Properties

 Label 1960.2.a.y.1.2 Level $1960$ Weight $2$ Character 1960.1 Self dual yes Analytic conductor $15.651$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.16448.2 Defining polynomial: $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-1.18398$$ of defining polynomial Character $$\chi$$ $$=$$ 1960.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.18398 q^{3} +1.00000 q^{5} -1.59819 q^{9} +O(q^{10})$$ $$q-1.18398 q^{3} +1.00000 q^{5} -1.59819 q^{9} +0.230234 q^{11} -2.27259 q^{13} -1.18398 q^{15} +6.53278 q^{17} -0.260186 q^{19} -8.87079 q^{23} +1.00000 q^{25} +5.44417 q^{27} +5.42662 q^{29} +4.87079 q^{31} -0.272593 q^{33} -1.15403 q^{37} +2.69070 q^{39} +4.43337 q^{41} +4.17723 q^{43} -1.59819 q^{45} +0.923793 q^{47} -7.73467 q^{51} -2.13487 q^{53} +0.230234 q^{55} +0.308055 q^{57} -5.07107 q^{59} +8.88833 q^{61} -2.27259 q^{65} +8.15968 q^{67} +10.5028 q^{69} -9.06556 q^{71} +6.00955 q^{73} -1.18398 q^{75} +0.112912 q^{79} -1.65120 q^{81} +8.72065 q^{83} +6.53278 q^{85} -6.42501 q^{87} +15.9170 q^{89} -5.76691 q^{93} -0.260186 q^{95} +4.29565 q^{97} -0.367959 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 4q^{5} + 6q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 4q^{5} + 6q^{9} + 2q^{11} + 10q^{13} + 2q^{15} + 6q^{17} - 4q^{23} + 4q^{25} + 14q^{27} - 2q^{29} - 12q^{31} + 18q^{33} + 14q^{39} + 12q^{41} - 8q^{43} + 6q^{45} - 2q^{47} + 2q^{51} - 4q^{53} + 2q^{55} - 8q^{57} + 8q^{59} + 20q^{61} + 10q^{65} - 8q^{67} + 24q^{69} + 4q^{71} + 16q^{73} + 2q^{75} + 22q^{79} - 20q^{81} + 36q^{83} + 6q^{85} - 18q^{87} + 40q^{89} - 32q^{93} + 26q^{97} + 12q^{99} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.18398 −0.683571 −0.341785 0.939778i $$-0.611032\pi$$
−0.341785 + 0.939778i $$0.611032\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −1.59819 −0.532731
$$10$$ 0 0
$$11$$ 0.230234 0.0694182 0.0347091 0.999397i $$-0.488950\pi$$
0.0347091 + 0.999397i $$0.488950\pi$$
$$12$$ 0 0
$$13$$ −2.27259 −0.630304 −0.315152 0.949041i $$-0.602055\pi$$
−0.315152 + 0.949041i $$0.602055\pi$$
$$14$$ 0 0
$$15$$ −1.18398 −0.305702
$$16$$ 0 0
$$17$$ 6.53278 1.58443 0.792216 0.610241i $$-0.208927\pi$$
0.792216 + 0.610241i $$0.208927\pi$$
$$18$$ 0 0
$$19$$ −0.260186 −0.0596908 −0.0298454 0.999555i $$-0.509501\pi$$
−0.0298454 + 0.999555i $$0.509501\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.87079 −1.84969 −0.924843 0.380348i $$-0.875804\pi$$
−0.924843 + 0.380348i $$0.875804\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.44417 1.04773
$$28$$ 0 0
$$29$$ 5.42662 1.00770 0.503849 0.863792i $$-0.331917\pi$$
0.503849 + 0.863792i $$0.331917\pi$$
$$30$$ 0 0
$$31$$ 4.87079 0.874819 0.437409 0.899262i $$-0.355896\pi$$
0.437409 + 0.899262i $$0.355896\pi$$
$$32$$ 0 0
$$33$$ −0.272593 −0.0474523
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.15403 −0.189721 −0.0948605 0.995491i $$-0.530240\pi$$
−0.0948605 + 0.995491i $$0.530240\pi$$
$$38$$ 0 0
$$39$$ 2.69070 0.430857
$$40$$ 0 0
$$41$$ 4.43337 0.692377 0.346188 0.938165i $$-0.387476\pi$$
0.346188 + 0.938165i $$0.387476\pi$$
$$42$$ 0 0
$$43$$ 4.17723 0.637021 0.318511 0.947919i $$-0.396817\pi$$
0.318511 + 0.947919i $$0.396817\pi$$
$$44$$ 0 0
$$45$$ −1.59819 −0.238245
$$46$$ 0 0
$$47$$ 0.923793 0.134749 0.0673745 0.997728i $$-0.478538\pi$$
0.0673745 + 0.997728i $$0.478538\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −7.73467 −1.08307
$$52$$ 0 0
$$53$$ −2.13487 −0.293247 −0.146623 0.989192i $$-0.546841\pi$$
−0.146623 + 0.989192i $$0.546841\pi$$
$$54$$ 0 0
$$55$$ 0.230234 0.0310448
$$56$$ 0 0
$$57$$ 0.308055 0.0408029
$$58$$ 0 0
$$59$$ −5.07107 −0.660197 −0.330098 0.943947i $$-0.607082\pi$$
−0.330098 + 0.943947i $$0.607082\pi$$
$$60$$ 0 0
$$61$$ 8.88833 1.13803 0.569017 0.822326i $$-0.307324\pi$$
0.569017 + 0.822326i $$0.307324\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.27259 −0.281880
$$66$$ 0 0
$$67$$ 8.15968 0.996864 0.498432 0.866929i $$-0.333909\pi$$
0.498432 + 0.866929i $$0.333909\pi$$
$$68$$ 0 0
$$69$$ 10.5028 1.26439
$$70$$ 0 0
$$71$$ −9.06556 −1.07588 −0.537942 0.842982i $$-0.680798\pi$$
−0.537942 + 0.842982i $$0.680798\pi$$
$$72$$ 0 0
$$73$$ 6.00955 0.703365 0.351682 0.936119i $$-0.385610\pi$$
0.351682 + 0.936119i $$0.385610\pi$$
$$74$$ 0 0
$$75$$ −1.18398 −0.136714
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0.112912 0.0127035 0.00635177 0.999980i $$-0.497978\pi$$
0.00635177 + 0.999980i $$0.497978\pi$$
$$80$$ 0 0
$$81$$ −1.65120 −0.183467
$$82$$ 0 0
$$83$$ 8.72065 0.957216 0.478608 0.878029i $$-0.341141\pi$$
0.478608 + 0.878029i $$0.341141\pi$$
$$84$$ 0 0
$$85$$ 6.53278 0.708579
$$86$$ 0 0
$$87$$ −6.42501 −0.688833
$$88$$ 0 0
$$89$$ 15.9170 1.68720 0.843601 0.536970i $$-0.180431\pi$$
0.843601 + 0.536970i $$0.180431\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.76691 −0.598001
$$94$$ 0 0
$$95$$ −0.260186 −0.0266945
$$96$$ 0 0
$$97$$ 4.29565 0.436157 0.218079 0.975931i $$-0.430021\pi$$
0.218079 + 0.975931i $$0.430021\pi$$
$$98$$ 0 0
$$99$$ −0.367959 −0.0369812
$$100$$ 0 0
$$101$$ 7.38712 0.735046 0.367523 0.930014i $$-0.380206\pi$$
0.367523 + 0.930014i $$0.380206\pi$$
$$102$$ 0 0
$$103$$ −10.5806 −1.04254 −0.521271 0.853391i $$-0.674542\pi$$
−0.521271 + 0.853391i $$0.674542\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −13.4661 −1.30182 −0.650910 0.759155i $$-0.725612\pi$$
−0.650910 + 0.759155i $$0.725612\pi$$
$$108$$ 0 0
$$109$$ 8.50568 0.814697 0.407348 0.913273i $$-0.366453\pi$$
0.407348 + 0.913273i $$0.366453\pi$$
$$110$$ 0 0
$$111$$ 1.36634 0.129688
$$112$$ 0 0
$$113$$ 18.3968 1.73063 0.865313 0.501232i $$-0.167120\pi$$
0.865313 + 0.501232i $$0.167120\pi$$
$$114$$ 0 0
$$115$$ −8.87079 −0.827205
$$116$$ 0 0
$$117$$ 3.63204 0.335782
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.9470 −0.995181
$$122$$ 0 0
$$123$$ −5.24902 −0.473288
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 8.41032 0.746295 0.373147 0.927772i $$-0.378279\pi$$
0.373147 + 0.927772i $$0.378279\pi$$
$$128$$ 0 0
$$129$$ −4.94575 −0.435449
$$130$$ 0 0
$$131$$ −1.78217 −0.155709 −0.0778546 0.996965i $$-0.524807\pi$$
−0.0778546 + 0.996965i $$0.524807\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 5.44417 0.468559
$$136$$ 0 0
$$137$$ −11.3137 −0.966595 −0.483298 0.875456i $$-0.660561\pi$$
−0.483298 + 0.875456i $$0.660561\pi$$
$$138$$ 0 0
$$139$$ 15.4390 1.30952 0.654761 0.755836i $$-0.272769\pi$$
0.654761 + 0.755836i $$0.272769\pi$$
$$140$$ 0 0
$$141$$ −1.09375 −0.0921105
$$142$$ 0 0
$$143$$ −0.523229 −0.0437546
$$144$$ 0 0
$$145$$ 5.42662 0.450656
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 23.2811 1.90726 0.953631 0.300978i $$-0.0973130\pi$$
0.953631 + 0.300978i $$0.0973130\pi$$
$$150$$ 0 0
$$151$$ 22.3398 1.81798 0.908992 0.416813i $$-0.136853\pi$$
0.908992 + 0.416813i $$0.136853\pi$$
$$152$$ 0 0
$$153$$ −10.4406 −0.844076
$$154$$ 0 0
$$155$$ 4.87079 0.391231
$$156$$ 0 0
$$157$$ −3.52603 −0.281407 −0.140704 0.990052i $$-0.544937\pi$$
−0.140704 + 0.990052i $$0.544937\pi$$
$$158$$ 0 0
$$159$$ 2.52764 0.200455
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −16.2522 −1.27297 −0.636485 0.771289i $$-0.719612\pi$$
−0.636485 + 0.771289i $$0.719612\pi$$
$$164$$ 0 0
$$165$$ −0.272593 −0.0212213
$$166$$ 0 0
$$167$$ −3.99714 −0.309308 −0.154654 0.987969i $$-0.549426\pi$$
−0.154654 + 0.987969i $$0.549426\pi$$
$$168$$ 0 0
$$169$$ −7.83532 −0.602717
$$170$$ 0 0
$$171$$ 0.415828 0.0317991
$$172$$ 0 0
$$173$$ 17.6214 1.33973 0.669865 0.742483i $$-0.266352\pi$$
0.669865 + 0.742483i $$0.266352\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00404 0.451291
$$178$$ 0 0
$$179$$ 25.2811 1.88960 0.944799 0.327650i $$-0.106257\pi$$
0.944799 + 0.327650i $$0.106257\pi$$
$$180$$ 0 0
$$181$$ 18.4950 1.37473 0.687363 0.726315i $$-0.258768\pi$$
0.687363 + 0.726315i $$0.258768\pi$$
$$182$$ 0 0
$$183$$ −10.5236 −0.777927
$$184$$ 0 0
$$185$$ −1.15403 −0.0848458
$$186$$ 0 0
$$187$$ 1.50407 0.109988
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.8658 1.43744 0.718719 0.695301i $$-0.244729\pi$$
0.718719 + 0.695301i $$0.244729\pi$$
$$192$$ 0 0
$$193$$ −11.4335 −0.823002 −0.411501 0.911409i $$-0.634995\pi$$
−0.411501 + 0.911409i $$0.634995\pi$$
$$194$$ 0 0
$$195$$ 2.69070 0.192685
$$196$$ 0 0
$$197$$ −4.56273 −0.325081 −0.162541 0.986702i $$-0.551969\pi$$
−0.162541 + 0.986702i $$0.551969\pi$$
$$198$$ 0 0
$$199$$ −14.4429 −1.02383 −0.511916 0.859036i $$-0.671064\pi$$
−0.511916 + 0.859036i $$0.671064\pi$$
$$200$$ 0 0
$$201$$ −9.66089 −0.681427
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4.43337 0.309640
$$206$$ 0 0
$$207$$ 14.1772 0.985385
$$208$$ 0 0
$$209$$ −0.0599037 −0.00414363
$$210$$ 0 0
$$211$$ 6.63651 0.456876 0.228438 0.973558i $$-0.426638\pi$$
0.228438 + 0.973558i $$0.426638\pi$$
$$212$$ 0 0
$$213$$ 10.7334 0.735443
$$214$$ 0 0
$$215$$ 4.17723 0.284884
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −7.11518 −0.480800
$$220$$ 0 0
$$221$$ −14.8463 −0.998673
$$222$$ 0 0
$$223$$ −20.3325 −1.36156 −0.680782 0.732486i $$-0.738360\pi$$
−0.680782 + 0.732486i $$0.738360\pi$$
$$224$$ 0 0
$$225$$ −1.59819 −0.106546
$$226$$ 0 0
$$227$$ 23.9312 1.58837 0.794185 0.607676i $$-0.207898\pi$$
0.794185 + 0.607676i $$0.207898\pi$$
$$228$$ 0 0
$$229$$ 2.59534 0.171505 0.0857523 0.996316i $$-0.472671\pi$$
0.0857523 + 0.996316i $$0.472671\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −15.1638 −0.993412 −0.496706 0.867919i $$-0.665457\pi$$
−0.496706 + 0.867919i $$0.665457\pi$$
$$234$$ 0 0
$$235$$ 0.923793 0.0602616
$$236$$ 0 0
$$237$$ −0.133685 −0.00868377
$$238$$ 0 0
$$239$$ 10.5656 0.683431 0.341716 0.939803i $$-0.388992\pi$$
0.341716 + 0.939803i $$0.388992\pi$$
$$240$$ 0 0
$$241$$ 19.6762 1.26745 0.633726 0.773557i $$-0.281525\pi$$
0.633726 + 0.773557i $$0.281525\pi$$
$$242$$ 0 0
$$243$$ −14.3775 −0.922318
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.591297 0.0376233
$$248$$ 0 0
$$249$$ −10.3251 −0.654325
$$250$$ 0 0
$$251$$ −25.0498 −1.58113 −0.790564 0.612380i $$-0.790212\pi$$
−0.790564 + 0.612380i $$0.790212\pi$$
$$252$$ 0 0
$$253$$ −2.04236 −0.128402
$$254$$ 0 0
$$255$$ −7.73467 −0.484364
$$256$$ 0 0
$$257$$ 1.71500 0.106979 0.0534894 0.998568i $$-0.482966\pi$$
0.0534894 + 0.998568i $$0.482966\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.67279 −0.536832
$$262$$ 0 0
$$263$$ −22.1173 −1.36381 −0.681906 0.731440i $$-0.738849\pi$$
−0.681906 + 0.731440i $$0.738849\pi$$
$$264$$ 0 0
$$265$$ −2.13487 −0.131144
$$266$$ 0 0
$$267$$ −18.8454 −1.15332
$$268$$ 0 0
$$269$$ −31.9261 −1.94657 −0.973283 0.229608i $$-0.926256\pi$$
−0.973283 + 0.229608i $$0.926256\pi$$
$$270$$ 0 0
$$271$$ −25.7767 −1.56582 −0.782910 0.622135i $$-0.786266\pi$$
−0.782910 + 0.622135i $$0.786266\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.230234 0.0138836
$$276$$ 0 0
$$277$$ −8.01755 −0.481728 −0.240864 0.970559i $$-0.577431\pi$$
−0.240864 + 0.970559i $$0.577431\pi$$
$$278$$ 0 0
$$279$$ −7.78445 −0.466043
$$280$$ 0 0
$$281$$ −3.13207 −0.186844 −0.0934218 0.995627i $$-0.529781\pi$$
−0.0934218 + 0.995627i $$0.529781\pi$$
$$282$$ 0 0
$$283$$ 3.17047 0.188465 0.0942325 0.995550i $$-0.469960\pi$$
0.0942325 + 0.995550i $$0.469960\pi$$
$$284$$ 0 0
$$285$$ 0.308055 0.0182476
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 25.6772 1.51042
$$290$$ 0 0
$$291$$ −5.08596 −0.298144
$$292$$ 0 0
$$293$$ −17.5264 −1.02390 −0.511952 0.859014i $$-0.671077\pi$$
−0.511952 + 0.859014i $$0.671077\pi$$
$$294$$ 0 0
$$295$$ −5.07107 −0.295249
$$296$$ 0 0
$$297$$ 1.25343 0.0727316
$$298$$ 0 0
$$299$$ 20.1597 1.16586
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −8.74620 −0.502456
$$304$$ 0 0
$$305$$ 8.88833 0.508944
$$306$$ 0 0
$$307$$ 2.11063 0.120460 0.0602300 0.998185i $$-0.480817\pi$$
0.0602300 + 0.998185i $$0.480817\pi$$
$$308$$ 0 0
$$309$$ 12.5273 0.712651
$$310$$ 0 0
$$311$$ 20.8149 1.18031 0.590153 0.807291i $$-0.299067\pi$$
0.590153 + 0.807291i $$0.299067\pi$$
$$312$$ 0 0
$$313$$ −21.3860 −1.20881 −0.604405 0.796678i $$-0.706589\pi$$
−0.604405 + 0.796678i $$0.706589\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.4909 0.870058 0.435029 0.900417i $$-0.356738\pi$$
0.435029 + 0.900417i $$0.356738\pi$$
$$318$$ 0 0
$$319$$ 1.24939 0.0699526
$$320$$ 0 0
$$321$$ 15.9436 0.889886
$$322$$ 0 0
$$323$$ −1.69974 −0.0945760
$$324$$ 0 0
$$325$$ −2.27259 −0.126061
$$326$$ 0 0
$$327$$ −10.0706 −0.556903
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −8.52360 −0.468499 −0.234250 0.972176i $$-0.575263\pi$$
−0.234250 + 0.972176i $$0.575263\pi$$
$$332$$ 0 0
$$333$$ 1.84436 0.101070
$$334$$ 0 0
$$335$$ 8.15968 0.445811
$$336$$ 0 0
$$337$$ −18.1246 −0.987309 −0.493655 0.869658i $$-0.664339\pi$$
−0.493655 + 0.869658i $$0.664339\pi$$
$$338$$ 0 0
$$339$$ −21.7814 −1.18301
$$340$$ 0 0
$$341$$ 1.12142 0.0607284
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 10.5028 0.565453
$$346$$ 0 0
$$347$$ 4.10226 0.220221 0.110110 0.993919i $$-0.464880\pi$$
0.110110 + 0.993919i $$0.464880\pi$$
$$348$$ 0 0
$$349$$ −8.67850 −0.464549 −0.232275 0.972650i $$-0.574617\pi$$
−0.232275 + 0.972650i $$0.574617\pi$$
$$350$$ 0 0
$$351$$ −12.3724 −0.660388
$$352$$ 0 0
$$353$$ 30.1162 1.60292 0.801462 0.598045i $$-0.204056\pi$$
0.801462 + 0.598045i $$0.204056\pi$$
$$354$$ 0 0
$$355$$ −9.06556 −0.481150
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 36.2843 1.91501 0.957505 0.288416i $$-0.0931285\pi$$
0.957505 + 0.288416i $$0.0931285\pi$$
$$360$$ 0 0
$$361$$ −18.9323 −0.996437
$$362$$ 0 0
$$363$$ 12.9610 0.680277
$$364$$ 0 0
$$365$$ 6.00955 0.314554
$$366$$ 0 0
$$367$$ −10.7115 −0.559134 −0.279567 0.960126i $$-0.590191\pi$$
−0.279567 + 0.960126i $$0.590191\pi$$
$$368$$ 0 0
$$369$$ −7.08539 −0.368850
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5.64335 0.292202 0.146101 0.989270i $$-0.453328\pi$$
0.146101 + 0.989270i $$0.453328\pi$$
$$374$$ 0 0
$$375$$ −1.18398 −0.0611404
$$376$$ 0 0
$$377$$ −12.3325 −0.635156
$$378$$ 0 0
$$379$$ −1.59944 −0.0821575 −0.0410787 0.999156i $$-0.513079\pi$$
−0.0410787 + 0.999156i $$0.513079\pi$$
$$380$$ 0 0
$$381$$ −9.95764 −0.510145
$$382$$ 0 0
$$383$$ −28.8894 −1.47618 −0.738089 0.674704i $$-0.764271\pi$$
−0.738089 + 0.674704i $$0.764271\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −6.67601 −0.339361
$$388$$ 0 0
$$389$$ −23.5226 −1.19265 −0.596323 0.802745i $$-0.703372\pi$$
−0.596323 + 0.802745i $$0.703372\pi$$
$$390$$ 0 0
$$391$$ −57.9509 −2.93070
$$392$$ 0 0
$$393$$ 2.11006 0.106438
$$394$$ 0 0
$$395$$ 0.112912 0.00568120
$$396$$ 0 0
$$397$$ −16.5806 −0.832159 −0.416079 0.909328i $$-0.636596\pi$$
−0.416079 + 0.909328i $$0.636596\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1.52161 −0.0759858 −0.0379929 0.999278i $$-0.512096\pi$$
−0.0379929 + 0.999278i $$0.512096\pi$$
$$402$$ 0 0
$$403$$ −11.0693 −0.551402
$$404$$ 0 0
$$405$$ −1.65120 −0.0820488
$$406$$ 0 0
$$407$$ −0.265697 −0.0131701
$$408$$ 0 0
$$409$$ −5.42927 −0.268460 −0.134230 0.990950i $$-0.542856\pi$$
−0.134230 + 0.990950i $$0.542856\pi$$
$$410$$ 0 0
$$411$$ 13.3952 0.660736
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8.72065 0.428080
$$416$$ 0 0
$$417$$ −18.2795 −0.895150
$$418$$ 0 0
$$419$$ 12.4199 0.606750 0.303375 0.952871i $$-0.401886\pi$$
0.303375 + 0.952871i $$0.401886\pi$$
$$420$$ 0 0
$$421$$ −20.6804 −1.00790 −0.503951 0.863732i $$-0.668121\pi$$
−0.503951 + 0.863732i $$0.668121\pi$$
$$422$$ 0 0
$$423$$ −1.47640 −0.0717850
$$424$$ 0 0
$$425$$ 6.53278 0.316886
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0.619492 0.0299093
$$430$$ 0 0
$$431$$ −31.6152 −1.52285 −0.761424 0.648254i $$-0.775500\pi$$
−0.761424 + 0.648254i $$0.775500\pi$$
$$432$$ 0 0
$$433$$ −31.1247 −1.49576 −0.747880 0.663834i $$-0.768928\pi$$
−0.747880 + 0.663834i $$0.768928\pi$$
$$434$$ 0 0
$$435$$ −6.42501 −0.308055
$$436$$ 0 0
$$437$$ 2.30805 0.110409
$$438$$ 0 0
$$439$$ 0.244398 0.0116645 0.00583223 0.999983i $$-0.498144\pi$$
0.00583223 + 0.999983i $$0.498144\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −3.45481 −0.164143 −0.0820716 0.996626i $$-0.526154\pi$$
−0.0820716 + 0.996626i $$0.526154\pi$$
$$444$$ 0 0
$$445$$ 15.9170 0.754540
$$446$$ 0 0
$$447$$ −27.5643 −1.30375
$$448$$ 0 0
$$449$$ −15.5329 −0.733044 −0.366522 0.930409i $$-0.619452\pi$$
−0.366522 + 0.930409i $$0.619452\pi$$
$$450$$ 0 0
$$451$$ 1.02071 0.0480636
$$452$$ 0 0
$$453$$ −26.4498 −1.24272
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.5379 −0.492943 −0.246471 0.969150i $$-0.579271\pi$$
−0.246471 + 0.969150i $$0.579271\pi$$
$$458$$ 0 0
$$459$$ 35.5655 1.66006
$$460$$ 0 0
$$461$$ 5.98972 0.278969 0.139485 0.990224i $$-0.455455\pi$$
0.139485 + 0.990224i $$0.455455\pi$$
$$462$$ 0 0
$$463$$ −33.3601 −1.55038 −0.775188 0.631731i $$-0.782345\pi$$
−0.775188 + 0.631731i $$0.782345\pi$$
$$464$$ 0 0
$$465$$ −5.76691 −0.267434
$$466$$ 0 0
$$467$$ 19.7076 0.911958 0.455979 0.889991i $$-0.349289\pi$$
0.455979 + 0.889991i $$0.349289\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.17474 0.192362
$$472$$ 0 0
$$473$$ 0.961740 0.0442209
$$474$$ 0 0
$$475$$ −0.260186 −0.0119382
$$476$$ 0 0
$$477$$ 3.41193 0.156222
$$478$$ 0 0
$$479$$ 0.325600 0.0148771 0.00743853 0.999972i $$-0.497632\pi$$
0.00743853 + 0.999972i $$0.497632\pi$$
$$480$$ 0 0
$$481$$ 2.62263 0.119582
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.29565 0.195055
$$486$$ 0 0
$$487$$ −16.9204 −0.766737 −0.383369 0.923595i $$-0.625236\pi$$
−0.383369 + 0.923595i $$0.625236\pi$$
$$488$$ 0 0
$$489$$ 19.2423 0.870165
$$490$$ 0 0
$$491$$ 15.0203 0.677859 0.338929 0.940812i $$-0.389935\pi$$
0.338929 + 0.940812i $$0.389935\pi$$
$$492$$ 0 0
$$493$$ 35.4509 1.59663
$$494$$ 0 0
$$495$$ −0.367959 −0.0165385
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 41.0431 1.83734 0.918670 0.395025i $$-0.129264\pi$$
0.918670 + 0.395025i $$0.129264\pi$$
$$500$$ 0 0
$$501$$ 4.73254 0.211434
$$502$$ 0 0
$$503$$ 8.29741 0.369963 0.184982 0.982742i $$-0.440777\pi$$
0.184982 + 0.982742i $$0.440777\pi$$
$$504$$ 0 0
$$505$$ 7.38712 0.328722
$$506$$ 0 0
$$507$$ 9.27686 0.412000
$$508$$ 0 0
$$509$$ 8.71266 0.386182 0.193091 0.981181i $$-0.438149\pi$$
0.193091 + 0.981181i $$0.438149\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.41650 −0.0625398
$$514$$ 0 0
$$515$$ −10.5806 −0.466239
$$516$$ 0 0
$$517$$ 0.212689 0.00935404
$$518$$ 0 0
$$519$$ −20.8634 −0.915800
$$520$$ 0 0
$$521$$ −27.1056 −1.18752 −0.593760 0.804642i $$-0.702357\pi$$
−0.593760 + 0.804642i $$0.702357\pi$$
$$522$$ 0 0
$$523$$ −1.37186 −0.0599870 −0.0299935 0.999550i $$-0.509549\pi$$
−0.0299935 + 0.999550i $$0.509549\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 31.8198 1.38609
$$528$$ 0 0
$$529$$ 55.6908 2.42134
$$530$$ 0 0
$$531$$ 8.10454 0.351707
$$532$$ 0 0
$$533$$ −10.0753 −0.436408
$$534$$ 0 0
$$535$$ −13.4661 −0.582191
$$536$$ 0 0
$$537$$ −29.9323 −1.29167
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.56559 0.368263 0.184132 0.982902i $$-0.441053\pi$$
0.184132 + 0.982902i $$0.441053\pi$$
$$542$$ 0 0
$$543$$ −21.8977 −0.939722
$$544$$ 0 0
$$545$$ 8.50568 0.364343
$$546$$ 0 0
$$547$$ 44.2056 1.89009 0.945046 0.326936i $$-0.106016\pi$$
0.945046 + 0.326936i $$0.106016\pi$$
$$548$$ 0 0
$$549$$ −14.2053 −0.606266
$$550$$ 0 0
$$551$$ −1.41193 −0.0601503
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 1.36634 0.0579981
$$556$$ 0 0
$$557$$ −13.3385 −0.565171 −0.282586 0.959242i $$-0.591192\pi$$
−0.282586 + 0.959242i $$0.591192\pi$$
$$558$$ 0 0
$$559$$ −9.49313 −0.401517
$$560$$ 0 0
$$561$$ −1.78079 −0.0751849
$$562$$ 0 0
$$563$$ 29.7320 1.25306 0.626528 0.779399i $$-0.284476\pi$$
0.626528 + 0.779399i $$0.284476\pi$$
$$564$$ 0 0
$$565$$ 18.3968 0.773960
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 23.5859 0.988774 0.494387 0.869242i $$-0.335393\pi$$
0.494387 + 0.869242i $$0.335393\pi$$
$$570$$ 0 0
$$571$$ 32.6434 1.36608 0.683042 0.730379i $$-0.260657\pi$$
0.683042 + 0.730379i $$0.260657\pi$$
$$572$$ 0 0
$$573$$ −23.5207 −0.982591
$$574$$ 0 0
$$575$$ −8.87079 −0.369937
$$576$$ 0 0
$$577$$ −28.8195 −1.19977 −0.599886 0.800085i $$-0.704788\pi$$
−0.599886 + 0.800085i $$0.704788\pi$$
$$578$$ 0 0
$$579$$ 13.5370 0.562580
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −0.491520 −0.0203567
$$584$$ 0 0
$$585$$ 3.63204 0.150166
$$586$$ 0 0
$$587$$ −1.82205 −0.0752039 −0.0376019 0.999293i $$-0.511972\pi$$
−0.0376019 + 0.999293i $$0.511972\pi$$
$$588$$ 0 0
$$589$$ −1.26731 −0.0522186
$$590$$ 0 0
$$591$$ 5.40218 0.222216
$$592$$ 0 0
$$593$$ 30.9369 1.27042 0.635212 0.772338i $$-0.280913\pi$$
0.635212 + 0.772338i $$0.280913\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 17.1001 0.699861
$$598$$ 0 0
$$599$$ −32.2912 −1.31938 −0.659691 0.751537i $$-0.729313\pi$$
−0.659691 + 0.751537i $$0.729313\pi$$
$$600$$ 0 0
$$601$$ −25.2829 −1.03131 −0.515655 0.856797i $$-0.672451\pi$$
−0.515655 + 0.856797i $$0.672451\pi$$
$$602$$ 0 0
$$603$$ −13.0407 −0.531060
$$604$$ 0 0
$$605$$ −10.9470 −0.445059
$$606$$ 0 0
$$607$$ −9.28852 −0.377010 −0.188505 0.982072i $$-0.560364\pi$$
−0.188505 + 0.982072i $$0.560364\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.09941 −0.0849329
$$612$$ 0 0
$$613$$ 40.0609 1.61805 0.809023 0.587777i $$-0.199997\pi$$
0.809023 + 0.587777i $$0.199997\pi$$
$$614$$ 0 0
$$615$$ −5.24902 −0.211661
$$616$$ 0 0
$$617$$ 27.9860 1.12667 0.563336 0.826228i $$-0.309518\pi$$
0.563336 + 0.826228i $$0.309518\pi$$
$$618$$ 0 0
$$619$$ −45.8048 −1.84105 −0.920525 0.390684i $$-0.872239\pi$$
−0.920525 + 0.390684i $$0.872239\pi$$
$$620$$ 0 0
$$621$$ −48.2940 −1.93797
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0.0709248 0.00283246
$$628$$ 0 0
$$629$$ −7.53901 −0.300600
$$630$$ 0 0
$$631$$ 22.5847 0.899082 0.449541 0.893260i $$-0.351588\pi$$
0.449541 + 0.893260i $$0.351588\pi$$
$$632$$ 0 0
$$633$$ −7.85749 −0.312307
$$634$$ 0 0
$$635$$ 8.41032 0.333753
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 14.4885 0.573157
$$640$$ 0 0
$$641$$ −33.5125 −1.32366 −0.661832 0.749652i $$-0.730221\pi$$
−0.661832 + 0.749652i $$0.730221\pi$$
$$642$$ 0 0
$$643$$ 46.4980 1.83370 0.916851 0.399231i $$-0.130723\pi$$
0.916851 + 0.399231i $$0.130723\pi$$
$$644$$ 0 0
$$645$$ −4.94575 −0.194739
$$646$$ 0 0
$$647$$ −1.57565 −0.0619453 −0.0309726 0.999520i $$-0.509860\pi$$
−0.0309726 + 0.999520i $$0.509860\pi$$
$$648$$ 0 0
$$649$$ −1.16753 −0.0458297
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 12.1311 0.474727 0.237364 0.971421i $$-0.423717\pi$$
0.237364 + 0.971421i $$0.423717\pi$$
$$654$$ 0 0
$$655$$ −1.78217 −0.0696352
$$656$$ 0 0
$$657$$ −9.60442 −0.374704
$$658$$ 0 0
$$659$$ 35.1682 1.36996 0.684979 0.728563i $$-0.259811\pi$$
0.684979 + 0.728563i $$0.259811\pi$$
$$660$$ 0 0
$$661$$ −1.62425 −0.0631759 −0.0315880 0.999501i $$-0.510056\pi$$
−0.0315880 + 0.999501i $$0.510056\pi$$
$$662$$ 0 0
$$663$$ 17.5778 0.682664
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −48.1384 −1.86393
$$668$$ 0 0
$$669$$ 24.0733 0.930726
$$670$$ 0 0
$$671$$ 2.04640 0.0790003
$$672$$ 0 0
$$673$$ −24.3194 −0.937443 −0.468721 0.883346i $$-0.655285\pi$$
−0.468721 + 0.883346i $$0.655285\pi$$
$$674$$ 0 0
$$675$$ 5.44417 0.209546
$$676$$ 0 0
$$677$$ 12.1553 0.467165 0.233582 0.972337i $$-0.424955\pi$$
0.233582 + 0.972337i $$0.424955\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −28.3341 −1.08576
$$682$$ 0 0
$$683$$ −32.5158 −1.24418 −0.622091 0.782945i $$-0.713717\pi$$
−0.622091 + 0.782945i $$0.713717\pi$$
$$684$$ 0 0
$$685$$ −11.3137 −0.432275
$$686$$ 0 0
$$687$$ −3.07282 −0.117236
$$688$$ 0 0
$$689$$ 4.85169 0.184834
$$690$$ 0 0
$$691$$ 20.3311 0.773432 0.386716 0.922199i $$-0.373609\pi$$
0.386716 + 0.922199i $$0.373609\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 15.4390 0.585636
$$696$$ 0 0
$$697$$ 28.9622 1.09702
$$698$$ 0 0
$$699$$ 17.9536 0.679068
$$700$$ 0 0
$$701$$ 7.35321 0.277727 0.138863 0.990312i $$-0.455655\pi$$
0.138863 + 0.990312i $$0.455655\pi$$
$$702$$ 0 0
$$703$$ 0.300262 0.0113246
$$704$$ 0 0
$$705$$ −1.09375 −0.0411931
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −11.6638 −0.438041 −0.219021 0.975720i $$-0.570286\pi$$
−0.219021 + 0.975720i $$0.570286\pi$$
$$710$$ 0 0
$$711$$ −0.180454 −0.00676757
$$712$$ 0 0
$$713$$ −43.2077 −1.61814
$$714$$ 0 0
$$715$$ −0.523229 −0.0195676
$$716$$ 0 0
$$717$$ −12.5094 −0.467173
$$718$$ 0 0
$$719$$ −5.68571 −0.212041 −0.106021 0.994364i $$-0.533811\pi$$
−0.106021 + 0.994364i $$0.533811\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −23.2962 −0.866394
$$724$$ 0 0
$$725$$ 5.42662 0.201540
$$726$$ 0 0
$$727$$ 18.6873 0.693074 0.346537 0.938036i $$-0.387357\pi$$
0.346537 + 0.938036i $$0.387357\pi$$
$$728$$ 0 0
$$729$$ 21.9763 0.813936
$$730$$ 0 0
$$731$$ 27.2889 1.00932
$$732$$ 0 0
$$733$$ 1.83694 0.0678488 0.0339244 0.999424i $$-0.489199\pi$$
0.0339244 + 0.999424i $$0.489199\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.87864 0.0692005
$$738$$ 0 0
$$739$$ −24.5634 −0.903579 −0.451789 0.892125i $$-0.649214\pi$$
−0.451789 + 0.892125i $$0.649214\pi$$
$$740$$ 0 0
$$741$$ −0.700083 −0.0257182
$$742$$ 0 0
$$743$$ −27.1926 −0.997601 −0.498800 0.866717i $$-0.666226\pi$$
−0.498800 + 0.866717i $$0.666226\pi$$
$$744$$ 0 0
$$745$$ 23.2811 0.852954
$$746$$ 0 0
$$747$$ −13.9373 −0.509939
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0.328457 0.0119856 0.00599278 0.999982i $$-0.498092\pi$$
0.00599278 + 0.999982i $$0.498092\pi$$
$$752$$ 0 0
$$753$$ 29.6584 1.08081
$$754$$ 0 0
$$755$$ 22.3398 0.813027
$$756$$ 0 0
$$757$$ −48.6331 −1.76760 −0.883801 0.467864i $$-0.845024\pi$$
−0.883801 + 0.467864i $$0.845024\pi$$
$$758$$ 0 0
$$759$$ 2.41811 0.0877718
$$760$$ 0 0
$$761$$ −8.02253 −0.290817 −0.145408 0.989372i $$-0.546450\pi$$
−0.145408 + 0.989372i $$0.546450\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −10.4406 −0.377482
$$766$$ 0 0
$$767$$ 11.5245 0.416125
$$768$$ 0 0
$$769$$ 35.6370 1.28510 0.642552 0.766242i $$-0.277876\pi$$
0.642552 + 0.766242i $$0.277876\pi$$
$$770$$ 0 0
$$771$$ −2.03053 −0.0731276
$$772$$ 0 0
$$773$$ 49.5955 1.78383 0.891914 0.452206i $$-0.149363\pi$$
0.891914 + 0.452206i $$0.149363\pi$$
$$774$$ 0 0
$$775$$ 4.87079 0.174964
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.15350 −0.0413285
$$780$$ 0 0
$$781$$ −2.08720 −0.0746859
$$782$$ 0 0
$$783$$ 29.5434 1.05580
$$784$$ 0 0
$$785$$ −3.52603 −0.125849
$$786$$ 0 0
$$787$$ 21.4434 0.764376 0.382188 0.924085i $$-0.375171\pi$$
0.382188 + 0.924085i $$0.375171\pi$$
$$788$$ 0 0
$$789$$ 26.1865 0.932262
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.1996 −0.717307
$$794$$ 0 0
$$795$$ 2.52764 0.0896461
$$796$$ 0 0
$$797$$ 35.1429 1.24482 0.622412 0.782690i $$-0.286153\pi$$
0.622412 + 0.782690i $$0.286153\pi$$
$$798$$ 0 0
$$799$$ 6.03494 0.213501
$$800$$ 0 0
$$801$$ −25.4385 −0.898825
$$802$$ 0 0
$$803$$ 1.38360 0.0488263
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 37.7998 1.33062
$$808$$ 0 0
$$809$$ −2.32597 −0.0817768 −0.0408884 0.999164i $$-0.513019\pi$$
−0.0408884 + 0.999164i $$0.513019\pi$$
$$810$$ 0 0
$$811$$ −11.8045 −0.414512 −0.207256 0.978287i $$-0.566453\pi$$
−0.207256 + 0.978287i $$0.566453\pi$$
$$812$$ 0 0
$$813$$ 30.5190 1.07035
$$814$$ 0 0
$$815$$ −16.2522 −0.569289
$$816$$ 0 0
$$817$$ −1.08686 −0.0380243
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −51.7790 −1.80710 −0.903550 0.428483i $$-0.859048\pi$$
−0.903550 + 0.428483i $$0.859048\pi$$
$$822$$ 0 0
$$823$$ −40.6259 −1.41613 −0.708064 0.706148i $$-0.750431\pi$$
−0.708064 + 0.706148i $$0.750431\pi$$
$$824$$ 0 0
$$825$$ −0.272593 −0.00949045
$$826$$ 0 0
$$827$$ 12.6201 0.438846 0.219423 0.975630i $$-0.429583\pi$$
0.219423 + 0.975630i $$0.429583\pi$$
$$828$$ 0 0
$$829$$ −50.4905 −1.75361 −0.876803 0.480849i $$-0.840328\pi$$
−0.876803 + 0.480849i $$0.840328\pi$$
$$830$$ 0 0
$$831$$ 9.49261 0.329295
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3.99714 −0.138327
$$836$$ 0 0
$$837$$ 26.5174 0.916574
$$838$$ 0 0
$$839$$ −4.41032 −0.152261 −0.0761305 0.997098i $$-0.524257\pi$$
−0.0761305 + 0.997098i $$0.524257\pi$$
$$840$$ 0 0
$$841$$ 0.448205 0.0154553
$$842$$ 0 0
$$843$$ 3.70831 0.127721
$$844$$ 0 0
$$845$$ −7.83532 −0.269543
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −3.75378 −0.128829
$$850$$ 0 0
$$851$$ 10.2371 0.350924
$$852$$ 0 0
$$853$$ 20.6785 0.708018 0.354009 0.935242i $$-0.384818\pi$$
0.354009 + 0.935242i $$0.384818\pi$$
$$854$$ 0 0
$$855$$ 0.415828 0.0142210
$$856$$ 0 0
$$857$$ −5.93055 −0.202584 −0.101292 0.994857i $$-0.532298\pi$$
−0.101292 + 0.994857i $$0.532298\pi$$
$$858$$ 0 0
$$859$$ 31.3832 1.07078 0.535390 0.844605i $$-0.320165\pi$$
0.535390 + 0.844605i $$0.320165\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 26.1710 0.890871 0.445435 0.895314i $$-0.353049\pi$$
0.445435 + 0.895314i $$0.353049\pi$$
$$864$$ 0 0
$$865$$ 17.6214 0.599145
$$866$$ 0 0
$$867$$ −30.4013 −1.03248
$$868$$ 0 0
$$869$$ 0.0259961 0.000881857 0
$$870$$ 0 0
$$871$$ −18.5436 −0.628327
$$872$$ 0 0
$$873$$ −6.86527 −0.232354
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −52.2304 −1.76369 −0.881847 0.471536i $$-0.843700\pi$$
−0.881847 + 0.471536i $$0.843700\pi$$
$$878$$ 0 0
$$879$$ 20.7509 0.699910
$$880$$ 0 0
$$881$$ 33.5930 1.13178 0.565888 0.824482i $$-0.308533\pi$$
0.565888 + 0.824482i $$0.308533\pi$$
$$882$$ 0 0
$$883$$ 30.5923 1.02951 0.514757 0.857336i $$-0.327882\pi$$
0.514757 + 0.857336i $$0.327882\pi$$
$$884$$ 0 0
$$885$$ 6.00404 0.201824
$$886$$ 0 0
$$887$$ 49.0305 1.64628 0.823141 0.567837i $$-0.192220\pi$$
0.823141 + 0.567837i $$0.192220\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −0.380163 −0.0127359
$$892$$ 0 0
$$893$$ −0.240358 −0.00804328
$$894$$ 0 0
$$895$$ 25.2811 0.845054
$$896$$ 0 0
$$897$$ −23.8686 −0.796951
$$898$$ 0 0
$$899$$ 26.4319 0.881553
$$900$$ 0 0
$$901$$ −13.9466 −0.464629
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.4950 0.614796
$$906$$ 0 0
$$907$$ −36.9013 −1.22529 −0.612644 0.790359i $$-0.709894\pi$$
−0.612644 + 0.790359i $$0.709894\pi$$
$$908$$ 0 0
$$909$$ −11.8060 −0.391582
$$910$$ 0 0
$$911$$ 34.6118 1.14674 0.573371 0.819296i $$-0.305636\pi$$
0.573371 + 0.819296i $$0.305636\pi$$
$$912$$ 0 0
$$913$$ 2.00779 0.0664483
$$914$$ 0 0
$$915$$ −10.5236 −0.347899
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −30.8170 −1.01656 −0.508279 0.861192i $$-0.669718\pi$$
−0.508279 + 0.861192i $$0.669718\pi$$
$$920$$ 0 0
$$921$$ −2.49894 −0.0823429
$$922$$ 0 0
$$923$$ 20.6023 0.678134
$$924$$ 0 0
$$925$$ −1.15403 −0.0379442
$$926$$ 0 0
$$927$$ 16.9099 0.555395
$$928$$ 0 0
$$929$$ −17.3387 −0.568865 −0.284433 0.958696i $$-0.591805\pi$$
−0.284433 + 0.958696i $$0.591805\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −24.6444 −0.806823
$$934$$ 0 0
$$935$$ 1.50407 0.0491883
$$936$$ 0 0
$$937$$ 44.3736 1.44962 0.724811 0.688948i $$-0.241927\pi$$
0.724811 + 0.688948i $$0.241927\pi$$
$$938$$ 0 0
$$939$$ 25.3206 0.826307
$$940$$ 0 0
$$941$$ 1.65934 0.0540929 0.0270465 0.999634i $$-0.491390\pi$$
0.0270465 + 0.999634i $$0.491390\pi$$
$$942$$ 0 0
$$943$$ −39.3275 −1.28068
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −37.4486 −1.21692 −0.608458 0.793586i $$-0.708211\pi$$
−0.608458 + 0.793586i $$0.708211\pi$$
$$948$$ 0 0
$$949$$ −13.6573 −0.443333
$$950$$ 0 0
$$951$$ −18.3409 −0.594746
$$952$$ 0 0
$$953$$ 37.5875 1.21758 0.608789 0.793332i $$-0.291656\pi$$
0.608789 + 0.793332i $$0.291656\pi$$
$$954$$ 0 0
$$955$$ 19.8658 0.642842
$$956$$ 0 0
$$957$$ −1.47926 −0.0478176
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −7.27545 −0.234692
$$962$$ 0 0
$$963$$ 21.5215 0.693519
$$964$$ 0 0
$$965$$ −11.4335 −0.368058
$$966$$ 0 0
$$967$$ −24.5757 −0.790302 −0.395151 0.918616i $$-0.629308\pi$$
−0.395151 + 0.918616i $$0.629308\pi$$
$$968$$ 0 0
$$969$$ 2.01245 0.0646494
$$970$$ 0 0
$$971$$ 48.6354 1.56078 0.780392 0.625290i $$-0.215019\pi$$
0.780392 + 0.625290i $$0.215019\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 2.69070 0.0861714
$$976$$ 0 0
$$977$$ 2.33743 0.0747811 0.0373906 0.999301i $$-0.488095\pi$$
0.0373906 + 0.999301i $$0.488095\pi$$
$$978$$ 0 0
$$979$$ 3.66465 0.117123
$$980$$ 0 0
$$981$$ −13.5937 −0.434014
$$982$$ 0 0
$$983$$ 28.7331 0.916442 0.458221 0.888838i $$-0.348487\pi$$
0.458221 + 0.888838i $$0.348487\pi$$
$$984$$ 0 0
$$985$$ −4.56273 −0.145381
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −37.0553 −1.17829
$$990$$ 0 0
$$991$$ −7.84787 −0.249296 −0.124648 0.992201i $$-0.539780\pi$$
−0.124648 + 0.992201i $$0.539780\pi$$
$$992$$ 0 0
$$993$$ 10.0918 0.320253
$$994$$ 0 0
$$995$$ −14.4429 −0.457871
$$996$$ 0 0
$$997$$ 34.4523 1.09112 0.545558 0.838073i $$-0.316318\pi$$
0.545558 + 0.838073i $$0.316318\pi$$
$$998$$ 0 0
$$999$$ −6.28272 −0.198776
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 1960.2.a.y.1.2 yes 4
4.3 odd 2 3920.2.a.cd.1.3 4
5.4 even 2 9800.2.a.cl.1.3 4
7.2 even 3 1960.2.q.x.361.3 8
7.3 odd 6 1960.2.q.y.961.2 8
7.4 even 3 1960.2.q.x.961.3 8
7.5 odd 6 1960.2.q.y.361.2 8
7.6 odd 2 1960.2.a.x.1.3 4
28.27 even 2 3920.2.a.ce.1.2 4
35.34 odd 2 9800.2.a.cs.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.a.x.1.3 4 7.6 odd 2
1960.2.a.y.1.2 yes 4 1.1 even 1 trivial
1960.2.q.x.361.3 8 7.2 even 3
1960.2.q.x.961.3 8 7.4 even 3
1960.2.q.y.361.2 8 7.5 odd 6
1960.2.q.y.961.2 8 7.3 odd 6
3920.2.a.cd.1.3 4 4.3 odd 2
3920.2.a.ce.1.2 4 28.27 even 2
9800.2.a.cl.1.3 4 5.4 even 2
9800.2.a.cs.1.2 4 35.34 odd 2