# Properties

 Label 1575.4.a.x.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: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.561553 q^{2} -7.68466 q^{4} +7.00000 q^{7} +8.80776 q^{8} +O(q^{10})$$ $$q-0.561553 q^{2} -7.68466 q^{4} +7.00000 q^{7} +8.80776 q^{8} +25.8078 q^{11} +15.5464 q^{13} -3.93087 q^{14} +56.5312 q^{16} -95.1619 q^{17} -143.447 q^{19} -14.4924 q^{22} +77.5227 q^{23} -8.73012 q^{26} -53.7926 q^{28} +204.170 q^{29} -40.6998 q^{31} -102.207 q^{32} +53.4384 q^{34} -95.6695 q^{37} +80.5530 q^{38} -24.7301 q^{41} -282.430 q^{43} -198.324 q^{44} -43.5331 q^{46} +257.417 q^{47} +49.0000 q^{49} -119.469 q^{52} +257.978 q^{53} +61.6543 q^{56} -114.652 q^{58} +651.365 q^{59} +451.304 q^{61} +22.8551 q^{62} -394.855 q^{64} -832.071 q^{67} +731.287 q^{68} +174.965 q^{71} +47.4166 q^{73} +53.7235 q^{74} +1102.34 q^{76} +180.654 q^{77} -1.16006 q^{79} +13.8873 q^{82} +1494.03 q^{83} +158.599 q^{86} +227.309 q^{88} -1309.13 q^{89} +108.825 q^{91} -595.736 q^{92} -144.553 q^{94} +1365.33 q^{97} -27.5161 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + O(q^{10})$$ $$2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.561553 −0.198539 −0.0992695 0.995061i $$-0.531651\pi$$
−0.0992695 + 0.995061i $$0.531651\pi$$
$$3$$ 0 0
$$4$$ −7.68466 −0.960582
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 8.80776 0.389252
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 25.8078 0.707394 0.353697 0.935360i $$-0.384924\pi$$
0.353697 + 0.935360i $$0.384924\pi$$
$$12$$ 0 0
$$13$$ 15.5464 0.331677 0.165838 0.986153i $$-0.446967\pi$$
0.165838 + 0.986153i $$0.446967\pi$$
$$14$$ −3.93087 −0.0750407
$$15$$ 0 0
$$16$$ 56.5312 0.883301
$$17$$ −95.1619 −1.35766 −0.678828 0.734297i $$-0.737512\pi$$
−0.678828 + 0.734297i $$0.737512\pi$$
$$18$$ 0 0
$$19$$ −143.447 −1.73205 −0.866026 0.499999i $$-0.833334\pi$$
−0.866026 + 0.499999i $$0.833334\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −14.4924 −0.140445
$$23$$ 77.5227 0.702809 0.351405 0.936224i $$-0.385704\pi$$
0.351405 + 0.936224i $$0.385704\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −8.73012 −0.0658507
$$27$$ 0 0
$$28$$ −53.7926 −0.363066
$$29$$ 204.170 1.30736 0.653681 0.756770i $$-0.273224\pi$$
0.653681 + 0.756770i $$0.273224\pi$$
$$30$$ 0 0
$$31$$ −40.6998 −0.235803 −0.117902 0.993025i $$-0.537617\pi$$
−0.117902 + 0.993025i $$0.537617\pi$$
$$32$$ −102.207 −0.564621
$$33$$ 0 0
$$34$$ 53.4384 0.269548
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −95.6695 −0.425080 −0.212540 0.977152i $$-0.568174\pi$$
−0.212540 + 0.977152i $$0.568174\pi$$
$$38$$ 80.5530 0.343880
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −24.7301 −0.0941999 −0.0471000 0.998890i $$-0.514998\pi$$
−0.0471000 + 0.998890i $$0.514998\pi$$
$$42$$ 0 0
$$43$$ −282.430 −1.00163 −0.500816 0.865554i $$-0.666967\pi$$
−0.500816 + 0.865554i $$0.666967\pi$$
$$44$$ −198.324 −0.679510
$$45$$ 0 0
$$46$$ −43.5331 −0.139535
$$47$$ 257.417 0.798895 0.399448 0.916756i $$-0.369202\pi$$
0.399448 + 0.916756i $$0.369202\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −119.469 −0.318603
$$53$$ 257.978 0.668604 0.334302 0.942466i $$-0.391499\pi$$
0.334302 + 0.942466i $$0.391499\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 61.6543 0.147123
$$57$$ 0 0
$$58$$ −114.652 −0.259562
$$59$$ 651.365 1.43730 0.718648 0.695374i $$-0.244761\pi$$
0.718648 + 0.695374i $$0.244761\pi$$
$$60$$ 0 0
$$61$$ 451.304 0.947271 0.473636 0.880721i $$-0.342941\pi$$
0.473636 + 0.880721i $$0.342941\pi$$
$$62$$ 22.8551 0.0468161
$$63$$ 0 0
$$64$$ −394.855 −0.771201
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −832.071 −1.51722 −0.758609 0.651546i $$-0.774121\pi$$
−0.758609 + 0.651546i $$0.774121\pi$$
$$68$$ 731.287 1.30414
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 174.965 0.292458 0.146229 0.989251i $$-0.453286\pi$$
0.146229 + 0.989251i $$0.453286\pi$$
$$72$$ 0 0
$$73$$ 47.4166 0.0760233 0.0380116 0.999277i $$-0.487898\pi$$
0.0380116 + 0.999277i $$0.487898\pi$$
$$74$$ 53.7235 0.0843950
$$75$$ 0 0
$$76$$ 1102.34 1.66378
$$77$$ 180.654 0.267370
$$78$$ 0 0
$$79$$ −1.16006 −0.00165211 −0.000826057 1.00000i $$-0.500263\pi$$
−0.000826057 1.00000i $$0.500263\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 13.8873 0.0187023
$$83$$ 1494.03 1.97580 0.987898 0.155107i $$-0.0495722\pi$$
0.987898 + 0.155107i $$0.0495722\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 158.599 0.198863
$$87$$ 0 0
$$88$$ 227.309 0.275354
$$89$$ −1309.13 −1.55919 −0.779593 0.626286i $$-0.784574\pi$$
−0.779593 + 0.626286i $$0.784574\pi$$
$$90$$ 0 0
$$91$$ 108.825 0.125362
$$92$$ −595.736 −0.675106
$$93$$ 0 0
$$94$$ −144.553 −0.158612
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1365.33 1.42916 0.714580 0.699553i $$-0.246618\pi$$
0.714580 + 0.699553i $$0.246618\pi$$
$$98$$ −27.5161 −0.0283627
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1155.99 −1.13887 −0.569434 0.822037i $$-0.692838\pi$$
−0.569434 + 0.822037i $$0.692838\pi$$
$$102$$ 0 0
$$103$$ 78.2074 0.0748156 0.0374078 0.999300i $$-0.488090\pi$$
0.0374078 + 0.999300i $$0.488090\pi$$
$$104$$ 136.929 0.129106
$$105$$ 0 0
$$106$$ −144.868 −0.132744
$$107$$ −94.0814 −0.0850018 −0.0425009 0.999096i $$-0.513533\pi$$
−0.0425009 + 0.999096i $$0.513533\pi$$
$$108$$ 0 0
$$109$$ −2128.25 −1.87018 −0.935088 0.354417i $$-0.884679\pi$$
−0.935088 + 0.354417i $$0.884679\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 395.719 0.333856
$$113$$ −596.493 −0.496578 −0.248289 0.968686i $$-0.579868\pi$$
−0.248289 + 0.968686i $$0.579868\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1568.98 −1.25583
$$117$$ 0 0
$$118$$ −365.776 −0.285359
$$119$$ −666.133 −0.513146
$$120$$ 0 0
$$121$$ −664.959 −0.499594
$$122$$ −253.431 −0.188070
$$123$$ 0 0
$$124$$ 312.764 0.226508
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2088.21 −1.45904 −0.729522 0.683957i $$-0.760257\pi$$
−0.729522 + 0.683957i $$0.760257\pi$$
$$128$$ 1039.39 0.717735
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −416.142 −0.277546 −0.138773 0.990324i $$-0.544316\pi$$
−0.138773 + 0.990324i $$0.544316\pi$$
$$132$$ 0 0
$$133$$ −1004.13 −0.654654
$$134$$ 467.252 0.301227
$$135$$ 0 0
$$136$$ −838.164 −0.528470
$$137$$ 619.252 0.386177 0.193089 0.981181i $$-0.438150\pi$$
0.193089 + 0.981181i $$0.438150\pi$$
$$138$$ 0 0
$$139$$ −266.994 −0.162922 −0.0814610 0.996677i $$-0.525959\pi$$
−0.0814610 + 0.996677i $$0.525959\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −98.2520 −0.0580643
$$143$$ 401.218 0.234626
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −26.6270 −0.0150936
$$147$$ 0 0
$$148$$ 735.187 0.408325
$$149$$ −2551.80 −1.40303 −0.701515 0.712655i $$-0.747493\pi$$
−0.701515 + 0.712655i $$0.747493\pi$$
$$150$$ 0 0
$$151$$ 1117.38 0.602195 0.301097 0.953593i $$-0.402647\pi$$
0.301097 + 0.953593i $$0.402647\pi$$
$$152$$ −1263.45 −0.674204
$$153$$ 0 0
$$154$$ −101.447 −0.0530833
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 456.104 0.231854 0.115927 0.993258i $$-0.463016\pi$$
0.115927 + 0.993258i $$0.463016\pi$$
$$158$$ 0.651435 0.000328009 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 542.659 0.265637
$$162$$ 0 0
$$163$$ 105.490 0.0506907 0.0253453 0.999679i $$-0.491931\pi$$
0.0253453 + 0.999679i $$0.491931\pi$$
$$164$$ 190.043 0.0904868
$$165$$ 0 0
$$166$$ −838.976 −0.392272
$$167$$ −2482.54 −1.15033 −0.575164 0.818038i $$-0.695062\pi$$
−0.575164 + 0.818038i $$0.695062\pi$$
$$168$$ 0 0
$$169$$ −1955.31 −0.889991
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2170.38 0.962150
$$173$$ 516.708 0.227079 0.113539 0.993534i $$-0.463781\pi$$
0.113539 + 0.993534i $$0.463781\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1458.94 0.624842
$$177$$ 0 0
$$178$$ 735.146 0.309559
$$179$$ −3847.77 −1.60668 −0.803341 0.595519i $$-0.796946\pi$$
−0.803341 + 0.595519i $$0.796946\pi$$
$$180$$ 0 0
$$181$$ −3026.41 −1.24282 −0.621411 0.783484i $$-0.713440\pi$$
−0.621411 + 0.783484i $$0.713440\pi$$
$$182$$ −61.1109 −0.0248892
$$183$$ 0 0
$$184$$ 682.802 0.273570
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2455.92 −0.960398
$$188$$ −1978.16 −0.767405
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1271.09 −0.481532 −0.240766 0.970583i $$-0.577399\pi$$
−0.240766 + 0.970583i $$0.577399\pi$$
$$192$$ 0 0
$$193$$ 1896.09 0.707170 0.353585 0.935402i $$-0.384963\pi$$
0.353585 + 0.935402i $$0.384963\pi$$
$$194$$ −766.707 −0.283744
$$195$$ 0 0
$$196$$ −376.548 −0.137226
$$197$$ 4067.89 1.47119 0.735597 0.677419i $$-0.236902\pi$$
0.735597 + 0.677419i $$0.236902\pi$$
$$198$$ 0 0
$$199$$ 78.1366 0.0278340 0.0139170 0.999903i $$-0.495570\pi$$
0.0139170 + 0.999903i $$0.495570\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 649.152 0.226110
$$203$$ 1429.19 0.494136
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −43.9176 −0.0148538
$$207$$ 0 0
$$208$$ 878.857 0.292970
$$209$$ −3702.05 −1.22524
$$210$$ 0 0
$$211$$ −1293.02 −0.421872 −0.210936 0.977500i $$-0.567651\pi$$
−0.210936 + 0.977500i $$0.567651\pi$$
$$212$$ −1982.47 −0.642250
$$213$$ 0 0
$$214$$ 52.8317 0.0168762
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −284.899 −0.0891253
$$218$$ 1195.12 0.371303
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1479.43 −0.450303
$$222$$ 0 0
$$223$$ 1786.65 0.536515 0.268257 0.963347i $$-0.413552\pi$$
0.268257 + 0.963347i $$0.413552\pi$$
$$224$$ −715.452 −0.213407
$$225$$ 0 0
$$226$$ 334.962 0.0985901
$$227$$ −4242.02 −1.24032 −0.620161 0.784475i $$-0.712933\pi$$
−0.620161 + 0.784475i $$0.712933\pi$$
$$228$$ 0 0
$$229$$ 3403.36 0.982096 0.491048 0.871132i $$-0.336614\pi$$
0.491048 + 0.871132i $$0.336614\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1798.29 0.508893
$$233$$ −3904.09 −1.09771 −0.548853 0.835919i $$-0.684935\pi$$
−0.548853 + 0.835919i $$0.684935\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −5005.51 −1.38064
$$237$$ 0 0
$$238$$ 374.069 0.101879
$$239$$ −6667.83 −1.80463 −0.902314 0.431079i $$-0.858133\pi$$
−0.902314 + 0.431079i $$0.858133\pi$$
$$240$$ 0 0
$$241$$ −6003.99 −1.60478 −0.802388 0.596803i $$-0.796437\pi$$
−0.802388 + 0.596803i $$0.796437\pi$$
$$242$$ 373.410 0.0991888
$$243$$ 0 0
$$244$$ −3468.12 −0.909932
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2230.08 −0.574481
$$248$$ −358.474 −0.0917869
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −728.467 −0.183189 −0.0915945 0.995796i $$-0.529196\pi$$
−0.0915945 + 0.995796i $$0.529196\pi$$
$$252$$ 0 0
$$253$$ 2000.69 0.497163
$$254$$ 1172.64 0.289677
$$255$$ 0 0
$$256$$ 2575.17 0.628703
$$257$$ −1479.86 −0.359187 −0.179593 0.983741i $$-0.557478\pi$$
−0.179593 + 0.983741i $$0.557478\pi$$
$$258$$ 0 0
$$259$$ −669.687 −0.160665
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 233.686 0.0551036
$$263$$ −6367.31 −1.49287 −0.746435 0.665458i $$-0.768236\pi$$
−0.746435 + 0.665458i $$0.768236\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 563.871 0.129974
$$267$$ 0 0
$$268$$ 6394.18 1.45741
$$269$$ −1787.18 −0.405079 −0.202539 0.979274i $$-0.564919\pi$$
−0.202539 + 0.979274i $$0.564919\pi$$
$$270$$ 0 0
$$271$$ 3907.65 0.875915 0.437957 0.898996i $$-0.355702\pi$$
0.437957 + 0.898996i $$0.355702\pi$$
$$272$$ −5379.62 −1.19922
$$273$$ 0 0
$$274$$ −347.743 −0.0766712
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3978.33 −0.862942 −0.431471 0.902127i $$-0.642005\pi$$
−0.431471 + 0.902127i $$0.642005\pi$$
$$278$$ 149.931 0.0323464
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6488.04 1.37738 0.688690 0.725055i $$-0.258186\pi$$
0.688690 + 0.725055i $$0.258186\pi$$
$$282$$ 0 0
$$283$$ 164.336 0.0345185 0.0172592 0.999851i $$-0.494506\pi$$
0.0172592 + 0.999851i $$0.494506\pi$$
$$284$$ −1344.55 −0.280930
$$285$$ 0 0
$$286$$ −225.305 −0.0465824
$$287$$ −173.111 −0.0356042
$$288$$ 0 0
$$289$$ 4142.79 0.843231
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −364.381 −0.0730266
$$293$$ 5004.57 0.997851 0.498925 0.866645i $$-0.333728\pi$$
0.498925 + 0.866645i $$0.333728\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −842.634 −0.165463
$$297$$ 0 0
$$298$$ 1432.97 0.278556
$$299$$ 1205.20 0.233105
$$300$$ 0 0
$$301$$ −1977.01 −0.378581
$$302$$ −627.470 −0.119559
$$303$$ 0 0
$$304$$ −8109.23 −1.52992
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 190.167 0.0353531 0.0176765 0.999844i $$-0.494373\pi$$
0.0176765 + 0.999844i $$0.494373\pi$$
$$308$$ −1388.27 −0.256831
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1182.57 0.215619 0.107810 0.994172i $$-0.465616\pi$$
0.107810 + 0.994172i $$0.465616\pi$$
$$312$$ 0 0
$$313$$ 4659.05 0.841358 0.420679 0.907209i $$-0.361792\pi$$
0.420679 + 0.907209i $$0.361792\pi$$
$$314$$ −256.127 −0.0460320
$$315$$ 0 0
$$316$$ 8.91467 0.00158699
$$317$$ −3694.55 −0.654595 −0.327297 0.944921i $$-0.606138\pi$$
−0.327297 + 0.944921i $$0.606138\pi$$
$$318$$ 0 0
$$319$$ 5269.18 0.924820
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −304.732 −0.0527392
$$323$$ 13650.7 2.35153
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −59.2379 −0.0100641
$$327$$ 0 0
$$328$$ −217.817 −0.0366675
$$329$$ 1801.92 0.301954
$$330$$ 0 0
$$331$$ −8632.79 −1.43354 −0.716769 0.697310i $$-0.754380\pi$$
−0.716769 + 0.697310i $$0.754380\pi$$
$$332$$ −11481.1 −1.89791
$$333$$ 0 0
$$334$$ 1394.08 0.228385
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8136.61 1.31522 0.657610 0.753358i $$-0.271567\pi$$
0.657610 + 0.753358i $$0.271567\pi$$
$$338$$ 1098.01 0.176698
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1050.37 −0.166806
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ −2487.58 −0.389887
$$345$$ 0 0
$$346$$ −290.159 −0.0450839
$$347$$ 8646.33 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$348$$ 0 0
$$349$$ −7455.43 −1.14350 −0.571748 0.820429i $$-0.693734\pi$$
−0.571748 + 0.820429i $$0.693734\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2637.74 −0.399410
$$353$$ 5450.21 0.821771 0.410886 0.911687i $$-0.365220\pi$$
0.410886 + 0.911687i $$0.365220\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10060.2 1.49773
$$357$$ 0 0
$$358$$ 2160.73 0.318989
$$359$$ −4775.79 −0.702107 −0.351053 0.936355i $$-0.614176\pi$$
−0.351053 + 0.936355i $$0.614176\pi$$
$$360$$ 0 0
$$361$$ 13718.0 2.00000
$$362$$ 1699.49 0.246749
$$363$$ 0 0
$$364$$ −836.281 −0.120420
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9636.30 −1.37060 −0.685301 0.728260i $$-0.740329\pi$$
−0.685301 + 0.728260i $$0.740329\pi$$
$$368$$ 4382.46 0.620792
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1805.85 0.252709
$$372$$ 0 0
$$373$$ −12180.4 −1.69082 −0.845412 0.534115i $$-0.820645\pi$$
−0.845412 + 0.534115i $$0.820645\pi$$
$$374$$ 1379.13 0.190676
$$375$$ 0 0
$$376$$ 2267.27 0.310971
$$377$$ 3174.11 0.433621
$$378$$ 0 0
$$379$$ 1689.39 0.228966 0.114483 0.993425i $$-0.463479\pi$$
0.114483 + 0.993425i $$0.463479\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 713.783 0.0956029
$$383$$ 1513.13 0.201872 0.100936 0.994893i $$-0.467816\pi$$
0.100936 + 0.994893i $$0.467816\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1064.76 −0.140401
$$387$$ 0 0
$$388$$ −10492.1 −1.37283
$$389$$ −12165.5 −1.58564 −0.792822 0.609453i $$-0.791389\pi$$
−0.792822 + 0.609453i $$0.791389\pi$$
$$390$$ 0 0
$$391$$ −7377.21 −0.954173
$$392$$ 431.580 0.0556074
$$393$$ 0 0
$$394$$ −2284.34 −0.292089
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −7353.40 −0.929613 −0.464807 0.885412i $$-0.653876\pi$$
−0.464807 + 0.885412i $$0.653876\pi$$
$$398$$ −43.8778 −0.00552612
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7481.62 −0.931707 −0.465853 0.884862i $$-0.654253\pi$$
−0.465853 + 0.884862i $$0.654253\pi$$
$$402$$ 0 0
$$403$$ −632.735 −0.0782104
$$404$$ 8883.42 1.09398
$$405$$ 0 0
$$406$$ −802.567 −0.0981053
$$407$$ −2469.02 −0.300699
$$408$$ 0 0
$$409$$ 9248.94 1.11817 0.559084 0.829111i $$-0.311153\pi$$
0.559084 + 0.829111i $$0.311153\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −600.997 −0.0718665
$$413$$ 4559.55 0.543247
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1588.96 −0.187272
$$417$$ 0 0
$$418$$ 2078.89 0.243258
$$419$$ 3363.39 0.392154 0.196077 0.980589i $$-0.437180\pi$$
0.196077 + 0.980589i $$0.437180\pi$$
$$420$$ 0 0
$$421$$ −3638.86 −0.421252 −0.210626 0.977567i $$-0.567550\pi$$
−0.210626 + 0.977567i $$0.567550\pi$$
$$422$$ 726.097 0.0837579
$$423$$ 0 0
$$424$$ 2272.21 0.260255
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3159.13 0.358035
$$428$$ 722.983 0.0816512
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −11243.2 −1.25653 −0.628264 0.778000i $$-0.716234\pi$$
−0.628264 + 0.778000i $$0.716234\pi$$
$$432$$ 0 0
$$433$$ −187.332 −0.0207912 −0.0103956 0.999946i $$-0.503309\pi$$
−0.0103956 + 0.999946i $$0.503309\pi$$
$$434$$ 159.986 0.0176948
$$435$$ 0 0
$$436$$ 16354.9 1.79646
$$437$$ −11120.4 −1.21730
$$438$$ 0 0
$$439$$ −11479.4 −1.24802 −0.624011 0.781415i $$-0.714498\pi$$
−0.624011 + 0.781415i $$0.714498\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 830.775 0.0894026
$$443$$ −9490.56 −1.01786 −0.508928 0.860809i $$-0.669958\pi$$
−0.508928 + 0.860809i $$0.669958\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −1003.30 −0.106519
$$447$$ 0 0
$$448$$ −2763.99 −0.291487
$$449$$ 5546.58 0.582982 0.291491 0.956574i $$-0.405849\pi$$
0.291491 + 0.956574i $$0.405849\pi$$
$$450$$ 0 0
$$451$$ −638.229 −0.0666364
$$452$$ 4583.85 0.477004
$$453$$ 0 0
$$454$$ 2382.12 0.246252
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1776.94 −0.181885 −0.0909426 0.995856i $$-0.528988\pi$$
−0.0909426 + 0.995856i $$0.528988\pi$$
$$458$$ −1911.16 −0.194984
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9059.81 0.915309 0.457655 0.889130i $$-0.348690\pi$$
0.457655 + 0.889130i $$0.348690\pi$$
$$462$$ 0 0
$$463$$ −14878.5 −1.49344 −0.746719 0.665139i $$-0.768372\pi$$
−0.746719 + 0.665139i $$0.768372\pi$$
$$464$$ 11542.0 1.15479
$$465$$ 0 0
$$466$$ 2192.35 0.217937
$$467$$ 14275.8 1.41458 0.707288 0.706926i $$-0.249918\pi$$
0.707288 + 0.706926i $$0.249918\pi$$
$$468$$ 0 0
$$469$$ −5824.50 −0.573455
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 5737.07 0.559470
$$473$$ −7288.89 −0.708548
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 5119.01 0.492919
$$477$$ 0 0
$$478$$ 3744.34 0.358289
$$479$$ 15377.8 1.46687 0.733433 0.679761i $$-0.237917\pi$$
0.733433 + 0.679761i $$0.237917\pi$$
$$480$$ 0 0
$$481$$ −1487.32 −0.140989
$$482$$ 3371.56 0.318610
$$483$$ 0 0
$$484$$ 5109.99 0.479901
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −19683.3 −1.83149 −0.915747 0.401756i $$-0.868400\pi$$
−0.915747 + 0.401756i $$0.868400\pi$$
$$488$$ 3974.98 0.368727
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −19585.1 −1.80013 −0.900065 0.435755i $$-0.856481\pi$$
−0.900065 + 0.435755i $$0.856481\pi$$
$$492$$ 0 0
$$493$$ −19429.3 −1.77495
$$494$$ 1252.31 0.114057
$$495$$ 0 0
$$496$$ −2300.81 −0.208285
$$497$$ 1224.75 0.110539
$$498$$ 0 0
$$499$$ −6888.47 −0.617977 −0.308988 0.951066i $$-0.599990\pi$$
−0.308988 + 0.951066i $$0.599990\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 409.073 0.0363701
$$503$$ 14878.0 1.31885 0.659423 0.751772i $$-0.270801\pi$$
0.659423 + 0.751772i $$0.270801\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −1123.49 −0.0987062
$$507$$ 0 0
$$508$$ 16047.2 1.40153
$$509$$ −1935.28 −0.168526 −0.0842629 0.996444i $$-0.526854\pi$$
−0.0842629 + 0.996444i $$0.526854\pi$$
$$510$$ 0 0
$$511$$ 331.917 0.0287341
$$512$$ −9761.22 −0.842557
$$513$$ 0 0
$$514$$ 831.019 0.0713126
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6643.35 0.565134
$$518$$ 376.064 0.0318983
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6892.28 0.579570 0.289785 0.957092i $$-0.406416\pi$$
0.289785 + 0.957092i $$0.406416\pi$$
$$522$$ 0 0
$$523$$ 1074.60 0.0898447 0.0449223 0.998990i $$-0.485696\pi$$
0.0449223 + 0.998990i $$0.485696\pi$$
$$524$$ 3197.91 0.266606
$$525$$ 0 0
$$526$$ 3575.58 0.296393
$$527$$ 3873.07 0.320140
$$528$$ 0 0
$$529$$ −6157.23 −0.506060
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 7716.39 0.628849
$$533$$ −384.464 −0.0312439
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −7328.69 −0.590580
$$537$$ 0 0
$$538$$ 1003.60 0.0804239
$$539$$ 1264.58 0.101056
$$540$$ 0 0
$$541$$ 8660.23 0.688230 0.344115 0.938928i $$-0.388179\pi$$
0.344115 + 0.938928i $$0.388179\pi$$
$$542$$ −2194.35 −0.173903
$$543$$ 0 0
$$544$$ 9726.25 0.766562
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 15346.1 1.19954 0.599771 0.800171i $$-0.295258\pi$$
0.599771 + 0.800171i $$0.295258\pi$$
$$548$$ −4758.74 −0.370955
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −29287.6 −2.26442
$$552$$ 0 0
$$553$$ −8.12042 −0.000624440 0
$$554$$ 2234.04 0.171328
$$555$$ 0 0
$$556$$ 2051.76 0.156500
$$557$$ 15544.9 1.18251 0.591257 0.806483i $$-0.298632\pi$$
0.591257 + 0.806483i $$0.298632\pi$$
$$558$$ 0 0
$$559$$ −4390.77 −0.332218
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −3643.38 −0.273464
$$563$$ −18511.9 −1.38576 −0.692881 0.721052i $$-0.743659\pi$$
−0.692881 + 0.721052i $$0.743659\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −92.2831 −0.00685326
$$567$$ 0 0
$$568$$ 1541.05 0.113840
$$569$$ −2157.36 −0.158948 −0.0794738 0.996837i $$-0.525324\pi$$
−0.0794738 + 0.996837i $$0.525324\pi$$
$$570$$ 0 0
$$571$$ 16010.3 1.17340 0.586700 0.809805i $$-0.300427\pi$$
0.586700 + 0.809805i $$0.300427\pi$$
$$572$$ −3083.22 −0.225378
$$573$$ 0 0
$$574$$ 97.2109 0.00706882
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2164.01 0.156134 0.0780668 0.996948i $$-0.475125\pi$$
0.0780668 + 0.996948i $$0.475125\pi$$
$$578$$ −2326.40 −0.167414
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 10458.2 0.746780
$$582$$ 0 0
$$583$$ 6657.84 0.472967
$$584$$ 417.635 0.0295922
$$585$$ 0 0
$$586$$ −2810.33 −0.198112
$$587$$ 27616.6 1.94184 0.970918 0.239412i $$-0.0769546\pi$$
0.970918 + 0.239412i $$0.0769546\pi$$
$$588$$ 0 0
$$589$$ 5838.26 0.408424
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −5408.32 −0.375474
$$593$$ −10205.4 −0.706718 −0.353359 0.935488i $$-0.614961\pi$$
−0.353359 + 0.935488i $$0.614961\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 19609.7 1.34773
$$597$$ 0 0
$$598$$ −676.783 −0.0462805
$$599$$ −11090.5 −0.756501 −0.378251 0.925703i $$-0.623474\pi$$
−0.378251 + 0.925703i $$0.623474\pi$$
$$600$$ 0 0
$$601$$ −26537.0 −1.80111 −0.900555 0.434742i $$-0.856839\pi$$
−0.900555 + 0.434742i $$0.856839\pi$$
$$602$$ 1110.20 0.0751631
$$603$$ 0 0
$$604$$ −8586.71 −0.578457
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 11234.5 0.751226 0.375613 0.926777i $$-0.377432\pi$$
0.375613 + 0.926777i $$0.377432\pi$$
$$608$$ 14661.3 0.977954
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4001.90 0.264975
$$612$$ 0 0
$$613$$ 10622.5 0.699900 0.349950 0.936768i $$-0.386199\pi$$
0.349950 + 0.936768i $$0.386199\pi$$
$$614$$ −106.789 −0.00701896
$$615$$ 0 0
$$616$$ 1591.16 0.104074
$$617$$ 20433.3 1.33325 0.666623 0.745395i $$-0.267739\pi$$
0.666623 + 0.745395i $$0.267739\pi$$
$$618$$ 0 0
$$619$$ −7963.57 −0.517097 −0.258549 0.965998i $$-0.583244\pi$$
−0.258549 + 0.965998i $$0.583244\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −664.077 −0.0428088
$$623$$ −9163.91 −0.589317
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2616.30 −0.167042
$$627$$ 0 0
$$628$$ −3505.01 −0.222715
$$629$$ 9104.09 0.577113
$$630$$ 0 0
$$631$$ −14703.1 −0.927608 −0.463804 0.885938i $$-0.653516\pi$$
−0.463804 + 0.885938i $$0.653516\pi$$
$$632$$ −10.2175 −0.000643088 0
$$633$$ 0 0
$$634$$ 2074.68 0.129963
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 761.773 0.0473824
$$638$$ −2958.92 −0.183613
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −3353.41 −0.206633 −0.103317 0.994649i $$-0.532945\pi$$
−0.103317 + 0.994649i $$0.532945\pi$$
$$642$$ 0 0
$$643$$ −31862.0 −1.95415 −0.977073 0.212906i $$-0.931707\pi$$
−0.977073 + 0.212906i $$0.931707\pi$$
$$644$$ −4170.15 −0.255166
$$645$$ 0 0
$$646$$ −7665.58 −0.466870
$$647$$ −8518.46 −0.517612 −0.258806 0.965929i $$-0.583329\pi$$
−0.258806 + 0.965929i $$0.583329\pi$$
$$648$$ 0 0
$$649$$ 16810.3 1.01673
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −810.651 −0.0486925
$$653$$ −28509.3 −1.70851 −0.854254 0.519856i $$-0.825986\pi$$
−0.854254 + 0.519856i $$0.825986\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −1398.02 −0.0832068
$$657$$ 0 0
$$658$$ −1011.87 −0.0599496
$$659$$ 27632.4 1.63339 0.816697 0.577066i $$-0.195803\pi$$
0.816697 + 0.577066i $$0.195803\pi$$
$$660$$ 0 0
$$661$$ −27052.8 −1.59188 −0.795941 0.605374i $$-0.793023\pi$$
−0.795941 + 0.605374i $$0.793023\pi$$
$$662$$ 4847.77 0.284613
$$663$$ 0 0
$$664$$ 13159.1 0.769082
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 15827.9 0.918826
$$668$$ 19077.5 1.10498
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 11647.1 0.670094
$$672$$ 0 0
$$673$$ 1569.99 0.0899235 0.0449618 0.998989i $$-0.485683\pi$$
0.0449618 + 0.998989i $$0.485683\pi$$
$$674$$ −4569.13 −0.261122
$$675$$ 0 0
$$676$$ 15025.9 0.854909
$$677$$ 13853.9 0.786482 0.393241 0.919435i $$-0.371354\pi$$
0.393241 + 0.919435i $$0.371354\pi$$
$$678$$ 0 0
$$679$$ 9557.33 0.540172
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 589.839 0.0331174
$$683$$ −28337.1 −1.58754 −0.793769 0.608219i $$-0.791884\pi$$
−0.793769 + 0.608219i $$0.791884\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −192.613 −0.0107201
$$687$$ 0 0
$$688$$ −15966.1 −0.884742
$$689$$ 4010.63 0.221760
$$690$$ 0 0
$$691$$ 16936.5 0.932410 0.466205 0.884677i $$-0.345621\pi$$
0.466205 + 0.884677i $$0.345621\pi$$
$$692$$ −3970.73 −0.218128
$$693$$ 0 0
$$694$$ −4855.37 −0.265572
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2353.37 0.127891
$$698$$ 4186.62 0.227028
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 29744.0 1.60259 0.801294 0.598271i $$-0.204145\pi$$
0.801294 + 0.598271i $$0.204145\pi$$
$$702$$ 0 0
$$703$$ 13723.5 0.736261
$$704$$ −10190.3 −0.545543
$$705$$ 0 0
$$706$$ −3060.58 −0.163154
$$707$$ −8091.96 −0.430452
$$708$$ 0 0
$$709$$ 34264.7 1.81500 0.907502 0.420047i $$-0.137986\pi$$
0.907502 + 0.420047i $$0.137986\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −11530.5 −0.606916
$$713$$ −3155.16 −0.165725
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 29568.8 1.54335
$$717$$ 0 0
$$718$$ 2681.86 0.139396
$$719$$ 25469.2 1.32106 0.660529 0.750801i $$-0.270332\pi$$
0.660529 + 0.750801i $$0.270332\pi$$
$$720$$ 0 0
$$721$$ 547.452 0.0282776
$$722$$ −7703.40 −0.397079
$$723$$ 0 0
$$724$$ 23256.9 1.19383
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 24294.6 1.23939 0.619695 0.784843i $$-0.287256\pi$$
0.619695 + 0.784843i $$0.287256\pi$$
$$728$$ 958.503 0.0487974
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 26876.6 1.35987
$$732$$ 0 0
$$733$$ 34870.8 1.75714 0.878569 0.477615i $$-0.158499\pi$$
0.878569 + 0.477615i $$0.158499\pi$$
$$734$$ 5411.29 0.272118
$$735$$ 0 0
$$736$$ −7923.40 −0.396821
$$737$$ −21473.9 −1.07327
$$738$$ 0 0
$$739$$ 7028.81 0.349877 0.174938 0.984579i $$-0.444027\pi$$
0.174938 + 0.984579i $$0.444027\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −1014.08 −0.0501725
$$743$$ −1368.74 −0.0675831 −0.0337915 0.999429i $$-0.510758\pi$$
−0.0337915 + 0.999429i $$0.510758\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6839.94 0.335694
$$747$$ 0 0
$$748$$ 18872.9 0.922541
$$749$$ −658.570 −0.0321276
$$750$$ 0 0
$$751$$ 37144.0 1.80480 0.902400 0.430900i $$-0.141804\pi$$
0.902400 + 0.430900i $$0.141804\pi$$
$$752$$ 14552.1 0.705665
$$753$$ 0 0
$$754$$ −1782.43 −0.0860907
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12042.4 0.578189 0.289095 0.957301i $$-0.406646\pi$$
0.289095 + 0.957301i $$0.406646\pi$$
$$758$$ −948.683 −0.0454588
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6112.92 −0.291187 −0.145593 0.989345i $$-0.546509\pi$$
−0.145593 + 0.989345i $$0.546509\pi$$
$$762$$ 0 0
$$763$$ −14897.7 −0.706860
$$764$$ 9767.88 0.462552
$$765$$ 0 0
$$766$$ −849.700 −0.0400795
$$767$$ 10126.4 0.476717
$$768$$ 0 0
$$769$$ −957.146 −0.0448837 −0.0224418 0.999748i $$-0.507144\pi$$
−0.0224418 + 0.999748i $$0.507144\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14570.8 −0.679295
$$773$$ −32867.9 −1.52934 −0.764668 0.644424i $$-0.777097\pi$$
−0.764668 + 0.644424i $$0.777097\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12025.5 0.556303
$$777$$ 0 0
$$778$$ 6831.58 0.314812
$$779$$ 3547.46 0.163159
$$780$$ 0 0
$$781$$ 4515.45 0.206883
$$782$$ 4142.69 0.189440
$$783$$ 0 0
$$784$$ 2770.03 0.126186
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 15320.0 0.693900 0.346950 0.937884i $$-0.387217\pi$$
0.346950 + 0.937884i $$0.387217\pi$$
$$788$$ −31260.4 −1.41320
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4175.45 −0.187689
$$792$$ 0 0
$$793$$ 7016.15 0.314188
$$794$$ 4129.32 0.184564
$$795$$ 0 0
$$796$$ −600.453 −0.0267368
$$797$$ −43629.0 −1.93904 −0.969522 0.245004i $$-0.921211\pi$$
−0.969522 + 0.245004i $$0.921211\pi$$
$$798$$ 0 0
$$799$$ −24496.3 −1.08463
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 4201.33 0.184980
$$803$$ 1223.72 0.0537784
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 355.314 0.0155278
$$807$$ 0 0
$$808$$ −10181.7 −0.443307
$$809$$ −127.735 −0.00555119 −0.00277560 0.999996i $$-0.500884\pi$$
−0.00277560 + 0.999996i $$0.500884\pi$$
$$810$$ 0 0
$$811$$ 16227.5 0.702618 0.351309 0.936260i $$-0.385737\pi$$
0.351309 + 0.936260i $$0.385737\pi$$
$$812$$ −10982.9 −0.474659
$$813$$ 0 0
$$814$$ 1386.48 0.0597005
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 40513.7 1.73488
$$818$$ −5193.77 −0.222000
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −34249.2 −1.45591 −0.727957 0.685623i $$-0.759530\pi$$
−0.727957 + 0.685623i $$0.759530\pi$$
$$822$$ 0 0
$$823$$ −6624.51 −0.280578 −0.140289 0.990111i $$-0.544803\pi$$
−0.140289 + 0.990111i $$0.544803\pi$$
$$824$$ 688.832 0.0291221
$$825$$ 0 0
$$826$$ −2560.43 −0.107856
$$827$$ 33786.8 1.42065 0.710327 0.703872i $$-0.248547\pi$$
0.710327 + 0.703872i $$0.248547\pi$$
$$828$$ 0 0
$$829$$ −30283.8 −1.26876 −0.634378 0.773023i $$-0.718744\pi$$
−0.634378 + 0.773023i $$0.718744\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −6138.57 −0.255789
$$833$$ −4662.93 −0.193951
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 28449.0 1.17695
$$837$$ 0 0
$$838$$ −1888.72 −0.0778578
$$839$$ −16810.9 −0.691750 −0.345875 0.938281i $$-0.612418\pi$$
−0.345875 + 0.938281i $$0.612418\pi$$
$$840$$ 0 0
$$841$$ 17296.6 0.709195
$$842$$ 2043.41 0.0836350
$$843$$ 0 0
$$844$$ 9936.39 0.405242
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4654.72 −0.188829
$$848$$ 14583.8 0.590579
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −7416.56 −0.298750
$$852$$ 0 0
$$853$$ 14875.8 0.597112 0.298556 0.954392i $$-0.403495\pi$$
0.298556 + 0.954392i $$0.403495\pi$$
$$854$$ −1774.02 −0.0710839
$$855$$ 0 0
$$856$$ −828.647 −0.0330871
$$857$$ 3987.55 0.158941 0.0794703 0.996837i $$-0.474677\pi$$
0.0794703 + 0.996837i $$0.474677\pi$$
$$858$$ 0 0
$$859$$ 39344.7 1.56278 0.781388 0.624046i $$-0.214512\pi$$
0.781388 + 0.624046i $$0.214512\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 6313.63 0.249470
$$863$$ 15627.0 0.616397 0.308198 0.951322i $$-0.400274\pi$$
0.308198 + 0.951322i $$0.400274\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 105.197 0.00412787
$$867$$ 0 0
$$868$$ 2189.35 0.0856122
$$869$$ −29.9386 −0.00116870
$$870$$ 0 0
$$871$$ −12935.7 −0.503226
$$872$$ −18745.1 −0.727969
$$873$$ 0 0
$$874$$ 6244.69 0.241682
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14519.0 −0.559034 −0.279517 0.960141i $$-0.590174\pi$$
−0.279517 + 0.960141i $$0.590174\pi$$
$$878$$ 6446.29 0.247781
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −24177.7 −0.924592 −0.462296 0.886726i $$-0.652974\pi$$
−0.462296 + 0.886726i $$0.652974\pi$$
$$882$$ 0 0
$$883$$ 10340.3 0.394089 0.197044 0.980395i $$-0.436866\pi$$
0.197044 + 0.980395i $$0.436866\pi$$
$$884$$ 11368.9 0.432553
$$885$$ 0 0
$$886$$ 5329.45 0.202084
$$887$$ 3222.62 0.121990 0.0609949 0.998138i $$-0.480573\pi$$
0.0609949 + 0.998138i $$0.480573\pi$$
$$888$$ 0 0
$$889$$ −14617.5 −0.551467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −13729.8 −0.515367
$$893$$ −36925.6 −1.38373
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 7275.74 0.271278
$$897$$ 0 0
$$898$$ −3114.70 −0.115745
$$899$$ −8309.70 −0.308280
$$900$$ 0 0
$$901$$ −24549.7 −0.907735
$$902$$ 358.399 0.0132299
$$903$$ 0 0
$$904$$ −5253.77 −0.193294
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −31692.8 −1.16025 −0.580123 0.814529i $$-0.696995\pi$$
−0.580123 + 0.814529i $$0.696995\pi$$
$$908$$ 32598.5 1.19143
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2403.15 −0.0873985 −0.0436993 0.999045i $$-0.513914\pi$$
−0.0436993 + 0.999045i $$0.513914\pi$$
$$912$$ 0 0
$$913$$ 38557.6 1.39767
$$914$$ 997.843 0.0361113
$$915$$ 0 0
$$916$$ −26153.6 −0.943385
$$917$$ −2912.99 −0.104902
$$918$$ 0 0
$$919$$ 19622.5 0.704339 0.352170 0.935936i $$-0.385444\pi$$
0.352170 + 0.935936i $$0.385444\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −5087.56 −0.181724
$$923$$ 2720.07 0.0970014
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8355.06 0.296506
$$927$$ 0 0
$$928$$ −20867.7 −0.738165
$$929$$ 12930.5 0.456660 0.228330 0.973584i $$-0.426673\pi$$
0.228330 + 0.973584i $$0.426673\pi$$
$$930$$ 0 0
$$931$$ −7028.90 −0.247436
$$932$$ 30001.6 1.05444
$$933$$ 0 0
$$934$$ −8016.64 −0.280848
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −18717.1 −0.652573 −0.326287 0.945271i $$-0.605797\pi$$
−0.326287 + 0.945271i $$0.605797\pi$$
$$938$$ 3270.76 0.113853
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 10152.9 0.351727 0.175863 0.984415i $$-0.443728\pi$$
0.175863 + 0.984415i $$0.443728\pi$$
$$942$$ 0 0
$$943$$ −1917.15 −0.0662045
$$944$$ 36822.4 1.26956
$$945$$ 0 0
$$946$$ 4093.09 0.140674
$$947$$ 15836.2 0.543408 0.271704 0.962381i $$-0.412413\pi$$
0.271704 + 0.962381i $$0.412413\pi$$
$$948$$ 0 0
$$949$$ 737.158 0.0252151
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −5867.15 −0.199743
$$953$$ 4847.86 0.164782 0.0823911 0.996600i $$-0.473744\pi$$
0.0823911 + 0.996600i $$0.473744\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 51240.0 1.73349
$$957$$ 0 0
$$958$$ −8635.44 −0.291230
$$959$$ 4334.76 0.145961
$$960$$ 0 0
$$961$$ −28134.5 −0.944397
$$962$$ 835.207 0.0279918
$$963$$ 0 0
$$964$$ 46138.6 1.54152
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −17153.8 −0.570454 −0.285227 0.958460i $$-0.592069\pi$$
−0.285227 + 0.958460i $$0.592069\pi$$
$$968$$ −5856.80 −0.194468
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −50352.2 −1.66414 −0.832070 0.554670i $$-0.812844\pi$$
−0.832070 + 0.554670i $$0.812844\pi$$
$$972$$ 0 0
$$973$$ −1868.96 −0.0615788
$$974$$ 11053.2 0.363623
$$975$$ 0 0
$$976$$ 25512.8 0.836725
$$977$$ 1510.03 0.0494474 0.0247237 0.999694i $$-0.492129\pi$$
0.0247237 + 0.999694i $$0.492129\pi$$
$$978$$ 0 0
$$979$$ −33785.7 −1.10296
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 10998.1 0.357396
$$983$$ −5310.89 −0.172321 −0.0861603 0.996281i $$-0.527460\pi$$
−0.0861603 + 0.996281i $$0.527460\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 10910.6 0.352396
$$987$$ 0 0
$$988$$ 17137.4 0.551836
$$989$$ −21894.7 −0.703956
$$990$$ 0 0
$$991$$ 35845.4 1.14901 0.574505 0.818501i $$-0.305195\pi$$
0.574505 + 0.818501i $$0.305195\pi$$
$$992$$ 4159.82 0.133140
$$993$$ 0 0
$$994$$ −687.764 −0.0219462
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 17857.5 0.567253 0.283627 0.958935i $$-0.408462\pi$$
0.283627 + 0.958935i $$0.408462\pi$$
$$998$$ 3868.24 0.122692
$$999$$ 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.x.1.1 2
3.2 odd 2 525.4.a.j.1.2 2
5.4 even 2 1575.4.a.o.1.2 2
15.2 even 4 525.4.d.m.274.3 4
15.8 even 4 525.4.d.m.274.2 4
15.14 odd 2 525.4.a.m.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.2 2 3.2 odd 2
525.4.a.m.1.1 yes 2 15.14 odd 2
525.4.d.m.274.2 4 15.8 even 4
525.4.d.m.274.3 4 15.2 even 4
1575.4.a.o.1.2 2 5.4 even 2
1575.4.a.x.1.1 2 1.1 even 1 trivial