# Properties

 Label 18.4.a.a.1.1 Level $18$ Weight $4$ Character 18.1 Self dual yes Analytic conductor $1.062$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 18.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.06203438010$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 18.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} -16.0000 q^{7} +8.00000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} -16.0000 q^{7} +8.00000 q^{8} -12.0000 q^{10} -12.0000 q^{11} +38.0000 q^{13} -32.0000 q^{14} +16.0000 q^{16} +126.000 q^{17} +20.0000 q^{19} -24.0000 q^{20} -24.0000 q^{22} -168.000 q^{23} -89.0000 q^{25} +76.0000 q^{26} -64.0000 q^{28} -30.0000 q^{29} -88.0000 q^{31} +32.0000 q^{32} +252.000 q^{34} +96.0000 q^{35} +254.000 q^{37} +40.0000 q^{38} -48.0000 q^{40} -42.0000 q^{41} -52.0000 q^{43} -48.0000 q^{44} -336.000 q^{46} +96.0000 q^{47} -87.0000 q^{49} -178.000 q^{50} +152.000 q^{52} -198.000 q^{53} +72.0000 q^{55} -128.000 q^{56} -60.0000 q^{58} +660.000 q^{59} -538.000 q^{61} -176.000 q^{62} +64.0000 q^{64} -228.000 q^{65} +884.000 q^{67} +504.000 q^{68} +192.000 q^{70} -792.000 q^{71} +218.000 q^{73} +508.000 q^{74} +80.0000 q^{76} +192.000 q^{77} -520.000 q^{79} -96.0000 q^{80} -84.0000 q^{82} +492.000 q^{83} -756.000 q^{85} -104.000 q^{86} -96.0000 q^{88} -810.000 q^{89} -608.000 q^{91} -672.000 q^{92} +192.000 q^{94} -120.000 q^{95} +1154.00 q^{97} -174.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ −6.00000 −0.536656 −0.268328 0.963328i $$-0.586471\pi$$
−0.268328 + 0.963328i $$0.586471\pi$$
$$6$$ 0 0
$$7$$ −16.0000 −0.863919 −0.431959 0.901893i $$-0.642178\pi$$
−0.431959 + 0.901893i $$0.642178\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 0 0
$$10$$ −12.0000 −0.379473
$$11$$ −12.0000 −0.328921 −0.164461 0.986384i $$-0.552588\pi$$
−0.164461 + 0.986384i $$0.552588\pi$$
$$12$$ 0 0
$$13$$ 38.0000 0.810716 0.405358 0.914158i $$-0.367147\pi$$
0.405358 + 0.914158i $$0.367147\pi$$
$$14$$ −32.0000 −0.610883
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 126.000 1.79762 0.898808 0.438342i $$-0.144434\pi$$
0.898808 + 0.438342i $$0.144434\pi$$
$$18$$ 0 0
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ −24.0000 −0.268328
$$21$$ 0 0
$$22$$ −24.0000 −0.232583
$$23$$ −168.000 −1.52306 −0.761531 0.648129i $$-0.775552\pi$$
−0.761531 + 0.648129i $$0.775552\pi$$
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ 76.0000 0.573263
$$27$$ 0 0
$$28$$ −64.0000 −0.431959
$$29$$ −30.0000 −0.192099 −0.0960493 0.995377i $$-0.530621\pi$$
−0.0960493 + 0.995377i $$0.530621\pi$$
$$30$$ 0 0
$$31$$ −88.0000 −0.509847 −0.254924 0.966961i $$-0.582050\pi$$
−0.254924 + 0.966961i $$0.582050\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 0 0
$$34$$ 252.000 1.27111
$$35$$ 96.0000 0.463627
$$36$$ 0 0
$$37$$ 254.000 1.12858 0.564288 0.825578i $$-0.309151\pi$$
0.564288 + 0.825578i $$0.309151\pi$$
$$38$$ 40.0000 0.170759
$$39$$ 0 0
$$40$$ −48.0000 −0.189737
$$41$$ −42.0000 −0.159983 −0.0799914 0.996796i $$-0.525489\pi$$
−0.0799914 + 0.996796i $$0.525489\pi$$
$$42$$ 0 0
$$43$$ −52.0000 −0.184417 −0.0922084 0.995740i $$-0.529393\pi$$
−0.0922084 + 0.995740i $$0.529393\pi$$
$$44$$ −48.0000 −0.164461
$$45$$ 0 0
$$46$$ −336.000 −1.07697
$$47$$ 96.0000 0.297937 0.148969 0.988842i $$-0.452405\pi$$
0.148969 + 0.988842i $$0.452405\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ −178.000 −0.503460
$$51$$ 0 0
$$52$$ 152.000 0.405358
$$53$$ −198.000 −0.513158 −0.256579 0.966523i $$-0.582595\pi$$
−0.256579 + 0.966523i $$0.582595\pi$$
$$54$$ 0 0
$$55$$ 72.0000 0.176518
$$56$$ −128.000 −0.305441
$$57$$ 0 0
$$58$$ −60.0000 −0.135834
$$59$$ 660.000 1.45635 0.728175 0.685391i $$-0.240369\pi$$
0.728175 + 0.685391i $$0.240369\pi$$
$$60$$ 0 0
$$61$$ −538.000 −1.12924 −0.564622 0.825350i $$-0.690978\pi$$
−0.564622 + 0.825350i $$0.690978\pi$$
$$62$$ −176.000 −0.360516
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ −228.000 −0.435076
$$66$$ 0 0
$$67$$ 884.000 1.61191 0.805954 0.591979i $$-0.201653\pi$$
0.805954 + 0.591979i $$0.201653\pi$$
$$68$$ 504.000 0.898808
$$69$$ 0 0
$$70$$ 192.000 0.327834
$$71$$ −792.000 −1.32385 −0.661923 0.749572i $$-0.730260\pi$$
−0.661923 + 0.749572i $$0.730260\pi$$
$$72$$ 0 0
$$73$$ 218.000 0.349520 0.174760 0.984611i $$-0.444085\pi$$
0.174760 + 0.984611i $$0.444085\pi$$
$$74$$ 508.000 0.798024
$$75$$ 0 0
$$76$$ 80.0000 0.120745
$$77$$ 192.000 0.284161
$$78$$ 0 0
$$79$$ −520.000 −0.740564 −0.370282 0.928919i $$-0.620739\pi$$
−0.370282 + 0.928919i $$0.620739\pi$$
$$80$$ −96.0000 −0.134164
$$81$$ 0 0
$$82$$ −84.0000 −0.113125
$$83$$ 492.000 0.650651 0.325325 0.945602i $$-0.394526\pi$$
0.325325 + 0.945602i $$0.394526\pi$$
$$84$$ 0 0
$$85$$ −756.000 −0.964703
$$86$$ −104.000 −0.130402
$$87$$ 0 0
$$88$$ −96.0000 −0.116291
$$89$$ −810.000 −0.964717 −0.482359 0.875974i $$-0.660220\pi$$
−0.482359 + 0.875974i $$0.660220\pi$$
$$90$$ 0 0
$$91$$ −608.000 −0.700393
$$92$$ −672.000 −0.761531
$$93$$ 0 0
$$94$$ 192.000 0.210673
$$95$$ −120.000 −0.129597
$$96$$ 0 0
$$97$$ 1154.00 1.20795 0.603974 0.797004i $$-0.293583\pi$$
0.603974 + 0.797004i $$0.293583\pi$$
$$98$$ −174.000 −0.179354
$$99$$ 0 0
$$100$$ −356.000 −0.356000
$$101$$ 618.000 0.608845 0.304422 0.952537i $$-0.401537\pi$$
0.304422 + 0.952537i $$0.401537\pi$$
$$102$$ 0 0
$$103$$ 128.000 0.122449 0.0612243 0.998124i $$-0.480499\pi$$
0.0612243 + 0.998124i $$0.480499\pi$$
$$104$$ 304.000 0.286631
$$105$$ 0 0
$$106$$ −396.000 −0.362858
$$107$$ 1476.00 1.33355 0.666777 0.745257i $$-0.267673\pi$$
0.666777 + 0.745257i $$0.267673\pi$$
$$108$$ 0 0
$$109$$ 1190.00 1.04570 0.522850 0.852425i $$-0.324869\pi$$
0.522850 + 0.852425i $$0.324869\pi$$
$$110$$ 144.000 0.124817
$$111$$ 0 0
$$112$$ −256.000 −0.215980
$$113$$ 462.000 0.384613 0.192307 0.981335i $$-0.438403\pi$$
0.192307 + 0.981335i $$0.438403\pi$$
$$114$$ 0 0
$$115$$ 1008.00 0.817361
$$116$$ −120.000 −0.0960493
$$117$$ 0 0
$$118$$ 1320.00 1.02980
$$119$$ −2016.00 −1.55300
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ −1076.00 −0.798496
$$123$$ 0 0
$$124$$ −352.000 −0.254924
$$125$$ 1284.00 0.918756
$$126$$ 0 0
$$127$$ −2536.00 −1.77192 −0.885959 0.463763i $$-0.846499\pi$$
−0.885959 + 0.463763i $$0.846499\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 0 0
$$130$$ −456.000 −0.307645
$$131$$ −2292.00 −1.52865 −0.764324 0.644832i $$-0.776927\pi$$
−0.764324 + 0.644832i $$0.776927\pi$$
$$132$$ 0 0
$$133$$ −320.000 −0.208628
$$134$$ 1768.00 1.13979
$$135$$ 0 0
$$136$$ 1008.00 0.635554
$$137$$ 726.000 0.452747 0.226374 0.974041i $$-0.427313\pi$$
0.226374 + 0.974041i $$0.427313\pi$$
$$138$$ 0 0
$$139$$ 380.000 0.231879 0.115939 0.993256i $$-0.463012\pi$$
0.115939 + 0.993256i $$0.463012\pi$$
$$140$$ 384.000 0.231814
$$141$$ 0 0
$$142$$ −1584.00 −0.936101
$$143$$ −456.000 −0.266662
$$144$$ 0 0
$$145$$ 180.000 0.103091
$$146$$ 436.000 0.247148
$$147$$ 0 0
$$148$$ 1016.00 0.564288
$$149$$ −1590.00 −0.874214 −0.437107 0.899410i $$-0.643997\pi$$
−0.437107 + 0.899410i $$0.643997\pi$$
$$150$$ 0 0
$$151$$ 2432.00 1.31068 0.655342 0.755332i $$-0.272524\pi$$
0.655342 + 0.755332i $$0.272524\pi$$
$$152$$ 160.000 0.0853797
$$153$$ 0 0
$$154$$ 384.000 0.200932
$$155$$ 528.000 0.273613
$$156$$ 0 0
$$157$$ 614.000 0.312118 0.156059 0.987748i $$-0.450121\pi$$
0.156059 + 0.987748i $$0.450121\pi$$
$$158$$ −1040.00 −0.523658
$$159$$ 0 0
$$160$$ −192.000 −0.0948683
$$161$$ 2688.00 1.31580
$$162$$ 0 0
$$163$$ −1852.00 −0.889938 −0.444969 0.895546i $$-0.646785\pi$$
−0.444969 + 0.895546i $$0.646785\pi$$
$$164$$ −168.000 −0.0799914
$$165$$ 0 0
$$166$$ 984.000 0.460080
$$167$$ 2136.00 0.989752 0.494876 0.868964i $$-0.335213\pi$$
0.494876 + 0.868964i $$0.335213\pi$$
$$168$$ 0 0
$$169$$ −753.000 −0.342740
$$170$$ −1512.00 −0.682148
$$171$$ 0 0
$$172$$ −208.000 −0.0922084
$$173$$ −1758.00 −0.772591 −0.386296 0.922375i $$-0.626246\pi$$
−0.386296 + 0.922375i $$0.626246\pi$$
$$174$$ 0 0
$$175$$ 1424.00 0.615110
$$176$$ −192.000 −0.0822304
$$177$$ 0 0
$$178$$ −1620.00 −0.682158
$$179$$ 540.000 0.225483 0.112742 0.993624i $$-0.464037\pi$$
0.112742 + 0.993624i $$0.464037\pi$$
$$180$$ 0 0
$$181$$ 1982.00 0.813928 0.406964 0.913444i $$-0.366588\pi$$
0.406964 + 0.913444i $$0.366588\pi$$
$$182$$ −1216.00 −0.495252
$$183$$ 0 0
$$184$$ −1344.00 −0.538484
$$185$$ −1524.00 −0.605658
$$186$$ 0 0
$$187$$ −1512.00 −0.591275
$$188$$ 384.000 0.148969
$$189$$ 0 0
$$190$$ −240.000 −0.0916391
$$191$$ 2688.00 1.01831 0.509154 0.860675i $$-0.329958\pi$$
0.509154 + 0.860675i $$0.329958\pi$$
$$192$$ 0 0
$$193$$ −2302.00 −0.858557 −0.429279 0.903172i $$-0.641232\pi$$
−0.429279 + 0.903172i $$0.641232\pi$$
$$194$$ 2308.00 0.854148
$$195$$ 0 0
$$196$$ −348.000 −0.126822
$$197$$ −4374.00 −1.58190 −0.790951 0.611880i $$-0.790414\pi$$
−0.790951 + 0.611880i $$0.790414\pi$$
$$198$$ 0 0
$$199$$ −1600.00 −0.569955 −0.284977 0.958534i $$-0.591986\pi$$
−0.284977 + 0.958534i $$0.591986\pi$$
$$200$$ −712.000 −0.251730
$$201$$ 0 0
$$202$$ 1236.00 0.430518
$$203$$ 480.000 0.165958
$$204$$ 0 0
$$205$$ 252.000 0.0858558
$$206$$ 256.000 0.0865843
$$207$$ 0 0
$$208$$ 608.000 0.202679
$$209$$ −240.000 −0.0794313
$$210$$ 0 0
$$211$$ 3332.00 1.08713 0.543565 0.839367i $$-0.317074\pi$$
0.543565 + 0.839367i $$0.317074\pi$$
$$212$$ −792.000 −0.256579
$$213$$ 0 0
$$214$$ 2952.00 0.942965
$$215$$ 312.000 0.0989685
$$216$$ 0 0
$$217$$ 1408.00 0.440467
$$218$$ 2380.00 0.739422
$$219$$ 0 0
$$220$$ 288.000 0.0882589
$$221$$ 4788.00 1.45736
$$222$$ 0 0
$$223$$ 2648.00 0.795171 0.397586 0.917565i $$-0.369848\pi$$
0.397586 + 0.917565i $$0.369848\pi$$
$$224$$ −512.000 −0.152721
$$225$$ 0 0
$$226$$ 924.000 0.271963
$$227$$ −2244.00 −0.656121 −0.328061 0.944657i $$-0.606395\pi$$
−0.328061 + 0.944657i $$0.606395\pi$$
$$228$$ 0 0
$$229$$ −5650.00 −1.63040 −0.815202 0.579177i $$-0.803374\pi$$
−0.815202 + 0.579177i $$0.803374\pi$$
$$230$$ 2016.00 0.577961
$$231$$ 0 0
$$232$$ −240.000 −0.0679171
$$233$$ −4698.00 −1.32093 −0.660464 0.750858i $$-0.729640\pi$$
−0.660464 + 0.750858i $$0.729640\pi$$
$$234$$ 0 0
$$235$$ −576.000 −0.159890
$$236$$ 2640.00 0.728175
$$237$$ 0 0
$$238$$ −4032.00 −1.09813
$$239$$ 1200.00 0.324776 0.162388 0.986727i $$-0.448080\pi$$
0.162388 + 0.986727i $$0.448080\pi$$
$$240$$ 0 0
$$241$$ −718.000 −0.191911 −0.0959553 0.995386i $$-0.530591\pi$$
−0.0959553 + 0.995386i $$0.530591\pi$$
$$242$$ −2374.00 −0.630605
$$243$$ 0 0
$$244$$ −2152.00 −0.564622
$$245$$ 522.000 0.136120
$$246$$ 0 0
$$247$$ 760.000 0.195780
$$248$$ −704.000 −0.180258
$$249$$ 0 0
$$250$$ 2568.00 0.649658
$$251$$ −6012.00 −1.51185 −0.755924 0.654659i $$-0.772812\pi$$
−0.755924 + 0.654659i $$0.772812\pi$$
$$252$$ 0 0
$$253$$ 2016.00 0.500968
$$254$$ −5072.00 −1.25294
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 2046.00 0.496599 0.248300 0.968683i $$-0.420128\pi$$
0.248300 + 0.968683i $$0.420128\pi$$
$$258$$ 0 0
$$259$$ −4064.00 −0.974999
$$260$$ −912.000 −0.217538
$$261$$ 0 0
$$262$$ −4584.00 −1.08092
$$263$$ 6072.00 1.42363 0.711817 0.702365i $$-0.247873\pi$$
0.711817 + 0.702365i $$0.247873\pi$$
$$264$$ 0 0
$$265$$ 1188.00 0.275390
$$266$$ −640.000 −0.147522
$$267$$ 0 0
$$268$$ 3536.00 0.805954
$$269$$ 6930.00 1.57074 0.785371 0.619025i $$-0.212472\pi$$
0.785371 + 0.619025i $$0.212472\pi$$
$$270$$ 0 0
$$271$$ 1352.00 0.303056 0.151528 0.988453i $$-0.451581\pi$$
0.151528 + 0.988453i $$0.451581\pi$$
$$272$$ 2016.00 0.449404
$$273$$ 0 0
$$274$$ 1452.00 0.320141
$$275$$ 1068.00 0.234192
$$276$$ 0 0
$$277$$ −1186.00 −0.257256 −0.128628 0.991693i $$-0.541057\pi$$
−0.128628 + 0.991693i $$0.541057\pi$$
$$278$$ 760.000 0.163963
$$279$$ 0 0
$$280$$ 768.000 0.163917
$$281$$ −2442.00 −0.518425 −0.259213 0.965820i $$-0.583463\pi$$
−0.259213 + 0.965820i $$0.583463\pi$$
$$282$$ 0 0
$$283$$ 2828.00 0.594018 0.297009 0.954875i $$-0.404011\pi$$
0.297009 + 0.954875i $$0.404011\pi$$
$$284$$ −3168.00 −0.661923
$$285$$ 0 0
$$286$$ −912.000 −0.188558
$$287$$ 672.000 0.138212
$$288$$ 0 0
$$289$$ 10963.0 2.23143
$$290$$ 360.000 0.0728963
$$291$$ 0 0
$$292$$ 872.000 0.174760
$$293$$ −4758.00 −0.948687 −0.474344 0.880340i $$-0.657315\pi$$
−0.474344 + 0.880340i $$0.657315\pi$$
$$294$$ 0 0
$$295$$ −3960.00 −0.781560
$$296$$ 2032.00 0.399012
$$297$$ 0 0
$$298$$ −3180.00 −0.618163
$$299$$ −6384.00 −1.23477
$$300$$ 0 0
$$301$$ 832.000 0.159321
$$302$$ 4864.00 0.926794
$$303$$ 0 0
$$304$$ 320.000 0.0603726
$$305$$ 3228.00 0.606016
$$306$$ 0 0
$$307$$ −8476.00 −1.57574 −0.787868 0.615844i $$-0.788815\pi$$
−0.787868 + 0.615844i $$0.788815\pi$$
$$308$$ 768.000 0.142081
$$309$$ 0 0
$$310$$ 1056.00 0.193473
$$311$$ −4632.00 −0.844555 −0.422278 0.906467i $$-0.638769\pi$$
−0.422278 + 0.906467i $$0.638769\pi$$
$$312$$ 0 0
$$313$$ −4822.00 −0.870785 −0.435392 0.900241i $$-0.643390\pi$$
−0.435392 + 0.900241i $$0.643390\pi$$
$$314$$ 1228.00 0.220701
$$315$$ 0 0
$$316$$ −2080.00 −0.370282
$$317$$ 3426.00 0.607014 0.303507 0.952829i $$-0.401842\pi$$
0.303507 + 0.952829i $$0.401842\pi$$
$$318$$ 0 0
$$319$$ 360.000 0.0631854
$$320$$ −384.000 −0.0670820
$$321$$ 0 0
$$322$$ 5376.00 0.930412
$$323$$ 2520.00 0.434107
$$324$$ 0 0
$$325$$ −3382.00 −0.577230
$$326$$ −3704.00 −0.629281
$$327$$ 0 0
$$328$$ −336.000 −0.0565625
$$329$$ −1536.00 −0.257393
$$330$$ 0 0
$$331$$ −2788.00 −0.462968 −0.231484 0.972839i $$-0.574358\pi$$
−0.231484 + 0.972839i $$0.574358\pi$$
$$332$$ 1968.00 0.325325
$$333$$ 0 0
$$334$$ 4272.00 0.699861
$$335$$ −5304.00 −0.865040
$$336$$ 0 0
$$337$$ 434.000 0.0701528 0.0350764 0.999385i $$-0.488833\pi$$
0.0350764 + 0.999385i $$0.488833\pi$$
$$338$$ −1506.00 −0.242354
$$339$$ 0 0
$$340$$ −3024.00 −0.482351
$$341$$ 1056.00 0.167700
$$342$$ 0 0
$$343$$ 6880.00 1.08305
$$344$$ −416.000 −0.0652012
$$345$$ 0 0
$$346$$ −3516.00 −0.546304
$$347$$ −6684.00 −1.03405 −0.517026 0.855970i $$-0.672961\pi$$
−0.517026 + 0.855970i $$0.672961\pi$$
$$348$$ 0 0
$$349$$ 2630.00 0.403383 0.201692 0.979449i $$-0.435356\pi$$
0.201692 + 0.979449i $$0.435356\pi$$
$$350$$ 2848.00 0.434949
$$351$$ 0 0
$$352$$ −384.000 −0.0581456
$$353$$ 7422.00 1.11907 0.559537 0.828805i $$-0.310979\pi$$
0.559537 + 0.828805i $$0.310979\pi$$
$$354$$ 0 0
$$355$$ 4752.00 0.710451
$$356$$ −3240.00 −0.482359
$$357$$ 0 0
$$358$$ 1080.00 0.159441
$$359$$ 10440.0 1.53482 0.767412 0.641154i $$-0.221544\pi$$
0.767412 + 0.641154i $$0.221544\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 3964.00 0.575534
$$363$$ 0 0
$$364$$ −2432.00 −0.350196
$$365$$ −1308.00 −0.187572
$$366$$ 0 0
$$367$$ 10424.0 1.48264 0.741319 0.671153i $$-0.234200\pi$$
0.741319 + 0.671153i $$0.234200\pi$$
$$368$$ −2688.00 −0.380765
$$369$$ 0 0
$$370$$ −3048.00 −0.428265
$$371$$ 3168.00 0.443327
$$372$$ 0 0
$$373$$ 3278.00 0.455036 0.227518 0.973774i $$-0.426939\pi$$
0.227518 + 0.973774i $$0.426939\pi$$
$$374$$ −3024.00 −0.418094
$$375$$ 0 0
$$376$$ 768.000 0.105337
$$377$$ −1140.00 −0.155737
$$378$$ 0 0
$$379$$ 6140.00 0.832165 0.416083 0.909327i $$-0.363403\pi$$
0.416083 + 0.909327i $$0.363403\pi$$
$$380$$ −480.000 −0.0647986
$$381$$ 0 0
$$382$$ 5376.00 0.720053
$$383$$ 3072.00 0.409848 0.204924 0.978778i $$-0.434305\pi$$
0.204924 + 0.978778i $$0.434305\pi$$
$$384$$ 0 0
$$385$$ −1152.00 −0.152497
$$386$$ −4604.00 −0.607092
$$387$$ 0 0
$$388$$ 4616.00 0.603974
$$389$$ −6150.00 −0.801587 −0.400794 0.916168i $$-0.631266\pi$$
−0.400794 + 0.916168i $$0.631266\pi$$
$$390$$ 0 0
$$391$$ −21168.0 −2.73788
$$392$$ −696.000 −0.0896768
$$393$$ 0 0
$$394$$ −8748.00 −1.11857
$$395$$ 3120.00 0.397428
$$396$$ 0 0
$$397$$ −106.000 −0.0134005 −0.00670024 0.999978i $$-0.502133\pi$$
−0.00670024 + 0.999978i $$0.502133\pi$$
$$398$$ −3200.00 −0.403019
$$399$$ 0 0
$$400$$ −1424.00 −0.178000
$$401$$ 1758.00 0.218929 0.109464 0.993991i $$-0.465086\pi$$
0.109464 + 0.993991i $$0.465086\pi$$
$$402$$ 0 0
$$403$$ −3344.00 −0.413341
$$404$$ 2472.00 0.304422
$$405$$ 0 0
$$406$$ 960.000 0.117350
$$407$$ −3048.00 −0.371213
$$408$$ 0 0
$$409$$ −3670.00 −0.443691 −0.221846 0.975082i $$-0.571208\pi$$
−0.221846 + 0.975082i $$0.571208\pi$$
$$410$$ 504.000 0.0607092
$$411$$ 0 0
$$412$$ 512.000 0.0612243
$$413$$ −10560.0 −1.25817
$$414$$ 0 0
$$415$$ −2952.00 −0.349176
$$416$$ 1216.00 0.143316
$$417$$ 0 0
$$418$$ −480.000 −0.0561664
$$419$$ 9660.00 1.12631 0.563153 0.826353i $$-0.309588\pi$$
0.563153 + 0.826353i $$0.309588\pi$$
$$420$$ 0 0
$$421$$ 8462.00 0.979602 0.489801 0.871834i $$-0.337069\pi$$
0.489801 + 0.871834i $$0.337069\pi$$
$$422$$ 6664.00 0.768717
$$423$$ 0 0
$$424$$ −1584.00 −0.181429
$$425$$ −11214.0 −1.27990
$$426$$ 0 0
$$427$$ 8608.00 0.975575
$$428$$ 5904.00 0.666777
$$429$$ 0 0
$$430$$ 624.000 0.0699813
$$431$$ −9792.00 −1.09435 −0.547174 0.837019i $$-0.684296\pi$$
−0.547174 + 0.837019i $$0.684296\pi$$
$$432$$ 0 0
$$433$$ −7342.00 −0.814859 −0.407430 0.913237i $$-0.633575\pi$$
−0.407430 + 0.913237i $$0.633575\pi$$
$$434$$ 2816.00 0.311457
$$435$$ 0 0
$$436$$ 4760.00 0.522850
$$437$$ −3360.00 −0.367805
$$438$$ 0 0
$$439$$ 10640.0 1.15676 0.578382 0.815766i $$-0.303684\pi$$
0.578382 + 0.815766i $$0.303684\pi$$
$$440$$ 576.000 0.0624085
$$441$$ 0 0
$$442$$ 9576.00 1.03051
$$443$$ 17412.0 1.86742 0.933712 0.358024i $$-0.116549\pi$$
0.933712 + 0.358024i $$0.116549\pi$$
$$444$$ 0 0
$$445$$ 4860.00 0.517722
$$446$$ 5296.00 0.562271
$$447$$ 0 0
$$448$$ −1024.00 −0.107990
$$449$$ 1710.00 0.179732 0.0898662 0.995954i $$-0.471356\pi$$
0.0898662 + 0.995954i $$0.471356\pi$$
$$450$$ 0 0
$$451$$ 504.000 0.0526218
$$452$$ 1848.00 0.192307
$$453$$ 0 0
$$454$$ −4488.00 −0.463948
$$455$$ 3648.00 0.375870
$$456$$ 0 0
$$457$$ −646.000 −0.0661239 −0.0330619 0.999453i $$-0.510526\pi$$
−0.0330619 + 0.999453i $$0.510526\pi$$
$$458$$ −11300.0 −1.15287
$$459$$ 0 0
$$460$$ 4032.00 0.408680
$$461$$ 6018.00 0.607996 0.303998 0.952673i $$-0.401678\pi$$
0.303998 + 0.952673i $$0.401678\pi$$
$$462$$ 0 0
$$463$$ −6712.00 −0.673722 −0.336861 0.941554i $$-0.609365\pi$$
−0.336861 + 0.941554i $$0.609365\pi$$
$$464$$ −480.000 −0.0480247
$$465$$ 0 0
$$466$$ −9396.00 −0.934037
$$467$$ −5364.00 −0.531512 −0.265756 0.964040i $$-0.585622\pi$$
−0.265756 + 0.964040i $$0.585622\pi$$
$$468$$ 0 0
$$469$$ −14144.0 −1.39256
$$470$$ −1152.00 −0.113059
$$471$$ 0 0
$$472$$ 5280.00 0.514898
$$473$$ 624.000 0.0606587
$$474$$ 0 0
$$475$$ −1780.00 −0.171941
$$476$$ −8064.00 −0.776498
$$477$$ 0 0
$$478$$ 2400.00 0.229652
$$479$$ −9840.00 −0.938624 −0.469312 0.883032i $$-0.655498\pi$$
−0.469312 + 0.883032i $$0.655498\pi$$
$$480$$ 0 0
$$481$$ 9652.00 0.914955
$$482$$ −1436.00 −0.135701
$$483$$ 0 0
$$484$$ −4748.00 −0.445905
$$485$$ −6924.00 −0.648253
$$486$$ 0 0
$$487$$ 1424.00 0.132500 0.0662501 0.997803i $$-0.478896\pi$$
0.0662501 + 0.997803i $$0.478896\pi$$
$$488$$ −4304.00 −0.399248
$$489$$ 0 0
$$490$$ 1044.00 0.0962513
$$491$$ 4548.00 0.418021 0.209011 0.977913i $$-0.432976\pi$$
0.209011 + 0.977913i $$0.432976\pi$$
$$492$$ 0 0
$$493$$ −3780.00 −0.345320
$$494$$ 1520.00 0.138437
$$495$$ 0 0
$$496$$ −1408.00 −0.127462
$$497$$ 12672.0 1.14370
$$498$$ 0 0
$$499$$ 6500.00 0.583126 0.291563 0.956552i $$-0.405825\pi$$
0.291563 + 0.956552i $$0.405825\pi$$
$$500$$ 5136.00 0.459378
$$501$$ 0 0
$$502$$ −12024.0 −1.06904
$$503$$ −12168.0 −1.07862 −0.539308 0.842108i $$-0.681314\pi$$
−0.539308 + 0.842108i $$0.681314\pi$$
$$504$$ 0 0
$$505$$ −3708.00 −0.326740
$$506$$ 4032.00 0.354238
$$507$$ 0 0
$$508$$ −10144.0 −0.885959
$$509$$ 21090.0 1.83654 0.918269 0.395957i $$-0.129587\pi$$
0.918269 + 0.395957i $$0.129587\pi$$
$$510$$ 0 0
$$511$$ −3488.00 −0.301957
$$512$$ 512.000 0.0441942
$$513$$ 0 0
$$514$$ 4092.00 0.351149
$$515$$ −768.000 −0.0657129
$$516$$ 0 0
$$517$$ −1152.00 −0.0979979
$$518$$ −8128.00 −0.689428
$$519$$ 0 0
$$520$$ −1824.00 −0.153822
$$521$$ 5238.00 0.440462 0.220231 0.975448i $$-0.429319\pi$$
0.220231 + 0.975448i $$0.429319\pi$$
$$522$$ 0 0
$$523$$ 8588.00 0.718025 0.359012 0.933333i $$-0.383114\pi$$
0.359012 + 0.933333i $$0.383114\pi$$
$$524$$ −9168.00 −0.764324
$$525$$ 0 0
$$526$$ 12144.0 1.00666
$$527$$ −11088.0 −0.916510
$$528$$ 0 0
$$529$$ 16057.0 1.31972
$$530$$ 2376.00 0.194730
$$531$$ 0 0
$$532$$ −1280.00 −0.104314
$$533$$ −1596.00 −0.129701
$$534$$ 0 0
$$535$$ −8856.00 −0.715660
$$536$$ 7072.00 0.569895
$$537$$ 0 0
$$538$$ 13860.0 1.11068
$$539$$ 1044.00 0.0834291
$$540$$ 0 0
$$541$$ 3062.00 0.243338 0.121669 0.992571i $$-0.461175\pi$$
0.121669 + 0.992571i $$0.461175\pi$$
$$542$$ 2704.00 0.214293
$$543$$ 0 0
$$544$$ 4032.00 0.317777
$$545$$ −7140.00 −0.561182
$$546$$ 0 0
$$547$$ −8476.00 −0.662537 −0.331268 0.943537i $$-0.607477\pi$$
−0.331268 + 0.943537i $$0.607477\pi$$
$$548$$ 2904.00 0.226374
$$549$$ 0 0
$$550$$ 2136.00 0.165599
$$551$$ −600.000 −0.0463899
$$552$$ 0 0
$$553$$ 8320.00 0.639787
$$554$$ −2372.00 −0.181907
$$555$$ 0 0
$$556$$ 1520.00 0.115939
$$557$$ 12546.0 0.954383 0.477191 0.878799i $$-0.341655\pi$$
0.477191 + 0.878799i $$0.341655\pi$$
$$558$$ 0 0
$$559$$ −1976.00 −0.149510
$$560$$ 1536.00 0.115907
$$561$$ 0 0
$$562$$ −4884.00 −0.366582
$$563$$ 12.0000 0.000898294 0 0.000449147 1.00000i $$-0.499857\pi$$
0.000449147 1.00000i $$0.499857\pi$$
$$564$$ 0 0
$$565$$ −2772.00 −0.206405
$$566$$ 5656.00 0.420034
$$567$$ 0 0
$$568$$ −6336.00 −0.468050
$$569$$ −19290.0 −1.42123 −0.710614 0.703582i $$-0.751583\pi$$
−0.710614 + 0.703582i $$0.751583\pi$$
$$570$$ 0 0
$$571$$ −12148.0 −0.890329 −0.445165 0.895449i $$-0.646855\pi$$
−0.445165 + 0.895449i $$0.646855\pi$$
$$572$$ −1824.00 −0.133331
$$573$$ 0 0
$$574$$ 1344.00 0.0977308
$$575$$ 14952.0 1.08442
$$576$$ 0 0
$$577$$ −10366.0 −0.747907 −0.373953 0.927447i $$-0.621998\pi$$
−0.373953 + 0.927447i $$0.621998\pi$$
$$578$$ 21926.0 1.57786
$$579$$ 0 0
$$580$$ 720.000 0.0515455
$$581$$ −7872.00 −0.562109
$$582$$ 0 0
$$583$$ 2376.00 0.168789
$$584$$ 1744.00 0.123574
$$585$$ 0 0
$$586$$ −9516.00 −0.670823
$$587$$ −7644.00 −0.537482 −0.268741 0.963213i $$-0.586607\pi$$
−0.268741 + 0.963213i $$0.586607\pi$$
$$588$$ 0 0
$$589$$ −1760.00 −0.123123
$$590$$ −7920.00 −0.552646
$$591$$ 0 0
$$592$$ 4064.00 0.282144
$$593$$ −8658.00 −0.599564 −0.299782 0.954008i $$-0.596914\pi$$
−0.299782 + 0.954008i $$0.596914\pi$$
$$594$$ 0 0
$$595$$ 12096.0 0.833425
$$596$$ −6360.00 −0.437107
$$597$$ 0 0
$$598$$ −12768.0 −0.873114
$$599$$ −25800.0 −1.75987 −0.879933 0.475098i $$-0.842413\pi$$
−0.879933 + 0.475098i $$0.842413\pi$$
$$600$$ 0 0
$$601$$ 16202.0 1.09966 0.549828 0.835278i $$-0.314693\pi$$
0.549828 + 0.835278i $$0.314693\pi$$
$$602$$ 1664.00 0.112657
$$603$$ 0 0
$$604$$ 9728.00 0.655342
$$605$$ 7122.00 0.478596
$$606$$ 0 0
$$607$$ −24136.0 −1.61392 −0.806960 0.590605i $$-0.798889\pi$$
−0.806960 + 0.590605i $$0.798889\pi$$
$$608$$ 640.000 0.0426898
$$609$$ 0 0
$$610$$ 6456.00 0.428518
$$611$$ 3648.00 0.241542
$$612$$ 0 0
$$613$$ −4642.00 −0.305854 −0.152927 0.988237i $$-0.548870\pi$$
−0.152927 + 0.988237i $$0.548870\pi$$
$$614$$ −16952.0 −1.11421
$$615$$ 0 0
$$616$$ 1536.00 0.100466
$$617$$ 6726.00 0.438863 0.219432 0.975628i $$-0.429580\pi$$
0.219432 + 0.975628i $$0.429580\pi$$
$$618$$ 0 0
$$619$$ −21220.0 −1.37787 −0.688937 0.724821i $$-0.741922\pi$$
−0.688937 + 0.724821i $$0.741922\pi$$
$$620$$ 2112.00 0.136806
$$621$$ 0 0
$$622$$ −9264.00 −0.597191
$$623$$ 12960.0 0.833437
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ −9644.00 −0.615738
$$627$$ 0 0
$$628$$ 2456.00 0.156059
$$629$$ 32004.0 2.02875
$$630$$ 0 0
$$631$$ 29792.0 1.87956 0.939779 0.341783i $$-0.111031\pi$$
0.939779 + 0.341783i $$0.111031\pi$$
$$632$$ −4160.00 −0.261829
$$633$$ 0 0
$$634$$ 6852.00 0.429223
$$635$$ 15216.0 0.950911
$$636$$ 0 0
$$637$$ −3306.00 −0.205633
$$638$$ 720.000 0.0446788
$$639$$ 0 0
$$640$$ −768.000 −0.0474342
$$641$$ 10158.0 0.625923 0.312962 0.949766i $$-0.398679\pi$$
0.312962 + 0.949766i $$0.398679\pi$$
$$642$$ 0 0
$$643$$ 29828.0 1.82940 0.914698 0.404138i $$-0.132429\pi$$
0.914698 + 0.404138i $$0.132429\pi$$
$$644$$ 10752.0 0.657901
$$645$$ 0 0
$$646$$ 5040.00 0.306960
$$647$$ −1944.00 −0.118124 −0.0590622 0.998254i $$-0.518811\pi$$
−0.0590622 + 0.998254i $$0.518811\pi$$
$$648$$ 0 0
$$649$$ −7920.00 −0.479025
$$650$$ −6764.00 −0.408163
$$651$$ 0 0
$$652$$ −7408.00 −0.444969
$$653$$ −26718.0 −1.60116 −0.800579 0.599227i $$-0.795475\pi$$
−0.800579 + 0.599227i $$0.795475\pi$$
$$654$$ 0 0
$$655$$ 13752.0 0.820359
$$656$$ −672.000 −0.0399957
$$657$$ 0 0
$$658$$ −3072.00 −0.182005
$$659$$ −4260.00 −0.251815 −0.125907 0.992042i $$-0.540184\pi$$
−0.125907 + 0.992042i $$0.540184\pi$$
$$660$$ 0 0
$$661$$ 22862.0 1.34528 0.672639 0.739971i $$-0.265161\pi$$
0.672639 + 0.739971i $$0.265161\pi$$
$$662$$ −5576.00 −0.327368
$$663$$ 0 0
$$664$$ 3936.00 0.230040
$$665$$ 1920.00 0.111962
$$666$$ 0 0
$$667$$ 5040.00 0.292578
$$668$$ 8544.00 0.494876
$$669$$ 0 0
$$670$$ −10608.0 −0.611676
$$671$$ 6456.00 0.371432
$$672$$ 0 0
$$673$$ −32542.0 −1.86390 −0.931948 0.362592i $$-0.881892\pi$$
−0.931948 + 0.362592i $$0.881892\pi$$
$$674$$ 868.000 0.0496055
$$675$$ 0 0
$$676$$ −3012.00 −0.171370
$$677$$ −14214.0 −0.806925 −0.403463 0.914996i $$-0.632193\pi$$
−0.403463 + 0.914996i $$0.632193\pi$$
$$678$$ 0 0
$$679$$ −18464.0 −1.04357
$$680$$ −6048.00 −0.341074
$$681$$ 0 0
$$682$$ 2112.00 0.118582
$$683$$ 7092.00 0.397317 0.198659 0.980069i $$-0.436341\pi$$
0.198659 + 0.980069i $$0.436341\pi$$
$$684$$ 0 0
$$685$$ −4356.00 −0.242970
$$686$$ 13760.0 0.765830
$$687$$ 0 0
$$688$$ −832.000 −0.0461042
$$689$$ −7524.00 −0.416026
$$690$$ 0 0
$$691$$ −13228.0 −0.728244 −0.364122 0.931351i $$-0.618631\pi$$
−0.364122 + 0.931351i $$0.618631\pi$$
$$692$$ −7032.00 −0.386296
$$693$$ 0 0
$$694$$ −13368.0 −0.731185
$$695$$ −2280.00 −0.124439
$$696$$ 0 0
$$697$$ −5292.00 −0.287588
$$698$$ 5260.00 0.285235
$$699$$ 0 0
$$700$$ 5696.00 0.307555
$$701$$ −28062.0 −1.51196 −0.755982 0.654592i $$-0.772840\pi$$
−0.755982 + 0.654592i $$0.772840\pi$$
$$702$$ 0 0
$$703$$ 5080.00 0.272540
$$704$$ −768.000 −0.0411152
$$705$$ 0 0
$$706$$ 14844.0 0.791305
$$707$$ −9888.00 −0.525992
$$708$$ 0 0
$$709$$ −27250.0 −1.44343 −0.721717 0.692188i $$-0.756647\pi$$
−0.721717 + 0.692188i $$0.756647\pi$$
$$710$$ 9504.00 0.502364
$$711$$ 0 0
$$712$$ −6480.00 −0.341079
$$713$$ 14784.0 0.776529
$$714$$ 0 0
$$715$$ 2736.00 0.143106
$$716$$ 2160.00 0.112742
$$717$$ 0 0
$$718$$ 20880.0 1.08529
$$719$$ 14400.0 0.746912 0.373456 0.927648i $$-0.378173\pi$$
0.373456 + 0.927648i $$0.378173\pi$$
$$720$$ 0 0
$$721$$ −2048.00 −0.105786
$$722$$ −12918.0 −0.665870
$$723$$ 0 0
$$724$$ 7928.00 0.406964
$$725$$ 2670.00 0.136774
$$726$$ 0 0
$$727$$ 17984.0 0.917455 0.458727 0.888577i $$-0.348305\pi$$
0.458727 + 0.888577i $$0.348305\pi$$
$$728$$ −4864.00 −0.247626
$$729$$ 0 0
$$730$$ −2616.00 −0.132634
$$731$$ −6552.00 −0.331511
$$732$$ 0 0
$$733$$ 16598.0 0.836373 0.418186 0.908361i $$-0.362666\pi$$
0.418186 + 0.908361i $$0.362666\pi$$
$$734$$ 20848.0 1.04838
$$735$$ 0 0
$$736$$ −5376.00 −0.269242
$$737$$ −10608.0 −0.530191
$$738$$ 0 0
$$739$$ 1460.00 0.0726752 0.0363376 0.999340i $$-0.488431\pi$$
0.0363376 + 0.999340i $$0.488431\pi$$
$$740$$ −6096.00 −0.302829
$$741$$ 0 0
$$742$$ 6336.00 0.313480
$$743$$ 30072.0 1.48484 0.742419 0.669936i $$-0.233678\pi$$
0.742419 + 0.669936i $$0.233678\pi$$
$$744$$ 0 0
$$745$$ 9540.00 0.469152
$$746$$ 6556.00 0.321759
$$747$$ 0 0
$$748$$ −6048.00 −0.295637
$$749$$ −23616.0 −1.15208
$$750$$ 0 0
$$751$$ −18088.0 −0.878882 −0.439441 0.898271i $$-0.644823\pi$$
−0.439441 + 0.898271i $$0.644823\pi$$
$$752$$ 1536.00 0.0744843
$$753$$ 0 0
$$754$$ −2280.00 −0.110123
$$755$$ −14592.0 −0.703387
$$756$$ 0 0
$$757$$ 24734.0 1.18755 0.593773 0.804633i $$-0.297638\pi$$
0.593773 + 0.804633i $$0.297638\pi$$
$$758$$ 12280.0 0.588430
$$759$$ 0 0
$$760$$ −960.000 −0.0458196
$$761$$ 22278.0 1.06120 0.530602 0.847621i $$-0.321966\pi$$
0.530602 + 0.847621i $$0.321966\pi$$
$$762$$ 0 0
$$763$$ −19040.0 −0.903400
$$764$$ 10752.0 0.509154
$$765$$ 0 0
$$766$$ 6144.00 0.289806
$$767$$ 25080.0 1.18069
$$768$$ 0 0
$$769$$ 16130.0 0.756388 0.378194 0.925726i $$-0.376545\pi$$
0.378194 + 0.925726i $$0.376545\pi$$
$$770$$ −2304.00 −0.107832
$$771$$ 0 0
$$772$$ −9208.00 −0.429279
$$773$$ −29718.0 −1.38277 −0.691386 0.722486i $$-0.742999\pi$$
−0.691386 + 0.722486i $$0.742999\pi$$
$$774$$ 0 0
$$775$$ 7832.00 0.363011
$$776$$ 9232.00 0.427074
$$777$$ 0 0
$$778$$ −12300.0 −0.566808
$$779$$ −840.000 −0.0386343
$$780$$ 0 0
$$781$$ 9504.00 0.435442
$$782$$ −42336.0 −1.93597
$$783$$ 0 0
$$784$$ −1392.00 −0.0634111
$$785$$ −3684.00 −0.167500
$$786$$ 0 0
$$787$$ 9524.00 0.431377 0.215689 0.976462i $$-0.430800\pi$$
0.215689 + 0.976462i $$0.430800\pi$$
$$788$$ −17496.0 −0.790951
$$789$$ 0 0
$$790$$ 6240.00 0.281024
$$791$$ −7392.00 −0.332275
$$792$$ 0 0
$$793$$ −20444.0 −0.915495
$$794$$ −212.000 −0.00947556
$$795$$ 0 0
$$796$$ −6400.00 −0.284977
$$797$$ 33906.0 1.50692 0.753458 0.657496i $$-0.228384\pi$$
0.753458 + 0.657496i $$0.228384\pi$$
$$798$$ 0 0
$$799$$ 12096.0 0.535577
$$800$$ −2848.00 −0.125865
$$801$$ 0 0
$$802$$ 3516.00 0.154806
$$803$$ −2616.00 −0.114965
$$804$$ 0 0
$$805$$ −16128.0 −0.706133
$$806$$ −6688.00 −0.292276
$$807$$ 0 0
$$808$$ 4944.00 0.215259
$$809$$ 630.000 0.0273790 0.0136895 0.999906i $$-0.495642\pi$$
0.0136895 + 0.999906i $$0.495642\pi$$
$$810$$ 0 0
$$811$$ −20788.0 −0.900081 −0.450040 0.893008i $$-0.648590\pi$$
−0.450040 + 0.893008i $$0.648590\pi$$
$$812$$ 1920.00 0.0829788
$$813$$ 0 0
$$814$$ −6096.00 −0.262487
$$815$$ 11112.0 0.477591
$$816$$ 0 0
$$817$$ −1040.00 −0.0445349
$$818$$ −7340.00 −0.313737
$$819$$ 0 0
$$820$$ 1008.00 0.0429279
$$821$$ 43098.0 1.83207 0.916036 0.401097i $$-0.131371\pi$$
0.916036 + 0.401097i $$0.131371\pi$$
$$822$$ 0 0
$$823$$ −14272.0 −0.604484 −0.302242 0.953231i $$-0.597735\pi$$
−0.302242 + 0.953231i $$0.597735\pi$$
$$824$$ 1024.00 0.0432921
$$825$$ 0 0
$$826$$ −21120.0 −0.889660
$$827$$ −13644.0 −0.573698 −0.286849 0.957976i $$-0.592608\pi$$
−0.286849 + 0.957976i $$0.592608\pi$$
$$828$$ 0 0
$$829$$ −2410.00 −0.100968 −0.0504842 0.998725i $$-0.516076\pi$$
−0.0504842 + 0.998725i $$0.516076\pi$$
$$830$$ −5904.00 −0.246905
$$831$$ 0 0
$$832$$ 2432.00 0.101339
$$833$$ −10962.0 −0.455955
$$834$$ 0 0
$$835$$ −12816.0 −0.531157
$$836$$ −960.000 −0.0397157
$$837$$ 0 0
$$838$$ 19320.0 0.796418
$$839$$ −23160.0 −0.953006 −0.476503 0.879173i $$-0.658096\pi$$
−0.476503 + 0.879173i $$0.658096\pi$$
$$840$$ 0 0
$$841$$ −23489.0 −0.963098
$$842$$ 16924.0 0.692684
$$843$$ 0 0
$$844$$ 13328.0 0.543565
$$845$$ 4518.00 0.183934
$$846$$ 0 0
$$847$$ 18992.0 0.770452
$$848$$ −3168.00 −0.128290
$$849$$ 0 0
$$850$$ −22428.0 −0.905028
$$851$$ −42672.0 −1.71889
$$852$$ 0 0
$$853$$ 32078.0 1.28761 0.643804 0.765190i $$-0.277355\pi$$
0.643804 + 0.765190i $$0.277355\pi$$
$$854$$ 17216.0 0.689835
$$855$$ 0 0
$$856$$ 11808.0 0.471483
$$857$$ 14406.0 0.574212 0.287106 0.957899i $$-0.407307\pi$$
0.287106 + 0.957899i $$0.407307\pi$$
$$858$$ 0 0
$$859$$ 30620.0 1.21623 0.608115 0.793849i $$-0.291926\pi$$
0.608115 + 0.793849i $$0.291926\pi$$
$$860$$ 1248.00 0.0494842
$$861$$ 0 0
$$862$$ −19584.0 −0.773821
$$863$$ −17568.0 −0.692957 −0.346478 0.938058i $$-0.612623\pi$$
−0.346478 + 0.938058i $$0.612623\pi$$
$$864$$ 0 0
$$865$$ 10548.0 0.414616
$$866$$ −14684.0 −0.576192
$$867$$ 0 0
$$868$$ 5632.00 0.220233
$$869$$ 6240.00 0.243587
$$870$$ 0 0
$$871$$ 33592.0 1.30680
$$872$$ 9520.00 0.369711
$$873$$ 0 0
$$874$$ −6720.00 −0.260077
$$875$$ −20544.0 −0.793730
$$876$$ 0 0
$$877$$ −21706.0 −0.835758 −0.417879 0.908503i $$-0.637226\pi$$
−0.417879 + 0.908503i $$0.637226\pi$$
$$878$$ 21280.0 0.817956
$$879$$ 0 0
$$880$$ 1152.00 0.0441294
$$881$$ 14958.0 0.572018 0.286009 0.958227i $$-0.407671\pi$$
0.286009 + 0.958227i $$0.407671\pi$$
$$882$$ 0 0
$$883$$ −32812.0 −1.25052 −0.625261 0.780415i $$-0.715008\pi$$
−0.625261 + 0.780415i $$0.715008\pi$$
$$884$$ 19152.0 0.728678
$$885$$ 0 0
$$886$$ 34824.0 1.32047
$$887$$ 38856.0 1.47086 0.735432 0.677598i $$-0.236979\pi$$
0.735432 + 0.677598i $$0.236979\pi$$
$$888$$ 0 0
$$889$$ 40576.0 1.53079
$$890$$ 9720.00 0.366084
$$891$$ 0 0
$$892$$ 10592.0 0.397586
$$893$$ 1920.00 0.0719489
$$894$$ 0 0
$$895$$ −3240.00 −0.121007
$$896$$ −2048.00 −0.0763604
$$897$$ 0 0
$$898$$ 3420.00 0.127090
$$899$$ 2640.00 0.0979410
$$900$$ 0 0
$$901$$ −24948.0 −0.922462
$$902$$ 1008.00 0.0372092
$$903$$ 0 0
$$904$$ 3696.00 0.135981
$$905$$ −11892.0 −0.436799
$$906$$ 0 0
$$907$$ −28276.0 −1.03516 −0.517579 0.855635i $$-0.673167\pi$$
−0.517579 + 0.855635i $$0.673167\pi$$
$$908$$ −8976.00 −0.328061
$$909$$ 0 0
$$910$$ 7296.00 0.265780
$$911$$ −8112.00 −0.295019 −0.147510 0.989061i $$-0.547126\pi$$
−0.147510 + 0.989061i $$0.547126\pi$$
$$912$$ 0 0
$$913$$ −5904.00 −0.214013
$$914$$ −1292.00 −0.0467566
$$915$$ 0 0
$$916$$ −22600.0 −0.815202
$$917$$ 36672.0 1.32063
$$918$$ 0 0
$$919$$ −26080.0 −0.936126 −0.468063 0.883695i $$-0.655048\pi$$
−0.468063 + 0.883695i $$0.655048\pi$$
$$920$$ 8064.00 0.288981
$$921$$ 0 0
$$922$$ 12036.0 0.429918
$$923$$ −30096.0 −1.07326
$$924$$ 0 0
$$925$$ −22606.0 −0.803547
$$926$$ −13424.0 −0.476393
$$927$$ 0 0
$$928$$ −960.000 −0.0339586
$$929$$ −49170.0 −1.73651 −0.868254 0.496120i $$-0.834757\pi$$
−0.868254 + 0.496120i $$0.834757\pi$$
$$930$$ 0 0
$$931$$ −1740.00 −0.0612526
$$932$$ −18792.0 −0.660464
$$933$$ 0 0
$$934$$ −10728.0 −0.375836
$$935$$ 9072.00 0.317311
$$936$$ 0 0
$$937$$ 48314.0 1.68447 0.842236 0.539110i $$-0.181239\pi$$
0.842236 + 0.539110i $$0.181239\pi$$
$$938$$ −28288.0 −0.984687
$$939$$ 0 0
$$940$$ −2304.00 −0.0799449
$$941$$ −34782.0 −1.20495 −0.602477 0.798137i $$-0.705819\pi$$
−0.602477 + 0.798137i $$0.705819\pi$$
$$942$$ 0 0
$$943$$ 7056.00 0.243664
$$944$$ 10560.0 0.364088
$$945$$ 0 0
$$946$$ 1248.00 0.0428922
$$947$$ 25116.0 0.861838 0.430919 0.902391i $$-0.358190\pi$$
0.430919 + 0.902391i $$0.358190\pi$$
$$948$$ 0 0
$$949$$ 8284.00 0.283361
$$950$$ −3560.00 −0.121581
$$951$$ 0 0
$$952$$ −16128.0 −0.549067
$$953$$ 15462.0 0.525565 0.262782 0.964855i $$-0.415360\pi$$
0.262782 + 0.964855i $$0.415360\pi$$
$$954$$ 0 0
$$955$$ −16128.0 −0.546481
$$956$$ 4800.00 0.162388
$$957$$ 0 0
$$958$$ −19680.0 −0.663708
$$959$$ −11616.0 −0.391137
$$960$$ 0 0
$$961$$ −22047.0 −0.740056
$$962$$ 19304.0 0.646971
$$963$$ 0 0
$$964$$ −2872.00 −0.0959553
$$965$$ 13812.0 0.460750
$$966$$ 0 0
$$967$$ −736.000 −0.0244759 −0.0122379 0.999925i $$-0.503896\pi$$
−0.0122379 + 0.999925i $$0.503896\pi$$
$$968$$ −9496.00 −0.315303
$$969$$ 0 0
$$970$$ −13848.0 −0.458384
$$971$$ 29268.0 0.967307 0.483653 0.875260i $$-0.339310\pi$$
0.483653 + 0.875260i $$0.339310\pi$$
$$972$$ 0 0
$$973$$ −6080.00 −0.200325
$$974$$ 2848.00 0.0936918
$$975$$ 0 0
$$976$$ −8608.00 −0.282311
$$977$$ −16674.0 −0.546007 −0.273003 0.962013i $$-0.588017\pi$$
−0.273003 + 0.962013i $$0.588017\pi$$
$$978$$ 0 0
$$979$$ 9720.00 0.317316
$$980$$ 2088.00 0.0680599
$$981$$ 0 0
$$982$$ 9096.00 0.295586
$$983$$ 31272.0 1.01467 0.507336 0.861749i $$-0.330630\pi$$
0.507336 + 0.861749i $$0.330630\pi$$
$$984$$ 0 0
$$985$$ 26244.0 0.848937
$$986$$ −7560.00 −0.244178
$$987$$ 0 0
$$988$$ 3040.00 0.0978900
$$989$$ 8736.00 0.280878
$$990$$ 0 0
$$991$$ −15928.0 −0.510565 −0.255282 0.966867i $$-0.582168\pi$$
−0.255282 + 0.966867i $$0.582168\pi$$
$$992$$ −2816.00 −0.0901291
$$993$$ 0 0
$$994$$ 25344.0 0.808715
$$995$$ 9600.00 0.305870
$$996$$ 0 0
$$997$$ 42014.0 1.33460 0.667300 0.744789i $$-0.267450\pi$$
0.667300 + 0.744789i $$0.267450\pi$$
$$998$$ 13000.0 0.412332
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 18.4.a.a.1.1 1
3.2 odd 2 6.4.a.a.1.1 1
4.3 odd 2 144.4.a.c.1.1 1
5.2 odd 4 450.4.c.e.199.2 2
5.3 odd 4 450.4.c.e.199.1 2
5.4 even 2 450.4.a.h.1.1 1
7.2 even 3 882.4.g.i.361.1 2
7.3 odd 6 882.4.g.f.667.1 2
7.4 even 3 882.4.g.i.667.1 2
7.5 odd 6 882.4.g.f.361.1 2
7.6 odd 2 882.4.a.n.1.1 1
8.3 odd 2 576.4.a.r.1.1 1
8.5 even 2 576.4.a.q.1.1 1
9.2 odd 6 162.4.c.f.109.1 2
9.4 even 3 162.4.c.c.55.1 2
9.5 odd 6 162.4.c.f.55.1 2
9.7 even 3 162.4.c.c.109.1 2
11.10 odd 2 2178.4.a.e.1.1 1
12.11 even 2 48.4.a.c.1.1 1
15.2 even 4 150.4.c.d.49.1 2
15.8 even 4 150.4.c.d.49.2 2
15.14 odd 2 150.4.a.i.1.1 1
21.2 odd 6 294.4.e.h.67.1 2
21.5 even 6 294.4.e.g.67.1 2
21.11 odd 6 294.4.e.h.79.1 2
21.17 even 6 294.4.e.g.79.1 2
21.20 even 2 294.4.a.e.1.1 1
24.5 odd 2 192.4.a.i.1.1 1
24.11 even 2 192.4.a.c.1.1 1
33.32 even 2 726.4.a.f.1.1 1
39.5 even 4 1014.4.b.d.337.2 2
39.8 even 4 1014.4.b.d.337.1 2
39.38 odd 2 1014.4.a.g.1.1 1
48.5 odd 4 768.4.d.n.385.2 2
48.11 even 4 768.4.d.c.385.1 2
48.29 odd 4 768.4.d.n.385.1 2
48.35 even 4 768.4.d.c.385.2 2
51.50 odd 2 1734.4.a.d.1.1 1
57.56 even 2 2166.4.a.i.1.1 1
60.23 odd 4 1200.4.f.j.49.2 2
60.47 odd 4 1200.4.f.j.49.1 2
60.59 even 2 1200.4.a.b.1.1 1
84.83 odd 2 2352.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 3.2 odd 2
18.4.a.a.1.1 1 1.1 even 1 trivial
48.4.a.c.1.1 1 12.11 even 2
144.4.a.c.1.1 1 4.3 odd 2
150.4.a.i.1.1 1 15.14 odd 2
150.4.c.d.49.1 2 15.2 even 4
150.4.c.d.49.2 2 15.8 even 4
162.4.c.c.55.1 2 9.4 even 3
162.4.c.c.109.1 2 9.7 even 3
162.4.c.f.55.1 2 9.5 odd 6
162.4.c.f.109.1 2 9.2 odd 6
192.4.a.c.1.1 1 24.11 even 2
192.4.a.i.1.1 1 24.5 odd 2
294.4.a.e.1.1 1 21.20 even 2
294.4.e.g.67.1 2 21.5 even 6
294.4.e.g.79.1 2 21.17 even 6
294.4.e.h.67.1 2 21.2 odd 6
294.4.e.h.79.1 2 21.11 odd 6
450.4.a.h.1.1 1 5.4 even 2
450.4.c.e.199.1 2 5.3 odd 4
450.4.c.e.199.2 2 5.2 odd 4
576.4.a.q.1.1 1 8.5 even 2
576.4.a.r.1.1 1 8.3 odd 2
726.4.a.f.1.1 1 33.32 even 2
768.4.d.c.385.1 2 48.11 even 4
768.4.d.c.385.2 2 48.35 even 4
768.4.d.n.385.1 2 48.29 odd 4
768.4.d.n.385.2 2 48.5 odd 4
882.4.a.n.1.1 1 7.6 odd 2
882.4.g.f.361.1 2 7.5 odd 6
882.4.g.f.667.1 2 7.3 odd 6
882.4.g.i.361.1 2 7.2 even 3
882.4.g.i.667.1 2 7.4 even 3
1014.4.a.g.1.1 1 39.38 odd 2
1014.4.b.d.337.1 2 39.8 even 4
1014.4.b.d.337.2 2 39.5 even 4
1200.4.a.b.1.1 1 60.59 even 2
1200.4.f.j.49.1 2 60.47 odd 4
1200.4.f.j.49.2 2 60.23 odd 4
1734.4.a.d.1.1 1 51.50 odd 2
2166.4.a.i.1.1 1 57.56 even 2
2178.4.a.e.1.1 1 11.10 odd 2
2352.4.a.e.1.1 1 84.83 odd 2