# Properties

 Label 675.4.a.r.1.3 Level $675$ Weight $4$ Character 675.1 Self dual yes Analytic conductor $39.826$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.8262892539$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.5637.1 Defining polynomial: $$x^{3} - x^{2} - 23x + 6$$ x^3 - x^2 - 23*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 135) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$5.20067$$ of defining polynomial Character $$\chi$$ $$=$$ 675.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.20067 q^{2} +19.0470 q^{4} -24.4013 q^{7} +57.4517 q^{8} +O(q^{10})$$ $$q+5.20067 q^{2} +19.0470 q^{4} -24.4013 q^{7} +57.4517 q^{8} +28.9839 q^{11} +65.3919 q^{13} -126.903 q^{14} +146.411 q^{16} +68.1718 q^{17} +104.424 q^{19} +150.736 q^{22} -154.807 q^{23} +340.082 q^{26} -464.772 q^{28} +205.658 q^{29} -18.2497 q^{31} +301.824 q^{32} +354.539 q^{34} +337.613 q^{37} +543.076 q^{38} +195.969 q^{41} -334.882 q^{43} +552.055 q^{44} -805.098 q^{46} -5.00398 q^{47} +252.425 q^{49} +1245.52 q^{52} -319.965 q^{53} -1401.90 q^{56} +1069.56 q^{58} -430.611 q^{59} +594.581 q^{61} -94.9106 q^{62} +398.396 q^{64} -195.876 q^{67} +1298.47 q^{68} -425.955 q^{71} -929.193 q^{73} +1755.82 q^{74} +1988.96 q^{76} -707.245 q^{77} +24.4296 q^{79} +1019.17 q^{82} -545.859 q^{83} -1741.61 q^{86} +1665.17 q^{88} +84.1332 q^{89} -1595.65 q^{91} -2948.60 q^{92} -26.0241 q^{94} -827.613 q^{97} +1312.78 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8}+O(q^{10})$$ 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8 $$3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8} + 38 q^{11} - 28 q^{13} - 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 122 q^{22} + 81 q^{23} + 416 q^{26} - 410 q^{28} + 160 q^{29} + 227 q^{31} + 569 q^{32} + 17 q^{34} - 78 q^{37} + 757 q^{38} - 338 q^{41} - 22 q^{43} + 1636 q^{44} - 1425 q^{46} + 472 q^{47} - 197 q^{49} + 1566 q^{52} - 521 q^{53} - 1254 q^{56} + 2096 q^{58} + 140 q^{59} + 595 q^{61} - 1407 q^{62} - 918 q^{64} - 878 q^{67} + 3053 q^{68} - 602 q^{71} - 1294 q^{73} + 2878 q^{74} + 525 q^{76} - 288 q^{77} + 629 q^{79} + 1682 q^{82} + 1287 q^{83} - 3730 q^{86} + 858 q^{88} + 2154 q^{89} - 440 q^{91} - 1959 q^{92} - 1108 q^{94} - 1392 q^{97} + 2693 q^{98}+O(q^{100})$$ 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8 + 38 * q^11 - 28 * q^13 - 108 * q^14 + 191 * q^16 + 19 * q^17 + 187 * q^19 - 122 * q^22 + 81 * q^23 + 416 * q^26 - 410 * q^28 + 160 * q^29 + 227 * q^31 + 569 * q^32 + 17 * q^34 - 78 * q^37 + 757 * q^38 - 338 * q^41 - 22 * q^43 + 1636 * q^44 - 1425 * q^46 + 472 * q^47 - 197 * q^49 + 1566 * q^52 - 521 * q^53 - 1254 * q^56 + 2096 * q^58 + 140 * q^59 + 595 * q^61 - 1407 * q^62 - 918 * q^64 - 878 * q^67 + 3053 * q^68 - 602 * q^71 - 1294 * q^73 + 2878 * q^74 + 525 * q^76 - 288 * q^77 + 629 * q^79 + 1682 * q^82 + 1287 * q^83 - 3730 * q^86 + 858 * q^88 + 2154 * q^89 - 440 * q^91 - 1959 * q^92 - 1108 * q^94 - 1392 * q^97 + 2693 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.20067 1.83871 0.919357 0.393424i $$-0.128709\pi$$
0.919357 + 0.393424i $$0.128709\pi$$
$$3$$ 0 0
$$4$$ 19.0470 2.38087
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −24.4013 −1.31755 −0.658774 0.752341i $$-0.728925\pi$$
−0.658774 + 0.752341i $$0.728925\pi$$
$$8$$ 57.4517 2.53903
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 28.9839 0.794451 0.397226 0.917721i $$-0.369973\pi$$
0.397226 + 0.917721i $$0.369973\pi$$
$$12$$ 0 0
$$13$$ 65.3919 1.39511 0.697556 0.716530i $$-0.254271\pi$$
0.697556 + 0.716530i $$0.254271\pi$$
$$14$$ −126.903 −2.42260
$$15$$ 0 0
$$16$$ 146.411 2.28768
$$17$$ 68.1718 0.972593 0.486296 0.873794i $$-0.338347\pi$$
0.486296 + 0.873794i $$0.338347\pi$$
$$18$$ 0 0
$$19$$ 104.424 1.26087 0.630435 0.776242i $$-0.282876\pi$$
0.630435 + 0.776242i $$0.282876\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 150.736 1.46077
$$23$$ −154.807 −1.40345 −0.701727 0.712446i $$-0.747587\pi$$
−0.701727 + 0.712446i $$0.747587\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 340.082 2.56521
$$27$$ 0 0
$$28$$ −464.772 −3.13691
$$29$$ 205.658 1.31689 0.658443 0.752631i $$-0.271215\pi$$
0.658443 + 0.752631i $$0.271215\pi$$
$$30$$ 0 0
$$31$$ −18.2497 −0.105734 −0.0528668 0.998602i $$-0.516836\pi$$
−0.0528668 + 0.998602i $$0.516836\pi$$
$$32$$ 301.824 1.66736
$$33$$ 0 0
$$34$$ 354.539 1.78832
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 337.613 1.50009 0.750044 0.661387i $$-0.230032\pi$$
0.750044 + 0.661387i $$0.230032\pi$$
$$38$$ 543.076 2.31838
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 195.969 0.746469 0.373234 0.927737i $$-0.378249\pi$$
0.373234 + 0.927737i $$0.378249\pi$$
$$42$$ 0 0
$$43$$ −334.882 −1.18765 −0.593826 0.804594i $$-0.702383\pi$$
−0.593826 + 0.804594i $$0.702383\pi$$
$$44$$ 552.055 1.89149
$$45$$ 0 0
$$46$$ −805.098 −2.58055
$$47$$ −5.00398 −0.0155299 −0.00776496 0.999970i $$-0.502472\pi$$
−0.00776496 + 0.999970i $$0.502472\pi$$
$$48$$ 0 0
$$49$$ 252.425 0.735934
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1245.52 3.32158
$$53$$ −319.965 −0.829256 −0.414628 0.909991i $$-0.636088\pi$$
−0.414628 + 0.909991i $$0.636088\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1401.90 −3.34529
$$57$$ 0 0
$$58$$ 1069.56 2.42138
$$59$$ −430.611 −0.950182 −0.475091 0.879937i $$-0.657585\pi$$
−0.475091 + 0.879937i $$0.657585\pi$$
$$60$$ 0 0
$$61$$ 594.581 1.24800 0.624002 0.781422i $$-0.285505\pi$$
0.624002 + 0.781422i $$0.285505\pi$$
$$62$$ −94.9106 −0.194414
$$63$$ 0 0
$$64$$ 398.396 0.778118
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −195.876 −0.357166 −0.178583 0.983925i $$-0.557151\pi$$
−0.178583 + 0.983925i $$0.557151\pi$$
$$68$$ 1298.47 2.31562
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −425.955 −0.711994 −0.355997 0.934487i $$-0.615859\pi$$
−0.355997 + 0.934487i $$0.615859\pi$$
$$72$$ 0 0
$$73$$ −929.193 −1.48978 −0.744889 0.667188i $$-0.767498\pi$$
−0.744889 + 0.667188i $$0.767498\pi$$
$$74$$ 1755.82 2.75824
$$75$$ 0 0
$$76$$ 1988.96 3.00197
$$77$$ −707.245 −1.04673
$$78$$ 0 0
$$79$$ 24.4296 0.0347917 0.0173959 0.999849i $$-0.494462\pi$$
0.0173959 + 0.999849i $$0.494462\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1019.17 1.37254
$$83$$ −545.859 −0.721877 −0.360938 0.932590i $$-0.617544\pi$$
−0.360938 + 0.932590i $$0.617544\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1741.61 −2.18375
$$87$$ 0 0
$$88$$ 1665.17 2.01714
$$89$$ 84.1332 0.100203 0.0501017 0.998744i $$-0.484045\pi$$
0.0501017 + 0.998744i $$0.484045\pi$$
$$90$$ 0 0
$$91$$ −1595.65 −1.83813
$$92$$ −2948.60 −3.34144
$$93$$ 0 0
$$94$$ −26.0241 −0.0285551
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −827.613 −0.866303 −0.433152 0.901321i $$-0.642599\pi$$
−0.433152 + 0.901321i $$0.642599\pi$$
$$98$$ 1312.78 1.35317
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −823.576 −0.811375 −0.405688 0.914012i $$-0.632968\pi$$
−0.405688 + 0.914012i $$0.632968\pi$$
$$102$$ 0 0
$$103$$ −1171.19 −1.12040 −0.560198 0.828359i $$-0.689275\pi$$
−0.560198 + 0.828359i $$0.689275\pi$$
$$104$$ 3756.87 3.54223
$$105$$ 0 0
$$106$$ −1664.03 −1.52477
$$107$$ 1023.21 0.924460 0.462230 0.886760i $$-0.347049\pi$$
0.462230 + 0.886760i $$0.347049\pi$$
$$108$$ 0 0
$$109$$ −403.647 −0.354700 −0.177350 0.984148i $$-0.556753\pi$$
−0.177350 + 0.984148i $$0.556753\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3572.63 −3.01413
$$113$$ 1082.20 0.900931 0.450465 0.892794i $$-0.351258\pi$$
0.450465 + 0.892794i $$0.351258\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3917.16 3.13534
$$117$$ 0 0
$$118$$ −2239.46 −1.74711
$$119$$ −1663.48 −1.28144
$$120$$ 0 0
$$121$$ −490.935 −0.368847
$$122$$ 3092.22 2.29473
$$123$$ 0 0
$$124$$ −347.601 −0.251738
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 774.132 0.540890 0.270445 0.962735i $$-0.412829\pi$$
0.270445 + 0.962735i $$0.412829\pi$$
$$128$$ −342.664 −0.236621
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1214.04 0.809702 0.404851 0.914383i $$-0.367323\pi$$
0.404851 + 0.914383i $$0.367323\pi$$
$$132$$ 0 0
$$133$$ −2548.09 −1.66126
$$134$$ −1018.69 −0.656726
$$135$$ 0 0
$$136$$ 3916.58 2.46944
$$137$$ 2300.15 1.43441 0.717207 0.696860i $$-0.245420\pi$$
0.717207 + 0.696860i $$0.245420\pi$$
$$138$$ 0 0
$$139$$ −1355.93 −0.827396 −0.413698 0.910414i $$-0.635763\pi$$
−0.413698 + 0.910414i $$0.635763\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2215.25 −1.30915
$$143$$ 1895.31 1.10835
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −4832.43 −2.73928
$$147$$ 0 0
$$148$$ 6430.51 3.57152
$$149$$ −259.845 −0.142868 −0.0714340 0.997445i $$-0.522758\pi$$
−0.0714340 + 0.997445i $$0.522758\pi$$
$$150$$ 0 0
$$151$$ −508.304 −0.273941 −0.136971 0.990575i $$-0.543737\pi$$
−0.136971 + 0.990575i $$0.543737\pi$$
$$152$$ 5999.34 3.20139
$$153$$ 0 0
$$154$$ −3678.15 −1.92463
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 23.3052 0.0118468 0.00592342 0.999982i $$-0.498115\pi$$
0.00592342 + 0.999982i $$0.498115\pi$$
$$158$$ 127.050 0.0639721
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3777.49 1.84912
$$162$$ 0 0
$$163$$ 4032.10 1.93754 0.968769 0.247964i $$-0.0797616\pi$$
0.968769 + 0.247964i $$0.0797616\pi$$
$$164$$ 3732.62 1.77725
$$165$$ 0 0
$$166$$ −2838.83 −1.32733
$$167$$ −671.911 −0.311341 −0.155671 0.987809i $$-0.549754\pi$$
−0.155671 + 0.987809i $$0.549754\pi$$
$$168$$ 0 0
$$169$$ 2079.10 0.946337
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −6378.49 −2.82765
$$173$$ 1633.53 0.717889 0.358944 0.933359i $$-0.383137\pi$$
0.358944 + 0.933359i $$0.383137\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4243.57 1.81745
$$177$$ 0 0
$$178$$ 437.549 0.184245
$$179$$ −341.260 −0.142497 −0.0712485 0.997459i $$-0.522698\pi$$
−0.0712485 + 0.997459i $$0.522698\pi$$
$$180$$ 0 0
$$181$$ −1695.92 −0.696447 −0.348223 0.937412i $$-0.613215\pi$$
−0.348223 + 0.937412i $$0.613215\pi$$
$$182$$ −8298.45 −3.37979
$$183$$ 0 0
$$184$$ −8893.90 −3.56341
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1975.88 0.772678
$$188$$ −95.3107 −0.0369747
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 726.451 0.275205 0.137603 0.990488i $$-0.456060\pi$$
0.137603 + 0.990488i $$0.456060\pi$$
$$192$$ 0 0
$$193$$ −4247.26 −1.58406 −0.792032 0.610479i $$-0.790977\pi$$
−0.792032 + 0.610479i $$0.790977\pi$$
$$194$$ −4304.14 −1.59288
$$195$$ 0 0
$$196$$ 4807.94 1.75216
$$197$$ 2678.52 0.968713 0.484357 0.874871i $$-0.339054\pi$$
0.484357 + 0.874871i $$0.339054\pi$$
$$198$$ 0 0
$$199$$ 1486.48 0.529517 0.264759 0.964315i $$-0.414708\pi$$
0.264759 + 0.964315i $$0.414708\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −4283.15 −1.49189
$$203$$ −5018.32 −1.73506
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −6090.97 −2.06009
$$207$$ 0 0
$$208$$ 9574.12 3.19157
$$209$$ 3026.62 1.00170
$$210$$ 0 0
$$211$$ −4827.41 −1.57504 −0.787519 0.616291i $$-0.788635\pi$$
−0.787519 + 0.616291i $$0.788635\pi$$
$$212$$ −6094.37 −1.97435
$$213$$ 0 0
$$214$$ 5321.37 1.69982
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 445.317 0.139309
$$218$$ −2099.23 −0.652193
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4457.88 1.35688
$$222$$ 0 0
$$223$$ −2774.48 −0.833153 −0.416576 0.909101i $$-0.636770\pi$$
−0.416576 + 0.909101i $$0.636770\pi$$
$$224$$ −7364.91 −2.19683
$$225$$ 0 0
$$226$$ 5628.19 1.65655
$$227$$ −5101.34 −1.49158 −0.745788 0.666184i $$-0.767927\pi$$
−0.745788 + 0.666184i $$0.767927\pi$$
$$228$$ 0 0
$$229$$ −4097.83 −1.18250 −0.591249 0.806489i $$-0.701365\pi$$
−0.591249 + 0.806489i $$0.701365\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 11815.4 3.34361
$$233$$ 357.613 0.100549 0.0502747 0.998735i $$-0.483990\pi$$
0.0502747 + 0.998735i $$0.483990\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −8201.83 −2.26226
$$237$$ 0 0
$$238$$ −8651.22 −2.35620
$$239$$ 351.682 0.0951818 0.0475909 0.998867i $$-0.484846\pi$$
0.0475909 + 0.998867i $$0.484846\pi$$
$$240$$ 0 0
$$241$$ −6165.53 −1.64795 −0.823976 0.566624i $$-0.808249\pi$$
−0.823976 + 0.566624i $$0.808249\pi$$
$$242$$ −2553.19 −0.678204
$$243$$ 0 0
$$244$$ 11325.0 2.97134
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6828.50 1.75906
$$248$$ −1048.48 −0.268461
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −3245.53 −0.816160 −0.408080 0.912946i $$-0.633802\pi$$
−0.408080 + 0.912946i $$0.633802\pi$$
$$252$$ 0 0
$$253$$ −4486.90 −1.11498
$$254$$ 4026.00 0.994543
$$255$$ 0 0
$$256$$ −4969.25 −1.21320
$$257$$ 3552.19 0.862178 0.431089 0.902309i $$-0.358129\pi$$
0.431089 + 0.902309i $$0.358129\pi$$
$$258$$ 0 0
$$259$$ −8238.22 −1.97644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6313.81 1.48881
$$263$$ 4416.59 1.03551 0.517754 0.855530i $$-0.326768\pi$$
0.517754 + 0.855530i $$0.326768\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −13251.8 −3.05458
$$267$$ 0 0
$$268$$ −3730.85 −0.850366
$$269$$ 3419.93 0.775155 0.387578 0.921837i $$-0.373312\pi$$
0.387578 + 0.921837i $$0.373312\pi$$
$$270$$ 0 0
$$271$$ 716.407 0.160585 0.0802927 0.996771i $$-0.474415\pi$$
0.0802927 + 0.996771i $$0.474415\pi$$
$$272$$ 9981.12 2.22498
$$273$$ 0 0
$$274$$ 11962.3 2.63748
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −657.529 −0.142625 −0.0713124 0.997454i $$-0.522719\pi$$
−0.0713124 + 0.997454i $$0.522719\pi$$
$$278$$ −7051.72 −1.52135
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1513.91 −0.321397 −0.160698 0.987004i $$-0.551375\pi$$
−0.160698 + 0.987004i $$0.551375\pi$$
$$282$$ 0 0
$$283$$ −3906.38 −0.820532 −0.410266 0.911966i $$-0.634564\pi$$
−0.410266 + 0.911966i $$0.634564\pi$$
$$284$$ −8113.16 −1.69517
$$285$$ 0 0
$$286$$ 9856.89 2.03794
$$287$$ −4781.91 −0.983509
$$288$$ 0 0
$$289$$ −265.611 −0.0540629
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −17698.3 −3.54697
$$293$$ −8048.76 −1.60483 −0.802413 0.596770i $$-0.796451\pi$$
−0.802413 + 0.596770i $$0.796451\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 19396.4 3.80877
$$297$$ 0 0
$$298$$ −1351.37 −0.262693
$$299$$ −10123.1 −1.95797
$$300$$ 0 0
$$301$$ 8171.57 1.56479
$$302$$ −2643.52 −0.503700
$$303$$ 0 0
$$304$$ 15288.9 2.88447
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −101.564 −0.0188814 −0.00944068 0.999955i $$-0.503005\pi$$
−0.00944068 + 0.999955i $$0.503005\pi$$
$$308$$ −13470.9 −2.49213
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7684.59 1.40113 0.700567 0.713586i $$-0.252930\pi$$
0.700567 + 0.713586i $$0.252930\pi$$
$$312$$ 0 0
$$313$$ 1345.15 0.242915 0.121457 0.992597i $$-0.461243\pi$$
0.121457 + 0.992597i $$0.461243\pi$$
$$314$$ 121.202 0.0217830
$$315$$ 0 0
$$316$$ 465.310 0.0828346
$$317$$ 7622.33 1.35051 0.675257 0.737583i $$-0.264033\pi$$
0.675257 + 0.737583i $$0.264033\pi$$
$$318$$ 0 0
$$319$$ 5960.76 1.04620
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 19645.5 3.40000
$$323$$ 7118.78 1.22631
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 20969.6 3.56258
$$327$$ 0 0
$$328$$ 11258.7 1.89531
$$329$$ 122.104 0.0204614
$$330$$ 0 0
$$331$$ −6585.09 −1.09350 −0.546751 0.837295i $$-0.684136\pi$$
−0.546751 + 0.837295i $$0.684136\pi$$
$$332$$ −10397.0 −1.71870
$$333$$ 0 0
$$334$$ −3494.39 −0.572468
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2946.94 0.476351 0.238175 0.971222i $$-0.423451\pi$$
0.238175 + 0.971222i $$0.423451\pi$$
$$338$$ 10812.7 1.74004
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −528.947 −0.0840002
$$342$$ 0 0
$$343$$ 2210.14 0.347919
$$344$$ −19239.5 −3.01548
$$345$$ 0 0
$$346$$ 8495.44 1.31999
$$347$$ 8493.48 1.31399 0.656994 0.753896i $$-0.271828\pi$$
0.656994 + 0.753896i $$0.271828\pi$$
$$348$$ 0 0
$$349$$ 5646.54 0.866053 0.433027 0.901381i $$-0.357446\pi$$
0.433027 + 0.901381i $$0.357446\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 8748.03 1.32464
$$353$$ −1221.93 −0.184240 −0.0921202 0.995748i $$-0.529364\pi$$
−0.0921202 + 0.995748i $$0.529364\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 1602.48 0.238571
$$357$$ 0 0
$$358$$ −1774.78 −0.262011
$$359$$ 4151.44 0.610319 0.305160 0.952301i $$-0.401290\pi$$
0.305160 + 0.952301i $$0.401290\pi$$
$$360$$ 0 0
$$361$$ 4045.41 0.589796
$$362$$ −8819.93 −1.28057
$$363$$ 0 0
$$364$$ −30392.3 −4.37635
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7038.71 1.00114 0.500569 0.865696i $$-0.333124\pi$$
0.500569 + 0.865696i $$0.333124\pi$$
$$368$$ −22665.5 −3.21065
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 7807.58 1.09259
$$372$$ 0 0
$$373$$ 7119.57 0.988303 0.494152 0.869376i $$-0.335479\pi$$
0.494152 + 0.869376i $$0.335479\pi$$
$$374$$ 10275.9 1.42073
$$375$$ 0 0
$$376$$ −287.487 −0.0394309
$$377$$ 13448.4 1.83720
$$378$$ 0 0
$$379$$ 3372.29 0.457053 0.228526 0.973538i $$-0.426609\pi$$
0.228526 + 0.973538i $$0.426609\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 3778.03 0.506024
$$383$$ 3958.63 0.528138 0.264069 0.964504i $$-0.414935\pi$$
0.264069 + 0.964504i $$0.414935\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22088.6 −2.91264
$$387$$ 0 0
$$388$$ −15763.5 −2.06256
$$389$$ −9654.01 −1.25830 −0.629148 0.777285i $$-0.716596\pi$$
−0.629148 + 0.777285i $$0.716596\pi$$
$$390$$ 0 0
$$391$$ −10553.4 −1.36499
$$392$$ 14502.3 1.86856
$$393$$ 0 0
$$394$$ 13930.1 1.78119
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10928.3 −1.38155 −0.690776 0.723068i $$-0.742731\pi$$
−0.690776 + 0.723068i $$0.742731\pi$$
$$398$$ 7730.70 0.973631
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −4085.57 −0.508787 −0.254393 0.967101i $$-0.581876\pi$$
−0.254393 + 0.967101i $$0.581876\pi$$
$$402$$ 0 0
$$403$$ −1193.38 −0.147510
$$404$$ −15686.6 −1.93178
$$405$$ 0 0
$$406$$ −26098.7 −3.19028
$$407$$ 9785.34 1.19175
$$408$$ 0 0
$$409$$ 10156.3 1.22786 0.613930 0.789361i $$-0.289588\pi$$
0.613930 + 0.789361i $$0.289588\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −22307.6 −2.66752
$$413$$ 10507.5 1.25191
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 19736.9 2.32615
$$417$$ 0 0
$$418$$ 15740.4 1.84184
$$419$$ −15878.8 −1.85139 −0.925693 0.378275i $$-0.876517\pi$$
−0.925693 + 0.378275i $$0.876517\pi$$
$$420$$ 0 0
$$421$$ −2279.85 −0.263926 −0.131963 0.991255i $$-0.542128\pi$$
−0.131963 + 0.991255i $$0.542128\pi$$
$$422$$ −25105.8 −2.89604
$$423$$ 0 0
$$424$$ −18382.5 −2.10551
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −14508.6 −1.64431
$$428$$ 19489.0 2.20102
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1947.38 0.217638 0.108819 0.994062i $$-0.465293\pi$$
0.108819 + 0.994062i $$0.465293\pi$$
$$432$$ 0 0
$$433$$ −12636.2 −1.40244 −0.701219 0.712946i $$-0.747361\pi$$
−0.701219 + 0.712946i $$0.747361\pi$$
$$434$$ 2315.95 0.256150
$$435$$ 0 0
$$436$$ −7688.25 −0.844496
$$437$$ −16165.6 −1.76957
$$438$$ 0 0
$$439$$ −15849.8 −1.72317 −0.861585 0.507614i $$-0.830528\pi$$
−0.861585 + 0.507614i $$0.830528\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 23184.0 2.49491
$$443$$ −17455.6 −1.87210 −0.936048 0.351872i $$-0.885545\pi$$
−0.936048 + 0.351872i $$0.885545\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14429.2 −1.53193
$$447$$ 0 0
$$448$$ −9721.41 −1.02521
$$449$$ 16068.1 1.68887 0.844435 0.535658i $$-0.179936\pi$$
0.844435 + 0.535658i $$0.179936\pi$$
$$450$$ 0 0
$$451$$ 5679.94 0.593033
$$452$$ 20612.7 2.14500
$$453$$ 0 0
$$454$$ −26530.4 −2.74258
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11891.7 −1.21722 −0.608612 0.793468i $$-0.708273\pi$$
−0.608612 + 0.793468i $$0.708273\pi$$
$$458$$ −21311.4 −2.17428
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2802.23 0.283108 0.141554 0.989931i $$-0.454790\pi$$
0.141554 + 0.989931i $$0.454790\pi$$
$$462$$ 0 0
$$463$$ 12933.3 1.29819 0.649096 0.760707i $$-0.275147\pi$$
0.649096 + 0.760707i $$0.275147\pi$$
$$464$$ 30110.6 3.01261
$$465$$ 0 0
$$466$$ 1859.83 0.184882
$$467$$ 5748.11 0.569573 0.284787 0.958591i $$-0.408077\pi$$
0.284787 + 0.958591i $$0.408077\pi$$
$$468$$ 0 0
$$469$$ 4779.65 0.470583
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −24739.3 −2.41254
$$473$$ −9706.17 −0.943531
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −31684.3 −3.05094
$$477$$ 0 0
$$478$$ 1828.98 0.175012
$$479$$ 11217.3 1.07000 0.535002 0.844851i $$-0.320311\pi$$
0.535002 + 0.844851i $$0.320311\pi$$
$$480$$ 0 0
$$481$$ 22077.2 2.09279
$$482$$ −32064.9 −3.03012
$$483$$ 0 0
$$484$$ −9350.83 −0.878177
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8905.12 0.828603 0.414301 0.910140i $$-0.364026\pi$$
0.414301 + 0.910140i $$0.364026\pi$$
$$488$$ 34159.7 3.16872
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6553.10 0.602316 0.301158 0.953574i $$-0.402627\pi$$
0.301158 + 0.953574i $$0.402627\pi$$
$$492$$ 0 0
$$493$$ 14020.1 1.28079
$$494$$ 35512.8 3.23440
$$495$$ 0 0
$$496$$ −2671.96 −0.241884
$$497$$ 10393.9 0.938087
$$498$$ 0 0
$$499$$ −4610.09 −0.413579 −0.206789 0.978385i $$-0.566302\pi$$
−0.206789 + 0.978385i $$0.566302\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −16878.9 −1.50069
$$503$$ 13069.1 1.15850 0.579249 0.815151i $$-0.303346\pi$$
0.579249 + 0.815151i $$0.303346\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −23334.9 −2.05012
$$507$$ 0 0
$$508$$ 14744.9 1.28779
$$509$$ −15930.8 −1.38727 −0.693635 0.720327i $$-0.743992\pi$$
−0.693635 + 0.720327i $$0.743992\pi$$
$$510$$ 0 0
$$511$$ 22673.6 1.96286
$$512$$ −23102.1 −1.99410
$$513$$ 0 0
$$514$$ 18473.8 1.58530
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −145.035 −0.0123378
$$518$$ −42844.3 −3.63411
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −3654.38 −0.307296 −0.153648 0.988126i $$-0.549102\pi$$
−0.153648 + 0.988126i $$0.549102\pi$$
$$522$$ 0 0
$$523$$ 5138.66 0.429633 0.214816 0.976654i $$-0.431085\pi$$
0.214816 + 0.976654i $$0.431085\pi$$
$$524$$ 23123.7 1.92780
$$525$$ 0 0
$$526$$ 22969.2 1.90400
$$527$$ −1244.11 −0.102836
$$528$$ 0 0
$$529$$ 11798.1 0.969681
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −48533.4 −3.95524
$$533$$ 12814.8 1.04141
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −11253.4 −0.906854
$$537$$ 0 0
$$538$$ 17785.9 1.42529
$$539$$ 7316.27 0.584664
$$540$$ 0 0
$$541$$ 6932.06 0.550892 0.275446 0.961317i $$-0.411175\pi$$
0.275446 + 0.961317i $$0.411175\pi$$
$$542$$ 3725.80 0.295271
$$543$$ 0 0
$$544$$ 20575.9 1.62166
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3423.11 0.267572 0.133786 0.991010i $$-0.457287\pi$$
0.133786 + 0.991010i $$0.457287\pi$$
$$548$$ 43810.8 3.41516
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 21475.6 1.66042
$$552$$ 0 0
$$553$$ −596.115 −0.0458398
$$554$$ −3419.59 −0.262246
$$555$$ 0 0
$$556$$ −25826.3 −1.96992
$$557$$ −24489.2 −1.86291 −0.931455 0.363856i $$-0.881460\pi$$
−0.931455 + 0.363856i $$0.881460\pi$$
$$558$$ 0 0
$$559$$ −21898.6 −1.65691
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −7873.37 −0.590957
$$563$$ −10053.1 −0.752552 −0.376276 0.926508i $$-0.622796\pi$$
−0.376276 + 0.926508i $$0.622796\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −20315.8 −1.50872
$$567$$ 0 0
$$568$$ −24471.8 −1.80777
$$569$$ 6670.45 0.491459 0.245729 0.969338i $$-0.420973\pi$$
0.245729 + 0.969338i $$0.420973\pi$$
$$570$$ 0 0
$$571$$ 4633.55 0.339594 0.169797 0.985479i $$-0.445689\pi$$
0.169797 + 0.985479i $$0.445689\pi$$
$$572$$ 36099.9 2.63884
$$573$$ 0 0
$$574$$ −24869.1 −1.80839
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −7045.15 −0.508307 −0.254154 0.967164i $$-0.581797\pi$$
−0.254154 + 0.967164i $$0.581797\pi$$
$$578$$ −1381.35 −0.0994062
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13319.7 0.951108
$$582$$ 0 0
$$583$$ −9273.82 −0.658804
$$584$$ −53383.7 −3.78259
$$585$$ 0 0
$$586$$ −41859.0 −2.95082
$$587$$ −8001.06 −0.562588 −0.281294 0.959622i $$-0.590764\pi$$
−0.281294 + 0.959622i $$0.590764\pi$$
$$588$$ 0 0
$$589$$ −1905.71 −0.133316
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 49430.4 3.43172
$$593$$ 6747.53 0.467265 0.233632 0.972325i $$-0.424939\pi$$
0.233632 + 0.972325i $$0.424939\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4949.26 −0.340150
$$597$$ 0 0
$$598$$ −52646.9 −3.60016
$$599$$ −21547.3 −1.46978 −0.734890 0.678186i $$-0.762766\pi$$
−0.734890 + 0.678186i $$0.762766\pi$$
$$600$$ 0 0
$$601$$ −12155.1 −0.824983 −0.412492 0.910961i $$-0.635341\pi$$
−0.412492 + 0.910961i $$0.635341\pi$$
$$602$$ 42497.6 2.87720
$$603$$ 0 0
$$604$$ −9681.64 −0.652219
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −16348.9 −1.09322 −0.546608 0.837388i $$-0.684081\pi$$
−0.546608 + 0.837388i $$0.684081\pi$$
$$608$$ 31517.7 2.10232
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −327.220 −0.0216660
$$612$$ 0 0
$$613$$ 29955.5 1.97372 0.986859 0.161581i $$-0.0516593\pi$$
0.986859 + 0.161581i $$0.0516593\pi$$
$$614$$ −528.202 −0.0347174
$$615$$ 0 0
$$616$$ −40632.4 −2.65767
$$617$$ 2159.74 0.140921 0.0704603 0.997515i $$-0.477553\pi$$
0.0704603 + 0.997515i $$0.477553\pi$$
$$618$$ 0 0
$$619$$ 22100.8 1.43507 0.717535 0.696523i $$-0.245270\pi$$
0.717535 + 0.696523i $$0.245270\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 39965.0 2.57629
$$623$$ −2052.96 −0.132023
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 6995.68 0.446651
$$627$$ 0 0
$$628$$ 443.893 0.0282058
$$629$$ 23015.7 1.45898
$$630$$ 0 0
$$631$$ 18360.1 1.15833 0.579164 0.815211i $$-0.303379\pi$$
0.579164 + 0.815211i $$0.303379\pi$$
$$632$$ 1403.52 0.0883372
$$633$$ 0 0
$$634$$ 39641.2 2.48321
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 16506.6 1.02671
$$638$$ 30999.9 1.92367
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 21064.5 1.29797 0.648985 0.760802i $$-0.275194\pi$$
0.648985 + 0.760802i $$0.275194\pi$$
$$642$$ 0 0
$$643$$ 10539.1 0.646381 0.323190 0.946334i $$-0.395245\pi$$
0.323190 + 0.946334i $$0.395245\pi$$
$$644$$ 71949.8 4.40251
$$645$$ 0 0
$$646$$ 37022.4 2.25484
$$647$$ 22553.4 1.37043 0.685213 0.728343i $$-0.259709\pi$$
0.685213 + 0.728343i $$0.259709\pi$$
$$648$$ 0 0
$$649$$ −12480.8 −0.754873
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 76799.4 4.61303
$$653$$ 22624.0 1.35582 0.677908 0.735147i $$-0.262887\pi$$
0.677908 + 0.735147i $$0.262887\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 28692.1 1.70768
$$657$$ 0 0
$$658$$ 635.022 0.0376227
$$659$$ −6376.60 −0.376930 −0.188465 0.982080i $$-0.560351\pi$$
−0.188465 + 0.982080i $$0.560351\pi$$
$$660$$ 0 0
$$661$$ −22097.5 −1.30029 −0.650146 0.759809i $$-0.725292\pi$$
−0.650146 + 0.759809i $$0.725292\pi$$
$$662$$ −34246.9 −2.01064
$$663$$ 0 0
$$664$$ −31360.5 −1.83287
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −31837.2 −1.84819
$$668$$ −12797.9 −0.741264
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 17233.3 0.991479
$$672$$ 0 0
$$673$$ −24033.0 −1.37653 −0.688264 0.725461i $$-0.741627\pi$$
−0.688264 + 0.725461i $$0.741627\pi$$
$$674$$ 15326.1 0.875873
$$675$$ 0 0
$$676$$ 39600.6 2.25311
$$677$$ 179.638 0.0101980 0.00509901 0.999987i $$-0.498377\pi$$
0.00509901 + 0.999987i $$0.498377\pi$$
$$678$$ 0 0
$$679$$ 20194.9 1.14140
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −2750.88 −0.154452
$$683$$ −30434.2 −1.70502 −0.852511 0.522709i $$-0.824921\pi$$
−0.852511 + 0.522709i $$0.824921\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 11494.2 0.639725
$$687$$ 0 0
$$688$$ −49030.5 −2.71696
$$689$$ −20923.1 −1.15691
$$690$$ 0 0
$$691$$ 9792.73 0.539122 0.269561 0.962983i $$-0.413121\pi$$
0.269561 + 0.962983i $$0.413121\pi$$
$$692$$ 31113.7 1.70920
$$693$$ 0 0
$$694$$ 44171.8 2.41605
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 13359.6 0.726010
$$698$$ 29365.8 1.59242
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8130.47 −0.438065 −0.219032 0.975718i $$-0.570290\pi$$
−0.219032 + 0.975718i $$0.570290\pi$$
$$702$$ 0 0
$$703$$ 35255.0 1.89142
$$704$$ 11547.1 0.618177
$$705$$ 0 0
$$706$$ −6354.86 −0.338765
$$707$$ 20096.4 1.06903
$$708$$ 0 0
$$709$$ −4859.95 −0.257432 −0.128716 0.991681i $$-0.541086\pi$$
−0.128716 + 0.991681i $$0.541086\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 4833.59 0.254419
$$713$$ 2825.17 0.148392
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6499.96 −0.339267
$$717$$ 0 0
$$718$$ 21590.3 1.12220
$$719$$ −19463.0 −1.00952 −0.504762 0.863259i $$-0.668420\pi$$
−0.504762 + 0.863259i $$0.668420\pi$$
$$720$$ 0 0
$$721$$ 28578.6 1.47618
$$722$$ 21038.8 1.08447
$$723$$ 0 0
$$724$$ −32302.2 −1.65815
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2432.66 −0.124102 −0.0620512 0.998073i $$-0.519764\pi$$
−0.0620512 + 0.998073i $$0.519764\pi$$
$$728$$ −91672.8 −4.66706
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −22829.5 −1.15510
$$732$$ 0 0
$$733$$ 17967.6 0.905386 0.452693 0.891666i $$-0.350463\pi$$
0.452693 + 0.891666i $$0.350463\pi$$
$$734$$ 36606.0 1.84081
$$735$$ 0 0
$$736$$ −46724.4 −2.34006
$$737$$ −5677.26 −0.283751
$$738$$ 0 0
$$739$$ 23473.0 1.16843 0.584214 0.811599i $$-0.301403\pi$$
0.584214 + 0.811599i $$0.301403\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 40604.6 2.00895
$$743$$ −33559.2 −1.65702 −0.828512 0.559971i $$-0.810812\pi$$
−0.828512 + 0.559971i $$0.810812\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 37026.5 1.81721
$$747$$ 0 0
$$748$$ 37634.6 1.83965
$$749$$ −24967.6 −1.21802
$$750$$ 0 0
$$751$$ −7782.75 −0.378158 −0.189079 0.981962i $$-0.560550\pi$$
−0.189079 + 0.981962i $$0.560550\pi$$
$$752$$ −732.640 −0.0355274
$$753$$ 0 0
$$754$$ 69940.5 3.37809
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38154.3 1.83189 0.915946 0.401300i $$-0.131442\pi$$
0.915946 + 0.401300i $$0.131442\pi$$
$$758$$ 17538.2 0.840390
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −19867.1 −0.946363 −0.473182 0.880965i $$-0.656895\pi$$
−0.473182 + 0.880965i $$0.656895\pi$$
$$762$$ 0 0
$$763$$ 9849.52 0.467335
$$764$$ 13836.7 0.655228
$$765$$ 0 0
$$766$$ 20587.5 0.971094
$$767$$ −28158.5 −1.32561
$$768$$ 0 0
$$769$$ −15710.8 −0.736730 −0.368365 0.929681i $$-0.620082\pi$$
−0.368365 + 0.929681i $$0.620082\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −80897.5 −3.77146
$$773$$ 25811.9 1.20102 0.600510 0.799617i $$-0.294964\pi$$
0.600510 + 0.799617i $$0.294964\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −47547.8 −2.19957
$$777$$ 0 0
$$778$$ −50207.3 −2.31365
$$779$$ 20463.9 0.941201
$$780$$ 0 0
$$781$$ −12345.8 −0.565645
$$782$$ −54885.0 −2.50982
$$783$$ 0 0
$$784$$ 36958.0 1.68358
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 29242.6 1.32450 0.662252 0.749281i $$-0.269601\pi$$
0.662252 + 0.749281i $$0.269601\pi$$
$$788$$ 51017.7 2.30638
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −26407.2 −1.18702
$$792$$ 0 0
$$793$$ 38880.8 1.74111
$$794$$ −56834.6 −2.54028
$$795$$ 0 0
$$796$$ 28313.0 1.26071
$$797$$ 32573.7 1.44771 0.723853 0.689955i $$-0.242370\pi$$
0.723853 + 0.689955i $$0.242370\pi$$
$$798$$ 0 0
$$799$$ −341.130 −0.0151043
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −21247.7 −0.935514
$$803$$ −26931.6 −1.18356
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −6206.39 −0.271229
$$807$$ 0 0
$$808$$ −47315.8 −2.06010
$$809$$ 34644.9 1.50562 0.752812 0.658236i $$-0.228697\pi$$
0.752812 + 0.658236i $$0.228697\pi$$
$$810$$ 0 0
$$811$$ −29057.9 −1.25815 −0.629077 0.777343i $$-0.716567\pi$$
−0.629077 + 0.777343i $$0.716567\pi$$
$$812$$ −95583.9 −4.13096
$$813$$ 0 0
$$814$$ 50890.3 2.19128
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −34969.8 −1.49748
$$818$$ 52819.3 2.25768
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −46709.5 −1.98560 −0.992798 0.119802i $$-0.961774\pi$$
−0.992798 + 0.119802i $$0.961774\pi$$
$$822$$ 0 0
$$823$$ 3468.10 0.146890 0.0734450 0.997299i $$-0.476601\pi$$
0.0734450 + 0.997299i $$0.476601\pi$$
$$824$$ −67286.8 −2.84472
$$825$$ 0 0
$$826$$ 54645.9 2.30191
$$827$$ 42454.9 1.78513 0.892564 0.450920i $$-0.148904\pi$$
0.892564 + 0.450920i $$0.148904\pi$$
$$828$$ 0 0
$$829$$ −3933.47 −0.164795 −0.0823975 0.996600i $$-0.526258\pi$$
−0.0823975 + 0.996600i $$0.526258\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 26051.9 1.08556
$$833$$ 17208.3 0.715764
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 57647.9 2.38492
$$837$$ 0 0
$$838$$ −82580.5 −3.40417
$$839$$ 32959.6 1.35625 0.678123 0.734948i $$-0.262794\pi$$
0.678123 + 0.734948i $$0.262794\pi$$
$$840$$ 0 0
$$841$$ 17906.1 0.734188
$$842$$ −11856.7 −0.485285
$$843$$ 0 0
$$844$$ −91947.6 −3.74996
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 11979.5 0.485974
$$848$$ −46846.5 −1.89707
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −52264.8 −2.10530
$$852$$ 0 0
$$853$$ −38845.8 −1.55927 −0.779634 0.626235i $$-0.784595\pi$$
−0.779634 + 0.626235i $$0.784595\pi$$
$$854$$ −75454.3 −3.02341
$$855$$ 0 0
$$856$$ 58785.0 2.34723
$$857$$ −29305.3 −1.16809 −0.584043 0.811723i $$-0.698530\pi$$
−0.584043 + 0.811723i $$0.698530\pi$$
$$858$$ 0 0
$$859$$ −909.659 −0.0361318 −0.0180659 0.999837i $$-0.505751\pi$$
−0.0180659 + 0.999837i $$0.505751\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10127.7 0.400173
$$863$$ 47998.0 1.89324 0.946622 0.322346i $$-0.104471\pi$$
0.946622 + 0.322346i $$0.104471\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −65716.6 −2.57868
$$867$$ 0 0
$$868$$ 8481.94 0.331677
$$869$$ 708.065 0.0276403
$$870$$ 0 0
$$871$$ −12808.7 −0.498286
$$872$$ −23190.2 −0.900594
$$873$$ 0 0
$$874$$ −84071.7 −3.25374
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −3258.01 −0.125445 −0.0627225 0.998031i $$-0.519978\pi$$
−0.0627225 + 0.998031i $$0.519978\pi$$
$$878$$ −82429.8 −3.16842
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 33380.2 1.27651 0.638256 0.769824i $$-0.279656\pi$$
0.638256 + 0.769824i $$0.279656\pi$$
$$882$$ 0 0
$$883$$ −33714.6 −1.28492 −0.642460 0.766319i $$-0.722086\pi$$
−0.642460 + 0.766319i $$0.722086\pi$$
$$884$$ 84909.2 3.23055
$$885$$ 0 0
$$886$$ −90780.6 −3.44225
$$887$$ −6218.80 −0.235408 −0.117704 0.993049i $$-0.537553\pi$$
−0.117704 + 0.993049i $$0.537553\pi$$
$$888$$ 0 0
$$889$$ −18889.8 −0.712649
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −52845.5 −1.98363
$$893$$ −522.537 −0.0195812
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8361.46 0.311760
$$897$$ 0 0
$$898$$ 83565.1 3.10535
$$899$$ −3753.19 −0.139239
$$900$$ 0 0
$$901$$ −21812.6 −0.806529
$$902$$ 29539.5 1.09042
$$903$$ 0 0
$$904$$ 62174.4 2.28749
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 22878.6 0.837566 0.418783 0.908086i $$-0.362457\pi$$
0.418783 + 0.908086i $$0.362457\pi$$
$$908$$ −97165.0 −3.55125
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 30144.8 1.09631 0.548157 0.836376i $$-0.315330\pi$$
0.548157 + 0.836376i $$0.315330\pi$$
$$912$$ 0 0
$$913$$ −15821.1 −0.573496
$$914$$ −61844.9 −2.23813
$$915$$ 0 0
$$916$$ −78051.2 −2.81538
$$917$$ −29624.1 −1.06682
$$918$$ 0 0
$$919$$ 3803.52 0.136525 0.0682625 0.997667i $$-0.478254\pi$$
0.0682625 + 0.997667i $$0.478254\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 14573.5 0.520555
$$923$$ −27854.0 −0.993312
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 67262.0 2.38700
$$927$$ 0 0
$$928$$ 62072.5 2.19572
$$929$$ −125.985 −0.00444934 −0.00222467 0.999998i $$-0.500708\pi$$
−0.00222467 + 0.999998i $$0.500708\pi$$
$$930$$ 0 0
$$931$$ 26359.3 0.927918
$$932$$ 6811.45 0.239395
$$933$$ 0 0
$$934$$ 29894.0 1.04728
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28107.9 −0.979984 −0.489992 0.871727i $$-0.663000\pi$$
−0.489992 + 0.871727i $$0.663000\pi$$
$$938$$ 24857.4 0.865268
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 49194.9 1.70426 0.852130 0.523330i $$-0.175311\pi$$
0.852130 + 0.523330i $$0.175311\pi$$
$$942$$ 0 0
$$943$$ −30337.3 −1.04763
$$944$$ −63046.3 −2.17371
$$945$$ 0 0
$$946$$ −50478.6 −1.73488
$$947$$ −14498.0 −0.497490 −0.248745 0.968569i $$-0.580018\pi$$
−0.248745 + 0.968569i $$0.580018\pi$$
$$948$$ 0 0
$$949$$ −60761.7 −2.07841
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −95569.8 −3.25361
$$953$$ 3201.79 0.108831 0.0544155 0.998518i $$-0.482670\pi$$
0.0544155 + 0.998518i $$0.482670\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6698.48 0.226616
$$957$$ 0 0
$$958$$ 58337.6 1.96743
$$959$$ −56126.7 −1.88991
$$960$$ 0 0
$$961$$ −29457.9 −0.988820
$$962$$ 114816. 3.84805
$$963$$ 0 0
$$964$$ −117435. −3.92356
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −57781.9 −1.92155 −0.960776 0.277326i $$-0.910552\pi$$
−0.960776 + 0.277326i $$0.910552\pi$$
$$968$$ −28205.1 −0.936513
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −2611.60 −0.0863133 −0.0431566 0.999068i $$-0.513741\pi$$
−0.0431566 + 0.999068i $$0.513741\pi$$
$$972$$ 0 0
$$973$$ 33086.4 1.09013
$$974$$ 46312.6 1.52356
$$975$$ 0 0
$$976$$ 87053.4 2.85503
$$977$$ −33598.3 −1.10021 −0.550104 0.835096i $$-0.685412\pi$$
−0.550104 + 0.835096i $$0.685412\pi$$
$$978$$ 0 0
$$979$$ 2438.51 0.0796067
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 34080.5 1.10749
$$983$$ −39484.8 −1.28115 −0.640575 0.767895i $$-0.721304\pi$$
−0.640575 + 0.767895i $$0.721304\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 72913.7 2.35501
$$987$$ 0 0
$$988$$ 130062. 4.18809
$$989$$ 51841.9 1.66681
$$990$$ 0 0
$$991$$ 39918.6 1.27957 0.639786 0.768553i $$-0.279023\pi$$
0.639786 + 0.768553i $$0.279023\pi$$
$$992$$ −5508.20 −0.176296
$$993$$ 0 0
$$994$$ 54055.2 1.72487
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 25670.3 0.815432 0.407716 0.913109i $$-0.366325\pi$$
0.407716 + 0.913109i $$0.366325\pi$$
$$998$$ −23975.6 −0.760454
$$999$$ 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 675.4.a.r.1.3 3
3.2 odd 2 675.4.a.q.1.1 3
5.2 odd 4 675.4.b.l.649.6 6
5.3 odd 4 675.4.b.l.649.1 6
5.4 even 2 135.4.a.f.1.1 3
15.2 even 4 675.4.b.k.649.1 6
15.8 even 4 675.4.b.k.649.6 6
15.14 odd 2 135.4.a.g.1.3 yes 3
20.19 odd 2 2160.4.a.bm.1.1 3
45.4 even 6 405.4.e.t.136.3 6
45.14 odd 6 405.4.e.r.136.1 6
45.29 odd 6 405.4.e.r.271.1 6
45.34 even 6 405.4.e.t.271.3 6
60.59 even 2 2160.4.a.be.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.1 3 5.4 even 2
135.4.a.g.1.3 yes 3 15.14 odd 2
405.4.e.r.136.1 6 45.14 odd 6
405.4.e.r.271.1 6 45.29 odd 6
405.4.e.t.136.3 6 45.4 even 6
405.4.e.t.271.3 6 45.34 even 6
675.4.a.q.1.1 3 3.2 odd 2
675.4.a.r.1.3 3 1.1 even 1 trivial
675.4.b.k.649.1 6 15.2 even 4
675.4.b.k.649.6 6 15.8 even 4
675.4.b.l.649.1 6 5.3 odd 4
675.4.b.l.649.6 6 5.2 odd 4
2160.4.a.be.1.1 3 60.59 even 2
2160.4.a.bm.1.1 3 20.19 odd 2