# Properties

 Label 1575.4.a.r.1.1 Level $1575$ Weight $4$ Character 1575.1 Self dual yes Analytic conductor $92.928$ Analytic rank $1$ 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: $$1$$ 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 315) 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

 $$f(q)$$ $$=$$ $$q-2.56155 q^{2} -1.43845 q^{4} +7.00000 q^{7} +24.1771 q^{8} +O(q^{10})$$ $$q-2.56155 q^{2} -1.43845 q^{4} +7.00000 q^{7} +24.1771 q^{8} +6.24621 q^{11} +56.3542 q^{13} -17.9309 q^{14} -50.4233 q^{16} -24.6004 q^{17} -90.7083 q^{19} -16.0000 q^{22} +69.8617 q^{23} -144.354 q^{26} -10.0691 q^{28} -228.847 q^{29} -67.8920 q^{31} -64.2547 q^{32} +63.0152 q^{34} +58.8466 q^{37} +232.354 q^{38} +19.2007 q^{41} -365.218 q^{43} -8.98485 q^{44} -178.955 q^{46} +195.153 q^{47} +49.0000 q^{49} -81.0625 q^{52} +511.201 q^{53} +169.240 q^{56} +586.203 q^{58} -284.000 q^{59} -123.460 q^{61} +173.909 q^{62} +567.978 q^{64} -144.968 q^{67} +35.3863 q^{68} -73.0284 q^{71} -638.850 q^{73} -150.739 q^{74} +130.479 q^{76} +43.7235 q^{77} +976.189 q^{79} -49.1837 q^{82} +484.466 q^{83} +935.525 q^{86} +151.015 q^{88} +1017.30 q^{89} +394.479 q^{91} -100.492 q^{92} -499.896 q^{94} -1806.67 q^{97} -125.516 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 7 q^{4} + 14 q^{7} + 3 q^{8}+O(q^{10})$$ 2 * q - q^2 - 7 * q^4 + 14 * q^7 + 3 * q^8 $$2 q - q^{2} - 7 q^{4} + 14 q^{7} + 3 q^{8} - 4 q^{11} + 22 q^{13} - 7 q^{14} - 39 q^{16} + 58 q^{17} - 32 q^{22} + 82 q^{23} - 198 q^{26} - 49 q^{28} - 334 q^{29} - 210 q^{31} + 123 q^{32} + 192 q^{34} - 6 q^{37} + 374 q^{38} - 176 q^{41} - 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 98 q^{49} + 110 q^{52} + 808 q^{53} + 21 q^{56} + 422 q^{58} - 568 q^{59} - 618 q^{61} - 48 q^{62} + 769 q^{64} - 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} - 374 q^{76} - 28 q^{77} + 600 q^{79} - 354 q^{82} - 268 q^{83} + 1434 q^{86} + 368 q^{88} + 72 q^{89} + 154 q^{91} - 168 q^{92} - 2 q^{94} - 1626 q^{97} - 49 q^{98}+O(q^{100})$$ 2 * q - q^2 - 7 * q^4 + 14 * q^7 + 3 * q^8 - 4 * q^11 + 22 * q^13 - 7 * q^14 - 39 * q^16 + 58 * q^17 - 32 * q^22 + 82 * q^23 - 198 * q^26 - 49 * q^28 - 334 * q^29 - 210 * q^31 + 123 * q^32 + 192 * q^34 - 6 * q^37 + 374 * q^38 - 176 * q^41 - 46 * q^43 + 48 * q^44 - 160 * q^46 + 514 * q^47 + 98 * q^49 + 110 * q^52 + 808 * q^53 + 21 * q^56 + 422 * q^58 - 568 * q^59 - 618 * q^61 - 48 * q^62 + 769 * q^64 - 694 * q^67 - 424 * q^68 - 814 * q^71 - 82 * q^73 - 252 * q^74 - 374 * q^76 - 28 * q^77 + 600 * q^79 - 354 * q^82 - 268 * q^83 + 1434 * q^86 + 368 * q^88 + 72 * q^89 + 154 * q^91 - 168 * q^92 - 2 * q^94 - 1626 * q^97 - 49 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.56155 −0.905646 −0.452823 0.891601i $$-0.649583\pi$$
−0.452823 + 0.891601i $$0.649583\pi$$
$$3$$ 0 0
$$4$$ −1.43845 −0.179806
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 24.1771 1.06849
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.24621 0.171209 0.0856047 0.996329i $$-0.472718\pi$$
0.0856047 + 0.996329i $$0.472718\pi$$
$$12$$ 0 0
$$13$$ 56.3542 1.20229 0.601147 0.799138i $$-0.294710\pi$$
0.601147 + 0.799138i $$0.294710\pi$$
$$14$$ −17.9309 −0.342302
$$15$$ 0 0
$$16$$ −50.4233 −0.787864
$$17$$ −24.6004 −0.350969 −0.175484 0.984482i $$-0.556149\pi$$
−0.175484 + 0.984482i $$0.556149\pi$$
$$18$$ 0 0
$$19$$ −90.7083 −1.09526 −0.547629 0.836721i $$-0.684470\pi$$
−0.547629 + 0.836721i $$0.684470\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −16.0000 −0.155055
$$23$$ 69.8617 0.633356 0.316678 0.948533i $$-0.397433\pi$$
0.316678 + 0.948533i $$0.397433\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −144.354 −1.08885
$$27$$ 0 0
$$28$$ −10.0691 −0.0679602
$$29$$ −228.847 −1.46537 −0.732685 0.680568i $$-0.761733\pi$$
−0.732685 + 0.680568i $$0.761733\pi$$
$$30$$ 0 0
$$31$$ −67.8920 −0.393347 −0.196674 0.980469i $$-0.563014\pi$$
−0.196674 + 0.980469i $$0.563014\pi$$
$$32$$ −64.2547 −0.354961
$$33$$ 0 0
$$34$$ 63.0152 0.317853
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 58.8466 0.261468 0.130734 0.991417i $$-0.458267\pi$$
0.130734 + 0.991417i $$0.458267\pi$$
$$38$$ 232.354 0.991916
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 19.2007 0.0731379 0.0365689 0.999331i $$-0.488357\pi$$
0.0365689 + 0.999331i $$0.488357\pi$$
$$42$$ 0 0
$$43$$ −365.218 −1.29524 −0.647618 0.761965i $$-0.724235\pi$$
−0.647618 + 0.761965i $$0.724235\pi$$
$$44$$ −8.98485 −0.0307845
$$45$$ 0 0
$$46$$ −178.955 −0.573596
$$47$$ 195.153 0.605661 0.302830 0.953044i $$-0.402068\pi$$
0.302830 + 0.953044i $$0.402068\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −81.0625 −0.216180
$$53$$ 511.201 1.32488 0.662442 0.749113i $$-0.269520\pi$$
0.662442 + 0.749113i $$0.269520\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 169.240 0.403850
$$57$$ 0 0
$$58$$ 586.203 1.32711
$$59$$ −284.000 −0.626672 −0.313336 0.949642i $$-0.601447\pi$$
−0.313336 + 0.949642i $$0.601447\pi$$
$$60$$ 0 0
$$61$$ −123.460 −0.259139 −0.129569 0.991570i $$-0.541359\pi$$
−0.129569 + 0.991570i $$0.541359\pi$$
$$62$$ 173.909 0.356233
$$63$$ 0 0
$$64$$ 567.978 1.10933
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −144.968 −0.264338 −0.132169 0.991227i $$-0.542194\pi$$
−0.132169 + 0.991227i $$0.542194\pi$$
$$68$$ 35.3863 0.0631062
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −73.0284 −0.122069 −0.0610344 0.998136i $$-0.519440\pi$$
−0.0610344 + 0.998136i $$0.519440\pi$$
$$72$$ 0 0
$$73$$ −638.850 −1.02427 −0.512135 0.858905i $$-0.671145\pi$$
−0.512135 + 0.858905i $$0.671145\pi$$
$$74$$ −150.739 −0.236797
$$75$$ 0 0
$$76$$ 130.479 0.196934
$$77$$ 43.7235 0.0647111
$$78$$ 0 0
$$79$$ 976.189 1.39025 0.695126 0.718888i $$-0.255349\pi$$
0.695126 + 0.718888i $$0.255349\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −49.1837 −0.0662370
$$83$$ 484.466 0.640687 0.320344 0.947301i $$-0.396202\pi$$
0.320344 + 0.947301i $$0.396202\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 935.525 1.17303
$$87$$ 0 0
$$88$$ 151.015 0.182935
$$89$$ 1017.30 1.21161 0.605806 0.795612i $$-0.292851\pi$$
0.605806 + 0.795612i $$0.292851\pi$$
$$90$$ 0 0
$$91$$ 394.479 0.454425
$$92$$ −100.492 −0.113881
$$93$$ 0 0
$$94$$ −499.896 −0.548514
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1806.67 −1.89113 −0.945564 0.325437i $$-0.894489\pi$$
−0.945564 + 0.325437i $$0.894489\pi$$
$$98$$ −125.516 −0.129378
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 483.053 0.475897 0.237948 0.971278i $$-0.423525\pi$$
0.237948 + 0.971278i $$0.423525\pi$$
$$102$$ 0 0
$$103$$ 339.049 0.324345 0.162172 0.986762i $$-0.448150\pi$$
0.162172 + 0.986762i $$0.448150\pi$$
$$104$$ 1362.48 1.28464
$$105$$ 0 0
$$106$$ −1309.47 −1.19987
$$107$$ −450.847 −0.407336 −0.203668 0.979040i $$-0.565286\pi$$
−0.203668 + 0.979040i $$0.565286\pi$$
$$108$$ 0 0
$$109$$ −1841.70 −1.61838 −0.809189 0.587548i $$-0.800093\pi$$
−0.809189 + 0.587548i $$0.800093\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −352.963 −0.297785
$$113$$ 1874.72 1.56069 0.780347 0.625347i $$-0.215042\pi$$
0.780347 + 0.625347i $$0.215042\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 329.184 0.263482
$$117$$ 0 0
$$118$$ 727.481 0.567543
$$119$$ −172.203 −0.132654
$$120$$ 0 0
$$121$$ −1291.98 −0.970687
$$122$$ 316.250 0.234688
$$123$$ 0 0
$$124$$ 97.6591 0.0707262
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 38.0984 0.0266196 0.0133098 0.999911i $$-0.495763\pi$$
0.0133098 + 0.999911i $$0.495763\pi$$
$$128$$ −940.868 −0.649702
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1551.51 1.03478 0.517389 0.855750i $$-0.326904\pi$$
0.517389 + 0.855750i $$0.326904\pi$$
$$132$$ 0 0
$$133$$ −634.958 −0.413969
$$134$$ 371.343 0.239396
$$135$$ 0 0
$$136$$ −594.765 −0.375005
$$137$$ −1203.24 −0.750364 −0.375182 0.926951i $$-0.622420\pi$$
−0.375182 + 0.926951i $$0.622420\pi$$
$$138$$ 0 0
$$139$$ 1897.00 1.15756 0.578781 0.815483i $$-0.303529\pi$$
0.578781 + 0.815483i $$0.303529\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 187.066 0.110551
$$143$$ 352.000 0.205844
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 1636.45 0.927626
$$147$$ 0 0
$$148$$ −84.6477 −0.0470135
$$149$$ −704.888 −0.387562 −0.193781 0.981045i $$-0.562075\pi$$
−0.193781 + 0.981045i $$0.562075\pi$$
$$150$$ 0 0
$$151$$ −3035.21 −1.63578 −0.817888 0.575378i $$-0.804855\pi$$
−0.817888 + 0.575378i $$0.804855\pi$$
$$152$$ −2193.06 −1.17027
$$153$$ 0 0
$$154$$ −112.000 −0.0586053
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2713.65 −1.37944 −0.689722 0.724074i $$-0.742267\pi$$
−0.689722 + 0.724074i $$0.742267\pi$$
$$158$$ −2500.56 −1.25908
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 489.032 0.239386
$$162$$ 0 0
$$163$$ 465.259 0.223570 0.111785 0.993732i $$-0.464343\pi$$
0.111785 + 0.993732i $$0.464343\pi$$
$$164$$ −27.6193 −0.0131506
$$165$$ 0 0
$$166$$ −1240.98 −0.580236
$$167$$ −4156.06 −1.92578 −0.962891 0.269892i $$-0.913012\pi$$
−0.962891 + 0.269892i $$0.913012\pi$$
$$168$$ 0 0
$$169$$ 978.792 0.445513
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 525.346 0.232891
$$173$$ 4241.17 1.86387 0.931936 0.362622i $$-0.118119\pi$$
0.931936 + 0.362622i $$0.118119\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −314.955 −0.134890
$$177$$ 0 0
$$178$$ −2605.87 −1.09729
$$179$$ −2940.35 −1.22778 −0.613889 0.789392i $$-0.710396\pi$$
−0.613889 + 0.789392i $$0.710396\pi$$
$$180$$ 0 0
$$181$$ −1986.35 −0.815716 −0.407858 0.913045i $$-0.633724\pi$$
−0.407858 + 0.913045i $$0.633724\pi$$
$$182$$ −1010.48 −0.411548
$$183$$ 0 0
$$184$$ 1689.05 0.676732
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −153.659 −0.0600891
$$188$$ −280.718 −0.108901
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1615.88 0.612150 0.306075 0.952007i $$-0.400984\pi$$
0.306075 + 0.952007i $$0.400984\pi$$
$$192$$ 0 0
$$193$$ 2052.44 0.765482 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$194$$ 4627.88 1.71269
$$195$$ 0 0
$$196$$ −70.4839 −0.0256866
$$197$$ 4468.58 1.61611 0.808054 0.589108i $$-0.200521\pi$$
0.808054 + 0.589108i $$0.200521\pi$$
$$198$$ 0 0
$$199$$ 1543.07 0.549675 0.274838 0.961491i $$-0.411376\pi$$
0.274838 + 0.961491i $$0.411376\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −1237.37 −0.430994
$$203$$ −1601.93 −0.553858
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −868.492 −0.293741
$$207$$ 0 0
$$208$$ −2841.56 −0.947245
$$209$$ −566.583 −0.187519
$$210$$ 0 0
$$211$$ 1284.02 0.418937 0.209469 0.977815i $$-0.432827\pi$$
0.209469 + 0.977815i $$0.432827\pi$$
$$212$$ −735.335 −0.238222
$$213$$ 0 0
$$214$$ 1154.87 0.368902
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −475.244 −0.148671
$$218$$ 4717.62 1.46568
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1386.33 −0.421968
$$222$$ 0 0
$$223$$ −3815.31 −1.14571 −0.572853 0.819658i $$-0.694163\pi$$
−0.572853 + 0.819658i $$0.694163\pi$$
$$224$$ −449.783 −0.134162
$$225$$ 0 0
$$226$$ −4802.18 −1.41344
$$227$$ 2271.53 0.664172 0.332086 0.943249i $$-0.392248\pi$$
0.332086 + 0.943249i $$0.392248\pi$$
$$228$$ 0 0
$$229$$ −2367.54 −0.683195 −0.341598 0.939846i $$-0.610968\pi$$
−0.341598 + 0.939846i $$0.610968\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −5532.84 −1.56573
$$233$$ −1617.71 −0.454849 −0.227425 0.973796i $$-0.573030\pi$$
−0.227425 + 0.973796i $$0.573030\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 408.519 0.112679
$$237$$ 0 0
$$238$$ 441.106 0.120137
$$239$$ −1935.29 −0.523781 −0.261891 0.965098i $$-0.584346\pi$$
−0.261891 + 0.965098i $$0.584346\pi$$
$$240$$ 0 0
$$241$$ 477.901 0.127736 0.0638679 0.997958i $$-0.479656\pi$$
0.0638679 + 0.997958i $$0.479656\pi$$
$$242$$ 3309.49 0.879099
$$243$$ 0 0
$$244$$ 177.591 0.0465947
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5111.79 −1.31682
$$248$$ −1641.43 −0.420286
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4769.70 −1.19944 −0.599722 0.800208i $$-0.704722\pi$$
−0.599722 + 0.800208i $$0.704722\pi$$
$$252$$ 0 0
$$253$$ 436.371 0.108436
$$254$$ −97.5910 −0.0241079
$$255$$ 0 0
$$256$$ −2133.74 −0.520933
$$257$$ 682.524 0.165660 0.0828302 0.996564i $$-0.473604\pi$$
0.0828302 + 0.996564i $$0.473604\pi$$
$$258$$ 0 0
$$259$$ 411.926 0.0988256
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −3974.27 −0.937142
$$263$$ 3029.11 0.710202 0.355101 0.934828i $$-0.384447\pi$$
0.355101 + 0.934828i $$0.384447\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1626.48 0.374909
$$267$$ 0 0
$$268$$ 208.529 0.0475295
$$269$$ −6187.33 −1.40241 −0.701205 0.712960i $$-0.747354\pi$$
−0.701205 + 0.712960i $$0.747354\pi$$
$$270$$ 0 0
$$271$$ −7558.90 −1.69436 −0.847178 0.531309i $$-0.821700\pi$$
−0.847178 + 0.531309i $$0.821700\pi$$
$$272$$ 1240.43 0.276516
$$273$$ 0 0
$$274$$ 3082.17 0.679564
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3685.36 −0.799393 −0.399697 0.916647i $$-0.630885\pi$$
−0.399697 + 0.916647i $$0.630885\pi$$
$$278$$ −4859.26 −1.04834
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −7969.11 −1.69180 −0.845902 0.533338i $$-0.820937\pi$$
−0.845902 + 0.533338i $$0.820937\pi$$
$$282$$ 0 0
$$283$$ 2479.73 0.520864 0.260432 0.965492i $$-0.416135\pi$$
0.260432 + 0.965492i $$0.416135\pi$$
$$284$$ 105.048 0.0219487
$$285$$ 0 0
$$286$$ −901.667 −0.186422
$$287$$ 134.405 0.0276435
$$288$$ 0 0
$$289$$ −4307.82 −0.876821
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 918.952 0.184170
$$293$$ −5950.02 −1.18636 −0.593181 0.805069i $$-0.702128\pi$$
−0.593181 + 0.805069i $$0.702128\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1422.74 0.279375
$$297$$ 0 0
$$298$$ 1805.61 0.350994
$$299$$ 3937.00 0.761480
$$300$$ 0 0
$$301$$ −2556.52 −0.489554
$$302$$ 7774.86 1.48143
$$303$$ 0 0
$$304$$ 4573.81 0.862915
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3129.90 0.581865 0.290933 0.956744i $$-0.406034\pi$$
0.290933 + 0.956744i $$0.406034\pi$$
$$308$$ −62.8939 −0.0116354
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −7261.25 −1.32395 −0.661973 0.749527i $$-0.730281\pi$$
−0.661973 + 0.749527i $$0.730281\pi$$
$$312$$ 0 0
$$313$$ 2310.83 0.417303 0.208652 0.977990i $$-0.433093\pi$$
0.208652 + 0.977990i $$0.433093\pi$$
$$314$$ 6951.16 1.24929
$$315$$ 0 0
$$316$$ −1404.20 −0.249975
$$317$$ 4701.40 0.832987 0.416494 0.909139i $$-0.363259\pi$$
0.416494 + 0.909139i $$0.363259\pi$$
$$318$$ 0 0
$$319$$ −1429.42 −0.250885
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1252.68 −0.216799
$$323$$ 2231.46 0.384401
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −1191.79 −0.202475
$$327$$ 0 0
$$328$$ 464.218 0.0781468
$$329$$ 1366.07 0.228918
$$330$$ 0 0
$$331$$ 1366.18 0.226864 0.113432 0.993546i $$-0.463816\pi$$
0.113432 + 0.993546i $$0.463816\pi$$
$$332$$ −696.879 −0.115199
$$333$$ 0 0
$$334$$ 10646.0 1.74408
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 740.632 0.119718 0.0598588 0.998207i $$-0.480935\pi$$
0.0598588 + 0.998207i $$0.480935\pi$$
$$338$$ −2507.23 −0.403477
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −424.068 −0.0673448
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ −8829.90 −1.38394
$$345$$ 0 0
$$346$$ −10864.0 −1.68801
$$347$$ −2605.56 −0.403094 −0.201547 0.979479i $$-0.564597\pi$$
−0.201547 + 0.979479i $$0.564597\pi$$
$$348$$ 0 0
$$349$$ 4665.07 0.715517 0.357758 0.933814i $$-0.383541\pi$$
0.357758 + 0.933814i $$0.383541\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −401.349 −0.0607726
$$353$$ 2964.75 0.447019 0.223509 0.974702i $$-0.428249\pi$$
0.223509 + 0.974702i $$0.428249\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −1463.33 −0.217855
$$357$$ 0 0
$$358$$ 7531.87 1.11193
$$359$$ −4267.55 −0.627389 −0.313695 0.949524i $$-0.601567\pi$$
−0.313695 + 0.949524i $$0.601567\pi$$
$$360$$ 0 0
$$361$$ 1369.00 0.199592
$$362$$ 5088.15 0.738749
$$363$$ 0 0
$$364$$ −567.437 −0.0817082
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9280.33 1.31997 0.659985 0.751279i $$-0.270562\pi$$
0.659985 + 0.751279i $$0.270562\pi$$
$$368$$ −3522.66 −0.498998
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3578.41 0.500759
$$372$$ 0 0
$$373$$ −10781.1 −1.49657 −0.748286 0.663376i $$-0.769123\pi$$
−0.748286 + 0.663376i $$0.769123\pi$$
$$374$$ 393.606 0.0544195
$$375$$ 0 0
$$376$$ 4718.24 0.647140
$$377$$ −12896.5 −1.76181
$$378$$ 0 0
$$379$$ −5914.16 −0.801557 −0.400779 0.916175i $$-0.631260\pi$$
−0.400779 + 0.916175i $$0.631260\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4139.15 −0.554391
$$383$$ −11513.9 −1.53612 −0.768059 0.640379i $$-0.778777\pi$$
−0.768059 + 0.640379i $$0.778777\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5257.44 −0.693256
$$387$$ 0 0
$$388$$ 2598.80 0.340036
$$389$$ −5399.73 −0.703797 −0.351898 0.936038i $$-0.614464\pi$$
−0.351898 + 0.936038i $$0.614464\pi$$
$$390$$ 0 0
$$391$$ −1718.62 −0.222288
$$392$$ 1184.68 0.152641
$$393$$ 0 0
$$394$$ −11446.5 −1.46362
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2622.13 −0.331488 −0.165744 0.986169i $$-0.553003\pi$$
−0.165744 + 0.986169i $$0.553003\pi$$
$$398$$ −3952.66 −0.497811
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 11119.1 1.38469 0.692344 0.721568i $$-0.256578\pi$$
0.692344 + 0.721568i $$0.256578\pi$$
$$402$$ 0 0
$$403$$ −3826.00 −0.472920
$$404$$ −694.846 −0.0855690
$$405$$ 0 0
$$406$$ 4103.42 0.501599
$$407$$ 367.568 0.0447658
$$408$$ 0 0
$$409$$ −6589.18 −0.796611 −0.398305 0.917253i $$-0.630402\pi$$
−0.398305 + 0.917253i $$0.630402\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −487.704 −0.0583191
$$413$$ −1988.00 −0.236860
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3621.02 −0.426767
$$417$$ 0 0
$$418$$ 1451.33 0.169825
$$419$$ 11871.6 1.38416 0.692081 0.721820i $$-0.256694\pi$$
0.692081 + 0.721820i $$0.256694\pi$$
$$420$$ 0 0
$$421$$ −1731.57 −0.200455 −0.100227 0.994965i $$-0.531957\pi$$
−0.100227 + 0.994965i $$0.531957\pi$$
$$422$$ −3289.09 −0.379409
$$423$$ 0 0
$$424$$ 12359.3 1.41562
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −864.222 −0.0979452
$$428$$ 648.519 0.0732415
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −10653.8 −1.19066 −0.595330 0.803481i $$-0.702979\pi$$
−0.595330 + 0.803481i $$0.702979\pi$$
$$432$$ 0 0
$$433$$ −2642.01 −0.293226 −0.146613 0.989194i $$-0.546837\pi$$
−0.146613 + 0.989194i $$0.546837\pi$$
$$434$$ 1217.36 0.134644
$$435$$ 0 0
$$436$$ 2649.19 0.290994
$$437$$ −6337.04 −0.693688
$$438$$ 0 0
$$439$$ 8858.21 0.963051 0.481525 0.876432i $$-0.340083\pi$$
0.481525 + 0.876432i $$0.340083\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3551.17 0.382153
$$443$$ −6621.73 −0.710176 −0.355088 0.934833i $$-0.615549\pi$$
−0.355088 + 0.934833i $$0.615549\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 9773.13 1.03760
$$447$$ 0 0
$$448$$ 3975.85 0.419288
$$449$$ −13081.7 −1.37497 −0.687487 0.726196i $$-0.741286\pi$$
−0.687487 + 0.726196i $$0.741286\pi$$
$$450$$ 0 0
$$451$$ 119.932 0.0125219
$$452$$ −2696.68 −0.280622
$$453$$ 0 0
$$454$$ −5818.65 −0.601504
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −12167.0 −1.24540 −0.622699 0.782461i $$-0.713964\pi$$
−0.622699 + 0.782461i $$0.713964\pi$$
$$458$$ 6064.59 0.618733
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −11283.8 −1.14000 −0.570000 0.821645i $$-0.693057\pi$$
−0.570000 + 0.821645i $$0.693057\pi$$
$$462$$ 0 0
$$463$$ −16542.9 −1.66051 −0.830254 0.557385i $$-0.811805\pi$$
−0.830254 + 0.557385i $$0.811805\pi$$
$$464$$ 11539.2 1.15451
$$465$$ 0 0
$$466$$ 4143.85 0.411932
$$467$$ −10266.9 −1.01734 −0.508668 0.860963i $$-0.669862\pi$$
−0.508668 + 0.860963i $$0.669862\pi$$
$$468$$ 0 0
$$469$$ −1014.77 −0.0999103
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −6866.29 −0.669590
$$473$$ −2281.23 −0.221757
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 247.704 0.0238519
$$477$$ 0 0
$$478$$ 4957.36 0.474360
$$479$$ 7967.98 0.760055 0.380027 0.924975i $$-0.375915\pi$$
0.380027 + 0.924975i $$0.375915\pi$$
$$480$$ 0 0
$$481$$ 3316.25 0.314362
$$482$$ −1224.17 −0.115683
$$483$$ 0 0
$$484$$ 1858.45 0.174535
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −9956.62 −0.926443 −0.463221 0.886243i $$-0.653306\pi$$
−0.463221 + 0.886243i $$0.653306\pi$$
$$488$$ −2984.91 −0.276886
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 18660.8 1.71518 0.857589 0.514336i $$-0.171962\pi$$
0.857589 + 0.514336i $$0.171962\pi$$
$$492$$ 0 0
$$493$$ 5629.71 0.514299
$$494$$ 13094.1 1.19258
$$495$$ 0 0
$$496$$ 3423.34 0.309904
$$497$$ −511.199 −0.0461377
$$498$$ 0 0
$$499$$ 4074.21 0.365504 0.182752 0.983159i $$-0.441499\pi$$
0.182752 + 0.983159i $$0.441499\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12217.8 1.08627
$$503$$ −4255.51 −0.377224 −0.188612 0.982052i $$-0.560399\pi$$
−0.188612 + 0.982052i $$0.560399\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −1117.79 −0.0982050
$$507$$ 0 0
$$508$$ −54.8025 −0.00478636
$$509$$ −10171.8 −0.885771 −0.442885 0.896578i $$-0.646045\pi$$
−0.442885 + 0.896578i $$0.646045\pi$$
$$510$$ 0 0
$$511$$ −4471.95 −0.387138
$$512$$ 12992.6 1.12148
$$513$$ 0 0
$$514$$ −1748.32 −0.150030
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1218.97 0.103695
$$518$$ −1055.17 −0.0895010
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1680.39 −0.141303 −0.0706517 0.997501i $$-0.522508\pi$$
−0.0706517 + 0.997501i $$0.522508\pi$$
$$522$$ 0 0
$$523$$ −13211.8 −1.10461 −0.552305 0.833642i $$-0.686251\pi$$
−0.552305 + 0.833642i $$0.686251\pi$$
$$524$$ −2231.76 −0.186059
$$525$$ 0 0
$$526$$ −7759.23 −0.643191
$$527$$ 1670.17 0.138053
$$528$$ 0 0
$$529$$ −7286.34 −0.598861
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 913.354 0.0744341
$$533$$ 1082.04 0.0879333
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −3504.90 −0.282441
$$537$$ 0 0
$$538$$ 15849.2 1.27009
$$539$$ 306.064 0.0244585
$$540$$ 0 0
$$541$$ 9650.84 0.766954 0.383477 0.923551i $$-0.374727\pi$$
0.383477 + 0.923551i $$0.374727\pi$$
$$542$$ 19362.5 1.53449
$$543$$ 0 0
$$544$$ 1580.69 0.124580
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 23864.3 1.86538 0.932689 0.360682i $$-0.117456\pi$$
0.932689 + 0.360682i $$0.117456\pi$$
$$548$$ 1730.80 0.134920
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 20758.3 1.60496
$$552$$ 0 0
$$553$$ 6833.33 0.525466
$$554$$ 9440.25 0.723967
$$555$$ 0 0
$$556$$ −2728.73 −0.208136
$$557$$ −2314.22 −0.176044 −0.0880221 0.996119i $$-0.528055\pi$$
−0.0880221 + 0.996119i $$0.528055\pi$$
$$558$$ 0 0
$$559$$ −20581.5 −1.55726
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 20413.3 1.53218
$$563$$ −7017.86 −0.525342 −0.262671 0.964885i $$-0.584603\pi$$
−0.262671 + 0.964885i $$0.584603\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −6351.95 −0.471718
$$567$$ 0 0
$$568$$ −1765.61 −0.130429
$$569$$ −5302.37 −0.390662 −0.195331 0.980737i $$-0.562578\pi$$
−0.195331 + 0.980737i $$0.562578\pi$$
$$570$$ 0 0
$$571$$ 17767.2 1.30216 0.651082 0.759008i $$-0.274315\pi$$
0.651082 + 0.759008i $$0.274315\pi$$
$$572$$ −506.333 −0.0370120
$$573$$ 0 0
$$574$$ −344.286 −0.0250352
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6089.57 0.439363 0.219681 0.975572i $$-0.429498\pi$$
0.219681 + 0.975572i $$0.429498\pi$$
$$578$$ 11034.7 0.794089
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3391.26 0.242157
$$582$$ 0 0
$$583$$ 3193.07 0.226833
$$584$$ −15445.5 −1.09442
$$585$$ 0 0
$$586$$ 15241.3 1.07442
$$587$$ −26543.9 −1.86641 −0.933205 0.359344i $$-0.883000\pi$$
−0.933205 + 0.359344i $$0.883000\pi$$
$$588$$ 0 0
$$589$$ 6158.37 0.430817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −2967.24 −0.206001
$$593$$ 16365.0 1.13327 0.566635 0.823969i $$-0.308245\pi$$
0.566635 + 0.823969i $$0.308245\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1013.94 0.0696859
$$597$$ 0 0
$$598$$ −10084.8 −0.689631
$$599$$ 17516.3 1.19482 0.597411 0.801935i $$-0.296196\pi$$
0.597411 + 0.801935i $$0.296196\pi$$
$$600$$ 0 0
$$601$$ 8693.80 0.590062 0.295031 0.955488i $$-0.404670\pi$$
0.295031 + 0.955488i $$0.404670\pi$$
$$602$$ 6548.67 0.443362
$$603$$ 0 0
$$604$$ 4365.99 0.294122
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 18096.9 1.21010 0.605051 0.796186i $$-0.293153\pi$$
0.605051 + 0.796186i $$0.293153\pi$$
$$608$$ 5828.44 0.388774
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10997.7 0.728183
$$612$$ 0 0
$$613$$ −4641.61 −0.305828 −0.152914 0.988239i $$-0.548866\pi$$
−0.152914 + 0.988239i $$0.548866\pi$$
$$614$$ −8017.40 −0.526964
$$615$$ 0 0
$$616$$ 1057.11 0.0691429
$$617$$ 14676.1 0.957600 0.478800 0.877924i $$-0.341072\pi$$
0.478800 + 0.877924i $$0.341072\pi$$
$$618$$ 0 0
$$619$$ −19645.3 −1.27563 −0.637813 0.770192i $$-0.720161\pi$$
−0.637813 + 0.770192i $$0.720161\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18600.1 1.19903
$$623$$ 7121.09 0.457946
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −5919.32 −0.377929
$$627$$ 0 0
$$628$$ 3903.44 0.248032
$$629$$ −1447.65 −0.0917671
$$630$$ 0 0
$$631$$ 26231.2 1.65491 0.827456 0.561531i $$-0.189788\pi$$
0.827456 + 0.561531i $$0.189788\pi$$
$$632$$ 23601.4 1.48546
$$633$$ 0 0
$$634$$ −12042.9 −0.754391
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2761.35 0.171756
$$638$$ 3661.55 0.227213
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30882.4 −1.90293 −0.951466 0.307754i $$-0.900423\pi$$
−0.951466 + 0.307754i $$0.900423\pi$$
$$642$$ 0 0
$$643$$ 6216.88 0.381290 0.190645 0.981659i $$-0.438942\pi$$
0.190645 + 0.981659i $$0.438942\pi$$
$$644$$ −703.447 −0.0430430
$$645$$ 0 0
$$646$$ −5716.00 −0.348132
$$647$$ −21210.4 −1.28882 −0.644410 0.764680i $$-0.722897\pi$$
−0.644410 + 0.764680i $$0.722897\pi$$
$$648$$ 0 0
$$649$$ −1773.92 −0.107292
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −669.251 −0.0401992
$$653$$ 32938.7 1.97395 0.986977 0.160864i $$-0.0514281\pi$$
0.986977 + 0.160864i $$0.0514281\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −968.165 −0.0576227
$$657$$ 0 0
$$658$$ −3499.27 −0.207319
$$659$$ −9543.51 −0.564131 −0.282066 0.959395i $$-0.591020\pi$$
−0.282066 + 0.959395i $$0.591020\pi$$
$$660$$ 0 0
$$661$$ 13274.5 0.781116 0.390558 0.920578i $$-0.372282\pi$$
0.390558 + 0.920578i $$0.372282\pi$$
$$662$$ −3499.55 −0.205459
$$663$$ 0 0
$$664$$ 11713.0 0.684565
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −15987.6 −0.928101
$$668$$ 5978.27 0.346267
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −771.159 −0.0443670
$$672$$ 0 0
$$673$$ 13575.3 0.777545 0.388772 0.921334i $$-0.372899\pi$$
0.388772 + 0.921334i $$0.372899\pi$$
$$674$$ −1897.17 −0.108422
$$675$$ 0 0
$$676$$ −1407.94 −0.0801058
$$677$$ 12020.7 0.682414 0.341207 0.939988i $$-0.389164\pi$$
0.341207 + 0.939988i $$0.389164\pi$$
$$678$$ 0 0
$$679$$ −12646.7 −0.714779
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 1086.27 0.0609905
$$683$$ 18391.9 1.03038 0.515188 0.857077i $$-0.327722\pi$$
0.515188 + 0.857077i $$0.327722\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −878.613 −0.0489003
$$687$$ 0 0
$$688$$ 18415.5 1.02047
$$689$$ 28808.3 1.59290
$$690$$ 0 0
$$691$$ 15594.1 0.858508 0.429254 0.903184i $$-0.358777\pi$$
0.429254 + 0.903184i $$0.358777\pi$$
$$692$$ −6100.69 −0.335135
$$693$$ 0 0
$$694$$ 6674.28 0.365061
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −472.346 −0.0256691
$$698$$ −11949.8 −0.648004
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 31093.7 1.67531 0.837656 0.546198i $$-0.183925\pi$$
0.837656 + 0.546198i $$0.183925\pi$$
$$702$$ 0 0
$$703$$ −5337.88 −0.286375
$$704$$ 3547.71 0.189928
$$705$$ 0 0
$$706$$ −7594.36 −0.404841
$$707$$ 3381.37 0.179872
$$708$$ 0 0
$$709$$ −2494.67 −0.132143 −0.0660714 0.997815i $$-0.521047\pi$$
−0.0660714 + 0.997815i $$0.521047\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 24595.3 1.29459
$$713$$ −4743.06 −0.249129
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4229.54 0.220762
$$717$$ 0 0
$$718$$ 10931.6 0.568192
$$719$$ −34467.1 −1.78777 −0.893885 0.448295i $$-0.852031\pi$$
−0.893885 + 0.448295i $$0.852031\pi$$
$$720$$ 0 0
$$721$$ 2373.34 0.122591
$$722$$ −3506.77 −0.180759
$$723$$ 0 0
$$724$$ 2857.27 0.146670
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −9314.97 −0.475204 −0.237602 0.971363i $$-0.576361\pi$$
−0.237602 + 0.971363i $$0.576361\pi$$
$$728$$ 9537.35 0.485547
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8984.49 0.454588
$$732$$ 0 0
$$733$$ −16146.3 −0.813611 −0.406805 0.913515i $$-0.633357\pi$$
−0.406805 + 0.913515i $$0.633357\pi$$
$$734$$ −23772.1 −1.19543
$$735$$ 0 0
$$736$$ −4488.95 −0.224816
$$737$$ −905.500 −0.0452571
$$738$$ 0 0
$$739$$ −36749.1 −1.82928 −0.914640 0.404268i $$-0.867526\pi$$
−0.914640 + 0.404268i $$0.867526\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −9166.27 −0.453510
$$743$$ −2527.09 −0.124778 −0.0623890 0.998052i $$-0.519872\pi$$
−0.0623890 + 0.998052i $$0.519872\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 27616.2 1.35536
$$747$$ 0 0
$$748$$ 221.031 0.0108044
$$749$$ −3155.93 −0.153959
$$750$$ 0 0
$$751$$ 15828.0 0.769070 0.384535 0.923111i $$-0.374362\pi$$
0.384535 + 0.923111i $$0.374362\pi$$
$$752$$ −9840.28 −0.477178
$$753$$ 0 0
$$754$$ 33035.0 1.59557
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 20845.9 1.00087 0.500435 0.865774i $$-0.333174\pi$$
0.500435 + 0.865774i $$0.333174\pi$$
$$758$$ 15149.4 0.725927
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2420.55 −0.115302 −0.0576511 0.998337i $$-0.518361\pi$$
−0.0576511 + 0.998337i $$0.518361\pi$$
$$762$$ 0 0
$$763$$ −12891.9 −0.611690
$$764$$ −2324.35 −0.110068
$$765$$ 0 0
$$766$$ 29493.5 1.39118
$$767$$ −16004.6 −0.753445
$$768$$ 0 0
$$769$$ 22646.3 1.06196 0.530980 0.847384i $$-0.321824\pi$$
0.530980 + 0.847384i $$0.321824\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2952.33 −0.137638
$$773$$ 27620.2 1.28516 0.642580 0.766219i $$-0.277864\pi$$
0.642580 + 0.766219i $$0.277864\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −43680.0 −2.02064
$$777$$ 0 0
$$778$$ 13831.7 0.637391
$$779$$ −1741.67 −0.0801049
$$780$$ 0 0
$$781$$ −456.151 −0.0208993
$$782$$ 4402.35 0.201314
$$783$$ 0 0
$$784$$ −2470.74 −0.112552
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −14767.1 −0.668859 −0.334429 0.942421i $$-0.608544\pi$$
−0.334429 + 0.942421i $$0.608544\pi$$
$$788$$ −6427.82 −0.290586
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 13123.0 0.589887
$$792$$ 0 0
$$793$$ −6957.50 −0.311561
$$794$$ 6716.71 0.300211
$$795$$ 0 0
$$796$$ −2219.62 −0.0988348
$$797$$ −7549.97 −0.335551 −0.167775 0.985825i $$-0.553658\pi$$
−0.167775 + 0.985825i $$0.553658\pi$$
$$798$$ 0 0
$$799$$ −4800.85 −0.212568
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −28482.1 −1.25404
$$803$$ −3990.39 −0.175365
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 9800.50 0.428298
$$807$$ 0 0
$$808$$ 11678.8 0.508489
$$809$$ −17920.0 −0.778783 −0.389391 0.921072i $$-0.627315\pi$$
−0.389391 + 0.921072i $$0.627315\pi$$
$$810$$ 0 0
$$811$$ 25536.6 1.10569 0.552843 0.833285i $$-0.313543\pi$$
0.552843 + 0.833285i $$0.313543\pi$$
$$812$$ 2304.29 0.0995869
$$813$$ 0 0
$$814$$ −941.545 −0.0405420
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 33128.3 1.41862
$$818$$ 16878.5 0.721447
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 13688.8 0.581904 0.290952 0.956738i $$-0.406028\pi$$
0.290952 + 0.956738i $$0.406028\pi$$
$$822$$ 0 0
$$823$$ 15102.9 0.639678 0.319839 0.947472i $$-0.396371\pi$$
0.319839 + 0.947472i $$0.396371\pi$$
$$824$$ 8197.22 0.346558
$$825$$ 0 0
$$826$$ 5092.37 0.214511
$$827$$ −36290.3 −1.52592 −0.762961 0.646445i $$-0.776255\pi$$
−0.762961 + 0.646445i $$0.776255\pi$$
$$828$$ 0 0
$$829$$ −39405.1 −1.65090 −0.825449 0.564477i $$-0.809078\pi$$
−0.825449 + 0.564477i $$0.809078\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 32007.9 1.33374
$$833$$ −1205.42 −0.0501384
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 815.000 0.0337170
$$837$$ 0 0
$$838$$ −30409.6 −1.25356
$$839$$ 33093.9 1.36177 0.680886 0.732389i $$-0.261595\pi$$
0.680886 + 0.732389i $$0.261595\pi$$
$$840$$ 0 0
$$841$$ 27981.8 1.14731
$$842$$ 4435.50 0.181541
$$843$$ 0 0
$$844$$ −1847.00 −0.0753274
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −9043.89 −0.366885
$$848$$ −25776.4 −1.04383
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4111.12 0.165602
$$852$$ 0 0
$$853$$ −29441.6 −1.18178 −0.590892 0.806750i $$-0.701224\pi$$
−0.590892 + 0.806750i $$0.701224\pi$$
$$854$$ 2213.75 0.0887037
$$855$$ 0 0
$$856$$ −10900.2 −0.435233
$$857$$ −16012.9 −0.638260 −0.319130 0.947711i $$-0.603391\pi$$
−0.319130 + 0.947711i $$0.603391\pi$$
$$858$$ 0 0
$$859$$ −13404.3 −0.532421 −0.266211 0.963915i $$-0.585772\pi$$
−0.266211 + 0.963915i $$0.585772\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 27290.2 1.07832
$$863$$ 2058.73 0.0812050 0.0406025 0.999175i $$-0.487072\pi$$
0.0406025 + 0.999175i $$0.487072\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 6767.66 0.265559
$$867$$ 0 0
$$868$$ 683.614 0.0267320
$$869$$ 6097.48 0.238024
$$870$$ 0 0
$$871$$ −8169.54 −0.317812
$$872$$ −44527.0 −1.72922
$$873$$ 0 0
$$874$$ 16232.7 0.628236
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 16477.0 0.634422 0.317211 0.948355i $$-0.397254\pi$$
0.317211 + 0.948355i $$0.397254\pi$$
$$878$$ −22690.8 −0.872183
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −43307.7 −1.65616 −0.828079 0.560612i $$-0.810566\pi$$
−0.828079 + 0.560612i $$0.810566\pi$$
$$882$$ 0 0
$$883$$ −15197.9 −0.579217 −0.289608 0.957145i $$-0.593525\pi$$
−0.289608 + 0.957145i $$0.593525\pi$$
$$884$$ 1994.17 0.0758723
$$885$$ 0 0
$$886$$ 16961.9 0.643168
$$887$$ 44953.6 1.70168 0.850842 0.525422i $$-0.176093\pi$$
0.850842 + 0.525422i $$0.176093\pi$$
$$888$$ 0 0
$$889$$ 266.689 0.0100613
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 5488.13 0.206005
$$893$$ −17702.0 −0.663355
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −6586.08 −0.245564
$$897$$ 0 0
$$898$$ 33509.5 1.24524
$$899$$ 15536.9 0.576400
$$900$$ 0 0
$$901$$ −12575.7 −0.464993
$$902$$ −307.212 −0.0113404
$$903$$ 0 0
$$904$$ 45325.2 1.66758
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38388.7 1.40537 0.702687 0.711499i $$-0.251983\pi$$
0.702687 + 0.711499i $$0.251983\pi$$
$$908$$ −3267.48 −0.119422
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 46222.1 1.68102 0.840508 0.541799i $$-0.182257\pi$$
0.840508 + 0.541799i $$0.182257\pi$$
$$912$$ 0 0
$$913$$ 3026.08 0.109692
$$914$$ 31166.3 1.12789
$$915$$ 0 0
$$916$$ 3405.59 0.122842
$$917$$ 10860.6 0.391109
$$918$$ 0 0
$$919$$ −30946.4 −1.11080 −0.555402 0.831582i $$-0.687436\pi$$
−0.555402 + 0.831582i $$0.687436\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 28904.1 1.03244
$$923$$ −4115.46 −0.146763
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 42375.6 1.50383
$$927$$ 0 0
$$928$$ 14704.5 0.520149
$$929$$ 1907.48 0.0673652 0.0336826 0.999433i $$-0.489276\pi$$
0.0336826 + 0.999433i $$0.489276\pi$$
$$930$$ 0 0
$$931$$ −4444.71 −0.156466
$$932$$ 2326.99 0.0817845
$$933$$ 0 0
$$934$$ 26299.2 0.921347
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3334.99 0.116275 0.0581374 0.998309i $$-0.481484\pi$$
0.0581374 + 0.998309i $$0.481484\pi$$
$$938$$ 2599.40 0.0904834
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −9632.00 −0.333681 −0.166841 0.985984i $$-0.553357\pi$$
−0.166841 + 0.985984i $$0.553357\pi$$
$$942$$ 0 0
$$943$$ 1341.40 0.0463223
$$944$$ 14320.2 0.493732
$$945$$ 0 0
$$946$$ 5843.48 0.200833
$$947$$ 53606.0 1.83945 0.919727 0.392559i $$-0.128410\pi$$
0.919727 + 0.392559i $$0.128410\pi$$
$$948$$ 0 0
$$949$$ −36001.9 −1.23148
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4163.36 −0.141739
$$953$$ −28433.6 −0.966481 −0.483240 0.875488i $$-0.660540\pi$$
−0.483240 + 0.875488i $$0.660540\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 2783.82 0.0941790
$$957$$ 0 0
$$958$$ −20410.4 −0.688341
$$959$$ −8422.70 −0.283611
$$960$$ 0 0
$$961$$ −25181.7 −0.845278
$$962$$ −8494.75 −0.284700
$$963$$ 0 0
$$964$$ −687.436 −0.0229676
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −33877.6 −1.12661 −0.563304 0.826250i $$-0.690470\pi$$
−0.563304 + 0.826250i $$0.690470\pi$$
$$968$$ −31236.4 −1.03717
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −10208.6 −0.337394 −0.168697 0.985668i $$-0.553956\pi$$
−0.168697 + 0.985668i $$0.553956\pi$$
$$972$$ 0 0
$$973$$ 13279.0 0.437517
$$974$$ 25504.4 0.839029
$$975$$ 0 0
$$976$$ 6225.27 0.204166
$$977$$ 35478.1 1.16177 0.580883 0.813987i $$-0.302707\pi$$
0.580883 + 0.813987i $$0.302707\pi$$
$$978$$ 0 0
$$979$$ 6354.27 0.207439
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −47800.7 −1.55334
$$983$$ −54435.8 −1.76626 −0.883130 0.469128i $$-0.844568\pi$$
−0.883130 + 0.469128i $$0.844568\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −14420.8 −0.465773
$$987$$ 0 0
$$988$$ 7353.04 0.236773
$$989$$ −25514.7 −0.820346
$$990$$ 0 0
$$991$$ −47319.6 −1.51681 −0.758404 0.651784i $$-0.774021\pi$$
−0.758404 + 0.651784i $$0.774021\pi$$
$$992$$ 4362.38 0.139623
$$993$$ 0 0
$$994$$ 1309.46 0.0417844
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26273.1 −0.834580 −0.417290 0.908773i $$-0.637020\pi$$
−0.417290 + 0.908773i $$0.637020\pi$$
$$998$$ −10436.3 −0.331017
$$999$$ 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.r.1.1 2
3.2 odd 2 1575.4.a.u.1.2 2
5.4 even 2 315.4.a.j.1.2 yes 2
15.14 odd 2 315.4.a.h.1.1 2
35.34 odd 2 2205.4.a.ba.1.2 2
105.104 even 2 2205.4.a.y.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.a.h.1.1 2 15.14 odd 2
315.4.a.j.1.2 yes 2 5.4 even 2
1575.4.a.r.1.1 2 1.1 even 1 trivial
1575.4.a.u.1.2 2 3.2 odd 2
2205.4.a.y.1.1 2 105.104 even 2
2205.4.a.ba.1.2 2 35.34 odd 2