# Properties

 Label 1575.4.a.m.1.1 Level $1575$ Weight $4$ Character 1575.1 Self dual yes Analytic conductor $92.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$0$$ 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 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.56155 q^{2} +22.9309 q^{4} +7.00000 q^{7} -83.0388 q^{8} +O(q^{10})$$ $$q-5.56155 q^{2} +22.9309 q^{4} +7.00000 q^{7} -83.0388 q^{8} +33.6155 q^{11} +38.3542 q^{13} -38.9309 q^{14} +278.378 q^{16} +65.7235 q^{17} +33.3996 q^{19} -186.955 q^{22} +207.447 q^{23} -213.309 q^{26} +160.516 q^{28} +189.170 q^{29} +202.108 q^{31} -883.902 q^{32} -365.525 q^{34} +16.5227 q^{37} -185.754 q^{38} -388.617 q^{41} -41.8144 q^{43} +770.833 q^{44} -1153.73 q^{46} +368.648 q^{47} +49.0000 q^{49} +879.494 q^{52} +458.172 q^{53} -581.272 q^{56} -1052.08 q^{58} -256.216 q^{59} -123.511 q^{61} -1124.03 q^{62} +2688.85 q^{64} +336.277 q^{67} +1507.10 q^{68} +453.312 q^{71} -22.0436 q^{73} -91.8920 q^{74} +765.882 q^{76} +235.309 q^{77} +385.417 q^{79} +2161.32 q^{82} +23.7501 q^{83} +232.553 q^{86} -2791.39 q^{88} +1482.81 q^{89} +268.479 q^{91} +4756.94 q^{92} -2050.25 q^{94} -51.9867 q^{97} -272.516 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8}+O(q^{10})$$ 2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 $$2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32} - 294 q^{34} + 132 q^{37} - 388 q^{38} - 200 q^{41} - 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 98 q^{49} + 1190 q^{52} + 34 q^{53} - 441 q^{56} - 826 q^{58} - 364 q^{59} + 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} - 68 q^{76} + 182 q^{77} + 408 q^{79} + 1890 q^{82} + 1136 q^{83} + 696 q^{86} - 2944 q^{88} - 36 q^{89} - 98 q^{91} + 4896 q^{92} - 1940 q^{94} + 498 q^{97} - 343 q^{98}+O(q^{100})$$ 2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 + 26 * q^11 - 14 * q^13 - 49 * q^14 + 297 * q^16 + 16 * q^17 + 174 * q^19 - 176 * q^22 + 184 * q^23 - 138 * q^26 + 119 * q^28 + 32 * q^29 + 330 * q^31 - 1071 * q^32 - 294 * q^34 + 132 * q^37 - 388 * q^38 - 200 * q^41 - 364 * q^43 + 816 * q^44 - 1120 * q^46 + 292 * q^47 + 98 * q^49 + 1190 * q^52 + 34 * q^53 - 441 * q^56 - 826 * q^58 - 364 * q^59 + 792 * q^61 - 1308 * q^62 + 2809 * q^64 + 788 * q^67 + 1802 * q^68 - 454 * q^71 - 778 * q^73 - 258 * q^74 - 68 * q^76 + 182 * q^77 + 408 * q^79 + 1890 * q^82 + 1136 * q^83 + 696 * q^86 - 2944 * q^88 - 36 * q^89 - 98 * q^91 + 4896 * q^92 - 1940 * q^94 + 498 * q^97 - 343 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.56155 −1.96631 −0.983153 0.182785i $$-0.941489\pi$$
−0.983153 + 0.182785i $$0.941489\pi$$
$$3$$ 0 0
$$4$$ 22.9309 2.86636
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ −83.0388 −3.66983
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 33.6155 0.921406 0.460703 0.887554i $$-0.347597\pi$$
0.460703 + 0.887554i $$0.347597\pi$$
$$12$$ 0 0
$$13$$ 38.3542 0.818272 0.409136 0.912474i $$-0.365830\pi$$
0.409136 + 0.912474i $$0.365830\pi$$
$$14$$ −38.9309 −0.743194
$$15$$ 0 0
$$16$$ 278.378 4.34965
$$17$$ 65.7235 0.937664 0.468832 0.883287i $$-0.344675\pi$$
0.468832 + 0.883287i $$0.344675\pi$$
$$18$$ 0 0
$$19$$ 33.3996 0.403284 0.201642 0.979459i $$-0.435372\pi$$
0.201642 + 0.979459i $$0.435372\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −186.955 −1.81177
$$23$$ 207.447 1.88068 0.940341 0.340234i $$-0.110506\pi$$
0.940341 + 0.340234i $$0.110506\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −213.309 −1.60897
$$27$$ 0 0
$$28$$ 160.516 1.08338
$$29$$ 189.170 1.21131 0.605656 0.795726i $$-0.292911\pi$$
0.605656 + 0.795726i $$0.292911\pi$$
$$30$$ 0 0
$$31$$ 202.108 1.17096 0.585478 0.810688i $$-0.300907\pi$$
0.585478 + 0.810688i $$0.300907\pi$$
$$32$$ −883.902 −4.88292
$$33$$ 0 0
$$34$$ −365.525 −1.84373
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 16.5227 0.0734141 0.0367070 0.999326i $$-0.488313\pi$$
0.0367070 + 0.999326i $$0.488313\pi$$
$$38$$ −185.754 −0.792980
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −388.617 −1.48029 −0.740144 0.672448i $$-0.765243\pi$$
−0.740144 + 0.672448i $$0.765243\pi$$
$$42$$ 0 0
$$43$$ −41.8144 −0.148294 −0.0741469 0.997247i $$-0.523623\pi$$
−0.0741469 + 0.997247i $$0.523623\pi$$
$$44$$ 770.833 2.64108
$$45$$ 0 0
$$46$$ −1153.73 −3.69800
$$47$$ 368.648 1.14410 0.572051 0.820218i $$-0.306148\pi$$
0.572051 + 0.820218i $$0.306148\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 879.494 2.34546
$$53$$ 458.172 1.18745 0.593725 0.804668i $$-0.297657\pi$$
0.593725 + 0.804668i $$0.297657\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −581.272 −1.38707
$$57$$ 0 0
$$58$$ −1052.08 −2.38181
$$59$$ −256.216 −0.565364 −0.282682 0.959214i $$-0.591224\pi$$
−0.282682 + 0.959214i $$0.591224\pi$$
$$60$$ 0 0
$$61$$ −123.511 −0.259246 −0.129623 0.991563i $$-0.541377\pi$$
−0.129623 + 0.991563i $$0.541377\pi$$
$$62$$ −1124.03 −2.30246
$$63$$ 0 0
$$64$$ 2688.85 5.25166
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 336.277 0.613175 0.306587 0.951843i $$-0.400813\pi$$
0.306587 + 0.951843i $$0.400813\pi$$
$$68$$ 1507.10 2.68768
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 453.312 0.757722 0.378861 0.925454i $$-0.376316\pi$$
0.378861 + 0.925454i $$0.376316\pi$$
$$72$$ 0 0
$$73$$ −22.0436 −0.0353426 −0.0176713 0.999844i $$-0.505625\pi$$
−0.0176713 + 0.999844i $$0.505625\pi$$
$$74$$ −91.8920 −0.144355
$$75$$ 0 0
$$76$$ 765.882 1.15596
$$77$$ 235.309 0.348259
$$78$$ 0 0
$$79$$ 385.417 0.548896 0.274448 0.961602i $$-0.411505\pi$$
0.274448 + 0.961602i $$0.411505\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2161.32 2.91070
$$83$$ 23.7501 0.0314085 0.0157043 0.999877i $$-0.495001\pi$$
0.0157043 + 0.999877i $$0.495001\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 232.553 0.291591
$$87$$ 0 0
$$88$$ −2791.39 −3.38140
$$89$$ 1482.81 1.76604 0.883020 0.469335i $$-0.155506\pi$$
0.883020 + 0.469335i $$0.155506\pi$$
$$90$$ 0 0
$$91$$ 268.479 0.309278
$$92$$ 4756.94 5.39071
$$93$$ 0 0
$$94$$ −2050.25 −2.24965
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −51.9867 −0.0544170 −0.0272085 0.999630i $$-0.508662\pi$$
−0.0272085 + 0.999630i $$0.508662\pi$$
$$98$$ −272.516 −0.280901
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1429.30 −1.40812 −0.704062 0.710138i $$-0.748632\pi$$
−0.704062 + 0.710138i $$0.748632\pi$$
$$102$$ 0 0
$$103$$ −434.212 −0.415381 −0.207690 0.978195i $$-0.566595\pi$$
−0.207690 + 0.978195i $$0.566595\pi$$
$$104$$ −3184.88 −3.00292
$$105$$ 0 0
$$106$$ −2548.15 −2.33489
$$107$$ 666.307 0.602003 0.301001 0.953624i $$-0.402679\pi$$
0.301001 + 0.953624i $$0.402679\pi$$
$$108$$ 0 0
$$109$$ −1199.51 −1.05406 −0.527029 0.849847i $$-0.676694\pi$$
−0.527029 + 0.849847i $$0.676694\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1948.64 1.64401
$$113$$ −81.5171 −0.0678627 −0.0339314 0.999424i $$-0.510803\pi$$
−0.0339314 + 0.999424i $$0.510803\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4337.84 3.47206
$$117$$ 0 0
$$118$$ 1424.96 1.11168
$$119$$ 460.064 0.354404
$$120$$ 0 0
$$121$$ −200.996 −0.151011
$$122$$ 686.915 0.509757
$$123$$ 0 0
$$124$$ 4634.51 3.35638
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 336.985 0.235453 0.117727 0.993046i $$-0.462439\pi$$
0.117727 + 0.993046i $$0.462439\pi$$
$$128$$ −7882.95 −5.44344
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2931.15 −1.95493 −0.977465 0.211097i $$-0.932297\pi$$
−0.977465 + 0.211097i $$0.932297\pi$$
$$132$$ 0 0
$$133$$ 233.797 0.152427
$$134$$ −1870.22 −1.20569
$$135$$ 0 0
$$136$$ −5457.60 −3.44107
$$137$$ −1585.07 −0.988477 −0.494238 0.869326i $$-0.664553\pi$$
−0.494238 + 0.869326i $$0.664553\pi$$
$$138$$ 0 0
$$139$$ −1298.85 −0.792569 −0.396284 0.918128i $$-0.629701\pi$$
−0.396284 + 0.918128i $$0.629701\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2521.12 −1.48991
$$143$$ 1289.30 0.753960
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 122.597 0.0694943
$$147$$ 0 0
$$148$$ 378.881 0.210431
$$149$$ 2003.29 1.10145 0.550724 0.834687i $$-0.314352\pi$$
0.550724 + 0.834687i $$0.314352\pi$$
$$150$$ 0 0
$$151$$ 2740.96 1.47719 0.738596 0.674148i $$-0.235489\pi$$
0.738596 + 0.674148i $$0.235489\pi$$
$$152$$ −2773.47 −1.47999
$$153$$ 0 0
$$154$$ −1308.68 −0.684783
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3644.22 −1.85249 −0.926243 0.376928i $$-0.876981\pi$$
−0.926243 + 0.376928i $$0.876981\pi$$
$$158$$ −2143.52 −1.07930
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1452.13 0.710831
$$162$$ 0 0
$$163$$ −2774.27 −1.33311 −0.666557 0.745454i $$-0.732233\pi$$
−0.666557 + 0.745454i $$0.732233\pi$$
$$164$$ −8911.33 −4.24304
$$165$$ 0 0
$$166$$ −132.087 −0.0617588
$$167$$ 1154.91 0.535149 0.267574 0.963537i $$-0.413778\pi$$
0.267574 + 0.963537i $$0.413778\pi$$
$$168$$ 0 0
$$169$$ −725.958 −0.330432
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −958.841 −0.425064
$$173$$ −3387.46 −1.48869 −0.744346 0.667794i $$-0.767239\pi$$
−0.744346 + 0.667794i $$0.767239\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 9357.82 4.00780
$$177$$ 0 0
$$178$$ −8246.73 −3.47258
$$179$$ −1603.32 −0.669486 −0.334743 0.942309i $$-0.608650\pi$$
−0.334743 + 0.942309i $$0.608650\pi$$
$$180$$ 0 0
$$181$$ 544.220 0.223489 0.111745 0.993737i $$-0.464356\pi$$
0.111745 + 0.993737i $$0.464356\pi$$
$$182$$ −1493.16 −0.608134
$$183$$ 0 0
$$184$$ −17226.2 −6.90179
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2209.33 0.863969
$$188$$ 8453.41 3.27941
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2993.44 −1.13402 −0.567010 0.823711i $$-0.691900\pi$$
−0.567010 + 0.823711i $$0.691900\pi$$
$$192$$ 0 0
$$193$$ −1309.32 −0.488325 −0.244163 0.969734i $$-0.578513\pi$$
−0.244163 + 0.969734i $$0.578513\pi$$
$$194$$ 289.127 0.107001
$$195$$ 0 0
$$196$$ 1123.61 0.409480
$$197$$ −1141.38 −0.412790 −0.206395 0.978469i $$-0.566173\pi$$
−0.206395 + 0.978469i $$0.566173\pi$$
$$198$$ 0 0
$$199$$ 2370.23 0.844327 0.422164 0.906520i $$-0.361271\pi$$
0.422164 + 0.906520i $$0.361271\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 7949.12 2.76880
$$203$$ 1324.19 0.457833
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2414.89 0.816765
$$207$$ 0 0
$$208$$ 10676.9 3.55920
$$209$$ 1122.75 0.371588
$$210$$ 0 0
$$211$$ −687.159 −0.224199 −0.112099 0.993697i $$-0.535758\pi$$
−0.112099 + 0.993697i $$0.535758\pi$$
$$212$$ 10506.3 3.40366
$$213$$ 0 0
$$214$$ −3705.70 −1.18372
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1414.76 0.442580
$$218$$ 6671.15 2.07260
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2520.77 0.767264
$$222$$ 0 0
$$223$$ −990.496 −0.297437 −0.148719 0.988880i $$-0.547515\pi$$
−0.148719 + 0.988880i $$0.547515\pi$$
$$224$$ −6187.32 −1.84557
$$225$$ 0 0
$$226$$ 453.362 0.133439
$$227$$ 1479.25 0.432517 0.216258 0.976336i $$-0.430615\pi$$
0.216258 + 0.976336i $$0.430615\pi$$
$$228$$ 0 0
$$229$$ 6704.47 1.93469 0.967345 0.253463i $$-0.0815696\pi$$
0.967345 + 0.253463i $$0.0815696\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −15708.5 −4.44531
$$233$$ −1749.09 −0.491789 −0.245895 0.969297i $$-0.579082\pi$$
−0.245895 + 0.969297i $$0.579082\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −5875.25 −1.62054
$$237$$ 0 0
$$238$$ −2558.67 −0.696866
$$239$$ 6320.89 1.71073 0.855365 0.518027i $$-0.173333\pi$$
0.855365 + 0.518027i $$0.173333\pi$$
$$240$$ 0 0
$$241$$ 3359.62 0.897975 0.448988 0.893538i $$-0.351785\pi$$
0.448988 + 0.893538i $$0.351785\pi$$
$$242$$ 1117.85 0.296935
$$243$$ 0 0
$$244$$ −2832.22 −0.743092
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1281.01 0.329996
$$248$$ −16782.8 −4.29721
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1330.50 0.334582 0.167291 0.985908i $$-0.446498\pi$$
0.167291 + 0.985908i $$0.446498\pi$$
$$252$$ 0 0
$$253$$ 6973.44 1.73287
$$254$$ −1874.16 −0.462973
$$255$$ 0 0
$$256$$ 22330.6 5.45182
$$257$$ −2476.95 −0.601197 −0.300599 0.953751i $$-0.597186\pi$$
−0.300599 + 0.953751i $$0.597186\pi$$
$$258$$ 0 0
$$259$$ 115.659 0.0277479
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 16301.8 3.84399
$$263$$ −5152.56 −1.20806 −0.604032 0.796960i $$-0.706440\pi$$
−0.604032 + 0.796960i $$0.706440\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1300.28 −0.299718
$$267$$ 0 0
$$268$$ 7711.11 1.75758
$$269$$ 1150.97 0.260876 0.130438 0.991456i $$-0.458362\pi$$
0.130438 + 0.991456i $$0.458362\pi$$
$$270$$ 0 0
$$271$$ 1838.32 0.412067 0.206034 0.978545i $$-0.433944\pi$$
0.206034 + 0.978545i $$0.433944\pi$$
$$272$$ 18296.0 4.07851
$$273$$ 0 0
$$274$$ 8815.43 1.94365
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −568.447 −0.123302 −0.0616510 0.998098i $$-0.519637\pi$$
−0.0616510 + 0.998098i $$0.519637\pi$$
$$278$$ 7223.62 1.55843
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6015.00 −1.27696 −0.638479 0.769640i $$-0.720436\pi$$
−0.638479 + 0.769640i $$0.720436\pi$$
$$282$$ 0 0
$$283$$ −3985.75 −0.837202 −0.418601 0.908170i $$-0.637479\pi$$
−0.418601 + 0.908170i $$0.637479\pi$$
$$284$$ 10394.8 2.17190
$$285$$ 0 0
$$286$$ −7170.48 −1.48252
$$287$$ −2720.32 −0.559497
$$288$$ 0 0
$$289$$ −593.424 −0.120787
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −505.479 −0.101305
$$293$$ −2490.01 −0.496478 −0.248239 0.968699i $$-0.579852\pi$$
−0.248239 + 0.968699i $$0.579852\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1372.03 −0.269417
$$297$$ 0 0
$$298$$ −11141.4 −2.16578
$$299$$ 7956.45 1.53891
$$300$$ 0 0
$$301$$ −292.701 −0.0560498
$$302$$ −15244.0 −2.90461
$$303$$ 0 0
$$304$$ 9297.72 1.75415
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 141.853 0.0263712 0.0131856 0.999913i $$-0.495803\pi$$
0.0131856 + 0.999913i $$0.495803\pi$$
$$308$$ 5395.83 0.998234
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2091.92 −0.381420 −0.190710 0.981646i $$-0.561079\pi$$
−0.190710 + 0.981646i $$0.561079\pi$$
$$312$$ 0 0
$$313$$ −5521.44 −0.997094 −0.498547 0.866863i $$-0.666133\pi$$
−0.498547 + 0.866863i $$0.666133\pi$$
$$314$$ 20267.5 3.64255
$$315$$ 0 0
$$316$$ 8837.94 1.57333
$$317$$ 5351.63 0.948195 0.474097 0.880472i $$-0.342774\pi$$
0.474097 + 0.880472i $$0.342774\pi$$
$$318$$ 0 0
$$319$$ 6359.06 1.11611
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −8076.09 −1.39771
$$323$$ 2195.14 0.378145
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 15429.2 2.62131
$$327$$ 0 0
$$328$$ 32270.3 5.43241
$$329$$ 2580.53 0.432430
$$330$$ 0 0
$$331$$ −4383.52 −0.727916 −0.363958 0.931415i $$-0.618575\pi$$
−0.363958 + 0.931415i $$0.618575\pi$$
$$332$$ 544.609 0.0900281
$$333$$ 0 0
$$334$$ −6423.11 −1.05227
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7124.57 1.15163 0.575817 0.817579i $$-0.304684\pi$$
0.575817 + 0.817579i $$0.304684\pi$$
$$338$$ 4037.46 0.649730
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6793.97 1.07893
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 3472.22 0.544214
$$345$$ 0 0
$$346$$ 18839.5 2.92722
$$347$$ 507.743 0.0785506 0.0392753 0.999228i $$-0.487495\pi$$
0.0392753 + 0.999228i $$0.487495\pi$$
$$348$$ 0 0
$$349$$ 6155.14 0.944060 0.472030 0.881582i $$-0.343521\pi$$
0.472030 + 0.881582i $$0.343521\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −29712.8 −4.49915
$$353$$ −6429.56 −0.969437 −0.484718 0.874670i $$-0.661078\pi$$
−0.484718 + 0.874670i $$0.661078\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 34002.1 5.06211
$$357$$ 0 0
$$358$$ 8916.97 1.31641
$$359$$ −10075.4 −1.48123 −0.740614 0.671931i $$-0.765465\pi$$
−0.740614 + 0.671931i $$0.765465\pi$$
$$360$$ 0 0
$$361$$ −5743.46 −0.837362
$$362$$ −3026.71 −0.439448
$$363$$ 0 0
$$364$$ 6156.46 0.886501
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −816.898 −0.116190 −0.0580950 0.998311i $$-0.518503\pi$$
−0.0580950 + 0.998311i $$0.518503\pi$$
$$368$$ 57748.6 8.18031
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3207.21 0.448814
$$372$$ 0 0
$$373$$ 3737.85 0.518870 0.259435 0.965761i $$-0.416464\pi$$
0.259435 + 0.965761i $$0.416464\pi$$
$$374$$ −12287.3 −1.69883
$$375$$ 0 0
$$376$$ −30612.1 −4.19866
$$377$$ 7255.47 0.991183
$$378$$ 0 0
$$379$$ 1950.47 0.264351 0.132176 0.991226i $$-0.457804\pi$$
0.132176 + 0.991226i $$0.457804\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16648.2 2.22983
$$383$$ 6762.06 0.902155 0.451077 0.892485i $$-0.351040\pi$$
0.451077 + 0.892485i $$0.351040\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 7281.84 0.960197
$$387$$ 0 0
$$388$$ −1192.10 −0.155979
$$389$$ 2551.98 0.332624 0.166312 0.986073i $$-0.446814\pi$$
0.166312 + 0.986073i $$0.446814\pi$$
$$390$$ 0 0
$$391$$ 13634.1 1.76345
$$392$$ −4068.90 −0.524262
$$393$$ 0 0
$$394$$ 6347.83 0.811672
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4097.93 0.518058 0.259029 0.965869i $$-0.416597\pi$$
0.259029 + 0.965869i $$0.416597\pi$$
$$398$$ −13182.2 −1.66021
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1046.81 0.130362 0.0651811 0.997873i $$-0.479238\pi$$
0.0651811 + 0.997873i $$0.479238\pi$$
$$402$$ 0 0
$$403$$ 7751.68 0.958161
$$404$$ −32775.1 −4.03619
$$405$$ 0 0
$$406$$ −7364.57 −0.900240
$$407$$ 555.420 0.0676441
$$408$$ 0 0
$$409$$ 6516.92 0.787876 0.393938 0.919137i $$-0.371113\pi$$
0.393938 + 0.919137i $$0.371113\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −9956.86 −1.19063
$$413$$ −1793.51 −0.213687
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −33901.3 −3.99555
$$417$$ 0 0
$$418$$ −6244.21 −0.730656
$$419$$ −12279.1 −1.43168 −0.715838 0.698267i $$-0.753955\pi$$
−0.715838 + 0.698267i $$0.753955\pi$$
$$420$$ 0 0
$$421$$ 10146.9 1.17465 0.587325 0.809351i $$-0.300181\pi$$
0.587325 + 0.809351i $$0.300181\pi$$
$$422$$ 3821.67 0.440844
$$423$$ 0 0
$$424$$ −38046.1 −4.35774
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −864.579 −0.0979858
$$428$$ 15279.0 1.72556
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7059.04 −0.788914 −0.394457 0.918914i $$-0.629067\pi$$
−0.394457 + 0.918914i $$0.629067\pi$$
$$432$$ 0 0
$$433$$ −6468.98 −0.717966 −0.358983 0.933344i $$-0.616876\pi$$
−0.358983 + 0.933344i $$0.616876\pi$$
$$434$$ −7868.24 −0.870248
$$435$$ 0 0
$$436$$ −27505.8 −3.02131
$$437$$ 6928.65 0.758449
$$438$$ 0 0
$$439$$ 4767.13 0.518275 0.259137 0.965840i $$-0.416562\pi$$
0.259137 + 0.965840i $$0.416562\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −14019.4 −1.50868
$$443$$ 2366.55 0.253810 0.126905 0.991915i $$-0.459496\pi$$
0.126905 + 0.991915i $$0.459496\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 5508.70 0.584853
$$447$$ 0 0
$$448$$ 18821.9 1.98494
$$449$$ −1814.17 −0.190681 −0.0953406 0.995445i $$-0.530394\pi$$
−0.0953406 + 0.995445i $$0.530394\pi$$
$$450$$ 0 0
$$451$$ −13063.6 −1.36395
$$452$$ −1869.26 −0.194519
$$453$$ 0 0
$$454$$ −8226.93 −0.850460
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8284.13 −0.847955 −0.423977 0.905673i $$-0.639366\pi$$
−0.423977 + 0.905673i $$0.639366\pi$$
$$458$$ −37287.3 −3.80419
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1384.62 −0.139888 −0.0699439 0.997551i $$-0.522282\pi$$
−0.0699439 + 0.997551i $$0.522282\pi$$
$$462$$ 0 0
$$463$$ 13210.3 1.32599 0.662994 0.748624i $$-0.269285\pi$$
0.662994 + 0.748624i $$0.269285\pi$$
$$464$$ 52660.9 5.26879
$$465$$ 0 0
$$466$$ 9727.67 0.967008
$$467$$ −4574.24 −0.453256 −0.226628 0.973981i $$-0.572770\pi$$
−0.226628 + 0.973981i $$0.572770\pi$$
$$468$$ 0 0
$$469$$ 2353.94 0.231758
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 21275.9 2.07479
$$473$$ −1405.61 −0.136639
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 10549.7 1.01585
$$477$$ 0 0
$$478$$ −35154.0 −3.36382
$$479$$ 11031.8 1.05231 0.526154 0.850389i $$-0.323634\pi$$
0.526154 + 0.850389i $$0.323634\pi$$
$$480$$ 0 0
$$481$$ 633.716 0.0600726
$$482$$ −18684.7 −1.76569
$$483$$ 0 0
$$484$$ −4609.02 −0.432853
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5194.06 0.483296 0.241648 0.970364i $$-0.422312\pi$$
0.241648 + 0.970364i $$0.422312\pi$$
$$488$$ 10256.2 0.951389
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −11954.7 −1.09880 −0.549398 0.835561i $$-0.685143\pi$$
−0.549398 + 0.835561i $$0.685143\pi$$
$$492$$ 0 0
$$493$$ 12432.9 1.13580
$$494$$ −7124.43 −0.648873
$$495$$ 0 0
$$496$$ 56262.4 5.09326
$$497$$ 3173.19 0.286392
$$498$$ 0 0
$$499$$ 2566.05 0.230205 0.115102 0.993354i $$-0.463280\pi$$
0.115102 + 0.993354i $$0.463280\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −7399.62 −0.657891
$$503$$ 21103.5 1.87069 0.935347 0.353731i $$-0.115087\pi$$
0.935347 + 0.353731i $$0.115087\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −38783.1 −3.40735
$$507$$ 0 0
$$508$$ 7727.36 0.674894
$$509$$ 781.732 0.0680740 0.0340370 0.999421i $$-0.489164\pi$$
0.0340370 + 0.999421i $$0.489164\pi$$
$$510$$ 0 0
$$511$$ −154.305 −0.0133582
$$512$$ −61129.5 −5.27650
$$513$$ 0 0
$$514$$ 13775.7 1.18214
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 12392.3 1.05418
$$518$$ −643.244 −0.0545609
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14013.0 −1.17835 −0.589176 0.808005i $$-0.700547\pi$$
−0.589176 + 0.808005i $$0.700547\pi$$
$$522$$ 0 0
$$523$$ 10310.7 0.862052 0.431026 0.902339i $$-0.358152\pi$$
0.431026 + 0.902339i $$0.358152\pi$$
$$524$$ −67213.8 −5.60353
$$525$$ 0 0
$$526$$ 28656.3 2.37542
$$527$$ 13283.2 1.09796
$$528$$ 0 0
$$529$$ 30867.2 2.53696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 5361.18 0.436911
$$533$$ −14905.1 −1.21128
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −27924.0 −2.25025
$$537$$ 0 0
$$538$$ −6401.16 −0.512962
$$539$$ 1647.16 0.131629
$$540$$ 0 0
$$541$$ −17562.9 −1.39572 −0.697862 0.716232i $$-0.745865\pi$$
−0.697862 + 0.716232i $$0.745865\pi$$
$$542$$ −10223.9 −0.810250
$$543$$ 0 0
$$544$$ −58093.1 −4.57853
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 19889.6 1.55469 0.777347 0.629072i $$-0.216565\pi$$
0.777347 + 0.629072i $$0.216565\pi$$
$$548$$ −36346.9 −2.83333
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6318.22 0.488503
$$552$$ 0 0
$$553$$ 2697.92 0.207463
$$554$$ 3161.45 0.242450
$$555$$ 0 0
$$556$$ −29783.8 −2.27179
$$557$$ 5579.54 0.424439 0.212219 0.977222i $$-0.431931\pi$$
0.212219 + 0.977222i $$0.431931\pi$$
$$558$$ 0 0
$$559$$ −1603.76 −0.121345
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 33452.8 2.51089
$$563$$ −24463.2 −1.83126 −0.915630 0.402022i $$-0.868307\pi$$
−0.915630 + 0.402022i $$0.868307\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 22167.0 1.64620
$$567$$ 0 0
$$568$$ −37642.5 −2.78071
$$569$$ 8582.14 0.632306 0.316153 0.948708i $$-0.397609\pi$$
0.316153 + 0.948708i $$0.397609\pi$$
$$570$$ 0 0
$$571$$ 17580.8 1.28850 0.644248 0.764816i $$-0.277170\pi$$
0.644248 + 0.764816i $$0.277170\pi$$
$$572$$ 29564.7 2.16112
$$573$$ 0 0
$$574$$ 15129.2 1.10014
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8692.57 0.627169 0.313585 0.949560i $$-0.398470\pi$$
0.313585 + 0.949560i $$0.398470\pi$$
$$578$$ 3300.36 0.237503
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 166.250 0.0118713
$$582$$ 0 0
$$583$$ 15401.7 1.09412
$$584$$ 1830.47 0.129701
$$585$$ 0 0
$$586$$ 13848.3 0.976227
$$587$$ 3584.61 0.252049 0.126024 0.992027i $$-0.459778\pi$$
0.126024 + 0.992027i $$0.459778\pi$$
$$588$$ 0 0
$$589$$ 6750.33 0.472228
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 4599.56 0.319326
$$593$$ −21853.6 −1.51335 −0.756676 0.653790i $$-0.773178\pi$$
−0.756676 + 0.653790i $$0.773178\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 45937.1 3.15714
$$597$$ 0 0
$$598$$ −44250.2 −3.02596
$$599$$ −9090.48 −0.620078 −0.310039 0.950724i $$-0.600342\pi$$
−0.310039 + 0.950724i $$0.600342\pi$$
$$600$$ 0 0
$$601$$ −19546.1 −1.32663 −0.663314 0.748341i $$-0.730851\pi$$
−0.663314 + 0.748341i $$0.730851\pi$$
$$602$$ 1627.87 0.110211
$$603$$ 0 0
$$604$$ 62852.6 4.23416
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 15726.0 1.05157 0.525783 0.850619i $$-0.323773\pi$$
0.525783 + 0.850619i $$0.323773\pi$$
$$608$$ −29522.0 −1.96920
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 14139.2 0.936186
$$612$$ 0 0
$$613$$ 13572.5 0.894269 0.447135 0.894467i $$-0.352444\pi$$
0.447135 + 0.894467i $$0.352444\pi$$
$$614$$ −788.921 −0.0518538
$$615$$ 0 0
$$616$$ −19539.8 −1.27805
$$617$$ 17378.5 1.13393 0.566964 0.823743i $$-0.308118\pi$$
0.566964 + 0.823743i $$0.308118\pi$$
$$618$$ 0 0
$$619$$ −25113.3 −1.63068 −0.815338 0.578985i $$-0.803449\pi$$
−0.815338 + 0.578985i $$0.803449\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 11634.3 0.749989
$$623$$ 10379.7 0.667501
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 30707.8 1.96059
$$627$$ 0 0
$$628$$ −83565.1 −5.30989
$$629$$ 1085.93 0.0688377
$$630$$ 0 0
$$631$$ −10814.4 −0.682276 −0.341138 0.940013i $$-0.610812\pi$$
−0.341138 + 0.940013i $$0.610812\pi$$
$$632$$ −32004.5 −2.01436
$$633$$ 0 0
$$634$$ −29763.4 −1.86444
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1879.35 0.116896
$$638$$ −35366.3 −2.19461
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 16359.0 1.00802 0.504010 0.863698i $$-0.331857\pi$$
0.504010 + 0.863698i $$0.331857\pi$$
$$642$$ 0 0
$$643$$ −8819.47 −0.540911 −0.270456 0.962732i $$-0.587174\pi$$
−0.270456 + 0.962732i $$0.587174\pi$$
$$644$$ 33298.6 2.03750
$$645$$ 0 0
$$646$$ −12208.4 −0.743549
$$647$$ −13828.8 −0.840290 −0.420145 0.907457i $$-0.638021\pi$$
−0.420145 + 0.907457i $$0.638021\pi$$
$$648$$ 0 0
$$649$$ −8612.83 −0.520930
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −63616.4 −3.82118
$$653$$ 23988.7 1.43760 0.718798 0.695219i $$-0.244693\pi$$
0.718798 + 0.695219i $$0.244693\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −108182. −6.43874
$$657$$ 0 0
$$658$$ −14351.8 −0.850289
$$659$$ −3109.28 −0.183794 −0.0918972 0.995769i $$-0.529293\pi$$
−0.0918972 + 0.995769i $$0.529293\pi$$
$$660$$ 0 0
$$661$$ 22695.0 1.33545 0.667726 0.744407i $$-0.267268\pi$$
0.667726 + 0.744407i $$0.267268\pi$$
$$662$$ 24379.2 1.43131
$$663$$ 0 0
$$664$$ −1972.18 −0.115264
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 39242.8 2.27809
$$668$$ 26483.2 1.53393
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4151.90 −0.238871
$$672$$ 0 0
$$673$$ −22073.4 −1.26429 −0.632145 0.774850i $$-0.717825\pi$$
−0.632145 + 0.774850i $$0.717825\pi$$
$$674$$ −39623.7 −2.26446
$$675$$ 0 0
$$676$$ −16646.9 −0.947136
$$677$$ −2489.50 −0.141328 −0.0706642 0.997500i $$-0.522512\pi$$
−0.0706642 + 0.997500i $$0.522512\pi$$
$$678$$ 0 0
$$679$$ −363.907 −0.0205677
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −37785.0 −2.12150
$$683$$ 7970.98 0.446561 0.223280 0.974754i $$-0.428323\pi$$
0.223280 + 0.974754i $$0.428323\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1907.61 −0.106171
$$687$$ 0 0
$$688$$ −11640.2 −0.645027
$$689$$ 17572.8 0.971656
$$690$$ 0 0
$$691$$ −23892.7 −1.31537 −0.657687 0.753292i $$-0.728465\pi$$
−0.657687 + 0.753292i $$0.728465\pi$$
$$692$$ −77677.4 −4.26712
$$693$$ 0 0
$$694$$ −2823.84 −0.154454
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −25541.3 −1.38801
$$698$$ −34232.1 −1.85631
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −12197.0 −0.657170 −0.328585 0.944474i $$-0.606572\pi$$
−0.328585 + 0.944474i $$0.606572\pi$$
$$702$$ 0 0
$$703$$ 551.853 0.0296067
$$704$$ 90387.0 4.83891
$$705$$ 0 0
$$706$$ 35758.4 1.90621
$$707$$ −10005.1 −0.532221
$$708$$ 0 0
$$709$$ −8982.28 −0.475792 −0.237896 0.971291i $$-0.576458\pi$$
−0.237896 + 0.971291i $$0.576458\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −123131. −6.48107
$$713$$ 41926.7 2.20220
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −36765.6 −1.91899
$$717$$ 0 0
$$718$$ 56035.0 2.91255
$$719$$ 6501.61 0.337231 0.168616 0.985682i $$-0.446070\pi$$
0.168616 + 0.985682i $$0.446070\pi$$
$$720$$ 0 0
$$721$$ −3039.49 −0.156999
$$722$$ 31942.6 1.64651
$$723$$ 0 0
$$724$$ 12479.4 0.640600
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 24228.7 1.23603 0.618013 0.786168i $$-0.287938\pi$$
0.618013 + 0.786168i $$0.287938\pi$$
$$728$$ −22294.2 −1.13500
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2748.19 −0.139050
$$732$$ 0 0
$$733$$ −36719.1 −1.85027 −0.925136 0.379636i $$-0.876049\pi$$
−0.925136 + 0.379636i $$0.876049\pi$$
$$734$$ 4543.22 0.228465
$$735$$ 0 0
$$736$$ −183363. −9.18321
$$737$$ 11304.1 0.564983
$$738$$ 0 0
$$739$$ 23304.5 1.16004 0.580021 0.814602i $$-0.303044\pi$$
0.580021 + 0.814602i $$0.303044\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −17837.0 −0.882505
$$743$$ 6875.35 0.339478 0.169739 0.985489i $$-0.445708\pi$$
0.169739 + 0.985489i $$0.445708\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −20788.3 −1.02026
$$747$$ 0 0
$$748$$ 50661.8 2.47644
$$749$$ 4664.15 0.227536
$$750$$ 0 0
$$751$$ 1182.65 0.0574640 0.0287320 0.999587i $$-0.490853\pi$$
0.0287320 + 0.999587i $$0.490853\pi$$
$$752$$ 102623. 4.97645
$$753$$ 0 0
$$754$$ −40351.7 −1.94897
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −25226.8 −1.21121 −0.605604 0.795766i $$-0.707068\pi$$
−0.605604 + 0.795766i $$0.707068\pi$$
$$758$$ −10847.7 −0.519795
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10909.2 0.519655 0.259827 0.965655i $$-0.416334\pi$$
0.259827 + 0.965655i $$0.416334\pi$$
$$762$$ 0 0
$$763$$ −8396.58 −0.398397
$$764$$ −68642.2 −3.25051
$$765$$ 0 0
$$766$$ −37607.6 −1.77391
$$767$$ −9826.95 −0.462621
$$768$$ 0 0
$$769$$ −12771.6 −0.598903 −0.299451 0.954112i $$-0.596804\pi$$
−0.299451 + 0.954112i $$0.596804\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −30023.8 −1.39972
$$773$$ 2199.06 0.102322 0.0511610 0.998690i $$-0.483708\pi$$
0.0511610 + 0.998690i $$0.483708\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 4316.92 0.199701
$$777$$ 0 0
$$778$$ −14193.0 −0.654041
$$779$$ −12979.7 −0.596977
$$780$$ 0 0
$$781$$ 15238.3 0.698170
$$782$$ −75827.0 −3.46748
$$783$$ 0 0
$$784$$ 13640.5 0.621379
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 19587.7 0.887201 0.443601 0.896225i $$-0.353701\pi$$
0.443601 + 0.896225i $$0.353701\pi$$
$$788$$ −26172.8 −1.18321
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −570.620 −0.0256497
$$792$$ 0 0
$$793$$ −4737.17 −0.212134
$$794$$ −22790.9 −1.01866
$$795$$ 0 0
$$796$$ 54351.4 2.42014
$$797$$ 21699.3 0.964401 0.482200 0.876061i $$-0.339838\pi$$
0.482200 + 0.876061i $$0.339838\pi$$
$$798$$ 0 0
$$799$$ 24228.8 1.07278
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −5821.89 −0.256332
$$803$$ −741.007 −0.0325649
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −43111.4 −1.88404
$$807$$ 0 0
$$808$$ 118687. 5.16758
$$809$$ 15649.4 0.680103 0.340051 0.940407i $$-0.389556\pi$$
0.340051 + 0.940407i $$0.389556\pi$$
$$810$$ 0 0
$$811$$ −33267.3 −1.44041 −0.720206 0.693760i $$-0.755953\pi$$
−0.720206 + 0.693760i $$0.755953\pi$$
$$812$$ 30364.9 1.31231
$$813$$ 0 0
$$814$$ −3089.00 −0.133009
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1396.59 −0.0598046
$$818$$ −36244.2 −1.54920
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −5158.58 −0.219288 −0.109644 0.993971i $$-0.534971\pi$$
−0.109644 + 0.993971i $$0.534971\pi$$
$$822$$ 0 0
$$823$$ −26333.6 −1.11535 −0.557674 0.830060i $$-0.688306\pi$$
−0.557674 + 0.830060i $$0.688306\pi$$
$$824$$ 36056.5 1.52438
$$825$$ 0 0
$$826$$ 9974.71 0.420175
$$827$$ 19572.7 0.822988 0.411494 0.911413i $$-0.365007\pi$$
0.411494 + 0.911413i $$0.365007\pi$$
$$828$$ 0 0
$$829$$ 9642.26 0.403968 0.201984 0.979389i $$-0.435261\pi$$
0.201984 + 0.979389i $$0.435261\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 103128. 4.29728
$$833$$ 3220.45 0.133952
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 25745.5 1.06511
$$837$$ 0 0
$$838$$ 68290.7 2.81511
$$839$$ 31081.1 1.27895 0.639475 0.768812i $$-0.279152\pi$$
0.639475 + 0.768812i $$0.279152\pi$$
$$840$$ 0 0
$$841$$ 11396.5 0.467278
$$842$$ −56432.3 −2.30972
$$843$$ 0 0
$$844$$ −15757.1 −0.642634
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1406.97 −0.0570770
$$848$$ 127545. 5.16499
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 3427.59 0.138068
$$852$$ 0 0
$$853$$ −25780.9 −1.03484 −0.517421 0.855731i $$-0.673108\pi$$
−0.517421 + 0.855731i $$0.673108\pi$$
$$854$$ 4808.40 0.192670
$$855$$ 0 0
$$856$$ −55329.3 −2.20925
$$857$$ 14452.6 0.576069 0.288035 0.957620i $$-0.406998\pi$$
0.288035 + 0.957620i $$0.406998\pi$$
$$858$$ 0 0
$$859$$ 889.366 0.0353257 0.0176628 0.999844i $$-0.494377\pi$$
0.0176628 + 0.999844i $$0.494377\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 39259.2 1.55125
$$863$$ 41460.3 1.63537 0.817685 0.575665i $$-0.195257\pi$$
0.817685 + 0.575665i $$0.195257\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 35977.6 1.41174
$$867$$ 0 0
$$868$$ 32441.6 1.26859
$$869$$ 12956.0 0.505756
$$870$$ 0 0
$$871$$ 12897.6 0.501744
$$872$$ 99606.0 3.86822
$$873$$ 0 0
$$874$$ −38534.1 −1.49134
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −21173.0 −0.815236 −0.407618 0.913153i $$-0.633640\pi$$
−0.407618 + 0.913153i $$0.633640\pi$$
$$878$$ −26512.6 −1.01909
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −9883.39 −0.377957 −0.188978 0.981981i $$-0.560518\pi$$
−0.188978 + 0.981981i $$0.560518\pi$$
$$882$$ 0 0
$$883$$ 45273.9 1.72547 0.862734 0.505658i $$-0.168750\pi$$
0.862734 + 0.505658i $$0.168750\pi$$
$$884$$ 57803.4 2.19925
$$885$$ 0 0
$$886$$ −13161.7 −0.499069
$$887$$ 644.388 0.0243928 0.0121964 0.999926i $$-0.496118\pi$$
0.0121964 + 0.999926i $$0.496118\pi$$
$$888$$ 0 0
$$889$$ 2358.89 0.0889930
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −22712.9 −0.852562
$$893$$ 12312.7 0.461398
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −55180.6 −2.05743
$$897$$ 0 0
$$898$$ 10089.6 0.374937
$$899$$ 38232.8 1.41839
$$900$$ 0 0
$$901$$ 30112.7 1.11343
$$902$$ 72653.8 2.68194
$$903$$ 0 0
$$904$$ 6769.09 0.249045
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 15065.2 0.551522 0.275761 0.961226i $$-0.411070\pi$$
0.275761 + 0.961226i $$0.411070\pi$$
$$908$$ 33920.5 1.23975
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −28789.9 −1.04704 −0.523520 0.852014i $$-0.675381\pi$$
−0.523520 + 0.852014i $$0.675381\pi$$
$$912$$ 0 0
$$913$$ 798.371 0.0289400
$$914$$ 46072.6 1.66734
$$915$$ 0 0
$$916$$ 153739. 5.54552
$$917$$ −20518.1 −0.738894
$$918$$ 0 0
$$919$$ −24163.8 −0.867345 −0.433673 0.901070i $$-0.642783\pi$$
−0.433673 + 0.901070i $$0.642783\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 7700.65 0.275062
$$923$$ 17386.4 0.620023
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −73469.5 −2.60730
$$927$$ 0 0
$$928$$ −167208. −5.91474
$$929$$ 35115.4 1.24015 0.620075 0.784542i $$-0.287102\pi$$
0.620075 + 0.784542i $$0.287102\pi$$
$$930$$ 0 0
$$931$$ 1636.58 0.0576120
$$932$$ −40108.2 −1.40964
$$933$$ 0 0
$$934$$ 25439.9 0.891239
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 15512.6 0.540849 0.270424 0.962741i $$-0.412836\pi$$
0.270424 + 0.962741i $$0.412836\pi$$
$$938$$ −13091.5 −0.455708
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 53283.8 1.84591 0.922956 0.384905i $$-0.125766\pi$$
0.922956 + 0.384905i $$0.125766\pi$$
$$942$$ 0 0
$$943$$ −80617.5 −2.78395
$$944$$ −71324.8 −2.45914
$$945$$ 0 0
$$946$$ 7817.39 0.268674
$$947$$ −55509.7 −1.90478 −0.952388 0.304890i $$-0.901380\pi$$
−0.952388 + 0.304890i $$0.901380\pi$$
$$948$$ 0 0
$$949$$ −845.464 −0.0289198
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −38203.2 −1.30060
$$953$$ −28080.6 −0.954480 −0.477240 0.878773i $$-0.658363\pi$$
−0.477240 + 0.878773i $$0.658363\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 144943. 4.90356
$$957$$ 0 0
$$958$$ −61353.9 −2.06916
$$959$$ −11095.5 −0.373609
$$960$$ 0 0
$$961$$ 11056.6 0.371140
$$962$$ −3524.44 −0.118121
$$963$$ 0 0
$$964$$ 77039.0 2.57392
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −56609.3 −1.88256 −0.941278 0.337634i $$-0.890374\pi$$
−0.941278 + 0.337634i $$0.890374\pi$$
$$968$$ 16690.5 0.554187
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6782.17 0.224151 0.112075 0.993700i $$-0.464250\pi$$
0.112075 + 0.993700i $$0.464250\pi$$
$$972$$ 0 0
$$973$$ −9091.95 −0.299563
$$974$$ −28887.0 −0.950309
$$975$$ 0 0
$$976$$ −34382.8 −1.12763
$$977$$ −45655.0 −1.49502 −0.747509 0.664252i $$-0.768750\pi$$
−0.747509 + 0.664252i $$0.768750\pi$$
$$978$$ 0 0
$$979$$ 49845.5 1.62724
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 66486.8 2.16057
$$983$$ −10102.3 −0.327785 −0.163893 0.986478i $$-0.552405\pi$$
−0.163893 + 0.986478i $$0.552405\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −69146.4 −2.23334
$$987$$ 0 0
$$988$$ 29374.8 0.945887
$$989$$ −8674.27 −0.278894
$$990$$ 0 0
$$991$$ 25416.2 0.814705 0.407353 0.913271i $$-0.366452\pi$$
0.407353 + 0.913271i $$0.366452\pi$$
$$992$$ −178644. −5.71768
$$993$$ 0 0
$$994$$ −17647.8 −0.563135
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −48152.5 −1.52959 −0.764797 0.644271i $$-0.777161\pi$$
−0.764797 + 0.644271i $$0.777161\pi$$
$$998$$ −14271.2 −0.452653
$$999$$ 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 1575.4.a.m.1.1 2
3.2 odd 2 525.4.a.p.1.2 2
5.4 even 2 315.4.a.m.1.2 2
15.2 even 4 525.4.d.i.274.4 4
15.8 even 4 525.4.d.i.274.1 4
15.14 odd 2 105.4.a.c.1.1 2
35.34 odd 2 2205.4.a.bh.1.2 2
60.59 even 2 1680.4.a.bk.1.2 2
105.104 even 2 735.4.a.k.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.1 2 15.14 odd 2
315.4.a.m.1.2 2 5.4 even 2
525.4.a.p.1.2 2 3.2 odd 2
525.4.d.i.274.1 4 15.8 even 4
525.4.d.i.274.4 4 15.2 even 4
735.4.a.k.1.1 2 105.104 even 2
1575.4.a.m.1.1 2 1.1 even 1 trivial
1680.4.a.bk.1.2 2 60.59 even 2
2205.4.a.bh.1.2 2 35.34 odd 2