# Properties

 Label 2352.4.a.cq.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.136768.1 Defining polynomial: $$x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7$$ Twist minimal: no (minimal twist has level 588) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.93153$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -19.1403 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -19.1403 q^{5} +9.00000 q^{9} -40.5951 q^{11} +50.4776 q^{13} -57.4210 q^{15} -51.9195 q^{17} +33.1331 q^{19} -62.8499 q^{23} +241.352 q^{25} +27.0000 q^{27} +129.921 q^{29} -242.415 q^{31} -121.785 q^{33} -389.385 q^{37} +151.433 q^{39} -470.110 q^{41} +125.003 q^{43} -172.263 q^{45} +386.624 q^{47} -155.758 q^{51} -611.436 q^{53} +777.004 q^{55} +99.3994 q^{57} -226.432 q^{59} +725.191 q^{61} -966.158 q^{65} -1045.11 q^{67} -188.550 q^{69} -169.839 q^{71} -381.731 q^{73} +724.057 q^{75} +1161.12 q^{79} +81.0000 q^{81} +808.448 q^{83} +993.756 q^{85} +389.764 q^{87} -319.867 q^{89} -727.245 q^{93} -634.179 q^{95} +1133.24 q^{97} -365.356 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 36q^{9} - 48q^{17} + 192q^{19} - 192q^{23} + 324q^{25} + 108q^{27} + 96q^{29} + 48q^{31} + 256q^{37} - 1008q^{41} + 112q^{43} + 864q^{47} - 144q^{51} - 648q^{53} + 2352q^{55} + 576q^{57} + 336q^{59} - 960q^{61} - 360q^{65} - 720q^{67} - 576q^{69} + 1344q^{71} - 672q^{73} + 972q^{75} + 1984q^{79} + 324q^{81} + 3120q^{83} + 680q^{85} + 288q^{87} - 2160q^{89} + 144q^{93} + 3744q^{95} - 2016q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −19.1403 −1.71196 −0.855982 0.517006i $$-0.827046\pi$$
−0.855982 + 0.517006i $$0.827046\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −40.5951 −1.11272 −0.556359 0.830942i $$-0.687802\pi$$
−0.556359 + 0.830942i $$0.687802\pi$$
$$12$$ 0 0
$$13$$ 50.4776 1.07692 0.538460 0.842651i $$-0.319006\pi$$
0.538460 + 0.842651i $$0.319006\pi$$
$$14$$ 0 0
$$15$$ −57.4210 −0.988402
$$16$$ 0 0
$$17$$ −51.9195 −0.740725 −0.370362 0.928887i $$-0.620766\pi$$
−0.370362 + 0.928887i $$0.620766\pi$$
$$18$$ 0 0
$$19$$ 33.1331 0.400066 0.200033 0.979789i $$-0.435895\pi$$
0.200033 + 0.979789i $$0.435895\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −62.8499 −0.569787 −0.284894 0.958559i $$-0.591958\pi$$
−0.284894 + 0.958559i $$0.591958\pi$$
$$24$$ 0 0
$$25$$ 241.352 1.93082
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 129.921 0.831924 0.415962 0.909382i $$-0.363445\pi$$
0.415962 + 0.909382i $$0.363445\pi$$
$$30$$ 0 0
$$31$$ −242.415 −1.40448 −0.702242 0.711938i $$-0.747818\pi$$
−0.702242 + 0.711938i $$0.747818\pi$$
$$32$$ 0 0
$$33$$ −121.785 −0.642428
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −389.385 −1.73012 −0.865061 0.501667i $$-0.832720\pi$$
−0.865061 + 0.501667i $$0.832720\pi$$
$$38$$ 0 0
$$39$$ 151.433 0.621761
$$40$$ 0 0
$$41$$ −470.110 −1.79071 −0.895353 0.445358i $$-0.853076\pi$$
−0.895353 + 0.445358i $$0.853076\pi$$
$$42$$ 0 0
$$43$$ 125.003 0.443321 0.221661 0.975124i $$-0.428852\pi$$
0.221661 + 0.975124i $$0.428852\pi$$
$$44$$ 0 0
$$45$$ −172.263 −0.570654
$$46$$ 0 0
$$47$$ 386.624 1.19989 0.599946 0.800040i $$-0.295189\pi$$
0.599946 + 0.800040i $$0.295189\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −155.758 −0.427658
$$52$$ 0 0
$$53$$ −611.436 −1.58466 −0.792332 0.610091i $$-0.791133\pi$$
−0.792332 + 0.610091i $$0.791133\pi$$
$$54$$ 0 0
$$55$$ 777.004 1.90493
$$56$$ 0 0
$$57$$ 99.3994 0.230978
$$58$$ 0 0
$$59$$ −226.432 −0.499644 −0.249822 0.968292i $$-0.580372\pi$$
−0.249822 + 0.968292i $$0.580372\pi$$
$$60$$ 0 0
$$61$$ 725.191 1.52215 0.761075 0.648664i $$-0.224672\pi$$
0.761075 + 0.648664i $$0.224672\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −966.158 −1.84365
$$66$$ 0 0
$$67$$ −1045.11 −1.90568 −0.952840 0.303474i $$-0.901853\pi$$
−0.952840 + 0.303474i $$0.901853\pi$$
$$68$$ 0 0
$$69$$ −188.550 −0.328967
$$70$$ 0 0
$$71$$ −169.839 −0.283889 −0.141945 0.989875i $$-0.545336\pi$$
−0.141945 + 0.989875i $$0.545336\pi$$
$$72$$ 0 0
$$73$$ −381.731 −0.612031 −0.306015 0.952027i $$-0.598996\pi$$
−0.306015 + 0.952027i $$0.598996\pi$$
$$74$$ 0 0
$$75$$ 724.057 1.11476
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1161.12 1.65362 0.826812 0.562479i $$-0.190152\pi$$
0.826812 + 0.562479i $$0.190152\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 808.448 1.06914 0.534570 0.845124i $$-0.320473\pi$$
0.534570 + 0.845124i $$0.320473\pi$$
$$84$$ 0 0
$$85$$ 993.756 1.26809
$$86$$ 0 0
$$87$$ 389.764 0.480311
$$88$$ 0 0
$$89$$ −319.867 −0.380964 −0.190482 0.981691i $$-0.561005\pi$$
−0.190482 + 0.981691i $$0.561005\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −727.245 −0.810880
$$94$$ 0 0
$$95$$ −634.179 −0.684899
$$96$$ 0 0
$$97$$ 1133.24 1.18622 0.593110 0.805121i $$-0.297900\pi$$
0.593110 + 0.805121i $$0.297900\pi$$
$$98$$ 0 0
$$99$$ −365.356 −0.370906
$$100$$ 0 0
$$101$$ −1045.74 −1.03025 −0.515126 0.857115i $$-0.672255\pi$$
−0.515126 + 0.857115i $$0.672255\pi$$
$$102$$ 0 0
$$103$$ 1898.21 1.81589 0.907943 0.419093i $$-0.137652\pi$$
0.907943 + 0.419093i $$0.137652\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 868.785 0.784940 0.392470 0.919765i $$-0.371621\pi$$
0.392470 + 0.919765i $$0.371621\pi$$
$$108$$ 0 0
$$109$$ 227.649 0.200044 0.100022 0.994985i $$-0.468109\pi$$
0.100022 + 0.994985i $$0.468109\pi$$
$$110$$ 0 0
$$111$$ −1168.16 −0.998886
$$112$$ 0 0
$$113$$ 581.462 0.484065 0.242032 0.970268i $$-0.422186\pi$$
0.242032 + 0.970268i $$0.422186\pi$$
$$114$$ 0 0
$$115$$ 1202.97 0.975455
$$116$$ 0 0
$$117$$ 454.299 0.358974
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 316.963 0.238139
$$122$$ 0 0
$$123$$ −1410.33 −1.03386
$$124$$ 0 0
$$125$$ −2227.02 −1.59353
$$126$$ 0 0
$$127$$ −1129.10 −0.788908 −0.394454 0.918916i $$-0.629066\pi$$
−0.394454 + 0.918916i $$0.629066\pi$$
$$128$$ 0 0
$$129$$ 375.010 0.255952
$$130$$ 0 0
$$131$$ −879.662 −0.586690 −0.293345 0.956007i $$-0.594769\pi$$
−0.293345 + 0.956007i $$0.594769\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −516.789 −0.329467
$$136$$ 0 0
$$137$$ −323.300 −0.201616 −0.100808 0.994906i $$-0.532143\pi$$
−0.100808 + 0.994906i $$0.532143\pi$$
$$138$$ 0 0
$$139$$ 1710.64 1.04385 0.521923 0.852993i $$-0.325215\pi$$
0.521923 + 0.852993i $$0.325215\pi$$
$$140$$ 0 0
$$141$$ 1159.87 0.692758
$$142$$ 0 0
$$143$$ −2049.14 −1.19831
$$144$$ 0 0
$$145$$ −2486.74 −1.42422
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −959.778 −0.527705 −0.263853 0.964563i $$-0.584993\pi$$
−0.263853 + 0.964563i $$0.584993\pi$$
$$150$$ 0 0
$$151$$ 1618.84 0.872449 0.436224 0.899838i $$-0.356315\pi$$
0.436224 + 0.899838i $$0.356315\pi$$
$$152$$ 0 0
$$153$$ −467.275 −0.246908
$$154$$ 0 0
$$155$$ 4639.90 2.40443
$$156$$ 0 0
$$157$$ 557.333 0.283312 0.141656 0.989916i $$-0.454757\pi$$
0.141656 + 0.989916i $$0.454757\pi$$
$$158$$ 0 0
$$159$$ −1834.31 −0.914906
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 366.034 0.175889 0.0879447 0.996125i $$-0.471970\pi$$
0.0879447 + 0.996125i $$0.471970\pi$$
$$164$$ 0 0
$$165$$ 2331.01 1.09981
$$166$$ 0 0
$$167$$ 2834.00 1.31318 0.656591 0.754247i $$-0.271998\pi$$
0.656591 + 0.754247i $$0.271998\pi$$
$$168$$ 0 0
$$169$$ 350.990 0.159759
$$170$$ 0 0
$$171$$ 298.198 0.133355
$$172$$ 0 0
$$173$$ 3570.39 1.56909 0.784543 0.620074i $$-0.212898\pi$$
0.784543 + 0.620074i $$0.212898\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −679.297 −0.288470
$$178$$ 0 0
$$179$$ 3677.38 1.53553 0.767767 0.640729i $$-0.221368\pi$$
0.767767 + 0.640729i $$0.221368\pi$$
$$180$$ 0 0
$$181$$ 1718.18 0.705588 0.352794 0.935701i $$-0.385232\pi$$
0.352794 + 0.935701i $$0.385232\pi$$
$$182$$ 0 0
$$183$$ 2175.57 0.878814
$$184$$ 0 0
$$185$$ 7452.96 2.96191
$$186$$ 0 0
$$187$$ 2107.68 0.824217
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1959.05 −0.742155 −0.371077 0.928602i $$-0.621012\pi$$
−0.371077 + 0.928602i $$0.621012\pi$$
$$192$$ 0 0
$$193$$ −2492.46 −0.929591 −0.464795 0.885418i $$-0.653872\pi$$
−0.464795 + 0.885418i $$0.653872\pi$$
$$194$$ 0 0
$$195$$ −2898.47 −1.06443
$$196$$ 0 0
$$197$$ 2174.45 0.786410 0.393205 0.919451i $$-0.371366\pi$$
0.393205 + 0.919451i $$0.371366\pi$$
$$198$$ 0 0
$$199$$ 3864.61 1.37666 0.688329 0.725399i $$-0.258345\pi$$
0.688329 + 0.725399i $$0.258345\pi$$
$$200$$ 0 0
$$201$$ −3135.33 −1.10024
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 8998.07 3.06562
$$206$$ 0 0
$$207$$ −565.649 −0.189929
$$208$$ 0 0
$$209$$ −1345.04 −0.445161
$$210$$ 0 0
$$211$$ 127.267 0.0415233 0.0207616 0.999784i $$-0.493391\pi$$
0.0207616 + 0.999784i $$0.493391\pi$$
$$212$$ 0 0
$$213$$ −509.516 −0.163904
$$214$$ 0 0
$$215$$ −2392.60 −0.758950
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1145.19 −0.353356
$$220$$ 0 0
$$221$$ −2620.77 −0.797702
$$222$$ 0 0
$$223$$ −4071.36 −1.22259 −0.611297 0.791401i $$-0.709352\pi$$
−0.611297 + 0.791401i $$0.709352\pi$$
$$224$$ 0 0
$$225$$ 2172.17 0.643606
$$226$$ 0 0
$$227$$ 5282.36 1.54450 0.772252 0.635316i $$-0.219130\pi$$
0.772252 + 0.635316i $$0.219130\pi$$
$$228$$ 0 0
$$229$$ 4580.19 1.32169 0.660846 0.750522i $$-0.270198\pi$$
0.660846 + 0.750522i $$0.270198\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2156.51 −0.606343 −0.303171 0.952936i $$-0.598045\pi$$
−0.303171 + 0.952936i $$0.598045\pi$$
$$234$$ 0 0
$$235$$ −7400.11 −2.05417
$$236$$ 0 0
$$237$$ 3483.36 0.954720
$$238$$ 0 0
$$239$$ 5755.76 1.55778 0.778889 0.627162i $$-0.215783\pi$$
0.778889 + 0.627162i $$0.215783\pi$$
$$240$$ 0 0
$$241$$ 2591.55 0.692683 0.346342 0.938108i $$-0.387424\pi$$
0.346342 + 0.938108i $$0.387424\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1672.48 0.430840
$$248$$ 0 0
$$249$$ 2425.34 0.617269
$$250$$ 0 0
$$251$$ −3809.47 −0.957974 −0.478987 0.877822i $$-0.658996\pi$$
−0.478987 + 0.877822i $$0.658996\pi$$
$$252$$ 0 0
$$253$$ 2551.40 0.634012
$$254$$ 0 0
$$255$$ 2981.27 0.732134
$$256$$ 0 0
$$257$$ −1357.69 −0.329534 −0.164767 0.986333i $$-0.552687\pi$$
−0.164767 + 0.986333i $$0.552687\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 1169.29 0.277308
$$262$$ 0 0
$$263$$ 4417.73 1.03578 0.517888 0.855448i $$-0.326718\pi$$
0.517888 + 0.855448i $$0.326718\pi$$
$$264$$ 0 0
$$265$$ 11703.1 2.71289
$$266$$ 0 0
$$267$$ −959.600 −0.219950
$$268$$ 0 0
$$269$$ 3627.32 0.822162 0.411081 0.911599i $$-0.365151\pi$$
0.411081 + 0.911599i $$0.365151\pi$$
$$270$$ 0 0
$$271$$ −5171.03 −1.15911 −0.579553 0.814935i $$-0.696773\pi$$
−0.579553 + 0.814935i $$0.696773\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −9797.72 −2.14845
$$276$$ 0 0
$$277$$ 1794.63 0.389274 0.194637 0.980875i $$-0.437647\pi$$
0.194637 + 0.980875i $$0.437647\pi$$
$$278$$ 0 0
$$279$$ −2181.74 −0.468162
$$280$$ 0 0
$$281$$ −9118.63 −1.93584 −0.967921 0.251254i $$-0.919157\pi$$
−0.967921 + 0.251254i $$0.919157\pi$$
$$282$$ 0 0
$$283$$ 5804.72 1.21927 0.609637 0.792681i $$-0.291315\pi$$
0.609637 + 0.792681i $$0.291315\pi$$
$$284$$ 0 0
$$285$$ −1902.54 −0.395426
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2217.37 −0.451327
$$290$$ 0 0
$$291$$ 3399.73 0.684865
$$292$$ 0 0
$$293$$ −1384.21 −0.275995 −0.137997 0.990433i $$-0.544067\pi$$
−0.137997 + 0.990433i $$0.544067\pi$$
$$294$$ 0 0
$$295$$ 4333.99 0.855372
$$296$$ 0 0
$$297$$ −1096.07 −0.214143
$$298$$ 0 0
$$299$$ −3172.51 −0.613616
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −3137.23 −0.594816
$$304$$ 0 0
$$305$$ −13880.4 −2.60587
$$306$$ 0 0
$$307$$ −337.112 −0.0626709 −0.0313355 0.999509i $$-0.509976\pi$$
−0.0313355 + 0.999509i $$0.509976\pi$$
$$308$$ 0 0
$$309$$ 5694.63 1.04840
$$310$$ 0 0
$$311$$ −1964.69 −0.358224 −0.179112 0.983829i $$-0.557322\pi$$
−0.179112 + 0.983829i $$0.557322\pi$$
$$312$$ 0 0
$$313$$ 3596.98 0.649563 0.324781 0.945789i $$-0.394709\pi$$
0.324781 + 0.945789i $$0.394709\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −215.450 −0.0381731 −0.0190865 0.999818i $$-0.506076\pi$$
−0.0190865 + 0.999818i $$0.506076\pi$$
$$318$$ 0 0
$$319$$ −5274.17 −0.925696
$$320$$ 0 0
$$321$$ 2606.35 0.453185
$$322$$ 0 0
$$323$$ −1720.25 −0.296339
$$324$$ 0 0
$$325$$ 12182.9 2.07934
$$326$$ 0 0
$$327$$ 682.947 0.115496
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1685.35 0.279865 0.139933 0.990161i $$-0.455311\pi$$
0.139933 + 0.990161i $$0.455311\pi$$
$$332$$ 0 0
$$333$$ −3504.47 −0.576707
$$334$$ 0 0
$$335$$ 20003.7 3.26245
$$336$$ 0 0
$$337$$ −7497.87 −1.21197 −0.605987 0.795475i $$-0.707222\pi$$
−0.605987 + 0.795475i $$0.707222\pi$$
$$338$$ 0 0
$$339$$ 1744.38 0.279475
$$340$$ 0 0
$$341$$ 9840.87 1.56279
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 3608.90 0.563179
$$346$$ 0 0
$$347$$ 6038.83 0.934241 0.467120 0.884194i $$-0.345291\pi$$
0.467120 + 0.884194i $$0.345291\pi$$
$$348$$ 0 0
$$349$$ 1992.86 0.305659 0.152830 0.988253i $$-0.451161\pi$$
0.152830 + 0.988253i $$0.451161\pi$$
$$350$$ 0 0
$$351$$ 1362.90 0.207254
$$352$$ 0 0
$$353$$ −6336.08 −0.955342 −0.477671 0.878539i $$-0.658519\pi$$
−0.477671 + 0.878539i $$0.658519\pi$$
$$354$$ 0 0
$$355$$ 3250.77 0.486008
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −904.510 −0.132975 −0.0664877 0.997787i $$-0.521179\pi$$
−0.0664877 + 0.997787i $$0.521179\pi$$
$$360$$ 0 0
$$361$$ −5761.20 −0.839947
$$362$$ 0 0
$$363$$ 950.890 0.137490
$$364$$ 0 0
$$365$$ 7306.46 1.04777
$$366$$ 0 0
$$367$$ −1252.76 −0.178184 −0.0890920 0.996023i $$-0.528397\pi$$
−0.0890920 + 0.996023i $$0.528397\pi$$
$$368$$ 0 0
$$369$$ −4230.99 −0.596902
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2565.91 −0.356187 −0.178094 0.984014i $$-0.556993\pi$$
−0.178094 + 0.984014i $$0.556993\pi$$
$$374$$ 0 0
$$375$$ −6681.06 −0.920022
$$376$$ 0 0
$$377$$ 6558.12 0.895916
$$378$$ 0 0
$$379$$ 3900.45 0.528634 0.264317 0.964436i $$-0.414853\pi$$
0.264317 + 0.964436i $$0.414853\pi$$
$$380$$ 0 0
$$381$$ −3387.29 −0.455476
$$382$$ 0 0
$$383$$ 1669.11 0.222683 0.111342 0.993782i $$-0.464485\pi$$
0.111342 + 0.993782i $$0.464485\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1125.03 0.147774
$$388$$ 0 0
$$389$$ 1189.74 0.155070 0.0775351 0.996990i $$-0.475295\pi$$
0.0775351 + 0.996990i $$0.475295\pi$$
$$390$$ 0 0
$$391$$ 3263.13 0.422056
$$392$$ 0 0
$$393$$ −2638.99 −0.338726
$$394$$ 0 0
$$395$$ −22224.2 −2.83094
$$396$$ 0 0
$$397$$ −802.190 −0.101412 −0.0507062 0.998714i $$-0.516147\pi$$
−0.0507062 + 0.998714i $$0.516147\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −10532.2 −1.31161 −0.655803 0.754932i $$-0.727670\pi$$
−0.655803 + 0.754932i $$0.727670\pi$$
$$402$$ 0 0
$$403$$ −12236.5 −1.51252
$$404$$ 0 0
$$405$$ −1550.37 −0.190218
$$406$$ 0 0
$$407$$ 15807.1 1.92514
$$408$$ 0 0
$$409$$ −3671.52 −0.443875 −0.221937 0.975061i $$-0.571238\pi$$
−0.221937 + 0.975061i $$0.571238\pi$$
$$410$$ 0 0
$$411$$ −969.899 −0.116403
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −15474.0 −1.83033
$$416$$ 0 0
$$417$$ 5131.92 0.602664
$$418$$ 0 0
$$419$$ 1668.29 0.194514 0.0972569 0.995259i $$-0.468993\pi$$
0.0972569 + 0.995259i $$0.468993\pi$$
$$420$$ 0 0
$$421$$ 16043.7 1.85730 0.928649 0.370961i $$-0.120972\pi$$
0.928649 + 0.370961i $$0.120972\pi$$
$$422$$ 0 0
$$423$$ 3479.62 0.399964
$$424$$ 0 0
$$425$$ −12530.9 −1.43020
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −6147.43 −0.691844
$$430$$ 0 0
$$431$$ −12986.7 −1.45138 −0.725692 0.688020i $$-0.758480\pi$$
−0.725692 + 0.688020i $$0.758480\pi$$
$$432$$ 0 0
$$433$$ 943.959 0.104766 0.0523831 0.998627i $$-0.483318\pi$$
0.0523831 + 0.998627i $$0.483318\pi$$
$$434$$ 0 0
$$435$$ −7460.21 −0.822275
$$436$$ 0 0
$$437$$ −2082.41 −0.227953
$$438$$ 0 0
$$439$$ 6427.32 0.698769 0.349384 0.936980i $$-0.386391\pi$$
0.349384 + 0.936980i $$0.386391\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7254.91 −0.778084 −0.389042 0.921220i $$-0.627194\pi$$
−0.389042 + 0.921220i $$0.627194\pi$$
$$444$$ 0 0
$$445$$ 6122.35 0.652196
$$446$$ 0 0
$$447$$ −2879.34 −0.304671
$$448$$ 0 0
$$449$$ 17381.9 1.82696 0.913478 0.406887i $$-0.133386\pi$$
0.913478 + 0.406887i $$0.133386\pi$$
$$450$$ 0 0
$$451$$ 19084.2 1.99255
$$452$$ 0 0
$$453$$ 4856.53 0.503708
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13800.8 −1.41264 −0.706319 0.707894i $$-0.749645\pi$$
−0.706319 + 0.707894i $$0.749645\pi$$
$$458$$ 0 0
$$459$$ −1401.83 −0.142553
$$460$$ 0 0
$$461$$ 13.6989 0.00138400 0.000691998 1.00000i $$-0.499780\pi$$
0.000691998 1.00000i $$0.499780\pi$$
$$462$$ 0 0
$$463$$ −13910.8 −1.39631 −0.698153 0.715949i $$-0.745994\pi$$
−0.698153 + 0.715949i $$0.745994\pi$$
$$464$$ 0 0
$$465$$ 13919.7 1.38820
$$466$$ 0 0
$$467$$ 7818.53 0.774729 0.387364 0.921927i $$-0.373386\pi$$
0.387364 + 0.921927i $$0.373386\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 1672.00 0.163570
$$472$$ 0 0
$$473$$ −5074.52 −0.493291
$$474$$ 0 0
$$475$$ 7996.75 0.772455
$$476$$ 0 0
$$477$$ −5502.92 −0.528221
$$478$$ 0 0
$$479$$ 18025.4 1.71942 0.859709 0.510784i $$-0.170645\pi$$
0.859709 + 0.510784i $$0.170645\pi$$
$$480$$ 0 0
$$481$$ −19655.2 −1.86320
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −21690.6 −2.03077
$$486$$ 0 0
$$487$$ 17431.4 1.62195 0.810977 0.585078i $$-0.198936\pi$$
0.810977 + 0.585078i $$0.198936\pi$$
$$488$$ 0 0
$$489$$ 1098.10 0.101550
$$490$$ 0 0
$$491$$ −7107.17 −0.653243 −0.326621 0.945155i $$-0.605910\pi$$
−0.326621 + 0.945155i $$0.605910\pi$$
$$492$$ 0 0
$$493$$ −6745.45 −0.616226
$$494$$ 0 0
$$495$$ 6993.03 0.634977
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −3499.61 −0.313956 −0.156978 0.987602i $$-0.550175\pi$$
−0.156978 + 0.987602i $$0.550175\pi$$
$$500$$ 0 0
$$501$$ 8501.99 0.758166
$$502$$ 0 0
$$503$$ 9115.24 0.808009 0.404005 0.914757i $$-0.367618\pi$$
0.404005 + 0.914757i $$0.367618\pi$$
$$504$$ 0 0
$$505$$ 20015.9 1.76375
$$506$$ 0 0
$$507$$ 1052.97 0.0922367
$$508$$ 0 0
$$509$$ 16049.7 1.39762 0.698811 0.715306i $$-0.253713\pi$$
0.698811 + 0.715306i $$0.253713\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 894.594 0.0769928
$$514$$ 0 0
$$515$$ −36332.4 −3.10873
$$516$$ 0 0
$$517$$ −15695.1 −1.33514
$$518$$ 0 0
$$519$$ 10711.2 0.905912
$$520$$ 0 0
$$521$$ 5574.48 0.468757 0.234379 0.972145i $$-0.424694\pi$$
0.234379 + 0.972145i $$0.424694\pi$$
$$522$$ 0 0
$$523$$ −458.890 −0.0383669 −0.0191834 0.999816i $$-0.506107\pi$$
−0.0191834 + 0.999816i $$0.506107\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12586.1 1.04034
$$528$$ 0 0
$$529$$ −8216.89 −0.675342
$$530$$ 0 0
$$531$$ −2037.89 −0.166548
$$532$$ 0 0
$$533$$ −23730.1 −1.92845
$$534$$ 0 0
$$535$$ −16628.8 −1.34379
$$536$$ 0 0
$$537$$ 11032.1 0.886541
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10972.7 −0.871999 −0.436000 0.899947i $$-0.643605\pi$$
−0.436000 + 0.899947i $$0.643605\pi$$
$$542$$ 0 0
$$543$$ 5154.54 0.407371
$$544$$ 0 0
$$545$$ −4357.28 −0.342468
$$546$$ 0 0
$$547$$ 19276.8 1.50679 0.753396 0.657567i $$-0.228414\pi$$
0.753396 + 0.657567i $$0.228414\pi$$
$$548$$ 0 0
$$549$$ 6526.72 0.507383
$$550$$ 0 0
$$551$$ 4304.70 0.332825
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 22358.9 1.71006
$$556$$ 0 0
$$557$$ −12191.4 −0.927410 −0.463705 0.885990i $$-0.653480\pi$$
−0.463705 + 0.885990i $$0.653480\pi$$
$$558$$ 0 0
$$559$$ 6309.87 0.477422
$$560$$ 0 0
$$561$$ 6323.03 0.475862
$$562$$ 0 0
$$563$$ 3236.06 0.242244 0.121122 0.992638i $$-0.461351\pi$$
0.121122 + 0.992638i $$0.461351\pi$$
$$564$$ 0 0
$$565$$ −11129.4 −0.828701
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 12167.8 0.896488 0.448244 0.893911i $$-0.352050\pi$$
0.448244 + 0.893911i $$0.352050\pi$$
$$570$$ 0 0
$$571$$ 24412.3 1.78918 0.894592 0.446884i $$-0.147466\pi$$
0.894592 + 0.446884i $$0.147466\pi$$
$$572$$ 0 0
$$573$$ −5877.14 −0.428483
$$574$$ 0 0
$$575$$ −15169.0 −1.10016
$$576$$ 0 0
$$577$$ −4346.85 −0.313625 −0.156812 0.987628i $$-0.550122\pi$$
−0.156812 + 0.987628i $$0.550122\pi$$
$$578$$ 0 0
$$579$$ −7477.37 −0.536700
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 24821.3 1.76328
$$584$$ 0 0
$$585$$ −8695.42 −0.614550
$$586$$ 0 0
$$587$$ 8752.61 0.615432 0.307716 0.951478i $$-0.400435\pi$$
0.307716 + 0.951478i $$0.400435\pi$$
$$588$$ 0 0
$$589$$ −8031.97 −0.561887
$$590$$ 0 0
$$591$$ 6523.34 0.454034
$$592$$ 0 0
$$593$$ 7788.77 0.539370 0.269685 0.962949i $$-0.413080\pi$$
0.269685 + 0.962949i $$0.413080\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11593.8 0.794814
$$598$$ 0 0
$$599$$ 1144.02 0.0780354 0.0390177 0.999239i $$-0.487577\pi$$
0.0390177 + 0.999239i $$0.487577\pi$$
$$600$$ 0 0
$$601$$ −24673.4 −1.67463 −0.837314 0.546723i $$-0.815875\pi$$
−0.837314 + 0.546723i $$0.815875\pi$$
$$602$$ 0 0
$$603$$ −9405.99 −0.635226
$$604$$ 0 0
$$605$$ −6066.78 −0.407686
$$606$$ 0 0
$$607$$ −6325.62 −0.422980 −0.211490 0.977380i $$-0.567832\pi$$
−0.211490 + 0.977380i $$0.567832\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 19515.9 1.29219
$$612$$ 0 0
$$613$$ 22976.2 1.51387 0.756935 0.653491i $$-0.226696\pi$$
0.756935 + 0.653491i $$0.226696\pi$$
$$614$$ 0 0
$$615$$ 26994.2 1.76994
$$616$$ 0 0
$$617$$ 8229.26 0.536949 0.268475 0.963287i $$-0.413480\pi$$
0.268475 + 0.963287i $$0.413480\pi$$
$$618$$ 0 0
$$619$$ 16852.8 1.09430 0.547150 0.837035i $$-0.315713\pi$$
0.547150 + 0.837035i $$0.315713\pi$$
$$620$$ 0 0
$$621$$ −1696.95 −0.109656
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 12456.9 0.797239
$$626$$ 0 0
$$627$$ −4035.13 −0.257014
$$628$$ 0 0
$$629$$ 20216.7 1.28154
$$630$$ 0 0
$$631$$ 8408.46 0.530484 0.265242 0.964182i $$-0.414548\pi$$
0.265242 + 0.964182i $$0.414548\pi$$
$$632$$ 0 0
$$633$$ 381.801 0.0239735
$$634$$ 0 0
$$635$$ 21611.3 1.35058
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −1528.55 −0.0946298
$$640$$ 0 0
$$641$$ 9122.52 0.562118 0.281059 0.959690i $$-0.409314\pi$$
0.281059 + 0.959690i $$0.409314\pi$$
$$642$$ 0 0
$$643$$ −19279.4 −1.18243 −0.591217 0.806513i $$-0.701352\pi$$
−0.591217 + 0.806513i $$0.701352\pi$$
$$644$$ 0 0
$$645$$ −7177.81 −0.438180
$$646$$ 0 0
$$647$$ −2177.86 −0.132335 −0.0661673 0.997809i $$-0.521077\pi$$
−0.0661673 + 0.997809i $$0.521077\pi$$
$$648$$ 0 0
$$649$$ 9192.05 0.555962
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6288.85 −0.376879 −0.188439 0.982085i $$-0.560343\pi$$
−0.188439 + 0.982085i $$0.560343\pi$$
$$654$$ 0 0
$$655$$ 16837.0 1.00439
$$656$$ 0 0
$$657$$ −3435.58 −0.204010
$$658$$ 0 0
$$659$$ −20346.5 −1.20271 −0.601357 0.798980i $$-0.705373\pi$$
−0.601357 + 0.798980i $$0.705373\pi$$
$$660$$ 0 0
$$661$$ 27669.2 1.62815 0.814074 0.580761i $$-0.197245\pi$$
0.814074 + 0.580761i $$0.197245\pi$$
$$662$$ 0 0
$$663$$ −7862.31 −0.460553
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8165.54 −0.474020
$$668$$ 0 0
$$669$$ −12214.1 −0.705865
$$670$$ 0 0
$$671$$ −29439.2 −1.69372
$$672$$ 0 0
$$673$$ 11862.6 0.679450 0.339725 0.940525i $$-0.389666\pi$$
0.339725 + 0.940525i $$0.389666\pi$$
$$674$$ 0 0
$$675$$ 6516.51 0.371586
$$676$$ 0 0
$$677$$ −13526.6 −0.767901 −0.383950 0.923354i $$-0.625437\pi$$
−0.383950 + 0.923354i $$0.625437\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 15847.1 0.891720
$$682$$ 0 0
$$683$$ 713.863 0.0399930 0.0199965 0.999800i $$-0.493634\pi$$
0.0199965 + 0.999800i $$0.493634\pi$$
$$684$$ 0 0
$$685$$ 6188.06 0.345159
$$686$$ 0 0
$$687$$ 13740.6 0.763079
$$688$$ 0 0
$$689$$ −30863.8 −1.70656
$$690$$ 0 0
$$691$$ −15816.1 −0.870727 −0.435364 0.900255i $$-0.643380\pi$$
−0.435364 + 0.900255i $$0.643380\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −32742.2 −1.78702
$$696$$ 0 0
$$697$$ 24407.9 1.32642
$$698$$ 0 0
$$699$$ −6469.54 −0.350072
$$700$$ 0 0
$$701$$ 28556.9 1.53863 0.769314 0.638871i $$-0.220598\pi$$
0.769314 + 0.638871i $$0.220598\pi$$
$$702$$ 0 0
$$703$$ −12901.5 −0.692163
$$704$$ 0 0
$$705$$ −22200.3 −1.18598
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1934.72 −0.102482 −0.0512411 0.998686i $$-0.516318\pi$$
−0.0512411 + 0.998686i $$0.516318\pi$$
$$710$$ 0 0
$$711$$ 10450.1 0.551208
$$712$$ 0 0
$$713$$ 15235.8 0.800258
$$714$$ 0 0
$$715$$ 39221.3 2.05146
$$716$$ 0 0
$$717$$ 17267.3 0.899384
$$718$$ 0 0
$$719$$ 18666.7 0.968218 0.484109 0.875008i $$-0.339144\pi$$
0.484109 + 0.875008i $$0.339144\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 7774.66 0.399921
$$724$$ 0 0
$$725$$ 31356.8 1.60629
$$726$$ 0 0
$$727$$ −25955.3 −1.32411 −0.662055 0.749455i $$-0.730315\pi$$
−0.662055 + 0.749455i $$0.730315\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −6490.10 −0.328379
$$732$$ 0 0
$$733$$ 2083.79 0.105002 0.0525010 0.998621i $$-0.483281\pi$$
0.0525010 + 0.998621i $$0.483281\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 42426.4 2.12048
$$738$$ 0 0
$$739$$ −28857.3 −1.43645 −0.718223 0.695813i $$-0.755044\pi$$
−0.718223 + 0.695813i $$0.755044\pi$$
$$740$$ 0 0
$$741$$ 5017.44 0.248745
$$742$$ 0 0
$$743$$ 899.017 0.0443900 0.0221950 0.999754i $$-0.492935\pi$$
0.0221950 + 0.999754i $$0.492935\pi$$
$$744$$ 0 0
$$745$$ 18370.5 0.903412
$$746$$ 0 0
$$747$$ 7276.03 0.356380
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 29009.3 1.40954 0.704770 0.709436i $$-0.251050\pi$$
0.704770 + 0.709436i $$0.251050\pi$$
$$752$$ 0 0
$$753$$ −11428.4 −0.553087
$$754$$ 0 0
$$755$$ −30985.2 −1.49360
$$756$$ 0 0
$$757$$ −10932.4 −0.524892 −0.262446 0.964947i $$-0.584529\pi$$
−0.262446 + 0.964947i $$0.584529\pi$$
$$758$$ 0 0
$$759$$ 7654.20 0.366047
$$760$$ 0 0
$$761$$ 13212.8 0.629387 0.314693 0.949193i $$-0.398098\pi$$
0.314693 + 0.949193i $$0.398098\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 8943.80 0.422698
$$766$$ 0 0
$$767$$ −11429.8 −0.538077
$$768$$ 0 0
$$769$$ −30129.7 −1.41288 −0.706440 0.707773i $$-0.749700\pi$$
−0.706440 + 0.707773i $$0.749700\pi$$
$$770$$ 0 0
$$771$$ −4073.06 −0.190257
$$772$$ 0 0
$$773$$ −11840.7 −0.550947 −0.275474 0.961309i $$-0.588835\pi$$
−0.275474 + 0.961309i $$0.588835\pi$$
$$774$$ 0 0
$$775$$ −58507.4 −2.71180
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −15576.2 −0.716401
$$780$$ 0 0
$$781$$ 6894.62 0.315889
$$782$$ 0 0
$$783$$ 3507.88 0.160104
$$784$$ 0 0
$$785$$ −10667.5 −0.485020
$$786$$ 0 0
$$787$$ 6529.10 0.295727 0.147863 0.989008i $$-0.452760\pi$$
0.147863 + 0.989008i $$0.452760\pi$$
$$788$$ 0 0
$$789$$ 13253.2 0.598006
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 36605.9 1.63924
$$794$$ 0 0
$$795$$ 35109.2 1.56628
$$796$$ 0 0
$$797$$ 41987.6 1.86609 0.933046 0.359757i $$-0.117140\pi$$
0.933046 + 0.359757i $$0.117140\pi$$
$$798$$ 0 0
$$799$$ −20073.3 −0.888790
$$800$$ 0 0
$$801$$ −2878.80 −0.126988
$$802$$ 0 0
$$803$$ 15496.4 0.681017
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10882.0 0.474676
$$808$$ 0 0
$$809$$ −13634.1 −0.592522 −0.296261 0.955107i $$-0.595740\pi$$
−0.296261 + 0.955107i $$0.595740\pi$$
$$810$$ 0 0
$$811$$ 20093.9 0.870025 0.435013 0.900424i $$-0.356744\pi$$
0.435013 + 0.900424i $$0.356744\pi$$
$$812$$ 0 0
$$813$$ −15513.1 −0.669210
$$814$$ 0 0
$$815$$ −7006.01 −0.301116
$$816$$ 0 0
$$817$$ 4141.75 0.177358
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −7636.29 −0.324615 −0.162307 0.986740i $$-0.551894\pi$$
−0.162307 + 0.986740i $$0.551894\pi$$
$$822$$ 0 0
$$823$$ 28788.4 1.21932 0.609660 0.792663i $$-0.291306\pi$$
0.609660 + 0.792663i $$0.291306\pi$$
$$824$$ 0 0
$$825$$ −29393.2 −1.24041
$$826$$ 0 0
$$827$$ 20515.4 0.862623 0.431312 0.902203i $$-0.358051\pi$$
0.431312 + 0.902203i $$0.358051\pi$$
$$828$$ 0 0
$$829$$ 16264.4 0.681407 0.340704 0.940171i $$-0.389335\pi$$
0.340704 + 0.940171i $$0.389335\pi$$
$$830$$ 0 0
$$831$$ 5383.89 0.224748
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −54243.7 −2.24812
$$836$$ 0 0
$$837$$ −6545.21 −0.270293
$$838$$ 0 0
$$839$$ 13770.2 0.566629 0.283314 0.959027i $$-0.408566\pi$$
0.283314 + 0.959027i $$0.408566\pi$$
$$840$$ 0 0
$$841$$ −7509.45 −0.307903
$$842$$ 0 0
$$843$$ −27355.9 −1.11766
$$844$$ 0 0
$$845$$ −6718.06 −0.273501
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 17414.2 0.703948
$$850$$ 0 0
$$851$$ 24472.8 0.985802
$$852$$ 0 0
$$853$$ −8031.48 −0.322383 −0.161191 0.986923i $$-0.551534\pi$$
−0.161191 + 0.986923i $$0.551534\pi$$
$$854$$ 0 0
$$855$$ −5707.61 −0.228300
$$856$$ 0 0
$$857$$ −42764.0 −1.70454 −0.852270 0.523102i $$-0.824775\pi$$
−0.852270 + 0.523102i $$0.824775\pi$$
$$858$$ 0 0
$$859$$ −8155.05 −0.323919 −0.161960 0.986797i $$-0.551781\pi$$
−0.161960 + 0.986797i $$0.551781\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −40660.0 −1.60380 −0.801901 0.597457i $$-0.796178\pi$$
−0.801901 + 0.597457i $$0.796178\pi$$
$$864$$ 0 0
$$865$$ −68338.5 −2.68622
$$866$$ 0 0
$$867$$ −6652.11 −0.260574
$$868$$ 0 0
$$869$$ −47135.8 −1.84002
$$870$$ 0 0
$$871$$ −52754.7 −2.05227
$$872$$ 0 0
$$873$$ 10199.2 0.395407
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −28260.6 −1.08813 −0.544067 0.839042i $$-0.683116\pi$$
−0.544067 + 0.839042i $$0.683116\pi$$
$$878$$ 0 0
$$879$$ −4152.64 −0.159346
$$880$$ 0 0
$$881$$ 15951.2 0.609998 0.304999 0.952353i $$-0.401344\pi$$
0.304999 + 0.952353i $$0.401344\pi$$
$$882$$ 0 0
$$883$$ 1750.48 0.0667137 0.0333569 0.999444i $$-0.489380\pi$$
0.0333569 + 0.999444i $$0.489380\pi$$
$$884$$ 0 0
$$885$$ 13002.0 0.493849
$$886$$ 0 0
$$887$$ −2007.02 −0.0759741 −0.0379871 0.999278i $$-0.512095\pi$$
−0.0379871 + 0.999278i $$0.512095\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3288.20 −0.123635
$$892$$ 0 0
$$893$$ 12810.1 0.480036
$$894$$ 0 0
$$895$$ −70386.3 −2.62878
$$896$$ 0 0
$$897$$ −9517.54 −0.354271
$$898$$ 0 0
$$899$$ −31494.9 −1.16842
$$900$$ 0 0
$$901$$ 31745.4 1.17380
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −32886.6 −1.20794
$$906$$ 0 0
$$907$$ 12069.7 0.441861 0.220930 0.975290i $$-0.429091\pi$$
0.220930 + 0.975290i $$0.429091\pi$$
$$908$$ 0 0
$$909$$ −9411.69 −0.343417
$$910$$ 0 0
$$911$$ −40167.1 −1.46081 −0.730404 0.683016i $$-0.760668\pi$$
−0.730404 + 0.683016i $$0.760668\pi$$
$$912$$ 0 0
$$913$$ −32819.0 −1.18965
$$914$$ 0 0
$$915$$ −41641.2 −1.50450
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 13503.5 0.484699 0.242349 0.970189i $$-0.422082\pi$$
0.242349 + 0.970189i $$0.422082\pi$$
$$920$$ 0 0
$$921$$ −1011.33 −0.0361831
$$922$$ 0 0
$$923$$ −8573.05 −0.305726
$$924$$ 0 0
$$925$$ −93978.9 −3.34055
$$926$$ 0 0
$$927$$ 17083.9 0.605295
$$928$$ 0 0
$$929$$ 31001.9 1.09488 0.547438 0.836846i $$-0.315603\pi$$
0.547438 + 0.836846i $$0.315603\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −5894.08 −0.206821
$$934$$ 0 0
$$935$$ −40341.6 −1.41103
$$936$$ 0 0
$$937$$ −34998.0 −1.22021 −0.610104 0.792322i $$-0.708872\pi$$
−0.610104 + 0.792322i $$0.708872\pi$$
$$938$$ 0 0
$$939$$ 10790.9 0.375025
$$940$$ 0 0
$$941$$ −11652.5 −0.403679 −0.201839 0.979419i $$-0.564692\pi$$
−0.201839 + 0.979419i $$0.564692\pi$$
$$942$$ 0 0
$$943$$ 29546.4 1.02032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2921.85 −0.100261 −0.0501306 0.998743i $$-0.515964\pi$$
−0.0501306 + 0.998743i $$0.515964\pi$$
$$948$$ 0 0
$$949$$ −19268.9 −0.659109
$$950$$ 0 0
$$951$$ −646.350 −0.0220392
$$952$$ 0 0
$$953$$ 24262.5 0.824702 0.412351 0.911025i $$-0.364708\pi$$
0.412351 + 0.911025i $$0.364708\pi$$
$$954$$ 0 0
$$955$$ 37496.8 1.27054
$$956$$ 0 0
$$957$$ −15822.5 −0.534451
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 28974.1 0.972577
$$962$$ 0 0
$$963$$ 7819.06 0.261647
$$964$$ 0 0
$$965$$ 47706.5 1.59143
$$966$$ 0 0
$$967$$ −10258.0 −0.341131 −0.170565 0.985346i $$-0.554559\pi$$
−0.170565 + 0.985346i $$0.554559\pi$$
$$968$$ 0 0
$$969$$ −5160.76 −0.171091
$$970$$ 0 0
$$971$$ −16635.7 −0.549809 −0.274905 0.961472i $$-0.588646\pi$$
−0.274905 + 0.961472i $$0.588646\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 36548.7 1.20051
$$976$$ 0 0
$$977$$ 12850.2 0.420791 0.210395 0.977616i $$-0.432525\pi$$
0.210395 + 0.977616i $$0.432525\pi$$
$$978$$ 0 0
$$979$$ 12985.0 0.423905
$$980$$ 0 0
$$981$$ 2048.84 0.0666814
$$982$$ 0 0
$$983$$ −42272.8 −1.37161 −0.685805 0.727785i $$-0.740550\pi$$
−0.685805 + 0.727785i $$0.740550\pi$$
$$984$$ 0 0
$$985$$ −41619.6 −1.34631
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −7856.44 −0.252599
$$990$$ 0 0
$$991$$ 11504.2 0.368761 0.184380 0.982855i $$-0.440972\pi$$
0.184380 + 0.982855i $$0.440972\pi$$
$$992$$ 0 0
$$993$$ 5056.06 0.161580
$$994$$ 0 0
$$995$$ −73969.9 −2.35679
$$996$$ 0 0
$$997$$ −17180.9 −0.545763 −0.272881 0.962048i $$-0.587977\pi$$
−0.272881 + 0.962048i $$0.587977\pi$$
$$998$$ 0 0
$$999$$ −10513.4 −0.332962
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 2352.4.a.cq.1.1 4
4.3 odd 2 588.4.a.j.1.1 4
7.6 odd 2 2352.4.a.cl.1.4 4
12.11 even 2 1764.4.a.bc.1.4 4
28.3 even 6 588.4.i.k.373.1 8
28.11 odd 6 588.4.i.l.373.4 8
28.19 even 6 588.4.i.k.361.1 8
28.23 odd 6 588.4.i.l.361.4 8
28.27 even 2 588.4.a.k.1.4 yes 4
84.11 even 6 1764.4.k.bb.1549.1 8
84.23 even 6 1764.4.k.bb.361.1 8
84.47 odd 6 1764.4.k.bd.361.4 8
84.59 odd 6 1764.4.k.bd.1549.4 8
84.83 odd 2 1764.4.a.ba.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.1 4 4.3 odd 2
588.4.a.k.1.4 yes 4 28.27 even 2
588.4.i.k.361.1 8 28.19 even 6
588.4.i.k.373.1 8 28.3 even 6
588.4.i.l.361.4 8 28.23 odd 6
588.4.i.l.373.4 8 28.11 odd 6
1764.4.a.ba.1.1 4 84.83 odd 2
1764.4.a.bc.1.4 4 12.11 even 2
1764.4.k.bb.361.1 8 84.23 even 6
1764.4.k.bb.1549.1 8 84.11 even 6
1764.4.k.bd.361.4 8 84.47 odd 6
1764.4.k.bd.1549.4 8 84.59 odd 6
2352.4.a.cl.1.4 4 7.6 odd 2
2352.4.a.cq.1.1 4 1.1 even 1 trivial