $$0.320162\pi$$
$$620$$ 24.0000 0.963863
$$621$$ 0 0
$$622$$ −2.14359 −0.0859503
$$623$$ −2.00000 −0.0801283
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 19.8564 0.792357
$$629$$ −74.6410 −2.97613
$$630$$ 0 0
$$631$$ 16.7846 0.668185 0.334092 0.942540i $$-0.391570\pi$$
0.334092 + 0.942540i $$0.391570\pi$$
$$632$$ −12.3923 −0.492939
$$633$$ 0 0
$$634$$ −7.85641 −0.312018
$$635$$ −60.4974 −2.40077
$$636$$ 0 0
$$637$$ −3.46410 −0.137253
$$638$$ 32.7846 1.29796
$$639$$ 0 0
$$640$$ 3.46410 0.136931
$$641$$ 15.8564 0.626290 0.313145 0.949705i $$-0.398617\pi$$
0.313145 + 0.949705i $$0.398617\pi$$
$$642$$ 0 0
$$643$$ −14.9282 −0.588711 −0.294355 0.955696i $$-0.595105\pi$$
−0.294355 + 0.955696i $$0.595105\pi$$
$$644$$ −5.46410 −0.215316
$$645$$ 0 0
$$646$$ −7.46410 −0.293671
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −21.8564 −0.857939
$$650$$ −24.2487 −0.951113
$$651$$ 0 0
$$652$$ 1.07180 0.0419748
$$653$$ 33.7128 1.31928 0.659642 0.751580i $$-0.270708\pi$$
0.659642 + 0.751580i $$0.270708\pi$$
$$654$$ 0 0
$$655$$ 17.5692 0.686486
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 10.9282 0.426026
$$659$$ −14.1436 −0.550956 −0.275478 0.961307i $$-0.588836\pi$$
−0.275478 + 0.961307i $$0.588836\pi$$
$$660$$ 0 0
$$661$$ 45.3205 1.76276 0.881382 0.472405i $$-0.156614\pi$$
0.881382 + 0.472405i $$0.156614\pi$$
$$662$$ 14.5359 0.564954
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ −3.46410 −0.134332
$$666$$ 0 0
$$667$$ 32.7846 1.26943
$$668$$ 18.9282 0.732354
$$669$$ 0 0
$$670$$ 32.7846 1.26658
$$671$$ −26.9282 −1.03955
$$672$$ 0 0
$$673$$ −16.9282 −0.652534 −0.326267 0.945278i $$-0.605791\pi$$
−0.326267 + 0.945278i $$0.605791\pi$$
$$674$$ 10.7846 0.415408
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ 19.8564 0.763144 0.381572 0.924339i $$-0.375383\pi$$
0.381572 + 0.924339i $$0.375383\pi$$
$$678$$ 0 0
$$679$$ −4.53590 −0.174072
$$680$$ −25.8564 −0.991548
$$681$$ 0 0
$$682$$ 37.8564 1.44960
$$683$$ −30.9282 −1.18343 −0.591717 0.806145i $$-0.701550\pi$$
−0.591717 + 0.806145i $$0.701550\pi$$
$$684$$ 0 0
$$685$$ 3.21539 0.122854
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ −6.92820 −0.264135
$$689$$ 6.92820 0.263944
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 0.928203 0.0352850
$$693$$ 0 0
$$694$$ −27.3205 −1.03707
$$695$$ 34.1436 1.29514
$$696$$ 0 0
$$697$$ −14.9282 −0.565446
$$698$$ 35.8564 1.35718
$$699$$ 0 0
$$700$$ −7.00000 −0.264575
$$701$$ −13.7128 −0.517926 −0.258963 0.965887i $$-0.583381\pi$$
−0.258963 + 0.965887i $$0.583381\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 5.46410 0.205936
$$705$$ 0 0
$$706$$ −5.32051 −0.200240
$$707$$ 4.53590 0.170590
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ −10.1436 −0.380682
$$711$$ 0 0
$$712$$ 2.00000 0.0749532
$$713$$ 37.8564 1.41773
$$714$$ 0 0
$$715$$ −65.5692 −2.45215
$$716$$ 9.85641 0.368351
$$717$$ 0 0
$$718$$ −35.3205 −1.31815
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 6.92820 0.258020
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −8.53590 −0.317234
$$725$$ 42.0000 1.55984
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 3.46410 0.128388
$$729$$ 0 0
$$730$$ 34.6410 1.28212
$$731$$ 51.7128 1.91267
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 5.46410 0.201409
$$737$$ 51.7128 1.90487
$$738$$ 0 0
$$739$$ −9.85641 −0.362574 −0.181287 0.983430i $$-0.558026\pi$$
−0.181287 + 0.983430i $$0.558026\pi$$
$$740$$ 34.6410 1.27343
$$741$$ 0 0
$$742$$ 2.00000 0.0734223
$$743$$ −27.7128 −1.01668 −0.508342 0.861155i $$-0.669742\pi$$
−0.508342 + 0.861155i $$0.669742\pi$$
$$744$$ 0 0
$$745$$ 10.6410 0.389857
$$746$$ 10.7846 0.394853
$$747$$ 0 0
$$748$$ −40.7846 −1.49123
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 20.3923 0.744126 0.372063 0.928208i $$-0.378651\pi$$
0.372063 + 0.928208i $$0.378651\pi$$
$$752$$ −10.9282 −0.398511
$$753$$ 0 0
$$754$$ −20.7846 −0.756931
$$755$$ 5.07180 0.184582
$$756$$ 0 0
$$757$$ 3.85641 0.140163 0.0700817 0.997541i $$-0.477674\pi$$
0.0700817 + 0.997541i $$0.477674\pi$$
$$758$$ −1.46410 −0.0531786
$$759$$ 0 0
$$760$$ 3.46410 0.125656
$$761$$ −4.53590 −0.164426 −0.0822131 0.996615i $$-0.526199\pi$$
−0.0822131 + 0.996615i $$0.526199\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 5.46410 0.197684
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 13.8564 0.500326
$$768$$ 0 0
$$769$$ −19.8564 −0.716040 −0.358020 0.933714i $$-0.616548\pi$$
−0.358020 + 0.933714i $$0.616548\pi$$
$$770$$ −18.9282 −0.682125
$$771$$ 0 0
$$772$$ −22.0000 −0.791797
$$773$$ −2.78461 −0.100155 −0.0500777 0.998745i $$-0.515947\pi$$
−0.0500777 + 0.998745i $$0.515947\pi$$
$$774$$ 0 0
$$775$$ 48.4974 1.74208
$$776$$ 4.53590 0.162829
$$777$$ 0 0
$$778$$ −29.7128 −1.06526
$$779$$ 2.00000 0.0716574
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ −40.7846 −1.45845
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 68.7846 2.45503
$$786$$ 0 0
$$787$$ −11.2154 −0.399785 −0.199893 0.979818i $$-0.564059\pi$$
−0.199893 + 0.979818i $$0.564059\pi$$
$$788$$ −4.92820 −0.175560
$$789$$ 0 0
$$790$$ −42.9282 −1.52732
$$791$$ 11.8564 0.421565
$$792$$ 0 0
$$793$$ 17.0718 0.606237
$$794$$ 19.8564 0.704677
$$795$$ 0 0
$$796$$ −24.7846 −0.878467
$$797$$ 44.6410 1.58127 0.790633 0.612290i $$-0.209752\pi$$
0.790633 + 0.612290i $$0.209752\pi$$
$$798$$ 0 0
$$799$$ 81.5692 2.88571
$$800$$ 7.00000 0.247487
$$801$$ 0 0
$$802$$ −19.8564 −0.701154
$$803$$ 54.6410 1.92824
$$804$$ 0 0
$$805$$ −18.9282 −0.667132
$$806$$ −24.0000 −0.845364
$$807$$ 0 0
$$808$$ −4.53590 −0.159572
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 33.0718 1.16131 0.580654 0.814150i $$-0.302797\pi$$
0.580654 + 0.814150i $$0.302797\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 0 0
$$814$$ 54.6410 1.91517
$$815$$ 3.71281 0.130054
$$816$$ 0 0
$$817$$ −6.92820 −0.242387
$$818$$ −31.1769 −1.09008
$$819$$ 0 0
$$820$$ 6.92820 0.241943
$$821$$ 48.9282 1.70761 0.853803 0.520596i $$-0.174290\pi$$
0.853803 + 0.520596i $$0.174290\pi$$
$$822$$ 0 0
$$823$$ −21.8564 −0.761866 −0.380933 0.924603i $$-0.624397\pi$$
−0.380933 + 0.924603i $$0.624397\pi$$
$$824$$ −6.92820 −0.241355
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −5.60770 −0.194763 −0.0973817 0.995247i $$-0.531047\pi$$
−0.0973817 + 0.995247i $$0.531047\pi$$
$$830$$ −27.7128 −0.961926
$$831$$ 0 0
$$832$$ −3.46410 −0.120096
$$833$$ −7.46410 −0.258616
$$834$$ 0 0
$$835$$ 65.5692 2.26912
$$836$$ 5.46410 0.188980
$$837$$ 0 0
$$838$$ −18.9282 −0.653864
$$839$$ 5.07180 0.175098 0.0875489 0.996160i $$-0.472097\pi$$
0.0875489 + 0.996160i $$0.472097\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 18.7846 0.647360
$$843$$ 0 0
$$844$$ 1.46410 0.0503965
$$845$$ −3.46410 −0.119169
$$846$$ 0 0
$$847$$ −18.8564 −0.647914
$$848$$ −2.00000 −0.0686803
$$849$$ 0 0
$$850$$ −52.2487 −1.79212
$$851$$ 54.6410 1.87307
$$852$$ 0 0
$$853$$ 35.0718 1.20084 0.600418 0.799687i $$-0.295001\pi$$
0.600418 + 0.799687i $$0.295001\pi$$
$$854$$ 4.92820 0.168640
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ −47.5692 −1.62493 −0.812467 0.583007i $$-0.801876\pi$$
−0.812467 + 0.583007i $$0.801876\pi$$
$$858$$ 0 0
$$859$$ −28.7846 −0.982118 −0.491059 0.871126i $$-0.663390\pi$$
−0.491059 + 0.871126i $$0.663390\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ 34.9282 1.18966
$$863$$ −18.1436 −0.617615 −0.308808 0.951125i $$-0.599930\pi$$
−0.308808 + 0.951125i $$0.599930\pi$$
$$864$$ 0 0
$$865$$ 3.21539 0.109327
$$866$$ 6.67949 0.226978
$$867$$ 0 0
$$868$$ −6.92820 −0.235159
$$869$$ −67.7128 −2.29700
$$870$$ 0 0
$$871$$ −32.7846 −1.11086
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ 5.46410 0.184826
$$875$$ −6.92820 −0.234216
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −4.78461 −0.161473
$$879$$ 0 0
$$880$$ 18.9282 0.638070
$$881$$ −43.1769 −1.45467 −0.727334 0.686284i $$-0.759241\pi$$
−0.727334 + 0.686284i $$0.759241\pi$$
$$882$$ 0 0
$$883$$ 48.4974 1.63207 0.816034 0.578004i $$-0.196168\pi$$
0.816034 + 0.578004i $$0.196168\pi$$
$$884$$ 25.8564 0.869645
$$885$$ 0 0
$$886$$ 11.3205 0.380320
$$887$$ 27.7128 0.930505 0.465253 0.885178i $$-0.345963\pi$$
0.465253 + 0.885178i $$0.345963\pi$$
$$888$$ 0 0
$$889$$ 17.4641 0.585727
$$890$$ 6.92820 0.232234
$$891$$ 0 0
$$892$$ −12.0000 −0.401790
$$893$$ −10.9282 −0.365698
$$894$$ 0 0
$$895$$ 34.1436 1.14129
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 7.85641 0.262172
$$899$$ 41.5692 1.38641
$$900$$ 0 0
$$901$$ 14.9282 0.497331
$$902$$ 10.9282 0.363869
$$903$$ 0 0
$$904$$ −11.8564 −0.394338
$$905$$ −29.5692 −0.982914
$$906$$ 0 0
$$907$$ −41.4641 −1.37679 −0.688396 0.725335i $$-0.741685\pi$$
−0.688396 + 0.725335i $$0.741685\pi$$
$$908$$ −14.9282 −0.495410
$$909$$ 0 0
$$910$$ 12.0000 0.397796
$$911$$ −22.6410 −0.750130 −0.375065 0.926998i $$-0.622380\pi$$
−0.375065 + 0.926998i $$0.622380\pi$$
$$912$$ 0 0
$$913$$ −43.7128 −1.44668
$$914$$ 12.1436 0.401674
$$915$$ 0 0
$$916$$ 3.07180 0.101495
$$917$$ −5.07180 −0.167485
$$918$$ 0 0
$$919$$ −49.5692 −1.63514 −0.817569 0.575831i $$-0.804679\pi$$
−0.817569 + 0.575831i $$0.804679\pi$$
$$920$$ 18.9282 0.624044
$$921$$ 0 0
$$922$$ −38.1051 −1.25493
$$923$$ 10.1436 0.333880
$$924$$ 0 0
$$925$$ 70.0000 2.30159
$$926$$ 24.0000 0.788689
$$927$$ 0 0
$$928$$ 6.00000 0.196960
$$929$$ −25.6077 −0.840161 −0.420081 0.907487i $$-0.637998\pi$$
−0.420081 + 0.907487i $$0.637998\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ −26.7846 −0.877359
$$933$$ 0 0
$$934$$ −35.7128 −1.16856
$$935$$ −141.282 −4.62042
$$936$$ 0 0
$$937$$ 20.9282 0.683695 0.341847 0.939756i $$-0.388947\pi$$
0.341847 + 0.939756i $$0.388947\pi$$
$$938$$ −9.46410 −0.309014
$$939$$ 0 0
$$940$$ −37.8564 −1.23474
$$941$$ 5.21539 0.170017 0.0850084 0.996380i $$-0.472908\pi$$
0.0850084 + 0.996380i $$0.472908\pi$$
$$942$$ 0 0
$$943$$ 10.9282 0.355871
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −37.8564 −1.23082
$$947$$ 28.1051 0.913294 0.456647 0.889648i $$-0.349050\pi$$
0.456647 + 0.889648i $$0.349050\pi$$
$$948$$ 0 0
$$949$$ −34.6410 −1.12449
$$950$$ 7.00000 0.227110
$$951$$ 0 0
$$952$$ 7.46410 0.241913
$$953$$ 31.8564 1.03193 0.515965 0.856610i $$-0.327433\pi$$
0.515965 + 0.856610i $$0.327433\pi$$
$$954$$ 0 0
$$955$$ 18.9282 0.612502
$$956$$ −30.2487 −0.978313
$$957$$ 0 0
$$958$$ −27.7128 −0.895360
$$959$$ −0.928203 −0.0299732
$$960$$ 0 0
$$961$$ 17.0000 0.548387
$$962$$ −34.6410 −1.11687
$$963$$ 0 0
$$964$$ 21.3205 0.686687
$$965$$ −76.2102 −2.45329
$$966$$ 0 0
$$967$$ −18.9282 −0.608690 −0.304345 0.952562i $$-0.598438\pi$$
−0.304345 + 0.952562i $$0.598438\pi$$
$$968$$ 18.8564 0.606068
$$969$$ 0 0
$$970$$ 15.7128 0.504508
$$971$$ 10.6410 0.341486 0.170743 0.985316i $$-0.445383\pi$$
0.170743 + 0.985316i $$0.445383\pi$$
$$972$$ 0 0
$$973$$ −9.85641 −0.315982
$$974$$ −39.3205 −1.25991
$$975$$ 0 0
$$976$$ −4.92820 −0.157748
$$977$$ 7.85641 0.251349 0.125674 0.992072i $$-0.459891\pi$$
0.125674 + 0.992072i $$0.459891\pi$$
$$978$$ 0 0
$$979$$ 10.9282 0.349267
$$980$$ 3.46410 0.110657
$$981$$ 0 0
$$982$$ 40.3923 1.28897
$$983$$ 32.7846 1.04567 0.522833 0.852435i $$-0.324875\pi$$
0.522833 + 0.852435i $$0.324875\pi$$
$$984$$ 0 0
$$985$$ −17.0718 −0.543953
$$986$$ −44.7846 −1.42623
$$987$$ 0 0
$$988$$ −3.46410 −0.110208
$$989$$ −37.8564 −1.20376
$$990$$ 0 0
$$991$$ 35.6077 1.13112 0.565558 0.824709i $$-0.308661\pi$$
0.565558 + 0.824709i $$0.308661\pi$$
$$992$$ 6.92820 0.219971
$$993$$ 0 0
$$994$$ 2.92820 0.0928770
$$995$$ −85.8564 −2.72183
$$996$$ 0 0
$$997$$ −47.8564 −1.51563 −0.757814 0.652471i $$-0.773732\pi$$
−0.757814 + 0.652471i $$0.773732\pi$$
$$998$$ −22.9282 −0.725780
$$999$$ 0 0
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 2394.2.a.x.1.2 2
3.2 odd 2 798.2.a.k.1.1 2
12.11 even 2 6384.2.a.br.1.1 2
21.20 even 2 5586.2.a.bd.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.k.1.1 2 3.2 odd 2
2394.2.a.x.1.2 2 1.1 even 1 trivial
5586.2.a.bd.1.2 2 21.20 even 2
6384.2.a.br.1.1 2 12.11 even 2