# Properties

 Label 1575.4.a.u.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 $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +O(q^{10})$$ $$q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +10.2462 q^{11} -34.3542 q^{13} -10.9309 q^{14} +11.4233 q^{16} -82.6004 q^{17} +90.7083 q^{19} -16.0000 q^{22} -12.1383 q^{23} +53.6458 q^{26} -38.9309 q^{28} +105.153 q^{29} -142.108 q^{31} -187.255 q^{32} +128.985 q^{34} -64.8466 q^{37} -141.646 q^{38} +195.201 q^{41} +319.218 q^{43} -56.9848 q^{44} +18.9545 q^{46} -318.847 q^{47} +49.0000 q^{49} +191.062 q^{52} -296.799 q^{53} +148.240 q^{56} -164.203 q^{58} +284.000 q^{59} -494.540 q^{61} +221.909 q^{62} +201.022 q^{64} -549.032 q^{67} +459.386 q^{68} +740.972 q^{71} +556.850 q^{73} +101.261 q^{74} -504.479 q^{76} +71.7235 q^{77} -376.189 q^{79} -304.816 q^{82} +752.466 q^{83} -498.475 q^{86} +216.985 q^{88} +945.299 q^{89} -240.479 q^{91} +67.5076 q^{92} +497.896 q^{94} +180.668 q^{97} -76.5161 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.56155 −0.552092 −0.276046 0.961144i $$-0.589024\pi$$
−0.276046 + 0.961144i $$0.589024\pi$$
$$3$$ 0 0
$$4$$ −5.56155 −0.695194
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 21.1771 0.935904
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 10.2462 0.280850 0.140425 0.990091i $$-0.455153\pi$$
0.140425 + 0.990091i $$0.455153\pi$$
$$12$$ 0 0
$$13$$ −34.3542 −0.732933 −0.366467 0.930431i $$-0.619433\pi$$
−0.366467 + 0.930431i $$0.619433\pi$$
$$14$$ −10.9309 −0.208671
$$15$$ 0 0
$$16$$ 11.4233 0.178489
$$17$$ −82.6004 −1.17844 −0.589222 0.807972i $$-0.700565\pi$$
−0.589222 + 0.807972i $$0.700565\pi$$
$$18$$ 0 0
$$19$$ 90.7083 1.09526 0.547629 0.836721i $$-0.315530\pi$$
0.547629 + 0.836721i $$0.315530\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −16.0000 −0.155055
$$23$$ −12.1383 −0.110044 −0.0550218 0.998485i $$-0.517523\pi$$
−0.0550218 + 0.998485i $$0.517523\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 53.6458 0.404647
$$27$$ 0 0
$$28$$ −38.9309 −0.262759
$$29$$ 105.153 0.673328 0.336664 0.941625i $$-0.390701\pi$$
0.336664 + 0.941625i $$0.390701\pi$$
$$30$$ 0 0
$$31$$ −142.108 −0.823334 −0.411667 0.911334i $$-0.635053\pi$$
−0.411667 + 0.911334i $$0.635053\pi$$
$$32$$ −187.255 −1.03445
$$33$$ 0 0
$$34$$ 128.985 0.650609
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −64.8466 −0.288127 −0.144064 0.989568i $$-0.546017\pi$$
−0.144064 + 0.989568i $$0.546017\pi$$
$$38$$ −141.646 −0.604684
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 195.201 0.743542 0.371771 0.928324i $$-0.378751\pi$$
0.371771 + 0.928324i $$0.378751\pi$$
$$42$$ 0 0
$$43$$ 319.218 1.13210 0.566049 0.824371i $$-0.308471\pi$$
0.566049 + 0.824371i $$0.308471\pi$$
$$44$$ −56.9848 −0.195245
$$45$$ 0 0
$$46$$ 18.9545 0.0607542
$$47$$ −318.847 −0.989544 −0.494772 0.869023i $$-0.664748\pi$$
−0.494772 + 0.869023i $$0.664748\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 191.062 0.509531
$$53$$ −296.799 −0.769217 −0.384609 0.923080i $$-0.625664\pi$$
−0.384609 + 0.923080i $$0.625664\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 148.240 0.353738
$$57$$ 0 0
$$58$$ −164.203 −0.371739
$$59$$ 284.000 0.626672 0.313336 0.949642i $$-0.398553\pi$$
0.313336 + 0.949642i $$0.398553\pi$$
$$60$$ 0 0
$$61$$ −494.540 −1.03802 −0.519011 0.854768i $$-0.673700\pi$$
−0.519011 + 0.854768i $$0.673700\pi$$
$$62$$ 221.909 0.454556
$$63$$ 0 0
$$64$$ 201.022 0.392621
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −549.032 −1.00112 −0.500559 0.865702i $$-0.666872\pi$$
−0.500559 + 0.865702i $$0.666872\pi$$
$$68$$ 459.386 0.819247
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 740.972 1.23855 0.619276 0.785174i $$-0.287426\pi$$
0.619276 + 0.785174i $$0.287426\pi$$
$$72$$ 0 0
$$73$$ 556.850 0.892800 0.446400 0.894834i $$-0.352706\pi$$
0.446400 + 0.894834i $$0.352706\pi$$
$$74$$ 101.261 0.159073
$$75$$ 0 0
$$76$$ −504.479 −0.761417
$$77$$ 71.7235 0.106151
$$78$$ 0 0
$$79$$ −376.189 −0.535755 −0.267877 0.963453i $$-0.586322\pi$$
−0.267877 + 0.963453i $$0.586322\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −304.816 −0.410504
$$83$$ 752.466 0.995107 0.497553 0.867433i $$-0.334232\pi$$
0.497553 + 0.867433i $$0.334232\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −498.475 −0.625023
$$87$$ 0 0
$$88$$ 216.985 0.262848
$$89$$ 945.299 1.12586 0.562930 0.826505i $$-0.309674\pi$$
0.562930 + 0.826505i $$0.309674\pi$$
$$90$$ 0 0
$$91$$ −240.479 −0.277023
$$92$$ 67.5076 0.0765016
$$93$$ 0 0
$$94$$ 497.896 0.546319
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 180.668 0.189114 0.0945572 0.995519i $$-0.469856\pi$$
0.0945572 + 0.995519i $$0.469856\pi$$
$$98$$ −76.5161 −0.0788703
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1463.05 1.44138 0.720689 0.693258i $$-0.243826\pi$$
0.720689 + 0.693258i $$0.243826\pi$$
$$102$$ 0 0
$$103$$ −535.049 −0.511844 −0.255922 0.966697i $$-0.582379\pi$$
−0.255922 + 0.966697i $$0.582379\pi$$
$$104$$ −727.521 −0.685955
$$105$$ 0 0
$$106$$ 463.468 0.424679
$$107$$ 327.153 0.295581 0.147790 0.989019i $$-0.452784\pi$$
0.147790 + 0.989019i $$0.452784\pi$$
$$108$$ 0 0
$$109$$ 1621.70 1.42506 0.712528 0.701644i $$-0.247550\pi$$
0.712528 + 0.701644i $$0.247550\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 79.9630 0.0674625
$$113$$ 450.716 0.375219 0.187610 0.982244i $$-0.439926\pi$$
0.187610 + 0.982244i $$0.439926\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −584.816 −0.468093
$$117$$ 0 0
$$118$$ −443.481 −0.345981
$$119$$ −578.203 −0.445410
$$120$$ 0 0
$$121$$ −1226.02 −0.921123
$$122$$ 772.250 0.573084
$$123$$ 0 0
$$124$$ 790.341 0.572377
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1710.10 −1.19486 −0.597428 0.801923i $$-0.703811\pi$$
−0.597428 + 0.801923i $$0.703811\pi$$
$$128$$ 1184.13 0.817683
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1584.49 −1.05678 −0.528388 0.849003i $$-0.677203\pi$$
−0.528388 + 0.849003i $$0.677203\pi$$
$$132$$ 0 0
$$133$$ 634.958 0.413969
$$134$$ 857.343 0.552710
$$135$$ 0 0
$$136$$ −1749.23 −1.10291
$$137$$ −2095.24 −1.30663 −0.653316 0.757085i $$-0.726623\pi$$
−0.653316 + 0.757085i $$0.726623\pi$$
$$138$$ 0 0
$$139$$ −1385.00 −0.845135 −0.422568 0.906331i $$-0.638871\pi$$
−0.422568 + 0.906331i $$0.638871\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1157.07 −0.683795
$$143$$ −352.000 −0.205844
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −869.551 −0.492908
$$147$$ 0 0
$$148$$ 360.648 0.200304
$$149$$ −2502.89 −1.37614 −0.688069 0.725645i $$-0.741541\pi$$
−0.688069 + 0.725645i $$0.741541\pi$$
$$150$$ 0 0
$$151$$ 395.212 0.212993 0.106496 0.994313i $$-0.466037\pi$$
0.106496 + 0.994313i $$0.466037\pi$$
$$152$$ 1920.94 1.02506
$$153$$ 0 0
$$154$$ −112.000 −0.0586053
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1732.35 −0.880615 −0.440308 0.897847i $$-0.645131\pi$$
−0.440308 + 0.897847i $$0.645131\pi$$
$$158$$ 587.439 0.295786
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −84.9678 −0.0415926
$$162$$ 0 0
$$163$$ −3303.26 −1.58731 −0.793654 0.608369i $$-0.791824\pi$$
−0.793654 + 0.608369i $$0.791824\pi$$
$$164$$ −1085.62 −0.516906
$$165$$ 0 0
$$166$$ −1175.02 −0.549391
$$167$$ 601.941 0.278920 0.139460 0.990228i $$-0.455463\pi$$
0.139460 + 0.990228i $$0.455463\pi$$
$$168$$ 0 0
$$169$$ −1016.79 −0.462809
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1775.35 −0.787028
$$173$$ 3741.17 1.64414 0.822068 0.569389i $$-0.192820\pi$$
0.822068 + 0.569389i $$0.192820\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 117.045 0.0501286
$$177$$ 0 0
$$178$$ −1476.13 −0.621578
$$179$$ −968.352 −0.404347 −0.202173 0.979350i $$-0.564800\pi$$
−0.202173 + 0.979350i $$0.564800\pi$$
$$180$$ 0 0
$$181$$ −1895.65 −0.778465 −0.389233 0.921139i $$-0.627260\pi$$
−0.389233 + 0.921139i $$0.627260\pi$$
$$182$$ 375.521 0.152942
$$183$$ 0 0
$$184$$ −257.053 −0.102990
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −846.341 −0.330966
$$188$$ 1773.28 0.687925
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2729.88 1.03417 0.517086 0.855933i $$-0.327017\pi$$
0.517086 + 0.855933i $$0.327017\pi$$
$$192$$ 0 0
$$193$$ −1460.44 −0.544689 −0.272345 0.962200i $$-0.587799\pi$$
−0.272345 + 0.962200i $$0.587799\pi$$
$$194$$ −282.123 −0.104409
$$195$$ 0 0
$$196$$ −272.516 −0.0993134
$$197$$ −4831.42 −1.74733 −0.873665 0.486527i $$-0.838263\pi$$
−0.873665 + 0.486527i $$0.838263\pi$$
$$198$$ 0 0
$$199$$ −873.070 −0.311006 −0.155503 0.987835i $$-0.549700\pi$$
−0.155503 + 0.987835i $$0.549700\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −2284.63 −0.795774
$$203$$ 736.074 0.254494
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 835.508 0.282585
$$207$$ 0 0
$$208$$ −392.438 −0.130820
$$209$$ 929.417 0.307603
$$210$$ 0 0
$$211$$ −5148.02 −1.67964 −0.839821 0.542863i $$-0.817340\pi$$
−0.839821 + 0.542863i $$0.817340\pi$$
$$212$$ 1650.66 0.534755
$$213$$ 0 0
$$214$$ −510.867 −0.163188
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −994.756 −0.311191
$$218$$ −2532.38 −0.786762
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2837.67 0.863720
$$222$$ 0 0
$$223$$ 2435.31 0.731304 0.365652 0.930752i $$-0.380846\pi$$
0.365652 + 0.930752i $$0.380846\pi$$
$$224$$ −1310.78 −0.390984
$$225$$ 0 0
$$226$$ −703.817 −0.207156
$$227$$ −3508.47 −1.02584 −0.512919 0.858437i $$-0.671436\pi$$
−0.512919 + 0.858437i $$0.671436\pi$$
$$228$$ 0 0
$$229$$ 3429.54 0.989653 0.494827 0.868992i $$-0.335232\pi$$
0.494827 + 0.868992i $$0.335232\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2226.84 0.630170
$$233$$ −3989.71 −1.12178 −0.560890 0.827890i $$-0.689541\pi$$
−0.560890 + 0.827890i $$0.689541\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1579.48 −0.435659
$$237$$ 0 0
$$238$$ 902.894 0.245907
$$239$$ 5934.71 1.60621 0.803105 0.595837i $$-0.203180\pi$$
0.803105 + 0.595837i $$0.203180\pi$$
$$240$$ 0 0
$$241$$ −6481.90 −1.73251 −0.866257 0.499598i $$-0.833481\pi$$
−0.866257 + 0.499598i $$0.833481\pi$$
$$242$$ 1914.49 0.508545
$$243$$ 0 0
$$244$$ 2750.41 0.721627
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3116.21 −0.802751
$$248$$ −3009.43 −0.770561
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5257.70 −1.32216 −0.661081 0.750314i $$-0.729902\pi$$
−0.661081 + 0.750314i $$0.729902\pi$$
$$252$$ 0 0
$$253$$ −124.371 −0.0309057
$$254$$ 2670.41 0.659671
$$255$$ 0 0
$$256$$ −3457.26 −0.844057
$$257$$ 4108.52 0.997209 0.498604 0.866830i $$-0.333846\pi$$
0.498604 + 0.866830i $$0.333846\pi$$
$$258$$ 0 0
$$259$$ −453.926 −0.108902
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2474.27 0.583438
$$263$$ −6236.89 −1.46229 −0.731146 0.682221i $$-0.761014\pi$$
−0.731146 + 0.682221i $$0.761014\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −991.521 −0.228549
$$267$$ 0 0
$$268$$ 3053.47 0.695972
$$269$$ 4356.67 0.987475 0.493738 0.869611i $$-0.335630\pi$$
0.493738 + 0.869611i $$0.335630\pi$$
$$270$$ 0 0
$$271$$ −63.0970 −0.0141434 −0.00707171 0.999975i $$-0.502251\pi$$
−0.00707171 + 0.999975i $$0.502251\pi$$
$$272$$ −943.568 −0.210339
$$273$$ 0 0
$$274$$ 3271.83 0.721382
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5804.64 −1.25909 −0.629543 0.776966i $$-0.716758\pi$$
−0.629543 + 0.776966i $$0.716758\pi$$
$$278$$ 2162.74 0.466593
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8430.89 1.78984 0.894920 0.446227i $$-0.147233\pi$$
0.894920 + 0.446227i $$0.147233\pi$$
$$282$$ 0 0
$$283$$ −7415.73 −1.55767 −0.778833 0.627232i $$-0.784188\pi$$
−0.778833 + 0.627232i $$0.784188\pi$$
$$284$$ −4120.95 −0.861034
$$285$$ 0 0
$$286$$ 549.667 0.113645
$$287$$ 1366.41 0.281033
$$288$$ 0 0
$$289$$ 1909.82 0.388728
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3096.95 −0.620669
$$293$$ −482.022 −0.0961094 −0.0480547 0.998845i $$-0.515302\pi$$
−0.0480547 + 0.998845i $$0.515302\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1373.26 −0.269659
$$297$$ 0 0
$$298$$ 3908.39 0.759755
$$299$$ 417.000 0.0806546
$$300$$ 0 0
$$301$$ 2234.52 0.427893
$$302$$ −617.144 −0.117592
$$303$$ 0 0
$$304$$ 1036.19 0.195492
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2757.90 −0.512709 −0.256354 0.966583i $$-0.582521\pi$$
−0.256354 + 0.966583i $$0.582521\pi$$
$$308$$ −398.894 −0.0737957
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5817.25 −1.06066 −0.530331 0.847791i $$-0.677932\pi$$
−0.530331 + 0.847791i $$0.677932\pi$$
$$312$$ 0 0
$$313$$ −2232.83 −0.403218 −0.201609 0.979466i $$-0.564617\pi$$
−0.201609 + 0.979466i $$0.564617\pi$$
$$314$$ 2705.16 0.486181
$$315$$ 0 0
$$316$$ 2092.20 0.372453
$$317$$ 81.4013 0.0144226 0.00721128 0.999974i $$-0.497705\pi$$
0.00721128 + 0.999974i $$0.497705\pi$$
$$318$$ 0 0
$$319$$ 1077.42 0.189104
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 132.682 0.0229629
$$323$$ −7492.54 −1.29070
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 5158.21 0.876341
$$327$$ 0 0
$$328$$ 4133.78 0.695884
$$329$$ −2231.93 −0.374012
$$330$$ 0 0
$$331$$ −6550.18 −1.08771 −0.543853 0.839181i $$-0.683035\pi$$
−0.543853 + 0.839181i $$0.683035\pi$$
$$332$$ −4184.88 −0.691792
$$333$$ 0 0
$$334$$ −939.963 −0.153990
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4124.63 −0.666715 −0.333358 0.942800i $$-0.608182\pi$$
−0.333358 + 0.942800i $$0.608182\pi$$
$$338$$ 1587.77 0.255513
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1456.07 −0.231233
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 6760.10 1.05954
$$345$$ 0 0
$$346$$ −5842.03 −0.907715
$$347$$ 1228.44 0.190047 0.0950233 0.995475i $$-0.469707\pi$$
0.0950233 + 0.995475i $$0.469707\pi$$
$$348$$ 0 0
$$349$$ 3320.93 0.509357 0.254678 0.967026i $$-0.418030\pi$$
0.254678 + 0.967026i $$0.418030\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1918.65 −0.290524
$$353$$ −4589.25 −0.691958 −0.345979 0.938242i $$-0.612453\pi$$
−0.345979 + 0.938242i $$0.612453\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −5257.33 −0.782691
$$357$$ 0 0
$$358$$ 1512.13 0.223237
$$359$$ −12381.6 −1.82026 −0.910130 0.414323i $$-0.864018\pi$$
−0.910130 + 0.414323i $$0.864018\pi$$
$$360$$ 0 0
$$361$$ 1369.00 0.199592
$$362$$ 2960.15 0.429785
$$363$$ 0 0
$$364$$ 1337.44 0.192585
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −5612.33 −0.798259 −0.399130 0.916894i $$-0.630688\pi$$
−0.399130 + 0.916894i $$0.630688\pi$$
$$368$$ −138.659 −0.0196416
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2077.59 −0.290737
$$372$$ 0 0
$$373$$ 3691.05 0.512374 0.256187 0.966627i $$-0.417534\pi$$
0.256187 + 0.966627i $$0.417534\pi$$
$$374$$ 1321.61 0.182724
$$375$$ 0 0
$$376$$ −6752.24 −0.926118
$$377$$ −3612.46 −0.493504
$$378$$ 0 0
$$379$$ 5350.16 0.725117 0.362559 0.931961i $$-0.381903\pi$$
0.362559 + 0.931961i $$0.381903\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4262.85 −0.570959
$$383$$ 7300.09 0.973936 0.486968 0.873420i $$-0.338103\pi$$
0.486968 + 0.873420i $$0.338103\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2280.56 0.300719
$$387$$ 0 0
$$388$$ −1004.80 −0.131471
$$389$$ 9102.27 1.18638 0.593192 0.805061i $$-0.297867\pi$$
0.593192 + 0.805061i $$0.297867\pi$$
$$390$$ 0 0
$$391$$ 1002.62 0.129680
$$392$$ 1037.68 0.133701
$$393$$ 0 0
$$394$$ 7544.51 0.964688
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11520.1 1.45637 0.728185 0.685381i $$-0.240364\pi$$
0.728185 + 0.685381i $$0.240364\pi$$
$$398$$ 1363.34 0.171704
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3812.93 −0.474834 −0.237417 0.971408i $$-0.576301\pi$$
−0.237417 + 0.971408i $$0.576301\pi$$
$$402$$ 0 0
$$403$$ 4882.00 0.603448
$$404$$ −8136.85 −1.00204
$$405$$ 0 0
$$406$$ −1149.42 −0.140504
$$407$$ −664.432 −0.0809205
$$408$$ 0 0
$$409$$ −4098.82 −0.495535 −0.247767 0.968820i $$-0.579697\pi$$
−0.247767 + 0.968820i $$0.579697\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 2975.70 0.355831
$$413$$ 1988.00 0.236860
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 6432.98 0.758180
$$417$$ 0 0
$$418$$ −1451.33 −0.169825
$$419$$ −14312.4 −1.66876 −0.834378 0.551193i $$-0.814173\pi$$
−0.834378 + 0.551193i $$0.814173\pi$$
$$420$$ 0 0
$$421$$ 6679.57 0.773259 0.386630 0.922235i $$-0.373639\pi$$
0.386630 + 0.922235i $$0.373639\pi$$
$$422$$ 8038.91 0.927317
$$423$$ 0 0
$$424$$ −6285.34 −0.719913
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3461.78 −0.392335
$$428$$ −1819.48 −0.205486
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −11503.8 −1.28566 −0.642828 0.766010i $$-0.722239\pi$$
−0.642828 + 0.766010i $$0.722239\pi$$
$$432$$ 0 0
$$433$$ −3243.99 −0.360037 −0.180019 0.983663i $$-0.557616\pi$$
−0.180019 + 0.983663i $$0.557616\pi$$
$$434$$ 1553.36 0.171806
$$435$$ 0 0
$$436$$ −9019.19 −0.990691
$$437$$ −1101.04 −0.120526
$$438$$ 0 0
$$439$$ −2744.21 −0.298346 −0.149173 0.988811i $$-0.547661\pi$$
−0.149173 + 0.988811i $$0.547661\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4431.17 −0.476853
$$443$$ 8180.27 0.877328 0.438664 0.898651i $$-0.355452\pi$$
0.438664 + 0.898651i $$0.355452\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −3802.87 −0.403747
$$447$$ 0 0
$$448$$ 1407.15 0.148397
$$449$$ −6725.70 −0.706916 −0.353458 0.935450i $$-0.614994\pi$$
−0.353458 + 0.935450i $$0.614994\pi$$
$$450$$ 0 0
$$451$$ 2000.07 0.208824
$$452$$ −2506.68 −0.260850
$$453$$ 0 0
$$454$$ 5478.65 0.566357
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10807.0 1.10619 0.553095 0.833118i $$-0.313447\pi$$
0.553095 + 0.833118i $$0.313447\pi$$
$$458$$ −5355.41 −0.546380
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17775.8 −1.79588 −0.897942 0.440113i $$-0.854938\pi$$
−0.897942 + 0.440113i $$0.854938\pi$$
$$462$$ 0 0
$$463$$ 4418.93 0.443554 0.221777 0.975097i $$-0.428814\pi$$
0.221777 + 0.975097i $$0.428814\pi$$
$$464$$ 1201.20 0.120182
$$465$$ 0 0
$$466$$ 6230.15 0.619326
$$467$$ 3373.08 0.334235 0.167117 0.985937i $$-0.446554\pi$$
0.167117 + 0.985937i $$0.446554\pi$$
$$468$$ 0 0
$$469$$ −3843.23 −0.378387
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6014.29 0.586505
$$473$$ 3270.77 0.317950
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3215.70 0.309646
$$477$$ 0 0
$$478$$ −9267.36 −0.886776
$$479$$ −4620.02 −0.440697 −0.220349 0.975421i $$-0.570720\pi$$
−0.220349 + 0.975421i $$0.570720\pi$$
$$480$$ 0 0
$$481$$ 2227.75 0.211178
$$482$$ 10121.8 0.956508
$$483$$ 0 0
$$484$$ 6818.55 0.640360
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9108.62 0.847538 0.423769 0.905770i $$-0.360707\pi$$
0.423769 + 0.905770i $$0.360707\pi$$
$$488$$ −10472.9 −0.971488
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2696.84 0.247875 0.123938 0.992290i $$-0.460448\pi$$
0.123938 + 0.992290i $$0.460448\pi$$
$$492$$ 0 0
$$493$$ −8685.71 −0.793478
$$494$$ 4866.12 0.443193
$$495$$ 0 0
$$496$$ −1623.34 −0.146956
$$497$$ 5186.80 0.468128
$$498$$ 0 0
$$499$$ −20054.2 −1.79910 −0.899549 0.436820i $$-0.856104\pi$$
−0.899549 + 0.436820i $$0.856104\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 8210.17 0.729956
$$503$$ 8106.49 0.718590 0.359295 0.933224i $$-0.383017\pi$$
0.359295 + 0.933224i $$0.383017\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 194.212 0.0170628
$$507$$ 0 0
$$508$$ 9510.80 0.830657
$$509$$ −12455.8 −1.08466 −0.542332 0.840164i $$-0.682458\pi$$
−0.542332 + 0.840164i $$0.682458\pi$$
$$510$$ 0 0
$$511$$ 3897.95 0.337447
$$512$$ −4074.36 −0.351686
$$513$$ 0 0
$$514$$ −6415.68 −0.550551
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3266.97 −0.277913
$$518$$ 708.830 0.0601239
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5539.61 0.465825 0.232913 0.972498i $$-0.425174\pi$$
0.232913 + 0.972498i $$0.425174\pi$$
$$522$$ 0 0
$$523$$ −15504.2 −1.29628 −0.648138 0.761523i $$-0.724452\pi$$
−0.648138 + 0.761523i $$0.724452\pi$$
$$524$$ 8812.24 0.734665
$$525$$ 0 0
$$526$$ 9739.23 0.807321
$$527$$ 11738.2 0.970252
$$528$$ 0 0
$$529$$ −12019.7 −0.987890
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −3531.35 −0.287789
$$533$$ −6705.96 −0.544967
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −11626.9 −0.936951
$$537$$ 0 0
$$538$$ −6803.17 −0.545178
$$539$$ 502.064 0.0401214
$$540$$ 0 0
$$541$$ 7853.16 0.624092 0.312046 0.950067i $$-0.398986\pi$$
0.312046 + 0.950067i $$0.398986\pi$$
$$542$$ 98.5293 0.00780848
$$543$$ 0 0
$$544$$ 15467.3 1.21904
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9177.75 0.717390 0.358695 0.933455i $$-0.383222\pi$$
0.358695 + 0.933455i $$0.383222\pi$$
$$548$$ 11652.8 0.908363
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 9538.29 0.737468
$$552$$ 0 0
$$553$$ −2633.33 −0.202496
$$554$$ 9064.25 0.695132
$$555$$ 0 0
$$556$$ 7702.73 0.587533
$$557$$ −25030.2 −1.90407 −0.952033 0.305996i $$-0.901010\pi$$
−0.952033 + 0.305996i $$0.901010\pi$$
$$558$$ 0 0
$$559$$ −10966.5 −0.829753
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −13165.3 −0.988157
$$563$$ 19486.1 1.45869 0.729345 0.684146i $$-0.239825\pi$$
0.729345 + 0.684146i $$0.239825\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 11580.0 0.859975
$$567$$ 0 0
$$568$$ 15691.6 1.15916
$$569$$ 5813.63 0.428331 0.214165 0.976797i $$-0.431297\pi$$
0.214165 + 0.976797i $$0.431297\pi$$
$$570$$ 0 0
$$571$$ 268.770 0.0196982 0.00984910 0.999951i $$-0.496865\pi$$
0.00984910 + 0.999951i $$0.496865\pi$$
$$572$$ 1957.67 0.143102
$$573$$ 0 0
$$574$$ −2133.71 −0.155156
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −8283.57 −0.597660 −0.298830 0.954306i $$-0.596596\pi$$
−0.298830 + 0.954306i $$0.596596\pi$$
$$578$$ −2982.29 −0.214614
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5267.26 0.376115
$$582$$ 0 0
$$583$$ −3041.07 −0.216035
$$584$$ 11792.5 0.835575
$$585$$ 0 0
$$586$$ 752.703 0.0530612
$$587$$ −6671.87 −0.469127 −0.234563 0.972101i $$-0.575366\pi$$
−0.234563 + 0.972101i $$0.575366\pi$$
$$588$$ 0 0
$$589$$ −12890.4 −0.901763
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −740.761 −0.0514275
$$593$$ −11409.0 −0.790071 −0.395035 0.918666i $$-0.629268\pi$$
−0.395035 + 0.918666i $$0.629268\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 13919.9 0.956683
$$597$$ 0 0
$$598$$ −651.167 −0.0445288
$$599$$ 13118.3 0.894826 0.447413 0.894327i $$-0.352345\pi$$
0.447413 + 0.894327i $$0.352345\pi$$
$$600$$ 0 0
$$601$$ 21970.2 1.49115 0.745577 0.666420i $$-0.232174\pi$$
0.745577 + 0.666420i $$0.232174\pi$$
$$602$$ −3489.33 −0.236237
$$603$$ 0 0
$$604$$ −2197.99 −0.148071
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 21115.1 1.41192 0.705959 0.708253i $$-0.250516\pi$$
0.705959 + 0.708253i $$0.250516\pi$$
$$608$$ −16985.6 −1.13299
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10953.7 0.725269
$$612$$ 0 0
$$613$$ 1963.61 0.129379 0.0646895 0.997905i $$-0.479394\pi$$
0.0646895 + 0.997905i $$0.479394\pi$$
$$614$$ 4306.60 0.283062
$$615$$ 0 0
$$616$$ 1518.89 0.0993474
$$617$$ 17352.1 1.13221 0.566103 0.824335i $$-0.308450\pi$$
0.566103 + 0.824335i $$0.308450\pi$$
$$618$$ 0 0
$$619$$ −3614.68 −0.234711 −0.117356 0.993090i $$-0.537442\pi$$
−0.117356 + 0.993090i $$0.537442\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 9083.94 0.585583
$$623$$ 6617.09 0.425535
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 3486.68 0.222613
$$627$$ 0 0
$$628$$ 9634.56 0.612199
$$629$$ 5356.35 0.339542
$$630$$ 0 0
$$631$$ 16368.8 1.03269 0.516347 0.856379i $$-0.327291\pi$$
0.516347 + 0.856379i $$0.327291\pi$$
$$632$$ −7966.59 −0.501415
$$633$$ 0 0
$$634$$ −127.112 −0.00796258
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1683.35 −0.104705
$$638$$ −1682.45 −0.104403
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 849.649 0.0523543 0.0261772 0.999657i $$-0.491667\pi$$
0.0261772 + 0.999657i $$0.491667\pi$$
$$642$$ 0 0
$$643$$ 1335.12 0.0818850 0.0409425 0.999162i $$-0.486964\pi$$
0.0409425 + 0.999162i $$0.486964\pi$$
$$644$$ 472.553 0.0289149
$$645$$ 0 0
$$646$$ 11700.0 0.712586
$$647$$ −5944.39 −0.361202 −0.180601 0.983556i $$-0.557804\pi$$
−0.180601 + 0.983556i $$0.557804\pi$$
$$648$$ 0 0
$$649$$ 2909.92 0.176001
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 18371.3 1.10349
$$653$$ −5561.29 −0.333277 −0.166639 0.986018i $$-0.553291\pi$$
−0.166639 + 0.986018i $$0.553291\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2229.84 0.132714
$$657$$ 0 0
$$658$$ 3485.27 0.206489
$$659$$ −27399.5 −1.61963 −0.809813 0.586688i $$-0.800431\pi$$
−0.809813 + 0.586688i $$0.800431\pi$$
$$660$$ 0 0
$$661$$ 3931.52 0.231344 0.115672 0.993287i $$-0.463098\pi$$
0.115672 + 0.993287i $$0.463098\pi$$
$$662$$ 10228.5 0.600514
$$663$$ 0 0
$$664$$ 15935.0 0.931324
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1276.38 −0.0740954
$$668$$ −3347.73 −0.193903
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −5067.16 −0.291528
$$672$$ 0 0
$$673$$ −11427.3 −0.654515 −0.327257 0.944935i $$-0.606124\pi$$
−0.327257 + 0.944935i $$0.606124\pi$$
$$674$$ 6440.83 0.368088
$$675$$ 0 0
$$676$$ 5654.94 0.321742
$$677$$ −16259.3 −0.923035 −0.461518 0.887131i $$-0.652695\pi$$
−0.461518 + 0.887131i $$0.652695\pi$$
$$678$$ 0 0
$$679$$ 1264.68 0.0714785
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 2273.73 0.127662
$$683$$ 10873.9 0.609192 0.304596 0.952482i $$-0.401479\pi$$
0.304596 + 0.952482i $$0.401479\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −535.613 −0.0298102
$$687$$ 0 0
$$688$$ 3646.52 0.202067
$$689$$ 10196.3 0.563785
$$690$$ 0 0
$$691$$ 21613.9 1.18991 0.594957 0.803758i $$-0.297169\pi$$
0.594957 + 0.803758i $$0.297169\pi$$
$$692$$ −20806.7 −1.14299
$$693$$ 0 0
$$694$$ −1918.28 −0.104923
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −16123.7 −0.876222
$$698$$ −5185.81 −0.281212
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20667.7 1.11357 0.556783 0.830658i $$-0.312036\pi$$
0.556783 + 0.830658i $$0.312036\pi$$
$$702$$ 0 0
$$703$$ −5882.12 −0.315574
$$704$$ 2059.71 0.110267
$$705$$ 0 0
$$706$$ 7166.36 0.382025
$$707$$ 10241.4 0.544790
$$708$$ 0 0
$$709$$ −22813.3 −1.20842 −0.604212 0.796824i $$-0.706512\pi$$
−0.604212 + 0.796824i $$0.706512\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 20018.7 1.05370
$$713$$ 1724.94 0.0906026
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 5385.54 0.281099
$$717$$ 0 0
$$718$$ 19334.4 1.00495
$$719$$ −9551.14 −0.495407 −0.247703 0.968836i $$-0.579676\pi$$
−0.247703 + 0.968836i $$0.579676\pi$$
$$720$$ 0 0
$$721$$ −3745.34 −0.193459
$$722$$ −2137.77 −0.110193
$$723$$ 0 0
$$724$$ 10542.7 0.541185
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 30003.0 1.53060 0.765302 0.643672i $$-0.222590\pi$$
0.765302 + 0.643672i $$0.222590\pi$$
$$728$$ −5092.65 −0.259267
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −26367.5 −1.33411
$$732$$ 0 0
$$733$$ −16863.7 −0.849762 −0.424881 0.905249i $$-0.639684\pi$$
−0.424881 + 0.905249i $$0.639684\pi$$
$$734$$ 8763.95 0.440713
$$735$$ 0 0
$$736$$ 2272.95 0.113834
$$737$$ −5625.50 −0.281164
$$738$$ 0 0
$$739$$ −30778.9 −1.53210 −0.766048 0.642783i $$-0.777780\pi$$
−0.766048 + 0.642783i $$0.777780\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 3244.27 0.160514
$$743$$ 28510.9 1.40776 0.703879 0.710320i $$-0.251450\pi$$
0.703879 + 0.710320i $$0.251450\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −5763.77 −0.282878
$$747$$ 0 0
$$748$$ 4706.97 0.230085
$$749$$ 2290.07 0.111719
$$750$$ 0 0
$$751$$ 12480.0 0.606395 0.303197 0.952928i $$-0.401946\pi$$
0.303197 + 0.952928i $$0.401946\pi$$
$$752$$ −3642.28 −0.176623
$$753$$ 0 0
$$754$$ 5641.04 0.272460
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 11272.1 0.541203 0.270601 0.962691i $$-0.412777\pi$$
0.270601 + 0.962691i $$0.412777\pi$$
$$758$$ −8354.56 −0.400332
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 28775.4 1.37071 0.685354 0.728210i $$-0.259647\pi$$
0.685354 + 0.728210i $$0.259647\pi$$
$$762$$ 0 0
$$763$$ 11351.9 0.538621
$$764$$ −15182.4 −0.718951
$$765$$ 0 0
$$766$$ −11399.5 −0.537702
$$767$$ −9756.58 −0.459309
$$768$$ 0 0
$$769$$ −17298.3 −0.811175 −0.405587 0.914056i $$-0.632933\pi$$
−0.405587 + 0.914056i $$0.632933\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8122.33 0.378665
$$773$$ 5348.18 0.248849 0.124425 0.992229i $$-0.460291\pi$$
0.124425 + 0.992229i $$0.460291\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 3826.03 0.176993
$$777$$ 0 0
$$778$$ −14213.7 −0.654994
$$779$$ 17706.3 0.814371
$$780$$ 0 0
$$781$$ 7592.15 0.347847
$$782$$ −1565.65 −0.0715954
$$783$$ 0 0
$$784$$ 559.741 0.0254984
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 21615.1 0.979030 0.489515 0.871995i $$-0.337174\pi$$
0.489515 + 0.871995i $$0.337174\pi$$
$$788$$ 26870.2 1.21473
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3155.01 0.141820
$$792$$ 0 0
$$793$$ 16989.5 0.760800
$$794$$ −17989.3 −0.804050
$$795$$ 0 0
$$796$$ 4855.62 0.216210
$$797$$ −38266.0 −1.70069 −0.850345 0.526225i $$-0.823607\pi$$
−0.850345 + 0.526225i $$0.823607\pi$$
$$798$$ 0 0
$$799$$ 26336.8 1.16612
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 5954.09 0.262152
$$803$$ 5705.61 0.250743
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −7623.50 −0.333159
$$807$$ 0 0
$$808$$ 30983.2 1.34899
$$809$$ −4724.05 −0.205301 −0.102651 0.994717i $$-0.532732\pi$$
−0.102651 + 0.994717i $$0.532732\pi$$
$$810$$ 0 0
$$811$$ 9687.39 0.419446 0.209723 0.977761i $$-0.432744\pi$$
0.209723 + 0.977761i $$0.432744\pi$$
$$812$$ −4093.71 −0.176923
$$813$$ 0 0
$$814$$ 1037.55 0.0446756
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 28955.7 1.23994
$$818$$ 6400.53 0.273581
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 32250.8 1.37096 0.685482 0.728090i $$-0.259592\pi$$
0.685482 + 0.728090i $$0.259592\pi$$
$$822$$ 0 0
$$823$$ −5858.93 −0.248153 −0.124076 0.992273i $$-0.539597\pi$$
−0.124076 + 0.992273i $$0.539597\pi$$
$$824$$ −11330.8 −0.479037
$$825$$ 0 0
$$826$$ −3104.37 −0.130768
$$827$$ 16375.7 0.688560 0.344280 0.938867i $$-0.388123\pi$$
0.344280 + 0.938867i $$0.388123\pi$$
$$828$$ 0 0
$$829$$ −15152.9 −0.634841 −0.317421 0.948285i $$-0.602817\pi$$
−0.317421 + 0.948285i $$0.602817\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −6905.94 −0.287765
$$833$$ −4047.42 −0.168349
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −5169.00 −0.213844
$$837$$ 0 0
$$838$$ 22349.6 0.921307
$$839$$ 8829.87 0.363338 0.181669 0.983360i $$-0.441850\pi$$
0.181669 + 0.983360i $$0.441850\pi$$
$$840$$ 0 0
$$841$$ −13331.8 −0.546630
$$842$$ −10430.5 −0.426911
$$843$$ 0 0
$$844$$ 28631.0 1.16768
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −8582.11 −0.348152
$$848$$ −3390.42 −0.137297
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 787.125 0.0317066
$$852$$ 0 0
$$853$$ 4227.64 0.169697 0.0848486 0.996394i $$-0.472959\pi$$
0.0848486 + 0.996394i $$0.472959\pi$$
$$854$$ 5405.75 0.216605
$$855$$ 0 0
$$856$$ 6928.15 0.276635
$$857$$ 32233.1 1.28479 0.642394 0.766375i $$-0.277941\pi$$
0.642394 + 0.766375i $$0.277941\pi$$
$$858$$ 0 0
$$859$$ 37392.3 1.48523 0.742613 0.669720i $$-0.233586\pi$$
0.742613 + 0.669720i $$0.233586\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 17963.8 0.709801
$$863$$ 8372.73 0.330256 0.165128 0.986272i $$-0.447196\pi$$
0.165128 + 0.986272i $$0.447196\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 5065.66 0.198774
$$867$$ 0 0
$$868$$ 5532.39 0.216338
$$869$$ −3854.52 −0.150467
$$870$$ 0 0
$$871$$ 18861.5 0.733753
$$872$$ 34343.0 1.33372
$$873$$ 0 0
$$874$$ 1719.33 0.0665416
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −38583.0 −1.48558 −0.742790 0.669524i $$-0.766498\pi$$
−0.742790 + 0.669524i $$0.766498\pi$$
$$878$$ 4285.23 0.164715
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −41611.7 −1.59130 −0.795650 0.605757i $$-0.792870\pi$$
−0.795650 + 0.605757i $$0.792870\pi$$
$$882$$ 0 0
$$883$$ −43408.1 −1.65436 −0.827180 0.561937i $$-0.810056\pi$$
−0.827180 + 0.561937i $$0.810056\pi$$
$$884$$ −15781.8 −0.600453
$$885$$ 0 0
$$886$$ −12773.9 −0.484366
$$887$$ 19523.6 0.739050 0.369525 0.929221i $$-0.379520\pi$$
0.369525 + 0.929221i $$0.379520\pi$$
$$888$$ 0 0
$$889$$ −11970.7 −0.451613
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −13544.1 −0.508398
$$893$$ −28922.0 −1.08381
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8288.92 0.309055
$$897$$ 0 0
$$898$$ 10502.5 0.390283
$$899$$ −14943.1 −0.554373
$$900$$ 0 0
$$901$$ 24515.7 0.906479
$$902$$ −3123.21 −0.115290
$$903$$ 0 0
$$904$$ 9544.84 0.351169
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −44114.7 −1.61500 −0.807499 0.589869i $$-0.799180\pi$$
−0.807499 + 0.589869i $$0.799180\pi$$
$$908$$ 19512.5 0.713156
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −49231.9 −1.79048 −0.895240 0.445585i $$-0.852996\pi$$
−0.895240 + 0.445585i $$0.852996\pi$$
$$912$$ 0 0
$$913$$ 7709.92 0.279476
$$914$$ −16875.7 −0.610719
$$915$$ 0 0
$$916$$ −19073.6 −0.688001
$$917$$ −11091.4 −0.399424
$$918$$ 0 0
$$919$$ −46993.6 −1.68681 −0.843403 0.537282i $$-0.819451\pi$$
−0.843403 + 0.537282i $$0.819451\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 27757.9 0.991494
$$923$$ −25455.5 −0.907775
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −6900.40 −0.244883
$$927$$ 0 0
$$928$$ −19690.5 −0.696521
$$929$$ 33023.5 1.16627 0.583135 0.812375i $$-0.301826\pi$$
0.583135 + 0.812375i $$0.301826\pi$$
$$930$$ 0 0
$$931$$ 4444.71 0.156466
$$932$$ 22189.0 0.779855
$$933$$ 0 0
$$934$$ −5267.25 −0.184529
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23431.0 0.816924 0.408462 0.912775i $$-0.366065\pi$$
0.408462 + 0.912775i $$0.366065\pi$$
$$938$$ 6001.40 0.208905
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −43688.0 −1.51348 −0.756742 0.653714i $$-0.773210\pi$$
−0.756742 + 0.653714i $$0.773210\pi$$
$$942$$ 0 0
$$943$$ −2369.40 −0.0818221
$$944$$ 3244.21 0.111854
$$945$$ 0 0
$$946$$ −5107.48 −0.175538
$$947$$ 34884.0 1.19702 0.598511 0.801115i $$-0.295759\pi$$
0.598511 + 0.801115i $$0.295759\pi$$
$$948$$ 0 0
$$949$$ −19130.1 −0.654363
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −12244.6 −0.416860
$$953$$ 34486.4 1.17222 0.586108 0.810233i $$-0.300659\pi$$
0.586108 + 0.810233i $$0.300659\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −33006.2 −1.11663
$$957$$ 0 0
$$958$$ 7214.40 0.243306
$$959$$ −14666.7 −0.493861
$$960$$ 0 0
$$961$$ −9596.33 −0.322122
$$962$$ −3478.75 −0.116590
$$963$$ 0 0
$$964$$ 36049.4 1.20443
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 46341.6 1.54110 0.770550 0.637379i $$-0.219981\pi$$
0.770550 + 0.637379i $$0.219981\pi$$
$$968$$ −25963.4 −0.862083
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 34551.4 1.14192 0.570962 0.820977i $$-0.306571\pi$$
0.570962 + 0.820977i $$0.306571\pi$$
$$972$$ 0 0
$$973$$ −9694.97 −0.319431
$$974$$ −14223.6 −0.467919
$$975$$ 0 0
$$976$$ −5649.27 −0.185275
$$977$$ 970.127 0.0317678 0.0158839 0.999874i $$-0.494944\pi$$
0.0158839 + 0.999874i $$0.494944\pi$$
$$978$$ 0 0
$$979$$ 9685.73 0.316198
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −4211.26 −0.136850
$$983$$ 18342.2 0.595142 0.297571 0.954700i $$-0.403824\pi$$
0.297571 + 0.954700i $$0.403824\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 13563.2 0.438073
$$987$$ 0 0
$$988$$ 17331.0 0.558068
$$989$$ −3874.75 −0.124580
$$990$$ 0 0
$$991$$ 48979.6 1.57002 0.785010 0.619484i $$-0.212658\pi$$
0.785010 + 0.619484i $$0.212658\pi$$
$$992$$ 26610.4 0.851694
$$993$$ 0 0
$$994$$ −8099.46 −0.258450
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −31492.9 −1.00039 −0.500196 0.865912i $$-0.666739\pi$$
−0.500196 + 0.865912i $$0.666739\pi$$
$$998$$ 31315.7 0.993268
$$999$$ 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.u.1.1 2
3.2 odd 2 1575.4.a.r.1.2 2
5.4 even 2 315.4.a.h.1.2 2
15.14 odd 2 315.4.a.j.1.1 yes 2
35.34 odd 2 2205.4.a.y.1.2 2
105.104 even 2 2205.4.a.ba.1.1 2

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