Properties

 Label 147.4.a.c.1.1 Level $147$ Weight $4$ Character 147.1 Self dual yes Analytic conductor $8.673$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 147.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +18.0000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +18.0000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} -54.0000 q^{10} -36.0000 q^{11} +3.00000 q^{12} +34.0000 q^{13} +54.0000 q^{15} -71.0000 q^{16} -42.0000 q^{17} -27.0000 q^{18} +124.000 q^{19} +18.0000 q^{20} +108.000 q^{22} +63.0000 q^{24} +199.000 q^{25} -102.000 q^{26} +27.0000 q^{27} +102.000 q^{29} -162.000 q^{30} +160.000 q^{31} +45.0000 q^{32} -108.000 q^{33} +126.000 q^{34} +9.00000 q^{36} +398.000 q^{37} -372.000 q^{38} +102.000 q^{39} +378.000 q^{40} +318.000 q^{41} -268.000 q^{43} -36.0000 q^{44} +162.000 q^{45} -240.000 q^{47} -213.000 q^{48} -597.000 q^{50} -126.000 q^{51} +34.0000 q^{52} -498.000 q^{53} -81.0000 q^{54} -648.000 q^{55} +372.000 q^{57} -306.000 q^{58} +132.000 q^{59} +54.0000 q^{60} -398.000 q^{61} -480.000 q^{62} +433.000 q^{64} +612.000 q^{65} +324.000 q^{66} +92.0000 q^{67} -42.0000 q^{68} -720.000 q^{71} +189.000 q^{72} +502.000 q^{73} -1194.00 q^{74} +597.000 q^{75} +124.000 q^{76} -306.000 q^{78} -1024.00 q^{79} -1278.00 q^{80} +81.0000 q^{81} -954.000 q^{82} +204.000 q^{83} -756.000 q^{85} +804.000 q^{86} +306.000 q^{87} -756.000 q^{88} -354.000 q^{89} -486.000 q^{90} +480.000 q^{93} +720.000 q^{94} +2232.00 q^{95} +135.000 q^{96} +286.000 q^{97} -324.000 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 1.00000 0.125000
$$5$$ 18.0000 1.60997 0.804984 0.593296i $$-0.202174\pi$$
0.804984 + 0.593296i $$0.202174\pi$$
$$6$$ −9.00000 −0.612372
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ 9.00000 0.333333
$$10$$ −54.0000 −1.70763
$$11$$ −36.0000 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$12$$ 3.00000 0.0721688
$$13$$ 34.0000 0.725377 0.362689 0.931910i $$-0.381859\pi$$
0.362689 + 0.931910i $$0.381859\pi$$
$$14$$ 0 0
$$15$$ 54.0000 0.929516
$$16$$ −71.0000 −1.10938
$$17$$ −42.0000 −0.599206 −0.299603 0.954064i $$-0.596854\pi$$
−0.299603 + 0.954064i $$0.596854\pi$$
$$18$$ −27.0000 −0.353553
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 18.0000 0.201246
$$21$$ 0 0
$$22$$ 108.000 1.04662
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 63.0000 0.535826
$$25$$ 199.000 1.59200
$$26$$ −102.000 −0.769379
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 102.000 0.653135 0.326568 0.945174i $$-0.394108\pi$$
0.326568 + 0.945174i $$0.394108\pi$$
$$30$$ −162.000 −0.985901
$$31$$ 160.000 0.926995 0.463498 0.886098i $$-0.346594\pi$$
0.463498 + 0.886098i $$0.346594\pi$$
$$32$$ 45.0000 0.248592
$$33$$ −108.000 −0.569709
$$34$$ 126.000 0.635554
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ 398.000 1.76840 0.884200 0.467109i $$-0.154704\pi$$
0.884200 + 0.467109i $$0.154704\pi$$
$$38$$ −372.000 −1.58806
$$39$$ 102.000 0.418797
$$40$$ 378.000 1.49418
$$41$$ 318.000 1.21130 0.605649 0.795732i $$-0.292913\pi$$
0.605649 + 0.795732i $$0.292913\pi$$
$$42$$ 0 0
$$43$$ −268.000 −0.950456 −0.475228 0.879863i $$-0.657634\pi$$
−0.475228 + 0.879863i $$0.657634\pi$$
$$44$$ −36.0000 −0.123346
$$45$$ 162.000 0.536656
$$46$$ 0 0
$$47$$ −240.000 −0.744843 −0.372421 0.928064i $$-0.621472\pi$$
−0.372421 + 0.928064i $$0.621472\pi$$
$$48$$ −213.000 −0.640498
$$49$$ 0 0
$$50$$ −597.000 −1.68857
$$51$$ −126.000 −0.345952
$$52$$ 34.0000 0.0906721
$$53$$ −498.000 −1.29067 −0.645335 0.763899i $$-0.723282\pi$$
−0.645335 + 0.763899i $$0.723282\pi$$
$$54$$ −81.0000 −0.204124
$$55$$ −648.000 −1.58866
$$56$$ 0 0
$$57$$ 372.000 0.864432
$$58$$ −306.000 −0.692755
$$59$$ 132.000 0.291270 0.145635 0.989338i $$-0.453477\pi$$
0.145635 + 0.989338i $$0.453477\pi$$
$$60$$ 54.0000 0.116190
$$61$$ −398.000 −0.835388 −0.417694 0.908588i $$-0.637162\pi$$
−0.417694 + 0.908588i $$0.637162\pi$$
$$62$$ −480.000 −0.983227
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ 612.000 1.16783
$$66$$ 324.000 0.604267
$$67$$ 92.0000 0.167755 0.0838775 0.996476i $$-0.473270\pi$$
0.0838775 + 0.996476i $$0.473270\pi$$
$$68$$ −42.0000 −0.0749007
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −720.000 −1.20350 −0.601748 0.798686i $$-0.705529\pi$$
−0.601748 + 0.798686i $$0.705529\pi$$
$$72$$ 189.000 0.309359
$$73$$ 502.000 0.804858 0.402429 0.915451i $$-0.368166\pi$$
0.402429 + 0.915451i $$0.368166\pi$$
$$74$$ −1194.00 −1.87567
$$75$$ 597.000 0.919142
$$76$$ 124.000 0.187155
$$77$$ 0 0
$$78$$ −306.000 −0.444201
$$79$$ −1024.00 −1.45834 −0.729171 0.684332i $$-0.760094\pi$$
−0.729171 + 0.684332i $$0.760094\pi$$
$$80$$ −1278.00 −1.78606
$$81$$ 81.0000 0.111111
$$82$$ −954.000 −1.28478
$$83$$ 204.000 0.269782 0.134891 0.990860i $$-0.456932\pi$$
0.134891 + 0.990860i $$0.456932\pi$$
$$84$$ 0 0
$$85$$ −756.000 −0.964703
$$86$$ 804.000 1.00811
$$87$$ 306.000 0.377088
$$88$$ −756.000 −0.915794
$$89$$ −354.000 −0.421617 −0.210809 0.977527i $$-0.567610\pi$$
−0.210809 + 0.977527i $$0.567610\pi$$
$$90$$ −486.000 −0.569210
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 480.000 0.535201
$$94$$ 720.000 0.790025
$$95$$ 2232.00 2.41051
$$96$$ 135.000 0.143525
$$97$$ 286.000 0.299370 0.149685 0.988734i $$-0.452174\pi$$
0.149685 + 0.988734i $$0.452174\pi$$
$$98$$ 0 0
$$99$$ −324.000 −0.328921
$$100$$ 199.000 0.199000
$$101$$ −414.000 −0.407867 −0.203933 0.978985i $$-0.565373\pi$$
−0.203933 + 0.978985i $$0.565373\pi$$
$$102$$ 378.000 0.366937
$$103$$ −56.0000 −0.0535713 −0.0267857 0.999641i $$-0.508527\pi$$
−0.0267857 + 0.999641i $$0.508527\pi$$
$$104$$ 714.000 0.673206
$$105$$ 0 0
$$106$$ 1494.00 1.36896
$$107$$ 12.0000 0.0108419 0.00542095 0.999985i $$-0.498274\pi$$
0.00542095 + 0.999985i $$0.498274\pi$$
$$108$$ 27.0000 0.0240563
$$109$$ 1478.00 1.29878 0.649389 0.760457i $$-0.275025\pi$$
0.649389 + 0.760457i $$0.275025\pi$$
$$110$$ 1944.00 1.68503
$$111$$ 1194.00 1.02099
$$112$$ 0 0
$$113$$ 402.000 0.334664 0.167332 0.985901i $$-0.446485\pi$$
0.167332 + 0.985901i $$0.446485\pi$$
$$114$$ −1116.00 −0.916868
$$115$$ 0 0
$$116$$ 102.000 0.0816419
$$117$$ 306.000 0.241792
$$118$$ −396.000 −0.308939
$$119$$ 0 0
$$120$$ 1134.00 0.862663
$$121$$ −35.0000 −0.0262960
$$122$$ 1194.00 0.886063
$$123$$ 954.000 0.699344
$$124$$ 160.000 0.115874
$$125$$ 1332.00 0.953102
$$126$$ 0 0
$$127$$ 1280.00 0.894344 0.447172 0.894448i $$-0.352431\pi$$
0.447172 + 0.894448i $$0.352431\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ −804.000 −0.548746
$$130$$ −1836.00 −1.23868
$$131$$ −1764.00 −1.17650 −0.588250 0.808679i $$-0.700183\pi$$
−0.588250 + 0.808679i $$0.700183\pi$$
$$132$$ −108.000 −0.0712136
$$133$$ 0 0
$$134$$ −276.000 −0.177931
$$135$$ 486.000 0.309839
$$136$$ −882.000 −0.556109
$$137$$ −2358.00 −1.47049 −0.735246 0.677800i $$-0.762934\pi$$
−0.735246 + 0.677800i $$0.762934\pi$$
$$138$$ 0 0
$$139$$ 52.0000 0.0317308 0.0158654 0.999874i $$-0.494950\pi$$
0.0158654 + 0.999874i $$0.494950\pi$$
$$140$$ 0 0
$$141$$ −720.000 −0.430035
$$142$$ 2160.00 1.27650
$$143$$ −1224.00 −0.715776
$$144$$ −639.000 −0.369792
$$145$$ 1836.00 1.05153
$$146$$ −1506.00 −0.853681
$$147$$ 0 0
$$148$$ 398.000 0.221050
$$149$$ −1746.00 −0.959986 −0.479993 0.877272i $$-0.659361\pi$$
−0.479993 + 0.877272i $$0.659361\pi$$
$$150$$ −1791.00 −0.974897
$$151$$ −232.000 −0.125032 −0.0625162 0.998044i $$-0.519913\pi$$
−0.0625162 + 0.998044i $$0.519913\pi$$
$$152$$ 2604.00 1.38955
$$153$$ −378.000 −0.199735
$$154$$ 0 0
$$155$$ 2880.00 1.49243
$$156$$ 102.000 0.0523496
$$157$$ −1694.00 −0.861120 −0.430560 0.902562i $$-0.641684\pi$$
−0.430560 + 0.902562i $$0.641684\pi$$
$$158$$ 3072.00 1.54681
$$159$$ −1494.00 −0.745169
$$160$$ 810.000 0.400226
$$161$$ 0 0
$$162$$ −243.000 −0.117851
$$163$$ −2932.00 −1.40891 −0.704454 0.709750i $$-0.748808\pi$$
−0.704454 + 0.709750i $$0.748808\pi$$
$$164$$ 318.000 0.151412
$$165$$ −1944.00 −0.917213
$$166$$ −612.000 −0.286147
$$167$$ −1176.00 −0.544920 −0.272460 0.962167i $$-0.587837\pi$$
−0.272460 + 0.962167i $$0.587837\pi$$
$$168$$ 0 0
$$169$$ −1041.00 −0.473828
$$170$$ 2268.00 1.02322
$$171$$ 1116.00 0.499080
$$172$$ −268.000 −0.118807
$$173$$ −870.000 −0.382340 −0.191170 0.981557i $$-0.561228\pi$$
−0.191170 + 0.981557i $$0.561228\pi$$
$$174$$ −918.000 −0.399962
$$175$$ 0 0
$$176$$ 2556.00 1.09469
$$177$$ 396.000 0.168165
$$178$$ 1062.00 0.447193
$$179$$ −2316.00 −0.967072 −0.483536 0.875324i $$-0.660648\pi$$
−0.483536 + 0.875324i $$0.660648\pi$$
$$180$$ 162.000 0.0670820
$$181$$ 106.000 0.0435299 0.0217650 0.999763i $$-0.493071\pi$$
0.0217650 + 0.999763i $$0.493071\pi$$
$$182$$ 0 0
$$183$$ −1194.00 −0.482312
$$184$$ 0 0
$$185$$ 7164.00 2.84707
$$186$$ −1440.00 −0.567666
$$187$$ 1512.00 0.591275
$$188$$ −240.000 −0.0931053
$$189$$ 0 0
$$190$$ −6696.00 −2.55673
$$191$$ −1128.00 −0.427326 −0.213663 0.976907i $$-0.568539\pi$$
−0.213663 + 0.976907i $$0.568539\pi$$
$$192$$ 1299.00 0.488267
$$193$$ 4034.00 1.50453 0.752263 0.658862i $$-0.228962\pi$$
0.752263 + 0.658862i $$0.228962\pi$$
$$194$$ −858.000 −0.317530
$$195$$ 1836.00 0.674250
$$196$$ 0 0
$$197$$ −1314.00 −0.475221 −0.237611 0.971360i $$-0.576364\pi$$
−0.237611 + 0.971360i $$0.576364\pi$$
$$198$$ 972.000 0.348874
$$199$$ −5096.00 −1.81531 −0.907653 0.419722i $$-0.862128\pi$$
−0.907653 + 0.419722i $$0.862128\pi$$
$$200$$ 4179.00 1.47750
$$201$$ 276.000 0.0968534
$$202$$ 1242.00 0.432608
$$203$$ 0 0
$$204$$ −126.000 −0.0432439
$$205$$ 5724.00 1.95015
$$206$$ 168.000 0.0568209
$$207$$ 0 0
$$208$$ −2414.00 −0.804715
$$209$$ −4464.00 −1.47742
$$210$$ 0 0
$$211$$ −3076.00 −1.00360 −0.501802 0.864982i $$-0.667330\pi$$
−0.501802 + 0.864982i $$0.667330\pi$$
$$212$$ −498.000 −0.161334
$$213$$ −2160.00 −0.694839
$$214$$ −36.0000 −0.0114996
$$215$$ −4824.00 −1.53020
$$216$$ 567.000 0.178609
$$217$$ 0 0
$$218$$ −4434.00 −1.37756
$$219$$ 1506.00 0.464685
$$220$$ −648.000 −0.198583
$$221$$ −1428.00 −0.434650
$$222$$ −3582.00 −1.08292
$$223$$ 1888.00 0.566950 0.283475 0.958980i $$-0.408513\pi$$
0.283475 + 0.958980i $$0.408513\pi$$
$$224$$ 0 0
$$225$$ 1791.00 0.530667
$$226$$ −1206.00 −0.354964
$$227$$ 4716.00 1.37891 0.689454 0.724330i $$-0.257851\pi$$
0.689454 + 0.724330i $$0.257851\pi$$
$$228$$ 372.000 0.108054
$$229$$ 1690.00 0.487678 0.243839 0.969816i $$-0.421593\pi$$
0.243839 + 0.969816i $$0.421593\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2142.00 0.606160
$$233$$ 138.000 0.0388012 0.0194006 0.999812i $$-0.493824\pi$$
0.0194006 + 0.999812i $$0.493824\pi$$
$$234$$ −918.000 −0.256460
$$235$$ −4320.00 −1.19917
$$236$$ 132.000 0.0364088
$$237$$ −3072.00 −0.841974
$$238$$ 0 0
$$239$$ 1896.00 0.513147 0.256573 0.966525i $$-0.417406\pi$$
0.256573 + 0.966525i $$0.417406\pi$$
$$240$$ −3834.00 −1.03118
$$241$$ 3598.00 0.961691 0.480846 0.876805i $$-0.340330\pi$$
0.480846 + 0.876805i $$0.340330\pi$$
$$242$$ 105.000 0.0278911
$$243$$ 243.000 0.0641500
$$244$$ −398.000 −0.104424
$$245$$ 0 0
$$246$$ −2862.00 −0.741766
$$247$$ 4216.00 1.08606
$$248$$ 3360.00 0.860323
$$249$$ 612.000 0.155759
$$250$$ −3996.00 −1.01092
$$251$$ 3060.00 0.769504 0.384752 0.923020i $$-0.374287\pi$$
0.384752 + 0.923020i $$0.374287\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −3840.00 −0.948595
$$255$$ −2268.00 −0.556971
$$256$$ 1513.00 0.369385
$$257$$ 6822.00 1.65582 0.827908 0.560864i $$-0.189531\pi$$
0.827908 + 0.560864i $$0.189531\pi$$
$$258$$ 2412.00 0.582033
$$259$$ 0 0
$$260$$ 612.000 0.145979
$$261$$ 918.000 0.217712
$$262$$ 5292.00 1.24787
$$263$$ 2592.00 0.607717 0.303858 0.952717i $$-0.401725\pi$$
0.303858 + 0.952717i $$0.401725\pi$$
$$264$$ −2268.00 −0.528734
$$265$$ −8964.00 −2.07794
$$266$$ 0 0
$$267$$ −1062.00 −0.243421
$$268$$ 92.0000 0.0209694
$$269$$ −8214.00 −1.86177 −0.930886 0.365311i $$-0.880963\pi$$
−0.930886 + 0.365311i $$0.880963\pi$$
$$270$$ −1458.00 −0.328634
$$271$$ 5344.00 1.19788 0.598939 0.800795i $$-0.295589\pi$$
0.598939 + 0.800795i $$0.295589\pi$$
$$272$$ 2982.00 0.664744
$$273$$ 0 0
$$274$$ 7074.00 1.55969
$$275$$ −7164.00 −1.57093
$$276$$ 0 0
$$277$$ −6514.00 −1.41295 −0.706477 0.707736i $$-0.749717\pi$$
−0.706477 + 0.707736i $$0.749717\pi$$
$$278$$ −156.000 −0.0336556
$$279$$ 1440.00 0.308998
$$280$$ 0 0
$$281$$ 6618.00 1.40497 0.702485 0.711698i $$-0.252074\pi$$
0.702485 + 0.711698i $$0.252074\pi$$
$$282$$ 2160.00 0.456121
$$283$$ −3260.00 −0.684759 −0.342380 0.939562i $$-0.611233\pi$$
−0.342380 + 0.939562i $$0.611233\pi$$
$$284$$ −720.000 −0.150437
$$285$$ 6696.00 1.39171
$$286$$ 3672.00 0.759195
$$287$$ 0 0
$$288$$ 405.000 0.0828641
$$289$$ −3149.00 −0.640953
$$290$$ −5508.00 −1.11531
$$291$$ 858.000 0.172841
$$292$$ 502.000 0.100607
$$293$$ −5118.00 −1.02047 −0.510233 0.860036i $$-0.670441\pi$$
−0.510233 + 0.860036i $$0.670441\pi$$
$$294$$ 0 0
$$295$$ 2376.00 0.468936
$$296$$ 8358.00 1.64121
$$297$$ −972.000 −0.189903
$$298$$ 5238.00 1.01822
$$299$$ 0 0
$$300$$ 597.000 0.114893
$$301$$ 0 0
$$302$$ 696.000 0.132617
$$303$$ −1242.00 −0.235482
$$304$$ −8804.00 −1.66100
$$305$$ −7164.00 −1.34495
$$306$$ 1134.00 0.211851
$$307$$ −452.000 −0.0840293 −0.0420147 0.999117i $$-0.513378\pi$$
−0.0420147 + 0.999117i $$0.513378\pi$$
$$308$$ 0 0
$$309$$ −168.000 −0.0309294
$$310$$ −8640.00 −1.58296
$$311$$ −5016.00 −0.914570 −0.457285 0.889320i $$-0.651178\pi$$
−0.457285 + 0.889320i $$0.651178\pi$$
$$312$$ 2142.00 0.388676
$$313$$ −5402.00 −0.975524 −0.487762 0.872977i $$-0.662187\pi$$
−0.487762 + 0.872977i $$0.662187\pi$$
$$314$$ 5082.00 0.913356
$$315$$ 0 0
$$316$$ −1024.00 −0.182293
$$317$$ 10086.0 1.78702 0.893511 0.449041i $$-0.148234\pi$$
0.893511 + 0.449041i $$0.148234\pi$$
$$318$$ 4482.00 0.790371
$$319$$ −3672.00 −0.644491
$$320$$ 7794.00 1.36156
$$321$$ 36.0000 0.00625958
$$322$$ 0 0
$$323$$ −5208.00 −0.897154
$$324$$ 81.0000 0.0138889
$$325$$ 6766.00 1.15480
$$326$$ 8796.00 1.49437
$$327$$ 4434.00 0.749849
$$328$$ 6678.00 1.12418
$$329$$ 0 0
$$330$$ 5832.00 0.972852
$$331$$ −8044.00 −1.33577 −0.667883 0.744267i $$-0.732799\pi$$
−0.667883 + 0.744267i $$0.732799\pi$$
$$332$$ 204.000 0.0337228
$$333$$ 3582.00 0.589467
$$334$$ 3528.00 0.577975
$$335$$ 1656.00 0.270080
$$336$$ 0 0
$$337$$ 4178.00 0.675342 0.337671 0.941264i $$-0.390361\pi$$
0.337671 + 0.941264i $$0.390361\pi$$
$$338$$ 3123.00 0.502570
$$339$$ 1206.00 0.193218
$$340$$ −756.000 −0.120588
$$341$$ −5760.00 −0.914726
$$342$$ −3348.00 −0.529354
$$343$$ 0 0
$$344$$ −5628.00 −0.882097
$$345$$ 0 0
$$346$$ 2610.00 0.405533
$$347$$ 156.000 0.0241341 0.0120670 0.999927i $$-0.496159\pi$$
0.0120670 + 0.999927i $$0.496159\pi$$
$$348$$ 306.000 0.0471360
$$349$$ 12418.0 1.90464 0.952321 0.305097i $$-0.0986888\pi$$
0.952321 + 0.305097i $$0.0986888\pi$$
$$350$$ 0 0
$$351$$ 918.000 0.139599
$$352$$ −1620.00 −0.245302
$$353$$ 7830.00 1.18059 0.590296 0.807187i $$-0.299011\pi$$
0.590296 + 0.807187i $$0.299011\pi$$
$$354$$ −1188.00 −0.178366
$$355$$ −12960.0 −1.93759
$$356$$ −354.000 −0.0527021
$$357$$ 0 0
$$358$$ 6948.00 1.02574
$$359$$ −9312.00 −1.36899 −0.684497 0.729016i $$-0.739978\pi$$
−0.684497 + 0.729016i $$0.739978\pi$$
$$360$$ 3402.00 0.498059
$$361$$ 8517.00 1.24173
$$362$$ −318.000 −0.0461705
$$363$$ −105.000 −0.0151820
$$364$$ 0 0
$$365$$ 9036.00 1.29580
$$366$$ 3582.00 0.511569
$$367$$ 3760.00 0.534797 0.267398 0.963586i $$-0.413836\pi$$
0.267398 + 0.963586i $$0.413836\pi$$
$$368$$ 0 0
$$369$$ 2862.00 0.403766
$$370$$ −21492.0 −3.01977
$$371$$ 0 0
$$372$$ 480.000 0.0669001
$$373$$ 5870.00 0.814845 0.407422 0.913240i $$-0.366428\pi$$
0.407422 + 0.913240i $$0.366428\pi$$
$$374$$ −4536.00 −0.627142
$$375$$ 3996.00 0.550273
$$376$$ −5040.00 −0.691272
$$377$$ 3468.00 0.473769
$$378$$ 0 0
$$379$$ −1852.00 −0.251005 −0.125502 0.992093i $$-0.540054\pi$$
−0.125502 + 0.992093i $$0.540054\pi$$
$$380$$ 2232.00 0.301314
$$381$$ 3840.00 0.516350
$$382$$ 3384.00 0.453247
$$383$$ −2160.00 −0.288175 −0.144087 0.989565i $$-0.546025\pi$$
−0.144087 + 0.989565i $$0.546025\pi$$
$$384$$ −4977.00 −0.661410
$$385$$ 0 0
$$386$$ −12102.0 −1.59579
$$387$$ −2412.00 −0.316819
$$388$$ 286.000 0.0374213
$$389$$ −6786.00 −0.884483 −0.442241 0.896896i $$-0.645817\pi$$
−0.442241 + 0.896896i $$0.645817\pi$$
$$390$$ −5508.00 −0.715150
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −5292.00 −0.679252
$$394$$ 3942.00 0.504048
$$395$$ −18432.0 −2.34788
$$396$$ −324.000 −0.0411152
$$397$$ 6514.00 0.823497 0.411748 0.911298i $$-0.364918\pi$$
0.411748 + 0.911298i $$0.364918\pi$$
$$398$$ 15288.0 1.92542
$$399$$ 0 0
$$400$$ −14129.0 −1.76612
$$401$$ 3330.00 0.414694 0.207347 0.978267i $$-0.433517\pi$$
0.207347 + 0.978267i $$0.433517\pi$$
$$402$$ −828.000 −0.102729
$$403$$ 5440.00 0.672421
$$404$$ −414.000 −0.0509833
$$405$$ 1458.00 0.178885
$$406$$ 0 0
$$407$$ −14328.0 −1.74499
$$408$$ −2646.00 −0.321070
$$409$$ 5398.00 0.652601 0.326301 0.945266i $$-0.394198\pi$$
0.326301 + 0.945266i $$0.394198\pi$$
$$410$$ −17172.0 −2.06845
$$411$$ −7074.00 −0.848990
$$412$$ −56.0000 −0.00669641
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3672.00 0.434341
$$416$$ 1530.00 0.180323
$$417$$ 156.000 0.0183198
$$418$$ 13392.0 1.56704
$$419$$ −13092.0 −1.52646 −0.763229 0.646128i $$-0.776387\pi$$
−0.763229 + 0.646128i $$0.776387\pi$$
$$420$$ 0 0
$$421$$ −322.000 −0.0372763 −0.0186381 0.999826i $$-0.505933\pi$$
−0.0186381 + 0.999826i $$0.505933\pi$$
$$422$$ 9228.00 1.06448
$$423$$ −2160.00 −0.248281
$$424$$ −10458.0 −1.19784
$$425$$ −8358.00 −0.953935
$$426$$ 6480.00 0.736988
$$427$$ 0 0
$$428$$ 12.0000 0.00135524
$$429$$ −3672.00 −0.413254
$$430$$ 14472.0 1.62303
$$431$$ 2616.00 0.292363 0.146181 0.989258i $$-0.453302\pi$$
0.146181 + 0.989258i $$0.453302\pi$$
$$432$$ −1917.00 −0.213499
$$433$$ −4322.00 −0.479681 −0.239841 0.970812i $$-0.577095\pi$$
−0.239841 + 0.970812i $$0.577095\pi$$
$$434$$ 0 0
$$435$$ 5508.00 0.607100
$$436$$ 1478.00 0.162347
$$437$$ 0 0
$$438$$ −4518.00 −0.492873
$$439$$ 9016.00 0.980205 0.490103 0.871665i $$-0.336959\pi$$
0.490103 + 0.871665i $$0.336959\pi$$
$$440$$ −13608.0 −1.47440
$$441$$ 0 0
$$442$$ 4284.00 0.461016
$$443$$ −5268.00 −0.564989 −0.282495 0.959269i $$-0.591162\pi$$
−0.282495 + 0.959269i $$0.591162\pi$$
$$444$$ 1194.00 0.127623
$$445$$ −6372.00 −0.678790
$$446$$ −5664.00 −0.601341
$$447$$ −5238.00 −0.554248
$$448$$ 0 0
$$449$$ −5310.00 −0.558117 −0.279058 0.960274i $$-0.590022\pi$$
−0.279058 + 0.960274i $$0.590022\pi$$
$$450$$ −5373.00 −0.562857
$$451$$ −11448.0 −1.19527
$$452$$ 402.000 0.0418329
$$453$$ −696.000 −0.0721875
$$454$$ −14148.0 −1.46255
$$455$$ 0 0
$$456$$ 7812.00 0.802260
$$457$$ 15770.0 1.61420 0.807100 0.590415i $$-0.201036\pi$$
0.807100 + 0.590415i $$0.201036\pi$$
$$458$$ −5070.00 −0.517261
$$459$$ −1134.00 −0.115317
$$460$$ 0 0
$$461$$ 5370.00 0.542529 0.271264 0.962505i $$-0.412558\pi$$
0.271264 + 0.962505i $$0.412558\pi$$
$$462$$ 0 0
$$463$$ −3328.00 −0.334050 −0.167025 0.985953i $$-0.553416\pi$$
−0.167025 + 0.985953i $$0.553416\pi$$
$$464$$ −7242.00 −0.724572
$$465$$ 8640.00 0.861657
$$466$$ −414.000 −0.0411549
$$467$$ −4548.00 −0.450656 −0.225328 0.974283i $$-0.572345\pi$$
−0.225328 + 0.974283i $$0.572345\pi$$
$$468$$ 306.000 0.0302240
$$469$$ 0 0
$$470$$ 12960.0 1.27192
$$471$$ −5082.00 −0.497168
$$472$$ 2772.00 0.270321
$$473$$ 9648.00 0.937876
$$474$$ 9216.00 0.893048
$$475$$ 24676.0 2.38361
$$476$$ 0 0
$$477$$ −4482.00 −0.430224
$$478$$ −5688.00 −0.544274
$$479$$ 8064.00 0.769214 0.384607 0.923080i $$-0.374337\pi$$
0.384607 + 0.923080i $$0.374337\pi$$
$$480$$ 2430.00 0.231070
$$481$$ 13532.0 1.28276
$$482$$ −10794.0 −1.02003
$$483$$ 0 0
$$484$$ −35.0000 −0.00328700
$$485$$ 5148.00 0.481977
$$486$$ −729.000 −0.0680414
$$487$$ 16616.0 1.54608 0.773042 0.634355i $$-0.218734\pi$$
0.773042 + 0.634355i $$0.218734\pi$$
$$488$$ −8358.00 −0.775305
$$489$$ −8796.00 −0.813433
$$490$$ 0 0
$$491$$ −7140.00 −0.656260 −0.328130 0.944633i $$-0.606418\pi$$
−0.328130 + 0.944633i $$0.606418\pi$$
$$492$$ 954.000 0.0874180
$$493$$ −4284.00 −0.391362
$$494$$ −12648.0 −1.15194
$$495$$ −5832.00 −0.529553
$$496$$ −11360.0 −1.02839
$$497$$ 0 0
$$498$$ −1836.00 −0.165207
$$499$$ −9124.00 −0.818530 −0.409265 0.912416i $$-0.634215\pi$$
−0.409265 + 0.912416i $$0.634215\pi$$
$$500$$ 1332.00 0.119138
$$501$$ −3528.00 −0.314610
$$502$$ −9180.00 −0.816182
$$503$$ 6552.00 0.580794 0.290397 0.956906i $$-0.406213\pi$$
0.290397 + 0.956906i $$0.406213\pi$$
$$504$$ 0 0
$$505$$ −7452.00 −0.656653
$$506$$ 0 0
$$507$$ −3123.00 −0.273565
$$508$$ 1280.00 0.111793
$$509$$ −2790.00 −0.242956 −0.121478 0.992594i $$-0.538763\pi$$
−0.121478 + 0.992594i $$0.538763\pi$$
$$510$$ 6804.00 0.590757
$$511$$ 0 0
$$512$$ 8733.00 0.753804
$$513$$ 3348.00 0.288144
$$514$$ −20466.0 −1.75626
$$515$$ −1008.00 −0.0862481
$$516$$ −804.000 −0.0685933
$$517$$ 8640.00 0.734984
$$518$$ 0 0
$$519$$ −2610.00 −0.220744
$$520$$ 12852.0 1.08384
$$521$$ 14862.0 1.24974 0.624871 0.780728i $$-0.285151\pi$$
0.624871 + 0.780728i $$0.285151\pi$$
$$522$$ −2754.00 −0.230918
$$523$$ −17660.0 −1.47652 −0.738258 0.674518i $$-0.764351\pi$$
−0.738258 + 0.674518i $$0.764351\pi$$
$$524$$ −1764.00 −0.147062
$$525$$ 0 0
$$526$$ −7776.00 −0.644581
$$527$$ −6720.00 −0.555461
$$528$$ 7668.00 0.632021
$$529$$ −12167.0 −1.00000
$$530$$ 26892.0 2.20399
$$531$$ 1188.00 0.0970900
$$532$$ 0 0
$$533$$ 10812.0 0.878649
$$534$$ 3186.00 0.258187
$$535$$ 216.000 0.0174551
$$536$$ 1932.00 0.155690
$$537$$ −6948.00 −0.558340
$$538$$ 24642.0 1.97471
$$539$$ 0 0
$$540$$ 486.000 0.0387298
$$541$$ −19834.0 −1.57621 −0.788106 0.615540i $$-0.788938\pi$$
−0.788106 + 0.615540i $$0.788938\pi$$
$$542$$ −16032.0 −1.27054
$$543$$ 318.000 0.0251320
$$544$$ −1890.00 −0.148958
$$545$$ 26604.0 2.09099
$$546$$ 0 0
$$547$$ 20972.0 1.63930 0.819651 0.572863i $$-0.194167\pi$$
0.819651 + 0.572863i $$0.194167\pi$$
$$548$$ −2358.00 −0.183812
$$549$$ −3582.00 −0.278463
$$550$$ 21492.0 1.66622
$$551$$ 12648.0 0.977900
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 19542.0 1.49866
$$555$$ 21492.0 1.64376
$$556$$ 52.0000 0.00396635
$$557$$ 21174.0 1.61072 0.805360 0.592786i $$-0.201972\pi$$
0.805360 + 0.592786i $$0.201972\pi$$
$$558$$ −4320.00 −0.327742
$$559$$ −9112.00 −0.689439
$$560$$ 0 0
$$561$$ 4536.00 0.341373
$$562$$ −19854.0 −1.49020
$$563$$ 17772.0 1.33037 0.665187 0.746677i $$-0.268352\pi$$
0.665187 + 0.746677i $$0.268352\pi$$
$$564$$ −720.000 −0.0537544
$$565$$ 7236.00 0.538798
$$566$$ 9780.00 0.726297
$$567$$ 0 0
$$568$$ −15120.0 −1.11694
$$569$$ 8250.00 0.607835 0.303917 0.952698i $$-0.401705\pi$$
0.303917 + 0.952698i $$0.401705\pi$$
$$570$$ −20088.0 −1.47613
$$571$$ 20756.0 1.52121 0.760606 0.649214i $$-0.224902\pi$$
0.760606 + 0.649214i $$0.224902\pi$$
$$572$$ −1224.00 −0.0894720
$$573$$ −3384.00 −0.246717
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 3897.00 0.281901
$$577$$ −2.00000 −0.000144300 0 −7.21500e−5 1.00000i $$-0.500023\pi$$
−7.21500e−5 1.00000i $$0.500023\pi$$
$$578$$ 9447.00 0.679833
$$579$$ 12102.0 0.868639
$$580$$ 1836.00 0.131441
$$581$$ 0 0
$$582$$ −2574.00 −0.183326
$$583$$ 17928.0 1.27359
$$584$$ 10542.0 0.746971
$$585$$ 5508.00 0.389278
$$586$$ 15354.0 1.08237
$$587$$ −26364.0 −1.85376 −0.926881 0.375354i $$-0.877521\pi$$
−0.926881 + 0.375354i $$0.877521\pi$$
$$588$$ 0 0
$$589$$ 19840.0 1.38793
$$590$$ −7128.00 −0.497382
$$591$$ −3942.00 −0.274369
$$592$$ −28258.0 −1.96182
$$593$$ −2298.00 −0.159136 −0.0795679 0.996829i $$-0.525354\pi$$
−0.0795679 + 0.996829i $$0.525354\pi$$
$$594$$ 2916.00 0.201422
$$595$$ 0 0
$$596$$ −1746.00 −0.119998
$$597$$ −15288.0 −1.04807
$$598$$ 0 0
$$599$$ 3072.00 0.209547 0.104773 0.994496i $$-0.466588\pi$$
0.104773 + 0.994496i $$0.466588\pi$$
$$600$$ 12537.0 0.853035
$$601$$ −24554.0 −1.66652 −0.833260 0.552881i $$-0.813528\pi$$
−0.833260 + 0.552881i $$0.813528\pi$$
$$602$$ 0 0
$$603$$ 828.000 0.0559184
$$604$$ −232.000 −0.0156290
$$605$$ −630.000 −0.0423358
$$606$$ 3726.00 0.249766
$$607$$ −16832.0 −1.12552 −0.562759 0.826621i $$-0.690260\pi$$
−0.562759 + 0.826621i $$0.690260\pi$$
$$608$$ 5580.00 0.372202
$$609$$ 0 0
$$610$$ 21492.0 1.42653
$$611$$ −8160.00 −0.540292
$$612$$ −378.000 −0.0249669
$$613$$ −2482.00 −0.163535 −0.0817676 0.996651i $$-0.526057\pi$$
−0.0817676 + 0.996651i $$0.526057\pi$$
$$614$$ 1356.00 0.0891266
$$615$$ 17172.0 1.12592
$$616$$ 0 0
$$617$$ −15798.0 −1.03080 −0.515400 0.856950i $$-0.672357\pi$$
−0.515400 + 0.856950i $$0.672357\pi$$
$$618$$ 504.000 0.0328056
$$619$$ 15460.0 1.00386 0.501930 0.864908i $$-0.332623\pi$$
0.501930 + 0.864908i $$0.332623\pi$$
$$620$$ 2880.00 0.186554
$$621$$ 0 0
$$622$$ 15048.0 0.970048
$$623$$ 0 0
$$624$$ −7242.00 −0.464603
$$625$$ −899.000 −0.0575360
$$626$$ 16206.0 1.03470
$$627$$ −13392.0 −0.852990
$$628$$ −1694.00 −0.107640
$$629$$ −16716.0 −1.05964
$$630$$ 0 0
$$631$$ −7720.00 −0.487050 −0.243525 0.969895i $$-0.578304\pi$$
−0.243525 + 0.969895i $$0.578304\pi$$
$$632$$ −21504.0 −1.35345
$$633$$ −9228.00 −0.579431
$$634$$ −30258.0 −1.89542
$$635$$ 23040.0 1.43987
$$636$$ −1494.00 −0.0931462
$$637$$ 0 0
$$638$$ 11016.0 0.683586
$$639$$ −6480.00 −0.401166
$$640$$ −29862.0 −1.84437
$$641$$ −17262.0 −1.06366 −0.531832 0.846850i $$-0.678496\pi$$
−0.531832 + 0.846850i $$0.678496\pi$$
$$642$$ −108.000 −0.00663928
$$643$$ 12220.0 0.749471 0.374735 0.927132i $$-0.377734\pi$$
0.374735 + 0.927132i $$0.377734\pi$$
$$644$$ 0 0
$$645$$ −14472.0 −0.883464
$$646$$ 15624.0 0.951576
$$647$$ −13560.0 −0.823955 −0.411977 0.911194i $$-0.635162\pi$$
−0.411977 + 0.911194i $$0.635162\pi$$
$$648$$ 1701.00 0.103120
$$649$$ −4752.00 −0.287415
$$650$$ −20298.0 −1.22485
$$651$$ 0 0
$$652$$ −2932.00 −0.176113
$$653$$ 23094.0 1.38398 0.691989 0.721908i $$-0.256735\pi$$
0.691989 + 0.721908i $$0.256735\pi$$
$$654$$ −13302.0 −0.795335
$$655$$ −31752.0 −1.89413
$$656$$ −22578.0 −1.34378
$$657$$ 4518.00 0.268286
$$658$$ 0 0
$$659$$ 22548.0 1.33285 0.666423 0.745574i $$-0.267825\pi$$
0.666423 + 0.745574i $$0.267825\pi$$
$$660$$ −1944.00 −0.114652
$$661$$ −17462.0 −1.02752 −0.513762 0.857933i $$-0.671748\pi$$
−0.513762 + 0.857933i $$0.671748\pi$$
$$662$$ 24132.0 1.41679
$$663$$ −4284.00 −0.250945
$$664$$ 4284.00 0.250379
$$665$$ 0 0
$$666$$ −10746.0 −0.625224
$$667$$ 0 0
$$668$$ −1176.00 −0.0681150
$$669$$ 5664.00 0.327329
$$670$$ −4968.00 −0.286464
$$671$$ 14328.0 0.824331
$$672$$ 0 0
$$673$$ −22462.0 −1.28655 −0.643274 0.765636i $$-0.722424\pi$$
−0.643274 + 0.765636i $$0.722424\pi$$
$$674$$ −12534.0 −0.716308
$$675$$ 5373.00 0.306381
$$676$$ −1041.00 −0.0592285
$$677$$ 25554.0 1.45069 0.725347 0.688383i $$-0.241679\pi$$
0.725347 + 0.688383i $$0.241679\pi$$
$$678$$ −3618.00 −0.204939
$$679$$ 0 0
$$680$$ −15876.0 −0.895319
$$681$$ 14148.0 0.796112
$$682$$ 17280.0 0.970213
$$683$$ 9276.00 0.519672 0.259836 0.965653i $$-0.416331\pi$$
0.259836 + 0.965653i $$0.416331\pi$$
$$684$$ 1116.00 0.0623850
$$685$$ −42444.0 −2.36745
$$686$$ 0 0
$$687$$ 5070.00 0.281561
$$688$$ 19028.0 1.05441
$$689$$ −16932.0 −0.936223
$$690$$ 0 0
$$691$$ −27380.0 −1.50736 −0.753679 0.657243i $$-0.771723\pi$$
−0.753679 + 0.657243i $$0.771723\pi$$
$$692$$ −870.000 −0.0477925
$$693$$ 0 0
$$694$$ −468.000 −0.0255980
$$695$$ 936.000 0.0510856
$$696$$ 6426.00 0.349967
$$697$$ −13356.0 −0.725817
$$698$$ −37254.0 −2.02018
$$699$$ 414.000 0.0224019
$$700$$ 0 0
$$701$$ 25830.0 1.39171 0.695853 0.718184i $$-0.255027\pi$$
0.695853 + 0.718184i $$0.255027\pi$$
$$702$$ −2754.00 −0.148067
$$703$$ 49352.0 2.64772
$$704$$ −15588.0 −0.834510
$$705$$ −12960.0 −0.692343
$$706$$ −23490.0 −1.25221
$$707$$ 0 0
$$708$$ 396.000 0.0210206
$$709$$ −6226.00 −0.329792 −0.164896 0.986311i $$-0.552729\pi$$
−0.164896 + 0.986311i $$0.552729\pi$$
$$710$$ 38880.0 2.05513
$$711$$ −9216.00 −0.486114
$$712$$ −7434.00 −0.391293
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −22032.0 −1.15238
$$716$$ −2316.00 −0.120884
$$717$$ 5688.00 0.296265
$$718$$ 27936.0 1.45204
$$719$$ 15072.0 0.781767 0.390884 0.920440i $$-0.372169\pi$$
0.390884 + 0.920440i $$0.372169\pi$$
$$720$$ −11502.0 −0.595353
$$721$$ 0 0
$$722$$ −25551.0 −1.31705
$$723$$ 10794.0 0.555233
$$724$$ 106.000 0.00544124
$$725$$ 20298.0 1.03979
$$726$$ 315.000 0.0161030
$$727$$ 32920.0 1.67942 0.839708 0.543038i $$-0.182726\pi$$
0.839708 + 0.543038i $$0.182726\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −27108.0 −1.37440
$$731$$ 11256.0 0.569519
$$732$$ −1194.00 −0.0602889
$$733$$ 6946.00 0.350009 0.175004 0.984568i $$-0.444006\pi$$
0.175004 + 0.984568i $$0.444006\pi$$
$$734$$ −11280.0 −0.567238
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3312.00 −0.165535
$$738$$ −8586.00 −0.428259
$$739$$ −2356.00 −0.117276 −0.0586379 0.998279i $$-0.518676\pi$$
−0.0586379 + 0.998279i $$0.518676\pi$$
$$740$$ 7164.00 0.355884
$$741$$ 12648.0 0.627039
$$742$$ 0 0
$$743$$ −23520.0 −1.16133 −0.580663 0.814144i $$-0.697207\pi$$
−0.580663 + 0.814144i $$0.697207\pi$$
$$744$$ 10080.0 0.496708
$$745$$ −31428.0 −1.54555
$$746$$ −17610.0 −0.864273
$$747$$ 1836.00 0.0899273
$$748$$ 1512.00 0.0739094
$$749$$ 0 0
$$750$$ −11988.0 −0.583653
$$751$$ 3008.00 0.146156 0.0730782 0.997326i $$-0.476718\pi$$
0.0730782 + 0.997326i $$0.476718\pi$$
$$752$$ 17040.0 0.826310
$$753$$ 9180.00 0.444273
$$754$$ −10404.0 −0.502508
$$755$$ −4176.00 −0.201298
$$756$$ 0 0
$$757$$ −20770.0 −0.997224 −0.498612 0.866825i $$-0.666157\pi$$
−0.498612 + 0.866825i $$0.666157\pi$$
$$758$$ 5556.00 0.266231
$$759$$ 0 0
$$760$$ 46872.0 2.23714
$$761$$ −11538.0 −0.549609 −0.274804 0.961500i $$-0.588613\pi$$
−0.274804 + 0.961500i $$0.588613\pi$$
$$762$$ −11520.0 −0.547671
$$763$$ 0 0
$$764$$ −1128.00 −0.0534157
$$765$$ −6804.00 −0.321568
$$766$$ 6480.00 0.305655
$$767$$ 4488.00 0.211281
$$768$$ 4539.00 0.213264
$$769$$ −8498.00 −0.398499 −0.199249 0.979949i $$-0.563850\pi$$
−0.199249 + 0.979949i $$0.563850\pi$$
$$770$$ 0 0
$$771$$ 20466.0 0.955986
$$772$$ 4034.00 0.188066
$$773$$ 32322.0 1.50393 0.751967 0.659200i $$-0.229105\pi$$
0.751967 + 0.659200i $$0.229105\pi$$
$$774$$ 7236.00 0.336037
$$775$$ 31840.0 1.47578
$$776$$ 6006.00 0.277839
$$777$$ 0 0
$$778$$ 20358.0 0.938136
$$779$$ 39432.0 1.81360
$$780$$ 1836.00 0.0842812
$$781$$ 25920.0 1.18757
$$782$$ 0 0
$$783$$ 2754.00 0.125696
$$784$$ 0 0
$$785$$ −30492.0 −1.38638
$$786$$ 15876.0 0.720456
$$787$$ −26228.0 −1.18796 −0.593982 0.804479i $$-0.702445\pi$$
−0.593982 + 0.804479i $$0.702445\pi$$
$$788$$ −1314.00 −0.0594027
$$789$$ 7776.00 0.350866
$$790$$ 55296.0 2.49031
$$791$$ 0 0
$$792$$ −6804.00 −0.305265
$$793$$ −13532.0 −0.605972
$$794$$ −19542.0 −0.873450
$$795$$ −26892.0 −1.19970
$$796$$ −5096.00 −0.226913
$$797$$ 43338.0 1.92611 0.963056 0.269302i $$-0.0867931\pi$$
0.963056 + 0.269302i $$0.0867931\pi$$
$$798$$ 0 0
$$799$$ 10080.0 0.446314
$$800$$ 8955.00 0.395759
$$801$$ −3186.00 −0.140539
$$802$$ −9990.00 −0.439849
$$803$$ −18072.0 −0.794206
$$804$$ 276.000 0.0121067
$$805$$ 0 0
$$806$$ −16320.0 −0.713210
$$807$$ −24642.0 −1.07489
$$808$$ −8694.00 −0.378532
$$809$$ −28902.0 −1.25604 −0.628022 0.778195i $$-0.716135\pi$$
−0.628022 + 0.778195i $$0.716135\pi$$
$$810$$ −4374.00 −0.189737
$$811$$ −27164.0 −1.17615 −0.588075 0.808807i $$-0.700114\pi$$
−0.588075 + 0.808807i $$0.700114\pi$$
$$812$$ 0 0
$$813$$ 16032.0 0.691595
$$814$$ 42984.0 1.85085
$$815$$ −52776.0 −2.26830
$$816$$ 8946.00 0.383790
$$817$$ −33232.0 −1.42306
$$818$$ −16194.0 −0.692188
$$819$$ 0 0
$$820$$ 5724.00 0.243769
$$821$$ −17202.0 −0.731247 −0.365624 0.930763i $$-0.619144\pi$$
−0.365624 + 0.930763i $$0.619144\pi$$
$$822$$ 21222.0 0.900489
$$823$$ −5992.00 −0.253789 −0.126894 0.991916i $$-0.540501\pi$$
−0.126894 + 0.991916i $$0.540501\pi$$
$$824$$ −1176.00 −0.0497183
$$825$$ −21492.0 −0.906976
$$826$$ 0 0
$$827$$ 25884.0 1.08836 0.544181 0.838968i $$-0.316841\pi$$
0.544181 + 0.838968i $$0.316841\pi$$
$$828$$ 0 0
$$829$$ 1474.00 0.0617541 0.0308770 0.999523i $$-0.490170\pi$$
0.0308770 + 0.999523i $$0.490170\pi$$
$$830$$ −11016.0 −0.460688
$$831$$ −19542.0 −0.815770
$$832$$ 14722.0 0.613454
$$833$$ 0 0
$$834$$ −468.000 −0.0194311
$$835$$ −21168.0 −0.877304
$$836$$ −4464.00 −0.184678
$$837$$ 4320.00 0.178400
$$838$$ 39276.0 1.61905
$$839$$ −33528.0 −1.37964 −0.689818 0.723983i $$-0.742310\pi$$
−0.689818 + 0.723983i $$0.742310\pi$$
$$840$$ 0 0
$$841$$ −13985.0 −0.573414
$$842$$ 966.000 0.0395375
$$843$$ 19854.0 0.811160
$$844$$ −3076.00 −0.125451
$$845$$ −18738.0 −0.762848
$$846$$ 6480.00 0.263342
$$847$$ 0 0
$$848$$ 35358.0 1.43184
$$849$$ −9780.00 −0.395346
$$850$$ 25074.0 1.01180
$$851$$ 0 0
$$852$$ −2160.00 −0.0868549
$$853$$ −1190.00 −0.0477665 −0.0238832 0.999715i $$-0.507603\pi$$
−0.0238832 + 0.999715i $$0.507603\pi$$
$$854$$ 0 0
$$855$$ 20088.0 0.803503
$$856$$ 252.000 0.0100621
$$857$$ −34578.0 −1.37825 −0.689126 0.724642i $$-0.742005\pi$$
−0.689126 + 0.724642i $$0.742005\pi$$
$$858$$ 11016.0 0.438322
$$859$$ 44404.0 1.76373 0.881865 0.471501i $$-0.156288\pi$$
0.881865 + 0.471501i $$0.156288\pi$$
$$860$$ −4824.00 −0.191276
$$861$$ 0 0
$$862$$ −7848.00 −0.310097
$$863$$ −38328.0 −1.51182 −0.755910 0.654676i $$-0.772805\pi$$
−0.755910 + 0.654676i $$0.772805\pi$$
$$864$$ 1215.00 0.0478416
$$865$$ −15660.0 −0.615556
$$866$$ 12966.0 0.508779
$$867$$ −9447.00 −0.370054
$$868$$ 0 0
$$869$$ 36864.0 1.43904
$$870$$ −16524.0 −0.643927
$$871$$ 3128.00 0.121686
$$872$$ 31038.0 1.20537
$$873$$ 2574.00 0.0997900
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 1506.00 0.0580856
$$877$$ −38842.0 −1.49555 −0.747777 0.663950i $$-0.768879\pi$$
−0.747777 + 0.663950i $$0.768879\pi$$
$$878$$ −27048.0 −1.03966
$$879$$ −15354.0 −0.589167
$$880$$ 46008.0 1.76242
$$881$$ 35046.0 1.34022 0.670108 0.742264i $$-0.266248\pi$$
0.670108 + 0.742264i $$0.266248\pi$$
$$882$$ 0 0
$$883$$ 14204.0 0.541339 0.270670 0.962672i $$-0.412755\pi$$
0.270670 + 0.962672i $$0.412755\pi$$
$$884$$ −1428.00 −0.0543313
$$885$$ 7128.00 0.270740
$$886$$ 15804.0 0.599262
$$887$$ 26136.0 0.989359 0.494679 0.869076i $$-0.335286\pi$$
0.494679 + 0.869076i $$0.335286\pi$$
$$888$$ 25074.0 0.947554
$$889$$ 0 0
$$890$$ 19116.0 0.719966
$$891$$ −2916.00 −0.109640
$$892$$ 1888.00 0.0708687
$$893$$ −29760.0 −1.11521
$$894$$ 15714.0 0.587869
$$895$$ −41688.0 −1.55696
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 15930.0 0.591972
$$899$$ 16320.0 0.605453
$$900$$ 1791.00 0.0663333
$$901$$ 20916.0 0.773377
$$902$$ 34344.0 1.26777
$$903$$ 0 0
$$904$$ 8442.00 0.310594
$$905$$ 1908.00 0.0700818
$$906$$ 2088.00 0.0765664
$$907$$ −9052.00 −0.331386 −0.165693 0.986177i $$-0.552986\pi$$
−0.165693 + 0.986177i $$0.552986\pi$$
$$908$$ 4716.00 0.172363
$$909$$ −3726.00 −0.135956
$$910$$ 0 0
$$911$$ 5016.00 0.182423 0.0912116 0.995832i $$-0.470926\pi$$
0.0912116 + 0.995832i $$0.470926\pi$$
$$912$$ −26412.0 −0.958979
$$913$$ −7344.00 −0.266211
$$914$$ −47310.0 −1.71212
$$915$$ −21492.0 −0.776507
$$916$$ 1690.00 0.0609598
$$917$$ 0 0
$$918$$ 3402.00 0.122312
$$919$$ 44552.0 1.59917 0.799584 0.600555i $$-0.205054\pi$$
0.799584 + 0.600555i $$0.205054\pi$$
$$920$$ 0 0
$$921$$ −1356.00 −0.0485144
$$922$$ −16110.0 −0.575439
$$923$$ −24480.0 −0.872989
$$924$$ 0 0
$$925$$ 79202.0 2.81529
$$926$$ 9984.00 0.354314
$$927$$ −504.000 −0.0178571
$$928$$ 4590.00 0.162364
$$929$$ −24234.0 −0.855858 −0.427929 0.903812i $$-0.640757\pi$$
−0.427929 + 0.903812i $$0.640757\pi$$
$$930$$ −25920.0 −0.913925
$$931$$ 0 0
$$932$$ 138.000 0.00485015
$$933$$ −15048.0 −0.528027
$$934$$ 13644.0 0.477993
$$935$$ 27216.0 0.951934
$$936$$ 6426.00 0.224402
$$937$$ 13894.0 0.484415 0.242208 0.970224i $$-0.422128\pi$$
0.242208 + 0.970224i $$0.422128\pi$$
$$938$$ 0 0
$$939$$ −16206.0 −0.563219
$$940$$ −4320.00 −0.149897
$$941$$ −46758.0 −1.61984 −0.809919 0.586542i $$-0.800489\pi$$
−0.809919 + 0.586542i $$0.800489\pi$$
$$942$$ 15246.0 0.527326
$$943$$ 0 0
$$944$$ −9372.00 −0.323128
$$945$$ 0 0
$$946$$ −28944.0 −0.994768
$$947$$ 13812.0 0.473949 0.236974 0.971516i $$-0.423844\pi$$
0.236974 + 0.971516i $$0.423844\pi$$
$$948$$ −3072.00 −0.105247
$$949$$ 17068.0 0.583826
$$950$$ −74028.0 −2.52820
$$951$$ 30258.0 1.03174
$$952$$ 0 0
$$953$$ −58518.0 −1.98907 −0.994535 0.104402i $$-0.966707\pi$$
−0.994535 + 0.104402i $$0.966707\pi$$
$$954$$ 13446.0 0.456321
$$955$$ −20304.0 −0.687981
$$956$$ 1896.00 0.0641433
$$957$$ −11016.0 −0.372097
$$958$$ −24192.0 −0.815875
$$959$$ 0 0
$$960$$ 23382.0 0.786095
$$961$$ −4191.00 −0.140680
$$962$$ −40596.0 −1.36057
$$963$$ 108.000 0.00361397
$$964$$ 3598.00 0.120211
$$965$$ 72612.0 2.42224
$$966$$ 0 0
$$967$$ 19640.0 0.653133 0.326567 0.945174i $$-0.394108\pi$$
0.326567 + 0.945174i $$0.394108\pi$$
$$968$$ −735.000 −0.0244047
$$969$$ −15624.0 −0.517972
$$970$$ −15444.0 −0.511213
$$971$$ 58308.0 1.92708 0.963539 0.267568i $$-0.0862200\pi$$
0.963539 + 0.267568i $$0.0862200\pi$$
$$972$$ 243.000 0.00801875
$$973$$ 0 0
$$974$$ −49848.0 −1.63987
$$975$$ 20298.0 0.666724
$$976$$ 28258.0 0.926759
$$977$$ −23550.0 −0.771168 −0.385584 0.922673i $$-0.626000\pi$$
−0.385584 + 0.922673i $$0.626000\pi$$
$$978$$ 26388.0 0.862776
$$979$$ 12744.0 0.416037
$$980$$ 0 0
$$981$$ 13302.0 0.432926
$$982$$ 21420.0 0.696069
$$983$$ −15768.0 −0.511619 −0.255809 0.966727i $$-0.582342\pi$$
−0.255809 + 0.966727i $$0.582342\pi$$
$$984$$ 20034.0 0.649045
$$985$$ −23652.0 −0.765092
$$986$$ 12852.0 0.415102
$$987$$ 0 0
$$988$$ 4216.00 0.135758
$$989$$ 0 0
$$990$$ 17496.0 0.561676
$$991$$ 35264.0 1.13037 0.565186 0.824964i $$-0.308805\pi$$
0.565186 + 0.824964i $$0.308805\pi$$
$$992$$ 7200.00 0.230444
$$993$$ −24132.0 −0.771204
$$994$$ 0 0
$$995$$ −91728.0 −2.92259
$$996$$ 612.000 0.0194698
$$997$$ 29338.0 0.931940 0.465970 0.884801i $$-0.345706\pi$$
0.465970 + 0.884801i $$0.345706\pi$$
$$998$$ 27372.0 0.868182
$$999$$ 10746.0 0.340329
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 147.4.a.c.1.1 1
3.2 odd 2 441.4.a.j.1.1 1
4.3 odd 2 2352.4.a.r.1.1 1
7.2 even 3 147.4.e.g.67.1 2
7.3 odd 6 147.4.e.i.79.1 2
7.4 even 3 147.4.e.g.79.1 2
7.5 odd 6 147.4.e.i.67.1 2
7.6 odd 2 21.4.a.a.1.1 1
21.2 odd 6 441.4.e.d.361.1 2
21.5 even 6 441.4.e.b.361.1 2
21.11 odd 6 441.4.e.d.226.1 2
21.17 even 6 441.4.e.b.226.1 2
21.20 even 2 63.4.a.c.1.1 1
28.27 even 2 336.4.a.f.1.1 1
35.13 even 4 525.4.d.c.274.2 2
35.27 even 4 525.4.d.c.274.1 2
35.34 odd 2 525.4.a.g.1.1 1
56.13 odd 2 1344.4.a.ba.1.1 1
56.27 even 2 1344.4.a.n.1.1 1
84.83 odd 2 1008.4.a.v.1.1 1
105.104 even 2 1575.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 7.6 odd 2
63.4.a.c.1.1 1 21.20 even 2
147.4.a.c.1.1 1 1.1 even 1 trivial
147.4.e.g.67.1 2 7.2 even 3
147.4.e.g.79.1 2 7.4 even 3
147.4.e.i.67.1 2 7.5 odd 6
147.4.e.i.79.1 2 7.3 odd 6
336.4.a.f.1.1 1 28.27 even 2
441.4.a.j.1.1 1 3.2 odd 2
441.4.e.b.226.1 2 21.17 even 6
441.4.e.b.361.1 2 21.5 even 6
441.4.e.d.226.1 2 21.11 odd 6
441.4.e.d.361.1 2 21.2 odd 6
525.4.a.g.1.1 1 35.34 odd 2
525.4.d.c.274.1 2 35.27 even 4
525.4.d.c.274.2 2 35.13 even 4
1008.4.a.v.1.1 1 84.83 odd 2
1344.4.a.n.1.1 1 56.27 even 2
1344.4.a.ba.1.1 1 56.13 odd 2
1575.4.a.b.1.1 1 105.104 even 2
2352.4.a.r.1.1 1 4.3 odd 2