Properties

 Label 4600.2.e.r.4049.6 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.24681024.1 Defining polynomial: $$x^{6} + 12x^{4} + 36x^{2} + 9$$ x^6 + 12*x^4 + 36*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 4049.6 Root $$2.66908i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.r.4049.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+2.66908i q^{3} -3.66908i q^{7} -4.12398 q^{9} +O(q^{10})$$ $$q+2.66908i q^{3} -3.66908i q^{7} -4.12398 q^{9} -1.21417 q^{11} +2.21417i q^{13} -1.21417i q^{17} +2.57889 q^{19} +9.79306 q^{21} +1.00000i q^{23} -3.00000i q^{27} -1.45490 q^{29} -6.46214 q^{31} -3.24073i q^{33} -4.00000i q^{37} -5.90981 q^{39} -10.9170 q^{41} +6.90981i q^{43} -5.45490i q^{47} -6.46214 q^{49} +3.24073 q^{51} +3.81962i q^{53} +6.88325i q^{57} +4.24797 q^{59} +6.78583 q^{61} +15.1312i q^{63} -12.8567i q^{67} -2.66908 q^{69} -6.91705 q^{71} -15.2214i q^{73} +4.45490i q^{77} -4.36471 q^{81} -15.5861i q^{83} -3.88325i q^{87} +10.9098 q^{89} +8.12398 q^{91} -17.2480i q^{93} -1.69563i q^{97} +5.00724 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^9 $$6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100})$$ 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ 2.66908i 1.54099i 0.637444 + 0.770497i $$0.279992\pi$$
−0.637444 + 0.770497i $$0.720008\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.66908i − 1.38678i −0.720562 0.693391i $$-0.756116\pi$$
0.720562 0.693391i $$-0.243884\pi$$
$$8$$ 0 0
$$9$$ −4.12398 −1.37466
$$10$$ 0 0
$$11$$ −1.21417 −0.366088 −0.183044 0.983105i $$-0.558595\pi$$
−0.183044 + 0.983105i $$0.558595\pi$$
$$12$$ 0 0
$$13$$ 2.21417i 0.614102i 0.951693 + 0.307051i $$0.0993422\pi$$
−0.951693 + 0.307051i $$0.900658\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 1.21417i − 0.294481i −0.989101 0.147240i $$-0.952961\pi$$
0.989101 0.147240i $$-0.0470391\pi$$
$$18$$ 0 0
$$19$$ 2.57889 0.591637 0.295819 0.955244i $$-0.404408\pi$$
0.295819 + 0.955244i $$0.404408\pi$$
$$20$$ 0 0
$$21$$ 9.79306 2.13702
$$22$$ 0 0
$$23$$ 1.00000i 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 3.00000i − 0.577350i
$$28$$ 0 0
$$29$$ −1.45490 −0.270169 −0.135084 0.990834i $$-0.543131\pi$$
−0.135084 + 0.990834i $$0.543131\pi$$
$$30$$ 0 0
$$31$$ −6.46214 −1.16063 −0.580317 0.814390i $$-0.697072\pi$$
−0.580317 + 0.814390i $$0.697072\pi$$
$$32$$ 0 0
$$33$$ − 3.24073i − 0.564139i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ 0 0
$$39$$ −5.90981 −0.946327
$$40$$ 0 0
$$41$$ −10.9170 −1.70496 −0.852478 0.522763i $$-0.824901\pi$$
−0.852478 + 0.522763i $$0.824901\pi$$
$$42$$ 0 0
$$43$$ 6.90981i 1.05374i 0.849947 + 0.526868i $$0.176634\pi$$
−0.849947 + 0.526868i $$0.823366\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 5.45490i − 0.795680i −0.917455 0.397840i $$-0.869760\pi$$
0.917455 0.397840i $$-0.130240\pi$$
$$48$$ 0 0
$$49$$ −6.46214 −0.923163
$$50$$ 0 0
$$51$$ 3.24073 0.453793
$$52$$ 0 0
$$53$$ 3.81962i 0.524665i 0.964978 + 0.262332i $$0.0844917\pi$$
−0.964978 + 0.262332i $$0.915508\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.88325i 0.911709i
$$58$$ 0 0
$$59$$ 4.24797 0.553038 0.276519 0.961008i $$-0.410819\pi$$
0.276519 + 0.961008i $$0.410819\pi$$
$$60$$ 0 0
$$61$$ 6.78583 0.868836 0.434418 0.900711i $$-0.356954\pi$$
0.434418 + 0.900711i $$0.356954\pi$$
$$62$$ 0 0
$$63$$ 15.1312i 1.90635i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.8567i − 1.57070i −0.619055 0.785348i $$-0.712484\pi$$
0.619055 0.785348i $$-0.287516\pi$$
$$68$$ 0 0
$$69$$ −2.66908 −0.321319
$$70$$ 0 0
$$71$$ −6.91705 −0.820902 −0.410451 0.911883i $$-0.634629\pi$$
−0.410451 + 0.911883i $$0.634629\pi$$
$$72$$ 0 0
$$73$$ − 15.2214i − 1.78153i −0.454463 0.890766i $$-0.650169\pi$$
0.454463 0.890766i $$-0.349831\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.45490i 0.507683i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ −4.36471 −0.484968
$$82$$ 0 0
$$83$$ − 15.5861i − 1.71080i −0.517968 0.855400i $$-0.673312\pi$$
0.517968 0.855400i $$-0.326688\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 3.88325i − 0.416329i
$$88$$ 0 0
$$89$$ 10.9098 1.15644 0.578219 0.815882i $$-0.303748\pi$$
0.578219 + 0.815882i $$0.303748\pi$$
$$90$$ 0 0
$$91$$ 8.12398 0.851625
$$92$$ 0 0
$$93$$ − 17.2480i − 1.78853i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 1.69563i − 0.172165i −0.996288 0.0860827i $$-0.972565\pi$$
0.996288 0.0860827i $$-0.0274349\pi$$
$$98$$ 0 0
$$99$$ 5.00724 0.503246
$$100$$ 0 0
$$101$$ −2.24797 −0.223681 −0.111841 0.993726i $$-0.535675\pi$$
−0.111841 + 0.993726i $$0.535675\pi$$
$$102$$ 0 0
$$103$$ − 2.78583i − 0.274495i −0.990537 0.137248i $$-0.956174\pi$$
0.990537 0.137248i $$-0.0438256\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 13.1578i − 1.27201i −0.771685 0.636005i $$-0.780586\pi$$
0.771685 0.636005i $$-0.219414\pi$$
$$108$$ 0 0
$$109$$ −3.42111 −0.327683 −0.163842 0.986487i $$-0.552389\pi$$
−0.163842 + 0.986487i $$0.552389\pi$$
$$110$$ 0 0
$$111$$ 10.6763 1.01335
$$112$$ 0 0
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 9.13122i − 0.844182i
$$118$$ 0 0
$$119$$ −4.45490 −0.408380
$$120$$ 0 0
$$121$$ −9.52578 −0.865980
$$122$$ 0 0
$$123$$ − 29.1385i − 2.62733i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 19.9508i − 1.77035i −0.465258 0.885175i $$-0.654038\pi$$
0.465258 0.885175i $$-0.345962\pi$$
$$128$$ 0 0
$$129$$ −18.4428 −1.62380
$$130$$ 0 0
$$131$$ −9.70287 −0.847744 −0.423872 0.905722i $$-0.639329\pi$$
−0.423872 + 0.905722i $$0.639329\pi$$
$$132$$ 0 0
$$133$$ − 9.46214i − 0.820472i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 2.82685i − 0.241514i −0.992682 0.120757i $$-0.961468\pi$$
0.992682 0.120757i $$-0.0385322\pi$$
$$138$$ 0 0
$$139$$ −2.36471 −0.200572 −0.100286 0.994959i $$-0.531976\pi$$
−0.100286 + 0.994959i $$0.531976\pi$$
$$140$$ 0 0
$$141$$ 14.5596 1.22614
$$142$$ 0 0
$$143$$ − 2.68840i − 0.224815i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 17.2480i − 1.42259i
$$148$$ 0 0
$$149$$ 11.9170 0.976282 0.488141 0.872765i $$-0.337675\pi$$
0.488141 + 0.872765i $$0.337675\pi$$
$$150$$ 0 0
$$151$$ 9.57889 0.779519 0.389759 0.920917i $$-0.372558\pi$$
0.389759 + 0.920917i $$0.372558\pi$$
$$152$$ 0 0
$$153$$ 5.00724i 0.404811i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.7439i 1.49593i 0.663740 + 0.747963i $$0.268968\pi$$
−0.663740 + 0.747963i $$0.731032\pi$$
$$158$$ 0 0
$$159$$ −10.1949 −0.808505
$$160$$ 0 0
$$161$$ 3.66908 0.289164
$$162$$ 0 0
$$163$$ − 19.3188i − 1.51317i −0.653896 0.756584i $$-0.726867\pi$$
0.653896 0.756584i $$-0.273133\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.81962i 0.140806i 0.997519 + 0.0704031i $$0.0224285\pi$$
−0.997519 + 0.0704031i $$0.977571\pi$$
$$168$$ 0 0
$$169$$ 8.09743 0.622879
$$170$$ 0 0
$$171$$ −10.6353 −0.813301
$$172$$ 0 0
$$173$$ 15.9581i 1.21327i 0.794981 + 0.606635i $$0.207481\pi$$
−0.794981 + 0.606635i $$0.792519\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 11.3382i 0.852228i
$$178$$ 0 0
$$179$$ −17.9508 −1.34171 −0.670854 0.741589i $$-0.734072\pi$$
−0.670854 + 0.741589i $$0.734072\pi$$
$$180$$ 0 0
$$181$$ 6.33092 0.470574 0.235287 0.971926i $$-0.424397\pi$$
0.235287 + 0.971926i $$0.424397\pi$$
$$182$$ 0 0
$$183$$ 18.1119i 1.33887i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.47422i 0.107806i
$$188$$ 0 0
$$189$$ −11.0072 −0.800659
$$190$$ 0 0
$$191$$ −15.1578 −1.09678 −0.548389 0.836223i $$-0.684759\pi$$
−0.548389 + 0.836223i $$0.684759\pi$$
$$192$$ 0 0
$$193$$ 18.9879i 1.36678i 0.730053 + 0.683390i $$0.239495\pi$$
−0.730053 + 0.683390i $$0.760505\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 19.1650i 1.36545i 0.730675 + 0.682725i $$0.239205\pi$$
−0.730675 + 0.682725i $$0.760795\pi$$
$$198$$ 0 0
$$199$$ 5.76651 0.408777 0.204388 0.978890i $$-0.434479\pi$$
0.204388 + 0.978890i $$0.434479\pi$$
$$200$$ 0 0
$$201$$ 34.3155 2.42043
$$202$$ 0 0
$$203$$ 5.33816i 0.374665i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 4.12398i − 0.286637i
$$208$$ 0 0
$$209$$ −3.13122 −0.216591
$$210$$ 0 0
$$211$$ −17.4057 −1.19826 −0.599130 0.800652i $$-0.704487\pi$$
−0.599130 + 0.800652i $$0.704487\pi$$
$$212$$ 0 0
$$213$$ − 18.4621i − 1.26501i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 23.7101i 1.60955i
$$218$$ 0 0
$$219$$ 40.6272 2.74533
$$220$$ 0 0
$$221$$ 2.68840 0.180841
$$222$$ 0 0
$$223$$ − 14.6763i − 0.982799i −0.870934 0.491399i $$-0.836486\pi$$
0.870934 0.491399i $$-0.163514\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 16.4283i − 1.09039i −0.838310 0.545194i $$-0.816456\pi$$
0.838310 0.545194i $$-0.183544\pi$$
$$228$$ 0 0
$$229$$ −19.3526 −1.27886 −0.639429 0.768850i $$-0.720829\pi$$
−0.639429 + 0.768850i $$0.720829\pi$$
$$230$$ 0 0
$$231$$ −11.8905 −0.782337
$$232$$ 0 0
$$233$$ − 19.4549i − 1.27453i −0.770643 0.637267i $$-0.780065\pi$$
0.770643 0.637267i $$-0.219935\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −23.7029 −1.53321 −0.766606 0.642118i $$-0.778056\pi$$
−0.766606 + 0.642118i $$0.778056\pi$$
$$240$$ 0 0
$$241$$ 0.233492 0.0150405 0.00752027 0.999972i $$-0.497606\pi$$
0.00752027 + 0.999972i $$0.497606\pi$$
$$242$$ 0 0
$$243$$ − 20.6498i − 1.32468i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.71011i 0.363325i
$$248$$ 0 0
$$249$$ 41.6006 2.63633
$$250$$ 0 0
$$251$$ 21.7511 1.37292 0.686460 0.727168i $$-0.259164\pi$$
0.686460 + 0.727168i $$0.259164\pi$$
$$252$$ 0 0
$$253$$ − 1.21417i − 0.0763345i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 16.3116i 1.01749i 0.860917 + 0.508745i $$0.169890\pi$$
−0.860917 + 0.508745i $$0.830110\pi$$
$$258$$ 0 0
$$259$$ −14.6763 −0.911942
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ − 2.53786i − 0.156491i −0.996934 0.0782455i $$-0.975068\pi$$
0.996934 0.0782455i $$-0.0249318\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 29.1191i 1.78206i
$$268$$ 0 0
$$269$$ 2.54510 0.155177 0.0775886 0.996985i $$-0.475278\pi$$
0.0775886 + 0.996985i $$0.475278\pi$$
$$270$$ 0 0
$$271$$ 24.1795 1.46880 0.734400 0.678717i $$-0.237464\pi$$
0.734400 + 0.678717i $$0.237464\pi$$
$$272$$ 0 0
$$273$$ 21.6836i 1.31235i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 21.2359i 1.27594i 0.770061 + 0.637970i $$0.220226\pi$$
−0.770061 + 0.637970i $$0.779774\pi$$
$$278$$ 0 0
$$279$$ 26.6498 1.59548
$$280$$ 0 0
$$281$$ −13.8872 −0.828441 −0.414220 0.910177i $$-0.635946\pi$$
−0.414220 + 0.910177i $$0.635946\pi$$
$$282$$ 0 0
$$283$$ − 29.5330i − 1.75556i −0.479068 0.877778i $$-0.659025\pi$$
0.479068 0.877778i $$-0.340975\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 40.0555i 2.36440i
$$288$$ 0 0
$$289$$ 15.5258 0.913281
$$290$$ 0 0
$$291$$ 4.52578 0.265306
$$292$$ 0 0
$$293$$ − 24.0821i − 1.40689i −0.710750 0.703444i $$-0.751644\pi$$
0.710750 0.703444i $$-0.248356\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.64252i 0.211361i
$$298$$ 0 0
$$299$$ −2.21417 −0.128049
$$300$$ 0 0
$$301$$ 25.3526 1.46130
$$302$$ 0 0
$$303$$ − 6.00000i − 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 19.9436i − 1.13824i −0.822254 0.569121i $$-0.807284\pi$$
0.822254 0.569121i $$-0.192716\pi$$
$$308$$ 0 0
$$309$$ 7.43559 0.422996
$$310$$ 0 0
$$311$$ −2.97345 −0.168609 −0.0843043 0.996440i $$-0.526867\pi$$
−0.0843043 + 0.996440i $$0.526867\pi$$
$$312$$ 0 0
$$313$$ 15.4887i 0.875473i 0.899103 + 0.437736i $$0.144220\pi$$
−0.899103 + 0.437736i $$0.855780\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8.82685i − 0.495765i −0.968790 0.247883i $$-0.920265\pi$$
0.968790 0.247883i $$-0.0797348\pi$$
$$318$$ 0 0
$$319$$ 1.76651 0.0989055
$$320$$ 0 0
$$321$$ 35.1191 1.96016
$$322$$ 0 0
$$323$$ − 3.13122i − 0.174226i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 9.13122i − 0.504958i
$$328$$ 0 0
$$329$$ −20.0145 −1.10343
$$330$$ 0 0
$$331$$ 17.4018 0.956489 0.478245 0.878227i $$-0.341273\pi$$
0.478245 + 0.878227i $$0.341273\pi$$
$$332$$ 0 0
$$333$$ 16.4959i 0.903972i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 26.9541i − 1.46828i −0.678995 0.734142i $$-0.737584\pi$$
0.678995 0.734142i $$-0.262416\pi$$
$$338$$ 0 0
$$339$$ 26.6908 1.44964
$$340$$ 0 0
$$341$$ 7.84617 0.424894
$$342$$ 0 0
$$343$$ − 1.97345i − 0.106556i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.1240i 0.543484i 0.962370 + 0.271742i $$0.0875997\pi$$
−0.962370 + 0.271742i $$0.912400\pi$$
$$348$$ 0 0
$$349$$ −7.38732 −0.395434 −0.197717 0.980259i $$-0.563353\pi$$
−0.197717 + 0.980259i $$0.563353\pi$$
$$350$$ 0 0
$$351$$ 6.64252 0.354552
$$352$$ 0 0
$$353$$ 13.8833i 0.738931i 0.929244 + 0.369466i $$0.120459\pi$$
−0.929244 + 0.369466i $$0.879541\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 11.8905i − 0.629312i
$$358$$ 0 0
$$359$$ −15.0902 −0.796430 −0.398215 0.917292i $$-0.630370\pi$$
−0.398215 + 0.917292i $$0.630370\pi$$
$$360$$ 0 0
$$361$$ −12.3493 −0.649965
$$362$$ 0 0
$$363$$ − 25.4251i − 1.33447i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.0289i 1.25430i 0.778898 + 0.627150i $$0.215779\pi$$
−0.778898 + 0.627150i $$0.784221\pi$$
$$368$$ 0 0
$$369$$ 45.0217 2.34374
$$370$$ 0 0
$$371$$ 14.0145 0.727595
$$372$$ 0 0
$$373$$ 30.6908i 1.58911i 0.607193 + 0.794554i $$0.292296\pi$$
−0.607193 + 0.794554i $$0.707704\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 3.22141i − 0.165911i
$$378$$ 0 0
$$379$$ −2.07087 −0.106374 −0.0531869 0.998585i $$-0.516938\pi$$
−0.0531869 + 0.998585i $$0.516938\pi$$
$$380$$ 0 0
$$381$$ 53.2504 2.72810
$$382$$ 0 0
$$383$$ − 1.58612i − 0.0810472i −0.999179 0.0405236i $$-0.987097\pi$$
0.999179 0.0405236i $$-0.0129026\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 28.4959i − 1.44853i
$$388$$ 0 0
$$389$$ −21.2142 −1.07560 −0.537801 0.843072i $$-0.680745\pi$$
−0.537801 + 0.843072i $$0.680745\pi$$
$$390$$ 0 0
$$391$$ 1.21417 0.0614035
$$392$$ 0 0
$$393$$ − 25.8977i − 1.30637i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.93965i 0.0973485i 0.998815 + 0.0486742i $$0.0154996\pi$$
−0.998815 + 0.0486742i $$0.984500\pi$$
$$398$$ 0 0
$$399$$ 25.2552 1.26434
$$400$$ 0 0
$$401$$ −5.69893 −0.284591 −0.142295 0.989824i $$-0.545448\pi$$
−0.142295 + 0.989824i $$0.545448\pi$$
$$402$$ 0 0
$$403$$ − 14.3083i − 0.712748i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.85670i 0.240738i
$$408$$ 0 0
$$409$$ −3.53786 −0.174936 −0.0874679 0.996167i $$-0.527878\pi$$
−0.0874679 + 0.996167i $$0.527878\pi$$
$$410$$ 0 0
$$411$$ 7.54510 0.372172
$$412$$ 0 0
$$413$$ − 15.5861i − 0.766943i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 6.31160i − 0.309081i
$$418$$ 0 0
$$419$$ −28.2624 −1.38071 −0.690355 0.723471i $$-0.742546\pi$$
−0.690355 + 0.723471i $$0.742546\pi$$
$$420$$ 0 0
$$421$$ 34.0974 1.66181 0.830904 0.556417i $$-0.187824\pi$$
0.830904 + 0.556417i $$0.187824\pi$$
$$422$$ 0 0
$$423$$ 22.4959i 1.09379i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 24.8977i − 1.20489i
$$428$$ 0 0
$$429$$ 7.17554 0.346438
$$430$$ 0 0
$$431$$ −13.2850 −0.639918 −0.319959 0.947431i $$-0.603669\pi$$
−0.319959 + 0.947431i $$0.603669\pi$$
$$432$$ 0 0
$$433$$ 11.0603i 0.531526i 0.964038 + 0.265763i $$0.0856239\pi$$
−0.964038 + 0.265763i $$0.914376\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.57889i 0.123365i
$$438$$ 0 0
$$439$$ 19.1916 0.915963 0.457982 0.888962i $$-0.348573\pi$$
0.457982 + 0.888962i $$0.348573\pi$$
$$440$$ 0 0
$$441$$ 26.6498 1.26904
$$442$$ 0 0
$$443$$ 24.2818i 1.15366i 0.816864 + 0.576831i $$0.195711\pi$$
−0.816864 + 0.576831i $$0.804289\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 31.8075i 1.50444i
$$448$$ 0 0
$$449$$ −14.8413 −0.700406 −0.350203 0.936674i $$-0.613887\pi$$
−0.350203 + 0.936674i $$0.613887\pi$$
$$450$$ 0 0
$$451$$ 13.2552 0.624163
$$452$$ 0 0
$$453$$ 25.5668i 1.20123i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 15.5861i − 0.729088i −0.931186 0.364544i $$-0.881225\pi$$
0.931186 0.364544i $$-0.118775\pi$$
$$458$$ 0 0
$$459$$ −3.64252 −0.170019
$$460$$ 0 0
$$461$$ −14.1312 −0.658157 −0.329078 0.944303i $$-0.606738\pi$$
−0.329078 + 0.944303i $$0.606738\pi$$
$$462$$ 0 0
$$463$$ 26.7584i 1.24357i 0.783189 + 0.621784i $$0.213592\pi$$
−0.783189 + 0.621784i $$0.786408\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 26.6908i − 1.23510i −0.786531 0.617551i $$-0.788125\pi$$
0.786531 0.617551i $$-0.211875\pi$$
$$468$$ 0 0
$$469$$ −47.1722 −2.17821
$$470$$ 0 0
$$471$$ −50.0289 −2.30521
$$472$$ 0 0
$$473$$ − 8.38972i − 0.385760i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 15.7520i − 0.721236i
$$478$$ 0 0
$$479$$ −4.90981 −0.224335 −0.112167 0.993689i $$-0.535779\pi$$
−0.112167 + 0.993689i $$0.535779\pi$$
$$480$$ 0 0
$$481$$ 8.85670 0.403831
$$482$$ 0 0
$$483$$ 9.79306i 0.445600i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 16.7786i − 0.760310i −0.924923 0.380155i $$-0.875871\pi$$
0.924923 0.380155i $$-0.124129\pi$$
$$488$$ 0 0
$$489$$ 51.5635 2.33178
$$490$$ 0 0
$$491$$ 41.4839 1.87214 0.936070 0.351814i $$-0.114435\pi$$
0.936070 + 0.351814i $$0.114435\pi$$
$$492$$ 0 0
$$493$$ 1.76651i 0.0795595i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 25.3792i 1.13841i
$$498$$ 0 0
$$499$$ 6.47751 0.289973 0.144987 0.989434i $$-0.453686\pi$$
0.144987 + 0.989434i $$0.453686\pi$$
$$500$$ 0 0
$$501$$ −4.85670 −0.216981
$$502$$ 0 0
$$503$$ 37.3825i 1.66680i 0.552669 + 0.833401i $$0.313610\pi$$
−0.552669 + 0.833401i $$0.686390\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 21.6127i 0.959853i
$$508$$ 0 0
$$509$$ 15.0410 0.666682 0.333341 0.942806i $$-0.391824\pi$$
0.333341 + 0.942806i $$0.391824\pi$$
$$510$$ 0 0
$$511$$ −55.8486 −2.47060
$$512$$ 0 0
$$513$$ − 7.73666i − 0.341582i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6.62321i 0.291288i
$$518$$ 0 0
$$519$$ −42.5934 −1.86964
$$520$$ 0 0
$$521$$ 15.9469 0.698646 0.349323 0.937002i $$-0.386412\pi$$
0.349323 + 0.937002i $$0.386412\pi$$
$$522$$ 0 0
$$523$$ 10.8567i 0.474730i 0.971420 + 0.237365i $$0.0762838\pi$$
−0.971420 + 0.237365i $$0.923716\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.84617i 0.341785i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −17.5185 −0.760240
$$532$$ 0 0
$$533$$ − 24.1722i − 1.04702i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 47.9122i − 2.06756i
$$538$$ 0 0
$$539$$ 7.84617 0.337958
$$540$$ 0 0
$$541$$ −30.3792 −1.30610 −0.653052 0.757313i $$-0.726512\pi$$
−0.653052 + 0.757313i $$0.726512\pi$$
$$542$$ 0 0
$$543$$ 16.8977i 0.725151i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 28.4090i − 1.21468i −0.794441 0.607341i $$-0.792236\pi$$
0.794441 0.607341i $$-0.207764\pi$$
$$548$$ 0 0
$$549$$ −27.9846 −1.19435
$$550$$ 0 0
$$551$$ −3.75203 −0.159842
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 36.7584i − 1.55750i −0.627333 0.778751i $$-0.715853\pi$$
0.627333 0.778751i $$-0.284147\pi$$
$$558$$ 0 0
$$559$$ −15.2995 −0.647101
$$560$$ 0 0
$$561$$ −3.93481 −0.166128
$$562$$ 0 0
$$563$$ − 3.03708i − 0.127998i −0.997950 0.0639989i $$-0.979615\pi$$
0.997950 0.0639989i $$-0.0203854\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 16.0145i 0.672545i
$$568$$ 0 0
$$569$$ 32.2093 1.35029 0.675143 0.737687i $$-0.264082\pi$$
0.675143 + 0.737687i $$0.264082\pi$$
$$570$$ 0 0
$$571$$ 27.7777 1.16246 0.581230 0.813739i $$-0.302572\pi$$
0.581230 + 0.813739i $$0.302572\pi$$
$$572$$ 0 0
$$573$$ − 40.4573i − 1.69013i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11.2890i − 0.469967i −0.971999 0.234984i $$-0.924496\pi$$
0.971999 0.234984i $$-0.0755036\pi$$
$$578$$ 0 0
$$579$$ −50.6803 −2.10620
$$580$$ 0 0
$$581$$ −57.1867 −2.37251
$$582$$ 0 0
$$583$$ − 4.63768i − 0.192073i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 23.6609i − 0.976592i −0.872678 0.488296i $$-0.837619\pi$$
0.872678 0.488296i $$-0.162381\pi$$
$$588$$ 0 0
$$589$$ −16.6651 −0.686675
$$590$$ 0 0
$$591$$ −51.1529 −2.10415
$$592$$ 0 0
$$593$$ − 17.8341i − 0.732358i −0.930544 0.366179i $$-0.880666\pi$$
0.930544 0.366179i $$-0.119334\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 15.3913i 0.629923i
$$598$$ 0 0
$$599$$ 23.8308 0.973700 0.486850 0.873486i $$-0.338146\pi$$
0.486850 + 0.873486i $$0.338146\pi$$
$$600$$ 0 0
$$601$$ 12.5297 0.511098 0.255549 0.966796i $$-0.417744\pi$$
0.255549 + 0.966796i $$0.417744\pi$$
$$602$$ 0 0
$$603$$ 53.0208i 2.15917i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23.1722i 0.940533i 0.882525 + 0.470266i $$0.155842\pi$$
−0.882525 + 0.470266i $$0.844158\pi$$
$$608$$ 0 0
$$609$$ −14.2480 −0.577357
$$610$$ 0 0
$$611$$ 12.0781 0.488628
$$612$$ 0 0
$$613$$ − 44.5780i − 1.80049i −0.435386 0.900244i $$-0.643388\pi$$
0.435386 0.900244i $$-0.356612\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 32.1385i 1.29385i 0.762556 + 0.646923i $$0.223944\pi$$
−0.762556 + 0.646923i $$0.776056\pi$$
$$618$$ 0 0
$$619$$ −31.5708 −1.26894 −0.634468 0.772949i $$-0.718781\pi$$
−0.634468 + 0.772949i $$0.718781\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ − 40.0289i − 1.60373i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 8.35748i − 0.333765i
$$628$$ 0 0
$$629$$ −4.85670 −0.193649
$$630$$ 0 0
$$631$$ 24.0145 0.956001 0.478001 0.878360i $$-0.341362\pi$$
0.478001 + 0.878360i $$0.341362\pi$$
$$632$$ 0 0
$$633$$ − 46.4573i − 1.84651i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 14.3083i − 0.566916i
$$638$$ 0 0
$$639$$ 28.5258 1.12846
$$640$$ 0 0
$$641$$ −10.4428 −0.412467 −0.206233 0.978503i $$-0.566121\pi$$
−0.206233 + 0.978503i $$0.566121\pi$$
$$642$$ 0 0
$$643$$ 4.39940i 0.173495i 0.996230 + 0.0867477i $$0.0276474\pi$$
−0.996230 + 0.0867477i $$0.972353\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 23.6498i 0.929768i 0.885372 + 0.464884i $$0.153904\pi$$
−0.885372 + 0.464884i $$0.846096\pi$$
$$648$$ 0 0
$$649$$ −5.15777 −0.202460
$$650$$ 0 0
$$651$$ −63.2842 −2.48030
$$652$$ 0 0
$$653$$ 40.9194i 1.60130i 0.599131 + 0.800651i $$0.295513\pi$$
−0.599131 + 0.800651i $$0.704487\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 62.7728i 2.44900i
$$658$$ 0 0
$$659$$ −48.8567 −1.90319 −0.951593 0.307360i $$-0.900555\pi$$
−0.951593 + 0.307360i $$0.900555\pi$$
$$660$$ 0 0
$$661$$ −12.1529 −0.472694 −0.236347 0.971669i $$-0.575950\pi$$
−0.236347 + 0.971669i $$0.575950\pi$$
$$662$$ 0 0
$$663$$ 7.17554i 0.278675i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 1.45490i − 0.0563341i
$$668$$ 0 0
$$669$$ 39.1722 1.51449
$$670$$ 0 0
$$671$$ −8.23918 −0.318070
$$672$$ 0 0
$$673$$ 9.20694i 0.354901i 0.984130 + 0.177451i $$0.0567850\pi$$
−0.984130 + 0.177451i $$0.943215\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 39.8196i 1.53039i 0.643797 + 0.765196i $$0.277358\pi$$
−0.643797 + 0.765196i $$0.722642\pi$$
$$678$$ 0 0
$$679$$ −6.22141 −0.238756
$$680$$ 0 0
$$681$$ 43.8486 1.68028
$$682$$ 0 0
$$683$$ − 5.60544i − 0.214486i −0.994233 0.107243i $$-0.965798\pi$$
0.994233 0.107243i $$-0.0342023\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 51.6537i − 1.97071i
$$688$$ 0 0
$$689$$ −8.45730 −0.322197
$$690$$ 0 0
$$691$$ −2.84223 −0.108123 −0.0540617 0.998538i $$-0.517217\pi$$
−0.0540617 + 0.998538i $$0.517217\pi$$
$$692$$ 0 0
$$693$$ − 18.3719i − 0.697893i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 13.2552i 0.502077i
$$698$$ 0 0
$$699$$ 51.9267 1.96405
$$700$$ 0 0
$$701$$ −24.4130 −0.922065 −0.461033 0.887383i $$-0.652521\pi$$
−0.461033 + 0.887383i $$0.652521\pi$$
$$702$$ 0 0
$$703$$ − 10.3155i − 0.389058i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.24797i 0.310197i
$$708$$ 0 0
$$709$$ −0.123983 −0.00465629 −0.00232814 0.999997i $$-0.500741\pi$$
−0.00232814 + 0.999997i $$0.500741\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 6.46214i − 0.242009i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 63.2648i − 2.36267i
$$718$$ 0 0
$$719$$ −47.1569 −1.75865 −0.879327 0.476218i $$-0.842007\pi$$
−0.879327 + 0.476218i $$0.842007\pi$$
$$720$$ 0 0
$$721$$ −10.2214 −0.380665
$$722$$ 0 0
$$723$$ 0.623208i 0.0231774i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 43.2962i − 1.60577i −0.596135 0.802884i $$-0.703298\pi$$
0.596135 0.802884i $$-0.296702\pi$$
$$728$$ 0 0
$$729$$ 42.0217 1.55636
$$730$$ 0 0
$$731$$ 8.38972 0.310305
$$732$$ 0 0
$$733$$ 11.5861i 0.427943i 0.976840 + 0.213972i $$0.0686400\pi$$
−0.976840 + 0.213972i $$0.931360\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15.6103i 0.575012i
$$738$$ 0 0
$$739$$ 30.3792 1.11752 0.558758 0.829331i $$-0.311278\pi$$
0.558758 + 0.829331i $$0.311278\pi$$
$$740$$ 0 0
$$741$$ −15.2407 −0.559882
$$742$$ 0 0
$$743$$ 35.5708i 1.30496i 0.757804 + 0.652482i $$0.226272\pi$$
−0.757804 + 0.652482i $$0.773728\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 64.2769i 2.35177i
$$748$$ 0 0
$$749$$ −48.2769 −1.76400
$$750$$ 0 0
$$751$$ −24.9774 −0.911438 −0.455719 0.890124i $$-0.650618\pi$$
−0.455719 + 0.890124i $$0.650618\pi$$
$$752$$ 0 0
$$753$$ 58.0555i 2.11566i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 9.35263i − 0.339927i −0.985450 0.169964i $$-0.945635\pi$$
0.985450 0.169964i $$-0.0543650\pi$$
$$758$$ 0 0
$$759$$ 3.24073 0.117631
$$760$$ 0 0
$$761$$ −18.1466 −0.657813 −0.328907 0.944362i $$-0.606680\pi$$
−0.328907 + 0.944362i $$0.606680\pi$$
$$762$$ 0 0
$$763$$ 12.5523i 0.454425i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 9.40574i 0.339622i
$$768$$ 0 0
$$769$$ −44.3445 −1.59910 −0.799552 0.600597i $$-0.794930\pi$$
−0.799552 + 0.600597i $$0.794930\pi$$
$$770$$ 0 0
$$771$$ −43.5370 −1.56795
$$772$$ 0 0
$$773$$ − 10.8036i − 0.388578i −0.980944 0.194289i $$-0.937760\pi$$
0.980944 0.194289i $$-0.0622400\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 39.1722i − 1.40530i
$$778$$ 0 0
$$779$$ −28.1538 −1.00872
$$780$$ 0 0
$$781$$ 8.39850 0.300522
$$782$$ 0 0
$$783$$ 4.36471i 0.155982i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 31.7810i 1.13287i 0.824107 + 0.566435i $$0.191678\pi$$
−0.824107 + 0.566435i $$0.808322\pi$$
$$788$$ 0 0
$$789$$ 6.77375 0.241152
$$790$$ 0 0
$$791$$ −36.6908 −1.30457
$$792$$ 0 0
$$793$$ 15.0250i 0.533554i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 35.0594i − 1.24187i −0.783862 0.620935i $$-0.786753\pi$$
0.783862 0.620935i $$-0.213247\pi$$
$$798$$ 0 0
$$799$$ −6.62321 −0.234312
$$800$$ 0 0
$$801$$ −44.9919 −1.58971
$$802$$ 0 0
$$803$$ 18.4815i 0.652196i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.79306i 0.239127i
$$808$$ 0 0
$$809$$ 51.4911 1.81033 0.905165 0.425060i $$-0.139747\pi$$
0.905165 + 0.425060i $$0.139747\pi$$
$$810$$ 0 0
$$811$$ −19.7705 −0.694235 −0.347117 0.937822i $$-0.612839\pi$$
−0.347117 + 0.937822i $$0.612839\pi$$
$$812$$ 0 0
$$813$$ 64.5370i 2.26341i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 17.8196i 0.623429i
$$818$$ 0 0
$$819$$ −33.5032 −1.17070
$$820$$ 0 0
$$821$$ 31.8486 1.11152 0.555761 0.831342i $$-0.312427\pi$$
0.555761 + 0.831342i $$0.312427\pi$$
$$822$$ 0 0
$$823$$ 9.34210i 0.325645i 0.986655 + 0.162823i $$0.0520598\pi$$
−0.986655 + 0.162823i $$0.947940\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30.9243i 1.07534i 0.843155 + 0.537671i $$0.180696\pi$$
−0.843155 + 0.537671i $$0.819304\pi$$
$$828$$ 0 0
$$829$$ 32.5701 1.13121 0.565603 0.824678i $$-0.308643\pi$$
0.565603 + 0.824678i $$0.308643\pi$$
$$830$$ 0 0
$$831$$ −56.6803 −1.96622
$$832$$ 0 0
$$833$$ 7.84617i 0.271854i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 19.3864i 0.670093i
$$838$$ 0 0
$$839$$ −48.5635 −1.67660 −0.838299 0.545210i $$-0.816450\pi$$
−0.838299 + 0.545210i $$0.816450\pi$$
$$840$$ 0 0
$$841$$ −26.8833 −0.927009
$$842$$ 0 0
$$843$$ − 37.0660i − 1.27662i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 34.9508i 1.20092i
$$848$$ 0 0
$$849$$ 78.8260 2.70530
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ − 33.6546i − 1.15231i −0.817340 0.576156i $$-0.804552\pi$$
0.817340 0.576156i $$-0.195448\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 24.4081i − 0.833766i −0.908960 0.416883i $$-0.863122\pi$$
0.908960 0.416883i $$-0.136878\pi$$
$$858$$ 0 0
$$859$$ 11.1394 0.380070 0.190035 0.981777i $$-0.439140\pi$$
0.190035 + 0.981777i $$0.439140\pi$$
$$860$$ 0 0
$$861$$ −106.911 −3.64353
$$862$$ 0 0
$$863$$ − 15.7173i − 0.535025i −0.963554 0.267512i $$-0.913798\pi$$
0.963554 0.267512i $$-0.0862016\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 41.4395i 1.40736i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 28.4670 0.964567
$$872$$ 0 0
$$873$$ 6.99276i 0.236669i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 33.2818i 1.12385i 0.827190 + 0.561923i $$0.189938\pi$$
−0.827190 + 0.561923i $$0.810062\pi$$
$$878$$ 0 0
$$879$$ 64.2769 2.16801
$$880$$ 0 0
$$881$$ 8.99187 0.302944 0.151472 0.988462i $$-0.451599\pi$$
0.151472 + 0.988462i $$0.451599\pi$$
$$882$$ 0 0
$$883$$ 58.5258i 1.96955i 0.173836 + 0.984775i $$0.444384\pi$$
−0.173836 + 0.984775i $$0.555616\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 32.8220i − 1.10206i −0.834487 0.551028i $$-0.814236\pi$$
0.834487 0.551028i $$-0.185764\pi$$
$$888$$ 0 0
$$889$$ −73.2012 −2.45509
$$890$$ 0 0
$$891$$ 5.29952 0.177541
$$892$$ 0 0
$$893$$ − 14.0676i − 0.470754i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 5.90981i − 0.197323i
$$898$$ 0 0
$$899$$ 9.40180 0.313567
$$900$$ 0 0
$$901$$ 4.63768 0.154504
$$902$$ 0 0
$$903$$ 67.6682i 2.25186i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 28.3300i − 0.940683i −0.882484 0.470341i $$-0.844131\pi$$
0.882484 0.470341i $$-0.155869\pi$$
$$908$$ 0 0
$$909$$ 9.27058 0.307486
$$910$$ 0 0
$$911$$ 15.3382 0.508176 0.254088 0.967181i $$-0.418225\pi$$
0.254088 + 0.967181i $$0.418225\pi$$
$$912$$ 0 0
$$913$$ 18.9243i 0.626302i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 35.6006i 1.17564i
$$918$$ 0 0
$$919$$ 38.8036 1.28001 0.640006 0.768370i $$-0.278932\pi$$
0.640006 + 0.768370i $$0.278932\pi$$
$$920$$ 0 0
$$921$$ 53.2310 1.75402
$$922$$ 0 0
$$923$$ − 15.3155i − 0.504117i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 11.4887i 0.377338i
$$928$$ 0 0
$$929$$ −4.01053 −0.131581 −0.0657906 0.997833i $$-0.520957\pi$$
−0.0657906 + 0.997833i $$0.520957\pi$$
$$930$$ 0 0
$$931$$ −16.6651 −0.546178
$$932$$ 0 0
$$933$$ − 7.93636i − 0.259825i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 43.6836i − 1.42708i −0.700615 0.713540i $$-0.747091\pi$$
0.700615 0.713540i $$-0.252909\pi$$
$$938$$ 0 0
$$939$$ −41.3406 −1.34910
$$940$$ 0 0
$$941$$ −24.8534 −0.810198 −0.405099 0.914273i $$-0.632763\pi$$
−0.405099 + 0.914273i $$0.632763\pi$$
$$942$$ 0 0
$$943$$ − 10.9170i − 0.355508i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 22.1690i 0.720394i 0.932876 + 0.360197i $$0.117291\pi$$
−0.932876 + 0.360197i $$0.882709\pi$$
$$948$$ 0 0
$$949$$ 33.7029 1.09404
$$950$$ 0 0
$$951$$ 23.5596 0.763971
$$952$$ 0 0
$$953$$ 15.9050i 0.515212i 0.966250 + 0.257606i $$0.0829337\pi$$
−0.966250 + 0.257606i $$0.917066\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 4.71495i 0.152413i
$$958$$ 0 0
$$959$$ −10.3719 −0.334928
$$960$$ 0 0
$$961$$ 10.7593 0.347073
$$962$$ 0 0
$$963$$ 54.2624i 1.74858i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 50.6272i − 1.62806i −0.580823 0.814030i $$-0.697269\pi$$
0.580823 0.814030i $$-0.302731\pi$$
$$968$$ 0 0
$$969$$ 8.35748 0.268481
$$970$$ 0 0
$$971$$ −52.4009 −1.68162 −0.840812 0.541327i $$-0.817922\pi$$
−0.840812 + 0.541327i $$0.817922\pi$$
$$972$$ 0 0
$$973$$ 8.67632i 0.278150i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 10.5258i − 0.336750i −0.985723 0.168375i $$-0.946148\pi$$
0.985723 0.168375i $$-0.0538519\pi$$
$$978$$ 0 0
$$979$$ −13.2464 −0.423357
$$980$$ 0 0
$$981$$ 14.1086 0.450453
$$982$$ 0 0
$$983$$ 9.44767i 0.301334i 0.988585 + 0.150667i $$0.0481421\pi$$
−0.988585 + 0.150667i $$0.951858\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 53.4202i − 1.70038i
$$988$$ 0 0
$$989$$ −6.90981 −0.219719
$$990$$ 0 0
$$991$$ 33.3252 1.05861 0.529305 0.848432i $$-0.322453\pi$$
0.529305 + 0.848432i $$0.322453\pi$$
$$992$$ 0 0
$$993$$ 46.4468i 1.47394i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 54.0289i − 1.71111i −0.517709 0.855557i $$-0.673215\pi$$
0.517709 0.855557i $$-0.326785\pi$$
$$998$$ 0 0
$$999$$ −12.0000 −0.379663
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 4600.2.e.r.4049.6 6
5.2 odd 4 920.2.a.g.1.3 3
5.3 odd 4 4600.2.a.y.1.1 3
5.4 even 2 inner 4600.2.e.r.4049.1 6
15.2 even 4 8280.2.a.bo.1.3 3
20.3 even 4 9200.2.a.cd.1.3 3
20.7 even 4 1840.2.a.t.1.1 3
40.27 even 4 7360.2.a.ca.1.3 3
40.37 odd 4 7360.2.a.cb.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.3 3 5.2 odd 4
1840.2.a.t.1.1 3 20.7 even 4
4600.2.a.y.1.1 3 5.3 odd 4
4600.2.e.r.4049.1 6 5.4 even 2 inner
4600.2.e.r.4049.6 6 1.1 even 1 trivial
7360.2.a.ca.1.3 3 40.27 even 4
7360.2.a.cb.1.1 3 40.37 odd 4
8280.2.a.bo.1.3 3 15.2 even 4
9200.2.a.cd.1.3 3 20.3 even 4