Properties

 Label 1232.4.a.s.1.4 Level $1232$ Weight $4$ Character 1232.1 Self dual yes Analytic conductor $72.690$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1232.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$72.6903531271$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.522072.1 Defining polynomial: $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ x^4 - x^3 - 12*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$0.555307$$ of defining polynomial Character $$\chi$$ $$=$$ 1232.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+5.49244 q^{3} -16.0955 q^{5} +7.00000 q^{7} +3.16692 q^{9} +O(q^{10})$$ $$q+5.49244 q^{3} -16.0955 q^{5} +7.00000 q^{7} +3.16692 q^{9} +11.0000 q^{11} +35.3712 q^{13} -88.4036 q^{15} +40.4757 q^{17} -118.159 q^{19} +38.4471 q^{21} +174.510 q^{23} +134.065 q^{25} -130.902 q^{27} -262.725 q^{29} +36.1894 q^{31} +60.4169 q^{33} -112.668 q^{35} +19.0464 q^{37} +194.274 q^{39} +156.996 q^{41} -287.182 q^{43} -50.9731 q^{45} -397.244 q^{47} +49.0000 q^{49} +222.311 q^{51} +272.483 q^{53} -177.050 q^{55} -648.984 q^{57} +507.466 q^{59} +35.5608 q^{61} +22.1684 q^{63} -569.317 q^{65} -979.229 q^{67} +958.484 q^{69} -750.404 q^{71} +395.594 q^{73} +736.344 q^{75} +77.0000 q^{77} +736.516 q^{79} -804.477 q^{81} -582.975 q^{83} -651.477 q^{85} -1443.00 q^{87} -806.201 q^{89} +247.598 q^{91} +198.768 q^{93} +1901.84 q^{95} -957.232 q^{97} +34.8361 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10})$$ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 $$4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100})$$ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 5.49244 1.05702 0.528510 0.848927i $$-0.322751\pi$$
0.528510 + 0.848927i $$0.322751\pi$$
$$4$$ 0 0
$$5$$ −16.0955 −1.43962 −0.719812 0.694169i $$-0.755772\pi$$
−0.719812 + 0.694169i $$0.755772\pi$$
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 0 0
$$9$$ 3.16692 0.117293
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 35.3712 0.754631 0.377316 0.926085i $$-0.376847\pi$$
0.377316 + 0.926085i $$0.376847\pi$$
$$14$$ 0 0
$$15$$ −88.4036 −1.52171
$$16$$ 0 0
$$17$$ 40.4757 0.577459 0.288730 0.957411i $$-0.406767\pi$$
0.288730 + 0.957411i $$0.406767\pi$$
$$18$$ 0 0
$$19$$ −118.159 −1.42672 −0.713359 0.700799i $$-0.752827\pi$$
−0.713359 + 0.700799i $$0.752827\pi$$
$$20$$ 0 0
$$21$$ 38.4471 0.399516
$$22$$ 0 0
$$23$$ 174.510 1.58208 0.791039 0.611766i $$-0.209541\pi$$
0.791039 + 0.611766i $$0.209541\pi$$
$$24$$ 0 0
$$25$$ 134.065 1.07252
$$26$$ 0 0
$$27$$ −130.902 −0.933040
$$28$$ 0 0
$$29$$ −262.725 −1.68230 −0.841152 0.540799i $$-0.818122\pi$$
−0.841152 + 0.540799i $$0.818122\pi$$
$$30$$ 0 0
$$31$$ 36.1894 0.209671 0.104836 0.994490i $$-0.466568\pi$$
0.104836 + 0.994490i $$0.466568\pi$$
$$32$$ 0 0
$$33$$ 60.4169 0.318704
$$34$$ 0 0
$$35$$ −112.668 −0.544127
$$36$$ 0 0
$$37$$ 19.0464 0.0846274 0.0423137 0.999104i $$-0.486527\pi$$
0.0423137 + 0.999104i $$0.486527\pi$$
$$38$$ 0 0
$$39$$ 194.274 0.797661
$$40$$ 0 0
$$41$$ 156.996 0.598017 0.299008 0.954250i $$-0.403344\pi$$
0.299008 + 0.954250i $$0.403344\pi$$
$$42$$ 0 0
$$43$$ −287.182 −1.01849 −0.509243 0.860623i $$-0.670074\pi$$
−0.509243 + 0.860623i $$0.670074\pi$$
$$44$$ 0 0
$$45$$ −50.9731 −0.168858
$$46$$ 0 0
$$47$$ −397.244 −1.23285 −0.616425 0.787413i $$-0.711420\pi$$
−0.616425 + 0.787413i $$0.711420\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 222.311 0.610386
$$52$$ 0 0
$$53$$ 272.483 0.706196 0.353098 0.935586i $$-0.385128\pi$$
0.353098 + 0.935586i $$0.385128\pi$$
$$54$$ 0 0
$$55$$ −177.050 −0.434063
$$56$$ 0 0
$$57$$ −648.984 −1.50807
$$58$$ 0 0
$$59$$ 507.466 1.11977 0.559885 0.828570i $$-0.310845\pi$$
0.559885 + 0.828570i $$0.310845\pi$$
$$60$$ 0 0
$$61$$ 35.5608 0.0746409 0.0373205 0.999303i $$-0.488118\pi$$
0.0373205 + 0.999303i $$0.488118\pi$$
$$62$$ 0 0
$$63$$ 22.1684 0.0443326
$$64$$ 0 0
$$65$$ −569.317 −1.08639
$$66$$ 0 0
$$67$$ −979.229 −1.78555 −0.892775 0.450502i $$-0.851245\pi$$
−0.892775 + 0.450502i $$0.851245\pi$$
$$68$$ 0 0
$$69$$ 958.484 1.67229
$$70$$ 0 0
$$71$$ −750.404 −1.25432 −0.627159 0.778891i $$-0.715782\pi$$
−0.627159 + 0.778891i $$0.715782\pi$$
$$72$$ 0 0
$$73$$ 395.594 0.634257 0.317129 0.948383i $$-0.397281\pi$$
0.317129 + 0.948383i $$0.397281\pi$$
$$74$$ 0 0
$$75$$ 736.344 1.13368
$$76$$ 0 0
$$77$$ 77.0000 0.113961
$$78$$ 0 0
$$79$$ 736.516 1.04892 0.524459 0.851436i $$-0.324268\pi$$
0.524459 + 0.851436i $$0.324268\pi$$
$$80$$ 0 0
$$81$$ −804.477 −1.10354
$$82$$ 0 0
$$83$$ −582.975 −0.770962 −0.385481 0.922716i $$-0.625964\pi$$
−0.385481 + 0.922716i $$0.625964\pi$$
$$84$$ 0 0
$$85$$ −651.477 −0.831325
$$86$$ 0 0
$$87$$ −1443.00 −1.77823
$$88$$ 0 0
$$89$$ −806.201 −0.960192 −0.480096 0.877216i $$-0.659398\pi$$
−0.480096 + 0.877216i $$0.659398\pi$$
$$90$$ 0 0
$$91$$ 247.598 0.285224
$$92$$ 0 0
$$93$$ 198.768 0.221627
$$94$$ 0 0
$$95$$ 1901.84 2.05394
$$96$$ 0 0
$$97$$ −957.232 −1.00198 −0.500991 0.865453i $$-0.667031\pi$$
−0.500991 + 0.865453i $$0.667031\pi$$
$$98$$ 0 0
$$99$$ 34.8361 0.0353652
$$100$$ 0 0
$$101$$ −996.143 −0.981386 −0.490693 0.871333i $$-0.663256\pi$$
−0.490693 + 0.871333i $$0.663256\pi$$
$$102$$ 0 0
$$103$$ −1338.55 −1.28050 −0.640248 0.768169i $$-0.721168\pi$$
−0.640248 + 0.768169i $$0.721168\pi$$
$$104$$ 0 0
$$105$$ −618.825 −0.575154
$$106$$ 0 0
$$107$$ −1449.25 −1.30939 −0.654693 0.755895i $$-0.727202\pi$$
−0.654693 + 0.755895i $$0.727202\pi$$
$$108$$ 0 0
$$109$$ −654.535 −0.575166 −0.287583 0.957756i $$-0.592852\pi$$
−0.287583 + 0.957756i $$0.592852\pi$$
$$110$$ 0 0
$$111$$ 104.611 0.0894529
$$112$$ 0 0
$$113$$ −1160.63 −0.966223 −0.483111 0.875559i $$-0.660493\pi$$
−0.483111 + 0.875559i $$0.660493\pi$$
$$114$$ 0 0
$$115$$ −2808.82 −2.27760
$$116$$ 0 0
$$117$$ 112.018 0.0885131
$$118$$ 0 0
$$119$$ 283.330 0.218259
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 862.293 0.632116
$$124$$ 0 0
$$125$$ −145.906 −0.104402
$$126$$ 0 0
$$127$$ 1055.41 0.737420 0.368710 0.929545i $$-0.379800\pi$$
0.368710 + 0.929545i $$0.379800\pi$$
$$128$$ 0 0
$$129$$ −1577.33 −1.07656
$$130$$ 0 0
$$131$$ −2657.40 −1.77235 −0.886175 0.463351i $$-0.846647\pi$$
−0.886175 + 0.463351i $$0.846647\pi$$
$$132$$ 0 0
$$133$$ −827.116 −0.539249
$$134$$ 0 0
$$135$$ 2106.93 1.34323
$$136$$ 0 0
$$137$$ −147.314 −0.0918676 −0.0459338 0.998944i $$-0.514626\pi$$
−0.0459338 + 0.998944i $$0.514626\pi$$
$$138$$ 0 0
$$139$$ −902.634 −0.550794 −0.275397 0.961331i $$-0.588809\pi$$
−0.275397 + 0.961331i $$0.588809\pi$$
$$140$$ 0 0
$$141$$ −2181.84 −1.30315
$$142$$ 0 0
$$143$$ 389.083 0.227530
$$144$$ 0 0
$$145$$ 4228.69 2.42189
$$146$$ 0 0
$$147$$ 269.130 0.151003
$$148$$ 0 0
$$149$$ 1212.63 0.666727 0.333363 0.942798i $$-0.391816\pi$$
0.333363 + 0.942798i $$0.391816\pi$$
$$150$$ 0 0
$$151$$ −2565.33 −1.38254 −0.691270 0.722597i $$-0.742948\pi$$
−0.691270 + 0.722597i $$0.742948\pi$$
$$152$$ 0 0
$$153$$ 128.183 0.0677320
$$154$$ 0 0
$$155$$ −582.486 −0.301848
$$156$$ 0 0
$$157$$ 702.237 0.356972 0.178486 0.983942i $$-0.442880\pi$$
0.178486 + 0.983942i $$0.442880\pi$$
$$158$$ 0 0
$$159$$ 1496.60 0.746464
$$160$$ 0 0
$$161$$ 1221.57 0.597969
$$162$$ 0 0
$$163$$ 1146.27 0.550814 0.275407 0.961328i $$-0.411187\pi$$
0.275407 + 0.961328i $$0.411187\pi$$
$$164$$ 0 0
$$165$$ −972.439 −0.458814
$$166$$ 0 0
$$167$$ 3255.04 1.50828 0.754140 0.656714i $$-0.228054\pi$$
0.754140 + 0.656714i $$0.228054\pi$$
$$168$$ 0 0
$$169$$ −945.879 −0.430532
$$170$$ 0 0
$$171$$ −374.201 −0.167344
$$172$$ 0 0
$$173$$ 4024.24 1.76854 0.884269 0.466977i $$-0.154657\pi$$
0.884269 + 0.466977i $$0.154657\pi$$
$$174$$ 0 0
$$175$$ 938.455 0.405374
$$176$$ 0 0
$$177$$ 2787.23 1.18362
$$178$$ 0 0
$$179$$ −706.090 −0.294836 −0.147418 0.989074i $$-0.547096\pi$$
−0.147418 + 0.989074i $$0.547096\pi$$
$$180$$ 0 0
$$181$$ −1268.90 −0.521087 −0.260544 0.965462i $$-0.583902\pi$$
−0.260544 + 0.965462i $$0.583902\pi$$
$$182$$ 0 0
$$183$$ 195.316 0.0788970
$$184$$ 0 0
$$185$$ −306.562 −0.121832
$$186$$ 0 0
$$187$$ 445.233 0.174110
$$188$$ 0 0
$$189$$ −916.313 −0.352656
$$190$$ 0 0
$$191$$ −4864.58 −1.84287 −0.921436 0.388529i $$-0.872983\pi$$
−0.921436 + 0.388529i $$0.872983\pi$$
$$192$$ 0 0
$$193$$ −2675.49 −0.997855 −0.498928 0.866644i $$-0.666273\pi$$
−0.498928 + 0.866644i $$0.666273\pi$$
$$194$$ 0 0
$$195$$ −3126.94 −1.14833
$$196$$ 0 0
$$197$$ 1627.73 0.588684 0.294342 0.955700i $$-0.404899\pi$$
0.294342 + 0.955700i $$0.404899\pi$$
$$198$$ 0 0
$$199$$ 2254.07 0.802947 0.401474 0.915871i $$-0.368498\pi$$
0.401474 + 0.915871i $$0.368498\pi$$
$$200$$ 0 0
$$201$$ −5378.36 −1.88736
$$202$$ 0 0
$$203$$ −1839.08 −0.635851
$$204$$ 0 0
$$205$$ −2526.93 −0.860920
$$206$$ 0 0
$$207$$ 552.657 0.185567
$$208$$ 0 0
$$209$$ −1299.75 −0.430172
$$210$$ 0 0
$$211$$ −3112.07 −1.01537 −0.507687 0.861542i $$-0.669499\pi$$
−0.507687 + 0.861542i $$0.669499\pi$$
$$212$$ 0 0
$$213$$ −4121.55 −1.32584
$$214$$ 0 0
$$215$$ 4622.34 1.46624
$$216$$ 0 0
$$217$$ 253.326 0.0792482
$$218$$ 0 0
$$219$$ 2172.78 0.670423
$$220$$ 0 0
$$221$$ 1431.67 0.435769
$$222$$ 0 0
$$223$$ 3558.38 1.06855 0.534275 0.845311i $$-0.320585\pi$$
0.534275 + 0.845311i $$0.320585\pi$$
$$224$$ 0 0
$$225$$ 424.572 0.125799
$$226$$ 0 0
$$227$$ 2330.51 0.681416 0.340708 0.940169i $$-0.389333\pi$$
0.340708 + 0.940169i $$0.389333\pi$$
$$228$$ 0 0
$$229$$ 676.106 0.195102 0.0975509 0.995231i $$-0.468899\pi$$
0.0975509 + 0.995231i $$0.468899\pi$$
$$230$$ 0 0
$$231$$ 422.918 0.120459
$$232$$ 0 0
$$233$$ 1620.20 0.455548 0.227774 0.973714i $$-0.426855\pi$$
0.227774 + 0.973714i $$0.426855\pi$$
$$234$$ 0 0
$$235$$ 6393.84 1.77484
$$236$$ 0 0
$$237$$ 4045.27 1.10873
$$238$$ 0 0
$$239$$ 2001.39 0.541669 0.270835 0.962626i $$-0.412700\pi$$
0.270835 + 0.962626i $$0.412700\pi$$
$$240$$ 0 0
$$241$$ −1586.39 −0.424018 −0.212009 0.977268i $$-0.568001\pi$$
−0.212009 + 0.977268i $$0.568001\pi$$
$$242$$ 0 0
$$243$$ −884.196 −0.233420
$$244$$ 0 0
$$245$$ −788.679 −0.205661
$$246$$ 0 0
$$247$$ −4179.44 −1.07665
$$248$$ 0 0
$$249$$ −3201.96 −0.814923
$$250$$ 0 0
$$251$$ −2612.23 −0.656902 −0.328451 0.944521i $$-0.606527\pi$$
−0.328451 + 0.944521i $$0.606527\pi$$
$$252$$ 0 0
$$253$$ 1919.61 0.477014
$$254$$ 0 0
$$255$$ −3578.20 −0.878727
$$256$$ 0 0
$$257$$ 4198.23 1.01898 0.509491 0.860476i $$-0.329834\pi$$
0.509491 + 0.860476i $$0.329834\pi$$
$$258$$ 0 0
$$259$$ 133.325 0.0319861
$$260$$ 0 0
$$261$$ −832.028 −0.197323
$$262$$ 0 0
$$263$$ 5170.31 1.21222 0.606112 0.795380i $$-0.292728\pi$$
0.606112 + 0.795380i $$0.292728\pi$$
$$264$$ 0 0
$$265$$ −4385.74 −1.01666
$$266$$ 0 0
$$267$$ −4428.01 −1.01494
$$268$$ 0 0
$$269$$ 1648.20 0.373578 0.186789 0.982400i $$-0.440192\pi$$
0.186789 + 0.982400i $$0.440192\pi$$
$$270$$ 0 0
$$271$$ −2562.79 −0.574459 −0.287230 0.957862i $$-0.592734\pi$$
−0.287230 + 0.957862i $$0.592734\pi$$
$$272$$ 0 0
$$273$$ 1359.92 0.301487
$$274$$ 0 0
$$275$$ 1474.72 0.323377
$$276$$ 0 0
$$277$$ 2762.94 0.599310 0.299655 0.954048i $$-0.403128\pi$$
0.299655 + 0.954048i $$0.403128\pi$$
$$278$$ 0 0
$$279$$ 114.609 0.0245930
$$280$$ 0 0
$$281$$ 6453.68 1.37009 0.685044 0.728502i $$-0.259783\pi$$
0.685044 + 0.728502i $$0.259783\pi$$
$$282$$ 0 0
$$283$$ −4540.78 −0.953787 −0.476893 0.878961i $$-0.658237\pi$$
−0.476893 + 0.878961i $$0.658237\pi$$
$$284$$ 0 0
$$285$$ 10445.7 2.17106
$$286$$ 0 0
$$287$$ 1098.97 0.226029
$$288$$ 0 0
$$289$$ −3274.72 −0.666541
$$290$$ 0 0
$$291$$ −5257.54 −1.05912
$$292$$ 0 0
$$293$$ 1916.34 0.382094 0.191047 0.981581i $$-0.438812\pi$$
0.191047 + 0.981581i $$0.438812\pi$$
$$294$$ 0 0
$$295$$ −8167.92 −1.61205
$$296$$ 0 0
$$297$$ −1439.92 −0.281322
$$298$$ 0 0
$$299$$ 6172.61 1.19388
$$300$$ 0 0
$$301$$ −2010.28 −0.384951
$$302$$ 0 0
$$303$$ −5471.26 −1.03735
$$304$$ 0 0
$$305$$ −572.369 −0.107455
$$306$$ 0 0
$$307$$ −6876.30 −1.27834 −0.639171 0.769064i $$-0.720722\pi$$
−0.639171 + 0.769064i $$0.720722\pi$$
$$308$$ 0 0
$$309$$ −7351.89 −1.35351
$$310$$ 0 0
$$311$$ −8469.32 −1.54422 −0.772108 0.635492i $$-0.780797\pi$$
−0.772108 + 0.635492i $$0.780797\pi$$
$$312$$ 0 0
$$313$$ −2882.28 −0.520498 −0.260249 0.965542i $$-0.583805\pi$$
−0.260249 + 0.965542i $$0.583805\pi$$
$$314$$ 0 0
$$315$$ −356.812 −0.0638224
$$316$$ 0 0
$$317$$ 9795.31 1.73552 0.867759 0.496985i $$-0.165560\pi$$
0.867759 + 0.496985i $$0.165560\pi$$
$$318$$ 0 0
$$319$$ −2889.98 −0.507234
$$320$$ 0 0
$$321$$ −7959.92 −1.38405
$$322$$ 0 0
$$323$$ −4782.59 −0.823871
$$324$$ 0 0
$$325$$ 4742.04 0.809357
$$326$$ 0 0
$$327$$ −3595.00 −0.607963
$$328$$ 0 0
$$329$$ −2780.71 −0.465974
$$330$$ 0 0
$$331$$ −1813.70 −0.301178 −0.150589 0.988596i $$-0.548117\pi$$
−0.150589 + 0.988596i $$0.548117\pi$$
$$332$$ 0 0
$$333$$ 60.3184 0.00992621
$$334$$ 0 0
$$335$$ 15761.2 2.57052
$$336$$ 0 0
$$337$$ −11964.2 −1.93393 −0.966964 0.254913i $$-0.917953\pi$$
−0.966964 + 0.254913i $$0.917953\pi$$
$$338$$ 0 0
$$339$$ −6374.71 −1.02132
$$340$$ 0 0
$$341$$ 398.083 0.0632182
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 0 0
$$345$$ −15427.3 −2.40747
$$346$$ 0 0
$$347$$ 2283.89 0.353330 0.176665 0.984271i $$-0.443469\pi$$
0.176665 + 0.984271i $$0.443469\pi$$
$$348$$ 0 0
$$349$$ −2472.48 −0.379223 −0.189612 0.981859i $$-0.560723\pi$$
−0.189612 + 0.981859i $$0.560723\pi$$
$$350$$ 0 0
$$351$$ −4630.15 −0.704101
$$352$$ 0 0
$$353$$ 10190.0 1.53642 0.768211 0.640197i $$-0.221147\pi$$
0.768211 + 0.640197i $$0.221147\pi$$
$$354$$ 0 0
$$355$$ 12078.1 1.80575
$$356$$ 0 0
$$357$$ 1556.17 0.230704
$$358$$ 0 0
$$359$$ 43.4294 0.00638473 0.00319236 0.999995i $$-0.498984\pi$$
0.00319236 + 0.999995i $$0.498984\pi$$
$$360$$ 0 0
$$361$$ 7102.66 1.03552
$$362$$ 0 0
$$363$$ 664.585 0.0960928
$$364$$ 0 0
$$365$$ −6367.28 −0.913093
$$366$$ 0 0
$$367$$ −8775.78 −1.24821 −0.624104 0.781342i $$-0.714536\pi$$
−0.624104 + 0.781342i $$0.714536\pi$$
$$368$$ 0 0
$$369$$ 497.194 0.0701433
$$370$$ 0 0
$$371$$ 1907.38 0.266917
$$372$$ 0 0
$$373$$ −2751.89 −0.382004 −0.191002 0.981590i $$-0.561174\pi$$
−0.191002 + 0.981590i $$0.561174\pi$$
$$374$$ 0 0
$$375$$ −801.379 −0.110355
$$376$$ 0 0
$$377$$ −9292.90 −1.26952
$$378$$ 0 0
$$379$$ −2605.59 −0.353141 −0.176570 0.984288i $$-0.556500\pi$$
−0.176570 + 0.984288i $$0.556500\pi$$
$$380$$ 0 0
$$381$$ 5796.77 0.779468
$$382$$ 0 0
$$383$$ 14360.5 1.91590 0.957949 0.286940i $$-0.0926380\pi$$
0.957949 + 0.286940i $$0.0926380\pi$$
$$384$$ 0 0
$$385$$ −1239.35 −0.164060
$$386$$ 0 0
$$387$$ −909.482 −0.119461
$$388$$ 0 0
$$389$$ −10607.9 −1.38262 −0.691311 0.722557i $$-0.742967\pi$$
−0.691311 + 0.722557i $$0.742967\pi$$
$$390$$ 0 0
$$391$$ 7063.40 0.913585
$$392$$ 0 0
$$393$$ −14595.6 −1.87341
$$394$$ 0 0
$$395$$ −11854.6 −1.51005
$$396$$ 0 0
$$397$$ 7362.35 0.930745 0.465373 0.885115i $$-0.345920\pi$$
0.465373 + 0.885115i $$0.345920\pi$$
$$398$$ 0 0
$$399$$ −4542.89 −0.569997
$$400$$ 0 0
$$401$$ −264.514 −0.0329406 −0.0164703 0.999864i $$-0.505243\pi$$
−0.0164703 + 0.999864i $$0.505243\pi$$
$$402$$ 0 0
$$403$$ 1280.06 0.158224
$$404$$ 0 0
$$405$$ 12948.5 1.58868
$$406$$ 0 0
$$407$$ 209.511 0.0255161
$$408$$ 0 0
$$409$$ −1244.03 −0.150400 −0.0751999 0.997168i $$-0.523959\pi$$
−0.0751999 + 0.997168i $$0.523959\pi$$
$$410$$ 0 0
$$411$$ −809.112 −0.0971060
$$412$$ 0 0
$$413$$ 3552.26 0.423233
$$414$$ 0 0
$$415$$ 9383.27 1.10990
$$416$$ 0 0
$$417$$ −4957.66 −0.582201
$$418$$ 0 0
$$419$$ −3974.76 −0.463436 −0.231718 0.972783i $$-0.574435\pi$$
−0.231718 + 0.972783i $$0.574435\pi$$
$$420$$ 0 0
$$421$$ −14910.2 −1.72608 −0.863041 0.505134i $$-0.831443\pi$$
−0.863041 + 0.505134i $$0.831443\pi$$
$$422$$ 0 0
$$423$$ −1258.04 −0.144605
$$424$$ 0 0
$$425$$ 5426.38 0.619336
$$426$$ 0 0
$$427$$ 248.926 0.0282116
$$428$$ 0 0
$$429$$ 2137.02 0.240504
$$430$$ 0 0
$$431$$ 10622.4 1.18715 0.593574 0.804779i $$-0.297716\pi$$
0.593574 + 0.804779i $$0.297716\pi$$
$$432$$ 0 0
$$433$$ 17440.5 1.93565 0.967823 0.251630i $$-0.0809667\pi$$
0.967823 + 0.251630i $$0.0809667\pi$$
$$434$$ 0 0
$$435$$ 23225.8 2.55999
$$436$$ 0 0
$$437$$ −20620.0 −2.25718
$$438$$ 0 0
$$439$$ −13627.1 −1.48151 −0.740756 0.671774i $$-0.765533\pi$$
−0.740756 + 0.671774i $$0.765533\pi$$
$$440$$ 0 0
$$441$$ 155.179 0.0167562
$$442$$ 0 0
$$443$$ −2135.29 −0.229008 −0.114504 0.993423i $$-0.536528\pi$$
−0.114504 + 0.993423i $$0.536528\pi$$
$$444$$ 0 0
$$445$$ 12976.2 1.38232
$$446$$ 0 0
$$447$$ 6660.28 0.704744
$$448$$ 0 0
$$449$$ 17780.8 1.86889 0.934443 0.356113i $$-0.115898\pi$$
0.934443 + 0.356113i $$0.115898\pi$$
$$450$$ 0 0
$$451$$ 1726.96 0.180309
$$452$$ 0 0
$$453$$ −14089.9 −1.46137
$$454$$ 0 0
$$455$$ −3985.22 −0.410615
$$456$$ 0 0
$$457$$ 10357.0 1.06013 0.530064 0.847957i $$-0.322168\pi$$
0.530064 + 0.847957i $$0.322168\pi$$
$$458$$ 0 0
$$459$$ −5298.35 −0.538792
$$460$$ 0 0
$$461$$ −19679.5 −1.98821 −0.994106 0.108410i $$-0.965424\pi$$
−0.994106 + 0.108410i $$0.965424\pi$$
$$462$$ 0 0
$$463$$ 7171.43 0.719838 0.359919 0.932984i $$-0.382804\pi$$
0.359919 + 0.932984i $$0.382804\pi$$
$$464$$ 0 0
$$465$$ −3199.27 −0.319059
$$466$$ 0 0
$$467$$ 12192.8 1.20817 0.604085 0.796920i $$-0.293539\pi$$
0.604085 + 0.796920i $$0.293539\pi$$
$$468$$ 0 0
$$469$$ −6854.61 −0.674875
$$470$$ 0 0
$$471$$ 3856.99 0.377327
$$472$$ 0 0
$$473$$ −3159.00 −0.307085
$$474$$ 0 0
$$475$$ −15841.1 −1.53018
$$476$$ 0 0
$$477$$ 862.930 0.0828319
$$478$$ 0 0
$$479$$ −8475.11 −0.808429 −0.404215 0.914664i $$-0.632455\pi$$
−0.404215 + 0.914664i $$0.632455\pi$$
$$480$$ 0 0
$$481$$ 673.695 0.0638624
$$482$$ 0 0
$$483$$ 6709.39 0.632066
$$484$$ 0 0
$$485$$ 15407.1 1.44248
$$486$$ 0 0
$$487$$ −2735.29 −0.254513 −0.127257 0.991870i $$-0.540617\pi$$
−0.127257 + 0.991870i $$0.540617\pi$$
$$488$$ 0 0
$$489$$ 6295.81 0.582222
$$490$$ 0 0
$$491$$ 2233.82 0.205317 0.102659 0.994717i $$-0.467265\pi$$
0.102659 + 0.994717i $$0.467265\pi$$
$$492$$ 0 0
$$493$$ −10634.0 −0.971462
$$494$$ 0 0
$$495$$ −560.704 −0.0509126
$$496$$ 0 0
$$497$$ −5252.83 −0.474088
$$498$$ 0 0
$$499$$ 18100.3 1.62381 0.811904 0.583791i $$-0.198431\pi$$
0.811904 + 0.583791i $$0.198431\pi$$
$$500$$ 0 0
$$501$$ 17878.1 1.59428
$$502$$ 0 0
$$503$$ −6149.06 −0.545076 −0.272538 0.962145i $$-0.587863\pi$$
−0.272538 + 0.962145i $$0.587863\pi$$
$$504$$ 0 0
$$505$$ 16033.4 1.41283
$$506$$ 0 0
$$507$$ −5195.18 −0.455081
$$508$$ 0 0
$$509$$ 14193.9 1.23602 0.618008 0.786172i $$-0.287940\pi$$
0.618008 + 0.786172i $$0.287940\pi$$
$$510$$ 0 0
$$511$$ 2769.16 0.239727
$$512$$ 0 0
$$513$$ 15467.3 1.33118
$$514$$ 0 0
$$515$$ 21544.6 1.84343
$$516$$ 0 0
$$517$$ −4369.68 −0.371718
$$518$$ 0 0
$$519$$ 22102.9 1.86938
$$520$$ 0 0
$$521$$ 10371.8 0.872163 0.436082 0.899907i $$-0.356366\pi$$
0.436082 + 0.899907i $$0.356366\pi$$
$$522$$ 0 0
$$523$$ 11369.4 0.950569 0.475285 0.879832i $$-0.342345\pi$$
0.475285 + 0.879832i $$0.342345\pi$$
$$524$$ 0 0
$$525$$ 5154.41 0.428489
$$526$$ 0 0
$$527$$ 1464.79 0.121076
$$528$$ 0 0
$$529$$ 18286.6 1.50297
$$530$$ 0 0
$$531$$ 1607.10 0.131341
$$532$$ 0 0
$$533$$ 5553.14 0.451282
$$534$$ 0 0
$$535$$ 23326.4 1.88503
$$536$$ 0 0
$$537$$ −3878.16 −0.311648
$$538$$ 0 0
$$539$$ 539.000 0.0430730
$$540$$ 0 0
$$541$$ 5386.88 0.428096 0.214048 0.976823i $$-0.431335\pi$$
0.214048 + 0.976823i $$0.431335\pi$$
$$542$$ 0 0
$$543$$ −6969.38 −0.550800
$$544$$ 0 0
$$545$$ 10535.1 0.828024
$$546$$ 0 0
$$547$$ −13890.8 −1.08579 −0.542894 0.839801i $$-0.682672\pi$$
−0.542894 + 0.839801i $$0.682672\pi$$
$$548$$ 0 0
$$549$$ 112.618 0.00875487
$$550$$ 0 0
$$551$$ 31043.5 2.40017
$$552$$ 0 0
$$553$$ 5155.61 0.396454
$$554$$ 0 0
$$555$$ −1683.77 −0.128779
$$556$$ 0 0
$$557$$ 17498.3 1.33111 0.665553 0.746351i $$-0.268196\pi$$
0.665553 + 0.746351i $$0.268196\pi$$
$$558$$ 0 0
$$559$$ −10158.0 −0.768581
$$560$$ 0 0
$$561$$ 2445.42 0.184038
$$562$$ 0 0
$$563$$ −147.373 −0.0110320 −0.00551600 0.999985i $$-0.501756\pi$$
−0.00551600 + 0.999985i $$0.501756\pi$$
$$564$$ 0 0
$$565$$ 18681.0 1.39100
$$566$$ 0 0
$$567$$ −5631.34 −0.417097
$$568$$ 0 0
$$569$$ −14155.3 −1.04292 −0.521459 0.853276i $$-0.674612\pi$$
−0.521459 + 0.853276i $$0.674612\pi$$
$$570$$ 0 0
$$571$$ 248.361 0.0182025 0.00910123 0.999959i $$-0.497103\pi$$
0.00910123 + 0.999959i $$0.497103\pi$$
$$572$$ 0 0
$$573$$ −26718.4 −1.94795
$$574$$ 0 0
$$575$$ 23395.6 1.69681
$$576$$ 0 0
$$577$$ 19364.6 1.39715 0.698576 0.715536i $$-0.253817\pi$$
0.698576 + 0.715536i $$0.253817\pi$$
$$578$$ 0 0
$$579$$ −14695.0 −1.05475
$$580$$ 0 0
$$581$$ −4080.83 −0.291396
$$582$$ 0 0
$$583$$ 2997.31 0.212926
$$584$$ 0 0
$$585$$ −1802.98 −0.127426
$$586$$ 0 0
$$587$$ 9135.18 0.642332 0.321166 0.947023i $$-0.395925\pi$$
0.321166 + 0.947023i $$0.395925\pi$$
$$588$$ 0 0
$$589$$ −4276.12 −0.299141
$$590$$ 0 0
$$591$$ 8940.20 0.622252
$$592$$ 0 0
$$593$$ −12287.6 −0.850916 −0.425458 0.904978i $$-0.639887\pi$$
−0.425458 + 0.904978i $$0.639887\pi$$
$$594$$ 0 0
$$595$$ −4560.34 −0.314211
$$596$$ 0 0
$$597$$ 12380.3 0.848732
$$598$$ 0 0
$$599$$ −12351.9 −0.842549 −0.421275 0.906933i $$-0.638417\pi$$
−0.421275 + 0.906933i $$0.638417\pi$$
$$600$$ 0 0
$$601$$ 21624.2 1.46767 0.733836 0.679327i $$-0.237728\pi$$
0.733836 + 0.679327i $$0.237728\pi$$
$$602$$ 0 0
$$603$$ −3101.14 −0.209433
$$604$$ 0 0
$$605$$ −1947.56 −0.130875
$$606$$ 0 0
$$607$$ 2086.03 0.139488 0.0697442 0.997565i $$-0.477782\pi$$
0.0697442 + 0.997565i $$0.477782\pi$$
$$608$$ 0 0
$$609$$ −10101.0 −0.672108
$$610$$ 0 0
$$611$$ −14051.0 −0.930347
$$612$$ 0 0
$$613$$ −8338.46 −0.549408 −0.274704 0.961529i $$-0.588580\pi$$
−0.274704 + 0.961529i $$0.588580\pi$$
$$614$$ 0 0
$$615$$ −13879.0 −0.910011
$$616$$ 0 0
$$617$$ −4771.20 −0.311315 −0.155657 0.987811i $$-0.549750\pi$$
−0.155657 + 0.987811i $$0.549750\pi$$
$$618$$ 0 0
$$619$$ 16609.4 1.07850 0.539248 0.842147i $$-0.318708\pi$$
0.539248 + 0.842147i $$0.318708\pi$$
$$620$$ 0 0
$$621$$ −22843.6 −1.47614
$$622$$ 0 0
$$623$$ −5643.40 −0.362919
$$624$$ 0 0
$$625$$ −14409.7 −0.922221
$$626$$ 0 0
$$627$$ −7138.82 −0.454700
$$628$$ 0 0
$$629$$ 770.918 0.0488688
$$630$$ 0 0
$$631$$ −17254.9 −1.08860 −0.544299 0.838891i $$-0.683204\pi$$
−0.544299 + 0.838891i $$0.683204\pi$$
$$632$$ 0 0
$$633$$ −17092.9 −1.07327
$$634$$ 0 0
$$635$$ −16987.3 −1.06161
$$636$$ 0 0
$$637$$ 1733.19 0.107804
$$638$$ 0 0
$$639$$ −2376.47 −0.147123
$$640$$ 0 0
$$641$$ −7350.77 −0.452945 −0.226473 0.974018i $$-0.572719\pi$$
−0.226473 + 0.974018i $$0.572719\pi$$
$$642$$ 0 0
$$643$$ −10117.8 −0.620542 −0.310271 0.950648i $$-0.600420\pi$$
−0.310271 + 0.950648i $$0.600420\pi$$
$$644$$ 0 0
$$645$$ 25387.9 1.54984
$$646$$ 0 0
$$647$$ 24590.9 1.49423 0.747116 0.664693i $$-0.231438\pi$$
0.747116 + 0.664693i $$0.231438\pi$$
$$648$$ 0 0
$$649$$ 5582.13 0.337624
$$650$$ 0 0
$$651$$ 1391.38 0.0837670
$$652$$ 0 0
$$653$$ 2339.03 0.140173 0.0700867 0.997541i $$-0.477672\pi$$
0.0700867 + 0.997541i $$0.477672\pi$$
$$654$$ 0 0
$$655$$ 42772.1 2.55152
$$656$$ 0 0
$$657$$ 1252.81 0.0743940
$$658$$ 0 0
$$659$$ 15735.7 0.930162 0.465081 0.885268i $$-0.346025\pi$$
0.465081 + 0.885268i $$0.346025\pi$$
$$660$$ 0 0
$$661$$ 4846.75 0.285199 0.142600 0.989780i $$-0.454454\pi$$
0.142600 + 0.989780i $$0.454454\pi$$
$$662$$ 0 0
$$663$$ 7863.39 0.460617
$$664$$ 0 0
$$665$$ 13312.8 0.776316
$$666$$ 0 0
$$667$$ −45848.1 −2.66154
$$668$$ 0 0
$$669$$ 19544.2 1.12948
$$670$$ 0 0
$$671$$ 391.169 0.0225051
$$672$$ 0 0
$$673$$ 15716.8 0.900205 0.450103 0.892977i $$-0.351387\pi$$
0.450103 + 0.892977i $$0.351387\pi$$
$$674$$ 0 0
$$675$$ −17549.4 −1.00070
$$676$$ 0 0
$$677$$ −856.968 −0.0486499 −0.0243249 0.999704i $$-0.507744\pi$$
−0.0243249 + 0.999704i $$0.507744\pi$$
$$678$$ 0 0
$$679$$ −6700.63 −0.378713
$$680$$ 0 0
$$681$$ 12800.2 0.720271
$$682$$ 0 0
$$683$$ −24804.0 −1.38960 −0.694801 0.719202i $$-0.744508\pi$$
−0.694801 + 0.719202i $$0.744508\pi$$
$$684$$ 0 0
$$685$$ 2371.09 0.132255
$$686$$ 0 0
$$687$$ 3713.47 0.206227
$$688$$ 0 0
$$689$$ 9638.04 0.532917
$$690$$ 0 0
$$691$$ −3475.97 −0.191364 −0.0956818 0.995412i $$-0.530503\pi$$
−0.0956818 + 0.995412i $$0.530503\pi$$
$$692$$ 0 0
$$693$$ 243.852 0.0133668
$$694$$ 0 0
$$695$$ 14528.3 0.792937
$$696$$ 0 0
$$697$$ 6354.54 0.345330
$$698$$ 0 0
$$699$$ 8898.85 0.481524
$$700$$ 0 0
$$701$$ 17461.0 0.940790 0.470395 0.882456i $$-0.344111\pi$$
0.470395 + 0.882456i $$0.344111\pi$$
$$702$$ 0 0
$$703$$ −2250.52 −0.120739
$$704$$ 0 0
$$705$$ 35117.8 1.87605
$$706$$ 0 0
$$707$$ −6973.00 −0.370929
$$708$$ 0 0
$$709$$ −18405.7 −0.974950 −0.487475 0.873137i $$-0.662082\pi$$
−0.487475 + 0.873137i $$0.662082\pi$$
$$710$$ 0 0
$$711$$ 2332.48 0.123031
$$712$$ 0 0
$$713$$ 6315.39 0.331716
$$714$$ 0 0
$$715$$ −6262.49 −0.327558
$$716$$ 0 0
$$717$$ 10992.5 0.572555
$$718$$ 0 0
$$719$$ 892.380 0.0462867 0.0231434 0.999732i $$-0.492633\pi$$
0.0231434 + 0.999732i $$0.492633\pi$$
$$720$$ 0 0
$$721$$ −9369.83 −0.483982
$$722$$ 0 0
$$723$$ −8713.15 −0.448196
$$724$$ 0 0
$$725$$ −35222.2 −1.80431
$$726$$ 0 0
$$727$$ −27919.2 −1.42430 −0.712149 0.702028i $$-0.752278\pi$$
−0.712149 + 0.702028i $$0.752278\pi$$
$$728$$ 0 0
$$729$$ 16864.5 0.856805
$$730$$ 0 0
$$731$$ −11623.9 −0.588134
$$732$$ 0 0
$$733$$ −4769.38 −0.240329 −0.120164 0.992754i $$-0.538342\pi$$
−0.120164 + 0.992754i $$0.538342\pi$$
$$734$$ 0 0
$$735$$ −4331.78 −0.217388
$$736$$ 0 0
$$737$$ −10771.5 −0.538364
$$738$$ 0 0
$$739$$ 5170.63 0.257381 0.128691 0.991685i $$-0.458923\pi$$
0.128691 + 0.991685i $$0.458923\pi$$
$$740$$ 0 0
$$741$$ −22955.3 −1.13804
$$742$$ 0 0
$$743$$ 29407.9 1.45205 0.726024 0.687669i $$-0.241366\pi$$
0.726024 + 0.687669i $$0.241366\pi$$
$$744$$ 0 0
$$745$$ −19517.8 −0.959836
$$746$$ 0 0
$$747$$ −1846.23 −0.0904286
$$748$$ 0 0
$$749$$ −10144.8 −0.494902
$$750$$ 0 0
$$751$$ 16956.5 0.823905 0.411952 0.911205i $$-0.364847\pi$$
0.411952 + 0.911205i $$0.364847\pi$$
$$752$$ 0 0
$$753$$ −14347.5 −0.694360
$$754$$ 0 0
$$755$$ 41290.3 1.99034
$$756$$ 0 0
$$757$$ −21322.0 −1.02373 −0.511864 0.859067i $$-0.671045\pi$$
−0.511864 + 0.859067i $$0.671045\pi$$
$$758$$ 0 0
$$759$$ 10543.3 0.504214
$$760$$ 0 0
$$761$$ −23548.0 −1.12170 −0.560851 0.827917i $$-0.689526\pi$$
−0.560851 + 0.827917i $$0.689526\pi$$
$$762$$ 0 0
$$763$$ −4581.75 −0.217392
$$764$$ 0 0
$$765$$ −2063.17 −0.0975087
$$766$$ 0 0
$$767$$ 17949.7 0.845014
$$768$$ 0 0
$$769$$ −17230.9 −0.808015 −0.404007 0.914756i $$-0.632383\pi$$
−0.404007 + 0.914756i $$0.632383\pi$$
$$770$$ 0 0
$$771$$ 23058.5 1.07709
$$772$$ 0 0
$$773$$ 12285.8 0.571655 0.285828 0.958281i $$-0.407732\pi$$
0.285828 + 0.958281i $$0.407732\pi$$
$$774$$ 0 0
$$775$$ 4851.73 0.224876
$$776$$ 0 0
$$777$$ 732.280 0.0338100
$$778$$ 0 0
$$779$$ −18550.6 −0.853202
$$780$$ 0 0
$$781$$ −8254.45 −0.378191
$$782$$ 0 0
$$783$$ 34391.2 1.56966
$$784$$ 0 0
$$785$$ −11302.8 −0.513906
$$786$$ 0 0
$$787$$ −663.152 −0.0300366 −0.0150183 0.999887i $$-0.504781\pi$$
−0.0150183 + 0.999887i $$0.504781\pi$$
$$788$$ 0 0
$$789$$ 28397.6 1.28135
$$790$$ 0 0
$$791$$ −8124.43 −0.365198
$$792$$ 0 0
$$793$$ 1257.83 0.0563264
$$794$$ 0 0
$$795$$ −24088.4 −1.07463
$$796$$ 0 0
$$797$$ 19216.3 0.854050 0.427025 0.904240i $$-0.359562\pi$$
0.427025 + 0.904240i $$0.359562\pi$$
$$798$$ 0 0
$$799$$ −16078.7 −0.711921
$$800$$ 0 0
$$801$$ −2553.17 −0.112624
$$802$$ 0 0
$$803$$ 4351.53 0.191236
$$804$$ 0 0
$$805$$ −19661.7 −0.860851
$$806$$ 0 0
$$807$$ 9052.65 0.394880
$$808$$ 0 0
$$809$$ −42881.8 −1.86359 −0.931794 0.362989i $$-0.881756\pi$$
−0.931794 + 0.362989i $$0.881756\pi$$
$$810$$ 0 0
$$811$$ 1205.73 0.0522058 0.0261029 0.999659i $$-0.491690\pi$$
0.0261029 + 0.999659i $$0.491690\pi$$
$$812$$ 0 0
$$813$$ −14076.0 −0.607215
$$814$$ 0 0
$$815$$ −18449.8 −0.792966
$$816$$ 0 0
$$817$$ 33933.3 1.45309
$$818$$ 0 0
$$819$$ 784.123 0.0334548
$$820$$ 0 0
$$821$$ −28577.6 −1.21482 −0.607408 0.794390i $$-0.707791\pi$$
−0.607408 + 0.794390i $$0.707791\pi$$
$$822$$ 0 0
$$823$$ −42524.8 −1.80112 −0.900561 0.434730i $$-0.856844\pi$$
−0.900561 + 0.434730i $$0.856844\pi$$
$$824$$ 0 0
$$825$$ 8099.79 0.341816
$$826$$ 0 0
$$827$$ −30768.7 −1.29375 −0.646876 0.762595i $$-0.723925\pi$$
−0.646876 + 0.762595i $$0.723925\pi$$
$$828$$ 0 0
$$829$$ −17583.3 −0.736661 −0.368330 0.929695i $$-0.620070\pi$$
−0.368330 + 0.929695i $$0.620070\pi$$
$$830$$ 0 0
$$831$$ 15175.3 0.633484
$$832$$ 0 0
$$833$$ 1983.31 0.0824942
$$834$$ 0 0
$$835$$ −52391.5 −2.17136
$$836$$ 0 0
$$837$$ −4737.25 −0.195631
$$838$$ 0 0
$$839$$ −19552.8 −0.804573 −0.402287 0.915514i $$-0.631784\pi$$
−0.402287 + 0.915514i $$0.631784\pi$$
$$840$$ 0 0
$$841$$ 44635.5 1.83015
$$842$$ 0 0
$$843$$ 35446.5 1.44821
$$844$$ 0 0
$$845$$ 15224.4 0.619805
$$846$$ 0 0
$$847$$ 847.000 0.0343604
$$848$$ 0 0
$$849$$ −24940.0 −1.00817
$$850$$ 0 0
$$851$$ 3323.78 0.133887
$$852$$ 0 0
$$853$$ −18524.1 −0.743557 −0.371779 0.928321i $$-0.621252\pi$$
−0.371779 + 0.928321i $$0.621252\pi$$
$$854$$ 0 0
$$855$$ 6022.95 0.240913
$$856$$ 0 0
$$857$$ 24439.0 0.974118 0.487059 0.873369i $$-0.338070\pi$$
0.487059 + 0.873369i $$0.338070\pi$$
$$858$$ 0 0
$$859$$ −11301.4 −0.448893 −0.224447 0.974486i $$-0.572057\pi$$
−0.224447 + 0.974486i $$0.572057\pi$$
$$860$$ 0 0
$$861$$ 6036.05 0.238918
$$862$$ 0 0
$$863$$ 26377.3 1.04043 0.520217 0.854034i $$-0.325851\pi$$
0.520217 + 0.854034i $$0.325851\pi$$
$$864$$ 0 0
$$865$$ −64772.1 −2.54603
$$866$$ 0 0
$$867$$ −17986.2 −0.704548
$$868$$ 0 0
$$869$$ 8101.68 0.316261
$$870$$ 0 0
$$871$$ −34636.5 −1.34743
$$872$$ 0 0
$$873$$ −3031.47 −0.117526
$$874$$ 0 0
$$875$$ −1021.34 −0.0394601
$$876$$ 0 0
$$877$$ −28425.0 −1.09446 −0.547232 0.836981i $$-0.684319\pi$$
−0.547232 + 0.836981i $$0.684319\pi$$
$$878$$ 0 0
$$879$$ 10525.4 0.403882
$$880$$ 0 0
$$881$$ 16897.1 0.646171 0.323086 0.946370i $$-0.395280\pi$$
0.323086 + 0.946370i $$0.395280\pi$$
$$882$$ 0 0
$$883$$ 25538.9 0.973331 0.486665 0.873588i $$-0.338213\pi$$
0.486665 + 0.873588i $$0.338213\pi$$
$$884$$ 0 0
$$885$$ −44861.8 −1.70397
$$886$$ 0 0
$$887$$ −6478.48 −0.245238 −0.122619 0.992454i $$-0.539129\pi$$
−0.122619 + 0.992454i $$0.539129\pi$$
$$888$$ 0 0
$$889$$ 7387.85 0.278718
$$890$$ 0 0
$$891$$ −8849.25 −0.332728
$$892$$ 0 0
$$893$$ 46938.1 1.75893
$$894$$ 0 0
$$895$$ 11364.9 0.424453
$$896$$ 0 0
$$897$$ 33902.7 1.26196
$$898$$ 0 0
$$899$$ −9507.86 −0.352731
$$900$$ 0 0
$$901$$ 11028.9 0.407799
$$902$$ 0 0
$$903$$ −11041.3 −0.406902
$$904$$ 0 0
$$905$$ 20423.6 0.750171
$$906$$ 0 0
$$907$$ 9356.17 0.342521 0.171260 0.985226i $$-0.445216\pi$$
0.171260 + 0.985226i $$0.445216\pi$$
$$908$$ 0 0
$$909$$ −3154.70 −0.115110
$$910$$ 0 0
$$911$$ 16574.0 0.602768 0.301384 0.953503i $$-0.402551\pi$$
0.301384 + 0.953503i $$0.402551\pi$$
$$912$$ 0 0
$$913$$ −6412.73 −0.232454
$$914$$ 0 0
$$915$$ −3143.70 −0.113582
$$916$$ 0 0
$$917$$ −18601.8 −0.669885
$$918$$ 0 0
$$919$$ −8214.92 −0.294870 −0.147435 0.989072i $$-0.547102\pi$$
−0.147435 + 0.989072i $$0.547102\pi$$
$$920$$ 0 0
$$921$$ −37767.7 −1.35123
$$922$$ 0 0
$$923$$ −26542.7 −0.946548
$$924$$ 0 0
$$925$$ 2553.46 0.0907646
$$926$$ 0 0
$$927$$ −4239.07 −0.150193
$$928$$ 0 0
$$929$$ 42653.9 1.50638 0.753192 0.657801i $$-0.228513\pi$$
0.753192 + 0.657801i $$0.228513\pi$$
$$930$$ 0 0
$$931$$ −5789.81 −0.203817
$$932$$ 0 0
$$933$$ −46517.2 −1.63227
$$934$$ 0 0
$$935$$ −7166.25 −0.250654
$$936$$ 0 0
$$937$$ −18484.8 −0.644473 −0.322237 0.946659i $$-0.604435\pi$$
−0.322237 + 0.946659i $$0.604435\pi$$
$$938$$ 0 0
$$939$$ −15830.7 −0.550177
$$940$$ 0 0
$$941$$ −7183.03 −0.248842 −0.124421 0.992230i $$-0.539707\pi$$
−0.124421 + 0.992230i $$0.539707\pi$$
$$942$$ 0 0
$$943$$ 27397.4 0.946109
$$944$$ 0 0
$$945$$ 14748.5 0.507692
$$946$$ 0 0
$$947$$ −41443.3 −1.42210 −0.711049 0.703143i $$-0.751779\pi$$
−0.711049 + 0.703143i $$0.751779\pi$$
$$948$$ 0 0
$$949$$ 13992.6 0.478630
$$950$$ 0 0
$$951$$ 53800.2 1.83448
$$952$$ 0 0
$$953$$ 7981.30 0.271290 0.135645 0.990757i $$-0.456689\pi$$
0.135645 + 0.990757i $$0.456689\pi$$
$$954$$ 0 0
$$955$$ 78297.8 2.65305
$$956$$ 0 0
$$957$$ −15873.0 −0.536157
$$958$$ 0 0
$$959$$ −1031.20 −0.0347227
$$960$$ 0 0
$$961$$ −28481.3 −0.956038
$$962$$ 0 0
$$963$$ −4589.65 −0.153582
$$964$$ 0 0
$$965$$ 43063.3 1.43654
$$966$$ 0 0
$$967$$ 18745.7 0.623394 0.311697 0.950182i $$-0.399103\pi$$
0.311697 + 0.950182i $$0.399103\pi$$
$$968$$ 0 0
$$969$$ −26268.1 −0.870849
$$970$$ 0 0
$$971$$ −3096.87 −0.102351 −0.0511757 0.998690i $$-0.516297\pi$$
−0.0511757 + 0.998690i $$0.516297\pi$$
$$972$$ 0 0
$$973$$ −6318.44 −0.208181
$$974$$ 0 0
$$975$$ 26045.4 0.855507
$$976$$ 0 0
$$977$$ 19960.2 0.653618 0.326809 0.945090i $$-0.394027\pi$$
0.326809 + 0.945090i $$0.394027\pi$$
$$978$$ 0 0
$$979$$ −8868.21 −0.289509
$$980$$ 0 0
$$981$$ −2072.86 −0.0674631
$$982$$ 0 0
$$983$$ 33434.5 1.08484 0.542419 0.840108i $$-0.317509\pi$$
0.542419 + 0.840108i $$0.317509\pi$$
$$984$$ 0 0
$$985$$ −26199.1 −0.847485
$$986$$ 0 0
$$987$$ −15272.9 −0.492544
$$988$$ 0 0
$$989$$ −50116.1 −1.61132
$$990$$ 0 0
$$991$$ −24855.2 −0.796722 −0.398361 0.917229i $$-0.630421\pi$$
−0.398361 + 0.917229i $$0.630421\pi$$
$$992$$ 0 0
$$993$$ −9961.63 −0.318351
$$994$$ 0 0
$$995$$ −36280.3 −1.15594
$$996$$ 0 0
$$997$$ −7810.38 −0.248102 −0.124051 0.992276i $$-0.539589\pi$$
−0.124051 + 0.992276i $$0.539589\pi$$
$$998$$ 0 0
$$999$$ −2493.21 −0.0789607
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 1232.4.a.s.1.4 4
4.3 odd 2 77.4.a.d.1.2 4
12.11 even 2 693.4.a.l.1.3 4
20.19 odd 2 1925.4.a.p.1.3 4
28.27 even 2 539.4.a.g.1.2 4
44.43 even 2 847.4.a.d.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.d.1.2 4 4.3 odd 2
539.4.a.g.1.2 4 28.27 even 2
693.4.a.l.1.3 4 12.11 even 2
847.4.a.d.1.3 4 44.43 even 2
1232.4.a.s.1.4 4 1.1 even 1 trivial
1925.4.a.p.1.3 4 20.19 odd 2