# Properties

 Label 825.6.a.j.1.1 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.34253.1 Defining polynomial: $$x^{3} - x^{2} - 52x + 48$$ x^3 - x^2 - 52*x + 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-7.17710$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} -121.000 q^{11} +46.7783 q^{12} +500.053 q^{13} -639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -419.345 q^{18} -932.948 q^{19} -1110.94 q^{21} +626.430 q^{22} +1225.18 q^{23} -1733.18 q^{24} -2588.83 q^{26} -729.000 q^{27} -641.576 q^{28} -2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} +1089.00 q^{33} -2188.00 q^{34} -421.005 q^{36} -5414.46 q^{37} +4829.97 q^{38} -4500.47 q^{39} -18066.7 q^{41} +5751.43 q^{42} -6815.47 q^{43} +628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} +7475.96 q^{48} -1570.23 q^{49} -3803.67 q^{51} -2599.07 q^{52} -15367.7 q^{53} +3774.11 q^{54} +23771.0 q^{56} +8396.53 q^{57} +10932.1 q^{58} +23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +9998.42 q^{63} +36221.0 q^{64} -5637.87 q^{66} -14507.1 q^{67} -2196.66 q^{68} -11026.6 q^{69} -28114.0 q^{71} +15598.6 q^{72} +28836.7 q^{73} +28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} +23299.4 q^{78} -8150.52 q^{79} +6561.00 q^{81} +93533.0 q^{82} +109864. q^{83} +5774.19 q^{84} +35284.4 q^{86} +19004.5 q^{87} -23301.7 q^{88} +69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} +1436.51 q^{93} -78168.5 q^{94} +16758.0 q^{96} -91551.4 q^{97} +8129.23 q^{98} -9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 $$3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100})$$ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 - 363 * q^11 - 225 * q^12 + 654 * q^13 - 728 * q^14 - 415 * q^16 + 2366 * q^17 + 567 * q^18 - 2872 * q^19 - 1548 * q^21 - 847 * q^22 - 2272 * q^23 - 2079 * q^24 + 3422 * q^26 - 2187 * q^27 - 4592 * q^28 - 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 3267 * q^33 + 2506 * q^34 + 2025 * q^36 + 9126 * q^37 - 13076 * q^38 - 5886 * q^39 - 8758 * q^41 + 6552 * q^42 + 14672 * q^43 - 3025 * q^44 - 28768 * q^46 + 19392 * q^47 + 3735 * q^48 - 26629 * q^49 - 21294 * q^51 + 61506 * q^52 + 4598 * q^53 - 5103 * q^54 + 2688 * q^56 + 25848 * q^57 - 8550 * q^58 - 9348 * q^59 - 60078 * q^61 + 14096 * q^62 + 13932 * q^63 - 7087 * q^64 + 7623 * q^66 + 38468 * q^67 - 59778 * q^68 + 20448 * q^69 - 74032 * q^71 + 18711 * q^72 + 44442 * q^73 + 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 30798 * q^78 - 108116 * q^79 + 19683 * q^81 + 92230 * q^82 + 81892 * q^83 + 41328 * q^84 + 126412 * q^86 + 69642 * q^87 - 27951 * q^88 + 167342 * q^89 - 31832 * q^91 - 72960 * q^92 - 5112 * q^93 + 12728 * q^94 + 9009 * q^96 - 159702 * q^97 - 163121 * q^98 - 29403 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.17710 −0.915191 −0.457596 0.889160i $$-0.651289\pi$$
−0.457596 + 0.889160i $$0.651289\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −5.19759 −0.162425
$$5$$ 0 0
$$6$$ 46.5939 0.528386
$$7$$ 123.437 0.952141 0.476071 0.879407i $$-0.342061\pi$$
0.476071 + 0.879407i $$0.342061\pi$$
$$8$$ 192.576 1.06384
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ 46.7783 0.0937759
$$13$$ 500.053 0.820649 0.410324 0.911940i $$-0.365415\pi$$
0.410324 + 0.911940i $$0.365415\pi$$
$$14$$ −639.048 −0.871392
$$15$$ 0 0
$$16$$ −830.662 −0.811194
$$17$$ 422.631 0.354682 0.177341 0.984150i $$-0.443251\pi$$
0.177341 + 0.984150i $$0.443251\pi$$
$$18$$ −419.345 −0.305064
$$19$$ −932.948 −0.592889 −0.296445 0.955050i $$-0.595801\pi$$
−0.296445 + 0.955050i $$0.595801\pi$$
$$20$$ 0 0
$$21$$ −1110.94 −0.549719
$$22$$ 626.430 0.275941
$$23$$ 1225.18 0.482926 0.241463 0.970410i $$-0.422373\pi$$
0.241463 + 0.970410i $$0.422373\pi$$
$$24$$ −1733.18 −0.614209
$$25$$ 0 0
$$26$$ −2588.83 −0.751051
$$27$$ −729.000 −0.192450
$$28$$ −641.576 −0.154651
$$29$$ −2111.62 −0.466251 −0.233126 0.972447i $$-0.574895\pi$$
−0.233126 + 0.972447i $$0.574895\pi$$
$$30$$ 0 0
$$31$$ −159.612 −0.0298306 −0.0149153 0.999889i $$-0.504748\pi$$
−0.0149153 + 0.999889i $$0.504748\pi$$
$$32$$ −1862.00 −0.321444
$$33$$ 1089.00 0.174078
$$34$$ −2188.00 −0.324601
$$35$$ 0 0
$$36$$ −421.005 −0.0541415
$$37$$ −5414.46 −0.650205 −0.325103 0.945679i $$-0.605399\pi$$
−0.325103 + 0.945679i $$0.605399\pi$$
$$38$$ 4829.97 0.542607
$$39$$ −4500.47 −0.473802
$$40$$ 0 0
$$41$$ −18066.7 −1.67849 −0.839244 0.543756i $$-0.817002\pi$$
−0.839244 + 0.543756i $$0.817002\pi$$
$$42$$ 5751.43 0.503098
$$43$$ −6815.47 −0.562114 −0.281057 0.959691i $$-0.590685\pi$$
−0.281057 + 0.959691i $$0.590685\pi$$
$$44$$ 628.908 0.0489729
$$45$$ 0 0
$$46$$ −6342.88 −0.441969
$$47$$ 15098.9 0.997012 0.498506 0.866886i $$-0.333882\pi$$
0.498506 + 0.866886i $$0.333882\pi$$
$$48$$ 7475.96 0.468343
$$49$$ −1570.23 −0.0934270
$$50$$ 0 0
$$51$$ −3803.67 −0.204775
$$52$$ −2599.07 −0.133294
$$53$$ −15367.7 −0.751484 −0.375742 0.926724i $$-0.622612\pi$$
−0.375742 + 0.926724i $$0.622612\pi$$
$$54$$ 3774.11 0.176129
$$55$$ 0 0
$$56$$ 23771.0 1.01293
$$57$$ 8396.53 0.342305
$$58$$ 10932.1 0.426709
$$59$$ 23400.5 0.875177 0.437588 0.899175i $$-0.355833\pi$$
0.437588 + 0.899175i $$0.355833\pi$$
$$60$$ 0 0
$$61$$ 10768.4 0.370532 0.185266 0.982688i $$-0.440685\pi$$
0.185266 + 0.982688i $$0.440685\pi$$
$$62$$ 826.328 0.0273007
$$63$$ 9998.42 0.317380
$$64$$ 36221.0 1.10538
$$65$$ 0 0
$$66$$ −5637.87 −0.159314
$$67$$ −14507.1 −0.394814 −0.197407 0.980322i $$-0.563252\pi$$
−0.197407 + 0.980322i $$0.563252\pi$$
$$68$$ −2196.66 −0.0576090
$$69$$ −11026.6 −0.278817
$$70$$ 0 0
$$71$$ −28114.0 −0.661876 −0.330938 0.943653i $$-0.607365\pi$$
−0.330938 + 0.943653i $$0.607365\pi$$
$$72$$ 15598.6 0.354614
$$73$$ 28836.7 0.633342 0.316671 0.948536i $$-0.397435\pi$$
0.316671 + 0.948536i $$0.397435\pi$$
$$74$$ 28031.2 0.595062
$$75$$ 0 0
$$76$$ 4849.08 0.0962998
$$77$$ −14935.9 −0.287081
$$78$$ 23299.4 0.433619
$$79$$ −8150.52 −0.146932 −0.0734662 0.997298i $$-0.523406\pi$$
−0.0734662 + 0.997298i $$0.523406\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 93533.0 1.53614
$$83$$ 109864. 1.75049 0.875246 0.483679i $$-0.160700\pi$$
0.875246 + 0.483679i $$0.160700\pi$$
$$84$$ 5774.19 0.0892879
$$85$$ 0 0
$$86$$ 35284.4 0.514442
$$87$$ 19004.5 0.269190
$$88$$ −23301.7 −0.320760
$$89$$ 69673.6 0.932380 0.466190 0.884685i $$-0.345626\pi$$
0.466190 + 0.884685i $$0.345626\pi$$
$$90$$ 0 0
$$91$$ 61725.2 0.781374
$$92$$ −6367.98 −0.0784390
$$93$$ 1436.51 0.0172227
$$94$$ −78168.5 −0.912457
$$95$$ 0 0
$$96$$ 16758.0 0.185586
$$97$$ −91551.4 −0.987952 −0.493976 0.869476i $$-0.664457\pi$$
−0.493976 + 0.869476i $$0.664457\pi$$
$$98$$ 8129.23 0.0855036
$$99$$ −9801.00 −0.100504
$$100$$ 0 0
$$101$$ 21299.3 0.207760 0.103880 0.994590i $$-0.466874\pi$$
0.103880 + 0.994590i $$0.466874\pi$$
$$102$$ 19692.0 0.187409
$$103$$ 6548.06 0.0608163 0.0304081 0.999538i $$-0.490319\pi$$
0.0304081 + 0.999538i $$0.490319\pi$$
$$104$$ 96298.1 0.873040
$$105$$ 0 0
$$106$$ 79560.3 0.687752
$$107$$ −127171. −1.07381 −0.536905 0.843643i $$-0.680407\pi$$
−0.536905 + 0.843643i $$0.680407\pi$$
$$108$$ 3789.04 0.0312586
$$109$$ 57285.2 0.461823 0.230912 0.972975i $$-0.425829\pi$$
0.230912 + 0.972975i $$0.425829\pi$$
$$110$$ 0 0
$$111$$ 48730.1 0.375396
$$112$$ −102535. −0.772371
$$113$$ 67774.2 0.499308 0.249654 0.968335i $$-0.419683\pi$$
0.249654 + 0.968335i $$0.419683\pi$$
$$114$$ −43469.7 −0.313274
$$115$$ 0 0
$$116$$ 10975.3 0.0757307
$$117$$ 40504.3 0.273550
$$118$$ −121147. −0.800954
$$119$$ 52168.4 0.337707
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −55749.1 −0.339108
$$123$$ 162600. 0.969075
$$124$$ 829.598 0.00484522
$$125$$ 0 0
$$126$$ −51762.9 −0.290464
$$127$$ 115352. 0.634622 0.317311 0.948322i $$-0.397220\pi$$
0.317311 + 0.948322i $$0.397220\pi$$
$$128$$ −127936. −0.690187
$$129$$ 61339.2 0.324537
$$130$$ 0 0
$$131$$ 43276.4 0.220329 0.110165 0.993913i $$-0.464862\pi$$
0.110165 + 0.993913i $$0.464862\pi$$
$$132$$ −5660.17 −0.0282745
$$133$$ −115161. −0.564514
$$134$$ 75104.6 0.361331
$$135$$ 0 0
$$136$$ 81388.4 0.377325
$$137$$ −401439. −1.82734 −0.913668 0.406461i $$-0.866763\pi$$
−0.913668 + 0.406461i $$0.866763\pi$$
$$138$$ 57085.9 0.255171
$$139$$ −302018. −1.32586 −0.662928 0.748683i $$-0.730686\pi$$
−0.662928 + 0.748683i $$0.730686\pi$$
$$140$$ 0 0
$$141$$ −135890. −0.575625
$$142$$ 145549. 0.605743
$$143$$ −60506.4 −0.247435
$$144$$ −67283.6 −0.270398
$$145$$ 0 0
$$146$$ −149290. −0.579629
$$147$$ 14132.1 0.0539401
$$148$$ 28142.1 0.105609
$$149$$ 326942. 1.20644 0.603218 0.797576i $$-0.293885\pi$$
0.603218 + 0.797576i $$0.293885\pi$$
$$150$$ 0 0
$$151$$ 164681. 0.587761 0.293881 0.955842i $$-0.405053\pi$$
0.293881 + 0.955842i $$0.405053\pi$$
$$152$$ −179663. −0.630740
$$153$$ 34233.1 0.118227
$$154$$ 77324.8 0.262734
$$155$$ 0 0
$$156$$ 23391.6 0.0769571
$$157$$ −248620. −0.804982 −0.402491 0.915424i $$-0.631856\pi$$
−0.402491 + 0.915424i $$0.631856\pi$$
$$158$$ 42196.1 0.134471
$$159$$ 138309. 0.433870
$$160$$ 0 0
$$161$$ 151233. 0.459813
$$162$$ −33967.0 −0.101688
$$163$$ 417656. 1.23126 0.615629 0.788036i $$-0.288902\pi$$
0.615629 + 0.788036i $$0.288902\pi$$
$$164$$ 93903.0 0.272628
$$165$$ 0 0
$$166$$ −568777. −1.60203
$$167$$ −704955. −1.95601 −0.978004 0.208588i $$-0.933113\pi$$
−0.978004 + 0.208588i $$0.933113\pi$$
$$168$$ −213939. −0.584814
$$169$$ −121240. −0.326535
$$170$$ 0 0
$$171$$ −75568.8 −0.197630
$$172$$ 35424.0 0.0913012
$$173$$ 171060. 0.434545 0.217272 0.976111i $$-0.430284\pi$$
0.217272 + 0.976111i $$0.430284\pi$$
$$174$$ −98388.5 −0.246361
$$175$$ 0 0
$$176$$ 100510. 0.244584
$$177$$ −210605. −0.505284
$$178$$ −360707. −0.853306
$$179$$ −166327. −0.387999 −0.193999 0.981002i $$-0.562146\pi$$
−0.193999 + 0.981002i $$0.562146\pi$$
$$180$$ 0 0
$$181$$ −584391. −1.32589 −0.662945 0.748668i $$-0.730694\pi$$
−0.662945 + 0.748668i $$0.730694\pi$$
$$182$$ −319558. −0.715107
$$183$$ −96915.5 −0.213927
$$184$$ 235940. 0.513756
$$185$$ 0 0
$$186$$ −7436.96 −0.0157621
$$187$$ −51138.3 −0.106941
$$188$$ −78477.8 −0.161939
$$189$$ −89985.8 −0.183240
$$190$$ 0 0
$$191$$ −715510. −1.41916 −0.709581 0.704624i $$-0.751116\pi$$
−0.709581 + 0.704624i $$0.751116\pi$$
$$192$$ −325989. −0.638189
$$193$$ 922088. 1.78188 0.890941 0.454118i $$-0.150046\pi$$
0.890941 + 0.454118i $$0.150046\pi$$
$$194$$ 473971. 0.904165
$$195$$ 0 0
$$196$$ 8161.40 0.0151749
$$197$$ 613632. 1.12653 0.563265 0.826276i $$-0.309545\pi$$
0.563265 + 0.826276i $$0.309545\pi$$
$$198$$ 50740.8 0.0919802
$$199$$ −378985. −0.678406 −0.339203 0.940713i $$-0.610157\pi$$
−0.339203 + 0.940713i $$0.610157\pi$$
$$200$$ 0 0
$$201$$ 130564. 0.227946
$$202$$ −110269. −0.190140
$$203$$ −260652. −0.443937
$$204$$ 19769.9 0.0332606
$$205$$ 0 0
$$206$$ −33900.0 −0.0556585
$$207$$ 99239.5 0.160975
$$208$$ −415375. −0.665705
$$209$$ 112887. 0.178763
$$210$$ 0 0
$$211$$ −473721. −0.732515 −0.366257 0.930514i $$-0.619361\pi$$
−0.366257 + 0.930514i $$0.619361\pi$$
$$212$$ 79875.1 0.122060
$$213$$ 253026. 0.382134
$$214$$ 658376. 0.982741
$$215$$ 0 0
$$216$$ −140388. −0.204736
$$217$$ −19702.1 −0.0284029
$$218$$ −296571. −0.422657
$$219$$ −259530. −0.365660
$$220$$ 0 0
$$221$$ 211338. 0.291069
$$222$$ −252281. −0.343559
$$223$$ 822747. 1.10791 0.553954 0.832547i $$-0.313118\pi$$
0.553954 + 0.832547i $$0.313118\pi$$
$$224$$ −229840. −0.306060
$$225$$ 0 0
$$226$$ −350874. −0.456962
$$227$$ 556097. 0.716285 0.358143 0.933667i $$-0.383410\pi$$
0.358143 + 0.933667i $$0.383410\pi$$
$$228$$ −43641.7 −0.0555987
$$229$$ −634919. −0.800073 −0.400036 0.916499i $$-0.631003\pi$$
−0.400036 + 0.916499i $$0.631003\pi$$
$$230$$ 0 0
$$231$$ 134423. 0.165747
$$232$$ −406646. −0.496017
$$233$$ 906561. 1.09397 0.546987 0.837141i $$-0.315775\pi$$
0.546987 + 0.837141i $$0.315775\pi$$
$$234$$ −209695. −0.250350
$$235$$ 0 0
$$236$$ −121626. −0.142150
$$237$$ 73354.7 0.0848314
$$238$$ −270081. −0.309066
$$239$$ 662586. 0.750322 0.375161 0.926960i $$-0.377587\pi$$
0.375161 + 0.926960i $$0.377587\pi$$
$$240$$ 0 0
$$241$$ −1.31823e6 −1.46200 −0.731001 0.682376i $$-0.760947\pi$$
−0.731001 + 0.682376i $$0.760947\pi$$
$$242$$ −75798.0 −0.0831992
$$243$$ −59049.0 −0.0641500
$$244$$ −55969.6 −0.0601836
$$245$$ 0 0
$$246$$ −841797. −0.886889
$$247$$ −466523. −0.486554
$$248$$ −30737.4 −0.0317350
$$249$$ −988775. −1.01065
$$250$$ 0 0
$$251$$ −11073.9 −0.0110947 −0.00554737 0.999985i $$-0.501766\pi$$
−0.00554737 + 0.999985i $$0.501766\pi$$
$$252$$ −51967.7 −0.0515504
$$253$$ −148247. −0.145608
$$254$$ −597188. −0.580800
$$255$$ 0 0
$$256$$ −496734. −0.473723
$$257$$ 788596. 0.744769 0.372384 0.928079i $$-0.378540\pi$$
0.372384 + 0.928079i $$0.378540\pi$$
$$258$$ −317560. −0.297013
$$259$$ −668346. −0.619087
$$260$$ 0 0
$$261$$ −171041. −0.155417
$$262$$ −224046. −0.201644
$$263$$ −211325. −0.188391 −0.0941956 0.995554i $$-0.530028\pi$$
−0.0941956 + 0.995554i $$0.530028\pi$$
$$264$$ 209715. 0.185191
$$265$$ 0 0
$$266$$ 596198. 0.516639
$$267$$ −627062. −0.538310
$$268$$ 75401.8 0.0641276
$$269$$ 552342. 0.465401 0.232701 0.972548i $$-0.425244\pi$$
0.232701 + 0.972548i $$0.425244\pi$$
$$270$$ 0 0
$$271$$ −1.49255e6 −1.23454 −0.617271 0.786751i $$-0.711762\pi$$
−0.617271 + 0.786751i $$0.711762\pi$$
$$272$$ −351063. −0.287715
$$273$$ −555527. −0.451126
$$274$$ 2.07829e6 1.67236
$$275$$ 0 0
$$276$$ 57311.8 0.0452868
$$277$$ −1.01431e6 −0.794275 −0.397138 0.917759i $$-0.629996\pi$$
−0.397138 + 0.917759i $$0.629996\pi$$
$$278$$ 1.56358e6 1.21341
$$279$$ −12928.6 −0.00994352
$$280$$ 0 0
$$281$$ −658216. −0.497282 −0.248641 0.968596i $$-0.579984\pi$$
−0.248641 + 0.968596i $$0.579984\pi$$
$$282$$ 703517. 0.526807
$$283$$ −1.65981e6 −1.23195 −0.615974 0.787767i $$-0.711237\pi$$
−0.615974 + 0.787767i $$0.711237\pi$$
$$284$$ 146125. 0.107505
$$285$$ 0 0
$$286$$ 313248. 0.226450
$$287$$ −2.23010e6 −1.59816
$$288$$ −150822. −0.107148
$$289$$ −1.24124e6 −0.874201
$$290$$ 0 0
$$291$$ 823963. 0.570394
$$292$$ −149881. −0.102870
$$293$$ 410864. 0.279594 0.139797 0.990180i $$-0.455355\pi$$
0.139797 + 0.990180i $$0.455355\pi$$
$$294$$ −73163.1 −0.0493655
$$295$$ 0 0
$$296$$ −1.04269e6 −0.691715
$$297$$ 88209.0 0.0580259
$$298$$ −1.69261e6 −1.10412
$$299$$ 612654. 0.396312
$$300$$ 0 0
$$301$$ −841283. −0.535212
$$302$$ −852570. −0.537914
$$303$$ −191694. −0.119950
$$304$$ 774965. 0.480948
$$305$$ 0 0
$$306$$ −177228. −0.108200
$$307$$ −1.34831e6 −0.816477 −0.408238 0.912875i $$-0.633857\pi$$
−0.408238 + 0.912875i $$0.633857\pi$$
$$308$$ 77630.7 0.0466291
$$309$$ −58932.6 −0.0351123
$$310$$ 0 0
$$311$$ −2.52585e6 −1.48083 −0.740417 0.672148i $$-0.765372\pi$$
−0.740417 + 0.672148i $$0.765372\pi$$
$$312$$ −866682. −0.504050
$$313$$ −2.23061e6 −1.28695 −0.643476 0.765466i $$-0.722509\pi$$
−0.643476 + 0.765466i $$0.722509\pi$$
$$314$$ 1.28713e6 0.736713
$$315$$ 0 0
$$316$$ 42363.0 0.0238654
$$317$$ 120908. 0.0675780 0.0337890 0.999429i $$-0.489243\pi$$
0.0337890 + 0.999429i $$0.489243\pi$$
$$318$$ −716043. −0.397074
$$319$$ 255506. 0.140580
$$320$$ 0 0
$$321$$ 1.14454e6 0.619964
$$322$$ −782948. −0.420817
$$323$$ −394292. −0.210287
$$324$$ −34101.4 −0.0180472
$$325$$ 0 0
$$326$$ −2.16225e6 −1.12684
$$327$$ −515566. −0.266634
$$328$$ −3.47920e6 −1.78564
$$329$$ 1.86377e6 0.949296
$$330$$ 0 0
$$331$$ 578480. 0.290214 0.145107 0.989416i $$-0.453647\pi$$
0.145107 + 0.989416i $$0.453647\pi$$
$$332$$ −571028. −0.284323
$$333$$ −438571. −0.216735
$$334$$ 3.64963e6 1.79012
$$335$$ 0 0
$$336$$ 922812. 0.445929
$$337$$ 1.78710e6 0.857186 0.428593 0.903498i $$-0.359009\pi$$
0.428593 + 0.903498i $$0.359009\pi$$
$$338$$ 627674. 0.298842
$$339$$ −609968. −0.288276
$$340$$ 0 0
$$341$$ 19313.1 0.00899425
$$342$$ 391228. 0.180869
$$343$$ −2.26844e6 −1.04110
$$344$$ −1.31249e6 −0.598000
$$345$$ 0 0
$$346$$ −885598. −0.397692
$$347$$ 1.46282e6 0.652179 0.326089 0.945339i $$-0.394269\pi$$
0.326089 + 0.945339i $$0.394269\pi$$
$$348$$ −98777.8 −0.0437231
$$349$$ 1.48501e6 0.652627 0.326314 0.945262i $$-0.394193\pi$$
0.326314 + 0.945262i $$0.394193\pi$$
$$350$$ 0 0
$$351$$ −364538. −0.157934
$$352$$ 225302. 0.0969189
$$353$$ −1.34528e6 −0.574616 −0.287308 0.957838i $$-0.592760\pi$$
−0.287308 + 0.957838i $$0.592760\pi$$
$$354$$ 1.09032e6 0.462431
$$355$$ 0 0
$$356$$ −362135. −0.151442
$$357$$ −469515. −0.194975
$$358$$ 861092. 0.355093
$$359$$ −1.83956e6 −0.753316 −0.376658 0.926352i $$-0.622927\pi$$
−0.376658 + 0.926352i $$0.622927\pi$$
$$360$$ 0 0
$$361$$ −1.60571e6 −0.648483
$$362$$ 3.02546e6 1.21344
$$363$$ −131769. −0.0524864
$$364$$ −320822. −0.126914
$$365$$ 0 0
$$366$$ 501742. 0.195784
$$367$$ 312079. 0.120948 0.0604741 0.998170i $$-0.480739\pi$$
0.0604741 + 0.998170i $$0.480739\pi$$
$$368$$ −1.01771e6 −0.391746
$$369$$ −1.46340e6 −0.559496
$$370$$ 0 0
$$371$$ −1.89695e6 −0.715519
$$372$$ −7466.38 −0.00279739
$$373$$ −3.38658e6 −1.26034 −0.630172 0.776455i $$-0.717016\pi$$
−0.630172 + 0.776455i $$0.717016\pi$$
$$374$$ 264748. 0.0978710
$$375$$ 0 0
$$376$$ 2.90768e6 1.06066
$$377$$ −1.05592e6 −0.382629
$$378$$ 465866. 0.167699
$$379$$ −1.62370e6 −0.580641 −0.290321 0.956929i $$-0.593762\pi$$
−0.290321 + 0.956929i $$0.593762\pi$$
$$380$$ 0 0
$$381$$ −1.03817e6 −0.366399
$$382$$ 3.70427e6 1.29881
$$383$$ 3.06187e6 1.06657 0.533285 0.845935i $$-0.320957\pi$$
0.533285 + 0.845935i $$0.320957\pi$$
$$384$$ 1.15142e6 0.398480
$$385$$ 0 0
$$386$$ −4.77375e6 −1.63076
$$387$$ −552053. −0.187371
$$388$$ 475847. 0.160468
$$389$$ 4.21840e6 1.41343 0.706715 0.707499i $$-0.250176\pi$$
0.706715 + 0.707499i $$0.250176\pi$$
$$390$$ 0 0
$$391$$ 517798. 0.171285
$$392$$ −302388. −0.0993915
$$393$$ −389487. −0.127207
$$394$$ −3.17684e6 −1.03099
$$395$$ 0 0
$$396$$ 50941.6 0.0163243
$$397$$ −513185. −0.163417 −0.0817085 0.996656i $$-0.526038\pi$$
−0.0817085 + 0.996656i $$0.526038\pi$$
$$398$$ 1.96205e6 0.620871
$$399$$ 1.03645e6 0.325922
$$400$$ 0 0
$$401$$ −2.18458e6 −0.678432 −0.339216 0.940709i $$-0.610162\pi$$
−0.339216 + 0.940709i $$0.610162\pi$$
$$402$$ −675942. −0.208614
$$403$$ −79814.5 −0.0244804
$$404$$ −110705. −0.0337453
$$405$$ 0 0
$$406$$ 1.34942e6 0.406287
$$407$$ 655149. 0.196044
$$408$$ −732496. −0.217849
$$409$$ 1.22307e6 0.361529 0.180764 0.983526i $$-0.442143\pi$$
0.180764 + 0.983526i $$0.442143\pi$$
$$410$$ 0 0
$$411$$ 3.61295e6 1.05501
$$412$$ −34034.1 −0.00987806
$$413$$ 2.88850e6 0.833292
$$414$$ −513774. −0.147323
$$415$$ 0 0
$$416$$ −931098. −0.263792
$$417$$ 2.71816e6 0.765483
$$418$$ −584426. −0.163602
$$419$$ 2.42879e6 0.675858 0.337929 0.941172i $$-0.390274\pi$$
0.337929 + 0.941172i $$0.390274\pi$$
$$420$$ 0 0
$$421$$ −727467. −0.200036 −0.100018 0.994986i $$-0.531890\pi$$
−0.100018 + 0.994986i $$0.531890\pi$$
$$422$$ 2.45250e6 0.670391
$$423$$ 1.22301e6 0.332337
$$424$$ −2.95945e6 −0.799460
$$425$$ 0 0
$$426$$ −1.30994e6 −0.349726
$$427$$ 1.32922e6 0.352799
$$428$$ 660981. 0.174413
$$429$$ 544557. 0.142857
$$430$$ 0 0
$$431$$ 6.18223e6 1.60307 0.801534 0.597949i $$-0.204018\pi$$
0.801534 + 0.597949i $$0.204018\pi$$
$$432$$ 605553. 0.156114
$$433$$ −2.42337e6 −0.621154 −0.310577 0.950548i $$-0.600522\pi$$
−0.310577 + 0.950548i $$0.600522\pi$$
$$434$$ 102000. 0.0259941
$$435$$ 0 0
$$436$$ −297745. −0.0750115
$$437$$ −1.14303e6 −0.286321
$$438$$ 1.34361e6 0.334649
$$439$$ −3.79993e6 −0.941054 −0.470527 0.882385i $$-0.655936\pi$$
−0.470527 + 0.882385i $$0.655936\pi$$
$$440$$ 0 0
$$441$$ −127188. −0.0311423
$$442$$ −1.09412e6 −0.266384
$$443$$ 5.58044e6 1.35101 0.675506 0.737354i $$-0.263925\pi$$
0.675506 + 0.737354i $$0.263925\pi$$
$$444$$ −253279. −0.0609736
$$445$$ 0 0
$$446$$ −4.25945e6 −1.01395
$$447$$ −2.94247e6 −0.696537
$$448$$ 4.47102e6 1.05247
$$449$$ 2.69609e6 0.631130 0.315565 0.948904i $$-0.397806\pi$$
0.315565 + 0.948904i $$0.397806\pi$$
$$450$$ 0 0
$$451$$ 2.18607e6 0.506083
$$452$$ −352263. −0.0810999
$$453$$ −1.48213e6 −0.339344
$$454$$ −2.87897e6 −0.655538
$$455$$ 0 0
$$456$$ 1.61697e6 0.364158
$$457$$ −1.14257e6 −0.255912 −0.127956 0.991780i $$-0.540842\pi$$
−0.127956 + 0.991780i $$0.540842\pi$$
$$458$$ 3.28704e6 0.732220
$$459$$ −308098. −0.0682585
$$460$$ 0 0
$$461$$ 5.37091e6 1.17705 0.588526 0.808478i $$-0.299708\pi$$
0.588526 + 0.808478i $$0.299708\pi$$
$$462$$ −695923. −0.151690
$$463$$ 3.80227e6 0.824311 0.412155 0.911114i $$-0.364776\pi$$
0.412155 + 0.911114i $$0.364776\pi$$
$$464$$ 1.75404e6 0.378220
$$465$$ 0 0
$$466$$ −4.69336e6 −1.00120
$$467$$ −9.41694e6 −1.99810 −0.999051 0.0435609i $$-0.986130\pi$$
−0.999051 + 0.0435609i $$0.986130\pi$$
$$468$$ −210525. −0.0444312
$$469$$ −1.79071e6 −0.375919
$$470$$ 0 0
$$471$$ 2.23758e6 0.464757
$$472$$ 4.50638e6 0.931049
$$473$$ 824672. 0.169484
$$474$$ −379765. −0.0776370
$$475$$ 0 0
$$476$$ −271150. −0.0548519
$$477$$ −1.24479e6 −0.250495
$$478$$ −3.43028e6 −0.686688
$$479$$ −4.69047e6 −0.934065 −0.467033 0.884240i $$-0.654677\pi$$
−0.467033 + 0.884240i $$0.654677\pi$$
$$480$$ 0 0
$$481$$ −2.70751e6 −0.533590
$$482$$ 6.82461e6 1.33801
$$483$$ −1.36110e6 −0.265473
$$484$$ −76097.9 −0.0147659
$$485$$ 0 0
$$486$$ 305703. 0.0587096
$$487$$ −7.07177e6 −1.35116 −0.675579 0.737288i $$-0.736106\pi$$
−0.675579 + 0.737288i $$0.736106\pi$$
$$488$$ 2.07373e6 0.394187
$$489$$ −3.75890e6 −0.710867
$$490$$ 0 0
$$491$$ 906051. 0.169609 0.0848045 0.996398i $$-0.472973\pi$$
0.0848045 + 0.996398i $$0.472973\pi$$
$$492$$ −845127. −0.157402
$$493$$ −892434. −0.165371
$$494$$ 2.41524e6 0.445290
$$495$$ 0 0
$$496$$ 132584. 0.0241984
$$497$$ −3.47031e6 −0.630199
$$498$$ 5.11899e6 0.924935
$$499$$ −7.13323e6 −1.28243 −0.641217 0.767360i $$-0.721570\pi$$
−0.641217 + 0.767360i $$0.721570\pi$$
$$500$$ 0 0
$$501$$ 6.34460e6 1.12930
$$502$$ 57330.9 0.0101538
$$503$$ 2.34927e6 0.414013 0.207006 0.978340i $$-0.433628\pi$$
0.207006 + 0.978340i $$0.433628\pi$$
$$504$$ 1.92545e6 0.337642
$$505$$ 0 0
$$506$$ 767489. 0.133259
$$507$$ 1.09116e6 0.188525
$$508$$ −599551. −0.103078
$$509$$ 1.97148e6 0.337286 0.168643 0.985677i $$-0.446062\pi$$
0.168643 + 0.985677i $$0.446062\pi$$
$$510$$ 0 0
$$511$$ 3.55952e6 0.603031
$$512$$ 6.66559e6 1.12373
$$513$$ 680119. 0.114102
$$514$$ −4.08264e6 −0.681606
$$515$$ 0 0
$$516$$ −318816. −0.0527128
$$517$$ −1.82697e6 −0.300610
$$518$$ 3.46010e6 0.566583
$$519$$ −1.53954e6 −0.250884
$$520$$ 0 0
$$521$$ 1.20074e7 1.93801 0.969004 0.247046i $$-0.0794599\pi$$
0.969004 + 0.247046i $$0.0794599\pi$$
$$522$$ 885497. 0.142236
$$523$$ −4.16772e6 −0.666260 −0.333130 0.942881i $$-0.608105\pi$$
−0.333130 + 0.942881i $$0.608105\pi$$
$$524$$ −224933. −0.0357869
$$525$$ 0 0
$$526$$ 1.09405e6 0.172414
$$527$$ −67456.9 −0.0105804
$$528$$ −904591. −0.141211
$$529$$ −4.93528e6 −0.766783
$$530$$ 0 0
$$531$$ 1.89544e6 0.291726
$$532$$ 598557. 0.0916910
$$533$$ −9.03428e6 −1.37745
$$534$$ 3.24637e6 0.492657
$$535$$ 0 0
$$536$$ −2.79371e6 −0.420020
$$537$$ 1.49694e6 0.224011
$$538$$ −2.85953e6 −0.425931
$$539$$ 189998. 0.0281693
$$540$$ 0 0
$$541$$ 1.01106e7 1.48519 0.742594 0.669741i $$-0.233595\pi$$
0.742594 + 0.669741i $$0.233595\pi$$
$$542$$ 7.72709e6 1.12984
$$543$$ 5.25952e6 0.765503
$$544$$ −786938. −0.114010
$$545$$ 0 0
$$546$$ 2.87602e6 0.412867
$$547$$ 7.39087e6 1.05615 0.528077 0.849196i $$-0.322913\pi$$
0.528077 + 0.849196i $$0.322913\pi$$
$$548$$ 2.08652e6 0.296805
$$549$$ 872239. 0.123511
$$550$$ 0 0
$$551$$ 1.97003e6 0.276435
$$552$$ −2.12346e6 −0.296617
$$553$$ −1.00608e6 −0.139900
$$554$$ 5.25119e6 0.726914
$$555$$ 0 0
$$556$$ 1.56977e6 0.215352
$$557$$ −1.52459e6 −0.208217 −0.104108 0.994566i $$-0.533199\pi$$
−0.104108 + 0.994566i $$0.533199\pi$$
$$558$$ 66932.6 0.00910023
$$559$$ −3.40809e6 −0.461299
$$560$$ 0 0
$$561$$ 460245. 0.0617421
$$562$$ 3.40765e6 0.455108
$$563$$ 1.40845e7 1.87271 0.936353 0.351059i $$-0.114178\pi$$
0.936353 + 0.351059i $$0.114178\pi$$
$$564$$ 706300. 0.0934957
$$565$$ 0 0
$$566$$ 8.59301e6 1.12747
$$567$$ 809872. 0.105793
$$568$$ −5.41407e6 −0.704131
$$569$$ −1.30473e7 −1.68942 −0.844712 0.535221i $$-0.820228\pi$$
−0.844712 + 0.535221i $$0.820228\pi$$
$$570$$ 0 0
$$571$$ −1.19873e7 −1.53862 −0.769312 0.638873i $$-0.779401\pi$$
−0.769312 + 0.638873i $$0.779401\pi$$
$$572$$ 314487. 0.0401895
$$573$$ 6.43959e6 0.819354
$$574$$ 1.15455e7 1.46262
$$575$$ 0 0
$$576$$ 2.93390e6 0.368459
$$577$$ −568904. −0.0711376 −0.0355688 0.999367i $$-0.511324\pi$$
−0.0355688 + 0.999367i $$0.511324\pi$$
$$578$$ 6.42603e6 0.800061
$$579$$ −8.29879e6 −1.02877
$$580$$ 0 0
$$581$$ 1.35613e7 1.66671
$$582$$ −4.26574e6 −0.522020
$$583$$ 1.85949e6 0.226581
$$584$$ 5.55325e6 0.673775
$$585$$ 0 0
$$586$$ −2.12708e6 −0.255882
$$587$$ −2.93323e6 −0.351358 −0.175679 0.984447i $$-0.556212\pi$$
−0.175679 + 0.984447i $$0.556212\pi$$
$$588$$ −73452.6 −0.00876120
$$589$$ 148910. 0.0176862
$$590$$ 0 0
$$591$$ −5.52269e6 −0.650402
$$592$$ 4.49758e6 0.527442
$$593$$ −5.90557e6 −0.689644 −0.344822 0.938668i $$-0.612061\pi$$
−0.344822 + 0.938668i $$0.612061\pi$$
$$594$$ −456667. −0.0531048
$$595$$ 0 0
$$596$$ −1.69931e6 −0.195955
$$597$$ 3.41087e6 0.391678
$$598$$ −3.17178e6 −0.362702
$$599$$ 1.07665e7 1.22605 0.613025 0.790064i $$-0.289953\pi$$
0.613025 + 0.790064i $$0.289953\pi$$
$$600$$ 0 0
$$601$$ −1.19746e7 −1.35231 −0.676154 0.736760i $$-0.736355\pi$$
−0.676154 + 0.736760i $$0.736355\pi$$
$$602$$ 4.35541e6 0.489822
$$603$$ −1.17507e6 −0.131605
$$604$$ −855944. −0.0954669
$$605$$ 0 0
$$606$$ 992418. 0.109777
$$607$$ 3.45506e6 0.380613 0.190307 0.981725i $$-0.439052\pi$$
0.190307 + 0.981725i $$0.439052\pi$$
$$608$$ 1.73715e6 0.190580
$$609$$ 2.34587e6 0.256307
$$610$$ 0 0
$$611$$ 7.55024e6 0.818197
$$612$$ −177929. −0.0192030
$$613$$ 8.22419e6 0.883979 0.441990 0.897020i $$-0.354273\pi$$
0.441990 + 0.897020i $$0.354273\pi$$
$$614$$ 6.98034e6 0.747233
$$615$$ 0 0
$$616$$ −2.87630e6 −0.305409
$$617$$ −9.83567e6 −1.04014 −0.520069 0.854124i $$-0.674094\pi$$
−0.520069 + 0.854124i $$0.674094\pi$$
$$618$$ 305100. 0.0321345
$$619$$ 267648. 0.0280762 0.0140381 0.999901i $$-0.495531\pi$$
0.0140381 + 0.999901i $$0.495531\pi$$
$$620$$ 0 0
$$621$$ −893156. −0.0929391
$$622$$ 1.30766e7 1.35525
$$623$$ 8.60032e6 0.887758
$$624$$ 3.73837e6 0.384345
$$625$$ 0 0
$$626$$ 1.15481e7 1.17781
$$627$$ −1.01598e6 −0.103209
$$628$$ 1.29222e6 0.130749
$$629$$ −2.28831e6 −0.230616
$$630$$ 0 0
$$631$$ −4.17135e6 −0.417065 −0.208532 0.978015i $$-0.566869\pi$$
−0.208532 + 0.978015i $$0.566869\pi$$
$$632$$ −1.56959e6 −0.156313
$$633$$ 4.26349e6 0.422918
$$634$$ −625951. −0.0618468
$$635$$ 0 0
$$636$$ −718876. −0.0704711
$$637$$ −785197. −0.0766708
$$638$$ −1.32278e6 −0.128658
$$639$$ −2.27723e6 −0.220625
$$640$$ 0 0
$$641$$ 8.24673e6 0.792751 0.396376 0.918088i $$-0.370268\pi$$
0.396376 + 0.918088i $$0.370268\pi$$
$$642$$ −5.92538e6 −0.567386
$$643$$ −1.07773e7 −1.02797 −0.513986 0.857799i $$-0.671832\pi$$
−0.513986 + 0.857799i $$0.671832\pi$$
$$644$$ −786046. −0.0746850
$$645$$ 0 0
$$646$$ 2.04129e6 0.192453
$$647$$ −7.89194e6 −0.741179 −0.370590 0.928797i $$-0.620844\pi$$
−0.370590 + 0.928797i $$0.620844\pi$$
$$648$$ 1.26349e6 0.118205
$$649$$ −2.83147e6 −0.263876
$$650$$ 0 0
$$651$$ 177319. 0.0163984
$$652$$ −2.17080e6 −0.199987
$$653$$ −1.47858e7 −1.35695 −0.678473 0.734625i $$-0.737358\pi$$
−0.678473 + 0.734625i $$0.737358\pi$$
$$654$$ 2.66914e6 0.244021
$$655$$ 0 0
$$656$$ 1.50073e7 1.36158
$$657$$ 2.33577e6 0.211114
$$658$$ −9.64891e6 −0.868788
$$659$$ −475088. −0.0426148 −0.0213074 0.999773i $$-0.506783\pi$$
−0.0213074 + 0.999773i $$0.506783\pi$$
$$660$$ 0 0
$$661$$ 2.73604e6 0.243567 0.121784 0.992557i $$-0.461139\pi$$
0.121784 + 0.992557i $$0.461139\pi$$
$$662$$ −2.99485e6 −0.265601
$$663$$ −1.90204e6 −0.168049
$$664$$ 2.11571e7 1.86224
$$665$$ 0 0
$$666$$ 2.27053e6 0.198354
$$667$$ −2.58711e6 −0.225165
$$668$$ 3.66407e6 0.317704
$$669$$ −7.40472e6 −0.639651
$$670$$ 0 0
$$671$$ −1.30297e6 −0.111720
$$672$$ 2.06856e6 0.176704
$$673$$ −1.52861e7 −1.30094 −0.650471 0.759531i $$-0.725428\pi$$
−0.650471 + 0.759531i $$0.725428\pi$$
$$674$$ −9.25202e6 −0.784489
$$675$$ 0 0
$$676$$ 630157. 0.0530374
$$677$$ 3.93225e6 0.329739 0.164869 0.986315i $$-0.447280\pi$$
0.164869 + 0.986315i $$0.447280\pi$$
$$678$$ 3.15787e6 0.263827
$$679$$ −1.13009e7 −0.940670
$$680$$ 0 0
$$681$$ −5.00487e6 −0.413547
$$682$$ −99985.7 −0.00823146
$$683$$ −3.42591e6 −0.281011 −0.140506 0.990080i $$-0.544873\pi$$
−0.140506 + 0.990080i $$0.544873\pi$$
$$684$$ 392776. 0.0320999
$$685$$ 0 0
$$686$$ 1.17439e7 0.952803
$$687$$ 5.71427e6 0.461922
$$688$$ 5.66135e6 0.455984
$$689$$ −7.68467e6 −0.616705
$$690$$ 0 0
$$691$$ 1.81896e7 1.44920 0.724599 0.689171i $$-0.242025\pi$$
0.724599 + 0.689171i $$0.242025\pi$$
$$692$$ −889102. −0.0705808
$$693$$ −1.20981e6 −0.0956938
$$694$$ −7.57316e6 −0.596868
$$695$$ 0 0
$$696$$ 3.65982e6 0.286376
$$697$$ −7.63552e6 −0.595328
$$698$$ −7.68804e6 −0.597279
$$699$$ −8.15905e6 −0.631607
$$700$$ 0 0
$$701$$ −1.15598e6 −0.0888494 −0.0444247 0.999013i $$-0.514145\pi$$
−0.0444247 + 0.999013i $$0.514145\pi$$
$$702$$ 1.88725e6 0.144540
$$703$$ 5.05141e6 0.385500
$$704$$ −4.38274e6 −0.333283
$$705$$ 0 0
$$706$$ 6.96468e6 0.525883
$$707$$ 2.62913e6 0.197817
$$708$$ 1.09464e6 0.0820705
$$709$$ 2.42664e7 1.81296 0.906481 0.422246i $$-0.138758\pi$$
0.906481 + 0.422246i $$0.138758\pi$$
$$710$$ 0 0
$$711$$ −660192. −0.0489775
$$712$$ 1.34174e7 0.991904
$$713$$ −195553. −0.0144059
$$714$$ 2.43073e6 0.178440
$$715$$ 0 0
$$716$$ 864499. 0.0630205
$$717$$ −5.96328e6 −0.433199
$$718$$ 9.52359e6 0.689429
$$719$$ −2.25503e7 −1.62679 −0.813394 0.581714i $$-0.802382\pi$$
−0.813394 + 0.581714i $$0.802382\pi$$
$$720$$ 0 0
$$721$$ 808276. 0.0579057
$$722$$ 8.31291e6 0.593486
$$723$$ 1.18641e7 0.844088
$$724$$ 3.03743e6 0.215357
$$725$$ 0 0
$$726$$ 682182. 0.0480351
$$727$$ 2.17941e7 1.52933 0.764667 0.644425i $$-0.222903\pi$$
0.764667 + 0.644425i $$0.222903\pi$$
$$728$$ 1.18868e7 0.831257
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −2.88043e6 −0.199372
$$732$$ 503727. 0.0347470
$$733$$ −1.73565e7 −1.19317 −0.596583 0.802551i $$-0.703476\pi$$
−0.596583 + 0.802551i $$0.703476\pi$$
$$734$$ −1.61567e6 −0.110691
$$735$$ 0 0
$$736$$ −2.28129e6 −0.155233
$$737$$ 1.75536e6 0.119041
$$738$$ 7.57617e6 0.512046
$$739$$ 1.16961e6 0.0787825 0.0393912 0.999224i $$-0.487458\pi$$
0.0393912 + 0.999224i $$0.487458\pi$$
$$740$$ 0 0
$$741$$ 4.19871e6 0.280912
$$742$$ 9.82071e6 0.654837
$$743$$ −2.98047e6 −0.198067 −0.0990336 0.995084i $$-0.531575\pi$$
−0.0990336 + 0.995084i $$0.531575\pi$$
$$744$$ 276637. 0.0183222
$$745$$ 0 0
$$746$$ 1.75327e7 1.15346
$$747$$ 8.89898e6 0.583497
$$748$$ 265796. 0.0173698
$$749$$ −1.56976e7 −1.02242
$$750$$ 0 0
$$751$$ −1.83578e7 −1.18774 −0.593868 0.804563i $$-0.702400\pi$$
−0.593868 + 0.804563i $$0.702400\pi$$
$$752$$ −1.25421e7 −0.808770
$$753$$ 99665.3 0.00640555
$$754$$ 5.46661e6 0.350178
$$755$$ 0 0
$$756$$ 467709. 0.0297626
$$757$$ −1.60050e7 −1.01511 −0.507557 0.861618i $$-0.669451\pi$$
−0.507557 + 0.861618i $$0.669451\pi$$
$$758$$ 8.40607e6 0.531398
$$759$$ 1.33422e6 0.0840665
$$760$$ 0 0
$$761$$ −1.67436e7 −1.04807 −0.524033 0.851698i $$-0.675573\pi$$
−0.524033 + 0.851698i $$0.675573\pi$$
$$762$$ 5.37469e6 0.335325
$$763$$ 7.07113e6 0.439721
$$764$$ 3.71892e6 0.230507
$$765$$ 0 0
$$766$$ −1.58516e7 −0.976116
$$767$$ 1.17015e7 0.718213
$$768$$ 4.47061e6 0.273504
$$769$$ −1.73863e7 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$770$$ 0 0
$$771$$ −7.09736e6 −0.429993
$$772$$ −4.79263e6 −0.289422
$$773$$ −1.79095e7 −1.07804 −0.539021 0.842292i $$-0.681206\pi$$
−0.539021 + 0.842292i $$0.681206\pi$$
$$774$$ 2.85804e6 0.171481
$$775$$ 0 0
$$776$$ −1.76306e7 −1.05102
$$777$$ 6.01511e6 0.357430
$$778$$ −2.18391e7 −1.29356
$$779$$ 1.68552e7 0.995157
$$780$$ 0 0
$$781$$ 3.40179e6 0.199563
$$782$$ −2.68070e6 −0.156758
$$783$$ 1.53937e6 0.0897301
$$784$$ 1.30433e6 0.0757874
$$785$$ 0 0
$$786$$ 2.01642e6 0.116419
$$787$$ 3.01291e7 1.73400 0.867000 0.498308i $$-0.166045\pi$$
0.867000 + 0.498308i $$0.166045\pi$$
$$788$$ −3.18941e6 −0.182976
$$789$$ 1.90192e6 0.108768
$$790$$ 0 0
$$791$$ 8.36587e6 0.475412
$$792$$ −1.88744e6 −0.106920
$$793$$ 5.38476e6 0.304077
$$794$$ 2.65681e6 0.149558
$$795$$ 0 0
$$796$$ 1.96981e6 0.110190
$$797$$ 5.51251e6 0.307400 0.153700 0.988118i $$-0.450881\pi$$
0.153700 + 0.988118i $$0.450881\pi$$
$$798$$ −5.36579e6 −0.298281
$$799$$ 6.38125e6 0.353622
$$800$$ 0 0
$$801$$ 5.64356e6 0.310793
$$802$$ 1.13098e7 0.620895
$$803$$ −3.48924e6 −0.190960
$$804$$ −678616. −0.0370241
$$805$$ 0 0
$$806$$ 413208. 0.0224043
$$807$$ −4.97108e6 −0.268700
$$808$$ 4.10173e6 0.221024
$$809$$ 3.11564e7 1.67369 0.836847 0.547436i $$-0.184396\pi$$
0.836847 + 0.547436i $$0.184396\pi$$
$$810$$ 0 0
$$811$$ −1.11336e7 −0.594406 −0.297203 0.954814i $$-0.596054\pi$$
−0.297203 + 0.954814i $$0.596054\pi$$
$$812$$ 1.35476e6 0.0721063
$$813$$ 1.34330e7 0.712763
$$814$$ −3.39178e6 −0.179418
$$815$$ 0 0
$$816$$ 3.15957e6 0.166113
$$817$$ 6.35848e6 0.333271
$$818$$ −6.33195e6 −0.330868
$$819$$ 4.99974e6 0.260458
$$820$$ 0 0
$$821$$ −3.44603e7 −1.78427 −0.892136 0.451767i $$-0.850794\pi$$
−0.892136 + 0.451767i $$0.850794\pi$$
$$822$$ −1.87046e7 −0.965539
$$823$$ 9.74417e6 0.501470 0.250735 0.968056i $$-0.419328\pi$$
0.250735 + 0.968056i $$0.419328\pi$$
$$824$$ 1.26100e6 0.0646989
$$825$$ 0 0
$$826$$ −1.49541e7 −0.762622
$$827$$ 9.63214e6 0.489733 0.244866 0.969557i $$-0.421256\pi$$
0.244866 + 0.969557i $$0.421256\pi$$
$$828$$ −515806. −0.0261463
$$829$$ −2.79310e7 −1.41156 −0.705780 0.708431i $$-0.749403\pi$$
−0.705780 + 0.708431i $$0.749403\pi$$
$$830$$ 0 0
$$831$$ 9.12879e6 0.458575
$$832$$ 1.81124e7 0.907126
$$833$$ −663626. −0.0331368
$$834$$ −1.40722e7 −0.700564
$$835$$ 0 0
$$836$$ −586739. −0.0290355
$$837$$ 116357. 0.00574090
$$838$$ −1.25741e7 −0.618540
$$839$$ −3.51475e7 −1.72381 −0.861906 0.507068i $$-0.830729\pi$$
−0.861906 + 0.507068i $$0.830729\pi$$
$$840$$ 0 0
$$841$$ −1.60522e7 −0.782610
$$842$$ 3.76617e6 0.183071
$$843$$ 5.92394e6 0.287106
$$844$$ 2.46221e6 0.118978
$$845$$ 0 0
$$846$$ −6.33165e6 −0.304152
$$847$$ 1.80725e6 0.0865583
$$848$$ 1.27654e7 0.609599
$$849$$ 1.49383e7 0.711265
$$850$$ 0 0
$$851$$ −6.63368e6 −0.314001
$$852$$ −1.31512e6 −0.0620680
$$853$$ −2.50498e7 −1.17878 −0.589389 0.807849i $$-0.700631\pi$$
−0.589389 + 0.807849i $$0.700631\pi$$
$$854$$ −6.88152e6 −0.322879
$$855$$ 0 0
$$856$$ −2.44900e7 −1.14236
$$857$$ 1.89631e7 0.881975 0.440987 0.897513i $$-0.354628\pi$$
0.440987 + 0.897513i $$0.354628\pi$$
$$858$$ −2.81923e6 −0.130741
$$859$$ −3.04605e7 −1.40849 −0.704245 0.709957i $$-0.748714\pi$$
−0.704245 + 0.709957i $$0.748714\pi$$
$$860$$ 0 0
$$861$$ 2.00709e7 0.922696
$$862$$ −3.20060e7 −1.46711
$$863$$ −3.46176e7 −1.58223 −0.791115 0.611667i $$-0.790499\pi$$
−0.791115 + 0.611667i $$0.790499\pi$$
$$864$$ 1.35740e6 0.0618619
$$865$$ 0 0
$$866$$ 1.25460e7 0.568475
$$867$$ 1.11712e7 0.504720
$$868$$ 102403. 0.00461333
$$869$$ 986213. 0.0443018
$$870$$ 0 0
$$871$$ −7.25430e6 −0.324004
$$872$$ 1.10317e7 0.491307
$$873$$ −7.41566e6 −0.329317
$$874$$ 5.91758e6 0.262039
$$875$$ 0 0
$$876$$ 1.34893e6 0.0593922
$$877$$ 3.61979e7 1.58922 0.794611 0.607119i $$-0.207675\pi$$
0.794611 + 0.607119i $$0.207675\pi$$
$$878$$ 1.96727e7 0.861245
$$879$$ −3.69777e6 −0.161424
$$880$$ 0 0
$$881$$ 2.40371e7 1.04338 0.521690 0.853135i $$-0.325302\pi$$
0.521690 + 0.853135i $$0.325302\pi$$
$$882$$ 658468. 0.0285012
$$883$$ −3.64938e7 −1.57513 −0.787567 0.616229i $$-0.788659\pi$$
−0.787567 + 0.616229i $$0.788659\pi$$
$$884$$ −1.09845e6 −0.0472768
$$885$$ 0 0
$$886$$ −2.88905e7 −1.23644
$$887$$ −2.29197e7 −0.978137 −0.489069 0.872245i $$-0.662663\pi$$
−0.489069 + 0.872245i $$0.662663\pi$$
$$888$$ 9.38424e6 0.399362
$$889$$ 1.42387e7 0.604250
$$890$$ 0 0
$$891$$ −793881. −0.0335013
$$892$$ −4.27630e6 −0.179952
$$893$$ −1.40865e7 −0.591117
$$894$$ 1.52335e7 0.637464
$$895$$ 0 0
$$896$$ −1.57920e7 −0.657156
$$897$$ −5.51389e6 −0.228811
$$898$$ −1.39579e7 −0.577605
$$899$$ 337039. 0.0139085
$$900$$ 0 0
$$901$$ −6.49487e6 −0.266538
$$902$$ −1.13175e7 −0.463163
$$903$$ 7.57155e6 0.309005
$$904$$ 1.30517e7 0.531184
$$905$$ 0 0
$$906$$ 7.67313e6 0.310565
$$907$$ 2.66664e7 1.07633 0.538167 0.842838i $$-0.319117\pi$$
0.538167 + 0.842838i $$0.319117\pi$$
$$908$$ −2.89036e6 −0.116342
$$909$$ 1.72524e6 0.0692533
$$910$$ 0 0
$$911$$ 1.73286e7 0.691778 0.345889 0.938276i $$-0.387577\pi$$
0.345889 + 0.938276i $$0.387577\pi$$
$$912$$ −6.97468e6 −0.277675
$$913$$ −1.32935e7 −0.527793
$$914$$ 5.91518e6 0.234208
$$915$$ 0 0
$$916$$ 3.30005e6 0.129952
$$917$$ 5.34192e6 0.209785
$$918$$ 1.59505e6 0.0624696
$$919$$ −1.72198e7 −0.672573 −0.336286 0.941760i $$-0.609171\pi$$
−0.336286 + 0.941760i $$0.609171\pi$$
$$920$$ 0 0
$$921$$ 1.21348e7 0.471393
$$922$$ −2.78058e7 −1.07723
$$923$$ −1.40585e7 −0.543168
$$924$$ −698677. −0.0269213
$$925$$ 0 0
$$926$$ −1.96848e7 −0.754402
$$927$$ 530393. 0.0202721
$$928$$ 3.93183e6 0.149874
$$929$$ 5.47137e6 0.207997 0.103998 0.994577i $$-0.466836\pi$$
0.103998 + 0.994577i $$0.466836\pi$$
$$930$$ 0 0
$$931$$ 1.46494e6 0.0553919
$$932$$ −4.71193e6 −0.177688
$$933$$ 2.27326e7 0.854960
$$934$$ 4.87525e7 1.82865
$$935$$ 0 0
$$936$$ 7.80014e6 0.291013
$$937$$ −8.04892e6 −0.299494 −0.149747 0.988724i $$-0.547846\pi$$
−0.149747 + 0.988724i $$0.547846\pi$$
$$938$$ 9.27072e6 0.344038
$$939$$ 2.00755e7 0.743022
$$940$$ 0 0
$$941$$ −1.13721e7 −0.418665 −0.209333 0.977844i $$-0.567129\pi$$
−0.209333 + 0.977844i $$0.567129\pi$$
$$942$$ −1.15842e7 −0.425341
$$943$$ −2.21349e7 −0.810584
$$944$$ −1.94379e7 −0.709938
$$945$$ 0 0
$$946$$ −4.26941e6 −0.155110
$$947$$ −4.97017e7 −1.80093 −0.900463 0.434932i $$-0.856772\pi$$
−0.900463 + 0.434932i $$0.856772\pi$$
$$948$$ −381267. −0.0137787
$$949$$ 1.44199e7 0.519751
$$950$$ 0 0
$$951$$ −1.08817e6 −0.0390162
$$952$$ 1.00464e7 0.359266
$$953$$ −2.95809e7 −1.05507 −0.527533 0.849534i $$-0.676883\pi$$
−0.527533 + 0.849534i $$0.676883\pi$$
$$954$$ 6.44438e6 0.229251
$$955$$ 0 0
$$956$$ −3.44385e6 −0.121871
$$957$$ −2.29955e6 −0.0811639
$$958$$ 2.42830e7 0.854849
$$959$$ −4.95526e7 −1.73988
$$960$$ 0 0
$$961$$ −2.86037e7 −0.999110
$$962$$ 1.40171e7 0.488337
$$963$$ −1.03008e7 −0.357937
$$964$$ 6.85161e6 0.237465
$$965$$ 0 0
$$966$$ 7.04654e6 0.242959
$$967$$ −3.10475e7 −1.06773 −0.533863 0.845571i $$-0.679260\pi$$
−0.533863 + 0.845571i $$0.679260\pi$$
$$968$$ 2.81950e6 0.0967128
$$969$$ 3.54863e6 0.121409
$$970$$ 0 0
$$971$$ −1.89426e7 −0.644751 −0.322376 0.946612i $$-0.604481\pi$$
−0.322376 + 0.946612i $$0.604481\pi$$
$$972$$ 306912. 0.0104195
$$973$$ −3.72803e7 −1.26240
$$974$$ 3.66113e7 1.23657
$$975$$ 0 0
$$976$$ −8.94489e6 −0.300573
$$977$$ −5.49466e7 −1.84164 −0.920819 0.389989i $$-0.872479\pi$$
−0.920819 + 0.389989i $$0.872479\pi$$
$$978$$ 1.94602e7 0.650580
$$979$$ −8.43050e6 −0.281123
$$980$$ 0 0
$$981$$ 4.64010e6 0.153941
$$982$$ −4.69072e6 −0.155225
$$983$$ 4.72203e7 1.55864 0.779318 0.626628i $$-0.215565\pi$$
0.779318 + 0.626628i $$0.215565\pi$$
$$984$$ 3.13128e7 1.03094
$$985$$ 0 0
$$986$$ 4.62022e6 0.151346
$$987$$ −1.67739e7 −0.548076
$$988$$ 2.42480e6 0.0790283
$$989$$ −8.35018e6 −0.271459
$$990$$ 0 0
$$991$$ 4.48844e7 1.45182 0.725908 0.687792i $$-0.241420\pi$$
0.725908 + 0.687792i $$0.241420\pi$$
$$992$$ 297198. 0.00958885
$$993$$ −5.20632e6 −0.167555
$$994$$ 1.79662e7 0.576753
$$995$$ 0 0
$$996$$ 5.13925e6 0.164154
$$997$$ −5.08797e7 −1.62109 −0.810543 0.585679i $$-0.800828\pi$$
−0.810543 + 0.585679i $$0.800828\pi$$
$$998$$ 3.69295e7 1.17367
$$999$$ 3.94714e6 0.125132
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 825.6.a.j.1.1 3
5.4 even 2 165.6.a.a.1.3 3
15.14 odd 2 495.6.a.e.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.3 3 5.4 even 2
495.6.a.e.1.1 3 15.14 odd 2
825.6.a.j.1.1 3 1.1 even 1 trivial