Properties

 Label 441.6.a.w.1.4 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 97 x^{2} + 7 x + 294$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$10.1812$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} +O(q^{10})$$ $$q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} -202.644 q^{10} -416.710 q^{11} +797.918 q^{13} +37.3245 q^{16} +1375.55 q^{17} +2313.03 q^{19} -1154.23 q^{20} -3825.91 q^{22} +955.402 q^{23} -2637.84 q^{25} +7325.87 q^{26} +7035.29 q^{29} +1261.19 q^{31} -5619.96 q^{32} +12629.2 q^{34} +9776.44 q^{37} +21236.4 q^{38} -4112.66 q^{40} +5400.95 q^{41} +19686.6 q^{43} -21791.8 q^{44} +8771.76 q^{46} -2056.56 q^{47} -24218.7 q^{50} +41727.1 q^{52} -18022.7 q^{53} +9197.45 q^{55} +64592.6 q^{58} -7435.68 q^{59} +3495.38 q^{61} +11579.3 q^{62} -52792.5 q^{64} -17611.3 q^{65} +15856.4 q^{67} +71934.4 q^{68} -58133.5 q^{71} +39110.7 q^{73} +89759.8 q^{74} +120960. q^{76} +9760.69 q^{79} -823.812 q^{80} +49587.4 q^{82} +70395.7 q^{83} -30360.6 q^{85} +180747. q^{86} -77646.6 q^{88} -144306. q^{89} +49962.7 q^{92} -18881.8 q^{94} -51052.2 q^{95} -79328.7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + O(q^{10})$$ $$4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 9.18123 1.62303 0.811514 0.584333i $$-0.198644\pi$$
0.811514 + 0.584333i $$0.198644\pi$$
$$3$$ 0 0
$$4$$ 52.2950 1.63422
$$5$$ −22.0716 −0.394829 −0.197414 0.980320i $$-0.563254\pi$$
−0.197414 + 0.980320i $$0.563254\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 186.333 1.02935
$$9$$ 0 0
$$10$$ −202.644 −0.640818
$$11$$ −416.710 −1.03837 −0.519184 0.854662i $$-0.673764\pi$$
−0.519184 + 0.854662i $$0.673764\pi$$
$$12$$ 0 0
$$13$$ 797.918 1.30948 0.654742 0.755853i $$-0.272777\pi$$
0.654742 + 0.755853i $$0.272777\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 37.3245 0.0364497
$$17$$ 1375.55 1.15439 0.577197 0.816605i $$-0.304146\pi$$
0.577197 + 0.816605i $$0.304146\pi$$
$$18$$ 0 0
$$19$$ 2313.03 1.46993 0.734965 0.678105i $$-0.237199\pi$$
0.734965 + 0.678105i $$0.237199\pi$$
$$20$$ −1154.23 −0.645236
$$21$$ 0 0
$$22$$ −3825.91 −1.68530
$$23$$ 955.402 0.376588 0.188294 0.982113i $$-0.439704\pi$$
0.188294 + 0.982113i $$0.439704\pi$$
$$24$$ 0 0
$$25$$ −2637.84 −0.844110
$$26$$ 7325.87 2.12533
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7035.29 1.55341 0.776707 0.629862i $$-0.216889\pi$$
0.776707 + 0.629862i $$0.216889\pi$$
$$30$$ 0 0
$$31$$ 1261.19 0.235709 0.117855 0.993031i $$-0.462398\pi$$
0.117855 + 0.993031i $$0.462398\pi$$
$$32$$ −5619.96 −0.970194
$$33$$ 0 0
$$34$$ 12629.2 1.87361
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9776.44 1.17402 0.587012 0.809579i $$-0.300304\pi$$
0.587012 + 0.809579i $$0.300304\pi$$
$$38$$ 21236.4 2.38574
$$39$$ 0 0
$$40$$ −4112.66 −0.406418
$$41$$ 5400.95 0.501777 0.250888 0.968016i $$-0.419277\pi$$
0.250888 + 0.968016i $$0.419277\pi$$
$$42$$ 0 0
$$43$$ 19686.6 1.62367 0.811837 0.583885i $$-0.198468\pi$$
0.811837 + 0.583885i $$0.198468\pi$$
$$44$$ −21791.8 −1.69692
$$45$$ 0 0
$$46$$ 8771.76 0.611213
$$47$$ −2056.56 −0.135799 −0.0678997 0.997692i $$-0.521630\pi$$
−0.0678997 + 0.997692i $$0.521630\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −24218.7 −1.37001
$$51$$ 0 0
$$52$$ 41727.1 2.13998
$$53$$ −18022.7 −0.881315 −0.440658 0.897675i $$-0.645255\pi$$
−0.440658 + 0.897675i $$0.645255\pi$$
$$54$$ 0 0
$$55$$ 9197.45 0.409978
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 64592.6 2.52123
$$59$$ −7435.68 −0.278093 −0.139047 0.990286i $$-0.544404\pi$$
−0.139047 + 0.990286i $$0.544404\pi$$
$$60$$ 0 0
$$61$$ 3495.38 0.120274 0.0601368 0.998190i $$-0.480846\pi$$
0.0601368 + 0.998190i $$0.480846\pi$$
$$62$$ 11579.3 0.382563
$$63$$ 0 0
$$64$$ −52792.5 −1.61110
$$65$$ −17611.3 −0.517022
$$66$$ 0 0
$$67$$ 15856.4 0.431537 0.215769 0.976445i $$-0.430774\pi$$
0.215769 + 0.976445i $$0.430774\pi$$
$$68$$ 71934.4 1.88653
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −58133.5 −1.36861 −0.684306 0.729195i $$-0.739895\pi$$
−0.684306 + 0.729195i $$0.739895\pi$$
$$72$$ 0 0
$$73$$ 39110.7 0.858990 0.429495 0.903069i $$-0.358692\pi$$
0.429495 + 0.903069i $$0.358692\pi$$
$$74$$ 89759.8 1.90547
$$75$$ 0 0
$$76$$ 120960. 2.40218
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 9760.69 0.175960 0.0879798 0.996122i $$-0.471959\pi$$
0.0879798 + 0.996122i $$0.471959\pi$$
$$80$$ −823.812 −0.0143914
$$81$$ 0 0
$$82$$ 49587.4 0.814397
$$83$$ 70395.7 1.12163 0.560816 0.827940i $$-0.310487\pi$$
0.560816 + 0.827940i $$0.310487\pi$$
$$84$$ 0 0
$$85$$ −30360.6 −0.455788
$$86$$ 180747. 2.63527
$$87$$ 0 0
$$88$$ −77646.6 −1.06885
$$89$$ −144306. −1.93112 −0.965562 0.260173i $$-0.916220\pi$$
−0.965562 + 0.260173i $$0.916220\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 49962.7 0.615427
$$93$$ 0 0
$$94$$ −18881.8 −0.220406
$$95$$ −51052.2 −0.580371
$$96$$ 0 0
$$97$$ −79328.7 −0.856053 −0.428027 0.903766i $$-0.640791\pi$$
−0.428027 + 0.903766i $$0.640791\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −137946. −1.37946
$$101$$ 84833.7 0.827495 0.413747 0.910392i $$-0.364220\pi$$
0.413747 + 0.910392i $$0.364220\pi$$
$$102$$ 0 0
$$103$$ 20332.3 0.188839 0.0944197 0.995532i $$-0.469900\pi$$
0.0944197 + 0.995532i $$0.469900\pi$$
$$104$$ 148678. 1.34792
$$105$$ 0 0
$$106$$ −165471. −1.43040
$$107$$ −6962.19 −0.0587877 −0.0293938 0.999568i $$-0.509358\pi$$
−0.0293938 + 0.999568i $$0.509358\pi$$
$$108$$ 0 0
$$109$$ 112651. 0.908177 0.454088 0.890957i $$-0.349965\pi$$
0.454088 + 0.890957i $$0.349965\pi$$
$$110$$ 84443.9 0.665406
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −112005. −0.825167 −0.412583 0.910920i $$-0.635373\pi$$
−0.412583 + 0.910920i $$0.635373\pi$$
$$114$$ 0 0
$$115$$ −21087.3 −0.148688
$$116$$ 367910. 2.53862
$$117$$ 0 0
$$118$$ −68268.7 −0.451353
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 12595.8 0.0782102
$$122$$ 32091.9 0.195207
$$123$$ 0 0
$$124$$ 65954.0 0.385200
$$125$$ 127195. 0.728108
$$126$$ 0 0
$$127$$ 82224.5 0.452368 0.226184 0.974085i $$-0.427375\pi$$
0.226184 + 0.974085i $$0.427375\pi$$
$$128$$ −304862. −1.64467
$$129$$ 0 0
$$130$$ −161694. −0.839140
$$131$$ 175812. 0.895097 0.447548 0.894260i $$-0.352297\pi$$
0.447548 + 0.894260i $$0.352297\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 145581. 0.700397
$$135$$ 0 0
$$136$$ 256310. 1.18828
$$137$$ −31330.3 −0.142614 −0.0713072 0.997454i $$-0.522717\pi$$
−0.0713072 + 0.997454i $$0.522717\pi$$
$$138$$ 0 0
$$139$$ 152234. 0.668305 0.334152 0.942519i $$-0.391550\pi$$
0.334152 + 0.942519i $$0.391550\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −533737. −2.22129
$$143$$ −332500. −1.35973
$$144$$ 0 0
$$145$$ −155280. −0.613332
$$146$$ 359084. 1.39416
$$147$$ 0 0
$$148$$ 511259. 1.91861
$$149$$ −362860. −1.33898 −0.669489 0.742822i $$-0.733487\pi$$
−0.669489 + 0.742822i $$0.733487\pi$$
$$150$$ 0 0
$$151$$ 205126. 0.732112 0.366056 0.930593i $$-0.380708\pi$$
0.366056 + 0.930593i $$0.380708\pi$$
$$152$$ 430992. 1.51308
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −27836.5 −0.0930648
$$156$$ 0 0
$$157$$ −77272.0 −0.250192 −0.125096 0.992145i $$-0.539924\pi$$
−0.125096 + 0.992145i $$0.539924\pi$$
$$158$$ 89615.1 0.285587
$$159$$ 0 0
$$160$$ 124042. 0.383060
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 184931. 0.545182 0.272591 0.962130i $$-0.412119\pi$$
0.272591 + 0.962130i $$0.412119\pi$$
$$164$$ 282442. 0.820012
$$165$$ 0 0
$$166$$ 646319. 1.82044
$$167$$ −129262. −0.358657 −0.179329 0.983789i $$-0.557393\pi$$
−0.179329 + 0.983789i $$0.557393\pi$$
$$168$$ 0 0
$$169$$ 265380. 0.714746
$$170$$ −278748. −0.739757
$$171$$ 0 0
$$172$$ 1.02951e6 2.65344
$$173$$ 507867. 1.29013 0.645067 0.764126i $$-0.276829\pi$$
0.645067 + 0.764126i $$0.276829\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −15553.5 −0.0378483
$$177$$ 0 0
$$178$$ −1.32491e6 −3.13427
$$179$$ 132589. 0.309296 0.154648 0.987970i $$-0.450576\pi$$
0.154648 + 0.987970i $$0.450576\pi$$
$$180$$ 0 0
$$181$$ −740060. −1.67908 −0.839538 0.543301i $$-0.817174\pi$$
−0.839538 + 0.543301i $$0.817174\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 178023. 0.387642
$$185$$ −215782. −0.463538
$$186$$ 0 0
$$187$$ −573205. −1.19869
$$188$$ −107548. −0.221926
$$189$$ 0 0
$$190$$ −468722. −0.941957
$$191$$ 582732. 1.15581 0.577904 0.816105i $$-0.303871\pi$$
0.577904 + 0.816105i $$0.303871\pi$$
$$192$$ 0 0
$$193$$ −400904. −0.774725 −0.387362 0.921928i $$-0.626614\pi$$
−0.387362 + 0.921928i $$0.626614\pi$$
$$194$$ −728335. −1.38940
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −671589. −1.23293 −0.616464 0.787383i $$-0.711436\pi$$
−0.616464 + 0.787383i $$0.711436\pi$$
$$198$$ 0 0
$$199$$ −455022. −0.814515 −0.407258 0.913313i $$-0.633515\pi$$
−0.407258 + 0.913313i $$0.633515\pi$$
$$200$$ −491517. −0.868887
$$201$$ 0 0
$$202$$ 778878. 1.34305
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −119208. −0.198116
$$206$$ 186675. 0.306492
$$207$$ 0 0
$$208$$ 29781.9 0.0477303
$$209$$ −963860. −1.52633
$$210$$ 0 0
$$211$$ −1.19545e6 −1.84852 −0.924260 0.381764i $$-0.875317\pi$$
−0.924260 + 0.381764i $$0.875317\pi$$
$$212$$ −942499. −1.44026
$$213$$ 0 0
$$214$$ −63921.4 −0.0954140
$$215$$ −434514. −0.641073
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 1.03428e6 1.47400
$$219$$ 0 0
$$220$$ 480980. 0.669993
$$221$$ 1.09758e6 1.51166
$$222$$ 0 0
$$223$$ 296529. 0.399305 0.199653 0.979867i $$-0.436019\pi$$
0.199653 + 0.979867i $$0.436019\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −1.02834e6 −1.33927
$$227$$ −218146. −0.280985 −0.140492 0.990082i $$-0.544869\pi$$
−0.140492 + 0.990082i $$0.544869\pi$$
$$228$$ 0 0
$$229$$ −1.22920e6 −1.54894 −0.774471 0.632609i $$-0.781984\pi$$
−0.774471 + 0.632609i $$0.781984\pi$$
$$230$$ −193607. −0.241324
$$231$$ 0 0
$$232$$ 1.31090e6 1.59901
$$233$$ 62944.2 0.0759567 0.0379784 0.999279i $$-0.487908\pi$$
0.0379784 + 0.999279i $$0.487908\pi$$
$$234$$ 0 0
$$235$$ 45391.7 0.0536175
$$236$$ −388849. −0.454465
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −219330. −0.248372 −0.124186 0.992259i $$-0.539632\pi$$
−0.124186 + 0.992259i $$0.539632\pi$$
$$240$$ 0 0
$$241$$ 433864. 0.481183 0.240592 0.970626i $$-0.422659\pi$$
0.240592 + 0.970626i $$0.422659\pi$$
$$242$$ 115645. 0.126937
$$243$$ 0 0
$$244$$ 182791. 0.196553
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.84561e6 1.92485
$$248$$ 235001. 0.242628
$$249$$ 0 0
$$250$$ 1.16781e6 1.18174
$$251$$ 1.71109e6 1.71431 0.857155 0.515059i $$-0.172230\pi$$
0.857155 + 0.515059i $$0.172230\pi$$
$$252$$ 0 0
$$253$$ −398125. −0.391037
$$254$$ 754922. 0.734206
$$255$$ 0 0
$$256$$ −1.10964e6 −1.05824
$$257$$ −984057. −0.929368 −0.464684 0.885477i $$-0.653832\pi$$
−0.464684 + 0.885477i $$0.653832\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −920984. −0.844926
$$261$$ 0 0
$$262$$ 1.61417e6 1.45277
$$263$$ −547095. −0.487724 −0.243862 0.969810i $$-0.578414\pi$$
−0.243862 + 0.969810i $$0.578414\pi$$
$$264$$ 0 0
$$265$$ 397791. 0.347969
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 829211. 0.705226
$$269$$ 641112. 0.540198 0.270099 0.962833i $$-0.412944\pi$$
0.270099 + 0.962833i $$0.412944\pi$$
$$270$$ 0 0
$$271$$ −548012. −0.453280 −0.226640 0.973979i $$-0.572774\pi$$
−0.226640 + 0.973979i $$0.572774\pi$$
$$272$$ 51341.8 0.0420774
$$273$$ 0 0
$$274$$ −287651. −0.231467
$$275$$ 1.09921e6 0.876498
$$276$$ 0 0
$$277$$ 1.67414e6 1.31097 0.655485 0.755208i $$-0.272464\pi$$
0.655485 + 0.755208i $$0.272464\pi$$
$$278$$ 1.39769e6 1.08468
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.81078e6 −1.36804 −0.684021 0.729462i $$-0.739770\pi$$
−0.684021 + 0.729462i $$0.739770\pi$$
$$282$$ 0 0
$$283$$ 2.51315e6 1.86531 0.932657 0.360764i $$-0.117484\pi$$
0.932657 + 0.360764i $$0.117484\pi$$
$$284$$ −3.04009e6 −2.23661
$$285$$ 0 0
$$286$$ −3.05276e6 −2.20687
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 472283. 0.332627
$$290$$ −1.42566e6 −0.995455
$$291$$ 0 0
$$292$$ 2.04529e6 1.40378
$$293$$ −107228. −0.0729691 −0.0364845 0.999334i $$-0.511616\pi$$
−0.0364845 + 0.999334i $$0.511616\pi$$
$$294$$ 0 0
$$295$$ 164117. 0.109799
$$296$$ 1.82167e6 1.20848
$$297$$ 0 0
$$298$$ −3.33150e6 −2.17320
$$299$$ 762332. 0.493136
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 1.88330e6 1.18824
$$303$$ 0 0
$$304$$ 86332.6 0.0535785
$$305$$ −77148.7 −0.0474875
$$306$$ 0 0
$$307$$ 1.49622e6 0.906042 0.453021 0.891500i $$-0.350346\pi$$
0.453021 + 0.891500i $$0.350346\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −255573. −0.151047
$$311$$ −861202. −0.504899 −0.252449 0.967610i $$-0.581236\pi$$
−0.252449 + 0.967610i $$0.581236\pi$$
$$312$$ 0 0
$$313$$ −503937. −0.290747 −0.145374 0.989377i $$-0.546438\pi$$
−0.145374 + 0.989377i $$0.546438\pi$$
$$314$$ −709452. −0.406068
$$315$$ 0 0
$$316$$ 510435. 0.287556
$$317$$ 480009. 0.268288 0.134144 0.990962i $$-0.457172\pi$$
0.134144 + 0.990962i $$0.457172\pi$$
$$318$$ 0 0
$$319$$ −2.93167e6 −1.61302
$$320$$ 1.16522e6 0.636109
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.18168e6 1.69688
$$324$$ 0 0
$$325$$ −2.10478e6 −1.10535
$$326$$ 1.69790e6 0.884845
$$327$$ 0 0
$$328$$ 1.00637e6 0.516505
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.19922e6 −1.10331 −0.551656 0.834072i $$-0.686004\pi$$
−0.551656 + 0.834072i $$0.686004\pi$$
$$332$$ 3.68134e6 1.83299
$$333$$ 0 0
$$334$$ −1.18678e6 −0.582110
$$335$$ −349977. −0.170383
$$336$$ 0 0
$$337$$ −1.35725e6 −0.651008 −0.325504 0.945541i $$-0.605534\pi$$
−0.325504 + 0.945541i $$0.605534\pi$$
$$338$$ 2.43652e6 1.16005
$$339$$ 0 0
$$340$$ −1.58771e6 −0.744857
$$341$$ −525550. −0.244753
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 3.66825e6 1.67133
$$345$$ 0 0
$$346$$ 4.66285e6 2.09392
$$347$$ −3.79941e6 −1.69392 −0.846959 0.531659i $$-0.821569\pi$$
−0.846959 + 0.531659i $$0.821569\pi$$
$$348$$ 0 0
$$349$$ −1.31753e6 −0.579024 −0.289512 0.957174i $$-0.593493\pi$$
−0.289512 + 0.957174i $$0.593493\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.34189e6 1.00742
$$353$$ 3.42505e6 1.46295 0.731477 0.681866i $$-0.238832\pi$$
0.731477 + 0.681866i $$0.238832\pi$$
$$354$$ 0 0
$$355$$ 1.28310e6 0.540367
$$356$$ −7.54649e6 −3.15588
$$357$$ 0 0
$$358$$ 1.21733e6 0.501997
$$359$$ −1.47084e6 −0.602322 −0.301161 0.953573i $$-0.597374\pi$$
−0.301161 + 0.953573i $$0.597374\pi$$
$$360$$ 0 0
$$361$$ 2.87399e6 1.16069
$$362$$ −6.79466e6 −2.72519
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −863235. −0.339154
$$366$$ 0 0
$$367$$ −4.98079e6 −1.93034 −0.965168 0.261630i $$-0.915740\pi$$
−0.965168 + 0.261630i $$0.915740\pi$$
$$368$$ 35659.9 0.0137265
$$369$$ 0 0
$$370$$ −1.98114e6 −0.752335
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.96047e6 1.47392 0.736962 0.675934i $$-0.236260\pi$$
0.736962 + 0.675934i $$0.236260\pi$$
$$374$$ −5.26273e6 −1.94550
$$375$$ 0 0
$$376$$ −383205. −0.139785
$$377$$ 5.61359e6 2.03417
$$378$$ 0 0
$$379$$ −1.75155e6 −0.626359 −0.313179 0.949694i $$-0.601394\pi$$
−0.313179 + 0.949694i $$0.601394\pi$$
$$380$$ −2.66977e6 −0.948452
$$381$$ 0 0
$$382$$ 5.35020e6 1.87591
$$383$$ −3.13834e6 −1.09321 −0.546604 0.837391i $$-0.684080\pi$$
−0.546604 + 0.837391i $$0.684080\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3.68079e6 −1.25740
$$387$$ 0 0
$$388$$ −4.14849e6 −1.39898
$$389$$ 1.05252e6 0.352661 0.176331 0.984331i $$-0.443577\pi$$
0.176331 + 0.984331i $$0.443577\pi$$
$$390$$ 0 0
$$391$$ 1.31420e6 0.434731
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −6.16601e6 −2.00108
$$395$$ −215434. −0.0694739
$$396$$ 0 0
$$397$$ 454724. 0.144801 0.0724005 0.997376i $$-0.476934\pi$$
0.0724005 + 0.997376i $$0.476934\pi$$
$$398$$ −4.17766e6 −1.32198
$$399$$ 0 0
$$400$$ −98456.3 −0.0307676
$$401$$ 2.88431e6 0.895739 0.447870 0.894099i $$-0.352183\pi$$
0.447870 + 0.894099i $$0.352183\pi$$
$$402$$ 0 0
$$403$$ 1.00633e6 0.308657
$$404$$ 4.43638e6 1.35231
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.07394e6 −1.21907
$$408$$ 0 0
$$409$$ −225914. −0.0667782 −0.0333891 0.999442i $$-0.510630\pi$$
−0.0333891 + 0.999442i $$0.510630\pi$$
$$410$$ −1.09447e6 −0.321548
$$411$$ 0 0
$$412$$ 1.06328e6 0.308605
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.55375e6 −0.442853
$$416$$ −4.48427e6 −1.27045
$$417$$ 0 0
$$418$$ −8.84942e6 −2.47727
$$419$$ −4.31027e6 −1.19941 −0.599707 0.800220i $$-0.704716\pi$$
−0.599707 + 0.800220i $$0.704716\pi$$
$$420$$ 0 0
$$421$$ 1.25088e6 0.343962 0.171981 0.985100i $$-0.444983\pi$$
0.171981 + 0.985100i $$0.444983\pi$$
$$422$$ −1.09757e7 −3.00020
$$423$$ 0 0
$$424$$ −3.35823e6 −0.907184
$$425$$ −3.62849e6 −0.974436
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −364087. −0.0960719
$$429$$ 0 0
$$430$$ −3.98937e6 −1.04048
$$431$$ 4.40793e6 1.14299 0.571494 0.820606i $$-0.306364\pi$$
0.571494 + 0.820606i $$0.306364\pi$$
$$432$$ 0 0
$$433$$ −1.60951e6 −0.412549 −0.206274 0.978494i $$-0.566134\pi$$
−0.206274 + 0.978494i $$0.566134\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 5.89110e6 1.48416
$$437$$ 2.20987e6 0.553558
$$438$$ 0 0
$$439$$ −4.42452e6 −1.09573 −0.547867 0.836566i $$-0.684560\pi$$
−0.547867 + 0.836566i $$0.684560\pi$$
$$440$$ 1.71379e6 0.422012
$$441$$ 0 0
$$442$$ 1.00771e7 2.45347
$$443$$ 3.61438e6 0.875034 0.437517 0.899210i $$-0.355858\pi$$
0.437517 + 0.899210i $$0.355858\pi$$
$$444$$ 0 0
$$445$$ 3.18507e6 0.762464
$$446$$ 2.72250e6 0.648084
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 467024. 0.109326 0.0546630 0.998505i $$-0.482592\pi$$
0.0546630 + 0.998505i $$0.482592\pi$$
$$450$$ 0 0
$$451$$ −2.25063e6 −0.521029
$$452$$ −5.85730e6 −1.34850
$$453$$ 0 0
$$454$$ −2.00285e6 −0.456046
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −601252. −0.134668 −0.0673342 0.997730i $$-0.521449\pi$$
−0.0673342 + 0.997730i $$0.521449\pi$$
$$458$$ −1.12856e7 −2.51398
$$459$$ 0 0
$$460$$ −1.10276e6 −0.242988
$$461$$ −2.87193e6 −0.629392 −0.314696 0.949192i $$-0.601903\pi$$
−0.314696 + 0.949192i $$0.601903\pi$$
$$462$$ 0 0
$$463$$ −2.91502e6 −0.631959 −0.315979 0.948766i $$-0.602333\pi$$
−0.315979 + 0.948766i $$0.602333\pi$$
$$464$$ 262589. 0.0566215
$$465$$ 0 0
$$466$$ 577906. 0.123280
$$467$$ 7.19376e6 1.52638 0.763192 0.646172i $$-0.223631\pi$$
0.763192 + 0.646172i $$0.223631\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 416751. 0.0870227
$$471$$ 0 0
$$472$$ −1.38551e6 −0.286256
$$473$$ −8.20358e6 −1.68597
$$474$$ 0 0
$$475$$ −6.10140e6 −1.24078
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −2.01372e6 −0.403115
$$479$$ 2.79649e6 0.556896 0.278448 0.960451i $$-0.410180\pi$$
0.278448 + 0.960451i $$0.410180\pi$$
$$480$$ 0 0
$$481$$ 7.80080e6 1.53736
$$482$$ 3.98340e6 0.780974
$$483$$ 0 0
$$484$$ 658698. 0.127812
$$485$$ 1.75091e6 0.337995
$$486$$ 0 0
$$487$$ −2.93247e6 −0.560289 −0.280144 0.959958i $$-0.590382\pi$$
−0.280144 + 0.959958i $$0.590382\pi$$
$$488$$ 651305. 0.123804
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3.06121e6 0.573046 0.286523 0.958073i $$-0.407501\pi$$
0.286523 + 0.958073i $$0.407501\pi$$
$$492$$ 0 0
$$493$$ 9.67740e6 1.79325
$$494$$ 1.69449e7 3.12408
$$495$$ 0 0
$$496$$ 47073.4 0.00859154
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −6.55154e6 −1.17786 −0.588928 0.808186i $$-0.700450\pi$$
−0.588928 + 0.808186i $$0.700450\pi$$
$$500$$ 6.65167e6 1.18989
$$501$$ 0 0
$$502$$ 1.57099e7 2.78237
$$503$$ 1.58524e6 0.279367 0.139684 0.990196i $$-0.455391\pi$$
0.139684 + 0.990196i $$0.455391\pi$$
$$504$$ 0 0
$$505$$ −1.87242e6 −0.326719
$$506$$ −3.65528e6 −0.634664
$$507$$ 0 0
$$508$$ 4.29993e6 0.739268
$$509$$ −6.77981e6 −1.15991 −0.579953 0.814650i $$-0.696929\pi$$
−0.579953 + 0.814650i $$0.696929\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −432315. −0.0728829
$$513$$ 0 0
$$514$$ −9.03486e6 −1.50839
$$515$$ −448766. −0.0745593
$$516$$ 0 0
$$517$$ 856990. 0.141010
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −3.28157e6 −0.532198
$$521$$ 1.01384e7 1.63634 0.818170 0.574977i $$-0.194989\pi$$
0.818170 + 0.574977i $$0.194989\pi$$
$$522$$ 0 0
$$523$$ 99997.3 0.0159858 0.00799289 0.999968i $$-0.497456\pi$$
0.00799289 + 0.999968i $$0.497456\pi$$
$$524$$ 9.19408e6 1.46278
$$525$$ 0 0
$$526$$ −5.02301e6 −0.791589
$$527$$ 1.73483e6 0.272102
$$528$$ 0 0
$$529$$ −5.52355e6 −0.858181
$$530$$ 3.65221e6 0.564763
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.30952e6 0.657068
$$534$$ 0 0
$$535$$ 153667. 0.0232111
$$536$$ 2.95457e6 0.444204
$$537$$ 0 0
$$538$$ 5.88620e6 0.876756
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −119743. −0.0175896 −0.00879481 0.999961i $$-0.502800\pi$$
−0.00879481 + 0.999961i $$0.502800\pi$$
$$542$$ −5.03143e6 −0.735687
$$543$$ 0 0
$$544$$ −7.73054e6 −1.11999
$$545$$ −2.48640e6 −0.358574
$$546$$ 0 0
$$547$$ 236568. 0.0338056 0.0169028 0.999857i $$-0.494619\pi$$
0.0169028 + 0.999857i $$0.494619\pi$$
$$548$$ −1.63842e6 −0.233063
$$549$$ 0 0
$$550$$ 1.00921e7 1.42258
$$551$$ 1.62728e7 2.28341
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 1.53707e7 2.12774
$$555$$ 0 0
$$556$$ 7.96107e6 1.09216
$$557$$ 4.83666e6 0.660553 0.330277 0.943884i $$-0.392858\pi$$
0.330277 + 0.943884i $$0.392858\pi$$
$$558$$ 0 0
$$559$$ 1.57083e7 2.12617
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.66252e7 −2.22037
$$563$$ 5.48295e6 0.729026 0.364513 0.931198i $$-0.381235\pi$$
0.364513 + 0.931198i $$0.381235\pi$$
$$564$$ 0 0
$$565$$ 2.47213e6 0.325800
$$566$$ 2.30738e7 3.02746
$$567$$ 0 0
$$568$$ −1.08322e7 −1.40878
$$569$$ 6.97660e6 0.903364 0.451682 0.892179i $$-0.350824\pi$$
0.451682 + 0.892179i $$0.350824\pi$$
$$570$$ 0 0
$$571$$ −1.17715e7 −1.51092 −0.755458 0.655197i $$-0.772585\pi$$
−0.755458 + 0.655197i $$0.772585\pi$$
$$572$$ −1.73881e7 −2.22209
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.52020e6 −0.317882
$$576$$ 0 0
$$577$$ 6.77292e6 0.846909 0.423454 0.905917i $$-0.360817\pi$$
0.423454 + 0.905917i $$0.360817\pi$$
$$578$$ 4.33614e6 0.539863
$$579$$ 0 0
$$580$$ −8.12037e6 −1.00232
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 7.51025e6 0.915130
$$584$$ 7.28760e6 0.884203
$$585$$ 0 0
$$586$$ −984484. −0.118431
$$587$$ 1.05020e7 1.25799 0.628996 0.777408i $$-0.283466\pi$$
0.628996 + 0.777408i $$0.283466\pi$$
$$588$$ 0 0
$$589$$ 2.91717e6 0.346476
$$590$$ 1.50680e6 0.178207
$$591$$ 0 0
$$592$$ 364901. 0.0427928
$$593$$ 7.59074e6 0.886436 0.443218 0.896414i $$-0.353837\pi$$
0.443218 + 0.896414i $$0.353837\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.89758e7 −2.18818
$$597$$ 0 0
$$598$$ 6.99915e6 0.800373
$$599$$ 1.30198e7 1.48265 0.741325 0.671146i $$-0.234198\pi$$
0.741325 + 0.671146i $$0.234198\pi$$
$$600$$ 0 0
$$601$$ 1.41821e7 1.60160 0.800801 0.598931i $$-0.204408\pi$$
0.800801 + 0.598931i $$0.204408\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1.07270e7 1.19643
$$605$$ −278010. −0.0308796
$$606$$ 0 0
$$607$$ 1.20772e7 1.33044 0.665221 0.746646i $$-0.268337\pi$$
0.665221 + 0.746646i $$0.268337\pi$$
$$608$$ −1.29991e7 −1.42612
$$609$$ 0 0
$$610$$ −708320. −0.0770735
$$611$$ −1.64097e6 −0.177827
$$612$$ 0 0
$$613$$ −2.72530e6 −0.292929 −0.146465 0.989216i $$-0.546789\pi$$
−0.146465 + 0.989216i $$0.546789\pi$$
$$614$$ 1.37371e7 1.47053
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −8.47094e6 −0.895816 −0.447908 0.894080i $$-0.647831\pi$$
−0.447908 + 0.894080i $$0.647831\pi$$
$$618$$ 0 0
$$619$$ 1.73835e7 1.82352 0.911759 0.410726i $$-0.134725\pi$$
0.911759 + 0.410726i $$0.134725\pi$$
$$620$$ −1.45571e6 −0.152088
$$621$$ 0 0
$$622$$ −7.90690e6 −0.819464
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 5.43586e6 0.556632
$$626$$ −4.62676e6 −0.471891
$$627$$ 0 0
$$628$$ −4.04093e6 −0.408868
$$629$$ 1.34480e7 1.35529
$$630$$ 0 0
$$631$$ −6.45149e6 −0.645040 −0.322520 0.946563i $$-0.604530\pi$$
−0.322520 + 0.946563i $$0.604530\pi$$
$$632$$ 1.81874e6 0.181124
$$633$$ 0 0
$$634$$ 4.40707e6 0.435438
$$635$$ −1.81483e6 −0.178608
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −2.69164e7 −2.61797
$$639$$ 0 0
$$640$$ 6.72879e6 0.649362
$$641$$ −1.77716e7 −1.70837 −0.854185 0.519969i $$-0.825943\pi$$
−0.854185 + 0.519969i $$0.825943\pi$$
$$642$$ 0 0
$$643$$ −9.34806e6 −0.891649 −0.445825 0.895120i $$-0.647090\pi$$
−0.445825 + 0.895120i $$0.647090\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 2.92118e7 2.75408
$$647$$ −5.34386e6 −0.501874 −0.250937 0.968003i $$-0.580739\pi$$
−0.250937 + 0.968003i $$0.580739\pi$$
$$648$$ 0 0
$$649$$ 3.09852e6 0.288764
$$650$$ −1.93245e7 −1.79401
$$651$$ 0 0
$$652$$ 9.67098e6 0.890946
$$653$$ 1.19701e6 0.109854 0.0549270 0.998490i $$-0.482507\pi$$
0.0549270 + 0.998490i $$0.482507\pi$$
$$654$$ 0 0
$$655$$ −3.88045e6 −0.353410
$$656$$ 201588. 0.0182896
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.17541e7 1.05433 0.527163 0.849764i $$-0.323256\pi$$
0.527163 + 0.849764i $$0.323256\pi$$
$$660$$ 0 0
$$661$$ −1.80115e7 −1.60341 −0.801706 0.597719i $$-0.796074\pi$$
−0.801706 + 0.597719i $$0.796074\pi$$
$$662$$ −2.01915e7 −1.79071
$$663$$ 0 0
$$664$$ 1.31170e7 1.15456
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.72153e6 0.584997
$$668$$ −6.75975e6 −0.586124
$$669$$ 0 0
$$670$$ −3.21322e6 −0.276537
$$671$$ −1.45656e6 −0.124888
$$672$$ 0 0
$$673$$ 1.40977e7 1.19981 0.599904 0.800072i $$-0.295206\pi$$
0.599904 + 0.800072i $$0.295206\pi$$
$$674$$ −1.24613e7 −1.05660
$$675$$ 0 0
$$676$$ 1.38780e7 1.16805
$$677$$ −7.65587e6 −0.641982 −0.320991 0.947082i $$-0.604016\pi$$
−0.320991 + 0.947082i $$0.604016\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −5.65718e6 −0.469167
$$681$$ 0 0
$$682$$ −4.82520e6 −0.397241
$$683$$ 1.08676e7 0.891415 0.445708 0.895179i $$-0.352952\pi$$
0.445708 + 0.895179i $$0.352952\pi$$
$$684$$ 0 0
$$685$$ 691510. 0.0563083
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 734791. 0.0591825
$$689$$ −1.43807e7 −1.15407
$$690$$ 0 0
$$691$$ −1.46118e7 −1.16415 −0.582073 0.813136i $$-0.697758\pi$$
−0.582073 + 0.813136i $$0.697758\pi$$
$$692$$ 2.65589e7 2.10836
$$693$$ 0 0
$$694$$ −3.48832e7 −2.74927
$$695$$ −3.36005e6 −0.263866
$$696$$ 0 0
$$697$$ 7.42928e6 0.579248
$$698$$ −1.20965e7 −0.939772
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.90104e7 −1.46115 −0.730577 0.682830i $$-0.760749\pi$$
−0.730577 + 0.682830i $$0.760749\pi$$
$$702$$ 0 0
$$703$$ 2.26132e7 1.72573
$$704$$ 2.19992e7 1.67292
$$705$$ 0 0
$$706$$ 3.14462e7 2.37441
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.15929e7 −0.866116 −0.433058 0.901366i $$-0.642565\pi$$
−0.433058 + 0.901366i $$0.642565\pi$$
$$710$$ 1.17804e7 0.877031
$$711$$ 0 0
$$712$$ −2.68890e7 −1.98781
$$713$$ 1.20494e6 0.0887653
$$714$$ 0 0
$$715$$ 7.33881e6 0.536859
$$716$$ 6.93374e6 0.505458
$$717$$ 0 0
$$718$$ −1.35041e7 −0.977585
$$719$$ 1.02861e7 0.742040 0.371020 0.928625i $$-0.379008\pi$$
0.371020 + 0.928625i $$0.379008\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2.63868e7 1.88384
$$723$$ 0 0
$$724$$ −3.87014e7 −2.74398
$$725$$ −1.85580e7 −1.31125
$$726$$ 0 0
$$727$$ 1.00970e7 0.708526 0.354263 0.935146i $$-0.384732\pi$$
0.354263 + 0.935146i $$0.384732\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −7.92556e6 −0.550456
$$731$$ 2.70799e7 1.87436
$$732$$ 0 0
$$733$$ 1.92402e6 0.132266 0.0661332 0.997811i $$-0.478934\pi$$
0.0661332 + 0.997811i $$0.478934\pi$$
$$734$$ −4.57298e7 −3.13299
$$735$$ 0 0
$$736$$ −5.36932e6 −0.365363
$$737$$ −6.60752e6 −0.448095
$$738$$ 0 0
$$739$$ 4.20273e6 0.283087 0.141543 0.989932i $$-0.454793\pi$$
0.141543 + 0.989932i $$0.454793\pi$$
$$740$$ −1.12843e7 −0.757522
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.99659e7 1.32684 0.663418 0.748249i $$-0.269105\pi$$
0.663418 + 0.748249i $$0.269105\pi$$
$$744$$ 0 0
$$745$$ 8.00891e6 0.528667
$$746$$ 3.63620e7 2.39222
$$747$$ 0 0
$$748$$ −2.99757e7 −1.95892
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −3.63975e6 −0.235490 −0.117745 0.993044i $$-0.537567\pi$$
−0.117745 + 0.993044i $$0.537567\pi$$
$$752$$ −76760.3 −0.00494985
$$753$$ 0 0
$$754$$ 5.15396e7 3.30151
$$755$$ −4.52745e6 −0.289059
$$756$$ 0 0
$$757$$ 1.73429e7 1.09997 0.549986 0.835174i $$-0.314633\pi$$
0.549986 + 0.835174i $$0.314633\pi$$
$$758$$ −1.60813e7 −1.01660
$$759$$ 0 0
$$760$$ −9.51270e6 −0.597406
$$761$$ 1.03568e7 0.648284 0.324142 0.946009i $$-0.394924\pi$$
0.324142 + 0.946009i $$0.394924\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 3.04740e7 1.88884
$$765$$ 0 0
$$766$$ −2.88138e7 −1.77431
$$767$$ −5.93306e6 −0.364159
$$768$$ 0 0
$$769$$ −1.81548e7 −1.10707 −0.553534 0.832826i $$-0.686721\pi$$
−0.553534 + 0.832826i $$0.686721\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.09653e7 −1.26607
$$773$$ 1.63566e7 0.984565 0.492282 0.870435i $$-0.336163\pi$$
0.492282 + 0.870435i $$0.336163\pi$$
$$774$$ 0 0
$$775$$ −3.32683e6 −0.198965
$$776$$ −1.47815e7 −0.881181
$$777$$ 0 0
$$778$$ 9.66346e6 0.572379
$$779$$ 1.24925e7 0.737576
$$780$$ 0 0
$$781$$ 2.42248e7 1.42112
$$782$$ 1.20660e7 0.705581
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.70552e6 0.0987829
$$786$$ 0 0
$$787$$ −4.44728e6 −0.255952 −0.127976 0.991777i $$-0.540848\pi$$
−0.127976 + 0.991777i $$0.540848\pi$$
$$788$$ −3.51207e7 −2.01487
$$789$$ 0 0
$$790$$ −1.97795e6 −0.112758
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2.78903e6 0.157496
$$794$$ 4.17493e6 0.235016
$$795$$ 0 0
$$796$$ −2.37953e7 −1.33110
$$797$$ 9.15303e6 0.510410 0.255205 0.966887i $$-0.417857\pi$$
0.255205 + 0.966887i $$0.417857\pi$$
$$798$$ 0 0
$$799$$ −2.82891e6 −0.156766
$$800$$ 1.48246e7 0.818950
$$801$$ 0 0
$$802$$ 2.64815e7 1.45381
$$803$$ −1.62978e7 −0.891948
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 9.23932e6 0.500959
$$807$$ 0 0
$$808$$ 1.58073e7 0.851784
$$809$$ −2.51923e7 −1.35331 −0.676654 0.736301i $$-0.736571\pi$$
−0.676654 + 0.736301i $$0.736571\pi$$
$$810$$ 0 0
$$811$$ −8.90585e6 −0.475470 −0.237735 0.971330i $$-0.576405\pi$$
−0.237735 + 0.971330i $$0.576405\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −3.74038e7 −1.97858
$$815$$ −4.08173e6 −0.215254
$$816$$ 0 0
$$817$$ 4.55355e7 2.38669
$$818$$ −2.07417e6 −0.108383
$$819$$ 0 0
$$820$$ −6.23396e6 −0.323765
$$821$$ −3.35490e7 −1.73709 −0.868543 0.495613i $$-0.834943\pi$$
−0.868543 + 0.495613i $$0.834943\pi$$
$$822$$ 0 0
$$823$$ −1.46001e7 −0.751376 −0.375688 0.926746i $$-0.622593\pi$$
−0.375688 + 0.926746i $$0.622593\pi$$
$$824$$ 3.78857e6 0.194382
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.97530e7 −1.00431 −0.502156 0.864777i $$-0.667460\pi$$
−0.502156 + 0.864777i $$0.667460\pi$$
$$828$$ 0 0
$$829$$ −4.06255e6 −0.205311 −0.102655 0.994717i $$-0.532734\pi$$
−0.102655 + 0.994717i $$0.532734\pi$$
$$830$$ −1.42653e7 −0.718762
$$831$$ 0 0
$$832$$ −4.21241e7 −2.10971
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.85302e6 0.141608
$$836$$ −5.04050e7 −2.49435
$$837$$ 0 0
$$838$$ −3.95735e7 −1.94668
$$839$$ 1.60509e7 0.787217 0.393609 0.919278i $$-0.371227\pi$$
0.393609 + 0.919278i $$0.371227\pi$$
$$840$$ 0 0
$$841$$ 2.89842e7 1.41309
$$842$$ 1.14846e7 0.558260
$$843$$ 0 0
$$844$$ −6.25159e7 −3.02088
$$845$$ −5.85737e6 −0.282202
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −672690. −0.0321237
$$849$$ 0 0
$$850$$ −3.33140e7 −1.58154
$$851$$ 9.34043e6 0.442123
$$852$$ 0 0
$$853$$ −1.23887e7 −0.582979 −0.291489 0.956574i $$-0.594151\pi$$
−0.291489 + 0.956574i $$0.594151\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −1.29728e6 −0.0605133
$$857$$ −2.63519e7 −1.22563 −0.612816 0.790226i $$-0.709963\pi$$
−0.612816 + 0.790226i $$0.709963\pi$$
$$858$$ 0 0
$$859$$ −1.21249e7 −0.560654 −0.280327 0.959905i $$-0.590443\pi$$
−0.280327 + 0.959905i $$0.590443\pi$$
$$860$$ −2.27229e7 −1.04765
$$861$$ 0 0
$$862$$ 4.04702e7 1.85510
$$863$$ −2.55952e6 −0.116985 −0.0584927 0.998288i $$-0.518629\pi$$
−0.0584927 + 0.998288i $$0.518629\pi$$
$$864$$ 0 0
$$865$$ −1.12094e7 −0.509382
$$866$$ −1.47773e7 −0.669578
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.06737e6 −0.182711
$$870$$ 0 0
$$871$$ 1.26521e7 0.565091
$$872$$ 2.09906e7 0.934834
$$873$$ 0 0
$$874$$ 2.02893e7 0.898439
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.65820e7 1.16705 0.583523 0.812096i $$-0.301674\pi$$
0.583523 + 0.812096i $$0.301674\pi$$
$$878$$ −4.06225e7 −1.77841
$$879$$ 0 0
$$880$$ 343290. 0.0149436
$$881$$ 8.32262e6 0.361260 0.180630 0.983551i $$-0.442186\pi$$
0.180630 + 0.983551i $$0.442186\pi$$
$$882$$ 0 0
$$883$$ 1.81133e7 0.781798 0.390899 0.920434i $$-0.372164\pi$$
0.390899 + 0.920434i $$0.372164\pi$$
$$884$$ 5.73977e7 2.47038
$$885$$ 0 0
$$886$$ 3.31845e7 1.42020
$$887$$ 2.04255e7 0.871694 0.435847 0.900021i $$-0.356449\pi$$
0.435847 + 0.900021i $$0.356449\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.92429e7 1.23750
$$891$$ 0 0
$$892$$ 1.55070e7 0.652552
$$893$$ −4.75689e6 −0.199616
$$894$$ 0 0
$$895$$ −2.92645e6 −0.122119
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 4.28785e6 0.177439
$$899$$ 8.87285e6 0.366154
$$900$$ 0 0
$$901$$ −2.47912e7 −1.01739
$$902$$ −2.06635e7 −0.845645
$$903$$ 0 0
$$904$$ −2.08702e7 −0.849388
$$905$$ 1.63343e7 0.662948
$$906$$ 0 0
$$907$$ −2.01197e7 −0.812089 −0.406044 0.913853i $$-0.633092\pi$$
−0.406044 + 0.913853i $$0.633092\pi$$
$$908$$ −1.14079e7 −0.459190
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −3.17075e7 −1.26580 −0.632902 0.774232i $$-0.718137\pi$$
−0.632902 + 0.774232i $$0.718137\pi$$
$$912$$ 0 0
$$913$$ −2.93345e7 −1.16467
$$914$$ −5.52023e6 −0.218571
$$915$$ 0 0
$$916$$ −6.42812e7 −2.53131
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −8.70727e6 −0.340090 −0.170045 0.985436i $$-0.554391\pi$$
−0.170045 + 0.985436i $$0.554391\pi$$
$$920$$ −3.92924e6 −0.153052
$$921$$ 0 0
$$922$$ −2.63678e7 −1.02152
$$923$$ −4.63857e7 −1.79217
$$924$$ 0 0
$$925$$ −2.57887e7 −0.991005
$$926$$ −2.67634e7 −1.02569
$$927$$ 0 0
$$928$$ −3.95381e7 −1.50711
$$929$$ −3.24042e7 −1.23186 −0.615931 0.787800i $$-0.711220\pi$$
−0.615931 + 0.787800i $$0.711220\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 3.29167e6 0.124130
$$933$$ 0 0
$$934$$ 6.60475e7 2.47736
$$935$$ 1.26516e7 0.473277
$$936$$ 0 0
$$937$$ −2.19155e7 −0.815458 −0.407729 0.913103i $$-0.633679\pi$$
−0.407729 + 0.913103i $$0.633679\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 2.37376e6 0.0876227
$$941$$ −1.46019e7 −0.537571 −0.268785 0.963200i $$-0.586622\pi$$
−0.268785 + 0.963200i $$0.586622\pi$$
$$942$$ 0 0
$$943$$ 5.16008e6 0.188963
$$944$$ −277533. −0.0101364
$$945$$ 0 0
$$946$$ −7.53189e7 −2.73638
$$947$$ 2.08378e7 0.755053 0.377527 0.925999i $$-0.376775\pi$$
0.377527 + 0.925999i $$0.376775\pi$$
$$948$$ 0 0
$$949$$ 3.12071e7 1.12483
$$950$$ −5.60184e7 −2.01382
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 942012. 0.0335988 0.0167994 0.999859i $$-0.494652\pi$$
0.0167994 + 0.999859i $$0.494652\pi$$
$$954$$ 0 0
$$955$$ −1.28618e7 −0.456346
$$956$$ −1.14698e7 −0.405894
$$957$$ 0 0
$$958$$ 2.56752e7 0.903857
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.70385e7 −0.944441
$$962$$ 7.16209e7 2.49518
$$963$$ 0 0
$$964$$ 2.26889e7 0.786359
$$965$$ 8.84860e6 0.305884
$$966$$ 0 0
$$967$$ −2.56570e7 −0.882346 −0.441173 0.897422i $$-0.645438\pi$$
−0.441173 + 0.897422i $$0.645438\pi$$
$$968$$ 2.34701e6 0.0805059
$$969$$ 0 0
$$970$$ 1.60755e7 0.548575
$$971$$ 2.14274e7 0.729324 0.364662 0.931140i $$-0.381184\pi$$
0.364662 + 0.931140i $$0.381184\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −2.69237e7 −0.909364
$$975$$ 0 0
$$976$$ 130464. 0.00438394
$$977$$ 2.04841e7 0.686562 0.343281 0.939233i $$-0.388462\pi$$
0.343281 + 0.939233i $$0.388462\pi$$
$$978$$ 0 0
$$979$$ 6.01338e7 2.00522
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 2.81057e7 0.930069
$$983$$ −2.77650e7 −0.916459 −0.458230 0.888834i $$-0.651516\pi$$
−0.458230 + 0.888834i $$0.651516\pi$$
$$984$$ 0 0
$$985$$ 1.48230e7 0.486796
$$986$$ 8.88504e7 2.91050
$$987$$ 0 0
$$988$$ 9.65159e7 3.14562
$$989$$ 1.88086e7 0.611456
$$990$$ 0 0
$$991$$ 3.46832e7 1.12185 0.560926 0.827866i $$-0.310445\pi$$
0.560926 + 0.827866i $$0.310445\pi$$
$$992$$ −7.08785e6 −0.228684
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1.00431e7 0.321594
$$996$$ 0 0
$$997$$ −1.48572e7 −0.473369 −0.236685 0.971587i $$-0.576061\pi$$
−0.236685 + 0.971587i $$0.576061\pi$$
$$998$$ −6.01512e7 −1.91169
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.6.a.w.1.4 4
3.2 odd 2 147.6.a.m.1.1 4
7.2 even 3 63.6.e.e.46.1 8
7.4 even 3 63.6.e.e.37.1 8
7.6 odd 2 441.6.a.v.1.4 4
21.2 odd 6 21.6.e.c.4.4 8
21.5 even 6 147.6.e.o.67.4 8
21.11 odd 6 21.6.e.c.16.4 yes 8
21.17 even 6 147.6.e.o.79.4 8
21.20 even 2 147.6.a.l.1.1 4
84.11 even 6 336.6.q.j.289.3 8
84.23 even 6 336.6.q.j.193.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.4 8 21.2 odd 6
21.6.e.c.16.4 yes 8 21.11 odd 6
63.6.e.e.37.1 8 7.4 even 3
63.6.e.e.46.1 8 7.2 even 3
147.6.a.l.1.1 4 21.20 even 2
147.6.a.m.1.1 4 3.2 odd 2
147.6.e.o.67.4 8 21.5 even 6
147.6.e.o.79.4 8 21.17 even 6
336.6.q.j.193.3 8 84.23 even 6
336.6.q.j.289.3 8 84.11 even 6
441.6.a.v.1.4 4 7.6 odd 2
441.6.a.w.1.4 4 1.1 even 1 trivial