# Properties

 Label 1856.4.a.bj.1.6 Level $1856$ Weight $4$ Character 1856.1 Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $12$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + 7106592 x^{4} - 7979328 x^{3} - 38912400 x^{2} + \cdots + 14751072$$ x^12 - 2*x^11 - 217*x^10 + 520*x^9 + 16022*x^8 - 37368*x^7 - 509640*x^6 + 989168*x^5 + 7106592*x^4 - 7979328*x^3 - 38912400*x^2 + 1083456*x + 14751072 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{17}$$ Twist minimal: no (minimal twist has level 928) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$0.613335$$ of defining polynomial Character $$\chi$$ $$=$$ 1856.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +O(q^{10})$$ $$q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +30.2948 q^{11} -19.9286 q^{13} -16.9919 q^{15} +105.498 q^{17} +79.5429 q^{19} -22.1701 q^{21} -14.8819 q^{23} -14.0732 q^{25} +82.9208 q^{27} -29.0000 q^{29} -61.9562 q^{31} -48.8756 q^{33} +144.731 q^{35} -6.43644 q^{37} +32.1516 q^{39} +351.682 q^{41} -473.240 q^{43} -256.955 q^{45} +492.253 q^{47} -154.163 q^{49} -170.203 q^{51} +668.911 q^{53} +319.070 q^{55} -128.329 q^{57} -12.7826 q^{59} -80.8937 q^{61} -335.261 q^{63} -209.892 q^{65} -529.153 q^{67} +24.0095 q^{69} -211.869 q^{71} +627.334 q^{73} +22.7048 q^{75} +416.305 q^{77} +915.349 q^{79} +524.944 q^{81} -411.330 q^{83} +1111.12 q^{85} +46.7867 q^{87} +1377.88 q^{89} -273.856 q^{91} +99.9561 q^{93} +837.761 q^{95} -649.398 q^{97} -739.106 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10})$$ 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9 $$12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9} - 46 q^{11} + 34 q^{13} - 50 q^{15} + 36 q^{17} - 148 q^{19} + 92 q^{21} + 328 q^{23} + 486 q^{25} - 326 q^{27} - 348 q^{29} + 374 q^{31} + 710 q^{33} - 204 q^{35} + 340 q^{37} - 122 q^{39} + 32 q^{41} - 462 q^{43} + 1132 q^{45} + 434 q^{47} + 1508 q^{49} - 440 q^{51} - 610 q^{53} + 46 q^{55} - 932 q^{57} - 1240 q^{59} + 1228 q^{61} + 4240 q^{63} + 730 q^{65} - 1672 q^{67} + 528 q^{69} + 3220 q^{71} + 564 q^{73} - 6032 q^{75} - 644 q^{77} + 1862 q^{79} + 3040 q^{81} - 3736 q^{83} + 808 q^{85} + 406 q^{87} + 584 q^{89} - 4844 q^{91} + 3226 q^{93} + 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100})$$ 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9 - 46 * q^11 + 34 * q^13 - 50 * q^15 + 36 * q^17 - 148 * q^19 + 92 * q^21 + 328 * q^23 + 486 * q^25 - 326 * q^27 - 348 * q^29 + 374 * q^31 + 710 * q^33 - 204 * q^35 + 340 * q^37 - 122 * q^39 + 32 * q^41 - 462 * q^43 + 1132 * q^45 + 434 * q^47 + 1508 * q^49 - 440 * q^51 - 610 * q^53 + 46 * q^55 - 932 * q^57 - 1240 * q^59 + 1228 * q^61 + 4240 * q^63 + 730 * q^65 - 1672 * q^67 + 528 * q^69 + 3220 * q^71 + 564 * q^73 - 6032 * q^75 - 644 * q^77 + 1862 * q^79 + 3040 * q^81 - 3736 * q^83 + 808 * q^85 + 406 * q^87 + 584 * q^89 - 4844 * q^91 + 3226 * q^93 + 2844 * q^95 + 904 * q^97 - 6832 * 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
$$3$$ −1.61334 −0.310486 −0.155243 0.987876i $$-0.549616\pi$$
−0.155243 + 0.987876i $$0.549616\pi$$
$$4$$ 0 0
$$5$$ 10.5322 0.942027 0.471013 0.882126i $$-0.343888\pi$$
0.471013 + 0.882126i $$0.343888\pi$$
$$6$$ 0 0
$$7$$ 13.7418 0.741988 0.370994 0.928635i $$-0.379017\pi$$
0.370994 + 0.928635i $$0.379017\pi$$
$$8$$ 0 0
$$9$$ −24.3972 −0.903598
$$10$$ 0 0
$$11$$ 30.2948 0.830384 0.415192 0.909734i $$-0.363714\pi$$
0.415192 + 0.909734i $$0.363714\pi$$
$$12$$ 0 0
$$13$$ −19.9286 −0.425170 −0.212585 0.977143i $$-0.568188\pi$$
−0.212585 + 0.977143i $$0.568188\pi$$
$$14$$ 0 0
$$15$$ −16.9919 −0.292487
$$16$$ 0 0
$$17$$ 105.498 1.50512 0.752558 0.658526i $$-0.228820\pi$$
0.752558 + 0.658526i $$0.228820\pi$$
$$18$$ 0 0
$$19$$ 79.5429 0.960442 0.480221 0.877147i $$-0.340556\pi$$
0.480221 + 0.877147i $$0.340556\pi$$
$$20$$ 0 0
$$21$$ −22.1701 −0.230377
$$22$$ 0 0
$$23$$ −14.8819 −0.134917 −0.0674586 0.997722i $$-0.521489\pi$$
−0.0674586 + 0.997722i $$0.521489\pi$$
$$24$$ 0 0
$$25$$ −14.0732 −0.112586
$$26$$ 0 0
$$27$$ 82.9208 0.591041
$$28$$ 0 0
$$29$$ −29.0000 −0.185695
$$30$$ 0 0
$$31$$ −61.9562 −0.358957 −0.179478 0.983762i $$-0.557441\pi$$
−0.179478 + 0.983762i $$0.557441\pi$$
$$32$$ 0 0
$$33$$ −48.8756 −0.257823
$$34$$ 0 0
$$35$$ 144.731 0.698972
$$36$$ 0 0
$$37$$ −6.43644 −0.0285985 −0.0142993 0.999898i $$-0.504552\pi$$
−0.0142993 + 0.999898i $$0.504552\pi$$
$$38$$ 0 0
$$39$$ 32.1516 0.132010
$$40$$ 0 0
$$41$$ 351.682 1.33960 0.669799 0.742543i $$-0.266380\pi$$
0.669799 + 0.742543i $$0.266380\pi$$
$$42$$ 0 0
$$43$$ −473.240 −1.67834 −0.839169 0.543871i $$-0.816958\pi$$
−0.839169 + 0.543871i $$0.816958\pi$$
$$44$$ 0 0
$$45$$ −256.955 −0.851214
$$46$$ 0 0
$$47$$ 492.253 1.52771 0.763856 0.645387i $$-0.223304\pi$$
0.763856 + 0.645387i $$0.223304\pi$$
$$48$$ 0 0
$$49$$ −154.163 −0.449454
$$50$$ 0 0
$$51$$ −170.203 −0.467318
$$52$$ 0 0
$$53$$ 668.911 1.73362 0.866811 0.498637i $$-0.166166\pi$$
0.866811 + 0.498637i $$0.166166\pi$$
$$54$$ 0 0
$$55$$ 319.070 0.782244
$$56$$ 0 0
$$57$$ −128.329 −0.298204
$$58$$ 0 0
$$59$$ −12.7826 −0.0282060 −0.0141030 0.999901i $$-0.504489\pi$$
−0.0141030 + 0.999901i $$0.504489\pi$$
$$60$$ 0 0
$$61$$ −80.8937 −0.169793 −0.0848965 0.996390i $$-0.527056\pi$$
−0.0848965 + 0.996390i $$0.527056\pi$$
$$62$$ 0 0
$$63$$ −335.261 −0.670459
$$64$$ 0 0
$$65$$ −209.892 −0.400521
$$66$$ 0 0
$$67$$ −529.153 −0.964870 −0.482435 0.875932i $$-0.660248\pi$$
−0.482435 + 0.875932i $$0.660248\pi$$
$$68$$ 0 0
$$69$$ 24.0095 0.0418899
$$70$$ 0 0
$$71$$ −211.869 −0.354144 −0.177072 0.984198i $$-0.556663\pi$$
−0.177072 + 0.984198i $$0.556663\pi$$
$$72$$ 0 0
$$73$$ 627.334 1.00581 0.502903 0.864343i $$-0.332265\pi$$
0.502903 + 0.864343i $$0.332265\pi$$
$$74$$ 0 0
$$75$$ 22.7048 0.0349563
$$76$$ 0 0
$$77$$ 416.305 0.616135
$$78$$ 0 0
$$79$$ 915.349 1.30361 0.651803 0.758389i $$-0.274013\pi$$
0.651803 + 0.758389i $$0.274013\pi$$
$$80$$ 0 0
$$81$$ 524.944 0.720088
$$82$$ 0 0
$$83$$ −411.330 −0.543967 −0.271984 0.962302i $$-0.587680\pi$$
−0.271984 + 0.962302i $$0.587680\pi$$
$$84$$ 0 0
$$85$$ 1111.12 1.41786
$$86$$ 0 0
$$87$$ 46.7867 0.0576559
$$88$$ 0 0
$$89$$ 1377.88 1.64107 0.820536 0.571595i $$-0.193675\pi$$
0.820536 + 0.571595i $$0.193675\pi$$
$$90$$ 0 0
$$91$$ −273.856 −0.315471
$$92$$ 0 0
$$93$$ 99.9561 0.111451
$$94$$ 0 0
$$95$$ 837.761 0.904762
$$96$$ 0 0
$$97$$ −649.398 −0.679756 −0.339878 0.940470i $$-0.610386\pi$$
−0.339878 + 0.940470i $$0.610386\pi$$
$$98$$ 0 0
$$99$$ −739.106 −0.750333
$$100$$ 0 0
$$101$$ 851.483 0.838869 0.419434 0.907786i $$-0.362228\pi$$
0.419434 + 0.907786i $$0.362228\pi$$
$$102$$ 0 0
$$103$$ −1923.60 −1.84017 −0.920085 0.391719i $$-0.871881\pi$$
−0.920085 + 0.391719i $$0.871881\pi$$
$$104$$ 0 0
$$105$$ −233.500 −0.217021
$$106$$ 0 0
$$107$$ −1648.55 −1.48945 −0.744726 0.667370i $$-0.767420\pi$$
−0.744726 + 0.667370i $$0.767420\pi$$
$$108$$ 0 0
$$109$$ −770.267 −0.676865 −0.338432 0.940991i $$-0.609897\pi$$
−0.338432 + 0.940991i $$0.609897\pi$$
$$110$$ 0 0
$$111$$ 10.3841 0.00887945
$$112$$ 0 0
$$113$$ 1429.70 1.19022 0.595111 0.803643i $$-0.297108\pi$$
0.595111 + 0.803643i $$0.297108\pi$$
$$114$$ 0 0
$$115$$ −156.739 −0.127096
$$116$$ 0 0
$$117$$ 486.202 0.384183
$$118$$ 0 0
$$119$$ 1449.73 1.11678
$$120$$ 0 0
$$121$$ −413.226 −0.310463
$$122$$ 0 0
$$123$$ −567.381 −0.415927
$$124$$ 0 0
$$125$$ −1464.74 −1.04809
$$126$$ 0 0
$$127$$ 2102.32 1.46890 0.734451 0.678662i $$-0.237440\pi$$
0.734451 + 0.678662i $$0.237440\pi$$
$$128$$ 0 0
$$129$$ 763.495 0.521101
$$130$$ 0 0
$$131$$ −1376.03 −0.917742 −0.458871 0.888503i $$-0.651746\pi$$
−0.458871 + 0.888503i $$0.651746\pi$$
$$132$$ 0 0
$$133$$ 1093.06 0.712637
$$134$$ 0 0
$$135$$ 873.337 0.556777
$$136$$ 0 0
$$137$$ 304.947 0.190171 0.0950853 0.995469i $$-0.469688\pi$$
0.0950853 + 0.995469i $$0.469688\pi$$
$$138$$ 0 0
$$139$$ 364.348 0.222328 0.111164 0.993802i $$-0.464542\pi$$
0.111164 + 0.993802i $$0.464542\pi$$
$$140$$ 0 0
$$141$$ −794.169 −0.474334
$$142$$ 0 0
$$143$$ −603.734 −0.353054
$$144$$ 0 0
$$145$$ −305.433 −0.174930
$$146$$ 0 0
$$147$$ 248.716 0.139549
$$148$$ 0 0
$$149$$ −200.018 −0.109974 −0.0549869 0.998487i $$-0.517512\pi$$
−0.0549869 + 0.998487i $$0.517512\pi$$
$$150$$ 0 0
$$151$$ −35.5937 −0.0191826 −0.00959130 0.999954i $$-0.503053\pi$$
−0.00959130 + 0.999954i $$0.503053\pi$$
$$152$$ 0 0
$$153$$ −2573.85 −1.36002
$$154$$ 0 0
$$155$$ −652.534 −0.338147
$$156$$ 0 0
$$157$$ −137.397 −0.0698437 −0.0349218 0.999390i $$-0.511118\pi$$
−0.0349218 + 0.999390i $$0.511118\pi$$
$$158$$ 0 0
$$159$$ −1079.18 −0.538266
$$160$$ 0 0
$$161$$ −204.504 −0.100107
$$162$$ 0 0
$$163$$ 867.825 0.417014 0.208507 0.978021i $$-0.433140\pi$$
0.208507 + 0.978021i $$0.433140\pi$$
$$164$$ 0 0
$$165$$ −514.767 −0.242876
$$166$$ 0 0
$$167$$ −66.2847 −0.0307142 −0.0153571 0.999882i $$-0.504889\pi$$
−0.0153571 + 0.999882i $$0.504889\pi$$
$$168$$ 0 0
$$169$$ −1799.85 −0.819230
$$170$$ 0 0
$$171$$ −1940.62 −0.867854
$$172$$ 0 0
$$173$$ 1798.76 0.790505 0.395253 0.918572i $$-0.370657\pi$$
0.395253 + 0.918572i $$0.370657\pi$$
$$174$$ 0 0
$$175$$ −193.391 −0.0835372
$$176$$ 0 0
$$177$$ 20.6226 0.00875759
$$178$$ 0 0
$$179$$ −1214.69 −0.507208 −0.253604 0.967308i $$-0.581616\pi$$
−0.253604 + 0.967308i $$0.581616\pi$$
$$180$$ 0 0
$$181$$ −643.623 −0.264310 −0.132155 0.991229i $$-0.542190\pi$$
−0.132155 + 0.991229i $$0.542190\pi$$
$$182$$ 0 0
$$183$$ 130.509 0.0527184
$$184$$ 0 0
$$185$$ −67.7898 −0.0269406
$$186$$ 0 0
$$187$$ 3196.03 1.24982
$$188$$ 0 0
$$189$$ 1139.48 0.438546
$$190$$ 0 0
$$191$$ 4085.97 1.54791 0.773953 0.633243i $$-0.218277\pi$$
0.773953 + 0.633243i $$0.218277\pi$$
$$192$$ 0 0
$$193$$ 864.326 0.322360 0.161180 0.986925i $$-0.448470\pi$$
0.161180 + 0.986925i $$0.448470\pi$$
$$194$$ 0 0
$$195$$ 338.626 0.124357
$$196$$ 0 0
$$197$$ 1935.81 0.700106 0.350053 0.936730i $$-0.386164\pi$$
0.350053 + 0.936730i $$0.386164\pi$$
$$198$$ 0 0
$$199$$ 3593.39 1.28005 0.640023 0.768356i $$-0.278925\pi$$
0.640023 + 0.768356i $$0.278925\pi$$
$$200$$ 0 0
$$201$$ 853.701 0.299579
$$202$$ 0 0
$$203$$ −398.512 −0.137784
$$204$$ 0 0
$$205$$ 3703.98 1.26194
$$206$$ 0 0
$$207$$ 363.076 0.121911
$$208$$ 0 0
$$209$$ 2409.74 0.797536
$$210$$ 0 0
$$211$$ −692.347 −0.225892 −0.112946 0.993601i $$-0.536029\pi$$
−0.112946 + 0.993601i $$0.536029\pi$$
$$212$$ 0 0
$$213$$ 341.816 0.109957
$$214$$ 0 0
$$215$$ −4984.25 −1.58104
$$216$$ 0 0
$$217$$ −851.390 −0.266342
$$218$$ 0 0
$$219$$ −1012.10 −0.312289
$$220$$ 0 0
$$221$$ −2102.43 −0.639930
$$222$$ 0 0
$$223$$ 1770.55 0.531680 0.265840 0.964017i $$-0.414351\pi$$
0.265840 + 0.964017i $$0.414351\pi$$
$$224$$ 0 0
$$225$$ 343.346 0.101732
$$226$$ 0 0
$$227$$ 2996.07 0.876017 0.438008 0.898971i $$-0.355684\pi$$
0.438008 + 0.898971i $$0.355684\pi$$
$$228$$ 0 0
$$229$$ 3790.17 1.09372 0.546860 0.837224i $$-0.315823\pi$$
0.546860 + 0.837224i $$0.315823\pi$$
$$230$$ 0 0
$$231$$ −671.640 −0.191301
$$232$$ 0 0
$$233$$ 6257.35 1.75937 0.879684 0.475558i $$-0.157754\pi$$
0.879684 + 0.475558i $$0.157754\pi$$
$$234$$ 0 0
$$235$$ 5184.50 1.43915
$$236$$ 0 0
$$237$$ −1476.77 −0.404752
$$238$$ 0 0
$$239$$ 5893.50 1.59506 0.797529 0.603280i $$-0.206140\pi$$
0.797529 + 0.603280i $$0.206140\pi$$
$$240$$ 0 0
$$241$$ −418.978 −0.111986 −0.0559932 0.998431i $$-0.517833\pi$$
−0.0559932 + 0.998431i $$0.517833\pi$$
$$242$$ 0 0
$$243$$ −3085.77 −0.814619
$$244$$ 0 0
$$245$$ −1623.67 −0.423398
$$246$$ 0 0
$$247$$ −1585.18 −0.408351
$$248$$ 0 0
$$249$$ 663.612 0.168894
$$250$$ 0 0
$$251$$ −7444.38 −1.87205 −0.936026 0.351932i $$-0.885525\pi$$
−0.936026 + 0.351932i $$0.885525\pi$$
$$252$$ 0 0
$$253$$ −450.844 −0.112033
$$254$$ 0 0
$$255$$ −1792.61 −0.440226
$$256$$ 0 0
$$257$$ 5332.46 1.29428 0.647140 0.762371i $$-0.275965\pi$$
0.647140 + 0.762371i $$0.275965\pi$$
$$258$$ 0 0
$$259$$ −88.4484 −0.0212197
$$260$$ 0 0
$$261$$ 707.517 0.167794
$$262$$ 0 0
$$263$$ 832.925 0.195286 0.0976432 0.995221i $$-0.468870\pi$$
0.0976432 + 0.995221i $$0.468870\pi$$
$$264$$ 0 0
$$265$$ 7045.09 1.63312
$$266$$ 0 0
$$267$$ −2222.99 −0.509531
$$268$$ 0 0
$$269$$ 4291.22 0.972640 0.486320 0.873781i $$-0.338339\pi$$
0.486320 + 0.873781i $$0.338339\pi$$
$$270$$ 0 0
$$271$$ −5967.70 −1.33768 −0.668841 0.743405i $$-0.733209\pi$$
−0.668841 + 0.743405i $$0.733209\pi$$
$$272$$ 0 0
$$273$$ 441.821 0.0979495
$$274$$ 0 0
$$275$$ −426.345 −0.0934893
$$276$$ 0 0
$$277$$ 2234.76 0.484742 0.242371 0.970184i $$-0.422075\pi$$
0.242371 + 0.970184i $$0.422075\pi$$
$$278$$ 0 0
$$279$$ 1511.55 0.324353
$$280$$ 0 0
$$281$$ −1672.59 −0.355083 −0.177542 0.984113i $$-0.556814\pi$$
−0.177542 + 0.984113i $$0.556814\pi$$
$$282$$ 0 0
$$283$$ 4837.14 1.01604 0.508018 0.861347i $$-0.330379\pi$$
0.508018 + 0.861347i $$0.330379\pi$$
$$284$$ 0 0
$$285$$ −1351.59 −0.280916
$$286$$ 0 0
$$287$$ 4832.75 0.993965
$$288$$ 0 0
$$289$$ 6216.79 1.26538
$$290$$ 0 0
$$291$$ 1047.70 0.211055
$$292$$ 0 0
$$293$$ 8937.62 1.78205 0.891027 0.453951i $$-0.149986\pi$$
0.891027 + 0.453951i $$0.149986\pi$$
$$294$$ 0 0
$$295$$ −134.629 −0.0265708
$$296$$ 0 0
$$297$$ 2512.07 0.490791
$$298$$ 0 0
$$299$$ 296.576 0.0573627
$$300$$ 0 0
$$301$$ −6503.18 −1.24531
$$302$$ 0 0
$$303$$ −1373.73 −0.260457
$$304$$ 0 0
$$305$$ −851.987 −0.159950
$$306$$ 0 0
$$307$$ −4175.09 −0.776173 −0.388086 0.921623i $$-0.626864\pi$$
−0.388086 + 0.921623i $$0.626864\pi$$
$$308$$ 0 0
$$309$$ 3103.40 0.571348
$$310$$ 0 0
$$311$$ 6986.98 1.27394 0.636970 0.770889i $$-0.280187\pi$$
0.636970 + 0.770889i $$0.280187\pi$$
$$312$$ 0 0
$$313$$ −7458.22 −1.34685 −0.673425 0.739256i $$-0.735177\pi$$
−0.673425 + 0.739256i $$0.735177\pi$$
$$314$$ 0 0
$$315$$ −3531.03 −0.631590
$$316$$ 0 0
$$317$$ 9362.43 1.65882 0.829411 0.558640i $$-0.188676\pi$$
0.829411 + 0.558640i $$0.188676\pi$$
$$318$$ 0 0
$$319$$ −878.549 −0.154198
$$320$$ 0 0
$$321$$ 2659.67 0.462455
$$322$$ 0 0
$$323$$ 8391.61 1.44558
$$324$$ 0 0
$$325$$ 280.460 0.0478680
$$326$$ 0 0
$$327$$ 1242.70 0.210157
$$328$$ 0 0
$$329$$ 6764.45 1.13354
$$330$$ 0 0
$$331$$ 7034.25 1.16809 0.584044 0.811722i $$-0.301469\pi$$
0.584044 + 0.811722i $$0.301469\pi$$
$$332$$ 0 0
$$333$$ 157.031 0.0258416
$$334$$ 0 0
$$335$$ −5573.13 −0.908933
$$336$$ 0 0
$$337$$ 3071.54 0.496490 0.248245 0.968697i $$-0.420146\pi$$
0.248245 + 0.968697i $$0.420146\pi$$
$$338$$ 0 0
$$339$$ −2306.59 −0.369548
$$340$$ 0 0
$$341$$ −1876.95 −0.298072
$$342$$ 0 0
$$343$$ −6831.91 −1.07548
$$344$$ 0 0
$$345$$ 252.872 0.0394614
$$346$$ 0 0
$$347$$ 3918.32 0.606186 0.303093 0.952961i $$-0.401981\pi$$
0.303093 + 0.952961i $$0.401981\pi$$
$$348$$ 0 0
$$349$$ −4880.00 −0.748482 −0.374241 0.927331i $$-0.622097\pi$$
−0.374241 + 0.927331i $$0.622097\pi$$
$$350$$ 0 0
$$351$$ −1652.50 −0.251293
$$352$$ 0 0
$$353$$ −1386.41 −0.209041 −0.104520 0.994523i $$-0.533331\pi$$
−0.104520 + 0.994523i $$0.533331\pi$$
$$354$$ 0 0
$$355$$ −2231.44 −0.333613
$$356$$ 0 0
$$357$$ −2338.90 −0.346745
$$358$$ 0 0
$$359$$ 1798.75 0.264441 0.132220 0.991220i $$-0.457789\pi$$
0.132220 + 0.991220i $$0.457789\pi$$
$$360$$ 0 0
$$361$$ −531.919 −0.0775506
$$362$$ 0 0
$$363$$ 666.673 0.0963946
$$364$$ 0 0
$$365$$ 6607.19 0.947497
$$366$$ 0 0
$$367$$ −8826.24 −1.25538 −0.627692 0.778462i $$-0.716000\pi$$
−0.627692 + 0.778462i $$0.716000\pi$$
$$368$$ 0 0
$$369$$ −8580.04 −1.21046
$$370$$ 0 0
$$371$$ 9192.04 1.28633
$$372$$ 0 0
$$373$$ 5324.16 0.739074 0.369537 0.929216i $$-0.379516\pi$$
0.369537 + 0.929216i $$0.379516\pi$$
$$374$$ 0 0
$$375$$ 2363.12 0.325416
$$376$$ 0 0
$$377$$ 577.930 0.0789521
$$378$$ 0 0
$$379$$ 486.170 0.0658915 0.0329458 0.999457i $$-0.489511\pi$$
0.0329458 + 0.999457i $$0.489511\pi$$
$$380$$ 0 0
$$381$$ −3391.74 −0.456074
$$382$$ 0 0
$$383$$ −10274.8 −1.37081 −0.685405 0.728162i $$-0.740375\pi$$
−0.685405 + 0.728162i $$0.740375\pi$$
$$384$$ 0 0
$$385$$ 4384.60 0.580415
$$386$$ 0 0
$$387$$ 11545.7 1.51654
$$388$$ 0 0
$$389$$ 8823.81 1.15009 0.575045 0.818122i $$-0.304985\pi$$
0.575045 + 0.818122i $$0.304985\pi$$
$$390$$ 0 0
$$391$$ −1570.01 −0.203066
$$392$$ 0 0
$$393$$ 2220.00 0.284946
$$394$$ 0 0
$$395$$ 9640.62 1.22803
$$396$$ 0 0
$$397$$ 7624.66 0.963906 0.481953 0.876197i $$-0.339928\pi$$
0.481953 + 0.876197i $$0.339928\pi$$
$$398$$ 0 0
$$399$$ −1763.48 −0.221264
$$400$$ 0 0
$$401$$ 5192.75 0.646667 0.323333 0.946285i $$-0.395196\pi$$
0.323333 + 0.946285i $$0.395196\pi$$
$$402$$ 0 0
$$403$$ 1234.70 0.152618
$$404$$ 0 0
$$405$$ 5528.80 0.678342
$$406$$ 0 0
$$407$$ −194.991 −0.0237477
$$408$$ 0 0
$$409$$ 15423.3 1.86463 0.932316 0.361645i $$-0.117785\pi$$
0.932316 + 0.361645i $$0.117785\pi$$
$$410$$ 0 0
$$411$$ −491.981 −0.0590454
$$412$$ 0 0
$$413$$ −175.656 −0.0209285
$$414$$ 0 0
$$415$$ −4332.20 −0.512432
$$416$$ 0 0
$$417$$ −587.815 −0.0690298
$$418$$ 0 0
$$419$$ 2112.52 0.246308 0.123154 0.992388i $$-0.460699\pi$$
0.123154 + 0.992388i $$0.460699\pi$$
$$420$$ 0 0
$$421$$ −2010.09 −0.232698 −0.116349 0.993208i $$-0.537119\pi$$
−0.116349 + 0.993208i $$0.537119\pi$$
$$422$$ 0 0
$$423$$ −12009.6 −1.38044
$$424$$ 0 0
$$425$$ −1484.69 −0.169454
$$426$$ 0 0
$$427$$ −1111.63 −0.125984
$$428$$ 0 0
$$429$$ 974.025 0.109619
$$430$$ 0 0
$$431$$ −10433.5 −1.16604 −0.583021 0.812457i $$-0.698129\pi$$
−0.583021 + 0.812457i $$0.698129\pi$$
$$432$$ 0 0
$$433$$ −6973.11 −0.773917 −0.386959 0.922097i $$-0.626474\pi$$
−0.386959 + 0.922097i $$0.626474\pi$$
$$434$$ 0 0
$$435$$ 492.766 0.0543134
$$436$$ 0 0
$$437$$ −1183.75 −0.129580
$$438$$ 0 0
$$439$$ −3247.81 −0.353096 −0.176548 0.984292i $$-0.556493\pi$$
−0.176548 + 0.984292i $$0.556493\pi$$
$$440$$ 0 0
$$441$$ 3761.13 0.406126
$$442$$ 0 0
$$443$$ −8227.84 −0.882430 −0.441215 0.897401i $$-0.645452\pi$$
−0.441215 + 0.897401i $$0.645452\pi$$
$$444$$ 0 0
$$445$$ 14512.1 1.54593
$$446$$ 0 0
$$447$$ 322.696 0.0341454
$$448$$ 0 0
$$449$$ −8823.99 −0.927460 −0.463730 0.885976i $$-0.653489\pi$$
−0.463730 + 0.885976i $$0.653489\pi$$
$$450$$ 0 0
$$451$$ 10654.1 1.11238
$$452$$ 0 0
$$453$$ 57.4245 0.00595594
$$454$$ 0 0
$$455$$ −2884.30 −0.297182
$$456$$ 0 0
$$457$$ −1162.68 −0.119010 −0.0595052 0.998228i $$-0.518952\pi$$
−0.0595052 + 0.998228i $$0.518952\pi$$
$$458$$ 0 0
$$459$$ 8747.97 0.889586
$$460$$ 0 0
$$461$$ −13275.8 −1.34124 −0.670622 0.741799i $$-0.733973\pi$$
−0.670622 + 0.741799i $$0.733973\pi$$
$$462$$ 0 0
$$463$$ 17666.1 1.77325 0.886625 0.462488i $$-0.153043\pi$$
0.886625 + 0.462488i $$0.153043\pi$$
$$464$$ 0 0
$$465$$ 1052.76 0.104990
$$466$$ 0 0
$$467$$ −19505.3 −1.93276 −0.966380 0.257117i $$-0.917227\pi$$
−0.966380 + 0.257117i $$0.917227\pi$$
$$468$$ 0 0
$$469$$ −7271.52 −0.715922
$$470$$ 0 0
$$471$$ 221.667 0.0216855
$$472$$ 0 0
$$473$$ −14336.7 −1.39366
$$474$$ 0 0
$$475$$ −1119.42 −0.108132
$$476$$ 0 0
$$477$$ −16319.5 −1.56650
$$478$$ 0 0
$$479$$ 9224.14 0.879878 0.439939 0.898028i $$-0.355000\pi$$
0.439939 + 0.898028i $$0.355000\pi$$
$$480$$ 0 0
$$481$$ 128.270 0.0121592
$$482$$ 0 0
$$483$$ 329.934 0.0310818
$$484$$ 0 0
$$485$$ −6839.57 −0.640348
$$486$$ 0 0
$$487$$ −9009.75 −0.838338 −0.419169 0.907908i $$-0.637679\pi$$
−0.419169 + 0.907908i $$0.637679\pi$$
$$488$$ 0 0
$$489$$ −1400.09 −0.129477
$$490$$ 0 0
$$491$$ 2822.52 0.259426 0.129713 0.991552i $$-0.458594\pi$$
0.129713 + 0.991552i $$0.458594\pi$$
$$492$$ 0 0
$$493$$ −3059.44 −0.279493
$$494$$ 0 0
$$495$$ −7784.40 −0.706834
$$496$$ 0 0
$$497$$ −2911.46 −0.262771
$$498$$ 0 0
$$499$$ −5407.39 −0.485106 −0.242553 0.970138i $$-0.577985\pi$$
−0.242553 + 0.970138i $$0.577985\pi$$
$$500$$ 0 0
$$501$$ 106.939 0.00953633
$$502$$ 0 0
$$503$$ −757.352 −0.0671345 −0.0335673 0.999436i $$-0.510687\pi$$
−0.0335673 + 0.999436i $$0.510687\pi$$
$$504$$ 0 0
$$505$$ 8967.97 0.790237
$$506$$ 0 0
$$507$$ 2903.76 0.254360
$$508$$ 0 0
$$509$$ −2637.36 −0.229664 −0.114832 0.993385i $$-0.536633\pi$$
−0.114832 + 0.993385i $$0.536633\pi$$
$$510$$ 0 0
$$511$$ 8620.70 0.746296
$$512$$ 0 0
$$513$$ 6595.77 0.567661
$$514$$ 0 0
$$515$$ −20259.7 −1.73349
$$516$$ 0 0
$$517$$ 14912.7 1.26859
$$518$$ 0 0
$$519$$ −2902.01 −0.245441
$$520$$ 0 0
$$521$$ −16495.4 −1.38709 −0.693546 0.720413i $$-0.743952\pi$$
−0.693546 + 0.720413i $$0.743952\pi$$
$$522$$ 0 0
$$523$$ 713.955 0.0596923 0.0298462 0.999555i $$-0.490498\pi$$
0.0298462 + 0.999555i $$0.490498\pi$$
$$524$$ 0 0
$$525$$ 312.005 0.0259372
$$526$$ 0 0
$$527$$ −6536.24 −0.540272
$$528$$ 0 0
$$529$$ −11945.5 −0.981797
$$530$$ 0 0
$$531$$ 311.859 0.0254869
$$532$$ 0 0
$$533$$ −7008.54 −0.569557
$$534$$ 0 0
$$535$$ −17362.8 −1.40310
$$536$$ 0 0
$$537$$ 1959.70 0.157481
$$538$$ 0 0
$$539$$ −4670.32 −0.373219
$$540$$ 0 0
$$541$$ −21514.0 −1.70972 −0.854861 0.518857i $$-0.826358\pi$$
−0.854861 + 0.518857i $$0.826358\pi$$
$$542$$ 0 0
$$543$$ 1038.38 0.0820648
$$544$$ 0 0
$$545$$ −8112.60 −0.637625
$$546$$ 0 0
$$547$$ −6782.38 −0.530153 −0.265077 0.964227i $$-0.585397\pi$$
−0.265077 + 0.964227i $$0.585397\pi$$
$$548$$ 0 0
$$549$$ 1973.58 0.153425
$$550$$ 0 0
$$551$$ −2306.75 −0.178350
$$552$$ 0 0
$$553$$ 12578.6 0.967260
$$554$$ 0 0
$$555$$ 109.368 0.00836468
$$556$$ 0 0
$$557$$ 11559.5 0.879337 0.439668 0.898160i $$-0.355096\pi$$
0.439668 + 0.898160i $$0.355096\pi$$
$$558$$ 0 0
$$559$$ 9431.04 0.713579
$$560$$ 0 0
$$561$$ −5156.27 −0.388053
$$562$$ 0 0
$$563$$ 11151.0 0.834736 0.417368 0.908737i $$-0.362953\pi$$
0.417368 + 0.908737i $$0.362953\pi$$
$$564$$ 0 0
$$565$$ 15057.9 1.12122
$$566$$ 0 0
$$567$$ 7213.68 0.534296
$$568$$ 0 0
$$569$$ 11955.9 0.880873 0.440436 0.897784i $$-0.354824\pi$$
0.440436 + 0.897784i $$0.354824\pi$$
$$570$$ 0 0
$$571$$ 18691.6 1.36991 0.684956 0.728585i $$-0.259822\pi$$
0.684956 + 0.728585i $$0.259822\pi$$
$$572$$ 0 0
$$573$$ −6592.03 −0.480604
$$574$$ 0 0
$$575$$ 209.436 0.0151897
$$576$$ 0 0
$$577$$ −11793.2 −0.850876 −0.425438 0.904987i $$-0.639880\pi$$
−0.425438 + 0.904987i $$0.639880\pi$$
$$578$$ 0 0
$$579$$ −1394.45 −0.100089
$$580$$ 0 0
$$581$$ −5652.41 −0.403617
$$582$$ 0 0
$$583$$ 20264.5 1.43957
$$584$$ 0 0
$$585$$ 5120.77 0.361910
$$586$$ 0 0
$$587$$ 759.837 0.0534273 0.0267137 0.999643i $$-0.491496\pi$$
0.0267137 + 0.999643i $$0.491496\pi$$
$$588$$ 0 0
$$589$$ −4928.18 −0.344757
$$590$$ 0 0
$$591$$ −3123.11 −0.217373
$$592$$ 0 0
$$593$$ −1286.39 −0.0890825 −0.0445412 0.999008i $$-0.514183\pi$$
−0.0445412 + 0.999008i $$0.514183\pi$$
$$594$$ 0 0
$$595$$ 15268.8 1.05203
$$596$$ 0 0
$$597$$ −5797.35 −0.397437
$$598$$ 0 0
$$599$$ 4463.66 0.304474 0.152237 0.988344i $$-0.451352\pi$$
0.152237 + 0.988344i $$0.451352\pi$$
$$600$$ 0 0
$$601$$ −13078.7 −0.887673 −0.443837 0.896108i $$-0.646383\pi$$
−0.443837 + 0.896108i $$0.646383\pi$$
$$602$$ 0 0
$$603$$ 12909.8 0.871855
$$604$$ 0 0
$$605$$ −4352.17 −0.292465
$$606$$ 0 0
$$607$$ −475.770 −0.0318137 −0.0159069 0.999873i $$-0.505064\pi$$
−0.0159069 + 0.999873i $$0.505064\pi$$
$$608$$ 0 0
$$609$$ 642.934 0.0427800
$$610$$ 0 0
$$611$$ −9809.93 −0.649537
$$612$$ 0 0
$$613$$ −12137.1 −0.799697 −0.399849 0.916581i $$-0.630937\pi$$
−0.399849 + 0.916581i $$0.630937\pi$$
$$614$$ 0 0
$$615$$ −5975.76 −0.391814
$$616$$ 0 0
$$617$$ 3751.56 0.244785 0.122392 0.992482i $$-0.460943\pi$$
0.122392 + 0.992482i $$0.460943\pi$$
$$618$$ 0 0
$$619$$ 14965.7 0.971763 0.485882 0.874025i $$-0.338499\pi$$
0.485882 + 0.874025i $$0.338499\pi$$
$$620$$ 0 0
$$621$$ −1234.02 −0.0797416
$$622$$ 0 0
$$623$$ 18934.6 1.21766
$$624$$ 0 0
$$625$$ −13667.8 −0.874739
$$626$$ 0 0
$$627$$ −3887.71 −0.247624
$$628$$ 0 0
$$629$$ −679.031 −0.0430441
$$630$$ 0 0
$$631$$ 4591.19 0.289655 0.144828 0.989457i $$-0.453737\pi$$
0.144828 + 0.989457i $$0.453737\pi$$
$$632$$ 0 0
$$633$$ 1116.99 0.0701363
$$634$$ 0 0
$$635$$ 22142.0 1.38375
$$636$$ 0 0
$$637$$ 3072.25 0.191094
$$638$$ 0 0
$$639$$ 5169.00 0.320004
$$640$$ 0 0
$$641$$ 25387.9 1.56437 0.782187 0.623044i $$-0.214104\pi$$
0.782187 + 0.623044i $$0.214104\pi$$
$$642$$ 0 0
$$643$$ −25270.1 −1.54986 −0.774928 0.632050i $$-0.782214\pi$$
−0.774928 + 0.632050i $$0.782214\pi$$
$$644$$ 0 0
$$645$$ 8041.27 0.490891
$$646$$ 0 0
$$647$$ 18802.7 1.14252 0.571259 0.820770i $$-0.306455\pi$$
0.571259 + 0.820770i $$0.306455\pi$$
$$648$$ 0 0
$$649$$ −387.246 −0.0234218
$$650$$ 0 0
$$651$$ 1373.58 0.0826955
$$652$$ 0 0
$$653$$ 16345.7 0.979564 0.489782 0.871845i $$-0.337076\pi$$
0.489782 + 0.871845i $$0.337076\pi$$
$$654$$ 0 0
$$655$$ −14492.6 −0.864537
$$656$$ 0 0
$$657$$ −15305.2 −0.908845
$$658$$ 0 0
$$659$$ −25185.3 −1.48874 −0.744370 0.667768i $$-0.767250\pi$$
−0.744370 + 0.667768i $$0.767250\pi$$
$$660$$ 0 0
$$661$$ −17987.6 −1.05845 −0.529227 0.848480i $$-0.677518\pi$$
−0.529227 + 0.848480i $$0.677518\pi$$
$$662$$ 0 0
$$663$$ 3391.92 0.198690
$$664$$ 0 0
$$665$$ 11512.3 0.671323
$$666$$ 0 0
$$667$$ 431.576 0.0250535
$$668$$ 0 0
$$669$$ −2856.49 −0.165079
$$670$$ 0 0
$$671$$ −2450.66 −0.140993
$$672$$ 0 0
$$673$$ 10249.1 0.587033 0.293516 0.955954i $$-0.405174\pi$$
0.293516 + 0.955954i $$0.405174\pi$$
$$674$$ 0 0
$$675$$ −1166.96 −0.0665428
$$676$$ 0 0
$$677$$ 4149.01 0.235538 0.117769 0.993041i $$-0.462426\pi$$
0.117769 + 0.993041i $$0.462426\pi$$
$$678$$ 0 0
$$679$$ −8923.90 −0.504371
$$680$$ 0 0
$$681$$ −4833.66 −0.271991
$$682$$ 0 0
$$683$$ −33571.1 −1.88076 −0.940382 0.340121i $$-0.889532\pi$$
−0.940382 + 0.340121i $$0.889532\pi$$
$$684$$ 0 0
$$685$$ 3211.75 0.179146
$$686$$ 0 0
$$687$$ −6114.82 −0.339585
$$688$$ 0 0
$$689$$ −13330.5 −0.737084
$$690$$ 0 0
$$691$$ 27656.6 1.52258 0.761292 0.648409i $$-0.224565\pi$$
0.761292 + 0.648409i $$0.224565\pi$$
$$692$$ 0 0
$$693$$ −10156.7 −0.556738
$$694$$ 0 0
$$695$$ 3837.37 0.209439
$$696$$ 0 0
$$697$$ 37101.7 2.01625
$$698$$ 0 0
$$699$$ −10095.2 −0.546260
$$700$$ 0 0
$$701$$ 14493.7 0.780912 0.390456 0.920622i $$-0.372317\pi$$
0.390456 + 0.920622i $$0.372317\pi$$
$$702$$ 0 0
$$703$$ −511.974 −0.0274672
$$704$$ 0 0
$$705$$ −8364.33 −0.446835
$$706$$ 0 0
$$707$$ 11700.9 0.622431
$$708$$ 0 0
$$709$$ 8279.81 0.438582 0.219291 0.975659i $$-0.429626\pi$$
0.219291 + 0.975659i $$0.429626\pi$$
$$710$$ 0 0
$$711$$ −22331.9 −1.17794
$$712$$ 0 0
$$713$$ 922.027 0.0484294
$$714$$ 0 0
$$715$$ −6358.63 −0.332586
$$716$$ 0 0
$$717$$ −9508.19 −0.495244
$$718$$ 0 0
$$719$$ −3180.45 −0.164966 −0.0824832 0.996592i $$-0.526285\pi$$
−0.0824832 + 0.996592i $$0.526285\pi$$
$$720$$ 0 0
$$721$$ −26433.7 −1.36538
$$722$$ 0 0
$$723$$ 675.952 0.0347703
$$724$$ 0 0
$$725$$ 408.123 0.0209066
$$726$$ 0 0
$$727$$ 11749.6 0.599409 0.299704 0.954032i $$-0.403112\pi$$
0.299704 + 0.954032i $$0.403112\pi$$
$$728$$ 0 0
$$729$$ −9195.10 −0.467160
$$730$$ 0 0
$$731$$ −49925.8 −2.52609
$$732$$ 0 0
$$733$$ 15005.6 0.756130 0.378065 0.925779i $$-0.376590\pi$$
0.378065 + 0.925779i $$0.376590\pi$$
$$734$$ 0 0
$$735$$ 2619.52 0.131459
$$736$$ 0 0
$$737$$ −16030.6 −0.801212
$$738$$ 0 0
$$739$$ −5555.46 −0.276537 −0.138269 0.990395i $$-0.544154\pi$$
−0.138269 + 0.990395i $$0.544154\pi$$
$$740$$ 0 0
$$741$$ 2557.43 0.126788
$$742$$ 0 0
$$743$$ −18358.9 −0.906493 −0.453246 0.891385i $$-0.649734\pi$$
−0.453246 + 0.891385i $$0.649734\pi$$
$$744$$ 0 0
$$745$$ −2106.63 −0.103598
$$746$$ 0 0
$$747$$ 10035.3 0.491528
$$748$$ 0 0
$$749$$ −22654.1 −1.10516
$$750$$ 0 0
$$751$$ −8818.75 −0.428496 −0.214248 0.976779i $$-0.568730\pi$$
−0.214248 + 0.976779i $$0.568730\pi$$
$$752$$ 0 0
$$753$$ 12010.3 0.581247
$$754$$ 0 0
$$755$$ −374.879 −0.0180705
$$756$$ 0 0
$$757$$ 1462.27 0.0702073 0.0351037 0.999384i $$-0.488824\pi$$
0.0351037 + 0.999384i $$0.488824\pi$$
$$758$$ 0 0
$$759$$ 727.363 0.0347847
$$760$$ 0 0
$$761$$ 15232.2 0.725581 0.362790 0.931871i $$-0.381824\pi$$
0.362790 + 0.931871i $$0.381824\pi$$
$$762$$ 0 0
$$763$$ −10584.9 −0.502225
$$764$$ 0 0
$$765$$ −27108.2 −1.28118
$$766$$ 0 0
$$767$$ 254.740 0.0119923
$$768$$ 0 0
$$769$$ −32110.5 −1.50577 −0.752883 0.658155i $$-0.771337\pi$$
−0.752883 + 0.658155i $$0.771337\pi$$
$$770$$ 0 0
$$771$$ −8603.05 −0.401856
$$772$$ 0 0
$$773$$ 13012.4 0.605465 0.302733 0.953076i $$-0.402101\pi$$
0.302733 + 0.953076i $$0.402101\pi$$
$$774$$ 0 0
$$775$$ 871.922 0.0404134
$$776$$ 0 0
$$777$$ 142.697 0.00658844
$$778$$ 0 0
$$779$$ 27973.8 1.28661
$$780$$ 0 0
$$781$$ −6418.52 −0.294075
$$782$$ 0 0
$$783$$ −2404.70 −0.109754
$$784$$ 0 0
$$785$$ −1447.09 −0.0657946
$$786$$ 0 0
$$787$$ 2738.98 0.124059 0.0620293 0.998074i $$-0.480243\pi$$
0.0620293 + 0.998074i $$0.480243\pi$$
$$788$$ 0 0
$$789$$ −1343.79 −0.0606338
$$790$$ 0 0
$$791$$ 19646.7 0.883131
$$792$$ 0 0
$$793$$ 1612.10 0.0721909
$$794$$ 0 0
$$795$$ −11366.1 −0.507061
$$796$$ 0 0
$$797$$ −43359.0 −1.92704 −0.963522 0.267628i $$-0.913760\pi$$
−0.963522 + 0.267628i $$0.913760\pi$$
$$798$$ 0 0
$$799$$ 51931.6 2.29938
$$800$$ 0 0
$$801$$ −33616.4 −1.48287
$$802$$ 0 0
$$803$$ 19004.9 0.835205
$$804$$ 0 0
$$805$$ −2153.88 −0.0943033
$$806$$ 0 0
$$807$$ −6923.17 −0.301992
$$808$$ 0 0
$$809$$ 400.518 0.0174060 0.00870300 0.999962i $$-0.497230\pi$$
0.00870300 + 0.999962i $$0.497230\pi$$
$$810$$ 0 0
$$811$$ −37589.7 −1.62756 −0.813782 0.581170i $$-0.802595\pi$$
−0.813782 + 0.581170i $$0.802595\pi$$
$$812$$ 0 0
$$813$$ 9627.90 0.415332
$$814$$ 0 0
$$815$$ 9140.09 0.392839
$$816$$ 0 0
$$817$$ −37642.9 −1.61195
$$818$$ 0 0
$$819$$ 6681.29 0.285059
$$820$$ 0 0
$$821$$ −17756.6 −0.754821 −0.377411 0.926046i $$-0.623185\pi$$
−0.377411 + 0.926046i $$0.623185\pi$$
$$822$$ 0 0
$$823$$ 38533.6 1.63207 0.816036 0.578001i $$-0.196167\pi$$
0.816036 + 0.578001i $$0.196167\pi$$
$$824$$ 0 0
$$825$$ 687.837 0.0290272
$$826$$ 0 0
$$827$$ −45875.2 −1.92895 −0.964473 0.264182i $$-0.914898\pi$$
−0.964473 + 0.264182i $$0.914898\pi$$
$$828$$ 0 0
$$829$$ −22725.1 −0.952082 −0.476041 0.879423i $$-0.657929\pi$$
−0.476041 + 0.879423i $$0.657929\pi$$
$$830$$ 0 0
$$831$$ −3605.41 −0.150506
$$832$$ 0 0
$$833$$ −16263.8 −0.676480
$$834$$ 0 0
$$835$$ −698.123 −0.0289336
$$836$$ 0 0
$$837$$ −5137.46 −0.212158
$$838$$ 0 0
$$839$$ 31480.4 1.29538 0.647690 0.761904i $$-0.275735\pi$$
0.647690 + 0.761904i $$0.275735\pi$$
$$840$$ 0 0
$$841$$ 841.000 0.0344828
$$842$$ 0 0
$$843$$ 2698.45 0.110249
$$844$$ 0 0
$$845$$ −18956.3 −0.771737
$$846$$ 0 0
$$847$$ −5678.48 −0.230360
$$848$$ 0 0
$$849$$ −7803.92 −0.315465
$$850$$ 0 0
$$851$$ 95.7866 0.00385843
$$852$$ 0 0
$$853$$ −14863.4 −0.596617 −0.298309 0.954470i $$-0.596422\pi$$
−0.298309 + 0.954470i $$0.596422\pi$$
$$854$$ 0 0
$$855$$ −20439.0 −0.817542
$$856$$ 0 0
$$857$$ 30070.7 1.19859 0.599297 0.800527i $$-0.295447\pi$$
0.599297 + 0.800527i $$0.295447\pi$$
$$858$$ 0 0
$$859$$ 16015.7 0.636146 0.318073 0.948066i $$-0.396964\pi$$
0.318073 + 0.948066i $$0.396964\pi$$
$$860$$ 0 0
$$861$$ −7796.84 −0.308613
$$862$$ 0 0
$$863$$ 4726.12 0.186418 0.0932092 0.995647i $$-0.470287\pi$$
0.0932092 + 0.995647i $$0.470287\pi$$
$$864$$ 0 0
$$865$$ 18944.9 0.744677
$$866$$ 0 0
$$867$$ −10029.8 −0.392882
$$868$$ 0 0
$$869$$ 27730.3 1.08249
$$870$$ 0 0
$$871$$ 10545.3 0.410234
$$872$$ 0 0
$$873$$ 15843.5 0.614226
$$874$$ 0 0
$$875$$ −20128.2 −0.777667
$$876$$ 0 0
$$877$$ −35681.6 −1.37387 −0.686933 0.726720i $$-0.741044\pi$$
−0.686933 + 0.726720i $$0.741044\pi$$
$$878$$ 0 0
$$879$$ −14419.4 −0.553303
$$880$$ 0 0
$$881$$ −14945.7 −0.571548 −0.285774 0.958297i $$-0.592251\pi$$
−0.285774 + 0.958297i $$0.592251\pi$$
$$882$$ 0 0
$$883$$ −39264.7 −1.49645 −0.748223 0.663447i $$-0.769093\pi$$
−0.748223 + 0.663447i $$0.769093\pi$$
$$884$$ 0 0
$$885$$ 217.201 0.00824988
$$886$$ 0 0
$$887$$ 19020.1 0.719991 0.359995 0.932954i $$-0.382778\pi$$
0.359995 + 0.932954i $$0.382778\pi$$
$$888$$ 0 0
$$889$$ 28889.7 1.08991
$$890$$ 0 0
$$891$$ 15903.1 0.597949
$$892$$ 0 0
$$893$$ 39155.3 1.46728
$$894$$ 0 0
$$895$$ −12793.3 −0.477803
$$896$$ 0 0
$$897$$ −478.477 −0.0178103
$$898$$ 0 0
$$899$$ 1796.73 0.0666566
$$900$$ 0 0
$$901$$ 70568.6 2.60930
$$902$$ 0 0
$$903$$ 10491.8 0.386651
$$904$$ 0 0
$$905$$ −6778.76 −0.248987
$$906$$ 0 0
$$907$$ −24945.4 −0.913228 −0.456614 0.889665i $$-0.650938\pi$$
−0.456614 + 0.889665i $$0.650938\pi$$
$$908$$ 0 0
$$909$$ −20773.8 −0.758000
$$910$$ 0 0
$$911$$ 2868.98 0.104340 0.0521698 0.998638i $$-0.483386\pi$$
0.0521698 + 0.998638i $$0.483386\pi$$
$$912$$ 0 0
$$913$$ −12461.1 −0.451702
$$914$$ 0 0
$$915$$ 1374.54 0.0496622
$$916$$ 0 0
$$917$$ −18909.1 −0.680953
$$918$$ 0 0
$$919$$ −20331.1 −0.729774 −0.364887 0.931052i $$-0.618892\pi$$
−0.364887 + 0.931052i $$0.618892\pi$$
$$920$$ 0 0
$$921$$ 6735.82 0.240991
$$922$$ 0 0
$$923$$ 4222.26 0.150571
$$924$$ 0 0
$$925$$ 90.5814 0.00321978
$$926$$ 0 0
$$927$$ 46930.2 1.66277
$$928$$ 0 0
$$929$$ 2069.37 0.0730827 0.0365413 0.999332i $$-0.488366\pi$$
0.0365413 + 0.999332i $$0.488366\pi$$
$$930$$ 0 0
$$931$$ −12262.6 −0.431675
$$932$$ 0 0
$$933$$ −11272.3 −0.395541
$$934$$ 0 0
$$935$$ 33661.2 1.17737
$$936$$ 0 0
$$937$$ −27162.9 −0.947038 −0.473519 0.880784i $$-0.657016\pi$$
−0.473519 + 0.880784i $$0.657016\pi$$
$$938$$ 0 0
$$939$$ 12032.6 0.418178
$$940$$ 0 0
$$941$$ −8127.05 −0.281546 −0.140773 0.990042i $$-0.544959\pi$$
−0.140773 + 0.990042i $$0.544959\pi$$
$$942$$ 0 0
$$943$$ −5233.70 −0.180735
$$944$$ 0 0
$$945$$ 12001.2 0.413122
$$946$$ 0 0
$$947$$ −9170.78 −0.314689 −0.157344 0.987544i $$-0.550293\pi$$
−0.157344 + 0.987544i $$0.550293\pi$$
$$948$$ 0 0
$$949$$ −12501.9 −0.427639
$$950$$ 0 0
$$951$$ −15104.7 −0.515042
$$952$$ 0 0
$$953$$ −25938.9 −0.881683 −0.440841 0.897585i $$-0.645320\pi$$
−0.440841 + 0.897585i $$0.645320\pi$$
$$954$$ 0 0
$$955$$ 43034.1 1.45817
$$956$$ 0 0
$$957$$ 1417.39 0.0478765
$$958$$ 0 0
$$959$$ 4190.52 0.141104
$$960$$ 0 0
$$961$$ −25952.4 −0.871150
$$962$$ 0 0
$$963$$ 40220.0 1.34587
$$964$$ 0 0
$$965$$ 9103.24 0.303672
$$966$$ 0 0
$$967$$ −6987.35 −0.232366 −0.116183 0.993228i $$-0.537066\pi$$
−0.116183 + 0.993228i $$0.537066\pi$$
$$968$$ 0 0
$$969$$ −13538.5 −0.448832
$$970$$ 0 0
$$971$$ −37865.2 −1.25144 −0.625722 0.780046i $$-0.715196\pi$$
−0.625722 + 0.780046i $$0.715196\pi$$
$$972$$ 0 0
$$973$$ 5006.80 0.164965
$$974$$ 0 0
$$975$$ −452.476 −0.0148624
$$976$$ 0 0
$$977$$ −47682.8 −1.56142 −0.780711 0.624893i $$-0.785143\pi$$
−0.780711 + 0.624893i $$0.785143\pi$$
$$978$$ 0 0
$$979$$ 41742.7 1.36272
$$980$$ 0 0
$$981$$ 18792.3 0.611614
$$982$$ 0 0
$$983$$ 10728.5 0.348103 0.174052 0.984737i $$-0.444314\pi$$
0.174052 + 0.984737i $$0.444314\pi$$
$$984$$ 0 0
$$985$$ 20388.3 0.659518
$$986$$ 0 0
$$987$$ −10913.3 −0.351950
$$988$$ 0 0
$$989$$ 7042.72 0.226436
$$990$$ 0 0
$$991$$ −33130.6 −1.06198 −0.530992 0.847377i $$-0.678181\pi$$
−0.530992 + 0.847377i $$0.678181\pi$$
$$992$$ 0 0
$$993$$ −11348.6 −0.362676
$$994$$ 0 0
$$995$$ 37846.3 1.20584
$$996$$ 0 0
$$997$$ 19283.2 0.612543 0.306272 0.951944i $$-0.400918\pi$$
0.306272 + 0.951944i $$0.400918\pi$$
$$998$$ 0 0
$$999$$ −533.715 −0.0169029
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 1856.4.a.bj.1.6 12
4.3 odd 2 1856.4.a.bl.1.7 12
8.3 odd 2 928.4.a.h.1.6 12
8.5 even 2 928.4.a.j.1.7 yes 12

By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.h.1.6 12 8.3 odd 2
928.4.a.j.1.7 yes 12 8.5 even 2
1856.4.a.bj.1.6 12 1.1 even 1 trivial
1856.4.a.bl.1.7 12 4.3 odd 2