Properties

 Label 7800.2.a.bv.1.4 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.10304.1 Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 8$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$1.69230$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.82843 q^{7} +1.00000 q^{9} +0.393270 q^{11} +1.00000 q^{13} -6.21302 q^{17} -7.34271 q^{19} -2.82843 q^{21} -1.44383 q^{23} -1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -0.393270 q^{33} +6.78654 q^{37} -1.00000 q^{39} +3.77786 q^{41} -2.51428 q^{43} +3.00868 q^{47} +1.00000 q^{49} +6.21302 q^{51} -9.55573 q^{53} +7.34271 q^{57} +0.393270 q^{59} +11.0414 q^{61} +2.82843 q^{63} +15.5557 q^{67} +1.44383 q^{69} +6.56440 q^{71} +13.6150 q^{73} +1.11233 q^{77} -14.9403 q^{79} +1.00000 q^{81} +16.4633 q^{83} +0.828427 q^{87} -9.67674 q^{89} +2.82843 q^{91} -1.65685 q^{93} +0.0418875 q^{97} +0.393270 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^9 $$4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{19} - 4 q^{23} - 4 q^{27} + 8 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{37} - 4 q^{39} - 4 q^{41} - 4 q^{43} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{57} - 8 q^{59} + 12 q^{61} + 24 q^{67} + 4 q^{69} - 12 q^{71} + 24 q^{73} + 8 q^{77} - 12 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{87} - 4 q^{89} + 16 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 + 4 * q^13 - 4 * q^17 - 12 * q^19 - 4 * q^23 - 4 * q^27 + 8 * q^29 - 16 * q^31 + 8 * q^33 + 8 * q^37 - 4 * q^39 - 4 * q^41 - 4 * q^43 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^57 - 8 * q^59 + 12 * q^61 + 24 * q^67 + 4 * q^69 - 12 * q^71 + 24 * q^73 + 8 * q^77 - 12 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^87 - 4 * q^89 + 16 * q^93 + 8 * q^97 - 8 * q^99

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.393270 0.118575 0.0592877 0.998241i $$-0.481117\pi$$
0.0592877 + 0.998241i $$0.481117\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.21302 −1.50688 −0.753440 0.657517i $$-0.771607\pi$$
−0.753440 + 0.657517i $$0.771607\pi$$
$$18$$ 0 0
$$19$$ −7.34271 −1.68453 −0.842266 0.539062i $$-0.818779\pi$$
−0.842266 + 0.539062i $$0.818779\pi$$
$$20$$ 0 0
$$21$$ −2.82843 −0.617213
$$22$$ 0 0
$$23$$ −1.44383 −0.301060 −0.150530 0.988605i $$-0.548098\pi$$
−0.150530 + 0.988605i $$0.548098\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −0.828427 −0.153835 −0.0769175 0.997037i $$-0.524508\pi$$
−0.0769175 + 0.997037i $$0.524508\pi$$
$$30$$ 0 0
$$31$$ 1.65685 0.297580 0.148790 0.988869i $$-0.452462\pi$$
0.148790 + 0.988869i $$0.452462\pi$$
$$32$$ 0 0
$$33$$ −0.393270 −0.0684595
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.78654 1.11570 0.557850 0.829942i $$-0.311626\pi$$
0.557850 + 0.829942i $$0.311626\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 3.77786 0.590003 0.295002 0.955497i $$-0.404680\pi$$
0.295002 + 0.955497i $$0.404680\pi$$
$$42$$ 0 0
$$43$$ −2.51428 −0.383424 −0.191712 0.981451i $$-0.561404\pi$$
−0.191712 + 0.981451i $$0.561404\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.00868 0.438860 0.219430 0.975628i $$-0.429580\pi$$
0.219430 + 0.975628i $$0.429580\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 6.21302 0.869997
$$52$$ 0 0
$$53$$ −9.55573 −1.31258 −0.656290 0.754509i $$-0.727875\pi$$
−0.656290 + 0.754509i $$0.727875\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 7.34271 0.972565
$$58$$ 0 0
$$59$$ 0.393270 0.0511994 0.0255997 0.999672i $$-0.491850\pi$$
0.0255997 + 0.999672i $$0.491850\pi$$
$$60$$ 0 0
$$61$$ 11.0414 1.41371 0.706856 0.707357i $$-0.250113\pi$$
0.706856 + 0.707357i $$0.250113\pi$$
$$62$$ 0 0
$$63$$ 2.82843 0.356348
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 15.5557 1.90043 0.950217 0.311588i $$-0.100861\pi$$
0.950217 + 0.311588i $$0.100861\pi$$
$$68$$ 0 0
$$69$$ 1.44383 0.173817
$$70$$ 0 0
$$71$$ 6.56440 0.779051 0.389526 0.921016i $$-0.372639\pi$$
0.389526 + 0.921016i $$0.372639\pi$$
$$72$$ 0 0
$$73$$ 13.6150 1.59351 0.796756 0.604302i $$-0.206548\pi$$
0.796756 + 0.604302i $$0.206548\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.11233 0.126762
$$78$$ 0 0
$$79$$ −14.9403 −1.68092 −0.840459 0.541875i $$-0.817714\pi$$
−0.840459 + 0.541875i $$0.817714\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 16.4633 1.80708 0.903540 0.428504i $$-0.140959\pi$$
0.903540 + 0.428504i $$0.140959\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0.828427 0.0888167
$$88$$ 0 0
$$89$$ −9.67674 −1.02573 −0.512866 0.858469i $$-0.671416\pi$$
−0.512866 + 0.858469i $$0.671416\pi$$
$$90$$ 0 0
$$91$$ 2.82843 0.296500
$$92$$ 0 0
$$93$$ −1.65685 −0.171808
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.0418875 0.00425303 0.00212652 0.999998i $$-0.499323\pi$$
0.00212652 + 0.999998i $$0.499323\pi$$
$$98$$ 0 0
$$99$$ 0.393270 0.0395251
$$100$$ 0 0
$$101$$ 12.1421 1.20819 0.604094 0.796913i $$-0.293535\pi$$
0.604094 + 0.796913i $$0.293535\pi$$
$$102$$ 0 0
$$103$$ −0.0173504 −0.00170958 −0.000854792 1.00000i $$-0.500272\pi$$
−0.000854792 1.00000i $$0.500272\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.10113 0.203123 0.101562 0.994829i $$-0.467616\pi$$
0.101562 + 0.994829i $$0.467616\pi$$
$$108$$ 0 0
$$109$$ 6.21302 0.595100 0.297550 0.954706i $$-0.403831\pi$$
0.297550 + 0.954706i $$0.403831\pi$$
$$110$$ 0 0
$$111$$ −6.78654 −0.644150
$$112$$ 0 0
$$113$$ 1.42648 0.134192 0.0670961 0.997747i $$-0.478627\pi$$
0.0670961 + 0.997747i $$0.478627\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −17.5731 −1.61092
$$120$$ 0 0
$$121$$ −10.8453 −0.985940
$$122$$ 0 0
$$123$$ −3.77786 −0.340639
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1.50307 0.133376 0.0666880 0.997774i $$-0.478757\pi$$
0.0666880 + 0.997774i $$0.478757\pi$$
$$128$$ 0 0
$$129$$ 2.51428 0.221370
$$130$$ 0 0
$$131$$ 3.85699 0.336987 0.168493 0.985703i $$-0.446110\pi$$
0.168493 + 0.985703i $$0.446110\pi$$
$$132$$ 0 0
$$133$$ −20.7683 −1.80084
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.5644 1.41519 0.707596 0.706617i $$-0.249780\pi$$
0.707596 + 0.706617i $$0.249780\pi$$
$$138$$ 0 0
$$139$$ 10.6854 0.906325 0.453163 0.891428i $$-0.350296\pi$$
0.453163 + 0.891428i $$0.350296\pi$$
$$140$$ 0 0
$$141$$ −3.00868 −0.253376
$$142$$ 0 0
$$143$$ 0.393270 0.0328869
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 4.05012 0.331799 0.165900 0.986143i $$-0.446947\pi$$
0.165900 + 0.986143i $$0.446947\pi$$
$$150$$ 0 0
$$151$$ 5.02856 0.409218 0.204609 0.978844i $$-0.434408\pi$$
0.204609 + 0.978844i $$0.434408\pi$$
$$152$$ 0 0
$$153$$ −6.21302 −0.502293
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 24.4961 1.95500 0.977499 0.210940i $$-0.0676526\pi$$
0.977499 + 0.210940i $$0.0676526\pi$$
$$158$$ 0 0
$$159$$ 9.55573 0.757819
$$160$$ 0 0
$$161$$ −4.08378 −0.321847
$$162$$ 0 0
$$163$$ 15.3137 1.19946 0.599731 0.800202i $$-0.295274\pi$$
0.599731 + 0.800202i $$0.295274\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −15.7770 −1.22086 −0.610430 0.792070i $$-0.709003\pi$$
−0.610430 + 0.792070i $$0.709003\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −7.34271 −0.561511
$$172$$ 0 0
$$173$$ −14.9706 −1.13819 −0.569095 0.822272i $$-0.692706\pi$$
−0.569095 + 0.822272i $$0.692706\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −0.393270 −0.0295600
$$178$$ 0 0
$$179$$ −4.84578 −0.362190 −0.181095 0.983466i $$-0.557964\pi$$
−0.181095 + 0.983466i $$0.557964\pi$$
$$180$$ 0 0
$$181$$ −7.99912 −0.594570 −0.297285 0.954789i $$-0.596081\pi$$
−0.297285 + 0.954789i $$0.596081\pi$$
$$182$$ 0 0
$$183$$ −11.0414 −0.816207
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2.44339 −0.178679
$$188$$ 0 0
$$189$$ −2.82843 −0.205738
$$190$$ 0 0
$$191$$ −20.4260 −1.47798 −0.738988 0.673718i $$-0.764696\pi$$
−0.738988 + 0.673718i $$0.764696\pi$$
$$192$$ 0 0
$$193$$ 24.0237 1.72926 0.864630 0.502408i $$-0.167553\pi$$
0.864630 + 0.502408i $$0.167553\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.80642 −0.342444 −0.171222 0.985233i $$-0.554771\pi$$
−0.171222 + 0.985233i $$0.554771\pi$$
$$198$$ 0 0
$$199$$ −14.9403 −1.05909 −0.529546 0.848281i $$-0.677638\pi$$
−0.529546 + 0.848281i $$0.677638\pi$$
$$200$$ 0 0
$$201$$ −15.5557 −1.09722
$$202$$ 0 0
$$203$$ −2.34315 −0.164457
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1.44383 −0.100353
$$208$$ 0 0
$$209$$ −2.88767 −0.199744
$$210$$ 0 0
$$211$$ −1.82799 −0.125844 −0.0629220 0.998018i $$-0.520042\pi$$
−0.0629220 + 0.998018i $$0.520042\pi$$
$$212$$ 0 0
$$213$$ −6.56440 −0.449786
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.68629 0.318126
$$218$$ 0 0
$$219$$ −13.6150 −0.920014
$$220$$ 0 0
$$221$$ −6.21302 −0.417933
$$222$$ 0 0
$$223$$ −7.07045 −0.473472 −0.236736 0.971574i $$-0.576078\pi$$
−0.236736 + 0.971574i $$0.576078\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5.43560 0.360773 0.180387 0.983596i $$-0.442265\pi$$
0.180387 + 0.983596i $$0.442265\pi$$
$$228$$ 0 0
$$229$$ 24.8147 1.63980 0.819899 0.572508i $$-0.194029\pi$$
0.819899 + 0.572508i $$0.194029\pi$$
$$230$$ 0 0
$$231$$ −1.11233 −0.0731863
$$232$$ 0 0
$$233$$ 23.4256 1.53466 0.767331 0.641251i $$-0.221584\pi$$
0.767331 + 0.641251i $$0.221584\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 14.9403 0.970478
$$238$$ 0 0
$$239$$ −21.7779 −1.40869 −0.704346 0.709856i $$-0.748760\pi$$
−0.704346 + 0.709856i $$0.748760\pi$$
$$240$$ 0 0
$$241$$ −15.2126 −0.979929 −0.489964 0.871742i $$-0.662990\pi$$
−0.489964 + 0.871742i $$0.662990\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −7.34271 −0.467205
$$248$$ 0 0
$$249$$ −16.4633 −1.04332
$$250$$ 0 0
$$251$$ −17.9399 −1.13236 −0.566178 0.824283i $$-0.691578\pi$$
−0.566178 + 0.824283i $$0.691578\pi$$
$$252$$ 0 0
$$253$$ −0.567816 −0.0356983
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3.88810 0.242533 0.121267 0.992620i $$-0.461304\pi$$
0.121267 + 0.992620i $$0.461304\pi$$
$$258$$ 0 0
$$259$$ 19.1952 1.19273
$$260$$ 0 0
$$261$$ −0.828427 −0.0512784
$$262$$ 0 0
$$263$$ −0.296797 −0.0183013 −0.00915064 0.999958i $$-0.502913\pi$$
−0.00915064 + 0.999958i $$0.502913\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 9.67674 0.592207
$$268$$ 0 0
$$269$$ −1.25447 −0.0764864 −0.0382432 0.999268i $$-0.512176\pi$$
−0.0382432 + 0.999268i $$0.512176\pi$$
$$270$$ 0 0
$$271$$ 9.81510 0.596225 0.298112 0.954531i $$-0.403643\pi$$
0.298112 + 0.954531i $$0.403643\pi$$
$$272$$ 0 0
$$273$$ −2.82843 −0.171184
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.35603 −0.261729 −0.130864 0.991400i $$-0.541775\pi$$
−0.130864 + 0.991400i $$0.541775\pi$$
$$278$$ 0 0
$$279$$ 1.65685 0.0991933
$$280$$ 0 0
$$281$$ 3.45207 0.205933 0.102967 0.994685i $$-0.467167\pi$$
0.102967 + 0.994685i $$0.467167\pi$$
$$282$$ 0 0
$$283$$ 2.25937 0.134306 0.0671528 0.997743i $$-0.478608\pi$$
0.0671528 + 0.997743i $$0.478608\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.6854 0.630740
$$288$$ 0 0
$$289$$ 21.6016 1.27068
$$290$$ 0 0
$$291$$ −0.0418875 −0.00245549
$$292$$ 0 0
$$293$$ 19.5358 1.14130 0.570648 0.821195i $$-0.306692\pi$$
0.570648 + 0.821195i $$0.306692\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −0.393270 −0.0228198
$$298$$ 0 0
$$299$$ −1.44383 −0.0834990
$$300$$ 0 0
$$301$$ −7.11146 −0.409898
$$302$$ 0 0
$$303$$ −12.1421 −0.697547
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −27.8142 −1.58744 −0.793721 0.608282i $$-0.791859\pi$$
−0.793721 + 0.608282i $$0.791859\pi$$
$$308$$ 0 0
$$309$$ 0.0173504 0.000987028 0
$$310$$ 0 0
$$311$$ 2.10113 0.119144 0.0595719 0.998224i $$-0.481026\pi$$
0.0595719 + 0.998224i $$0.481026\pi$$
$$312$$ 0 0
$$313$$ 9.64397 0.545109 0.272555 0.962140i $$-0.412131\pi$$
0.272555 + 0.962140i $$0.412131\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −16.1201 −0.905397 −0.452698 0.891664i $$-0.649539\pi$$
−0.452698 + 0.891664i $$0.649539\pi$$
$$318$$ 0 0
$$319$$ −0.325795 −0.0182410
$$320$$ 0 0
$$321$$ −2.10113 −0.117273
$$322$$ 0 0
$$323$$ 45.6204 2.53839
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −6.21302 −0.343581
$$328$$ 0 0
$$329$$ 8.50982 0.469161
$$330$$ 0 0
$$331$$ 13.9537 0.766962 0.383481 0.923549i $$-0.374725\pi$$
0.383481 + 0.923549i $$0.374725\pi$$
$$332$$ 0 0
$$333$$ 6.78654 0.371900
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 9.66974 0.526744 0.263372 0.964694i $$-0.415165\pi$$
0.263372 + 0.964694i $$0.415165\pi$$
$$338$$ 0 0
$$339$$ −1.42648 −0.0774759
$$340$$ 0 0
$$341$$ 0.651591 0.0352856
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.2014 0.547638 0.273819 0.961781i $$-0.411713\pi$$
0.273819 + 0.961781i $$0.411713\pi$$
$$348$$ 0 0
$$349$$ 28.7402 1.53843 0.769214 0.638992i $$-0.220648\pi$$
0.769214 + 0.638992i $$0.220648\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ 20.9067 1.11275 0.556375 0.830931i $$-0.312192\pi$$
0.556375 + 0.830931i $$0.312192\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 17.5731 0.930066
$$358$$ 0 0
$$359$$ 19.3162 1.01947 0.509736 0.860331i $$-0.329743\pi$$
0.509736 + 0.860331i $$0.329743\pi$$
$$360$$ 0 0
$$361$$ 34.9153 1.83765
$$362$$ 0 0
$$363$$ 10.8453 0.569233
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 37.6957 1.96770 0.983851 0.178990i $$-0.0572828\pi$$
0.983851 + 0.178990i $$0.0572828\pi$$
$$368$$ 0 0
$$369$$ 3.77786 0.196668
$$370$$ 0 0
$$371$$ −27.0277 −1.40321
$$372$$ 0 0
$$373$$ 22.4607 1.16297 0.581487 0.813556i $$-0.302471\pi$$
0.581487 + 0.813556i $$0.302471\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −0.828427 −0.0426662
$$378$$ 0 0
$$379$$ −24.3133 −1.24889 −0.624444 0.781069i $$-0.714675\pi$$
−0.624444 + 0.781069i $$0.714675\pi$$
$$380$$ 0 0
$$381$$ −1.50307 −0.0770046
$$382$$ 0 0
$$383$$ 37.2498 1.90338 0.951688 0.307065i $$-0.0993471\pi$$
0.951688 + 0.307065i $$0.0993471\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2.51428 −0.127808
$$388$$ 0 0
$$389$$ −0.467931 −0.0237250 −0.0118625 0.999930i $$-0.503776\pi$$
−0.0118625 + 0.999930i $$0.503776\pi$$
$$390$$ 0 0
$$391$$ 8.97056 0.453661
$$392$$ 0 0
$$393$$ −3.85699 −0.194559
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 34.5263 1.73282 0.866412 0.499329i $$-0.166420\pi$$
0.866412 + 0.499329i $$0.166420\pi$$
$$398$$ 0 0
$$399$$ 20.7683 1.03972
$$400$$ 0 0
$$401$$ 31.8781 1.59192 0.795958 0.605351i $$-0.206967\pi$$
0.795958 + 0.605351i $$0.206967\pi$$
$$402$$ 0 0
$$403$$ 1.65685 0.0825338
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.66894 0.132294
$$408$$ 0 0
$$409$$ −8.01735 −0.396432 −0.198216 0.980158i $$-0.563515\pi$$
−0.198216 + 0.980158i $$0.563515\pi$$
$$410$$ 0 0
$$411$$ −16.5644 −0.817062
$$412$$ 0 0
$$413$$ 1.11233 0.0547344
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −10.6854 −0.523267
$$418$$ 0 0
$$419$$ 8.40151 0.410440 0.205220 0.978716i $$-0.434209\pi$$
0.205220 + 0.978716i $$0.434209\pi$$
$$420$$ 0 0
$$421$$ −11.7514 −0.572728 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$422$$ 0 0
$$423$$ 3.00868 0.146287
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 31.2299 1.51132
$$428$$ 0 0
$$429$$ −0.393270 −0.0189872
$$430$$ 0 0
$$431$$ −20.5818 −0.991388 −0.495694 0.868497i $$-0.665086\pi$$
−0.495694 + 0.868497i $$0.665086\pi$$
$$432$$ 0 0
$$433$$ −5.12522 −0.246303 −0.123151 0.992388i $$-0.539300\pi$$
−0.123151 + 0.992388i $$0.539300\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 10.6016 0.507145
$$438$$ 0 0
$$439$$ −12.8868 −0.615053 −0.307526 0.951540i $$-0.599501\pi$$
−0.307526 + 0.951540i $$0.599501\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −35.1279 −1.66898 −0.834489 0.551024i $$-0.814237\pi$$
−0.834489 + 0.551024i $$0.814237\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4.05012 −0.191564
$$448$$ 0 0
$$449$$ 18.2213 0.859914 0.429957 0.902849i $$-0.358529\pi$$
0.429957 + 0.902849i $$0.358529\pi$$
$$450$$ 0 0
$$451$$ 1.48572 0.0699598
$$452$$ 0 0
$$453$$ −5.02856 −0.236262
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −23.4118 −1.09516 −0.547580 0.836753i $$-0.684451\pi$$
−0.547580 + 0.836753i $$0.684451\pi$$
$$458$$ 0 0
$$459$$ 6.21302 0.289999
$$460$$ 0 0
$$461$$ −14.2913 −0.665611 −0.332805 0.942996i $$-0.607995\pi$$
−0.332805 + 0.942996i $$0.607995\pi$$
$$462$$ 0 0
$$463$$ −23.7152 −1.10214 −0.551070 0.834459i $$-0.685780\pi$$
−0.551070 + 0.834459i $$0.685780\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 21.8989 1.01336 0.506680 0.862134i $$-0.330873\pi$$
0.506680 + 0.862134i $$0.330873\pi$$
$$468$$ 0 0
$$469$$ 43.9982 2.03165
$$470$$ 0 0
$$471$$ −24.4961 −1.12872
$$472$$ 0 0
$$473$$ −0.988790 −0.0454646
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −9.55573 −0.437527
$$478$$ 0 0
$$479$$ −17.0916 −0.780934 −0.390467 0.920617i $$-0.627686\pi$$
−0.390467 + 0.920617i $$0.627686\pi$$
$$480$$ 0 0
$$481$$ 6.78654 0.309440
$$482$$ 0 0
$$483$$ 4.08378 0.185818
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 1.87434 0.0849343 0.0424672 0.999098i $$-0.486478\pi$$
0.0424672 + 0.999098i $$0.486478\pi$$
$$488$$ 0 0
$$489$$ −15.3137 −0.692510
$$490$$ 0 0
$$491$$ −10.7704 −0.486063 −0.243031 0.970018i $$-0.578142\pi$$
−0.243031 + 0.970018i $$0.578142\pi$$
$$492$$ 0 0
$$493$$ 5.14704 0.231811
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 18.5669 0.832841
$$498$$ 0 0
$$499$$ 33.5259 1.50082 0.750412 0.660971i $$-0.229855\pi$$
0.750412 + 0.660971i $$0.229855\pi$$
$$500$$ 0 0
$$501$$ 15.7770 0.704864
$$502$$ 0 0
$$503$$ −7.58473 −0.338186 −0.169093 0.985600i $$-0.554084\pi$$
−0.169093 + 0.985600i $$0.554084\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ 38.0145 1.68497 0.842483 0.538724i $$-0.181093\pi$$
0.842483 + 0.538724i $$0.181093\pi$$
$$510$$ 0 0
$$511$$ 38.5089 1.70354
$$512$$ 0 0
$$513$$ 7.34271 0.324188
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.18322 0.0520380
$$518$$ 0 0
$$519$$ 14.9706 0.657135
$$520$$ 0 0
$$521$$ −18.2420 −0.799197 −0.399599 0.916690i $$-0.630851\pi$$
−0.399599 + 0.916690i $$0.630851\pi$$
$$522$$ 0 0
$$523$$ 10.1720 0.444791 0.222396 0.974957i $$-0.428612\pi$$
0.222396 + 0.974957i $$0.428612\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −10.2941 −0.448417
$$528$$ 0 0
$$529$$ −20.9153 −0.909363
$$530$$ 0 0
$$531$$ 0.393270 0.0170665
$$532$$ 0 0
$$533$$ 3.77786 0.163637
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4.84578 0.209111
$$538$$ 0 0
$$539$$ 0.393270 0.0169393
$$540$$ 0 0
$$541$$ 1.98923 0.0855236 0.0427618 0.999085i $$-0.486384\pi$$
0.0427618 + 0.999085i $$0.486384\pi$$
$$542$$ 0 0
$$543$$ 7.99912 0.343275
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.31817 0.0563609 0.0281804 0.999603i $$-0.491029\pi$$
0.0281804 + 0.999603i $$0.491029\pi$$
$$548$$ 0 0
$$549$$ 11.0414 0.471238
$$550$$ 0 0
$$551$$ 6.08290 0.259140
$$552$$ 0 0
$$553$$ −42.2576 −1.79698
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.0804 0.511861 0.255931 0.966695i $$-0.417618\pi$$
0.255931 + 0.966695i $$0.417618\pi$$
$$558$$ 0 0
$$559$$ −2.51428 −0.106343
$$560$$ 0 0
$$561$$ 2.44339 0.103160
$$562$$ 0 0
$$563$$ −22.9930 −0.969039 −0.484519 0.874781i $$-0.661005\pi$$
−0.484519 + 0.874781i $$0.661005\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.82843 0.118783
$$568$$ 0 0
$$569$$ 29.0370 1.21729 0.608647 0.793441i $$-0.291713\pi$$
0.608647 + 0.793441i $$0.291713\pi$$
$$570$$ 0 0
$$571$$ 5.56950 0.233076 0.116538 0.993186i $$-0.462820\pi$$
0.116538 + 0.993186i $$0.462820\pi$$
$$572$$ 0 0
$$573$$ 20.4260 0.853310
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −20.2822 −0.844357 −0.422179 0.906513i $$-0.638734\pi$$
−0.422179 + 0.906513i $$0.638734\pi$$
$$578$$ 0 0
$$579$$ −24.0237 −0.998389
$$580$$ 0 0
$$581$$ 46.5652 1.93185
$$582$$ 0 0
$$583$$ −3.75798 −0.155640
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.82377 0.364196 0.182098 0.983280i $$-0.441711\pi$$
0.182098 + 0.983280i $$0.441711\pi$$
$$588$$ 0 0
$$589$$ −12.1658 −0.501283
$$590$$ 0 0
$$591$$ 4.80642 0.197710
$$592$$ 0 0
$$593$$ 10.2213 0.419737 0.209868 0.977730i $$-0.432696\pi$$
0.209868 + 0.977730i $$0.432696\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 14.9403 0.611467
$$598$$ 0 0
$$599$$ −18.4925 −0.755582 −0.377791 0.925891i $$-0.623316\pi$$
−0.377791 + 0.925891i $$0.623316\pi$$
$$600$$ 0 0
$$601$$ 26.0129 1.06109 0.530544 0.847657i $$-0.321988\pi$$
0.530544 + 0.847657i $$0.321988\pi$$
$$602$$ 0 0
$$603$$ 15.5557 0.633478
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 24.1227 0.979109 0.489554 0.871973i $$-0.337159\pi$$
0.489554 + 0.871973i $$0.337159\pi$$
$$608$$ 0 0
$$609$$ 2.34315 0.0949491
$$610$$ 0 0
$$611$$ 3.00868 0.121718
$$612$$ 0 0
$$613$$ 34.2056 1.38155 0.690775 0.723070i $$-0.257270\pi$$
0.690775 + 0.723070i $$0.257270\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.3622 −0.497682 −0.248841 0.968544i $$-0.580050\pi$$
−0.248841 + 0.968544i $$0.580050\pi$$
$$618$$ 0 0
$$619$$ −11.6685 −0.468997 −0.234498 0.972117i $$-0.575345\pi$$
−0.234498 + 0.972117i $$0.575345\pi$$
$$620$$ 0 0
$$621$$ 1.44383 0.0579390
$$622$$ 0 0
$$623$$ −27.3700 −1.09655
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 2.88767 0.115322
$$628$$ 0 0
$$629$$ −42.1649 −1.68123
$$630$$ 0 0
$$631$$ 3.79775 0.151186 0.0755930 0.997139i $$-0.475915\pi$$
0.0755930 + 0.997139i $$0.475915\pi$$
$$632$$ 0 0
$$633$$ 1.82799 0.0726560
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 6.56440 0.259684
$$640$$ 0 0
$$641$$ −6.91008 −0.272932 −0.136466 0.990645i $$-0.543574\pi$$
−0.136466 + 0.990645i $$0.543574\pi$$
$$642$$ 0 0
$$643$$ −2.50139 −0.0986452 −0.0493226 0.998783i $$-0.515706\pi$$
−0.0493226 + 0.998783i $$0.515706\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −30.3124 −1.19170 −0.595852 0.803095i $$-0.703185\pi$$
−0.595852 + 0.803095i $$0.703185\pi$$
$$648$$ 0 0
$$649$$ 0.154661 0.00607098
$$650$$ 0 0
$$651$$ −4.68629 −0.183670
$$652$$ 0 0
$$653$$ −33.4537 −1.30915 −0.654573 0.755999i $$-0.727151\pi$$
−0.654573 + 0.755999i $$0.727151\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 13.6150 0.531170
$$658$$ 0 0
$$659$$ −31.2942 −1.21905 −0.609525 0.792767i $$-0.708640\pi$$
−0.609525 + 0.792767i $$0.708640\pi$$
$$660$$ 0 0
$$661$$ 40.1119 1.56017 0.780086 0.625672i $$-0.215175\pi$$
0.780086 + 0.625672i $$0.215175\pi$$
$$662$$ 0 0
$$663$$ 6.21302 0.241294
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.19611 0.0463136
$$668$$ 0 0
$$669$$ 7.07045 0.273359
$$670$$ 0 0
$$671$$ 4.34227 0.167631
$$672$$ 0 0
$$673$$ 4.99386 0.192499 0.0962496 0.995357i $$-0.469315\pi$$
0.0962496 + 0.995357i $$0.469315\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −16.0174 −0.615597 −0.307798 0.951452i $$-0.599592\pi$$
−0.307798 + 0.951452i $$0.599592\pi$$
$$678$$ 0 0
$$679$$ 0.118476 0.00454668
$$680$$ 0 0
$$681$$ −5.43560 −0.208292
$$682$$ 0 0
$$683$$ −35.5400 −1.35990 −0.679951 0.733258i $$-0.737999\pi$$
−0.679951 + 0.733258i $$0.737999\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −24.8147 −0.946738
$$688$$ 0 0
$$689$$ −9.55573 −0.364044
$$690$$ 0 0
$$691$$ 13.0393 0.496040 0.248020 0.968755i $$-0.420220\pi$$
0.248020 + 0.968755i $$0.420220\pi$$
$$692$$ 0 0
$$693$$ 1.11233 0.0422541
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −23.4720 −0.889064
$$698$$ 0 0
$$699$$ −23.4256 −0.886038
$$700$$ 0 0
$$701$$ −0.502632 −0.0189841 −0.00949207 0.999955i $$-0.503021\pi$$
−0.00949207 + 0.999955i $$0.503021\pi$$
$$702$$ 0 0
$$703$$ −49.8316 −1.87943
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 34.3431 1.29161
$$708$$ 0 0
$$709$$ 42.0712 1.58002 0.790009 0.613095i $$-0.210076\pi$$
0.790009 + 0.613095i $$0.210076\pi$$
$$710$$ 0 0
$$711$$ −14.9403 −0.560306
$$712$$ 0 0
$$713$$ −2.39222 −0.0895893
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 21.7779 0.813309
$$718$$ 0 0
$$719$$ 32.7848 1.22267 0.611333 0.791373i $$-0.290634\pi$$
0.611333 + 0.791373i $$0.290634\pi$$
$$720$$ 0 0
$$721$$ −0.0490743 −0.00182762
$$722$$ 0 0
$$723$$ 15.2126 0.565762
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −49.5246 −1.83677 −0.918383 0.395692i $$-0.870505\pi$$
−0.918383 + 0.395692i $$0.870505\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 15.6213 0.577774
$$732$$ 0 0
$$733$$ −2.11848 −0.0782477 −0.0391238 0.999234i $$-0.512457\pi$$
−0.0391238 + 0.999234i $$0.512457\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.11760 0.225345
$$738$$ 0 0
$$739$$ −47.7014 −1.75473 −0.877363 0.479827i $$-0.840699\pi$$
−0.877363 + 0.479827i $$0.840699\pi$$
$$740$$ 0 0
$$741$$ 7.34271 0.269741
$$742$$ 0 0
$$743$$ 32.8055 1.20352 0.601759 0.798677i $$-0.294467\pi$$
0.601759 + 0.798677i $$0.294467\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 16.4633 0.602360
$$748$$ 0 0
$$749$$ 5.94288 0.217148
$$750$$ 0 0
$$751$$ −0.852087 −0.0310931 −0.0155465 0.999879i $$-0.504949\pi$$
−0.0155465 + 0.999879i $$0.504949\pi$$
$$752$$ 0 0
$$753$$ 17.9399 0.653766
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6.68454 −0.242954 −0.121477 0.992594i $$-0.538763\pi$$
−0.121477 + 0.992594i $$0.538763\pi$$
$$758$$ 0 0
$$759$$ 0.567816 0.0206104
$$760$$ 0 0
$$761$$ −2.92578 −0.106059 −0.0530297 0.998593i $$-0.516888\pi$$
−0.0530297 + 0.998593i $$0.516888\pi$$
$$762$$ 0 0
$$763$$ 17.5731 0.636188
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0.393270 0.0142001
$$768$$ 0 0
$$769$$ −12.6274 −0.455356 −0.227678 0.973736i $$-0.573113\pi$$
−0.227678 + 0.973736i $$0.573113\pi$$
$$770$$ 0 0
$$771$$ −3.88810 −0.140027
$$772$$ 0 0
$$773$$ 15.9454 0.573517 0.286758 0.958003i $$-0.407422\pi$$
0.286758 + 0.958003i $$0.407422\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −19.1952 −0.688625
$$778$$ 0 0
$$779$$ −27.7398 −0.993880
$$780$$ 0 0
$$781$$ 2.58158 0.0923763
$$782$$ 0 0
$$783$$ 0.828427 0.0296056
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 48.5825 1.73178 0.865890 0.500234i $$-0.166753\pi$$
0.865890 + 0.500234i $$0.166753\pi$$
$$788$$ 0 0
$$789$$ 0.296797 0.0105662
$$790$$ 0 0
$$791$$ 4.03470 0.143457
$$792$$ 0 0
$$793$$ 11.0414 0.392093
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 19.8971 0.704792 0.352396 0.935851i $$-0.385367\pi$$
0.352396 + 0.935851i $$0.385367\pi$$
$$798$$ 0 0
$$799$$ −18.6930 −0.661310
$$800$$ 0 0
$$801$$ −9.67674 −0.341911
$$802$$ 0 0
$$803$$ 5.35436 0.188951
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.25447 0.0441595
$$808$$ 0 0
$$809$$ −12.5487 −0.441189 −0.220595 0.975366i $$-0.570800\pi$$
−0.220595 + 0.975366i $$0.570800\pi$$
$$810$$ 0 0
$$811$$ 30.9149 1.08557 0.542785 0.839872i $$-0.317370\pi$$
0.542785 + 0.839872i $$0.317370\pi$$
$$812$$ 0 0
$$813$$ −9.81510 −0.344231
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 18.4616 0.645890
$$818$$ 0 0
$$819$$ 2.82843 0.0988332
$$820$$ 0 0
$$821$$ −29.9308 −1.04459 −0.522296 0.852765i $$-0.674924\pi$$
−0.522296 + 0.852765i $$0.674924\pi$$
$$822$$ 0 0
$$823$$ −16.1831 −0.564109 −0.282055 0.959398i $$-0.591016\pi$$
−0.282055 + 0.959398i $$0.591016\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 54.2022 1.88479 0.942397 0.334497i $$-0.108566\pi$$
0.942397 + 0.334497i $$0.108566\pi$$
$$828$$ 0 0
$$829$$ −42.2929 −1.46889 −0.734447 0.678666i $$-0.762558\pi$$
−0.734447 + 0.678666i $$0.762558\pi$$
$$830$$ 0 0
$$831$$ 4.35603 0.151109
$$832$$ 0 0
$$833$$ −6.21302 −0.215268
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.65685 −0.0572693
$$838$$ 0 0
$$839$$ 36.7873 1.27004 0.635020 0.772496i $$-0.280992\pi$$
0.635020 + 0.772496i $$0.280992\pi$$
$$840$$ 0 0
$$841$$ −28.3137 −0.976335
$$842$$ 0 0
$$843$$ −3.45207 −0.118896
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −30.6753 −1.05401
$$848$$ 0 0
$$849$$ −2.25937 −0.0775414
$$850$$ 0 0
$$851$$ −9.79863 −0.335893
$$852$$ 0 0
$$853$$ −15.4148 −0.527794 −0.263897 0.964551i $$-0.585008\pi$$
−0.263897 + 0.964551i $$0.585008\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0.0628221 0.00214596 0.00107298 0.999999i $$-0.499658\pi$$
0.00107298 + 0.999999i $$0.499658\pi$$
$$858$$ 0 0
$$859$$ 28.6577 0.977787 0.488893 0.872344i $$-0.337401\pi$$
0.488893 + 0.872344i $$0.337401\pi$$
$$860$$ 0 0
$$861$$ −10.6854 −0.364158
$$862$$ 0 0
$$863$$ −5.96189 −0.202945 −0.101473 0.994838i $$-0.532355\pi$$
−0.101473 + 0.994838i $$0.532355\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −21.6016 −0.733630
$$868$$ 0 0
$$869$$ −5.87558 −0.199315
$$870$$ 0 0
$$871$$ 15.5557 0.527086
$$872$$ 0 0
$$873$$ 0.0418875 0.00141768
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −6.70276 −0.226336 −0.113168 0.993576i $$-0.536100\pi$$
−0.113168 + 0.993576i $$0.536100\pi$$
$$878$$ 0 0
$$879$$ −19.5358 −0.658928
$$880$$ 0 0
$$881$$ 39.8624 1.34300 0.671500 0.741005i $$-0.265651\pi$$
0.671500 + 0.741005i $$0.265651\pi$$
$$882$$ 0 0
$$883$$ 21.7131 0.730704 0.365352 0.930869i $$-0.380949\pi$$
0.365352 + 0.930869i $$0.380949\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 51.3757 1.72503 0.862513 0.506035i $$-0.168889\pi$$
0.862513 + 0.506035i $$0.168889\pi$$
$$888$$ 0 0
$$889$$ 4.25133 0.142585
$$890$$ 0 0
$$891$$ 0.393270 0.0131750
$$892$$ 0 0
$$893$$ −22.0918 −0.739275
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.44383 0.0482082
$$898$$ 0 0
$$899$$ −1.37258 −0.0457782
$$900$$ 0 0
$$901$$ 59.3700 1.97790
$$902$$ 0 0
$$903$$ 7.11146 0.236654
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −21.3966 −0.710463 −0.355231 0.934778i $$-0.615598\pi$$
−0.355231 + 0.934778i $$0.615598\pi$$
$$908$$ 0 0
$$909$$ 12.1421 0.402729
$$910$$ 0 0
$$911$$ −51.0044 −1.68985 −0.844925 0.534884i $$-0.820355\pi$$
−0.844925 + 0.534884i $$0.820355\pi$$
$$912$$ 0 0
$$913$$ 6.47451 0.214275
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 10.9092 0.360254
$$918$$ 0 0
$$919$$ 11.3664 0.374942 0.187471 0.982270i $$-0.439971\pi$$
0.187471 + 0.982270i $$0.439971\pi$$
$$920$$ 0 0
$$921$$ 27.8142 0.916510
$$922$$ 0 0
$$923$$ 6.56440 0.216070
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −0.0173504 −0.000569861 0
$$928$$ 0 0
$$929$$ 47.9344 1.57268 0.786338 0.617797i $$-0.211975\pi$$
0.786338 + 0.617797i $$0.211975\pi$$
$$930$$ 0 0
$$931$$ −7.34271 −0.240648
$$932$$ 0 0
$$933$$ −2.10113 −0.0687878
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −23.3266 −0.762047 −0.381023 0.924565i $$-0.624428\pi$$
−0.381023 + 0.924565i $$0.624428\pi$$
$$938$$ 0 0
$$939$$ −9.64397 −0.314719
$$940$$ 0 0
$$941$$ −39.6316 −1.29195 −0.645977 0.763357i $$-0.723550\pi$$
−0.645977 + 0.763357i $$0.723550\pi$$
$$942$$ 0 0
$$943$$ −5.45460 −0.177626
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −19.7770 −0.642666 −0.321333 0.946966i $$-0.604131\pi$$
−0.321333 + 0.946966i $$0.604131\pi$$
$$948$$ 0 0
$$949$$ 13.6150 0.441961
$$950$$ 0 0
$$951$$ 16.1201 0.522731
$$952$$ 0 0
$$953$$ −45.2847 −1.46692 −0.733458 0.679735i $$-0.762095\pi$$
−0.733458 + 0.679735i $$0.762095\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0.325795 0.0105315
$$958$$ 0 0
$$959$$ 46.8512 1.51290
$$960$$ 0 0
$$961$$ −28.2548 −0.911446
$$962$$ 0 0
$$963$$ 2.10113 0.0677078
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 13.3903 0.430603 0.215301 0.976548i $$-0.430927\pi$$
0.215301 + 0.976548i $$0.430927\pi$$
$$968$$ 0 0
$$969$$ −45.6204 −1.46554
$$970$$ 0 0
$$971$$ −39.6150 −1.27130 −0.635652 0.771975i $$-0.719269\pi$$
−0.635652 + 0.771975i $$0.719269\pi$$
$$972$$ 0 0
$$973$$ 30.2229 0.968902
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −19.2100 −0.614584 −0.307292 0.951615i $$-0.599423\pi$$
−0.307292 + 0.951615i $$0.599423\pi$$
$$978$$ 0 0
$$979$$ −3.80557 −0.121627
$$980$$ 0 0
$$981$$ 6.21302 0.198367
$$982$$ 0 0
$$983$$ −32.3960 −1.03327 −0.516636 0.856205i $$-0.672816\pi$$
−0.516636 + 0.856205i $$0.672816\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −8.50982 −0.270871
$$988$$ 0 0
$$989$$ 3.63020 0.115434
$$990$$ 0 0
$$991$$ 10.5107 0.333883 0.166942 0.985967i $$-0.446611\pi$$
0.166942 + 0.985967i $$0.446611\pi$$
$$992$$ 0 0
$$993$$ −13.9537 −0.442806
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 41.0753 1.30087 0.650433 0.759563i $$-0.274587\pi$$
0.650433 + 0.759563i $$0.274587\pi$$
$$998$$ 0 0
$$999$$ −6.78654 −0.214717
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 7800.2.a.bv.1.4 4
5.2 odd 4 1560.2.l.e.1249.6 yes 8
5.3 odd 4 1560.2.l.e.1249.2 8
5.4 even 2 7800.2.a.bw.1.2 4
15.2 even 4 4680.2.l.f.2809.6 8
15.8 even 4 4680.2.l.f.2809.5 8
20.3 even 4 3120.2.l.o.1249.6 8
20.7 even 4 3120.2.l.o.1249.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.e.1249.2 8 5.3 odd 4
1560.2.l.e.1249.6 yes 8 5.2 odd 4
3120.2.l.o.1249.2 8 20.7 even 4
3120.2.l.o.1249.6 8 20.3 even 4
4680.2.l.f.2809.5 8 15.8 even 4
4680.2.l.f.2809.6 8 15.2 even 4
7800.2.a.bv.1.4 4 1.1 even 1 trivial
7800.2.a.bw.1.2 4 5.4 even 2