# Properties

 Label 1521.4.a.t.1.2 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.56155 q^{2} +12.8078 q^{4} -2.80776 q^{5} +9.56155 q^{7} +21.9309 q^{8} +O(q^{10})$$ $$q+4.56155 q^{2} +12.8078 q^{4} -2.80776 q^{5} +9.56155 q^{7} +21.9309 q^{8} -12.8078 q^{10} -39.4233 q^{11} +43.6155 q^{14} -2.42329 q^{16} -2.01515 q^{17} -60.1922 q^{19} -35.9612 q^{20} -179.831 q^{22} -4.46876 q^{23} -117.116 q^{25} +122.462 q^{28} -140.693 q^{29} +136.155 q^{31} -186.501 q^{32} -9.19224 q^{34} -26.8466 q^{35} -185.708 q^{37} -274.570 q^{38} -61.5767 q^{40} -310.231 q^{41} +427.471 q^{43} -504.924 q^{44} -20.3845 q^{46} +258.617 q^{47} -251.577 q^{49} -534.233 q^{50} -612.656 q^{53} +110.691 q^{55} +209.693 q^{56} -641.779 q^{58} +517.885 q^{59} -161.311 q^{61} +621.080 q^{62} -831.348 q^{64} -49.8987 q^{67} -25.8096 q^{68} -122.462 q^{70} -279.963 q^{71} +467.732 q^{73} -847.118 q^{74} -770.928 q^{76} -376.948 q^{77} +37.5379 q^{79} +6.80403 q^{80} -1415.14 q^{82} +76.1553 q^{83} +5.65808 q^{85} +1949.93 q^{86} -864.587 q^{88} -202.806 q^{89} -57.2348 q^{92} +1179.70 q^{94} +169.006 q^{95} -1174.37 q^{97} -1147.58 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8}+O(q^{10})$$ 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8 $$2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8} - 5 q^{10} - 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} - 141 q^{19} - 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} + 80 q^{28} - 34 q^{29} - 140 q^{31} - 105 q^{32} - 39 q^{34} + 70 q^{35} - 190 q^{37} - 310 q^{38} - 185 q^{40} - 538 q^{41} + 455 q^{43} - 680 q^{44} - 82 q^{46} - 60 q^{47} - 565 q^{49} - 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} - 595 q^{58} + 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} - 475 q^{67} + 505 q^{68} - 80 q^{70} - 127 q^{71} + 585 q^{73} - 849 q^{74} - 140 q^{76} - 255 q^{77} + 240 q^{79} + 1065 q^{80} - 1515 q^{82} - 260 q^{83} - 1205 q^{85} + 1962 q^{86} - 1020 q^{88} - 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} - 415 q^{97} - 1285 q^{98}+O(q^{100})$$ 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8 - 5 * q^10 - 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 - 141 * q^19 - 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 + 80 * q^28 - 34 * q^29 - 140 * q^31 - 105 * q^32 - 39 * q^34 + 70 * q^35 - 190 * q^37 - 310 * q^38 - 185 * q^40 - 538 * q^41 + 455 * q^43 - 680 * q^44 - 82 * q^46 - 60 * q^47 - 565 * q^49 - 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 - 595 * q^58 + 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 - 475 * q^67 + 505 * q^68 - 80 * q^70 - 127 * q^71 + 585 * q^73 - 849 * q^74 - 140 * q^76 - 255 * q^77 + 240 * q^79 + 1065 * q^80 - 1515 * q^82 - 260 * q^83 - 1205 * q^85 + 1962 * q^86 - 1020 * q^88 - 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 - 415 * q^97 - 1285 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.56155 1.61275 0.806376 0.591403i $$-0.201426\pi$$
0.806376 + 0.591403i $$0.201426\pi$$
$$3$$ 0 0
$$4$$ 12.8078 1.60097
$$5$$ −2.80776 −0.251134 −0.125567 0.992085i $$-0.540075\pi$$
−0.125567 + 0.992085i $$0.540075\pi$$
$$6$$ 0 0
$$7$$ 9.56155 0.516275 0.258138 0.966108i $$-0.416891\pi$$
0.258138 + 0.966108i $$0.416891\pi$$
$$8$$ 21.9309 0.969217
$$9$$ 0 0
$$10$$ −12.8078 −0.405017
$$11$$ −39.4233 −1.08060 −0.540299 0.841473i $$-0.681689\pi$$
−0.540299 + 0.841473i $$0.681689\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 43.6155 0.832624
$$15$$ 0 0
$$16$$ −2.42329 −0.0378639
$$17$$ −2.01515 −0.0287498 −0.0143749 0.999897i $$-0.504576\pi$$
−0.0143749 + 0.999897i $$0.504576\pi$$
$$18$$ 0 0
$$19$$ −60.1922 −0.726792 −0.363396 0.931635i $$-0.618383\pi$$
−0.363396 + 0.931635i $$0.618383\pi$$
$$20$$ −35.9612 −0.402058
$$21$$ 0 0
$$22$$ −179.831 −1.74274
$$23$$ −4.46876 −0.0405131 −0.0202565 0.999795i $$-0.506448\pi$$
−0.0202565 + 0.999795i $$0.506448\pi$$
$$24$$ 0 0
$$25$$ −117.116 −0.936932
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 122.462 0.826542
$$29$$ −140.693 −0.900899 −0.450449 0.892802i $$-0.648736\pi$$
−0.450449 + 0.892802i $$0.648736\pi$$
$$30$$ 0 0
$$31$$ 136.155 0.788845 0.394423 0.918929i $$-0.370945\pi$$
0.394423 + 0.918929i $$0.370945\pi$$
$$32$$ −186.501 −1.03028
$$33$$ 0 0
$$34$$ −9.19224 −0.0463663
$$35$$ −26.8466 −0.129654
$$36$$ 0 0
$$37$$ −185.708 −0.825142 −0.412571 0.910925i $$-0.635369\pi$$
−0.412571 + 0.910925i $$0.635369\pi$$
$$38$$ −274.570 −1.17214
$$39$$ 0 0
$$40$$ −61.5767 −0.243403
$$41$$ −310.231 −1.18171 −0.590853 0.806779i $$-0.701209\pi$$
−0.590853 + 0.806779i $$0.701209\pi$$
$$42$$ 0 0
$$43$$ 427.471 1.51602 0.758008 0.652246i $$-0.226173\pi$$
0.758008 + 0.652246i $$0.226173\pi$$
$$44$$ −504.924 −1.73000
$$45$$ 0 0
$$46$$ −20.3845 −0.0653375
$$47$$ 258.617 0.802622 0.401311 0.915942i $$-0.368555\pi$$
0.401311 + 0.915942i $$0.368555\pi$$
$$48$$ 0 0
$$49$$ −251.577 −0.733460
$$50$$ −534.233 −1.51104
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −612.656 −1.58783 −0.793913 0.608031i $$-0.791960\pi$$
−0.793913 + 0.608031i $$0.791960\pi$$
$$54$$ 0 0
$$55$$ 110.691 0.271375
$$56$$ 209.693 0.500383
$$57$$ 0 0
$$58$$ −641.779 −1.45293
$$59$$ 517.885 1.14276 0.571381 0.820685i $$-0.306408\pi$$
0.571381 + 0.820685i $$0.306408\pi$$
$$60$$ 0 0
$$61$$ −161.311 −0.338585 −0.169293 0.985566i $$-0.554148\pi$$
−0.169293 + 0.985566i $$0.554148\pi$$
$$62$$ 621.080 1.27221
$$63$$ 0 0
$$64$$ −831.348 −1.62373
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −49.8987 −0.0909865 −0.0454933 0.998965i $$-0.514486\pi$$
−0.0454933 + 0.998965i $$0.514486\pi$$
$$68$$ −25.8096 −0.0460276
$$69$$ 0 0
$$70$$ −122.462 −0.209100
$$71$$ −279.963 −0.467965 −0.233982 0.972241i $$-0.575176\pi$$
−0.233982 + 0.972241i $$0.575176\pi$$
$$72$$ 0 0
$$73$$ 467.732 0.749916 0.374958 0.927042i $$-0.377657\pi$$
0.374958 + 0.927042i $$0.377657\pi$$
$$74$$ −847.118 −1.33075
$$75$$ 0 0
$$76$$ −770.928 −1.16357
$$77$$ −376.948 −0.557886
$$78$$ 0 0
$$79$$ 37.5379 0.0534600 0.0267300 0.999643i $$-0.491491\pi$$
0.0267300 + 0.999643i $$0.491491\pi$$
$$80$$ 6.80403 0.00950892
$$81$$ 0 0
$$82$$ −1415.14 −1.90580
$$83$$ 76.1553 0.100712 0.0503562 0.998731i $$-0.483964\pi$$
0.0503562 + 0.998731i $$0.483964\pi$$
$$84$$ 0 0
$$85$$ 5.65808 0.00722006
$$86$$ 1949.93 2.44496
$$87$$ 0 0
$$88$$ −864.587 −1.04733
$$89$$ −202.806 −0.241544 −0.120772 0.992680i $$-0.538537\pi$$
−0.120772 + 0.992680i $$0.538537\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −57.2348 −0.0648602
$$93$$ 0 0
$$94$$ 1179.70 1.29443
$$95$$ 169.006 0.182522
$$96$$ 0 0
$$97$$ −1174.37 −1.22927 −0.614634 0.788812i $$-0.710696\pi$$
−0.614634 + 0.788812i $$0.710696\pi$$
$$98$$ −1147.58 −1.18289
$$99$$ 0 0
$$100$$ −1500.00 −1.50000
$$101$$ −970.697 −0.956316 −0.478158 0.878274i $$-0.658695\pi$$
−0.478158 + 0.878274i $$0.658695\pi$$
$$102$$ 0 0
$$103$$ −1899.70 −1.81731 −0.908654 0.417550i $$-0.862889\pi$$
−0.908654 + 0.417550i $$0.862889\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2794.66 −2.56077
$$107$$ 1906.49 1.72250 0.861251 0.508180i $$-0.169682\pi$$
0.861251 + 0.508180i $$0.169682\pi$$
$$108$$ 0 0
$$109$$ −896.004 −0.787354 −0.393677 0.919249i $$-0.628797\pi$$
−0.393677 + 0.919249i $$0.628797\pi$$
$$110$$ 504.924 0.437660
$$111$$ 0 0
$$112$$ −23.1704 −0.0195482
$$113$$ 334.882 0.278788 0.139394 0.990237i $$-0.455484\pi$$
0.139394 + 0.990237i $$0.455484\pi$$
$$114$$ 0 0
$$115$$ 12.5472 0.0101742
$$116$$ −1801.96 −1.44231
$$117$$ 0 0
$$118$$ 2362.36 1.84299
$$119$$ −19.2680 −0.0148428
$$120$$ 0 0
$$121$$ 223.196 0.167690
$$122$$ −735.827 −0.546054
$$123$$ 0 0
$$124$$ 1743.84 1.26292
$$125$$ 679.806 0.486430
$$126$$ 0 0
$$127$$ 620.893 0.433822 0.216911 0.976191i $$-0.430402\pi$$
0.216911 + 0.976191i $$0.430402\pi$$
$$128$$ −2300.23 −1.58839
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1331.70 0.888180 0.444090 0.895982i $$-0.353527\pi$$
0.444090 + 0.895982i $$0.353527\pi$$
$$132$$ 0 0
$$133$$ −575.531 −0.375225
$$134$$ −227.616 −0.146739
$$135$$ 0 0
$$136$$ −44.1941 −0.0278648
$$137$$ −622.015 −0.387900 −0.193950 0.981011i $$-0.562130\pi$$
−0.193950 + 0.981011i $$0.562130\pi$$
$$138$$ 0 0
$$139$$ 330.580 0.201723 0.100861 0.994900i $$-0.467840\pi$$
0.100861 + 0.994900i $$0.467840\pi$$
$$140$$ −343.845 −0.207573
$$141$$ 0 0
$$142$$ −1277.07 −0.754711
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 395.033 0.226246
$$146$$ 2133.58 1.20943
$$147$$ 0 0
$$148$$ −2378.51 −1.32103
$$149$$ 1810.54 0.995470 0.497735 0.867329i $$-0.334165\pi$$
0.497735 + 0.867329i $$0.334165\pi$$
$$150$$ 0 0
$$151$$ −423.239 −0.228097 −0.114049 0.993475i $$-0.536382\pi$$
−0.114049 + 0.993475i $$0.536382\pi$$
$$152$$ −1320.07 −0.704419
$$153$$ 0 0
$$154$$ −1719.47 −0.899732
$$155$$ −382.292 −0.198106
$$156$$ 0 0
$$157$$ 1322.17 0.672105 0.336052 0.941843i $$-0.390908\pi$$
0.336052 + 0.941843i $$0.390908\pi$$
$$158$$ 171.231 0.0862178
$$159$$ 0 0
$$160$$ 523.651 0.258739
$$161$$ −42.7283 −0.0209159
$$162$$ 0 0
$$163$$ −3606.39 −1.73297 −0.866486 0.499201i $$-0.833627\pi$$
−0.866486 + 0.499201i $$0.833627\pi$$
$$164$$ −3973.37 −1.89188
$$165$$ 0 0
$$166$$ 347.386 0.162424
$$167$$ 3415.43 1.58260 0.791300 0.611429i $$-0.209405\pi$$
0.791300 + 0.611429i $$0.209405\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 25.8096 0.0116442
$$171$$ 0 0
$$172$$ 5474.94 2.42710
$$173$$ −2342.23 −1.02934 −0.514671 0.857388i $$-0.672086\pi$$
−0.514671 + 0.857388i $$0.672086\pi$$
$$174$$ 0 0
$$175$$ −1119.82 −0.483715
$$176$$ 95.5342 0.0409157
$$177$$ 0 0
$$178$$ −925.110 −0.389550
$$179$$ 666.891 0.278468 0.139234 0.990260i $$-0.455536\pi$$
0.139234 + 0.990260i $$0.455536\pi$$
$$180$$ 0 0
$$181$$ −701.037 −0.287888 −0.143944 0.989586i $$-0.545978\pi$$
−0.143944 + 0.989586i $$0.545978\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −98.0037 −0.0392659
$$185$$ 521.425 0.207221
$$186$$ 0 0
$$187$$ 79.4440 0.0310670
$$188$$ 3312.31 1.28497
$$189$$ 0 0
$$190$$ 770.928 0.294363
$$191$$ −1300.88 −0.492819 −0.246409 0.969166i $$-0.579251\pi$$
−0.246409 + 0.969166i $$0.579251\pi$$
$$192$$ 0 0
$$193$$ −519.333 −0.193691 −0.0968457 0.995299i $$-0.530875\pi$$
−0.0968457 + 0.995299i $$0.530875\pi$$
$$194$$ −5356.94 −1.98251
$$195$$ 0 0
$$196$$ −3222.14 −1.17425
$$197$$ −3121.05 −1.12876 −0.564379 0.825516i $$-0.690884\pi$$
−0.564379 + 0.825516i $$0.690884\pi$$
$$198$$ 0 0
$$199$$ 1237.06 0.440667 0.220333 0.975425i $$-0.429285\pi$$
0.220333 + 0.975425i $$0.429285\pi$$
$$200$$ −2568.47 −0.908090
$$201$$ 0 0
$$202$$ −4427.89 −1.54230
$$203$$ −1345.25 −0.465112
$$204$$ 0 0
$$205$$ 871.056 0.296767
$$206$$ −8665.57 −2.93087
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2372.98 0.785369
$$210$$ 0 0
$$211$$ −2531.67 −0.826006 −0.413003 0.910730i $$-0.635520\pi$$
−0.413003 + 0.910730i $$0.635520\pi$$
$$212$$ −7846.76 −2.54206
$$213$$ 0 0
$$214$$ 8696.57 2.77797
$$215$$ −1200.24 −0.380723
$$216$$ 0 0
$$217$$ 1301.86 0.407261
$$218$$ −4087.17 −1.26981
$$219$$ 0 0
$$220$$ 1417.71 0.434463
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1195.53 −0.359008 −0.179504 0.983757i $$-0.557449\pi$$
−0.179504 + 0.983757i $$0.557449\pi$$
$$224$$ −1783.24 −0.531909
$$225$$ 0 0
$$226$$ 1527.58 0.449617
$$227$$ 869.192 0.254142 0.127071 0.991894i $$-0.459442\pi$$
0.127071 + 0.991894i $$0.459442\pi$$
$$228$$ 0 0
$$229$$ 4684.64 1.35183 0.675916 0.736978i $$-0.263748\pi$$
0.675916 + 0.736978i $$0.263748\pi$$
$$230$$ 57.2348 0.0164085
$$231$$ 0 0
$$232$$ −3085.52 −0.873166
$$233$$ 4868.99 1.36900 0.684502 0.729011i $$-0.260020\pi$$
0.684502 + 0.729011i $$0.260020\pi$$
$$234$$ 0 0
$$235$$ −726.137 −0.201566
$$236$$ 6632.95 1.82953
$$237$$ 0 0
$$238$$ −87.8920 −0.0239378
$$239$$ −4807.53 −1.30114 −0.650572 0.759444i $$-0.725471\pi$$
−0.650572 + 0.759444i $$0.725471\pi$$
$$240$$ 0 0
$$241$$ 5875.96 1.57056 0.785278 0.619143i $$-0.212520\pi$$
0.785278 + 0.619143i $$0.212520\pi$$
$$242$$ 1018.12 0.270443
$$243$$ 0 0
$$244$$ −2066.03 −0.542065
$$245$$ 706.368 0.184197
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 2986.00 0.764562
$$249$$ 0 0
$$250$$ 3100.97 0.784490
$$251$$ −5806.27 −1.46011 −0.730057 0.683386i $$-0.760506\pi$$
−0.730057 + 0.683386i $$0.760506\pi$$
$$252$$ 0 0
$$253$$ 176.173 0.0437783
$$254$$ 2832.24 0.699647
$$255$$ 0 0
$$256$$ −3841.83 −0.937947
$$257$$ 1195.86 0.290256 0.145128 0.989413i $$-0.453641\pi$$
0.145128 + 0.989413i $$0.453641\pi$$
$$258$$ 0 0
$$259$$ −1775.66 −0.426001
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6074.64 1.43241
$$263$$ −234.184 −0.0549066 −0.0274533 0.999623i $$-0.508740\pi$$
−0.0274533 + 0.999623i $$0.508740\pi$$
$$264$$ 0 0
$$265$$ 1720.19 0.398757
$$266$$ −2625.32 −0.605145
$$267$$ 0 0
$$268$$ −639.091 −0.145667
$$269$$ 2668.27 0.604785 0.302393 0.953183i $$-0.402215\pi$$
0.302393 + 0.953183i $$0.402215\pi$$
$$270$$ 0 0
$$271$$ 5701.28 1.27796 0.638982 0.769222i $$-0.279356\pi$$
0.638982 + 0.769222i $$0.279356\pi$$
$$272$$ 4.88331 0.00108858
$$273$$ 0 0
$$274$$ −2837.35 −0.625587
$$275$$ 4617.12 1.01245
$$276$$ 0 0
$$277$$ −7152.49 −1.55145 −0.775725 0.631072i $$-0.782615\pi$$
−0.775725 + 0.631072i $$0.782615\pi$$
$$278$$ 1507.96 0.325329
$$279$$ 0 0
$$280$$ −588.769 −0.125663
$$281$$ 6132.87 1.30198 0.650990 0.759086i $$-0.274354\pi$$
0.650990 + 0.759086i $$0.274354\pi$$
$$282$$ 0 0
$$283$$ 3377.15 0.709367 0.354683 0.934986i $$-0.384589\pi$$
0.354683 + 0.934986i $$0.384589\pi$$
$$284$$ −3585.70 −0.749198
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2966.29 −0.610086
$$288$$ 0 0
$$289$$ −4908.94 −0.999173
$$290$$ 1801.96 0.364879
$$291$$ 0 0
$$292$$ 5990.60 1.20059
$$293$$ 4704.77 0.938073 0.469037 0.883179i $$-0.344601\pi$$
0.469037 + 0.883179i $$0.344601\pi$$
$$294$$ 0 0
$$295$$ −1454.10 −0.286986
$$296$$ −4072.75 −0.799742
$$297$$ 0 0
$$298$$ 8258.86 1.60545
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4087.28 0.782681
$$302$$ −1930.62 −0.367864
$$303$$ 0 0
$$304$$ 145.863 0.0275192
$$305$$ 452.922 0.0850303
$$306$$ 0 0
$$307$$ 5130.49 0.953787 0.476894 0.878961i $$-0.341763\pi$$
0.476894 + 0.878961i $$0.341763\pi$$
$$308$$ −4827.86 −0.893159
$$309$$ 0 0
$$310$$ −1743.84 −0.319496
$$311$$ −7948.94 −1.44933 −0.724667 0.689099i $$-0.758006\pi$$
−0.724667 + 0.689099i $$0.758006\pi$$
$$312$$ 0 0
$$313$$ −8521.87 −1.53893 −0.769465 0.638689i $$-0.779477\pi$$
−0.769465 + 0.638689i $$0.779477\pi$$
$$314$$ 6031.14 1.08394
$$315$$ 0 0
$$316$$ 480.776 0.0855879
$$317$$ 6662.46 1.18044 0.590222 0.807241i $$-0.299040\pi$$
0.590222 + 0.807241i $$0.299040\pi$$
$$318$$ 0 0
$$319$$ 5546.59 0.973509
$$320$$ 2334.23 0.407773
$$321$$ 0 0
$$322$$ −194.907 −0.0337322
$$323$$ 121.297 0.0208951
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −16450.8 −2.79486
$$327$$ 0 0
$$328$$ −6803.64 −1.14533
$$329$$ 2472.78 0.414374
$$330$$ 0 0
$$331$$ −3911.77 −0.649578 −0.324789 0.945786i $$-0.605293\pi$$
−0.324789 + 0.945786i $$0.605293\pi$$
$$332$$ 975.379 0.161238
$$333$$ 0 0
$$334$$ 15579.7 2.55234
$$335$$ 140.104 0.0228498
$$336$$ 0 0
$$337$$ −627.211 −0.101384 −0.0506919 0.998714i $$-0.516143\pi$$
−0.0506919 + 0.998714i $$0.516143\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 72.4674 0.0115591
$$341$$ −5367.69 −0.852424
$$342$$ 0 0
$$343$$ −5685.08 −0.894943
$$344$$ 9374.80 1.46935
$$345$$ 0 0
$$346$$ −10684.2 −1.66007
$$347$$ 3823.02 0.591442 0.295721 0.955274i $$-0.404440\pi$$
0.295721 + 0.955274i $$0.404440\pi$$
$$348$$ 0 0
$$349$$ 3410.67 0.523120 0.261560 0.965187i $$-0.415763\pi$$
0.261560 + 0.965187i $$0.415763\pi$$
$$350$$ −5108.10 −0.780112
$$351$$ 0 0
$$352$$ 7352.48 1.11332
$$353$$ 5587.64 0.842492 0.421246 0.906946i $$-0.361593\pi$$
0.421246 + 0.906946i $$0.361593\pi$$
$$354$$ 0 0
$$355$$ 786.070 0.117522
$$356$$ −2597.49 −0.386704
$$357$$ 0 0
$$358$$ 3042.06 0.449100
$$359$$ −2230.14 −0.327861 −0.163931 0.986472i $$-0.552417\pi$$
−0.163931 + 0.986472i $$0.552417\pi$$
$$360$$ 0 0
$$361$$ −3235.89 −0.471774
$$362$$ −3197.82 −0.464292
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1313.28 −0.188330
$$366$$ 0 0
$$367$$ −8699.14 −1.23731 −0.618653 0.785664i $$-0.712321\pi$$
−0.618653 + 0.785664i $$0.712321\pi$$
$$368$$ 10.8291 0.00153398
$$369$$ 0 0
$$370$$ 2378.51 0.334197
$$371$$ −5857.94 −0.819756
$$372$$ 0 0
$$373$$ 10964.2 1.52199 0.760997 0.648755i $$-0.224710\pi$$
0.760997 + 0.648755i $$0.224710\pi$$
$$374$$ 362.388 0.0501033
$$375$$ 0 0
$$376$$ 5671.70 0.777914
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 13910.1 1.88526 0.942631 0.333835i $$-0.108343\pi$$
0.942631 + 0.333835i $$0.108343\pi$$
$$380$$ 2164.58 0.292213
$$381$$ 0 0
$$382$$ −5934.03 −0.794794
$$383$$ 494.753 0.0660071 0.0330035 0.999455i $$-0.489493\pi$$
0.0330035 + 0.999455i $$0.489493\pi$$
$$384$$ 0 0
$$385$$ 1058.38 0.140104
$$386$$ −2368.97 −0.312376
$$387$$ 0 0
$$388$$ −15041.0 −1.96802
$$389$$ 4140.47 0.539666 0.269833 0.962907i $$-0.413032\pi$$
0.269833 + 0.962907i $$0.413032\pi$$
$$390$$ 0 0
$$391$$ 9.00524 0.00116474
$$392$$ −5517.30 −0.710881
$$393$$ 0 0
$$394$$ −14236.8 −1.82041
$$395$$ −105.398 −0.0134256
$$396$$ 0 0
$$397$$ −1881.79 −0.237895 −0.118948 0.992901i $$-0.537952\pi$$
−0.118948 + 0.992901i $$0.537952\pi$$
$$398$$ 5642.90 0.710686
$$399$$ 0 0
$$400$$ 283.807 0.0354759
$$401$$ −421.765 −0.0525236 −0.0262618 0.999655i $$-0.508360\pi$$
−0.0262618 + 0.999655i $$0.508360\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −12432.5 −1.53103
$$405$$ 0 0
$$406$$ −6136.41 −0.750110
$$407$$ 7321.23 0.891646
$$408$$ 0 0
$$409$$ 2550.22 0.308313 0.154157 0.988046i $$-0.450734\pi$$
0.154157 + 0.988046i $$0.450734\pi$$
$$410$$ 3973.37 0.478611
$$411$$ 0 0
$$412$$ −24330.9 −2.90946
$$413$$ 4951.79 0.589980
$$414$$ 0 0
$$415$$ −213.826 −0.0252923
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 10824.5 1.26661
$$419$$ −12384.8 −1.44400 −0.722002 0.691891i $$-0.756778\pi$$
−0.722002 + 0.691891i $$0.756778\pi$$
$$420$$ 0 0
$$421$$ 10463.0 1.21124 0.605622 0.795752i $$-0.292924\pi$$
0.605622 + 0.795752i $$0.292924\pi$$
$$422$$ −11548.3 −1.33214
$$423$$ 0 0
$$424$$ −13436.1 −1.53895
$$425$$ 236.008 0.0269366
$$426$$ 0 0
$$427$$ −1542.38 −0.174803
$$428$$ 24417.9 2.75767
$$429$$ 0 0
$$430$$ −5474.94 −0.614012
$$431$$ 3962.39 0.442834 0.221417 0.975179i $$-0.428932\pi$$
0.221417 + 0.975179i $$0.428932\pi$$
$$432$$ 0 0
$$433$$ −8394.14 −0.931632 −0.465816 0.884882i $$-0.654239\pi$$
−0.465816 + 0.884882i $$0.654239\pi$$
$$434$$ 5938.48 0.656812
$$435$$ 0 0
$$436$$ −11475.8 −1.26053
$$437$$ 268.984 0.0294446
$$438$$ 0 0
$$439$$ 10174.5 1.10616 0.553079 0.833129i $$-0.313453\pi$$
0.553079 + 0.833129i $$0.313453\pi$$
$$440$$ 2427.56 0.263021
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 5880.74 0.630705 0.315353 0.948975i $$-0.397877\pi$$
0.315353 + 0.948975i $$0.397877\pi$$
$$444$$ 0 0
$$445$$ 569.431 0.0606598
$$446$$ −5453.48 −0.578990
$$447$$ 0 0
$$448$$ −7948.97 −0.838289
$$449$$ −10664.9 −1.12095 −0.560475 0.828172i $$-0.689381\pi$$
−0.560475 + 0.828172i $$0.689381\pi$$
$$450$$ 0 0
$$451$$ 12230.3 1.27695
$$452$$ 4289.10 0.446332
$$453$$ 0 0
$$454$$ 3964.87 0.409869
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −14828.9 −1.51787 −0.758933 0.651169i $$-0.774279\pi$$
−0.758933 + 0.651169i $$0.774279\pi$$
$$458$$ 21369.2 2.18017
$$459$$ 0 0
$$460$$ 160.702 0.0162886
$$461$$ 9711.70 0.981169 0.490585 0.871394i $$-0.336783\pi$$
0.490585 + 0.871394i $$0.336783\pi$$
$$462$$ 0 0
$$463$$ 11353.5 1.13962 0.569809 0.821777i $$-0.307017\pi$$
0.569809 + 0.821777i $$0.307017\pi$$
$$464$$ 340.941 0.0341116
$$465$$ 0 0
$$466$$ 22210.1 2.20787
$$467$$ −6451.31 −0.639252 −0.319626 0.947544i $$-0.603557\pi$$
−0.319626 + 0.947544i $$0.603557\pi$$
$$468$$ 0 0
$$469$$ −477.109 −0.0469741
$$470$$ −3312.31 −0.325076
$$471$$ 0 0
$$472$$ 11357.7 1.10758
$$473$$ −16852.3 −1.63820
$$474$$ 0 0
$$475$$ 7049.50 0.680954
$$476$$ −246.780 −0.0237629
$$477$$ 0 0
$$478$$ −21929.8 −2.09842
$$479$$ 9566.46 0.912531 0.456266 0.889844i $$-0.349187\pi$$
0.456266 + 0.889844i $$0.349187\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 26803.5 2.53292
$$483$$ 0 0
$$484$$ 2858.64 0.268467
$$485$$ 3297.35 0.308711
$$486$$ 0 0
$$487$$ 4917.11 0.457527 0.228764 0.973482i $$-0.426532\pi$$
0.228764 + 0.973482i $$0.426532\pi$$
$$488$$ −3537.68 −0.328162
$$489$$ 0 0
$$490$$ 3222.14 0.297064
$$491$$ −2950.82 −0.271220 −0.135610 0.990762i $$-0.543299\pi$$
−0.135610 + 0.990762i $$0.543299\pi$$
$$492$$ 0 0
$$493$$ 283.519 0.0259007
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −329.944 −0.0298688
$$497$$ −2676.88 −0.241599
$$498$$ 0 0
$$499$$ 13430.1 1.20484 0.602418 0.798180i $$-0.294204\pi$$
0.602418 + 0.798180i $$0.294204\pi$$
$$500$$ 8706.79 0.778759
$$501$$ 0 0
$$502$$ −26485.6 −2.35480
$$503$$ −1320.29 −0.117035 −0.0585175 0.998286i $$-0.518637\pi$$
−0.0585175 + 0.998286i $$0.518637\pi$$
$$504$$ 0 0
$$505$$ 2725.49 0.240164
$$506$$ 803.623 0.0706036
$$507$$ 0 0
$$508$$ 7952.25 0.694536
$$509$$ −20916.4 −1.82143 −0.910713 0.413041i $$-0.864467\pi$$
−0.910713 + 0.413041i $$0.864467\pi$$
$$510$$ 0 0
$$511$$ 4472.24 0.387163
$$512$$ 877.105 0.0757089
$$513$$ 0 0
$$514$$ 5454.98 0.468110
$$515$$ 5333.90 0.456388
$$516$$ 0 0
$$517$$ −10195.5 −0.867311
$$518$$ −8099.77 −0.687033
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 10104.2 0.849661 0.424831 0.905273i $$-0.360334\pi$$
0.424831 + 0.905273i $$0.360334\pi$$
$$522$$ 0 0
$$523$$ 7131.22 0.596227 0.298113 0.954530i $$-0.403643\pi$$
0.298113 + 0.954530i $$0.403643\pi$$
$$524$$ 17056.2 1.42195
$$525$$ 0 0
$$526$$ −1068.24 −0.0885507
$$527$$ −274.374 −0.0226792
$$528$$ 0 0
$$529$$ −12147.0 −0.998359
$$530$$ 7846.76 0.643097
$$531$$ 0 0
$$532$$ −7371.27 −0.600724
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −5352.98 −0.432579
$$536$$ −1094.32 −0.0881856
$$537$$ 0 0
$$538$$ 12171.5 0.975369
$$539$$ 9917.98 0.792575
$$540$$ 0 0
$$541$$ 16831.7 1.33762 0.668809 0.743435i $$-0.266805\pi$$
0.668809 + 0.743435i $$0.266805\pi$$
$$542$$ 26006.7 2.06104
$$543$$ 0 0
$$544$$ 375.828 0.0296204
$$545$$ 2515.77 0.197731
$$546$$ 0 0
$$547$$ −9560.55 −0.747312 −0.373656 0.927567i $$-0.621896\pi$$
−0.373656 + 0.927567i $$0.621896\pi$$
$$548$$ −7966.62 −0.621017
$$549$$ 0 0
$$550$$ 21061.2 1.63282
$$551$$ 8468.64 0.654766
$$552$$ 0 0
$$553$$ 358.920 0.0276001
$$554$$ −32626.5 −2.50210
$$555$$ 0 0
$$556$$ 4234.00 0.322952
$$557$$ 22827.9 1.73653 0.868267 0.496097i $$-0.165234\pi$$
0.868267 + 0.496097i $$0.165234\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 65.0571 0.00490922
$$561$$ 0 0
$$562$$ 27975.4 2.09977
$$563$$ −21629.7 −1.61916 −0.809578 0.587013i $$-0.800304\pi$$
−0.809578 + 0.587013i $$0.800304\pi$$
$$564$$ 0 0
$$565$$ −940.271 −0.0700133
$$566$$ 15405.1 1.14403
$$567$$ 0 0
$$568$$ −6139.83 −0.453559
$$569$$ 10589.9 0.780229 0.390114 0.920766i $$-0.372436\pi$$
0.390114 + 0.920766i $$0.372436\pi$$
$$570$$ 0 0
$$571$$ −1757.27 −0.128791 −0.0643954 0.997924i $$-0.520512\pi$$
−0.0643954 + 0.997924i $$0.520512\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −13530.9 −0.983917
$$575$$ 523.365 0.0379580
$$576$$ 0 0
$$577$$ −13580.6 −0.979840 −0.489920 0.871767i $$-0.662974\pi$$
−0.489920 + 0.871767i $$0.662974\pi$$
$$578$$ −22392.4 −1.61142
$$579$$ 0 0
$$580$$ 5059.49 0.362214
$$581$$ 728.163 0.0519953
$$582$$ 0 0
$$583$$ 24152.9 1.71580
$$584$$ 10257.8 0.726831
$$585$$ 0 0
$$586$$ 21461.0 1.51288
$$587$$ 957.326 0.0673136 0.0336568 0.999433i $$-0.489285\pi$$
0.0336568 + 0.999433i $$0.489285\pi$$
$$588$$ 0 0
$$589$$ −8195.49 −0.573327
$$590$$ −6632.95 −0.462838
$$591$$ 0 0
$$592$$ 450.026 0.0312431
$$593$$ −6729.49 −0.466015 −0.233007 0.972475i $$-0.574857\pi$$
−0.233007 + 0.972475i $$0.574857\pi$$
$$594$$ 0 0
$$595$$ 54.1000 0.00372754
$$596$$ 23188.9 1.59372
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −2281.52 −0.155626 −0.0778132 0.996968i $$-0.524794\pi$$
−0.0778132 + 0.996968i $$0.524794\pi$$
$$600$$ 0 0
$$601$$ 6401.42 0.434475 0.217237 0.976119i $$-0.430295\pi$$
0.217237 + 0.976119i $$0.430295\pi$$
$$602$$ 18644.4 1.26227
$$603$$ 0 0
$$604$$ −5420.74 −0.365177
$$605$$ −626.682 −0.0421128
$$606$$ 0 0
$$607$$ 2779.24 0.185841 0.0929207 0.995674i $$-0.470380\pi$$
0.0929207 + 0.995674i $$0.470380\pi$$
$$608$$ 11225.9 0.748800
$$609$$ 0 0
$$610$$ 2066.03 0.137133
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 22620.8 1.49045 0.745226 0.666812i $$-0.232342\pi$$
0.745226 + 0.666812i $$0.232342\pi$$
$$614$$ 23403.0 1.53822
$$615$$ 0 0
$$616$$ −8266.80 −0.540712
$$617$$ −21974.0 −1.43378 −0.716889 0.697187i $$-0.754435\pi$$
−0.716889 + 0.697187i $$0.754435\pi$$
$$618$$ 0 0
$$619$$ 7145.19 0.463957 0.231979 0.972721i $$-0.425480\pi$$
0.231979 + 0.972721i $$0.425480\pi$$
$$620$$ −4896.30 −0.317162
$$621$$ 0 0
$$622$$ −36259.5 −2.33742
$$623$$ −1939.14 −0.124703
$$624$$ 0 0
$$625$$ 12730.8 0.814773
$$626$$ −38873.0 −2.48191
$$627$$ 0 0
$$628$$ 16934.0 1.07602
$$629$$ 374.231 0.0237227
$$630$$ 0 0
$$631$$ −18883.2 −1.19133 −0.595666 0.803232i $$-0.703112\pi$$
−0.595666 + 0.803232i $$0.703112\pi$$
$$632$$ 823.239 0.0518144
$$633$$ 0 0
$$634$$ 30391.1 1.90376
$$635$$ −1743.32 −0.108947
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 25301.1 1.57003
$$639$$ 0 0
$$640$$ 6458.50 0.398898
$$641$$ 3631.08 0.223743 0.111871 0.993723i $$-0.464316\pi$$
0.111871 + 0.993723i $$0.464316\pi$$
$$642$$ 0 0
$$643$$ −10772.0 −0.660660 −0.330330 0.943866i $$-0.607160\pi$$
−0.330330 + 0.943866i $$0.607160\pi$$
$$644$$ −547.253 −0.0334857
$$645$$ 0 0
$$646$$ 553.301 0.0336987
$$647$$ −15148.3 −0.920464 −0.460232 0.887799i $$-0.652234\pi$$
−0.460232 + 0.887799i $$0.652234\pi$$
$$648$$ 0 0
$$649$$ −20416.7 −1.23487
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −46189.8 −2.77444
$$653$$ −7358.89 −0.441004 −0.220502 0.975387i $$-0.570770\pi$$
−0.220502 + 0.975387i $$0.570770\pi$$
$$654$$ 0 0
$$655$$ −3739.11 −0.223052
$$656$$ 751.780 0.0447441
$$657$$ 0 0
$$658$$ 11279.7 0.668282
$$659$$ −28333.3 −1.67482 −0.837411 0.546574i $$-0.815932\pi$$
−0.837411 + 0.546574i $$0.815932\pi$$
$$660$$ 0 0
$$661$$ 1109.68 0.0652975 0.0326488 0.999467i $$-0.489606\pi$$
0.0326488 + 0.999467i $$0.489606\pi$$
$$662$$ −17843.8 −1.04761
$$663$$ 0 0
$$664$$ 1670.15 0.0976121
$$665$$ 1615.96 0.0942317
$$666$$ 0 0
$$667$$ 628.724 0.0364982
$$668$$ 43744.1 2.53369
$$669$$ 0 0
$$670$$ 639.091 0.0368511
$$671$$ 6359.39 0.365874
$$672$$ 0 0
$$673$$ 20979.1 1.20161 0.600806 0.799395i $$-0.294846\pi$$
0.600806 + 0.799395i $$0.294846\pi$$
$$674$$ −2861.06 −0.163507
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −30941.9 −1.75656 −0.878282 0.478142i $$-0.841310\pi$$
−0.878282 + 0.478142i $$0.841310\pi$$
$$678$$ 0 0
$$679$$ −11228.8 −0.634641
$$680$$ 124.087 0.00699780
$$681$$ 0 0
$$682$$ −24485.0 −1.37475
$$683$$ −5426.21 −0.303995 −0.151997 0.988381i $$-0.548571\pi$$
−0.151997 + 0.988381i $$0.548571\pi$$
$$684$$ 0 0
$$685$$ 1746.47 0.0974150
$$686$$ −25932.8 −1.44332
$$687$$ 0 0
$$688$$ −1035.89 −0.0574023
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −33792.7 −1.86040 −0.930199 0.367056i $$-0.880366\pi$$
−0.930199 + 0.367056i $$0.880366\pi$$
$$692$$ −29998.7 −1.64795
$$693$$ 0 0
$$694$$ 17438.9 0.953849
$$695$$ −928.192 −0.0506595
$$696$$ 0 0
$$697$$ 625.164 0.0339738
$$698$$ 15557.9 0.843662
$$699$$ 0 0
$$700$$ −14342.3 −0.774413
$$701$$ −6905.96 −0.372089 −0.186045 0.982541i $$-0.559567\pi$$
−0.186045 + 0.982541i $$0.559567\pi$$
$$702$$ 0 0
$$703$$ 11178.2 0.599707
$$704$$ 32774.5 1.75459
$$705$$ 0 0
$$706$$ 25488.3 1.35873
$$707$$ −9281.37 −0.493723
$$708$$ 0 0
$$709$$ −2007.13 −0.106318 −0.0531589 0.998586i $$-0.516929\pi$$
−0.0531589 + 0.998586i $$0.516929\pi$$
$$710$$ 3585.70 0.189534
$$711$$ 0 0
$$712$$ −4447.71 −0.234108
$$713$$ −608.445 −0.0319585
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 8541.38 0.445819
$$717$$ 0 0
$$718$$ −10172.9 −0.528759
$$719$$ 12787.4 0.663270 0.331635 0.943408i $$-0.392400\pi$$
0.331635 + 0.943408i $$0.392400\pi$$
$$720$$ 0 0
$$721$$ −18164.1 −0.938231
$$722$$ −14760.7 −0.760854
$$723$$ 0 0
$$724$$ −8978.72 −0.460900
$$725$$ 16477.5 0.844081
$$726$$ 0 0
$$727$$ −6090.70 −0.310717 −0.155359 0.987858i $$-0.549653\pi$$
−0.155359 + 0.987858i $$0.549653\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −5990.60 −0.303729
$$731$$ −861.420 −0.0435852
$$732$$ 0 0
$$733$$ −38846.5 −1.95747 −0.978737 0.205117i $$-0.934243\pi$$
−0.978737 + 0.205117i $$0.934243\pi$$
$$734$$ −39681.6 −1.99547
$$735$$ 0 0
$$736$$ 833.427 0.0417399
$$737$$ 1967.17 0.0983198
$$738$$ 0 0
$$739$$ 14457.5 0.719661 0.359830 0.933018i $$-0.382835\pi$$
0.359830 + 0.933018i $$0.382835\pi$$
$$740$$ 6678.29 0.331755
$$741$$ 0 0
$$742$$ −26721.3 −1.32206
$$743$$ 1277.80 0.0630929 0.0315464 0.999502i $$-0.489957\pi$$
0.0315464 + 0.999502i $$0.489957\pi$$
$$744$$ 0 0
$$745$$ −5083.56 −0.249996
$$746$$ 50013.7 2.45460
$$747$$ 0 0
$$748$$ 1017.50 0.0497373
$$749$$ 18229.0 0.889285
$$750$$ 0 0
$$751$$ −13007.9 −0.632042 −0.316021 0.948752i $$-0.602347\pi$$
−0.316021 + 0.948752i $$0.602347\pi$$
$$752$$ −626.706 −0.0303904
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1188.35 0.0572829
$$756$$ 0 0
$$757$$ −10723.2 −0.514850 −0.257425 0.966298i $$-0.582874\pi$$
−0.257425 + 0.966298i $$0.582874\pi$$
$$758$$ 63451.7 3.04046
$$759$$ 0 0
$$760$$ 3706.44 0.176904
$$761$$ −13621.8 −0.648870 −0.324435 0.945908i $$-0.605174\pi$$
−0.324435 + 0.945908i $$0.605174\pi$$
$$762$$ 0 0
$$763$$ −8567.19 −0.406491
$$764$$ −16661.4 −0.788988
$$765$$ 0 0
$$766$$ 2256.84 0.106453
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −8495.15 −0.398365 −0.199183 0.979962i $$-0.563829\pi$$
−0.199183 + 0.979962i $$0.563829\pi$$
$$770$$ 4827.86 0.225953
$$771$$ 0 0
$$772$$ −6651.50 −0.310094
$$773$$ −34262.5 −1.59423 −0.797113 0.603830i $$-0.793641\pi$$
−0.797113 + 0.603830i $$0.793641\pi$$
$$774$$ 0 0
$$775$$ −15946.0 −0.739094
$$776$$ −25754.9 −1.19143
$$777$$ 0 0
$$778$$ 18887.0 0.870348
$$779$$ 18673.5 0.858854
$$780$$ 0 0
$$781$$ 11037.1 0.505681
$$782$$ 41.0779 0.00187844
$$783$$ 0 0
$$784$$ 609.644 0.0277717
$$785$$ −3712.33 −0.168788
$$786$$ 0 0
$$787$$ −12642.6 −0.572629 −0.286315 0.958136i $$-0.592430\pi$$
−0.286315 + 0.958136i $$0.592430\pi$$
$$788$$ −39973.6 −1.80711
$$789$$ 0 0
$$790$$ −480.776 −0.0216522
$$791$$ 3202.00 0.143932
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −8583.89 −0.383666
$$795$$ 0 0
$$796$$ 15843.9 0.705494
$$797$$ −19084.4 −0.848184 −0.424092 0.905619i $$-0.639407\pi$$
−0.424092 + 0.905619i $$0.639407\pi$$
$$798$$ 0 0
$$799$$ −521.154 −0.0230752
$$800$$ 21842.3 0.965304
$$801$$ 0 0
$$802$$ −1923.90 −0.0847075
$$803$$ −18439.5 −0.810357
$$804$$ 0 0
$$805$$ 119.971 0.00525269
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −21288.2 −0.926878
$$809$$ −11610.0 −0.504558 −0.252279 0.967655i $$-0.581180\pi$$
−0.252279 + 0.967655i $$0.581180\pi$$
$$810$$ 0 0
$$811$$ 9613.36 0.416240 0.208120 0.978103i $$-0.433266\pi$$
0.208120 + 0.978103i $$0.433266\pi$$
$$812$$ −17229.6 −0.744630
$$813$$ 0 0
$$814$$ 33396.2 1.43800
$$815$$ 10125.9 0.435208
$$816$$ 0 0
$$817$$ −25730.4 −1.10183
$$818$$ 11632.9 0.497233
$$819$$ 0 0
$$820$$ 11156.3 0.475115
$$821$$ 26481.5 1.12571 0.562856 0.826555i $$-0.309703\pi$$
0.562856 + 0.826555i $$0.309703\pi$$
$$822$$ 0 0
$$823$$ 13814.5 0.585107 0.292553 0.956249i $$-0.405495\pi$$
0.292553 + 0.956249i $$0.405495\pi$$
$$824$$ −41662.0 −1.76136
$$825$$ 0 0
$$826$$ 22587.8 0.951491
$$827$$ 44401.0 1.86696 0.933479 0.358633i $$-0.116757\pi$$
0.933479 + 0.358633i $$0.116757\pi$$
$$828$$ 0 0
$$829$$ 24337.4 1.01963 0.509815 0.860284i $$-0.329714\pi$$
0.509815 + 0.860284i $$0.329714\pi$$
$$830$$ −975.379 −0.0407902
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 506.966 0.0210868
$$834$$ 0 0
$$835$$ −9589.73 −0.397445
$$836$$ 30392.5 1.25735
$$837$$ 0 0
$$838$$ −56494.0 −2.32882
$$839$$ 24680.1 1.01556 0.507778 0.861488i $$-0.330467\pi$$
0.507778 + 0.861488i $$0.330467\pi$$
$$840$$ 0 0
$$841$$ −4594.43 −0.188381
$$842$$ 47727.4 1.95344
$$843$$ 0 0
$$844$$ −32425.0 −1.32241
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2134.10 0.0865744
$$848$$ 1484.65 0.0601214
$$849$$ 0 0
$$850$$ 1076.56 0.0434421
$$851$$ 829.885 0.0334290
$$852$$ 0 0
$$853$$ −10151.7 −0.407490 −0.203745 0.979024i $$-0.565311\pi$$
−0.203745 + 0.979024i $$0.565311\pi$$
$$854$$ −7035.65 −0.281914
$$855$$ 0 0
$$856$$ 41811.1 1.66948
$$857$$ −2028.92 −0.0808713 −0.0404357 0.999182i $$-0.512875\pi$$
−0.0404357 + 0.999182i $$0.512875\pi$$
$$858$$ 0 0
$$859$$ 6655.76 0.264367 0.132184 0.991225i $$-0.457801\pi$$
0.132184 + 0.991225i $$0.457801\pi$$
$$860$$ −15372.3 −0.609526
$$861$$ 0 0
$$862$$ 18074.6 0.714182
$$863$$ −45690.8 −1.80224 −0.901121 0.433568i $$-0.857254\pi$$
−0.901121 + 0.433568i $$0.857254\pi$$
$$864$$ 0 0
$$865$$ 6576.42 0.258503
$$866$$ −38290.3 −1.50249
$$867$$ 0 0
$$868$$ 16673.9 0.652014
$$869$$ −1479.87 −0.0577688
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −19650.1 −0.763117
$$873$$ 0 0
$$874$$ 1226.99 0.0474868
$$875$$ 6500.00 0.251132
$$876$$ 0 0
$$877$$ 30447.5 1.17234 0.586168 0.810189i $$-0.300636\pi$$
0.586168 + 0.810189i $$0.300636\pi$$
$$878$$ 46411.6 1.78396
$$879$$ 0 0
$$880$$ −268.237 −0.0102753
$$881$$ −32542.0 −1.24446 −0.622230 0.782835i $$-0.713773\pi$$
−0.622230 + 0.782835i $$0.713773\pi$$
$$882$$ 0 0
$$883$$ 27641.9 1.05348 0.526741 0.850026i $$-0.323414\pi$$
0.526741 + 0.850026i $$0.323414\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 26825.3 1.01717
$$887$$ −40099.9 −1.51795 −0.758976 0.651119i $$-0.774300\pi$$
−0.758976 + 0.651119i $$0.774300\pi$$
$$888$$ 0 0
$$889$$ 5936.70 0.223971
$$890$$ 2597.49 0.0978293
$$891$$ 0 0
$$892$$ −15312.1 −0.574761
$$893$$ −15566.8 −0.583339
$$894$$ 0 0
$$895$$ −1872.47 −0.0699328
$$896$$ −21993.8 −0.820044
$$897$$ 0 0
$$898$$ −48648.4 −1.80781
$$899$$ −19156.1 −0.710670
$$900$$ 0 0
$$901$$ 1234.60 0.0456497
$$902$$ 55789.3 2.05940
$$903$$ 0 0
$$904$$ 7344.26 0.270206
$$905$$ 1968.35 0.0722984
$$906$$ 0 0
$$907$$ −36824.9 −1.34812 −0.674062 0.738674i $$-0.735452\pi$$
−0.674062 + 0.738674i $$0.735452\pi$$
$$908$$ 11132.4 0.406874
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −34520.5 −1.25545 −0.627725 0.778435i $$-0.716014\pi$$
−0.627725 + 0.778435i $$0.716014\pi$$
$$912$$ 0 0
$$913$$ −3002.29 −0.108830
$$914$$ −67642.6 −2.44794
$$915$$ 0 0
$$916$$ 59999.8 2.16424
$$917$$ 12733.2 0.458545
$$918$$ 0 0
$$919$$ 23522.8 0.844336 0.422168 0.906518i $$-0.361269\pi$$
0.422168 + 0.906518i $$0.361269\pi$$
$$920$$ 275.171 0.00986101
$$921$$ 0 0
$$922$$ 44300.4 1.58238
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 21749.5 0.773102
$$926$$ 51789.7 1.83792
$$927$$ 0 0
$$928$$ 26239.4 0.928180
$$929$$ 24563.2 0.867482 0.433741 0.901038i $$-0.357193\pi$$
0.433741 + 0.901038i $$0.357193\pi$$
$$930$$ 0 0
$$931$$ 15143.0 0.533073
$$932$$ 62360.9 2.19174
$$933$$ 0 0
$$934$$ −29428.0 −1.03096
$$935$$ −223.060 −0.00780197
$$936$$ 0 0
$$937$$ −12115.6 −0.422411 −0.211206 0.977442i $$-0.567739\pi$$
−0.211206 + 0.977442i $$0.567739\pi$$
$$938$$ −2176.36 −0.0757576
$$939$$ 0 0
$$940$$ −9300.19 −0.322701
$$941$$ −14898.3 −0.516123 −0.258062 0.966128i $$-0.583084\pi$$
−0.258062 + 0.966128i $$0.583084\pi$$
$$942$$ 0 0
$$943$$ 1386.35 0.0478745
$$944$$ −1254.99 −0.0432695
$$945$$ 0 0
$$946$$ −76872.7 −2.64201
$$947$$ −7434.32 −0.255103 −0.127552 0.991832i $$-0.540712\pi$$
−0.127552 + 0.991832i $$0.540712\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 32156.7 1.09821
$$951$$ 0 0
$$952$$ −422.564 −0.0143859
$$953$$ 23528.6 0.799754 0.399877 0.916569i $$-0.369053\pi$$
0.399877 + 0.916569i $$0.369053\pi$$
$$954$$ 0 0
$$955$$ 3652.56 0.123764
$$956$$ −61573.8 −2.08309
$$957$$ 0 0
$$958$$ 43637.9 1.47169
$$959$$ −5947.43 −0.200263
$$960$$ 0 0
$$961$$ −11252.7 −0.377723
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 75257.9 2.51441
$$965$$ 1458.17 0.0486425
$$966$$ 0 0
$$967$$ 23558.0 0.783427 0.391713 0.920087i $$-0.371882\pi$$
0.391713 + 0.920087i $$0.371882\pi$$
$$968$$ 4894.88 0.162528
$$969$$ 0 0
$$970$$ 15041.0 0.497875
$$971$$ 262.338 0.00867026 0.00433513 0.999991i $$-0.498620\pi$$
0.00433513 + 0.999991i $$0.498620\pi$$
$$972$$ 0 0
$$973$$ 3160.86 0.104144
$$974$$ 22429.7 0.737878
$$975$$ 0 0
$$976$$ 390.903 0.0128202
$$977$$ −33144.4 −1.08535 −0.542673 0.839944i $$-0.682588\pi$$
−0.542673 + 0.839944i $$0.682588\pi$$
$$978$$ 0 0
$$979$$ 7995.28 0.261011
$$980$$ 9047.00 0.294894
$$981$$ 0 0
$$982$$ −13460.3 −0.437410
$$983$$ 4866.80 0.157911 0.0789557 0.996878i $$-0.474841\pi$$
0.0789557 + 0.996878i $$0.474841\pi$$
$$984$$ 0 0
$$985$$ 8763.17 0.283470
$$986$$ 1293.28 0.0417714
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1910.26 −0.0614184
$$990$$ 0 0
$$991$$ 12533.9 0.401770 0.200885 0.979615i $$-0.435618\pi$$
0.200885 + 0.979615i $$0.435618\pi$$
$$992$$ −25393.1 −0.812733
$$993$$ 0 0
$$994$$ −12210.7 −0.389639
$$995$$ −3473.37 −0.110666
$$996$$ 0 0
$$997$$ −3560.92 −0.113115 −0.0565574 0.998399i $$-0.518012\pi$$
−0.0565574 + 0.998399i $$0.518012\pi$$
$$998$$ 61262.1 1.94310
$$999$$ 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 1521.4.a.t.1.2 2
3.2 odd 2 169.4.a.f.1.1 2
13.3 even 3 117.4.g.d.100.1 4
13.9 even 3 117.4.g.d.55.1 4
13.12 even 2 1521.4.a.l.1.1 2
39.2 even 12 169.4.e.g.147.1 8
39.5 even 4 169.4.b.e.168.4 4
39.8 even 4 169.4.b.e.168.1 4
39.11 even 12 169.4.e.g.147.4 8
39.17 odd 6 169.4.c.f.146.1 4
39.20 even 12 169.4.e.g.23.1 8
39.23 odd 6 169.4.c.f.22.1 4
39.29 odd 6 13.4.c.b.9.2 yes 4
39.32 even 12 169.4.e.g.23.4 8
39.35 odd 6 13.4.c.b.3.2 4
39.38 odd 2 169.4.a.j.1.2 2
156.35 even 6 208.4.i.e.81.2 4
156.107 even 6 208.4.i.e.113.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.2 4 39.35 odd 6
13.4.c.b.9.2 yes 4 39.29 odd 6
117.4.g.d.55.1 4 13.9 even 3
117.4.g.d.100.1 4 13.3 even 3
169.4.a.f.1.1 2 3.2 odd 2
169.4.a.j.1.2 2 39.38 odd 2
169.4.b.e.168.1 4 39.8 even 4
169.4.b.e.168.4 4 39.5 even 4
169.4.c.f.22.1 4 39.23 odd 6
169.4.c.f.146.1 4 39.17 odd 6
169.4.e.g.23.1 8 39.20 even 12
169.4.e.g.23.4 8 39.32 even 12
169.4.e.g.147.1 8 39.2 even 12
169.4.e.g.147.4 8 39.11 even 12
208.4.i.e.81.2 4 156.35 even 6
208.4.i.e.113.2 4 156.107 even 6
1521.4.a.l.1.1 2 13.12 even 2
1521.4.a.t.1.2 2 1.1 even 1 trivial